IEEE Transactions on Antennas and Propagation [volume 59 number 11]

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NOVEMBER 2011

VOLUME 59

NUMBER 11

IETPAK

(ISSN 0018-926X)

PAPERS

Antennas A Novel Dual-Antenna Structure for UHF RFID Tags . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Y.-S. Chen, S.-Y. Chen, and H.-J. Li Embedded Singularity Chipless RFID Tags . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. T. Blischak and M. Manteghi External-Excitation Curl Antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H. Nakano, S. Kirita, N. Mizobe, and J. Yamauchi Investigation on the EM-Coupled Stacked Square Ring Antennas With Ultra-Thin Spacing . . . . . . . . . . . . . . . . . . . . S. I. Latif and L. Shafai Choosing Dielectric or Magnetic Material to Optimize the Bandwidth of Miniaturized Resonant Antennas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. O. Karilainen, P. M. T. Ikonen, C. R. Simovski, and S. A. Tretyakov Rectilinear Leaky-Wave Antennas With Broad Beam Patterns Using Hybrid Printed-Circuit Waveguides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J. L. Gómez-Tornero, A. R. Weily, and Y. J. Guo Vivaldi Antenna With Integrated Switchable Band Pass Resonator . . . . . . . . . . . . . . . M. R. Hamid, P. Gardner, P. S. Hall, and F. Ghanem A Seamless Integration of 3-D Vertical Filters With Highly Efficient Slot Antennas . . . . . . . . . . . . . . . Y. Yusuf, H. T. Cheng, and X. Gong Rigorous MoM Analysis of Finite Conductivity Effects in RLSA Antennas . . . . . . . . . . . . . . . . . . . . . M. Albani, A. Mazzinghi, and A. Freni A UWB Unidirectional Antenna With Dual-Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Wu and K.-M. Luk Flat-Shaped Dielectric Lens Antenna for 60-GHz Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Rolland, R. Sauleau, and L. Le Coq Spherical Near-Field Scanning With Higher-Order Probes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . T. B. Hansen Arrays The Banyan Tree Antenna Array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S. S. Holland and M. N. Vouvakis Wide-Angle Scanning Phased Array With Pattern Reconfigurable Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Y.-Y. Bai, S. Xiao, M.-C. Tang, Z.-F. Ding, and B.-Z. Wang Microstrip Grid and Comb Array Antennas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . L. Zhang, W. Zhang, and Y. P. Zhang A Novel Deterministic Synthesis Technique for Constrained Sparse Array Design Problems . . . . . . . . . . D. Caratelli and M. C. Viganó Evolutionary Design of Wide-Band Parasitic Dipole Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G. A. Casula, G. Mazzarella, and N. Sirena The Placement of Antenna Elements in Aperture Synthesis Microwave Radiometers for Optimum Radiometric Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J. Dong, Q. Li, R. Shi, L. Gui, and W. Guo Fast and Accurate Array Calibration Using a Synthetic Array Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W. P. M. N. Keizer Numerical and Analytical Techniques Application of Analytical Retarded-Time Potential Expressions to the Solution of Time Domain Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H. A. Ülkü and A. A. Ergin Hierarchical Matrix Techniques Based on Matrix Decomposition Algorithm for the Fast Analysis of Planar Layered Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . T. Wan, Z. N. Jiang, and Y. J. Sheng Fast Frequency Sweep of FEM Models via the Balanced Truncation Proper Orthogonal Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W. Wang, G. N. Paraschos, and M. N. Vouvakis Modeling of Nanophotonic Resonators With the Finite-Difference Frequency-Domain Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. M. Ivinskaya, A. V. Lavrinenko, and D. M. Shyroki

3950 3961 3969 3978 3991 3999 4008 4016 4023 4033 4041 4049 4060 4071 4077 4085 4094 4103 4115

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(Contents Continued on p. 3949)

(Contents Continued from Front Cover) An Efficient Scattered-Field Formulation for Objects in Layered Media Using the FVTD Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. Isleifson, I. Jeffrey, L. Shafai, J. Lovetri, and D. G. Barber Analysis of Seismic Electromagnetic Phenomena Using the FDTD Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Y. Wang and Q. Cao The Resonance Mode Theory for Exterior Problems of Electrodynamics and Its Application to Discrete Antenna Modeling in a Frequency Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I. P. Kovalyov and D. M. Ponomarev Wave Propagation and Wireless Compact and Multiband Dielectric Resonator Antenna With Pattern Diversity for Multistandard Mobile Handheld Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . L. Huitema, M. Koubeissi, M. Mouhamadou, E. Arnaud, C. Decroze, and T. Monediere A Novel Multiband Planar Antenna for GSM/UMTS/LTE/Zigbee/RFID Mobile Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . T. Zhang, R. L. Li, G. P. Jin, G. Wei, and M. M. Tentzeris Internal Coupled-Fed Dual-Loop Antenna Integrated With a USB Connector for WWAN/LTE Mobile Handset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F.-H. Chu and K.-L. Wong Experimental Analysis of a Wideband Pattern Diversity Antenna With Compact Reconfigurable CPW-to-Slotline Transition Feed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Y. Li, Z. Zhang, J. Zheng, Z. Feng, and M. F. Iskander Near- and Far-Field Models for Scattering Analysis of Buildings in Wireless Communications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Y. B. Ouattara, S. Mostarshedi, E. Richalot, J. Wiart, and O. Picon Modeling Propagation in Multifloor Buildings Using the FDTD Method . . . . . . . . . . . . . . . . . A. C. M. Austin, M. J. Neve, and G. B. Rowe Analysis and Modeling on co- and Cross-Polarized Urban Radio Propagation for Dual-Polarized MIMO Wireless Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V. Degli-Esposti, V.-M. Kolmonen, E. M. Vitucci, and P. Vainikainen Propagation Parameter Estimation, Modeling and Measurements for Ultrawideband MIMO Radar . . . . . . . . J. Salmi and A. F. Molisch Estimation of Wall Parameters From Time-Delay-Only Through-Wall Radar Measurements . . . . P. Protiva, J. Mrkvica, and J. Macháˇc Accuracy Evaluation of Ultrawideband Time Domain Systems for Microwave Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . X. Zeng, A. Fhager, M. Persson, P. Linner, and H. Zirath MultiEXCELL: A New Rain Field Model for Propagation Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . L. Luini and C. Capsoni The Physical Basis of Atmospheric Depolarization in Slant Paths in the V Band: Theory, Italsat Experiment and Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Paraboni, A. Martellucci, C. Capsoni, and C. Riva

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COMMUNICATIONS

Bandwidth Enhancement of Low-Profile PEC-Backed Equiangular Spiral Antennas Incorporating Metallic Posts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. Veysi and M. Kamyab Parasitic Current Reduction on Electrically Long Coaxial Cables Feeding Dipoles of a Collinear Array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. R. Guraliuc, A. A. Serra, P. Nepa, and G. Manara Enhanced Return-Loss and Flat-Gain Bandwidths for Microstrip Patch Antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . T.-N. Chang and J.-M. Lin K-Band Varactor Diode-Tuned Elliptical Slot Antenna for Wideband Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S. Kharkovsky, M. T. Ghasr, M. A. Abou-Khousa, and R. Zoughi Compact Circularly-Polarized Patch Antenna Loaded With Metamaterial Structures . . . . . . . . . . . . . . . . . . . Y. Dong, H. Toyao, and T. Itoh Effect of the Longitudinal Component of the Aperture Electric Field on the Analysis of Waveguide Longitudinal Slots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G. Montisci and G. Mazzarella Higher-Order MoM Analysis of the Rectangular Waveguide Edge Slot Arrays . . . . . . . . . . B. Lai, X.-W. Zhao, Z.-J. Su, and C.-H. Liang Design of Periodic Antenna Arrays With the Excitation Phases Synthesized for Optimum Near-Field Patterns via Steepest Descent Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H.-T. Chou, K.-L. Hung, and H.-H. Chou Subwavelength Array of Planar Monopoles With Complementary Split Rings Based on Far-Field Time Reversal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G.-D. Ge, B.-Z. Wang, D. Wang, D. Zhao, and S. Ding Closed-Loop Feed Architectures for RCS Beam Broadening of Retro-Reflective Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J. A. Vitaz, A. M. Buerkle, and K. Sarabandi A New Formulation of Pocklington’s Equation for Thin Wires Using the Exact Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E. Forati, A. D. Mueller, P. Gandomkar Yarandi, and G. W. Hanson Accelerated Source-Sweep Analysis Using a Reduced-Order Model Approach . . . . P. Bradley, C. Brennan, M. Condon, and M. Mullen Extended Mie Theory for a Gyrotropic-Coated Conducting Sphere: An Analytical Approach . . . . . . . . . . . . . . Y.-L. Geng and C.-W. Qiu Scattering by a Multilayered Infinite Cylinder Arbitrarily Illuminated With a Shaped Beam . . . . . . . . . . H. Zhang, Y. Sun, and Z. Huang Scattering of a Gaussian Beam by a Conducting Spheroidal Particle With Non-Confocal Dielectric Coating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H. Zhang, Z. Huang, and Y. Sun Electromagnetic Wave Scattering of a High-Order Bessel Vortex Beam by a Dielectric Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F. G. Mitri Absorption Loss Reduction in a Mobile Terminal With Switchable Monopole Antennas . . . . . . . M. Berg, M. Sonkki, and E. T. Salonen Radiation Improvement of Printed, Shorted Monopole Antenna for USB Dongle by Integrating Choke Sleeves on the System Ground . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S.-W. Su and T.-C. Hong Characterization of the Body-Area Propagation Channel for Monitoring a Subject Sleeping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. B. Smith, D. Miniutti, and L. W. Hanlen

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Digital Object Identifier 10.1109/TAP.2011.2173830

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A Novel Dual-Antenna Structure for UHF RFID Tags Yen-Sheng Chen, Shih-Yuan Chen, and Hsueh-Jyh Li

Abstract—A novel dual-antenna tag structure for UHF radio-frequency identification (RFID) systems is proposed. It is formed by two linearly tapered meander dipole antennas that are perpendicular to each other and connected to the slightly modified tag chip. One of the antennas is for receiving, while the other is for backscattering. The input impedance of the receiving antenna is designed to be conjugate matched to the highly capacitive chip impedance for the maximum power transfer. Meanwhile, the backscattering antenna is alternatively terminated by an open or a short circuit to modulate the backscattered field. By making the input impedance of the backscattering antenna real-valued, the maximum differential RCS may be achieved leading to a longer read range and better reliability. With the aid of the design of experiments (DOE) technique, the proposed dual-antenna structure is designed to fit within a compact area of 32 8 32 8 mm2 while keeping relatively low mutual coupling between the two antennas. The impedance, receiving, and backscattering performances of the proposed dual-antenna structure are measured and simulated, and they agree very well. Also, it is demonstrated that the proposed dual-antenna structure outperforms the conventional single-antenna tag design in every respect. Index Terms—Loaded antennas, optimization methods, radiofrequency identification (RFID), scattering, UHF antennas.

I. INTRODUCTION

I

N the past years, passive ultra-high frequency (UHF) radiofrequency identification (RFID) systems have been widely deployed in inventory systems, logistics, and retail sales [1]. This emergent technology has drawn more and more attention due to its longer read range and higher data transfer rate [2] as compared to the low-frequency (LF) and high-frequency (HF) RFID systems. A typical passive UHF RFID system consists of a reader and a passive tag. A successful communication between the reader and the tag involves two essentials. First, in the forward link, the reader sends out continuous wave (CW) and commands to the tag. Only as the tag chip receives sufficient

power from the CW, it can be turned on and respond to the command. Secondly, in the backward link, the tag antenna is alternatively connected to two different load impedances according to the data stored in the chip. The CW is modulated in this manner and scattered back to the reader. Only when the difference between the high and low levels of the backscattered wave is large enough, the reader could demodulate the backscattered signal correctly. Therefore, an optimal tag design should continuously provide sufficient power for the tag chip while exhibiting maximal level difference in the backscattered fields [3], [4]. However, this is not the case in conventional tag design, in which a single antenna is used for both receiving and backscattering. To create two impedance loading states for the backscattering modulation, the tag chip alternatively changes its impedance between a short circuit and a highly capacitive value, to which the tag antenna is made conjugate matched. This leads to two major limitations. First, when the impedance of the tag chip is switched to the short circuit, there would be no power received by the chip due to the total reflection at the antenna-chip junction [1]. Therefore, the efficiency of energy absorption drops significantly. Second, the two impedance states, namely the short and conjugate matched states, would not provide the maximum level difference in the backscattered signals resulting in a shorter read range [5]–[9]. In this paper, a novel tag structure is proposed to solve the above problems of the conventional design. The proposed tag structure consists of two independent antennas, one exclusively for receiving and the other for backscattering, such that the requirements for the maximum and continuous power supply and maximum level difference in the antenna backscattering can be achieved simultaneously. The design guidelines for the proposed tag structure and design methodologies used are presented in detail. In addition, the performances of this dual-antenna structure supplemented with simulated and measured results are also presented and compared to those of the conventional design. II. PROPOSED TAG STRUCTURE

Manuscript received November 02, 2010; revised March 20, 2011; accepted May 01, 2011. Date of publication August 12, 2011; date of current version November 02, 2011. This work was supported by the National Science Council, Taiwan, under Contracts NSC 96-2221-E-002-013MY2 and NSC 99-2221-E002-059, and in part by the National Taiwan University under Excellent Research Project NTU-ERP-98R0062-01. Y.-S. Chen and S.-Y. Chen are with the Graduate Institute of Communication Engineering, National Taiwan University, Taipei 10617, Taiwan (e-mail: [email protected]). H.-J. Li is with the Department of Electrical Engineering and the Graduate Institute of Communication Engineering, National Taiwan University, Taipei 10617, Taiwan (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2164199

A. Reception and Backscattering in RFID Tags Fig. 1 shows a tag antenna of input impedance being con. As the tag is illunected to a tag chip of input impedance minated by a plane wave , part of the power collected by the antenna is transferred to power up the chip, while the other part is scattered or reradiate into free space. The power received by is given as [10] the chip

0018-926X/$26.00 © 2011 IEEE

(1)

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Fig. 1. Configuration and equivalent circuit of a passive RFID tag.

where is the power collected by the antenna, and the complex reflection coefficient given as [11]

Fig. 2. Block diagram of proposed dual-antenna tag.

is

(2) The electric field scattered by the antenna by [12], [13]

can be expressed

(3) and are the electric field scattered and the current where flowing into the load, respectively, when the tag antenna is conis the field radiated by the tag antenna when jugate matched. it is excited by a unit current. The first term on the right-hand side of (3) is referred to as the structural mode of the scattered field, which is independent of the load impedance , while the second one is referred to as the antenna mode, which, through , is a function of . switches between two impedance states, denoted When and , the difference in the scattered field strength as between them can be written as

(4) is detected by the reader to differentiate the This quantity high and low levels of the backscattered signals modulated in accordance with the data stored in the tag chip [8]. The term could be any positive real number between [0, 2] and should be maximized to increase , and hence the read range of the tag. One can see from (1) that the tag antenna should be conju, for the most gate matched to the chip impedance, i.e. power to drive the chip. Whereas, according to (4), one should for any real argument choose such that the level difference in the backscattered signal is max. A conventional passive RFID imized, namely tag typically uses a single antenna for both reception and scattering. When the tag is in response, the tag chip alternatively switches between two different impedance states. Therefore, conventional tags cannot have simultaneously maximum power reception and maximum level difference in the backscattered signal because there is no power supply for the chip as or . In contrast, if

one of the impedance states is chosen to be conjugate matched to the input impedance of the tag antenna regarding the power supply for the tag chip, which is the case in conventional passive RFID tags, the level difference thus obtained is obviously not the optimum. B. Dual-Antenna Tag Structure To simultaneously maximize the power supply for the chip and the level difference in the backscattered signals , we propose a new dual-antenna structure for passive RFID tags as shown in Fig. 2. The two ports of the tag chip are formed by three terminals, one of which is the common ground. Each port is connected to an independent antenna. One antenna, denoted as the receiving antenna, is constantly used for reception, while the other, denoted as the backscattering antenna, is used for signal backscattering and modulation. The control circuit within the chip is responsible for switching the impedance states according to the data stored. For the maximum power reception, the input impedance of should be conjugate matched to that the receiving antenna , or more specifically, the input impedance of the tag chip of the rectifier connected to the receiving antenna. Therefore, . For the the design goal for the receiving antenna is maximum signal level difference , the backscattering antenna should be alternatively connected to two impedance loading . A specific and states satisfying simple choice is and . Since , in order to make , the input impedance of must be a purely the backscattering antenna . real number, namely Fig. 3 depicts the block diagram of the proposed dual-antenna tag, including the receiving antenna, the backscattering antenna, and the modified tag IC. The modified tag IC is composed of circuit blocks identical to those used in conventional tag ICs [14]–[16], and no extra circuit components are required. However, the modified tag IC has three terminals instead of two in conventional tag ICs, denoted as RF1, RF2, and common ground, respectively. RF1 and RF2 are connected to the signal terminals of the receiving antenna and the backscattering antenna, respectively, while the common ground is connected to the ground terminals of both antennas. In this proposed tag design, the only modification needed is that the backscattering antenna should be connected directly to the switching transistor, which is controlled by the backscatter modulator and provides either an open- or a short-circuited impedance loading for the antenna. More specifically, the bias condition of the gate of the switching transistor is altered according to the output of the

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Fig. 4. Loci of the input impedance of the transistor. (a) OFF and (b) ON states.

should be designed and optimized simultaneously. Detailed design procedures and the performances of the proposed dual-antenna structure are presented in the following sections. III. DESIGN METHODOLOGY

Fig. 3. Schematics of (a) conventional tag and (b) proposed dual-antenna tag.

backscatter modulator, and thus, the ON and OFF states of the switching transistor can be controlled. The loading impedance for the backscattering antenna is relatively high (open-circuited) as the transistor is turned OFF, and is approximately zero (shortcircuited) as the transistor is turned ON. To further verify the feasibility of this modified configuration, the input impedance of the switching transistor under various bias voltages is simulated by Agilent ADS. An NMOS transistor is chosen as the switching transistor, and the associated DC bias voltage is set to be 3 V and 0 V to turn it ON and OFF, respectively. Fig. 4 shows the loci of the simulated input impedance on the Smith Chart. Obviously, when the transistor is turned OFF and ON, its input impedance at 900 MHz is equal and 0. , respectively. to These impedance values can be considered as satisfactory open and short circuits for terminating the backscattering antenna. Most important of all, such a modification to the tag chip is very simple and can be readily realized; however, the co-design of the receiving and backscattering antennas within a relatively small area remains a challenging task because the mutual coupling between these two closely-spaced antennas may severely affect the performances of the individual antenna. In particular, the loading condition of the backscattering antenna may affect the power reception of the receiving antenna. To achieve the design goals for both antennas while keeping a low mutual coupling, the receiving and backscattering antennas in the proposed tag

The benchmarking configuration of the proposed dual-antenna structure is shown in Fig. 5. It consists of two linearly tapered meander dipoles printed on the opposite sides of the and loss tangent FR4 substrate (dielectric constant ) of thickness . In practical RFID aptan plications, the antennas should be printed on the same metallic layer for easier integration with the chip; however, in our test pieces, they are placed on the opposite sides of the substrate to facilitate soldering of the SMA connectors and testing. The two antennas are perpendicular to each other to reduce the mutual coupling. In addition, both antennas are anti-symmetric and of uniform strip width of 0.2 mm. The geometric parameters for the receiving (backscattering) antenna include the length of the , the length of the first transverse segcentral segments , the spacing between adjacent transverse segments , and the uniform increment in length . Furments thermore, the maximum available area is set to be at 915 MHz, namely , suitable for tagging most small objects in UHF RFID applications. To communicate with the proposed dual-antenna tag structure, the reader antenna must be circularly polarized. However, this is not a problem in most passive UHF RFID applications, where circularly polarized reader antennas are adopted [17]–[20] because the orientations of the tags or objects being tagged are unknown [2], [21]–[23]. Therefore, it is straightforward to replace the conventional tags with the proposed dual-antenna tag in nowadays passive UHF RFID systems. Also, the backscattering antenna and the receiving antenna are made spatially orthogonal to each other, and thus the polarization of the backscattered field would be nearly orthogonal to that of the receiving antenna. Due to the large polarization loss, the interference signal at the receiving antenna is insignificant. Therefore, the signal-to-noise ratio of the receiving antenna of the proposed dual-antenna structure is almost time-invariant, and its effect on the performance of the entire RFID system is negligible. factorial design of experiments (DOE) In this work, the technique [24] is adopted to investigate the dependence of each

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TABLE I RANGES FOR THE INPUT GEOMETRIC PARAMETERS (mm)

Fig. 5. Geometry of the proposed dual-antenna structure.

of the antenna performances on the geometric parameters based on only sixty-five full-wave simulations. The resultant statistical models greatly simplify the optimization process. The full-wave simulations are all carried out by the Ansoft HFSS. Also, an Impinj Monza Gen2 tag chip with a complex impedance [25] at 915 MHz is chosen as an example. To facilitate the performance analysis of the proposed dual-antenna structure, the frequency variation of the chip impedance can be modeled by either a parallel or a series RC equivalent circuit [26], [27]. For the tag IC chosen in this work, the element values of and the series RC equivalent is modeled as in the antenna design. Such lumped element model was also used in [27] to simulate the frequency response of the impedance of a commercial tag chip and to estimate the performance of conventional tags. A. Statistical Model Development Compared to the commonly-used one-parameter-at-a-time design approach, which is time-consuming and without any guarantee of success, the DOE technique is an efficient statistical analysis, in which multiple parameters are varied at the same time, and is capable of identifying the parameters or the interactions between parameters that have more significant effects on the responses. Statistical models for each of the responses are then developed with only the significant parameters being taken into account. By applying the optimization constraints to these models, the values of the input parameters that best satisfy all the constraints can be determined, and an optimum design can thus be obtained [28]–[33].

Assume that there are totally significant parameters for the proposed dual-antenna structure and that each varies at two levels, namely the lower and upper limits of its interval. To begin with, we select a full factorial design with a center point as the experimental method for the DOE. The center point is used to check the linearity of the statistical models we obtained. The selected response variables are the input resistance and reactance of the receiving antenna, the input reactance of the backscat, tering antenna, and the isolation between them, namely , , and , all at the central frequency of the operational band (902–928 MHz). The optimization goals are, at 915 , , , and a miniMHz, mized . Although there are totally eight geometric parameters indicated in Fig. 5, preliminary simulations showed that the effects of the lengths of the central segments and on the response variables are insignificant. So and are both fixed at 7 mm, and a factorial design with a central point is chosen for the DOE design. The ranges of the six input parameters are listed in Table I. Also, note that only one replicate of each experimental combination should be conducted since the full-wave simulations are deterministic. After performing sixty-four simulations, the behavior patterns for the proposed dual-antenna structure can be analyzed. Through the particular-effect calculations [24], the effects of the input parameters and their interactions on each response variable can be obtained. The statistical significances of these effects are further examined by the Lenth method [34], [35], which is particularly used in unreplicated factorial designs. The effects that are considered as statistically significant are then confirmed via the analysis of variance (ANOVA) [24] leading to the final regression models in terms of the geometrical pa, , rameters for the response variables. The models for , and at 915 MHz are obtained as

(5)

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are then utilized to optimize the performance of the proposed dual-antenna structure. B. Optimization of Multiple Responses

(6)

(7)

(8) The residual analyses of these models (5)–(8) are all satisfactory, and the test for lack-of-fit is also performed showing that all the models satisfy the linearity assumption in the DOE [24], [36]. Due to the mutual coupling between the two closely-spaced antennas, the geometric parameters of the backscattering anand , are involved in the models of the input resistenna, tance and reactance of the receiving antenna (5) and (6), while and , are also inthe parameters of the receiving antenna, cluded in the model of the input reactance of the backscattering antenna (7). Also, as one may expect, the model for the isolation (8) is more complicated than the others. These models (5)–(8)

To simultaneously optimize all the responses, we formulate the problem into a constrained optimization problem, in which should be minimized subject to the constraints of , , and . Recall that, for should be equal to 33 at conjugate matching, the value of 915 MHz; however, the input resistance of the tapered meander is found to dipole designed within a small area of be around 10 [37], [38]. Therefore, the optimization strategy above an acceptable limit, say is modified to maximizing 12 . Although using inductive loop as antenna feed may provide better conjugate match condition for the receiving antenna [39]–[41], it is accompanied by two drawbacks in the proposed dual-antenna structure. First, the mutual coupling between the two element antennas would increase drastically, which would degrade the performance of the proposed dual-antenna tag and make it difficult, if not impossible, to design the backscattering antenna with a purely real input impedance. Second, feeding through inductive loops would complicate the feeding network of the proposed dual-antenna structure, in which the element antennas should be orthogonal to each other to reduce the mutual coupling, and inevitably increase the total area occupied. Therefore, the inductive loop is not adopted in this proposed design. The above constrained optimization problem is then solved by the Derringer’s desirability functions [42]. As a series of de, , and satisfying the constraints signs with their are found in a direct-search manner, the Derringer’s desirability functions are applied respectively to the responses of all the designs. Each response is transformed into a dimensionless desirability such that the desirability values obtained from different responses for the same design can be combined for inter-design comparison. The value of ranges between zero, which represents a completely undesirable response, and unity for a fully desired response. The overall desirability for a design is defined as the geometric mean of the desirability values of all the responses of that design, and the design having the largest is thus the optimal. However, it is optimum only within the specified ranges for the geometric parameters listed in Table I. The associated parameters for this optimal dual-antenna design are , , , , , and . The design is then verified by the HFSS, and the simulated responses at 915 MHz , , , and are . The total computational time for finding the optimal design parameters is about twenty-six hours, which is determined mainly by the sixty-five full-wave simulations (all simulations were performed on a computer with a 2.33 GHz processor and 8 GB of RAM). Obviously, with the design methodology presented above, the optimum performances for the proposed dual-antenna structure are obtained in a simple and very efficient manner. IV. EXPERIMENTAL VERIFICATION For verification, the optimum dual-antenna structure is fabricated and tested, and a photograph of the test piece is shown in

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Fig. 6. Photograph of the prototype of the proposed dual-antenna structure.

= 1)

Fig. 8. Simulated and measured input impedances of the receiving antenna with the backscattering antenna being open-circuited (Z .

Fig. 7. Simulated and measured isolation between the receiving and backscattering antennas.

Fig. 6. All measurements are conducted in an anechoic chamber via a HP 8753D vector network analyzer (VNA). A. Isolation and Antenna Impedances The simulated and measured isolation responses for the prototype dual-antenna structure are plotted in Fig. 7, and they show very good agreement. The simulated and measured isolation levels at 915 MHz are 48.4 dB and 46.1 dB, respectively, both indicating a relatively low mutual coupling between the two constituent antennas. Due to such a low mutual coupling, the input impedance of the receiving antenna in the proposed dual-antenna structure is almost unaffected by the change in the impedance loading (either open or short circuit) of the backscattering antenna. The input impedance of the receiving antenna is simulated and measured with the backscattering antenna terminated in turn by an and a short . Figs. 8 and 9 illusopen trate the input impedance responses of the receiving antenna as the backscattering antenna is open- and short-circuited, respectively. Good agreements between the simulated and measured results can be observed in both cases. At 915 MHz, the simulated and measured input impedances of the receiving antenna as the backscattering antenna is open-circuited (short-cirand cuited) are , respectively. One can see that the impedance values are comparable in the two scenarios because of the low mutual coupling between the receiving and the backscattering antennas. This further implies that, in the

Fig. 9. Simulated and measured input impedances of the receiving antenna with . the backscattering antenna being short-circuited Z

(

= 0)

proposed dual-antenna structure, the power received by the receiving antenna would remain unchanged during the process of backscattering modulation. As for the backscattering antenna, the input impedance is measured while the receiving antenna is terminated in a series and , which is used to RC circuit with simulate the tag chip. The resistance and capacitance are realized by 0402 chip resistor and capacitor. Fig. 10 shows the simulated and measured input impedance responses of the backscattering antenna. Good agreement between them can be observed. The simulated and measured input impedances at 915 MHz are and , respectively. The reactance values as expected are close to zero. B. Receiving Performance To facilitate assessing the receiving and the backscattering performances of the proposed dual-antenna structure, a conis ventional tag antenna of the same size also designed and fabricated on the same dielectric substrate (1.6-mm-thick FR4). The conventional design consists of only one linearly tapered meander dipole, which is used for both receiving and backscattering. Its input impedance is designed

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Fig. 10. Simulated and measured input impedances of the backscattering antenna with the receiving antenna being conjugate matched.

to be conjugate matched to the chosen tag chip of impedance at 915 MHz. Initially, the receiving radiation patterns are measured in an anechoic chamber for three cases: the conventional single-antenna design and the proposed dual-antenna structure but with its backscattering antenna being open- and short- circuited, respectively. It must be mentioned that the mismatch losses between the highly inductive receiving antennas and the 50measurement system is compensated for in the measurements. The measured E- and H-plane patterns for the three cases are plotted in Figs. 11(a) and (b), respectively. Clearly, the radiation pattern of the receiving antenna in the proposed dual-antenna structure remains unchanged no matter which impedance state the backscattering antenna is connected to. The receiving patterns resemble that of the conventional design except for the slightly larger cross-polarized radiation components. Moreover, the antenna gain of the proposed dual-antenna structure remains approximately 0.5 dB lower than that of the conventional design, which may be attributed to the larger cross-polarized radiation. The dipole-like patterns make the proposed dual-antenna structure suitable for use in RFID applications. We also setup a measurement system, which is shown in Fig. 12, to further investigate the receiving capability of the proposed dual-antenna structure. Port 1 of the VNA is connected to a double-ridged horn antenna to generate the illuminating continuous wave, while port 2 is connected to the receiving antenna of the dual-antenna structure. Meanwhile, the backscattering antenna is connected to an open circuit and a short circuit via a one-pole-two-throw switch, which is controlled by a personal computer through a GPIB interface and is switched from one state to the other for every 200 ms. For a given measurement frequency, say 915 MHz, the time response of the transmission is recorded. The measured time responses of coefficient or at 915 MHz for various spacing between the horn antenna , 100, 150, and 200 cm) and the dual-antenna structure ( are plotted in Fig. 13. For each spacing , the time response remains flat and stable regardless of the periodically changing loading states for the backscattering antenna. In contrast, the time response for a conventional tag antenna, though not shown here, never holds constant because its loading impedance is

Fig. 11. (a) E- and (b) H-plane patterns measured at 915 MHz for the conventional tag antenna and the proposed dual-antenna structure with the backscattering antenna being open- and short-circuited.

Fig. 12. Experimental setup for measuring the receiving performance of the proposed dual-antenna tag structure.

alternatively switched between a short circuit and a conjugate matched load. During the short-circuited state, no power is supplied to the tag chip. C. Backscattering Performance By alternating the loading impedances for the backscattering antenna, the field intensity backscattered from the tag varies accordingly, and the data stored in the tag chip is thus delivered to the reader. The performance of the backscattering antenna depends on the detection method used in the reader receiver being incoherent or coherent. If the incoherent detection method is employed, the performance of the backscattering antenna is mainly determined by its radar cross sections (RCS) at the two loading states, hence the scalar differential RCS [1],

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Fig. 14. Experimental setup for measuring the backscattering performance.

Fig. 13. Measured receiving performance of the proposed dual-antenna tag for various antenna spacing d as the backscattering antenna is alternatively switched between open and short circuits. (Frequency: 915 MHz).

[4]–[6], [40]. Many RFID tag antennas can be treated as the minimum-scattering antennas [43], and the backscattering RCS as is given as [44], [45] the tag antenna is terminated by

(9) and are where is wavelength and the gain and input impedance of the tag antenna, respectively. Note that (9) is valid only when the reader and tag antennas are polarization-matched. The measurement setup to acquire the differential RCS of the antenna under test (AUT) is shown in Fig. 14. The AUT is illuminated by a plane wave from the transmitting horn antenna, and its backscattered field is received by the receiving horn antenna next to the transmitting one. The AUT and the transmitting and receiving antennas are all placed at the same height and polarization-matched. The proposed dual-antenna structure and the conventional design are used respectively as the AUT. After are measured, the the received-to-transmitted power ratio backscattered RCS of the AUT can be extracted via [44], [46]

(10) where and are the gains of the transmitting and receiving horn antennas, respectively, and is the antenna separation between the AUT and the transmitting/receiving antenna. Here, is chosen. The measured RCSs for the backscattering antenna in the proposed dual-antenna structure when it and open-circuited are is short-circuited plotted in Fig. 15. The associated results predicted by (9) and those simulated via HFSS are also shown in the figure, and they are all in great agreement. Please notice that the receiving antenna of the dual-antenna structure is conjugate matched in the experiments and full-wave simulations. In addition, according to (9), should vanish when the AUT is open-circuited . In the measurement, it is observed that the RCS at open is

Fig. 15. Simulated, measured, and theoretically calculated backscattering RCS of the proposed dual-antenna structure when the receiving antenna is conjugate matched.

much smaller than that at short, and this verifies the effectiveness of (9). Likewise, the measured, simulated, and theoretically predicted RCSs for the conventional tag antenna when it is short-circuited and conjugate matched are shown in Fig. 16. The conjugate matched load is implemented by a series RC and to obtain the frecircuit with quency responses for RCS. The slight differences between the measured and simulated responses may be attributed to the fabrication errors and the ambiguity in the distance between the AUT and the transmitting/receiving antenna. However, reasonably good agreements among them are achieved. The simulated and measured RCSs at 915 MHz for the proposed dual-antenna structure and the conventional design are summarized in Table II. Clearly, the proposed dual-antenna structure exhibits a much larger scalar differential RCS than the conventional tag antenna. The maximum detection range of the proposed dual-antenna structure and the conventional tag can be accurately predicted through the classical radar range equation [12], [47], and it is given by:

(11)

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Fig. 16. Simulated, measured, and theoretically calculated backscattering RCS of the conventional tag structure.

Fig. 17. Calculated detection ranges of the proposed dual-antenna tag and the conventional tag.

TABLE II SIMULATED AND MEASURED BACKSCATTERING RCS AT 915 MHz

where denotes the power transmitted by the reader circuitry, is the gain of the reader antenna, is the free-space waveis the sensitivity of the reader circuitry, and length, is the difference between the RCS of the backscattering antenna under the two loading states. Consider a typical passive RFID system, in which the reader has an EIRP of 4 W and a sensitivity of 80 dBm. The reader antenna and the tag antenna are assumed to be polarization-matched. The maximum detection range can be calculated by substituting into (11) the measured RCS data. The results thus obtained for the proposed dual-antenna structure and the conventional tag are shown in Fig. 17. Obviously, the detection range for the proposed dual-antenna structure remains larger than that of the conventional design throughout the 900–930 MHz band. Also, since the termination impedance for the backscattering antenna of the proposed dual-antenna tag switches between “short” and “open” instead of “short” and “conjugate-matched load,” the backscattering performance would remain unchanged even if the chip impedance varies with the absorbed power or operation frequency. This is thus another advantage of the proposed dual-antenna tag over conventional tags. In the above, the backscattering performances are discussed when the incoherent detection method is employed, which is equivalent to the amplitude shift keying (ASK). If the coherent detection method is employed in the reader receiver [7], [8], both the amplitude and phase of the backscattered field should be considered. The detection capability is proportional to the difference between the complex-valued backscattered fields under the two loading impedances. Thus, system designers in (4). The difference beshould maximize the quantity tween the complex-valued backscattered fields obtained as the backscattering antenna is open- and short-circuited ( and ), namely , is computed and normalized to that of

Fig. 18. Difference between backscattered fields of the proposed dual-antenna structure normalized to that of the conventional design.

the conventional design as being short-circuited and conjugate matched . The simulated and measured frequency responses of the normalized difference between the backscattered fields are plotted in Fig. 18. Throughout the frequency range of 902–928 MHz, the proposed dual-antenna structure exhibits a larger difference between the backscattered fields than that of the conventional design. Therefore, the detection capability or the detection range of UHF RFID systems can be significantly enhanced by the proposed dual-antenna tag structure utilizing either coherent or incoherent readers. V. CONCLUSION A novel tag structure, in which a slightly modified tag chip is connected to a dual-antenna structure, has been proposed. One of the antennas is for receiving and designed to have the maximum power transfer to the tag chip, while the other is for backscattering and designed for the maximum differential backscattering. The design considerations for the dual-antenna structure are discussed intensively. In addition, a compact dualat 915 MHz) with very low muantenna structure ( tual coupling between the constituent antennas has been proposed. The design procedure is greatly simplified by the DOE technique and an optimization method is presented as well. Due to the very low mutual coupling, the receiving performances of the receiving antenna in the proposed dual-antenna structure

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is almost unaffected by the loading condition of the backscattering antenna. The differential RCS of the dual-antenna structure is demonstrated to be much larger than that of the conventional tag design. All the results are verified via measurements and full-wave simulations. With the proposed dual-antenna tag structure, the detection range and reliability of UHF RFID systems may be significantly increased.

REFERENCES [1] K. Finkenzeller, RFID Handbook: Radio-Frequency Identification Fundamentals and Applications, 2nd ed. New York: Wiley, 2004. [2] V. Chawla and D.-S. Ha, “An overview of passive RFID,” IEEE Commun. Mag., vol. 45, no. 9, pp. 11–17, Sep. 2007. [3] G. De Vita and G. Iannaccone, “Design criteria for the RF section of UHF and microwave passive RFID transponders,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 9, pp. 2978–2990, Sep. 2005. [4] A. Bletsas, A. G. Dimitriou, and J. N. Sahalos, “Improving backscatter radio tag efficiency,” IEEE Trans. Microw. Theory Tech., vol. 58, no. 6, pp. 1502–1509, Jun. 2010. [5] K. Penttila, M. Keskilammi, L. Sydanheimo, and M. Kivikoski, “Radar cross-section analysis for passive RFID systems,” Inst. Elect. Eng. Proc., Micorw. Antennas Propag., vol. 153, no. 1, pp. 103–109, Feb. 2006. [6] N. Kim, H. Kwon, J. Lee, and B. Lee, “Performance analysis of RFID tag antenna at UHF (911 MHz) band,” in Proc. IEEE Antennas Propag. Symp., Jul. 2006, pp. 3275–3278. [7] P. V. Nikitin, K. V. S. Rao, and R. Martinez, “Differential RCS of RFID tag,” Electron. Lett., vol. 43, no. 8, pp. 431–432, Apr. 2007. [8] H.-J. Li, C.-T. Lo, and J.-Y. Chen, “Impedance loading state determination for UHF Passive RFID Applications,” in Proc. IEEE Ant. and Propag. Int. Symp., Jun. 2007, pp. 1213–1216. [9] C.-C. Yen, A. E. Gutierrez, D. Veeramani, and D. van der Weide, “Radar cross-section analysis of backscattering RFID tags,” IEEE Antennas Wireless Propag. Lett., vol. 6, pp. 279–281, Jun. 2007. [10] C. A. Balanis, Antenna Theory: Analysis and Design, 3rd ed. Hoboken, NJ: Wiley, 2005. [11] P. V. Nikitin, K. V. S. Rao, S. F. Lam, V. Pillai, and H. Heintich, “Power reflection coefficient analysis for complex impedance in RFID tag design,” IEEE Trans. Microw. Theory Tech., vol. 53, Sep. 2005. [12] G. T. Ruck, D. E. Barrick, W. D. Stuart, and C. K. Krichbaum, Radar Cross Section Handbook. : Plenum Press, 1970, vol. 2. [13] R. B. Green, “The General Theory of Antenna Scattering,” Ph.D. dissertation, Dept. Elect. Comput. Eng., Ohio State Univ, Columbus, OH, 1963. [14] U. Karthaus and M. Fischer, “Fully integrated passive UHF RFID transponder IC with 16.7-  minimum RF input power,” IEEE J. Solid-State Circuits, vol. 38, no. 10, pp. 1602–1608, Oct. 2003. [15] A. Ricci, M. Grisanti, I. D. Munari, and P. Ciapolini, “Design of a low-power digital core for passive UHF RFID transponder,” in Proc 9th Euromicro. Conf. Digital Syst. Design, Sep. 2006, pp. 561–568. [16] J.-P. Curty, N. Joehl, C. Dehollain, and M. J. Declercq, “Remotely powered addressable UHF RFID integrated system,” IEEE J. Solid-State Circuits, vol. 40, no. 11, pp. 2193–2202, Nov. 2005. [17] Intermec Technologies Corp. West Everett, WA [Online]. Available: www.intermec.com [18] Applied Wireless Identifications Group, Inc.. Monsey, NY [Online]. Available: www.awid.com [19] Symbol Technologies, Inc. Holtsville, NY, 11742-1300 [Online]. Available: www.symbol.com [20] SAMSys Technologies, Inc. Ontario, Canada [Online]. Available: www.samsys.com [21] X. Chen, G. Gu, S.-X. Gong, Y.-L. Yan, and W. Zhao, “Circularly polarized stacked annular-ring microstrip antenna with integrated feeding network for UHF RFID readers,” IEEE Antennas Wireless Propag. Lett., vol. 9, pp. 542–545, Jun. 2010. [22] Z. N. Chen, X. M. Qing, and H. L. Chung, “A universal UHF RFID reader antenna,” IEEE Trans. Microw. Theory Tech., vol. 57, no. 5, pp. 1275–1282, May 2009. [23] P. V. Nikitin and K. V. S. Rao, “Helical antenna for handheld UHF RFID reader,” in Proc. IEEE Int. Conf. on RFID, May 2010, pp. 166–173.

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[24] D. C. Montgometry, Design and Analysis of Experiments. New York: Wiley, 1997. [25] Impinj Inc, The RFID Antenna: Maximum Power Transfer ser. Impinj RFID Technology Series, pp. 1–3, 2005 [Online]. Available: www.impinj.com [26] A. Ghiotto, T. P. Vuong, and K. Wu, “Novel design strategy for passive UHF RFID tags,” presented at the Int. Symp. Antenna Technology and Applied Electromagnetics and the American Electromagnetics Conf., Aug. 2010. [27] Monza 4 tag chip datasheet [Online]. Available: www.impinj.com [28] D. Staiculescu, N. Bushyager, A. Obatoyinbo, L. J. Matin, and M. M. Tentzeris, “Design and optimization of 3-D compact stripline and microstrip Bluetooth/WLAN balun architectures using the design of experiments technique,” IEEE Trans. Antennas Propag., vol. 53, no. 5, pp. 1805–1812, May 2005. [29] N. Bushyager, D. Staiculescu, L. Martin, N. Vasiloglou, and M. M. Tentzers, “Design and optimization of 3-D RF modules, microsystems and packages using electromagnetic, statistical and genetic tools,” in Proc. IEEE ECTC Conf. Dog., Jun. 2004, pp. 1412–1415. [30] D. Staiculescu, J. Laskar, and M. M. Tentzers, “Design rule development for microwave flip-chip applications,” IEEE Trans. Microwave Thoery Tech., vol. 48, no. 9, pp. 1476–1481, Sep. 2000. [31] L. Yang, L. J. Martin, D. Staiculescu, C. P. Wong, and M. M. Tentzeris, “Conformal magnetic composite RFID for wearable RF and bio-monitoring applications,” IEEE Trans. Microw. Theory Tech., vol. 56, no. 12, pp. 3223–3230, Dec. 2008. [32] A. Amador-Perez and R. A. Rodriguez-Solis, “Analysis of a CPW fed annular slot ring antenna using DOE,” in Proc. IEEE Antennas and Propag. Int. Symp., Jul. 2006, pp. 4301–4304. [33] F. Placentino, D. Staiculescu, S. Nikolaou, L. Martin, A. Scarponi, F. Alimenti, L. Roselli, and M. M. Tentzeris, “Concurrent circuit-level/ system level optimization of a 24 GHz mixer for automotive applications using hybrid electromagnetic/statistical technique,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2007, pp. 1217–1220. [34] K. Ye and M. S. Hamada, “Critical values of the length method for unreplicated factorial designs,” J. Qual. Technol., vol. 32, no. 1, pp. 57–66, Jan. 2000. [35] M. S. Hamada and N. Balakrishnan, “Analyzing unreplicated factorial experiments: A review with some new proposals,” Statistica Sinica, vol. 8, pp. 1–41, 1998. [36] J. Neter et al., Applied Linear Statistical Models, 4th ed. Chicago: McGraw-Hill, 1996. [37] T. J. Warnagiris and T. J. Minardo, “Performance of a meander line as electrically small transmitting antenna,” IEEE Trans. Antennas Propag., vol. 46, pp. 1797–1876, Dec. 1998. [38] G. Marrocco, “Gain-optimized self-resonant meander line antennas for RFID applications,” IEEE Antennas Wireless Propag. Lett., vol. 2, pp. 302–305, 2003. [39] H.-W. Son and C.-S. Pyo, “Design of RFID tag antennas using an inductively coupled feed,” Electron. Lett., vol. 41, no. 18, pp. 994–996, Sep. 2005. [40] J. Ahn, H. Jang, H. Moon, J.-W. Lee, and B. Lee, “Inductively coupled compact RFID tag antenna at 910 MHz with near-isotropic radar cross section (RCS) patterns,” IEEE Antenna Wireless Propag. Lett., vol. 6, pp. 518–520, 2007. [41] C. Cho and I. Park, “Design of UHF small passive tag antennas,” in Proc. IEEE Antennas and Propagation Society Int. Symp., Jul. 2005, vol. 2, pp. 349–352. [42] G. Derringer and R. Suich, “Simultaneous optimization of several response variables,” J. Qual. Technol., vol. 12, no. 4, pp. 214–219, Oct. 1980. [43] W. Kahn and H. Kurss, “Minimum-scattering antennas,” IEEE Trans. Antennas Propag., vol. 13, no. 5, pp. 671–675, Sep. 1965. [44] P. V. Nikitin and K. V. S. Rao, “Theory and measurement of backscattering from RFID tags,” IEEE Antennas Propag. Mag., vol. 48, no. 6, pp. 212–218, Dec. 2006. [45] R. Harrington, “Electromagnetic scattering by antennas,” IEEE Trans. Antennas Propag., vol. 11, no. 5, pp. 595–596, Sep. 1963. [46] M. O. White, “Radar cross-section: Measurement, prediction, control,” Electron. Commun. Eng. J., vol. 10, no. 4, pp. 169–180, Aug. 1998. [47] A. Aleksieieva and M. Vossiek, “Design and optimization of amplitude-modulated microwave backscatter transponders,” in Proc. German Microwave Conf., July 2010, pp. 134–137.

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Yen-Sheng Chen was born in Taichung, Taiwan, on January 4, 1985. He received the B.S. degree in electrical engineering in 2007, and the M.S. degree in communication engineering in 2009, all from the National Taiwan University, Taipei, Taiwan, where he is currently pursuing the Ph.D. degree. His research interests include the area of near-field communication systems, RFID antennas design, selfstructuring devices, and optimization techniques in electromagnetics.

Shih-Yuan Chen was born in Changhua, Taiwan, in May 1978. He received the B.S. degree in electrical engineering in 2000, and the M.S. and Ph.D. degrees in communication engineering in 2002 and 2005, respectively, all from the National Taiwan University, Taipei, Taiwan. From 2005 to 2006, he was a Postdoctoral Research Fellow with the Graduate Institute of Communication Engineering, National Taiwan University, working on the 60-GHz switched-beam circularly-polarized antenna module. In July 2006, he joined the faculty of the Department of Electrical Engineering and Graduate

Institute of Communication Engineering, National Taiwan University, and served as an Assistant Professor. From August 2008 to July 2009, he visited the Department of Electrical and Computer Engineering, Michigan State University, East Lansing, MI. His current research interests include the design and analysis of slot antennas/arrays, microstrip antennas, RFID tag/reader antennas, near-field communication systems, self-structuring devices, and metamaterial-inspired antennas.

Hsueh-Jyh Li was born in Yun-Lin, Taiwan, China, on August 11, 1949. He received the B.S.E.E. degree from National Taiwan University, Taipei, Taiwan, in 1971, the M.S.E.E. degree in 1980, and the Ph.D. degree from the University of Pennsylvania, Philadelphia, in 1987. Since 1973, he has been with the Department of Electrical Engineering, National Taiwan University, where he is a Professor. He is the Director of the Communication Research Center at NTU from 1995 to 2000. He was the Chairman of the Graduate Institute of Communication Engineering at NTU from 1997 to 2000. He received the Distinguished Research Award from the National Science Council, Republic of China, in 1992. His main research interests are in microstrip antennas, radar scattering, microwave imaging, and wireless communication.

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Embedded Singularity Chipless RFID Tags Andrew T. Blischak and Majid Manteghi, Member, IEEE

Abstract—Every structure scatters an impulse plane wave in a unique fashion. The structural information of an object can be extracted by analyzing the late-time scattered field as the impulse-response of the structure. The late-time scattered field, which represents the source-free response of the structure, contains a summation of damped sinusoids. The frequency and damping factor of these damped sinusoids are uniquely associated with the structural information, and can be used to identify an unknown object. We propose to create uniquely identifiable scattered fields from an object by incorporating “notches” in the structure giving rise to specific damped sinusoids in the source-free scattered field of the structure. In this manner, data can be embedded into the structure of an object which is detectable using electromagnetic waves, allowing a metallic object to serve as a chipless radio-frequency identification tag (RFID). Data is encoded as complex natural resonant frequencies (referred to as poles) in the structure and is retrieved from the scattered field. Data retrieval is based on Singularity Expansion Method (SEM) analysis using target identification techniques. Each complex-frequency pole provides two-dimensional data (real and imaginary) which can be extracted from the late-time impulse response of the structure using a numerical technique such as the Matrix Pencil Method. We have designed and prototyped a 6-bit (3-pole) tag. The tag is analyzed using simulations and measurements. The tag is successfully read remotely via its scattered fields. The measured data are compared with simulation, and are in close agreement. Index Terms—Chipless RFID, matrix pencil, pole singularities, SEM, singularity expansion method.

I. INTRODUCTION

R

ADIO-FREQUENCY identification (RFID) is an automatic identification method using devices called tags and radio waves to store and retrieve remote data [1]. RFID tags can be attached or incorporated in any product for identification purposes. These tags are typically chip-based, and generally contain silicon chips and antennas. There are two subtypes of RFID tags: active tags (which require an internal power source), and passive tags (which do not require an internal power source). In Passive RFID tags, the electrical current induced in the antenna by the radio frequency excitation typically provides the power for the integrated circuit (ASIC) used in the tag. The IC transmits its signal via modulation of the backscattered signal to the reader [2]. Passive tags can be read from a distance ranging from

Manuscript received September 06, 2010; revised November 08, 2010; accepted April 07, 2011. Date of publication August 12, 2011; date of current version November 02, 2011. A. T. Blischak is with the Virginia Tech Antenna Group, Department of Electrical and Computer Engineering, Virginia Tech, Blacksburg, VA 24060 USA. M. Manteghi is with the Bradley Department of Electrical and Computer Engineering, Virginia Tech, Blacksburg, VA 24061 USA (e-mail: manteghi@vt. edu). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2164191

Fig. 1. Required power to read a RFID tag versus distance.

10 cm to several meters. They operate in the frequency band of 13.56 MHz (HF) or the UHF ISM band (902–928 MHz). Because these tags do not require a battery, they can have an unlimited life span and are compact. This paper considers a new class of RFID tags which eliminate the need for a chip. A “chipless” RFID is not an entirely new concept, as several concepts have been presented [3]–[13]. A surface acoustic wave (SAW) device has been used as a wireless passive sensor [12], [13]. SAW devices operate on a timegating technique, which separates the sensed-data from environmental reflections [5], [6], [8]. Due to the costly process of making the SAW and attaching it to the antenna, this type of RFID tag is more expensive than the silicon based tags. The SAW based wireless sensor, based on mismatch techniques, inspired another group to design a chipless RFID using a transmission line-network as the mismatched device to store information [11]. Using the mismatch technique for wireless applications has a fundamental difficulty due to ambiguity between multiple-reflections in the environment and multiple reflections from the mismatched network. In addition, transmission line networks are often large and lossy. To avoid this loss, an ideal design would not use any intermediate stage or extra components other than the individual scattering body. In short, SAW devices are expensive and inherently lossy. As a result, SAW based tags have not found their way into practical RFID applications. The tag presented in this paper differs from currently proposed chipless RFID tags in that it relies solely on the scattering properties of a passive object and requires no auxiliary components. A significant problem with conventional RFID tags is the power-scavenging or self-power requirement [14]. One needs to power a monolithic chip (typically requiring 15 dBm at the chip terminals) to start the communication. In general, the chipless RFID tags do not have such an issue. Fig. 1 compares the estimated power required to read the conventional and chipless RFID tag systems. The curve for the conventional RFID is computed based on the required power to turn on the tag, 15 dBm,

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physical features of the structure and retrieved via the scattered fields. The investigation presented in this paper considers the notched elliptical dipole tag for analysis. The purpose of this article is not to present a mature chipless RFID design, but rather to investigate the feasibility of the chipless RFID, with specific emphasis on Singularity Expansion Method (SEM) based concepts with complex frequency-domain singularities as the method for encoding and retrieving data. To this end, we will use the notched, elliptical-dipole tag to explore the use of SEM derived methods. First, we will introduce SEM concepts that are applicable to our investigation in Section II. We will consider simulated results for the elliptical dipole tag in Section III and then present measured results in Section IV. II. BACKGROUND Fig. 2. Photograph of the notched elliptical dipole tag prototype tag used for the measurements presented in this paper.

at the tag’s antenna port. The reader’s transmitted power is limited by the Federal Communications Commission, FCC, Section 15.247 to an effective isotropic radiated power (EIRP) of 1 Watt [15]. The gain of the antennas at the reader and tag are assumed to be 6 dB (patch antenna) and 1.76 dB (dipole antenna) respectively. The curve for the chipless RFID is computed for a frequency sweeping system with a receiver sensitivity of 100 dBm at 10 GHz. The radar cross section of the tag is considered to be 30 dBsm (a typical value from simulation results). At practical operating distances, the chipless RFID has a clear advantage, requiring less radiated power. This lower radiated power advantage also applies when comparing the chipless RFID tag concept presented in this paper to other proposed chipless RFIDs because it does not include any extra components such as: transmit or receive antennas, transmission lines, or filters. All of these components are sources of loss, and by eliminating them the power needed to detect the tag can approach the theoretical minimum limits. Another advantage, in addition to requiring less radiated power, is that a chipless RFID is more robust in extreme environmental conditions. It will operate until its structure is compromised. At temperatures where chips are rendered inoperable, a chipless RFID has no difficulties operating. There are a few drawbacks, which are significant, associated with chipless RFIDs. No protocol is followed with this type of RFID; therefore, collision is a major challenge. The required reader would need more sophisticated architecture and will be more expensive than existing commercial readers. The basic concept of a chipless RFID is obvious from the name. The question then arises “how are we to store and retrieve data with no chip?”. The chipless RFID presented in this paper is a structure that, when electromagnetically excited, scatters a unique signature containing the desired data. Data is encoded in the structure and retrieved from the scattered fields. An existing example of such a structure is the planar tag of Fig. 2. The notched elliptical dipole tag was presented as a multi-bit scatterer [9]. It is a chipless RFID with bits of data encoded as

A. Singularity Based Encoding The original motivation for the Singularity Expansion Method (SEM) dates back to the 1970s when it was introduced as a means for characterizing the electromagnetic response or scattering behavior of objects subjected to EMP phenomena [16]–[18]. It arises from practical observations noting that electromagnetic responses to impulsive excitations generally take the form of a summation of damped sinusoids. In this section, SEM concepts are briefly reviewed with most emphasis placed on the features and benefits of an SEM based approach to chipless RFID tags. The basic premise of SEM is that an object response to electromagnetic stimuli can be completely characterized by singularities in the complex frequency plane (the double-sided Laplace transform plane) [18]. In terms of singularities, a damped sinusoid corresponds to a complex conjugate pair of pole singularities in the complex-frequency plane. Pole Singularities: In the context of our intended application to chipless RFID tags, passive scattering objects, poles are limited to the left half of the complex-frequency plane. This is attributable to the passive nature of the tags and the decaying nature of any excited and observed scattering response. A pole lying on the imaginary axis represents a sinusoid with no decay or growth. This would be a lossless scenario. In practice, some loss mechanism will always be present (radiation, Ohmic losses, etc.), and so no poles lying exactly on the imaginary axis should be observed. An SEM type solution cast in terms of pole singularities would take the form

(1) where: observed time response; residues or complex amplitudes; complex frequency; poles;

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damping factors; angular frequencies. As mentioned before, the solution is expanded in a form with the poles singularities being only dependent on the geometry of , is dependent on the obthe object. The residue term, servation position, as well as the manner of excitation. Note that the pole terms, , are independent of observation parameters. One particularly useful property of an SEM based solution, specifically when poles are the singularity of interest, is that it can be manipulated into a factored form separating the solution dependencies on various parameters, including parameters of the incident field, observation location, and characteristics of the scattering body [18]. Most important for our purposes, the pole singularities, specifically the pole location in the complex-frequency plane, are found to be aspect independent and depend only on characteristics of the scattering body. The aspect independent nature was immediately recognized as particularly useful for target identification purposes [19]. In essence, the process of creating a chipless RFID tag is a form of target identification with the exception that the target is created in such a way as to be uniquely identifiable. The pole singularities are inherent to the structure and represent the source-free solution or natural oscillations of the tag. Therefore, poles can be created and or manipulated by changing the structure. If a structure can be created or modified such that it has a distinct set of poles, the poles can be used as the parameters in which the data is encoded and retrieved. This encoding is the basis for our chipless RFID concept; to create a structure with a distinct signature of embedded singularities that are retrievable via the scattered fields. Extracting Poles: If we are to retrieve the data from a pole signature, we must first extract the poles from the scattered fields. This extraction is accomplished by considering the scattered, time-domain waveform and using one of several methods for extracting poles. Prony’s method is the first such method for extracting poles from a waveform, however it is too susceptible to noise to be very practical [20]. The Matrix Pencil Algorithm has an acceptable ability to cope with noise, and as a result was used for the purposes of this investigation [21], [22].

Fig. 3. Simulated frequency-domain scattering response of the notched elliptical dipole tag.

III. SIMULATED RESPONSE A. Simulation Setup We simulated the structure to examine the scattering behavior of the notched, planar, elliptical dipole tag shown in Fig. 2. The scattered fields of the tag were simulated in FEKO, a full-wave simulator based on the method of moments. The simulation was run for 802 frequencies from 25 MHz to 20.05 GHZ (25 MHz steps between points). In order to facilitate comparison with measured results, a dielectric of thickness 0.127 mm, , and was incorporated in the simulation. These parameters were chosen to model RT/duroid 5880 material, which was used to fabricate the prototype tag. The tag was assumed to have zero response at 0 MHz. Excitation was in the form of a plane wave, normally incident to the surface of the tag. The plane wave was polarized such that the electric field component is in line with the minor axis of the tag’s constitutive ellipses (the axis as defined in Fig. 2). The tag is sensitive to polarization, with the electric field polarized perpendicular to the slots being the ideal excitation to elicit the desired scattered response from the tag. A purely y-polarized incident plane wave does not excite the notches strongly.

B. Notched Elliptical Dipole

B. Frequency-Domain Scattered Fields

In order to explore the concept of using embedded singularities as a data-encoding method, as well as a mechanism for retrieval, we consider an existing chipless RFID tag, the notched elliptical dipole. The prototype tag used for the measurements can be seen in Fig. 2. The concept of a chipless RFID based on the scattering properties of an object was first introduced in [9]. We described our chipless RFID as a Notched Elliptical Dipole Tag with multi-bit data scattering properties. The notched planar elliptical dipole tag is a planar scattering body which originated as a modification of an existing antenna design [23]. The premise of our work was that we were able to create nulls in the RCS spectrum of the tag by creating physical notches in the structure. Each null was regarded as a bit of data giving the tag 3 bits of data storage.

We now consider the frequency-domain scattered electric fields shown in Fig. 3. The plot is of the -polarized portion of the scattered far fields at an observation point located normal to the surface (on the axis as defined in Fig. 2). The physical slots are 6.32 mm, 7.95 mm, and 9.74 mm in length. They are a half-wave length slot cut in the middle, and thus are quarter-wave resonant structures. In free space assuming quarter-wave length resonance, from shortest to longest, the notch lengths correspond to frequencies of 11.87 GHz, 9.44 GHz, and 7.70 GHz. In practice, these frequencies will be higher than those at which resonant behaviors are observed in measurements due to two factors: the slots have a width and thus an effective length that is longer, and the presence of a dielectric substrate.

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Fig. 4. Simulated scattered time-domain response of notched elliptical dipole tag to impulsive plane wave excitation. The red box indicates the time window used for pole extraction.

Considering the scattered electric field of Fig. 3, it is clear that there are three nulls present in the frequency-domain scattered electric field at 5.65, 7.08, and 8.45 GHz. These nulls are created by the three physical notches, and as expected are at lower frequencies than would be expected based solely on the length of the notches. The differences are within reason to be attributable to the effects of the notch widths and dielectric substrate. The peak at 3.00 GHz and null at 10.6 GHz are inherent to the base dipole tag structure. We are able to observe the effect of the notches in the frequency-domain form of the scattered field. If the data can be recovered from the frequency-domain response, what is the advantage of a singularity-based analysis? By considering a pole signature rather than interpreting the frequency-domain response, we are analyzing discrete parameters rather than interpreting a continuous curve. Due to the nature of poles, they do not change in location due to changes in observation position and/or excitation. This is one of the main differentiators between our detection technique and other chipless RFID’s detection techniques. Considering a signature which is based on pole location (damping, frequency) is the most simplistic approach. A pole singularity and associated residue provide four dimensions of data; frequency, damping, phase, and magnitude. Using all four available parameters associated with each pair of poles, an embedded singularity tag, in addition to being an RFID tag, has the potential to act as a remote sensor. Acting as a remote sensor, the tag could detect rotations, translation, deformations, and possibly more. C. Simulated Pole Signature We now consider the notched elliptical dipole tag in terms of its complex frequency-plane singularities. The process of extracting the poles is carried out on the time-domain form of the scattered fields using the Matrix Pencil Algorithm [21]. The time-domain form of the simulated scattered electric field, shown in Fig. 4, represents the response to an impulsive planewave illumination. Due to the absence of noise in the simulations, the poles can be extracted from a time-window much

Fig. 5. Simulated Pole signature for notched elliptical dipole tag extracted from simulated time-domain scattered fields resulting from an impulsive plane wave excitation.

later in time than would be possible in practice. As a result, for the simulation results, the most important consideration when applying a time window is the starting point such that it is sufficiently later in time from the onset of the excitation. The time window in Fig. 4 indicated by the rectangular box was found to be sufficiently late time. The set of poles show in Fig. 5 were extracted from this portion of the time-domain signal using the matrix pencil algorithm. This set of four poles is the tag signature, and is consistently extracted for time windows of sufficient duration starting after the peak of the response. The three poles at 5.72 GHz, 7.27 GHz, and 8.84 GHz are attributable to the notches. This was validated by selectively removing notches from the structure and noting that the presence or absence of a particular pole corresponds to a particular notch. The pole at 2.98 GHz was present for all combinations of notches, including the tag structure with no notches. This indicates that the 2.98 GHz pole is inherent to the base dipole tag structure. These four poles are the signature of the elliptical dipole tag with all notches present. Since the Matrix Pencil method (which uses SVD for noise reduction) is a nonlinear numerical technique, the resolution of extracted poles is not directly tied to the frequency step of the measured or simulated data. IV. MEASURED RESPONSE A. Measurement Setup The basic measurement setup is that of a bi-static radar designed to capture the scattered response of a metallic body. Fig. 6 shows the measurement arrangement, consisting of two horn antennas placed next to each other pointed in the same direction, with a piece of absorbing foam between them to reduce coupling. A quad-rigid horn (model A6100, EM systems Inc.) with 2–20 GHz bandwidth and 5–19 dB gain in addition to the DRG-118/A (1–18 GHz bandwidth and 6–14 dB gain) is used for this measurement. The measurement plane of the objects for evaluation was defined 1 meter away from the physical apertures of the horns. An HP8510 Vector Network Analyzer was used to measure of the antenna link for a frequency range

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Fig. 7. IFFT of the reference frequency-domain measurement taken with no tag present.

Before extracting the poles and residues, the measured data requires extra treatment due to the presence of undesired elements. These undesired elements include: the contribution of the horns, coupling between the horns, and scattering from background objects. In order to remove these undesired components, three different measurements are required. The necessary measurements and processing are the same for both orientations. Reference Measurement: The first measurement is done with no objects present in the plane of measurement. (2)

Fig. 6. Three configurations used to obtain measurements necessary to process data. (a) Reference measurement with no objects present. (b) Measurement with large ground plane. (c) Measurement with tag present.

of 50 MHz to 20.05 GHz in 25 MHz steps, for a total of 801 points and 20 dBm output power. The tag was measured for two different orientations. First measurement oriented the tags with the surface normal ( axis as defined in Fig. 2) pointed directly at the horns. This configuration will be referred to as the “normal” orientation. The second configuration is with the surface normal rotated by 90 , such that the tag is being read on edge. This measurement will be referred to as the “on edge” measurement. For both orientations, the tag was oriented to be co-polarized with the radiated fields of the horns.

Equation (2) reflects the expected responses in this measurement. Fig. 7 shows this measurement in the time-domain, and the coupling between the horns is clearly evident as an early time waveform at 10 ns. Scattering from background objects is also present in the form of a late time response, but is less distinct than the coupling. We do not care about characterizing these particular responses, but simply removing them. Ground Plane Measurement: The second measurement places a large ground plane in the plane of measurement . Ideally the ground plane acts as an “infinite” ground plane, allowing a pure observation of the horn contributions to the measurement. The contribution of the horns to the measurement is reflected in (3) by the gain terms, and .

(3) Evident in the time-domain form of the measurement in Fig. 8 is a large response due to the presence of the ground plane at 15 ns. The intent of this measurement is to capture the response of the horns so it can later be deconvolved from the response of the tag. Tag Response Measurement: The final measurement is with the tag present. This is the measurement that contains the response of the tag which we want to isolate. This measurement

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accomplished via a subtraction operation in the frequency-domain as shown in (5).

(5) The results of this subtraction can be clearly seen in the timedomain. In Fig. 9, the early time response at 10 ns attributable to the coupling and also the background scattering are removed as desired. Remove Horn Response: The ground plane measurement contains the desired response of the horns, , but also coupling between the horns and background scattering. Removing the unwanted elements via a subtraction of the reference is not valid as it would be subtracting responses that due to the presence of the large ground-plane are not present. To remove the coupling between the horns and background scattering we employed a time window as seen in Fig. 8. The process is represented by (6), and the result is the isolated contribution of the horns and also the free space path losses due to the observation distance. The early and late time responses are windowed out, and the reflection from the ground plane at 15 ns is retained. Using the horn response that was isolated from the ground-plane measurement, we can now extract the tag response.

Fig. 8. IFFT of the frequency-domain ground plane measurement.

(6) Fig. 9. IFFT of the measured frequency-domain response of normal oriented notched elliptical dipole tag before and after subtracting reference measurement.

is represented by (4), and the primary term of interest is the desired tag response, .

Going back to the frequency-domain, the horn response is deconvolved from the tag response via a division of the subtracted tag measurement by the windowed ground-plane measurement. This operation is shown in (7), and the result is , the transfer function representing the impulse response of the tag: (7) C. Isolated Response of the Tag

(4) The response of the horns, the response due to coupling between the horns, and background scattering are also present. Looking at actual measurement results in Fig. 9, the response of the tag is not immediately evident due being lost in the coupling between the horns. B. Processing the Measured Data The following details the results for each data processing step only for the normal oriented case. The processing details for the case with the tag on edge are omitted as they are similar and would be repetitive. The post-processing results are presented for both orientations. Subtract Reference: The first step in processing the data, in order to isolate the response of the tag, is to remove the coupling between the horns and also background scattering. This is

The process of subtraction, windowing, and deconvolution has had several effects upon the data and the combination of these processes can be seen in both the frequency-domain response of Fig. 11, and also in the time-domain response of Fig. 10. Also evident in the time-domain response is that the deconvolution operation has removed the linear phase progression representing the distance between the calibration reference and the location of the tag. This is evident in the time-domain as a shift in the onset of response of the tag. An issue that becomes obvious after deconvolution is that the horns have a limited measurement range, and measurements below 2 GHz are invalid. This behavior is evident in the subtracted and deconvolved response shown in Fig. 11. The measured data below 2 GHz was replaced with extrapolated points using a (frequency) relationship. This was based on assuming Rayleigh type scattering behavior. Rayleigh type behavior is

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Fig. 10. IFFT of the measured frequency response of normal oriented notched elliptical dipole tag after subtracting the reference measurement and deconvolving the ground plane measurement.

Fig. 12. Comparison of the frequency-domain response for simulated versus measured notched elliptical dipole tag pole signatures. Measured results for the tag positioned in two different orientations are presented.

Fig. 13. Comparison of simulated versus measured pole signatures. Fig. 11. Measured frequency-domain response of normal oriented notched elliptical dipole tag at various stages of processing.

dent. If the poles are independent of changes in orientation, then only a single pole signature is needed for all orientations. valid for only a portion of the range below 2 GHz, but it was found unnecessary to assume a more complex approximation of the scattering behavior. Extracting a pole signature is not sensitive to the form of extrapolation as long as it does not introduce any discontinuities at the 2 GHz splice point, and approaches a zero response at 0 Hz. The phase was also extrapolated to assume a linear phase progression in accordance with the retained valid measured points above 2 GHz. The subtracted, time-gated, and deconvolved response in Fig. 11 has frequencies points below 2 GHz extrapolated, and is the response used for the pole extraction. The end result of all the processing is that we now have a frequency-domain response in Fig. 12 which represents the isolated response of the tag. This response shares similar nulls and peaks to the simulated data for the normal orientation. However, the frequency-domain response with the tag on edge is not a clear match. The highest frequency null is not well defined, or even arguably present. This is not unexpected as the simulated results are for the normal orientation of tag, and leads to one of the advantages of an embedded singularity based tag. Embedded singularities in the form of poles are aspect indepen-

D. Measured Pole Signature The time-domain waveform in Fig. 10 is the scattered response of the normal oriented tag to an impulsive plane wave excitation. To ensure that we are considering the source free response of the tag, we must use a time window that considers the late time and is free of the excitation. The time window indicated by the box in Fig. 10 was experimentally found suitable, and application of the matrix pencil algorithm resulted in the poles seen in Fig. 13. These poles represent the measured pole signature of the tag for both a normal orientation and with the tag on edge. In Fig. 13, the simulated poles are also plotted for comparison. The measured pole signatures show close agreement with the simulated signature indicating that the tag has successfully been read. The fact that both orientations were successfully read displays the inherent advantage of singularities arising from their aspect independent nature. V. CONCLUSION Successfully measuring and extracting the pole signature proves the feasibility of an embedded singularity tag. The tag

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considered in this paper only incorporates three pole singularities, and thus has extremely limited practicality. The next step in the evolution of these tags would be to devise tags with practical levels of data density and investigate the issues that arise. These issues could include the potential need for a more intelligent data encoding scheme, the practical limits of data density, dealing with noise and other undesired signals, and undoubtedly more. ACKNOWLEDGMENT The authors would like to thank Dr. W. Davis at Virginia Tech for his kind help and insightful comments, and R. Nealy and T. Y. Yang for their help throughout the measurement process. REFERENCES [1] D. D. Mawhinney, “Microwave tag identification systems,” RCA Rev., vol. 44, pp. 589–610, Dec. 1983. [2] K. V. S. Rao, “An overview of backscattered radio frequency identification system (RFID),” in Proc. 1999 Asia Pacific Microwave Conf., 1999, vol. 3, pp. 746–749. [3] A. Blischak and M. Manteghi, “Pole residue techniques for chipless RFID detection,” in Proc. IEEE Antennas and Propagation Society Int. Symp. (APSURSI’09), 2009, pp. 1–4. [4] A. Blischak and M. Manteghi, “Pole-residue analysis of a notched UWB elliptical dipole tag,” presented at the URSI-USNC National Radio Science Meeting, Boulder, CO, 2009. [5] M. Brandl et al., “A new anti-collision method for SAW tags using linear block codes,” in Proc. 2008 IEEE Int. Frequency Control Symp., 2008, pp. 284–289. [6] S. Harma et al., “Inline SAW RFID tag using time position and phase encoding,” IEEE Trans. Ultrason., Ferroelectr. Freq. Contr., vol. 55, pp. 1840–1846, 2008. [7] S. Preradovic et al., “A novel chipless RFID system based on planar multiresonators for barcode replacement,” in Proc. 2008 IEEE Int. Conf. RFID, 2008, pp. 289–296. [8] M. Pasternak and J. Pietrasinski, “Quasi-loop antenna for SAW RFID device,” in Proc. Microwaves, Radar and Remote Sensing Symp. (MRRS 2008), 2008, pp. 201–203. [9] M. Manteghi and Y. Rahmat-Samii, “Frequency notched UWB elliptical dipole tag with multi-bit data scattering properties,” in Proc. 2007 IEEE Antennas and Propagation Society Int. Symp., San Diego, CA, 2007, pp. 789–792. [10] S. Mukherjee, “Chipless radio frequency identification by remote measurement of complex impedance,” in Proc. 2007 Eur. Conf. Wireless Technologies, 2007, pp. 249–252. [11] Z. Lu et al., “An innovative fully printable RFID technology based on high speed time-domain reflections,” in Proc. Conf. High Density Microsystem Design and Packaging and Component Failure Analysis (HDP’06), 2006, pp. 166–170. [12] L. Reindl et al., “SAW devices as wireless passive sensors,” in Proc. 1996 IEEE Ultrasonics Symp., 1996, vol. 1, pp. 363–367. [13] L. Reindl and W. Ruile, “Programmable reflectors for SAW-ID-tags,” in Proc. 1993 IEEE Ultrasonics Symp., 1993, vol. 1, pp. 125–130. [14] D. M. Dobkin, The RF in RFID: Passive UHF RFID in Practice. Boston, MA: Elsevier/Newnes, 2008.

[15] S. Dayhoff, “New policies for part 15 devices,” presented at the TCBC Workshop, May 13–15, 2005, Online. [16] C. E. Baum, “On the singularity expansion method for the solution of electromagnetic interaction problems,” Interaction Notes, vol. EMP-3, pp. 1–111, 1971. [17] C. Baum, “The singularity expansion method: Background and developments,” IEEE Antennas and Propagation Soc. Newslett., vol. 28, pp. 14–23, 1986. [18] L. B. Felsen and C. E. Baum, Transient Electromagnetic Fields. New York: Springer-Verlag, 1976. [19] C. E. Baum et al., “The singularity expansion method and its application to target identification,” Proc. IEEE, vol. 79, pp. 1481–1492, Oct. 1991. [20] M. VanBlaricum and R. Mittra, “Problems and solutions associated with Prony’s method for processing transient data,” IEEE Trans. Antennas Propag., vol. 26, pp. 174–182, 1978. [21] T. K. Sarkar and O. Pereira, “Using the matrix pencil method to estimate the parameters of a sum of complex exponentials,” IEEE Antennas Propag. Mag., vol. 37, pp. 48–55, 1995. [22] R. S. Adve et al., “Extrapolation of time-domain responses from threedimensional conducting objects utilizing the matrix pencil technique,” IEEE Trans. Antennas Propag., vol. 45, pp. 147–156, 1997. [23] H. Schantz, The Art and Science of Ultrawideband Antennas. Boston, MA: Artech House, 2005. Andrew T. Blischak was born in Pittsburgh, PA, in 1980. He received the B.S. degree in electrical engineering from Virginia Tech, Blacksburg, VA, in 2007, where he is currently working towards the M.S. degree. He currently resides in Annapolis, MD, and works as a Staff Engineer for ARINC Engineering Services. His interests include antennas, propagation, and simulation of communication systems using Matlab/Simulink.

Majid Manteghi (M’05) received the B.S. and M.S. degrees from the University of Tehran, Tehran, Iran, in 1994 and 1997, respectively, and the Ph.D. degree from the University of California, Los Angeles, in 2005, all in electrical engineering. From 1997 to 2000, he worked in the telecommunication industry in Tehran, where he served as the head of an RF group for a GSM base transceiver station project. In the fall of 2000, he joined the Antenna Research, Analysis, and Measurement Laboratory (ARAM) of the University of California, Los Angeles (UCLA). His research areas have included ultra-wideband (UWB) impulse-radiating antennas (IRAs), phased-array design, miniaturized patch antennas, multi-port antennas, dual-frequency dual-polarized stacked-patch array designs, mobile TV antennas, RFID circuits and systems, and miniaturized multi-band antennas for MIMO applications. Dr. Manteghi was a research engineer and lecturer with the Electrical Engineering Department of UCLA until spring 2007, while collaborating with Mojix Inc. as a research scientist. He has been with the Bradley Department of Electrical and Computer Engineering at Virginia Tech as assistant professor and a faculty member of the Virginia Tech Antenna Group (VTAG) since fall 2007. His research focuses on nonlinear time-variant techniques in antennas and microwave devices, chipless RFID, SEM based space-time-frequency detection techniques, and wireless power transportation.

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External-Excitation Curl Antenna Hisamatsu Nakano, Life Fellow, IEEE, Shigetsugu Kirita, Naoki Mizobe, and Junji Yamauchi, Senior Member, IEEE

Abstract—A strip curl antenna is investigated for obtaining a circularly-polarized (CP) tilted beam. This curl is excited through a strip line (called the excitation line) that connects the curl arm to a coaxial feed line. The antenna structure has the following features: a small circumference not exceeding two wavelengths and a small antenna height of less than 0.42 wavelength. The antenna arm is printed on a dielectric hollow cylinder, leading to a robust structure. The investigation reveals that an external excitation for the curl using a straight line (ST-line) is more effective for generating a tilted beam than an internal excitation. It is found that the axial ratio of the radiation field from the external-excitation curl is improved by transforming the ST-line into a wound line (WD-line). It is also found that a modification to the end area of the WD-line leads to an almost constant input impedance (50 ohms). Note that these results are demonstrated for the Ku-band (from 11.7 GHz to 12.75 GHz, 8.6% bandwidth). Index Terms—Circularly polarized wave radiation, curl antenna, tilted beam.

I. INTRODUCTION

B

ASE station antennas are often required to have a tilted beam for better directivity toward a wide or a narrow region, depending on the requirements. In most cases, the tilted beam is generated by using an array technique [1], [2]. For example, a tilted beam in the x-z plane of the rectangular coordinate system (x, y, z) is generated by arraying radiation elements in the x direction and adjusting the excitation phase of each of the elements. Suppose that such an array is composed of two elements (a minimum array) and fabricated along the x-direction of the x-y plane (installation plane). This minimum array needs a dimenin the x direction, where is the dimension sion of is the of the array element in the x direction (for example, side length of a rectangular patch or the diameter of a circular patch) and S is the spacing between the two elements. In contrast, if a single element itself has a small dimension of in the x direction and radiates a tilted beam in the x-z plane, as desired, the antenna installation area is smaller than that for the minimum array. The antenna composed of two bent leaf-like conductors (BeToL) in [3] is one such antenna, where one of the two conductors is excited with the other being parasitic. The side length D of the BeToL is small: 0.22 wavelength. The BeToL

Manuscript received October 23, 2010; revised April 06, 2011; accepted May 09, 2011. Date of publication August 12, 2011; date of current version November 02, 2011. The authors are with the College of Engineering, Hosei University, Koganei, Tokyo 184-8584, Japan (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2164196

forms a linearly polarized (LP) tilted beam with the help of the radiation from the current induced on the parasitic conductor. Note that the antenna composed of four bent leaf-like conductors (BeFoL) in [4] also radiates a LP tilted beam, with a small . antenna dimension of In addition to LP tilted beam antennas, circularly polarized (CP) tilted beam antennas have also been developed [5]–[7]. The antenna in [5] is composed of two loops above a ground plane; one loop has a circumference of one wavelength and the other has a circumference of two wavelengths. Each of these loops radiates a CP wave using perturbation elements attached to the loop. A tilted beam is formed by superimposing these CP radiated waves. The antennas in [6], [7] consist of a single spiral arm, unlike the conventional two spiral arms, and form a CP tilted beam by superimposing second-mode radiation onto first-mode radiation [8]. It is emphasized that these loop- and spiral-based antennas used for generating a CP tilted beam need a minimum antenna circumference of two wavelengths. In other words, the antenna dimension (diameter D) cannot be less than , where is the free-space wavelength. This paper presents a new CP tilted beam antenna, using a curl element, where the antenna diameter D is smaller than for the conventional loop- and spiral-based CP tilted beam antennas. Realization of this CP tilted beam is demonstrated using a frequency range of 11.7 to 12.75 GHz (Ku-band, used for satellite communication services), where the antenna height is made small: less than 0.42 wavelength within the range. Note that the curl is made of a single arm whose circumference is slightly larger than one wavelength but less than two wavelengths [9]–[11]. Six sections constitute this paper, including this section. Section II describes the design specifications and investigates a curl when its respective outermost and innermost points are excited through a straight line (ST-line) connected to the inner conductor of a coaxial feed line. The excitations are referred to as the external excitation and internal excitation, respectively. Section III is devoted to improving the axial ratio in the beam direction for the external-excitation curl. For this, the ST-line is transformed into a wound line (WD-line). A detailed analysis of the antenna characteristics for the external-excitation curl having a WD-line is performed in Section IV. Section V presents a technique for matching the impedance of the external-excitation curl to a 50-ohm feed line and analyzes the frequency response of the antenna characteristics for the impedance-matched curl. Finally, the obtained results are summarized in Section VI. II. EXTERNAL- AND INTERNAL-EXCITATION CURLS When elements, each radiating a tilted CP beam, are arrayed side by side in a row, this one-dimensional linear array forms a high-gain CP tilted beam, which is applied to a base station

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antenna for satellite communications systems [this one-dimensional linear array has a smaller installation area than conventional two-dimensional phased arrays using non-tilted beam elements]. For such applications (market requirements), we develop an element radiating a CP tilted radiation beam. The antenna design and consideration is performed using a practical satellite communications frequency range of 11.7 GHz to 12.75 GHz , which is referred to as the design frequency range. The angular separation between the ground and satellite is assumed to be around 30 and hence the direction of the beam radiated from a CP element/antenna to be de, measured from the veloped is needed to be around zenith. The following three requirements are imposed on the antenna design: (1) a small antenna circumference not exceeding , i.e., a small antenna diameter not extwo wavelengths , (2) a low-profile structure ceeding not exceeding half the wavelength, and (3) a robust structure. Meeting these requirements contributes to realization of a robust low-profile antenna with small dimensions [Requirement (1) accommodates suppression of grating lobes when this antenna is used as an array element. Currently, there are no elements meeting these three requirements]. A center-fed single-arm spiral antenna backed by a conducting reflector radiates a quasi-axial CP beam (whose maximum radiation appears near the antenna axis normal to the spiral plane) when the antenna circumference is between one wavelength and two wavelengths. As the antenna circumference is increased from two wavelengths, the CP radiation beam becomes tilted from the antenna axis [6], as desired. However, this structure does not meet requirement (1). A helical antenna is also a CP antenna. When the circumference of a helical antenna is between 3/4 wavelength and 4/3 wavelengths, the antenna radiates an axial CP beam [12]. As the antenna circumference is further increased from 4/3 wavelengths, the radiation becomes tilted. For this tilted beam to be circularly polarized, the antenna height above the ground plane (longitudinal length) must be greater than one wavelength. It follows that this helical antenna does not meet requirement (2). To achieve requirements (1) and (2), an antenna structure different from the above-mentioned spiral and helical antennas must be created. For this, we first consider a curl located at height h above a ground plane of infinite extent, shown in Fig. 1, where the radial distance from the center point to a point on the curl, , is defined by an equiangular function:

(1) being the maximum radial distance of the curl (anwith tenna radius), measured from the center point , “a” being the curl growth rate, and being the winding angle . The curl is excited at the outermost point P (and hence the antenna is referred to as an external-excitation curl) through a vertical straight strip line (ST-line), whose bottom end is connected to the inner conductor of a coaxial feed line. Note that . both the curl and the ST-line have the same strip width The analysis is performed using the finite-difference time-domain method (FDTDM) [13], [14], where the antenna analysis

Fig. 1. External-excitation curl antenna with a vertical straight excitation strip line (ST-line). (a) Perspective view, (b) curl on top, (c) x-z plane cut, and (d) modeling for junction between the excitation strip line and curl.

space is divided into numerous cubes. Fig. 2 reveals the radiation pattern (elevation pattern) of the external-excitation curl antenna at the center frequency of the design frequency range, , where the axis shows the beam direction in the azimuth plane [see Fig. 1(a)]. This radiation pattern is calculated using the equivalence principle [15] with the obtained FDTDM electric and magnetic fields. The configura, tion parameters used for this calculation are , and , meeting requirements (1) and (2). For requirement (3), the curl and the excitation ST-line are supported by (printed on) a dielectric (Teflon) hollow cylinder of thickness t and relative permittivity . In order to reduce the weight of the dielectric cylinder (important when an array composed of numerous curls is constructed), the thickness t is se. The parameters that remain fixed lected to be small: in Table I is throughout this paper are summarized in Table I; not used for Fig. 2, which will be defined in Section III. For modeling the cylinder, 5 and 34 cubes (each having a side length of ) are used for thickness of and a height , respectively; the junction P between the exciof for a width of ) tation strip line and the curl ( is shown in Fig. 1(d).

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Fig. 2. Radiation from the external-excitation curl at f = 12:225 GHz. The = 6:8 mm,  = 4 rad: , and h = configuration parameters are r 6:8 mm. Other parameters are shown in Table I.

TABLE I FIXED PARAMETERS

Fig. 4. External-excitation curl antenna with a wound excitation strip line (WD-line). (a) Perspective, (b) curl on top, and (c) x-z plane cut.

III. REDUCTION OF THE CROSS-POLARIZATION COMPONENT

Fig. 3. Radiation from the internal-excitation curl at f = 6:8 mm,  configuration parameters are r 6:8 mm. Other parameters are shown in Table I.

= 12:225 GHz. = 4 rad,

and

The

h =

As seen from Fig. 2, a tilted right-handed CP wave is obtained in the elevation plane, as desired. However, the cross-po) is undesirlarization component (left-handed electric field ably large. This is attributed to the radiation from the excitation ST-line, which radiates a linearly polarized wave, affecting the radiation from the curl located at the top. For comparison, Fig. 3 shows the radiation pattern for the internal-excitation curl, where the curl is excited at the innermost point T through the same excitation ST-line, as shown in the inset. The principal component and the cross-polarization comand , respecponent for the internal-excitation curl are tively, as opposed to those for the external-excitation curl, due to the progressive direction of the current. It is found that the beam tilt angle from the z-axis for the internal-excitation curl is small, compared with that for the external-excitation curl. Therefore, we focus on the external-excitation curl for our goal and continue our discussion in the following sections.

In this section we reduce the undesirable high cross-polarshown in Fig. 2 for the external-excitaization component tion curl above an infinite ground plane. For this, the excitation ST-line is replaced with a wound excitation line (WD-line), ( as shown in Fig. 4, where a small segment of length , see Table I) is added to the bottom of the WD-line in order to facilitate soldering the WD-line to the inner conductor and of a coaxial feed line. The WD-line, having strip width , is wound with an angle of around the z-axis radius is called “the rotation angle”, measured from the x-axis ( to a projection of the outermost point P onto the x-y plane), with a slope of angle . Note that the winding sense of the WD-line is the same as that of the curl in order to strengthen the principal , resulting in a decrease in the cross-poradiation component . larization component Due to the rotation angle , (1) is transformed into

(2) starts at and ends at , where angle being the winding angle of the curl. Note that requirewith appropriately. The rotation angle ment (1) is met by setting and slope are chosen such that the antenna height meets

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Fig. 6. Current distributions at f

;  ; r ;  ; r

( (

; ;

) = ) = (8

;

(4

; ;r

=

;r ;

:

12 225

;

GHz. (a) ). (b)

For For

).

TABLE II ORIGINAL PARAMETERS

Fig. 5. Radiation pattern for (;  ; r ;  ) = (variable;  ; ; ) at f = 12:225 GHz. (a)  = 4 . (b)  = 8 . (c)  = 12 .

r

requirement (2). The WD-line is printed on the dielectric hollow cylinder, as is the curl, meeting requirement (3). , with Fig. 5 shows the radiation pattern at the slope as a parameter, where the following configuration parameters are used together with the parameters shown in , maxTable I: curl winding angle , and rotation angle imum radius . This situation is expressed as . It is found that an appropriate selection of slope leads to a small cross-polarization component , which is attributed to the fact that the out-going current flowing toward the arm end T (the innermost point of the curl) is dominant, compared with the in-coming current flowing toward the feed point F (the reflected current from the arm end T). For a better understanding of this, and are shown the current distributions for in Fig. 6, where the current ( with being the amplitude) is calculated by integrating the obtained FDTDM magnetic field around the antenna arm. The current for exhibits smoother decay from the feed point F than that for ; the current for has ripples in the amplitude due to the presence of the in-coming current. Note that at the antenna height h (including ) is and 0.28 wavelength approximately 0.15 wavelength at

at , meeting requirement (2). Also, note that the slope controls the beam direction angle in the elevation plane: for and for . IV. VARIATION IN THE RADIATION CHARACTERISTICS Section III reveals that the cross-polarization component for the external-excitation curl above an infinite ground plane is small when configuration parameters of are used together with the fixed parameters shown in Table I. The will now be parameters referred to as the original parameters (see Table II). In this section, the radiation characteristics are investigated when each of these original parameters, except for slope , is varied from its original value. First, analysis is performed by varying the curl , i.e., winding angle . Fig. 7 and Fig. 8 show at the center frequency the beam direction and the axial ratio (AR) in the beam direction at , respectively. It is found that the beam direction is not sensitive to the curl winding angle. It is also found that the axial ratio in the beam direction is improved, with a wave-like variation as the curl winding angle is increased; a suitable

NAKANO et al.: EXTERNAL-EXCITATION CURL ANTENNA

Fig. 7. Beam direction ( ; ) at f ; ) = ( ; variable; r (;  ; r

=

;

Fig. 8. Axial ratio in the beam direction ( ; ; ) = ( ; variable; r where (;  ; r

Fig. 9. Beam direction ( ; ) at f = (;  ; r ; ) = ( ;  ; variable; 

12:225 GHz, where ).

) at f = 12:225 GHz, ; ).

12:225 GHz, where ).

axial ratio is obtained around . Note that an leads to a longer curl arm, which is needed for increase in an out-going current traveling to decay from the feed point F toward the arm end T (as a result, the undesirable in-coming current toward the feed point F is reduced). Also, note that each of the out-going and in-coming currents do not necessarily exhibit a smooth decay, due to mutual effects between the currents flowing on neighboring curl conductors; this causes a non-monotonous change in the radiation pattern (and hence the axial ratio). Second, the radiation for is analyzed, where only

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Fig. 10. Axial ratio in the beam direction ( ; ) at f = 12:225 GHz, where (;  ; r ; ) = ( ;  ; variable;  ).

; ) at f = 12:225 GHz, where Fig. 11. Beam direction ( ; ) = ( ;  ;r ; variable). (;  ; r

the antenna radius among the original parameters is varied. Figs. 9 and 10 show, respectively, the beam direction and the axial ratio in the beam direction, both at . The largest analysis the center frequency corresponds to 0.3 wavelength at radius of , meeting requirement (1). The frequency analysis results in Fig. 9 reveal that the beam direction remains . almost unchanged with a change in the antenna radius It is also revealed from Fig. 10 that the axial ratio in the beam direction is less sensitive to the antenna radius, as opposed to is varied the previous case where the curl winding angle (see Fig. 8). This derives from the fact that the variation in the antenna radius does not generate any significant change in the current distribution on the antenna arm conductor from the feed point F to the arm end T. among the Third, only the WD-line rotation angle original parameters is varied, i.e., , to investigate its effects on the radiation beam. Note that the antenna height including for is 0.34 wavethe largest analysis value of , meeting requirement (2). length at frequency As the rotation angle is increased (or decreased), the length of the WD-line becomes longer (or shorter). This variation in the WD-line length changes the amount that the current along the WD-line contributes to the radiation beam. Over an angle to , the beam azimuth range of angle varies approximately 105 , as shown in Fig. 11, and the axial ratio in the beam direction varies between 2.5 dB

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Fig. 14. Transformed arm.

Fig. 12. Axial ratio in the beam direction ( ; ; ) = ( ;  ;r where (;  ; r

) at f = 12:225 GHz, ; variable).

Fig. 13. Frequency response of the input impedance, where the configuration parameters shown in Tables I and II are used. The input impedance before transforming the arm is shown by lines without dots. The input impedance after transforming the arm is shown by lines with dots. The ground plane is of infinite extent.

and 0.5 dB, as shown in Fig. 12. It can be said that the rotation controls the CP beam azimuth angle . angle V. INPUT IMPEDANCE MATCHING AND FREQUENCY RESPONSES OF THE RADIATION PATTERN AND GAIN Section IV reveals the effects of the configuration parameters on a CP tilted radiation beam, including the beam direction and the axial ratio. Based on these results, we select an external-excitation curl antenna having the parameters shown in Tables I and II and investigate the frequency response of the input impedance. In addition, an impedance matching technique and the frequency response of the other antenna characteristics (radiation pattern and gain) are discussed. The lines without dots in Fig. 13 show the input impedance for the curl with an infinite ground plane, which is calculated by using the input current (obtained by integrating the FDTDM magnetic field around the antenna arm at the feed point F) for a given applied voltage. It is found that, to within the design frequency range (from ), the input reactance is small, and the input resistance is approximately . Next we transform the input impedance into 50 ohms, to accommodate most applications in the microwave frequency range. For this, the antenna arm near the antenna input F is transformed, as shown in Fig. 14, where the transformed sec-

Fig. 15. Frequency response of the VSWR, where the transformed arm section has parameters of (L ; W ) = (5:2 mm; 2:4 mm). (a) Theoretical result = ) and experimental result (D = 200 mm). (b) Photo of proto(D type antenna used for the experiment.

1

tion (matching section) is expressed by length and width . It is found that there exists an optimal arm transformation; provide an almost parameters resistive value of 50 ohms, as desired, as shown by the lines is approximately “after trans.” in Fig. 13. This length at the design center frequency of , where is and is the the guided wavelength given by free-space wavelength. The theoretical VSWR calculated from the input impedance after transforming the arm is shown in Fig. 15, together with experimental results (dots). The ground plane used for this experiment (and for the following investigations) is chosen to be large, in order to approximate the theoretical ground plane of infinite extent; the ground plane is circular with a diameter of , where is the free-space wavelength at the center frequency [see a proto-

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Fig. 17. Frequency response of the axial ratios before and after transforming the arm. An infinite ground plane is used for the theoretical axial ratio.

Fig. 18. Frequency response of the gain for the curl having a transformed arm section. An infinite ground plane is used for the theoretical gain.

Fig. 16. Radiation pattern for the curl having a transformed arm section. (a) f = 11:75 GHz. (b) f = 12:225 GHz. (c) f = 12:75 GHz. An infinite ground plane is used for the theoretical radiation pattern.

type curl shown in Fig. 15(b)]. The theoretical and experimental results are in good agreement; the VSWR is less than 2 within the design frequency range of to , as desired. Further calculation beyond this frequency range reveals that the impedance ) is wide: 54% from bandwidth (for a 8.9 GHz to 15.5 GHz. This bandwidth is narrower than that obtained by inserting a conventional L-network matching circuit [16] (77%: from 7.5 GHz to 16.9 GHz), but the simple arm transformation provides a sufficient wide bandwidth exceeding to the design frequency range (8.6%: from ), without using any matching circuits. The radiation patterns before and after transforming the arm (i.e., without and with the impedance matching section) are almost the same; in other words, the matching section does not remarkably affect the radiation pattern. A finding is that the

radiation pattern around the z-axis [the radiation pat] is wide, tern in the horizontal (azimuth) plane at as shown in Fig. 16, where an infinite ground plane is used for calculating the theoretical radiation pattern. The radiation patis confirmed by the experimental work tern at (dots) using the prototype shown in Fig. 15(b). For additional information, the theoretical axial ratios (ARs) before and after transforming the arm are shown by the dashed and solid lines in Fig. 17, respectively; the ground plane is assumed to be infinite extent. These ARs are calculated in the fixed beam direction, which is chosen to be the beam direction observed at the center : . It is frequency found that the AR after transforming the arm is less than 3 dB within the design frequency range, as desired. The dots in this figure show the ARs measured using the prototype. radiation pattern shown It is expected that the wide (for the in Fig. 16 will lead to the fact that the gain curl having the transformed arm) observed in the fixed beam diremains almost unchanged rection within the design frequency range. This is confirmed in Fig. 18, where the theoretical gain (solid line) is in good agreement with experimental results (dots) obtained using the prototype shown is the gain for a right-handed in Fig. 15(b). Note that relative to an isotropic antenna; the theoretradiation field is calculated under the condition that the ground ical plane is of infinite extent and the antenna materials, including

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Fig. 19. Frequency response of the gain for the curl having a transformed arm as a parameter, where the ground plane with the ground plane diameter :  . size is expressed using 

D = 24 5 mm =

from the external-excitation curl is tilted in the elevation plane with a large cross-polarization component. It is found that this large cross-polarization component is reduced by winding the ST-line; this wound line is designated as the WD-line. Effects of the configuration parameters, including the , the antenna WD-line slope , the curl winding angle , and the WD-line rotation angle , on the radius radiation beam are analyzed. The analysis shows that the beam is sensitive to ; the axial ratio in the direction angle ; and both the beam direction beam direction depends on . and the axial ratio are less sensitive to The input impedance is also analyzed and its transformation to 50 ohms for impedance matching is realized by modifying the antenna strip arm near the input terminal. As a result, a VSWR of less than 2 (relative to 50 ohms) is obtained over a design frequency range of 11.7 GHz to 12.75 GHz. A wide radiation pattern over this design frequency range leads to a relatively constant gain in the fixed direction (approximately 9 dBi).

ACKNOWLEDGMENT The authors thank V. Shkawrytko and H. Mimaki for their assistance in the preparation of this manuscript.

REFERENCES

Fig. 20. Frequency response of the axial ratio for the curl having a transformed as a parameter, where the ground plane arm with the ground plane diameter size is expressed using  . : 

D = 24 5 mm =

the arm conductor, cylindrical dielectric, and ground plane conductor, are lossless. Note also that impedance mismatch to the . feed line is not included in this Lastly, the effects of the ground plane size on the antenna characteristics are revealed in Figs. 19 and 20, where the frequency responses of the gain and axial ratio are obtained with of a circular ground plane as a parameter. the diameter It is found that the gain and axial ratio for the finite ground plane approach those for an infinite ground plane, as the ground plane diameter is increased. It can be said that the ground plane of the prototype shown in Fig. 15(b) well approximates an infinite ground plane. Note that a change in the VSWR is also . negligibly small for

VI. CONCLUSIONS Two kinds of curl antennas fed through a straight excitation line (ST-line) are investigated: external-excitation and internal-excitation curls. The principal component of radiation

[1] W. L. Stutzman and G. A. Thiele, Antenna Theory and Design. New York: Wiley, 1981, ch. 3. [2] C. Balanis, Antenna Theory. New York: Harper, 1982, ch. 6. [3] H. Nakano, Y. Ogino, and J. Yamauchi, “Bent two-leaf antenna radiating a tilted, linearly polarized, wide beam,” IEEE Trans. Antennas Propag., vol. 58, no. 11, pp. 3721–3725, Nov. 2010. [4] H. Nakano, Y. Ogino, and J. Yamauchi, “Bent four-leaf antenna,” IEEE Antennas Wireless Propag. Lett., vol. 10, pp. 223–226, 2011. [5] K. Hirose, S. Okazaki, and H. Nakano, “Double loop antennas for a circularly polarized tilted beam,” (in Japanese) Trans. IEICE, vol. J85-B, no. 11, pp. 1934–1943, Nov. 2002. [6] H. Nakano, Y. Shinma, and J. Yamauchi, “A monofilar spiral antenna and its array above a ground plane—Formation of a circularly polarized tilted fan beam,” IEEE Trans. Antennas Propag., vol. 45, no. 10, pp. 1506–1511, October 1997. [7] H. Nakano, Y. Okabe, H. Mimaki, and J. Yamauchi, “Monofilar spiral antenna excited through a helical wire,” IEEE Trans. Antennas Propag., vol. 51, no. 3, pp. 661–664, March 2003. [8] J. A. Kaiser, “The Archimedean two-wire spiral antenna,” IRE Trans. Antennas Propag., vol. AP-8, no. 3, pp. 312–323, May 1960. [9] H. Nakano, S. Okuzawa, K. Ohishi, H. Mimaki, and J. Yamauchi, “A curl antenna,” IEEE Trans. Antennas Propag., vol. 41, no. 11, pp. 1570–1575, March 1993. [10] H. Nakano and H. Mimaki, “Axial ratio of a curl antenna,” in Inst. Elect. Eng. Proc. Microwaves, Antennas Propag., 1997, vol. 144, no. 6, pp. 488–490. [11] F. Yang and Y. Rahmat-Samii, “Curl antennas over electromagnetic band-gap surface: A low profiled design for CP applications,” in Proc. IEEE Antennas Propag. Society Int. Symp., Jul. 2001, vol. 3, pp. 372–375. [12] J. D. Kraus and R. J. Marhefka, Antennas, 3rd ed. New York: McGraw Hill, 2003, ch. 8. [13] K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag., vol. 14, no. 3, pp. 302–307, May 1966. [14] A. Taflove, Computational Electrodynamics. Norwood, MA: Artech House, 1995. [15] R. F. Harrington, Time-Harmonic Electromagnetic Fields. New York: McGraw-Hill, 1961, pp. 106–110. [16] D. Pozar, Microwave Engineering. New York: Wiley, 1998, p. 252.

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Hisamatsu Nakano (M’75–SM’87–F’92–LF’11) received the B.E., M.E., and Dr.E. degrees in electrical engineering from Hosei University, Tokyo, in 1968, 1970, and 1974, respectively. Since 1973, he has been a member of the faculty of Hosei University, where he is now a Professor of electronic informatics. His research topics include numerical methods for low- and high-frequency antennas and optical waveguides. Prof. Nakano received the IEE (currently IET) International Conference on Antennas and Propagation Best Paper Award and the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION Best Application Paper Award (H. A. Wheeler Award) in 1989 and 1994, respectively. In 1992, he was elected an IEEE Fellow for contributions to the design of spiral and helical antennas. He was also the recipient of the Chen-To Tai Distinguished Educator Award (from the IEEE Antennas and Propagation Society) in 2006. He is an Associate Editor of several journals and magazines, such as Electromagnetics, IEEE Antennas and Propagation Magazine, and IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS.

Shigetsugu Kirita was born in Kanagawa, Japan, on July 27, 1987. He is currently working toward the M.E. degree in electronic informatics from Hosei University, Tokyo, Japan. Mr. Kirita is a Member of the Institute of Electronics, Information and Communication Engineers (IEICE) of Japan.

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Naoki Mizobe was born in Kanagawa, Japan, on March 3, 1989. He is currently working toward the M.E. degree in electronic informatics at Hosei University, Tokyo, Japan. Mr. Mizobe is a Member of the Institute of Electronics, Information and Communication Engineers (IEICE) of Japan.

Junji Yamauchi (M’84–SM’08) was born in Nagoya, Japan, on August 23, 1953. He received the B.E., M.E., and Dr.E. degrees from Hosei University, Tokyo, Japan, in 1976, 1978, and 1982, respectively. From 1984 to 1988, he served as a Lecturer in the Electrical Engineering Department, Tokyo Metropolitan Technical College. Since 1988, he has been a member of the faculty of Hosei University, where he is now a Professor of electronic informatics. His research interests include optical waveguides and circularly polarized antennas. He is the author of the book Propagating Beam Analysis of Optical Waveguides (Baldock, Hertfordshire, U.K.: Research Studies Press, 2003). Dr. Yamauchi is a member of the Optical Society of America and the Institute of Electronics, Information and Communication Engineers of Japan.

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Investigation on the EM-Coupled Stacked Square Ring Antennas With Ultra-Thin Spacing Saeed I. Latif, Member, IEEE, and Lotfollah Shafai, Life Fellow, IEEE

Abstract—Stacked electromagnetically coupled square ring antennas with extremely thin spacing are studied. The separation between rings is kept very small so that they do not increase the overall antenna height, yet can provide multiband operation. The coupling effects among these very closely placed rings are studied based on substrate parameters so that they can be practically implemented using commercially available microwave substrates. It is observed that with an asymmetric arrangement of the stacked rings, different polarizations can be obtained for different resonances. However, if a fixed polarization is desired, a concentric stacked rings configuration has to be used. Experimental investigations are conducted to confirm simulation results. Index Terms—Microstrip antennas, microstrip ring antennas, polarization, stacked microstrip ring antennas.

I. INTRODUCTION

C

URRENT portable wireless devices require multiband operations to access various wireless communication systems, e.g., mobile network, Bluetooth, GPS, etc. In some mobile terminals, to receive several radio signals, e.g., the terrestrial digital broadcasting and the cellular phone whose frequency bands are largely different with different polarizations, by a single antenna, switched polarization at different bands is desired. In satellite communications, also multiband, or at least dual-band, operations are needed, which normally use orthogonal polarizations to simplify the electronics. Microstrip antennas are suitable for such devices because of their light weight, low cost, and ease of integration with other electronics [1]. Since several techniques are available to miniaturize microstrip antennas [2], they are also attractive as internal antennas. The microstrip square ring antenna is a miniaturized antenna and is a perturbed form of a square patch antenna, where the central metal portion is removed from the square patch. A typical square patch geometry is shown in Fig. 1(a), where and determine the width of the ring , which is , and the resonant frequency as well. The input impedance of the antenna is dependent on . To achieve a miniaturized antenna, the width has to be narrow, i.e., ratio has to be larger, which results in a high input impedance [3], [4]. As such, it becomes impractical to feed the Manuscript received November 08, 2010; revised March 14, 2011; accepted April 04, 2011. Date of publication August 12, 2011; date of current version November 02, 2011. This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC). The authors are with the Department of Electrical and Computer Engineering, The University of Manitoba, Winnipeg, MB R3T 5V6, Canada (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2164203

Fig. 1. (a) Top and side view of the single-layer square ring antenna on a microwave substrate, its simulated (b) reflection coefficients, (c) gain patterns at 2.28 GHz, and (d) current distribution. Simulation: Ansoft Designer, ver. 4.0; , thickinfinite ground plane. Substrate parameters: dielectric constant mm, and loss tangent . Ring #1 size: ness mm, mm, mm, . The antenna is capacitively mm. The antenna is -polarized at this fed: feed-line size 2 23 mm , is the copolarization pattern in the plane, which is frequency as also demonstrated in the current distributions on the ring at 2.28 GHz, obtained from Ansoft HFSS, ver. 12.0 (finite-element-method-based software).

antenna with a regular SMA probe. It was observed that the ratio has to be less than 0.4 if it is intended to be used with an SMA probe. A solution to this problem is the capacitive feeding technique, which allows us to have a narrow width yet with good impedance match [5]–[7]. In this paper, an electromagnetically fed square ring antenna, having more square rings added on top of it with very small spacing, is investigated to achieve multiband performance. The stacked patch configuration is not new, proposed earlier for dual-frequency or wideband performance [8]–[12]. However, the configuration proposed in this paper is different from a regular stacked patch antenna in the sense that the spacing between rings is chosen extremely small and in order of 0.001 wavelength, i.e., the stacking of rings will

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The average ring length is about one wavelength in the dielectric, and it resonates at GHz, as can be noticed in its simulated reflection coefficient plot in Fig. 1(b). Its simulated radiation patterns are shown in Fig. 1(c), which indicate that it is -polarized, as is the copolarization pattern in the plane. This is due to strong -directed currents along the ring arms because of the feed line along the -axis, which can also be noticed in the current distributions on the ring, shown in Fig. 1(d). Peak cross polarization is only 6 dB below the copolarization, showing higher cross polarization due to large -directed currents on the ring. The simulation has been carried out in method of moments (MoM)-based software Ansoft Designer, ver. 4, considering an infinite ground plane [16]. The current distribution has been obtained from finite-element-method-based software Ansoft HFSS, ver. 12.0 [16]. A. Two Stacked Rings in Asymmetric Configuration

Fig. 2. (a) Side view and (b) 3-D view of the stacked microstrip square ring antenna with ultra-thin spacing, excited capacitively by a feed line.

add a negligible height to the antenna other than the main dielectric substrate. Thus, the proposed stacked ring antenna will have an overall antenna height much smaller than that of a regular stacked patch antenna. One possible configuration is an asymmetric configuration, where all rings are aligned along one of its edges, and a feed line is used to electromagnetically excite all rings. Offset dual- and tri-patch configurations are studied in [13]–[15] for broadband operation. However, in this paper, a closer look at the polarization is given for the off-centered dual and triple stacked rings, and a detailed study is conducted to understand the effects of substrate parameters on their resonances in Section II. It is observed that the polarization at the resonance due to the top ring is different from others. Prototypes were fabricated, tested, and compared in this section as well to confirm simulation results. Sections III and IV discuss other possible configurations, the concentric dual- and triple-rings arrangements with symmetric and asymmetric feeds, which show a fixed polarization at all resonances. The effect of substrate parameters on the resonances and the coupling effects due to the close proximity of the rings are also investigated. Finally, in Section V, the conclusions are drawn. II. MULTIPLE SQUARE RINGS IN ASYMMETRIC ARRANGEMENT FOR MULTIBAND OPERATION The detailed geometry of the stacked square ring antenna is shown in Fig. 2. Without Ring #2 and Ring #3, the configuration is a single-layer square ring antenna with a capacitive feed line, as shown in Fig. 1(a). The main antenna substrate is a dielectric material with permittivity , and thickness mm. As mentioned before, the advantage of using the capacitive feeding is that a narrower width of the ring can be chosen, thus a miniaturized antenna can be found. In this example, the width of the ring is only 6 mm .

In a typical stacked patch antenna, the separation between two patches is usually in the order of 0.05 or so [6]. However, in this paper, to obtain dual-frequency operation using two stacked rings as shown in Fig. 3(a), the spacing between Rings #1 and #2 is chosen to be extremely small ( is in the order of 0.001 ). At first, an asymmetric configuration is chosen where the rings are right-aligned, i.e., they are off-centered, as shown in Fig. 3(a) and (b). Fig. 3(c) shows that with a very small air gap between the two rings mm , slightly larger sizes of Ring #2 than Ring #1 (with the same inner dimension as the Ring #1, mm) generate another resonance at a lower frequency. As the separation between the two rings is very small, the feed line in the first layer can electromagnetically excite Ring #2 as well. However, the feed line needs to be located closer to the rings, and its width has to be made narrower for better impedance match at resonances, with respect to those in the single-layer case. It appears that the effective size of Ring #2 is not predominantly determined by the permittivity of the separating substrate, the foam in this case . Rather, the main substrate of the antenna plays a role in it, as the spacing between the two rings is very small, and as such, the effective size of Ring #2 is dependent on the combined effect of the main dielectric substrate and the substrate separating the two rings. It is noticed that the effective permittivity of the substrate separating the two rings is because of the close proximity of the two rings. Fig. 3(b) shows the effects of changing the Ring #2 size, for a fixed , on the reflection coefficient of the stacked square ring antenna. As increases, its resonant wavelength increases, and thus the first resonance shifts to the lower frequency. Since both rings are located very closely, the second resonance , which is due to Ring #1, also shifts to the lower frequency, however at a slower rate. Therefore, the frequency span between and increases with the increase in . 1) Effects of Substrate Parameters Separating the Two Rings: Fig. 4(a) shows the effects of varying substrate thickness , considering foam as the substrate separating the two rings, on the resonances of the antenna in Fig. 3(a). As increases, both and move to the higher frequency region. With larger , the coupling between the two resonances decreases, and they

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Fig. 4. Effects of varying (a) substrate thickness considering foam as the substrate between two rings, and (b) the substrate permittivity with mm on the reflection coefficient of the antenna in Fig. 3. The other antenna parameters are kept the same as mentioned in Fig. 3 except mm. Simulation: Ansoft Designer, ver. 4.0; infinite ground plane.

Fig. 3. (a) Side view and (b) ring parameters of the ultra-thin profile dual-layer square ring antenna (asymmetric configuration) on a microwave substrate: di, thickness mm, and loss tangent electric constant . Ring #1 size: mm, mm, mm, (same as before). (c) Effects of varying Ring #2 size on the return loss: is mm. The rings are separated by foam substrate: mm. varied, mm. SimThe antenna is capacitively fed: feed-line size 1.5 26 mm , ulation: Ansoft Designer, ver. 4.0; infinite ground plane.

come closer to each other. It can be noticed that mainly moves to the higher frequency and approaches with larger values. This indicates that the effective size of the Ring #2 is now mainly determined by the permittivity of the separating substrate. Thus, the configuration behaves as a regular stacked microstrip antenna with wide impedance bandwidth, as both resonances merge with one another due to weaker coupling. However, the most important point to note here is that for extremely small thicknesses (0.1–0.4 mm), the two rings give two resonances. Thus, dual-frequency operation can be achieved using ultra-thin substrate thickness separating the two rings.

Fig. 4(b) shows the effects of varying the substrate permittivity , separating the two rings, on the reflection coefficients of the dual-layer square ring antenna. The separation between two rings is considered very small, mm. As increases, both and shift to the lower frequencies, with slightly larger shift for compared to ; i.e., because of the extremely thin spacing between the two rings, both resonances are affected. However, the effective size of Ring #2 changes to a larger extent. Therefore, as increases, two resonances move further away from each other. These plots confirm that even for microwave substrates, two distinct resonances can be obtained with extremely small spacing between the two rings. 2) Experimental Study: In order to verify the simulation results showed previously, a prototype of the dual-layer stacked square ring antenna with ultra-thin separation between the rings was fabricated and tested in the Antenna Lab at the University of Manitoba, Winnipeg, MB, Canada. The antenna parameters are tabulated in Table I. Simulated and measured reflection

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DIMENSIONS

OF THE FOR

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TABLE I STACKED SQUARE RING ANTENNAS ASYMMETRIC CASE

Fig. 5. (a) Simulated and (b) measured reflection coefficients of the antenna mm mm , and (c) its in Fig. 3 with simulated impedance plot on a Smith chart. Other parameters are given in the caption of Fig. 3. Simulation: Ansoft Designer, ver. 4.0. Finite ground plane size: 32 32 cm .

coefficient plots are shown in Fig. 5, which shows two resonances. The simulated impedance plot on the Smith chart indicates the dual-band operation and the potential for wideband performance. In both simulation and measurement, a ground plane of 32 32 cm was used. When compared to the simulated one, a slight discrepancy can be noticed in terms of resonance frequencies. This can be explained with the help of the study on shown in Fig. 4(b), where it can be noticed that when is larger, and move further away from each other. In simulation, the foam substrate was considered between the two rings, whereas in fabrication, the paper layer that comes with the commercial copper tape was used as the separation, which had a different dielectric constant, definitely larger than that of the foam. Moreover, the paper and the glue present with the copper tape contributed to a larger value. Therefore, the frequency span between measured resonances and is slightly larger than the simulated one. Measured gain patterns are compared to the simulated ones in Fig. 6 for the ultra-thin spaced dual-layer stacked ring configuration in Fig. 3(a). As mentioned before, is due to Ring #1 and at this frequency, it is -polarized because is the copolarization pattern in the plane; see Fig. 6(b) and (d). Since the feed line is along the -axis, it generates strong -directed

currents on Ring #1 and provides the -polarization. However, at , which is due to Ring #2, the antenna is -polarized; see Fig. 6(a) and (c). This is different from the regular stacked patch antenna, where the antenna has the same polarization at both frequencies. In our case, this is happening because of the asymmetric arrangement of the rings. The rings are off-centered only along the -axis, but are symmetric along the -axis, which produces large -directed currents on Ring #2, and as such, is -polarized. This polarization switch at two frequencies is also evident in current distribution plots for the rings, as shown in Fig. 6(e) and (f). For the same reason, the cross-polarization performance at both frequencies is better than the single-layer case. Considering a handmade configuration and different permittivity of the ring spanning, it can be concluded that the simulated and measured gain patterns are in good agreement. The dips along the boresight in both simulated and measured copolarization patterns are due to the finite ground plane. Later, we will show that when concentric ring configurations are chosen, the polarization does not switch from to , in the multifrequency operation, from this ultra-thin profile antenna fed by a capacitive feed line. B. Three Stacked Rings in Asymmetric Configuration In order to study the possibility of obtaining tri-frequency operation with ultra-thin spacing, another ring, which is larger than Rings #1 and #2, is stacked on top of the dual square rings, aligned at the right edge, as shown in Fig. 7(a) and (b). Here also, the spacing between Rings #2 and #3 is only 0.1 mm, with the same ultra-thin spacing between Rings #1 and #2. The effects of the Ring #3 size (the inner dimension is the same as Rings #1 and 2: mm) is shown in Fig. 7(c). It can be noticed that a much wider Ring #3 , or , is at least required in this case

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Fig. 6. (a), (b) Simulated and (c), (d) measured gain patterns, and (e), (f) current and with distributions on Rings #2 and #1, of the antenna in Fig. 3 at mm mm . The antenna parameters are as is the given in the caption of Fig. 3. The antenna is -polarized at plane, and -polarized at as is copolarization pattern in the plane, also evident in the current distribution plots. the co-pol in the Simulation: Ansoft Designer, ver. 4.0. Finite ground plane size 32 32 cm . Current distributions were obtained from Ansoft HFSS, ver. 12.0.

for good impedance matching at all three resonances. Moreover, a small adjustment of the feed line is also required for the impedance matching. Although, Ring #1 and Ring #2 dimensions are kept the same as before, their frequency of operation changes, compared to the dual-layer case, due to very strong interaction among the rings. The effective permittivity of the substrate separating the rings is observed as the following: Ring #3 Ring #2 . This also indicates the sensitivity of the third ring size on three resonances. As the rings are progressively larger, the frequencies of operation shift to lower frequencies, compared to the previous case. 1) Effects of Substrate Parameters Separating the Three Rings: Fig. 8 shows the effects of varying the dielectric constant of the substrate separating the three rings, on the reflection coefficients of the antenna in Fig. 7. In this case, mm. As increases, all three resonances

Fig. 7. (a) Side view and (b) ring parameters of the ultra-thin profile triple-layer square ring antenna (asymmetric configuration) on a microwave substrate: di, thickness mm, and loss tangent electric constant . Rings #1 and #2 are kept the same as before. Ring #1 size: mm, mm, mm, . Ring #2 size: mm, mm, mm, . (c) Effects of varying is varied, mm. The rings are sepaRing #3 size on the return loss: mm. The antenna is capacitively fed: rated by foam substrate: mm. Simulation: Ansoft Designer, feed-line size 0.75 28 mm , . ver. 4.0; infinite ground plane.

shift to the lower frequency region. This study suggests that ultra-thin microwave substrate materials can also be used for multifrequency operations.

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Fig. 8. Effects of varying the substrate permittivity (separating rings) with mm on the reflection coefficient of the antenna in Fig. 7. The other antenna parameters are kept the same as mentioned in Fig. 7, except mm and mm. Simulation: Ansoft Designer, ver. 4.0; infinite ground plane.

Fig. 10. (a) Simulated and (d) measured reflection coefficients, and (c) its simulated impedance plot on a Smith chart of the antenna in Fig. 7 with mm mm . Other parameters are given in the caption of Fig. 7. Simulation: Ansoft Designer, ver. 4.0; infinite ground plane. Measurement: ground plane size 32 32 cm .

Fig. 9. Effects of varying substrate thickness (both and ) considering foam as the substrate material separating three rings on the reflection coefficient of the antenna in Fig. 7. The other antenna parameters are kept the same as mm and mm. Simulation: mentioned in Fig. 7, except Ansoft Designer, ver. 4.0; infinite ground plane.

Next, the effects of the substrate thickness are studied, considering foam as the substrate material separating the three rings from each other. The other resonances move to the higher frequency region. Fig. 9 shows the effects of changing both and on the reflection coefficients of the antenna. It shows a very sensitive behavior of the antenna, as the third resonance disappears with a small increase in the thickness. The other resonances move to the higher-frequency region. 2) Experimental Study: A prototype of the EM-coupled asymmetric triple-square ring antenna was also fabricated to compare to simulation results. The antenna with the following Ring #3 dimensions was chosen for fabrication on foam substrates separating the three rings: mm and mm mm . The simulated and measured reflection coefficient plots are shown in Fig. 10. In the case of simulation, Fig. 10(a) shows that Rings #1 and #2 are operating at GHz and GHz,

respectively, and Ring #3 is operating at GHz. However, from the measurement result in Fig. 10(b), we can notice that the operating frequencies are slightly different: GHz, GHz, and GHz. The reason for this discrepancy can be explained with the substrate parameters study discussed before, where it was mentioned that with increased substrate permittivity and thickness, the frequencies move away from each other. In the experimental prototypes, copper tapes with a thin paper backing was used. Thus, the spacing between the rings was due to the paper, plus a thin air gap between the paper and the lower ring. The paper had a higher permittivity, and the air gap increased the spacing. Moreover, the presence of the glue on the paper had resulted in a higher substrate permittivity. The simulated impedance plot on the Smith chart shows three loops, which were created because of mutual resonances produced by the three rings, and indicates three resonances. However, as the loops are on the low-impedance side of the chart, the antenna shows narrow impedance bandwidths at those bands. The radiation patterns at these frequencies are shown in Fig. 11. In this case, the antenna is -polarized at and . However, it is -polarized at , which is due to Ring #3. The current distributions on the rings at three frequencies confirm this polarization switch, which also indicate that in both dual- and triple-ring asymmetric configurations, the top ring has strong -directed currents due to the asymmetry along -axis, and thus is -polarized at this frequency. The cross polarization is at least 11 dB below the copolarization for all three frequencies. The simulated and measured results are in good agreement, as before.

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Fig. 11. (a), (c), (e) Simulated and (b), (d), (f) measured gain patterns at , , and of the antenna in Fig. 7, and (g) current distributions on Rings #3, #2 and #1 at , , and , respectively. The antenna parameters are given in the capis the co-pol in the tion of Figs. 7 and 10. The antenna is -polarized at as plane, and -polarized at and as is the co-pol in the plane. Simulation: Ansoft Designer, ver. 4.0; infinite ground plane. Measurement: ground plane size 32 32 cm . Current distributions were obtained from GHz, -polarized. (b) GHz, Ansoft HFSS, ver. 12.0. (a) -polarized. (c) GHz, -polarized. (d) GHz, -polarGHz, -polarized. (f) GHz, -polarized. ized. (e)

III. MULTIPLE CONCENTRIC SQUARE RINGS FOR MULTIBAND OPERATION WITH SINGLE POLARIZATION A. Two Concentric Stacked Rings for Dual-Frequency Operation In Section II, it was observed that due to the off-centered configuration, the polarization was not the same at different

Fig. 12. (a) Side view and (b) ring parameters of the ultra-thin profile dual-layer square ring antenna (concentric rings configuration) on a microwave substrate: , thickness mm, and loss tangent dielectric constant . Ring #1 size: mm, mm, mm, (same as before). (c) Effects of varying Ring #2 size on the return is varied, mm. The rings are separated by foam substrate: loss: mm. The antenna is capacitively fed: feed line size 2 21 mm , mm. Simulation: Ansoft Designer, ver. 4.0; infinite ground plane. Measurement: ground plane size 32 32 cm .

resonances of the multiring antenna. When the same polarization is desired at all frequencies, a concentric ring configuration is required, which is studied in detail in this section. Such a configuration is shown in Fig. 12(a) and (b), which is a dual-layer capacitively fed ring antenna with ultra-thin spacing. It is observed that it needs much larger width of the Ring #2 ( , i.e., ) compared to the asymmetric dualsquare-ring case for the same Ring #1 size, to achieve a good impedance match, as can be seen in Table II. The feed line has to be adjusted slightly to excite both rings properly as well. For a fixed value, the effects of varying on the reflection coefficients of this antenna are shown in Fig. 12(c). It shows that as increases, both and move to the lower frequency

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TABLE II DIMENSIONS OF THE STACKED SQUARE RING ANTENNAS FOR SYMMETRIC CASE WITH ASYMMETRIC FEED

region. However, the change is faster for , which indicates that it is due to Ring #2. It is also important to note that the two resonances are farther away from each other, as shown in Fig. 13(a), compared to the asymmetric case [Fig. 5(a)]. This is due to the stronger coupling between the two rings in this concentric arrangement, as can be noticed in the Smith chart in Fig. 13(b), which has two looser loops compared to the asymmetric case in Fig. 5(b). The gain patterns at GHz and GHz are plotted in Fig. 13(c) and (d), which confirms that the polarization at two operating frequencies is in the -direction, as determined by the feed line along the -axis. i.e., the polarization does not switch from to at two frequencies as the asymmetric case. This is also evident in the current distributions on two rings, as shown in Fig. 13(e) and (f). The antenna shows high cross-polarization levels at both frequencies. In fact, at , which is due to Ring #1, the antenna shows elliptical polarization because of the feed-point location and the symmetry among the rings. This discussion suggests that due to the off-centered arrangement, discussed in Section II, the polarization switches from to and has an advantage in low cross polarization. 1) Effects of Substrate Parameters Separating the Two Rings: Fig. 14 shows the effects of varying the parameters ( and ) of the substrate separating the two concentric rings on the reflection coefficients of this antenna. It shows a very stable performance: As increases, increases but decreases, thus

Fig. 13. Simulated (a) reflection coefficient, (b) impedance plot on a Smith GHz and (d) GHz of the chart, and gain patterns at (c) antenna in Fig. 12. The antenna parameters are given in the caption of Fig. 12 mm mm . The antenna is -polarized with and , as is the co-pol in the plane. Simulation: Ansoft at both Designer, ver. 4.0; infinite ground plane. Current distributions on Rings #2 and and (f) , respectively, are also presented, obtained from Ansoft #1, at (e) HFSS, ver. 12.0.

increasing the frequency span between and . On the contrary, as increases, both and drop, with a much larger change for , i.e., moves closer to . This is opposite of what is observed for the asymmetric case. B. Three Concentric Rings for Tri-Frequency Operation For the concentric case also, three stacked rings are studied mm ; see with a very small spacing Fig. 15(a) and (b). Because of the extremely strong coupling, the resonances are very sensitive to the ring dimensions and the substrate parameters separating the three rings. Fig. 15(c) shows the effects of varying the Ring #3 size with a fixed mm, keeping Ring #1 and #2 sizes the same as the concentric dual-ring case, on the reflection coefficients of the antenna. With increased , only moves to the lower

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Fig. 14. Effects of varying (a) the substrate permittivity with mm and (b) substrate thickness considering foam as the substrate on the reflection coefficient of the antenna in Fig. 12. The other antenna parameters are kept mm. Simulation: Ansoft the same as mentioned in Fig. 12, except Designer, ver. 4.0; infinite ground plane.

frequency, which is excited due to the Ring #3. However, a much wider width of the Ring #3 is required to obtain impedance match at all three resonances ( , , i.e., , ). Moreover, the feed line width has to be increased and moved farther away from the bottom ring to properly excite all three rings. Unlike the asymmetric case, when is increased, the frequency span between and decreases, as can be noticed in Fig. 16. The antenna shows much stable performance against the change in the substrate thickness, as can be noticed in Fig. 17, compared to its asymmetric counterpart in Fig. 9. When both and are increased, as long as the separations are small, all three resonances can be obtained easily. If they are much larger, they move farther away from each other. For a particular case mm mm with very small separations of the rings mm , the reflection coefficient, impedance plot on a Smith chart, and gain patterns are plotted in Fig. 18. The Smith chart has three loose loops, which indicates that the coupling is stronger, and therefore, the frequency span among resonances are large, compared to the asymmetric case in Fig. 10(c). As the rings are concentric, the polarization of the antennas does not change at different

Fig. 15. (a) Side view and (b) ring parameters of the ultra-thin profile triplelayer square ring antenna (concentric rings configuration) on a microwave sub, thickness mm, and loss tangent strate: dielectric constant . Rings #1 and #2 are kept the same as before. Ring #1 size: mm, mm, mm, . Ring #2 size: mm, mm, mm, . (c) Effects of is varied, mm. The rings varying Ring #3 size on the return loss: mm. The antenna is capaciare separated by foam substrate: mm. Simulation: Ansoft Designer, tively fed: feed-line size 3 21 mm , ver. 4.0; infinite ground plane.

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Fig. 16. Effects of varying the substrate permittivity (separating rings) with mm on the reflection coefficient of the antenna in Fig. 15. The other antenna parameters are kept the same as mentioned in Fig. 15, except mm. Simulation: Ansoft Designer, ver. 4.0, infinite ground plane.

Fig. 17. Effects of varying substrate thickness (both and ) considering foam as the substrate material separating three rings on the reflection coefficient of the antenna in Fig. 15(a). The other antenna parameters are kept the same mm. Simulation: Ansoft Designer, as mentioned in Fig. 15, except ver. 4.0; infinite ground plane.

resonant frequencies. However, the cross-polarization level is much higher in all three cases, which is not the same for the asymmetric case. IV. MULTIPLE CONCENTRIC SQUARE RINGS WITH SYMMETRIC FEEDING FOR LOW CROSS-POLARIZATION In Section III, it can be noticed that the cross-polar level is very high for all three frequencies, even higher than those in the off-centered case in Section I. The reason for this high cross polarization is the feed line and the feed-point location for that line. When a symmetric feed line is used with the feed point at the center as shown in Figs. 19(a) and 20, the crosspolarization levels are much lower at both frequencies in the case of dual concentric square rings, and at all three frequencies in the case of triple-concentric rings, which can be noticed in Figs. 19(c) and (d) and Figs. 21(c) and (e), respectively. It is important to note that the polarization at all frequencies is -polarized, as the feed point is symmetric about the -axis, which is evident in the current distribution plots of the antennas

Fig. 18. Simulated (a) reflection coefficient, (b) impedance plot on a Smith GHz, (d) GHz, and chart, and gain patterns at (c) GHz of the antenna in Fig. 15, and (f) current distributions on (e) Rings #3, #2, and #1 at , , and , respectively. The antenna parameters are mm mm given in the caption of Fig. 15 with . The antenna is -polarized at all three frequencies as is the co-pol in plane. Simulation: Ansoft Designer, ver. 4.0; infinite ground plane. the Current distributions were obtained from Ansoft HFSS, ver. 12.0.

shown in Figs. 19(e) and 21(f). However, special care is required for the feed-line design, in terms of spacing and length, in the case of three-concentric-ring antenna, for its good impedance matching. The impedance plots on the Smith chart for both cases indicate that loose loops are present; see Figs. 19(b) and 21(b). However, the centers of the loops are widely separated, especially in the triple-ring case, making it difficult to match all three resonances at the same time. The effects of different parameters are the same as the concentric rings with the asymmetric feed

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Fig. 20. Geometry of the tri-stacked concentric ring antenna with a symmetric feed.

V. CONCLUSION

Fig. 19. (a) Geometry of the dual stacked concentric ring antenna with a symmetric feed, and its simulated (b) reflection coefficient, (c) impedance GHz, and plot on a Smith chart, and gain patterns at (d) GHz. Simulation: Ansoft Designer, ver. 4.0; infinite (e) ground plane. The antenna parameters are the same as in Fig. 12 with mm mm , except the symmetric feed line: mm. The antenna is -polarized at both and , as 2 27 mm , is the co-pol in the plane. Current distributions on Rings #2 and and (g) , respectively, are also presented, obtained from Ansoft #1 at (f) HFSS, ver. 12.0.

line discussed in Section II and are not included here again for brevity.

In this paper, multiple stacked microstrip ring antennas with extremely small spacing were investigated and discussed. A detailed study was conducted to understand the effects of small spacing between rings using different substrate materials. It showed that multiband performance can be achieved even for separations as small as 0.001 . Because of the extremely small separation, the overall height of the antenna is much smaller than a regular stacked patch antenna. However, the resonance due to each ring is strongly dependant on the main substrate parameters. Although, in this paper, every parameter was not optimized for large frequency ratios, from the results it can be concluded that, in the case of dual-ring antennas with ultra-thin spacing, a wide range of ratio can be obtained by choosing appropriate ring sizes with larger ratio for the Ring #2, as long as the feed line can excite both rings. In the case of triple-ring antennas, the range of the frequency ratio is smaller, and larger width of Ring #3 is needed. In both cases, the adjustment of the feed line is required for good impedance matching. Thus, the design approach in this paper can be summarized as the following. At first, dimensions of Ring #1 were selected for a particular frequency. Then, a larger Ring #2 was chosen with a larger width, which resonated at a lower frequency. After optimization for a good impedance match, the third ring was chosen, again for a lower frequency. In the case of off-centered configuration, different polarization can be achieved at different frequencies, which is sometimes useful in multiband wireless applications. In order to have a fixed polarization at all resonances, a concentric arrangement of the rings is essential. In this case, a symmetric feed line can give low cross-polarization at all operating frequencies. Depending on the feed-line orientation, linear polarizations (vertical or horizontal) can be obtained for both asymmetric (Section II)

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REFERENCES [1] R. Garg, P. Bhartia, I. J. Bahl, and A. Ittipibon, Microstrip Antenna Design Handbook. Norwood, MA: Artech House, 2001. [2] A. K. Skrivervik, J. F. Zurcher, O. Staub, and J. R. Mosig, “PCS antenna design: The challenge of miniaturization,” IEEE Antennas Propag. Mag., vol. 43, no. 4, pp. 12–23, Aug. 2001. [3] P. M. Bafrooei and L. Shafai, “Characteristics of single- and doublelayer microstrip square ring antennas,” IEEE Trans. Antennas Propag., vol. 47, no. 10, pp. 1633–1639, Oct. 1999. [4] R. Garg and V. S. Reddy, “Edge feeding of microstrip ring antennas,” IEEE Trans. Antennas Propag., vol. 51, no. 8, pp. 1941–1946, Aug. 2003. [5] G. Mayhew-Ridgers, J. W. Odondaal, and J. Joubert, “New feeding mechanism for annular-ring microstrip antenna,” Electron. Lett., vol. 36, pp. 605–606, Mar. 2000. [6] G. Mayhew-Ridgers, J. W. Odendaal, and J. Joubert, “Single-layer capacitive feed for wideband probe-fed microstrip antenna elements,” IEEE Trans. Antennas Propag., vol. 51, no. 6, pp. 1405–1407, Jun. 2003. [7] S.-S. Oh and L. Shafai, “Antenna miniaturization using open squarering microstrip geometries,” Electron. Lett., vol. 42, pp. 500–502, Apr. 2006. [8] R. Q. Lee and K. F. Lee, “Characteristics of a two-layer electromagnetically coupled rectangular patch antenna,” Electron. Lett., vol. 23, pp. 180–181, Sep. 1987. [9] F. Croq and D. M. Pozar, “Millimeter-wave design of wideband aperture-coupled stacked microstrip antennas,” IEEE Trans. Antennas Propag., vol. 39, no. 12, pp. 1770–1776, Dec. 1991. [10] R. B. Waterhouse, “Design of probe-fed stacked patches,” IEEE Trans. Antennas Propag., vol. 47, no. 12, pp. 1780–1784, Dec. 1999. [11] R. B. Waterhouse, “Stacked patches using high and low dielectric constant material combinations,” IEEE Trans. Antennas Propag., vol. 47, pp. 1767–1771, Dec. 1999. [12] S. D. Targonski, R. B. Waterhouse, and D. M. Pozar, “Design of wideband aperture-stacked patch microstrip antennas,” IEEE Trans. Antennas Propag., vol. 46, no. 9, pp. 1245–1251, Sep. 1998. [13] K. F. Tong and K. M. Luk, “Offset tri-patch microstrip antenna,” Microw. Opt. Technol. Lett., vol. 7, pp. 873–875, Dec. 1994. [14] K. M. Luk, K. F. Tong, and T. M. Au, “Offset dual patch microstrip antenna,” Electron. Lett., vol. 29, pp. 1635–1636, Sep. 1993. [15] T. M. Au, K. F. Tong, and K. M. Luk, “Analysis of offset dual-patch microstrip antenna,” Microw. Antennas Propag., vol. 141, pp. 523–526, Dec. 1994. [16] Ansoft. Ansoft Corporation, Canonsburg, PA.

Fig. 21. Simulated (a) reflection coefficient, (b) impedance plot on a Smith GHz, (d) GHz, and (e) chart, and gain patterns at (c) GHz of the antenna in Fig. 20, and (f) current distributions on Rings #3, #2, and #1 at , , and , respectively. The antenna parameters are the mm ( mm, , except same as in Fig. 15 with mm. The antenna is -polarized the symmetric feed line: 1.5 23 mm , as is the co-pol in the plane. Simulation: Ansoft at , , and Designer, ver. 4.0; infinite ground plane. Current distributions were obtained from Ansoft HFSS, ver. 12.0.

and concentric configurations (Section IV). The concentric configuration can be designed for circular polarization using conventional techniques, i.e., dual orthogonal feeds at phase quadrature or a single feed with ring corner cuts. However, circular polarization was not investigated in this paper. ACKNOWLEDGMENT The authors would like to thank B. Tabachnick and C. Smit for their help in antenna fabrication and measurements.

Saeed I. Latif (S’03–M’08) received the B.Sc. degree in electrical and electronics engineering from Bangladesh University of Engineering and Technology, Dhaka, Bangladesh, in 2000, and the M.Sc. and Ph.D. degrees in electrical engineering from the University of Manitoba, Winnipeg, MB, Canada, in 2004 and 2009, respectively. During his graduate studies, he was a Research Assistant and was involved in various microstrip and array antenna projects. He was also a Teaching Assistant during that period and contributed to the undergraduate teaching in the Department of Electrical and Computer Engineering, University of Manitoba. From 2009 to 2010, he was a Postdoctoral Fellow with the same department. Currently, he is a Natural Sciences and Engineering Research Council (NSERC) Postdoctoral Fellow with CancerCare Manitoba, Winnipeg, MB, Canada, and is involved in the design of ultrawideband antenna probes for microwave imaging. Dr. Latif is a registered professional engineer with APEGM. He was one of the 15 finalists in the Student Paper Competition at the 2004 IEEE Antennas and Propagation Society (AP-S) International Symposium in Monterey, CA. He was the recipient of the Young Scientist Award at the International Symposium on Electromagnetic Theory (EMTS), held in Ottawa, ON, Canada, in 2007.

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Lotfollah Shafai (S-67–M’69–SM’75–F’87–LF’07) received the B.Sc. degree from the University of Tehran, Tehran, Iran, in 1963, and the M.Sc. and Ph.D. degrees from the University of Toronto, Toronto, ON, Canada, in 1966 and 1969, all in electrical engineering. In November 1969, he joined the Department of Electrical and Computer Engineering, University of Manitoba, Winnipeg, MB, Canada, as a Sessional Lecturer, becoming an Assistant Professor in 1970, Associate Professor in 1973, and Professor in 1979. Since 1975, he has made special efforts to link the University research to the industrial development, by assisting industries in the development of new products or establishing new technologies. To enhance the University of Manitoba contact with industry, in 1985 he assisted in establishing the Institute for Technology Development and was its Director until 1987, when he became the Head of the Electrical Engineering Department. His assistance to industry was instrumental in establishing an Industrial Research Chair in Applied Electromagnetics at the University of Manitoba in 1989, which he held until July 1994. Dr. Shafai was elected a Fellow of the Royal Society of Canada in 1998. In 2009, he was elected a Fellow of the Engineering Institute of Canada. He holds a Canada Research Chair in Applied Electromagnetics. He has been a participant in nearly all Antennas and Propagation Symposia and participates on the Review Committees. He is a member of the International Union of Radio Science (URSI) Commission B and was its Chairman from 1985 to 1988 and In-

ternational Chair from 2005 to 2008. In 1986, he established the Symposium on Antenna Technology and Applied Electromagnetics, ANTEM, at the University of Manitoba that is currently held every two years. He has been the recipient of numerous awards. In 1978, his contribution to the design of a small ground station for the Hermus satellite was selected as the 3rd Meritorious Industrial Design. In 1984, he received the Professional Engineers Merit Award and in 1985, “The Thinker” Award from Canadian Patents and Development Corporation. From the University of Manitoba, he received the “Research Awards” in 1983, 1987, and 1989, the Outreach Award in 1987, and the Sigma Xi, Senior Scientist Award in 1989. In 1990, he received the Maxwell Premium Award from the Institution of Engineering and Technology (London, U.K.), and in 1993 and 1994, the Distinguished Achievement Awards from Corporate Higher Education Forum. In 1998, he received the Winnipeg RH Institute Foundation Medal for Excellence in Research. In 1999 and 2000, he received the University of Manitoba Faculty Association Research Award. He was a recipient of the IEEE Third Millennium Medal in 2000, and in 2002 was elected a Fellow of the Canadian Academy of Engineering and Distinguished Professor at the University of Manitoba. in 2003, he received the IEEE Canada “Reginald A. Fessenden Medal” for “Outstanding Contributions to Telecommunications and Satellite Communications” and a Natural Sciences and Engineering Research Council (NSERC) Synergy Award for “Development of Advanced Satellite and Wireless Antennas.” In 2009, he was the recipient of an IEEE Chen-To-Tai Distinguished Educator Award. In 2011, he received a Killam Prize in Engineering from the Canada Council for the Arts for his “outstanding Canadian career achievements in engineering, and his work in antenna research.”

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Choosing Dielectric or Magnetic Material to Optimize the Bandwidth of Miniaturized Resonant Antennas Antti O. Karilainen, Member, IEEE, Pekka M. T. Ikonen, Member, IEEE, Constantin R. Simovski, and Sergei A. Tretyakov, Fellow, IEEE

Abstract—We address the question of the optimal choice of loading material for antenna miniaturization. A new approach to identify the optimal loading material (dielectric or magnetic) for maximization of bandwidth of resonant antennas is introduced. Instead of equivalent resonant circuits or transmission-line resonators, we use the analysis of radiation mechanism to identify the fields contributing mostly to the stored energy and determine the beneficial material type. The formulated rule is qualitatively illustrated using a dipole and a patch antenna, as well as a planar inverted-L antenna where the conventional analysis of circuit or a transmission-line resonator leads to incorrect conclusions. Guidelines are presented for miniaturizing different antenna types. Index Terms—Equivalent circuit, material loading, quality factor, resonant antennas, transmission line.

I. INTRODUCTION

D

EMAND towards smaller and smaller portable communication devices has continuously challenged antenna engineers to come up with more compact antenna designs. However, the task has not been trivial, especially because these small antennas should still have enough bandwidth, as required by system specifications. Resonant antennas, that used, e.g., in most if not all hand-held devices, can be made geometrically smaller by inserting some material in the structure, so that the propagation constant inside the antenna increases, and the resonant electric size of the antenna corresponds to a smaller geometrical size. Of course, filling an antenna with a material usually changes also the field distribution, and the operation of the antenna is changed not only in view of the resonance frequency, but the bandwidth and also matching, efficiency, etc. are affected. Miniaturization of patch antennas using materials with various material parameters and studied in [1] aroused discussion of the benefits of different materials with magnetic response. Various loading scenarios have been considered: For example, dielectric coated dipole antennas were experimentally and theoretically studied in [2], dipole antennas covered with metamaterials were studied in [3], and double negative materials for electrically small antennas in [4]. Recent analytical Manuscript received June 29, 2009; revised September 13, 2010; accepted May 05, 2011. Date of publication August 12, 2011; date of current version November 02, 2011. This work was supported in part by Intel Corporation and Nokia Corporation and in part by the Academy of Finland and Nokia through the Center-of-Excellence program. A. O. Karilainen, C. R. Simovski, and S. A. Tretyakov are with the Department of Radio Science and Engineering, Aalto University, FI-00076 Aalto, Finland (see http://radio.tkk.fi/en/contact/). P. M. T. Ikonen is with TDK-EPC, P.O.Box 275, FI-02601 Espoo, Finland (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2011.2164170

results from a top-loaded dipole antenna are presented in [5]. In this article, we discuss how to determine the proper type of the loading material for a generic small resonant antenna. The analysis here assumes lossless non-dispersive materials, because the goal is to determine the optimal loading type (dielectric or magnetic) for maximization of bandwidth. Effects of material dispersion (reduced bandwidth) and losses (reduced efficiency) are well known. Antenna applications of metamaterials with tailored magnetic response as well as magneto-dielectric materials with natural magnetic response have been under study in the recent literature. Magneto-dielectric meta-substrates (metamaterials) were used to miniaturize patch antennas in [6], [7]. A patch antenna with a metamaterial substrate was further studied with the dispersion included, and challenged against reference antennas with dielectric substrates in [8]. It was theoretically and experimentally seen that the negative effect of material dispersion is stronger than the beneficial effect of the artificial magnetic response. Magneto-dielectric substrates with natural magnetic inclusions were however seen to provide wider bandwidths than reference dielectric substrates [9]. The reduced stored energy, while preserving the radiation fields in patch antennas with ideal dielectric and magnetic substrates, was clearly illustrated in [10]. The explanation for this effect was the fringing electric field at patch slots as the main radiation mechanism, whereas the magnetic field in the antenna vicinity contributed mostly to the stored energy. Here, we extend the approach established in [10] and propose a general rule for determination of the beneficial substrate type from the analysis of the radiation mechanism. Some preliminary results of this study were presented in [11]. Resonance circuits are sometimes used to model the behavior of resonant antennas near the resonance frequency. An equivalent ladder circuit model for small omni-directional antennas was used in [12]. For finite sized dipole antennas equivalent circuits are discussed, e.g., in [13] and [14]. Patch antenna is analyzed in [9]. Below we show that equivalent circuits are not always to be trusted when considering approaches to antenna miniaturization. In the following sections, we first discuss the conventional equivalent circuit models: how the material loading affects the antenna bandwidth and what is the role of the feeding type. Also, the use of the transmission-line theory is discussed with material loading in mind. Next, we introduce the original field approach and formulate rules. We present examples with a patch antenna, a dipole antenna, and a planar inverted-L antenna. Finally, partial filling and material dispersion is discussed.

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II. ANTENNA MINIATURIZATION MODELED USING RESONANCE EQUIVALENT CIRCUITS AND TRANSMISSION LINES When discussing miniaturization of small resonant antennas, the quality factor plays an important role, and it is written as: (1) is the average stored enwhere is the angular frequency, is the radiated power. Since the bandwidth is ergy and inversely proportional to the quality factor: , a small is usually desired. The decrease in can be achieved in difshould be minimized, ferent ways. Ideally, the stored energy should be maximized. while the radiated power The antenna quality factor can be described also in terms of of the anthe circuit theory: The slope of the reactive part should be minimized, while tenna impedance are not dethe desired losses due to the radiation resistance creased (assuming no losses due to resistive dissipation to heat), according to (e.g., [8]) (2) in case of a series-type resonance of . One has to remember affects the that the change in radiating fields and hence in , which has to be considered when the radiation resistance antenna is connected to the feeding waveguide. The main reason that the circuit theory analogy has been widely used (e.g. in [9]) is its simplicity. The input impedance or admittance and the quality factor of a resonator can be neatly written using a parallel or series type circuit in the vicinity of the resonances. At series resonance, the quality circuit is factor for the series resonance (3) where is the resonance angular frequency and is the half-power bandwidth [15]. Similar equation can be derived for circuit: the parallel (4) The resonance angular frequency for both circuits is . Now, let us assume that a resonant antenna behaves like one of these resonators in the vicinity of its resonance frequency. When we load or fill the antenna with a magnetic material, it is circuits with, e.g., a feranalogous to filling the coils in the rite core. Let us assume further, that the increase in inductance is accordingly compensated by decreasing the capacitance to maintain the same resonance angular frequency . For the series resonator the results can be clearly seen from increases and the bandwidth (3): The quality factor decreases. For the parallel resonator the effect is opposite, as seen from (4). As is decreased, the quality factor decreases and the bandwidth increases. In other words, if an antenna operating at the parallel resonance can be loaded (completely or

partially filled) with a magnetic material, the size decrease does not harm the bandwidth, opposite to the dielectric loading. description of antennas is as The main drawback of this follows: An antenna impedance dispersion changes with simple impedance transformers and moreover with matching networks while the antenna remains the same. Also, this simple theory assumes that material loading does not change the current disof the antenna at tribution mode nor the radiation resistance the resonance frequency. Our approach to the description of antenna loading will be free of this vagueness. Another option is to analyze the series and parallel resonance with transmission lines (TLs). If we consider the TL as an arbitrary TEM waveguide, it is impossible to see how the material filling parameters would affect the quality factors, since is proportional to both material parameters and of the medium. This can be seen, e.g., from -long the well-known equation for the quality factor of a open-ended TL resonator [15]: (5) where is the attenuation constant due to small losses. and can be related to the characteristic impedance and . The general transmission-line theory does not describe the antenna geometry through , not to mention the radiation, in enough detail in order to make definite conclusions about the bandfor different loading materials. More detailed anwidth or tenna models must be used to describe the fields and especially the radiation. III. ANTENNA MINIATURIZATION MODELED USING EQUIVALENT RADIATING CURRENTS From the resonant-circuit model of small resonant antennas (the previous section) it appears that the choice of the loading material (dielectric or magnetic) for maximum bandwidth is determined by the resonance type (series or parallel). However, there are antennas that apparently do not obey this rule. One of such antennas is the planar inverted-L antenna (described below). This calls for a development of a more advanced approach to this problem. The challenge is to describe the matching-independent effect of decreasing the stored energy (or enhancing the radiation) in practical antennas, that can be achieved by the material loading. Because practical antennas are three-dimensional structures and can have various shapes and field distributions, the one-dimensional equivalent circuit or transmission-line representation are not accurate enough. We will see that quite often for resonator antennas one field (electric or magnetic) provides the main contribution to the radiation, whereas the other field contributes mainly to the reactive energy storage. The strategy for miniaturization of an antenna is then to fill or cover the antenna to suppress the second field, whereas the radiating field is not diminished. To determine the material type which will be most effective for miniaturization of a particular antenna, we propose to model the antenna radiation using the equivalent surface currents on a conveniently chosen surface encompassing the antenna. If the antenna radiates mostly from electric field (equivalent magnetic

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Fig. 2. Cavity model of a patch antenna.

B. Patch Antenna Fig. 1. Dipole antenna with the surface S brought away from the surface of the antenna.

surface current , see, e.g., [16]) or magnetic field (equivalent electric surface current ), the non-radiating field can be modified using magnetic or dielectric material loading, respectively, without significantly affecting the radiation process. The material loading should, of course, diminish the amplitude of the field which is less contributing to radiation.

IV. EXAMPLES Next, we will illustrate the proposed radiating-surface model on examples of some simple antennas and compare the conclusions with those given by the conventional resonant-circuit approach. The example of the planar inverted-L antenna is analyzed in detail, because in that case the conventional and the new models give different results.

A. Dipole Antenna Let us consider a dipole antenna, as shown in Fig. 1. This structure radiates from the electric surface current or, by using the Huygens’ principle, from the equivalent current on created . The electric field is normal by the magnetic field: to the conducting cylinder and therefore also mostly normal to a virtual surface located close to the antenna. This lead to an insignificant equivalent magnetic current on , and we see that the electric surface current is the main radiation mechanism. If we want to miniaturize the antenna, caused by the magnetic field is preserved, when the volume bounded by is filled with a dielectric material. Therefore, miniaturization is possible without disturbing the radiating electric current. Magnetic loading also leads to miniaturization, but simultaneously with the decrease of bandwidth. In practice, the bandwidth also somewhat decreases with a dielectric coating [2], but this effect is much stronger with magnetic coating (simulation results not presented here). The antenna impedance resonance of a dipole antenna in its fundamental mode is of a series type, and the dipole antenna behaves as expected in terms of miniaturization also according to the equivalent-circuit model.

For an opposite example in terms of the radiation mechanism, assume a simple cavity model for the patch antenna with a PEC ground plane and patch, as seen in Fig. 2. By neglecting the fringing fields using the PMC boundary condition at the aperture surrounding the cavity, we have vertical electric field across the sidewalls of the cavity, and magnetic equivalent surface current at the walls [17]. The electric surface current density in the top patch is relatively weak when the patch height is small. Also, is parallel to the ground plane, thus leading to an out-of-phase image current, and hence the electric surface current contributes weakly to radiation. Since the electric surface current does not produce much radiation, this structure radiates mostly from the magnetic equivat the aperture. Now, magnetic filling can be alent current used for miniaturization without affecting the radiating electric patch antenna, the electric field. As it happens, in case of a fields at the patch ends actually do not suffer at all from magnetic filling when compared to dielectric filling, as shown in [10] in theory and with simulation. The current and hence the magnetic field under the patch is on the other hand reduced. Unlike the previous case, the optimal material loading not only miniaturizes the antenna (reduces the wavelength of the resonance mode) but also increases the relative bandwidth. Again, as the antenna impedance (neglecting the reactance of the probe) behaves as a parallel circuit, the equivalent-circuit model gives the same conclusion. C. Planar Inverted-L Antenna Next we consider an example of an antenna, where the equivalent-circuit model fails. A planar inverted-L antenna (PILA) is a quarter-wavelength patch antenna with a series-type resonance. This behavior can be understood from a simple open-ended transmission line [15]. The real part of the input impedance of a lossless transmission line is nearly zero, and the imaginary part crosses zero at the resonance, just dipole antenna. However, it will be like in the case of a shown below that when the permittivity of a dielectric substrate is comparable with the permeability of a magnetic substrate, the magnetic response of the substrate increases the relative bandwidth. We will study a patch antenna at the frequency where its . Here is the effective wavelength length is is the effective wavenumber. The antenna is in the patch and illustrated in Fig. 3. We will consider the case when the distance , and let the inductance from the edge to the feed point

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formula yields the standard radiation conductance of an electrically short narrow slot in a PEC plane:

(9)

Fig. 3. Illustration of a planar inverted-L patch antenna on a substrate with ground plane below.

This approximate formula works up to [18]. For should be corrected, howwider patches1 the factor ever it is important that it never depends on the material parameters of the substrate [19], [20]. Substituting (9) into (7) we come to the result for the radiation -patch antenna: quality factor of a (10)

and resistance of a presumed feeding pin be negligible. We also neglect the conductivity losses and losses in the substrate (mod). eled by material parameters at the feed point is the sum of The antenna admittance the radiation conductance of the left edge and the input admittance of the microstrip line with the length loaded by the same radiation conductance :

Let us now compare two antennas of the same size, miniaturized , and a dielectric one using a magnetic substrate , (here can be treated as the refraction index of the substrate). In the presented model both miniaturized antennas experience the series resonance at the same frequency. The ratio of the radiation quality factors of the antennas with dielectric and magnetic substrates, i.e. the inverse ratio for their bandwidth follows from (10):

(6)

(11)

is the radiation conductance of the slot, is the characteristic admittance of the line and is . We related with the free-space wavenumber as can write the radiation quality factor directly, when we assume lossless antenna materials, through the real and imaginary part from (2). We set of the antenna impedance as the frequency at which and the patch is in the (series-type) resonance. Substituting (6) into (2) and applying we easily for the derivative the resonance condition find the radiation quality factor

where and are the quality factor and bandwidth and of the antenna with the magnetic substrate and are the respective values for the dielectric substrate. This shows that the magnetic substrate always provides lower than the dielectric one, despite the fact that the antenna works in the series resonance. This result is similar to the relative radiation quality factor derived for a patch in [8], [9], [21]. By inwas below unity, i.e. the magnetic creasing , . The same effect is seen in (10), substrate provided lower where is directly proportional to , not to . Simulations were made to validate the presented model for PILA. A structure similar to Fig. 3 was studied. The the , dimensions of the patch were and the ground plane and the patch were modeled as PEC boundaries, zero-volume in the case of the patch. First the or substrate was simulated using structure with either Ansoft HFSS full-wave simulator [22] and the quality factor was calculated using (2). This equation does not rely on any specific impedance-matching condition. Thus in reality the low needs to be matched to, e.g., , that in its turn due to the stored energy in the matching affects the total network (and may change the resonance type of the input resonance frequency with impedance). When the was found, the realized and the chosen were substituted in (10) and the theoretical was calculated. The results for the aforementioned dimensions are presented in Table I and Fig. 4. The simulated and theoretical results, and respectively, do not give the same results in magni-

Here

(7) The permeability of the substrate cancels out and does not enter into this expression. A useful intermediate result in this derivation is as follows: (8) In papers [18], [19] it was indicated (and confirmed by the validation of the whole model) that the radiation conductance of an edge of microstrip antennas weakly depends on its length . There is no practical difference between of a standard patch antenna (experiencing the parallel resonance) and of a -patch antenna. In both cases the distribution of (i.e. the voltage distribution) along the axis (see in Fig. 3) is decoupled with the voltage distribution along . In other words, depends . only on the electrical width of the slot The general formula for arbitrary was accurately derived and this involved in [20, formula (40)]. For

ad

1The current distribution corresponding to the series resonant mode can be axis for when . excited along the

X

w>d

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TABLE I QUARTER-WAVELENGTH PATCH SIMULATION RESULTS

Fig. 4. Simulated and theoretical quality factors for a =4 patch. Simulations for dielectric and magnetic substrates, using (2) to calculate the respective quality factors Q and Q . Theoretical quality factors were obtained using (10) with the realized resonance frequencies ! from the simulations.

tude, but the difference between the and -loaded ( and , respectively) antennas is clear when the material parameters change. As the permittivity of the substrate is increased, start to increase, but as the permeability increases, holds at the same level in contradiction to the TL-model prediction. The difference between the simulated and theoretical results are due to the approximations made in the derivation. For exis not completely accuample, the approximation rate. In fact, the distributions of the electric and magnetic fields across the patch are strongly different. As a result, the ratio of the effective permittivity of the microstrip line formed by the patch and the physical substrate is different from the corresponding ratio for the effective permeability and . Therefore, the contribution of into differs from that of . This factor is not taken into account in the approximate theory above. Also, the effective length of the patch was not compensated in the calculations. This means that the total volume of the antenna in simulations was greater due to fringing fields outside the patch edges, and hence the simulated quality factors were lower. As an example of quality factor’s effect on the achievable bandwidth, we have matched the simulated antenna impedances with lumped elements (e.g. [15]) to a 50- transmission line with critical coupling and calculated the relative bandwidth at ) and dB bandwidth criteria. These results can ( be seen in Table I. The patch antenna with a dielectric substrate

undergoes a serious degradation of bandwidth as increases whereas with a magnetic substrate the bandwidth saturates to levels comparable to air filling, as predicted by the quality factor (Fig. 4). What happens in the antenna with dielectric and magnetic loading that explains the observed behavior? Let us examine the field distributions in the simulated antenna with dielectric and magnetic substrates (material parameters according to Table I) as they produce approximately the same . Fig. 5 shows magand Fig. 6 the magnetic nitudes of the electric field strength field strength (with 90 degrees phase shift from ), i.e. also , at the patch level. As opposed to the half-wavelength patch antenna analyzed decreases in [10], we see that with the dielectric loading along the whole patch length. The magnetic field and surface current is virtually the same for all close to the feed where the infield strength is at its strongest. With magnetic loading exhibits only moderated increase. creases now strongly but As we have discussed above with the patch antenna, the radiation mechanism is the fringing electric field at the open end of the patch. Now we see that the dielectric loading was harmful for the radiation and the magnetic loading was, on the contrary, beneficial. can be understood from (8), where the This difference in radiation resistance is proportional to but inversely proportional to . The electric field contributes to the radiated power in (1), and in case of the magnetic loading this overcomes the and we see the beneficial behavior change in stored energy in . This can be seen also in the antenna impedance, which is or plotted in Fig. 7 for the loading parameter values . The higher electric field is seen as the higher for and the average stored enthe magnetic filling. The slope of ergy are higher for the magnetic filling, but not enough higher to destroy the improvement in due to enhanced radiation. We can conclude that the magnetic loading can be really useful also for microstrip antennas with a series resonance, at least when the losses can be neglected. It is seen from this example that the resonance type of the antenna impedance as such is not enough to characterize resonant antennas for either magnetic or dielectric loading. An example of a similar behavior was recently seen in [5], where a top-loaded dipole antenna was seen to benefit from magnetic material. In fact, after loading the dipole from the top in [5], the field distribution on the antenna boundary resembles the patch antenna of Fig. 2 more than a wire dipole antenna of Fig. 1, and the outcome is in line with the analysis of this study.

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Fig. 5. The electric field distributions simulated for the square =4 patch at the patch level along the center line (x-direction in Fig. 3) with (a) dielectric and (b) magnetic loading. The feed is at the edge (a = 0) and the curves with similar line styles correspond to the same resonance frequency.

Fig. 6. The magnetic field distributions simulated for the square =4 patch at the patch level along the center line (x-direction in Fig. 3) with (a) dielectric and (b) magnetic loading. The feed is at the edge (a = 0) and the curves with similar line styles correspond to the same resonance frequency.

Finally, we note that antennas that benefit from magnetic materials can be further distinguished from the effect seen in the radiation resistance . The half-wavelength patch maintained the same in miniaturization [10], but the PILA above . Generally speaking, in some cases the showed increase in magnetic filling can bring additional advantage increasing the radiating fields, thus eliminating the need for a more complex matching network. V. GUIDELINES In the above examples we have considered quite simple antennas in which one of the equivalent surface currents strongly dominates in its contribution to radiation. We formulated a rule as follows: For miniaturization of small resonant antennas magnetic material should be used in case of radiating magnetic equivalent current and dielectric material in case of radiating electric equivalent current. However, complex antennas may radiate via both of the equivalent currents. For example, a planar inverted-F antenna (PIFA) with a considerable height may radiate also through the shorting element, usually a pin or a narrow strip, due to the high current in this element. In this case, the radiation from the vertical electric current may become comparable with the radiation via the electric field at the

Fig. 7. The real and imaginary part of the antenna impedance Z . R and R are the antenna resistances for dielectric and magnetic loading and X and X the reactances, respectively. The dielectric material has  = 3:5 and the magnetic material  = 3:2.

open end of the patch. By following our suggestions, the most beneficial loading material would actually be a combination of both dielectric and magnetic materials. In reality, one should of

KARILAINEN et al.: CHOOSING DIELECTRIC OR MAGNETIC MATERIAL TO OPTIMIZE THE BANDWIDTH OF MINIATURIZED RESONANT ANTENNAS

course use the filling materials at the position where the effect is the strongest, as it is presented in [23] for partial dielectric resonance. loading of microstrip antennas working at the If an antenna radiates from both equivalent currents, partial material loading using both types of substrates may prove to be most beneficial. The frequency dispersion of natural magnetic materials is also an important limitation. The benefit of magnetic loading has to be significant because the dispersion increases the stored energy [9] and also the material losses reduce the radiation efficiency. This has to be taken into account when calculating the radiation quality factor. Also, the conductor losses are significant when the antenna itself is small compared to the free-space wavelength. The radiation efficiency may be different with the same antenna utilizing dielectric or magnetic materials due to the different current distributions inside the antenna. Dielectric and magnetic substrates as such are hardly equal in their potential for antenna miniaturization. Even when a magnetic material might provide lower radiation quality factor, the radiation efficiency due to material losses may be lower than with the dielectric material filling, and the comparison must be made between the obtained impedance bandwidth and efficiency. Also, the price of magnetic materials plays maybe the most restricting role in industrial applications. VI. CONCLUSION We have introduced a simple and intuitive rule based on the analysis of the radiating fields instead of conventional equivalent circuits or transmission-line theory for determining the beneficial filling material type for small resonant antennas. The method can be applied to small resonant antennas and is based on the analysis of equivalent radiating currents: if the radiation mechanism is an (equivalent) electric current, it is better to use dielectric material loading instead of magnetic material, and vice versa, if the equivalent magnetic radiating current is dominant. The respective typical examples for these antennas are the usual electric dipole and patch antenna, and the models for these antenna can be assumed to work at qualitative level. An example of quarter-wavelength patch antenna has been presented, where the miniaturization can be explained by using our method, whereas the conventional circuit approach would lead to a wrong conclusion about the optimal material loading. The effects of material dispersion as well as partial loading of resonant antennas have been also discussed.

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[4] S. Tretyakov, S. Maslovski, A. Sochava, and C. Simovski, “The influence of complex material coverings on the quality factor of simple radiating systems,” IEEE Trans. Antennas Propag., vol. 53, no. 3, pp. 965–970, 2005. [5] H. R. Stuart and A. D. Yaghjian, “Approaching the lower bounds on Q for electrically small electric-dipole antennas using high permeability shells,” IEEE Trans. Antennas Propag., vol. 58, no. 12, pp. 3865–3872, Dec. 2010. [6] H. Mosallaei and K. Sarabandi, “Magneto-dielectrics in electromagnetics: Concept and applications,” IEEE Trans. Antennas Propag., vol. 52, no. 6, pp. 1558–1567, 2004. [7] H. Mosallaei and K. Sarabandi, “Antenna miniaturization and bandwidth enhancement using a reactive impedance substrate,” IEEE Trans. Antennas Propag., vol. 52, no. 9, pp. 2403–2414, Sep. 2004. [8] P. M. T. Ikonen, S. I. Maslovski, C. R. Simovski, and S. A. Tretyakov, “On artificial magnetodielectric loading for improving the impedance bandwidth properties of microstrip antennas,” IEEE Trans. Antennas Propag., vol. 54, no. 6, pp. 1654–1662, Jun. 2006. [9] P. M. T. Ikonen, K. N. Rozanov, A. V. Osipov, P. Alitalo, and S. A. Tretyakov, “Magnetodielectric substrates in antenna miniaturization: Potential and limitations,” IEEE Trans. Antennas Propag., vol. 54, no. 11, pp. 3391–3399, Nov. 2006. [10] P. Ikonen and S. Tretyakov, “On the advantages of magnetic materials in microstrip antenna miniaturization,” Microwave Opt. Technol. Lett., vol. 50, no. 12, pp. 3131–3134, 2008. [11] A. O. Karilainen, P. M. T. Ikonen, C. R. Simovski, and S. A. Tretyakov, “Benefits of material loading of electrically small resonant antennas,” in Proc. Progress in Electromagnetics Research Symp. (PIERS 2009), Moscow, Russia, Aug. 2009, p. 635 [Online]. Available: http://piers. mit.edu/piers/, PIERS 2009 Moscow Abstracts [12] L. J. Chu, “Physical limitations of omni-directional antennas,” J. Appl. Phys, vol. 19, no. 12, pp. 1163–1175, 1948, 12. [13] T. G. Tang, Q. M. Tieng, and M. W. Gunn, “Equivalent circuit of a dipole antenna using frequency-independent lumped elements,” IEEE Trans. Antennas Propag., vol. 41, no. 1, pp. 100–103, 1993. [14] M. Hamid and R. Hamid, “Equivalent circuit of dipole antenna of arbitrary length,” IEEE Trans. Antennas Propag., vol. 45, no. 11, pp. 1695–1696, 1997. [15] D. M. Pozar, Microwave Engineering, 3rd ed. New York: Wiley, 2005. [16] R. F. Harrington, Time-Harmonic Electromagnetic Fields, ser. IEEE Press Classic Reissue. New York: Wiley, 2001. [17] C. A. Balanis, Antenna Theory: Analysis and Design, 3rd ed. : WileyInterscience, 2005. [18] S. Pinhas and S. Shtrikman, “Comparison between computed and measured bandwidth of quarter-wave microstrip radiators,” IEEE Trans. Antennas Propag., vol. 36, no. 11, pp. 1615–1616, 1988. [19] A. Derneryd, “Linearly polarized microstrip antennas,” IEEE Trans. Antennas Propag., vol. 24, no. 6, pp. 846–851, 1976. [20] D. Jackson and N. Alexopoulos, “Simple approximate formulas for input resistance, bandwidth, and efficiency of a resonant rectangular patch,” IEEE Trans. Antennas Propag., vol. 39, no. 3, pp. 407–410, 1991. [21] P. M. T. Ikonen, P. Alitalo, and S. A. Tretyakov, “On impedance bandwidth of resonant patch antennas implemented using structures with engineered dispersion,” IEEE Antennas Wireless Propag. Lett., vol. 6, pp. 186–190, Nov. 2007. [22] HFSS 11 Ansoft Corporation [Online]. Available: http://www.ansoft. com/ [23] O. Luukkonen, P. Ikonen, and S. Tretyakov, “Microstrip antenna miniaturization using partial dielectric material filling,” Microwave Opt. Technol. Lett., vol. 49, no. 1, pp. 155–159, Jan. 2007.

ACKNOWLEDGMENT A. O. Karilainen would like to thank R. Valkonen for the help in data processing and calculations. REFERENCES [1] R. C. Hansen and M. Burke, “Antennas with magneto-dielectrics,” Microwave Opt. Technol. Lett., vol. 26, no. 2, pp. 75–78, 2000. [2] D. Lamensdorf, “An experimental investigation of dielectric-coated antennas,” IEEE Trans. Antennas Propag., vol. 15, no. 6, pp. 767–771, Nov. 1967. [3] R. Ziolkowski and A. Kipple, “Application of double negative materials to increase the power radiated by electrically small antennas,” IEEE Trans. Antennas Propag., vol. 51, no. 10, pp. 2626–2640, 2003.

Antti O. Karilainen (M’11) was born in Tuusula, Finland, in September 1981. He received the M.Sc. (Tech.) degree in electrical engineering from the Helsinki University of Technology (TKK), in 2008, where he is currently working toward the D.Sc. degree. He worked as a trainee at VTI Technologies in Vantaa, Finland during 2004–2006. In 2007 he joined the Radio Laboratory at Helsinki University of Technology (TKK) as a Research Assistant where he is currently working as a Research Engineer in the Department of Radio Science and Engineering (partly established from the former Radio Laboratory in 2008 at TKK and now a part of Aalto University). His main interests are low-profile and small resonant antennas, high-impedance surface applications, and other electromagnetic composite materials.

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Pekka M. T. Ikonen (S’04–M’07) was born on December 30, 1981, in Mäntyharju, Finland. He received M.Sc. and Ph.D. degrees in communications engineering (both with distinction) from the TKK Helsinki University of Technology, Espoo, Finland, in 2005 and 2007. From 2007 to 2008, he was with Nokia Research Center, and from 2008 to 2009, with Nokia Devices R&D working as Antenna Researcher and Antenna Technology Manager, respectively. Since August 2009, he has been with TDK-EPC working as Antenna Technology Manager responsible for strategy creation and implementation for new antenna and RF-systems.

Constantin R. Simovski was born on December 7, 1957, in Leningrad, Russian Federative Republic of the Soviet Union (now St. Petersburg, Russia) and graduated from Leningrad Polytechnic Institute in 1980 as an engineer-physicist (in radiophysics). He received the Candidate (Ph.D.) degree in physical and mathematical sciences in 1986 and the Doctor in Physical and Mathematical of Sciences in 2000. Currently, he is a Visiting Professor (2008–2013) with the Department of Radio Science and Engineering, Aalto University, School of Science and Technology (formerly Helsinki University of Technology). From 1980 to 1992, he was with the Soviet scientific and industrial firm “Impulse,” from 1992 to 2008, he was with the State Institute of Fine Mechanics and Optics (now University ITMO). He has pursued research in the fields of electromagnetics for microwave applications (non-reflective antenna shields and radomes, polarisation transformers and frequency selective surfaces, theory and applications of photonic crystals and other electromagnetic band-gap structures, electrodynamic problems of scattering in lattices, resonant, radiating and waveguide properties of metal nanoparticles and nano-arrays, a general theory of homogenization for meta-materials. He has authored and coauthored 114 papers in refereed journals, 27 book chapters, a monograph, 97 conference papers, a U.S. patent and three study books for students.

Sergei A. Tretyakov (M’92–SM’98–F’08) received the Dipl. Engineer-Physicist, the Candidate of Sciences (Ph.D.), and the Doctor of Sciences degrees (all in radiophysics) from the St. Petersburg State Technical University (Russia), in 1980, 1987, and 1995, respectively. From 1980 to 2000, he was with the Radiophysics Department, St. Petersburg State Technical University. Presently, he is a Professor of radio engineering at the Department of Radio Science and Engineering, Aalto University, Finland, and the President of the Virtual Institute for Artificial Electromagnetic Materials and Metamaterials (Metamorphose VI). His main scientific interests are electromagnetic field theory, complex media electromagnetics and microwave engineering. Prof. Tretyakov served as Chairman of the St. Petersburg IEEE ED/MTT/AP Chapter from 1995 to 1998.

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Rectilinear Leaky-Wave Antennas With Broad Beam Patterns Using Hybrid Printed-Circuit Waveguides José Luis Gómez-Tornero, Member, IEEE, Andrew R. Weily, Member, IEEE, and Yingjie Jay Guo, Senior Member, IEEE

Abstract—A theoretical study on the design of broadbeam leakywave antennas (LWAs) of uniform type and rectilinear geometry is presented. A new broadbeam LWA structure based on the hybrid printed-circuit waveguide is proposed, which allows for the necessary flexible and independent control of the leaky-wave phase and leakage constants. The study shows that both the real and virtual focus LWAs can be synthesized in a simple manner by tapering the printed-slot along the LWA properly, but the real focus LWA is preferred in practice. Practical issues concerning the tapering of these LWA are investigated, including the tuning of the radiation pattern asymmetry level and beamwidth, the control of the ripple level inside the broad radiated main beam, and the frequency response of the broadbeam LWA. The paper provides new insight and guidance for the design of this type of LWAs. Index Terms—Antenna synthesis, broadbeam antennas, leakywave antennas (LWAs).

I. INTRODUCTION

D

IVERGING of electromagnetic waves radiated by leakywave antennas (LWAs) has been proposed in recent years to realize broadbeam antennas with low radiation outside of the main beam [1]–[7]. The first designs by Ohtera [2]–[5] made use of dielectric grating LWAs [8], which were curved following an equiangular spiral to make the emitted rays converge/diverge and thus obtain the desired broadbeam patterns. Burghignoli et al. [6] proposed an interesting mechanism to obtain broadbeam LWAs without the need of curving the leaky-wave line source structure, thus leading to simpler manufacturing and volume reduction of the structure. To achieve this, the rectilinear LWA must be tapered in a nonstandard way, so that the beam angle is varied along the rectilinear geometry of the LWA to make the emitted rays converge/diverge in a specified fashion, as illustrated in Fig. 1. A conformal planar microstrip LWA has also been proposed in [7] to obtain diverging-focusing patterns. Diverging tapered LWAs present a broad beamwidth in the scanned direction of the LWA, with high rejection out of the prescribed angular region. This type of radiation pattern may Manuscript received January 14, 2010; revised February 16, 2011; accepted April 09, 2011. Date of publication August 12, 2011; date of current version November 02, 2011. This work was supported by the Spanish National project TEC2007-67630-C03-02/TCM, Regional Seneca project 08833/PI/08, and by the Spanish scholarship “Salvador de Madariaga” (ref. PR2009-0336). J. L. Gómez-Tornero is with the Department of Communication and Information Technologies, Technical University of Cartagena, Cartagena 30202, Spain (e-mail: [email protected]). A. Weily and Y. J. Guo are with the CSIRO ICT Centre, Epping, NSW 1710, Australia. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2164173

Fig. 1. The two approaches proposed in [6] to taper rectilinear LWAs in order to synthesize broad radiated main beams: (a) virtual focus; (b) real focus.

find applications in frequency scanning radar (FSR) [9], or high efficiency primary feeds of low parabolic reflectors [10]. Regarding the latter application, Fig. 2 shows in red lines the radiation pattern of a corrugated chaparral horn antenna presented in [11], which is optimized to illuminate a parabolic dish with a maximum feasible efficiency. Also shown in long Fig. 2 with black lines is the radiation pattern of a tapered LWA with a broadbeam. It is seen that the broadbeam LWA can further reduce the spillover and illumination losses, thus increasing the antenna efficiency by an estimated 10%. It should be pointed out that the one-dimensional tapering studied in this paper is able to shape the beam in only one plane. To obtain full shaping of the radiated fields and the aforementioned 10% efficiency enhancement, a two-dimensional LWA structure should be tapered in both transverse dimensions. In [10], the excitation of several TE/TM leaky-modes in a 2D EBG structure was used to shape the radiation pattern in both planes. However, this approach is totally different to the unconventional tapering of the pointing angle of a single leaky-mode presented in the paper. As shown later, our approach grants a more flexible control on the shaping of the broadbeam at the expense of a lower bandwidth. Although the relevant theory of tapered rectilinear LWAs for broadbeam pattern synthesis was described in [6], and a possible realization using the stepped-waveguide technology

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Fig. 2. Potential application of broadbeam tapered LWAs to feed low parabolic reflectors, comparing with optimized chaparral horn [11].

leaky-waves. The results presented are of general applicability to many applications, but no attempt has been made to study any particular ones in depth which is beyond the scope of the paper. The rest of the paper is organized as follows. Section II presents the analysis and design principles of broadbeam tapered LWA employing the proposed hybrid technology. It is shown that the real focus version is preferred to the virtual focus alternative for practical reasons. Then, design considerations regarding the optimization and tuning of the radiation pattern including the asymmetry level, the beamwidth, and the range of scanned angles are discussed in Section III. An amplitude illumination tapering technique is proposed and applied in Section IV to reduce the ripple inside the broadbeam. Finally, the frequency response of this type of tapered LWA is analyzed in Section V by means of the frequency dispersion curves of the constituent tapered leaky-mode. All the theoretical results are validated by comparing the leaky-mode theory with 3D full-wave analysis of the designed LWAs using the commercial package HFSS [25]. II. DESIGN OF BROADBEAM LWAS IN HYBRID TECHNOLOGY

Fig. 3. Leaky-wave antenna in hybrid waveguide printed-slot-circuit technology. (a) Cross section dimensions; (b) non-tapered LWA; (c) tapered LWA.

[12], [13] was also proposed in [6], no attempt has been reported to examine the practical design issues, and to validate this nonstandard tapering approach. In this paper, we present a comprehensive study on the practical design of rectilinear tapered LWA. The unusual tapering concept proposed in [6] is realized using hybrid waveguide printed-circuit technology [14]–[16]. The proposed hybrid technology provides a simple mechanism for the simultaneous tapering of the phase and leakage rates of the constituent leaky-mode by properly varying the width and the position of a printed-circuit slot. This mechanism is much simpler compared to previous rectilinear LWAs in pure waveguide technology, such as the aforementioned stepped-waveguide [12], [13] or other similar waveguide LWA, like the trough waveguide [17], the NRD guide [18], [19], the ridge waveguide [20]–[22], and the offset groove guide [23] also known as the stub loaded rectangular waveguide [24]. Although those LWAs presented flexible and quasi independent control of the phase and leakage rates, they all require that the bulky leaky waveguide cross-section dimensions be varied along the LWA length using complicated and most likely expensive mechanization processes. In contrast, the proposed hybrid technology can be tapered using standard photolithography techniques for printed circuits, thus providing less costly, much easier manufacturing and tuning of the response of the antenna. A scheme of the main geometrical parameters of this LWA technology is illustrated in Fig. 3. The paper demonstrates for the first time the mechanism to taper the angle of radiation from a practical one-dimensional rectilinear LWA. Moreover, it is shown that the resulting broadbeam pattern can be shaped by properly tapering the associated leaky-wave, thus presenting a simpler solution than the one presented in [10], which needed the interaction of several TE/TM

The synthesis of broadbeam radiation patterns using rectilinear LWAs is achieved by a nonstandard tapering of the crosssection dimensions of the straight LWA along its length [6], so that the phase constant and the leakage rate of the leakywave are functions of the longitudinal position (see Figs. 1 and 3): (1) and can be found The theoretical expression for in [6] for both the real focus (Fig. 1(a)) and the virtual focus (Fig. 1(b)) alternatives. To investigate some real issues related operating to the proposed LWA, a LWA of length at , with a prescribed 10 dB beamwidth in the is designed, where stands region for the elevation angle as illustrated in Figs. 1–3. In order to obtain the desired wide beam, a real focus located at or a virtual focus located at is chosen. The radiation patterns obtained by Fourier transforming the theoretical tapered leaky-mode illumination are plotted in Fig. 4(a), while the near-fields in the proximity of the LWA obtained in a similar way to [26] are shown in Fig. 4(b) (real focus) and Fig. 4(c) (virtual focus). Fig. 4(a) illustrates how both the real and virtual focus alternatives theoretically provide identical broadbeam farfield radiation patterns covering the prescribed 10 dB region , even if the near field distributions are remarkably different. Once the necessary tapered functions and are known, the most difficult task in the design of a tapered LWA is to relate these functions with the longitudinal tapering of the cross-section dimension of the LWA [1]. To this end, a LWA technology with flexible and independent control of and is of vital importance, even more so in the nonstandard tapering mechanism presented here, since it involves the unusual simultaneous tapering of and . Much research has been conducted in recent decades to develop LWAs with independent and flexible control of and [1], [12]–[24] and, in our opinion, the hybrid waveguide printed-circuit LWA

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Fig. 5. Two-dimensional dispersion plot showing constant pointing angle and constant leakage rate contour curves as a function of the and for 5.5 GHz [27]. geometry design parameters

Fig. 4. Leaky-wave radiation pattern and near fields for a broadbeam tapered at 5.5 GHz with real focus at LWA with and virtual focus at . (a) Far field patterns for both cases. (b) Near fields for real focus. (c) Near fields for virtual focus.

technology serves as one of the most convenient mechanisms for achieving this. The main cross-section parameters involved in this hybrid technology are shown in Fig. 3(a). Basically, it consists of a dielectric rectangular waveguide perturbed by a longitudinal slot printed on its upper broad wall, and loaded with a parallel-plate waveguide. It allows flexible control of leaky-mode complex propagation constant the perturbed and position . by simply modifying the printed-slot width , and therefore The leaky-mode normalized phase constant

the elevation pointing angle related by [1], can be easily controlled by simply changing the slot width . Higher values of are obtained by narrower slots, whereas wider slots result in the leaky-pointing angle closer to broadin Fig. 1) [14]–[16]. On the other hand, the side ( slot position with respect to one parallel plate (see Fig. 3) determines the leaky-mode leakage rate by means of the asymmetry radiation mechanism widely used in stub-loaded LWAs [1], [12]–[24]. If the slot is located in a centered position no rameasured diation would occur. The greater the slot offset [15]), the higher the from the center ( leakage-level. The offset of the slot has also a small effect on , but it mainly determines the leakage rate . Under the above principle, the basic procedure to simultaneously and quasi-independently control the leaky-mode phase and the leakage rate starts with the selection of the printed-slot which provides a given angle of radiation . After width that, the slot position can be derived from the offset to obtain the desired value of . Since also affects , an iterative procedure is normally needed [1], [20]–[24]. An alternative approach was proposed in [27] for the design of tapered LWAs, which makes use of two-dimensional dispersion data oband , and then tained by sweeping all possible values of identifying the desired contour curves which provide constant values of and , as shown in Fig. 5 for a hybrid LWA with waveguide cross section dimensions [see Fig. 3(a)] given in the inset of Fig. 5 and operating at 5.5 GHz. Using the 2D dispersion data shown in Fig. 5, one can accurately obtain the taper slot geometry and which provides the desired leaky-mode taper functions and [27]. Fig. 6(a) shows the leaky-mode taper functions , (obtained with the theoretical equations of [6]), and Fig. 6(b) represents the corresponding slot tapered geometries , along the hybrid LWA length (obtained from the two-dimensional dispersion data of Fig. 5 [27]), for the design of the real focus broadbeam rectilinear LWA of Fig. 4(b). As can be seen in Fig. 6, the real focus design needs the slot width to be increased along the antenna length from mm

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Fig. 6. Real focus design: (a) variation of tapered leaky-wave properties and along the LWA length, (b) corresponding tapered slot and , and (c) resulting printed-slot layout. geometry

mm, so that the beginning of the LWA radiates while the far end provides radiation around , thus generating converging/diverging rays. In order to synthesize an equiangular amplitude illumination, the leakage rate must be exponentially increased along the LWA length, as shown with dashed red line in Fig. 6(a). This is achieved by keeping a constant value of the slot asymmetry around mm along all the LWA length, as shown also in Fig. 6(b). One might think that higher values of require higher values of asymmetry , but one must bear in mind that, in the real focus design, higher values of correspond to lower values of (higher values of ). Due to the inherent nature of leaky-waves, which always provide higher leakage rates for lower pointing angles [1], there is no need of increasing the asymmetry level while is decreased along the LWA length. For this reason, the real focus design shown in Fig. 6 enables use of a constant value of mm for all the length of the LWA while providing exponentially increasing leakage, as is decreased from 45 to 10 and is increased from 8 mm to 15 mm along the LWA. Moreover, it is interesting to note that tapered geometry functions and shown in Fig. 6(b) can be very well approximated by straight lines. This dramatically simplifies the design and tuning process, as will be shown in this paper. For comparison, the results for the virtual focus design of Fig. 4(c) are presented in Fig. 7. This alternative design, although theoretically equivalent, has some practically important drawbacks. As shown in Fig. 7(b), the slot width must be shortened from mm to mm, so that the pointing angle is mirrored with respect to the real focus design, providing a tapered function from at the beginning of the LWA and at the end [solid black curve in Fig. 7(a)], which creates the diverging phase-front pattern.

Fig. 7. Virtual focus design: (a) variation of tapered leaky-wave properties and along the LWA length, (b) corresponding tapered slot and , and (c) resulting printed-slot layout. geometry

to around

Fig. 8. Slot dimensions

contour plots for the real and the virtual focus.

However, the leakage rate must be exponentially tapered [see dashed red curve in Fig. 7(a)] as done in the real focus design, since both cases require equiangular amplitude illumination. Since the virtual focus LWA needs both the angle and the leakage to be increasing functions, this design requires an exponential taper function for the slot offset instead of a constant one as in the case of the real focus design. This fact has two practical implications. First, it makes the slot more difficult to realize. Second, there is no guarantee that the large values of and required at the end of the LWA can always be obtained using the hybrid technology. This difference between the real focus and the virtual focus designs is illustrated in Fig. 8, where the contour lines for the variation of the slot dimensions for each design are plotted and the arrow indicates direction towards the end of the LWA length.

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Fig. 10. Control of the leaky-mode amplitude distribution to modify the radiation amplitude symmetry within the prescribed beamwidth [15 , 45 ].

(5) Fig. 9. (a) Illumination and (b) radiation patterns obtained for the designed real and virtual focus LWA, validated with 3D full-wave simulations.

As can be seen, the real focus design can be obtained with a linear variation of the slot dimensions, whereas the virtual focus design needs a curved tapering of the slot, which also must provide high leakage rates for high pointing angles at the end of the LWA, which is not always possible. Therefore, it can be concluded that the real focus design provides more flexible and easier tuning than the virtual focus design, and consequently it is the generally preferred option. To clarify the relation between the radiation patterns and the tapered leaky-mode, the radiation and the normalized amplitude illuminaangle tion functions are obtained from the full-wave (FW) HFSS simulations for each design: (2) (3) where is the phase distribution along the LWA radiating is the y-directed radiated electric field aperture, and amplitude distribution (this type of LWAs radiate with pure horizontal polarization provided that the stub height is high enough [15]). The full-wave data are compared in Fig. 9(a) with the theoretical leaky-wave (LW) illumination, which is extracted from the tapered complex propagation constant: (4)

As shown in Fig. 9(a), the desired functions covand equiangular ering the scanning range amplitude are achieved for the real focus design. However, the virtual focus design cannot provide the requested illuat the far end of the LWA, due to the fact that a mination too high leakage rate a high angle of is required. As a result, the radiation pattern of the virtual focus design shows in Fig. 9(b). lower radiated power around III. OPTIMIZATION OF RADIATION PATTERN As illustrated earlier, the hybrid LWA technology allows the control of the real-focus leaky-mode amplitude and phase illumination in a simple way by choosing the slot width and position at the beginning and the end of the LWA length, and then by tapering the slot using a rectilinear printed circuit geometry. This feature makes the adjustment of the broadbeam LWA very simple. Fig. 10 illustrates how the radiation asymmetry level can be tuned and optimized by simply unbalancing the leaky-mode amplitude distribution. This control of the symmetry of the radiation pattern is very well predicted by the leaky-mode theory, as demonstrated in Fig. 10(b). The illumination functions (4)–(5) are also plotted in Fig. 10(c) to illustrate that the tuning of the radiation pattern amplitude balance is obtained by modifying the leaky-wave illumination distribution while keeping the same angular distribution from 45 to 15 , thus creating an unbalance with respect to the symmetric case (which can be useful,

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Fig. 11. Slot dimensions contour plots used to adjust the radiation pattern symmetry of real focus broadbeam LWA designs as shown in Fig. 10.

for instance, to obtain a cosecant square beam [4], or for correcting the effect of tapering manufacturing errors or ohmic losses [28], [29]). Fig. 11 illustrates this procedure, showing the tapering of the slot dimensions along the LWA length for the three cases shown in Fig. 10. As can be seen, a simple straight line determines the tapering of the slot dimensions; when higher leakage is requested at the beginning of the LWA (for higher radiation at ), the initial slot position and width must be located in the pointing angle contour curve with higher value of , while the final slot dimensions must be in the prescribed final pointing angle contour curve but with a lower value of to keep the radiation efficiency at the same level . For the complementary case (higher leakage at the end of the LWA), it must be proceeded in a similar way, choosing the initial dimensions to lie in the curve with lower leakage, and the final dimensions in the curve with higher leakage to keep . The layouts obtained are also plotted in Fig. 11, showing how the slot asymmetry level is increased or decreased at the beginning and end of the LWA, according to the described procedure. Furthermore, the illumination can be tapered to adjust the range of scanned angles (beamwidth and beam center). In this case, the amplitude illumination function must be kept constant while the angle distribution must be varied, so that the range of scanned angles is adjusted. This procedure is illustrated in Fig. 12, showing the agreement between full-wave results and leaky-mode theory for three adjusted beam widths ( , , and ) centered around , covering the ranges , and . This adjustment is performed by modifying the leaky-mode phase distribution , while keeping the same leakage function to provide equiangular amplitude illumination with , as shown in Fig. 12(c). The control of the scanning angle range is mainly performed by modifying the range of values for the tapered slot width , as previously explained [15]. The tapering of the slot dimensions to obtain the radiation patterns of Fig. 12 are illustrated in Fig. 13. Again, a simple straight line can be used to determine the tapered slot dimensions.

Fig. 12. Control of the leaky-mode phase distribution to adjust the beamwidth (range of scanned angles) for a fixed center pointing angle of 30 .

Fig. 13. Slot dimensions contour plots used to adjust the beamwidth of real focus broadbeam LWA designs as shown in Fig. 12.

IV. RIPPLE REDUCTION All the broadbeam radiation patterns reported by previous authors [3]–[6] show a significant ripple level inside the wide beam radiation range, as can also be seen in Figs. 4, 9, 10, and 12 of this paper. A theoretical 3 dB ripple-level corresponds to a perfect equiangular amplitude distribution [6]. However, the illumination amplitude can be tapered to control the ripple level, so that a flatter response is obtained across the wide angular range of the beam. This is illustrated in Fig. 14, where it can be seen that the ripple level is reduced when the amplitude illumination is tapered at the edges of the antenna, as customarily done to avoid diffraction and reduce the sidelobe level in conventional LWAs [1], [13], [15], [16], [21], [22]. Since the effective length of the LWA is shortened at the edges, the angular range and the beamwidth are consequently reduced, as can also be seen in Fig. 14(c). However, the reduced beamwidth can be readjusted as described in Section III, by increasing the tapering range of the slot width.

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Fig. 14. Control of the ripple level inside the broadbeam: (a) illumination , (b) detail of radiation diagram, and (c) radiation amplitude tapering diagram. Fig. 16. Radiation pattern for different frequencies around 5.5 GHz for the real focus broadbeam LWA design of Fig. 6.

Fig. 15. Printed slot layout design to reduce the ripple level and radiation pattern obtained with full-wave simulations.

The tapering of the slot dimensions for this low-ripple design is illustrated in Fig. 15, where it can be seen that the slot tapered geometry is no longer rectilinear, since the leakage rate must be varied in a more complicated fashion so that very low leakage is induced in both the beginning and the end of the LWA to avoid diffraction at the edges. The radiation patterns obtained with full-wave simulations are also plotted in the figure, showing the effective reduction of the ripple level and beamwidth, observing good agreement with the leaky-mode theory. V. FREQUENCY SCANNING ANALYSIS It is well known that LWAs provide frequency scanning due to the dispersive frequency response of the leaky-mode phase constant, [1]. This makes the main beam pointing angle shift to higher values as the frequency is increased. Broadbeam LWAs, like the one studied in this paper, present a particularly interesting scenario since many emitted rays contribute to the total focusing/diverging radiation pattern. The frequency response of the broadbeam radiation pattern depends on the particular frequency dispersion response of the tapered leaky-mode complex propagation constant along the LWA length: (6) For a given tapered LWA geometry and , the tapered leaky-mode dispersion (6) can be obtained from a leakywave analysis of the structure. Then, the illumination functions for the radiation angle (4) and the normalized amplitude (5) can be extracted for any particular frequency

of interest, . Finally, the radiation pattern can be obtained for each frequency by Fourier transforming the complex illumination function. Fig. 16 shows the radiation patterns for different frequencies around the design frequency (5.5 GHz), for the real-focus broadbeam LWA presented in Fig. 6. Good agreement is observed between the theoretical results obtained from the leaky-mode dispersion (6) and full-wave simulations. As expected, the broadbeam is shifted from lower to higher angles as the frequency is increased. However, the beamwidth is not constant for all frequencies, but becomes narrower as the frequency is increased. This can be explained from the converging-pointing-angle response of the leaky-mode along the LWA length for different frequencies , shown in Fig. 17 with black lines. As can be seen, the range of scanned angles is reduced from at 5.5 GHz, to at 6.5 GHz. Above 7 GHz, the leaky-mode radiates close to endfire , eventually entering the surface-wave regime and drastically reducing the radiation efficiency of the LWA [1]. On the other hand, for frequencies below 5 GHz, the last parts of the LWA are below cut-off, also reducing the antenna effective length and creating a radiation pattern with low directivity close to broadside, observed at 5 GHz in Fig. 16. Moreover, the symmetry of the broadbeam level cannot be kept constant for all frequencies, since the amplitude illumination function is also perturbed from the equiangular distribution synthesized at 5.5 GHz. A similar frequency dependence analysis is performed for the case of the virtual-focus broadbeam LWA presented in Fig. 7. The obtained tapered leaky-mode diverging-pointing-angle distributions along the LWA length for different frequencies are plotted with red curves in Fig. 17. The resulting leaky-mode theoretical frequency-scanning broadbeam radiation patterns are plotted in Fig. 18, and are compared with full-wave results, showing very good agreement. As can be seen by comparing Figs. 16 and 18, the real and the virtual focus broadbeam LWA present quite similar frequency-scanning behaviors, due to the mirrored dispersion curves shown in Fig. 17. A similar compression of the virtual focus beamwidth is observed when frequency is increased, due to the same reason previously explained for the case of the

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Fig. 17. Frequency dispersion of the tapered radiation angle distribution for both real (Fig. 6) and virtual (Fig. 7) focus broadbeam LWA designs.

Fig. 18. Radiation pattern for different frequencies around 5.5 GHz for the virtual focus broadbeam LWA design of Fig. 7.

real focus LWA. However, for frequencies below 5.5 GHz, the virtual focus design presents a different response than its real focus equivalent. As illustrated in Fig. 18, the virtual focus broadbeam at 5 GHz is wider than the one of the real focus at the same frequency in Fig. 16, although the dispersion curves in Fig. 17 are symmetrically mirrored and should provide a similar range of conjugated broadbeam scanned angles. This can be explained from the following fact: whilst the input sections of the real focus LWA length are radiative and the last part is below cut-off (see black curve in Fig. 17 for 5 GHz), the virtual focus LWA has a mirrored response (input sections are below cut-off and last section is radiative), which results in a poorer radiation efficiency and reduced effective length. As a result, the virtual focus LWA has worse radiation performance in the lower range of the scanning band than the real focus design, resulting in a narrower frequency-scanning bandwidth. VI. CONCLUSION A comprehensive theoretical study on the design of practical rectilinear tapered LWAs of uniform type to produce broadbeam patterns has been presented for the first time. The hybrid printed-circuit waveguide LWA has been chosen due to the ability to independently control the leaky-mode phase and leakage constants. It has been demonstrated how a simple linear tapering of the printed-slot dimensions (width and position)

provides broadbeam radiation patterns with flexible tuning of the scanning range (beamwidth and beam center) and the broad beam shape (symmetry and ripple level), while keeping high radiation efficiency. Also, the frequency response of this type of antenna has been investigated, showing that the frequency beam shifting and the degradation of the broad radiated beam can be accurately predicted from the tapered leaky-mode frequency-dispersion curves. This rectilinear technology avoids the use of more complicated curved leaky structures. Also, compared to previous rectilinear LWAs, the hybrid technology avoids the tapering of the waveguide cross section, so only the planar printed-slot dimensions need to be tapered. Therefore, the proposed technology presents a cheaper and more flexible approach for frequency-steering broadbeam LWAs. REFERENCES [1] A. A. Oliner, “Leaky-wave antennas,” in Antenna Engineering Handbook, R. C. Johnson, Ed., 3rd ed. New York: McGraw-Hill, 1993, ch. 10. [2] I. Ohtera, “Focusing properties of a microwave radiator utilizing a slotted rectangular waveguide,” IEEE Trans. Antennas Propagat., vol. 38, pp. 121–124, Jan. 1990. [3] I. Ohtera, “Diverging/focusing of electromagnetic waves by utilizing the curved leaky wave structure: Application to broad-beam antenna for radiating within specified wide-angle,” IEEE Trans. Antennas Propagat., vol. 47, pp. 1470–1475, Sep. 1999. [4] I. Ohtera, “On a forming of cosecant square beam using a curved leakywave structure,” IEEE Trans. Antennas Propagat., vol. 49, pp. 1004–1006, Jun. 2001. [5] I. Ohtera, “Estimation of the radiation patterns of diverging/focusing type of leakywave antennas,” Microwave Opt. Technol. Lett., vol. 33, pp. 358–360, Jun. 2002. [6] P. Burghignoli, F. Frezza, A. Galli, and G. Schettini, “Synthesis of broadbeam patterns through leaky-wave antennas with rectilinear geometry,” IEEE Antennas Wireless Prop. Lett., vol. 2, pp. 136–139, 2003. [7] O. Losito, “The diverging-focusing properties of a tapered leaky wave antenna,” in Proc. 3rd Eur. Conf. Antennas and Propagation, EUCAP2009, Mar. 2009, pp. 1304–1307. [8] F. K. Schwering and S. T. Peng, “Design of dielectric grating antennas for millimeter-wave applications,” IEEE Trans. Microw. Theory Tech., vol. 31, pp. 199–209, Feb. 1983. [9] K. Van Caekenberghe, K. F. Brakora, and K. Sarabandi, “A 94 GHz OFDM frequency scanning radar for autonomous landing guidance,” in Proc. IEEE Radar Conf., Apr. 2007, pp. 248–253. [10] A. Neto, N. Lombart, G. Gerini, D. M. Bonnedal, and P. de Maagt, “EBG enhanced feeds for the improvement of the aperture efficiency of reflector antennas,” IEEE Trans. Antennas Propagat., vol. 55, no. 8, pp. 2185–2193, Aug. 2007. [11] L. Shafai, “Broadening of primary feed patterns by small E-plane slots,” Electron. Lett., vol. 13, no. 4, pp. 102–103, Feb. 1977. [12] C. Di Nallo, F. Frezza, A. Galli, G. Gerosa, and P. Lampariello, “Stepped leaky-wave antennas for microwave and millimeter-wave applications,” Ann. Telecommun., vol. 52, pp. 202–208, Mar.–Apr. 1997. [13] C. Di Nallo, F. Frezza, A. Galli, and P. Lampariello, “Theoretical and experimental investigations on the ‘stepped’ leaky-wave antennas,” in Dig. 1997 IEEE Antennas and Propagat. Int. Symp., Canada, Jul. 13–18, 1997, pp. 1446–1449. [14] P. Lampariello and A. A. Oliner, “A novel phased array of printedcircuit leaky-wave line sources,” in Proc. 17th Eur. Microwave Conf., Rome, Italy, Sep. 7–11, 1987, pp. 550–560. [15] J. L. Gómez, A. de la Torre, D. Cañete, M. Gugliemi, and A. A. Melcón, “Design of tapered leaky-wave antennas in hybrid waveguide-planar technology for millimeter waveband applications,” IEEE Trans. Antennas Propagat., vol. 53, no. 8, pt. I, pp. 2563–2577, Aug. 2005. [16] J. L. Gómez,, G. Goussetis, A. Feresidis, and A. A. Melcón, “Control of leaky-mode propagation and radiation properties in hybrid dielectric-waveguide printed-circuit technology: Experimental results,” IEEE Trans. Antennas Propagat., vol. 54, no. 11, pp. 3383–3390, Nov. 2006. [17] W. Rotman and A. A. Oliner, “Asymmetrical trough waveguide antennas,” IRE Trans.Antennas Propagat., vol. AP-7, pp. 153–162, Apr. 1959.

GÓMEZ-TORNERO et al.: RECTILINEAR LEAKY-WAVE ANTENNAS WITH BROAD BEAM PATTERNS USING HYBRID PRINTED-CIRCUIT WAVEGUIDES

[18] Q. Han, A. A. Oliner, and A. Sanchez, “A new leaky waveguide for millimeter waves using nonradiative dielectric (NRD) waveguide—Part II: Comparison with experiments,” IEEE Trans. Microw. Theory Tech., vol. MTT-35, pp. 748–752, Aug. 1987. [19] A. A. Oliner, S. T. Peng, and K. M. Sheng, “Leakage from a gap in NRD guide,” in Dig. 1985 IEEE Int. Microwave Symp., St. Louis, MO, Jun. 3–7, 1985, pp. 619–62. [20] F. Frezza, M. Guglielmi, and P. Lampariello, “Millimetre-wave leakywave antennas based on slitted asymmetric ridge waveguides’,” in IEE Proc. H, 1994, vol. 141, no. 3, pp. 175–180. [21] M. Tsuji, T. Harada, H. Deguchi, and H. Shigesawa, “Frequency-scanning antennas with low sidelobes using stub-loaded ridge-rectangular leaky waveguides,” in Proc. 2003 IEEE Topical Conf. Wireless Communication Technology, 2003, pp. 352–353. [22] M. Tsuji and H. Deguchi, “Stub-loaded ridge waveguides for frequencyscanning antenna application,” in Dig. 2007 IEEE Antennas and Propagat. Int. Symp., Honolulu, HI, Jun. 10–15, 2007, pp. 449–452. [23] P. Lampariello, F. Frezza, H. Shigesawa, M. Tsuji, and A. A. Oliner, “Guidance and leakage properties of offset groove guide,” in Dig. IEEE Int. Microwave Symp., Las Vegas, NV, Jun. 9–11, 1987, pp. 731–734. [24] P. Lampariello, F. Frezza, H. Shigesawa, M. Tsuji, and A. A. Oliner, “A versatile leaky-wave antenna based on stub-loaded rectangular waveguide: Part I—Theory,” IEEE Trans. Antennas Propagat., vol. 44, no. 7, pp. 1032–1041, Jul. 1998. [25] “High Frequency Structure Simulator (HFSS),” ver. v11, Ansoft Corp. [26] J. L. Gómez, F. D. Quesada, A. A. Melcón, G. Goussetis, A. R. Weily, and Y. Jay Guo, “Frequency steerable two dimensional focusing using rectilinear leaky-wave lenses,” IEEE Trans. Antennas Propagat., vol. 59, no. 2, pp. 407–415, Feb. 2011. [27] J. L. Gómez, J. Pascual, and A. A. Melcón, “A novel full-wave CAD for the design of tapered leaky-wave antennas in hybrid waveguide printed-circuit technology,” Int. J. RF Microw. Comput. Aid. Eng., vol. 16, no. 4, pp. 297–308, Jul. 2006. [28] C. Di Nallo, F. Frezza, A. Galli, and P. Lampariello, “Rigorous evaluation of ohmic-loss effects for accurate design of traveling wave antennas,” J. Electrom. Waves Appl., vol. 12, pp. 39–58, 1998. [29] J. L. Gómez, G. Goussetis, and A. A. Melcón, “Correction of dielectric losses in leaky-wave antenna designs,” J. Electrom. Waves Appl., vol. 21, no. 8, pp. 1025–1036, 2007.

José Luis Gómez-Tornero (M’06) was born in Murcia, Spain, in 1977. He received the Telecommunications Engineer degree from the Polytechnic University of Valencia (UPV), Valencia, Spain, in 2001, and the Laurea cum laude Ph.D. degree in telecommunication engineering from the Technical University of Cartagena (UPCT), Cartagena, Spain, in 2005. In 1999, he joined the Radio Communications Department, UPV, as a research student, where he was involved in the development of analytical and numerical tools for the automated design of microwave filters in waveguide technology for space applications. In 2000, he joined the Radio Frequency Division, Industry Alcatel Espacio, Madrid, Spain, where he was involved with the development of microwave active circuits for telemetry, tracking and control (TTC) transponders for space applications. In 2001, he joined the Technical University of Cartagena (UPCT), Spain, as an Assistant Professor. From October 2005 to February 2009, he held the position of Vice Dean for Students and Lectures affairs in the Telecommunication Engineering Faculty at the UPCT. Since 2008, he has been an Associate Professor at the Department of Communication and Information Technologies, UPCT. His current research interests include analysis and design of leaky-wave antennas and the development of numerical methods for the analysis of novel passive radiating structures in planar and waveguide technologies.

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Dr. Gómez Tornero received in July 2004 the national award from the foundation EPSON-Ibérica to the best Ph.D. project in the field of Technology of Information and Communications (TIC). In June 2006, he received the Vodafone foundation-COIT/AEIT (Colegio Oficial de Ingenieros de Telecomunicación) award to the best Spanish Ph.D. thesis in the area of Advanced Mobile Communications Technologies. This thesis was also awarded in December 2006 as the best thesis in the area of Electrical Engineering, by the Technical University of Cartagena. In February 2010, he was appointed CSIRO Distinguished Visiting Scientist by the CSIRO ICT Centre, Sydney.

Andrew R. Weily (S’96–M’01) received the B.E. degree in electrical engineering from the University of New South Wales, Australia, in 1995, and the Ph.D. degree in electrical engineering from the University of Technology Sydney (UTS), Australia, in 2001. From 2000 to 2001, he was a research assistant at UTS. He was a Macquarie University Research Fellow, then an ARC Linkage Postdoctoral Research Fellow from 2001 to 2006 with the Department of Electronics, Macquarie University, Sydney, NSW, Australia. In October 2006, he joined the Wireless Technology Laboratory at CSIRO ICT Centre, Sydney. His research interests are in the areas of reconfigurable antennas, EBG antennas and waveguide components, leaky wave antennas, frequency selective surfaces, dielectric resonator filters, and numerical methods in electromagnetics.

Yingjie Jay Guo (SM’96) received the Bachelor and Master degrees from Xidian University, China, in 1982 and 1984, respectively, and the Ph.D. degree from Xian Jiaotong University in 1987. He was also awarded a Ph.D. degree by the University of Bradford, U.K., in 1997 for his contribution to the field of Fresnel zone antennas. His research interest ranges from electromagnetics and antennas, signal processing to mobile and wireless communications and positioning networks. From 1989 to 1997, he was a Research Fellow and later a Senior Fellow at the University of Bradford, conducting and managing research on Fresnel zone antennas and signal processing for mobile and wireless communications. From 1997 to 2005, he held various senior positions in the European wireless industry managing strategic planning and the development of advanced technologies for the third generation (3G) mobile communications systems in Fujitsu, Siemens, and NEC. From 2005 to January 2010, he served as the Director of the Wireless Technologies Laboratory in CSIRO ICT Centre, Australia, managing over 60 research scientists and engineers on antennas and propagation, millimeter-wave systems, signal processing, and wireless communications. Currently, he is the Leader of the Broadband for Australia Theme in CSIRO, Australia, and the Director of the Australia China Research Centre for Wireless Communications. He is an Adjunct Professor at Macquarie University, Australia, and a Guest Professor at the Chinese Academy of Science (CAS). Dr. Guo has played active roles in the organizing committees of a number of international conferences. He served as Chair of the Technical Program Committee (TPC) of 2010 IEEE WCNC and 2007 IEEE ISCIT. He was Executive Chair of Australia China ICT Summit in 2009 and 2010. He was a Guest Editor of the special issue on Antennas and Propagation Aspects of 60–90 GHz Wireless Communications of the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION. He has been a recipient of the Australian Engineering Excellence Award and CSIRO Chairman’s Medal. He has published three technical books: Fresnel Zone Antennas, Advances in Mobile Radio Access Networks, and Ground-Based Wireless Positioning, and has authored and co-authored over 50 journal papers and over 80 refereed international conference papers. He holds 14 patents in wireless communications and antennas. He is a Fellow of IET.

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Vivaldi Antenna With Integrated Switchable Band Pass Resonator M. R. Hamid, Peter Gardner, Senior Member, IEEE, Peter S. Hall, Fellow, IEEE, and F. Ghanem

Abstract—A novel reconfigurable wideband to narrowband Vivaldi antenna is presented. A single pair of ring slot resonators is located in the Vivaldi to realize frequency reconfiguration, maintaining the original size unchanged. The proposed antenna is capable of switching six different narrow pass bands within a wide operating band of 1–3 GHz, offering added prefiltering functionality. A fully functional prototype has been developed. PIN diode switches were employed at specific locations in the resonator to change its effective electrical length, hence forming different filter configurations. Antenna performance obtained from simulation and measurement results shows good agreement, which verifies the proposed design concepts. The antenna is potentially suitable for applications requiring dynamic band switching such as cognitive radio. Index Terms—Antenna, cognitive radio, reconfigurable, Vivaldi, wideband.

I. INTRODUCTION IDEBAND to narrowband reconfigurable antennas have received great attention recently, as they are important for systems that combine wideband and multiradio applications [1]. The approach offers additional prefiltering, which reduces the interference levels at the receiver giving them advantages over fixed, nonreconfigurable transceivers. Common issues for most reconfigurable antennas are moderate performance due to additional loss arising from switches, the need for switch biasing networks and nonlinearity of the switches. In general, there are two ways to achieve wideband to narrowband reconfiguration. One approach is to combine a wideband and a narrowband antenna using separate excitation ports [2]–[4]. A second approach is to reconfigure an inherently wideband antenna either by switching parts of the antenna structure [5], [6] or altering the loading of the antenna internally

W

Manuscript received October 25, 2010; revised March 15, 2011; accepted April 30, 2011. Date of publication August 12, 2011; date of current version November 02, 2011. M. R. Hamid was supported by the Universiti Teknologi Malaysia (UTM). This work was supported in part by EPSRC, grant reference number: EP/FOl 7502/1. M. R. Hamid is with the School of Electronics, Electrical, and Computer Engineering, University of Birmingham, Birmingham B15 2TT, U.K., and also with the Faculty of Electrical Engineering (FKE), Universiti Teknologi Malaysia (UTM), Johor Bahru Campus 81310, Malaysia (e-mail: [email protected]). P. Gardner, P. S. Hall, and F. Ghanem are with the School of Electronics, Electrical and Computer Engineering, University of Birmingham, Edgbaston, Birmingham B15 2TT, U.K. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2164197

or externally [7], [8]. Recently, a number of combined narrowband to wideband reconfigurable antennas have been presented. In [2], a switched sub-band antenna was integrated with a wideband antenna into the same substrate. A motor is used to transform the antenna geometry, achieving two sub bands by means of a 180 rotation. In [3], a single disc monopole excited using two ports at opposite sides is proposed. One side was fed with microstrip line and the other with coplanar waveguide. The first port is operated in narrow band mode and can be reconfigured by varying the meandered rectangular slot length inserted in the ground plane. The second port is kept wideband. In [4] a wideband monopole has been integrated with a planar inverted F antenna (PIFA). The PIFA can be reconfigured to various frequencies by external matching circuits. In the log periodic dipole array described in [5], ideal switches are used to control parts of the antenna structure. This can switch from a wideband of 1–3 GHz, to several narrow bands. In [6] a reconfigurable planar monopole to microstrip patch antenna is proposed. The transformation from wide to narrowband operation is demonstrated by switching “in” and “out” the ground plane beneath the patch. In [7] a Vivaldi antenna loaded with multiple ring slots shows wideband to three narrowband reconfiguration. In [8] an external stub loads a wideband antenna to switch the operating band. Other antennas that have been proposed include an L-shaped slot antenna [9], the switchable quad-band antenna [10] and switched wideband-dual band [11] types. Previously, we have demonstrated a Vivaldi antenna incorporating multiple rings [7]. The antenna shows band switching between a wideband mode and three narrowband modes. In that configuration, the number of frequency bands controlled depends on the number of resonators incorporated. This means that if more bands were desired, the antenna has to be much longer to incorporate more resonators. In this paper we demonstrate a new design method for band switching in a Vivaldi antenna that allows better control of the narrow operating bands. By incorporating only a single pair of slot resonators, six different narrow frequency bands can be switched within the wideband range, double the number reported in [7]. To achieve switched band properties, the proposed approach reconfigures the operating band by varying the electrical length of the slot resonators by means of PIN diode switches. This effectively stops, or passes, current at different frequencies. The design principles of the proposed antenna are described. Sections II and III discuss the initial design of the Vivaldi antenna and the switchable slot line resonator. In Section IV, the switchable narrowband antenna configuration is explained. The fabricated antenna and the results are discussed in Section V. Finally, Section VI gives conclusions.

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Fig. 3. Simulated surface current distribution at 2 GHz. Fig. 1. Vivaldi antenna, (a) front view and (b) rear view showing microstrip feed.

Fig. 2. S

of Vivaldi antenna in Fig. 1.

II. WIDEBAND ANTENNA DESIGN The reconfigurable antenna is based on a wideband Vivaldi operating from 1 to 3 GHz. The structure is based on the antenna in [12], scaled down in frequency by a factor of two, as shown in Fig. 1. It consists of a tapered slot with an elliptic shape, fed by a microstrip line on the back of the structure. The tapered slot has three main parts, comprised of a circular slot stub at the lower end, with radius 4.6 mm and an elliptically shaped slot with horizontal radius 40 mm and vertical radius 80 mm respectively. The tapered slot is terminated with a semi circular disk of 30 mm radius. A 1.6 mm thick FR4 substrate, with dielectric constant of mm wide 4.9 is used. The tapered slot is fed with a mm radius quarter circle. feed line terminated in a Detailed dimensions are shown in Fig. 1. The design guideline , is shown in Fig. 2. can be found in [12]. The simulated III. SLOT LINE RESONATOR DESIGN In principle, distorting the current distributions along its current path will changes the radiation properties. The Vivaldi has currents that propagate near the edge of the tapered slot, as shown in Fig. 3. The narrower end is acting as a transmission region whilst the wider end is acting as a radiation region. It is thus appropriate to locate the filter in the narrower end. A slot ring resonator as shown in Fig. 4 is chosen for this. The basic principle of the reconfigurable ring slot band pass resonator is now described. The slot has an inner radius of 8 mm and an outer radius of 12 mm. Small gaps, of size 3.67 mm 4 mm, are used to

Fig. 4. Single stop band filter, (a) the configuration, (P is slot length, indicated by dotted lines) and (b) the responses.

couple the resonator to the slot line, as shown in Fig. 4(a). The ring slot is acting as an open circuit stub where it presents an open circuit to the coupling slot in the edge. The slot length, , indicated by dotted lines, is 72 mm, which is approximately a half wavelength at 1.69 GHz. If the slot is now bridged at some point, two short circuited stubs are created that can be used to form a pass band filter. Fig. 5 shows the configuration to obtain a band pass response at 1.68 GHz. Instead of acting as single open circuit stub, the slot now has two short circuit stubs which and , which each give a stop band have different lengths, response. mm represents a quarter wavelength at 3.2 mm represents a quarter wavelength at GHz while 1.13 GHz. To switch to a different band, the bridges are moved to other and specific positions, changing the electrical length of . Alternatively an extra bridge can be added. For example, Fig. 6 shows an alternative band pass configuration operating is reduced to 36 mm at 2.3 GHz using an extra bridge. is representing a quarter wavelength at 1.68 GHz, whilst kept to 18 mm. The different pass band configurations, as a function of bridge position, are summarized in Fig. 7.

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Fig. 7. Bridge position for various pass band center frequencies.

Fig. 5. Band pass filter, (a) the configuration, (P and P are slot lengths, indicated by dotted lines) and (b) the responses.

Fig. 8. (a) Band 2 configuration, x 2 configuration.

Fig. 6. Band pass filter, (a) the configuration and (b) the responses.

IV. SEVEN MODE RECONFIGURABLE CONFIGURATION The design and simulation of a reconfigurable Vivaldi antenna with seven modes is now described. To simulate real switches, ON state s2p-data for Infineon PIN diode switches, model BAR50-02V was used. To simplify the simulation process, the OFF state is represented by a gap. Also to reduce computation time, there are neither DC lines nor DC decoupling components included in the simulation. An example of Band 2 (Fig. 7) configuration is shown in Fig. 8(a). The ring slot is mm above the circular slot stub, at the lower placed end of the tapered slot, to avoid strong coupling from the feed line located on the reverse side of the substrate. By using ideal

= 21 3 mmand (b) simulated S :

of band

switches, consisting of metal pads of size 1 mm 4 mm, the simulated resonance of Band 2 configuration occurs at 1.7 GHz. However, with Infineon PIN diode switches, the simulated resonance is shifted down around 14.7% to 1.45 GHz. This can be explained by the package parasitic effects. The simulated results are shown in Fig. 8(b). Fig. 9 shows the current distributions of the Band 2 configuration using ideal switches when excited at the resonance frequency 1.7 GHz [Fig. 9(b)] and at out of band frequencies 1 GHz [Fig. 9(a)] and 3 GHz [Fig. 9(c)]. It is observed that the ring slot passes the current at 1.7 GHz, but significantly reduces the current on the upper part of the antenna at 1 and 3 GHz. The effect of the position of the ring slot was examined by mm of the moving it up toward the wider end tapered slot as shown in Fig. 10(a). The pass band became wider

HAMID et al.: VIVALDI ANTENNA WITH INTEGRATED SWITCHABLE BAND PASS RESONATOR

Fig. 11. Band 6 simulated S

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when ring slot at x = 21:3 mm and 36.3 mm.

Fig. 9. Current distributions (a) excited at 1.0 GHz, (b) excited at 1.7 GHz, and (c) excited at 3.0 GHz.

Fig. 12. Effects of on state resistance on S .

TABLE I SIMULATED TOTAL EFFICIENCY AND GAIN FOR BAND2 CONFIGURATION WITH DIFFERENT ON STATE RESISTANCE VALUES EXCITED AT 1.65 GHz

Fig. 10. (a) Band 2 configurations, x = 36:26 mm (b) simulated S ring slot at x = 21:3 mm and 36.3 mm.

when

at the upper end, as can be seen in Fig. 10(b). It is presumed that the ring slot has little current interception at high frequencies, thus reducing the quality factor. Similar behavior is also noticed for Band 1, Band 3, and Band 4 after the ring slot is positioned mm. Bands 5 and 6, however, show higher . at mm position is Fig. 11 shows this. Therefore, the chosen for a relatively high Q and good impedance matching.

The switch ON state resistance has an effect on the impedance match and antenna total efficiency. Fig. 12 shows the simulated as a function of the ON state resistance between 0 and 8 . The return loss degrades as the resistance is increased. Table I compares the simulated total efficiency and gain for different ON state resistances. The gain decrease is mainly due to the power lost in the resistance, which thus reduces radiation efficiency. The mismatch loss can be seen by comparing radiation efficiency and total efficiency. In general it is much lower than the radiation efficiency component. The PIN diodes noted previously and MEMS switch model RMSW100 from Radant Mems including package parasitics, of Band 2 and have also been compared. The simulated Band 5 are shown in Fig. 13. It is observed that the MEMS switch has lower ON state resistances than PIN diode switches, thus giving the higher total efficiency and gain as noted in value is 3 to 4.5 for PIN diode switches Table II. The [13] while less than 2 for MEMS switches, RMSW100 [14].

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Fig. 13. Effects of PIN diode and MEMs switches on S

responses.

Fig. 15. Wideband mode S

response of the proposed antenna.

TABLE II SIMULATED TOTAL EFFICIENCY AND GAIN FOR BAND2 (EXCITED AT 1.45 GHz) AND BAND5 (EXCITED AT 2.1 GHz) CONFIGURATION FOR DIFFERENT TYPE OF SWITCHES

Fig. 16. Measured narrow band mode S

Fig. 14. (a) Pproposed antenna diagram (b) The fabricated antenna model.

A MEMS switch is thus preferable from the point of view of efficiency. V. ANTENNA CONSTRUCTION AND RESULTS The fabricated reconfigurable Vivaldi antenna including switch bias is presented in Fig. 14. Switches are placed at specific locations along the ring resonators in order to achieve specific pass bands as summarized in Fig. 7.

of the proposed antenna.

There are seven switches on the left ring and eight switches on the right. Two switches in the small gaps at each side are used to decouple the resonator from the tapered slot, whilst the rest of the switches are used to vary the effective electrical lengths of the ring resonator by using different combinations of the ON and OFF states. Infineon PIN diodes switches, model BAR50-02V, are used because no suitable MEMS devices were available. The diodes were forward biased appropriately with DC voltage to obtain 100 mA ON state bias current. To obtain the OFF state, the diodes were left unbiased. In order to bias the switches appropriately, several 0.3 mm DC lines were printed parallel to the beam direction on either side of the antenna. The DC line is isolated from the RF signal by using 27nH SMD inductors. On the antenna radiating part itself, 0.3 mm width slots are created for DC isolation. The RF continuity is preserved by bridging the slots every 10 mm with 22pF SMD capacitors. Two switches, each 1.2 mm long, were needed to bridge the slot gaps of 4 mm on each side. Fig. 15 shows the simulated and measured wideband responses of the antenna. To select the wideband mode, the slot resonators are decoupled from the tapered slot edges. Wideband operation is obtained over a 1–3 GHz bandwidth with a very good agreement between them. To select the narrowband mode, the slot resonators are coupled through the gaps. By short circuiting a specific set of diodes as indicated in Fig. 7, the narrowband modes are achieved. The measured return loss for the six narrow band states are shown in Fig. 16. The for Band 2 and Band 3 are shown simulated and measured

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Fig. 19. Simulated narrow band mode S , Band 2, using lossless and lossy FR4.

Fig. 17. Measured and simulated narrow band mode S (b) Band 3 of the proposed antenna.

(a) Band 2 and Fig. 20. Gain of wideband, Band 2 and Band 4 operations.

Fig. 18. Measured gain of wideband mode.

in Fig. 17. In general, a good agreement has been achieved. Small frequency shifts can be accounted for by the fabrication tolerance, the diode package parasitics and bias components. The effects of the PIN diode on the gain of the antenna in wideband mode have been analyzed by comparing two antenna configurations, one with the ring slots using ideal switches and one with the ring slots using PIN diode switches. It is observed that there is gain degradation due to the loss in the PIN diode switches as shown in Fig. 18. plot is relatively poor. The out of band rejection in the This can only be expected as the filtering or reconfiguration is achieved with a single resonator, which has relatively low Q due

Fig. 21. Wideband radiation pattern. (a) excited at 1.45 GHz, left: xy -plane ), right: yz -plane (normalized gain(theta); (normalized gain(phi); theta ), (b) exited at 1.92 GHz, left: xy -plane (normalized gain(phi); phi theta ), right: yz -plane (normalized gain(theta); phi ). Measured Co-, Measured Cross-, Simulated Co, Simulated Cross.

= 90 = 90

= 90

0000

00x00

= 90 00o00 0

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TABLE III GAIN IN WIDEBAND MODE

TABLE IV GAIN IN NARROW BAND MODE

TABLE V BEAMWIDTH IN NARROW AND WIDEBAND MODE AT 2.2 GHz

Fig. 22. Narrow band radiation pattern. (a) Band 1 excited at 1.45 GHz, left: xy -plane (normalized gain(phi); theta ), right: yz -plane (normalized ). (b) Band 2 exited at 1.53 GHz, left: xy -plane gain(theta); (normalized gain(phi); theta ), right: yz -plane (normalized gain(theta); ). (c) Band 5 excited at 2.1 GHz left: xy -plane (normalized gain(phi); phi theta ), right: yz -plane (normalized gain(theta); phi ). (d) Band 6 ), right: excited at 2.2 GHz, left: xy -plane (normalized gain(phi); theta yz -plane (normalized gain(theta); phi ). Measured Co-, Measured Cross-, Simulated Co, : Simulated Cross.

phi = 90

= 90 = 90

0 0 00

= 90

= 90

= 90 = 90 = 90 0 0 o 0 0 00x00

to being mounted in an antenna, as well as the switch and substrates losses. It was found that one of the main factors determining out of band rejection comes from the dielectric losses of the FR4 substrate. Fig. 19 shows the improved rejection when

of the FR4 substrate is put to zero for Band 2. Higher the out of band rejection is, therefore, achievable with lower loss is also substrate. The effect of the ideal and real switches on compared in Fig. 8(b). The loss has little effect on rejection. A better measure of rejection is gain as shown in Fig. 20 where Band 2 and Band 4 operation are shown. Rejection of around 20 dB is seen at the zeroes of the slot resonators, such as at 1.0 and 2.2 GHz for Band 2 case. Of course rejection is much worse at 3 GHz, as might be expected from a single resonator configuration. The simulated and measured -plane (normalized gain(phi); ) and -plane (normalized gain(theta); phi ) theta radiation patterns for the wideband mode excited at 1.45 GHz and 1.925 GHz are shown in Fig. 21 and four examples of the narrowband modes are presented in Fig. 22. Well behaved radiation patterns are obtained and good agreement between measured and simulated is seen. The cross-polarization is less than 20 dB at the peak direction for most of the bands. Finally, the measured and simulated gains are compared, wideband mode in Table III and narrowband mode in Table IV. Gain in the narrowband mode is lower than in the wideband state. There are two factors to account for the reduction in gain. One is switch loss. The other one is the change in current distribution in the narrowband mode, which reduces the effective aperture. From a gain-beamwidth relationship, wider beamwidth will result in low gain. For example, it is observed that the simulated beamwidth in the Band 6 is wider than in the wideband mode when excited at the same frequency, 2.2 GHz. Table V shows this. The estimated gain loss from the beamwidths in Table V was found to be 1.9 dB where the gain loss is given by the ratio of the - and -plane beamwidths for the wideband and narrowband modes [15]. The difference between the gains in Table III and IV is 2.8 dB, indicating that 0.9 dB gain loss is due to switch loss.

HAMID et al.: VIVALDI ANTENNA WITH INTEGRATED SWITCHABLE BAND PASS RESONATOR

VI. CONCLUSION A new concept of switchable filtering in a Vivaldi antenna has been demonstrated. The Vivaldi described here has added prefiltering functionality by reconfiguring wideband frequency response to six different narrower pass bands while maintaining the original size unchanged. Seven modes are presented, in which six are narrow bands and one is wideband. As with any filtering arrangement, a small excess loss is incurred. To demonstrate the reconfigurability, single pairs of switchable length slot resonators were employed within the antenna. The example shown has good return loss for each mode and well controlled patterns. The antenna may be useful for multiband or agile systems and also in cognitive radio.

REFERENCES [1] S. Yang, C. Zhang, H. Pan, A. Fathy, and V. Nair, “Frequency-reconfigurable antennas for multiradio wireless platforms,” IEEE Microw. Mag., vol. 10, no. 2, pp. 66–83, Feb. 2009. [2] Y. Tawk and C. G. Christodoulou, “A new reconfigurable antenna design for cognitive radio,” IEEE Antennas Wireless Propagat. Lett., vol. 8, no. 2, pp. 1378–1381, Dec. 2009. [3] F. Ghanem, P. S. Hall, and J. R. Kelly, “Two port frequency reconfigurable antenna for cognitive radios,” IET Electron. Lett., vol. 45, no. 11, pp. 534–536, May 2009. [4] E. Ebrahimi and P. S. Hall, “Integrated wide-narrow band antenna for multiband applications,” Microw. Opt. Tech. Lett., vol. 52, no. 2, pp. 425–430, 2010. [5] A. Mirkamali and P. S. Hall, “Wideband frequency reconfiguration of a printed log periodic dipole array,” Microw. Opt. Tech. Lett., vol. 52, no. 4, pp. 861–864, 2010. [6] J. R. Kelly, P. S. Hall, and P. Gardner, “Integrated wide-narrow band antenna for switched operation,” in Proc. IEEE EuCAP Conf. Antennas and Propagation, Berlin, Germany, 2009, pp. 3757–3760. [7] M. R. Hamid, P. Gardner, P. S. Hall, and F. Ghanem, “Switched-band Vivaldi antenna,” IEEE Trans. Antennas Propagat., vol. 59, no. 5, pp. 1472–1480, May 2011. [8] Z. Zhou and K. L. Melde, “Frequency agility of broadband antennas integrated with a reconfigurable RF impedance tuner,” IEEE Antennas Wireless Propagat. Lett., vol. 9, no. 2, pp. 56–59, Feb. 2007. [9] L. Zidong, K. Boyle, J. Krogerus, M. de Jongh, K. Reimann, R. Kaunisto, and J. Ollikainen, “MEMS-switched, frequency-tunable hybrid slot/PIFA antenna,” IEEE Antennas Wireless Propagat. Lett., vol. 8, no. 2, pp. 311–314, Feb. 2009. [10] T. Wu, R. L. Li, S. Y. Eom, S. S. Myoung, K. Lim, J. Laskar, S. I. Jeon, and M. M. Tentzeris, “Switchable quad-band antennas for cognitive radio base station applications,” IEEE Trans. Antennas Propagat., vol. 58, no. 5, pp. 1468–1476, May 2010. [11] H. F. A. Tarboush, S. Khan, R. Nilavalan, H. S. Al-Raweshidy, and D. Budimir, “Reconfigurable wideband patch antenna for cognitive radio,” in Proc. LAPC Conf. Antennas and Propagation, Loughborough, U.K., 2009, pp. 141–144. [12] P. Li, J. Liang, and X. Chen, “UWB tapered-slot-fed antenna,” in Proc. IET Seminar on Ultra Wideband Systems, Technologies and Applications, 2006, pp. 235–238. [13] BAR50 Series Infineon PIN Diode Datasheet. [14] RMSW100 Radant MEMS Datasheet. [15] C. A. Balanis, Antenna Theory: Analysis and Design, 3rd ed. Hoboken, NJ: Wiley, 2005.

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M. R. Hamid received the M.Sc. degrees in communication engineering from the Universiti Teknologi Malaysia (UTM), Skudai, Johor, Malaysia, in 2001. He is currently pursuing the Ph.D. degree at the University of Birmingham, Birmingham, U.K. He has been with the Faculty of Electrical Engineering (FKE), UTM, since 2001. His major research interest is reconfigurable antenna design for multimode wireless applications. He was awarded a scholarship from the UTM to further study in the U.K

Peter Gardner (M’99–SM’00) received the B.A. degree in physics from the University of Oxford, Oxford, U.K., in 1980, and the M.Sc. and Ph.D. degrees in electronic engineering from the University of Manchester Institute of Science and Technology (UMIST), Manchester, U.K., in 1990 and 1992, respectively. In 1994, he became a Lecturer at the School of Electronic and Electrical Engineering, University of Birmingham, Birmingham, U.K. He was promoted to Senior Lecturer in 2002 and to Reader in Microwave Engineering in 2009. His current research interests are in the areas of microwave and millimetric integrated active antennas and beamformers, microwave amplifier linearization techniques, and reconfigurable broadband and multiband antennas.

Peter S. Hall (M’88–SM’93–F’01) is Professor of communications engineering, leader of the Antennas and Applied Electromagnetics Laboratory, and Head of the Devices and Systems Research Centre in the Department of Electronic, Electrical and Computer Engineering at the University of Birmingham, Birmingham, U.K. He joined The University of Birmingham in 1994. He has researched extensively in the areas of microwave antennas and associated components and antenna measurements. He has published five books, over 250 learned papers, and taken various patents. These publications have earned six IEE premium awards, including the 1990 IEE Rayleigh Book Award for the Handbook of Microstrip Antennas. Prof. Hall is a Fellow of the IEE and the IEEE and a past IEEE Distinguished Lecturer. He is a past Chairman of the IEE Antennas and Propagation Professional Group and past coordinator for Premium Awards for the IEE Proceedings on Microwave, Antennas, and Propagation and is currently a member of the Executive Group of the IEE Professional Network in Antennas and Propagation. He was Honorary Editor of the IEE Proceedings Part H from 1991 to 1995 and currently on the editorial board of Microwave and Optical Technology Letters. He is a member of the Executive Board of the EC Antenna Network of Excellence.

F. Ghanem received the Bachelor degree in electronics engineering from the Ecole Nationale Polytechnique of Algiers in 1996 and the M.Sc. and Ph.D. degrees from the Institut National de la Recherche Scientifique (INRS) in 2007. He was an honorary Research Fellow in the Department of Electrical Engineering and Electronics at the University of Birmingham, Birmingham, U.K., from 2007 until October 2009. In 2010, he became an Assistant Professor at the Prince Mohammed Bin Fahd University in Al-Khobar, Saudi Arabia. His current research interests are in the areas of antenna and RF passive and active circuits design. He is also interested in wireless signal processing.

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A Seamless Integration of 3-D Vertical Filters With Highly Efficient Slot Antennas Yazid Yusuf, Student Member, IEEE, Haitao Cheng, Student Member, IEEE, and Xun Gong, Senior Member, IEEE

Abstract—A seamless integration of 3-D vertical cavity filters with highly efficient slot antennas is presented herein. This integration technique enables low-loss filtering and reduces co-site interference within phased arrays. A vertical two-pole cavity filter inteband. The center grated with a slot antenna is demonstrated at frequency and fractional bandwidth of the filter/antenna system are 10.16 GHz and 3.0%, respectively. Due to the near-zero transition loss achieved by this seamless integration, the efficiency of the integrated slot antenna is shown to be as high as 97%. Equivalent circuit models are developed to identify the losses in the filter/antenna, which are verified by full-wave simulations. The measured impedance matching and radiation patterns closely agree with simulation results. This technique can be extended for higher-order filters seamlessly integrated with slot antennas. In addition, this technique can be applied in all microwave, millimeter-wave, and submillimeter-wave frequency regions. Index Terms—Bandpass filter, cavity resonator, packaging, slot antenna, time-domain analysis, vertical integration.

I. INTRODUCTION

H

IGH-QUALITY (Q)-factor filters are critical elements in RF front ends since their low insertion loss is the key to improve the signal-to-noise ratio, and therefore the sensitivity of communications or radar systems. In order to maximize the Q factor of a resonator, 3-D structures such as waveguide cavities are often implemented. However, they are bulky in comparison to their planar structure counterparts. This complicates integration into compact RF front ends or phased arrays where the spacing between antenna elements is generally limited to less than one wavelength. Therefore, high-Q-factor filters with small footprints are necessary. Vertically integrated high-Q-factor 3-D cavity filters using silicon micromachining [1], low-temperature cofired ceramics (LTCC) [2]–[5], and polymer stereolithography [6], [7] have been demonstrated with significantly reduced footprints. To realize a low-loss RF front end, highly efficient antennas and low-loss connections between antennas and filters are also necessary. However, these connections are usually achieved

Manuscript received August 13, 2010; revised April 08, 2011; accepted April 30, 2011. Date of publication August 12, 2011; date of current version November 02, 2011. This work was supported in part by the National Science Foundation (NSF) Faculty Early CAREER Award under Grant 0846672. The authors are with the Antenna, RF and Microwave Integrated (AMRI) System Laboratory, Department of Electrical Engineering and Computer Science, University of Central Florida (UCF), Orlando, FL 32816 USA (e-mail: [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2164186

using either lossy slot-to-microstrip transitions [1]–[5] or bulky coaxial connectors [6]. Therefore, it is desirable to integrate filters with antennas in a seamless manner by avoiding those aforementioned transitions. Abbaspour-Tamijani et al. showed an integration of a coplanar waveguide (CPW) filter and a patch antenna [8]. They used the same technique to realize high-order frequency selective surfaces (FSSs) [9]–[11]. However, the achievable Q factor using this approach is rather limited ( 150) due to the planar transmission line structure. In addition, the equivalent circuit models developed in [8]–[11] can only be used for planar structures. The integration of a four-pole 3-D cavity filter with a highly efficient slot antenna was presented in [12], where all four cavity resonators were designed on a single-layer substrate. In order to significantly reduce the footprint of the integrated system, a compact vertical integration of a two-pole cavity filter with a slot antenna is demonstrated in this paper. A more generic synthesis approach is used herein and can be extended for integrated systems with very different filter or antenna structures. It is noted that vertically integrated filters and filter/antenna systems exhibit increased thickness and therefore extra care should be taken to minimize the potential warpage by selecting metal and dielectric materials with similar coefficients of thermal expansion (CTE) and using appropriate fabrication techniques. The efficiency of the slot antenna is shown to be as high as 97%, compared with 86% efficiency for a standalone substrate integrated waveguide (SIW) cavity-backed slot antenna reported in [13] at the same frequency. In addition, the integrated slot antenna presented herein is embedded inside the filter and therefore represents zero added volume. Bulky or lossy transitions between the filter and antenna are thus avoided. Fig. 1 shows the schematic of a phased array using the proposed integrated filter/antenna, exhibiting compact size, high-Q filtering characteristics, and reduced co-site interference. It should be noted here that this technique can be applied to other high-Q 3-D filter structures and different antennas as well. The organization of this paper is as follows. Section II presents the synthesis of a vertical two-pole cavity filter, which is used as a reference for the filter/antenna system. The filter/antenna synthesis procedure is described in Section III. In Section IV, equivalent circuit models are developed to identify the losses in the antenna, which are also verified by full-wave simulations. The measured results are shown and discussed in Section V. II. FILTER SYNTHESIS A vertical two-pole cavity filter is first designed using the approach presented in [7] to serve as a reference for the

0018-926X/$26.00 © 2011 IEEE

YUSUF et al.: SEAMLESS INTEGRATION OF 3-D VERTICAL FILTERS

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Fig. 1. Schematic of a phased array with integrated filter/antenna.

Fig. 2. Exploded view of a vertical two-pole cavity filter with CPW feeding. SMA connectors are soldered to the CPW lines for measurement purposes.

filter/antenna system. The filter structure is illustrated in Fig. 2. The sidewalls of each resonator are formed by closely-spaced metallic vias. The via diameter and spacing are 1 and 1.3 mm, respectively. The gap between the vias, 300 m, is much smaller than the wavelength at band. Therefore, the leakage loss through the sidewalls is insignificant compared with the metallic and dielectric losses [14]. The internal coupling is achieved through a slot in the ground plane between the two resonators, representing magnetic coupling. The external coupling is realized through the magnetic coupling from the short-ended CPW feeding lines [7]. The center frequency and fractional bandwidth are 10.18 GHz and 2.9%, respectively. The design parameters of the vertical two-pole cavity filter are shown as (1) (2) where is the internal coupling coefficient between the two is the external coupling coefficient of the resonators and filter [15].

Fig. 3. (a) Stack and (b) top view of the vertical filter. (Dimensions are : ;h : ;h : ;h : ; in millimeters. h : ;L : ;L : ;g : ;L ; W : ;D : ;L : ;W : ;Y : ). ;S

= 1 524 prepreg thickness = 0 1 = 12 5 = 1 = 1 3

= 0 762 = 0 508 = 0 254 =56 =12 =04 =9 =39 =05 =05

This filter is designed in a multi-layer RO4350B stack as shown in Fig. 3(a). The dielectric constant and loss tangent for RO4350B are specified as 3.48 and 0.004, respectively, from Rogers Corp. The prepreg which is used to bond the RO4350B substrates exhibits a dielectric constant of 3.2 and a loss tangent of 0.004. The filter dimensions are shown in Fig. 3(b). III. FILTER/ANTENNA SYNTHESIS The slot antenna is formed by etching the top side of the upper resonator as shown in Fig. 4. As a result, this slot antenna does not occupy any additional volume or require any transition structure between the antenna and filter. The slot antenna has a much wider fractional bandwidth than the 2.9% filter bandwidth. Therefore, the slot antenna can act as an equivalent load to the filter within the filter pass band as long as the coupling between the antenna and upper resonator is identical to that between the port and resonator; and the frequency loading from the slot antenna is the same as that from the port. As a result, the filtering function of the filter/antenna system should be identical to that of the same standalone filter. It will be shown in both simulation and measurement results that the slot antenna exhibits the same radiating characteristics in terms of both radiation patterns

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Fig. 6. Schematics of a cavity resonator terminated with (a) a slot antenna or (b) a 50-ohm port excited by a waveguide port.

Fig. 4. Exploded view of a vertical two-pole cavity filter seamlessly integrated with a slot antenna.

Fig. 5. Magnetic field distribution at the top of (a) an unperturbed resonator and (b) the upper resonator of the filter antenna.(Dimensions are in millimeters. L ;W : ;L ;W : ;Y : ;L : ;W :; X : ;X : ;Y :)

= 9 = 12 5 = 4 = 0 5 =33 =22 =07

=05

=95

=05

and gain as a standalone slot antenna. Moreover, the efficiency of this antenna within the integrated structure is much higher than that of the same standalone antenna. This improved efficiency is partly due to the near-zero transition loss between the antenna and filter. Furthermore, since the antenna and filter in the integrated structure share the same substrate, the dielectric and metallic losses are significantly reduced as compared to the discrete case. A. Frequency-Domain Slot Antenna Synthesis The antenna length and width are chosen to be 9.5 and 0.5 mm, respectively. It is noted here that the choice of these two parameters is not unique. Since the slot antenna bandwidth is much wider than the filter bandwidth, slightly different antenna dimensions still cover the filter frequencies. controls the coupling between The slot antenna position the upper resonator and slot antenna. This coupling is stronger is smaller, since the magnetic field is stronger at the when perimeter of the cavity as shown in Fig. 5(a) and the slot antenna predominantly couples to the magnetic field.

Any external coupling detunes the resonant frequency of the upper cavity, so does the slot antenna. It is necessary that this frequency loading effect from the slot antenna be the same as that from the filter port. Therefore, a metallic via inside the upper cavity is used here to adjust the frequency loading caused by the slot antenna as shown in Fig. 5(b). The resonant frequency value. of the upper cavity can be increased with a larger Due to the position and orientation of the coupling slot with respect to the slot antenna in Fig. 5, the approach used in [12] to design the external coupling to the slot antenna cannot be readily used herein. For this purpose, a more generic method that is amenable to use with other resonator and antenna structures is employed. A waveguide port is used to excite the upper resonator terminated with either a slot antenna as shown in Fig. 6(a) or a filter port as shown in Fig. 6(b). To achieve reasonably small cross-sectional sizes, i.e., 8 by 1.6 mm herein, the waveguide port is loaded with a dielectric material of . Within 9–11 GHz, which is much wider than the filter bandwidth, this waveguide operates in single-mode condition. , which is the parallel Using the method described in [16], of the upper resonator and from the combination of slot antenna or the filter port, can be extracted from the reflection coefficient of the waveguide port. Ansoft High Frequency Structure Simulator (HFSS) full-wave simulations are used to is the minimum reflection coeffimodel these structures. cient occurring at the resonant frequency , as shown in Fig. 7. and correspond to the frequencies when , where is defined as (3)

The coupling coefficient between the waveguide port and the resonator can be found using (4) (5) The

can be calculated using (6)

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Fig. 7. Reflection coefficients of the waveguide port for the upper cavity terminated with either a slot antenna (Fig. 6(a)) or a filter port (Fig. 6(b)).

It should be noted that is independent of since the total loss of the upper cavity due to radiation, metal, and dielectric material is independent of the level of coupling from is found to be 42.6 when the upper the waveguide port. This cavity is terminated by a filter port. To match this for the slot is performed. It should antenna case, a parametric sweep of be noted that the frequency loading effect is not constant for different antenna positions. Therefore, a parameter sweep of for each is necessary to achieve the same resonant frequency . The reflection coefficient of the waveguide port for every mm and mm is shown corresponding to for the slot antenna in Fig. 7. The resonant frequency and are 10.18 GHz and 45, respectively, compared with 10.18 GHz and 42.6 for the filter port. B. Time-Domain (TD) Filter/Antenna Synthesis The frequency-domain responses of the reference filter and filter/antenna are shown in Fig. 8(a). The gain of the filter/anof the reference filter. The gain of tenna closely follows the the filter/antenna and the insertion loss of the filter are found to be 4.9 and 2.18 dB, respectively, at the center frequency. Howof the filter/antenna is noticeably different from ever, the that of the filter. Fine tuning the filter/antenna dimensions in frequency domain using full-wave simulation tools such as HFSS can be very time consuming. In order to close the loop for this filter/antenna synthesis, a time-domain technique [17] is used here. Using an inverse Chirp-Z transform software package developed in the response is plotted in the time ARMI laboratory, the filter domain as shown in Fig. 8(b). It is observed that the filter responses from the different sections of the filter are isolated in the time domain. The peaks in the time-domain response correspond to the external coupling between Port 1 and the first , the internal coupling between resonators 1 resonator , and the external coupling between the second resand 2 , respectively, from left to right. The onator and Port 2 dips correspond to the resonators 1 and 2, respectively. It is obof the served in Fig. 8(b) that the peak corresponding to filter/antenna is higher than the filter case, which implies that the reflection from the coupling slot is a little bit too high. Therefore, the coupling slot dimensions need to be increased to allow

Fig. 8. Simulated filter and filter/antenna responses in (a) frequency domain and (b) time domain.

more coupling between the two resonators. A parametric sweep of is then performed to match the peaks corresponding to between the filter and filter/antenna cases. It is found that an value of 4 mm provides the best match in the time-domain responses as shown in Fig. 8(b). The return losses for both filter and filter/antenna in the frequency domain are shown to be close to each other as illustrated in Fig. 8(a). The final dimensions of the filter/antenna are listed in Fig. 5(b). IV. Q BREAKDOWN AND LOSS ANALYSIS Section III presented the synthesis of a filter/antenna system exhibiting the same filtering characteristics as the reference filter. In this section, equivalent circuits are developed to model the loss associated with this filter/antenna integration. The structures shown in Fig. 6 can be modeled by an equivacalculent circuit illustrated in Fig. 9(a). It is noted that the lated by (6) is expressed by (7) is equal to when is infinity. This can be realized by removing the metal and dielectric losses in HFSS simulations of the structures shown in Fig. 6. It is found that for the , and are 42.6, 196 and 54.4, respectively. filter, For the filter/antenna, the three parameters are 45, 187, and 59.4, is very different between respectively. It is noted that the

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Fig. 9. Equivalent circuits of the structures (a) in Fig. 6 and (b) in Fig. 2 and L nH, C : ,C : , Fig. 4. (Filter: L R R : ,K K : ,K : ,Z ) L nH, C C : ,R : , (Filter/Antenna: L : ,K : ,K : ,K : ,Z ). R

= =3 = = 0 979

= = =3 = 1 03

= 13 3

= 0 0815 pF = 0 0816 pF = 13 3

= 4 02 = 50

= = 0 0815 pF = 0 979

=39

= 12 7 = 50

the two cases. In addition, the is degraded from 196 to 187 for the upper resonator loaded with the slot antenna. However, in order to realize the same reflection coefficient, the same not is necessary. This has been proven in the synthesis procedure described in Section III. The equivalent circuit model for the entire filter or filter/anand values tenna is illustrated in Fig. 9(b). Using the found through HFSS simulations, the values of the circuit eleand ments are calculated and presented in Fig. 9. The of the filter using both HFSS simulations and equivalent circuit model are compared in Fig. 10(a) and closely match each other, particularly within the pass band. Using the equivalent circuit response in the passmodel, it is found that the asymmetric band is due to a 0.05% frequency difference between the two resonators, which is caused by the different number of prepreg layers within the two cavities. The insertion loss of the filter/antenna system from the equivalent circuit model is found to be 2.3 dB, compared with 2.18 dB for the filter. This 0.12-dB difference is due to the removal of one CPW feeding and addition of one slot antenna. Using HFSS simulations, it is found that the attenuation of one CPW feeding line is 0.035 dB. Therefore, the efficiency of the slot antenna is calculated to be 97%, equivalent to a 0.155-dB loss. To verify this loss from the equivalent circuit model, HFSS simulations are performed to find the overall efficiency of the filter/antenna system. This overall efficiency is shown to be 58.9%, which is equivalent to a 2.3-dB loss. From both equivalent circuit model and full-wave simulations, it is observed that (1) the slot antenna has a very high (97%) efficiency, and (2) near zero loss occurs in the transition between the filter and antenna. Very close agreeis observed as shown in Fig. 10(b). The from ment in the equivalent circuit closely follows the roll off of the gain obtained from HFSS simulations. The 7.2-dB difference between the two curves is due to the fact that the equivalent circuit model does not take the 7.2-dB antenna directivity into consideration.

S S

S

Fig. 10. (a) Filter and using HFSS simulation and equivalent circuit. and gain using HFSS simulation as well as filter/antenna (b) Filter/antenna and using equivalent circuit.

S

S

V. RESULTS AND DISCUSSION An integrated filter/antenna system is fabricated and measured. As shown in Fig. 11(a), SMA connectors are soldered to the CPW lines for measurement purposes. The measured filter/antenna center frequency of 10.16 GHz is very close to the simulated 10.18 GHz as shown in Fig. 11(b). The measured bandwidth of 3.0% is slightly larger than the simulated 2.7% for the filter/antenna. The measured impedance matching is better than 28 dB within the passband. The measured filtering shape matches the simulated result very well. The radiation patterns and gain are measured in an anechoic chamber built by TDK Inc. in ARMI Lab. As shown in Fig. 11(c) and Fig. 11(d), the measured radiation patterns agree very well with the simulation results in both H- and E-planes at the center frequency. The small discrepancy in the backside lobes is caused by the finite ground plane size and scattering effects from the connectors and cables. Though the radiations patterns shown in Fig. 11(c) and Fig. 11(d) correspond to the filter/antenna center frequency , similar radiation patterns are observed across the entire passband of the filter/antenna. The measured maximum gain at the boresight of the radiation pattern is shown to be 4.9 dB, which is identical to the simulated gain. Given the 7.2-dB directivity from HFSS simulations, the overall efficiency of the filter/antenna from the measurement is

YUSUF et al.: SEAMLESS INTEGRATION OF 3-D VERTICAL FILTERS

Fig. 11. (a) Photo of fabricated filter/antenna. Simulated and measured (b) S

calculated to be 58.9% as well, which implies that a 97% efficient slot antenna is achieved based on the calculations shown in Section IV. 3-D cavity resonators loaded with air dielectric have the potential to achieve unloaded Q factors up to several thousand [6]. SIW cavity resonators loaded with RT/Duroid 5880 substrate material were reported to achieve unloaded Q factors as high as 1,000 [18]. Using these low-loss cavity resonators, the filter insertion loss can be greatly improved. As a result, the noise figure of the RF front end is significantly reduced. In this paper, the choice of the substrate material and stack is not optimum for loss performance. However, even with the relatively lossy RO4350B substrates, antenna efficiency as high as 97% is still achieved. To demonstrate the possibility of realizing a highly efficient filter/antenna system, we reduce the loss tangent of RO4350B and prepreg in HFSS to emulate a resonator with an unloaded Q of 1,000. It is found that the filter insertion loss is reduced to 0.5 dB and the antenna efficiency is still as high as 97%, leading to an 86% overall efficiency for the filter/antenna system. Using this low-loss substrate, the simulated gain of the filter/antenna system is as high as 6.58 dB. VI. CONCLUSION A synthesis approach to seamlessly integrate 3-D vertical filters with highly efficient antennas was presented. The measured results verified the presented synthesis approach. It has been

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and gain; (c) H plane radiation pattern at f ; and (d) E plane radiation pattern at f .

shown that the integrated slot antenna is a 97% efficient radiator with expected radiation patterns and gain. In addition, this slot antenna acts as an equivalent load to the filter, without compromising the filtering characteristics. This compact lowloss integration technique enables low-loss filtering and reduces co-site interference within phased arrays. Following the same synthesis procedure, higher-order filters can also be seamlessly integrated with highly efficient slot antennas. Furthermore, this technique can be applied to all microwave, millimeter wave, and submillimeter wave frequencies with appropriate fabrication techniques and materials. ACKNOWLEDGMENT The authors wish to thank Rogers Corporation for kind donation of substrate materials and Dr. S. Ebadi from ARMI Lab at UCF for helpful comments and suggestions. REFERENCES [1] L. Harle and L. P. B. Katehi, “A vertically integrated micromachined filter,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 9, pp. 2063–2068, Sep. 2002. [2] J.-H. Lee, S. Pinel, J. Papapolymerou, J. Laskar, and M. M. Tentzeris, “Low-loss LTCC cavity filters using system-on-package technology at 60 GHz,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 12, pp. 3817–3824, Dec. 2005. [3] T.-M. Shen, C.-F. Chen, T.-Y. Huang, and R.-B. Wu, “Design of vertically stacked waveguide filters in LTCC,” IEEE Trans. Microw. Theory Tech., vol. 55, no. 8, pp. 1771–1779, Aug. 2007.

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[4] J.-H. Lee, S. Pinel, J. Laskar, and M. M. Tentzeris, “Design and development of advanced cavity-based dual-mode filters using low-temperature co-fired ceramic technology for V-band gigabit wireless systems,” IEEE Trans. Microw. Theory Tech., vol. 55, no. 9, pp. 1869–1879, Sep. 2007. [5] K. Ahn and I. Yom, “A Ka-band multilayer LTCC 4-pole bandpass filter using dual-mode cavity resonators,” in 2008 IEEE MTT-S Int. Microwave Symp. Dig., Atlanta, GA, Jun. 2008, pp. 1235–1238. [6] B. Liu, X. Gong, and W. J. Chappell, “Applications of layer-by-layer polymer stereolithography for three-dimensional high-frequency components,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 11, pp. 2567–2575, Nov. 2004. [7] X. Gong, B. Liu, L. P. B. Katehi, and W. J. Chappell, “Laser-based polymer stereolithography of vertically integrated narrow bandpass filters operating in K band,” in Proc. 2004 IEEE MTT-S Int. Microwave Symp. Dig., Fort Worth, TX, Jun. 2004, pp. 425–428. [8] A. Abbaspour-Tamijani, J. Rizk, and G. Rebeiz, “Integration of filters and microstrip antennas,” in 2002 IEEE AP-S Digest, Jun. 2002, pp. 874–877. [9] A. Abbaspour-Tamijami, K. Sarabandi, and G. M. Rebeiz, “Antennas-filter-antenna arrays as a class of bandpass frequency-selective surfaces,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 8, pp. 1781–1789, Aug. 2004. [10] C.-C. Cheng, B. Lakshminarayanan, and A. Abbaspour-Tamijani, “A programmable lens-array antenna with monolithically integrated MEMS switches,” IEEE Trans. Microw. Theory Tech., vol. 57, no. 8, pp. 1874–1884, Aug. 2009. [11] A. Abbaspour-Tamijani, K. Sarabandi, and G. M. Rebeiz, “A millimeter-wave bandpass filter-lens array,” IET Microwaves, Antennas, Propagat., vol. 1, no. 2, pp. 388–395, 2007. [12] Y. Yusuf and X. Gong, “Compact low-loss integration of high-Q 3-D filters with highly efficient antennas,” IEEE Trans. Microw. Theory Tech., vol. 59, no. 4, pp. 857–865, Apr. 2011. [13] G. Q. Luo, Z. F. Hu, L. X. Dong, and L. L. Sun, “Planar slot antenna backed by substrate integrated waveguide cavity,” IEEE Antennas Wireless Propag. Lett., vol. 7, pp. 236–239, 2008. [14] D. Deslandes and K. Wu, “Single-substrate integration technique of planar circuits and waveguide filters,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 2, pp. 593–596, Feb. 2003. [15] J.-S. Hong and M. J. Lancaster, Microstrip Filters for RF/Microwave Applications. : Wiley, 2001. [16] R. J. Cameron, C. M. Kudsia, and R. R. Mansour, Microwave Filters for Communication Systems: Fundamentals, Design, and Applications. New York: Wiley, 2007, ch. 11. [17] Application Note 1287-8: Simplified Filter Tuning Using Time Domain Agilent Technologies Corp., 2001. Palo Alto, CA. [18] W. J. Chappell and X. Gong, “Wide bandgap composite EBG substrates,” IEEE Trans. Antennas Propag., vol. 51, no. 10, pp. 2744–2750, Oct. 2003. Yazid Yusuf (S’05) received the B.S. degree in electrical engineering from Jordan University of Science and Technology, Irbid, Jordan, in 2005. He is currently pursuing the Ph.D. degree in the Department of Electrical Engineering and Computer Science at University of Central Florida (UCF). His research interests include high- resonators and filters, antennas, phased arrays, and integrated RF front-end. Mr. Yusuf is a member of the International Microelectronics and Packaging Society (IMAPS).

Q

Haitao Cheng (S’10) received the B.S. degree in electrical engineering from University of Electronic Science and Technology of China (UESTC), Chengdu, China, and M.S. degree in microelectronics from Shanghai Institute of Microsystem and Information Technology, Chinese Academy of Sciences, Beijing, China, in 2006 and 2009, respectively. He is currently pursuing the Ph.D. degree in the Department of Electrical Engineering and Computer Science at University of Central Florida (UCF). His research interests include high- resonators and filters, antennas, phased arrays, and wireless passive sensors.

Q

Xun Gong (S’02–M’05–SM’11) received the B.S. and M.S. degrees in electrical engineering from FuDan University, Shanghai, China, in 1997 and 2000, respectively, and the Ph.D. degree in electrical engineering from The University of Michigan at Ann Arbor in 2005. He is currently an Associate Professor of Electrical Engineering and Computer Science (EECS) at University of Central Florida (UCF) and Director of the Antenna, RF and Microwave Integrated Systems (ARMI) Laboratory. He joined UCF as an Assistant Professor in 2005. He worked at Air Force Research Laboratory (AFRL) in Hanscom, MA in 2009 under the support of Air Force Office of Scientific Research (AFOSR) Summer Faculty Fellowship Program (SFFP). He was with the Birck Nanotechnology Center at Purdue University, West Lafayette, IN, as a Post-Doctoral Research Associate in 2005. His research interests include high-Q resonators and filters, microwave sensors, antennas, phased arrays, integrated RF front-end, flexible electronics, and packaging. Dr. Gong is a member of the International Microelectronics and Packaging Society (IMAPS). He has served on the Editorial Boards of IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES (MTT), IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION (AP), IEEE MICROWAVE AND WIRELESS COMPONENT LETTERS (MWCL), and IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS (AWPL). He was the IEEE AP/MTT Orlando Chapter Chair in 2007–2010. He has been the recipient of the NSF Faculty Early CAREER Award in 2009. He received the Research Incentive Award (RIA) at UCF in 2011. He is the recipient of the Teaching Incentive Program (TIP) Award at UCF in 2010. He received the Outstanding Graduate Teaching Award in 2010 and Outstanding Undergraduate Teaching Award in 2009, respectively, within EECS Department at UCF. He is the recipient of Outstanding Engineer Awards from IEEE Florida Council and Orlando Section, respectively, in 2009. He is the recipient of the Third Place Award in the Student Paper Competition presented at the 2004 IEEE MTT-S International Microwave Symposium (IMS), Fort Worth, TX.

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Rigorous MoM Analysis of Finite Conductivity Effects in RLSA Antennas Matteo Albani, Senior Member, IEEE, Agnese Mazzinghi, and Angelo Freni, Senior Member, IEEE

Abstract—Although slot antennas are usually modeled as perfectly electric conductors, for accurate antenna design and optimization, ohmic loss effects cannot be neglected. This is especially true in millimeter and submillimeter-wave applications, and in low-cost technology for mass production, where highly conductive surfaces are out of budget. This paper presents a rigorous but efficient method-of-moments (MoM) formulation for the analysis of radial line slot array (RLSA) antennas, which includes the finite conductivity of metals. First, by using equivalence and reciprocity theorems, effective magnetic currents are defined on each slot aperture, instead of standard electric and magnetic equivalent currents. This choice halves the number of unknowns of the MoM linear system, still preserving the rigor of the electromagnetic formulation. Next, proper Green’s functions accounting for the finite conductivity of metals are derived analytically and used in the MoM admittance matrix expressions. A numerical check of self and mutual admittances for a couple of slots etched in a nonperfectly conducting structure is provided against results from a finite-element method. Finally, a few RLSA realizations are analyzed to investigate the effect of ohmic losses in a practical antenna design. Index Terms—Antenna arrays, Green’s function, impedance boundary condition, moment methods, radial line slot array (RLSA).

I. INTRODUCTION HE GROWING interest in radial line slot arrays (RLSAs) for applications at millimeter and sub-millimeter frequencies suggests the necessity of accounting for the influence of the finite conductivity of the metal plates in the study of the antenna. As a matter of fact, the losses due to the Joule effect may not be negligible, for example, when the metallizations are not realized with high-conductivity metal, such as gold or copper [1]. Indeed, when low-cost technologies are adopted, such as metal painting or metal foils backed by adhesive films, the conductivity decreases so that the antenna gain is appreciably reduced. Furthermore, for high-frequency applications, the roughness of the metal plates has to be considered by introducing an effective conductivity that can be several times lower than the standard values [2]. Moreover, losses can be significantly affected

T

Manuscript received August 18, 2010; revised January 27, 2011; accepted April 23, 2011. Date of publication August 12, 2011; date of current version November 02, 2011. M. Albani is with the D.I.I., University of Siena, Siena 53100, Italy (e-mail: [email protected]). A. Mazzinghi and A. Freni are with the D.E.T., University of Florence, Florence 50139, Italy (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2164187

by specific antenna arrangements (e.g., in connection to current hot spots flowing between two very closely etched slots). Usually, in order to have a low ohmic loss design, it is preferable to choose thick dielectric substrates, high conductivity metals [3], and a low slot density on the radiating surface. However, such requirements are in tradeoff with other specifications, such as low profile, low cost, or high aperture efficiency. Therefore, an analysis tool accounting for the finite conductivity is mandatory to find an optimum compromise. The aim of this paper is two-fold: 1) to introduce a rigorous but efficient method-of-moments (MoM) formulation for the analysis of RLSAs made of good conductors and 2) to investigate the influence of their finite conductivity on the antenna performances. To achieve these purposes, the same analytical procedure is applied twice to either the parallel plate waveguide (PPW) internal region or the grounded half-space (GHS) external region. By modeling the conductors via the impedance boundary condition, the procedure yields an analytical form for both admittance Green’s functions involved in the MoM formulation. According to the equivalence theorem [4, Sec. 7.8], the electric and magnetic equivalent currents have to be considered to describe the field in each slot. However, neglecting the equivalent electric currents is common in MoM applications [5], because of the high values of the conductivity usually considered. This approximation enables convenient simplification of the numerical formulation by halving the number of unknowns of the MoM linear system. Here, the description of the electromagnetic phenomenon is kept rigorous but only an effective magnetic equivalent current distribution is considered, thus maintaining the same halved size of the MoM linear system. The results obtained with the rigorous and the approximated approaches are compared in Section VII. In the same section, the analysis of a complete RLSA demonstrates the accuracy and the efficiency of the method against a general-purpose commercial analysis tool. II. FORMULATION OF THE PROBLEM IN TERMS OF EFFECTIVE MAGNETIC CURRENTS ONLY Let us consider an RLSA made of conductors with finite conductivity [Fig. 1(a)]. We model the metallic surfaces as impedance surfaces by imposing the Leontovich boundary on them [6]–[8]. condition In this paper, denotes the outward unit vector normal to the surface. An RLSA basically consists of two metallic plates: a top and a bottom plate. Between the two plates, a feeding PPW is excited. The metal layer is assumed to be thicker than several skin depths . In such a case, the fields on the two sides of the

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Fig. 1. (a) Schematic lateral view of an RLSA: two metal plates, characterized by an arbitrary impedance Z , form a feeding parallel plate waveguide (PPW); slots are etched in the upper plate and radiate in the external region (i.e., in a grounded half space (GHS). (b) The two region problems are separated by around the slots, thus inapplying the equivalent principle to the volumes troducing equivalent surface currents J and M. (c) Equivalent currents of both types are then replaced by effective magnetic currents only, which are different in the internal and external regions and are backed by a homogeneous metallic plane.

Green’s function, and they are coupled by unknown electric and magnetic currents on the slot apertures [Fig. 1(b)]. As a last step, we can assume only effective magnetic surface on the slot apertures [Fig. 1(c)]. In Appendix A, currents it is demonstrated that an effective magnetic current on an impedance surface radiates the same field as the equivalent sources and . This choice makes the formulation very similar to that of the perfectly conducting case, where only magnetic currents are considered, and simplifies the extension of existing MoM codes. In this paper, we consider an infinitesimal top-plate thickness. A strategy to introduce the plate thickness effect is presented in [9]. Note that for vanishing slot thickness, equivalent currents just swap direction from the internal to the external problem and ), as a consequence of ( field continuity through the slot apertures, and of the change of sign of the normal vectors of the internal and external problem . Conversely, the effective magnetic currents (1) are different on the two sides.

conductor are isolated from each other, and the metal surface impedance is equal to the metal characteristic impedance (i.e., , with and denoting the metal permeability and conductivity). When the metal-layer thickness is comparable to the skin depth, the impedance boundary condition approach can still be applied by assuming [11],1 provided that the characteristic impedance of the metal is much smaller than that of the surrounding medium. slots are etched in the top plate [Fig. 1(a)] and radiate in the GHS above the plate. By applying the equivalence prinbetween the upper and ciple to volumes the lower apertures of the slots, electric and magequivalent surface currents are distributed netic on both apertures of the slots, thus providing a vanishing field [Fig. 1(b)]. Since no field is present, inside each volume volumes can be filled with any type of impenetrable material without affecting the field distribution. The material has not to be necessarily impenetrable, but this choice allows the separation of the external (GHS) and internal (PPW) problems. In a standard MoM formulation, apertures are usually filled with a perfect electric conductor (PEC) which short-circuits the equivalent electric currents , whose contribution vanishes, and only are introduced. However, when unknown magnetic currents dealing with non-perfectly conducting walls, such magnetic currents would radiate on a nonhomogenous ground plane (i.e., an impedance surface with perfectly conducting patches, for which an analytic expression for the Green’s function is not attainable). Therefore, in order to achieve a homogenous boundary condition throughout the plates, we chose to fill the volumes with the material that the RLSA plates are made of. So far, the external and internal problems are separated. They require homogenous boundary conditions thus admitting an analytic

t

1The

formulation can be also applied to superconductors, when the thickness , where  is the London pentretation depth, by defining the surface j!  t= . impedance as Z

=

coth(

)

III. SOLUTION OF THE PROBLEM BY THE MOM To calculate the unknown effective magnetic current distributions, we resort to a MoM strategy. We consider only the external effective magnetic current as unknown and we expand it basis functions on each slot (i.e., by using ), with denoting the total number of basis functions. The internal effective magnetic current distribution is then derived by (1) in terms of the external one as (2) Furthermore, the tangential component of the magnetic field on slot apertures is also expanded by using the same basis functions . The magnetic field radiated either in the internal or in the external region can be related to the effective magnetic currents by means of the pertinent admitas follows: tance dyadic Green’s function (3)

By combining the magnetic-field integral equation (MFIE) (which imposes the magnetic-field continuity through the slots) with (2) and (3), according to the Galerkin strategy, we can write the following MoM algebraic system of equations:

(4) where (5) (6)

ALBANI et al.: RIGOROUS MoM ANALYSIS OF FINITE CONDUCTIVITY EFFECTS

and (7) with denoting the feeding magnetic field impinging on the slots in the PPW. Note that for orthonormal basis functions (e.g., current basis functions derived from rectreduces to the identity maangular-waveguide modes), trix and (4) simplifies accordingly. By solving the MoM linear system (4), one determines the external effective magnetic currents, and from those the field in both regions. For evaluating the far field in the GHS region (40)–(46) in [10] can be conveniently used. To build the linear system (4), one needs to calculate each term (6) of the MoM admittance matrix. This involves knowledge of the dyadic admittance Green’s functions of the external and the internal problems, where an impedance boundary condition is imposed on the metallic plates. These Green’s functions will be evaluated analytically in the next two sections. Note that the entire linear system matrix between braces in (4) involves matrix multiplications which renders its numerical calculation significantly more intense than that associated 0. The reason is that in the PEC with the PEC case case, the electric-field continuity results in a trivial relation between the internal and external magnetic currents (and, in turn, between their associated voltage ). On the other hand, in the finite-conunknowns ductivity case, their relation is not straightforward, and the evaluation of the internal and the external magnetic currents linear system. To keep requires the solution of a the linear system size to , we inserted relation (2) directly in the MFIE, expressing all of the quantities in terms of just the external magnetic currents. This matrix reduction introduces the aforementioned extra effort, which reflects a hidden linear system solution. It also worth noting that if the linear system is solved with an iterative algorithm (e.g., the conjugate gradient in (4) method), the matrix multiplication can be avoided by cascading the matrix-vector multiplications with considerable time savings. IV. CALCULUS OF THE MoM ADMITTANCE

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, it is assumed that the planes 0 and 0 for the TM or TE cases, denotes the spatial differential respectively. Also, in (8), operator with respect to the observation (source) position ; is the transverse spatial differential whereas operator with respect to the Hertz potential direction , which, in our case, is assumed normal to RLSA plates. coordinate system local to each slot with Consider a along the longest side of the slot, and . Assuming the width of each slot is much smaller than its length (i.e., narrow slot approximation), we can assume that the effective magnetic currents are flowing only along . Thus, when calculating the transverse to MoM admittance (6), only the component of the dyadic Green’s function is needed, with and denoting the unit vectors in the direction of the weighting function and basis function , re0 spectively. Integrating by parts and assuming that at the slot ends, we can rewrite (6) as

(9) This expression only requires the calculation of scalar functions. In case of stratified structures, the 3D scalar Green’s functions are simply expressed via the characteristic Green’s function method ([12], Sec. 3.3 and 5.1) as (10) where the observation point (source) coordinate is split into its transverse and longitudinal parts, i.e., . For the sake of simplicity, we consider here the geometry to be radially unbounded, so that the transverse 2D Green’s in (10) is the same for the TM and TE cases and function admits the simple closed form ([12], Sec. 5.1) (11)

According to [12], Secs. 2.3 and 5.2 and the notation therein, the dyadic admittance Green’s function is calculated via a scalarization process as

(8) where the TM and TE scalar functions are related by to the respective TM and TE scalar Green’s functions, which are the solutions of the scalar Helmholtz wave , subjected to proper equation boundary conditions (the omission of prime and double prime denotes relations valid for both TM and TE cases). Namely, on

In [13], a more complicated expression is given which accounts for the presence of a closing rim in the internal problem. However, for a well-designed RLSA, most of the power flowing in the PPW radiates through the slots, and the PPW truncation effect becomes negligible; also, in some practical designs, the PPW rim is closed by an absorber. Therefore, the unbounded structure assumption is not just an idealization but it is useful for practical antenna design. ([12], Sec. 1.1.), the expression of By inverting functions is found to be analogous to that for G in the scalar is replaced by (10), where

(12)

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with denoting an arbitrary constant. In (10), is the characteristic Green’s function calculated by resorting to an equivalent transmission line with propagation constant and characteristic impedance or for the corresponds TM or TE case, respectively. Indeed, to the current at on a transmission line fed by a unit series voltage generator located at (TM case); whereas corresponds to the voltage at on a transmission line fed by a unit shunt current generator at (TE case). Finally, in (10), is a closed (to infinity) path that encloses in the positive sense all singularities of the characteristic Green’s function in the complex plane, but no others. Namely, must exclude the singularities of the transverse 2-D Green’s (i.e., a branch cut on the negative function defined by real axis starting from the branch point at ) (see Figs. 3 and 5). We use standard transmission-line theory to define the dual TE characteristic Green’s function

Fig. 2. Equivalent z transmission line for the calculation of the characteristic Green’s function in the PPW (internal region).

(13) which in the TE case corresponds to the current in a transmission line fed by a unit series voltage generator (as in the TM case). Employing (13), (9) finally reduces to

Fig. 3. Topology of singularities in the  complex plane for the PPW Green’s function spectral representation.

ysis. With reference to the case of interest (i.e., source and observation at the PPW top plate), the expression (16)

(14) where

.

The dual TE scalar functions and are also defined by (10)–(12) but in combination with (13). In summary, (14) provides a general expression for any MoM admittance matrix entry in terms of 3-D scalar functions, which are obtained by (10) in connection with the proper characteristic Green’s function in the specific region. The internal and external characteristic in an equivGreen’s functions are calculated as the current alent transmission line, accounting for the stratification of the region, and excited by a series voltage generator at corresponding to the magnetic surface current (see Figs. 2 and 4), i.e.,

is obtained by (15). The PPW characteristic Green’s function (16), as a func, exhibits an infinite set of tion of the spectral variable , for , where the pole singularities at denominator of (16) vanishes (Fig. 3). Conversely, the branch is fictitious because (16) is an even function of cut . For a good conductor and ; therefore, the pole locations can be approximately but accurately found in closed form via the Taylor expansion of the tangent function around its zeros. Thus, we obtain and (17)

(15)

V. GREEN’S FUNCTION REPRESENTATION FOR THE INTERNAL REGION (PPW) With reference to the internal problem (PPW), the equivalent transmission line is a section of length (PPW height) terminated on the metal impedance at both ends , which corresponds to the lower and upper plate (Fig. 2). The current in the transmission line is calculated by standard circuit anal-

for the TM and TE cases, respectively, with denoting the 2 or 1, for 0 or 0, reNeumann number spectively. Hence, the characteristic Green’s function exhibits complex poles located very close to the real pole singularities , relevant to the PEC case 0). To calculate the scalar functions involved in (14), their integral expressions in (10) can be closed as the sum of residues at poles, leading to a radial PPW mode series representation (18)

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with either (19) or

(20) or the TE “dual” for the calculation of either the TM scalar Green’s function. The PPW mode series representation (18), with (19) and (20), also allows us to calculate the scalar once (11) is replaced by (12). Since in an functions 0 mode is propagating, while higher RLSA, only the first are in cutoff and, therefore, radially evanesorder modes cent, the series representation (18) is rapidly convergent except ). To correctly when one observes close to the source (i.e., reconstruct the field singularity at the source, as required for the self-admittance calculation, we resort to a singularity extraction strategy as the one detailed in [13]. VI. GREEN’S FUNCTION REPRESENTATION FOR THE EXTERNAL PROBLEM (GHS)

Fig. 4. Equivalent z transmission line for the calculation of the characteristic Green’s function in the GHS (external region).

Fig. 5. Topology of singularities in the  complex plane for the GHS Green’s function spectral representation.

The external problem (GHS) leads to a semi-infinite equivtransmission-line section terminated on the metal alent impedance at 0 (Fig. 4). With reference to the case (i.e., source and observation on the of interest ground plate), the expression (21) is obtained by (15), where the voltage reflection coefficient is defined by TM case TE case.

(22)

The GHS characteristic Green’s function (21), as a function of , exhibits a branch singularity given the spectral variable . In order to ensure the convergence at by in the top Riemann sheet of the complex plane, the branch is chosen, taking the branch cut on the positive real axis. Furthermore, the characteristic Green’s function exhibits pole singularities (Zenneck pole) at , so that and , for the TM and TE case, respectively. For a good conductor case, the TM pole is in the top Riemann sheet, very close to the origin whereas the TE pole is far outside the visible region and lies in the bottom Riemann sheet (Fig. 5). In order to conveniently calculate the scalar functions involved in (14), we introduce the change of variable and, thus, obtain

Fig. 6. Topology of singularities in the  complex plane for the GHS Green’s function spectral representation.

The integration path around the singularities in the complex plane is mapped onto slightly below the real axis (and, in the TM case, also below the Zenneck pole). In the complex plane, the branch cut associated with the 2-D and is still present and defined by transverse functions . Furthermore, a saddle point arises at 0. A representation analogous to (23), but with (12) in place of (11), holds for the scalar functions . To obtain a closed-form expression for the scalar functions in (14), the integration path is deformed onto the steepest descent path (SDP) and evaluated by using the Van der Waerden method, as detailed in Appendix B. By this evaluation, it follows that:

(23) (24)

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Fig. 7. Real and imaginary parts of the quantity  , where  is the freespace wavelength, for  S/m (above). Relative error of the approximate solution (25), together with either (26) or (27), with respect to its accurate numerical evaluation (below).

= 10

Moreover, the function follows:

where the function

appearing in (14), is rewritten as

(25) can be asymptotically approximated as

(26) for

, and

(27) otherwise, and denote the Struve and the Neumann functions of order , respectively. The threshold has been identified empirically in order to minimize the error for the asymptotic and the small argument approximations. The aforementioned approximations were checked against the numerical calculation of the integral along the SDP and were found to be very accurate in all the ranges of interest of the various parameters involved. As an example, Fig. 7 shows the quanfor S/m and tity H/m calculated by (25), together with either (26) or (27), and calculated by a numerical integration of (37). Considering the second calculation as a reference, the figure also shows the relative error of the first calculation. Note that the above proposed switch between (26) and (27) results in a relative error of lower than 0.2%. VII. NUMERICAL EXAMPLES To check the formulation correctness and accuracy, in Fig. 8 1.6 mm, filled with we have considered a PPW of height

3.38), on whose top plate two identical para dielectric 5.7 mm allel slots are etched. The slot length and width are 0.4 mm, respectively, while the centers are 4.05 mm and apart. Furthermore, the fairly extreme case of S/m. This simple geometry has been analyzed with Ansoft HFSS [14] (dotted line) has been considered and the proposed formulation (continuous line). In particular, the two slots are considered radiating in either the PPW or in the GHS region, and the scattering parameters relevant to the first characteristic modes have been calculated. However, for the sake of brevity, only mode on the two the effect of the interaction between the slots is reported here, since it is the most significant interaction. Fig. 8 shows the relative variation of the scattering parameters ), with respect to the case of PEC plates (i.e., and are the scattering coefficients calculated for where finite conductivity and PEC plates). We observe good agreement between the two techniques along the whole frequency range, for what concerns the reflectance and the transmission. The largest variation with respect to the perfectly electric case occurs around 25 GHz, where the transition of the slot fundamental mode from the evanescent to the propagating state is present. The same two-slot configuration previously analyzed has also been considered to show the variation of admittances versus conductivity and frequency. Specifically, Figs. 9 and 10 show the real and imaginary part of the self and mutual admittance, respectively. It is evident that the variation of the real and the imaginary part of the admittances is not proportional to the conductivity and that the non-proportionality is more pronounced for the low conductivity values. Concerning the external region, we report the values of the self and mutual admittance in Tables I and II. We note that for the external and internal regions, the variation of the self admittance is more evident than that for the mutual admittance and can exceed 10%. Fig. 11 shows the calculated gain of the CP-RLSA antenna versus frequency for various conductivity values of the antenna plates. The antenna consists of 1866 slots spirally arranged as shown in the inset of Fig. 12 and similar to that described in 5 mm [15, Sec. V]. The slot length ranges linearly from to 6.3 mm as a function of the distance from the 0.4 mm. center; whereas their width is kept constant to S/m) It is worth noting that, when copper plates are used, the gain is very close to that of the ideal PEC case. However, significant gain reduction is observed when low-cost production techniques are employed: more than 2 dB is expected from a prototype realized by making use of adhesive aluminum S/m) and about 3.5 dB for steel metal foil painting techniques (i.e., for a conductivity value of approxiS/m [16]). Fig. 11 also shows the result obmately tained when the effect of the equivalent electric currents in the slots is considered negligible [5] [i.e., when the matrix products in the left-hand side of (4) are neglected (dots)]. The almost perfect matching with the rigorous solutions (solid curve) demonstrates, for the first time to the best of our knowledge, that the simplified approach of neglecting the effect of the equivalent electric currents in the slots can be used even for complex antennas, such as high-gain RLSA and the quite extreme case of an effective conductivity of S/m.

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Fig. 10. PPW mutual admittance (Y ) behavior versus frequency and conductivity for two identical parallel slots with centers 4.05 mm far apart, length L = 5.7 mm and width w = 0.4 mm. To better appreciate the differences between the mutual admittances Y , their values at f = 17 GHz are shown in the inset.

TABLE I GHS SELF-ADMITTANCE (Y ) BEHAVIOR VERSUS FREQUENCY AND CONDUCTIVITY

Fig. 8. Relative variation of the scattering parameters with respect to the case of perfect electric conductor plates. A comparison between the MoM (continuous line) and HFSS results (dotted line) for (a) the real part and (b) the imaginary part. TABLE II GHS MUTUAL ADMITTANCE (Y ) BEHAVIOR VERSUS FREQUENCY AND CONDUCTIVITY

Fig. 9. PPW self admittance (Y ) behavior versus frequency and conductivity for a slot of length L = 5.7 mm and width w = 0.4 mm. To better appreciate the differences between the self admittances Y , their values at f = 17 GHz are shown in the inset.

Fig. 12 shows the calculated gain versus normalized frequency of the CP-RLSA antenna for the two specific nominal operating frequencies of 17.2 and 60 GHz for a stainless-steel

prototype. Specifically, the dimensions of the antenna have been properly scaled in order to obtain the same radiative characteristics at 60 GHz, as those of perfect conductors. Comparing the results for the two prototypes, the gain reduction with respect . However, by to the PEC case is roughly predictable by the further decreasing the conductivity value, the maximum gain will reduce more slowly since the high attenuation constant in the PPW makes the effective antenna dimension smaller than the physical one. Thus, we have two conflicting effects: 1) a reduction of the antenna directivity due to a smaller effective size and 2) lower dissipation due to ohmic losses.

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The effect of ohmic losses in a practical antenna design has also been investigated, showing a reduction in the antenna efficiency of about 40% when low-cost materials (e.g., adhesive aluminum foil or painted steel) are used. Moreover, the results of this rigorous approach have been compared with those obtained when the effect of the equivalent electric currents in the slots is considered negligible [5]. They confirm that the latter approximation can be conveniently used also for conductors having efS/m. fective conductivity APPENDIX A

Fig. 11. Calculated gain of the RLSA antenna versus frequency for five different metal plates conductors: perfect electric conductor (PEC), copper ( = 5:8 1 10 S/m), stainless steel ( = 1:1 1 10 S/m), adhesive aluminum foil ( = 3:5 1 10 S/m) and steel metal painting ( = 5 1 10 S/m). The last case has been calculated with the rigorous and the approximated approaches.

In this appendix, we demonstrate that an effective magnetic , flowing on a surface surface current with impedance boundary condition imposed on it, radiates the same fields of the actual surat any point face currents and , flowing on . Let us consider an unit impulsive electric test source located at an arbitrary test point and directed along the unit vector . The reciprocity principle between the actual and the test source reads sources

(28) where the subscript “0” denotes the fields radiated by the electric test source, and the surface integral is extended over the impedance surface, on which the actual currents flow. Analogously, the reciprocity principle applied to the effective magand the test source , yields netic current (29)

Fig. 12. Calculated gain versus frequency normalized to the working nominal frequency f for two stainless-steel prototypes designed to work equally at f = 17.2 GHz and at f = 60.0 GHz when PEC plates are used. In the former case, the maximum gain reduction with respect to the PEC case is around 0.7 dB while the latter one is around 1.4 dB.

VIII. CONCLUSION In this paper, a rigorous MoM formulation for the analysis of RLSA antennas made of conductors with finite conductivity has been presented. Contrary to classical approaches where electric and magnetic currents are considered to equivalently describe the field in each aperture, only one effective magnetic equivalent current distribution is introduced. The latter maintains a rigorous description of the electromagnetic phenomenon (i.e., no inaccuracy is introduced in the formulation), but halves the number of unknowns in the MoM final linear system. Analytic expressions for the proper Green’s functions, in both the parallel plate waveguide and the grounded half-space, accounting for the finite conductivity of metals have also been derived. This results in a very simple, accurate, and fast evaluation of the MoM admittance matrix entries.

where is the field radiated by at . Since the field radiated by the test source satisfies the impedance boundary con(i.e., ), dition at any point ), then and is tangent to the surface (i.e., (30) Equation (30) readily follows that the integrands of (28) and (29) are equal at any integration point , where

(31) that is, the component of the electric field radiated by is . equal to the component of the electric field radiated by and , it holds that Due to the arbitrary choice of at any point. The same equivalence can be demonstrated by repeating the same for the magnetic field . steps with a magnetic test source APPENDIX B We first consider the representation (23) of the TM Green’s and, by using the Van der Waerden procedure, we function evaluate it asymptotically, resorting to a singularity extraction

ALBANI et al.: RIGOROUS MoM ANALYSIS OF FINITE CONDUCTIVITY EFFECTS

and a path deformation. To this end, the reflection coefficient in (22) is rearranged as

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Next, by using (11), (12), and (22) in the representation (23) apfor and in the analogous representation for , the pearing in (14) reduces to

(32) By using (32) and the relation

(33)

(23) reduces to

(37) (34) where the Green’s function is expressed as a perturbation with respect to the PEC case, for which only the first (spherical) term . In the remaining integral term, the integraremains as tion path is deformed onto the SDP through the saddle point 0 dictated by the phase variation of the Hankel function in (11). The SDP detours counterclockwise around the pole singularity at , not to include any additional residue contribution in the path deformation. The Van der Waerden uniform asymptotic evaluation [12, Sec. 4.4] is adopted to treat the pole singularity at , which is close to the saddle point. The SDP is parameterized by the variable through , thus leading to

The first integral term in (37) vanishes when closing the integration path clockwise at infinity without capturing any singularity (see Fig. 6). By deforming the integration path of the second integral term in (37) onto the SDP, we obtain an expression analogous to (34), which is asymptotically equal to (38) Finally, the last integral term in (37) can be asymptotically evaluated by assuming since , thus leading to (26). , the same integral can be rearConversely, for ranged as (39)

(35)

where

is a Fresnel function, and is the Sommerfeld’s “numerical distance” [17], with being the ambient intrinsic impedance and being the ground plane surface resistance. The last integral term can be easily calculated either asymptotically in closed form or numerically, since its integrand function

(36) is regular after the pole extraction. However, for practical RLSA analysis, even for a very high frequency, low surface conduc, tivity, and large directive antennas, it always holds that so that the third integral term in (35) is negligible with respect to the second one, which can be calculated by invoking in the approximate closed form (35).

which is the spatial convolution between a spherical wave and a unit step, which are the Fourier transform counterparts of the Hankel functions and of the simple pole appearing in its spectral original form in (37). Then, by using the second-order for the spherical Maclaurin approximation function phase delay, (39) can be calculated in terms of special functions as in (27). REFERENCES [1] P. W. Davis and M. E. Bialkowski, “Linearly polarized radial-line slot-array antennas with improved return-loss performance,” IEEE Antennas Propag. Mag., vol. 41, no. 1, pp. 52–61, Feb. 1999. [2] J. R. Mosig, R. C. Hall, and F. E. Gardiol, “Numerical analysis of microstrip patch antennas,” in Handbook of Microstrip Antennas, J. R. James and P. S. Hall, Eds. London, U.K.: Inst. Eng. Technol., 1989, ch. 8. [3] A. Akiyama, T. Yamamoto, J. Hirokawa, M. Ando, E. Takeda, and Y. Arai, “High gain radial line slot antennas for millimetre wave applications,” Proc. IREE Microw. Antennas Propag., vol. 147, no. 2, pp. 134–138, Apr. 2000. [4] C. A. Balanis, Advanced Engineering Electromagnetics. New York: Wiley, 1989. [5] R. C. Hall and J. R. Mosig, “The analysis of arbitrarily shaped aperturecoupled patch antennas via a mixed-potential integral equation,” IEEE Trans. Antennas Propag., vol. 44, no. 5, pp. 608–614, May 1996. [6] S. A. Schelkunoff, “The impedance concept and its applications to problems of reflection, radiation, shielding and power absorption,” Bell Syst. Tech. J., vol. 17, 1938.

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[7] M. A. Leontovich, “Approximate boundary conditions for electromagnetic field on the surface of conducting bodies,” in Collection Book Investigation of Radio Waves Propagation (in Russian). Moscow, Russia: Academy of Sciences, 1948. [8] T. B. A. Senior and J. L. Volakis, Approximate Boundary Conditions in Electromagnetics, ser. IEE Electromagnetic Waves Series. London, U.K.: IEE, 1995. [9] A. Mazzinghi, A. Freni, and M. Albani, “Influence of the finite slot thickness on RLSA antenna design,” IEEE Trans. Antennas Propag., vol. 58, no. 1, pp. 215–218, Jan. 2010. [10] K. Yoshitomi, “Radiation from a slot in an impedance surface,” IEEE Trans. Antennas Propag., vol. 49, no. 10, pp. 1370–1376, Oct. 2001. [11] R. E. Matick, Transmission Lines for Digital and Communication Networks. New York: McGraw-Hill, 1969. [12] L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves. Englewood Cliffs, NJ: Prentice-Hall, 1973, reprinted by IEEE Press, 1994. [13] M. Albani, G. Lo Cono, R. Gardelli, and A. Freni, “An efficient fullwave method of moments analysis for RLSA antennas,” IEEE Trans. Antennas Propag., vol. 54, no. 8, pp. 2326–2336, Aug. 2006. [14] ANSYS, Inc. [Online]. Available: http://www.ansoft.com/products/hf/ hfss/ [15] M. Albani, A. Mazzinghi, and A. Freni, “Asymptotic approximation of mutual admittance involved in MoM analysis of RLSA antennas,” IEEE Trans. Antennas Propag., vol. 57, no. 4, pp. 1057–1063, Apr. 2009. [16] T. Y. Otoshi and M. M. Franco, “The electrical conductivities of candidate beam-waveguide antenna shroud materials,” May 1994, pp. 35–41, Telecommun. Data Acqusit. Rep. [17] A. N. Sommerfed, Partial Differential Equations in Physics, 2nd ed. New York: Academic Press, 1953. Matteo Albani (M’98–SM’10) received the Laurea degree in electronic engineering and the Ph.D. degree in telecommunications engineering from the University of Florence, Florence, Italy, in 1994 and 1999, respectively. Currently, he is an Adjunct Professor in the Information Engineering Department, University of Siena, Italy, where he is also Director of the Applied Electromagnetics Lab. His research interests are in the areas of high-frequency methods for electromagnetic scattering and propagation, numerical methods for array antennas, antenna analysis, and design. Dr. Albani received the “Giorgio Barzilai” Prize for the Best Young Scientist Paper at the Italian National Conference on Electromagnetics in 2002 (XIV RiNEm).

Agnese Mazzinghi received the Laurea degree (Hons.) in electronic engineering from the University of Florence, Florence, Italy, in 2006 and the Ph.D. degree in information engineering from the University of Sienna, Sienna, in 2010. Currently, she is a Research Assistant in the Applied Electromagnetic Laboratory, University of Florence. She spent six months working on her master thesis project at the Defence, Security and Safety Institute of the Netherlands Organization for Applied Scientific Research (TNO), The Hague, The Netherlands. Moreover, she has been a Visiting Independent Advisor at Jet Propulsion Laboratory (JPL), Pasadena, CA , where she worked with the Submillimeter-Wave Advanced Technology Group for six months. Her research interests include numerical and asymptotic methods in electromagnetic antenna problems with a particular attention to the design of radial line slot-array antennas, Electromagnetic Band Gap structures, and dielectric waveguides.

Angelo Freni (S’90–M’91–SM’03) received the Laurea (Doctors) degree in electronics engineering from the University of Florence, Florence, Italy, in 1987. Since 1990, he has been with the Department of Electronic Engineering, University of Florence, first as Assistant Professor and since 2002, as Associate Professor of Electromagnetism. From 1995 to 1999, he was an Adjunct Professor at the University of Pisa, Pisa, Italy, and in 2010, a Visiting Professor at the TU Delft University of Technology, Delft, The Netherlands. In 1994, he was involved in research at the Engineering Department, University of Cambridge, Cambridge, U.K., concerning the extension and the application of the finite–element method to the electromagnetic scattering from periodic structures. From 2009 to 2010, he spent one year as a Researcher at the TNO Defence, Security and Safety, The Hague, The Netherlands, working on the electromagnetic modeling of kinetic inductance devices and their coupling with an array of slots in the tetrahertz range. His research interests include meteorological radar systems, radio-wave propagation, numerical and asymptotic methods in electromagnetic scattering and antenna problems, as well as electromagnetic interaction with moving media and remote sensing. In particular, part of his research focused on numerical techniques based on the integral equation, with a focus on domain decomposition and fast solution methods.

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A UWB Unidirectional Antenna With Dual-Polarization Biqun Wu and Kwai-Man Luk, Fellow, IEEE

Abstract—A novel 45 dual-polarized unidirectional antenna element is presented, consisting of two cross center-fed tapered mono-loops and two cross electric dipoles located against a reflector for ultrawideband applications. The operation principle of the antenna including the use of elliptically tapered transmission line for transiting the unbalanced energy to the balanced energy is described. Designs with different reflectors—planar or conical—are investigated. A measured overlapped impedance bandwidth of 126% (SWR 2) is demonstrated. Due to the complementary nature of the structure, the antenna has a relatively stable broadside radiation pattern with low cross polarization and low back lobe radiation over the operating band. The measured gain of the proposed antenna varies from 4 to 13 dBi and 7 to 14.5 dBi for port 1 and port 2, respectively, over the operating band, when mounted against a conical backed reflector. The measured coupling between the two ports is below 25 dB over the operating band. Index Terms—Dipole antenna, directional antennas, microstrip antennas, ultrawideband (UWB) antennas.

I. INTRODUCTION

U

LTRAWIDEBAND (UWB) radio is an emerging wireless communication technology with many attractive features. It is a license free usage of a wide frequency spectrum covering from 3.1 GHz to 10.6 GHz defined by FCC leading to many new possible applications [1]. The crucial component of a UWB system is the radiator. Various designs of UWB antennas with conical radiation pattern are now available with good performance. On the other hand, only few designs of UWB antennas with unidirectional radiation pattern are proposed in literature [1], [2]. which however are not suitable for practical applications due to their unstable radiation patterns, low gain and efficiency. Recently, a UWB antenna element backed by a conical reflector was proposed [3]. The antenna exhibits good characteristics such as ultrawide bandwidth, stable radiation pattern over the operating band and high gain. However, this antenna structure cannot be modified for dual-polarization radiation. In radar sensing and imaging applications, the system performance can be significantly improved by employing polarization

Manuscript received September 01, 2010; revised April 09, 2011; accepted April 30, 2011. Date of publication August 12, 2011; date of current version November 02, 2011. This work was supported in part by a grant from the Research Grant Commerce of the Hong Kong Special Administration Region, China under Project No. CityU 119008. The authors are with the State Key Laboratory of Millimeter Waves, Department of Electronic Engineering, City University of Hong Kong, Kowloon Tong, Hong Kong (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2164189

diversity. The polarization diversity technique is very effective to mitigate the multipath-fading effect in irregular environments [4]. For that purpose, dual-polarized antennas with overlapped operating frequency band are necessary. There are various publications on the study of dual-polarized unidirectional antennas. Dual-polarized broadside patterns can be produced by a microstrip patch antenna with two orthogonal L-probe feeds [5], stacked microstrip antenna with aperture coupling [6], or two orthogonal magneto-electric dipoles with two L-probe feeds [7]. However, none of them can exhibit a bandwidth wide enough for UWB applications. Very recently, a dual-polarized UWB antenna embedded in a dielectric is proposed based on the use of tapered slot antennas as feeds [8]. This antenna has good performance with stable antenna gain. But the antenna is too long in size and the construction of antenna is too complicated and expensive for commercial applications [8]. There are several ways to design an antenna with a unidirectional radiation pattern. The directed electric dipole is simple in structure but poor in radiation pattern stability over the operating frequency band. The microstrip patch antenna is low in profile but narrow in bandwidth. Although techniques are available to improve its bandwidth, the radiation pattern and gain of the patch antenna is also unstable over the operating band. To overcome the weaknesses of these antennas, an approach to excite an electric dipole and a magnetic dipole simultaneously to produce a symmetrical radiation pattern was proposed by Clavin [9]. Since then, several similar investigations [10], [11] have been conducted to demonstrate the advantages of this complementary antenna concept. In 1974, Clavin finally presented a simple practical structure of a complementary antenna [12], in which two parasitic inverted-L wires are placed beside a waveguide slot antenna. The two parasitic inverted-L wires form an electric dipole, and the rectangular slot acts like a magnetic dipole. This antenna however is not very wide in bandwidth. Recently, two wideband complementary antennas—the magneto-electric dipole [13] and the shorted bowtie patch antenna with an electric dipole [14]—were proposed by Luk et al.. Each of these antennas composes of a patch antenna and an electric dipole antenna, which functions as a combination of a magnetic dipole and an electric dipole. Good electrical characteristics, such as low back radiation, stable antenna gain, and symmetrical radiation pattern were demonstrated. In this paper, a novel ultrawideband unidirectional antenna designated as the “UWB magneto-electric dipole antenna” designed based on an integration of a bowtie electric dipole and a center-fed tapered mono-loop is proposed [15]. Since the radiation due to the mono-loop is equivalent to that of a magnetic current, it is reasonable to call this type of antenna as a UWB magneto-electric dipole antenna. In comparison to most UWB

0018-926X/$26.00 © 2011 IEEE

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Fig. 2. Photo of prototype. (a) Top view of prototype; (b) side view of prototype.

II. LINEARLY POLARIZED UWB ANTENNA A. Antenna Description and Design Geometry

Fig. 1. Geometry of the proposed antenna. (a) Perspective view; (b) top view; -axis); (d) Side view (from -axis). (c) side view (from

+X

0X

unidirectional antennas, the proposed antenna element is low in profile, high in antenna gain and relatively stable in radiation pattern. By adopting the proposed linear polarized design, a dual-polarized ultrawideband unidirectional antenna can be formed. In this design, the two linearly polarized elements are planes and orthogonal to each other at the center located at of a ground reflector. A planar reflector and a conical backed reflector are tested separately for enhancing the radiation pattern and gain stability over the operating frequency band [3].

The geometry of the ultrawideband magneto-electric dipole antenna is shown in Fig. 1 and Fig. 2, with detailed dimensions for operation at around 4 GHz. The dimensions were selected after a detailed parametric study for good performance. The configuration of the proposed magneto-electric dipole element is a combination of a bowtie-shaped dipole antenna [3], which is horizontally oriented, and a center-fed elliptically tapered mono-loop antenna [15], which is vertically oriented. Each arm of the planar dipole antenna has a bowtie-shaped arm with a radius R of 25 mm, and the ends of the two arms are connected together through circular thin strips. The bowtie antenna works as an electric dipole whereas the mono-loop antenna radiates like a magnetic antenna. The two elliptically tapered strips of mm. The gap between the mono-loop has a height mm. The two horizontal arms of the planar dipole is antenna is excited through an elliptically tapered transmission line of the mono-loop printed on the dielectric substrate supporting by one tapered strip of the mono-loop. The tapered transmission line with one strip of the mono-loop is used as a transition from the unbalanced energy to the balanced energy for exciting the bowtie dipole and the center-fed mono-loop simultaneously. From Fig. 1(c), one end ‘ ’ of the tapered ’ is connected to the other tapered strip transmission line ‘ ’ and an arm of the bowtie of the mono-loop antenna ‘ antenna ‘ ’, while the other end ‘A’ of the line is connected to an SMA connector located under the ground reflector. From the Fig. 1(d), one end ‘ ’ of the other tapered strip of mono-loop antenna is connected to the other arm of the bowtie antenna ‘ ’. An annular ring encloses the bowtie dipole to stabilize the radiation patterns across the operating band, as shown in Fig. 1.

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Fig. 4. SWR and gain versus frequency.

Fig. 3. Current distribution of the proposed antenna. (a) Top view, t = 0; (b) top view, t = T=4; (c) side view, t = 0; (d) side view, t = T=4.

B. Operation Principle The proposed UWB antenna is composed of an elliptically tapered vertical mono-loop and a horizontal bowtie-shaped dipole, which is equivalent to a combination of a magnetic dipole and an electric dipole to form a radiating element. Fig. 3 illustrates the current distribution of the antenna at a reference and respectively, where T represents the time period of time. It is found that, at a , the current densities are very high at the planar dipole, whereas the current densities . It can be are high at the tapered mono-loop when concluded that two degenerate resonant modes are excited with 90 degree phase difference to provide a wideband impedance bandwidth and stable radiation patterns over the operating band. C. Simulation and Measurement Results Simulation results of SWR, radiation pattern and gain were obtained by the commercial solver HFSS Ver.12.30 [16]. The performance of the prototype was measured by an Agilent E5071C Network Analyzer and SATIMO Near-field Measurement System. Fig. 4 depicts the SWR and the antenna gain versus frequency. The simulated impedance bandwidth is about from 3.25 GHz to 11.2 GHz), whereas 113% (with the measured impedance bandwidth is 128.7% (with from 2.6 GHz to 12.0 GHz). The measured impedance bandwidth is slightly wider than the simulated result. This may be due to the effect of material losses. The simulated gain of antenna is 5—10.8 dBi over the operating band from 3.25 GHz to 11.2 GHz while the measured gain is 2–11.3 dBi over the operating band from 2.6 GHz to 9 GHz. A substantial gain drop between 8.5 and 10 GHz in the measurement. The separation between the bowtie dipole and the ground plane is changing

Fig. 5. Measured input impedance versus frequency.

in terms of wavelength due to the variation of the frequency. At the particular frequency, the radiation is changed to the higher order mode, and introduces the cancellation effect to the broadside direction. The gain at the broadside direction across the higher operating band is no longer stable and varying randomly. So the resulting gain drop and radiation can be observed at that frequency band. The disadvantage of this antenna is that the measured broadside gain deteriorates within the frequency range from 9 GHz to 12 GHz The broadside radiation pattern also cannot be maintained. For commercial UWB applications, the proposed antenna is required to scale down to cover the frequency band of 3.1–10.6 GHz with a bandwidth of 110%. It can be observed from Fig. 5 that the input impedance is oscillatory over the operating band, which reveals the multiple resonance nature of the antenna. The measured radiation patterns of the antenna at 3 GHz, 5 GHz, 7 GHz, 9 GHz, 10 GHz and 11 GHz are plotted in Fig. 6. As shown in the figures, the levels of cross polarization are generally less than 10 dB. The back lobe radiation is generally less than 20 dB except at 7 GHz and 8 GHz.

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Fig. 6. Measured radiation patterns at (a) 3 GHz; (b) 5 GHz; (c) 7 GHz; (d) 9 GHz (e) 10 GHz and (d) 11 GHz.

III. A DUAL-POLARIZED UWB UNIDIRECTIONAL ANTENNA A. Antenna Description and Design Geometry Based on the previous proposed linearly polarized antenna structure, a UWB dual-polarized magneto-electric dipole antenna was designed and constructed by integrating two linearly polarized elements described previously with orthogonal orientation. Because the antenna gain at the Z-direction varies substantially, a conical backed reflector was employed to replace the planar reflector for stabilizing the antenna gain and radiation pattern of the antenna over the operating frequency band [3].

The geometry of the dual-polarized UWB unidirectional antenna is shown in Figs. 7, 8. The upper layer of the antenna is made up of four identical bowtie arms, each with an apex angle of 41.7 and radius of 22 mm. The ends of the sectors are connected by arc shaped strips each with a width of 0.3 mm. This metal layer is printed on a substrate with thickness of 1.57 mm and for ease of construction and support. The arm of the two bowtie dipoles are connected to the vertical part of the antenna consisting of two cross elliptically tapered mono-loops. The height of the horizontal sectors is mm. The detailed dimensions of the proposed antenna are shown in Table II. The two cross elliptically tapered mono-loops are printed on substrates with thickness of 0.79 mm and . The two SMA

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TABLE I DIMENSIONS OF PROPOSED ANTENNA

TABLE II DIMENSIONS OF PROPOSED ANTENNA

Fig. 7. Prototype of the proposed antennas. (a) Proposed antenna with ground reflector; (b) proposed antenna with conical backed reflector.

connectors located under the ground reflector are connected to the ends of the two cross elliptically tapered transmission line of the mono-loops. To study the antenna gain and the radiation pattern performance, the proposed antenna is first mounted against a planar square reflector with a thickness of 2 mm and a side of 120 mm. It is then mounted against a conical backed reflector. A parametric study on the conical reflector dimensions are performed in [3]. The height of the conical reflector has some influence on the reflection coefficient and gain flatness of the proposed antenna. So the and are optimized to be 20 mm and 10 mm for good reflection coefficient and flat gain in the operating band. The diameter of the conical backed reflector will also affect the return loss and the antenna gain of the proposed antenna. A large diameter of the conical will lead to the large gain variation, while a small diameter of the conical will cause bad return loss. So the and are optimized to be 130 mm and 70 mm for stable antenna gain and good return loss.

B. Simulation and Measurement Results The performance of the antenna was simulated and optimized with Ansoft HFSS Ver.12 [16]. The prototype was measured by a Agilent E5071C Network Analyzer and a SATIMO Near-field Measurement System. Fig. 9(a) depicts the SWR and antenna gain versus frequency, when the proposed antenna is mounted against a conducting planar reflector. Table III tabulates the

simulated and measured SWR bandwidths of the proposed antenna. It is observed that, the measured impedance bandwidth and 126.2% at port 1 and is 127.0% port 2 respectively, while the simulated impedance bandwidth is 133.3% and 131.0% respectively. Good agreement between simulation and measurement of impedance bandwidth can be observed. The overlapped impedance bandwidths in simulation and measurement are 131.0% and 126.2% respectively, which satisfies the requirement of UWB applications. On the other hand, the impedance bandwidths of the proposed antenna mounted against a conical backed reflector is shown in Fig. 9(b). It can be observed that, the simulated and measured impedance bandwidths of the two ports remain unchanged comparing with the planar reflector case. Fig. 9 also illustrates the antenna gains over the operating band. When the proposed antenna is mounted against the planar reflector, both simulated and measured gains vary from about 7.5 dBi to10.5 dBi over the operating band, which indicates that the main beam of radiation pattern is not always pointing to the z-direction. To improve the patterns performance, the prototype is mounted against the conical backed reflector. The antenna gains in simulation and measurement are shown in Fig. 9(b). It can be observed that, the measured antenna gains are 4–13 dBi and 7–14.5 dBi, while the simulated antenna

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TABLE III SUMMARY OF SWR BANDWIDTH OF DUAL-POLARIZED ANTENNA MOUNTED AGAINST A PLANAR REFLECTOR

Fig. 8. Geometry of the proposed antenna. (a) Top view; (b) side view at Y Z plane; (c) side view at XZ plane; (d) mounted against a planar reflector; (e) mounted against a conical backed reflector.

gains are 3.8–14.8 dBi and 7.1–16 dBi at port 1 and port 2 respectively, which represents a significant improvement in gain stability. The measured radiation patterns at port1 and port 2 are plotted in Fig. 10 and Fig. 11 respectively. Comparison between the planar and conical backed reflector cases is also studied and plotted in Figs. 10 and 11. It can be observed that, when the prototype is mounted against the planar reflector, the radiation pat-

Fig. 9. Prototype of the proposed antennas. (a) Mounted against a planar reflector; (b) proposed antenna with conical backed reflector.

tern in broadside direction varies substantially with frequency at both ports. The main beam of the radiation cannot be fixed in the broadside direction when the operating frequency exceeds 6 GHz. On the other hand, when the conical backed reflector is used, the main beam of the radiation is always fixed in the broadside Z-direction. The back radiation can be kept 15 dB smaller than the main beam over the operating band. It can be observed from Figs. 10 and 11 that the conical backed reflector can suppress the side lobes and enhance the gain of the main beam.

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Fig. 11. Measured radiation pattern of the proposed antenna at port 2. Fig. 10. Measured radiation pattern of the proposed antenna at port 1.

IV. CONCLUSION The simulated and measured coupling with planar and conical backed reflectors versus frequency are plotted in Fig. 12. It is found that the measured coupling is less than 25 dB over the operating band when the conical backed reflector is employed for the proposed antenna, which is about 2 dB better than the planar reflector case. The simulation results agree well with measurements.

A novel unidirectional antenna composed of two orthogonal elliptically tapered mono-loops and two electric dipoles and excited by an elliptically tapered transmission line has been designed, constructed and tested. The antenna can be designed with linear or dual polarization radiation. Two types of reflector–planar reflector and a conical backed reflector—have been studied. Measurement results reveal that the proposed antenna when mounted against the a conical backed reflector

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[9] A. Clavin, “A new antenna feed having equal E- and H-plane patterns,” IRE Trans. Antenna Propogat., vol. AP-2, pp. 113–119, 1954. [10] K. Itoh and D. K. Cheng, “A novel slots-and-monopole antenna with a steerable cardioids pattern,” IEEE Trans. Aerosp. Electron. Syst., vol. AES-8, no. 2, pp. 130–134, Mar. 1972. [11] W. W. Black and A. Clavin, “Dipole Augmented Slot Radiating Element,” U. S. Patent 3594806, Jul. 1971. [12] A. Clavin, D. A. Huebner, and F. J. Kilburg, “An improved element for use in array antennas,” IEEE Trans. Antennas Propogat., vol. AP-22, no. 4, pp. 521–526, Jul. 1974. [13] K. M. Luk and H. Wong, “A new wideband unidirectional antenna element,” Int. J. Microw. Opt. Technol., vol. 1, no. 1, pp. 35–44, Jun. 2006. [14] H. Wong, K. M. Mak, and K. M. Luk, “Wideband shorted bowtie patch antenna with electric dipole,” IEEE Trans. Antennas Propogat., vol. 56, no. 7, pp. 2098–2101, Jul. 2008. [15] H. G. Schantz, “UWB magnetic antennas,” Proc. IEEE APS Int. Symp. Dig., vol. 3, pp. 604–607, 2003. [16] Ansoft HFSS [Online]. Available: http://www.ansoft.com

Fig. 12. Coupling versus frequency.

features an overlapped impedance bandwidth of 126.2% (2.4–10.6 GHz) and gains of 4–13 dBi and 7–14.5 dBi at port 1 and port 2 respectively. The coupling is less than 25 dB over the operating band. The antenna is attractive for various UWB wireless communication systems. REFERENCES [1] H. Schantz, The Art and Science of Ultrawideband Antennas. Norwood, MA: Artech House, 2005. [2] Y.-S. Hu, M. Li, G.-P. Gao, J.-S. Zhang, and M.-K. Yang, “A double-printed trapezoidal patch dipole antenna for uwb applications with band-notched characteristic,” Progr. Electromagn. Res., vol. 103, pp. 259–269, 2010. [3] S.-W. Qu, C.-H. Chan, and Q. Xue, “Ultrawideband composite cavitybacked folded sectorial bowtie antenna with stable pattern and high gain,” IEEE Trans. Antennas Propag., vol. 57, no. 8, pp. 2478–2483, Aug. 2009. [4] P. Li, K. M. Luk, and K. L. Lau, “A dual-feed dual-band L-probe patch antenna,” IEEE Trans. Antennas Propag., vol. 53, no. 7, pp. 2321–2323, Jul. 2005. [5] H. Wong, K. L. Lau, and K. M. Luk, “Design of dual-polarized L-probe patch antenna arrays with high isolation,” IEEE Trans. Antennas Propag., vol. 52, no. 1, pp. 45–52, Jan. 2004. [6] S. B. Chakrabarty, F. Klefenz, and A. Dreher, “Dual polarized wideband stacked microstrip antenna with aperture coupling for SAR applications,” in Proc. IEEE Antennas Propag. Soc. Int. Symp., 2000, vol. 4, pp. 2216–2219, vol. 4. [7] B. Q. Wu and K. M. Luk, “A broadband dual-polarized magneto-electric dipole antenna with simple feeds,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 60–63, 2009. [8] G. Adamiuk, T. Zwick, and W. Wiesbeck, “Compact, dual-polarized UWB-antenna, embedded in a dielectric,” IEEE Trans. Antennas Propag., vol. 58, no. 2, pp. 279–286, Feb. 2010.

Biqun Wu was born in Guangdong, China. He received the B.Eng. degree (first class honors) in electronic engineering from City University of Hong Kong, in 2007, where he is currently working towards the Ph.D. degree. His research interest focuses on patch antenna and diversity antenna design. Mr. Wu was awarded the Second Prize in the IEEE Region 10 Student Paper Contest (Postgraduate Category) 2010 and the First Prize in 2009 IEEE Hong Kong Section (Postgraduate) Student Paper Contest.

Kwai-Man Luk (M’79–SM’94–F’03) was born and educated in Hong Kong. He received the B.Sc.Eng. and Ph.D. degrees in electrical engineering from The University of Hong Kong, Hong Kong, in 1981 and 1985, respectively. He joined the Department of Electronic Engineering at City University of Hong Kong in 1985 as a Lecturer. Two years later, he moved to the Department of Electronic Engineering at The Chinese University of Hong Kong where he spent four years. He returned to the City University of Hong Kong in 1992, and he is currently Chair Professor of Electronic Engineering and Director of State Key Laboratory in Millimeter waves (Hong Kong). His recent research interests include design of patch, planar and dielectric resonator antennas, and microwave measurements. He is the author of 3 books, 9 research book chapters, over 260 journal papers and 200 conference papers. He was awarded 2 US and more than 10 PRC patents on the design of a wideband patch antenna with an L-shaped probe feed. Prof. Luk was the Technical Program Chairperson of the 1997 Progress in Electromagnetics Research Symposium (PIERS 1997), and the General ViceChairperson of the 1997 and 2008 Asia-Pacific Microwave Conference, and the General Chairman of the 2006 IEEE Region Ten Conference. Professor Luk received the Japan Microwave Prize, at the 1994 Asia Pacific Microwave Conference held in Chiba in December 1994 and the Best Paper Award at the 2008 International Symposium on Antennas and Propagation held in Taipei in October 2008. He was awarded the very competitive 2000 Croucher Foundation Senior Research Fellow in Hong Kong. He is a deputy editor-in-chief of JEMWA. Professor Luk is a Fellow of the Chinese Institute of Electronics, PRC, a Fellow of the Institution of Engineering and Technology, U.K., and a Fellow of the Electromagnetics Academy.

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Flat-Shaped Dielectric Lens Antenna for 60-GHz Applications Anthony Rolland, Ronan Sauleau, Senior Member, IEEE, and Laurent Le Coq, Member, IEEE

Abstract—We describe the performance of a flat shaped dielectric lens antenna designed to produce a flat-top beam in H-plane and a nearly omni-directional pattern in E-plane in the 60-GHz band. Such radiation characteristics may be useful for access points or user terminals in high data rate wireless local area networks. For the antenna design, a specific two-stage methodology combining 2-D and 3-D modeling has been implemented. First, the lens shape is optimized in 2-D using a 2-D FDTD kernel coupled to a genetic algorithm. Second, all building blocks of the final antenna (3-D lens with a finite thickness, antenna feed) are optimized using the solution of the 2-D problem as an initial guess. This strategy has been validated experimentally: a 2.5-mm-thick flat lens in Rexolite with a shaped profile in H-plane has been fabricated and measured. It is shielded by two 1-mm-thick half metallic disks. The radiation patterns are very stable from 57 to 63 GHz, and the total antenna efficiency is better than 50%. Index Terms—60-GHz band, flat dielectric lenses, flat-top beam, millimeter waves, optimization.

I. INTRODUCTION N the last decade, the growing demand for broadband wireless communication systems has led to the development of emerging applications and multi-gigabits services [1] at millimeter waves, like wireless high-definition video transmission, wireless personal and local area networks (WPANs and WLANs), and radio-over-fiber networking solutions. In this context, the assignment of a large worldwide license-free bandwidth around 60 GHz has created new opportunities and standards for 60-GHz radio front-ends [2]–[4]. For indoor WLANs, dense nodes, usually called radio access points, are necessary to mesh efficiently the in-building areas, and it becomes essential to control efficiently the distribution of the radiated signals uniquely in the desired coverage zone to improve the radio link performance and avoid possible interferences with other signals coming from other nodes. To this end, base station or user-terminal antennas with properly shaped beams (fan beams, secant-squared beams) are considered as very attractive solutions, as shown in [2] and [5] for instance.

I

In this frame, reflector [6] and dielectric lens antennas [5], [7], [8] have been identified as promising candidates due to their ability to produce a wide variety of shaped beams at a low cost while maintaining very good performances (pattern quality, bandwidth, efficiency). In both cases, the shape of the antenna radiation patterns is controlled by properly designing the reflector and lens shape. In particular, over the last decade, numerous lens-based demonstrators have been designed and characterized experimentally at millimeter waves to produce various kinds of patterns, like flat-top beams [7], [9]–[11], secantsquared patterns [7], [12], elliptical Gaussian beams [9], [13], or fan beams (using 3-D shaped or cylindrical lenses [14]). Nevertheless, despite several attempts aiming at reducing the size of dielectric lens antennas (e.g., [15], [16]), the latter remain often bulky especially for shaped beam applications at 60 GHz, e.g., [5]. In this frame, flat lenses become attractive since the total antenna thickness can be reduced down to a couple of millimeters when operating in -band. Several approaches have been proposed in the literature to design flat devices with beam collimating capabilities, namely metamaterial-based configurations, e.g., [17], [18], holographic antennas [19], or flat dielectric lenses (extended hemielliptical lenses [20], cylindrical Luneburg lenses with a variable thickness [21] or hole density [22], [23]), and integrated horns in SIW technology [24], [25]. To our best knowledge, the capabilities of optimized shaped flat lenses have never been studied although these devices might be very useful for applications where the antenna beam needs to be shaped only in one plane. This constitutes the main goal of this work. This paper is organized as follows. First the design methodology is described in Section II. Then it is applied to optimize a 10% bandwidth shaped flat dielectric lens radiating a flat-top pattern in H-plane and a nearly omni-directional beam in E-plane. The numerical results are given in Section III, and the experimental data obtained in the 60-GHz band are discussed in Section IV. Finally, conclusions are drawn in Section V.

II. DESCRIPTION OF THE DESIGN TECHNIQUE Manuscript received December 22, 2010; manuscript revised April 06, 2011; accepted April 14, 2011. Date of publication August 12, 2011; date of current version November 02, 2011. This work was supported in part by the “Université Européenne de Bretagne” (UEB), France under the “International chair” program and the OPTIMISE and GRAPPAS projects, in part by the Conseil Régional de Bretagne (CREATE/CONFOCAL project), and in part by CNRS. The authors are with the Institute of Electronics and Telecommunications of Rennes (IETR), UMR CNRS 6164, University of Rennes 1 and INSA de Rennes, Rennes, France (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2164218

Due to the peculiar features of the lens body studied here (flat device), the geometrical optics/physical optics (GO/PO) method that has been employed very frequently to analyze and optimize lens shapes in the past [9]–[11], [16] cannot be used here in a straightforward way. Therefore, we have implemented an alternative method of analysis to optimize flat lenses. It consists of two successive stages as represented in the flow chart of Fig. 1. In the first stage, a 2-D CAD tool combining a 2-D FDTD solver (in TE and TM modes) and genetic algorithms (GA) is used to

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upper and lower template [orange lines in Fig. 4(a)]: the maximum ripple level equals 2 dB in the flat top part of the radiated beam, and the side lobe rejection is fixed to 20 dB below the maximum. The optimization bandwidth extends from 57 to 63 GHz. The antenna dimensions are optimized in 2-D so as to minimize the global fitness function

(1)

In this equation, is the total number of frequency points where the antenna is optimized, and is the weight coefficient at . The fitness function at a given frequency is defined as follows:

(2)

Fig. 1. General flow chart of the design technique.

define the optimized shape of a 2-D antenna configuration complying with the desired specifications in a pre-defined radiation plane. Then, this 2-D solution is transformed in a 3-D one to design a real 3-D flat device with the expected performances. Both stages are described in Sections II-A and II-B, respectively. This strategy has been briefly described in [26]. A. First Stage: Optimization in 2-D and Specifications in Radiation The CAD tool implemented here is based on an in-house 2-D-FDTD solver combined with a real-valued genetic algorithm. This choice offers several advantages, namely: 1) fast analysis of the antenna configuration over a broadband in a single run; 2) reduced implementation complexity; 3) accuracy of the numerical results (compared to the GO/PO scheme which does not take into account properly multiple and total reflections within the dielectric body); and 4) ability to model any kind of material and shape. This tool allows a flexible specification of the desired performance and an easy assignment of the parameters to optimize (e.g., dimensions and shape of the antenna profile, ground plane size, and dielectric materials). Here, the geometry of each interface between two different materials is defined using a set of control points whose locations in space is optimized using a differential or absolute encoding, as previously described in 3-D with the GO/PO and BoR-FDTD methods [10], [27]. The continuous contour of each interface is reconstructed using cubic splines. In this work, the radiation pattern desired in one plane is a flat-top beam, as explained in Section III-A. It is defined by an

where is the total number of observation points used to evaluate the radiation pattern in the far-field zone, is a direction of is the difference between the raobservation, and diation pattern (in magnitude) of the lens under optimization and denotes the angular the power mask template [Fig. 4(a)]. are user-defined weight coefficients discretization step, and defined to assign more importance to predefined directions. In this paper, the following parameters have been used: GHz , to , , , and to . B. Second Stage: Extension to 3-D Flat Structures Several issues must be considered when extending in 3-D the 2-D lens configuration defined in stage 1, namely: • Replacement of the 2-D numerical feed (stage 1) by a real 3-D feed (stage 2) whose radiation characteristics are close to those of the 2-D feed. of the flat lens. • Selection of the optimum thickness To this end, two cases should be distinguished in practice depending on the lens configuration [26]: 1) In the first case, the two flat surfaces of the lens are coated by PEC walls, as illustrated in Fig. 2. For E-plane structures (i.e., antennas where the E-field is parallel to the PEC planes, TE mode), the permittivity values must be corrected to guarantee that the effective permittivity is similar to the one used for the 2-D modeling [28]. On the contrary, for H-plane configurations (i.e., antennas where the E-field is perpendicular to the PEC planes, TM mode, as in Fig. 2), no permittivity correction is needed. 2) In the second case, both flat sides of the lens are in direct contact with air, and the corresponding effective permittivity is lower than the permittivity of the lens material.

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Fig. 2. Generic configuration studied in this paper (H-plane flat lenses). The lens is encapsulated between two metallic sheets (of arbitrary shape in this figure).

Fig. 4. Far-field radiation of the optimized 2-D lens. (a) Radiation pat: 57 GHz. : 58.5 GHz. terns computed at five frequency points. : 60 GHz. : 61.5 GHz. : 63 GHz. (b) 2-D plot of the far-field patterns versus frequency. Fig. 3. 2-D optimized lens at 60 GHz.

For the configurations requiring permittivity corrections, the corrective terms are available in [28]. They depend on the ratio , where is the wavelength at the operating frequency. This constitutes one drawback of these configurations since, for a given lens thickness, the correction is applied at only one frequency point at a time, thus making it not suitable for broadband applications. Here we focus our attention on the unique configuration which does not imply any permittivity correction, namely the H-plane flat lenses encapsulated between two parallel metallic plates as illustrated in Fig. 2. For such configurations, the lens thickness is optimized in order to 1) keep the same desired pattern (here a flat-top beam) in H-plane as for the 2-D case (stage 1), and 2) produce a nearly omni-directional pattern in E-plane. III. NUMERICAL RESULTS A. Optimization in 2-D (Stage 1) The antenna configuration studied here is represented in , ) Fig. 3. The lens is in Rexolite ( and is fed by an open-ended metallic WR-15 waveguide loaded is finite, and its aperture size with Rexolite. Its flange

mm coincides with the normalized dimension of a WR-15 waveguide loaded with Rexolite. The antenna is symmetric, and its geometrical parameters are optimized to produce a symmetric flat-top pattern defined by the two orange lines plotted in Fig. 4(a). For symmetry reasons, only half of the lens profile is optimized here. Its contour is represented (Fig. 3). The flange with the help of ten control nodes has been also optimized to better control the ripple size and diameter and side lobe levels. The height of the lens are bounded to 30 mm and 45 mm, respectively (Fig. 3) to keep an antenna size of reasonable size. The optimized antenna geometry is represented in Fig. 3, and all its dimensions are provided in Table I. The antenna height is equal to 28.86 mm, and its size along -direction mm slightly exceed the boundaries defined above because of the cubic splines interpolation. Note also that mm is much smaller than the the flange size lens diameter. The radiation patterns of the optimized antenna in 2-D are plotted in Fig. 4(a) for the five frequency points . They are in good agreement with the mask template, although the ripple and side lobe level are very slightly higher than the specified values, especially at 63 GHz. Fig. 4(b) represents the variations of the 2-D patterns versus frequency. It shows that the antenna radiation is very stable over the [57–63] GHz bandwidth and that it complies very well with the predefined template. Moreover,

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TABLE I DIMENSIONS OF THE OPTIMIZED 2-D LENS

TABLE II DIMENSIONS OF THE FEED PART

Fig. 6. Radiation patterns of the flat lens antenna with two 1-mm-thick half-disk walls. The five frequency values correspond to the frequency points where the 2-D lens contour has been optimized (Section III-A). (a) H-plane. : 57 GHz. : 58.5 GHz. : 60 GHz. : 61.5 GHz. (b) E-plane. : 63 GHz.

Fig. 5. 3-D flat lens analyzed with HFSS. (a) Perspective view. (b) Zoom on the feed part of the prototype. (c) Cut-view of the feed part in H-plane. (d) Cut-view of the feed part in E-plane.

the back-radiation level (not shown here) remains lower than 20 dB. B. Extension in 3-D (Stage 2): Flat Lens Based on the previous results a thin flat lens has been designed. All simulations have been carried out using HFSS. The final antenna configuration is represented in Fig. 5: • The lens contour is the same as the one derived in 2-D (Fig. 3 and Table I).

• The lens thickness has been fixed to mm [Fig. 5(a)]. This value has been adjusted to reduce the ripple level in the transverse plane (E-plane), especially to minimize diffraction phenomena along the edges of the parallel metallic plates. Such a value also ensures a good mechanical robustness of the prototype, especially around the feed part where the presence of “sharp edges” and the use of thinner Rexolite would make the lens much more fragile. • As explained in Section II-B, both flat interfaces of the lens are shielded with two PEC walls to keep the same permittivity values both in 2-D and 3-D. To facilitate their fabrication, both walls have a half-disk shape; their thickness is fixed to 1 mm, and their radius is equal to the maximum mm , as represented in lens radius Fig. 5(a).

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Fig. 7. Reflection coefficient of the optimized flat lens antenna ( ), and comparison with the S of several building blocks: transition between the air), 2) the filled WR-15 waveguide and the dielectric-loaded waveguide ( ), and 3) the full feed system rasame transition up to the horn aperture ( diating in an infinite dielectric slab encapsulated between two infinite parallel ). plates (

Fig. 9. Co-polarization components measured and computed in H-plane from 57 to 63 GHz. (a) Numerical results (Ansoft HFSS). (b) Measurements.

The radiation patterns computed at 57 GHz, 58.5 GHz, 60 GHz, 61.5 GHz, and 63 GHz are represented in Fig. 6(a) and 6(b) in H-plane and E-plane, respectively. In H-plane, we obtain a very good agreement with the 2-D flat-top beams given in Fig. 4(a). These results demonstrate the reliability and relevance of the 2-D optimization to design flat shaped lenses. Moreover, in the orthogonal plane (E-plane), the antenna radiates a smooth and wide beam over the whole bandwidth, as expected. The reflection coefficient of the antenna is plotted in Fig. 7. It remains lower than 13 dB over the bandwidth. The impact of all building blocks of the feed system is also given. The antenna reflection coefficient drop arising around 59.4 GHz is mainly due to the internal reflection phenomena occurring at the dielectric/air lens interface. Fig. 8. Antenna prototype. (a) Dielectric lens fabricated in a single bloc. (b) Prototype after assembly.

• The antenna is fed by an air-filled standard WR-15 rectangular waveguide (3.76 mm 1.88 mm). The transition between this feed waveguide and the lens base consists of three sections in Rexolite: 1) a pyramidal taper used to ensure a smooth transition between the air-filled waveguide and the Rexolite-loaded waveguide (2.36 mm 1.18 mm); 2) a Rexolite-loaded 3 mm-long waveguide section; 3) a pyramidal transition tapered in only one plane (E-plane) to feed the lens with a “dielectric” horn of same width as the lens thickness [Fig. 5(b)]. All the dimensions of the feed part have been optimized based on a parametric study (using HFSS). They are provided in Table II according to the notations defined in Fig. 5(c) and (d).

IV. EXPERIMENTAL RESULTS A. Prototyping The antenna optimized in Section III (Fig. 5) has been fabricated using a 3-D-axis milling machine. The shaped lens with its tapered transitions has been fabricated in a single block [Fig. 8(a)]. Due to the tiny dimensions of the feed parts, the prototype has been built as follows. • The three metallic waveguide sections surrounding the airfilled section, the pyramidal taper, and dielectric-loaded section (Figs. 5(c) and 5(d)) have been fabricated into four self-assembled blocks. • A standard flange (UG-385/U) has been manufactured at the extremity of the WR-15 feed waveguide to connect the antenna to our measurement system [Figs. 8(b)].

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Fig. 11. Co-polarization components measured and computed at 61.5 GHz. ). Numerical results (Ansoft (a) H-plane. (b) E-plane. Measurements ( ). Measured cross-polarization component ( ). HFSS) (

Fig. 10. Co-polarization components measured and computed in 3-D at 61.5 GHz. (a) Numerical results (Ansoft HFSS). (b) Measurements.

• Six Nylon screws have been used to assemble all the metallic pieces and hold firmly the dielectric lens [Fig. 8(b)]. B. Experimental Results The numerical results provided in this section take into account all features of the antenna prototype, including the flange and screws. Therefore, comparison with those provided in Section III-B highlights the influence of the metallic feed block upon the antenna performance. The radiation patterns measured and computed in H-plane from 57 to 63 GHz are represented in Fig. 9. They confirm the very good quality of the flat-top beam over the whole bandwidth. The ripple level does not exceed 5 dB, and the side lobe level remains always below 12 dB. Since the agreement between simulations and measurements is very satisfactory, we focus our attention at only one frequency point (61.5 GHz) and provide a comparison between the measured and computed 3-D and 2-D patterns in Figs. 10 and 11, respectively. The “openeye” shape of the 3-D pattern is very well predicted. The discrepancies between the experimental and numerical data can be better appreciated in Fig. 11. Those observed in H-plane might

be due to a slight bending of the prototype, the tiny parasitic air-gaps between the dielectric lens body and the Aluminum half-disks (less than 0.1 mm), and some inaccuracies in the fabrication and assembly of the antenna (less than 0.1 mm). In E-plane, the agreement between the measured and simulated co-polarization components is acceptable, especially if we keep in mind that measuring broad radiation patterns at millimeter wave is challenging due to the influence of all surrounding objects (like feed connectors and flanges for instance). Comparison between the simulation data given in Figs. 6 and 11 evidences the impact of the metallic supports upon the ripple levels in E- and H-planes. Smaller undulations would be expected if the external mechanical parts were less bulky. One solution consists in using the metallized-foam technology that would lead to a much lighter prototype and very thin PEC walls ( 0.1 mm thick). This technology would also facilitate the antenna assembly and improve its integration. Nevertheless, such a work is out of the scope of this paper. The measured and computed reflection coefficients are represented in Fig. 12. Experimentally, the antenna is well matched, is smaller than 14 dB over the entire bandwidth. and the might The discrepancy between the simulated and measured also come from fabrication uncertainties, as discussed above. The antenna gain has been measured between 57 and 63 GHz with the comparison method using a 20-dBi standard gain horn. It is plotted in Fig. 13. The 2.5-dB difference (in average over

ROLLAND et al.: FLAT SHAPED DIELECTRIC LENS ANTENNA FOR 60-GHz APPLICATIONS

Fig. 12. Reflection coefficients (S ). Measurements ( ). tion (

Fig. 13. Measured gain ( ) and directivity ( loss varies between 2 and 3 dB.

), HFSS simula-

) at broadside. The total

the entire bandwidth) between the measured values of gain and directivity can be decomposed as follows: 0.5-dB dielectric loss , less than 0.20-dB inside the lens itself return loss, and around 1.5-dB insertion loss in the WR-15 feed waveguide and connecting parts (Fig. 12). V. CONCLUSION The capabilities of shaped flat lenses for beam shaping applications in the 60-GHz band have been investigated, with emphasis on H-plane configurations. A two-stage methodology has been proposed. The first one consists of a 2-D optimization process based on a 2-D FDTD kernel coupled to a genetic algorithm, and, in the second stage, the solution of the 2-D problem is extended and optimized in 3-D. This methodology has been validated experimentally in V-band. In particular, we have shown that the flat-top beam radiated by the antenna prototype (lens in Rexolite) is very stable between 57 and 63 GHz. The impact of the bulky mechanical parts (needed for the antenna measurement) has been also discussed. The total antenna efficiency is better than 50%. REFERENCES [1] R. C. Daniels, J. N. Murdock, T. S. Rappaport, and R. W. Heath, Jr., “60 GHz wireless: Up close and personal,” IEEE Microw. Mag., vol. 11, no. 7, pp. 44–50, Dec. 2010.

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[2] P. F. M. Smulders, “Exploiting the 60 GHz band for local wireless multimedia access: Prospects and future directions,” IEEE Commun. Mag., vol. 40, no. 1, pp. 140–147, Jan. 2002. [3] High Rate 60 GHz PHY, MAC and HDMI PAL, Standard ECMA-387, Dec. 2008 [Online]. Available: http://www.ecma-international.org/ publications/standards/Ecma-387.htm [4] “IEEE 802.11 Working Group,”, Very High Throughput in 60 GHz [Online]. Available: http://www.ieee802.org/11/Reports/tgad_update.htm [5] C. A. Fernandes and L. M. Anunciada, “Constant flux illumination of square cells for millimeter-wave wireless communications,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 11, pp. 2137–2141, Nov. 2001. [6] P. F. M. Smulders, S. Khusial, and M. H. A. J. Herben, “A shaped reflector antenna for 60-GHz indoor wireless LAN access points,” IEEE Trans. Veh. Tech., vol. 50, no. 2, pp. 584–591, Mar. 2001. [7] C. A. Fernandes, “Shaped dielectric lenses for wireless millimeterwave communications,” IEEE Antennas Propagat. Mag., vol. 41, no. 5, pp. 141–150, Oct. 1999. [8] G. Godi, R. Sauleau, and D. Thouroude, “Performance of reduced size substrate lens antennas for millimetre-wave communications,” IEEE Trans. Antennas Propagat., vol. 53, no. 4, pp. 1278–1286, Apr. 2005. [9] R. Sauleau and B. Barès, “A complete procedure for the design and optimization of arbitrarily-shaped integrated lens antennas,” IEEE Trans. Antennas Propagat., vol. 54, no. 4, pp. 1122–1133, Apr. 2006. [10] G. Godi, R. Sauleau, L. Le Coq, and D. Thouroude, “Design and optimization of three-dimensional integrated lens antenna with genetic algorithm,” IEEE Trans. Antennas Propagat., vol. 55, no. 3, pp. 770–775, Mar. 2008. [11] N.-T. Nguyen, R. Sauleau, and L. Le Coq, “Reduced-size double-shell lens antenna with flat-top radiation pattern for indoor communications at millimeter waves,” IEEE Trans. Antennas Propagat., vol. 59, no. 6, pp. 2424–2429, Jun. 2011. [12] J. R. Costa, C. A. Fernandes, G. Godi, R. Sauleau, L. Le Coq, and H. Legay, “Compact Ka-band lens antennas for LEO satellites,” IEEE Trans. Antennas Propagat., vol. 56, no. 5, pp. 1251–1258, May 2008. [13] B. Chantraine-Barès, R. Sauleau, L. L. Coq, and K. Mahdjoubi, “A new accurate design method for millimeter-wave homogeneous dielectric substrate lens antennas of arbitrary shape,” IEEE Trans. Antennas Propagat., vol. 53, no. 3, pp. 1069–1082, Mar. 2005. [14] T. Komljenovic, R. Sauleau, Z. Sipus, and L. Le Coq, “Layered circular-cylindrical dielectric lens antennas—Synthesis and height reduction technique,” IEEE Trans. Antennas Propagat., vol. 58, no. 5, pp. 1783–1788, May 2010. [15] J. R. Costa, E. B. Lima, and C. A. Fernandes, “Compact beam-steerable lens antenna for 60-GHz wireless communications,” IEEE Trans. Antennas Propagat., vol. 57, no. 10, pp. 2926–2933, Oct. 2009. [16] B. Barès and R. Sauleau, “Electrically-small shaped integrated lens antennas: A study of feasibility in Q-band,” IEEE Trans. Antennas Propagat., vol. 55, no. 4, pp. 1038–1044, Apr. 2007. [17] M. Casaletti, F. Caminita, and S. Maci, “A Luneburg lens designed by using a variable artificial surface,” in Proc. IEEE Antennas Propagat. Symp., APS, Toronto, ON, Canada, Jul. 2010. [18] C. Pfeiffer and A. Grbic, “A printed, broadband Luneburg lens antenna,” IEEE Trans. Antennas Propagat., vol. 58, no. 9, pp. 3055–3059, Sep. 2010. [19] G. Minatti, F. Caminita, and S. Maci, “A circularly polarized dielectric lens antennas designed by holographic principle,” in Proc. IEEE Antennas Propagat. Symp., APS 2010, Toronto, ON, Canada, Jul. 2010, pp. 1–4. [20] L. Xue and V. Fusco, “Patch fed planar dielectric slab waveguide extended hemielliptical lens antenna,” IEEE Trans. Antennas Propagat., vol. 56, no. 3, pp. 661–666, Mar. 2008. [21] X. Wu and J.-J. Laurin, “Fan-beam millimeter-wave antenna design based on the cylindrical Luneburg lens,” IEEE Trans. Antennas Propagat., vol. 55, no. 8, pp. 2147–2155, Aug. 2007. [22] L. Xue and V. Fusco, “Patch fed 2D planar Luneburg lens,” Microw. Opt. Tech. Lett., vol. 49, no. 12, pp. 2922–2924, Dec. 2007. [23] K. Sato and H. Ujiie, “A plate Luneburg lens with the permittivity distribution controlled by hole density,” Electro. Commun. Jpn. Lett., vol. 85, no. 9, pp. 1–12, Apr. 2002.

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[24] M. Wong, A. R. Sebak, and T. A. Denidni, “A broadside substrate integrated horn antenna,” in Proc. IEEE Int. Symp. Antennas Propagat., APS’08, San Diego, CA, Jul. 2008, pp. 1–4. [25] H. Wang, D.-G. Fang, B. Zhang, and W.-Q. Che, “Dielectric loaded substrate integrated waveguide (SIW) H-plane horn antennas,” IEEE Trans. Antennas Propagat., vol. 58, no. 3, pp. 640–647, Mar. 2010. [26] A. Rolland, R. Sauleau, and M. Drissi, “Design of H-plane shaped flat lenses using a 2-D approach based on FDTD and genetic algorithm,” in Proc. Eur. Conf. Antennas Propagat., EuCAP’10, Barcelona, Spain, Apr. 12–16, 2010. [27] A. Rolland, M. Ettorre, M. Drissi, L. Le Coq, and R. Sauleau, “Optimization of reduced-size smooth-walled conical horns using BoRFDTD and genetic algorithm,” IEEE Trans. Antennas Propagat., vol. 58, no. 9, pp. 3094–3100, Sep. 2010. [28] J. F. Lotspeich, “Explicit general eigenvalue solutions for dielectric slab waveguides,” Appl. Electron., vol. 14, no. 2, pp. 327–335, Feb. 1975.

Ronan Sauleau (M’04–SM’06) graduated in electrical engineering and radio communications from the Institut National des Sciences Appliquées, Rennes, France, in 1995 and received the Agrégation degree from the Ecole Normale Supérieure de Cachan, Cachan, France, in 1996 and the Doctoral degree in signal processing and telecommunications and the “Habilitation à Diriger des Recherche” degree from the University of Rennes 1, France, in 1999 and 2005, respectively. He was an Assistant Professor and Associate Professor at the University of Rennes 1, between September 2000 and November 2005, and between December 2005 and October 2009. He was appointed as a full Professor at the same University in November 2009. His current research fields are numerical modeling (mainly FDTD), millimeter-wave printed and reconfigurable (MEMS) antennas, lens-based focusing devices, periodic and non-periodic structures (electromagnetic bandgap materials, metamaterials, reflectarrays, and transmitarrays) and biological effects of millimeter waves. He has received five patents and is the author or coauthor of 86 journal papers and more than 210 publications in national and international conferences and workshops. Prof. Sauleau received the 2004 ISAP Conference Young Researcher Scientist Fellowship (Japan) and the first Young Researcher Prize in Brittany, France, in 2001 for his research work on gain-enhanced Fabry–Perot antennas. In September 2007, he was elevated to Junior member of the “Institut Universitaire de France.” He was awarded the Bronze medal by CNRS in 2008.

Anthony Rolland received the electronic engineering and the M.S. degree in electronics and radio communications from the Institut National des Sciences Appliquées (INSA), Rennes, France, in 2005 and the Ph.D. degree in signal processing and telecommunications from the University of Rennes 1, Rennes, France. Currently, he is a Postdoctoral Fellow at the Institut d’Electronique et de Télécommunications de Rennes (IETR), University of Rennes 1, France. His main fields of interest include the numerical modeling, the analysis and optimization of lens-based focusing devices for microwave and millimeter-wave applications.

Laurent Le Coq received the electronic engineering and radio communications degree and the French DEA degree (M.Sc.) in electronics and the Ph.D. from the National Institute of Applied Science (INSA), Rennes, France, in 1995 and 1999, respectively. In 1999, he joined Institute of Electronics and Telecommunications of Rennes (IETR), University of Rennes 1, as a Research Lab Engineer, where he is responsible for measurement technical facilities up to 110 GHz. He has been involved in more than 20 research contracts of national or European interest. He is an author and coauthor of 20 journal papers and 30 papers in conference proceedings.

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Spherical Near-Field Scanning With Higher-Order Probes Thorkild B. Hansen, Member, IEEE

Abstract—A general method for higher-order probe correction in spherical scanning is obtained from a renormalized least-squares approach. The renormalization causes the normal matrix of the least-squares problem to closely resemble the identity matrix when most of the energy of the probe pattern resides in the first-order modes. The normal equation can be solved either with a linear iterative solver (leading to an iterative scheme), or with a Neumann series (leading to a direct scheme). The computation scheme can handle non-symmetric probes, requires only the output of two independent ports of a dual-polarized probe, and works for both and scans. The probe can be characterized either by a complex dipole model or by a standard spherical-wave representation. The theory is validated with experimental data. Index Terms—Antenna measurements, antenna theory.

I. INTRODUCTION

P

ROBE-CORRECTED near-field techniques [1] have been used widely for the past 50 years to characterize antennas from measurements on planar, cylindrical, and spherical scanning surfaces. The output of a known probe is recorded as the probe moves along the scanning surface in the near field of the antenna under test (AUT). Probe-corrected formulas subsequently determine the desired far field of the AUT from the probe output. The measurements are typically performed in anechoic chambers. The probe-corrected formulas take into account the fact that the incident field at the position of the probe is a near field that consists of many different wave components, each of which is received in accordance with the probe receiving characteristic. The standard theories for spherical near-field scanning of electromagnetic fields [2]–[4] hold for first-order probes that azimuthal pattern dependence only. In addition, have one must assume that the patterns of the two probe ports are identical, except for a phase and amplitude factor and a 90 degree rotation. For such first-order probes one can define a vector output that can be expressed in terms of the transverse vector-wave functions, so that the unknown expansion coefficients of the AUT can be determined straightforwardly from orthogonality. Wacker was the first to recognize the advantage of using first-order probes in spherical scanning [5]. Manuscript received December 19, 2010; revised April 11, 2011; accepted May 01, 2011. Date of publication August 12, 2011; date of current version November 02, 2011. The Air Force Office of Scientific Research supported this work. The author is with Seknion Inc., Boston, MA USA (e-mail: thorkild. [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2164217

For higher-order probes, which have patterns that contain with , one cannot express the vector modes output in terms of the transverse vector-wave functions, and the AUT expansion coefficients cannot be determined from orthogonality.1 Therefore, first-order probes have been preferred in spherical near-field scanning over the past 30 years. Unfortunately, first-order probes are inherently narrowbanded, and thus a large number of different probes (each covering a narrow frequency band) are needed to measure the AUT over a broad band of frequencies. In practice, this means that wide-band characterizations become very time consuming because the spherical scan must be repeated many times. In addition, it takes a considerable amount of time to calibrate and change probes because precise alignment procedures are required. Broadband probes that overcome all these problems do indeed exist, but their patterns contain higher-order modes. Typically, most of the probe-pattern energy is still in the first-order modes, but the higher-order modes are nevertheless strong enough to prevent a first-order correction scheme from being accurate (see Fig. 5 of this paper). Therefore, a higher-order probe-correction method is required if one is to cover the entire frequency band of an anechoic chamber with one single broadband probe. With such a method available, one would avoid ever having to changes probes and thereby significantly improve the efficiency of spherical near-field scanning. An effort to solve the problem of higher-order probe correction was initiated some years ago at the Technical University of Denmark. The result of that effort is the FFT matrix inversion method [8], which computes the AUT spherical exoperations. Here, is the elecpansion coefficients in trical radius of the AUT, so that the total number of AUT expan. Schmidt et al. [9] developed a sion coefficients is higher-order probe-correction method based on the multilevel fast multipole method. This method is iterative and requires operations per iteration. Since a comprehensive comparative study of higher-order probe-correction methods is currently in progress [10], we shall not present a full review here.2 1We assume that the probe has two independent ports and that the probe is not rotated during a scan. Therefore, neither -scanning [2, Sec.4.3] nor test-zone field compensation [6] apply. For higher-order acoustic probes that exhibit symmetry it was shown in [7] that orthogonality can be restored by employing a complex point-source representation of the probe with point sources residing on one single circular arc. 2For N < 50 the AUT expansion coefficients can be obtained on a modern PC running Matlab in less than one minute as the brute-force least-squares solution to Jensen’s transmission formula (see (38) below). For larger values of N this simple approach is too slow and requires a prohibitively large amount of storage. Hence, the need for a faster method.

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In this paper we present a solution to the higher-order probe problem that is based on a renormalized least-squares approach and comes in two alternative versions: (i) an iterative method that employs the conjugate gradient method, and (ii) a direct method where the normal matrix is inverted explicitly.3 Both operations with very little overhead. versions require Indeed, we demonstrate with experimental data that the method can determine 29280 AUT expansion coefficients in 8 seconds on a PC running Matlab. Moreover, since the least-squares normal matrix employed here closely resembles the identity matrix, it is easy to impose tight control on the errors of the computed AUT expansion coefficients. To obtain a higher-order correction scheme that is useful for practical measurements, we assume the following. The higherorder probe is non-symmetric, has two independent ports, and is not rotated during a scan. The measurement sphere is cov, ) or a -scan ered by either a -scan ( ( , ), and the scan points are obtained by sampling. However, the method can be comrectangular bined with the excellent work of Wittmann et al. [11] to deal with non-ideal probe positions. To ensure a well-posed system of equations for determining the AUT expansion coefficients, the higher-order probe must produce a least-squares normal matrix that is non-singular. We show that this latter condition is also the condition that first-order probes must satisfy for first-order theory to produce reliable results. The paper is organized as follows. In Section II we introduce the transverse spherical vector-wave functions that will form the basis for this paper. Section III describes a matrix formulation for first-order probes that will guide the development of the higher-order computation scheme in Section IV. In Section V we compute a first-order correction factor that greatly accelerates the higher-order computation scheme. Section VI describes how the Fourier coefficients that determine the higher-order probe output can be computed from either a complex dipole model or from Jensen’s transmission formula. The higher-order computation scheme is validated with experimental data in Section VII, and Section VIII presents conclusions. Appendix A proves that the standard first-order probe-corrected AUT expansion coefficients are the solution to a least-squares problem. Appendix B explains how the transverse electric field can be computed from a Fourier series. Finally, Appendix C presents the formulas for first-order probe correction based on a complex dipole model of the probe. Throughout, we assume time-harmonic fields that have time dependence with . II. VECTOR-WAVE EXPANSION OF THE ELECTRIC FIELD OF THE AUT In this section we present the spherical-wave expansions of the AUT electric field that form the basis for the paper. We with unit vecuse the standard spherical coordinates , tors given by , and

N

3For large antennas is on the order of a few hundred, so the number of iterations could be on the same order as . Hence, eliminating (or significantly reducing) the number of iterations can reduce the computational complexity by almost one order. Also, the word “preconditioning” can be used to describe the method instead of “renormalization.”

N

Fig. 1. A probe with its reference point on the scan sphere of radius normal pointing towards the origin measures the field of an AUT.

R and it

. Moreover, , , and are the unit vectors for the rectangular coordinates . The geometry is shown in Fig. 1 in a global coordinate system. The scan sphere has radius , and is the AUT minimum-sphere radius defined such that the maximum (supremum) value of the coordinate for all points on the AUT equals . of the AUT in The transverse electric field Fig. 1 can be expressed in terms of vector spherical harmonics outside the minimum sphere as ( is the wavenumber)

(1) where the AUT, and

and

are the spherical expansion coefficients of is the spherical Hankel function [12, p. 740]

(2) The truncation number the minimum sphere as

is determined from the radius of (3)

where the constant determines the number of digits of accuracy achieved [13, Sec. 3.4.2]. The spherical vector-wave functions for the transverse fields [12, pp. and 742–746] satisfy with (4) is a normalized Legendre function dewhere fined such that the spherical harmonics are ; see Jackson [12, p. 99]. The normalized Legendre function has a Fourier expansion of the form (5)

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where the expansion coefficients can be determined from recursion relations [2, pp. 319–320]. The transverse vector-wave functions satisfy the orthogonality relations [12, pp. 742–746]

the two-dimensional arrays and are collapsed into one dimensional arrays and combined in a column vector. Similarly, we let the sampled vector probe output be given by the column vector of length where and are integers. Hence, the sphere is discretized through and . In a -scan with , and with . with , and In a -scan with . The sampling requirements and be less than or equal to are satthat isfied if , , , and . that contains the We can now introduce a probe matrix and products , so that the probe output is

(6) (7) where , indicates complex for and for . conjugation, and A far-field formula is readily obtained by inserting the large-arand gument approximations into (1). The orthogonality relations (6), (7) can be used to get expressions for the AUT expansion coefficients in terms of the transverse electric field on the scan sphere. III. MATRIX FORMULATION FOR FIRST-ORDER PROBES First-order probe-correction theory provides important clues that will help construct an efficient computation scheme for higher-order probes. When multiple interactions between probe and AUT are negligible, the vector output of a linearly-polar, ized first-order probe with being the output of port , can be expressed in terms of the transverse vector-wave functions and AUT expansion coefficients as [3], [4]4

(8) where and depend on the probe and scan radius. and in terms Appendix C presents expressions for of the parameters of an electric dipole model. For a Hertzian is proportional to the transverse electric dipole probe and , field, and (1) gives where is a constant. Here, accounts for positioning errors, reflections from chamber walls, inaccuracies in the probe model, etc. Application of the orthogonality relations (6), (7) immediately produces the standard probe-corrected expressions for the AUT expansion coefficients (9) In Appendix A it is shown that these expansion coefficients are the solution to a least-squares problem. We shall next present a matrix-formulation for the discrete version of this least-squares problem. and Let the unknown AUT expansion coefficients be given by the column vector with length . Hence, 4In [2, Eq.(3.38)] the expression (8) emerges from Jensen’s transmission forcan be expressed in terms of mula by noting that the d-functions for  and ; see Section VI-B. In [3] the formula (8) is obtained by noting that the linear-differential operator acts only on the radial variable for first-order probes. In [4] the formula (8) follows from the fact that first-order probes can be modeled by dipoles displaced only in the radial direction.

M

N

= 61

(10) Equation (10) is a non-square matrix equation with more equa. Typitions than unknowns for determining for a given cally (10) has no exact solution because of the error term in (8). It follows from Appendix A that the least-squares solution given by (9) can be obtained by solving the following normal equation (11) is the transpose complex conjugate of . Moreover, where is a diagonal matrix that contains the surface area elements . We refer on the unit sphere given by to as the normal matrix for the least-squares problem. Equation (11) was used by Wittmann et al. [11] to deal with non-ideal probe positions. As will now be demonstrated, we can renormalize to get a normal equation in which the normal matrix almost equals the identity matrix. To see this, we express the probe matrix as so that and (12) where is a matrix that contains the transverse vector-wave and , and is a diagonal matrix that confunctions tains the probe constants and . We refer to as the is a column vector first-order correction factor and note that and . The normal equation now that contains becomes (13) contains discretzed versions of where the normal matrix closely the orthogonality integrals (6) and (7). Hence, and are less than resembles the identity matrix when or equal to . Moreover, converges to the . Similarly, contains identity matrix in the limit discretized versions of the integrals in (9). We shall use a similar renormalization for higher-order probes to obtain a normal matrix that almost equals the identity

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matrix, when most of the energy in the probe pattern is confined to the first-order modes. is almost the identity matrix, we see that the Since in (11) is non-singular if least-squares normal matrix and only if the diagonal matrix is non-singular. Hence, for the normal equation (11) to have a unique solution we must require and all are nonzero. This is that the correction factors precisely the requirement stated in [14] that all first-order probes must meet to be suitable for spherical near-field scanning. We shall employ the same requirement, stated in terms of the normal matrix, for higher-order probes in Section IV. The orthogonality relations (6) and (7) hold only approximately when the integrals are replaced by summations over dis. Therefore, in stancrete points dard first-order theory the expansion coefficients are not computed by applying the discretization to the integrals (9). Instead, a Fourier series expansion for the probe output is used in conjunction with the fact that the first-order probe satisfies the symmetry relations ; see [2], [5], [15], and Footnote 5 below. However, the least-squares solution obtained from the matrix equation (13) is identical to the solution obtained from standard first-order algorithms, since the two solutions solve the same least-squares problem. IV. MATRIX FORMULATION FOR HIGHER-ORDER PROBES As explained in the Introduction, the vector output of a higher-order probe cannot be expressed in terms of the transverse vector-wave functions. Instead we employ a Fourier , , 2, of the expansion for the two outputs higher-order probe. Wacker [5] appears to be the first to have used a Fourier series in spherical scanning. Recently, Pogorzelski [16] used such a Fourier series for the formulation of a calibration technique. The Fourier expansion (5) for the Associated Legendre function shows that the and components of and can be expanded in terms of with and ; see Appendix B for details. Therefore, if the incident field consists of just one single spherical wave , the higher-order probe output can be expanded in terms of with and . We introduce the probe Fourier coefficients as the Fourier coefficients for the probe output when the incident field is a single spherical wave . Similarly, are the probe Fourier coefficients of the probe output when the incident field is . Section VI explains how the probe Fourier coefficients can be computed. The Fourier expansion of the higher-order probe output due to the AUT field therefore is

and when either or Note that . Also, and when , and (14) defines a function that is -periodic with respect to both and . The formula (14) defines a probe matrix of the form (15) performs the summation where performs a 2D FFT and over in the square brackets of (14). Hence, the element of with index is computed as (16)

We note that are the Fourier coefficients in the Fourier expansion (14) of the higher-order probe output. For given we can use the 2D FFT to rapidly compute the probe output (14) over the entire sphere. However, (14) cannot be used with from the probe output, since Fourier theory to determine , or the probe output is known only for , .5 for Instead we determine a least-squares solution by solving the normal equation (17) is the column vector that contains the output where is a matrix with eleof the higher-order probe. Since ments , the matrix has elements . Hence, we can use an inverse 2D FFT algorithm to compute matrix vector products involving , even though the data is known only for , or for , . When the matrix is multiplied by a vector with elements , the result is a vector containing

(18) As discussed in Section III, we consider only higher-order probes for which the normal matrix in (17) is non-singular. To obtain an efficient method this matrix should not be computed explicitly; see the discussion after (22). The higher-order probes that are used in spherical scanning have most of their energy located in the first-order modes. This fact can be used to determine a first-order correction factor (see Section III) that converts (17) into an equation where the normal matrix closely resembles . We describe how to compute in Section V below. 5For probes that satisfy the symmetry relation V (; ) = 0V (2 0 ;  + ), one can straightforwardly extend the probe output into the doublesphere domain 0    2 , 0    2 . The Fourier coefficients 0

(14)

can then be computed directly with the inverse 2D FFT. Standard first-order algorithms use this trick. Of course, this approach does not work for the nonsymmetric probes considered in this paper. (This extension can be performed also for non-symmetric first-order probes; see [2, p.112].)

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Writing and multiplying from the left with converts (17) to

When the two ports of the higher-order probe are different (the pattern of port 2 is not just a rotated version of the pattern of port 1), must be computed from an average of the patterns of the two ports, or simply from the pattern of one of the two ports. A simpler first-order correction factor, which does not require the use of any first-order theory, is obtained from the Hertziandipole approximation of the higher-order probe. This correction and , factor is found from (1) to contain where is a constant. A closed-form estimate for in terms of the probe Fourier coefficients will now be obtained. First note that

(19) where the square matrix

is (20)

Section III confirms that for first-order probes, so we can expect that for higher-order probes with most of the energy in the first-order modes (this is confirmed in Section VII). Note that the introduction of has not changed the least-squares solution . In other words, (17) and (19) have the same unique solution. using a linear Equation (19) can be solved with respect to iterative solver such as the conjugate gradient method, regardresembles the identity matrix. However, less of how closely when we can obtain a closed-form solution by using the Neumann series (21) and where the truncation limit depends on the norm of the desired precision. The AUT expansion coefficients can thus be expressed explicitly in terms of the output of the higher-order probe as (22) The computation scheme presented here has computational . This follows from the facts that (i) it takes complexity operations to perform multiplications with the diagonal , (ii) it takes operations (using matrices and 2D FFT algorithms) to perform matrix multiplications with and , and (iii) it takes operations to perform multiplications with the matrices and using the summations in is never computed explicitly. The (16) and (18). The matrix scheme is iterative when the conjugate gradient method is used to solve (19), and direct when the Neumann expression (22) is used. The computation scheme has been designed to be especially efficient when most of the probe-pattern energy is in the firstorder modes. However, the scheme produces the exact leastsquares solution even when most of the energy is in higher-order is non-singular as assumed. modes, provided the matrix V. FIRST-ORDER CORRECTION FACTOR The best first-order correction factor is obtained by comand as if first-order probe correction were to be puting used in measurements taken with the higher-order probe. (Reis a column vector that call that is a diagonal matrix and and .) Hence, can be obtained from contains any of the three first-order probe-correction theories [2]–[4]. For and are given explicitly in Appendix C in terms example, of the parameters of an electric dipole model.

(23) In the probe coordinate system we approximate port by an directed dipole. When the probe is at on the scan sphere, the direction of the probe coordinate system direction of the global coordinate system. corresponds to the where Hence (23) gives denotes the higher-order probe output due to a source and . Using (14) gives with (24) A similar relation can be derived for in terms of the probe Fourier coefficients for port 2. When the two ports are different, different values for would be obtained. One could use averages or simply, as we will do in this paper, use the value in (24). We emphasize that the choice of first-order correction factor does not change the outcome of the computation of the AUT and of expansion coefficients. It merely affects the matrix course the left side of (19). A good correction factor results in being close to the identity matrix, so that the solution can be obtained from just a few iterations with the conjugate gradient method of from just a few terms of the Neumann expression (22). Regardless of the correction factor, the conjugate gradient method will always converge to produce the exact least-squares solution when the normal equation is non-singular. VI. PROBE FOURIER COEFFICIENTS In this section we explain how the probe Fourier coefficients and occurring in (14) can be computed from the probe far-field pattern. Let denote the higher-order probe output due to a source with and . Similarly, is the higher-order probe and . output due to a source with The Fourier expansion (14) implies that and can be computed at through (25) and (26)

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A. Probe Fourier Coefficients Obtained From Complex Dipole Model

Fig. 2. A probe with its reference point at the origin of the probe coordinate system and its axis aligned with the positive z -axis.

We extend the variable to the entire range by noting that the spherical unit vector satisfies . Hence, the points on the half circle determined by the , intersection of the scan sphere and the half plane are given by , , and the points on the half circle determined by the intersection of the scan sphere and the half , are given by , . plane The outputs and are computed as foland lows. The probe is first placed at the north pole , towards the south pole then scanned along . The scan is continued past the south pole (without rotating the probe) into the region , until the north pole is reached . The probe outputs and are thus spatially bandlimited periodic functions on the interval . The standard sampling theorem for bandlimited periodic functions gives the following exact expressions for the probe Fourier coefficients

(27) and

(28) where . We shall next determine and in terms of the probe far-field pattern (typically obtained from measurements) given in the probe coordinate system where the reference point of the probe is the origin (see Fig. 2). The exact probe , where or far-field pattern is denoted by corresponds to the two ports of the probe. We assume no relation between the two ports, so we need to determine indepenand . As usual, the electric dent probe models for through far field is determined from the far-field pattern . Section VI-A shows how a complex dipole model can be used to obtain and in terms of the probe pattern . This method has been implemented and will be used in the experimental validation of Section VII. In Section VI-B we present an alternative approach that employs Jensen’s transmission formula.6

A

In this section we employ a complex dipole representation for the higher-order probe that is similar to the representation used in [4] for first-order probes; the reader is referred to [4] for a detailed introduction to the subject of complex dipoles. directed electric dipole located First consider a . The pattern of this dipole is on the -axis at

(29) is the identity dyad and and are the permittivity and has permeability, respectively, for free space. The constant dimension ampere/meter and depends on the coaxial waveguide and have assumed to be attached to the probe. Moreover, dimension meter. The reference point of the dipole is chosen to be the origin, even though the physical location of the dipole is . at The output corresponding to the pattern (29) is given in terms [4] of the incident electric field at the complex point (30) The constants and were introduced in (29) to make the expression (30) for the probe output as simple as possible. and To facilitate the computation of solely in terms of the and components of the electric field, we employ a complex dipole representation with the electric dipoles located along two circular arcs that conform to the scan sphere. Hence, the probe model depends on the radius of the scan sphere. plane with radius We first consider the circular arc in the and center point , as shown in Fig. 3. The pattern is that of the electric dipole in (29) after its reference point has been moved through the angle along this arc. The new location of the reference point is (31) The dipole location and orientation change during the movement along the circular arc: An original source point is moved to

(32) and an original dipole orientation

becomes (33)

The orientation of a directed dipole remains unchanged. The complex displacement creates a perpendicular source disk on which the currents of the complex dipole reside in real space; see Fig. 3. The explicit expression for the pattern is

B

6The probe Fourier coefficients and can also be computed from the plane-wave receiving characteristic of the probe [17, Eq.(6.41)] by first employing [17, Eq.(3.180)] to get the plane-wave spectrum for each of the transverse vector-wave functions.

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HANSEN: SPHERICAL NEAR-FIELD SCANNING WITH HIGHER-ORDER PROBES

Fig. 3. The probe is pointing in the positive z -direction with its reference point at the origin. The black dot represents one of the complex-point dipoles placed near a circle in the x z plane with radius R and center point (x; y; z ) = (0; 0; R).

0

The probe output corresponding to (34) is given in terms of the as electric field at the point

(35) plane with We next consider the circular arc in the radius and center point . This configuration is shown by Fig. 3 with the axis replaced by a axis. The pattern is that of the electric dipole in (29) after its reference point has been moved through the angle along this arc. In this case the orientation of an directed dipole remains unchanged, and the formulas for , , are obtained from (31)–(33) by replacing with . and Moreover, are obtained from (34), (35) by replacing with and with . The actual higher-order probe used in spherical scanning is modeled by a collection of dipoles along the two arcs as

(36)

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, , , 2) Determine the remaining parameters by solving the linear system of equations and using the standard closed-form least-squares formula. The equality is enforced only for , where is determined by the size of the region occupied by the AUT.7 The higher-order probe outputs and required in (27) and (28) can now be computed as follows. Place the probe model on the scan sphere and illuminate the probe with one vector-wave function at a time. Compute the probe and output at the required points on the circle by adding the values of the electric field of the vector-wave function at the complex-dipole locations in accordance with (37). The electric field of each vector-wave function can be computed efficiently from the Fourier expansions in Appendix B. In this section the probe model consisted solely of electric dipoles placed on only two circular arcs. This simple model led to extremely high accuracy for the SATf3000 probe used for experimental validation in Section VII: the model error was 80 dB below the peak value of the pattern throughout the angular re. Of course, it is possible to use more than gion two arcs and to include magnetic dipoles, but this appears to be unnecessary. It follows from Huygen’s principle that a model consisting of electric and magnetic dipoles placed on circular arcs can always be constructed to produce an exact representation of the probe. A major advantage of dipole models is that the probe output is obtained simply by adding field values sampled at a few points in complex space. There is no need for either a plane wave, cylindrical wave, or spherical wave expansion of the incident field. We have found that the use of complex dipoles rather than real dipoles significantly reduces the number of dipoles required. Moreover, it has recently been discovered that complex point sources (the scalar analogs to complex dipoles) automatically lead to fast computation methods due to their directivity [18]. In future work we shall explore the use of the electromagnetic analogs of the scalar results in [18] for spherical scanning. B. Probe Fourier Coefficients Obtained From Jensen’s Transmission Formula Jensen’s transmission formula states that the higher-order probe output is [19], [2, Eq. (4.40)]

with the corresponding probe output given by

(37) , , , , We have one set of model parameters , , , , , and for each port of the probe. The parameters are determined by the following two-step procedure. 1) Determine the range for each of the parameters , , , and from the physical size of the probe. Choose the number of dipoles and , and distribute , , , and evenly within the ranges.

(38) where a rotation angle has been omitted since we consider two independent ports and therefore do not need to rotate the probe around its axis. and are the probe response constants, which can be expressed in terms of the spherical translation coefficients and the spherical expansion coeffiare the spherical rotation cocients for the probe pattern. 7The complex dipole models in [4] were obtained with a nonlinear optimizer to minimize the difference between the exact and model patterns. That approach is overly complicated. The nonlinear optimizer can be replaced by the closedform least squares formulas used here.

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efficients (also called the -functions), which can be expressed in terms of the Jacobi polynomial. unless For first-order probes, , and since can be expressed in terms of the transverse vector-wave functions, the first-order probe output can be written in the form (8) if the two ports are identical; see [2, Eq.(3.38)] for details. For higher-order probes the transmission formula (38) immeand , which diately gives expressions for can be used in (27) and (28) if the -functions and the probe response constants have been programmed. We can go one step further and use the fact that the -functions have the Fourier expansion [2, Eq. (A2.12)] (39) where the “deltas” can be obtained from recursion relations [2, Eq.(A.2.4)]. Combining (38) and (39) with (14) shows that the probe Fourier coefficients can be expressed in terms of the deltas and the probe response constants as (40) and (41)

Fig. 4. TOP: The  component of the far-field pattern of the log-periodic AUT in four different ways: (1) reference pattern obtained computed for  from near-field data collected with a first-order probe and processed with a first-order correction scheme, (2) pattern obtained by assuming that the higherorder probe is an ideal electric dipole, (3) pattern obtained by applying the first-order correction scheme of Appendix C to the output of the higher-order probe, and (4) pattern obtained by applying the higher-order correction scheme of Section IV to the output of the higher-order probe. BOTTOM: Error curves based on the reference pattern.

=0

VII. EXPERIMENTAL VALIDATION We shall now employ the higher-order probe-correction scheme in Section IV and the complex dipole model in Section VI-A to process near-field data collected with the non-symmetric higher-order SATf3000 probe. The AUT is an offset log-periodic antenna that is placed near the edge of the quiet zone to increase the need for probe correction. To validate the higher-order probe-correction scheme, we use a reference solution computed from near-field data collected with a first-order probe, also with the AUT offset. The complex dipole model for the SATf3000 probe was discussed at the end of Section VI-A. The spherical expansion was , and we used 151 and 300 sample points truncated at in and , respectively. Hence, we have 29280 unknown AUT expansion coefficients and 90600 sampled probe output values. The top part of Fig. 4 shows the component of the far-field in four difpattern of the log-periodic AUT computed for ferent ways as indicated in the figure caption. The corresponding error curves (based on the assumption that the reference pattern is exact) are shown in the bottom part of Fig. 4. The corresponding plots of the component of the AUT far-field pattern are shown in Fig. 5. Both plots are normalized by the same constant, so that the maximum value of the component is 0 dB. These plots demonstrate that the first-order probe-correction scheme is incapable of producing an accurate AUT far-field pattern from the near-field data collected with the SATf3000 probe. In particular, the error of the first-order probe-corrected AUT pattern is greater than 10 dB in parts of the -component plot of Fig. 5. Figs. 4 and 5 validate the higher-order probe-correction scheme.

Fig. 5. This figure shows the  components that correspond to each of the  components in Fig. 4.

The following table shows the Matlab runtime for various computations. For example, it takes 8 seconds to compute the AUT expansion coefficients using the Neumann series solution and a first-order correction factor. With the (22) with same correction factor it takes the conjugate gradient method 16 seconds using 4 iterations. Of course, the four alternative higherorder probe-correction computations in the table produce the

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and from (9). Equation (44) where we introduced immediately gives us the desired result that the error integral and (42) attains its minimum value when . APPENDIX B FOURIER EXPANSION OF THE ELECTRIC FIELD

TABLE I MATLAB RUNTIME ON 3 GHz PC

same . The runtime can likely be reduced by a factor of 10 by using Fortran or C instead of Matlab. is not computed explicitly, we used a Since the matrix is to the set of random unit vectors to investigate how close identity matrix. With obtained from the first-order correction factor, we found from the random samples that the magnitude of the largest off-diagonal element is below 0.12. The diagonal elements have magnitudes in the range from 1 to 1.2 and zero phases.

We explain how the transverse electric field can be computed with the FFT over an entire sphere from a given set of AUT and . The resulting formulas expansion coefficients are used both in Section VI-A to compute the higher-order probe output due to a single vector-wave function and in Section VII to compute various AUT far-field patterns. in conjunction Using the Fourier expansion (5) of with (1) proves that the following Fourier expansions exist8 (45) and (46)

VIII. CONCLUSIONS We presented a solution to the higher-order probe problem that comes in two alternative versions: (i) an iterative method that employs the conjugate gradient method, and (ii) a direct method that relies on the Neumann series. Both versions require operations with very little overhead. The key step is to solve a matrix equation involving a square matrix that closely resembles the identity matrix when most of the probe-pattern energy resides in the first-order modes. Using experimental data we validated both versions of the method and the claims about the square matrix. APPENDIX A LEAST-SQUARES PROPERTY OF STANDARD SOLUTION FOR FIRST-ORDER PROBES

Multiplying (1) by produces

, and integrating over

and

(47) and

We shall now show that the AUT expansion coefficients (9) minimize the error integral

(48)

(42) where with (49)

(43)

and

where and are arbitrary complex numbers. From the orthogonality relations (6) and (7) it follows that

(50) are constants. Inserting the Fourier series (5) for the normalized Legendre function into (49) gives (51)

(44)

8 The expression [12, Eq.(3.49), p.98] for the associated Legendre function shows that the fraction  (cos ) sin can indeed be expanded in terms of with = ... .

e

mP q 0n; ;n

= 

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To get an expression for we note that the normalized Legendre functions satisfy the recursion relation

Combining the expression (1) for the transverse electric field with the expressions (8) and (56) for the first-order output shows that the correction factors are (57) and

(52)

(58)

.

The first-order probe-corrected AUT expansion coefficients can then be determined by inserting (57) and (58) into (9).

since which holds also for Inserting (52) into (50) gives

ACKNOWLEDGMENT

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The author is grateful for discussions with S. Pivnenko, O. Breinbjerg, and T. Laitinen. In addition, O. Breinbjerg arranged a TUD contract and visit, and S. Pivnenko provided the measured data. An anonymous reviewer provided numerous valuable comments.

Notice that (45), (46), (47), and (48) hold for any complex with . REFERENCES APPENDIX C FIRST-ORDER PROBE CORRECTION In this appendix we derive expressions for the first-order corand in terms of the parameters of an elecrection factors tric-dipole model. The model parameters are determined from the method described in Section VI-A. Reference [4] presents a version of this first-order scheme that also includes magnetic dipoles. denoting the scan point on the sphere, the comWith plex source points of the electric dipoles are when is the the probe is on the scan sphere. Furthermore, output of the probe when the electric dipole directions are parallel to . Hence, (30) gives (54) where are model parameters and is the number of is the probe output model parameters. Similarly, when the electric dipole directions are parallel to , (55) We combine the two scalar probe outputs into a single vector output as

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[1] A. D. Yaghjian, “An overview of near-field antenna measurements,” IEEE Trans. Antenna Propag., vol. 34, pp. 30–45, Jan. 1986. [2] J. Hald, F. Jensen, and F. H. Larsen, , J. E. Hansen, Ed., Spherical Near-Field Antenna Measurements. London: Peter Peregrinus, 1988. [3] A. D. Yaghjian and R. C. Wittmann, “The receiving antenna as a linear differential operator: Application to spherical near-field measurements,” IEEE Trans. Antenna Propag., vol. AP-33, pp. 1175–1185, Nov. 1985. [4] T. B. Hansen, “Complex-point dipole formulation of probe-corrected cylindrical and spherical near-field Scanning of electromagnetic fields,” IEEE Trans. Antennas Propag., vol. 57, pp. 728–741, Mar. 2009. [5] P. F. Wacker, Non-Planar Near-Field Measurements: Spherical Scanning NBS Internal Rep. 75-809, 1975. [6] D. N. Black and E. B. Joy, “Test zone field compensation,” IEEE Trans. Antennas Propag., vol. 43, pp. 362–368, April 1995. [7] T. B. Hansen, “Complex point receiver formulation of spherical nearfield scanning using higher-order probes,” Wave Motion, vol. 46, pp. 498–510, Dec. 2009. [8] T. Laitinen, S. Pivnenko, J. M. Nielsen, and O. Breinbjerg, “Theory and practice of the FFT/matrix inversion technique for probe-corrected spherical near-field antenna measurements with higher-order probes,” IEEE Trans. Antennas Propag., vol. 58, pp. 2623–2631, Aug. 2010. [9] C. H. Schmidt and T. F. Eibert, “Multilevel plane wave based near-field far-field transformation for electrically large antennas in free-space or above material halfspace,” IEEE Trans. Antennas Propag., vol. 57, pp. 1382–1390, May 2009. [10] O. Breinbjerg et al., Comparative Study of Higher-Order Probe-Correction Methods In progress. [11] R. C. Wittmann, B. K. Alpert, and M. H. Francis, “Near-field spherical scanning antenna measurements with non-ideal probe locations,” IEEE Trans. Antenna Propag., vol. AP-52, pp. 2184–2186, Aug. 2004. [12] J. D. Jackson, Classical Electrodynamics, 2nd ed. New York: John Wiley and Sons, 1975. [13] , W. C. Chew, J. M. Jin, E. Michielssen, and J. Song, Eds., Fast and Efficient Algorithms in Computational Electromagnetics. Boston: Artech House, 2006. [14] P. C. Hansen and F. H. Larsen, “Suppression of reflections by directive probes in spherical near-field measurements,” IEEE Trans. Antennas Propag., vol. 32, pp. 119–125, Feb. 1984. [15] T. B. Hansen, “Formulation of spherical near-field scanning for timedomain electromagnetic fields,” IEEE Trans. Antennas Propag., vol. 45, pp. 620–630, Apr. 1997.

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[16] R. J. Pogorzelski, “Experimental demonstration of the extended probe instrument calibration (EPIC) technique,” IEEE Trans. Antennas Propag., vol. 58, pp. 2093–2097, Jun. 2010. [17] T. B. Hansen and A. D. Yaghjian, Plane-Wave Theory of Time-Domain Fields. New York: IEEE Press, 1999. [18] T. B. Hansen, “Efficient field computation using Gaussian beams for both transmission and reception,” Wave Motion, 2010, accepted for publication. [19] F. Jensen, “Electromagnetic Near-Field Far-Field Correlations,” Ph.D. dissertation, Technical University of Denmark, Lyngby, Denmark, 1970.

Thorkild B. Hansen (M’91) received the Ph.D. degree in electromagnetics from the Technical University of Denmark in 1991. From 1991 to 1997, he worked at the Air Force Research Laboratory (formerly, Rome Laboratory) of Hanscom Air Force Base, MA, on techniques for analyzing electromagnetic waves and antennas. He joined Schlumberger’s underground radar project in 1997 and transferred with the project to Witten Technologies in 2000. He is currently developing techniques for RFID and near-field communications at Seknion, Inc. Dr. Hansen received the R.W.P. King Prize in 1992 and the S.A. Schelkunoff Prize in 1995 for publications on electromagnetic wave propagation. He is coauthor of Plane-Wave Theory of Time-Domain Fields, a featured book of IEEE Press in 1999. The underground radar imaging technology he helped develop at Sclumberger and Witten Technologies won the 2002 NOVA Award for innovation in construction and Wall Street Journal’s 2004 Technology Innovation Award in Software.

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The Banyan Tree Antenna Array Steven S. Holland, Student Member, IEEE, and Marinos N. Vouvakis, Member, IEEE

Abstract—A new wideband, wide-scan array is introduced, called the Banyan Tree Antenna (BTA) array, that employs modular, low-profile, low-cost elements fed directly from standard unbalanced RF interfaces. The elements consist of vertically-integrated, flared metallic fins over a ground plane that are excited by a vertical two conductor unbalanced transmission line. The antenna resembles the bunny-ear or balanced antipodal Vivaldi antenna (BAVA) designs, but most importantly uses metallic shorting posts between the fins and the ground plane that suppress a mid-band catastrophic common-mode resonance that occurs in 2D arrays of balanced radiators fed with unbalanced feeds. This work introduces simple circuit models that describe key performance attributes of the BTA array, leading to unique physical insights and design guidelines. Simulations of infinite singleand dual-polarized BTA arrays have achieved approximately 2.2 at broadside and two octaves of bandwidth for VSWR VSWR 2.8 at scans out to 45 , while maintaining better 45 in the D-plane. than 14 dB polarization purity at Index Terms—Antenna array feeds, antenna array mutual coupling, phased arrays, ultrawideband antennas, Vivaldi antennas.

I. INTRODUCTION

W

IDEBAND phased antenna arrays will be critical components of future multi-functional communication/sensing/countermeasure systems [1], which will utilize one or two wideband phased arrays to replace multiple antennas. Additional wideband array applications include wideband radar [2], radio telescopes [3], wideband communication systems [4], power combiners [5], and wideband reflector feeds [6]. These phased arrays have stringent electrical and manufacturing specifications that include wideband and wide-scan performance, multi-beam ability, and polarization agility. Currently, arrays of tapered-slot end-fire radiators have been one of the most popular choices of wideband arrays in multifunctional systems [7]. The Vivaldi array [8], [9], has demonstrated bandwidths greater than 10:1 [10], [11], and wide-scan performance, while providing a direct connection to standard RF interfaces. When designed carefully, these arrays do not support common-modes, avoiding catastrophic resonances. Despite their desirable impedance performance, the Vivaldi array ele(where is ments are long—namely, a length of the wavelength at the highest frequency) is needed for a 10:1 bandwidth and yields only a 2:1 bandwidth. This high

Manuscript received April 24, 2010; revised October 20, 2010; accepted April 14, 2011. Date of publication August 12, 2011; date of current version November 02, 2011. This work was supported by the Naval Research Laboratory through Grant PG#11320000000008. The authors are with the Department of Electrical and Computer Engineering, University of Massachusetts Amherst, Amherst, MA 01003 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2164177

profile leads to high cross-polarization levels when scanned in the D-plane. In addition, their elements are vertically-integrated and require electrical connection between them, making modular designs difficult. Nevertheless, some variations can be manufactured modularly, such as the body-of-revolution (BOR) [12] and Mecha-Notch [13] Vivaldi arrays, albeit with complicated fabrication. The outstanding impedance performance of Vivaldi arrays has inspired the development of alternative low-profile tapered-slot configurations such as the Antipodal Vivaldi Antenna (AVA) [14], and the Balanced Antipodal Vivaldi Antenna (BAVA) [15], [16]. These elements are modular, thus easier deep, leading to better to manufacture, and are only D-plane cross-pol levels. However, two-dimensional (2D) arrays of these elemants can support a broadside common-mode that splits the operating band into two disconnected bands with approximately an octave of bandwidth. Balanced excitation through external baluns was identified as the key to eliminating the broadside common-mode resonance in 2D arrays. Differential feeding forces the push-pull mode currents to exist on the radiating fins, removing any common-mode components of the current altogether. One example of such an array is the bunny ear element [17], which is modular, low-profile, and has achieved bandwidths up to 5:1 [18]. The Doubly-Mirrored BAVA (DmBAVA) [19] incorporates mirroring that rotates elements 180 in the E- and H-planes of the array and uses a wideband hybrid to properly phase adjacent elements, achieving up to 5:1 bandwidths. Although balanced excitation eliminates the broadside common-mode, these elements support common-modes when scanned in the E-plane [20] and [21]. Several methods of suppressing these scan-induced common-modes have been proposed in the literature. One method places metallic walls along the H-plane, effectively shielding adjacent elements, [22], [23], at the cost of increased fabrication complexity. Another approach is to place chip resistors between the fins and the ground plane near the end of the fins [17], or at the base of the element feed, as in [21], to dissipate the common-mode currents. This resistive loading lowers efficiency and power handling, and increases the noise temperature and fabrication/maintenance difficulty. From the above discussion it is apparent that none of the current ultrawideband tapered-slot antenna array technologies are simultaneously modular and low-profile with low cross-pol while maintaining a direct connecttion to standard unbalanced RF interfaces. In an attempt to develop such a design, this paper introduces the Banyan Tree Antenna (BTA) array, which modularly achieves wideband/wide-scan performance without the need of external baluns, resistive loading, or metal H-plane walls. This is achieved with the addition of a novel shorting post arrangement at the tapered fins that is used to control the broadside common-mode resonance frequency. One can

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HOLLAND AND VOUVAKIS: THE BANYAN TREE ANTENNA ARRAY

view this modification as an integrated balun, thus alleviating the need for external baluns/hyrbids, allowing a direct connection to standard RF interfaces. Additionally, co-design of the integrated balun and antenna leads to better combined performance than that of an independently designed external balun and antenna. Thanks to the simple and insightful theory developed herein, BTA arrays can be designed to operate free of both the broadside common-mode and the scan-induced common-modes over 4:1 bandwidths in both singleand dual-polarized configurations with good matching out to in all planes. Due to the low profile of the elements, the cross-pol levels are below 20 dB in the principle planes and better than 14 dB in the diagonal plane for dual-polarized designs. The remainder of this paper is organized as follows. Section II discusses the broadside common-mode of 2D arrays of balanced elements fed with unbalanced feeds, and presents a simple model to predict its resonant frequency. Section III presents the topology of the Banyan Tree Antenna. Section IV develops the theory of operation of the Banyan Tree Antenna, using simple circuit models to provide physical insights and design guidelines. Section V presents results for a single- and a dual-polarized BTA. The paper concludes in Section VI.

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Fig. 1. Uneven differential-mode (push-pull) current densities due to the unbalanced excitation of balanced tapered-slot elements.

II. COMMON-MODE RESONANCE This section addresses the common-mode problem encountered in balanced arrays fed directly by unbalanced feeds. 1) Single-Polarized Arrays: Consider a single-polarized, doubly-periodic infinite array of balanced fins over a ground plane fed by unbalanced feed lines, as shown in Fig. 1, that is excited at broadside. This unbalanced feeding will induce differential, but unequal, currents on the two vertical fins, as shown by the arrows in Fig. 1. As a result, the currents on the vertical feed lines no longer cancel and a net flow of upward (in this case) current results on one conductor. This unequal current distribution between neighboring elements can excite a -directed electric field much like the probe feeding arrangement of a cavity or a waveguide, shown in Fig. 9(a). When the distance between two vertical grounded conductors of neighboring elements in the array is equal to one half-wavelength, a field mode with dominant -directed electric fields can be excited. This mode’s fields do not radiate (reactive fields) and in turn induce -directed (common-mode) electric currents on the metallic fins. Because the array is doubly periodic, a half-wavelength length between neighboring elements can be along the E-, H- or D-planes, as shown in Fig. 2. Any other resonant length between grounded conductors of non-neighboring elements is precluded due to the periodic nature of the array. Along the H-plane , no resonant mode can be excited because in this direction there is no probe-like feed conductor between grounded conductors. Along the E-plane , a resonant mode could be excited, but the un-equal currents at this frequency contribute to dominant radiated fields similar to those encountered in most dipole arrays. Both resonant lengths above correspond to frequencies close to the grating lobe frequency, and for scanning arrays they occur out-of-band. The resonant mode along the D-plane , is responsible for the dominant, problematic common mode, because the

Fig. 2. Top view sketch of the array of Fig. 1. This model will be the basis for theoretically predicting the frequency of the common mode.

probe current has a component along this direction, as shown in Fig. 3, and the components of the horizontal radiated fields along this direction are small. Using as the resonant length, where and are the E- and H-plane spacings, respectively, the following common-mode resonance frequency is obtained: cm where is the speed of light in a vacuum, and fective relative permittivity given by

(1) is the ef-

(2) is the thickness of the dielectric used to support the where metallic fins (Fig. 2). The resulting common-mode currents on the fin conductors and the field distribution near the ground plane are plotted at cm in Fig. 4. To validate (1), the common mode frequency of a singlepolarized array of elements shown in Fig. 8 (without shorting posts) with the parameters of Table I is considered. The array is analyzed using Ansoft/Ansys HFSS [24] and is compared to the theoretically calculated cm from (1), shown in Table II. The results show very good agreement, having less than 2% error for

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TABLE I PARAMETERS USED FOR THE COMMON-MODE STUDIES

TABLE II THEORETICAL VERSUS NUMERICAL cm

IN

SINGLE-POL ARRAYS

Fig. 3. Sketch of the resonant electric fields developed in the array of Fig. 1 at the common mode frequency (only the fields along the resonant length are shown).

Fig. 4. Fields and currents in the unit cell of single-polarized, unbalanced fed fins scanned to broadside. (a) Electric currents on the element fins. (b) Overhead along the plane cut in view of the -polarized electric field magnitude, Fig. 4(a).

various element spacings and , dielectric constants , and dielectric thicknesses , validating the proposed theory. These insights can provide design guidelines for moving this resonance frequency out of band. One solution is to decrease

cm and move it out of band by increasing either the E- or H-plane spacing of the elements (or both), but since the element spacings are constrained by grating lobe onset at and , this option is deemed impractical. Another option is to move the common mode resonance up in frequency, by decreasing the element spacing. However, minimum spacing is limited by element size, and small element spacing results in a large T/R module count. Unlike the element spacing, which has an inverse relationship to cm , the dielectric constant has an inverse square root relationship to cm , resulting in only a modest shift in resonant frequency for a . It is clear that these parameters alone cannot successfully move the resonance out of the operating band. The BTA modifies the element topology to introduce additional design degrees of freedom that are used to shift the common mode out of band, as will be elaborated in Sections III and IV. This common-mode resonance is excited strongly at broadside where the fields of the array are in phase at each element. Scanning off broadside in any plane introduces a phase progression in the fields that, since that the fields are no longer in phase at neighboring elements, reduces the strength of the resonance excitation. Fig. 5 shows the impedance for scanning along the E-plane near cm of a single-polarized array with parameters in Table I. At broadside, the impedance swings through a series resonance with . As increases, the resonance is excited weakly and the loci quickly contract toward the center of the Smith chart. When the elements are embedded in a dielectric, the strength of the common mode excitation falls off slower with , and cm increases with as a function of (substituting with the effective length, , in (1), where the E-plane is assumed to be the -plane). The common-mode resonance disappears along D- or H-plane scanning, for both dielectric and dielectric-free arrays. As a result, moving cm above the operating band at broadside implies common-mode free scanning operation. It is important to note that the common-mode described in this section is unique to unbalanced fed arrays only. Balanced fed elements do not support this particular mode, but instead suffer

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Fig. 6. Top view sketch of the the dual-polarized array of Fig. 1.

TABLE III THEORETICAL VERSUS NUMERICAL cm

IN

DUAL-POL ARRAYS

Fig. 5. Impedance variation of the arrays in Figs. 1 and 3 near cm for various scanning angles in the E-plane.

from other common-modes that, while not excited at broadside, appear with E-plane scan [20], [21], [23], [25]. 2) Dual-Polarized Arrays: In dual-polarized arrays the topology is altered by the presence of the orthogonal set of fins. A top view of a dual-polarized unit cell and some important dimensions are shown in Fig. 6. Because the orthogonal fins provide vertical grounded conductors that lie exactly in the middle of , the common mode described in the previous subsection is shifted to twice the frequency, well above the operating band. In this case, the resonant length responsible for the problematic common mode is that of , for the reasons described in the previous subsection. Thus, in dual-pol arrays cm can be predicted by cm

Fig. 7. BTA array arrangements: (a) single-polarized, and (b) dual-polarized (egg-crate grid).

(3)

. Again, the effective relative permitwhere tivity, , is found using the same weighted average method, and results in (4) The common mode now occurs at the grating lobe frequency for dielectric-free arrays, and inside the operating band of arrays with dielectric. Most practical dual-polarized arrays of balanced radiators with unbalanced feeding use dielectrics to support the conductors and do have a common-mode inside the operating band. Table III shows good agreement between numerical and theoretical calculations of cm , with less than 2% error. III. BANYAN TREE ANTENNA (BTA) ARRAY The common-mode was shown to limit the performance of balanced elements fed with an unbalanced feed, and in response the Banyan Tree Antenna (BTA) array has been developed to alleviate this problem.

Single- and dual-polarized arrangements of the BTA elements on a rectangular, egg-crate lattice are shown in Fig. 7(a) and (b), respectively. Elements do not require electrical continuity and are arranged with gaps between neighbors, allowing for modular construction. The topology, along with the main geometric design parameters of the BTA element, is shown in Fig. 8. Vertical shorting posts connect the radiating fins to the ground plane, and are the key to controlling the common-mode resonance frequency, which will be discussed in detail in Section III-A. These shorting posts resemble the vertical root system of the distinctive Banyan tree, hence the origin of the array’s name. This paper focuses on printed BTAs, namely a single metal layer embedded between two equal thickness dielectric slabs, as shown in the side view in Fig. 8. Other variations on the fabrication method are possible and can be found in [26], [27]. An individual element consists of two exponentially tapered fins with inner and outer flare rates of and , respectively, oriented vertically over a ground plane. Together, the fins effectively form a tapered slot structure in the void between the fins. However, due to the extremely

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Fig. 10. Model used to theoretically predict the common mode frequency in a BTA array.

trolling the common mode resonance frequency. This will be thoroughly explained in Section IV-A. IV. THEORY OF OPERATION

Fig. 8. Element topology and main geometrical design parameters of the Banyan Tree Antenna (BTA) element.

The BTA has a complex geometry that requires rigorous fullwave numerical analysis for evaluation and optimization of the array. Nevertheless, a simple approximate theory will be developed to predict the occurrence of important performance-limiting resonances as well as to provide physical insight into controlling the BTA operation. A. Common-Mode Control With Shorting Posts

Fig. 9. Mode excitation and mode suppression principles. (a) Fundamental mode excitation with current source (see Fig. 3 for BTA analogy). (b) Control of fundamental mode through shorting via.

short length of this tapered slot (total element height typically ), the fins operate more like fat, capacitive dipoles with an exponentially flared matching section over a ground plane. At the aperture of the array, the impedance is typically 150–200 . Matching this large impedance to 50 poses a formidable challenge, especially over a wide bandwidth. The exponentially tapered fins and the short vertical feeds act as an integrated impedance matching circuit that allows the input impedance at the ground plane to be well matched to 50 . The vertical feed consists of a pair of printed strips that form an unbalanced feed line of length , where one line is connected to the ground plane, and the other is fed directly from an unbalanced T-line, i.e., a standard RF interface. Thus, the BTA connects directly to unbalanced feed network T-lines or T/R modules below the ground plane, without the need of an external balun or hybrid. The use of unbalanced feed lines is possible because of the additional printed lines, referred to as shorting posts, which directly short the fins to the ground plane. Prudent placement of the shorting posts forces electric field nulls that modify the common-mode resonant length , providing a means of con-

1) Single-Polarized Arrays: As shown in Section II, modifications of the geometry or materials in the traditional structure of Fig. 1 are unable to shift the common-mode frequency out of band. A solution emerges when one considers modifying the topology in order to introduce additional degrees-of-freedom into the length expression. This is done in the BTA by inserting shorting posts between the fins and the ground plane, strategically forcing the electric field to zero. Much like the PCB mode suppression vias used to shift troublesome cavity resonances [Fig. 9(b)] out of the desired range, the shorting posts of the BTA constrain the resonant fields to smaller resonant lengths, shifting the resonant frequency upwards. A top view of the BTA unit cell is shown in Fig. 10, showing the location of the BTA shorting posts. It is clear that the diagonal length is now shorter, and the common-mode resonant frequency is approximately cm

(5)

and are the distances between the center of the where element and the shorting post on the fed and grounded fins, respectively, as shown in Fig. 8. For comparison, an exemplary BTA design with design parameters shown in Table I and with the addition of shorting posts of width mm is considered for various shorting post locations , and with mm. The fins are printed on Rogers 5880 of thickness mm. The broadside VSWR for these cases is shown in Fig. 11. As the shorting post spacing is increased, the frequency of the large spike in the VSWR due to the common-mode resonance increases; in this case, the common-mode is shifted from 4.9 GHz to nearly 7 GHz. The numerical results are compared to the theoretical predictions from (5) in Table IV. The analytic results are within 4% of the numerical results, showing excellent agreement.

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TABLE VI COMPARISON OF LOOP-MODE RESONANCE THEORY WITH NUMERICAL SIMULATION

B. Transmission Line Model of Shorting Posts

Fig. 11. Variation of the common-mode resonance as a function of the shorting post position for a single-pol BTA array.

TABLE IV THEORETICAL VERSUS NUMERICAL PREDICTION IN SINGLE-POL BTA

OF

cm

TABLE V THEORETICAL VERSUS NUMERICAL PREDICTION IN DUAL-POL BTA ARRAYS

OF

cm

Having explored the effect of the shorting posts on the common-mode resonance frequency, their impact on the array impedance over the remainder of the band is now considered using elementary transmission line models. Modeling the impedance seen at the beginning of the fins (and absent of shorts) as , four distinct transmission lines can be identified due to the two feed lines and the two shorting posts. The first transmission line is formed by the vertical feeds, as in Fig. 12(a). The remainder of the transmission lines can be identified as shown in Fig. 12(b)–(d). A full transmission line model is shown in Fig. 12(e), where the lines are shown in parallel with , which is fed via transmission line . The shorting post transmission lines are approximately a quarter-wavelength at mid-band, appearing as an open circuit at the location of , leaving the impedance unaffected. Therefore, despite providing a direct DC path between the fins and the ground plane, the shorting posts do not short out the radiating currents at mid-band and high frequencies. At low frequencies, the shorting post transmission lines are electrically short, and they look like short circuits in parallel with ; this results in the excitation of a loop mode that is explored in the next section. C. Low Frequency Loop-Mode Resonance

2) Dual-Polarized Arrays: In dual-polarized arrays, (3) is modified to account for the shorting posts, and results in cm

(6)

where . Table V shows good agreement between numerical and theoretical calculations from (6) of cm in a dual-pol BTA array, using the element of Section IV-A1. In dual-polarized BTA arrays, the common-mode is readily moved out of the operating band, above , since the common-mode is already near the high end of the operating band. In fact, the shorting posts move cm well above , which can further increase the bandwidth for applications requiring only a limited scan volume. While the shorting posts are effective in controlling the common-mode resonance, their presence affects the operation over the rest of the band, as shown in Fig. 11. The remaining sections discuss the theory of operation over these bands and highlight design implications and compromises.

While the upper limit of the operating band is primarily dictated by the grating lobe frequency, the BTA has a low frequency limit defined by a sharp loop-mode resonance that has a large spike (ideally infinite in height) in the resistance and a large resonant swing of the reactance typical of a parallel resonance. This resonance is a consequence of the shorting posts, which at low frequencies provide a low impedance path for currents, thus allowing a current path to form a resonant loop between elements along the E-plane. The effects of this resonance can be seen in Fig. 11 as a shift of the low end band-edge. The resonant loop can be seen by examining the currents on the fins of the BTA at the loop-mode resonance frequency, as shown in Fig. 13(a). The generator excites currents on the vertical feed stem, which flow onto the fin section of the element. The current, instead of continuing to flow along the inner and outer edges of the fin, is shunted to the ground plane through the shorting post, and additional current flows from the top of the fin down into the shorting post. On the grounded fin, the currents are similar to those in regular operation, with currents on the shorting posts and feed lines flowing in the same direction. Considering image theory in Fig. 13(b), two distinct loop paths can be identified as

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Fig. 13. The loop-mode resonance in the BTA. (a) Current distribution around the loop-mode resonance (low frequency of operating band). (b) Current distribution using image theory. (c) Effective resonant current loop paths on the fins.

Fig. 12. Transmission line models of the BTA. (a) The feed lines are modeled , and the shorting posts are modeled as three T-lines with T-line impedance connected in parallel with the feed lines, as shown in (e). (b) , (c) , (d)

in Fig. 13(c), where two loops are shown to share close proximity along one arm of each loop, producing high coupling. From a loop antenna perspective, these coupled loops can be modeled as shown in Fig. 14, where a small driver loop couples strongly to a large resonant loop. The capacitive coupling between neighboring elements is represented by a lumped capacitance inside the resonant loop. The length of the resonant loop was estimated using known lengths in the element geometry. By expressing the approximate length of the loop as , the approximate loop-mode resonant frequency is given by (7) is a non-negative function of the capacitance used where to account for the effects of capacitive coupling between neighboring fins. Table VI shows theoretical and numerical results (Ansoft/Ansys HFSS [24]) predicting for the design of

Fig. 14. Loop-mode circuit model consisting of a non-resonant driving loop (left) coupling energy into a large half-wave resonant loop.

Table I, with the addition of shorting posts of width mm at locations mm, printed on Rogers 5880 of thickness mm, and with ; the results demonstrate good agreement with less than 7% error, even without using . Since , the loop-mode resonance frequency is not a strong function of the shorting post separation. However, the loop-mode resonant width (inversely proportional to the quality factor ) increases directly with and , such that a larger frequency range around is affected by the resonance. This is evident in Fig. 11, where the low frequency limit of the operating band increases as the common-mode is shifted upward in frequency. This indicates an inherent compromise between moving the common-mode out of band and maintaining operation at low frequencies. To help alleviate the low-frequency degradation, strategies for decreasing or increasing are required.

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TABLE VII SINGLE-POL BTA ARRAY DIMENSIONS

Fig. 15. Effect of shorting post asymmetry on the VSWR of the single-pol BTA.

Equation (7) indicates that there are three distinct ways of lowering the loop resonance frequency . The first is to increase the E-plane spacing to add length to the loop, but this is a poor option since it lowers the grating lobe onset frequency. The second method is to increase the vertical feed line height , which can significantly increase the length of the resonant loop. Finally, the third method is to increase the capacitive coupling between E-plane neighbors, which is primarily controlled through the separation of neighboring fins, and the area of the fins (e.g., large values of ). However, the minimum fin separation is limited by fabrication constraints. Also, high capacitive coupling results in large resonant impedance swings at the second order resonance of the loop mode, which constitutes itself as a VSWR hump around 3.2 GHz in Fig. 11. Among these methods, it was found that the width of the feed lines and and the height of the fins over the ground plane, along with asymmetric shorting post placement, provide a good design compromise. To demonstrate the effect of this asymmetric shorting post placement on the BTA, Fig. 15 shows three example cases, where the shorting posts are placed symmetrically with mm, asymmetrically with mm and mm, and asymmetrically with mm and mm. In the latter two cases, cm occurs at the same frequency, since the overall separation between the two shorting posts is equal. However, when is equal to 6 mm, the low frequency limit increases, and the match worsens over the low frequencies. When is increased to 6 mm, the match improves at the low frequencies. For asymmetric placement of the shorting posts, (5) remains valid. More importantly, the cross-polarization studies in the next section indicate that the asymmetric short locations do not lead to increased cross-pol levels. V. DESIGN EXAMPLES The theory developed in the previous section leads to unique physical insights that can be used (along with the tuning via full-wave analysis) to design a single- and dual-polarized BTA array operating over the band of 2–7.5 GHz. It is noted that no extensive bandwidth optimization was attempted and it is

believed that the bandwidth enhancement techniques described in [19] can be readily applied in the BTA. All numerical results were obtained using infinite array analysis with Ansys/Ansoft HFSS 11 [24]. All results are referenced to a 50 waveport impedance, and use real dielectric materials, but infinitely thin perfect electric conductors. A long air box terminated with a thick PML is used for the simulations. A. Single-Polarized BTA A simple rectangular grid, similar to that depicted in Fig. 7(a), is utilized for the single-polarized BTA array. The array element is depicted in Fig. 8, where the E-plane element spacing is cm, and the H-plane spacing cm. This grid leads to a grating lobe onset frequency of GHz (assuming scan at ) in the E-plane, and GHz in the H-plane. The elements are printed on Rogers 5880 dielectric , and a detailed description of the element dimensions can be found in Table VII. The loop mode resonance described in Section IV-C has a that decreases as and are increased, and thus and should be minimized when possible. In addition, the shorting posts are not symmetrically arranged on each fin of the element. Instead, the shorting post on the fed fin is located at mm from the center, while the shorting post on the grounded fin is located at mm. 1) Scan Impedance: The infinite array VSWR performance of the single-pol BTA array is shown in Fig. 16, at scans out to in the E- and H-planes (note that the vertical lines at 2 and 7.5 GHz indicate the edges of the operating band). The impedance is well behaved with scan, exhibiting the highest VSWR when scanned to , VSWR 2.2, in the E-plane, while the maximum VSWR 2.9 in the H-plane. The higher VSWR in the H-plane is typical for this type of element. Although not shown here, the D-plane VSWR results are an approximate average of the principal plane VSWRs. 2) Cross-Polarization: Infinite arrays radiate a discrete spectrum of plane waves (Floquet modes). The main lobe of radiation can be decomposed into two orthogonally polarized plane waves propagating in the scan direction. In order to obtain the radiated power carried by these polarizations, a surface parallel to the ground plane is defined some distance from the array. Power flowing through this surface can be calculated by integrating the Poynting vector over the surface, using field components defined by Ludwig’s third definition of co- and cross-polarization [28], providing the radiated power per unit cell area. The co- and cross-polarized radiated powers per unit cell are shown in Fig. 17 versus frequency for scanning in the E- and

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Fig. 16. VSWR versus frequency and scan angle of the infinite single-polarized BTA array. (a) E-plane; (b) H-plane. The D-plane impedance (not shown here) is approximately the average of the two.

D-planes, plotted in dB and normalized to the incident power at the input port; therefore, these levels include mismatch losses. For all scan angles, the co-polarized power is nearly 0.1 dB down from the input power over the full frequency band, indicating high efficiency. The co-polarization level is shown to slightly decrease near 3 GHz, where the VSWR approaches 2 (an impedance match of VSWR 2 has an insertion loss of 0.5 dB due to mismatch). When scanning along the E-plane, the cross-polarization is well below 50 dB, since this plane preserves symmetry and the currents on the vertical feed lines radiate the same polarization as the currents along the fins. Scanning along the D-plane shows higher cross-polarization levels since the vertical components of the current on the feed lines and fins radiate power that is orthogonally polarized to the main beam, substantially adding to the cross-polarization level. The H-plane, although not shown for the sake of brevity, is below 20 dB for all angles. B. Dual-Polarized BTA Next, an egg-crate, rectangular grid dual-polarized BTA array is analyzed, as shown in Fig. 7(b). The design is based upon the

Fig. 17. Co- and cross-polarization radiated power versus frequency and scan angle of the infinite single-polarized BTA array. (a) E-plane; and (b) D-plane. The H-plane polarization levels (not shown here) are approximately the same as the E-plane.

single-polarized element in Section V-A with some modifications due to both mechanical and electrical considerations introduced by the dual-polarized arrangement. The element spacing is equal in both planes, chosen as cm for a grating lobe frequency GHz. The elements are printed on Rogers 5880 dielectric , and a detailed description of all design parameters is shown in Table VIII. The shorting posts are now symmetrically placed on each of the element fins, with mm. This is possible since the dual-polarized arrays have an cm that occurs at a higher frequency than in the single-polarized arrays. 1) Scan Impedance: Fig. 18 shows the infinite array VSWR performance, with the -polarized elements excited and the -polarized elements terminated in 50 loads. The coupling between the two polarizations was found to be very low, better than 25 dB in the principle planes and 15 dB in the D-plane, but is not included due to space limitations. The D-plane VSWR is approximately an average of the principle plane VSWRs, thus are omitted for brevity. The E-plane VSWRs have a VSWR 2.0 over all scan angles, for the full band of 2–7.5 GHz (3.75:1

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TABLE VIII DUAL-POL BTA ARRAY DIMENSIONS

Fig. 19. Co- and cross-polarization radiated power versus frequency and scan angle of the infinite dual-polarized BTA array. (a) E-plane; (b) D-plane. The H-plane polarization levels (not shown here) are approximately the the same as the E-plane.

Fig. 18. VSWR versus frequency and scan angle of the infinite dual-polarized BTA array. (a) E-plane; (b) H-plane. The D-plane impedance (not shown here) is approximately the average of the two.

bandwidth). Along the H-plane, the VSWR again shows the characteristic increase with , reaching a maximum VSWR 2.8 at 3 GHz. 2) Cross-Polarization: The co- and cross-polarization levels for the dual-polarized BTA array are plotted in Fig. 19 for scanning along the E- and D-planes. The -polarized elements are excited and the -polarized elements are terminated in 50 loads. As in the single-polarized BTA, the efficiency of the dualpolarized array is very high, decreasing slightly near 3 GHz due to the mismatch loss of 0.5 dB at VSWR 2. This high efficiency also indicates very low coupling of power between the - and -polarized elements.

The E-plane scan has the best cross-polarization performance, with levels between 27 dB and 40 dB. The H-plane (not shown here) had similar cross-polarization behavior with levels below 25 dB. The D-plane exhibits higher cross-polarization levels than the principle planes, as observed in the single-pol BTA of Section V-A2, but this dual-pol array has a notably better performance, reaching a maximum level of 14 dB for . These results are consistent with observations reported in [29], which found single-polarized arrays to have a much higher change in their polarization state when they are scanned away from broadside, as compared to dual-polarized arrays. VI. CONCLUSION The BTA array achieves wideband operation over a wide scan volume, while maintaining good polarization purity. The array uses modular, low-profile, vertically integrated PCB elements that are fed directly by standard unbalanced RF interfaces. The conception of the array was based on insights gained from a simple resonant model, which was developed to predict

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the common-mode resonance encountered in balanced fin arrays fed by unbalanced feeds. The BTA uses shorting posts that connect each element fin to the ground, effectively tuning the common mode out-of-band. The addition of the shorting posts has other performance implications, which were also described using simple models that provide insight into the low frequency operation and offer design guidelines. Asymmetric placement of the shorting posts was found to result in the best performance and led to single- and dual-polarized infinite BTA arrays with 3.75:1 bandwidths out to . The maximum cross-polarization level at in the D-plane was 10 dB and 14 dB for the single- and dual-polarized designs, respectively. REFERENCES [1] G. C. Tavik, C. L. Hilterbrick, J. B. Evins, J. J. Alter, J. G. Crnkovich, J. W. de Graaf, W. Habicht, G. P. Hrin, S. A. Lessin, D. C. Wu, and S. M. Hagewood, “The advanced multifunction RF concept,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 3, pp. 1009–1020, Mar. 2005. [2] K. Trott, B. Cummings, R. Cavener, M. Deluca, J. Biondi, and T. Sikina, “Wideband phased array radiator,” in Proc. IEEE Int. Symp. Phased Array Systems and Technology, 2003, pp. 383–386. [3] P. J. Hall, R. T. Schilizzi, P. E. F. Dewdney, and T. J. W. Lazio, “The square kilometer array (SKA) radio telescope: Progress and technical directions,” URSI Radio Science Bulletin, no. 326, pp. 4–19, Sep. 2008. [4] S. Balling, M. Hein, M. Hennhofer, G. Sommerkorn, R. Stephan, and R. Thoma, “Broadband dual polarized antenna arrays for mobile communication applications,” in Proc. 33rd Eur. Microwave Conf., Oct. 2003, vol. 3, pp. 927–930. [5] , R. A. York and Z. B. Popovic, Eds., Active and Quasi-Optical Arrays for Solid-State Power Combining, ser. Wiley Series in Microwave and Optical Engineering. New York: Wiley, 1997. [6] M. V. Ivashina, M. Kehn, P. S. Kildal, and R. Maaskant, “Decoupling efficiency of a wideband Vivaldi focal plane array feeding a reflector antenna,” IEEE Trans. Antennas Propag., vol. 57, pp. 373–382, Feb. 2009. [7] W. Croswell, T. Durham, M. Jones, D. Schaubert, P. Friederich, and J. Maloney, “Wideband antenna arrays,” in Modern Antenna Handbook, C. A. Balanis, Ed. New York: Wiley, 2008. [8] T. Chio and D. Schaubert, “Parameter study and design of wide-band widescan dual-polarized tapered slot antenna arrays,” IEEE Trans. Antennas Propag., vol. 48, no. 6, pp. 879–886, Jun. 2000. [9] M. Kragalott, W. R. Pickles, and M. Kluskens, “Design of a 5:1 bandwidth stripline notch array from FDTD analysis,” IEEE Trans. Antennas Propag., vol. 48, no. 11, pp. 1733–1741, Nov. 2000. [10] N. Schuneman, J. Irion, and R. Hodges, “Decade bandwidth tapered notch antenna array element,” in 2001 Antenna Applications Symp., Allerton Park, Monticello, IL, Sep. 2001. [11] M. Stasiowski and D. H. Schaubert, “Broadband array antenna,” in 2008 Antenna Applications Symp., Allerton Park, Monticello, IL, Sep. 2008. [12] H. Holter, “Dual-polarized broadband array antenna with BOR-elements, mechanical design and measurements,” IEEE Trans. Antennas Propag., vol. 55, no. 2, pp. 305–312, Feb. 2007. [13] E. W. Lucas, M. A. Mongilio, K. M. Leader, C. P. Stieneke, and J. W. Cassen, “Notch radiator elements,” U.S. patent 5,175,560, Dec. 29, 1992. [14] H. Loui, J. P. Weem, and Z. Popovic, “A dual-band dual-polarized nested Vivaldi slot array with multi-level ground planes,” IEEE Trans. Antennas Propag., vol. 51, pp. 2168–2175, Sep. 2003. [15] J. D. S. Langley, P. S. Hall, and P. Newham, “Balanced antipodal Vivaldi antenna for wide bandwidth phased arrays,” IEE Proc.—Microwaves, Antennas and Propagation, vol. 143, no. 2, pp. 97–102, Apr. 1996.

[16] M. W. Elsallal and D. H. Schaubert, “Reduced-height array of balanced antipodal vivaldi antennas (bava) with greater than octave bandwidth,” in 2005 Antenna Applications Symp., Allerton Park, Monticello, IL, Sep. 2005, pp. 226–242. [17] J. J. Lee, S. Livingston, and R. Koenig, “Performance of a wideband (3–14 GHz) dual-pol array,” in Proc. 2004 IEEE Antennas and Propagation Society Int. Symp., Jun. 2004, vol. 1, pp. 551–554. [18] J. J. Lee, S. Livingston, and R. Koenig, “A low-profile wide-band (5:1) dual-pol array,” IEEE Antennas Wireless Propag. Lett., vol. 2, pp. 46–49, 2003. [19] M. Elsallal, “Doubly-Mirrored Balanced Antipodal Vivaldi Antenna (Dm-BAVA) for high performance arrays of electrically short, modular elements,” Ph.D. dissertation, Dept. Electr. Comput. Eng., Univ. Massachusetts, Amherst, MA, 2007. [20] S. G. Hay and J. D. O’Sullivan, “Analysis of common-mode effects in a dual polarized planar connected-array antenna,” Radio Science, RS6S04, vol. 43, Dec. 2008. [21] E. de Lera Acedo, E. Garcia, V. Gonzalez-Posadas, J. L. Vazquez-Roy, R. Maaskant, and D. Segovia, “Study and design of a differentially-fed tapered slot antenna array,” IEEE Trans. Antennas Propag., vol. 58, no. 1, pp. 68–78, Jan. 2010. [22] S. Edelberg and A. A. Oliner, “Mutual coupling effects in large antenna arrays II: Compensation effects,” IRE Trans. Antennas Propag., vol. 8, no. 4, pp. 360–367, Jul. 1960. [23] J. R. Bayard, D. H. Schaubert, and M. E. Cooley, “E-plane scan performance of infinite arrays of dipoles printed on protruding dielectric substrates: Coplanar feed line and E-plane metallic wall effects,” IEEE Trans. Antennas Propag., vol. 41, no. 6, pp. 837–841, Jun. 1993. [24] Ansoft HFSS. ver. 11.2. [Online]. Available: www.ansoft.com [25] D. Cavallo, A. Neto, and G. Gerini, “PCB slot based transformers to avoid common-mode resonances in connected arrays of dipoles,” IEEE Trans. Antennas Propag., vol. 58, pp. 2767–2771, Aug. 2010. [26] S. S. Holland, M. N. Vouvakis, and D. H. Schaubert, “Modular wideband antenna array,” U.S. patent application 61/230,768, Aug. 3, 2009. [27] S. S. Holland, M. N. Vouvakis, and D. H. Schaubert, “A new modular wideband array topology,” in 2009 Antenna Applications Symp., Allerton Park, Monticello, IL, Sep. 2009. [28] A. Ludwig, “The definition of cross polarization,” IEEE Trans. Antennas Propag., vol. 21, pp. 116–119, Jan. 1973. [29] D. T. McGrath, N. Schuneman, T. H. Shively, and J. Irion, II, “Polarization properties of scanning arrays,” in Proc. IEEE Int. Symp. Phased Array Systems and Technology, Oct. 2003, pp. 295–299. Steven S. Holland (S’05) was born in Chicago, IL, in 1984. He received the B.S. degree in electrical engineering from the Milwaukee School of Engineering (MSOE), Milwaukee, WI, in 2006. Since 2006, he has been with the Antennas and Propagation Laboratory at the University of Massachusetts Amherst, where he received the M.S. degree in 2008 and is currently working towards the Ph.D. degree, both in electrical engineering. HIs research interests include ultra-wideband phased arrays and electrically small antennas. Mr. Holland is a member of Tau Beta Pi.

Marinos N. Vouvakis (S’99–M’05) received the Diploma degree in electrical engineering, from Democritus University of Thrace (DUTH), Xanthi, Greece, in 1999. He received the M.S. degree from Arizona State University (ASU), Tempe, AZ, and the Ph.D. degree from The Ohio State University (OSU), Columbus, OH, both in electrical and computer engineering. Currently he is an Assistant Professor with the Center for Advanced Sensor and Communication Antennas in the Electrical and Computer Engineering Department, University of Massachusetts at Amherst. His research interests are in the area of computational electromagnetics with emphasis on domain decomposition, fast finite element and integral equation methods, hybrid methods, model order reduction and unstructured meshing for electromagnetic radiation and scattering applications. His interests extend to the design and manufacturing of ultra-wideband phased array systems.

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Wide-Angle Scanning Phased Array With Pattern Reconfigurable Elements Yan-Ying Bai, Shaoqiu Xiao, Member, IEEE, Ming-Chun Tang, Zhuo-Fu Ding, and Bing-Zhong Wang, Member, IEEE

Abstract—A novel phased array is presented to extend array scanning range by using pattern reconfigurable antenna elements and weighted thinned synthesis technology in this paper. The pattern reconfigurable microstrip Yagi antenna element is used as a basic element in array and it is capable of reconfiguring its patterns from broadside to quasi-endfire radiation by shifting states of the PIN diode switches integrated on parasitic strips. A weighted thinned linear array synthesis technique is analyzed and some interesting conclusions have been made. A linear array composed of eight pattern reconfigurable antenna elements is manufactured to demonstrate the excellent performance of the array. The active element pattern of each element is measured and pre-stored. Based on active element patterns and weighted thinned linear array synthesis technique, the pattern scanning performance of the novel array is synthesized. The results indicate that the array can scan its main beam from to in -plane with gain fluctuation less than 3 dB while maintaining low side lobes, to and the 3 dB beam width coverage is about from . The performance is superior to the traditional phased array made of wide-beam elements. Index Terms—Active element pattern, pattern reconfigurable antenna, thinned array, wide-angle scanning.

I. INTRODUCTION

T

HASED array antennas have been widely used in radar and communications for their unique non-inertial beam scanning, convenient beam controlling and energy management advantages, etc. [1]. Microstrip antenna is one of the important element formats in phased arrays due to its low profile, small size, light weight, conformal structure and ease of integration with radio frequency (RF) integrated circuits (ICs). Previous studies have indicated that, typically, microstrip phased array can effectively scan its main beam from to with array gain reductions of 4–5 dB compared with the maximum gain of the phased array as well as an increase in active reflection coefficient resulted from element mutual coupling [2]. These disadvantages limit its commercial and military applications.

Manuscript received April 11, 2010; manuscript revised February 23, 2011; accepted April 14, 2011. Date of publication August 12, 2011; date of current version November 02, 2011. This work was supported in part by the New-Century Talent Program of the Education Department of China under Grant NCET070154, in part by the Hi-Tech Research, Development Program of China under Grant 2009AA01Z231, and in part by National Defense Research Funding of China under Grants 08DZ0229 and 09DZ0204. The authors are with the Institute of Applied Physics, University of Electronic Science and Technology of China, Chengdu, 610054, China (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2164176

According to phased array theory, the scanning angle and radiation gain of phased array are associated with antenna elements. Gain decreases by at least 3 dB when scanning angle of the array increases beyond 3-dB beam angle of the isolated element. This phenomenon inhibits the extension of the scanning angle of the phased array. To solve this problem, many technologies have been developed by enlarging 3-dB beam width of the isolated element. For example, a wide-beam antenna is designed by combining microstrip antenna and dielectric antenna in reference [3] and another wide-beam antenna is designed in [4] and [5] by changing the excitation model of antenna and loading U-slot on antenna patch. However, in phased array application, large beam width of the elements can cause high side lobes and large grating lobes and decrease the main beam gain of array when the main beam of phased array scans to the larger angles. Aperiodic arrangements of antenna elements are adopted in arrays in order to reduce side lobes [6], [7], in which element spacings are optimized by genetic algorithms (GAs) and particle swarm optimization (PSO) and vary from a few tenths of the wavelength to several wavelengths. To obtain larger scanning angles, some element spacings need to be arranged less than quarter wavelength. Obviously, a traditional microstrip antenna cannot be applied in this case. In [8], Zhang proposed a weighted thinned linear array, and the array composed of omni-directional pattern elements maintains almost the same side lobe level when the array scans its main beam, but when the number of the array element is small, the side lobes of the array become higher and the array cannot scan to a large angle. These attempts imply that the aperiodic thinned array is one of the potential methods for phased array application. However, it is difficult for aperiodic thinned array to reduce side lobes while extending scanning angle range simultaneously. Therefore, some new techniques are still desired eagerly to improve the performances of phased arrays. Pattern reconfigurable antenna can reconfigure its radiation beams at certain frequencies by shifting the states of the switches integrated in the radiation aperture of the element [9]–[11]. If the pattern reconfigurable antenna element is applied into the array as a basic element, a novel phased array is structured and some surprising characteristics may be obtained because the element provides a new freedom degree besides excitation amplitude and phase. By using pattern reconfigurable elements, some attempts have been made to reduce the side lobes of phased array antenna within interesting angle range in literature [12], [13], but much work remains to be done to enhance the performance of the phased array.

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Fig. 1. Structure of pattern reconfigurable microstrip Yagi antenna element. (a) Antenna geometry. (b) Antenna prototype.

In this paper, a novel phased array is developed. The proposed array is a weighted thinned microstrip linear array in which the pattern reconfigurable microstrip Yagi antennas are used as the radiation elements. The weighted thinned linear array synthesis technique is analyzed in details. In order to perform the design concept, a novel phased array of eight pattern reconfigurable elements arranged with weighted thinned linear distribution is studied and fabricated. The active pattern of each element is measured in anechoic chamber and pre-stored. Finally, based on these active element patterns, the performance of the proposed eight-element phased array is synthesized, and the results indicate that the proposed array can scan its main beam within a wide-angle range with gain fluctuation of less than 3 dB while maintaining low side lobes level. II. PATTERN RECONFIGURABLE ANTENNA ELEMENT DESIGN The pattern reconfigurable antenna element IS A microstripYagi antenna, as shown in Fig. 1(a). The element operates at the frequency of about 5.8 GHz, and is printed on a dielectric substrate with permittivity of 2.2 and thickness (H) of 8 mm. Four PIN diode switches SMV2019, i.e., k1, k2, k3, and k4, are installed in four gaps, respectively. The layout of the DC-bias circuits of four PIN switches can be found in Fig. 1(a). DC voltage is applied to PIN diodes through metal pillars at point V and . ( is the guided-wave wavelength at the operation frequency) high impedance lines (width: 0.2 mm) are used to connect the antenna strips to the ground. The inductances, L1–L6 with 110 nH, are used to block high-frequency currents. The detailed configuration parameters shown in Fig. 1(a) are: mm, mm, mm, mm, mm, mm, and mm, respectively. The pattern reconfigurable antenna can operate in three states by shifting states of PIN diodes. In Fig. 1(a), when voltage of

Fig. 2. Measured results of three modes. (a) Reflection coefficients. (b) Patterns in -plane.

3 V is applied to the point V and , all of the switches are open and the antenna operates in -mode. The radiation pattern of the antenna directs to broadside and is similar with that of the conventional microstrip antenna. When voltages of 3 V and 3 V are applied to the point V and , respectively, the switches k1 and k2 are open, k3 and k4 are closed, and the antenna operates in -mode. In this mode, effective length of the left parasitic strip is longer than the right one and the antenna deflects its radiation pattern in -plane from broadside to the positive -axis. By symmetry, when the voltage of the point V and is 3 V and 3 V, respectively, the switch k1 and k2 are closed, k3 and k4 are open, the so called -mode can be constructed and it deflects radiation pattern in -plane from broadside to the negative -axis. The antenna scheme has been presented for microwave application and the ideal switch model is adopted in reference [14]. However, based on our studies, designing a DC-bias network with good electromagnetic compatibility (EMC) performance is very challenging in the design. The model of the antenna prototype is displayed in Fig. 1(b). The measured reflection coefficients are shown in Fig. 2(a), and the measured patterns of the three modes in -plane ( -plane) are displayed in Fig. 2(b). It can be observed that the antenna element can work well in three modes around

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Fig. 3. Configuration of the weighted thinned linear phased array.

5.8 GHz. The measured main beam directions and the corresponding half power beam coverage in -plane are: ( -mode, , 27 -78 ), ( -mode, , ), and ( -mode, , -48 ), respectively. By shifting the three modes, the 3 dB beam of the antenna can cover 140 uninterruptedly. The antenna in -mode has wider 3 dB beam coverage and a lower gain compared with -mode and -mode. It should be mentioned that the side lobe levels of -mode and -mode are less 9 dB than their respective main beam, which contributes to obtain low side lobes when the antenna element is used in phased array to extend scanning angle. III. WEIGHTED THINNED LINEAR ARRAY An N-element linear array formed by isotropic elements is plane is described shown in Fig. 3, and its array factor in by the following formula: (1) is the array factor, , is the where is the phase maximum radiation angle of the array, is the wavelength in free space at the operation constant and represent the excitafrequency, , and tion amplitude and the position of the th element in the array, respectively. In uniform array with element spacing , grating scans lobe (i.e. high side lobes) occurs when array factor to the angle (2) Generally, the physical sizes of practical antenna elements are not small enough to constitute easily an array with element spacing less than half wavelength, but large element spacing may lead to the appearance of the large grating lobes even if the scanning angle is small under proper condition. In [8], Zhang proposed a weighted thinned linear array and in this array the location of every element is determined by (3) where

(4) is the array length, i.e., , . The excitation amplitude of the th element can be determined by (5) where

is the weight factor.

Fig. 4. Scanned array factors with different maximum radiation angle.

We study the scanning characteristics of the array factor in details for its potential appliance. The studied results demonstrate that there exists a zone with ultra-low side lobe level beside the main beam and the width of this zone is denoted by . In this zone, the side lobe level called nearby side lobe (NSL) level is much less than main beam level. The peak side lobe (PSL) is the is mainly affected side lobe level out of the zone . Here, by the average distance between array elements and it decreases depends on when the average distance increases. The and it rapidly decreases as increases. The is associated increases. Howwith and . When or increases, the are changeless when ever, for the fixed and , , and the array scans its main beam, which has a potential value for extending antenna array scanning range. , , To perform intuitively the array factors, , and are chosen and the curves with difare shown in Fig. 4. In Fig. 4, it is indicated that the ferent level of the weighted thinned linear array with omnidirectional elements is still not low enough compared with the main beam level when the number of antenna elements is small. IV. NOVEL PHASED ARRAY WITH EXTENDED SCANNING ANGLE AND LOW SIDE LOBES In order to extend the scanning angle range of the phased array and reduce the side lobes, a novel weighted thinned linear phased array is proposed using the pattern reconfigurable antenna elements. To perform the novel phased array, we make a tradeoff between array radiation performance, array size and mutual coupling between elements. The studied phased array is composed of eight antenna elements. The array parameters are chosen as , and , just like the case mentioned in Fig. 4. All of the elements are arranged along -axis at respective location of . is determined according to (3) and listed in Table I. The proposed phased array is studied and manufactured, and its photograph and the adopted Cartesian coordinates are shown in Fig. 5. Active element pattern for each element is measured in the anechoic chamber. These active element patterns have already taken into account the contribution of the mutual couplings between the elements. When the active pattern of one element is measured, all of the other elements operate in the same mode. For examples, the active gain patterns in -plane

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AND

TABLE I WEIGHTED VALUES

IN

PROPOSED ARRAY

Fig. 5. Photograph of the weighted thinned linear array.

( -plane) of four elements, named No.02, No.04, No.06, and No.08, are displayed in Fig. 6. In Fig. 6(a), (b), and (c), all of the elements operate in -mode, -mode, and -mode, respectively. From this figure, it can be seen that the active element pattern of each element is different from each other. The difference comes from the different related position of each element and the variable mutual couplings between elements. With the increase of element spacing, such as the elements of No.06 and No.08, the active element patterns match well with each other. It implies that when the element spacing is large enough the active element pattern should be close to that of the isolated element. The active element patterns are adopted to synthesize the array pattern. This technique has been validated in [15] and [16]. The pattern for th element is noted as . Then the array pattern is obtained by a superposition of pattern of each element: (6) where is the expected main beam direction, the weighted amplitude coefficient is calculated by (5) and also shown in Table I. Fig. 7 shows the synthesized patterns of the proposed phased array in -plane. Fig. 7(a) depicts the array patterns whose main beam points to 0 , 20 , 40 , and 60 , respectively, when all of the elements operate in -mode and with prospective progressive phases. Similarly, Fig. 7(b) depicts the array patterns whose main beam points to , , , and ,

Fig. 6. -plane active patterns of the elements in the weighted thinned linear array. (a) -mode. (b) -mode. (c) -mode.

respectively, when all of the elements operate in -mode, and Fig. 7(c) shows the array pattern when all of the elements operate in -mode. The array performances in -plane, including the scanning angles, array gains, levels, and operating modes, are listed in Table II. It can be observed that, when all of the elements operate in -mode, the array can scan its main beam from 7 to 60 with a gain fluctuation less than 3 dB and the maximum gain

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TABLE II PATTERN CHARACTERISTICS OF THE NOVEL PHASED ARRAY

characteristic and the inertial broadside of the element patterns. These results demonstrate that the array can scan from to 60 by adjusting progressive phase and shifting corresponding operating state of the elements. The operation guideline of the array is as follows. 1) When the scanning angle ranges from 7 to 60 , all of the elements should be in -mode. 2) When the scanning angle ranges from 7 to 7 all of the elements should be in -mode. 3) When the scanning angle ranges from 60 to 7 all of the elements should be in -mode. Based on the results mentioned above, we can draw a conclusion that the novel phased array can improve dramatically its operation performance in two aspects compared with the traditional microstrip phased array. First, the novel array can scan its main beam with wider angle coverage and gain fluctuation less than 3 dB. The second advantage is that the element can obtain low side lobe level in -mode and -mode, which enables the novel phased array to scan in wide angle with low side lobe levels. V. CONCLUSION

Fig. 7. Pattern scanning characteristics of the weighted thinned linear array in -plane. (a) All elements operate in -mode. (b) All elements operate in -mode. (c) All elements operate in -mode.

of 13.2 dBi occurs at the scanning angle of . When the array scans its main beam to , the 3 dB beam coverage can get to and the side lobe level is low. Based on symmetry, the array can scan its main beam from to when all of the elements operate in -mode. When all of the elements operate in -mode, the array can scan its main beam from to with an almost invariable gain of about 11 dBi despite the large side lobe levels due to the wide-beam

A weighted thinned linear phased array with pattern reconfigurable antenna elements is proposed to extend the scanning angle coverage in this paper. The studies show that the novel array can scan its main beam from 60 to 60 in -plane with the gain fluctuation less than 3 dB and low side lobe level by shifting states of the elements and adjusting the excitations of the elements. These performances are superior to those of the array composed of wide-beam antenna elements. REFERENCES [1] R. J. Mailloux, Phased Array Antenna Handbook, 2nd ed. Beijing, China: Publishing House of Electronics Industry, 2008. [2] Z. Zhang and L. Jin, Radar Antenna Technology. Beijing, China: Publishing House of Electronics Industry, 2007.

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[3] S. Chattopadhyay, “Rectangular microstrip patch on a composite dielectric substrate for high-gain wide-beam radiation patterns, antennas and propagation,” IEEE Trans. Antennas Propagat., vol. 57, no. 10, pt. 2, pp. 3325–3328, Oct. 2009. [4] T. K.Wuang, “Low-cost antennas for direct broadcast satellite radio,” Microw. Opt. Technol. Lett., vol. 7, no. 10, pp. 440–444, Jul. 1994. [5] G. Lin, “A wide-beam antenna element for phased-array,” Radar Sci. Technol., vol. 5, no. 2, pp. 157–160, Apr. 2007. [6] P. J. Bevelacqua, “Minimum sidelobe levels for linear arrays,” IEEE Trans. Antennas Propagat., vol. 55, no. 12, pp. 3442–3449, Dec. 2007. [7] M. G. Bray, “Optimization of thinned aperiodic linear phased arrays using genetic algorithms to reduce grating lobes during scanning,” IEEE Trans. Antennas Propagat., vol. 50, no. 12, pp. 1732–1742, Dec. 2002. [8] Y. H. Zhang, “Optimum thinning of weighted linear arrays,” Acta Electronica Sinica, vol. 18, no. 5, pp. 34–39, Sep. 1990. [9] J. Zhang and A. Wang, “A survey on reconfigurable antennas,” in Proc. Microw. Millimeter Wave Technol. Int. Conf. (ICMMT), 2008, Apr. 2008, vol. 3, pp. 1156–1159. [10] S. Xiao and Z. Shao, “Pattern reconfigurable leaky-wave antenna design by FDTD method and Floquet’s theorem,” IEEE Trans. Antenna Propagat., vol. 53, no. 5, pp. 1845–1848, May 2005. [11] X.-S. Yang, B.-Z. Wang, W. Wu, and S. Xiao, “Yagi patch antenna with dual-band and pattern reconfigurable characteristics,” IEEE Antennas Wireless Propagat. Lett., vol. 6, pp. 168–171, 2007. [12] T. L. Roach and J. T. Bernhard, “Investigation of sidelobe level performance in phased arrays with pattern reconfigurable elements,” in Proc. IEEE Antennas Propagat. Soc. Int. Symp., Jun. 2007, pp. 105–108. [13] T. L. Roach and J. T. Bernhard, “On the applications for a radiation reconfigurable antenna,” in Proc. Adaptive Hardware Syst., AHS . 2nd NASA/ESA Conf., Aug. 2007, pp. 7–13. [14] S. Zhang, “A pattern reconfigurable microstrip parasitic array,” IEEE Trans. Antennas Propagat., vol. 52, no. 10, pp. 2773–2776, Oct. 2004. [15] D. F. Kelley, “Array antenna pattern modeling methods that include mutual coupling effects,” IEEE Trans. Antennas Propagat., vol. 41, no. 12, pp. 1625–1632, Dec. 1993. [16] J. L. Allen, “Gain and impedance variation in scanned dipole arrays,” IRE Trans. Antennas Propagat., vol. AP-10, pp. 566–573, Sep. 1962.

Yan-Ying Bai was born in Hunan Province, China, in 1978. She received the B.S. degree in physics from Jishou University, Jishou, China, in 2000 and the M.S. degrees in electromagnetic field and microwave engineering from the University of Electronic Science and Technology of China (UESTC), Chengdu, in 2005, where she is currently working toward the Ph.D. degree in radio engineering. Her current research interests include antenna elements, phased arrays, and RF circuits. She is also interested in electromagnetic theory and computational electromagnetics.

Shaoqiu Xiao (M’05) received the Ph.D. degree in electromagnetic field and microwave engineering from the University of Electronic Science and Technology of China (UESTC), Chengdu, in 2003. From January 2004 to June 2004, he was with UESTC as an Assistant Professor. From July 2004 to March 2006, he was with the Wireless Communications Laboratory, National Institute of Information and Communications Technology of Japan (NICT), Singapore, as a Researcher with a focus on the planar antenna and smart antenna design and optimization. From July 2006 to June 2010, he was with UESTC as an Associate Professor. He is now a Professor with UESTC. His current research interests include planar antenna and arrays, microwave passive circuits, and electromagnetics in ultrawide band communication. He has authored/coauthored more than 100 technical journals, conference papers, books, and book chapters.

Ming-Chun Tang received the B.S. degree in physics from the Neijiang Normal University, Neijiang, China, in 2005. He is currently working toward the Ph.D. degree in radio physics at the University of Electronic Science and Technology of China (UESTC). His research interest includes miniature antennas, RF circuits, and metamaterial design and its application. He has authored or coauthored more than 30 international referred journal and conference papers. Mr. Tang was a recipient of the Best Student Paper Award at the 2010 International Symposium on Signals, Systems, and Electronics (ISSSE2010). He serves as a reviewer for the IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS.

Zhuo-Fu Ding, photograph and biography not available at the time of publication.

Bing-Zhong Wang (M’06) received the Ph.D. degree in electrical engineering from the University of Electronic Science and Technology of China (UESTC), Chengdu, China, in 1988. He joined the UESTC in 1984 and is currently a Professor there. He has been a Visiting Scholar at the University of Wisconsin-Milwaukee, a Research Fellow at the City University of Hong Kong, and a Visiting Professor in the Electromagnetic Communication Laboratory, Pennsylvania State University, University Park. His current research interests are in the areas of computational electromagnetics, antenna theory and techniques, electromagnetic compatibility analysis, and computer-aided design for passive microwave integrated circuits.

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Microstrip Grid and Comb Array Antennas Lin Zhang, Wenmei Zhang, Member, IEEE, and Y. P. Zhang, Fellow, IEEE

Abstract—The design of a high-gain microstrip grid array antenna (MGAA) for 24-GHz automotive radar sensor applications is first presented. An amplitude tapering technique utilizing variable line width on the individual radiating element is then applied to lower sidelobe level. Next, the MGAA is simplified to a microstrip comb array antenna (MCAA). The MCAA shows broader impedance bandwidth and lower cross-polarization radiation as compared with those of the MGAA. The MCAA is designed not as a travelling-wave but a standing-wave antenna. As a result, the match load and the reflection-cancelling structure can be avoided, which is important, especially in the millimeter-wave frequencies. Finally, an emphasis is given to 45 linearly-polarized MCAA because the radiation with the orthogonal polarization from cars coming from the opposite direction does not affect the radar operation. Index Terms—24 GHz, automotive radar, comb array antenna, grid array antenna.

I. INTRODUCTION HE grid array antenna was proposed by Kraus in 1964 as a linearly-polarized travelling-wave (non-resonant) antenna with the main lobe of radiation in a backward angle-fire direction. The grid array antenna allows a high radiating element density to be achieved while minimizing the number of feed lines external to the antenna structure [1]. The grid array antenna was implemented in microstrip technology by Conti, etal in 1981 as a linearly-polarized standing-wave (resonant) antenna with the main beam of radiation in the broadside direction. The grid array antenna in microstrip technology has all the usual benefits of conventional patch type radiators plus broad bandwidth, high gain, adequate cross polarization control, and beam scan ability with frequency [2]. However, the grid array antenna has not found wide applications or received enough attention. A literature survey shows that there have been only about 20 papers published over the last 45 years and the majority of them have dealt with microstrip grid array antennas at lower microwave frequencies [3]–[7]. Recently, Zhang and Sun have advocated reviving microstrip grid array antennas. They have demonstrated that a microstrip grid array antenna is the antenna

T

Manuscript received December 17, 2010; revised March 16, 2011; accepted May 05, 2011. Date of publication August 12, 2011; date of current version November 02, 2011. This work was supported by the National Science Foundation of China (No. 60771052) Program for the Top Young Academic Leaders of Higher Learning Institutions of Shanxi. L. Zhang and W. Zhang are with the College of Physics and Electronics, Shanxi University, Taiyuan 030006, China (e-mail: [email protected]). Y. P. Zhang is with the Integrated Systems Research Lab, School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798 (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2164216

of choice for a highly-integrated 60-GHz radio for short-range and high-speed wireless communications [8], [9]. The automotive radar is developed for detection of vehicles moving at high speed from the opposite direction. The radar first reacts to object moving at high speed and checks its distance. If the distance is critical, i.e., the estimation is that collision cannot be avoided; the radar gives the signal for activation of the protection system. Antennas for automotive radars should have a narrow beamwidth about 5 in the azimuth plane and a wider beamwidth around 60 in the elevation plane [10]–[15]. In addition, the wave with the orthogonal polarization from cars coming from the opposite direction does not affect the radar operation. In this case, the 45 linearly polarized antenna for the automotive radar is required [16], [17]. In this paper, we extend the application of microstrip grid array antennas to automotive radar sensors for the first time. We design a high-gain MGAA in the 24-GHz band and apply an amplitude tapering technique utilizing variable line width on the individual radiating element to low side-lobe level in Section II. We simplify the MGAA to a MCAA and modify the MCAA to radiate 45 linearly polarized waves in Section III. Different from the travelling-wave MCAA in [12], [13], the new MCAA is a standing wave antenna, which avoids the problem associated with the match load and simplifies the antenna design. Also, a wider impedance bandwidth and gain bandwidth are obtained. It should be mentioned that these array antennas are simulated with the High-Frequency Structure Simulator (HFSS) from Ansoft and measured with an HP 8510C network analyzer from Agilent in an anechoic chamber. Finally, we draw conclusions in Section IV. II. MICROSTRIP GRID ARRAY ANTENNA The MGAA is shown in Fig. 1. In the operation, the MGAA can function as either a travelling-wave or a standing-wave antenna. This paper concentrates on the latter type that requires and short side length approximately grid long side length , where is the guided wavelength at the center frequency of operation [2]. The radiation is essentially from the short sides with the long sides acting mainly as guiding or transmission lines. This is because the current on each short side is basically in phase, while each long side supports a full wavelength current. With such a current distribution, the maximum radiation would be broadside to the array. A number of variables are available in designing the MGAA. Given substrate and manufacturability, one can change the grid dimensions, the number of loops, and feed location to meet the specification. For the grid dimensions, the lengths of the long and short sides of the grid ( and ) are governed by the resonance at the center frequency of operation, while the widths of the long and short sides of the grid ( and ) can be chosen for better transmission and the desired amplitude taper on the array,

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Fig. 3. Non-amplitude-tapered MGAA: (a) radiator, (b) photo.

Fig. 1. The configuration of MGAA.

Fig. 4. The simulated and measured MGAA.

j

S

j

for the non-amplitude-tapered

and Fig. 2 shows the simulated directivities for different with , , , and . and are selected to meet the requirement of radiation beamwidth, that is, about 5 in the azimuth plane and 60 in the elevation plane. For the feed point, it is typically located at the joint of the long and short sides near the center of the antenna. The feed point is labelled as ’F’ in Sections II and III. In addition, considering the gain and bandwidth, we choose the RT/duriod 5880 substrate , loss tangent , with dielectric constant . The dimensions of all realized and thickness . antennas are

N = 1), (b) for different N (M =

Fig. 2. The directivity: (a) for different M ( 16).

A. Non-Amplitude-Tapered MGAA

respectively. Generally, the impedance of long sides should less than 250 in order to reduce the cross polarization and transmission loss. Considering the precision of process, the minis 0.1 mm in our design. imum of The number of loops controls the directivity and functional bandwidth. As the number of loops increases, the directivity inand creases but the functional bandwidth decreases. Let denote the number of loops along - and - axis, respectively.

Fig. 3 shows the radiator and photo of the non-amplitude-tapered MGAA. It has , , , and . Fig. 4 shows the simulated results for the MGAA. It is evident from and measured the figure that the simulated 10-dB impedance bandwidth is 400 MHz from 23.90 to 24.31 GHz (or 1.6% at 24.105 GHz) covering the allocated band from 24.05 to 24.25 GHz, and the measured 10-dB impedance bandwidth is 200 MHz (0.9% at 24.505 GHz) from 24.40 to 24.61 GHz. The center frequency has shifted 400 MHz (or 1.6% at 24.15 GHz).

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Fig. 6. Amplitude-Tapered MGAA: (a) radiator, (b) photo.

TABLE I CALCULATED PARAMETER BY TAYLOR SYNTHESIS METHOD

Fig. 7. The simulated and measured j

Fig. 5. The simulated and measured radiation patterns for the non-amplitude-tapered MGAA: (a) co-polar in the -plane, (b) co-polar in the -plane, (c) cross-polar in the -plane, (d) cross-polar in the -plane.

H

H

E

E

Fig. 5(a) and (b) show the simulated and measured co-polar radiation patterns of the non-amplitude-tapered MGAA in the

S

j

for the amplitude-tapered MGAA.

and planes at 24.5 GHz, respectively. It has a narrow beam plane and the 3-dB beamwidth is 5 ; a broad beam in the in the plane and 3-dB beamwidth is about 60 . The simuat 7 and at 353 lated side-lobe levels are in the plane, respectively. The measured side-lobe levels are at 12 and at 348 in the plane respectively. Fig. 5(c) and (d) show the simulated and measured cross-polar radiation patterns of the non-amplitude-tapered MGAA in the and planes at 24.5 GHz, respectively. The cross-polar radiation level is low. The gain of the non-amplitude-tapered MGAA was also measured (since it is a narrowband antenna, the gain curve is not shown). The gain values are larger than 18.2 dBi from 24 to 24.61 GHz and the maximum gain is 20.6 dBi at 24.5 GHz.

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Fig. 9. 90 linearly polarized MCAA: (a) radiator, (b) photo.

Taylor synthesis method is applied to suppress the side-lobe level. Fig. 6 shows the radiator and photo of the amplitude-tapered MGAA. It consist 33 short sides and symmetric to the ‘symmetric axis’. The array is divided into 11 units and short sides in every unit have same width. According to the Taylor synthesis method, the width of short sides can be obtained as shown in , , and Table I. The other dimensions are . Fig. 7 shows the simulated and measured results. The simulated frequency range for is from 24.0 to 24.3 GHz (1.2% at 24.15 GHz). However, the is from 24.27 to measured frequency range for 24.67 GHz (1.7% at 24.47 GHz) and the center frequency has shifted 320 MHz (1.3% at 24.15 GHz). Fig. 8(a) and (b) show the simulated and measured co-polar radiation patterns of the amplitude-tapered MGAA in the and planes at 24.5 GHz, respectively. It has a narrow beam in the plane and the 3-dB beamwidth is 5 ; a broad beam in the plane and 3-dB beamwidth is about 60 . The simulated at both 10 and 350 in the side-lobe levels are plane. The measured side-lobe levels are at 20 and at 342 in the plane, respectively. Compared with results in Fig. 5(a), the maximum side-lobe-level is improved by 6 dB. Fig. 8(c) and (d) show the simulated and measured crosspolar radiation patterns of the non-amplitude-tapered MGAA and planes at 24.5 GHz, respectively. The crossin the polar radiation level is low. The gain of the amplitude-tapered MGAA was also measured. The gain values are larger than 19 dBi from 24 to 24.67 GHz and the maximum gain is 19.85 dBi at 24.47 GHz. III. MICROSTRIP COMB ARRAY ANTENNA A. 90 Linearly Polarized MCAA

Fig. 8. The simulated and measured radiation pattern for the amplitude-tapered MGAA: (a) co-polar in -plane, (b) co-polar in the -plane, (c) cross-polar in the -plane, (d) cross-polar in the -plane.

H

H

E

E

B. Amplitude-Tapered MGAA As shown previously, the non-amplitude-tapered MGAA has a larger side-lobe level of . In this section, the

For the MGAA, the radiation is essentially from the short sides and the long sides act mainly as guiding or transmission lines. Also, the currents at the end of each short side approach to zero [3]. In this case, the long sides at the edge of the MGAA can be removed and the MGAA is simplified as a MCAA, as shown in Fig. 9. The design of MCAA is similar as the design and . of MGAA, that is, Fig. 10(a) shows the input reactance of the MCAA for the , 13 and 33. It can be seen that number of short sides the bandwidth of the MCAA is related with Ns. As Ns increases, the number of zeros of the input reactance increases and a wider

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Fig. 10. The results for the 90 linearly polarized MCAA: (a) simulated impedance, (b) simulated current amplitude, (c) simulated and measured jS j.

impedance bandwidth can be achieved. Fig. 10(b) shows the , , current distribution along the MCAA for , , and . It clearly indicates that this MCAA is a standing wave antenna, which is different from the travelling-wave MCAAs in [12], [13]. Thus, the match load and reflection-cancelling structure can be avoided, leading to a simpler antenna solution, which is important especially in millimeter-wave frequencies. , The optimized dimensions for the MCAA are , , , . Fig. 10(c) results for the MCAA. shows the simulated and measured The measured impedance bandwidth of the MCAA for is from 17.33 to 25.34 GHz (38% at 21.33 GHz). Compared with the MGAA in Section II and travelling-wave antennas in [12], [13], a wider bandwidth is obtained because more resonances appear. Fig. 11(a) and (b) show the simulated and measured co-polar radiation patterns of the MCAA at 24.15 GHz in the and planes, respectively. It has a narrow beam in the plane and

Fig. 11. The simulated and measured radiation pattern for the 90 linearly polarized MCAA: (a) co-polar in the H -plane, (b) co-polar in the E -plane, (c) cross-polar in the H -plane, (d) cross-polar in the E -plane.

the 3-dB beamwidth is 5 ; a broad beam in the plane and 3-dB beamwidth is about 60 . Fig. 11(a) indicates the side-lobe level

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Fig. 12. Measured Gain of the MCAA.

Fig. 13. Half-Touched 45 (b) photo.

linearly polarized MCAA. (a) Configuration,

Fig. 14. The simulated and Measured jS polarized MCAA.

j

for the half-touched 45 linearly

is suppressed below . Fig. 11(c) and (d) show the simulated and measured cross-polar radiation patterns of the MCAA in the and planes, respectively. The cross-polar levels in the - and -plane are below . The lower cross-polar radiation is because the long sides on the edge of the MGAA have been removed, leading to less sides that cause the cross-polar radiation. The measured gain of the MCAA is shown in Fig. 12. The maximum gain is 21.2 dBi. The gain at the 24.15 GHz is 21 dBi. The 3-dB gain bandwidth is from 23.5 to 25.5 GHz (8.2% at

Fig. 15. The Simulated and Measured Radiation Pattern for the Half-Touched 45 linearly polarized MCAA: (a) co-polar in the E -plane, (b) co-polar in the H -plane, (c) cross-polar in the E -plane, (d) cross-polar in the H -plane.

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using the match load and reflection-cancellation structure and achieved a wider impedance bandwidth of 38% and gain bandwidth of 8.2% at 24 GHz. Also, the 45 linearly polarized MCAA was designed and realized to meet the demand of automotive radars.

ACKNOWLEDGMENT The authors would like to thank Mr. Z. Bing for his help in fabricating and testing the antennas.

REFERENCES Fig. 16. Measured gains for half-touched 45 linearly polarized MCAA.

24.5 GHz) and is better than that of the travelling-wave MCAA in [12].

B. 45 Linearly Polarized MCAA The wave with the orthogonal polarization from cars coming from the opposite direction does not affect the radar operation. In this case, a 45 linearly polarized antenna for the automotive radar is required. In this Section, we design a 45 linearly polarized MCAA. In order to improve the match and reduce the reflection from the long side, the short sides are arranged to be half-touch of the long side as shown in Fig. 13. The optimized dimensions for the 45 linearly polarized MCAA are , , , . Fig. 14 shows the simulated and measured results. The measured is from 22.6 to 24.9 impedance bandwidth for GHz (9.8% at 23.75 GHz). Fig. 15(a) and (b) show the simulated and measured co-polar radiation patterns of the 45 linearly polarized MCAA in the and planes at 24.15 GHz, respectively. The 3-dB bandwidth in the - and - plane are 12 and 14 , respectively. Fig. 15(c) and (d) show the simulated and measured cross-polar radiation patterns of the 45 linearly polarized MCAA in the and planes at 24.15 GHz, respectively. The 45 linearly . polarized MCAA has a large cross-polar level high up to The measured gain of the 45 linearly polarized MCAA is shown in Fig. 16. The maximum gain is 17.4 dBi. The gain at the 24.15 GHz is 16.2 dBi. The 3-dB gain bandwidth is from 22.6 to 24.55 GHz (8.3% at 23.575 GHz).

IV. CONCLUSION A microstrip grid array antenna that achieved the maximum gain of 20.6 dBi in the 24-GHz band was designed. The Taylor synthesis method was applied to suppress the side-lobe level . The MGAA was simplified into the MCAA. below Unlike the conventional travelling-wave MCAA, this new MCAA was designed to be a standing-wave antenna that avoids

[1] J. D. Kraus, “A backward angle-fire array antenna,” IEEE Trans. Antennas Propag., vol. AP-12, pp. 48–50, Jan. 1964. [2] R. Conti, J. Toth, T. Dowling, and J. Weiss, “The wire-grid microstrip antenna,” IEEE Trans. Antennas Propag., vol. AP-29, pp. 157–166, Jan. 1981. [3] H. Nakano, I. Oshima, H. Mimaki, K. Hirose, and J. Yamauchi, “A fast MoM calculation technique using sinusodial basis and testing functions for a wire on a dielectric substrate and its application to meander loop and grid array antennas,” IEEE Trans. Antennas Propag., vol. 53, no. 10, pp. 3300–3307, Oct. 2005. [4] H. Nakano, I. Oshima, H. Mimaki, K. Hirose, and J. Yamauchi, “Center-fed grid array antennas,” in Proc. IEEE Int. Symp. Antennas and Propagation, 1995, vol. 4, pp. 2010–2013. [5] H. Nakano and T. Kawano, “Grid array antennas,” in Proc. IEEE Int. Symp. Antennas and Propagation, 1997, vol. 1, pp. 236–239. [6] H. Nakano, H. Osada, H. Mimaki, Y. Iitsaka, and J. Yamauchi, “A modified grid array antenna radiating a circularly polarized wave,” in Proc. Int. Symp. Microwave, Antennas, Propagation and EMC Technologies for Wireless Communications, Aug. 2007, pp. 527–4530. [7] Y. Iitsuka, J. Yamauchi, and H. Nakano, “Grid array antenna composed of V-shaped and rhombic elements for beam scanning,” in Proc. IEEE Int. Symp. Antennas and Propagation, Jun. 2009. [8] M. Sun and Y. P. Zhang, “Design and integration of 60-GHz grid array antenna in chip package,” in Proc. Asia-Pacific Microwave Conf., Dec. 2008, pp. 1–4. [9] M. Sun, Y. P. Zhang, Y. X. Guo, K. M. Chua, and L. L. Wai, “Integration of grid array antenna in chip package for highly integrated 60-GHz radios,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 1364–1366, 2009. [10] R. Kulke et al., “24 GHz radar sensor integrates patch antenna and frontend module in single multilayer LTCC substrate,” in Proc. 15th Eur. Microelectronics and Packaging Conf., Brugge, Belgium, Jun. 12–15, 2005. [11] C. Morhart, M. O. Olbrich, and E. M. Biebl, “High gain crank line antenna for 24 GHz,” in Proc. Eur. Radar Conf., 2005, pp. 201–204. [12] H. Moheb, L. Shafai, and M. Barakat, “Design of 24 GHz microstrip travelling wave antenna for radar application,” in Proc. IEEE AP-S Int. Symp. Dig., 1995, pp. 350–353. [13] K. Sakakibara, S. Sugawa, N. Kikuma, and H. Hirayama, “MillimeterWave microstrip array antenna with matching-circuit-integrated radiating-elements for travelling-wave excitation,” in Proc. IEEE AP-S Int. Symp. Dig., (EuCAP), 2010, pp. 1–5. [14] K. Wincza, S. Gruszczynski, J. Borgosz, J. G. Hallatt, and I. Aldred, “Design of electromagnetically coupled corner-series-fed antenna arrays with the application for 24 GHz Doppler sensors,” in Proc. IEEE AP-S Int. Symp. Dig., 2007, pp. 2140–2143. [15] M. Slovic, B. Jokanovic, and B. Kolundzjia, “High efficiency patch antenna for 24 GHz anticollision radar,” in Proc. IEEE Int. Conf., 2005, pp. 20–23. [16] J. Hirokawa and M. Ando, “45 linearly polarized post-wall waveguide-fed parallel plate slot arrays,” IEE Proc.-Microw. Antennas Propag., vol. 147, no. 6, pp. 515–519, Dec. 2000. [17] Q. F. Zhang and Y. Lu, “Design of 45-degree linearly polarized substrate integrated waveguide-fed slot array antennas,” Int J. Infrared Milli Waves, vol. 29, pp. 1019–1027, 2008.

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Lin Zhang received the B.E. degree from Taiyuan University of Technology and M.E. degree from Shanxi University, China, in 2008 and 2011, respectively, all in electronic engineering. She was a project officer in Nanyang Technological University, Singapore in 2010. Her research interests are planar antenna arrays and microwave circuits.

Wenmei Zhang (M’05) was born in 1969. She received the B.S. and M.S. degrees from Nanjing University of Science and Technology (China) in 1992 and 1995, respectively, and the Ph.D. degree in electronic engineering from Shanghai Jiao Tong University, China, in 2004. Currently, she is a Professor at the College of Physics and Electronics, Shanxi University. She was a Tan Chin Tuan Fellow and Senior Research Fellow in Nanyang Technological University, Singapore in 2008 and 2010, respectively. Her research interests include microwave and millimeter-wave integrated circuits, EMC and microstrip antenna.

Y. P. Zhang (M’03–SM’07–F’10) received the B.E. and M.E. degrees from Taiyuan Polytechnic Institute and Shanxi Mining Institute of Taiyuan University of Technology, Shanxi, China, in 1982 and 1987, respectively and the Ph.D. degree from the Chinese University of Hong Kong, Hong Kong, in 1995, all in electronic engineering. From 1982 to 1984, he worked at Shanxi Electronic Industry Bureau, from 1990 to 1992, the University of Liverpool, Liverpool, U. K., and from 1996 to 1997, City University of Hong Kong. From 1987 to 1990, he taught at Shanxi Mining Institute and from 1997 to 1998, the University of Hong Kong. He was promoted to a Full Professor at Taiyuan University of Technology in 1996. He is now an Associate Professor and the Deputy Supervisor of Integrated Circuits and Systems Laboratories with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore. He has broad interests in radio science and technology and published widely across seven IEEE societies. He has delivered scores of invited papers/keynote addresses at international scientific conferences. He has organized/chaired dozens of technical sessions of international symposia. Dr. Zhang received the Sino-British Technical Collaboration Award in 1990 for his contribution to the advancement of subsurface radio science and technology. He received the Best Paper Award from the Second International Symposium on Communication Systems, Networks and Digital Signal Processing, 18-20th July 2000, Bournemouth, UK and the Best Paper Prize from the Third IEEE International Workshop on Antenna Technology, 21-23rd March 2007, Cambridge, UK. He was awarded a William Mong Visiting Fellowship from the University of Hong Kong in 2005. He was a Guest Editor of the International Journal of RF and Microwave Computer-Aided Engineering and an Associate Editor of the International Journal of Microwave Science and Technology . He serves as an Editor of ETRI Journal, an Associate Editor of the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, and an Associate Editor of the International Journal of Electromagnetic Waves and Applications . He also serves on the Editorial Boards of a large number of Journals including the IEEE Transactions on Microwave Theory and Techniques and the IEEE Microwave and Wireless Components Letters.

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A Novel Deterministic Synthesis Technique for Constrained Sparse Array Design Problems Diego Caratelli and Maria Carolina Viganó, Member, IEEE

Abstract—A novel analytical approach to the synthesis of linear sparse arrays with non-uniform amplitude excitation is presented and thoroughly discussed in this paper. The proposed technique, based on the concept of auxiliary array factor, is aimed at the deterministic determination of the optimal array element density and excitation tapering distributions useful to mimic a desired radiation pattern. In particular, the developed antenna placement method does not require any iterative or stochastic optimization procedure, resulting in a dramatic reduction of antenna design times. Selected examples are included in order to assess the effectiveness and versatility of the proposed approach in the framework of aperiodic array synthesis problems. Index Terms—Aperiodic array, auxiliary array factor, deterministic antenna placement technique.

I. INTRODUCTION

I

N recent years, a renewed scientific interest has been put on the design of sparse antenna arrays [1]–[11]. That is mainly due to the appealing advantages resulting from their usage. In particular, the proper shaping of the radiation pattern with a reduced number of antenna elements is one of the well-known properties of non-uniformly spaced arrays [12]–[15]. This typically implies a reduced weight, cost, and complexity of the feeding network [16], as well as a larger average inter-element distance resulting in a smaller parasitic coupling level. Furthermore, thanks to the aperiodic spacing between the radiators, no replicas of the antenna main lobe occur in the visible space, even where a pattern scanning is performed. Several techniques for the synthesis of sparse antenna arrays are available in the scientific literature, but most of them rely on the use of iterative multidimensional optimization algorithms [17], [18], and evolutionary stochastic methods [19]–[24]. In some cases, deterministic optimization techniques can fail to provide the global optimum of the specified objective function, being trapped in a local minimum solution. To alleviate this problem, stochastic approaches based on genetic algorithms (GAs), simulated annealing (SA) and, more recently, particle

Manuscript received September 23, 2010; revised April 04, 2011; accepted May 06, 2011. Date of publication August 12, 2011; date of current version November 02, 2011. D. Caratelli is with the Microwave Technology and Systems for Radar (MTSR) Group, Delft University of Technology-Mekelweg 4, 2628 CD, Delft, The Netherlands (e-mail: [email protected]). M. C. Viganó is with JAST Antenna Systems, ViaSat Inc.-PSE-C, CH-1015, Lausanne, Switzerland. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2164193

swarm optimization (PSO) may be considered [23], [24]. However, it is to be pointed out that, especially for large array design problems characterized by a significant number of unknowns, the adoption of the aforementioned techniques usually results in a significantly large synthesis time. In this context, deterministic methods are surely to be preferred. Some attempts have been made in the past to develop semi-analytical synthesis techniques, mainly based on the perturbation of uniformly spaced and/or illuminated linear arrays [17], [18]. Unfortunately, most of the presented procedures are unable to handle constraints on the minimum inter-element spacing as required in real-life applications, and can not take the illumination phase distribution into account, restricting the antenna designer to what is called power pattern synthesis (PPS). In this paper, a novel non-uniform array design method, based on the concept of the auxiliary array factor (AAF) function, is developed in order to determine analytically the optimal element density and excitation tapering distributions useful to mimic a given radiation pattern. In this way, the array sparseness can be conveniently tuned in order to meet the design requirements in terms of minimum spacing between the antenna elements, and maximal array aperture size. To the best knowledge of the authors at the time of writing, the proposed approach is the only deterministic array synthesis technique available in the scientific literature, capable of exploiting both the illumination distribution and the antenna element density as additional degrees of freedom in the antenna design cycle. In the framework of the presented array synthesis technique, it is required to specify a target radiation mask and a desired phase distribution in order to derive the continuous positioning and illumination functions which provide, respectively, the actual locations and excitation coefficients of the array elements. In particular, the combined controllable tapering of the excitation and element positioning allows handling constrained array design problems with severe requirements in terms of maximum number of power levels to be operated in the feeding network. Moreover, the computational burden of the proposed approach is significantly reduced in comparison with other array synthesis techniques since no use of optimization or iterative procedures is made. The paper is organized as follows. In Section II the proposed sparse array synthesis procedure is detailed with emphasis defunction, and the deterministic voted to the concept of method for the evaluation of the array element density and excitation tapering distributions. The application of the proposed formulation to the design of aperiodic antenna arrays with complex radiation pattern masks is then presented in Section III. The concluding remarks are summarized in Section IV.

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II. ARRAY SYNTHESIS PROCEDURE In this section, emphasis is put on the analytical details of the proposed design technique for the deterministic synthesis of a general linear non-uniform-amplitude sparse phased array. A. Auxiliary Array Factor Function Let us first consider the array factor relevant to a radiating antennas deployed over a line as structure consisting of shown in Fig. 1: Fig. 1. Antenna element distribution in a general linear aperiodic array.

(1) being the propagation constant in free space, and the excitation coefficient of the -th array . As it can be readily inferred, element located at the expression in (1) may be regarded as the Riemann’s sum defined below: approximating the (2) denote, respectively, the continuous norwhere malized positioning, amplitude and phase distributions generalappearing in izing the discrete quantities (1). Similarly, is the continuous version of the index relevant to the general antenna element forming the array, and ranging (typically specified to be the from 0 to the maximum value unity for the purpose of normalization). After setting for short, the can be written as: ness

Fig. 2. Piecewise linear approximation of the normalized positioning  (q ) and phase (q ) distributions as functions of the continuous index q relevant to the general antenna element forming the array structure. A similar discretization procedure is applied to the amplitude tapering function A(q ) not sketched here for the sake of clarity.

(3) with: (6) (4) . being the contribution pertinent to the -th interval In each interval, the positioning, amplitude and phase functions are assumed to be linearized (see Fig. 2) according to the following expression:

In order to achieve a fully analytical formulation useful to mimic in a deterministic a given objective array factor mask way, it is convenient to directly carry out the synthesis procewith respect dure in the domain of the Fourier transform to the variable [25]. By making judicious use of the shift prop, one can readily obtain, after simple erty of the operator mathematical manipulations:

(5) for , and . As it is obvious, the larger the of intervals which the domain is divided number into, the better the accuracy of the described discretization procan be cedure on the array tapering functions. Heuristically, with . selected to be Under the mentioned assumptions, the general term can be evaluated in a closed form as follows: where

,

(7) where:

(8)

(9)

CARATELLI AND VIGANÓ: A NOVEL DETERMINISTIC SYNTHESIS TECHNIQUE FOR CONSTRAINED SPARSE ARRAY DESIGN PROBLEMS

denoting the pulse distribution having width and cen. It is to be pointed out that, by tered at the origin virtue of the said piecewise linearization of the normalized positioning, amplitude and phase functions, each term in (3), (4) is Fourier-transformed into a function with compact support centered at and having width . Upon noticing that are not the supports of the modulated-pulse functions overlapping, it is straightforward to show that the following equality holds true: (10)

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from the given array factor mask. Such parasitic effect is neglected within the proposed research, but it may be handled by means of suitable compensation techniques based on the concept of active element pattern (AEP), already available in the scientific literature [26]–[30]. Despite of that, it is to be emphasized that the proposed array design approach already provides reasonably robust results for reduced inter-element spacings as pointed out in the following section. The synthesis procedure relies on the solution, in a sequenand , tial way, of (12) involving the unknown quantities . To this starting from given values and end, the following parameter depending on the assigned mask and the minimal normalized inter-spacing is evaluated for :

A similar property is featured by the argument of the Fouriertransformed auxiliary function, namely: (14)

is larger than the amplitude-related threshold , the spacing is taken to be exactly , and the excitation amplitude tapering is determined in such a way that (12) is satisfied, namely: Where

(11) Thanks to that, the proposed array synthesis technique turns to be analytically rigorous and fully deterministic, without resorting to any optimization procedure, as it will be shown in the next subsection. B. Deterministic Evaluation of the Array Element Density and Excitation Tapering Distributions The array synthesis is carried out by enforcing or, equivalently, within each interval for . So, by making use of (8) it follows that:

(12) is assumed to be where the continuous amplitude function , denoting the greater or equal to a given threshold minimum power gain level to be operated in the array feeding network (typically set to unity). Furthermore, in order to address bea possible physical constraint on the minimal distance tween the antenna elements, an additional condition is to be enforced on the first derivative of the normalized positioning func, that is: tion (13) being the free-space wavelength. As it can be readily inferred, the minimal inter-element spacing directly affects the antenna mutual coupling level, a reduced value of potentially resulting in a severe degradation of the structure performance in terms of deviation of the relevant radiation pattern

(15) On the other hand, if , the amplitude can be selected to be equal to the reference value coefficient , and the normalized antenna element position is derived by solving the modified equation:

(16)

turns to be larger than , reIn this case, the spacing sulting in a mask-driven sparseness of the array. The solution of (16) can be carried out by using different numerical or analytical methods. In particular, a combined technique based on the Gauss-Kronrod quadrature formula [31] and the NewtonRaphson root-finding method [32] has been specifically developed within the proposed research activity. The considered approach is stable by construction and computationally efficient, resulting in extremely reduced solving times which are typically in the order of few tens of milliseconds. It is worth noting that the balanced combination between amplitude and position tapering is strongly affected by the requirement on the minimal , and the number of antenna elements. inter-spacing Once the piecewise linear approximation of the posiis derived, the phase quantities tioning distribution can be easily evaluated by enforcing the point-matching or, equivalently, condition at each node , so yielding [see equation (9)]: (17) for

.

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Let us now focus the attention on the selection of the parameter . To this end, a suitable energy criterion can be adopted which satisfies: by determining the value

(18)

being a given threshold controlling the extension of the visibility region relevant to the assigned radiation mask, and hence affecting the accuracy of the proposed antenna placement procedure. Under the assumption that the maximum array aper, the mentioned parameter can be ture is required to be taken as: (19) On the other hand, if no specification on the maximum array . size is provided, can be readily selected to be and , appearing in the exFinally, the quantities pression of the actual array factor in (1) can be computed in a straightforward manner by uniformly sampling the normalized positioning, amplitude and phase functions, respectively, at the indicial barycenters:

Fig. 3. Angular behavior of the synthesized array factor useful to mimic a 20-element Dolph-Chebyshev pattern mask with side-lobe level SLL . The array aperture is specified to be D : .

= 020 dB

= 9 725

TABLE I SYNTHESIS OF A DOLPH-CHEBYSHEV ARRAY WITH SIDE-LOBE LEVEL SLL

= 020 dB

(20) . Thanks to the outlined approach, the for proposed synthesis technique features an excellent versatility in handling aperiodic array design problems with given constraints on the maximum aperture size and the minimum inter-element spacing, with no need for any optimization procedure. III. APPLICATION TO THE SYNTHESIS OF SPARSE ANTENNA ARRAYS A. Validation of the Synthesis Technique The developed technique has been validated by application to the synthesis of a 20-element linear uniform array with Dolphhaving side-lobe level Chebyshev pattern distribution (see Fig. 3) [33]. Under the constraint that , the excitation cothe inter-element spacings are set to be efficients of the array elements have been evaluated by means of the numerical procedure detailed in Section II. In this way, an excellent agreement with the reference amplitude coefficients has been achieved, the maximum relative deviation on the synthesized values being found equal to about 1.25% (see Table I) [33]. In order to verify the effectiveness of the proposed antenna placement algorithm, the radiation pattern mask shown in Fig. 3 has been, also, synthesized by using a uniformly fed aperiodic array topology. In doing so, the antenna aperture has been speci. As outlined in the previous section, fied to be the array size is to be chosen in such a way that all or most of the energy content relevant to the spatial Fourier transform of the mask lies in the spatial range . Subject to the mentioned constraints, the developed deterministic

design method has been applied selecting the number of an. The resulting element positions are listed tennas as in Table I, and plotted in Fig. 4. In particular, the minimum , whereas the avinter-element spacing is . As erage antenna distance has been found to be about it appears from Fig. 3, the array sparseness can be usefully exploited to accurately mimic the desired radiation pattern without using any amplitude tapering of the excitation coefficients. That in turn is important in order to allow all the amplifiers in the

CARATELLI AND VIGANÓ: A NOVEL DETERMINISTIC SYNTHESIS TECHNIQUE FOR CONSTRAINED SPARSE ARRAY DESIGN PROBLEMS

Fig. 4. Antenna element positions relevant to the uniformly fed linear sparse array featuring the radiation pattern shown in Fig. 3. The number of antenna elements is selected to be .

N = 24

TABLE II COMPUTATIONAL BURDEN OF DIFFERENT DESIGN METHODOLOGIES FOR SYNTHESIZING A UNIFORMLY FED APERIODIC ARRAY (SEE FIG. 4)

array feeding network to be operated under the same optimal condition, thus gaining efficiency and reducing the manufacturing costs significantly. The selected example clearly shows the versatility of the developed technique which provides the antenna engineer with an effective array synthesis tool allowing for combined controllable tapering of the illumination distribution and element positioning density and, hence, capable of yielding a number of beneficial degrees of freedom in constrained design problems for a great variety of applications with different requirements on the number of power levels, antenna spacing, number of radiating elements, and array sparseness characteristics (uniform/aperiodic element placement). An additional favorable feature of the considered methodology lies in the relevant fully deterministic formulation, which typically results in reduced computational times and negligible memory usage. In order to point out this important aspect, the problem of synthesizing the 24-element uniformly fed linear sparse array with the desired radiation pattern mask corresponding to the mentioned Dolph-Chebyshev distribution (see Fig. 3) has been, also, addressed by means of the global optimization method discussed in [34], as well as -based procedure detailed in [19]. In doing the evolutionary so, the needed numerical computations have been performed in double-precision floating-point arithmetic on the same workstation equipped with a 2.99 GHz Intel Core Duo processor [35], and 3.25 GByte memory. Notably, the considered approaches yield essentially the same results in terms of antenna positions and excitation coefficients (see Table I) although, as it can be noticed in Table II, the proposed analytical technique, featuring a nearly real-time computing capability, outperforms the alternative ones. The achieved performance is in full accordance with the Nemirovsky-Yudin theorem which states that the complexity and, therefore, the computational burden of numerical optimization algorithms increase dramatically (at exponential rate) with the number of problem unknowns [36]. The degradation of the array characteristics due to the spurious mutual coupling is neglected in the presented research. However, in order to assess the sensitivity of the proposed design tool to such antenna non-ideality, a dedicated numerical

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investigation has been carried out. To this end, the radiation properties of the synthesized uniformly fed linear sparse array identical perfectly conducting dipoles (see consisting of Fig. 4) have been analyzed in a rigorous way by using the full-wave electromagnetic field solver CST Microwave Studio [37] based on the finite integration technique (FIT). In particular, the length, radius, and feeding delta gap of the individual , , dipole have been selected to be , respectively, in such a way as to achieve and a reasonably good performance in terms of return loss (with ) in the array conrespect to the reference impedance figuration, namely at the , denoting the speed central working frequency of light in free space. Under such assumption, the maximum parasitic coupling level between the radiating elements has , hence resulting been found to be in a significant deviation from ideal antenna operation. Despite of that, as it can be noticed in Fig. 3, the angular behavior of the synthesized array factor is not very severely impacted, except for the first side-lobe level, showing that the presented antenna placement technique yields pretty accurate results in realistic operative scenarios too. It is, however, to be stressed out that the rigorous modeling of mutual coupling in sparse arrays is surely not trivial and strongly problem-dependent. This important point is to be carefully addressed in future research by exploring suitable extensions of the proposed analytical formu[27]. lation based on the concept of approximate/average In this respect, one can also readily infer that, in array design contexts adversely affected by a large parasitic coupling level and/or number of radiating elements, the developed algorithm may be usefully adopted as an effective preconditioner in advanced hybrid deterministic/metaheuristic antenna placement methodologies in order to derive a well-conditioned initial array configuration useful to enhance convergence in terms of the number of iterations within the metaheuristic procedure and, consequently, to reduce the total computational time required to obtain a converged solution of the problem. B. Constrained Design of a Planar Antenna Array for FMCW Radar Applications The presented synthesis technique has been, also, applied to the design of a planar rectangular array for frequency-modulated continuos-wave (FMCW) radar applications. The configuration at hand is considered with respect to a background Cartesian reference frame , the array antenna being located in the plane. A polar reference frame is conveniently adopted for analyzing the far-field radiation properties. The beamwidth of the considered array system at 3 dB level is required to and be 0.9 and 6 along the two principal cut-planes respectively, with a side-lobe level meeting the con. Furthermore, a minimum dition is required. The aim of this inter-element spacing study is to assess the versatility and effectiveness of the proposed method when handling complex antenna problems. The array design has been accomplished by making judicious use of the projection-based methodology discussed in [38]. In and doing so, two linear sparse sub-distributions of antennas located in the plane along the and

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Fig. 5. Total quadratic error  relevant to the planar aperiodic array for radar applications as function of the number of radiators N and normalized aperture size D = . The optimal array configuration is marked with a red star.

directions, respectively, have been introduced. Afterwards, a planar orthogonal lattice has been generated by a parallel replication of the considered sub-arrays, elementary radiators being located at each of the nodes in the grid. As heuristically proven in [38], provided that each coordinate sub-array meets the mask specifications along the relevant cut-plane, the radiation pattern of the complete array assembled according to the orthogonal projection procedure turns to comply to the design requirements along any angular direcin terms of main lobe features and tion. By virtue of such property, the antenna element positions and excitations can be optimized by matching the prescribed along the principal azimuthal planes mask and . In this respect, the following relative quadratic error:

(21) has been thoroughly investigated as a function of the antenna and number of elements of each aperture size sub-array, under the hypothesis of negligible parasitic coupling is the usual Euclidean norm, whereas effects. In (21), denotes the radiation pattern of the individual antenna element which is a resonant microstrip radiator operating at 9.4 GHz [39], whose electromagnetic characteristics are not discussed here for the sake of brevity. In this way, one can where readily determine the parameter domain is minimized (see Fig. 5). As for the considered array structure, the optimal architecture has been found to feature and elements along the and coordinate directions, with aperture size and , respectively. The resulting antenna positions are shown in Fig. 6. Due to the reduced number of radiators in the array aperture, the additional degree of freedom provided by a suitable amplitude tapering in the direction has been exploited in the design procedure in order to meet the specified requirements in terms of

Fig. 6. Antenna element positions relevant to the planar aperiodic array for and N radar applications. The number of radiators is N along the x and y coordinate direction, respectively.

= 66

= 17

Fig. 7. Non-uniform amplitude tapering distribution featured by the planar sparse array for radar applications along the y direction. The quantization into levels is outlined.

3

beamwidth and minimum inter-element spacing. However, as it can be noticed in Fig. 7, the continuous amplitude tapering distribution has been quantified in such a way as to conveniently reduce the number of power levels in the array beam-forming network. The normalized radiation pattern featured by the designed and planar array along the principal cut-planes has been finally analyzed. As it appears from Fig. 8, an excellent agreement between the main beam of the synthesized antenna structure and the desired mask has been achieved. Furthermore, has the design requirement on the side-lobe level been successfully met. C. Synthesis and Experimental Verification of a Linear Aperiodic Array With Uniform-Amplitude Excitations Let us finally consider the problem of synthesizing a linear aperiodic array with uniform-amplitude excitations and matching the radiation mask plotted in Fig. 9. As it can be readily observed, the considered reference mask features a complex angular behavior with a wide radiation null useful to reduce, along the desired spatial direction, possible electromagnetic interference problems with other users or sensitive

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Fig. 9. Angular behavior of the pattern mask and the measured array factor synthesized by means of the proposed antenna placement technique.

Fig. 10. Antenna positions relevant to the 22-element equi-amplitude sparse array featuring the radiation pattern shown in Fig. 9.

Fig. 8. Angular behavior of the array radiation pattern along the vertical cut(a) and ' (b). plane '

=0

= 90

electronic devices, so enhancing the quality of service in the radio link. In order to achieve a good accuracy in the design procedure, the array aperture has been selected to be with a number of radiators . Under such assumptions, the element positions plotted in Fig. 10 have been determined, resulting in an average antenna spacing of about . It is worth noting that the application of the proposed deterministic antenna placement technique to the synthesis of the considered radiation mask leads to a non-uniform phase distribution of the array excitation coefficients, as it can be inferred from the visual inspection of Fig. 11. In this context, the phase tapering may be regarded as a degree of freedom in the design useful to match the antenna specifications. The performance of the synthesized array has been experimentally assessed at the facility of Delft University of Technology. The measurement setup adopted in the acquirement of the antenna radiation patterns is shown in Fig. 12. The array under test, consisting of ultra-wideband antipodal Vivaldi antennas [40], is placed inside an electromagnetic anechoic chamber optimized for -band operations, at close distance from an electromagnetic probe sensor. In this way, the angular behavior of the array factor can be conveniently evaluated by carrying out a conventional near-to-far-field transformation,

Fig. 11. Non-uniform phase tapering of the equi-amplitude linear sparse array featuring the radiation pattern shown in Fig. 9.

and de-embedding the effect of the probe and the individual element pattern. As it appears in Fig. 9, a good agreement with the required mask has been achieved thus demonstrating the effectiveness of the developed design methodology. It is to be pointed out that the deviation on the measured array factor occurring around the observation angle is due to a deep radiation null featured by the adopted field sensor along that specific spatial direction. On the other hand, one can notice that the array characteristics are only marginally affected by the parasitic antenna coupling, whose level has been found to be below 23 dB where the minimum electrical separation within the measurement frequency range [7, 10] GHz is considered.

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REFERENCES

Fig. 12. The experimental setup for the measurement of the array factor pattern. Ultra-wideband antipodal Vivaldi antennas are used as radiating elements.

IV. CONCLUSION A general and computationally efficient antenna placement method for linear aperiodic arrays has been presented. Such concept, allows for the determinmethod, based on the istic derivation of the optimal element density and excitation tapering distributions useful to mimic a desired radiation pattern mask, without resorting to any iterative procedure. In this way, antenna array synthesis problems subject to demanding requirements in terms of maximum aperture size and minimum inter-element spacing can be addressed in a straightforward and computationally inexpensive manner. The developed technique has been successfully validated and assessed by application to the synthesis of complex array factor functions arising in wireless communication and radar applications. In doing so, the impact of the parasitic antenna coupling has been investigated by carrying out dedicated full-wave analyses and experimental measurements. In this way, it has been found out that the proposed design methodology yields reasonably accurate results even in operative scenarios where a significant deviation from ideal antenna operation occurs. A generalization of the presented synthesis approach, useful to handle fully two-dimensional sparse array design problems, is currently under development.

ACKNOWLEDGMENT This research is conducted as part of the Sensor Technology Applied in Reconfigurable systems for sustainable Security (STARS) project. For further information: http://www.starsproject.nl/. Ir. P. J. Aubry from Delft University of Technology is gratefully acknowledged for his technical support in the antenna measurements. The authors would like to thank the anonymous reviewers and associate editor for contributing with their constructive remarks to the improvement of the quality of this work.

[1] M. I. Skolnik, “Nonuniform arrays,” in Antenna Theory, R. E. Collin and F. Zucker, Eds. New York: McGraw-Hill, 1969, pt. I. [2] B. P. Kumar and G. R. Branner, “Design of unequally spaced arrays for performance improvement,” IEEE Trans. Antennas Propag., vol. 47, no. 3, pp. 511–523, Mar. 1999. [3] I. N. Prudyus, E. I. Klepfer, L. Grigoryeva, and L. V. Lazko, “Investigation of the sparse antenna array properties in the radio-imaging systems,” in Proc. 11th Conf. on Microwave and Telecommunication Technology, Sevastopol, Ukraine, Sep. 10–14, 2001, pp. 329–330. [4] B. P. Kumar and G. R. Branner, “Generalized analytical technique for the synthesis of unequally spaced arrays with linear, planar, cylindrical or spherical geometry,” IEEE Trans. Antennas Propag., vol. 53, no. 2, pp. 621–634, Feb. 2005. [5] K. Chen, Z. He, and C. Han, “Design of linear sparse antenna arrays for performance improvement,” in Proc. 4th IEEE Workshop on Sensor Array and Multichannel Processing, Waltham, MA, Jul. 12–14, 2006, pp. 166–170. [6] M. C. Viganó, G. Toso, G. Caille, C. Mangenot, and I. E. Lager, “Sunflower array antenna with adjustable density taper,” Int. J. Antennas Propag., vol. 2009, 10.1155/2009/624035, ID 624035. [7] M. C. Viganó, G. Toso, P. Angeletti, I. E. Lager, A. Yarovoy, and D. Caratelli, “Sparse antenna array for Earth-coverage satellite applications,” presented at the Proc. 4th Eur. Conf. on Antennas and Propagation, Barcelona, Spain, Apr. 12–16, 2010. [8] O. M. Bucci, M. D’Urso, T. Isernia, P. Angeletti, and G. Toso, “Deterministic synthesis of uniform amplitude sparse arrays via new density taper techniques,” IEEE Trans. Antennas Propag., vol. 58, no. 6, pp. 1949–1958, Jun. 2010. [9] M. C. Viganó and D. Caratelli, “Analytical synthesis technique for uniform-amplitude linear sparse arrays,” presented at the IEEE AP-S/ URSI Symp., Toronto, Canada, Jul. 11–17, 2010, 10.1109/APS.2010. 5561703. [10] D. Caratelli, M. C. Viganó, G. Toso, and P. Angeletti, “Analytical placement technique for sparse arrays,” presented at the 32nd ESA Antenna Workshop, Noordwijk, The Netherlands, Oct. 5–8, 2010. [11] D. Caratelli and M. C. Viganó, “Analytical synthesis technique for linear uniform-amplitude sparse arrays,” Radio Sci., 2011, 10.1029/ 2010RS004522. [12] H. Unz, “Linear arrays with arbitrarily distributed elements,” IRE Trans. Antennas Propag., vol. 8, no. 2, pp. 222–223, Mar. 1960. [13] A. L. Maffett, “Array factors with nonuniform spacing parameter,” IRE Trans. Antennas Propag., vol. 10, no. 2, pp. 131–136, Mar. 1962. [14] R. E. Wiley, “Space tapering of linear and planar arrays,” IRE Trans. Antennas Propag., vol. 10, no. 4, pp. 369–377, Jul. 1962. [15] A. Ishimaru, “Theory of unequally-spaced arrays,” IRE Trans. Antennas Propag., vol. 10, no. 6, pp. 691–702, Nov. 1962. [16] G. Caille, Y. Cailloce, C. Guiraud, D. Auroux, T. Touya, and M. Masmousdi, “Large multibeam array antennas with reduced number of active chains,” in Proc. 2nd Eur. Conf. on Antennas and Propagation, Edinburgh, UK, Nov. 11–16, 2007, pp. 1–7. [17] R. Harrington, “Sidelobe reduction by nonuniform element spacing,” IRE Trans. Antennas Propag., vol. 9, no. 2, pp. 187–192, Mar. 1961. [18] F. Hodjat and S. A. Hovanessian, “Nonuniformly spaced linear and planar array antennas for sidelobe reduction,” IEEE Trans. Antennas Propag., vol. 26, no. 2, pp. 198–204, Mar. 1978. [19] R. L. Haupt, “Thinned arrays using genetic algorithms,” IEEE Trans. Antennas Propag., vol. 41, no. 2, pp. 993–999, Feb. 1993. [20] V. Murino, A. Trucco, and C. S. Regazzoni, “Synthesis of unequally spaced arrays by simulated annealing,” IEEE Trans. Signal Processing, vol. 44, no. 1, pp. 119–123, Jan. 1996. [21] F. J. Ares-Pena, J. A. Rodriguez-Gonzalez, E. Villanueva-Lopez, and S. R. Rengarajan, “Genetic algorithms in the design and optimization of antenna array patterns,” IEEE Trans. Antennas Propag., vol. 47, no. 3, pp. 506–510, Mar. 1999. [22] K. Chen, X. Yun, Z. He, and C. Han, “Synthesis of sparse planar arrays using modified real genetic algorithm,” IEEE Trans. Antennas Propag., vol. 55, no. 4, pp. 1067–1073, Apr. 2007. [23] M. Donelli, A. Martini, and A. Massa, “A hybrid approach based on PSO and Hadamard difference sets for the synthesis of square thinned arrays,” IEEE Trans. Antennas Propag., vol. 57, no. 8, pp. 2491–2495, Aug. 2009. [24] G. Oliveri, M. Donelli, and A. Massa, “Linear array thinning exploiting almost difference sets,” IEEE Trans. Antennas Propag., vol. 57, no. 12, pp. 3800–3812, Dec. 2009.

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[25] R. N. Bracewell, The Fourier Transform and Its Applications, 3rd ed. New York: McGraw-Hill, 1999. [26] Y.-W. Kang and D. M. Pozar, “Correction of error in reduced sidelobe synthesis due to mutual coupling,” IEEE Trans. Antennas Propag., vol. 33, no. 9, pp. 1025–1028, Sep. 1985. [27] D. F. Kelley and W. L. Stutzman, “Array antenna pattern modeling methods that include mutual coupling effects,” IEEE Trans. Antennas Propag., vol. 41, no. 12, pp. 1625–1632, Dec. 1993. [28] D. M. Pozar, “The active element pattern,” IEEE Trans. Antennas Propag., vol. 42, no. 8, pp. 1176–1178, Aug. 1994. [29] G. D’Elia, F. Soldovieri, and G. Di Massa, “Mutual coupling compensation in a power synthesis technique of planar array antennas,” IEE Proc. Microw. Antennas Propag., vol. 147, no. 2, pp. 95–99, Apr. 2000. [30] Q.-Q. He, B.-Z. Wang, and W. Shao, “Radiation pattern calculation for arbitrary conformal arrays that include mutual-coupling effects,” IEEE Antennas Propag. Mag., vol. 52, no. 2, pp. 57–63, Apr. 2010. [31] P. K. Kythe and M. R. Schaferkotter, Handbook of Computational Methods for Integration. Boca Raton: CRC Press, 2004. [32] C. T. Kelley, Solving Nonlinear Equations with Newton’s Method. Philadelphia: SIAM, 2003. [33] C. L. Dolph, “A current distribution for broadside arrays which optimizes the relationship between beamwidth and sidelobe level,” Proc. IRE, vol. 34, no. 6, pp. 335–348, Jun. 1946. [34] W. Doyle, “On approximating linear array factors,” RAND Corp. Memorandum RM-3530-PR. Santa Monica, CA, Feb. 1963. [35] Intel Corporation, “Core Duo Processor,” in [Online]. Available: http:// www.intel.com/ [36] A. S. Nemirovsky and D. B. Yudin, Problem Complexity and Method Efficiency in Optimization. New York: Wiley-Interscience, 1983. [37] CST Computer Simulation Technology, Microwave Studio Suite [Online]. Available: http://www.cst.com/ [38] M. C. Viganó, I. E. Lager, G. Toso, C. Mangenot, and G. Caille, “Projection based methodology for designing non-periodic, planar arrays,” in Proc. 38th Eur. Microwave Conf., Amsterdam, The Netherlands, Oct. 27–31, 2008, pp. 1624–1627. [39] I. E. Lager, C. Trampuz, M. Simeoni, and L. P. Ligthart, “Interleaved array antennas for FMCW radar applications,” IEEE Trans. Antennas Propag., vol. 57, no. 8, pp. 2486–2490, Aug. 2009. [40] X. Zhuge, A. Yarovoy, and L. P. Ligthart, “Circularly tapered antipodal Vivaldi antenna for array-based ultra-wideband near-field imaging,” in Proc. 6th Eur. Radar Conf., Rome, Italy, Sep. 30–Oct. 2 2009, pp. 250–253.

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Diego Caratelli was born in Latina, Italy, on May 2, 1975. He received the Laurea (summa cum laude) and Ph.D. degrees in electronic engineering from “La Sapienza” University of Rome, Italy, in 2000 and 2004, respectively. In 2005 he joined as a Contract Researcher the Department of Electronic Engineering, “La Sapienza” University of Rome. Since 2007 he is with the Microwave Technology and Systems for Radar (MTSR) Group of Delft University of Technology, the Netherlands, as a Senior Researcher. His main research activities include the design, analysis and experimental verification of printed microwave and millimeter-wave passive devices and wideband antennas for satellite, WLAN and GPR applications, the development of analytically based numerical techniques devoted to the modeling of electromagnetic field propagation and diffraction processes, as well as the analysis of EMC/EMI problems in sensitive electronic equipment. Dr. Caratelli was the recipient of the Young Antenna Engineer Prize at the 32th European Space Agency Antenna Workshop. He received the 2010 Best Paper Award from the Applied Computational Electromagnetics Society (ACES). He serves as reviewer for several international journals, and is a member of ACES and the Italian Electromagnetic Society (SIEm).

Maria Carolina Viganó (M’07) was born in Florence, Italy on February 15, 1982. She received the Laurea (summa cum laude) degree in telecommunication engineering from the University of Florence, Italy, in 2006. Since October 2005, she has been working at the European Space Agency first as a Stagier, then as YGT, and as Contractor. Her research interest includes phased array, satellite communication antennas and synthesis techniques for non-regular arrays. She was co-recipient of the 2010 Young Antenna Engineer Prize at the 32th European Space Agency Antenna Workshop. In January 2011 she completed her Ph.D. degree with the title “Sunfower array antenna for multi-beam satellite applications” cosponsored by Delft University of Technology, Thales Alenia Space Toulouse, and ESA-ESTEC. She is currently working as R&D antenna engineer at JAST, ViaSat Inc.

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Evolutionary Design of Wide-Band Parasitic Dipole Arrays Giovanni Andrea Casula, Member, IEEE, Giuseppe Mazzarella, Senior Member, IEEE, and Nicola Sirena

Abstract—An innovative antenna design technique, based on evolutionary programming, has been devised and applied to the design of broadband parasitic wire arrays for VHF-UHF bands with a significant gain. The chosen fitness function includes far-field requirements, as well as wideband input matching specifications. The latter requirements, which must be present in every useful antenna design, allow to stabilize the algorithm, and to design both optimal and robust antennas. The designed antennas show significant improvements over existing solutions (Yagi and LPDA) for the same frequency bands. Index Terms—Automatic design, evolutionary programming, parasitic dipole array.

I. INTRODUCTION INCE their first introduction in 1926 [1], [2], [36], parasitic dipole arrays (PDAs), universally known as Yagi or Yagi-Uda antennas, have been very popular as VHF-UHF antennas because of their advantages. A PDA is thin and easy to realize, and therefore amenable to low-cost mass production. Its transverse size is around in the E-Plane and significantly smaller in the H-Plane. On the other hand, its length can be relevant, but directly linked to the required gain. Despite of its simplicity, a PDA allows a gain from 7 dB up to 18 dB (for array ranging from 3 to about 20 elements) [3]. This quite high gain (and an even larger Front-toBack ratio) is the reason for the ubiquitous use of Yagi antennas. The main drawback of Yagi antennas is their bandwidth. According to [3], the bandwidth of an array of Yagis is at best 10%. Actually, the bandwidth can be larger than this, up to 20% (see, for examples [4], [5]) but also this value is typical of a narrowband antenna. Note that we consider here the gain bandwidth as the primary limitation of the array, since no simple countermeasure can be used to improve it. It is well known that a Yagi antenna has a reduced input bandwidth, too, but it is usually possible to devise a suitable matching network to increase the input bandwidth to be larger than the gain one (i.e., up to 20%–30%) without affecting the other antenna performances. Therefore the input bandwidth of actual antennas is, including the matching network, usually larger than the gain bandwidth.

S

Manuscript received September 21, 2010; revised April 13, 2011; accepted May 06, 2011. Date of publication August 12, 2011; date of current version November 02, 2011. This work was supported in part by Regioe Autonoma della Sardegna, under Contract CRP1_511 titled “Valutazione e utilizzo della Genetic Programming nel progetto di strutture a radiofrequenza e microonde.” The authors are with the Dipartimento di Ingegneria Elettrica ed Elettronica, Cagliari 09123, Italy (e-mail: [email protected]; mazzarella@diee. unica.it; [email protected]). Digital Object Identifier 10.1109/TAP.2011.2164185

Considerable work has been spent (and reported in the literature) to optimize Yagi antenna behavior, using many different approaches. Apart some applications of standard design techniques [4]–[6], Yagi antennas have been optimized using gradient techniques [7]–[9], computational intelligence [10], particle swarm optimization (PSO) [11]–[13], differential evolution [14]–[18], and genetic algorithms [19], [20]. In [21] an evolutionary design based on an automated antenna optimization system that uses a fixed Yagi-Uda topology and a byte-encoded antenna representation is presented. It is worth noting that, for the same VHF-UHF band and antenna size, another antenna concept is available, with opposite gain and bandwidth performances, namely log-periodic dipole antenna (LPDA), proposed by DuHamel in the 1957 [22]. LPDA can be designed with a bandwidth up to a decade (the ratio between the lower and the upper frequency is about 1:10), incomparably larger than a PDA, but with a typical gain around 10 dBi [3]. Moreover, LPDA realization is more difficult than PDA, since all dipoles must be fed (and with alternate sign), and some dipoles should be very thin. What is still lacking is an antenna with an intermediate behavior, i.e. a significant gain, but with a bandwidth considerably larger than a typical Yagi. Aim of this paper is to fill the gap between PDA and LPDA. This is obtained starting from the Yagi side, i.e., looking for a PDA with a significantly larger bandwidth in order to retain both the high gain and the relative easiness of realization of PDA. However, this can be achieved only with a different approach to the array optimization, which allows to explore more general, and unusual, PDA configurations than standard optimizations used so far [4]–[21]. This latter requirement can be fulfilled resorting to a different optimization strategy, first proposed by Koza [23] in a paper titled “Genetic Programming: On the Programming of Computers by Means of Natural Selection” and called by him “genetic programming”. Actually this name resembles too closely another optimization approach, but with marked differences with the Koza approach, namely the genetic algorithm (GA), already widely used in antenna design [20], [21], [24]. In order to better grasp the differences between them, we can say that GA works on the “nucleotide” (i.e. bit) level, in the sense that the antenna structure is completely defined from the beginning, and only an handful of parameters remains to be optimized. The approach proposed by Koza, and adopted in this paper, assumes no “a priori” structure. Instead, it builds up the structure of the individuals as the procedure evolves. Therefore it operates at the “organ” (i.e. physical structure) level, a far more powerful level (and a level not attained by the biological evolution of individuals, of course). As a consequence, its solution space has the power of the continuum, while the GA solution space is a dis-

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CASULA et al.: EVOLUTIONARY DESIGN OF WIDE-BAND PARASITIC DIPOLE ARRAYS

crete one, so it is a very small subspace of the former. Moreover, the typical evolution operators work on actual physical structures, rather than on sequences of bits with no intuitive link to the antenna shape. In order to avoid any misunderstanding, we speak, in this paper, of structure-based evolutionary design (SED) to name the Koza strategy. The enormous power of SED fully allows the exploration of more general PDA shapes. The main drawback is the ill-posedness of the SED, which calls for a regularization procedure. As we will show in the following, the regularization can be obtained at virtually no cost, since it can be provided by suitable input impedance constraints, which are already among the optimization requirements. It is of no surprise that, since its first proposal [23], SED has been widely used in different optimization problems [25], but antenna design is not among them. As a matter of fact, apart from the pioneeristic work of Jones [26], this is, to the best of our knowledge, the first use of genetic programming to antenna design. II. STRUCTURE-BASED EVOLUTIONARY DESIGN OF WIRE ANTENNAS The SED approach mimics the behavior of the natural evolution for the search of the individual showing the best adaptation to the local environment (in our case, to the requirement we set). Each individual is a “computer1 program”, i.e., a sequential set of unambiguous instructions completely (and uniquely) describing the physical structure of an admissible antenna, and its realization. This is a marked difference with GA, where an individual is only a set of physical dimensions and other parameters. In the practical implementation of SED, populations of thousands of computer programs, which describe antennas (and are traditionally stored as tree structures) are genetically bred: this breeding is made using the Darwinian principle of survival and reproduction of the fittest, along with recombination operations appropriate for mating computer1 programs. Tree structures can be easily evaluated in a recursive manner; every tree node has an operator function and every terminal node has an operand, making mathematical expressions easy to evolve and to be evaluated. The starting point is an initial population of randomly generated computer programs composed of functions and terminals appropriate to the problem domain. The functions may be standard arithmetic operations, standard programming operations, mathematical or logical functions, or even domain-specific functions. The main (meta)-operators used in SED are crossover and mutation. Crossover is applied on an individual by simply switching one of its nodes with another node from another individual in the population. With a tree-based representation, replacing a node means the replacement of the whole branch. This adds greater effectiveness to the crossover operation, since it exchanges two actual subantennas with different dimensions. The expressions resulting from crossover can be either quite close or very different from their initial parents. The sudden jump from an individual to a 1The word “computer” has been used here to enforce the unambiguous nature of the instruction set, which is a key to the use of this description in an automatic optimization procedure.

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very different one is a powerful trap-escaping mechanism. As a matter of fact, we experimented no trap problems in our extensive testing of SED. Mutation affects an individual in the population, replacing a whole node in the selected individual, or just the node’s information. To maintain integrity, operations must be fail-safe, i.e. the type of information the node holds must be taken into account. The performance, in the particular problem environment, of each individual computer program in the population is measured by its “fitness”. The nature of the fitness measure depends on the problem at hand. In antenna design, the most intuitive fitness function can be built as the “distance” between actual and required far-field behavior [27] or, even more simply, as the antenna gain or SNR [28]. However, this is not the case for SED. The solution space, i.e., the set of admissible solution in which the procedure looks for the optimum, is composed, in our case, of every PDA antenna with no limit on the number of wire segments, nor on the size or orientation, represented as real numbers. As a consequence, its size has the power of the continuum. This is the main advantage of SED, since it allows exploring, and evaluating, general wire antenna configurations. But it poses also very hard problem, since it can lead to a severely ill-conditioned synthesis problem. As a consequence, a naive implementation usually does not work, since different starting populations lead to completely different final populations, possibly containing only local minima. A suitable stabilization is therefore needed. This role can be accomplished by suitable antenna requirements, or forced by imposing further constraints, not included in the antenna requirements. It is clear that, whenever possible, the former one are the better choice, and should be investigated first. Even at a first look, it appears that far-field requirements are unable to stabilize the problem, since the far-field degrees of freedom are order of magnitude less than those of the solution space, so that a huge number of different antennas gives the same far field. As a matter of fact, a wire segment whose length is a small fraction of the wavelength can be added or eliminated without affecting the far field. We must therefore revert to near-field requirements. Among them, the easiest to implement, and probably the most important, is the requirement on the input impedance over the required bandwidth. Since this constraint is a “must-be” in order to get a usable solution, we get the required stabilization at virtually no additional cost. As a further advantage, a low input reactance over the bandwidth prevents from superdirective solutions even when a reduced size is forced as a constraint. The SED is a strategy to devise the best individual, in terms of their closeness to the constraints set in the design. This closeness is evaluated as a “fitness” function. The details of the fitness function we have chosen for PDA design are widely described in the next section. However, at this point it must be stressed that the fitness function depends in an essential way on the electromagnetic behavior of the individual. Since we are interested in assess SED as a viable, and very effective, design tool, we accurately try to avoid any side-effect stemming out from the e.m. analysis of our individuals. Therefore we rely on known, well-established and widely used

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antenna analysis programs. Since our individuals are wire antennas, our choice has fallen on NEC-2. The Numerical Electromagnetics Code (NEC-2) is a MoMbased, user-oriented computer code for the analysis of the electromagnetic response of wire antennas and other metallic structures [29]. It is built around the numerical solution of the integral equations for the currents induced on the structure. This approach allows taking well into account the main second-order effects, such as conductor losses and the effect of lossy ground on the far field. This allows to evaluate the actual gain, and not the array directivity, with a two-fold advantage. First of all, the gain is the far-field parameter of interest and, second, this prevents from considering superdirective antennas, either during the evolution and as final solution, which is even worse. NEC has been successfully used to model a wide range of antennas, with high accuracy [30]–[33] and is now considered as one of the reference e.m. software [29], [34], [35]. However, since SED is by no means linked, or tailored, to NEC, a different, and most effective, EM software could be used, to reduce the total computational time.

III. PDA DESIGN AND FITNESS FUNCTION The goal of the design process is to develop a PDA which fulfills the desired requirements for both Gain and VSWR in a frequency band as wide as possible, and with the smallest size. Each PDA is composed of a driven element and a fixed number of parasitic elements. In order to get transverse dimensions close to those of Yagi and LPDA, and to ease the realization, the centers of the elements are arranged on a line, with the driven element at the second place of the row. In Yagi terminology, we use a single reflector. We actually have experimented with more reflectors but, exactly as in standard Yagi, without any advantage over the single-reflector configuration. Each element is symmetric w.r.t its center, and the upper part is represented, in the algorithm, as a tree. Each node of the tree is an operator belonging to one of the following classes: a) add a wire according to the present default directions and length; b) set the end of the last added wire as a branching point; c) modify the present default directions and length; d) stretch (or shrink) the last added wire. This mixed representation largely increases the power of the standard genetic operation (mutation and cross-over). Each element can evolve independently from the others. Of course, after each complete PDA is generated, we test its geometrical coherency. Incoherent antennas (e.g., an antenna with two elements too close, or even intersecting) are discarded. The SED approach has been implemented in Java, while the analysis of each individual has been implemented in C++ (using the freeware source code Nec2cpp) and checked using the freeware tool 4nec2. The integration with NEC-2 has mainly been achieved through three classes: 1) a parser for the conversion of the s-expressions, represented as n-ary trees, in the equivalent NEC input files;

Fig. 1. Flowchart of the whole evolutionary design.

2) a NecWrapper which writes the NEC listing to a file, launches a NEC2 instance in a separate process, and parses the output generated by NEC; 3) an Evaluator which calculates the fitness using the output data generated by NEC. The main steps of the whole evolutionary design can be summarized by the flowchart in Fig. 1. The evaluation procedure for each individual (i.e. for each antenna) can be described by the flowchart in Fig. 2. The process requires, as inputs, the minimum and maximum operational frequencies of the antenna, the number of frequency points NF to be evaluated, the metal conductivity and the maximum size of the antenna. Actually, the generated antenna can overcome the bounding box dimensions, but with a penalty directly proportional to the excess size. To achieve the design goal a fitness has been developed, able to direct the evolution process toward a structure with reduced size, with the highest endfire gain, and with an input match as better as possible in the widest frequency range. The increase in a parameter (i.e. the gain) usually results in a reduction in the other ones (i.e. frequency bandwidth and input matching), thus we have to devise an elaborate trade-off in reaching such a design goal. The resulting fitness function is quite complicate, and it is composed by several secondary objectives overlapped to the main goal; these objectives are expressed by suitable weights modeling trade-offs between different goals. These relative weights have been modeled by linear relations to avoid discontinuities and thus reducing the probability of local maxima of the fitness, which trap the evolution process. It is worth noting, moreover, that cross-over

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Fig. 2. Flowchart of the evaluation procedure for each individual.

(and sometimes mutation) in SED results in sibling antennas significantly different from the parents. This careful choice of the relative weights should therefore minimize the risk that the procedure ends up in a local trap. We choose to maximize gain as the main goal of the fitness. Since we want to maximize the endfire gain, the radiation pattern has been divided into 4 regions: 1) The endfire direction:

2) The back direction:

frequency points, spanwherein the average is taken over the ning the whole bandwidth of interest. In (1) is the endfire depends on the input impedance of the PDA: gain and If If Otherwise is a weight proportional to the difference between the imagiand the real part of the array input impedance. nary part in the The average gains over all other regions, namely in the front region and in the rear back direction, region, are then computed. An “effective” endfire gain is then obtained properly weighting each gain:

3) The FRONT region: (2) (where and take into account the desired main lobe amplitude) 4) The REAR region:

Our goal is the maximization of the gain in the region 1 while minimizing the gains in the other 3 regions, with all the gains expressed in dB. Since we want to optimize the antenna in a certain frequency bandwidth, we start computing a suitable weighted on region 1: average gain (1)

The weigths , and are chosen through a local tuning in order to get the maximum gain in the endfire direction and an acceptable radiation pattern in the rest of the space. In our case, we obtained the following values: , and . To design a wideband antenna, we must add some parameters that take into account the antenna input matching, and therefore we have to introduce appropriate weights related to the antenna input impedance. Holding gain weights fixed, the other parameters concerning input matching are added one by one choosing each weight through a further local tuning. is therefore furthermore modified taking into The account: (averaged over the BW), and a) The values of their normalized variance;

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Fig. 3. Designed antenna structure.

b) The SWR over all the required bandwidth according to the following guidelines: if 1) A step is introduced, with a weight , and otherwise, to boost up structures with ; is introduced, related to , 2) A weight forcing the evolution process to structures with an as small as possible; is introduced, related to 3) A weight , to advantage structures with a low Q factor; 4) A weight is introduced, related to , to boost up structures with a high real part of the input impedance (as long as it is lower than 300 ); are introduced, inversely 5) Weights related to the normalized variance of and , to advantage structures with a regular impedance behaviour; 6) A sequence of small steps, related to the SWR (with a between 30 for an and 0.005 weight for an ), is introduced to first boost up and then hold the evolution in areas of the evolution space with good SWR values. , expressed At this point we have a modified average gain by:

Fig. 4. Plot of convergence of the designed antenna in Fig. 3.

rors and bad weather conditions (for example movements due to wind effect). On the other hand, this robustness test is quite time-consuming. Therefore it is performed only on antennas already showing good performances. The final population is graded according to their value. Like all like all evolutionary algorithms, SED is able to evolve and optimize only the parameters that can be observed and controlled by the designer. For this reason, a high number of parameters in the fitness pushes the evolution process to more refined individuals with better performances. It is obviously possible to use a fitness function which is significantly simpler than (3), but the results are very far from the one shown here, in terms of performances. IV. RESULTS

(3) and are the normalized variance of and of where , respectively. (where is a suitably high gain, The difference needed only to work with positive fitness values) is then modulated taking into account both the Q factor (obtained as the ratio between the imaginary part and the real part of the array input impedance at the central frequency) and the structure size to get a particular fitness . The individual generated by the genetic process associated to a fitness higher or very close to the best fitness obtained as yet, are then perturbed (assigning random relocations to array elements) and analyzed. Two different perturbed antennas are considered for each individual, and the final fitness is the partial fitness averaged over all the initial and perturbed configurations. This random relocation allows getting robust structures respect to both constructive er-

The automated design of PDAs using SED has been applied to several antennas, with different maximum sizes, number of elements, and operation frequencies, always obtaining very good results. In this section we present a PDA with 20 elements: 1 reflector, 1 driven element and 18 directors. The operation frequency is 500 MHz, and the requested bandwidth is of 70 MHz (i.e. 14%, from 475 MHz to 545 MHz). The best antenna is represented in Fig. 3, and its shape is typical of all antennas designed using our SED optimization technique. The antenna size is very small, since it fits in a box large , being the space wavelength at the operation frequency of 500 MHz. Its SWR is less than 2 in the whole bandwidth of 70 MHz, and its gain is above 18 dB. The antenna has been designed using a population size of 1000 individuals, with a crossover rate set to 60%, and a mutation rate set to 40%. Its convergence plot is shown in Fig. 4, and it appears that 300 generations are enough to reach convergence.

CASULA et al.: EVOLUTIONARY DESIGN OF WIDE-BAND PARASITIC DIPOLE ARRAYS

Fig. 5. (a) Gain comparison between the PDA Designed Antenna and a standard Yagi with the same size (and 9 elements). (b) SWR comparison between the PDA Designed Antenna and a standard Yagi with the same size (and 9 elements).

The comparison of our designed PDA with existing Yagis is difficult. It is worth noting that, for a parasitic antenna, an increase in the number of elements adds little to the antenna complexity. Therefore we think that the most significant comparison is with a standard Yagi with the same size of our PDA (about in the endfire direction). This standard Yagi is composed of only 9 elements, and its gain and SWR, compared to our optimized PDA, are shown in Fig. 5. It is clear that the standard is about 35 MHz (7% compared Yagi bandwidth to 14%) with a gain between 12 and 13 dB, i.e. at least 5 dB less than ours, over the whole bandwidth. To better evaluate the performances of our designed PDA, however, we have compared it also with a standard Yagi antenna with the same number of elements, i.e., 20. Though this antenna ), it has (see Fig. 6) a is very large (its size is about quite narrow bandwidth (its gain is above 15 dB in a bandwidth smaller than 10%, and even its SWR is less than 2 in a bandwidth of about 9%) if compared with our PDA. The PDA antenna of Figs. 5 and 6 has been designed choosing a fitness which pushes individuals toward higher Gain giving lesser importance to input matching. It is possible, by suitably choosing the fitness weights, to design a PDA antenna which favors individuals with better

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Fig. 6. (a) Gain comparison between the PDA Designed Antenna and a standard Yagi with the same number of elements, 20, and a far larger size (6 versus 1:72 ). (b) SWR comparison between the PDA Designed Antenna and a standard Yagi with the same number of elements, 20, and a far larger size (6 versus 1:72 ).

input matching. The performances of such an antenna are shown in Fig. 7. The bandwidth (with ) has increased to 150 MHZ (30%), and its gain is only a few dB less than the first optimized PDA antenna. It is worth noting that the size of the antenna with a larger input bandwidth is the same of the antenna with a higher gain. In Fig. 8 is shown the F/B ratio of both the PDA designed antennas, which is very close also to standard Yagis’ F/B. This comparison shows that, though the PDA we have designed appear to be more difficult to realize than a standard Yagi, they allow significantly better performances in a larger bandwidth, both on input matching, gain and F/B ratio. Furthermore, it is significantly smaller than standard Yagis. The computational cost of SED, like that of many other random optimization techniques, is the computational cost required to evaluate each individual. Therefore different techniques, such as SED and standard GA, can have different cost as long as they evaluate a different number of individuals, or more complex ones. For the example presented in Fig. 4, SED requires NEC evaluations of individuals. GA with comparable antenna

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Fig. 9. Standard Deviation of SWR and Gain of the PDA Designed Antenna in Fig. 3, considering 100 randomly perturbed configurations.

Fig. 7. (a) Gain of the PDA Designed Antenna with a fitness pushing towards a larger SWR bandwidth. (b) SWR of the PDA Designed Antenna with a fitness pushing towards a larger SWR bandwidth.

NEC unknown is more or less the same for both approaches, depending essentially on the antenna size, we can conclude that SED has a computational cost comparable, or slightly larger than standard GA. On the other hand, SED allows to explore a far larger solution space, so we can consider SED significantly more effective than standard GA, if we take into account both the final result and the corresponding computational cost. A comparison between SED and real-based algorithms like PSO ([13]) and DE ([14]–[16]), shows that both the computational cost and the complexity are of the same order of magnitude, but the performances obtained by them are not as good as the ones obtained using SED. In fact we are able to get a wideband antenna with a very high gain, i.e. we both maximize antenna gain and minimize SWR and antenna size within the whole bandwidth (which is a wide bandwidth, equal to 30%), while the procedures based on PSO and DE described in [13]–[16] work only at the center frequency. Nonetheless, the results of the latter have lower performances w.r.t. SED. Finally, in order to demonstrate the robustness of the final design obtained by SED to fabrication tolerances, a hundred random perturbations of the reference antenna of Fig. 3 have been obtained perturbing the ends of each arm of the antenna with a random value between 2 and 2 mm. The standard deviations of the SWR and gain are shown in Fig. 9 and are expressed in percentage with respect to the values of the unperturbed antenna shown in Fig. 3. Despite of such huge perturbation, the designed PDA is so robust that the behavior is essentially the same. V. CONCLUSION

Fig. 8. F/B ratio comparison between the PDA Designed Antenna with a fitness pushing towards a larger Gain bandwidth and one towards a larger SWR bandwidth.

size (such as the one described in [20]) requires a likely, or even larger, number of NEC evaluations. Since also the number of

Standard solutions for VHF-UHF antennas, i.e Yagi and logperiodic dipole antennas, are known since the 50’s, and have an orthogonal behavior (large gain—small bandwidth versus small gain—large bandwidth). In order to overcome this situation, a new design procedure for parasitic dipole arrays, based on the evolutionary programming concept, has been devised. Its main advantage is the ability to explore a far larger solution space than standard optimization algorithms. Inclusion of input matching requirements prevents from ill-posedness, a danger always present when the solution

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space is so large. The presented results show that a PDA with high gain, reduced size and a moderate bandwidth can be designed, and its performances are robust w.r.t. tolerances or atmospheric events. APPENDIX In this section we show an example of implementation of an individual of the population. The S-expression of the individual is the following: S-expression: Tree 0: (StretchAlongZ 1.3315124586134857 (Wire 0.42101090906114413 1.0

Fig. 1-A. Antenna structure corresponding to the S-expression of the example.

(StretchAlongX 0.5525837649288541 (StretchAlongY 1.4819461053740617 (RotateWithRespectTo_Y 0.3577743384222999 END)))))

allows transverse dimensions to get close to those of Yagi and LPDA, and to ease the realization of the designed antenna. REFERENCES

Tree 1: (Wire 0.5581593081319647 1.0 (RotateWithRespectTo_X 0.44260816356142224 (RotateWithRespectTo_Z 0.08068272691709244 (StretchAlongZ 0.7166185389610261 (StretchAlongX 1.42989629787443 (StretchAlongZ 1.346598788775623 END)))))) Tree 2: (Wire 0.3707701115469606 1.0 (RotateWithRespectTo_X 0.5262591815805174 (RotateWithRespectTo_Z 0.7423883999218206 (RotateWithRespectTo_Z 0.07210315212202911 END)))) The corresponding NEC-2 input file is the following: GW 1 17 0.00E00 0.00E00 0.00E00 1.44E-02 1.33E-01 1.36E-03

01.34E-02

GW 2 22 01.38E-01 0.00E00 0.00E00 0.00E00 1.66E-01 1.36E-03

01.25E-01

GW 3 15 1.21E-01 0.00E00 0.00E00 1.21E-01 0.00E00 1.18E-01 1.36E-03 GX 4 001 GE

The corresponding antenna, composed by three elements, is shown in Fig. 1-A. The antenna structure shown in Fig. 1-A is composed of three wires: one reflector, one driven elements, and one director. We choose to arrange the centers of the elements on a line, with the driven element at the second place of the row. This choice

[1] S. Uda, “Wireless beam of short electric waves,” J. IEE (Japan), no. 452, pp. 273–282, Mar. 1926. [2] H. Yagi, “Beam transmission of ultra short waves,” IRE Proc., vol. 16, pp. 715–741, Jun. 1928. [3] R. C. Johnson and H. Jasik, Antenna Engineering Handbook—Second Edition. New York: Mc Graw-Hill, 1984. [4] L. B. Cebik, Wide-Band Yagi Notes vol. 1 and 2, W4RNL [Online]. Available: http://www.antennex.com/Sshack/books.htm [5] L. B. Cebik, Long-Boom Yagi Studies W4RNL [Online]. Available: http://www.antennex.com/Sshack/books.htm [6] B. H. Sun, S. G. Zhou, Y. F. Wei, and Q. Z. Liu, “Modified two-element Yagi-Uda antenna with tunable beams,” Progr. Electromagn.s Res., vol. 100, pp. 175–187, 2010. [7] D. K. Cheng and C. A. Chen, “Optimum element spacings for Yagi-Uda arrays,” IEEE Trans. Antennas Propag., vol. AP-21, pp. 615–623, Sep. 1973. [8] C. A. Chen and D. K. Cheng, “Optimum element lengths for Yagi-Uda arrays,” IEEE Trans. Antennas Propag., vol. AP-23, pp. 8–15, Jan. 1975. [9] D. K. Cheng, “Gain optimization for Yagi-Uda arrays,” IEEE Antennas Propag. Mag., vol. 33, pp. 42–45, Jun. 1991. [10] N. V. Venkatarayalu and T. Ray, “Optimum design of Yagi-Uda antennas using computational intelligence,” IEEE Trans. Antennas Propag., vol. 52, pp. 1811–1818, 2004. [11] Bayraktar, P. L. Werner, and D. H. Werner, “The design of miniature three-element stochastic Yagi-Uda arrays using particle swarm optimization,” IEEE Antennas Wireless Propag. Lett., vol. 5, no. 1, pp. 22–26, Dec. 2006. [12] M. Rattan, M. S. Patterh, and B. S. Sohi, “Optimization of gain, impedance, and bandwidth of Yagi-Uda array using particle swarm optimization,” Int. J. Antennas Propag., vol. 2008. [13] S. Baskar, A. Alphones, P. N. Suganthan, and J. J. Liang, “Design of Yagi-Uda antennas using comprehensive learning particle swarm optimization,” in IEE Proc. Microwaves, Antennas Propag., vol. 152, no. 5. [14] S. K. Goudos, K. Siakavara, E. E. Vafiadis, and J. N. Sahalos, “Pareto optimal Yagi-Uda antenna design using multi-objective differential evolution,” Progr. Electromagn. Res., vol. 105, pp. 231–251, 2010. [15] J. Y. Li and J. L. Guo, “Optimization technique using differential evolution for Yagi-Uda antennas,” J. Electromagn. Waves Applicat., vol. 23, pp. 449–461, 2009. [16] Y. Yan, G. Fu, S. Gong, X. Chen, and D. Li, “Design of a wide-band Yagi-Uda antenna using differential evolution algorithm,” presented at the Int. Symp. on Signals Systems and Electronics (ISSSE), Sep. 2010. [17] J.-Y. Li, “Optimizing design of antenna using differential evolution,” presented at the Proc. Microwave Conf. Asia-Pacific, 2007.

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[18] L. Zhang, Y.-C. Jiao, H. Li, and F.-S. Zhang, “Antenna optimization by hybrid differential evolution,” Int. J. RF Microw. Comput.-Aided Engrg., vol. 20, no. 1, Jan. 2010. [19] Y. Kuwahara, “Multiobjective optimization design of Yagi-Uda antenna,” IEEE Trans. Antennas Propag., vol. 53, pp. 1984–1992, 2005. [20] E. A. Jones and W. T. Joines, “Design of Yagi-Uda antennas using genetic algorithms,” IEEE Trans. Antennas Propag., vol. 45, no. 9, pp. 1386–1392, Sep. 1997. [21] J. D. Lohn, W. F. Kraus, D. S. Linden, and S. Colombano, “Evolutionary optimization of Yagi-Uda antennas,” in Proc. 4th Int. Conf. on Evolvable Systems: From Biology to Hardware, 2001, pp. 236–243. [22] R. H. DuHamel and D. E. Isbell, “Broadband logarithmically periodic antenna structures,” in IRE Nat. Conv. Rec., 1957, pp. 119–128, part I. [23] J. R. Koza, Genetic Programming: On the Programming of Computers by Means of Natural Selection. Cambridge, MA: MIT Press, 1992. [24] E. A. Jones and W. T. Joines, “Genetic design of linear antenna arrays,” Antennas Propag. Mag., vol. 42, no. 3, Jun. 2000. [25] J. R. Koza, M. A. Keane, M. J. Streeter, W. Mydlowec, J. Yu, and G. Lanza, Genetic Programming IV: Routine Human-Competitive Machine Intelligence. Berlin: Springer, 2003. [26] E. A. Jones, “Genetic design of antennas and electronic circuits,” Ph.D. dissertation, Department of Electrical and Computer Engineering, Duke University, Durham, NC, 1999. [27] O. M. Bucci, G. D’Elia, G. Mazzarella, and G. Panariello, “Antenna pattern synthesis: A new general approach,” Proc. IEEE, vol. 82, pp. 358–371, Mar. 1994. [28] Y. T. Lo, S. W. Lee, and Q. H. Lee, “Optimization of directivity and signal-to-noise ratio of an arbitrary antenna array,” Proc. IEEE, vol. 54, pp. 1033–1045, Aug. 1966. [29] J. D. Lohn, G. S. Hornby, and D. S. Linden, An Evolved Antenna for Deployment on NASA’s Space Technology 5 Mission, in Genetic Programming Theory and Practice II. Berlin: Springer, 2005. [30] G. J. Burke and A. J. Poggio, Computer Analysis of the Twin-Whip Lawrence Livermore Lab., CA, UCRL-52080, June 1, 1976. [31] G. J. Burke and A. J. Poggio, “Computer Analysis of the Bottom-Fed FM Antenna,” Lawrence Livermore Lab., CA, UCRL-52109, August 19, 1976. [32] J. Deadrick, G. J. Burke, and A. J. Poggio, “Computer Analysis the Trussed-Whip and Discone-Cage Antennas,” UCRL-52201, Jan. 6, 1977. [33] G. J. Burke and A. J. Poggio, “Numerical Electromagnetics Code-Method of Moments,” Lawrence Livermore National Lab., Tech. Rep. UCID-18834, Jan. 1981. [34] D. Linden and E. Altshuler, “Automating wire antenna design using genetic algorithms,” Microw. J., vol. 39, no. 3, pp. 74–86, 1996. [35] D. Linden and E. Altshuler, “Wire-antenna designs using genetic algorithms,” IEEE Antennas Propag. Mag., vol. 39, no. 2, pp. 33–43, 1997. [36] S. Uda, “Wireless beam of short electric waves,” J. IEE (Japan), no. 472, pp. 1209–1219, Nov. 1927.

Giovanni Andrea Casula (M’04) was born in Sassari, Italy, in 1974. He received the Laurea degree (summa cum laude) in electronic engineering and the Ph.D. degree in electronic engineering and computer science from the Università di Cagliari. Italy, in 2000 and 2004, respectively. Since March 2006, he is an Assistant Professor of electromagnetic field and microwave engineering at the Department of Electrical and Electronic Engineering, Università di Cagliari. His current research interests are in the field of synthesis, analysis and design of wire, patch and slot antennas. Prof. Casula serves as reviewer for several international journals, and is a member of the Italian Electromagnetic Society (SIEm).

Giuseppe Mazzarella (S’82–M’90–SM’99) graduated (summa cum laude) in electronic engineering from the Università “Federico II” of Naples in 1984 where he received the Ph.D. degree in electronic engineering and computer science in 1989. In 1990, he became an Assistant Professor at the DIpartimento di Ingegneria Elettronica at the Università “Federico II” of Naples. Since 1992 he is with the Dipartimento di Ingegneria Elettrica ed Elettronica, Università di Cagliari, first as Associate Professor and then, since 2000, as Full Professor, teaching courses in electromagnetics, microwave, antennas and remote sensing. Since 2005, he is (Executive) President of the CyberSar Consortium, which has realized, from 2006 to 2009, the CyberSar project (about 14.5 MEuro cost), funded by EU through “PON ricerca” and that continues to carry out research activities in the supercomputing and computational modelling areas. His research activity has focused mainly on: efficient synthesis of large arrays of slots, power synthesis of array factor, with emphasis on inclusion of constraints, microwave holography techniques for the diagnosis of large reflector antennas, use of evolutionary programming for inverse problems solving, in particular problems of synthesis of antennas and periodic structures. He is the author (or coauthor) of about 40 papers in international journals, and is a reviewer for many EM journals.

Nicola Sirena was born in Cagliari, in 1972. He received the Laurea degree in electronic engineering in 2007 from the Università di Cagliari. Since June 2007, he works in collaboration with the Department of Electrical and Electronic Engineering, Università di Cagliari. His current research interests are in the field of automated antenna design.

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The Placement of Antenna Elements in Aperture Synthesis Microwave Radiometers for Optimum Radiometric Sensitivity Jian Dong, Member, IEEE, Qingxia Li, Member, IEEE, Ronghua Shi, Liangqi Gui, and Wei Guo

Abstract—Antenna array configurations have significant influence on the radiometric sensitivity of aperture synthesis microwave radiometers. In this paper, we propose a minimum degradation array (MDA) for optimum sensitivity. First, the degradation factor (DF) is defined to characterize the effect of redundant spatial frequency samples formed by an array on the sensitivity. Aiming at minimizing DF, a simulated annealing (SA) based method is proposed to search for an MDA, which, combining with a concept of augmented maximum baseline (CAMB), can effectively locate the true global minimum of DF. Numerical results validate the effectiveness of the proposed method as well as CAMB in optimizing DF. Further, the lower bound of DF is discussed. Finally, simulation and experiment results demonstrate that the proposed method as well as CAMB is of significance in achieving optimum sensitivity. Index Terms—Aperture synthesis radiometer, degradation factor (DF), minimum degradation array (MDA), sensitivity, simulated annealing (SA), thinned antenna arrays.

I. INTRODUCTION

I

NTERFEROMETRIC aperture synthesis technique, first used in the radio astronomy community [1], has been introduced into passive microwave remote sensing of the Earth with high spatial resolution since the 1980s [2]–[5]. Different spatial frequencies are sampled by the cross-correlation of antenna pairs with different separations, and all sampled spatial frequencies can then be inverted to estimate the original brightness temperature distribution of a scene. As a crucial technique for aperture synthesis microwave radiometers, antenna array design plays an important role in radiometric imaging. Conventionally, the design of the antenna array in aperture synthesis radiometers aims to find the minimum redundancy array (MRA) [6]–[9] for high spatial resolution, which can achieve a given uniform coverage in the Fourier plane with the least number of antenna Manuscript received August 17, 2009; revised April 05, 2010; accepted April 14, 2011. Date of publication August 12, 2011; date of current version November 02, 2011. This work was supported in part by the Central South University under Grant 721500238 and 74341015812 and in part by the Huazhong University of Science and Technology under Grant NSF60705018. J. Dong and R. Shi are with the School of Information Science and Engineering, Central South University, 410083 Changsha, China (e-mail: [email protected]) Q. Li, L. Gui, and W. Guo are with the Department of Electronics and Information Engineering, Huazhong University of Science and Technology, 430074 Wuhan, China (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2164172

elements. However, this advantage of thinning may be a problem for some imaging applications from space concerning the radiometric sensitivity, which is one of the most crucial specifications determining the viability of aperture synthesis for Earth remote sensing. For example, for measurements from a moving platform such as low Earth orbit satellite and aircraft, the integration time available for imaging is limited by the rate of forward motion of the platform and the size of the spot on the ground. This property limits the practical sensitivity of the radiometer. In the case of a maximally thinned (i.e., minimum redundancy) array, the sensitivity is further limited due to the very little physical collecting area [4], [10]. So a tradeoff must be made between thinning of an array and obtaining the required sensitivity [3], [10]. Besides, the inclusion of some additional antennas ensures sufficient redundancy to mitigate the effect of antenna/receiver failure and to enhance the robustness to system errors such as array distortion [11]–[14]. As examples, the airborne ESTAR [5] and the spaceborne MeoSTAR [15], [16] used a modestly thinned array instead of a maximally thinned array. The optimum redundant topologies were also considered in MIRAS [12] and GeoSTAR [13], [14] to reduce the degradation of radiometric accuracy and sensitivity caused by antenna failure. As far as the antenna array is concerned, the radiometric sensitivity of aperture synthesis microwave radiometers depends not only on the physical collecting area (i.e., the number and the size of the antenna elements), but also on the particular array coverage and difconfigurations because of their different ferent levels of redundancy [3], [17]–[19]. Ruf et al. [3] first investigated the effect of redundant visibility samples on the sensitivity for thinned linear array radiometers under the assumption that the errors in the visibility samples are uncorrelated. Later, the correlation between visibility sample errors existing in the actual radiometer systems was studied in [18], [19]. In this paper, following the developments of [3] and taking into account the correlation between errors in the visibility samples, we will address the subject of how to properly arrange antenna elements to achieve the optimum sensitivity of aperture synthesis microwave radiometers, on which no dedicated research has been found up to date. In this study, the ideal “analog” correlation (i.e., disregarding quantization errors) is assumed. The paper is organized as follows. In Section II, we define the degradation factor (DF) of an antenna array to characterize the effect of redundant spatial frequency samples on the sensitivity of aperture synthesis radiometers, and formulate the problem of the minimum degradation array (MDA) design for optimum

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sensitivity. In Section III, a simulated annealing (SA) based method is proposed for the MDA design. In Section IV, the MDA design for Earth observation is mainly addressed following with design examples of both linear and planar arrays. To overcome the low improvement of DF for a densely populated array, a concept of augmented maximum baseline (CAMB) is proposed in Section V and its efficiency is validated by numerical examples. In Section VI, the lower bound of DF is discussed. In Sections VII and VIII, simulation and experiment are performed, respectively, to demonstrate the effectiveness of the MDA design. Section IX sums up the paper and presents conclusions. II. PROBLEM FORMULATION FOR THE MDA DESIGN The radiometric sensitivity is defined as the smallest change in the average brightness temperature that can be detected by the instrument ([20], p.359). For an aperture synthesis radiometer, is influenced by many system factors, such the sensitivity as array configuration, channel filter’s type, correlator’s type, quantization level, and so on [17]. In this paper, only the effect will be addressed of redundant spatial frequency samples on . and used to guide the antenna array design for optimum First, we will derive the relationship between redundant spatial frequency samples and the sensitivity considering the correlation between visibility sample errors. Taking any group of four antennas , , , and , the crossmeasured by the first correlation between the visibility measured by the other two antennas and the visibility two antennas is given by [18] (1) where denotes ensemble average; is the pre-detection bandwidth (in Hz); is the integration time (in s). Since the brightness temperature image is obtained by means of a discrete Fourier transform (DFT) of the visibility samples, of the image can be calculated the cross-correlation matrix by [19] (2) where is the cross-correlation matrix of the visibility with each element computed as (1); is the discrete Fourier is the weighting matrix (here the uniform transform matrix; ). weighting is assumed, so The variance of the image is then (3) where are the direction cosines defined with respect to the axes; denotes the diagonal of a matrix. Note that the variance of the image is direction dependent, i.e., there are different values of the variance in different directions. Also, the sensitivity , i.e., the standard deviation of the image, is different from pixel to pixel and called as the pixel sensitivity [18]. To evaluate the global radiometric performance of an aperture synthesis radiometer, the average sensitivity [18] is adopted in

this paper, which is the average value of the image variance over all directions and expressed by

(4) is the variance of a given where ; is the total number of pixel nonredundant visibility samples, where and are the maximum baselines (i.e., the largest multiple of the unit spacing) on the axes, respectively. Taking into account that a DFT preserves the Euclidean norm, (4) can be rewritten as (5) is the elementary area in the plane ( where for rectangular sampling planar arrays; for for linear arhexagonal sampling planar arrays [21]; rays, where is the minimum spacing normalized to the waveis the variance of a given length); , which is the same for each visibility visibility sample sample and given by [3] (6) is the antenna temperature (in K); is the receiver’s where is the sensitivity of a real apernoise temperature (in K), ture total power radiometer [20]. To describe the effect of the spatial sampling performance on the sensitivity, we define the degradation factor (DF) of a thinned array radiometer as (7) which gives the degree to which the sensitivity of a thinned array radiometer is degraded compared to that of a real aperture radiometer. Note that, in (7), it is assumed that the ratio of the two radiometric sensitivities is defined for the same integration time in order to emphasize the effect of array configurations. In fact, a synthetic aperture radiometer often has a longer integration time per pixel for its snap-shot imaging. is measured by one baseline (i.e., it If a visibility sample is nonredundant), by substituting (5) and (6) into (7), we have (8) is measured by baselines (i.e., it If a visibility sample , we can take is -fold redundant), each yielding a value its average to reduce the visibility error variance and then (9) From (5), (7), and (9), we can see that redundant samples formed by an antenna array play an important role in DF because redundant measurements of the same visibility sample can be averaged together to reduce the noise in the estimate of that sample . Moreover, the degree to which the and hence improve

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variance is reduced depends on the correlation between the errors of redundant samples, which depends on a particular scene under observation and receiver’s noise temperature [17]–[19]. For a given scene and given system parameters (including reis constant, we will ceiver’s noise temperature), since address how to properly arrange the antenna elements to mini. In particular, we call mize DF, i.e., to achieve optimum the antenna array with minimum DF as the minimum degradation array (MDA). coverage For a given number of elements and a given , the problem of the MDA design can be generalized as a multivariable minimization problem mathematically expressed as

(10) where the objective function DF to be minimized is expressed by (7) and can be computed from (1)–(7); the constraint indicates that there are no missing samples within the given coverage for imaging purpose; is the ith element’s position region normalized by the minimum spacing; denotes a plane, where and are the maximum baselines in the (i.e., the largest multiple of the unit spacing) on the and axes, respectively. Note that, since an MRA is the maximally thinned [6]–[9], the element number of an array satisfying MDA will be no less than that of an MRA. III. SA-BASED METHOD FOR SEARCHING AN MDA A. Method Description Since an MRA is the maximally thinned array satisfying the in (10), a fast constructive procedure for constraint an MDA can be naturally developed as follows: starting from an MRA, a complete MDA is constructed by adding one antenna element at a time. At each construction step, the placement of the element is chosen to minimize DF. This algorithm is a so-called “greedy” algorithm ([22], chap. 16) because it just takes the best location at each step. In spite of its simplicity, the algorithm tends to run into the traps of local minima because of the large number of local minima in the solution space. Moreover, the seeking of an MRA is in itself difficult for any given coverage , which restricts this algorithm from the practical use. One effective method for locating the global minimum in the multidimensional nonlinear optimization problem is simulated annealing (SA) [23], [24]. SA has been successfully used to locate efficient distributions of antenna elements lying in a linear MRA [7] or in a ring [25] manner for imaging applications. The fundamental concept of SA is based on the Metropolis algorithm for simulating the behavior of an ensemble of atoms that are cooled slowly from their melted state to their low-energy ground state. The ground state corresponds to the global optimum in topological optimization. In topological optimization, array configurations with a given number of elements are tried at random using the perturbation mechanism and accepted according to the Metropolis rule [23]:

(11)

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where is the energy of the configuration at the th iteration, is a variable called the “temperature” of the annealing process, and is a random number drawn from a uniform distribution ranging from 0 to 1. In this application, the energy is to be identified with the objective function DF in (7), and element positions are to be varied. From (11), it is observed that if , then the new, th configuration is always accepted. Otherwise, the probability of accepting a worse configuration is dependent on annealing temperature . More specifically, a higher temperature corresponds to a higher probability of accepting a worse array. At the beginning of the random search, the temperature variable is set sufficiently high. High temperatures provide the agitation needed to free the array configuration from local minimum wells and drop it into the global minimum. As the performance of the array improves, the temperature is gradually lowered and the selection of worse arrays becomes less likely. A critical part of the SA algorithm is the annealing schedule, which determines the manner in which the temperature is decreased. A fast cooling schedule is adopted here [26], [27]: (12) where is the initial temperature, is the rate of annealing. This schedule cools rapidly at initial higher temperatures and cools very slowly at lower temperatures. The chief virtue of the SA-based method exists in its generality. The method does not require analyticity or differentiation of the objective function and is hence less coupled to the solution space. This property makes the method very suitable for the MDA design. Strictly speaking, DF in (7) depends on the particular scene and receiver’s noise temperature, and hence has not a general analytical expression. Even so, for any given scene and any given receiver’s noise temperature, DF can be optimized by the numerical procedures described above combining with (1)–(7). A second virtue of the SA-based method is its versatility. One could easily apply the above procedures to more complicated situations. For example, to improve the sensitivity, nonuniform weighting [17] of visibility samples can be easily incorporated into DF optimization and result in a modified MDA design by introducing a weighting matrix into (2). B. Numerical Example A one-dimensional scene was selected as the test scene: a two-step scene at 150 K and 300 K extending from to 30 incidence. The system parameters of aperture synthesis radiometers are set as follows: system bandwidth MHz, integration time , receiver’s noise temperature . Given antenna element number , the maximumbaseline , and the minimum spacing of the array , we will use the SA-based method to search for an MDA. The assumed parameters for the SA-based procedures are , and . To verify that consistent true minima had been obtained, all of the annealings were performed a few times with different random initial conditions. It

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TABLE I OBTAINED BY THE SA-BASED METHOD AND THE COMPARISON OF GREEDY ALGORITHM, RESPECTIVELY, FOR LINEAR ARRAYS ( , )

Fig. 1. Convergence curve of

using the SA-based method.

is concluded that multiple trials with different initial configurations and with different randomly perturbation do not affect the minima obtained but merely the rate of convergence. Also, various array configurations which are equally good in DF measure may be generated by the SA-based method due to the heuristic nature of the SA. The optimum array configuration obtained by the SA-based method is {0 1 5 8 10 14 16 17}, and the optimum value of DF is . Fig. 1 shows the convergence curve of using the SA-based method. Starting from a 7-element MRA with its location set {0 1 2 6 10 14 17} [7], an 8-element MDA can also be constructed by the greedy algorithm. As a result, the optimum array configuration obtained by the greedy algorithm is {0 1 2 6 10 11 14 17} with . By comparing the results obtained by both methods, it is concluded that the SA-based method can effectively escape from local minima of DF and approach a global minimum by accepting a worse configuration with a probability dependent on annealing temperature. One problem for the SA-based method could be the computational complexity of the objective function DF. Since DF is computed from (1)–(7), the SA-based optimization procedures will be time-consuming, especially for an array with large numbers of elements or redundancies.

IV. MDA DESIGN FOR EARTH OBSERVATION In Earth remote sensing, the imaging scene is an extended, radiometrically bright, scene with a smooth temperature distribution, and the correlation between redundant sample errors is small because the amplitude of the visibility function decays rapidly [17]. In this case, the averaging of redundancies is of more significance in improving the sensitivity. When neglecting the correlation, the visibility error variance in (9) is reduced by , and DF in (7) can be analytically approximated by

(13)

Note that, in (13), DF does not depend on the scene and receiver’s noise temperature, only depends on the particular array configuration represented by the redundancy set . In the following sections, if no special statement, the optimization and discussion on DF will be based on (13). With this simplified objective function, the optimization procedures will be greatly speeded up. A. Linear Array Case For a linear array, (13) can be simplified due to the Hermitian symmetry of visibility samples as (14) In (14), it is assumed that the zero baseline (self-correlation) is measured by as many total power radiometers as receiving elements. Given and , Table I presents of linear arrays with different values of using the SA-based method described in Section III-A. The results obtained by the greedy algorithm are also listed for comparison, which start from a known 16-element linear MRA [7], [8] with . The theoretical limits of DF in Table I are referred to (28) in the Appendix. From Table I, it is observed that for a small value of , better results can be obtained by the SA-based method than by the greedy algorithm due to the probabilistic nature of the SA; but for a relatively large value of , there is only a slight difference between the results obtained by both methods because of the little flexibility of placing elements. Fig. 2 shows the practical distribution of the redundancy set for , which is different from the ideal uniform distribution and can be fitted by a three-polyline model. This distribution implies that by reducing redundant measurements in low spatial frequency samples, the number of correlators will be considerably reduced with small loss of system radiometric performance. Specifically, as sampled by the desirable curve in Fig. 2, the number of correlators is reduced by 18.62% at the cost of DF loss 5.38%. Given different values of , Fig. 3 depicts the improvement of with the increasing percentage of filled aperture. It can be clearly seen that the greatest improvement occurs at a relatively small percentage of filled aperture. For the assumed system parameters (e.g., , , and so on), the thinned array with

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Fig. 2. Distribution of the redundancy set for ( and ). Only right-hand side is given considering the Hermitian symmetry of samples.

Fig. 4. Thinned planar array with , , and dancy set (b) (

Fig. 3. aperture.

(a) and its distribution of the redun).

improvement with the increasing percentage of the filled

TABLE II OBTAINED BY THE SA-BASED METHOD AND THE COMPARISON OF GREEDY ALGORITHM, RESPECTIVELY, FOR RECTANGULAR PLANAR ARRAYS ( , , AND )

a proper percentage of filled aperture can be determined from Fig. 3 to meet the required sensitivity. B. Planar Array Case The SA-based method can also be used to design planar arrays, provided that antenna elements are located at uniform rectangular lattices. Given , , and , Table II presents of rectangular planar arrays with different values of using the

Fig. 5. aperture.

improvement with the increasing percentage of the filled

SA-based method. The results obtained by the greedy algorithm are also listed for comparison, which start from a known 16-element U-shape array [29] with . The distribution of the redundancy set similar to the one-dimensional case is shown in Fig. 4. The improvement with the increasing percentage of filled aperture is depicted in Fig. 5 and conclusion similar to the one-dimensional case can be drawn.

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V. CONCEPT

AUGMENTED MAXIMUM BASELINE AND ITS APPLICATION

OF

TABLE III BETWEEN THE RESTRICTED LINEAR MDA AND THE COMPARISON OF GENERAL LINEAR MDA ( , )

A. Concept of Augmented Maximum Baseline (CAMB) From Figs. 3 and 5, it is observed that for an array with a large percentage of the filled aperture, the DF improvement by the SA-based method is limited. This is because of the very few redundancy times in the high spatial frequency samples, which is unavoidable even in the optimum distribution of the redundancy set shown in Figs. 2 and 4. For example, the maximum baseline of an array is sampled only once in any case. One possible way to overcome this problem is to move those high spatial frequency components with few redundancies outside of the concerned spatial frequency coverage as many as possible. Inspired from this, a concept of augmented maximum baseline (CAMB) is proposed. In this concept, antenna elements are allowed to be placed outside of (i.e., removing the restriction in (10)), naturally following with the augmented maximum baseline. With the increased flexibility of placing elements, the optimization capability of SA-based method is enhanced. As far as the initial is concerned, it could be expected that a better distribution of the redundancy set is achieved at the cost of the reduced total number of samples within it, and therefore a better DF may be achieved. For the convenience of expression, we call the array resulting from CAMB as the general MDA, distinguished from the MDA in the restricted case. In some applications, there may be sufficient space to provide the extra array length, and the general array will be attractive. The key point of finding out such a general MDA is to choose the proper value of the augmented maximum baseline, or . Taking the one-dimensional case as an example, an effective iterative procedure is developed as follows to determine : Step 0. Set an initial value of the augmented maximum baseline , where is the maximum baseline in the restricted case. Step 1. Search the general MDA using the SA-based method described in Section III-A, and observe the actual maximum baseline of the resulting MDA. Step 2. If , then ; otherwise, increase and repeat Step 1–2. It can be further concluded from a large number of numerical tests that there always exists a limiting value, , for the augmented maximum baseline in which case is achieved with respect to , i.e., any value larger than will not yield a better DF. B. Numerical Examples and 1) Application of CAMB to Linear Arrays: Given , Table III and Fig. 6 present comparisons of between the restricted linear MDA and the general linear MDA. It is observed from them that is significantly improved by introducing CAMB, especially for a densely populated array. The best improvement of is predicted to be 32.60%. Moreover, with the increasing number of elements, in the general MDA approaches the theoretical limit of DF given by (28).

TABLE IV BETWEEN THE RESTRICTED PLANAR MDA AND THE COMPARISON OF GENERAL PLANAR MDA ( , , AND )

Fig. 6.

Improvement for linear arrays by CAMB (

and

).

As a limiting case, Fig. 7(a) and (b) depict the distributions of the redundancy set for a 91-element filled linear array (i.e., a 91-element restricted linear MDA) and a 91-element general linear MDA, respectively, and a better DF with respect to is achieved in the latter array. 2) Application of CAMB to Planar Arrays: Given , , and , Table IV and Fig. 8 present comparisons of between the restricted planar MDA and the general planar MDA. It is observed from them that the improvement of is significant by introducing CAMB, especially for a densely populated array. The best improvement of is predicted to be 18.82%.

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Fig. 9. Distribution of the redundancy set for a 36-element filled planar array (a) and a 36-element general planar MDA (b).

VI. DISCUSSION ON THE LOWER BOUND OF DF Fig. 7. Distribution of the redundancy set for a 91-element filled linear array (a) and a 91-element general linear MDA (b). Only right-hand side is given considering the Hermitian symmetry of samples.

Due to the correlation between redundant samples errors, the actual DF will be no less than that computed by (13) in any case. So, in this section, we will derive the lower bound of DF from (13). According to the property of the mean inequalities, DF satisfies (see the Appendix) (15)

Fig. 8. and

improvement for planar arrays by CAMB (

,

).

As a limiting case, Fig. 9(a) and (b) show the distributions of the redundancy set for a 36-element filled planar array (i.e., a 36-element restricted planar MDA) and a 36-element general planar MDA, respectively, and a better DF with respect to and is achieved in the latter array.

are where the sign of equality holds if and only if all the equal, for rectangular planar arrays; for linear arrays. From (15), the lower bound of DF that a filled array provides approaches for the two-dimensional case and for the one-dimensional case, respectively. This lower bound only serves as a theoretical one because the condition that all the are equal can not be satisfied in the placement of array elements. This bound indicates that of a filled array radiometer with a half-wavelength minimum spacing is comparable to that of a real aperture total power radiometer in the ideal case. For a filled array, the actual redundancy set is

(16)

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TABLE V IMPROVEMENT BY CAMB FOR THE ONE-DIMENSIONAL CASE

TABLE VI IMPROVEMENT BY CAMB FOR THE TWO-DIMENSIONAL CASE Fig. 10. The simulated scene and the inversed image.

VII. SIMULATION VALIDATION FOR MDA DESIGN For redundancy case, the cross-correlation between the -fold redundant visibility and the -fold redundant visibility is computed by By substituting (16) into (13), the actual lower bound of DF can be estimated by (19)

(17) where the harmonic series , for is used, is Euler’s Constant. This bound can be interpreted as a two-dimensional generalization of (39) in [3] by considering the Hermitian symmetry of visibility samples. For the one-dimensional case, (17) can be simplified as

(18) By comparing (17) and (18), it can be seen that of a planar array grows in the square order of that of a linear array. Further, a lower than (17) and (18) can be achieved by introducing CAMB. Although difficult to express it mathematically, it is possible to approach this by a piecewise linear approximation (as in Fig. 2). Tables V and VI present some numerical examples of the improvement by CAMB, for the one-dimensional case and the two-dimensional case, respectively. From both tables, a more slowly growing is observed in the general case than those in the restricted case given by (17) and (18).

Based on (1)–(5) and (19), we will evaluate the effect of redundant spatial frequency samples on the sensitivity to justify the approximation of (13) in the MDA design for Earth observation. The simulation parameters of the first example are set as follows: MHz, , (representative value at high frequency of microwave band which corresponds to noise figure dB). The antenna arrays with respect to are a 16-element linear MRA [7], [8], a 50-element random thinned linear array satisfying , a 50-element restricted linear MDA, and a 50-element general linear MDA, respectively. The latter two arrays are obtained by the SA-based method according to (13). The minimum spacing of the arrays is . The simulated scene shown in Fig. 10 is a three-step uniform brightness temperature scene with 3 K (representative value for the cosmic background), 90 K (idem for sea), and 250 K (idem for land), respectively. No instrumental imperfections are included in the simulations. The simulation results are shown in Figs. 10–12, and Table VII. The simulation parameters of the second example are set as follows: MHz, (representative value at low frequency of microwave band which corresponds to noise figure dB), other system parameters and the simulated scene are the same with those of the first example. The simulation results are summarized in Table VIII. In Figs. 11(b) and 12(b), the pixel sensitivity, i.e., the standard deviation of the image, basically follows the brightness temperature distribution of the scene. This confirms the conclusion presented by Bará et al. [18] and Butora and Camps [19]. In Tables VII and VIII, for a maximally thinned array with a few redundancies, the estimated value of DF by (13) excellently agrees with the exact value; even for a densely populated array

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TABLE VIII SIMULATION RESULTS ON DF (

,

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MHz)

Fig. 11. (a) A 16-element linear MRA. (b) Simulated pixel sensitivity and average sensitivity with and without the averaging of redundancies.

Fig. 13. Photograph of the imaging scene and the instrument HUST-ASR.

Fig. 12. (a) A 50-element random thinned linear array satisfying , a 50-element restricted linear MDA, and a 50-element general linear MDA, respectively. (b) Simulated pixel sensitivity and average sensitivity with different array configurations by averaging redundancies.

TABLE VII SIMULATION RESULTS ON DF (

,

a smooth brightness temperature distribution (as in the case of Earth observation), the correlation between the errors of redundant samples (i.e., the case of and in (19)) is small and hence the approximation in (13) is reasonable. The simulated results show that for Earth observation, the average sensitivity of a synthetic aperture radiometer with a particular array configuration can be quantitatively characterized by DF in (13), and hence the MDA design obtained by the proposed method as well as CAMB can ensure optimum sensitivity.

MHz)

VIII. EXPERIMENT RESULTS

The value in the parenthesis denotes the ratio of the approximated DF by (13) to the exact DF by (7). This notation means that 60 redundant baselines are located in 38 visibility samples with different redundancy levels (considering the Hermitian symmetry of samples).

with large numbers of redundancies, the estimated value of DF still agrees with the exact value well. This fact indicates that for

To quantitatively assess the actual effect that redundant baselines have on , experiments were also performed on HUST-ASR [30], which is a prototype of aperture synthesis radiometer working at 8-mm wave band. The system parameters of HUST-ASR are as follows: MHz, , receiver’s equivalent noise temperature (corresponding to noise figure dB). The physical antenna array available is a 16-element linear MRA with minimum spacing . In our experiments, the imaging scene is the wall, i.e., a stationary scene with a smooth temperature distribution. The wall was measured 500 times and hence 500 images of the wall were generated from the measured visibilities with and without the averaging of redundancies. By calculating the standard deviation of each pixel in the inversed images, the sensitivity was obtained. The imaging scene and the instrument HUST-ASR are shown in Fig. 13. The experimental results are shown in Fig. 14.

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Fig. 14. Experiment sensitivities with and without the averaging of redundancies, respectively.

From Fig. 14, the improvement of the sensitivity by averaging redundant baselines is (20) and are the average sensitivities with and where without the averaging of redundancies, respectively. From (13), the improvement of the sensitivity by redundancies is predicted as (21) where and are the values of DF with and without the averaging of redundancies, respectively. Experimental results show that for a smooth brightness temperature scene, the averaging of redundant baselines does improve the sensitivity, and the measured improvement of approximately agrees with that predicted by DF in (13). In view of its effectiveness on minimizing DF, the proposed method as well as CAMB is of significance for the optimum sensitivity of aperture synthesis microwave radiometers. IX. CONCLUSION Antenna array design is an important issue of aperture synthesis microwave radiometers. Different from the conventional MRA design for high spatial resolution, this paper addresses the MDA design for optimum radiometric sensitivity, which is of significance in some applications especially for observations from a moving platform such as a low Earth orbit satellite and an aircraft. In this paper, the degradation factor (DF) is defined to characterize the effect of redundant spatial frequency samples formed by array configurations on the sensitivity, and used to guide the proper placement of the antenna elements. Aiming at minimizing DF, a simulated annealing (SA) based method is proposed to search for an MDA. The method provides a general solution for any given scene and any given system parameters, and its effectiveness in locating global optimum is validated by a numerical example.

The MDA design for Earth observation is mainly addressed with a simplified analytical expression of DF. Numerical studies show that for a sparsely populated array, significant improvement of DF can be achieved by the SA-based method; but for a densely populated array, the DF improvement is small because of the little flexibility of placing elements. To overcome this problem, a concept of augmented maximum baseline (CAMB) is proposed. The adding baselines provide more options for the SA-based method to reach a minimum in DF. Numerical examples show that for a linear array with respect to the maximum baseline , the best improvement by CAMB is predicted to be 32.60%; for a rectangular planar array with respect to the maximum baseline and , the best improvement by CAMB is predicted to be 18.82%. Further, the lower bound of DF is derived. Finally, both simulation and experiment results are presented to justify the analytical approximation of DF in the MDA design for Earth observation. The proposed method as well as CAMB for the MDA design is of significance in reaching the optimum sensitivity of aperture synthesis radiometers. Further work will be aimed at multiobjective optimization design of the antenna array in aperture synthesis radiometers for both high spatial resolution and high radiometric sensitivity. APPENDIX THEORETICAL LOWER BOUND OF DF DERIVED FROM THE MEAN INEQUALITIES For any positive values, (22) and positive weights (23) the mean order , or the norm, of the values is defined by ([28], chap. 1.16)

with weights

(24)

In particular, the means of order 1, 0, 1, and 2 are the harmonic mean, the geometric mean, the arithmetic mean, and the root-mean-square, respectively. Beckenbach [28] has shown that, for positive values , is a nondecreasing function of for , and is strictly increasing unless all the are equal. In particular, assuming , for and 1, we have the following mean inequality

(25)

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that is

(26)

In our problem, is identified with the redundancy times , is identified with the total number of visibility samples . Note that , where n is the number of antennas, we have (27)

Therefore, DF in (13) satisfies (28) the sign of equality holding if and only if all the

are equal.

ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers and editors, whose comments made this paper much improved. The authors also acknowledge the work of Dr. Liangbin Chen, Dr. Ke Chen and Dr. Rong Jin, in Prof. Qingxia Li’s group, in helping to develop the simulation code and perform experiments. REFERENCES [1] A. R. Thompson, J. M. Morgan, and J. G. W. Swenson, Interferometry and Synthesis in Radio Astronomy, 2nd ed. New York: Wiley, 2001. [2] D. M. Le Vine and J. C. Good, “Aperture Synthesis for Microwave Radiometry in Space,” NASA Technical Memorandum 85033, Aug. 1983. [3] C. S. Ruf, C. T. Swift, A. B. Tanner, and D. M. Le Vine, “Interferometric synthetic aperture microwave radiometry for the remote sensing of the Earth,” IEEE Trans. Geosci. Remote Sens., vol. 26, no. 5, pp. 597–611, Sep. 1988. [4] A. S. Milman, “Sparse-aperture microwave radiometers for Earth remote sensing,” Radio Sci., vol. 23, no. 2, pp. 193–206, 1988. [5] C. T. Swift, D. M. Le Vine, and C. S. Ruf, “Aperture synthesis concepts in microwave remote sensing of the Earth,” IEEE Trans. Micro. Theory Techn., vol. 39, no. 12, pp. 1931–1935, Dec. 1991. [6] A. T. Moffet, “Minimum-redundancy linear arrays,” IEEE Trans. Antennas Propagat., vol. AP-16, no. 2, pp. 172–175, Mar. 1968. [7] C. S. Ruf, “Numerical annealing of low-redundancy linear arrays,” IEEE Trans. Antennas Propagat., vol. 41, no. 1, pp. 85–90, Jan. 1993. [8] A. Camps, A. Cardama, and D. Infantes, “Synthesis of large low-redundancy linear arrays,” IEEE Trans. Antennas Propagat., vol. 49, no. 12, pp. 1881–1883, Dec. 2001. [9] J. Dong, Q. Li, R. Jin, Y. Zhu, Q. Huang, and L. Gui, “A method for seeking low-redundancy large linear arrays for aperture synthesis microwave radiometers,” IEEE Trans. Antennas Propagat., vol. 58, no. 6, pp. 1913–1921, Jun. 2010. [10] D. M. LeVine, “The sensitivity of synthetic aperture radiometers for remote sensing applications from space,” Radio Sci., vol. 25, no. 4, pp. 441–453, 1990. [11] M. Vall-llossera, N. Duffo, A. Camps, I. Corbella, F. Torres, and J. Bara, “Reliability analysis in aperture synthesis interferometric radiometers: Application to L-band microwave imaging radiometer with aperture synthesis instrument,” Radio Sci., vol. 36, no. 1, pp. 107–117, Jan. 2001.

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[12] I. Corbella, A. Camps, N. Duffo, and M. Vall-llossera, “Optimum redundant array configurations for Earth observation aperture synthesis microwave radiometers,” Electron. Lett., vol. 38, no. 20, pp. 1205–1207, Sep. 2002. [13] F. Torres, A. B. Tanner, S. T. Brown, and B. H. Lambrigsten, “Robust array configuration for a microwave interferometric radiometer: Application to the GeoSTAR project,” IEEE Geosci. Remote Sens. Lett., vol. 4, no. 1, pp. 97–101, Jan. 2007. [14] F. Torres, A. B. Tanner, S. T. Brown, and B. H. Lambrigsten, “Analysis of array distortion in a microwave interferometric radiometer: Application to the GeoSTAR project,” IEEE Trans. Geosci. Remote Sens., vol. 45, no. 7, pp. 1958–1966, Jul. 2007. [15] T. A. Doiron and J. R. Piepmeier, “Hybrid synthetic/real aperture antenna for high resolution microwave imaging,” in Proc. IEEE IGARSS, 2003, vol. 7, pp. 4368–4370. [16] W. J. Wilson, A. B. Tanner, B. H. Lambrigtsen, T. A. Doiron, J. R. Piepmeier, and C. S. Ruf, “STAR concept for passive microwave temperature sounding from middle Earth orbit (MeoSTAR),” in Proc. IEEE IGARSS, 2004, vol. 2, pp. 789–790. [17] A. Camps, I. Corbella, J. Bará, and F. Torres, “Radiometric sensitivity computation in aperture synthesis interferometric radiometry,” IEEE Trans. Geosci. Remote Sens., vol. 36, no. 2, pp. 680–685, Mar. 1998. [18] J. Bará, A. Camps, F. A. Torres, and I. Corbella, “The correlation of visibility noise and its impact on the radiometric resolution of an aperture synthesis interferometric radiometry,” IEEE Trans. Geosci. Remote Sens., vol. 38, no. 5, pp. 2423–2426, Sep. 2000. [19] R. Butora and A. Camps, “Noise maps in aperture synthesis radiometric images due to cross-correlation of visibility noise,” Radio Sci., vol. 38, no. 4, pp. 1067–1074, 2003. [20] F. T. Ulaby, R. K. Moore, and A. K. Fung, Microwave Remote Sensing Fundamentals and Radiometry. Boston, MA: Addison-Wesley, 1981, vol. 1. [21] A. Camps, J. Baŕa, I. Corbella, and F. Torres, “The processing of hexagonally sampled signals with standard rectangular techniques: Application to aperture synthesis interferometer radiometers,” IEEE Trans. Geosci. Remote Sens., vol. 35, no. 1, pp. 183–190, Jan. 1997. [22] T. H. Cormen, C. E. Leiserson, and R. L. Rivest, Introduction to Algorithms. Cambridge, MA: MIT Press, 1990. [23] S. Kirkpatrick, C. D. Gelatt, , and M. P. Vecchi, “Optimization by simulated annealing,” Sci., vol. 220, pp. 671–680, May 1983. [24] W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes, the Art of Scientific Computing, 3rd ed. New York: Cambridge Univ. Press, 2007. [25] T. J. Cornwell, “A novel principle for optimization of the instantaneous Fourier plane coverage of correlation arrays,” IEEE Trans. Antennas Propagat., vol. 36, no. 8, pp. 1165–1167, Aug. 1988. [26] M. Lundy and A. Mees, “Convergence of an annealing algorithm,” Math. Programming, vol. 34, no. 1, pp. 111–124, Jan. 1986. [27] H. Cohn and M. Fielding, “Simulated annealing: Searching for an optimal temperature schedule,” Soc. Industr. Appl. Mathemat. J. Optimization, vol. 9, no. 3, pp. 779–802, 1999. [28] E. F. Beckenbach and R. Bellman, Inequalities. New York: SpringerVerlag, 1983. [29] U. R. Kraft, “Two-dimensional aperture synthesis radiometers in a low earth orbit mission and instrument analysis,” in Proc. IEEE IGARSS, 1996, vol. 2, pp. 866–868. [30] Q. Li, F. Hu, W. Guo, K. Chen, L. Lang, J. Zhang, Y. Zhu, and Z. Zhang, “A general platform for millimeter wave synthetic aperture radiometers,” in Proc. IEEE IGARSS, 2008, vol. 2, pp. 1156–1159.

Jian Dong (M’11) received the B.S. degree in electrical engineering from Hunan University, Changsha, China, in 2004, and the Ph.D. degree in electrical engineering from Huazhong University of Science and Technology (HUST), Wuhan, China, in 2010. He is now a postdoctoral researcher with the School of Information Science and Engineering, Central South University (CSU), Changsha, China. His research interests include antenna arrays, microwave remote sensing, and numerical optimization techniques for electromagnetic problems.

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Qingxia Li (M’08) received the B.S., M.S., and Ph.D. degrees in electrical engineering from Huazhong University of Science and Technology, Wuhan, China, in 1987, 1990, and 1999, respectively. He is presently a Professor in the Department of Electronics and Information Engineering, Huazhong University of Science and Technology. His research interests include microwave techniques, microwave remote sensing, antennas, and wireless communication.

Ronghua Shi received the B.S., M.S., and Ph.D. degrees in electrical engineering from Central South University (CSU), Changsha, China, in 1986, 1989, and 2007, respectively. He is presently a Professor and the Vice Dean of the School of Information Science and Engineering at Central South University. His research interests include wireless communication, antennas, satellite navigation and positioning.

Liangqi Gui received the B.S. degree in electrical engineering from Wuhan University, Wuhan, China, in 1998, and the Ph.D. degree in electrical engineering from Huazhong University of Science and Technology (HUST), Wuhan, China, in 2005. He is presently an Associate Professor at the Department of Electronics and Information Engineering, Huazhong University of Science and Technology. His research interests include wireless communication, microwave remote sensing, EMC, and passive THz imaging technology.

Wei Guo received the B.S. and M.S. degrees in electrical engineering from University of Science and Technology of China (USTC), Hefei, China, in 1982, and 1987, respectively. He is presently a Professor at the Department of Electronics and Information Engineering, Huazhong University of Science and Technology. His research interests include microwave remote sensing, antennas, and microwave and millimeter-wave techniques.

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Fast and Accurate Array Calibration Using a Synthetic Array Approach Will P. M. N. Keizer, Member, IEEE

Abstract—A new method for the calibration of arrays is prearray elesented that allows the simultaneous calibration of ments. It comprises the measurement of the array output signal at phase settings applied to the elements involved in the calibration. These phase settings correspond to a linear phase taper that is unique for each of the involved elements and reveals theredifferent phase tapers. The complex measurements of fore the array signal are thought to be the excitation coefficients of a synthetic -element linear phased array. The array factor of this -element array comprises a superposition of synthetic array factors all pointing with their main beam into different diarray factors rections. By converting this superposition of into a set of simultaneous linear equations, the signals of the individual elements to be calibrated including the combined signal contribution of the static elements, can be solved by standard matrix inversion techniques. Computer simulations are presented to demonstrate the capabilities of the new calibration method.

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Index Terms—Calibration, measurement errors, phased arrays, synthetic linear array.

I. INTRODUCTION

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ODERN large phased array antennas are equipped with electronic components of which the parameters vary with temperature, drift over time or can suffer from aging effects resulting in unwanted phase and/or amplitude errors for the array elements. Periodic array calibration during operational use of these antennas is therefore necessary to establish phase and amplitude corrections for each array element in order to maintain the original array performance. Phased arrays are typically calibrated using an external far-field or using near-field test source(s) positioned along the aperture to illuminate the array aperture with a test signal. Some phased arrays employ a built-in calibration network for injection of a test or calibration signal to all elements. Among existing calibration methods, two major categories can be distinguished, single element measurement and simultaneous measurement of a number of array elements. Single element calibration performs the characterization of each array element, in terms of amplitude and phase, one at

Manuscript received August 03, 2009; revised March 07, 2011; accepted May 05, 2011. Date of publication August 12, 2011; date of current version November 02, 2011. The author, retired, was with TNO Physics and Electronics Laboratory, 2597 AK, The Hague, Netherlands. He is now at Clinckenburgh 32, 2343 JH Oegstgeest, The Netherlands (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2164171

a time. Usually this carried out by isolating an individual element followed by characterizing that element. Isolation of an individual element can be arranged by turning off all remaining elements except the one to be calibrated or their influence is suppressed by proper phase setting. Having only one element activated at a time with the remaining ones turned off, leads to prime power supply loads that differ from normal operating conditions, and is therefore not always acceptable. The best-known single element measurement techniques are the rotating element electric field vector (REV) method [1], the method proposed by Sorace [2] and the phase toggling method of Lee et al. [3]. In the REV method, the relative amplitude/ phase of the elements is determined by measuring the maximum and minimum power ratio of the power composite array signal and the phase that provides maximum power. Sorace’s method, using power only measurements, characterizes each array element by changing its phase settings in four orthogonal states, while the others are operating in nominal conditions. The simplest single element calibration method is to switch its phase between 0 and 180 states, measures the complex array signal at both states and by subtracting these two array signals, the complex signal contribution of the involved element can be determined [3]. Measurement techniques that calibrate simultaneously a number of array elements are the methods [4]–[8]. The multiple REV technique [4] is an extension of the single element REV method and shifts the phase of several elements simultaneously using different intervals while the array power variations are measured. The measured power variations are expanded into a Fourier series that can be rearranged into an expansion to be identical with the conventional REV method. In the multi-element phase toggle method [5] the phase of several elements are also shifted simultaneously by using different intervals. The complex signals of the involved elements are obtained by applying an inverse fast Fourier transform (FFT) on the measured complex array signals. In [6] and [7], phase control with mutually orthogonal (Hadamard) codes is employed to all array elements, so that they can be calibrated simultaneously. The complex weights of the individual elements are determined by performing a set of cross-correlations between the cumulative coded array signal and each of the code sequences applied to the array elements. Both methods cycle one bit of the phase shifter of each element for the orthogonal phase coding, that must be the 180 bit for the pseudo noise (P/N) gating method [7] while for the control circuit encoding (CCE) algorithm [6] any phase bit can be used. In [8] variants of the CCE method are described that allow the simultaneous determination of the individual embedded element patterns

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in the array environment as well as the determination of the non-linear characteristics of the element amplifiers. The duration of the array calibration depends on the number of array signal measurements established by the number of used phase cycles and on the total number of array elements. For the calibration methods [1]–[6] the total number of array signal measurements is at least a factor two higher than that of the array elements while for method [7] these measurements correspond with the number of array elements. This paper presents a new calibration technique requiring a number of measurements just about 18 percent higher than the number elements of the array and that supports the correction of phase shifter imperfections to improve calibration accuracy which is not possible with the multi-element calibration methods [4], [5] and [7]. In the proposed calibration method a series of successive phase shifts is carried out for multiple antenna elements while at each phase shift the complex array signal variation is measured. The successive phase shifts applied to an individual element constitute a linear phase taper having a unique slope. The data collected by the array signal measurements are regarded as the excitation coefficients of a synthetic uniform linear array. The array factor of this synthetic linear array is in fact a superposition of separate array factors each having a beam pointing to a different direction due to the application of unique linear phase slopes. By converting this superposition of array factors into a series of simultaneous linear equations the unknown complex signals of the elements involved in the calibration can be solved using standard matrix inversion techniques. The proposed calibration method, called synthetic array calibration (SAC), is verified through a number of simulations carried out for a large array. In these simulations, 40 elements are calibrated simultaneously using 47 phase toggling measurements. Additional validation concerns the comparison of the proposed method with two earlier reported methods [3] and [5]. This comparison will not involve the methods [6], [7] which will be the subject of a future publication.

Fig. 1. Schematic diagram of the array calibration system.

phase shifter. Since the calibration comprises measurements, involved EUCs have to be simultaneously the phases of the phase states. The key characteristic of the cycled through phase proposed calibration technique is that the successive states employed for each individual EUC correspond to a linear phase taper having a unique slope. The incremental phase shift between two successive measurements is for each EUC fixed . At the th measurement and equal to the phase setting at the th EUC is given by . for the th measurement observed at The array signal the output of the beamforming network can be written as

(1) II. SAC CALIBRATION THEORY The proposed SAC technique is based on a procedure that allows the simultaneous characterization of the amplitude and array elements using complex phase characteristics of array signal measurements with all elements turned on. When array signal the calibration occurs in the receive mode the measurements are performed at the composite port of the beammeaforming network. For a transmit mode calibration the surements occur at the composite port of a built-in calibration network. In this paper only the calibration of the array operating in the receive mode of the array is discussed. The test signal delivered to all elements, which number is equal to , is provided by an external source located in the far-field of the array, Fig. 1, or injected through a built-in calibration network. The pick-up horns positioned along the aperture in Fig. 1 provide the reference signal needed for making complex measurements. Prior to a calibration measurement, any element under calibration (EUC) gets an individual phase setting arranged by its

where is the to be determined signal for zero phase state provided by th EUC to the beamformer output and the comarray bined fixed signal contribution of the remaining elements not involved in the calibration. In the above expression, it is assumed that a change in phase setting of the th EUC does not affect meaning that the phase shifters are not troubled by phase and/or amplitude imperfections. By assuming that the calibration signals are the excitation coefficients of a synthetic linear array having elements, the array factor of this synthetic array can be written as

(2) , the wavelength, the interwhere the wavenumber element spacing of the synthetic linear array, and the angle between the radiation direction and the array normal.

KEIZER: FAST AND ACCURATE ARRAY CALIBRATION USING A SYNTHETIC ARRAY APPROACH

By inserting (1) in (2), the last equation can be expressed as array factors the superposition of

(3) where

(4) and

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which parameter has beamwidth is inverse proportional to as was found experimenalso to satisfy the condition , the proposed tally. For an array with elements and calibration has to be repeated times to get all elements calibrated. The derivation of (3) assumes perfect phase and amplitude control of the element signal. Actual phase shifters are unlikely to give exact phase settings and furthermore each new phase setting is frequently accompanied by a limited variation in insertion loss. The occurrence of such phase shifter imperfections affects the calibration process. Phase shifter errors can be taken repreinto account by inserting in (4) the quantity senting the complex signal imperfection of phase shifter of the th EUC when switched to the phase state in the following way:

(5) The first term on the right hand side of (3) represents array scanned main beams all pointing in diffactors generating ferent directions due to the unique slope of the applied phase are tapers. The incremental phase shifts selected in such a way that there is a limited overlap between on the adjacent scanned main beams. The array factor right hand side of (3) has its main beam positioned at broadside due to the zero slope phase taper and represents the contribution of the static elements not involved in the calibration of the EUCs. The pointing in different directions of the main beams of the array factors in (3) provides the opportunity to uniquely determine the signals and . The paand are all known since these can be calcurameters , lated from (2), (4) and (5), respectively. Since (3) is valid for any far-field direction , the unknown signals and can be solved from (3) by converting this expression in a simultaneous linear equations. This conversion series of known far-field directions can be arranged by selecting for . Suitable choices are the scanned main beam peak positions of given by together with the main beam peak position located at broadside direction . Using these of values for in (3) results in a set of simultaneous linear equations from which the unknown element signals tocan be solved using standard gether with the unknown signal matrix inversion techniques. To avoid that this system of simultaneous equations becomes ill conditioned, there must be sufficient angular separation between neighboring main beam peaks. This condition is met scanned when for an even value of , the positions of the main beam peaks are derived from , (6) is the angular separation between two where guarantees neighboring beams. Using this expression for that for any value of , the used angular separation always exceeds the 3 dB beamwidth of the synthetic arrays since this

(7) For modern phased array antennas equipped with transmit/receive (T/R) modules constructed in monolithic microwave integrated circuit (MMIC) technology, the amplitude variation with phase state change is in general quite small meaning that . Also the phase deviation from nominal phase shift is for most MMIC phase shifters less than a few degrees meaning that . Phase shifters free from production tolerances will for any value have equal imperfections resulting in of . III. SIMULATIONS The SAC method was validated by simulations performed on a large array equipped with elements operating with 6-bit phase shifters. The performance of the phase shifters used in the simulations is derived from that of the MMIC phase shifter TGP2103 developed by TriQuint Semiconductor. This control device features a maximum phase deviation from nominal phase setting of 4 degrees and the maximum amplitude variation over all phase states is of the order of 0.5 dB. It is assumed that in each element this phase shifter is embedded in a three-port multifunction control MMIC along with an amplitude control device, two T/R switches and various buffer amplifiers as shown in Fig. 2. The reverse isolation of the matched buffer amplifiers located at both ports of the phase shifter prevents any VSWR interaction between the phase shifter and the other components of the T/R module. Due to absence of this VSWR interaction and the construction in MMIC technology, all phase shifters of the elements will feature for any phase setting nearly the same predictable performance. The accuracy of the proposed calibration method depends on the way the phase imperfections are taken into account in (7). Simulation results will be given for all three possibilities: individual correction, equal correction corresponding to zero production spread for the phase shifters which represents perfect phase shifters. and It must be noted that when RF coupling exists in the T/R modules between the phase shifter and the attenuator due to impedance mismatches, this coupling can induce unpredictable

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Fig. 2. Diagram of a 3-port multifunction control MMIC containing T/R switches, a 6-bit phase shifter, a 6-bit attenuator and various buffer amplifiers.

phase and amplitude errors which the result a degradation of the accuracy of the SAC method. However, when the effect of this coupling is known through testing of the T/R modules, good accuracy with the SAC method is still achievable under the condition that the coupling errors are taken into account by the phase which are also then a function of the setting imperfections of the associated attenuator. A. Array Calibration The array involved in the simulation to verify the SAC method is equipped with elements. The number of elements . to be calibrated simultaneously was set equal to 40 had For this number of EUCs, the number of phase cycles to be at least 47 as was found experimentally. A lower value results in an ill-conditioned series of simultaneous linear for and , and is equations to solve both was used therefore unacceptable. An element spacing of in the calculation of the synthetic array factor. The phase settings of the 40 EUCs were computed from where is given by (6). The amplitudes of 40 EUC signals employed during the simulations were in the range between 29.4 dB and 0.2 dB. The amplitude of , comprising the contribution of all static elements, can be obtained was set equal to 8 dB. Such a low value for by applying random phase settings to the phase shifters of static elements and/or by lowering their gains the using the controllable attenuator contained in these elements. favors the accuracy of the array A low amplitude value for are degraded by meacalibration when the array test signals surement uncertainty due to dynamic accuracy of the network analyzer (NA). Dynamic accuracy is a measure for the linearity of the NA [9]. Dynamic accuracy was taken into account into the simulations of the calibration method by introducing for a random error of 0.2 the amplitude of the test signal dB, and for the phase; both uniformly distributed over the indicated intervals. Both values for dynamic accuracy are more or less representative for modern automatic NA’s. With respect to additive white noise, the signal-to-noise (S/N) ratio of the measured calibration signal is not an item for the SAC calibration method due to the coherent integration of the signals at the beam peaks of the synthetic arrays. Due to this coherent integration, the S/N ratio improves with a factor

Fig. 3. Simultaneous calibration of 40 array elements using 47phase cycles. (a) Behavior of the complex array signal. (b) Synthetic array factor having as excitation coefficients the simulated 47 calibration signals.

compared to that of the single element calibration methods [1]–[3]. Fig. 3(a) shows the simulated measured array signals at the beamformer output when the calibration of the 40 EUCs involved 47 phase cycles. Fig. 3(b) depicts of the synthetic linear array consisting the array factor of 47 array elements and illuminated by the signal shown in Fig. 3(a). Calculation of the array factor of Fig. 3(b) was done using (2). The phase shifters of the 40 EUCs suffer from amplitude and phase imperfections including production tolerances as is illustrated by Fig. 4. Taking the individual phase shifter imperfections into account in the array calibration by using (7) for reveals the calibration the calculation of the array factors results for the 40 EUCs as shown in Fig. 5. The result for element #41 corresponds to the fixed contribution . Fig. 5(a) provides a comparison of the amplitudes of 40 EUCs obtained from the calibration versus the actual ones. One can see that the maximum amplitude calibration error is of the order of 1.15 dB (RMS 0.29 dB). The corresponding results for the calibrated phases and the actual ones are shown in Fig. 5(b); the maximum calibration phase error is equal to 4.18 (RMS

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Fig. 4. Phase and amplitude imperfections of the 40 phase shifters involved in the array calibration. (a) Phase imperfections. (b) Amplitude imperfections.

1.59 ). Fig. 5(c) shows the calibration phase and amplitude errors versus the amplitude level of the 40 EUCs. As can be expected, the lowest calibration errors occur for elements having the highest amplitude level. In absence of measurement errors, the same calibration reveals a 100% perfect calibration result. Fig. 6 illustrates what the effect is on the calibration accuracy array factors are computed using common when the for all 40 considered EUCs. phase shifter imperfections Compared to the corresponding results of Fig. 5, no substantial difference in calibration accuracy is noted. This comparison reveals that array calibration accuracy is mainly dominated by the dynamic accuracy of the NA and to a lesser extent by the way the phase shifter imperfections, individual or common, are taken into account. More pronounced calibration errors can be is observed when the computation of the array factors for all 40 EUCs as based on error free phase shifters or is demonstrated by Fig. 7. The maximum phase and amplitude errors of the calibration are increased with a factor between 3 and 5 compared to the results of Figs. 5 and 6. This is also the case with the corresponding RMS errors.

Fig. 5. Calibration results for the 40 EUCs using individual corrections for phase shifter imperfections. (a) Comparison of calibrated and actual amplitudes. (b) Comparison calibrated and actual phases. (c) Calibration errors versus amplitude level. Array signals corrupted by random uniformly distributed measurement errors due to dynamic accuracy: amplitude 0.2 dB and phase 0:2 .

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Fig. 6. Calibration results for the 40 EUCs using common phase shifter imperfections. Calibration errors versus amplitude level. Array signals corrupted by random uniformly distributed measurement errors due to dynamic accuracy: amplitude 0.2 dB and phase 0:2 .

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Fig. 8. Calibration errors versus amplitude level for method [3]. Array signals corrupted by random uniformly distributed measurement errors due to dynamic accuracy: amplitude 0.2 dB and phase 0:2 .

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and after switching to the 180 phase state this signal . By subtracting the complex element signal follows from . Fig. 8 shows the calibration results of method [3] for the same 40 EUCs as used for the previous simulations with the same complex signals and . As can be seen, method [3] reveals fairly large calibration errors, the maximum error for the amplitude is 4.71 dB and for the phase 28.52 . Responsible for these in combination large errors is the 8 dB amplitude level of with the random measurement errors. Therefore, better calibrais tion results can be obtained when the amplitude level for lowered to 30 dB. Under this circumstance the maximum amplitude error reduces to 0.37 dB, (RMS 0.11 dB) and the maximum phase error to 1.14 , (RMS 0.26 ). These results have been obtained without using any correction for the amplitude induced by cycling the 180 bit of each and phase error of the 40 phase shifters involved in the simulation. Method [5] allows the calibration of multiple elements and makes use of a comparable phase cycling scheme for the elements involved in the calibration, as with the proposed method. Therefore (1) is applicable to describe the measured array signals when using method [5] as calibration tool. Since (1) represents a finite Fourier series, an inverse FFT carried out on this equation reveals the individual signals of the involved EUCs. For elements equipped with 6-bit phase shifters this FFT approach will only work when following conditions are met. The may not exnumber of simultaneously calibrated elements must be equal to 64 and ceed 31, the number of phase cycles for the EUCs has be equal the incremental phase shifts to . Meeting these requirements, the application of the inverse FFT on the 64 measured array signals provides 64 spectral samples of which the ones with an odd index starting at index 3, represent the complex signals . Applying method [5] to calibrate the 40 EUCs having the same complex element signals as used before, reveals the calibration results as shown in Fig. 9. The largest amplitude error for method [5] is of the order of 3.79 dB (RMS 0.93 dB) and the

becomes from

Fig. 7. Calibration results for the 40 EUCs without phase shifter correction. Calibration errors versus amplitude level. Array signals corrupted by random uniformly distributed measurement errors due to dynamic accuracy: amplitude 0.2 dB and phase 0:2 .

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The results of Figs. 5–7 demonstrate that independent of the way phase shifter imperfections are taken into account, that for , the maxall elements with amplitude levels of imum calibration error is 0.9 dB for the amplitude and for the phase. Smallest errors are noted when individual phase shifter corrections are applied. B. Comparison With Other Array Calibration Methods The evaluation of the SAC method comprises in addition a comparison of its performance capabilities with those of two earlier reported calibration methods [3] and [5]. Method [3] represents a single element calibration. It consists of the phase cycling of the 180 phase bit of the element phase shifter and requires only two signal measurements per EUC at the beamformer output. For zero phase state of its phase shifter the signal of th element at the beamformer output is given by

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KEIZER: FAST AND ACCURATE ARRAY CALIBRATION USING A SYNTHETIC ARRAY APPROACH

Fig. 9. Calibration errors versus amplitude level for method [5]. Array signals corrupted by random uniformly distributed measurement errors due to dynamic accuracy: amplitude 0.2 dB and phase 0:2 .

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maximum phase error amounts 19.76 (RMS 4.42 ). The largest 10 dB. errors apply to the signals with an amplitude level In absence of measurement errors, method [5] gives only 100% calibration accuracy when the phase shifters are not troubled by amplitude and phase imperfections. However, there is no possibility to include in method [5] a compensation scheme for these errors like for the SAC method. The comparative evaluation of the three calibration methods with respect to accuracy reveals that the SAC method performs equally or even better than the two other methods [3] and [5] but having the advantage of a reduction in measurement by at least a factor 1.7 for the same number of elements. C. Calibration Accuracy and Measurement Errors The accuracy of the SAC method is governed mainly by two due to factors: measurement uncertainty in the array signal dynamic accuracy of the NA and phase and amplitude imperfections of the element phase shifters. Maximum calibration accuracy, no errors, is achieved in absence of any measurement error and noise in combination with a complete (individual) correction for phase shifter imperfections. In the presence of measurement errors due to dynamic accuracy, the degree of calibration accuracy depends on the way phase and amplitude imperfections of the phase shifters are corrected in the SAC method as is demonstrated by the results of Figs. 5–7. Two possibilities exist to reduce the effect of measurement errors on calibration to a level accuracy. One way is to lower the magnitude of less than 10 dB above the maximum amplitude level of the EUCs. The second possibility to raise calibration accuracy, is to select the EUCs such that the ratio of the maximum and is smaller than minimum amplitude level of 15 dB. The rationale for this selection follows from the results of Figs. 5–7 which indicate small calibration errors for ampli15 dB. Such a selection can be tude levels of the elements realized by the formation of two groups of elements A and B, 15 with group A containing only elements with signal levels dB, group B the remaining elements and then to calibrate both groups separately. The formation of the groups A and B is part of

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Fig. 10. Calibration results for the 40 EUCs with common phase shifter correction. Calibration errors versus amplitude level. Array signals corrupted by random uniformly distributed measurement errors; amplitude: 0.2 dB and phase: 0:2 .

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the initial array calibration that is performed immediately after the successful electrical alignment of the T/R modules at the near-field antenna test range to match the pattern requirements. After the initial calibration, a second and final one needs to be accomplished for both groups separately in order to validate the element selection in the two groups and to get accurate reference data for the complex signals of the array elements in both groups. Fig. 10 illustrates the improvement in accuracy that can be obtained when the calibration of the 40 EUCs involves only amin the range between 30 dB and 15 dB. The applitudes plication of more than two groups of elements to get a lower ratio of the maximum and minimum levels within the groups, will improve the calibration accuracy even further. D. Individual Phase Shifter Failure Verification For a good accuracy of the SAC method, it is essential that all involved element phase shifters perform correctly. An unnoof a phase shifter ticed stuck phase bit element, degrades not only the calibration accuracy of that element but eventually also that of some other EUCs with failure free phase shifters. Therefore, the correct behavior of the phase bits, in particular the largest ones, of all phase shifters has to be checked periodically. Testing of possible phase bit failures can be arranged by performing an SAC calibration that involves only one EUC. meaWith a single element array calibration, the signal sured at the output of the array is given by (8) and refer to the element of which the where , phase shifter is calibrated. Since as outcome from this single , , and element array calibration, the parameters are all known, the imperfections of the phase states involved in the calibration phase shifter for the can then be determined using (8).

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Verification of the performance of the individual the bits of a 6-bit phase shifter requires only five phase cycles when the . At this scan position main beam is scanned to the involved linear phase taper is not corrupted by discretization errors due to the finite phase resolution of the 6-bit phase shifter. Furthermore all phase bits are cycled. Therefore, at scan any stuck bit can be detected except the position smallest bit, as has been verified. Detection of a failing 5.625 bit is troubled by the maximum phase error of the phase shifter, Fig. 3(a). IV. DISCUSSION AND CONCLUSION The minimum number of elements that can be calibrated simultaneously is two and requires at least three phase cycles. The highest value of that was applied to verify the SAC method was 80 and provided good results comparable to those obtained . For any even number of in the range bewith tween 2 and 80, the required minimum number of measurements to get good calibration results, can be calculated from , where is the nearest integer and , called the calibration efficiency, is a constant value equal to 1.18. A smaller value of than causes that the series of simultaneous linear equations becomes ill-conditioned. There is in principle no upper limit for the number of elements that can be calibrated simultaneously. implies more scanned beams but also A larger number for less angular spacing between these beams due to their smaller beamwidth that is inversely proportional to . The smaller angular spacing compensates therefore the increase in the number of scanned beams. larger than to improve calibration accuMaking since this racy, turns out to be hardly effective unless improvement concerns mainly the element signals having amplitude levels in excess of 20 dB and for these levels the errors are already quite small. is equal For the calibration methods [1]–[6] the value of to two or higher. Compared to these methods, the duration of an array calibration performed with the SAC approach is at least a factor 1.7 smaller. Short calibration times are of great relevance when the array consists of million radiating elements as is the case with various very large phased array radio telescopes now worldwide under investigation [5]. A novel approach, called the SAC method, has been described for the calibration of phased array antennas. The key characteristic of the SAC method is the simultaneous calibration of a large number of elements resulting in a serious reduction of calibration time by a factor 1.7 compared to corresponding values from earlier reported procedures. The accuracy offered by the SAC method is comparable or even better than that of current

array calibration methods. Unlike the multi-element calibration methods [4]–[7], the phase cycling of the SAC method has not to generate EUC signals that are orthogonal to each other. This gives the SAC method a greater flexibility compared to the other multi-element calibration methods. An example of this greater flexibility is the opportunity to correct for phase shifter imperfections to enhance calibration accuracy. REFERENCES [1] N. Takemura, H. Deguchi, R. Yonezawa, and I. Chiba, “Phased array calibration method with evaluating phase shifter error,” presented at the Antennas and Propag. Int. Symp., Fukuoka, Japan, Aug. 21–25, 2000. [2] R. Sorace, “Phased array calibration,” IEEE Trans. Antennas Propag., vol. 49, no. 4, pp. 517–525, Apr. 2001. [3] K. M. Lee, R. S. Chu, and S. C. Liu, “A built-in performance-monitoring/fault isolation and correction (PM/FIC) system for active phased-array antennas,” IEEE Trans. Antennas Propag., vol. 41, no. 11, pp. 1530–1540, Nov. 1993. [4] T. Takahashi, Y. Konishi, S. Makino, H. Ohmine, and H. Nakaguro, “Fast measurement technique for phased array calibration,” IEEE Trans. Antennas Propag., vol. 56, no. 7, pp. 1888–1899, Jul. 2008. [5] G. A. Hampson and A. B. Smolders, “A fast and accurate scheme for calibration of active phased-array antennas,” in IEEE AP-S Int. Symp. Digest, 1999, pp. 1040–1043. [6] S. D. Silverstein, “Remote calibration of active phased array antennas for communication satellites,” in Proc. IEEE Int. Conf. on Acoustics, Speech, Signal Processing, Munich, Germany, Apr. 21–24, 1997, vol. 5, pp. 4057–4060. [7] B. Bräutigam, M. Schwerdt, and M. Bachmann, “An efficient method for performance monitoring of active phased array antennas,” IEEE Trans. Geosci. Remote Sensing, vol. 47, no. 4, pp. 1236–1243, Apr. 2009. [8] E. Lier, M. Zemlyansky, D. Farina, and D. Purdy, “Efficient phased array calibration and characterization based on orthogonal coding: Theory and experimental validation,” presented at the IEEE Int. Symp. on Phased Array Systems and Technology, Boston, MA, Oct. 12–15, 2010. [9] Agilent PNA Microwave Network Analyzers, Data Sheet, E8362B, E8363B, E8364B, E8361A Jan. 22, 2007, 5988-7988EN. Will P. M. N. Keizer (M’99) received the M.S. degree in electrical and electronics engineering (cum laude) from the Eindhoven University of Technology, The Netherlands, in 1970. From 1971 to 2002, he was with TNO Physics and Electronics Laboratory, The Hague, The Netherlands, as Research Scientist and Technical Manager in the field of microwave components, antennas, propagation and radar. He has been involved in the NATO Anti Air Warfare System (NAAWS) study directed by the US Navy and aiming at an advanced sensor suite for future warships. He was one of the key persons in the initiation of the APAR naval multi-function active phased array radar based on the NAAWS concept that is now in operational use with the Netherlands and German navies. He was responsible for the design of the APAR antenna aperture consisting of waveguide radiators featuring a 60 deg scan angle capability in all directions over a 30% bandwidth at X-band. His research interests include phased array antennas, near-field antenna testing, microwaves, low-angle radio-wave propagation, multi-function phased array radar and synthetic aperture radar. He authored or coauthored more than 40 papers in journals and conference proceedings. Since 2003 he is retired.

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Application of Analytical Retarded-Time Potential Expressions to the Solution of Time Domain Integral Equations H. Arda Ülkü, Student Member, IEEE, and A. Arif Ergin

Abstract—Recently, exact closed-form expressions of the electric and magnetic fields and potentials due to impulsively excited Rao–Wilton–Glisson basis functions have been presented. In this work, the application of these expressions in the solution of the combined field integral equation (CFIE) is presented. Solutions via analytical expressions of the electric and magnetic fields are verified and compared with the conventional marching-on-in time (MOT) algorithm solutions that employ numerical basis integrations. It is shown that the accuracy and stability of the solutions obtained with the proposed (analytical) approach are better when compared to those obtained with the conventional (numerical) method. In addition, a discussion section about the effect of the proposed approach in solving the electric field integral equation (EFIE) is added. In this section, it is demonstrated that the dependency of the solution via analytical expressions of the fields are less sensitive to the time step size in comparison to the conventional solution, and that the increase in accuracy is a necessary but not sufficient condition for stability. Index Terms—Combined field integral equation (CFIE), electric field integral equation (EFIE), integral equations, marching-on-intime (MOT) method, Rao–Wilton–Glisson (RWG) basis, time domain analysis, transient analysis.

I. INTRODUCTION

T

HE marching-on-in-time (MOT) method is a powerful tool for solving time-domain surface integral equations. However, in the MOT method, stability is a major problem [1]. It has been conjectured that the stability of the solution primarily depends on the accurate computation of the impedance matrix elements and the behavior of the integral equation (i.e., whether it is well-conditioned or not) [2]–[5]. Recent studies show that accurate calculation of the MOT matrix elements contributes to the stability of the MOT solutions [6]–[10]. Accurate computation of the matrix elements is achieved by using analytical expressions of the electric and magnetic fields [6]–[10]. In [9], it is shown that with the analytical expressions, the stability of the MOT solutions of the magnetic field integral equation (MFIE) is improved although the inner resonance problem still remains. A similar result is shown in [10] for the integral equation obtained by the PMCHWT formulation. For a perfect electric conductor (PEC) scatterer, Manuscript received July 21, 2010; revised February 22, 2011; accepted April 14, 2011. Date of publication August 12, 2011; date of current version November 02, 2011. This work was supported in part by the GEBIP program of the Turkish Academy of Sciences (TÜBA). The authors are with the Department of Electronics Engineering, Gebze Institute of Technology, Kocaeli 41400, Turkey (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TAP.2011.2164180

the PMCHWT formulation reduces to either the EFIE or the MFIE, and therefore the solutions might still be susceptible to internal resonance problem if special precautions are not practiced. However, to the best knowledge of the authors, the benefit of the application of the closed-form expressions in the solution of the CFIE has not been shown so far. Furthermore, it is the authors’ experience that the instability of the MOT solutions of the time-domain integral equations cannot be attributed to a single rule or a scheme—let alone accuracy. For example, for the same scatterer and MOT formulation, the solution can be stable or unstable for different spatial discretization or time step sizes. The main aim of this paper is to show the benefit(s) of using the closed-form expressions in the MOT solution of the CFIE. Accurate calculation of the MOT matrix elements is achieved by using the closed-form expressions for the electric and magnetic fields (and/or potentials). The basic idea is that the spatial bases integrals alone can be regarded as a form of Radon transformation. This idea was first applied to the evaluation of the physical optics integral for triangular patches in [11] and for NURBS surfaces in [12]. There, the result of the Radon transform interpretation is an intersection of a plane wave with the scatterer. Hence, this can be called a planar Radon transform. With this approach, the physical optics integral can be evaluated analytically in the time-domain without any approximations. The second application of the idea is seen within the evaluation of integral equations as a spherical Radon transform [6] performed for the well-known Rao–Wilton–Glisson (RWG) basis functions [13]. In this context, the idea of spherical Radon transform provides a means to analytically determine the electric field radiated by an impulsively excited RWG basis. However, the algorithm drawn in [6] is not convenient to determine the convolution of the result with the temporal basis functions analytically. With the analytical formulae and algorithm given in [8] it is possible to determine both the spatial and temporal (convolution) integrals for both magnetic and electric fields analytically, as long as the temporal basis functions are piecewise polynomial functions. In order to constitute the clarity of the formulation for the CFIE, the formulae developed in [6] and [8] are revisited, and compact forms of the analytical expressions for impulsively excited fields are presented in the first part of this paper. Furthermore, an algorithm to evaluate these expressions within the context of the MOT formulation is elucidated. As was shown for the MFIE solutions in [9], it is expected that using the analytical expressions in the solution of the time-domain CFIE will improve the accuracy and hence the stability of the solution. This will be demonstrated through a numerical example.

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The proposed Radon transform-based approach has various advantages over alternative approaches such as that outlined in [10], which uses spatial partitioning with respect to time step size to obtain the line integrals that are evaluated analytically. The proposed method does not require any spatial partitioning, and the analytical expressions for the fields/potentials at any time instant can be found independent of the time step size. Furthermore, logarithmic or weakly singular terms that appear in the exact expressions in [10] do not appear in the analytical expressions given in the proposed approach. Once the main aim, i.e., the improvement of the stability of the MOT solution of the CFIE using analytical expressions has been shown, a discussion section that explores whether improved accuracy is sufficient for stability of the EFIE will be presented. In this section, it will be emphasized that neither the increase in the accuracy of the MOT matrix elements guarantees stability nor the inaccurateness of the solution guarantees instability. To this end, first, the dependence of the MOT matrix elements on the time step size will be studied for the EFIE. In general, it is expected that decreasing the time step size would increase the accuracy of the temporal interpolation process yielding better solutions. However, contrary to this expectation, it will be shown that the matrix elements obtained via the conventional numerical algorithm show drastic differences from those computed by the analytical approach with decreasing time step size. As a result, it will be shown that the solutions obtained by the conventional (numerical) MOT scheme may become corrupt (although stable) as the time step size is decreased while those obtained by the proposed (analytical) scheme are not affected much. This shows a clear advantage of using the analytical formulae besides exemplifying that “inaccurateness of the solution does not guarantee instability.” As a second discussion point, it is shown with a single example that instabilities can prevail even though the proposed (analytical) scheme is used and the solution agrees well until the onset of the instability. Hence, “the increase in the accuracy of the solutions does not guarantee stability.” Note that, these two adverse cases have been demonstrated by using the EFIE (which is not as well-conditioned as the CFIE), and that the authors have not run into any similar cases while solving the CFIE using the analytical closed-form expressions. This paper is organized as follows. In Section II, formulation of the EFIE, MFIE, and CFIE via analytical expressions of the electric and magnetic fields due to RWG basis and piece-wise polynomial temporal basis functions is presented. In Section III, numerical examples are presented and improvements obtained by using the analytic formulae of the potentials and fields are shown. In Section IV, a discussion on two adverse cases that appear in the solution of the EFIE is presented. Conclusions are drawn in Section V. II. IMPLEMENTATION OF ANALYTICAL RETARDED-TIME POTENTIALS In the MOT method, the unknown current density, , is discretized with spatial and temporal basis functions as follows: (1)

Fig. 1. Definition of the RWG basis.

where is the th spatial basis function, is the th temare the unknown coefficients. In poral basis function, and this paper, all formulations are based on the RWG basis functions [13] in space and temporal basis functions are chosen as piece-wise polynomial interpolation functions given in [14]. In accordance with Fig. 1, the RWG basis functions are defined as elsewhere

(2)

is the area of the where is the common edge length and patch. In Sections II-A, II-B, and II-C, time domain EFIE, MFIE, and CFIE are investigated, respectively, and the integrals that must be evaluated analytically are determined. In these sections, the spatial integrals, separate from the temporal convolutions, can be determined in terms of arc length and bisecting vector functions. The results of the spatial integrals can be regarded as closed-form expressions of the potentials and magnetic field due to impulsively excited RWG function given in (2). Analytical expressions of the arc length and bisecting vector functions and their derivatives are drawn in Section II-D. To obtain the electric and magnetic fields, convolution of these functions with temporal basis functions must be evaluated, which is explained in Section II-E. A. Electric Field Integral Equation (EFIE) The EFIE for perfect electric conductor (PEC) surfaces can be written as

(3) where is the incident electric field, denotes temporal derivative, denotes convolution with respect to time, is the outward unit normal vector of the scatterer surface at the observation point , is the permittivity, is the permeability of the surrounding medium, and is the speed of light. denotes the distance between the source (integration) and observation points.

ÜLKÜ AND ERGIN: APPLICATION OF ANALYTICAL RETARDED-TIME POTENTIAL EXPRESSIONS

In the conventional MOT algorithm, the convolution operations are performed analytically—because convolution with Dirac delta function is easy to perform—but surface integral with respect to basis functions is calculated numerically. However, to use the closed-form expressions of the potentials developed in [6] and [8], the spatial integral over the basis triangle must be evaluated first via Radon transform interpretation. After using (1) in (3) and testing the resulting equation with testing at , the MOT matrix elements of the refunctions sulting equation system can be written as

(4)

and are the magnetic vector potential where and electric scalar potential due to impulsively excited th basis function, given respectively as

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(8) is the bisecting vector and is the arc where . Closed-form expressions of and length for patch are derived in [8] the results of which will be repeated in Section II-D for convenience. Once these potential functions are evaluated, their convolution with the temporal basis functions have to be performed as indicated in (4). As mentioned before, convolutions can be found analytically if the temporal basis functions are chosen as piece-wise polynomial functions. With the analytical evaluation of the convolution integrals, the development of the closed-form expression of the electric field due to RWG spatial basis and piece-wise polynomial temporal basis will be completed. Testing of the electric field with will be performed numerically as done in the conventional MOT schemes. B. Magnetic Field Integral Equation (MFIE) The MFIE for PEC surfaces can be written as

(5) (9)

(6)

The closed-form expressions of the vector and scalar potentials are developed in [6] in terms of geometric quantities formed by the intersection of a triangular patch and a sphere that is centered at the observation point with radius . According to [6] and [8], analytical expressions of scalar and vector potentials can be written as

where is the incident magnetic field. After using (1) in (9) and testing the resulting equation with testing functions at , the MOT matrix elements for the resulting equation system can be written as

(10)

where is the magnetic field due to impulsively excited RWG function

(11)

By using the vector definitions given in Fig. 2 together with (2), can be written as

(7)

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Fig. 2. Vector definitions for a triangular patch. Fig. 3. Definitions of the geometric parameters for ith edge.

D. Arc Length and Bisecting Vector

(12) where

(13)

(14) denotes derivative with respect to . The In (13) and (14), and are shown in [8]. Their evaluaevaluation of tion will become clear by the formulae that will be presented in Section II-D. With the end of this section, the first step to determine the electric and magnetic fields is completed. The final expressions derived in Sections II-A and B are the potentials (needed for the electric field) and magnetic field due to impulsively excited RWG bases. In Section II-D, the analytical expressions of the , and arc length, , and their derivabisecting vector, and ) will be drawn. tives with respect to (

This section is organized to illustrate the closed-form expresand bisecting vector that sions of the arc length can be used directly in implementation. The evaluation of the integrals given in (7), (8), (13), and (14) is explained in [8], in detail. In this section, the resulting functions given in [8] are briefly presented. The bisecting vector and arc length functions are geometric quantities and can be determined analytically. The analytical expressions of the bisecting vector and arc length are expressed in terms of the barycentric coordinates. These analytical expressions lead to the analytical determination of the temporal convolution of the resulting functions with the temporal basis functions, which will be explained in the next section. In this section, an algorithm, also given in [8], will be drawn to , , , and . It must be determine noted that the RWG function consists of a pair of triangles de. For brevity, formulations will be presented only noted as , omitting the “ .” The results can be easily extended to for with appropriate sign changes. It must be emphasized that all functions are used as defined in [8]. Before moving to the expressions of arc length and bisecting vector, it is convenient to elucidate some of the definitions used in the upcoming formulae with reference to Fig. 3: is the length of the th edge and , respectively, denote the smaller and larger and distances to the ends of the th line segment from the point , , and denote the angles measured from the axis and , respectively. Also, let denote the perpento dicular distance from to the extension of the th line segment. , arc length can be written as With these definitions, for

C. Combined Field Integral Equation (CFIE)

(16)

The CFIE in this paper consists of the linear combination of the EFIE and MFIE expressed as (15) is the combining constant and is where the intrinsic impedance. Although CFIE is proposed as a remedy for the inner resonance problem, sometimes the behavior of the EFIE affects the CFIE solution [5].

, is the barycentric coordinates of and is the arc length function for the point related to the th is the radius of the circle edge of the triangle, that emerges due to the intersection of the sphere with radius and the plane that contains the triangular surface . It is obvious that if there is no intersection of the sphere and . For , to evaluate triangular surface, so two cases depending on the relative positions of the th edge and where

ÜLKÜ AND ERGIN: APPLICATION OF ANALYTICAL RETARDED-TIME POTENTIAL EXPRESSIONS

have to be considered. In the first case, if a single arc is formed, and

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,

(17)

(25) where and are the contributions due to the th edge of the triangle. In order to evaluate the derivative of with respect to , the chain rule can be used as

. In the second case, for , two arcs emerge. So is denoted by the sum of two arc lengths as

(26) where

and

can be evaluated as

(18) where

and

are given by (27)

. The derivative of with can be determined as

where If

(19) with respect to

(28)

(20)

The closed-form expressions for , , , and are presented in Table I. The development of these expressions are described in [8] in detail. In Table I, is defined as

can be determined from (17)–(19) as follows. ,

elsewhere and if

(21)

assuming that

is in the interval ,

, (22)

where

elsewhere.

(29)

m

, ,

, and . With the analytical expressions presented in this and the previous sections, it is seen that the temporal behavior of the impulsively excited electric field in EFIE depends on four functions of time: and

(23)

(30) The behavior of is investigated in [8] in detail. It must be are very important emphasized that the discontinuities in and determinative of the singular behavior of the magnetic field [15]. has components along the and The bisecting vector directions as defined in Fig. 2. Following the formulation drawn in [8], the components of the , i.e., and , sphere are found via projections of the intersection of the and the triangle surface to and axes. It is obvious that if , then , because the sphere does not intersect with the triangle surface. For , and are given by (24)

Similarly, the temporal behavior of the impulsively excited magnetic field in MFIE can be written as a function of and (31) Time dependency of the fields radiated from an impulsively excited RWG function behaves as the functions given in (30) and (31). Since, in the MOT method, the temporal behavior of the currents are expressed as a superposition of weighted and shifted temporal basis functions, the fields due to these currents can be readily evaluated by temporally convolving (7), (8), (13),

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TABLE I CLOSED FORM EXPRESSIONS FOR THE BISECTING VECTOR AND ITS DERIVATIVE

and (14) with the temporal basis functions. Next section will elucidate how the temporal convolutions can be evaluated. E. Evaluation of the Convolution Integrals In the previous sections, the analytical expressions of the fields due to impulsively excited RWG functions are presented. With the temporal behavior of the spatial integrals given in can be used to Section II-D, any temporal basis function develop an MOT solution by determining the convolution of with the functions given in (30) and (31). The approach presented here is different than the conventional MOT approach where the convolution of the temporal with the Green’s function is performed analytically basis first, and then the spatial basis integral is calculated numerically. Here, the spatial basis integral is performed first (as outlined in Section II-D) and then the temporal convolution is evaluated, which can be performed either analytically or numerically de. As shown in [7] and [9], the accuracy of the pending on impedance matrix elements is better than the accuracy of those obtained by the conventional approach even if the temporal convolution is evaluated numerically. Obviously, (if the choice of permits) analytical evaluation of the temporal convolution will be devoid of any numerical errors and it is expected that the solution of the MOT algorithm will be better. In this paper, because of the second order derivative with respect to time in EFIE, the third-order piece-wise polynomial temporal basis functions introduced in [14] will be used. These basis functions are expressed as linear combinations of the monomial functions: 1, , , and . To evaluate the temporal convolution analytically, the convolution of the monomial functions with the functions given in (30) and (31) must be evaluated. This can be achieved by using standard integral tables such as those given in [16]. Some of the convolution integrals are readily derived in [8] for the MFIE. The integration limits of the temporal convolution must be determined carefully. These limits depend on both the intervals

of the temporal basis function and the intervals of the arc length and bisecting vector functions. With the intervals of these functions given as in Table I, the integration limits can be found easily. With the analytical expressions of the temporal convolution, the exact expression to the electric and magnetic fields temporal due to RWG spatial basis functions excited with basis will have been found. Hence, only the testing integrals that appear in (4) and (10) must be calculated numerically to find the MOT matrix elements. III. NUMERICAL RESULTS In this section, verification of the developed method will be demonstrated via numerical examples. In all examples, the incident field is chosen as a modulated Gaussian plane wave: (32) where is the center frequency, is the direction of propagation, and is the polarization of the incident wave. In (32), , , and will be referred to as the effective bandwidth. In all examples, time step size is chosen , unless another value is indicated. as All spatial numerical integrals are implemented by subdividing the integration domain to sub-triangles whose edges are much smaller than the minimum wavelength of the incident field and using a seven-point Gauss quadrature rule for the small triangular domains. While obtaining the conventional (numerical) time domain EFIE results, the singularities in the vector and scalar potentials are extracted by the formulae given in [17]. In the presented results, “A” is used to denote the solutions obtained by the proposed method based on analytical expressions and “N” is used to denote solutions obtained by the conventional MOT algorithm based on numerical integration. In the first example, scattering from a cube with dimensions m m discretized with 192 triangles is investiof m with gated. The incident field propagates along

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unstable, it can be concluded that the A-CFIE is more accurate and hence less prone to instabilities, which is the main point of this study. IV. DISCUSSION

Fig. 4. Comparison of the current observed on the unit cube computed using the proposed method (A-CFIE, A-EFIE) and the conventional method (N-CFIE, N-EFIE).

Fig. 5. Comparison of the current observed on the almond computed with analytical and numerical based CFIEs ( = 0:2).

the polarization and the cube is investigated at MHz and MHz. The first resonance frequency of the scatterer is at 150 MHz and the upper limit of effective spectrum of the incident field is at 140 MHz. Therefore the effect of the inner resonance problem must not be seen in the MOT solutions. As seen in Fig. 4, the current densities obtained by both analytical and numerical based methods coincide. Therefore, we conclude that the formulation presented in this paper is correct. As expected, both EFIE results show a linearly increasing behavior [18]. Also, the results do not show any growing oscillations because the effective bandwidth does not contain any resonance frequencies of the unit cube. As will be seen in the next example, the conventional MOT scheme for the CFIE does not always produce such accurate (and stable) results in a more general setting. In the second example, a NASA almond [19] with maximum height of 0.0575 m and maximum width of 1.15 m, discretized with 704 triangular patches is analyzed. The properties of the , , incident field are chosen as MHz, and MHz. As seen in Fig. 5, the CFIE results do not coincide and N-CFIE result is growing exponentially with respect to time. Since the N-CFIE solution is

In the previous section, the proposed approach to employ exact potential/field expressions in an MOT implementation has been verified. It has also been shown—albeit with a single example—that the use of exact expressions contributes to the stability of the results as mentioned in the literature (e.g., [9] and [10]). When the analytical expressions are used in EFIE, beside the accurate calculation of the matrix elements, one of the advantages is that there is no need to perform any singularity extraction, because the analytical expressions of the retarded-time vector and scalar potentials indeed do not have any apparent singularities. However, in the conventional MOT algorithm, which uses numerical integration, the extraction of the singularities for the scalar and vector potentials depends on the time step and patch sizes [17]. This section is prepared to discuss the effects of the use of the analytical expressions in the MOT solution of the EFIE. This is investigated via two examples. In the first example, the dependence of the matrix elements and hence the solution to time step size is investigated. In the second example, it is shown that neither the increase in the accuracy of the solutions guarantees stability nor the inaccurateness of the solution guarantees instability. That is, accurate calculation of the matrix elements (by presented approach) is a necessary but not sufficient condition for the stability of the solution. , the It is expected that by decreasing the time step size interpolation points will increase and the current density will be interpolated more accurately in time. To investigate the convergence of the MOT matrix elements (and hence the solution of the EFIE) with respect to time step size, scattering from a sphere with radius 1 m, discretized with 290 triangular patches, is studied. The frequency properties of the incident wave are MHz and MHz to exclude the resonance frequencies of the sphere. The other properties of the incident wave are chosen as in the first example in the previous section. In Table II, proportional error in one of the self term elements of the impedance matrix of EFIE is given for different time step sizes. The proportional error is calculated by (33) where and are matrix elements obtained with conventional numerical-based method and analytical-based method proposed in this paper, respectively. It is seen in Table II that as the time step size is decreased, the values of the matrix element obtained via the conventional numerical method diverge from those obtained by the analytical-based method proposed in this paper. This difference is mainly caused by the discontinuities in the derivative of the temporal basis function and the numerical accuracy of the spatial integration in the numerical-based method. In Fig. 6, solutions of the same problem with different time step sizes are shown. (We note that the first

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TABLE II COMPARISON OF THE PROPORTIONAL ERROR IN THE SELF TERM ELEMENT OF THE EFIE IMPEDANCE MATRIX OBTAINED WITH PROPOSED AND CONVENTIONAL METHOD FOR DIFFERENT t

1

Fig. 7. Comparison of the current observed on the sphere with 1-m radius com: . puted with analytical and numerical based EFIE and CFIE 

( = 0 2)

Fig. 6. Comparison of current observed on sphere obtained using proposed and conventional method to solve EFIE with different t.

1

order formulation introduced in [18], which does not change the MOT matrix elements, is used to obtain the EFIE results that are devoid of a linearly increasing residue.) As a reference value is chosen. As seen in Fig. 6, the numerical and analytical-based results obtained with coincide indistinguishably. However, contrary to the expecta, tion mentioned above, if the time step size is decreased to the N-EFIE solution is erroneous and shows an oscillating behavior, whereas, the A-EFIE solution is still accurate and coin. In conclusion, cides well with the results obtained with once a time step size (e.g., ) yields an acceptable solution in the conventional numerical approach, it would be wrong to assume that the matrix elements have converged with respect to time step size and that the solution will be better for smaller . Contrarily, the proposed analytical algorithm is immune to the effect of the time step size. In the second example, scattering from the sphere investigated in the previous example is analyzed. The properties of the incident wave are as in the previous example except MHz and MHz. This time, the spectrum of the

incident wave includes several resonant frequencies of the unit sphere. It is known that the solution of the conventional EFIE (N-EFIE) for the given parameters and scatterer is not accurate and do not converge as shown in [1], especially because of the excited resonant modes. In Fig. 7, the current density associated with a basis function is shown. The solutions in Fig. 7 can be inspected in two parts: the “main response” appears for ns and a “residual response” after 55 ns. In the “main response” part, the A-EFIE result closely follows the A-CFIE result as seen in the inset. The N-EFIE result is not accurate in the “main response” section, which clearly shows the advantage of using the analytical formulation. It must be noted that (although not shown here) using more sample points to increase the accuracy of the spatial basis integrals does not change the inaccurate nature of the N-EFIE result. In the “residual response” part, neither of the EFIE results is correct. The A-EFIE shows an unstable behavior emphasizing that the increased accuracy does not guarantee stability. The instability of the A-EFIE can be attributed to the poor conditioning of the EFIE [2], [3], [5], which implies that a little error in the matrix elements disproportionately affects the solution. Note that although the analytical formulae are used in evaluating the basis integrals, the testing integrals are still determined by numerical integration. Whether evaluating the testing integrals analytically too (if possible at all) would improve the stability of the solution is yet to be answered. V. CONCLUSION In this paper, application of the analytical expressions of the retarded-time potentials to the MOT solution of the CFIE is demonstrated. To explain the developed and applied algorithm, analytical expressions of the electric and magnetic fields due to impulsively excited RWG bases are revisited and presented in compact form as used in the implementation. In the numerical results, it is shown that using the analytical expressions in the solution of the time-domain CFIE improves the accuracy and hence the stability of the solution. It can be concluded that the CFIE solutions obtained by using analytical expressions of the electric and magnetic fields is always stable. In addition a discussion section is prepared to investigate the effects of using analytical expressions in MOT solution of EFIE.

ÜLKÜ AND ERGIN: APPLICATION OF ANALYTICAL RETARDED-TIME POTENTIAL EXPRESSIONS

First, the dependency of the conventional EFIE to time step size is investigated. It is shown that the conventional EFIE results become corrupted due to discontinuous behavior of the derivatives of the temporal basis function, when the time step size is chosen smaller, whereas the results obtained by the developed method remain accurate. It can be concluded that the developed method is more immune to time step size selection. Second, in some cases, especially when the incident wave includes many resonance frequencies, the EFIE solution obtained by using the analytical formulae developed in this paper, shows unstable behavior, even if the solution coincides with the CFIE results at early times. For the same cases, conventional EFIE solution is not accurate and does not coincide with the CFIE solution. Hence, accurate calculation of the matrix elements (obtained by the presented approach) is a necessary but not sufficient condition for the stability of the solution. REFERENCES [1] B. Shanker, A. A. Ergin, K. Aygün, and E. Michielssen, “Analysis of transient electromagnetic scattering from closed surfaces using a combined field integral equation,” IEEE Trans. Antennas Propag., vol. 48, no. 7, pp. 1064–1074, Jul. 2000. [2] K. Cools, F. P. Andriulli, F. Olyslager, and E. Michielssen, “Time-domain Calderon identities and their application to the integral equation analysis of scattering by PEC objects Part I: Preconditioning,” IEEE Trans. Antennas Propag., vol. 57, no. 8, pp. 2365–2375, Jul. 2009. [3] F. P. Andriulli, K. Cools, F. Olyslager, and E. Michielssen, “Time-domain Calderon identities and their application to the integral equation analysis of scattering by PEC objects Part II: Stability,” IEEE Trans. Antennas Propag., vol. 57, no. 8, pp. 2352–2364, Jul. 2009. [4] M. Lu and E. Michielssen, “Closed form evaluation of time domain fields due to Rao–Wilton–Glisson sources for use in marching-on-intime based EFIE solvers,” in Proc. IEEE AP-S Int. Symp., San Antonio, TX, vol. 1, pp. 74–77. [5] H. Contopanagos, B. Dembart, M. Epton, J. J. Ottusch, V. Rokhlin, J. L. Visher, and S. M. Wandzura, “Well-conditioned boundary integral equations for three-dimensional electromagnetic scattering,” IEEE Trans. Antennas Propag., vol. 50, no. 12, pp. 1824–1830, Dec. 2002. [6] A. C. Yücel and A. A. Ergin, “Exact evaluation of retarded-time potential integrals for the RWG bases,” IEEE Trans. Antennas Propag., vol. 54, no. 5, pp. 1496–1502, May 2006. [7] J. Pingenot, S. Chakraborty, and V. Jandhyala, “Polar integration for exact space-time quadrature in time-domain integral equations,” IEEE Trans. Antennas Propag., vol. 54, no. 10, pp. 3037–3042, Oct. 2006. [8] H. A. Ülkü and A. A. Ergin, “Analytical evaluation of transient magnetic fields due to RWG current bases,” IEEE Trans. Antennas Propag., vol. 55, no. 12, pp. 3565–3575, Dec. 2007. [9] H. A. Ülkü and A. A. Ergin, “Application of analytical expressions of transient potentials to the MOT solution of integral equations,” in Proc. IEEE AP-S Int. Symp. and URSI Radio Sci. Meeting, San Diego, CA, 2008, pp. 1–4. [10] B. Shanker, M. Lu, and E. Michielssen, “Time domain integral equation analysis of scattering from composite bodies via exact evaluation of radiation fields,” IEEE Trans. Antennas Propag., vol. 57, no. 5, pp. 1506–1520, May 2009. [11] D. Bolukbas and A. A. Ergin, “A radon transformation interpretation of the physical optics integral,” Microw. Opt. Tech. Lett., vol. 44, p. 284, Feb. 2005.

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[12] H. A. Serim and A. A. Ergin, “Computation of the physical optics integral on NURBS surfaces using a radon transform interpretation,” IEEE Antenna Wireless Propag. Lett., vol. 7, pp. 70–73, 2008. [13] S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag., vol. 30, no. 3, pp. 408–418, May 1982. [14] K. Aygün, S. Balasubramaniam, A. A. Ergin, and E. Michielssen, “A two-level plane wave time-domain algorithm for fast analysis of EMC/EMI problems,” IEEE Trans. Electromagn. Compat., vol. 44, no. 1, pp. 152–164, Feb. 2002. [15] H. A. Ülkü and A. A. Ergin, “On the singularity of the closed-form expression of the magnetic field in time domain,” IEEE Trans. Antennas Propag., vol. 59, no. 2, pp. 691–694, Feb. 2011. [16] J. J. Tuma, Engineering Mathematics Handbook. New York: McGraw-Hill, 1987. [17] D. Wilton, S. Rao, A. Glisson, D. Schaubert, O. Al-Bundak, and C. Butler, “Potential integrals for uniform and linear source distributions on polygonal and polyhedral domains,” IEEE Trans. Antennas Propag., vol. 32, no. 3, pp. 276–281, Mar. 1984. [18] H. A. Ülkü, “Radon transform interpretation of radiation integrals and time domain scattering problems,” (in Turkish) Ph.D. dissertation, Dept. Electronics Eng., Gebze Inst. of Technol., Kocaeli, Turkey, 2011. [19] A. C. Woo, H. T. G. Wang, M. J. Schuh, and M. L. Sanders, “Benchmark radar targets for the validation of computational electromagnetics programs,” IEEE Antennas Propag. Mag., vol. 35, no. 1, pp. 84–89, Feb. 1993.

H. Arda Ülkü (S’06) was born in Antalya, Turkey, in 1984. He received the B.S., M.S., and Ph.D. degree in electronics engineering from the Gebze Institute of Technology (GIT), Kocaeli, Turkey, in 2006, 2008, and 2011, respectively. He has been a Research and Teaching Assistant at the Electromagnetic Fields and Microwave Division, Department of Electronics Engineering, GIT, since 2007. His graduate study was partially supported by the Scientific and Technical Research Council of Turkey (TUBITAK). His research interests include computational electromagnetics, especially time domain integral equations. Dr. Arda Ülkü was awarded for Excellence in Electromagnetics (for Turkish students/researchers) by the Leopold B. Felsen Award in 2009.

A. Arif Ergin was born in Ankara, Turkey, in 1970. He received the B.S. degree in electrical and electronics engineering from the Middle East Technical University, Ankara, in 1992, and the M.S. and Ph.D. degrees in electrical and computer engineering from the University of Illinois at Urbana-Champaign, in 1995 and 2000, respectively. Until mid-1993, he was with ASELSAN Military Electronics, Inc. Since 2000, he has been with the Electronics Engineering Department, Gebze Institute of Technology, Gebze, Kocaeli, Turkey. He has also assumed part-time positions on various projects with the Marmara Research Center of The Scientific and Technical Research Council of Turkey (TUBITAK) as well as several R&D companies. He is interested in the numerical analysis, simulation, and measurement of electromagnetic wave phenomena. Dr. Ergin is a member of Phi Kappa Phi.

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Hierarchical Matrix Techniques Based on Matrix Decomposition Algorithm for the Fast Analysis of Planar Layered Structures Ting Wan, Zhao Neng Jiang, and Yi Jun Sheng

Abstract—The matrix decomposition algorithm (MDA) provides an efficient matrix-vector product for the iterative solution of the integral equation (IE) by a blockwise compression of the impedance matrix. The MDA with a singular value decomposition (SVD) recompression scheme, i.e., so-called MDA-SVD method, shows strong ability for the analysis of planar layered structures. However, iterative solution faces the problem of convergence rate. An efficient hierarchical ( -) LU decomposition algorithm based on the -matrix techniques is proposed to handle this problem. Exploiting the data-sparse representation of the MDA-SVD compressed impedance matrix, -LU decomposition can be efficiently implemented by -matrix arithmetic. -matrix techniques provide a flexible way to control the accuracy of the approximate -LU-factors. -LU decomposition with low accuracy can be used as an efficient preconditioner for the iterative solver due to its low computational cost, while -LU decomposition with high accuracy can be used as a direct solver for dealing with multiple right-hand-side (RHS) vector problems particularly. Numerical examples demonstrate that the proposed method is very robust for the analysis of various planar layered structures. Index Terms—Fast direct solver, hierarchical matrices ( -matrices), matrix decomposition algorithm (MDA), planar layered structures, preconditioning technique.

I. INTRODUCTION MONG the numerical methods for the full-wave analysis, the method of moments (MoM) [1], [2] for the solution of integral equation (IE) is widely used. MoM discretization of IE yields a dense complex linear system, which is a serious handicap especially for electrically large problems. Direct solution of memory this system is basically impractical due to the computational complexity, where requirement and refers to the number of unknowns. This difficulty can be circumvented by the use of iterative methods, in which the required matrix-vector product (MVP) operations can be accelerated by fast numerical algorithms, such as the multilevel fast multipole algorithm (MLFMA) [3], [4]. Another attractive fast algorithm is

A

Manuscript received January 19, 2011; revised March 30, 2011; accepted May 01, 2011. Date of publication August 12, 2011; date of current version November 02, 2011. This work was supported in part by the Major State Basic Research Development Program of China (973 Program: 2009CB320201), in part by Natural Science Foundation under Grants 60871013, 60701004, and in part by Jiangsu Natural Science Foundation under Grant BK2008048. The authors are with the Department of Communication Engineering, Nanjing University of Science and Technology, Nanjing 210094, China (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2164222

the matrix decomposition algorithm (MDA) [5]–[7]. Exploiting the rank-deficiency feature of IE, the MDA reduces the storage requirement and speeds up the MVPs of iterative solution by a blockwise low-rank compression of the impedance matrix. It is well-known that the MLFMA is highly efficient for 3D structures in free space. However, the MDA has shown to be much more efficient for planar or piece-wise planar structures. Another important advantage of the MDA is that it does not rely on specific mathematical properties of the Green’s functions. Thus, the whole procedure can be easily applied to interesting configurations governed by special Green’s functions like layered media [7], [8]. Hence, the MDA is a powerful approach for the analysis of planar layered structures, such as frequency selective surfaces (FSS), planar microstrip antennas, electromagnetic band gap (EBG) materials, etc. [7]–[10]. To further improve the performance of MDA, a singular value decomposition (SVD) scheme is introduced to recompress the MDA compressed matrices, i.e., the resulting MDA-SVD method [11]. Although the MDA-SVD method significantly reduces the computational requirements, iterative solution usually encounters the following three obstacles: (1) the number of iterations will increase with the unknown number, leading to an increased complexity; (2) the convergence rate will be slow down when the impedance matrix is ill-conditioned; and (3) for problems with multiple right-hand-side (RHS) vectors, such as the calculation of monostatic RCS, the iterative solution needs to be resumed for each independent RHS vector, leading to redundant computational consumption. Direct solvers can be employed to circumvent these obstacles. The latest and attractive works on the fast direct solvers are the compressed block decomposition (CBD) algorithm and its successor named the multiscale CBD (MS-CBD) algorithm [12]–[15]. The CBD algorithm has been successfully applied to solve not only electrostatic problems but also electromagnetic scattering and radiation problems, while the MS-CBD algorithm adopts a recursive binary subdivision scheme and further improves the ability of CBD algorithm. In this paper, to overcome the above mentioned obstacles of the iterative solution, an efficient hierarchical ( -) LU decomposition algorithm based on the MDA-SVD compression and -matrix techniques is proposed, which produces an efficient preconditioner for the iterative solution of problems with a single RHS-vector and a fast direct solver for problems with multiple RHS-vectors. -matrix theory has first been introduced by W. Hackbusch [16], and subsequently widely applied to analysis different problems [17]–[20]. -matrix techniques are based on a data-sparse representation, which is

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WAN et al.: HIERARCHICAL MATRIX TECHNIQUES BASED ON MDA FOR THE FAST ANALYSIS OF PLANAR LAYERED STRUCTURES

an inexpensive but sufficiently accurate way to approximate a fully populated matrix. The key idea is that certain sub-blocks of a reordered matrix can be approximated by a product of two low-rank matrices [21]–[24], which can be represented with , , as and , . The MDA-SVD compressed matrices are just data-sparse since they have hierarchical low-rank matrix representations, or they are -matrices. By the -matrix arithmetic, the essential operations of -matrices, such as matrix-vector and matrix-matrix multiplication, addition, inversion and LU comdecomposition, can be performed in plexity with appropriate blockwise rank and parameters , [25], [26]. -LU decomposition provides a flexible way to control the accuracy of the approximate -LU-factors. The -LU-factors with low accuracy have much less storage requirement than MDA-SVD compressed matrix itself, and hence can be used as an efficient preconditioner, which enters the MVPs of iterative solution through fast -matrix formatted forward and backward substitutions ( -FBS). The -LU decomposition with high accuracy can be used as a direct solver for handling problems with multiple RHS-vectors particularly, since the manipulation of -LU decomposition need to be implemented only once for as many RHS-vectors as needed and only -FBS is required for each independent RHS-vector. Numerical examples will show that the -LU-based preconditioning technique and direct solver can significantly improve the performance of MDA-SVD for the analysis of planar layered structures. The remainder of this paper is organized as follows: Section II describes the theory and implementation of -LU decomposition and the MDA-SVD method in detail. Numerical examples with several planar layered structures are presented to demonstrate the efficiency of the proposed method in Section III. Section IV gives some conclusions. II. THEORY A. MoM Discretization of MPIE Most of the implementations proposed for the planar layered structures are on the basis of planar layered media with different levels of metallization that behave as an effective medium. Since the layered media Green’s function is adopted, only the metallic surface has to be meshed. The layered media Green’s function is solved by discrete complex image method. The induced current on the metallic surface can be obtained by the solution of the mixed potential integral equation (MPIE) [27] as follows (1) where denotes the metallic surface, is an outwardly directed normal and is the magnetic permeability. The vector and scalar potentials can be expressed as (2) (3)

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is the magnetic vector potential dyadic Green’s funcwhere is the Green’s function of the electric scalar potion and tential. To solve the (1), the current on the metallic surface is expanded by the Rao-Wilton-Glisson (RWG) basis functions [2]. Applying the Galerkin’s method to (1), the resulting linear system can be symbolically rewritten as (4) where is the impedance matrix with the size being the number of RWG basis functions, denotes the unknown vector of the surface current coefficients and is the RHS-vector. B. Admissible Partitioning to Form an

-Matrix Structure

Both the -matrix techniques and the MDA-SVD method are based on the hierarchical partitioning of the impedance matrix into blocks. An admissible partitioning is required for generating an -matrix structure of and implementing the MDA-SVD compression. The basic steps for building an -matrix structure are: (1) construction of an cluster tree of index based on a block set , (2) admissible partitioning of . To be able to uniquely localize an element cluster tree of in block-matrices on arbitrary levels, it is necessary to apply a partitioning to the index set , which is a finite index set of all the RWG basis functions. A cluster tree of should satisfy: • Root ; with , then • If ; with , then ; • If where denotes the cardinality of the cluster and is predetermined threshold parameter to control the depth of the cluster tree. A cluster tree is usually generated by recursive geometry-based subdivision of the index set . In classical -matrix theory, binary tree is adopted by subdividing an index set into two subsets recursively as shown in Fig. 1. This is because the binary tree is simple and flexible, further, -matrix operations such as -inversion and -LU are recursively implemented on the basis of 2 2 block matrices. The cluster tree shown in Fig. 2(a) is a simplified case for including only eight elements. In the MDA-SVD method, to build a cluster tree for the coming compression, the object is first entirely enclosed in a large cube which is subdivided into eight cubes as shown in Fig. 1. Then the box-wise partitioning is recursively carried out until the size of the smallest box is generally the order of half a wavelength. The resulting cluster tree is an octal tree and its leaves are referred to as leaf boxes. Since the binary tree is usually used in -matrix techniques, in order to implementing the -matrix techniques based on the MDA-SVD, the tree structures of the MDA-SVD and -matrix methods should be uniformized, i.e., the binary tree of -matrix should include all the leaf boxes in the octal tree of the MDA-SVD. As can be seen from Fig. 1, in the MDA-SVD, the manipulation of partitioning a box into eight children can be decomposed by the geometrically balanced divisions in ordinal directions, where each division is bisection. Based on this, one level of an octal tree can be divided into three levels of a binary

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Fig. 1. Hierarchical Subdivision of an object using bounding boxes in the MDA-SVD and

H-matrix methods.

with , in full matrix representation, and is usually much smaller than and . Based on the cluster tree and the block cluster tree, the class of -matrices for an admissible partitioning and the number of the impedance matrix can be defined as (10) Fig. 2. (a) Cluster tree T . (b) Block cluster tree T light gray.

. Admissible blocks are

tree. Thus, when the octal tree of the MDA-SVD is constructed, with its all leaf boxes included, a binary tree for implementing -matrix techniques is also obtained. arises from the grouping of pairs A block cluster tree of clusters from the cluster tree , as depicted in Fig. 2(b), the index set corresponding to the impedance matrix is split into a partition (5) To approximate a matrix by a block-wise low-rank approximation, the sub-blocks have to fulfill a so-called admissibility condition as follows (6) denotes the minimal bounding box for the supwhere and port of cluster and , diam and dist denote the Euclidean diameter and distance of cluster and respectively, and controls the trade-off between the number of admissible blocks. A partitioning is called admissible, if all blocks are admissible. Based on an admissible partitioning of , a low-rank approximation of each admissible matrix-block can be applied. A given block with disjoint bounding boxes can be approximated by low-rank representations, separating the partitioning into admissible (“far-field”) and inadmissible (“near-field”) blocks described as: (7) Typical MDA-SVD implementations usually use a single buffer box to determine near or far interactions, leading to a partitioning of described as (8) where admissible blocks can be approximated by low-rank representation in the following Rk-matrices as follows (9)

C. Hierarchical Compression Based on the MDA-SVD The MDA method provides an efficient way to compress the far-field blocks to low-rank representations described in (9) with little loss of accuracy, while the near-field blocks are uncompressed and to be computed via MoM. The MDA is based on the observation that the number of degrees of freedom (DoF) required to represent interactions between well-separated groups of panels is generally much lower than the typical number of panels in those groups. For an interaction between a pair of non-touching boxes in the cluster tree, the RWG basis functions inside the boxes can be represented by the equivalent RWG basis functions defined in the boundary of the boxes. For planar structures, all the original RWG basis functions in the box are contained in a plane. Thus, the equivalent RWG sources are necessary only at the vertices and edges of the rectangle resulting from the intersection between the box and the plane. using MDA, the origFor the computation of inal and equivalent RWG functions produce the same field tested by the equivalent RWG functions at the observation box (11) then (12) Thus, the field produced by the equivalent RWG functions in the source box can be expressed as (13) Therefore, the sub-matrix which represents the interaction of the well-separated source and observation boxes can be represented as (14) Since the number of the equivalent RWG basis functions is always much larger than rank of the sub-matrix , a recompression scheme based on SVD is introduced to further reduce the computational requirement of the MDA with little loss of accuracy. The resulting MDA-SVD method has been proved to

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and are vector, upper triangular solver is similar. When and are the hierarthe processes of solving chical forward and backward substitutions ( -FBS). Algorithm I Recursive LU Decomposition Procedure

-LU

if

then

calculate the LU decomposition Fig. 3. A typical

else

H-matrix in practice. Far-field blocks are white.

be comparable to the MLFMA and the adaptive cross approximation (ACA) algorithm in computational cost. The SVD recompression can be implemented by (15) (16) (17) Equation (15) and (16) represent QR decompositions, and (17) represents a truncated SVD. Threshold is used to control the for an optimum compression. Let truncation accuracy and , the MDA compressed is further recompressed to sub-matrix (18) To be applied in -matrix techniques, the following form

exactly

is transformed to (19)

call

-LU

call LowerTrigsolver call UpperTrigsolver call

-LU

end end Algorithm II Recursive Lower Triangular Solver Procedure LowerTrigsolver if calculate the

then exactly

else

with (20) of is generated Thus, the -matrix representation based on the hierarchical compression using the MDA-SVD. In practice, the structure of an -matrix is usually complicated, as shown in Fig. 3. D.

call LowerTrigsolver call LowerTrigsolver call LowerTrigsolver

-LU Decomposition Algorithm

-matrix arithmetic provides an efficient way to implement the LU decomposition of an -matrix. An -matrix generated from the MDA-SVD method has a structure of a quad block cluster tree based on a binary cluster tree, so the -LU decomposition can be recursively computed in the form of a 2 2 block-matrix as follows: (21) The process of the recursive LU decomposition is described by the pseudo-code presented in Algorithm I. In Algorithm I, a trior is required for a given angular solver lower or upper triangular matrix and a given RHS-matrix . The lower triangular solver can be recursively implemented by the pseudo-code presented in Algorithm II and the case of the

call LowerTrigsolver end end In Algorithm I and II, the addition and multiplication are replaced by the faster formatted -matrix counterparts ( and ) [25]. Truncation operator based on truncated versions -decomposition and SVD is used to define -matrix of the and -matrix multiaddition [12], [14], [26]. In this plication paper, an adaptive truncation scheme with a relative truncation is adopted, similarly defined as in Section II-C. error

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Fig. 4. Schematic drawing of a typical

Fig. 5. Flow chart of converting an

H-matrix multiplication.

H-matrix to an Rk-matrix. “F” and “Rk” denote full matrices and Rk-matrices, respectively.

determines the computational requirements and the accuracy -LU decomposition. -matrix addition is defined between two -matrices with the same block cluster tree by adding the corresponding subblocks of them, and the target -matrix also has the same tree-structure. There are two cases: and are both full matrices. Exact matrix addition (1) is used and generates a block in the full matrix format. and are both Rk-matrices. Let (2) , then the adaptive truncated operator is used to define , the Rk-matrix addition like the SVD recompression scheme of MDA described in Section II-B. -matrix multiplication, as depicted in Fig. 4, is a fundamental operation required in the -LU decomposition. It is more complicated since it involves the conversion of different tree-structures. -matrix multiplication is defined between two -matrices, i.e., with and with , and the target matrix has a tree-structure . There are four cases, as follows: , and all have subblocks. The multiplication is (1) done recursively in the subblocks. has subblocks, but or does not have. The (2) product of and , which is an Rk-matrix or full matrix, is partitioned and added to . is a full matrix. The product is added to the target (3) directly. is an Rk-matrix. The hierarchical multiplication and (4) truncation is required by multiplying two subdivided matrices, truncating the product to the set of Rk-matrices and adding the result to , as shown in Fig. 5. -LU decomposition can be performed with computational complexity and memory requirement [25], [26], which is the same as the -matrix multiplication, and the computational complexity of -FBS of

, where denotes the blockwise rank of is low-rank blocks generated from the MDA-SVD compression. For geometrically complicated problems, a type of numerical complexity can be reported by varying the discretization sizes with fixed frequency. Here, the number of unknowns increases since finer and finer discretization is employed for better accuracy. In this case, a constant blockwise rank can be adopted for the increasing problem sizes without loss of accuracy. Hence, the computational complexity and memory requirement of -LU decomposition can be evaluated as and respectively by discarding the item of rank . Besides, another case is varying the frequencies with fixed discretization size. In this case, the problem size electrically increases while the discretization size keeps constant to the wavelength. Here, since the kernel function of integral operator is oscillatory, the blockwise rank will grow with the electric for a given accuracy. size as well as the unknown number Hence, the computational complexity and memory requireand , ment would be higher than respectively. E.

-LU-Based Preconditioning and Direct Solution Since the impedance matrix has been approximated by an -matrix , (4) can be written as (22)

When (22) is iteratively solved by the Krylov subspace method, such as GMRES method, the GMRES iteration usually encounters slow convergence rate. In order to accelerate the convergence of the GMRES iteration, a preconditioner should be introduced to transform (22) into an equivalent one as follows (23)

WAN et al.: HIERARCHICAL MATRIX TECHNIQUES BASED ON MDA FOR THE FAST ANALYSIS OF PLANAR LAYERED STRUCTURES

The product matrix has much better spectral property , which leads to a greatly reduced number of than matrix is used to iterations. In this paper, construct a preconditioner. Here, and are the lower and upper triangular factors from the approximate -LU decompo, which enters the MVPs of GMRES iterasition tions by -FBS. Define the accuracy of -LU decomposition as and the as . Let 2-norm based condition number of such that and we have

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Fig. 6. The configuration of the Y loop FSS.

(24) From (24),

(25) Then, (26) According to the Neumman series, we have

(27)

Fig. 7. Frequency dependence of transmission coefficients of the 20 loop FSS.

2 20 Y

Combining (26) and (27), we conclude that

(28) It can be seen from (28) that has an explicit bound. For instance, if the accuracy satisfies , the precondtioned matrix has condition number at most 3. Hence, the -LU decomposition can be computed with lower accuracy compared with building the -matrix representation of by the MDA-SVD compression. Due to the logarithmic-linear complexity, -LU decomposition combined with -FBS also provides an efficient direct solver for the solution of (22). Here, the accuracy of the -LU decomposition has to be of the same order as the accuracy of apbased on the MDA-SVD compression. After the proximate completion of -LU decomposition, -FBS are implemented to obtain the solution of (22) without any iterative process. Hence, the original -matrix can be overwritten that requires no extra storage. For problems with multiple RHS-vectors, once the manipulation of -LU decomposition is accomplished, the -LU factors are reserved and only -FBS is required for each independent RHS-vector. Hence, the overall complexity of the -LU-based direct solver might be less than the GMRES iterative solution and even the -LU preconditioned GMRES iterative solution. In conclusion, -LU decomposition with low accuracy leads to much smaller -LU factors compared with the memory requirement of the MDA-SVD compressed -matrix, and can be used as an efficient preconditioning for the iterative solution. Otherwise, -LU decomposition with relatively high accuracy

can be used as a direct solver to deal with multiple RHS-vector problems particularly, while it may be defeated by the -LU preconditioned iterative solution for a single RHS-vector. III. NUMERICAL RESULTS To demonstrate the efficiency and accuracy of the proposed method for the analysis of planar layered structures, several numerical experiments are presented. First, two structures of frequency selective surfaces (FSS) with dielectric substrates are tested by using the GMRES iterative solver with the MDASVD-based -LU preconditioner. The residual norm is set to for the GMRES iteration. The relative truncation error be of the MDA-SVD is fixed at . Then, a planar microstrip antenna array is simulated for testing the MDA-SVDbased -LU direct solver compared with the iterative solver. All the computations in this section are performed on a common PC with Intel Core2 2.8 GHz CPU in double precision. A.

Loop FSS

An example of the loop FSS is first investigated. The length and width of “ ” arm are 4 mm and 1 mm, respectively, the diand , mensions of the unit are the skew angle is 60 , the substrate has a relative permittivity and a thickness of . Fig. 6 shows the of configuration of this loop FSS. The incident plane wave is TM polarization wave propagating from the direction and . As a criterion, the Ansoft Designer software is employed to simulate an infinite array FSS, which has been proved having the similar characteristic of the finite FSS that is large enough. As shown in Fig. 7, the transmission coefficients of the

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Fig. 8. The computational costs of -LU decomposition with the number of unknowns increasing by varying the discretization sizes with fixed frequency for the Y loop FSS example. (a) Time required for the -LU decomposition. (b) Memory required for -LU factors.

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TABLE I

PERFORMANCE OF THE MDA-SVD COMPRESSION AND

H-LU PRECONDITIONING FOR THE Y LOOP FSS WITH DIFFERENT SIZE

20 20 loop array FSS computed by the -LU preconditioned GMRES agree well with the results simulated by Ansoft Designer. To test the performance of the -LU preconditioner, GMRES with and without -LU preconditioner are applied to the computation at the frequency of 7 GHz. The relative truncation error , which is of -matrix arithmetic is fixed at relatively low and can be used to construct an effective preconditioner cheaply. Table I shows the computational cost of the MDA-SVD compression and compares the performance of GMRES solver with and without the -LU preconditioner. It can be seen from the fourth and fifth columns that the -LU preconditioner has an excellent compression which is nearly 70% of the original -matrix. Since the GMRES iteration with no preconditioner has an extremely slow convergence rate for this example, the sixth column only presents the residual of the GMRES iteration with no preconditioner after 1000 steps. As shown in the last column, the -LU preconditioned GMRES improves the convergence rate and reduces the overall solution time significantly. Then, we fixed the frequency at 7 GHz, and increase the number of unknowns by decreasing the discretization size to test the computational costs of the -LU decompo-

sition. The CPU time and memory requirements are validated and , respectively, to scale close to as can be seen from Fig. 8(a) and (b). B. Circular Loop FSS The second example deals with a circular loop FSS, which consists of two layers of metallizations and three layers of dielectric substrates, as shown in Fig. 9. Metallizations on the top and bottom are both 10 10 circular loop arrays with the diand . The inner mension of units and outer radiuses of the circular loop are and . From the top down, the thicknesses of , and the layered substrates are , and the corresponding relative permittivities are , and , respectively. The incident plane wave is TE polarization wave propagating and . The number of unfrom the direction knowns after the discretization with RWG functions is 10,720. In Fig. 10, the transmission coefficients computed by the -LU preconditioned GMRES are plotted together with the results calculated by the method of moments (MoM). Table II shows the computational requirements for the -matrix construction and

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TABLE II

H-LU PRECONDITIONING WITH DIFFERENT " THE 10 2 10 CIRCULAR LOOP FSS

THE PERFORMANCE OF

FOR

Fig. 9. The configuration of the circular loop FSS.

Fig. 12. The configuration of the microstrip patch antenna.

Fig. 10. Frequency dependence of transmission coefficients of the 10 cular loop FSS.

2 10 cir-

Fig. 11. Number of iterations at different frequency in the sweeping range.

the behavior of -LU preconditioned GMRES with different -LU truncation accuracy . Obviously, the number of itexponentially improved, while the erations decreases with time and memory requirements increase gradually. As can be seen from Table II, -LU decomposition with low accuracy is cheap but fully competent for an efficient preconditioning. is reIn Fig. 11, the number of iterations for ported at different frequency within the sweeping range. It is obvious that the number of iterations keeps almost constant for different frequency, which validates the stability of the -LU preconditioning. C. Microstrip Patch Antenna Array The last example considers the problem of scattering from a 10 10 rectangular microstrip patch antenna array. The monostatic radar cross section (RCS) of the microstrip patch antenna

arrays is attractive, since it is directly related to the radiation efficiency of the conductive body [28], [29]. The computation of monostatic RCS can be classified as the solution of a multiple RHS-vector problem, that is, only the RHS-vector need to be modified to calculate each new solution for different excitations, while the impedance matrix is invariable. -LU decomposition provides an efficient direct solver with almost linear complexity to deal with this multiple-RHS-vector problems. This is because the process of -LU decomposition only needs to be performed once, and then multiple RHS vectors can be handled efficiently using only -FBS for each excitation. The layout of the microstrip antennas is shown in Fig. 12, the configura, , tion of the unit of the arrays is , the relative permittivity and the thickness and respecof the substrate are tively. The number of unknowns after the discretization with RWG functions is 20,812. The relative truncation error is set to for the -LU preconditioner and be for the -LU direct solver. The monostatic RCS computed by the -LU direct solver is plotted in Fig. 13. 172 sample points of excitations are chosen here for the monostatic RCS curve. To compare the -LU direct solver with the -LU preconditioned GMRES iterative solver and the GMRES solver with no preconditioning, the computational costs of them are presented in Table III. Though the -LU preconditioner can be constructed more easily than the -LU decomposition of the -LU direct solver, it has much longer solution time as shown in the fourth column in Table III. This is because a process of iterations, including some steps of -FBS and -matrix-based MVP, should be done for each RHS-vector in the -LU preconditioned GMRES solver, but only one step of -FBS is needed for each RHS-vector in the -LU direct solver. The advantage of -LU direct solver will be more and more obvious with the

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Fig. 13. The monostatic RCS of the microstrip patch antenna.

TABLE III THE PERFORMANCE OF DIFFERENT SOLVER FOR COMPUTING THE MONOSTATIC RCS OF THE MICROSTRIP PATCH ANTENNA

number of RHS-vectors increasing. Since the GMRES with no preconditioning always fails to converge, the approximate lower bound solution time within the bounded 1000 steps is presented. In the last column, the total solution time is presented, including the -LU decomposition and the ensuing solution for all the 172 excitations. IV. CONCLUSION A new method for matrix compression and solution based on the MDA-SVD method and -matrix techniques is presented for the IE-based electromagnetic analysis. The MDA-SVD method provides an efficient compression of the impedance matrix to significantly reduce the memory requirement, especially for planar structures. Since the MDA-SVD method is purely algebraic and independent of the Green’s functions, it is suitable for the analysis of layered media. The MDA-SVD compressed impedance matrix has an -matrix representation. -LU decomposition provides a cheap but efficient preconditioner to accelerate the iteration convergence and a fast direct solver to deal with multiple RHS-vector problems particularly. Numerical results demonstrate the proposed method is robust for the analysis of planar layered structures. REFERENCES [1] S. M. Rao and R. F. Harrington, Field Computation by Moment Methods. New York: Macmillan, 1968. [2] S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag., vol. 30, pp. 409–418, 1982. [3] W. C. Chew, J. M. Jin, E. Midielssen, and J. M. Song, Fast and Efficient Algorithms in Computational Electromagnetics. Boston, MA: Artech House, 2001.

[4] J. M. Song, C. C. Lu, and W. C. Chew, “Multilevel fast multipole algorithm for electromagnetic scattering by large complex objects,” IEEE Trans. Antennas Propag., vol. 45, pp. 1488–1493, 1997. [5] E. Michielssen and A. Boag, “A multilevel matrix decomposition algorithm for analyzing scattering from large structures,” IEEE Trans. Antennas Propag, vol. 44, no. 8, pp. 1086–1093, Aug. 1996. [6] J. M. Rius, J. Parron, E. Ubeda, and J. Mosig, “Multilevel matrix decomposition algorithm for analysis of electrically large electromagnetic problems in 3-D,” Microw Opt. Technol. Lett, vol. 22, no. 3, pp. 177–182, Aug. 1999. [7] J. Parron, J. M. Rius, and J. Mosig, “Application of the multilevel decomposition algorithm to the frequency analysis of large microstrip antenna arrays,” IEEE Trans. Magn, vol. 38, no. 2, pp. 721–724, Mar. 2002. [8] J. Parron, G. Junkin, and J. M. Rius, “Improving the performance of the multilevel matrix decomposition algorithm for 2.5-D structures application to metamaterials,” in Proc. Antennas Propag. Soc. Int. Symp, Jul. 2006, pp. 2941–2944. [9] R. Kastener and R. Mittra, “Iterative analysis of finite-Sized planar frequency selective surfaces with rectangular patches or perforatios,” IEEE Trans. Antennas Propag., vol. 35, pp. 372–377, 1987. [10] E. F. Knott, J. F. Shaeffer, and M. T. Tuley, Radar Cross Section, 2nd ed. Norwood, MA: Artech House, Inc., 1993. [11] J. M. Rius, J. Parron, A. Heldring, J. M. Tamayo, and E. Ubeda, “Fast iterative solution of integral equations with method of moments and matrix decomposition algorithm—Singular value decomposition,” IEEE Trans. Antennas Propag., vol. 56, no. 8, pp. 2314–2324, Aug. 2008. [12] A. Heldring, J. M. Rius, J. M. Tamayo, J. Parrón, and E. úbeda, “Fast direct solution of method of moments linear system,” IEEE Trans. Antennas Propag., vol. 55, no. 11, pp. 3220–3228, Nov. 2007. [13] A. Heldring, J. M. Rius, J. M. Tamayo, and J. Parrón, “Compressed block-decomposition algorithm for fast capacitance extraction,” IEEE Trans. on Computer Aided Design., vol. 27, no. 2, pp. 265–271, Feb. 2008. [14] A. Heldring, J. M. Rius, J. M. Tamayo, J. Parrón, and E. Ubeda, “Multiscale compressed block decomposition for fast direct solution of method of moments linear system,” IEEE Trans. Antennas Propag., vol. 59, no. 2, pp. 526–536, Feb. 2011. [15] A. Heldring, J. M. Rius, J. M. Tamayo, J. Parrón, and E. úbeda, “Multiscale CBD for fast direct solution of MoM linear system,” in Antennas and Propag. Society International Symposium, 2008. [16] W. Hackbusch, “A sparse matrix arithmetic based on -matrices. Part I. Introduction to -matrices,” Computing, vol. 62, no. 2, pp. 89–108, 1999. [17] W. Hackbusch and B. Khoromskij, “A sparse -matrix arithmetic. Part II: Application to multi-dimensional problems,” Computing, vol. 64, pp. 21–47, 2000. [18] S. L. Borne, “ -matrices for convection-diffusion problems with constant convection,” Computing, vol. 70, pp. 261–274, 2003. [19] W. Chai and D. Jiao, “An -matrix-based method for reducing the complexity of integral-equation-based solutions of electromagnetic problems,” presented at the IEEE Int. Symp. on Antennas and Propagation, 2008. -matrix-based integral-equation solver [20] W. Chai and D. Jiao, “An of reduced complexity and controlled accuracy for solving electrodynamic problems,” IEEE Trans. Antennas Propag., vol. 57, no. 10, pp. 3147–3159, Oct. 2009. [21] S. Börm and L. Grasedyck, “Low-rank approximation of integral operators by interpolation,” Computing, vol. 72, pp. 325–332, 2004. [22] M. Bebendorf and S. Rjasanow, “Adaptive low-rank approximation of collocation matrices,” Computing, vol. 70, pp. 1–24, 2003. [23] M. Bebendorf, “Approximation of boundary element matrices,” Numer. Math., no. 86, pp. 565–589, 2000. [24] M. Bebendorf, Hierarchical Matrices: A Means to Efficiently Solve Elliptic Boundary Value Problems. Berlin: Springer, 2008, vol. 63, Lecture Notes in Computational Science and Engineering, pp. 49–98. [25] L. Grasedyck and W. Hackbusch, “Construction and arithmetics of -matrices,” Computing, vol. 70, no. 4, pp. 295–344, 2003. [26] S. Börm, L. Grasedyck, and W. Hackbusch, “Induction to hierarchical matrices with applications,” Engrg. Anal. Bound. Elem., no. 27, pp. 405–422, 2003. [27] K. A. Michalski and D. L. Zheng, “Electromagnetic scattering and radiation by surfaces of arbitrary shape in layered media, part I: Theory,” IEEE Trans. Antennas Propag, vol. 38, no. 3, pp. 335–344, 1990. [28] A. S. King and W. J. Bow, “Scattering from a finite array of microstrip patches,” IEEE Trans. Antennas Propag., vol. 40, no. 2, pp. 770–774, 1992. [29] D. R. Jackson, “The RCS of a rectangular microstrip patch in a substrate- superstrate geometry,” IEEE Trans. Antennas Propag., vol. 38, pp. 2–8, Jan. 1990.

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Ting Wan was born in Huanggang, Hubei, China, in 1981. He received the B.S. and M.S. degrees in electrical engineering from Nanjing University of Science and Technology (NJUST), Nanjing, China, in 2003 and 2006, respectively, where he is currently working toward the Ph.D. degree. His research interests include computational electromagnetics, electromagnetic scattering and radiation, and electromagnetic modeling of micro- wave integrated circuits.

Zhao Neng Jiang was born in Jiangsu Province, China. He received the B.S. degree in physics from Huaiyin Normal College in 2007, and is currently working toward the Ph.D. degree at Nanjing University of Science and Technology (NJUST), Nanjing, China. His current research interests include computational electromagnetics, antennas and electromagnetic scattering and propagation, and electromagnetic modeling of microwave integrated circuits.

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Yi Jun Sheng was born in Jiangsu Province, China. He received the M.S. degree in physics from Nanjing University of Science and Technology (NJUST) in 2003, where he is currently working toward the Ph.D. degree. His research interests include computational electromagnetics, antennas and electromagnetic scattering and propagation, and electromagnetic modeling of microwave integrated circuits.

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Fast Frequency Sweep of FEM Models via the Balanced Truncation Proper Orthogonal Decomposition Wei Wang, Student Member, IEEE, Georgios N. Paraschos, Student Member, IEEE, and Marinos N. Vouvakis, Member, IEEE

Abstract—A fast frequency sweep method for wideband antennas and infinite arrays based on a singular value decomposition (SVD)-Krylov model reduction method for frequency-domain tangential vector finite elements (TVFEMs) is presented. Reduced models are constructed using balanced congruence transformations constructed from the dominant invariant subspace of the system’s Hankel matrix. Traditionally, forming such matrix requires the intensive computation of Gramians; the proposed method only forms their low-rank Cholesky factors via a novel adaptive proper orthogonal decomposition (POD) sampling strategy, leading to significant savings. Unlike some other model reduction methods, balanced truncation POD (BT-POD) is directly applicable to lossy and dispersive electromagnetic models. Numerical studies on large-scale wideband antennas and infinite arrays show that the method is stable, error controllable and, without memory overheads capable of up to two orders-of-magnitude speed-ups. Index Terms—Finite element method, infinite arrays, Karhunen-Loeve expansion, model order reduction, principal component analysis, singular value decomposition.

I. INTRODUCTION ULL-WAVE computational electromagnetic (CEM) methods such as the tangential vector finite element method (TVFEM) [1] play important role in the design of antennas, microwave devices and RF integrated circuits (RFICs). Many of these devices require long runs because they are broadband, or are connected to non-linear loads or drives necessitating co-simulation with circuit and behavioral simulators. Therefore, numerical methods that speed-up broadband TVFEM simulations, or generate compact TVFEM macro-models are key in accelerating antenna and RF system design. Time-domain (TD) CEM methods [2] are quite efficient for the analysis of broadband EM systems, but are less effective on highly resonant or dispersive problems, and the temporal axis is

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Manuscript received July 24, 2010; revised April 18, 2011; accepted May 01, 2011. Date of publication August 15, 2011; date of current version November 02, 2011. The authors are with the Electrical and Computer Engineering Department, University of Massachusetts at Amherst, Amherst, MA 01003 USA (e-mail: [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2164184

difficult to parallelize. Frequency interpolation schemes of -parameters or other observable quantities [3], [4] are very efficient on wideband systems, but do not provide a full EM system representation. Contrary, model order reduction (MOR) techniques [5], [6] provide a full EM system description, but are most efficient on resonant structures or problems with perturbative losses and dispersion. Among the first MOR methods to be used for large-scale EM problems were the model-based parameter estimation MBPE [7] and the closely related asymptotic waveform evaluation (AWE) method [8]. These methods are applicable to lossy and dispersive problems because they deal with polynomial frequency dependence matrix systems. Ill-conditioning of the moment-matching process has prompted multipoint alternatives such as the complex frequency hopping (CFH) [9] or the multipoint Galerkin AWE (MGAWE) [10], and well-conditioned formulations such as the well-conditioned AWE (WCAWE) [11]. For systems with linear frequency behavior (lossless and dispersionless) many extremely efficient methods based on the Lanczos [12], Arnoldi [13] or rational Krylov [14] methods have emerged. These methods have been extended to lossy and dispersive problems [15], at the price of a larger system-of-equations from linearization. Krylov or moment-based methods, although quite efficient, are known in some cases to produce non-optimal order reduced models, and the error theory is still work in progress. Contrary, SVD or Gramian-based methods [16] use information from the dominant invariant subspace of system Gramians to produce nearly optimal reduced problems, with inherent error bounds. These methods are computationally expensive because they require solution of the Lyapunov equations [5], and there are not reported applications to CEM. Prakash et al. [17] and Knockaert et al. [18] have used SVD for MOR in microwaves and electromagnetics, but their work merely used SVD as a more robust alternative to modified Gram-Schmidt for orthogonalizing the reduced basis set. The major computational bottleneck of SVD methods is encountered at the exact solution of the Lyapunov equations. Methods that seek only approximate or low-rank solutions to these equations significantly reduce computational time and memory. Among the most popular methods is the low-rank Smith method of Penzl [19], and the low-rank ADI [20], where a spectral transformation is used to convert the Lyapunov

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WANG et al.: FAST FREQUENCY SWEEP OF FEM MODELS VIA THE BALANCED TRUNCATION POD

equation to a Stein equation that admits an infinite series solution. Truncated versions of this solution lead to low-rank Gramian representations, where each summation term requires the solution of a sparse matrix system, often achieved via efficient preconditioned Krylov solvers. These methods belong to the SVD-Krylov MOR category, and are the only viable SVD-based methods for large-scale problems. To the best of our knowledge, SVD-Krylov MOR methods have never been used for CEM radiation problems. This paper presents an alternative SVD-Krylov MOR method for wideband radiation problems that is based on the frequencydomain representation of the Gramians. Numerical integration is used to approximate the integral in the representation, leading to a freqency-domain proper orthogonal decomposition (POD) sampling strategy. This approach shares the spirit of the lowrank POD [21] and the poor man’s truncated balanced realization (PMTBR) [22], but several important technical differences make the proposed method more efficient. Firstly, goal-oriented error estimators are used in conjunction with an adaptive sampling strategy to guide the POD. Moreover, instead of computing the eigendecomposition or SVD of the forward and adjoint problem POD samples to form the controllability and observability Gramians, this work uses the samples to first form the approximate Hankel matrix, that is significantly smaller in size, thus computing its SVD is considerably more efficient. Finally, this method is applied to electromagnetic radiation problems that have non-linear frequency behavior. In particular, the proposed MOR method is used in conjunction with a new version of the infinite array formulation from [23] that consumes less memory and has better iterative convergence. The rest of the paper is organized as follows. Section II begins with the formulation of the Floquet-cell TVFEM model for infinite arrays. The core principles of BT-POD are presented in Sections II-Bthrough II-D, followed by the error estimation and adaptive sampling strategy in Section II-E. A summary of the BT-POD algorithm and its complexity are presented in Section II-F. Numerical results from a wideband antenna and two ultrawideband infinite arrays are presented in Section III. Section IV concludes the paper.

II. THEORY BT-POD will be presented in the context of TVFEM models arising from general radiation and infinite array problems. The proposed Floquet cell TVFEM formulation for infinite arrays is new, thus a short section is included.

A. Non-Conforming Finite Elements for Infinite Arrays Consider a generic EM radiation system, with multiple ports, perfect electric conductors (PECs), materials and boundary conditions, as shown in Fig. 1. The time-harmonic boundary value problem (BVP) over a frequency band (where is the Laplace frequency, is the wave number

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Fig. 1. A generic EM system used to set-up the TVFEM models.

and and reads as

are the frequency and speed of light in vacuum),

where (1) is the space of admissible electric fields , is the computational domain, are the perfect electric conductor (PEC) surfaces, is the surface of the radiation boundary condition, is the surface of the wave port. The relative perand mittivity and permeability of the media are denoted by , whereas , and are the free-space, material and port wave impedances, respectively. Each wave port , is excited with an incident modal guided wave with tangential magnetic field . The outward pointing unit vector on a surface is denoted by . The second equation in (1) is the first order absorbing boundary condition (ABC) used to truncate the computational domain. The BT-POD presented here in the context of FEM is not restricted to absorbing or periodic boundary conditions alone. It could be used for reduction of FEM-PML or even boundary element method (BEM) and other frequency domain CEM methods and their hybrids. In the case of infinite arrays, radiators are arranged periodically in space, extending to infinity and they have a periodic excitation. The periodic arrangement could be one, two or three dimensional, in rectangular, hexagonal or other more complicated lattices. Under these assumptions, one can invoke the Floquet

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Fig. 2. Infinite periodic structure with nonconforming mesh on the master and slave periodic boundaries.

theorem [24] to convert the boundary value problem over the infinite domain into one inside a period (Floquet cell), amended by appropriate periodic boundary conditions (PBC) at the boundaries between cells. To illustrate the situation, the structure in Fig. 1, is assumed to be periodic in one dimension (laterally), as shown in Fig. 2. The surfaces where the PBCs are enforced are , and . Similar shown with red lines and are denoted with to [23], the Robin-type transmission conditions will be used to weakly enforce the PBC between master and slave boundaries, through

(2) where (3) (4) are the tangential electric trace and the normalized electric current at each surface. Moreover, , is the phase difference between fields on opposite master-slave boundaries, where is the wave-vector dictated by the array scan excitation and is the displacement vector between master and slave surfaces. The complex constants and are the Robin constants, whose values affect iterative convergence, and specific choices will be given below. In doubly or triply periodic structures, each pair of master-slave surfaces has its own PBCs set similar to (2).

In order to avoid using periodic meshes (identical triangulations between master and slave boundaries), the work in [23] chose to solved the the mixed FEM problem arising from (1) and (2), instead of using a standard variational formulation. Although successful, the solution of the resulting matrix equation could not be solved using highly efficient Schur complement (finite element tearing and interconnecting procedure) solvers for the Lagrange multipliers (in this case ), because internal resonances lead to singularities during the inversion of the primal matrix block. To avoid this problem a less efficient FETI-like solution was suggested that required the factorization of a larger matrix, and iterates over both and , leading to increased memory and solution time. The proposed formulation alleviates these complications by modifying the Lagrange multiplier space leading to an internal resonance free FETI solution process. The key idea here is to redefine the Lagrange multiplier space into a linear combination of and as follows (5) where is a complex variable usually function of . This type of transformation was first proposed by de la Bourdonnaie et al. in [25] for scalar Helmholtz problems. Inserting (5) into (2) and assuming , leads to

(6) The transformed Robin PBCs of (6) and the BVP of (1) form the basis for the new mixed FEM formulation, that involve unknowns , instead of in [23]. Without going into technical details, after setting the mixed variational statement and choosing the appropriate basis of the trial and testing spaces, the final matrix equation of (7) is obtained (shown at the bottom of the page). In (7), , is vector with the interior port and ABC unknown electric field coefficients, and represent the electric field unknowns on master and slave surfaces, and the Lagrange multiplier unknown coefficients on the master and slave surfaces are denoted by and . The various sub-matrices are given by (8)

(7)

WANG et al.: FAST FREQUENCY SWEEP OF FEM MODELS VIA THE BALANCED TRUNCATION POD

where

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preserves the dominant dynamical behavior, and then efficiently solve that macro-model at multiple frequencies using a direct solver. The oblique projection is performed using two congruence matrices

(13)

(9) with being the basis of the discrete trial and test spaces for . In this work the hierarchical, tangentially continuous vector tetrahedral finite elements from the second Nedelec family are used. The subscripts and in the last equation could be or . The matrix blocks associated with the PBCs are of the form

where and each represents a block-wise partitioning analogous to . The key challenge in this type of MOR is to find and with small that preserve the dynamical properties of the system and satisfy the relation . These issues will be addressed in the balancing transformation section. Given and , the reduction is achieved by pre- and post-multiand plying the system matrix by

(10)

also bewhere the basis of the subspaces associated with long in the second Nedelec family, but defined over triangles. Some nuances about the basis choices can be found in [26]. Subscripts represent master or slave boundaries, , and . The right-hand side vector due to port is given by (11)

The form of (7) is quite general and can be used for standard (non-infinitely periodic) radiation problems by simply dropping , , and (solving for the first matrix block ). In either case, no material or conductor losses have been included for simplicity, but as seen in (8) this model includes “losses” due to radiation and ports. Further abstracting (7), leads to the input-to-output system

(12) is the matrix in (7), and are the scaled input current excitation coefficients, and output power vectors, respectively; the matrix , where denotes its transpose-conjugate. where

,

B. Reduction via Oblique Projection The goal of the fast frequency sweep is to solve (7) for multiple, closely spaced frequencies in a wideband range. A bruteforce approach requires the time consuming solution of multiple sparse systems of equations. In this MOR work, an oblique projection (Petrov-Galerkin) approach is used to first reduce the dynamical system into a compact macro-model that

(14) where

or

and

or , and

(15) all with size , where the reduced order size is typically less that 30. Using the same process, the matrix is reduced via , leading to the following reduced system representation

(16) where . The system in (16) is fully parametrized with respect to frequency , and its size is orders of magnitude smaller than (12), therefore it can be repeatedly solved extremely fast using direct dense matrix solvers. C. Gramians and Their Low-Rank Cholesky Factors This and the following sections will introduce the key concepts behind the BT-POD algorithm, starting with that of the notions of reachability, observability and gramian or autocorrelation matrix, of a input-output system [5]. Reachability is the concept that measures the extent to which a state of a system can be manipulated through the input , accordantly, observability measures the extent to which the states of a system can be inferred by knowledge of its outputs . Because in MOR, the resulting macro-model is valid with respect to the inputs and outputs of the system, these two concepts are central to the reduction. In control theory the Gramians are the autocorrelation matrices of the reachability or observability. The reachability Gramian and observability Gramian

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are symmetric positive definite matrices (symmetric positive semi-definite in case of non-controlable or non-obserbable systems) that encode the energy related to the reachability and observability of a system. More specifically the inner products, [5]

denote the minimum energy required to steer the system from state to , and the maximum energy obtained by observing the system output at state with zero excitation, respectively. Therefore, if the eigenvalue decomposition (EVD) of the Gramian is known, one could identify the modes (states) that contribute least to the output energy and are least “energized” by the input; such modes can then be ignored without significant loss of information about the system, leading to a reduced representation. This is the basic principle behind any SVD-based MOR. The Gramians are not known a-priori, but can be computed using the above definitions as, [5], [27]

the benefits of so doing, the simplest numerical integration using the Riemannian integral definition will be first used. The continuous-frequency Gramian expressions in (18), are via approximated over the band of interest

(20) where in this section indicates approximate quantity, due to numerical integration, is the total number of frequency samples, is the location of the samples and . Each sample evaluation requires the solution of the forward and adjoint problems times at frequency . More importantly, (20) suggest the following matrix decomposition (21) where

(17)

and , state-to-input and where output-to-state impulse responses of the system. Using Parseval’s theorem and spectral symmetries due to causality, their frequency domain representation becomes

(22) are the low-rank Cholesky factors of the Gramians. These factors contain the same “information” as the approximate Gramians, but require significantly smaller storage since . Therefore, only and will be used in the development of BT-POD, alleviating the overhead of storing the Gramians.

(18) D. Balancing and Truncation where,

(19) are the state-to-input transfer function associated with the forward, and adjoint problem solutions, respectively. Several difficulties arise when computing the Gramians using the above equation. First, computing the reachability Gramian requires solutions of the forward problem, while the observability Gramian requires solutions of the adjoint problem. Secondly, the exact evaluation of these integrals requires closed-form expressions in frequency of the forward and adjoint transfer functions; and more importantly, these Gramians are dense matrices that require prohibitively large storage. In an attempt to resolve these difficulties, this paper will approximately compute the system Gramians in (18) using an adaptive numerical integration scheme that will be fully elaborated in Section II-ER. To demonstrate the concept and

Having found a computationally efficient representation for the Gramians, the next step is to use and to identify and ignore states (modes) with small reachability and observability energies. This requires ranking the states according to their ability to be simultaneously reachable and observable. However, this task is complicated since states may have large reachability energy but are not observable, or vise versa. Since the reachability and observability energy of a state is basis dependent, the task becomes equivalent to finding a non-singular similarity transformation matrix that produces the same eigenspectrum for both the reachability and observability. Such transformation is called balancing transformation and is critical in producing optimal order reduced models. Because the Gramian matrices are SPD and must be unitary (see Section II-B), one can use the Schur decomposition to diagonalize the Gramians, (23) where and are diagonal matrices (the inverse in the second transformation is used to reflect the inverse relation-

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ship between observability and reachability energies). There. fore, balancing amounts to finding a that leads to To construct such , one starts by forming the approximate Hankel matrix using the low-rank Cholesky factors (24) Physically the Hankel operator in time-domain, maps past inputs to future outputs, and is related to the system’s cross Gramian. Computationally, working with its low-rank approximation is quite advantageous because it is considerably smaller in size than both low-rank Gramians or Cholesky factors. The Hankel matrix is non-symmetric and can be diagonalized by the SVD (25) are the approximate Hankel where singular values (HSVs) of the system. The decay of the HSVs is an important metric because it is related to the “information” capacity, or the electromagnetic degrees-of-freedoms (EM-DOFs) [28] of the system. Through experiments, it was observed that HSVs of complex, ultrawideband systems decay less rapidly than simpler or narrower-band systems, indicating that MOR for ultrawideband systems such as arrays is more challenging. In the context of this work, the decay of HSVs is used to rank states according to their ability to be simultaneously reached and observed. The SVD in (25) also provides two orthogonal matrices used in constructing via

(26) Because the HSVs are already ranked, the truncation is simply achieved by partitioning the SVD matrices as follows, (27) where , and are the blocks that correspond to HSVs that are grater than a user-defined relative tolerance . Larger ’s give rise to smaller, but lower fidelity reduced models, thus the judicious choice of is important input parameter. Using the truncated HSVs, the approximate balancing transformation becomes

(28) The transformation in (28) is a truncated and diagonally scaled version of the one advocated in (26), and sometimes leads to better conditioning. A proof of the balancing property of these transformations can be found in the Appendix. These transformation are computed once at the off-line state and remain the same for all frequency sweep points, thus they are used once to reduce the system.

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E. Error Estimates and Adaptive Sampling The accuracy, robustness and efficiency of the proposed approach relies on the reliable and accurate numerical integration of the Gramians. To achieve this an adaptive mid-point integration rule based on a greedy algorithm [29] guided by goal-oriented error estimators [30] is presented. Efficient adaptive MOR algorithms have been proposed in the past, e.g. [31], [32], in the context of multipoint Krylov methods, but they have never been viewed from the perspective of adaptive integration and error estimation. The centerpiece of the proposed greedy algorithm is a goaloriented error estimator that is used to guide sample selection. The proposed indicator is based on the reaction of the residual error field with the wave-port modal fields. The goal-functional for this error estimate is tailored around the -parameters of the problem, but can be similarly extended to other quantities such as far-field, etc. To determine the accuracy of the reduced soin (16), one finds the lution residual error through (29) is the residual matrix, with being the residual vector of the exited port. An appropriate norm of this matrix can provide a metric of the error, but is likely to lead to inefficient iterations since in MOR only input/output quantities are of interest. A more efficient approach : that gives the is to define a functional port influence of a residual current on the where

(30)

is the electric modal fields of the wave-port. where Expanding the residual field due to the excited port into divconforming functions gives , where is a matrix of similar in form to (10) but involving the ports. For all ports the final matrix representation becomes (31) where is a port impedance normalization matrix, and is a matrix with all the port magnetic field projections on tangental curl-conforming elements. To establish must an error indicator, the norm of the matrix be taken, where the could be an induced or the Frobenius matrix norm. Empirically it was found that the Frobenius norm lead to most effective adaptation. Having found an error estimate expression, the goal is to select the frequencies that would provide maximal error reduction, and sample them in the next iteration step. Therefore, at each adaptive POD pass the following optimization problem must be solved, (32)

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This will be achieved with a greedy algorithm [29], where a freset of simple local error estimates are defined, and the quency regions with the largest error are chosen for subdivision by sampling at their midpoints. This approach may not guaranty optimal sample placement, but it is considerably faster than formally solving the optimization problem in (32). The local error is estimate in a frequency segment (33) where is the mid-frequency of the segment. This approximation is simple and effective, because it requires the solution of additional reduced problems per segment. A global error estimate, is defined by (34) where and is the number of POD samples at the current iteration. At the end of each iteration, if the global error is larger than a user-defined global relative , the segments with the largest local errors are tolerance subdivided by solving the forward and adjoint full-order problems at their midpoints. A pictorial view of an adaptive integration step and the greedy sample selection are depicted in Fig. 3 . This algorithm could be interpreted as an adaptive for midpoint integration rule for the frequency-domain Gramian representation. F. Adaptive BT-POD Algorithm and Complexity Algorithm 1 summarizes the most important steps of the adaptive BT-POD. Algorithm 1 BT-POD for TVFEM Fig. 3. Overview of the adaptive integration of the reachability Gramian using a greedy algorithm and simple mid-point integration rule. (a) Mid-point integration rule at the k adaptive step. (b) Error estimation stage; (c) Greedy selection showing the new integration (sample) points.

INPUTS: : frequency band. : number of uniformly distributed sweep points in band.

i. Solve at

: preconditioned Krylov solver residual relative tolerance. : HSVs truncation relative tolerance. : global relative error tolerance. : number of samples added per pass. OUTPUTS: ,

: port voltage at sweep frequencies.

: unknowns field coefficients at frequency

.

1: Initialization i. Assemble global FEM matrices , , , and ii. Set , , , and . 2: Adaptive POD Sampling while do ( : at first iteration)

.

and with precond. Krylov solver

. ii. Update , via (22), and via (24). iii. Perform SVD on and truncate singular vectors with . iv. Form and via (??) v. Form , , , , , and via (15). vi. Set , . vii. Form via (14) and solve at with direct solver. viii. Compute via (29) and via (31). ix. Compute and via (33) and (34). x. Find new frequency samples such that , set . end while

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TABLE I COMPUTATIONAL COMPLEXITY OF BT-POD

3: Fast Frequency Sweep for do i. Form and solve with a direct solver. ii. Compute iii. (optional) Recover fields end for

. .

The computational complexity estimates for memory and time are listed in Table I. The POD sampling is the most time time since consuming part, and requires a preconditioned Krylov solver [33] with time complexity is used to solve the forward and adjoint problems for each port and frequency sample. Constant is associated with the sparsity of the FEM matrix, and the user defined is the block size used in the out-of-core assembly of and . Typical values for range between 20–50, and depend on problem size and system memory. III. NUMERICAL RESULTS One wideband antenna and two infinite array numerical examples were used to study the accuracy, efficiency and versatility of BT-POD. All computations were performed on a 2.4 GHz quad-core Intel Pentium 4 processor with 2 4 MB L2 cach and 4 GB RAM. The codes were implemented in C++ optimization. The iterative sousing GNU compilers and , the global relative error lution tolerance was set to and the HSV truncation to , tolerance to except when stated otherwise.

Fig. 4. Geometry of the BAVA antenna.

A. BAVA Antenna Element In this example a Balanced Antipodal Vivaldi Antenna (BAVA) is considered. The geometry and dimensions of the antenna are shown in Fig. 4. This antenna is excited by a wave port from the back-side of a finite ground plane and is enclosed in a rectangular computational domain that is approximately (at 8.5 GHz) away from the structure. The bounding box is terminated in first-order ABC on all sides. The computational mesh was produced through 6 passes of adaptive mesh refinement, leading to 462,298 second-order TVFEM unknowns. The , and the bandwidth of interest is . sweep point number was Four passes of BT-POD were required to produce 30 dB ac. Each pass added one additional sample curacy on point, and the initial sample number was 3, leading to a total of 6 samples at the end of the fourth pass. The decay of HSVs at each pass are plotted in Fig. 5 along with the reference HSV curve, obtained through an very precise numerical integration

Fig. 5. Decay of HSVs for BAVA example.

(45 point quadrature). It is seen that the HSVs of the last adaptive pass closely follow the reference, indicating that the adaptive mid-point integration is effective and accurate.

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Fig. 6. Comparison of the BT-POD predicted js point-by-point TVFEM for the BAVA example.

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j

vs. frequency with the

The resulting frequency response is compared with the point-by-point TVFEM results in Fig. 6, where arrows indicated the locations of the sampling points. The two methods virtually overlap throughout the band. A closer look at accuracy reveals versus frequency is around 30 dB for the that the error of error final pass, as shown in Fig. 7(a). In the same plot the of the third pass shows that significant improvement is achieved by appropriately placing the last sample. The convergence of error in the frethe Hankel norm of the quency band is plotted in Fig. 7(b), for various iterative residual tolerances and sampling strategies. Blue curves represent relawhile the red represents . tive solver residual equal to Lower iterative residuals lead to faster BT-POD convergence at lower errors, but as expected, stagnation is observed at around the iterative solver residual level. This observation, is important for setting the inputs of the BT-POD algorithm. One of the advantages of BT-POD over some interpolation based fast frequency sweeps is that it efficiently provides a complete EM macro-model, that can be used to recover quantities other than -parameters. Even-though, the BT-POD algorithm is not tailored to optimize such other quantities it was found that . To demonstrate they could be evaluated as accurately as that point, the antenna broadside directivity versus frequency is plotted in Fig. 8. Throughout the band BT-POD results match well with the reference results of the point-by-point TVFEM. The detailed computational statistics for this example as well as the following examples are listed in Table II. It can be observed that BT-POD is an order-of-magnitude faster than the point-by-point TVFEM sweep. B. Infinite Single-Polarized Vivaldi Array This section will be used to demonstrate the validity of BT-POD on highly dispersive models resulting from the PBC enforcement. The test array is a infinite 5:1 single-polarized Vivaldi array previously studied in [34]. The infinite Vivaldi array is modeled with the new non-conforming mesh periodic boundary conditions approach described in Section II-A. The array is simulated from 1–6 GHz, and is expected to operate

Fig. 7. Error study on BAVA example: (a) js j error vs. frequency; (b) global convergence for uniform and adaptive POD sampling and error ks (s)k different iterative residual tolerances.

well in the band of 1.2–5.5 GHz. The sweep point number is , and the computational model of the Floquet-cell is truncated at the top boundary with ABC. The ABCs are placed (at 6 GHz) above the top of the array. The array unit-cell was discretized with 158,024 second-order tetrahedral TVFEM unknowns. The frequency response of the broadside active reflection coefficient of the array is plotted in Fig. 9. The solid red curve is obtained via 6 adaptive passes of BT-POD (total of 8 adaptively selected samples used to reach global relative error ), whereas the reference point-by-point TVFEM is plotted with circles. In the same figure the dashed blue curve represents the 8 point uniformly sampled BT-POD result. The adaptive BT-POD result almost perfectly match the reference, whereas the uniformly sampled case shows discrepancies at the error for the adaptive low frequency end. Specifically the BT-POD is well below 25 dB, whereas the uniform case exceeds 15 dB. More importantly, the adaptive BT-POD provides an automated way of terminating the sampling process,

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TABLE II SUMMARY OF THE COMPUTATIONAL STATISTICS OF BT-POD

Fig. 8. Broadside directivity vs. frequency of BAVA. Comparison of BT-POD and point-by-point TVFEM results.

Fig. 10. True and estimated absolute error convergence for the adaptive frequency sampling for the single-polarized Vivaldi array example.

of using the residual and the goal-oriented functionals. As expected the estimate follows approximately the same slope as the true error, and always provides an upper-bound. Therefore, it is concluded that it is reliable and can be used to terminate the process. To demonstrate the superiority of the proposed goal-oriented adaptation strategy, over the more traditional residual-based, -norm error convergence of the two methods are plotted the in Fig. 11. Both goal and residual based adaptive methods con, but the goal-oriented strategy verge to relative residual of reaches the desired relative error tolerance using fewer POD samples (i.e. 8 vs. 10), leading to 20% speed-up. The error stagnation at low values is associated with the choice to the iterative solver residual documented in the previous section. Fig. 9. Active reflection coefficient prediction of BT-POD and point-by-point TVFEM for the single-polarized Vivaldi array example.

whereas the uniform sampling approach requires several “trial runs” to determine the number of samples. The price paid for this extra reliability and accuracy is a slightly longer run-time (39 min vs. 38 min) and lower parallel scalability. To study the accuracy of the error estimates and therefore the reliability of the termination process, the global error estimate versus the “true” error at each adaptive pass are plotted in Fig. 10. The true BT-POD error at each pass is found by solving the full TVFEM model at all mid-segment frequencies, instead

C. Infinite Dual-Polarized Vivaldi Array In the last example, a 10:1 bandwidth dual-polarized infinite Vivaldi antenna array is studied. Fig. 12 shows the three-dimensional structure and element dimensions. This problem is significantly more challenging because the bandwidth is larger, and the structures involve multi-scale features. The Floquet cell away from the is truncated on the top with ABCs placed structure. The final mesh is obtained through 8 step of adaptive mesh refinement and leads to 290,516 second-order TVFEM unknowns. In this simulation, the bandwidth of interest is 2–20 sweep points are used. Each Floquet-cell GHz, and includes two wave-ports, and the BT-POD gives the full 2 2 -matrix. For this example the iterative solver tolerance is

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Fig. 11. Convergence of the true error for the proposed goal-oriented, and a residual-based adaptation strategy.

Fig. 13. Convergence of HSVs using the proposed adaptive midpoint integration.

Fig. 14. BT-POD and HFSS comparison of the active VSWR vs. frequency of the 10:1 Vivaldi array example.

Fig. 12. Infinite 10:1 dual-polarized Vivaldi array example: (a) array configuration; (b) element geometry.

, and the adaptive process starts with 4 equi. distant samples, and The BT-POD converged into 8 adaptive passes leading to 18 samples. To show the effectiveness of the adaptive mid-point integration approach, the convergence history of select HSVs are plotted in Fig. 13. This figure shows that the HSVs converge fast to a stable value, indicating that the Hankel matrix and Gramian integration is accurate. Recording the relative HSV variation from pass-to-pass, can be used as an alternative or a boot-strap error estimate or termination criterion.

Finally, the converged (with ) active VSWR results are given in Fig. 14. The result is compared with simulations using ANSOFT/ANSYS HFSS. Overall, the agreement of the methods is very good. Once again, to verify the reliability of the global error estimate, Fig. 15 plots the convergence history of the true and estimated global error. Both plots follow the same pattern and the estimate provides an upper-bound of the error. IV. CONCLUSIONS The balanced truncation proper orthogonal decomposition (BT-POD) model reduction method was introduced in the context of tangential vector finite elements. The method belongs to the SVD-Krylov family and uses approximate Gramians to construct a nearly optimal reduced space. The method is general, so it could be used on lossy and dispersive problems with wide frequency response. The complexity of the proposed algorithm is bounded by the complexity of the iterative

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REFERENCES

Fig. 15. True and estimated absolute error convergence of the adaptive POD sampling for the 10:1 Vivaldi array example.

solver used to solve the frequency-domain FEM problem at each sample. The method was used in conjunction with a new computationally efficient non-conforming periodic FEM formulation to study infinite arrays. Results on antenna and infinite array radiation problems show that the method is reliable, robust, and resulted up to 16 times speed-up over the frequency-domain point-by-point TVFEM. Comparisons with commercial software demonstrated the accuracy of the method. APPENDIX The balancing property of transformations in the from of in (28) is proven as follows:

similarly

Hence,

. ACKNOWLEDGMENT

The authors would like to thank S. S. Holland for the HFSS results of Fig. 14.

[1] Y. Zhu and A. Cangellaris, Multigrid Finite Element Methods for Electromagnetic Field Modeling. New York: Wiley, 2006. [2] A. Taflove, Advances in Computational Electrodynamics: The Finite-Difference Time-Domain Method. Norwood, MA: Artech House, 1998. [3] E. Miller, “Model-based parameter estimation in electromagnetics. II. Applications to EM observables,” IEEE Antennas Propag. Mag., vol. 40, no. 2, pp. 51–65, 1998. [4] B. Anderson, J. E. Bracken, J. B. Manges, G. Peng, and Z. Cendes, Full-Wave Analysis in SPICE via Model-Order Reduction vol. 52, no. 9, pp. 2314–2320, Sep. 2004. [5] A. C. Antoulas, Approximation of Large-Scale Dynamical System. Philadelphia, PA: SIAM, Jul. 2005. [6] S. X.-D. Tan and L. He, Advanced Model Order Reduction Techniques in VLSI Design. Cambridge, U.K.: Cambridge Univ. Press, 2007. [7] G. J. Burke, E. K. Miller, S. Chakrabarti, and K. Demarest, “Using model-based parameter estimation to increase the efficiency of computing electromagnetic transfer functions,” IEEE Trans. Magn., vol. 25, no. 4, pp. 2807–2809, Jul. 1989. [8] L. T. Pillage and R. A. Rohrer, “Asymptotic waveform evaluation for timing analysis,” IEEE Trans. Comput.-Aided Design Integr. Circ. Syst., vol. 9, no. 4, pp. 352–366, Apr. 1990. [9] M. A. Kolbehdari, M. Srinivasan, M. S. Nakhla, Q.-J. Zhang, and R. Achar, “Simultaneous time and frequency domain solutions of EM problems using finite element and CFH techniques,” IEEE Trans. Micro. Theory Tech., vol. 44, no. 9, pp. 1526–1534, Sep. 1996. [10] R. Slone, R. Lee, and J.-F. Lee, “Multipoint Galerkin asymptotic waveform evaluation for model order reduction of frequency domain FEM electromagnetic radiation problems,” IEEE Trans. Antennas Propag., vol. 49, no. 10, pp. 1504–1513, Oct. 2001. [11] R. Slone, R. Lee, and J.-F. Lee, “Well-conditioned asymptotic waveform evaluation for finite elements,” IEEE Trans. Antennas Propag., vol. 51, no. 9, pp. 2442–2447, Sep. 2003. [12] K. Gallivan, E. J. Grimme, and P. Van Dooren, “Asymptotic waveform evaluation via a Lanczos method,” Appl. Math. Lett., vol. 7, no. 5, pp. 75–80, 1994. [13] A. Odabasioglu, M. Celik, and L. T. Pileggi, “PRIMA: Passive reduced-order interconnect macromodeling algorithm,” in Proc. IEEE/ACM Int. Conf. on Computer-Aided Design, San Jose, CA, 1997, pp. 58–65. [14] E. J. Grimme, “Krylov Projection Methods for Model Reduction,” Ph.D. dissertation, electrical Engineering, University of Illinois at Urbana-Champaign, Urbana-Champaign, 1997. [15] A. C. Cangellaris, M. Celik, S. Pasha, and L. Zhao, “Electromagnetic model order reduction for system-level modeling,” IEEE Trans. Micro. Theory Tech., vol. 47, no. 6, pp. 840–850, 1999. [16] B. Moore, “Principal component analysis in linear systems: Controllability, observability, and model reduction,” IEEE Trans. Automat. Contr., vol. AC-26, no. 1, pp. 17–31, 1981. [17] V. V. S. Prakash, H. Yeo, and R. Mittra, “An adaptive algorithm for fast frequency response computation of planar microwave structures,” IEEE Trans. Micro. Theory Tech., vol. 52, no. 3, pp. 920–926, Jan. 2004. [18] L. Knockaert and D. D. Zutter, “Laguerre-SVD reduced-order modeling,” IEEE Trans. Micrw. Theory Tech., vol. 48, no. 9, Sept. 2000. [19] T. Penzl, “A cyclic low-rank Smith method for large sparse Lyapunov equations,” SIAM J. Sci. Comp., vol. 21, no. 4, pp. 1401–1418, 2000. [20] J.-R. Li and J. White, “Low rank solution of Lyapunov equations,” SIAM J. Matrix Anal. Appl., vol. 24, no. 1, pp. 260–280, 2002. [21] K. Willcox and J. Peraire, “Balanced model reduction via the proper orthogonal decomposition,” AIAA J., vol. 40, no. 11, pp. 2323–2330, Nov. 2002. [22] J. Phillips and L. Silveira, “Poor man’s TBR: A simple model reduction scheme,” IEEE Trans. Comput.-Aided Design Integr. Circ. Syst., vol. 24, pp. 43–55, 2005. [23] M. N. Vouvakis, K. Z. Zhao, and J.-F. Lee, “Finite-element analysis of infinite periodic structures with nonmatching triangulations,” IEEE Trans. Magn., vol. 42, no. 4, pp. 691–694, Apr. 2006. [24] M. G. Floquet, “Sur les equations differentielles lineaires a coefficients periodiques,” Ann. Ecole Normale Super., vol. 47–88, 1883. [25] A. de La Bourdonnaie, C. Farhat, A. Macedo, F. Magoules, and F.-X. Roux, “A non-overlapping domain decomposition method for the exterior helmholtz problem,” Contemp. Math., vol. 218, pp. 42–66, 1998.

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[26] G. N. Paraschos and M. N. Vouvakis, “On the accuracy of -based FETI method for electromagnetic problems,” in Proc. IEEE Antennas and Prop. Soc. Intern. Symp., Jul. 11–17, 2010, pp. 1–4. [27] D. C. Sorensen and A. C. Antoulas, “On model reduction of structured systems,” in Dimension Reduction of Large-Scale Systems, P. Benner, V. Mehrmann, and D. C. Sorensen, Eds. Berlin: Springer, 2005, pp. 117–130. [28] M. D. Migliore, “On electromagnetics and information,” IEEE Trans. Antennas Propag., vol. 56, no. 10, pp. 3188–3320, Oct. 2008. [29] T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein, Introduction to Algorithms, 3rd ed. Cambridge, MA: MIT Press, 2009. [30] R. Becker and R. Rannacher, “An optimal control approach to a posteriori error estimation in finite element methods,” Acta Numer., vol. 10, no. 1, pp. 1–102, 2001. [31] S.-H. Lee and J.-M. Jin, “Adaptive solution space projection for fast and robust wideband finite-element simulation of microwave components,” IEEE Microw. Wireless Compon. Lett., vol. 17, no. 7, pp. 474–476, Jul. 2007. [32] A. Schultschik, O. Farle, and R. Dyczij-Edlinger, “An adaptive multipoint fast frequency sweep for large-scale finite element models,” IEEE Trans. Magn., vol. 45, no. 3, pp. 1108–1111, Mar. 2009. [33] J.-F. Lee and D.-K. Sun, “p-Type multiplicative Schwarz (pMUS) method with vector finite elements for modeling three-dimensional waveguide discontinuities,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 3, pp. 864–870, 2004. [34] D. Schaubert and T. Chio, “Wideband Vivaldi arrays for large aperture antennas,” in Proc. Conf. at the ASTRON Netherlands Institute for Radio Astronomy, Apr. 1999, pp. 49–57. Wei Wang (S’07) received the B.S. degree in electronics science and technology from Zhejiang University, Hangzhou, China, in 2007 and the M.S. degree in electrical engineering from University of Massachusetts, Amherst, (UMASS), in 2009, where he is currently working toward the Ph.D. degree. From September 2006 to August 2007, he was a developer of a commercial 3D full-wave EM simulation software at EMdesigner Inc., Hangzhou, China. In September 2007, he joined the Antenna and Propagation Lab, ECE, UMASS, as a Research Assistant. His research interests include finite element method and model reduction for large-scale full-wave simulation problems.

Georgios N. Paraschos (S’06) was born in Agios Ioannis Monemvasias, Greece, in August 1982. He received the Diploma Degree in electrical engineering from Democritus University of Thrace (DUTH), Xanthi, Greece, in 2006 and is currently working toward the Ph.D. degree at the University of Massachusetts, Amherst (UMASS). In fall 2006, he joined the Center for Advanced Sensor and Communication Antennas (CASCA), Electrical and Computer Engineering Department, UMASS, as a Graduate Research Assistant. His research interests include computational electromagnetics, parallel processing, finite element and domain decomposition methods.

Marinos N. Vouvakis (S’99–M’05) received the Diploma degree in electrical engineering, from Democritus University of Thrace (DUTH), Xanthi, Greece, in 1999, the M.S. degree from Arizona State University (ASU), Tempe, in 2002, and the Ph.D. degree from The Ohio State University (OSU), Columbus, in 2005, both in electrical and computer engineering. Currently he is an Assistant Professor with the Center for Advanced Sensor and Communication Antennas, Electrical and Computer Engineering Department, University of Massachusetts at Amherst. His research interests are in the area of computational electromagnetics with emphasis on domain decomposition, fast finite element and integral equation methods, hybrid methods, model order reduction and unstructured meshing for electromagnetic radiation and scattering applications. His interests extend to the design and manufacturing of ultra-wideband phased array systems.

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Modeling of Nanophotonic Resonators With the Finite-Difference Frequency-Domain Method Aliaksandra M. Ivinskaya, Andrei V. Lavrinenko, and Dzmitry M. Shyroki

Abstract—Finite-difference frequency-domain method with perfectly matched layers and free-space squeezing is applied to model open photonic resonators of arbitrary morphology in three dimensions. Treating each spatial dimension independently, nonuniform mesh of continuously varying density can be built easily to better resolve mode features. We explore the convergence of the eigenmode wavelength and quality factor of an open dielectric sphere and of a very-high- photonic crystal cavity calculated with different mesh density distributions. On a grid having, for example, 10 nodes per lattice constant in the region of high field intensity, we are able to find the eigenwavelength with a half-percent precision and the -factor with an order-of-magnitude accuracy. We also suggest the rule (where is the cavity refractive index) for the optimal cavity-to-PML distance. Index Terms—Finite-difference frequency-domain (FDFD) method, perfectly matched layer (PML), -factor.

I. INTRODUCTION

N

ANOCAVITIES with high quality factor and low mode are attractive passive optical components in volume the rapidly developing area of cavity quantum electrodynamics [1]–[3]. Astonishing quantum phenomena are possible to observe with this kind of cavity, such as the enhancement of luminescence, alternation of emitter lifetime, Rabi oscillations, single-photon emission, and the enhancement of slow down factor in electromagnetically induced transparency. A variety of high- , low- cavity designs were proposed based on structural modifications in photonic crystal (PhC) matrices [3]–[7]. A workhorse for finding the eigenmodes in such complicated, multiple-interface structures is the finite-difference time-domain (FDTD) method [8], [9]. There are many variations in after the FDTD run, most of which require how to extract additional, sometimes vigorous, post-processing—see [6], [7], and [10]–[15], to list a few. As a natural alternative to the FDTD cavity analysis with -factor evaluation dependent on a choice of a specific fitting procedure, the finite-difference frequency-domain (FDFD) method can be used. In the frequency domain, Manuscript received December 17, 2010; revised February 12, 2011; accepted March 09, 2011. Date of publication September 08, 2011; date of current version November 02, 2011. A. M. Ivinskaya and A. V. Lavrinenko are with the Department of Photonics Engineering, Technical University of Denmark, 2800 Kongens Lyngby, Denmark (e-mail: [email protected]; [email protected]). D. M. Shyroki is with the Institute of Optics, Information and Photonics, University Erlangen-Nürnberg, 91058 Erlangen, Germany (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2164215

the (many-at-a-time) resonance wavelength(s) and quality factor(s) appear straightforwardly from real and imaginary parts of complex eigenvalue(s) of the discretized source-free Maxwell equations. From the early 1980s, three-dimensional (3-D) finite-difference and finite-integral techniques in the frequency domain were applied to modeling of closed cavities and metal structures [16]–[19]. Numerical accuracy of the FDFD solution for various Cartesian discretization meshes was analyzed by Smith [20], [21]. In particular, different offsets in location of field components within the primary cell were discussed with the conclusion that the most accurate is the classical Yee staggering [8]. Today, 3-D frequency-domain method is used widely in microwave cavity analysis [22]–[24] and, to a lesser extent, in photonic bandgap computations where Bloch-periodic boundary eigenproblem is solved [25], [26]. Analysis of open photonic resonators with the 3-D FDFD method was problematic, in contrast to photonic band calculations, for two main reasons. First, much larger, i.e., multiple-lattice-constant pieces of PhC matrices must be considered. Second, nontrivial absorbing boundaries like the perfectly matched layers (PMLs) placed at sufficient distance from the modeled resonator are to be used instead of simple Bloch-periodic walls. Addressing these issues is worthwhile as the FDFD method could be very convenient for many typical cavity problems in nanophotonics, such as optimization of the PhC resonators in 2-D and 3-D by tracing the wavelengths, -factors, and field maps of one or several spectrally close resonances. In this paper, we describe 3-D FDFD method on a logically Cartesian, physically nonuniform grid of continuously varying density, with the PMLs and buffer space squeezing [27], [28]. After recasting basic FDFD matrices, we focus on application of the method to modeling open, high- PhC resonators. In particular, we discuss the necessary thickness of the PMLs and free-space buffers for reliable calculations, the choice of optimal free-space squeezing function to reduce computational domain size, and the robustness of the method with respect to adaptive grid density variation inside the modeled structure. We report that the method can give good estimates for the PhC cavity -factor on a personal computer with 4 Gb of RAM within 10 min in 3-D.

II. FDFD METHOD The source-free Maxwell equations in the frequency domain can be combined into the eigenvalue problem

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(1)

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for the vector; is the magnetic permeability, and is the dielectric permittivity. In the discretized form, (1) reads

(2)

where

are column-vectors of length each, and is the total number of grid nodes in the domain. It is assumed that is represented by a diagonal matrix (which is not always true for upon its polarization-sensitive averaging at material boundaries; see [29]) so that can be split in its square roots trivially in order to get symmetrized eigenmatrix . We define 1-D forward of size acting along on the difference matrix electric field component normal to , for example .. .

..

.

..

.

.. .

(3) being the grid step (which is constant in computawith tional space, but may vary in physical coordinates). The maof the size are trices

Fig. 1. (Color online) Yee scheme version we are using (xy -projected). Here, the locations of electric (red) and magnetic (blue) nodes correspond to the PMC boundaries at the start and PEC walls at the end of each of the domain extensions. E -grid nodes are framed; other grids are obtained by half-cell shifts in appropriate directions.

that prevents PMLs from being constructed starting immediately from the boundaries of the structure. This is bad news since increased computation domain size amounts to increased computer memory consumption, especially in 3-D modeling. To squeeze the buffer physical space into the interval (where is the most extended point of the modeled object, and the computational domain boundary) most efficiently, we combine standard PMLs with free-space mapping as described in [27] and [28] so that the wavelength-dependent buffer “stretch function” in (and similarly and ) direction is (5) is the derivative of the Here, chosen real space squeezing function; the normalized maximum conductivity is (6)

(4) is the unit diagonal matrix of the size , and where is the Kronecker product. Backward-difference matrices are equal to the minus transposed. Thus, the final size of . The combination of the eigenvalue matrix is forward and backward differentials acting on the Yee-staggered electric and magnetic fields allows second-order-accurate discretization of (1), at least locally. The chosen arrangement of electric and magnetic field components assumes placing of the perfect magnetic conductor (PMC) boundary at the beginning and the perfect electric conductor (PEC) boundary at the end of each of the domain extensions, as shown in Fig. 1. This is very convenient in modeling open symmetric systems: By centering such a system in an appropriate corner of computational domain, we cut the system by a combination of the PMC and PEC planes according to the expected mode symmetry while covering three remaining domain walls with absorbing boundary layers. Construction of efficient absorbing boundaries for modeling an open photonic system is not a trivial task. Profound research was done on optimizing the PMLs—their thickness, conductivity profile, and frequency dispersion—for better absorption of oscillatory waves. In many real-life simulations, however, one deals routinely with an admixture of evanescent field spreading from a photonic structure; it is this evanescent field

where the PML starts from the coordinate , the logarithmic damping by an idealized (continuous) PML can be taken equal to 20, and is the wavelength. The conductivity profile function is zero everywhere except inside the PML, (7) where is the PML conductivity profile order, 2 being a common choice. We found that, contrary to our previous claim [28], the choice of space squeezing function (see Fig. 2 for a few examples) does matter in real-life simulations. We function, which alstick to the lows to represent on a given computational coordinate interval a considerably wider physical interval . For translates to , which means that if the example, PMLs cover half of the squeezed free-space buffer in computational space, the actual “physical” distance between the PMLs and the modeled object is double that. More precisely, our on , while on mapping function is , we define (8)

IVINSKAYA et al.: MODELING OF NANOPHOTONIC RESONATORS WITH FDFD METHOD

=

Fig. 2. Examples of the mapping function x x x (plotted in bold). to the identity line x

=

(x

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) compared

and thus in (5) we put

(9) , and 1 otherwise. An even steeper mapfor ping function might not be a good option since upon discretization with equidistant steps in the computational domain space, the physical coordinates rapidly become too poorly sampled to represent outcoming oscillatory waves adequately. Due to the presence of the PMLs, the eigenmatrix is complex nonhermitian even if the constitutive materials are nonabsorptive. Such nonhermitian problem leads to complex eigenwith their real parts giving the resofrequencies nance frequencies and the imaginary parts directly connected to . For a sufficiently good resonator, the -factors: , so that comparable absolute errors we always have , lead to much higher relative errors in in and , the -factor values: . This is exactly what we see in the numeric examples that follow. Furthermore, the -factor is more sensitive to the PML parameters (their placement, absorption profile, etc.) as it arises entirely due to the introduction of artificial imaginary parts in and within the PMLs, while for the eigenwavelength, the PMLs can be seen as just a perturbation. Additional complication arises through the frequency dependence of material tensors and/or and hence the eigenmatrix . When the resonator contains metal components or any other materials whose dispersion should accurately be taken into for the eigenfreaccount, one can start with some guess quency and attempt repetitive solving of the linearized eigenfor the eigenfrequency , then problem with for , etc., until a convergence criterion with is met. However, unlike material dispersion, the PML dispersion is a mathematical artifact inessential to the user of a numerical tool and, due to certain robustness of PML performance with respect to the values of PML conductivity, the PML dispersion can be ignored in the frequency domain by in the eigenproblem. If this putting used to define , (6), is not too far from exact eigenfrequency, one can expect that an error introduced due to nonoptimal PML conductivity value is negligible.

Fig. 3. (Color online) (top) Electric and magnetic vector field distributions of (“ ”) mode in an octant of the sphere, and (bottom) z cuts the of the E and H components restored over the extended domain.

TE

TE

=0

III. SIMULATION EXAMPLES We generate the permittivity and permeability arrays and assemble the eigenmatrix in MATLAB [30], which normally takes seconds. The eigenmatrix assembly time is linear with , while the array generation time scales in 3-D as the , because its major overhead is linear measure squared, treating grid cells crossed by material boundaries. We pass the , which assembled eigenmatrix to the MATLAB function provides an interface to the Fortran-based ARPACK [31] library for solving sparse eigenvalue problems iteratively. However, we rely on direct LU factorization at the shift-and-invert step. This is more memory-consuming, but also more reliable and universal than iterative algebraic methods that require very careful and problem-dependent preconditioning. The solution time and the maximal resolution for the given memory limit depend strongly on the structure of the eigenmatrix. Thus, elongated cavities give eigenmatrices with much smaller bandwidth than the structures defined on a and hence larger maximal cubic domain. Characteristic run-time in 3-D modeling is about an hour if we are close to the memory limit on a 16-Gb machine. A. Dielectric Sphere Analytic solution for the resonance frequencies and -factors of a homogeneous dielectric sphere of radius is well known [32]. For the sphere of relatively high refractive index in the air [33], we consider two lowest TE modes, ) having the eigenwavelength a dipole one (we label it as and the quality factor , with a comparable quality factor and the next one , but nearly twice smaller eigenwavelength . The boundary conditions corresponding to the symmetry of these modes are one PMC and two PEC planes dissecting the sphere. The dipole mode is sketched in , , and , which Fig. 3. The dominant components are corresponds to the TE polarization of this mode. We see that

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Fig. 5. (Color online) (left) PhC membrane cavity with six rings of holes in the xy -plane. (right) H -field of the hexapole mode.

Fig. 4. Relative errors in (top)  and (bottom) Q for the TE and TE modes of a sphere of radius a, centered in the corner of a cubic computational domain of size 1:5a or 2:0a, thus giving the squeezed-space buffers of 0:5a or 1:0a, in computational coordinates. Half of each buffer layer is covered with the PML. The sphere is discretized uniformly: 1x = 1y = 1z .

the electric field forms a toroid near the sphere border, and the magnetic field flows round this toroid having its maximum at the very center of the resonator. The convergence of numerically computed and to their analytic values is shown in Fig. 4. All the curves demon. As expected from strate second-order convergence rate in or 70 for the modes considered, the relative errors in are an order of magnitude lower than the relative -factor errors—note the difference in the -scales in the upper and lower plots in the figure. From the upper plot, we see that the values (black and gray curves) are largely insensitive to varying buffer thickness; this is what we generally observe for any modes and structures calculated with the FDFD method. At the relative a grid resolution of 10 voxels per error stays within 0.25% accuracy, while the error is just below 1%. The reason is that the resolution of 10 voxels per amounts to the impressive 120 voxels per , while it translates to around 60 voxels per . With the second-order method, this would account for the four-times-poorer ratio of

results in both and numerical values than those we have and . for From the lower plot in Fig. 4, we can see the effect of the computational domain size on the -factor accuracy. Note that the widths of squeezed-space buffer layers are given in computational coordinates, which are translated to the physical and or . Thus, the coordinates via (8) with buffer is sphere-to-PML distance with the narrower , and with the wider buffer, . The difference between the two it is curves in the plot is huge: The narrower buffer is clearly mode -factor simulations, while inadequate for the sphere-to-PML distance is a good choice for the the mode in the sense that further increasing this distance gives only marginal improvement to the accuracy, and the makes no sense. sphere-to-PML distance in excess of mode, the curve is already smooth for the For the (for which it is plotted), sphere-to-PML distance of yet a safer choice appears to be . Thus, for both modes, . By modeling spheres of other the safe buffer size equals rule holds quite refractive indices, we found that this generally for this geometry. In the following section, we will see that this rule is also helpful in modeling of an entirely different, PhC-membrane-based resonator. B. PhC Cavity Here, we consider a high- PhC membrane cavity shown in , thickthe left side of Fig. 5: In a slab of refractive index , with the hexagonal array of air holes of diameter ness ( is the lattice constant), a defect is formed by excluding from one central hole and shifting the six holes next to it by . The hexapole the center while shrinking their diameters by mode in this membrane cavity was reported having the quality and the wavelength [34]. factor The right side of Fig. 5 represents the standing wave pattern of this mode. For actual computations, the structure can be cut , and the PMC planes though its center by the PEC wall at at and , so that an octant of what is shown in Fig. 5 has to be modeled. To resolve mode features better in the area of field hotspots while avoiding extra-fine sampling in low-intensity region, physically nonuniform mesh can be introduced. To retain mesh orthogonality, each direction should be resampled indepenfunction that allows building of plateaus dently. We use the of different but roughly equidistant meshing, connected through transition region with faster variation of sampling. With the ratio between grid steps on the two plateaus, the width and

IVINSKAYA et al.: MODELING OF NANOPHOTONIC RESONATORS WITH FDFD METHOD

Fig. 6. (top) Versions of (10) plotted for the start grid step (bottom) xy grid obtained via s4 stretch of x and y .

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the center of the transition region, the nonuniform grid step as a function of grid coordinate is defined via

Fig. 7. (top) Eigenwavelength  and (bottom) quality factor Q of the hexapole mode versus computational grid step, calculated with : a-wide squeezed-space buffer layers (half covered by PMLs). The PMLs in x- and y -directions extend over three grid cells. Solid black curves are obtained with x y z , while the other curves with the physically cubic grid nonuniform meshes in the xy -plane are like those shown in Fig. 6.

10

(1 = 1 = 1 )

(10) At , we have , provided sufficiently narrow width and sufficiently distant . Physically nononuniform coordinates of grid nodes are obtained as (11) The black curves in Fig. 6 are plotted for stretch ratio in (10), i.e., the mesh is nearly twice as coarse at the outskirts . Two sets of than in the center; the gray curves are for , transition region parameters are considered: giving relatively slow transfer to a sparser mesh (solid curves); , with a faster jump to a coarser gridding and (dotted curves) so that the larger membrane area is covered with a rough mesh. Note that the grid metric information can be transferred to and via the standard ideology of generally covariant electrodynamics, hence we can still use simple matrices (4) for logically equidistant grid. In Fig. 7, we see how and depend on the grid step size in the case of the physically uniform grid ( everywhere within the membrane) compared to the grids stretched and equal , but in and such that at the center, toward the periphery, they gradually increase as in Fig. 6. The

values obtained are consistent with the FDTD results [34], the discrepancy in being around 0.2% while the -factors lie between to provided the grid step is smaller than (thus the PMLs comprise five or more grid steps). Prominently, smooth nonequidistant discretization gives results very similar to those obtained on the uniform grid, while the reduction in is almost twofold for the s4 the total number of grid nodes stretch. Thus, applying S4 function in and with gives the total number of grid nodes , the eigen, and the quality factor ; wavelength the run-time on a laptop with 4 Gb of RAM and 2.2-GHz CPU frequency is 10 min. A slightly different function with gives , , and in a 2-min run-time. In all these calculations, we used -thick squeezed-space buffer layer (half of which is covered by the PML) in -direcrule. To see how sensitive tion, in line with the proposed to the -buffer thickness the results are, we chose a wider and a smaller domain and calculated the eigenwavelength (upper plot in Fig. 8) and -factor convergence (lower plot in Fig. 8) of the same hexapole mode and of the quadrupole mode reported and [34]. While curves having

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Fig. 8. (top) Eigenwavelength  and (bottom) quality factor Q of the hexapole mode (curves with circles) and quadrupole mode (curves with crosses) versus computational grid step, calculated with either 0:8a- or 1:2a-wide buffer layers in z (half covered by PMLs). The PMLs in x- and y -directions extend over three grid cells. All the results are obtained with nonuniform s4-stretched grid as shown in Fig. 6.

are seen to be little affected by changing the domain size, there curves of the hexapole is a striking contrast between the mode calculated with and buffers, the latter giving clearly erroneous results. The quadrupole mode -factor is also problematic to define correctly at low resolution when buffers. On the other hand, if the buffer layers using the are thick enough, an order-of-magnitude accuracy in is achieved already at a very rough resolution of eight grid points per . Thus, as in the previous example of a sphere, the eigenwavelength accuracy is primarily limited here by the grid step while the -factor is mostly affected by the overall thickness of buffer layers no matter what the actual grid cell size is. Note that -wide squeezed-space buffers is a reasonable choice also for other modes of interest in this membrane, and generally for most PhC resonators that typically have band-gaps with the . defect modes around IV. CONCLUSION We applied the FDFD method to calculate the quality factor and the resonance wavelength of three-dimensional open dielectric resonators and addressed the minimum thickness of freespace buffer layers required for getting reliable -factors, the

possibility to squeeze those buffer layers onto a tighter computational domain, and the nonuniform meshing within the structure for accurate calculations. We found the following. 1) For a given eigenmode in an open cavity, an optimal cavity-to-PML distance (in physical space) equals —the ratio between the (expected) wavelength of that eigenmode and the refractive index contrast between cavity material and the surrounding medium. This rule is convenient to use for automated construction of absorbing buffer layers in the finite-difference- or finite-element-based software. 2) To further squeeze the cavity-to-PML distance in commapputational domain, the ping is suggested. With the PMLs covering one half of the squeezed-space buffer layer (the cavity-to-PML distance in physical coordinates thus being equal to the total squeezed-space buffer size in the computational space) and comprising at least five grid-cell sublayers, this would give robust and efficient absorbing buffers for the finite-difference simulations. 3) Building an orthogonal, nonuniform grid of continuously varying density—for example, approaching a ratio 1:3 between its high- and low-density regions—gives stable convergence for both and -factor and saves computer memory a lot. For high- PhC cavities, at sampling of 30 pixels per , an error in the eigenwavelength is well below 0.5%, and the -factor order of magnitude can easily be obtained. This is a sufficient accuracy for many resonator design and optimization tasks. REFERENCES [1] S. Noda, M. Fujita, and T. Asano, “Spontaneous-emission control by photonic crystals and nanocavities,” Nature Photon., vol. 1, pp. 449–458, 2007. [2] K. Aoki, D. Guimard, M. Nishioka, M. Nomura, S. Iwamoto, and Y. Arakawa, “Coupling of quantum-dot light emission with a three-dimentional photonic-crystal nanocavity,” Nature Photon., vol. 2, pp. 688–692, 2008. [3] Y. Gong, B. Ellis, G. Shambat, T. Sarmiento, J. S. Harris, and J. Vuckovic, “Nanobeam photonic crystal cavity quantum dot laser,” Opt. Exp., vol. 18, pp. 8781–8789, 2010. [4] B. S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high- Q photonic double-heterostructure nanocavity,” Nature Mater., vol. 4, pp. 207–210, 2005. [5] Y. Zhang, M. W. McCutcheon, I. B. Burgess, and M. Loncar, “Ulta-high-Q TE/TM dual-polarized photonic crystal nanocavities,” Opt. Lett., vol. 34, pp. 2694–2696, 2009. [6] O. Painter, J. Vuckovic, and A. Scherer, “Defect modes of a two-dimensional photonic crystal in an optically thin dielectric slab,” J. Opt. Soc. Amer. B, vol. 16, pp. 275–285, 1999. [7] Z. Zhang and M. Qiu, “Small-volume waveguide-section high Q microcavities in 2D photonic crystal slabs,” Opt. Exp., vol. 12, pp. 3988–3995, 2004. [8] K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag., vol. AP-14, no. 3, pp. 302–307, May 1966. [9] A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. Norwood, MA: Artech House, 2005. [10] Y. Hua and T. K. Sarkar, “Generalized pencil-of-function method for extracting poles of an EM system from its transient response,” IEEE Trans. Antennas Propag., vol. 37, no. 2, pp. 229–234, Feb. 1989. [11] P. Kozakowski, A. Lamecki, and M. Mrozowski, “Provisional model technique in the FDTD analysis of high-Q resonators,” IEEE Microw. Wireless Compon. Lett, vol. 14, no. 11, pp. 501–503, Nov. 2004.

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[12] J. A. Pereda, J. E. F. de Rio, F. Wysocka-Schillak, A. Prieto, and A. Vegas, “On the use of linear-prediction techniques to improve the computational efficiency of the FDTD method for the analysis of resonant structures,” IEEE Trans. Microw. Theory Tech., vol. 46, no. 7, pp. 1027–1032, Jul. 1998. [13] C. Wang, B. Q. Gao, and C. P. Deng, “Accurate study of -factor of resonator by a finite-difference time-domain method,” IEEE Trans. Microw. Theory Tech., vol. 43, no. 7, pp. 1524–1529, Jul. 1995. [14] W. H. Guo, W. J. Li, and Y. Z. Huang, “Computation of resonant frequencies and quality factors of cavities by FDTD technique and pade approximation,” IEEE Microw. Guided Wave Lett., vol. 11, no. 5, pp. 223–225, May 2001. [15] “Harminv,” 2006 [Online]. Available: http://ab-initio.mit.edu/wiki/ index.php/Harminv [16] M. Albani and P. Bernardi, “A numerical method based on the discretization of Maxwell equations in integral form,” IEEE Trans. Microw. Theory Tech., vol. 22, no. 4, pp. 446–450, Apr. 1974. [17] T. Weiland, “A discretization method for the solution of Maxwell’s equations for six-component fields,” Electron. Commun. AEUE, vol. 31, no. 3, pp. 116–120, 1977. [18] W. Wilhelm, “Three dimensional resonator mode computation by finite difference method,” IEEE Trans. Magn., vol. MAG-21, no. 6, pp. 2340–2343, Nov. 1985. [19] A. Christ and H. L. Hartnagel, “Three-dimensional finite-difference method for the analysis of microwave-device embedding,” IEEE Trans. Microw. Theory Tech., vol. MTT-35, no. 8, pp. 688–696, Aug. 1987. [20] J. T. Smith, “Conservative modeling of 3-D electromagnetic fields, part I: Properties and error analysis,” Geophysics, vol. 61, pp. 1308–1318, 1995. [21] J. T. Smith, “Conservative modeling of 3-D electromagnetic fields, part II: Biconjugate gradient solution and an accelerator,” Geophysics, vol. 61, pp. 1319–1324, 1995. [22] I. Munteanu, M. Timm, and T. Weiland, “It’s about time,” IEEE Microw. Mag., vol. 11, no. 2, pp. 60–69, Apr. 2010. [23] T. Weiland, M. Timm, and I. Munteanu, “A practical guide to 3-D simulations,” IEEE Microw. Mag., vol. 9, no. 6, pp. 62–75, Dec. 2008. [24] R. M. Makinen, H. De Gersem, T. Weiland, and M. A. Kivikoski, “Modelling of lossy curved surfaces in the 3-D frequency-domain finite-difference methods,” Int. J. Numer. Model, vol. 14, pp. 1741–1746, 2006. [25] G. Lubkowski, B. Bandlow, R. Schuhmann, and T. Weiland, “Effective modeling of double negative metamaterial macrostructures,” IEEE Trans. Microw. Theory Tech., vol. 57, no. 7, pp. 1136–1146, Jul. 2009. [26] T. Terao, “Computing interior eigenvalues of nonsymmetric matrices: Application to three-dimensional metamaterial composites,” Phys. Rev. E, vol. 82, pp. 026702-1–6, 2010. [27] D. M. Shyroki and A. V. Lavrinenko, “Perfectly matched layer method in the finite-difference time-domain and frequency-domain calculations,” Phys. Status Solidi B, vol. 244, pp. 3506–3514, 2007. [28] D. M. Shyroki, A. M. Ivinskaya, and A. V. Lavrinenko, “Free-space squeezing assists perfectly matched layers in simulations on a tight domain,” IEEE Antennas Wireless Propag. Lett, vol. 9, pp. 389–392, 2010. [29] D. M. Shyroki, “Modeling of sloped interfaces on a Yee grid,” IEEE Trans. Antennas Propag., vol. 59, no. 9, pp. 3290–3295, Sep. 2011. [30] MATLAB R2010b. MathWorks, Natick, MA, 2010 [Online]. Available: http://www.mathworks.com

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[31] R. B. Lehoucq, D. C. Sorensen, and C. Yang, ARPACK Users’ Guide: Solution of Large-Scale Eigenvalue Problems With Implicitly Restarted Arnoldi Methods. Philadelphia, PA: SIAM, 1998. [32] H. A. van de Hulst, Light Scattering by Small Particles. New York: Dover, 1984. [33] D. M. Shyroki, “Efficient Cartesian-grid-based modeling of rotationally symmetric bodies,” IEEE Trans. Microw. Theory Tech., vol. 55, no. 6, pp. 1132–1138, Jun. 2007. [34] S.-H. Kim, S.-K. Kim, and Y.-H. Lee, “Vertical beaming of wavelength-scale photonic crystal resonators,” Phys. Rev. B, vol. 73, p. 235117, 2006.

Aliaksandra M. Ivinskaya was born in Minsk, Belarus, in 1981. She received the Specialist degree from Byelorussian State University, Minsk, Belarus, in 2004 and is currently pursuing the Ph.D. degree in photonics engineering at the Technical University of Denmark, Lyngby, Denmark.

Andrei V. Lavrinenko received the M.Sc., Ph.D., and D.Sci. degrees from the Belarussian State University (BSU), Minsk, Belarus, in 1982, 1989, and 2004, respectively. He was an Assistant Professor and Associate Professor with the Physics Department, BSU, from 1990 to 2004. Since 2004, he has been an Associate Professor with the Department of Photonics Engineering, Technical University of Denmark, Lyngby, Denmark. Since 2008, he has been leading the Metamaterials Group. He is the author or coauthor of more than 300 publications, including 10 textbooks and book chapters. His research interests are in metamaterials, plasmonics, photonic crystals, quasicrystals and photonic circuits, slow light, and numerical methods in electromagnetics and photonics. Prof. Lavrinenko is a member of the OSA and Danish Optical Society.

Dzmitry M. Shyroki was born in Minsk, Belarus, in 1981. He received the Specialist degree from Byelorussian State University, Minsk, Belarus, in 2004, and the Ph.D. degree from the Technical University of Denmark, Lyngby, Denmark, in 2008. Currently, he is with the University Erlangen-Nürnberg and the Max Planck Institute for the Science of Light, Erlangen, Germany, where he is involved in the modeling of plasmonic devices.

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An Efficient Scattered-Field Formulation for Objects in Layered Media Using the FVTD Method Dustin Isleifson, Student Member, IEEE, Ian Jeffrey, Member, IEEE, Lotfollah Shafai, Life Fellow, IEEE, Joe Lovetri, Senior Member, IEEE, and David G. Barber

Abstract—A technique for efficiently simulating the scattering from objects in multilayered media is presented. The efficiency of the formulation comes from the fact that the sources for the scattered fields (SFs) only occur at the inhomogeneities and, therefore, the SFs impinging on the boundaries are more easily absorbed. To demonstrate the technique, a 1-D-finite-difference time-domain solution to the plane-wave propagation through a multilayered medium is used as an incident-field source for an SF formulation of the finite-volume time-domain method. Practical aspects of the application are discussed and numerical examples for scattering from canonical objects are presented to show the validity of the proposed technique. The simulation scheme described herein can be used for simulations of geophysical media with appropriate specifications of the dielectric properties of the media and the inhomogeneities. Index Terms—Finite difference time domain (FDTD), finite volume time domain (FVTD), numerical methods.

I. INTRODUCTION

T

HE STUDY of electromagnetic wave scattering from objects in multilayered media is a widespread problem with diverse applications including the remote sensing of earth environments [1] and buried object detection [2], [3]. In contrast to the problem where the target or object of interest lies in free space, the formulation and subsequent analysis of the multilayered media are complicated due to the layer interfaces that govern the propagation of the interrogating incident field. Determining a simple way to account for the layers is a nontrivial task, which has led to a variety of techniques in the literature for modeling subsurface problems [4] (and references therein) [5]. As stated in [6], a variety of analytic methods has been developed and is capable of describing the propagation through multilayered media; however, it is challenging to utilize these methods in an existing numerical solver. Differential-equation-based techniques are particularly well suited for modeling wave interactions with inhomogeneities in

Manuscript received October 27, 2010; revised March 01, 2011; accepted April 23, 2011. Date of publication August 12, 2011; date of current version November 02, 2011. This work was supported by an NSERC graduate scholarship to the first author, and NSERC operating grants and a Canada Research Chairs grant to the third author. D. Isleifson, I. Jeffrey, L. Shafai, and J. LoVetri are with the Department of Electrical and Computer Engineering, University of Manitoba, Winnipeg, MB R3T 2N2, Canada (e-mail: [email protected]; joe_lovetri@ umanitoba.ca; [email protected]; [email protected]). D. G. Barber is with the Centre for Earth Observation Science and the Faculty of Environment, University of Manitoba, Winnipeg, MB R3T 2N2, Canada (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2011.2164198

a multilayered medium since the incorporation of an inhomogeneity does not increase the number of unknowns that must be solved [6]. The ubiquitous finite-difference time-domain (FDTD) method [7] has been used in this respect, and while it is appropriate for many geometries, it is deficient in that it requires a high level of discretization in order to resolve objects with curved features. In many subsurface scattering problems, the inhomogeneities exhibit curved features (e.g., brine inclusions in sea ice, landmines, cancerous tumors) and so, a method that takes the curved features into account would be useful. One such method is the finite-volume time-domain (FVTD) method [8], [9], which is particularly well suited to modeling curved features due to its ability to use a conforming irregular grid. With appropriate interpolation methods, the solutions for the multilayered problem that have been developed for the FDTD method can be used with the FVTD method. When using a scattered-field (SF) technique to perform calculations, knowledge of the incident field at the inhomogeneity for the entire time history of the simulation is required. When the medium is multilayered, the incident field at the inhomogeneity is the resultant of various reflections from and transmissions across the layer interfaces. The SFs generated by the inhomogeneity then propagate in the multilayered medium and are absorbed at the boundaries. In time-domain differential-equation-based methods, the boundary conditions are more efficient at absorbing normally incident SFs, and so, the incorporation of the incident field into the numerical method and solving for only the SF allows the boundary condition to be more effective. A method for generating a total-field/SF (TF/SF) source for general layered media was presented in [10]. Continued interest in developing the technique is evident in the variations that have been presented in the literature (for example, [11] and [12]). Previously, [6] presented an FDTD method for modeling scatterers in stratified media, but they limited their study to time-harmonic plane waves. In this paper, we adapt the general TF/SF source for use in an SF formulation of the FVTD method and show how it can be used for scattering from objects within a multilayered medium. The focus of [10] is mostly on computational aspects, such as stability, dispersion, and evanscent waves; their examples are strictly multilayered media without any inhomogeneities. In contrast, our work focuses on the utilization of the method for providing an incident field and calculating the scattering from inhomogeneities lying within the medium. Furthermore, our work considers an interpolation scheme that permits an irregular mesh, such as that utilized in the FVTD calculation. To validate our method, we compare with other published data for canonical shapes lying within a half-space medium and below a lossy layer. Incorporation of the SF formulation for

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multilayered media in the FVTD engine provides a new tool for modeling complex layered media and has future potential to be used for modeling electromagnetic interactions in remote sensing studies. The organization of this paper is as follows. In Section II. we formulate the total-field/SF background theory. Details of the practical application of this general concept in an FVTD numerical solver are given in Section III. We present numerical results for several canonical shapes for validation in Section IV. We conclude with several remarks on the implementation and applicability of the method, along with some suggestions for our future work. II. FORMULATION Let us denote the time-domain electric-field intensity as , the time-domain magnetic-field intensity as , the relative permittivity as , the relative permeability as , and the impressed time-domain electric current as . Using the standard incident-field-SF decomposition, we have (1) and (2) where the incident fields are defined to exist in a background and , and are produced by the impressed media, with . That is current (3) (4) The SF is then produced by the difference between the true and , and the background. After some algebraic media manipulation (5) (6) where the contrast magnetic and electric contrast sources are defined as (7) (8) For nonmagnetic media, . If we take the incident field to propagate in the layered background medium with no inhomogeneities, then it is clear from (8) that the SF is generated by equivalent sources at the inhomogeneities. The fact that the sources for the SF occur only at the inhomogeneities means that the field impinging on the boundaries is more easily

absorbed by whatever absorbing boundary condition is being used. The incident field still contains all of the information on the interactions of the waves as they propagate from one layer to another and scatter from the layer interfaces. One of the benefits of using the SF formulation is that the incident-field can be specified either analytically or numerically. III. NUMERICAL IMPLEMENTATION In order to demonstrate the utility of the SF formulation as a general concept, we apply the method into the framework of an existing FVTD numerical solver. This is similar to the approach used in [6], where a time-harmonic plane-wave source was applied to a scatterer in a finely stratified layered medium. In our work, we focus on the FVTD method for numerically solving Maxwell’s equations and utilize a numerically defined (as opposed to an analytically defined) incident-field source term. A technique similar to the one presented here can be used with any time-domain field solver. A. FVTD Computations The FVTD method is a robust and flexible scheme for numerically simulating 3-D electromagnetic problems [8], [9]. It is an solver, which means that it scales linearly with the number of elements in the mesh. One of the advantages FVTD has over the FDTD algorithm is that structured and arbitrary unstructured meshes are equally suitable discretizations for FVTD simulations. This implies that the volumetric mesh can be created to naturally follow oblique surfaces and no alteration to the algorithm is required to compensate for an inaccurate physical model. The FVTD formulation used for the numerical simulations produced in this work is a cell-centered, upwind, characteristic-based numerical engine for meshes consisting of first-order polyhedral elements [9]. It is second order accurate in time and space. The engine is capable of solving Maxwell’s equations in the time domain using either a total- or SF formulation, the latter permitting arbitrary (i.e., nonhomogeneous) background media [13]. The numerical implementation has been parallelized for distributed parallel environments by decomposing the computational domain into subdomains by using orthogonal-recursive bisection (ORB). Each subdomain is assigned to a unique processor and the underlying system of partial differential equations is solved locally on each processor by introducing a halo/ghost duplication of elements lying on the boundary of a processor’s domain [9]. The upwind formulation explicitly imposes the electromagnetic boundary conditions at the facets of each mesh element. Not only does this achieve accurate modeling of irregularly shaped inhomogeneous objects, but also allows for very simple, but effective absorbing boundary conditions (ABC) at the edges of the computational domain without having to introduce perfectly matched layers (PML). These ABCs are known as the Silver–Müller conditions. PML absorbing boundary conditions have also been implemented [14]. Due to its advantages, FVTD is an excellent candidate for solving field problems with a large number of small inhomogeneities.

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Fig. 1. Hypothetical geometry illustrating the decomposition of the fields in the SF formulation. (a) Total-field geometry. (b) Incident-field geometry. (c) SF geometry.

B. Plane-Wave Injector

(13)

When simulating multilayered media, truncating a plane wave in a total-field formulation is very inefficient because the incident-wave vector is impinging tangentially (or close to tangentially) to the ABC. This is the worst possible case for the ABC to absorb the wave. In addition, spurious reflections occur where the layer interfaces meet the boundary conditions, corrupting the desired SFs from objects within the layered medium. In effect, it is impossible to distinguish between SFs from the objects and erroneous SFs from the ABC. One way to mitigate this problem is through the use of the SF formulation. A total-field simulation can be performed to obtain the total fields for a wave propagating in layered media. We then use this total field as the incident field in an SF formulation. A hypothetical SF decomposition is shown in Fig. 1 for the case of multilayered media. Following [10], we utilize a 1-D-FDTD solution for a wave propagating through layered media at an arbitrary incidence angle, with appropriate modifications for the FVTD method. For completeness, the derived expressions for TE and TM wave propagation are summarized here. The TE equations for plane-wave propagation through multilayered media are given as (9) (10) (11) where is the incidence angle measured between the direction is the intrinsic of propagation and the axis, is the dielectric constant of admittance of free space, and the uppermost (or th) layer. Similarly, the TM equations are given as (12)

(14) is the intrinsic impedance of free space. where These equations resemble the familiar transmission-line equations and with appropriate descriptions of the dielectric profile in the z-direction, they describe the propagation of a plane wave through layered media. In order to include lossy material, a variation to the TM equations must be made (not demonstrated in this paper, but described in [10]). To solve the TE or TM equations, we discretize the equations, following the approach of [15], where the electric- and magnetic-field components are interleaved, and we use a central-difference approximation. We set the temporal discretization to obey , where is the the Courant stability criterion velocity of propagation in the layer with the lowest dielectric constant. We apply a basic PML boundary condition at the upper and lower bounds of the 1-D solution grid, but also pad the solution domain to minimize the error created by the boundary (i.e., add more distance for the wave to travel than is necessary for the FVTD solution). This has no impact on the memory required to store the 1-D solution since we only keep the portion which will correspond to the variation required within the FVTD domain, but it does remove artificial reflections created from the boundaries in the 1-D solution. In practice, the 1-D solution is interpolated into the 3-D irregular mesh that is used in the FVTD computations. A principle plane containing the time history of the plane-wave propagation through layered media is made to coincide with the plane. To find the field values along this principle plane, we need only a time shift, since this is a property of plane-wave propagation. This concept is illustrated in Fig. 2, where the principle plane is plane, and ( shown to coincide with the is the speed of light in a vacuum). The solution for wave propagation along the negative axis is calculated and the time-delayed result is utilized to find the field value at a location .

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Fig. 3. Geometry for a dielectric sphere in free space, with a radius of 0.1 , 0.25. and centered at z

=0

Fig. 2. Calculation of field values on the principle plane using the time-delay factor  . The fields calculated at x are the fields calculated along the z axis with an appropriate time delay.

To obtain a 3-D representation, we utilize the invariance of the solution along the other coordinate axis ( axis). C. Interpolation of the 1-D FDTD Solution to the FVTD Grid Many practical implementation issues exist, such as choosing an appropriate interpolation method and ensuring that the spatial discretization of the auxiliary 1D-FDTD simulation does not cause dispersion in the main FVTD simulation. If the FVTD grid were regular (as would be used in a 3-D-FDTD grid), we could follow the method of [10] and increase the spatial sampling by an odd factor and thereby ensure that the interfaces were preserved. Moreover, since the auxiliary FDTD simulation takes a small fraction of the time needed for the overall FVTD scattering simulation, we choose the spatial discretization to be much smaller than that of the overall FVTD mesh. Cubic spline spatial interpolations and nearest-neighbor temporal interpolations are performed to find the corresponding field values in the elements (tetrahedrons) that make up the centroids of the FVTD mesh. The results of the incident-field propagation are calculated and stored in arrays in memory. They are potentially accessed only during the update scheme of the FVTD simulations, which checks to see whether the contrast between the background mesh (that which is seen by the incident field) and the scattered field mesh are nonzero. When they are nonzero, this indicates ) and a the presence of a contrast source (i.e., scattered field is generated at that particular mesh element. This means that results of the incident-field interpolations are only used when needed and they are not calculated and stored throughout the entire mesh, which would be inefficient. IV. NUMERICAL RESULTS AND DISCUSSION In this section, we calculate the scattered near-field values for a variety of objects located within a planarly layered medium. Canonical examples are chosen so that comparisons can be made with other published data in the literature. We have found that there is a dearth of examples that compute and provide graphical results for the near-field scattering from

objects within multilayered media, despite the fact that this problem is conceptually well known. In particular, we provide comparisons with the work in [5] (which provides a method based on the Born approximation) and [16] (which introduced a method based on the method of moments). The computational geometry was created by using a freeware mesh-generating program GMSH [17], which was also used to transform the physical description into a mesh for FVTD computations. The time function of the input waveform was a Gaussian derivative (15) where 1, 0.2 ns, and 70 ps. The constants in (15) were chosen so that sufficient energy would propagate at the frequency of interest (specifically 6 GHz). A. Scattering From a Dielectric Sphere For our first example, we consider a dielectric sphere in a free-space background. In this case, we are able to compare our results with the commercially available software program 4, FEKO. The permittivity value of the sphere is set to and the radius of the sphere is 0.1 . The geometry of the problem (for FVTD) is given in Fig. 3, with the sphere centered 0.25 . The results of our computations using FVTD at 0.5 , 0 , and 0.01 are given in and FEKO at Fig. 4. The excellent agreement between the results shows the validity of the scheme. B. Scattering From a Dielectric Cube For our next example, we consider a dielectric cube buried in a half-space medium. Both the half-space and the dielectric and cube are lossy, with permittivity values of . To calculate the conductivity that must be used in the FVTD simulations, we used a frequency of 6 GHz, which 0.03338 [S/m] and 0.01669 gives conductivities as [S/m]. Our results are normalized to the free-space wavelength, as in [5]. The geometry of the problem is given in Fig. 5, and the 0.1 are given in Fig. 6. In results of our computations at comparison with [5], our results are slightly higher; however, in comparison with [16], our results are very similar. For example,

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Fig. 7. Geometry for a lossy dielectric box buried in a lossy half-space. The box dimensions are 0.6 , 0.2  and 0.066  for the x, y , and z dimensions, respectively. Fig. 4. Scattered electric fields for a dielectric sphere in free space, centered at z 0.25. FVTD results are compared with FEKO.

=0

Fig. 5. Geometry for a lossy dielectric cube buried in a lossy half-space.

Fig. 8. Scattered electric fields for various incidence angles for a lossy dielectric box buried in a lossy half-space.

C. Scattering From a Dielectric Box To show the variation of the scattered electric fields as a function of incidence angle, we consider another one of the examples presented in [5]. In this case, the buried object is a rectangular box, with a size given by 0.6 , 0.2 , and 0.066 for the , , and dimensions, respectively. The geometry of the problem is given in Fig. 7, and the results of our computations are given in Fig. 8. For , the peak of the scattered field has shifted toward the specular direction, and the magnitude of the peak decreases as the incidence angle increases. This is similar to the result observed in [5], although our magnitudes are slightly different because those in [5] are obtained by using an approximate technique. Fig. 6. Scattered electric fields for a lossy dielectric cube buried in a lossy half-space at z 0.1. FVTD results are compared with [16] (Cui, MoM) and [5] (Hill, Born Approximation).

=

our peak value at 0.2 is 0.002, while [5] reports 0.0017 and [16] reports 0.002. The difference is associated with the error in using the Born approximation [16]. The decay of the curve appears to be in agreement as well.

D. Scattering From a Dielectric Slab in a Half-Space The previous examples considered were weak scatterers (i.e., the dielectric contrast is not very large). For this example, we consider a stronger scatterer, following the examples in [16]. Both the half-space and the dielectric slab are lossy, with permittivity values of and . At the frequency of 6 GHz, the conductivities are 0.16689 [S/m] and 1.6689 [S/m]. The geometry of the problem is given in

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Fig. 9. Geometry for a lossy dielectric slab buried in a lossy half-space. Slab dimensions are 0.3 , 0.3 , and 0.05  for the x, y , and z dimensions, respectively. Fig. 11. Scattered electric fields for a lossy dielectric slab buried in a lossy half-space. FVTD results are normalized to [16] (Cui, MoM).

Fig. 10. Scattered electric fields for a lossy dielectric slab buried in a lossy half-space. FVTD results are compared with [16] (Cui, MoM).

Fig. 9, and the results of our computations are given in Fig. 10. In comparison with [16], our results are similar, but slightly lower, similar to the example of the dielectric cube. For example, our 0.002 is 0.081, while [16] reports 0.092 peak value at 0, yielding a relative error of about 12%. As another exat 0.01 is 0.0599, while [16] reports ample, our peak value at 0.01 , yielding a relative error of about 9%. We 0.066 at consider this relative error to be acceptable considering the differences in the computational methods. As an additional test, we rescaled our simulation results to the peak value of the result at 0, and these results are presented in Fig. 11. It is clear that our results match up extremely well with those of [16] as long the as they are normalized. E. Scattering From a Dielectric Slab in Multilayered Media As a modification to the dielectric slab example, we consider the same slab buried under a lossy dielectric layer. Again, the half-space and the dielectric slab are lossy, with permittivity values of and , with the lossy layer of . The conductivity of the lossy layer is 0.66756 [S/m]. The geometry of the problem is given in Fig. 12, and the results of our computations are given in Fig. 13. In comparison with [16], our results are very similar. For ex-

Fig. 12. Geometry for a lossy dielectric slab buried in a lossy half-space beneath another lossy layer. Slab dimensions are 0.3 , 0.3 , and 0.05  for the x, y , and z dimensions, respectively.

0.002 is 0.0387, while the corample, our peak value at responding peak value reported in [16] is 0.042, yielding a relative error of about 8%. As another example, our peak value at 0.01 is 0.0283, while [16] reports 0.031 corresponding to the same height above the surface, yielding a relative error of about 9%. We consider this to be acceptable considering the differences in the computational methods. Again, as an additional test, we rescaled our simulation results to the peak value of the 0, and these results are presented in Fig. 14. This result at scale factor was the same value as in the previous example (slab in a half-space). It is clear that our results match up extremely well with those of [16] as long as they are normalized. Since the slab buried in a half-space and the slab buried in a multilayered medium had the same scale factor, we hypothesize that this is a constant difference between the two methods. The exact nature of the difference cannot be determined at this point; however, we are confident that our methodology is sound due to our accurate comparison with the sphere in free space using FEKO. F. Scattering From Dielectric Spheres In most of our previous examples, we used shapes with a cubic geometry, yet in our introduction, we suggested that one of

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Fig. 15. Geometry for dielectric spheres buried in a half-space. Sphere radii 0.25. are 0.1, separation is 1.0 , and sphere centers are z

=0

Fig. 13. Scattered electric fields for a lossy dielectric slab buried in a lossy multilayered medium. FVTD results are compared with [16] (Cui, MoM).

Fig. 16. Scattered electric fields for spheres buried in a half-space. Solid line: simultaneous scattering from both spheres, dotted line: superposition of sphere 1 and sphere 2 scattered fields. Fig. 14. Scattered electric fields for a lossy dielectric slab buried in a lossy multilayered medium. FVTD results are normalized to [16] (Cui, MoM).

the major benefits of using the FVTD method versus the FDTD method was the ability to have a mesh conform to an irregular surface. In this example, we present the scattering for multiple spheres buried in a dielectric half-space. This type of problem is very common in scattering simulations of geophysical media, where the subsurface can be populated with regions of dielectric discontinuity (for example, brine pockets in sea ice). The proximity of spheres is also an issue in studies involving the homogenization of random media [18]. The geometry of the computational domain for the case of two spheres is shown in Fig. 15. We simulated the scattering from both spheres simultaneously, from sphere 1 only (the left-hand sphere in Fig. 15), and from sphere 2 only (the right-hand sphere in Fig. 15). The simulation results are presented in Fig. 16, where we have also plotted the superposition of the scattering from sphere 1 and sphere 2. It is clear from the plotted results that multiple spheres must be simulated simultaneously since an attempt to approximate the scattering by superposition does not apply when the spheres are in close proximity. The importance of the proximity effect was also discussed in [19], particularly with regards to discrete

modeling in remote sensing studies. Since our future work includes modeling electromagnetic scattering for remote sensing, it is important that we should examine and consider these effects before embarking on such modeling studies.

V. CONCLUSION Through incorporating the SF formulation for multilayered media in an FVTD engine, an efficient method of modeling complex layered media has been developed. In this paper, we have presented details of the method used to calculate electromagnetic scattering from objects buried in multilayered media, which has a wide range of potential applications. Our method is capable of calculating the scattering from multiple objects with a minimal increase in the number of unknowns in the computation. Comparisons with other published data in the literature provided good agreement, giving us confidence in the FVTD implementation that we have developed. In our future research, we intend to use the proposed method for modeling electromagnetic scattering from geophysical media, with an application to remote sensing studies.

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ACKNOWLEDGMENT The authors would like to thank V. Okhmatovski for the use of his computing cluster. REFERENCES [1] S. V. Nghiem, R. Kwok, S. H. Yueh, and M. R. Drinkwater, “Polarimetric signatures of sea ice 1. theoretical model,” J. Geophys. Res., vol. 100, no. C7, 1995. [2] M. El-Shenawee, “Polarimetric scattering from two-layered two-dimensional random rough surfaces with and without buried objects,” IEEE Trans. Geosci. Remote Sens., vol. 42, no. 1, pp. 67–76, Jan. 2004. [3] C. D. Moss, F. L. Teixeira, Y. E. Yang, and J. A. Kong, “Finite-difference time-domain simulation of scattering from objects in continuous random media,” IEEE Trans. Geosci. Remote Sens., vol. 40, no. 1, pp. 178–186, Jan. 2002. [4] T. J. Cui and W. C. Chew, “Fast evaluation of sommerfeld integrals for electromagnetic scattering and radiation by three-dimensional buried objects,” IEEE Trans. Geosci. Remote Sens., vol. 37, no. 2, pp. 887–900, Mar. 1999. [5] D. Hill, “Electromagnetic scattering by buried objects of low contrast,” IEEE Trans. Geosci. Remote Sens., vol. 26, no. 2, pp. 195–203, Mar. 1988. [6] K. Demarest, R. Plumb, and Z. Huang, “FDTD modeling of scatterers in stratified media,” IEEE Trans. Antennas Propag., vol. 43, no. 10, pp. 1164–1168, 1995. [7] A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 2nd ed. Norwood, MA: Artech House, 2000. [8] P. Bonnet, X. Ferrieres, P. L. Michielsen, and P. Klotz, Finite Volume Time Domain Method, S. M. Rao, Ed. San Diego, CA: Academic Press, 1999. [9] D. Firsov, J. LoVetri, I. Jeffrey, V. Okhmatovski, C. Gilmore, and W. Chamma, “High-order FVTD on unstructured grids using an objectoriented computational engine,” . Proc. Appl. Comput. Electromagn. Soc., vol. 22, no. 1, pp. 71–82, 2007. [10] I. Capoglu and G. Smith, “A total-field/scattered-field plane-wave source for the FDTD analysis of layered media,” IEEE Trans. Antennas Propag., vol. 56, no. 1, pp. 158–169, Jan. 2008. [11] Y. Fang, L. Wu, and J. Zhang, “Excitation of plane waves for FDTD analysis of anisotropic layered media,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 414–417, 2009. [12] T. Tan and M. Potter, “FDTD discrete planewave (FDTD-DPW) formulation for a perfectly matched source inTFSF simulations,” IEEE Trans. Antennas Propag., vol. 58, no. 8, pp. 2641–2648, Aug. 2010. [13] D. Isleifson, L. Shafai, J. LoVetri, and D. G. Barber, “On the development of a scattered-field formulation for objects in layered media using the FVTD method,” presented at the Int. Rev. Progr. Appl. Comput. Electromagn. Symp., Williamsburg, VA, 2011. [14] K. Sankaran, C. Fumeaux, and R. Vahldieck, “Cell-centered finite-volume-based perfectly matched layer for time-domain Maxwell system,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 3, pp. 1269–1276, Mar. 2006. [15] K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag., vol. 14, no. 3, pp. 302–307, May 1966. [16] T. J. Cui, W. Wiesbeck, and A. Herschlein, “Electromagnetic scattering by multiple three-dimensional scatterers buried under multilayered media. II numerical implementations and results,” IEEE Trans. Geosci. Remote Sens., vol. 36, no. 2, pp. 535–546, Mar. 1998. [17] C. Geuzaine and J.-F. Remacle, “Gmsh: A three-dimensional finite element mesh generator with built-in pre- and post-processing facilities,” Int. J. Num. Meth. Eng., vol. 79, no. 11, pp. 1309–1331, 2009. [18] D. Isleifson and L. Shafai, “Numerical homogenization of heterogeneous media using FVTD simulations,” in Proc. 14th Int. Symp. Antenna Technol. Appl. Electromagn. Amer. Electromagn. Conf., 2010, pp. 1–4. [19] K. Sarabandi and P. Polatin, “Electromagnetic scattering from two adjacent objects,” IEEE Trans. Antennas Propag., vol. 42, no. 4, pp. 510–517, Apr. 1994.

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Dustin Isleifson (S’01) received the B.Sc. degree in electrical engineering (Hons.) from the University of Manitoba, Winnipeg, MB, Canada, in 2005, where he is currently pursuing the Ph.D. degree in electrical engineering. He conducted research in the Canadian Arctic through ArcticNet and the Circumpolar Flaw Lead System Study (CFL). He held a Canadian NSERC Canada Graduate Scholarship CGS-M in 2006 and currently holds an NSERC Canada Graduate Scholarship CGS-D3. His current research interests are in the areas of microwave remote sensing, computational electromagnetics, and arctic science.

Ian Jeffrey (M’11) received the B.S. degree in computer engineering (Hons.), and the M.S. and Ph.D. degrees in electrical and computer engineering from the University of Manitoba, Winnipeg, MB, Canada, in 2002, 2004, and 2011, respectively. In 2008, he was an Intern at Cadence Design Systems, Inc., Tempe, AZ, where he worked with the Department of Custom Integrated Circuits Advanced Research and Development. He currently holds a MITACS industrial postdoctoral fellowship with the National Research Council Canada’s Institute for Biodiagnostics (NRC-IBD). His current research interests include finite-volume and discontinuous galerkin time-domain methods, fast algorithms for computational electromagnetics, high-performance computing, and inverse problems.

Lotfollah Shafai (LF’07) received the B.Sc. degree in electrical engineering from the University of Tehran, Tehran, Iran, in 1963 and the M.Sc. and Ph.D. degrees in electrical engineering from the Faculty of Applied Sciences and Engineering, University of Toronto, Toronto, ON, Canada, in 1966 and 1969, respectively. In 1969, he joined the Department of Electrical and Computer Engineering, University of Manitoba as a Sessional Lecturer, and became Assistant Professor in 1970, Associate Professor in 1973, and Professor in 1979. Since 1975, he has made special effort to link the university research to the industrial development by assisting industries in the development of new products or establishing new technologies. To enhance the University of Manitoba contact with industry, in 1985 he assisted in establishing “The Institute for Technology Development,” and was its Director until 1987, when he became the Head of the Electrical Engineering Department. His assistance to industry was instrumental in establishing an Industrial Research Chair in Applied Electromagnetics at the University of Manitoba in 1989, which he held until 1994. He has been a participant in nearly all Antennas and Propagation symposia and participates in the review committees. In 1986, he established the symposium on Antenna Technology and Applied Electromagnetics (ANTEM) at the University of Manitoba that is currently held every two years. Dr. Shafai has been the recipient of numerous awards. In 1978, his contribution to the design of a small ground station for the Hermus satellite was selected as the 3rd Meritorious Industrial Design. In 1984, he received the Professional Engineers Merit Award and in 1985, “The Thinker” Award from Canadian Patents and Development Corporation. From the University of Manitoba, he received the “Research Awards” in 1983, 1987, and 1989; the Outreach Award in 1987; and the Sigma Xi, Senior Scientist Award in 1989. In 1990, he received the Maxwell Premium Award from the Institute of Electrical Engineers (London) and in 1993 and 1994, the Distinguished Achievement Awards from the Corporate Higher Education Forum. In 1998, he received the Winnipeg RH Institute Foundation Medal for Excellence in Research. In 1999 and 2000, he received the University of Manitoba, Faculty Association Research Award. He was elected a Fellow of The Royal Society of Canada in 1998. He was a recipient of the IEEE Third Millennium Medal in 2000 and in 2002, was elected a Fellow of The Canadian Academy of Engineering and Distinguished Professor at The University of Manitoba. In 2003, he received an IEEE Canada “Reginald A. Fessenden Medal” for “Outstanding Contributions to Telecommunications and Satellite Communications” and a Natural Sciences and Engineering Research Council (NSERC) Synergy Award for the “Development of Advanced Satellite and Wireless Antennas.” He holds a Canada Research Chair in Ap-

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plied Electromagnetics and was the International Chair of Commission B of the International Union of Radio Science (URSI) for 2005-2008. In 2009, he was elected a Fellow of the Engineering Institute of Canada, and was the recipient of an IEEE Chen-To-Tai Distinguished Educator Award. In 2011, he received a Killam Prize in Engineering from The Canada Council for the Arts, for his “outstanding Canadian career achievements in engineering, and his work in antenna research.” He is a member of URSI Commission B and was its chairman during 1985-1988.

Joe Lovetri (SM’09) received the B.Sc. (Hons.) and M.Sc. degrees in electrical engineering from the University of Manitoba, Winnipeg, MB, Canada, in 1984 and 1987, the Ph.D. degree in electrical engineering from the University of Ottawa, Ottawa, ON, in 1991, and the M.A. in philosophy from the University of Manitoba in 2006. From 1984 to 1986, he was an EMI/EMC Engineer at Sperry Defence Division, Winnipeg, MB, Canada, and from 1986 to 1988, he held the position of TEMPEST Engineer at the Communications Security Establishment, Ottawa. From 1988 to 1991, he was a Research Officer at the Institute for Information Technology of the National Research Council of Canada. His academic career began in 1991 when he joined the Department of Electrical and Computer Engineering, The University of Western Ontario, London, where he remained until 1999. In 1997–1998, he spent a sabbatical year at the TNO Physics and Electronics Laboratory, The Netherlands, conducting research in time-domain computational methods and ground penetrating radar. In 1999, he joined the University of Manitoba, where he is currently a Professor in the Department of Electrical and Computer Engineering. From 2004 to 2009, he was the Associate Dean (Research and Graduate Programs) for the Faculty of Engineering. His main research interests lie in the areas of time-domain computational electromagnetics, modeling of electromagnetic-compatibility problems, inverse problems, and biomedical imaging.

David G. Barber received the B.Sc. and M.Sc. degrees from the University of Manitoba, Winnipeg, MB, Canada, in 1981 and 1987, respectively, and the Ph.D. degree from the University of Waterloo, Waterloo, ON, Canada, in 1982. He is a Canada Research Chair (CRC) Professor of Environment and Geography and Canada Research Chair in Arctic System Science with the Clayton H. Riddell (CHR) Faculty of Environment Earth and Resources, University of Manitoba. He was appointed to a faculty position at the University of Manitoba in 1993 and received a CRC in Arctic System Science in 2002. Currently, he is Director of the Centre for Earth Observation Science and Associate Dean (Research), CHR Faculty of Environment, Earth and Resources, University of Manitoba. He has extensive experience in the examination of the Arctic marine environment as a “system,” and the effect climate change has on this system. He has published many articles in the peer-reviewed literature pertaining to sea ice, climate change, and physical-biological coupling in the Arctic marine system. He led the largest International Polar Year (IPY) project in the world, known as the circumpolar flaw lead (CFL) system study. He is recognized internationally through scientific leadership in large network programs (e.g., NOW, CASES, ArcticNet, the Canadian Research Icebreaker (Amundsen), and CFL), as an invited member of several Natural Sciences and Engineering Research Council (NSERC) national committees (e.g., NSERC GSC 09; NSERC IPY, NSERC northern supplements, etc.), international committees (GEWEX, IAPP, CNC-SCOR, IARC, etc.), and invitations to national and international science meetings (e.g., American Geophysical Union (AGU), Canadian Meteorological and Oceanographic Society (CMOS), American Meteorological Society (AMS), American Society for Limnology and Oceanography (Spain), IMPACTS (Russia), European Space Agency (ESA, Italy), Arctic Frontiers (Norway), etc). He currently supervises nine M.Sc. students; nine Ph.D. students, four postdoctoral fellows, and nine full-time research staff.

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Analysis of Seismic Electromagnetic Phenomena Using the FDTD Method Yi Wang and Qunsheng Cao

Abstract—Anomalous electromagnetic (EM) phenomena of the ultra-low frequency (ULF) and the extremely low frequency (ELF) prior to or during earthquakes are analyzed using the geodesic finite-difference time-domain (FDTD) algorithm and accurate geological EM data. Firstly, the underground EM eruption caused by the earthquake physical process is simulated, and its effects on the surface EM field are analyzed.Based on a series of simulations, a new explanation to the former observed ULF and ELF anomalies prior to and during earthquakes is proposed. Thereafter, to explain the enhancement of the Schumann resonances (SRs) observed in recent years, localized D-region ionospheric anomalies and background lightning noises are added simultaneously into the earthquake EM simulations, and the results are discussed. Finally, the general rules of seismic EM phenomena in the frequency range 0–40 Hz are summarized, and potential earthquake prediction methods are suggested. Index Terms—Earthquake precursors, extremely low frequency (ELF), finite-difference time-domain (FDTD), ionosphere anomaly, Schumann resonances (SRs).

I. INTRODUCTION

A

NOMALOUS electromagnetic (EM) phenomena prior to or during earthquakes, especially large earthquakes, have been observed and reported by many researchers over the years [1]–[4]. These phenomena include the substantial increase in background EM noise in the different frequency ranges: the ultra-low frequency (ULF: 0–3 Hz) and the extremely low frequency (ELF: 3–3000 Hz) [1], [5], [6]; the anomalous propagation behavior of EM waves in the very low frequency range (VLF: 3–3000 kHz) [7]; the anomalous enhancement of the Schumann resonances (SRs) [8], [9]; and so on [10]. In recent years, many observation stations have been established in the hope of detecting precursors of earthquakes. However, researchers are still confronted by much difficulty in understanding the process of these EM anomalies, which blocks the study of potential earthquake prediction. Some theories have been proposed to explain these anomalous seismic EM phenomena [2], [11]–[14]. In the 1990s,

Manuscript received November 08, 2010; revised April 25, 2011; accepted May 01, 2011. Date of publication August 12, 2011; date of current version November 02, 2011. This work is supported by the Natural Science Foundation of Jiangsu Province (China) under Grant BK2009368 and the Innovation Fund Project for Graduate Student of Jiangsu Province (China) under Grant CX09B_080Z. The authors are with the College of Electronic and Information Engineering, Nanjing University of Aeronautics and Astronautics (NUAA), Nanjing 210016, China (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2164204

Fenoglio [11] and Majaeva [12] separately modeled electrokinetic currents caused by compartment failure to explain the behavior of the anomalous precursory EM data reported in [5]. Thereafter Simpson and Taflove [13], and Simpson [14] reported a more rigorous analysis based on the former theory. By using the finite-difference time-domain (FDTD) [15], [16] algorithm, they studied the magnetic field at the Earth’s surface excited by underground electric currents and proposed theories to explain the former observations [5]. Pulinets [2] introduced a seismo-ionospheric coupling model in detail, and he pointed out the possible physical mechanism for the electron density changes in the ionosphere prior to or during earthquakes. These ionospheric changes have been observed during many earthquakes (such as the Wenchuan earthquake in China) [2], [17], [18]. Although many researchers have proposed different theories on this subject, a fully rigorous description of the seismic EM phenomena has yet to be completed, and the existence of some pre-seismic EM signatures is still very controversial. Before the 1980s, most methods in solving the global EM phenomena were based on the mode theory or the transmission line theory [19], [20]. In 1983, Holland [21] initially introduced a time-domain solution, based on the FDTD algorithm and spherical geometry, to solve the EM propagation problems in a sphere. In his model, the sphere was divided into grid cells using the artificial longitude and latitude lines. From the 1990s, several papers have reported the application of this technology to simulate global ELF EM propagation problems [13], [14], [22]–[32]. Among these works, Cummer [27], [28] has discussed the EM propagation in small parts of the Earth-ionosphere system in the VLF and ELF ranges, and his simulations were based on the traditional FDTD algorithm. From 2002, Simpson and Taflove have been initially applying and developing the idea of Holland [21] to solve ELF propagation problems in the Earth-ionosphere system [13], [14]. They have also developed a geodesic FDTD model [16], [23], which has many advantages [16], [33] over the former [21]. Thereafter, they have applied the two models to application studies, including hypothesized earthquake precursors, remote sensing of ionospheric disturbances, and detection of deep underground resource formations [13], [14], [22]–[26]. In [14], Simpson and Taflove employed the three-dimensional (3D) latitude-longitude FDTD model to investigate the anomalous pre-seismic EM signals observed during the Loma Prieta earthquake [5]. In their experiments, they monitored surface wave propagation, which is excited by electric current sources buried at various depths and with different orientations in the Earth’s crust. They found that EM fields recorded at the Earth’s surface, which are due to any hypothesized electric currents occurring in the Earth’s crust at 5 km from the surface or deeper, will

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only have significant spectra below 1 Hz [19]. In [25], they first simulated the localized D-region (60 km to 90 km above the Earth’s surface) ionospheric anomalies using the geodesic FDTD model and proposed an ELF radar system according to the simulation results. Recently, Simpson [19] has summarized the recent development of the FDTD algorithm for solving the ELF problems in detail. We intend to establish a time-domain model of the seismic EM phenomena in the ULF and the low ELF ( 40 Hz) range using the FDTD algorithm [15], [16]. Because of its simplicity and flexibility in treating medium with arbitrary inhomogeneities, the FDTD algorithm provides straightforward time-domain description of the EM wave propagation process. For this reason, the FDTD algorithm is considered to be the technique of the future [19], [28], for solving ELF EM problems. We apply the geodesic FDTD algorithm proposed by Simpson [19] to simulate the possible ELF EM phenomena in the Earth-ionosphere system and use the simulation results to discuss and explain the observed ELF EM anomalies during or prior to earthquakes. The paper is organized as the follows. In Section II, the geodesic FDTD algorithm and simulation details are briefly introduced. In Section III, the two major seismic EM phenomena in the frequency range 0–40 Hz, including the substantial increase in the background noise and the SR enhancement, are studied by simulating underground sources and ionospheric anomalies with detailed environmental parameters. Then, the relations between these anomalous EM phenomena and the possible physical processes, such as earthquake locations, lightning activities, electron density changes, and the observation locations, are studied in detail. Finally, we draw conclusions and discuss possible future works in this field. II. EM MODEL AND SIMULATION PARAMETERS A. 3D FDTD Model We apply the 3D geodesic FDTD algorithm proposed by Simpson and Taflove [16], [19], [23] in this work. In the geodesic FDTD algorithm, the Earth-ionosphere system model is constructed by the alternating planes of transverse-magnetic (TM) and transverse-electric (TE) field components, which are composed of triangular cells and hexagonal cells (including 12 pentagonal cells), respectively. Fig. 1 illustrates only the model of the surface plane consisting of hexagonal cells and pentagonal cells. In the present 3D study, the model has a total of 10230 hexagonal cells and 12 pentagonal cells comprising the TM plane (20480 triangular cells comprising the TE plane) in the horizontal direction (tangential direction of the Earth’s surface), and the average distance between adjacent cells (nodes) is about 250 km; in radial (vertical direction), the model extends to a depth of 100 km into the lithosphere and to an altitude of 100 km above sea level, and the average distance between adjacent planes is 5 km. It is worth noting that the resolution used here is not very high; however, this resolution can provide high precision for EM waves with a wavelength more than 5000 km (20 times the cell size) or with a frequency below 60 Hz, which is sufficient for the study of EM phenomena in the ULF and the low ELF ( 40 Hz) range.

Fig. 1. The geodesic FDTD model of the Earth’s surface mapped with topographic data (at sea level, tangential resolution is 250 km per grid, east hemisphere). Only the sphere comprised of hexagonal grid-cells and 12 pentagon grid-cells are illustrated.

B. EM Parameters in the Earth-Ionosphere System To describe the Earth-ionosphere system with sufficient EM parameters, topographic data generated by the GEODAS Grid Translator from the NOAA-NGDC [34] are used. Fig. 1 illustrates the geodesic FDTD model of the Earth’s surface mapped with topographic data at sea level; the color differences stand for the topographies with different EM parameters. In the present study, the Earth’s crust is supposed to fill with granite (top of the crust) and basalt; an average relative permittivity of 6 is used [35]. The same treatment is used for the rest of the lithosphere (the uppermost mantle) because EM waves decay very fast in the lossy medium. For the seas and oceans, an average relative permittivity of 80 is used [35]. The conductivity profile in different locations plays a key role in absorbing and reflecting EM waves in the Earth-ionosphere system. In this work, the conductivity value depends on both the location and altitude of the FDTD grid. For the lithosphere, the conductivity profiles are assigned according to [36]; for the ionosphere, the data are obtained from the International Reference Ionosphere (IRI) model [37], [38] to account for a realistic conductivity distribution. The method for relating the vertical profile of the ion number density and electron number density of the ionosphere to the conductivity profile is described in [39], [40]. A perfect electric conductor (PEC) boundary condition is used both at the bottom of the lithosphere and at the top of the ionosphere D region (about 95–100 km above the Earth’s surface); the reasons for these assumptions are explained in [14]. C. Underground Current Sources and Background Lightning Noises Electric current sources are believed to arise due to the seismic physical process prior to or during earthquakes [1]. Both analytical and numerical models [11]–[14] are proposed to

WANG AND CAO: ANALYSIS OF SEISMIC ELECTROMAGNETIC PHENOMENA USING THE FDTD METHOD

Fig. 2. Comparisons of the time-domain waveforms of the surface horizontal magnetic field excited by currents at different depths (all values are normalized).

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Fig. 3. Comparisons of the frequency-domain waveforms of the surface horizontal magnetic field excited by currents at different depths (all values are normalized).

TABLE I RELATION OF SOURCE DEPTH AND SURFACE FIELD PARAMETERS

III. RESULTS A. Surface Response of Underground Current Sources

explain the behavior of the precursory anomalous EM observation data [5]. In the present study, the currents [11]–[14] buried underground are used to simulate real seismic EM sources. Note that those currents with frequencies higher than ELF are ignored as they decay very fast in the Earth’s lossy crust. During earthquakes, many other sources also have effects on the surface EM observation data, such as global lightning activities, the variety of artificial EM signals, ionospheric interferences, and so on. In the present study, are considered to have the most important effects on the earthquake EM observation because these activities are continuous and possess tremendous power. According to [32], [39], [41], lightning currents from the three lightning centers (Southeast Asia, Africa, and South America [32]) are modeled as background noises. The form of the lightning currents is described in [32]. D. Localized D-Region Ionospheric Anomalies Ionospheric anomalies have many times been observed [2], [17], [18] and proposed to be precursors of earthquakes [2]. In the present study, localized D-region ionospheric anomalies, which include the increase and decrease in the ion number density and electron number density, are simulated and discussed to possibly explain the observed anomalous SR enhancement [8], [9]. To quantitatively simulate the ionospheric anomalies occurring 0–100 km above the Earth’s surface, the ion and electron number densities are adjusted by 20 per cent; this assumption was also used in [25]. Additionally, because the observed ionospheric anomalies only appear in a small area near the earthquake center (the exact measure of which is yet to be determined), we assume that these phenomena are restricted to the earthquake preparation zone [2].

A substantial increase in the background noise in the ULF and low ELF range is observed at the Earth’s surface prior to or during earthquakes in different places [1], [5], [6]. Underground EM sources triggered by seismic physical process are suggested to be the most possible reasons for this phenomenon [11]–[13]. To study the surface EM field response, a horizontal electric current source is buried deep underground to simulate the seismic EM source and then observed at the Earth’s surface. The source is assumed to linearly increase to its maximum value at 900 time steps (about 0.015 s) and to linearly decrease to zero at 1800 time steps (about 0.03 s). In this simulation, the sources are located in a continental environment and are buried at depths of 7.5 km, 12.5 km, and 17.5 km, respectively. Environmental EM parameters are assigned according to the previous section, but the ionospheric anomalies and the lightning noises are not added into this simulation. Fig. 2 depicts the distribution of the surface magnetic field in the time domain, which is excited by underground currents at different depths. The amplitude of each wave is normalized to its maximum value to give a better view of the attenuation time. The results in Fig. 2 clearly show the relation between the source depth and the period of the surface magnetic wave. Table I lists the numerical comparison of these periods (the amplitude of the wave attenuates to 10% of its maximum amplitude). The results shown in this table prove that the attenuation rate of the surface response waveform is directly related to the depth of the underground source (a faster attenuation rate of the surface waveform implies a deeper source). Similar results were also proposed in [14]. Fig. 3 shows the waveforms of the surface response in the frequency domain. In the figure, the surface responses excited by sources at different depths almost coincide after normalization. It is noted, however, that the attenuation rate of waveforms in Fig. 3 is very small compared to the results reported in [14]. The reasons for this phenomenon will be discussed in the next section. According to these results, the surface EM fields excited by underground electric currents in the continental area

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Fig. 4. The time-domain waveforms of the surface horizontal magnetic field simulated under different conditions: (A) without ionospheric anomaly; (B) with ionospheric anomalies; (C) with ionospheric anomalies and underground current source. The observation location is 250 km away from the earthquake center. Lightning noises are included in all three cases.

Fig. 5. The frequency-domain waveforms of the surface horizontal magnetic field simulated under different conditions: (A) without ionospheric anomaly; (B) with ionospheric anomalies; (C) with ionospheric anomalies and underground current source. The observation location is 250 km away from the earthquake center. Lightning noises are included in all three cases. Note that line B and line C almost coincide. Frequency range is 5–40 Hz.

will have significant frequency spectra in the low ELF range. In real earthquake observations, such EM surface responses often last for a long time (even several days) because of the accumulation of abundant underground currents. B. Effects of Ionospheric Anomalies and Background Lightning Noises Anomalous Schumann resonance phenomena are observed prior to or during many earthquakes [8], [9]. These phenomena include 1) the enhancement of some SR harmonics and 2) a frequency shift of SRs in the peak frequency. In the present study, we propose a research model to relate these phenomena to the widely observed ionospheric anomalies [2], [17], [18] and the seismic EM sources. In order to prove our assumption, we use not only the real world geometry and the real ionosphere conductivity profile but also lightning noises from the three lightning centers [32], [39], [41], to assure the correctness and accuracy of results. The ionospheric anomalies are placed only in the earthquake propagation zone [2] (the size of which changes with the earthquake magnitude; in this case, the radius of the area is ). 840 km, corresponding to earthquake magnitude The ionospheric conductivity value is increased by 20 percent (this assumption is also used in [25]). Fig. 4 shows the waveform of horizontal magnetic field in the time domain under different conditions. Lightning noises are included in all three cases, and the maximum amplitudes of both lightning noises and underground sources are assumed to be equal for simplicity. Comparisons of the line A (with a normal ionosphere) and the lines B and C (with ionospheric anomalies) clearly show the surface field enhancement due to the existence of the ionospheric anomalies. This enhancement can also be used to explain the observed background ELF noises during earthquakes. Though the underground current source is added in the simulation (line C), it has no obvious effect on the surface EM waveform. The reason for this is that the underground EM waves have to go through long distances in the lossy medium (the Earth’s crust) before it reaches the Earth’s surface.

Fig. 6. Frequency domain waveforms of the surface horizontal magnetic field under different conditions: (A) with no anomalous ionosphere, (B) with the anomalous ionosphere, (C) with the anomalous ionosphere and the underground current source. The observation place is 250 km away from the earthquake center. Lightning noises are included in all three cases. Frequency range is 0–5 Hz.

Fig. 5 plots the Fourier transform of the time-domain results shown in Fig. 4 (5 Hz to 40 Hz, in which the first 5 SRs lie). In the figure, the amplitude enhancements of the SR harmonics in line B and line C are clearly shown, which are simulated under conditions of ionospheric anomalies. However, no distinct frequency shift of SR peaks is observed for these results. It is noted that the results under conditions B and C almost coincide. We believe this is because in the frequency range 5–40 Hz, the effects of ionospheric anomalies are much larger than the effects of underground current sources. The effects of underground sources are obvious in Fig. 6, which shows the frequency-domain waveforms in the frequency range from 0 Hz to 5 Hz. Fig. 6 also clearly shows the power enhancement in the results of line C (in which underground sources are added to the simulation) in the ULF range ( 1 Hz), while no obvious power enhancements are observed in the results of lines A and B. Through Fig. 4 to Fig. 6, the following results are obtained: 1) the existences of ionospheric anomalies contribute much to

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the enhancement of the observed low ELF background noises and some of the SR harmonics; and 2) when assuming that the maximum amplitude of both the underground EM sources and the lightning noises are almost the same, the underground current sources do have effects on the surface response waveform, but only in the ULF range. It is noted that the peak frequency shifts of SRs are not observed in all the simulations. IV. DISCUSSION A. Observations of the 1989 Loma Prieta Earthquake The observations in [5] are considered to be the most prominent ones that prove the existence of seismic EM anomalies. Although a previous paper [42] has discussed the possible measurement error in [5], our work focuses on studying possible surface observation results in real seismic EM conditions. In [14], a time-domain solution was proposed to explain the observed anomalous eruption of ELF background noise during the 1989 Loma Prieta earthquake [5], and the differences between the simulation results and the observations were supposed to be caused by the existence of other types of seismic EM sources. In the present study, a more rigorous discussion is given to explain the observation data using only the seismic underground current sources caused by the electrokinetic effect. Fig. 7 and Fig. 8 show the time-domain and frequency-domain waveforms of the surface magnetic field, respectively. The current sources are placed in three different underground locations: at the seaside (line A), within the continent (line B), and in the deep sea (or ocean, line C). The depths of the sources are all 17.5 km, corresponding to the center of the Loma Prieta earthquake, and the source type is set according to [11], with periods less than 1 s. Fig. 7 clearly displays that the surface magnetic field lasted for several hundreds of seconds under conditions A and C (similar to each other). However, under condition B, the waveform only lasted for several tens of seconds. It is noted that in the experiments, only the first 30 seconds are simulated. Prony’s method has been used to extend the waveform to its ultimate decay. The differences among these simulation results are more evident in the frequency domain. From Fig. 8, it has been observed that the frequency spectra (0.01–10 Hz) of the surface magnetic field are largely different because of the different source locations: the surface field attenuates much slower when the sources are excited in the continental area than when they are excited at the seaside or in the sea (or ocean). It is noted that the attenuation rates (versus the frequency) of the seaside and the oceanic simulation results (line A and line C) at 10 Hz). The are similar to those reported in [14] (about observed attenuation rate (versus the frequency) of the surface response in the 1989 earthquake [5] (spectrum was provided in [14]) is faster than that in the continental simulation result (line B) but slower than in the seaside and the oceanic simulation results (line A and line C). According to these findings, the spectra of surface responses caused by underground sources can have significant effects above 1 Hz (line B in Fig. 8), which implies that the recorded anomalous EM data (below or above 1 Hz) of the 1989 Loma Prieta earthquake or of any earthquakes are possibly excited by underground current sources. The reasons for this are: 1) the real geometry conditions of the Loma Prieta

Fig. 7. The time-domain waveforms of the surface horizontal magnetic field excited by underground current source at depth 17.5 km: (A) the source is at the seaside, (B) the source is within the continent, (C) the source is in the sea (or ocean).

0

Fig. 8. The frequency-domain waveforms of the surface horizontal magnetic field excited by underground current source at depth 17.5 km: (A) the source is at the seaside, (B) the source is in the continent, (C) the source is in the sea (or ocean).

0

earthquake location are much like those between the assumed seaside condition and the assumed continental condition; 2) before the earthquake, more than one electric current has been excited at different depths and at different periods, which makes the observations at the Earth’s surface the accumulation of these excitations. B. Vertical and Horizontal Variations of the Magnetic Field Fig. 9 shows the vertical (or the radial direction of the Earth) variation of the maximum amplitude of the horizontal magnetic field. The sample interval is 5 km in the radial direction. In this simulation, the electric current source has been excited at 17.5 km below the earth’s surface. Fig. 9 shows clearly that the horizontal magnetic field excited by an underground source has a much larger transient amplitude underground than in the air, which means that the magnetic field anomalies are more obvious below the Earth’s surface. This phenomenon should be more obvious and much easier to observe during earthquakes, however, still much work have to be improved to make underground observations of EM field practical. It is noted that the magnetic amplitudes underground and in the air shown in Fig. 9

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Fig. 9. Distribution of the maximum magnetic field amplitude in the radial direction, source is a 17.5-km-deep electric current source.

Fig. 10. Distribution of the maximum magnetic field amplitude along the equator, source is excited underground (at 0 Mm in the figure).

do not appear at the same time; the magnetic field in the underground has a larger maximum amplitude but a shorter period because EM waves attenuate faster in the lossy underground than in the air. For the same reason, the amplitude of the horizontal magnetic field nearly does not change with altitude in the air. However, it changes rapidly with depth under the surface. Fig. 10 displays the variation of the maximum amplitude of the horizontal magnetic field along the equator. The sampling interval is 250 km. It has been found that as the distance from the source increases, the amplitude of the surface field decreases rapidly. According to this phenomenon, the amplitude of the magnetic field along the equator has become very small except for the field near the source location, which is the possible reason for many failured observations of seismic EM background noise anomalies. C. Vertical Distribution of Other Field Quantities Fig. 11 and Fig. 12 present the waveforms of the maximum amplitude of the electric field (E) and the magnetic field (H), respectively. At the same time, both vertical (r) and horizontal (t) field components are considered. In the experiment, the source is a horizontal electric current and has been excited underground. All the amplitudes of the surface field responses are taken as absolute values. From the figures, results similar

Fig. 11. Comparison of the maximum amplitude of horizontal magnetic field and radial magnetic field in the radial direction, source is a 17.5-kmdeep electric current source.

Ht

Hr

Et

Fig. 12. Comparison of maximum amplitude of horizontal electric field and radial electric field in the radial direction, source is a 17.5-km-deep electric current source.

Er

to those in the previous section have been obtained except for: 1) the discontinuity of the radial electric field near the altitude 0 (the Earth’s surface), which is because of the change in field directions (in the air, the vertical electric field points down to the ground); 2) the attenuations (versus height) of all field quantities are obvious in the ionosphere except for that of the horizontal magnetic field, which nearly does not attenuate. D. Effects of Underground Sources on SR Observations The above analyses indicate that the underground electric current sources have direct effects on the surface responses in the frequency range higher than 1 Hz, which brings in the possibility that the observed seismic SR (5–40 Hz) anomalies are results of the underground sources. In the present study, underground sources are excited under different conditions, and surface waveforms in the frequency range of 5–40 Hz have been obtained. It is noted that the SR frequencies are related to the distance from the observation point to the source, so different observation locations have been selected to assure the correctness of the results. In following simulations, the ionospheric conductivity has been increased by 20 percent (unless specified) in the earthquake preparation zone to simulate possible ionospheric anomalies prior to or during earthquakes.

WANG AND CAO: ANALYSIS OF SEISMIC ELECTROMAGNETIC PHENOMENA USING THE FDTD METHOD

Fig. 13. Comparison of the frequency-domain (5–40 Hz) waveforms (vertical electric field at surface, 250 km surface distance from source center) excited by underground sources at different depths with different ionosphere conditions: (A) the source is at 12.5 km with normal ionosphere; (B) the source is at 2.5 km with normal ionosphere; (C) the source is at 12.5 km with ionospheric anomalies; (D) the source is at 2.5 km with ionospheric anomalies.

0

0

0

0

Fig. 13 compares the frequency-domain waveforms of the vertical electric field observed at the Earth’s surface (250 km from the source center) under different ionospheric conditions and at different source depths. Note that the observed SR frequency values depend on the distance between the source and the observation stations, and these values are easily disturbed by noises. The distance and the noises can largely decrease the quality factor Q value of the observed SR frequencies. However, this does not upset the following discussions because we mainly focus on the power distribution of the field spectrum in the region of 5–40 Hz. In the figure, the results under condition B have the lowest power distribution, which is simulated with the deepest source and no ionospheric anomalies; the results under condition D have the highest power distribution, which is simulated with the shallowest source and anomalous ionospheric conditions; and the results under conditions B and C are between the above two cases. These results prove that the power of the observed SR frequencies (not only the harmonics but also the frequencies near them) near the earthquake center can be enhanced by either the underground sources or the existence of anomalous ionospheric conditions, and when the underground sources are excited, the shallower ones provide more power enhancements. It is noted that the power of the underground sources attenuates very fast in the lossy crust, and the effects of the source power on the surface field are not very obvious in the previous experiment. The horizontal magnetic field at the surface (375 km away from the source center), which is depicted in Fig. 14, is also excited by the underground current sources but with a different source power. In the figure, the frequency-domain waveforms almost coincide under the same ionospheric condition (the results under conditions A and B almost coincide, as do the results under conditions C and D) despite the source power differences. This means the influence of the underground source power change on the SR frequencies is not obvious. Note that distinct power enhancements are also observed under the condition of ionospheric anomalies.

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Fig. 14. Comparison of the frequency-domain (5–40 Hz) waveforms (horizontal magnetic field at surface, 375 km surface distance from source center) excited by underground sources with different powers and different ionosphere conditions: (A) 100 times of unit source power, normal ionosphere; (B) unit source power, normal ionosphere; (C) 100 times of unit source power, anomalous ionosphere; (D) unit source power, anomalous ionosphere. Notice line A and line B almost coincide; line C and line D almost coincide.

The above experiments reveal that the underground sources, which occur during earthquake preparing processes, affect the surface EM field in the SR frequency range. Their effects are related to the source depth, the ionospheric conditions, and the observation locations. The power distribution of the surface field is directly affected by the depth of the underground source but not by its power. Despite these, the changes in conductivity profiles in part of the global ionosphere also enhance the spectra in the ULF and low ELF range, especially the SR harmonics. It is noted that the frequency shifts observed in [8], [9] are not observed in our simulations. E. Effects of Lightning Source on SR Observations In the assumption of Fenoglio et al. [11], the generation of underground sources was limited to a series of geologic conditions that were hard to reach in some earthquake preparing processes. At the same time, some factors, such as ionospheric processes or lightning, were also possible causes of the excitation of anomalous seismic EM phenomena during earthquake processes. In the present study, to examine the effects of lightning on field observations at the Earth’s surface, lightning currents are added as additional sources at the center of the earthquake zone. Also, lightning currents from the three lightning centers of the world are added and considered as background noises. Fig. 15 shows the frequency-domain waveforms of the surface vertical electric field observed 500 km away from the source center. It is noted that lightning currents significantly increase the SR power under both ionospheric conditions (lines B and D), and the effects of the ionospheric anomalies on the enhancement of the SR frequencies are very obvious. The results also show that some SR frequencies are excited under the condition of ionospheric anomalies (line D) rather than under the condition of a normal ionosphere (line B). F. Long-Distance Observation of the SR With Ionospheric Anomalies In previous discussions, all observation locations are within the earthquake preparation zone. The frequency-domain wave-

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Fig. 15. Comparison of the frequency-domain (5–40 Hz) waveforms (vertical electric field at surface, 500 km surface distance from source center) excited by different type sources and with different ionosphere conditions: (A) only noises, normal ionosphere; (B) lightning source and normal ionosphere; (C) only noises, anomalous ionosphere; (D) lightning source and anomalous ionosphere.

Fig. 16. Comparison of the frequency-domain (5–40 Hz) waveforms (vertical electric field at surface, 2000 km surface distance from source center) excited by underground sources with different ionosphere conditions: (A) only noise, normal ionosphere; (B) the source depth is 2.5 km, normal ionosphere; (C) only noise, anomalous ionosphere; (D) the source depth is 2.5 km, anomalous ionosphere.

0

0

forms of the horizontal magnetic field observed at long distance are plotted in Fig. 16. All the frequency-domain waveforms in the figure are similar to each other under different ionospheric conditions, and no obvious power enhancement is observed. The results prove that the power increase in the spectrum in the range of 5–40 Hz can only be observed near the earthquake center. We believe this is because in the Earth-ionosphere system, the EM field in the low-frequency range forms standing waves in the horizontal direction; thus, the power of the EM field in the long distance is very little compared to that near the source center. This is also the reason why during many earthquakes, long-distance observations of SR anomalies still fail although ionospheric anomalies and underground currents occur. G. Effects of Ionosphere Conductivity Change on SRs Observations In real observations, the ionospheric anomalies during earthquakes not only involve the increase of the ionospheric conduc-

Fig. 17. Comparison of the frequency-domain (5–40 Hz) waveforms (vertical electric field at surface, 500 km surface distance from source center) excited by lightning sources with different ionosphere conditions: (A) normal ionosphere; (B) conductivity increased ionosphere; (C) conductivity decreased ionosphere.

tivity but also its decrease. The effects of different types of ionospheric anomalies on SR observations are depicted in Fig. 17. In the figure, both increased and decreased ionospheric conductivity conditions are considered. In both simulations, the conductivity anomalies only occur in the earthquake preparation zone, and the anomalies increase or decrease the original conductivity by 20 percent. At the same time, lightning currents are used as excitation sources. In Fig. 17, power enhancements in the frequency range 5–40 Hz under both increased (line B) and decreased (line C) conductivity conditions are observed, with the results showing that the enhancements are more obvious when the ionospheric conductivity is increased. We believe this is because both the increase and the decrease in conductivity profile in parts of the global ionosphere have broken the unity of the global conductivity distribution and changed the EM environment locally. V. CONCLUSIONS In this work, the earthquake EM phenomena in the frequency range 0.01–40 Hz are fully investigated using the 3D geodesic FDTD algorithm. With the aid of rigorous geometry information on the Earth-ionosphere system, we have validated and discussed the anomalous ULF and ELF observations prior to or during earthquakes. At the same time, we have proposed possible reasons for the observed ULF and ELF anomalies according to the possible physical processes that happen during earthquake processes. In summary, the following conclusions have been obtained to describe the earthquake EM phenomena: 1) The current sources caused by underground physical processes have effects on the surface observations of EM fields; the effects are not obvious above 1 Hz in the sea (or ocean) and seaside area but are obvious above 1 Hz in the continental area. 2) After the excitation of underground sources, the EM field below the Earth’s surface has much more power than that in the air, and when standing waves form, in the horizontal direction, most of the EM power caused by the underground sources are limited to a small area, which makes long distance observation impossible.

WANG AND CAO: ANALYSIS OF SEISMIC ELECTROMAGNETIC PHENOMENA USING THE FDTD METHOD

3) Both the underground source and the air source have effects on the resulting SR frequencies observed near the source center, especially when ionospheric anomalies happen, and the SR power is enhanced only in a small area near the source center. 4) Both the increase and the decrease in the localized ionospheric conductivity profile enhance the observed SR frequencies. 5) The frequency shift observed in [8], [9] is not caused by underground current sources or ionospheric anomalies. Our results offer the first full analysis of seismic EM phenomena in the ULF and lower ELF range that includes both the underground sources caused by the earthquake physical process and the ionospheric anomalies caused by the seismo-ionospheric coupling process. Because these processes cause anomalous EM behavior on the surface EM field, the observation of the horizontal EM field in the ULF (0–3 Hz) range and of SRs are two possible methods for studying earthquake activities. However, it is noted that using only EM field observation data is not enough to predict the occurrence of potential earthquakes because the observation of anomalous seismic EM phenomena is not always successful. The possible reasons for this are: 1) the underground physical process and the seismo-ionospheric coupling process are not fully understood, which possibly causes erroneous results; 2) these seismic physical processes do not happen during all earthquakes, and neither do the anomalous EM phenomena; and 3) other factors, such as the observation distance from earthquake center, the earthquake magnitude and depth, or artificial noises, possibly disrupt the earthquake process. Further works on this subject should focus on the improvement of the time-domain models and algorithms, gaining better understanding of earthquake physical processes and their effects on the EM field, and the application of the anomalous EM phenomena to earthquake prediction. REFERENCES [1] K. Park, M. J. S. Johnston, T. R. Madden, F. D. Morgan, and H. F. Morrison, “Electromagnetic precursors to earthquakes in the ULF band: A review of observations and mechanisms,” Rev. Geophys., vol. 31, no. 2, pp. 117–132, 1993. [2] S. Pulinets, “Ionospheric precursors of earthquakes; recent advances in theory and practical applications,” Terrest. Atmos. Ocean. Sci., vol. 15, no. 3, pp. 413–435, 2004. [3] D. Jianhai, S. Xuhvi, P. Weiyan, Z. Jing, Y. Surong, L. Gang, and G. Huwping, “Seismo-electromagnetism precursor research progress (in Chinese),” Chinese J. Radio Sci., vol. 21, no. 5, 2006. [4] Z. Kobylinski and S. Michnowski, “Atmospheric electric and electromagnetic field rapid changes as possible precursors of earthquakes and volcano eruption: A brief review,” Sun Geosphere, vol. 2, no. 1, pp. 43–47, 2007. [5] A. C. Fraser-Smith, A. Bernardi, P. R. McGill, M. E. Ladd, R. A. Helliwell, and J. O. G. Villard, “Low-frequency magnetic field measurements near the epicenter of the ms 7.1 Loma Prieta earthquake,” Geophys. Res. Lett., vol. 17, no. 9, pp. 1465–1468, 1990. [6] M. Hata and S. Yabashi, “Pre- and after-sign detection of earthquake through ELF radiation,” Geosci. Remote Sensing, pp. 1376–1381, 2002. [7] O. Molchanov, “Precursory effects in the subionospheric VLF signals for the Kobe earthquake,” Phys. Earth Planet. Inter., vol. 105, no. 3–4, pp. 239–248, Jan. 1998. [8] K. Ohta, N. Watanabe, and M. Hayakawa, “Survey of anomalous Schumann resonance phenomena observed in Japan, in possible association with earthquakes in Taiwan,” Phys. Chem. Earth, vol. 31, no. 4–9, pp. 397–402, 2006.

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[9] K. Ohta, J. Izutsu, and M. Hayakawa, “Anomalous excitation of Schumann resonances and additional anomalous resonances before the 2004 Mid-Niigata prefecture earthquake and the 2007 Noto Hantou Earthquake,” Phys. Chem. Earth, vol. 34, no. 6–7, pp. 441–448, 2009. [10] T. Ondoh, “Investigation of precursory phenomena in the ionosphere, atmosphere and groundwater before large earthquakes of ,” Adv. Space Res., vol. 43, no. 2, pp. 214–223, Jan. 2009. [11] M. A. Fenoglio, M. J. S. Johnston, and J. D. Byerlee, “Magnetic and electric fields associated with changes in high pore pressure in fault zones: Application to the Loma Prieta ULF emissions,” J. Geophys., vol. 100, no. B7, pp. 12,951–12,958, 1995. [12] O. Majaeva, Y. Fujinawa, and M. E. Zhitomirsky, “Modeling of nonstationary electrokinetic effect in a conductive crust,” J. Geomag. Geoelectr., vol. 49, pp. 1317–1326, 1997. [13] J. J. Simpson and A. Taflove, “Electrokinetic effect of the Loma Prieta earthquake calculated by an entire-earth FDTD solution of Maxwell’s equations,” Geophys. Res. Lett., vol. 32, no. L09302, 2005. [14] J. J. Simpson, “Global FDTD Maxwell’s equations modeling of electromagnetic propagation from currents in the lithosphere,” IEEE Trans. Antennas Propag., vol. 56, no. 1, pp. 199–203, 2008. [15] K. Yee, “Numerical solution of initial boundary value problems involving maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag., vol. AP-14, no. 3, pp. 302–307, 1966. [16] A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. Boston, MA: Artech House, 2005. [17] Z. Biqiang, W. Min, Y. Tao, W. Weixing, L. Jiuhou, L. Libo, and N. Baiql, “Is an unusual large enhancement of ionospheric electron density linked with the 2008 great Wenchuan earthquake?,” J. Geophys. Res., vol. 113, no. A11, 2008. [18] Y. Tao, M. Tian, W. Yungang, and W. Jinsong, “Study of the ionospheric anomaly before the Wenchuan earthquake,” Chinese Sci. Bulletin, vol. 54, no. 4, pp. 493–499, Jan. 2009. [19] J. J. Simpson, “Current and future applications of 3-D global earth-ionosphere models based on the full-vector Maxwell’s equations FDTD method,” Surveys GeoPhys., vol. 30, no. 2, pp. 105–130, Mar. 2009. [20] L. Sevgi, F. Akleman, and L. Felsen, “Groundwave propagation modeling: Problem-matched analytical formulations and direct numerical techniques,” IEEE Antennas Propag. Mag. vol. 44, no. 1, pp. 55–75, 2002 [Online]. Available: http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=997903 [21] R. Holland, “THREDS: A finite-difference time-domain EMP code in 3D spherical coordinates,” IEEE Trans. Nucl. Sci., vol. 30, no. 6, pp. 4592–4595, 1983. [22] J. J. Simpson and A. Taflove, “Two-dimensional FDTD model of antipodal ELF propagation and Schumann resonance of the earth,” Antennas Wireless Propag. Lett., vol. 1, no. 1, pp. 53–56, 2002. [23] J. J. Simpson and A. Taflove, “Efficient modeling of impulsive ELF antipodal propagation about the earth sphere using an optimized twodimensional geodesic FDTD grid,” IEEE Antennas Wireless Propag. Lett., vol. 3, no. 11, pp. 215–218, 2004. [24] J. J. Simpson and A. Taflove, “Three-dimensional FDTD modeling of impulsive ELF propagation about the earth-sphere,” IEEE Trans. Antennas Propag., vol. 52, no. 2, pp. 443–451, Feb. 2004. [25] J. J. Simpson and A. Taflove, “ELF radar system proposed for localized D-region ionospheric anomalies,” IEEE Geosci. Remote Sensing Lett., vol. 3, no. 4, pp. 500–503, Oct. 2006. [26] J. J. Simpson, R. Heikes, and A. Taflove, “FDTD modeling of a novel ELF radar for major oil deposits using a three-dimensional geodesic grid of the earth-ionosphere waveguide,” IEEE Trans. Antennas Propag., vol. 54, no. 6, pp. 1734–1741, Jun. 2006. [27] S. A. Cummer, “An analysis of new and existing FDTD methods for isotropic cold plasma and a method for improving their accuracy,” IEEE Trans. Antennas Propag., vol. 45, no. 3, pp. 392–400, Mar. 1997. [28] S. A. Cummer, “Modeling electromagnetic propagation in the earthionosphere waveguide,” IEEE Trans. Antennas Propag., vol. 48, no. 9, pp. 1420–1429, 2000. [29] J.-P. Berenger, “Long range propagation of lightning pulses using the FDTD method,” IEEE Trans. Electromagn. Compat., vol. 47, no. 4, pp. 1008–1012, 2005. [30] A. Soriano, E. Navarro, D. Paul, and J. Porti, “Finite difference time domain simulation of the earth-ionosphere resonant cavity: Schumann resonances,” IEEE Trans. Antennas Propag., vol. 53, no. 4, pp. 1535–1541, Apr. 2005.

M > 6:5

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[31] H. Yang, “Three-dimensional finite difference time domain modeling of the earth-ionosphere cavity resonances,” Geophys. Res. Lett., vol. 32, no. 3, pp. 4–7, 2005. [32] H. Yang, V. P. Pasko, and G. Sátori, “Seasonal variations of global lightning activity extracted from Schumann resonances using a genetic algorithm method,” J. Geophys. Res., vol. 114, no. D1, pp. 1–10, Jan. 2009. [33] Y. Wang, H. Xia, and Q. Cao, “Analysis of ELF propagation along the earth surface using the FDTD model based on the spherical triangle meshing,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 1017–1020, 2009. [34] GEODAS Grid Translator—Design-a-Grid 2011 [Online]. Available: http://www.ngdc.noaa.gov/mgg/gdas/gd_designagrid.html [35] A. Martinez, A. P. Byrnes, K. G. Survey, and C. Avenue, “Modeling dielectric-constant values of geologic materials: An aid to ground-penetrating radar data collection and interpretation,” Current Res. Earth Sci. , vol. Bulletin 2, no. Part 1, 2001. [36] J. Hermance, “Electrical conductivity models of the crust and mantle,” Global Earth Physics: A Handbook of Physical, pp. 190–205, 1995. [37] D. Bilitza, “International reference ionosphere 2000,” Radio Sci., vol. 36, no. 2, pp. 261–275, 2001. [38] International Reference Ionosphere—IRI-2007 2010 [Online]. Available: http://ccmc.gsfc.nasa.gov/modelweb/models/iri_vitmo.php [39] H. Yang, V. P. Pasko, and Y. Yair, “Three-dimensional finite difference time domain modeling of the Schumann resonance parameters on titan, venus, and mars,” Radio Sci., vol. 42, no. 2, Sep. 2006. [40] V. P. Pasko, U. S. Inan, and T. F. Bell, “Sprites produced by quasieletrostatic heating and ionization in the lower ionosphere,” J. Geophys. Res., vol. 102, no. A3, pp. 4529–4561, 1997. [41] D. D. Sentman and B. J. Fraser, “Simulation observations of Schumann resonances in California and Australia: Evidence for intensity modulation by the local height of the D region,” J. Geophys. Res., vol. 96, no. Sep. 1989, 1991. [42] J. N. Thomas, J. J. Love, and M. J. Johnston, “On the reported magnetic precursor of the 1989 Loma Prieta earthquake,” Phys. Earth Planet. Inter., vol. 173, no. 3–4, pp. 207–215, Apr. 2009.

Yi Wang received the B.S. degree from Nanjing University of Aeronautics and Astronautics, Nanjing, China in 2006, where he is currently working toward the Ph.D. degree. His research interests include computational electromagnetics, especially the finite-difference time-domain (FDTD) method, the FDTD modeling of the entire Earth-ionosphere system, and the earthquake electromagnetics. His current research focuses on the FDTD simulation of the ELF electromagnetic waves to study earthquake electromagnetic phenomena.

Qunsheng Cao received the Ph.D. degree in electronic engineering from The Hong Kong Polytechnic University, in 2001. From 2001 to 2005, he was a Research Associate with the Department of Electrical Engineering, University of Illinois at Urbana-Champaign, and with the Army High Performance Computing Research Center (AHPCRC), University of Minnesota, respectively. In 2006, he joined the University of Aeronautics and Astronautics (NUAA), Nanjing, China, as a Professor of electronic engineering. He has authored or coauthored over 70 papers in refereed journals and conference proceedings. He coauthored Multiresolution Time Domain Scheme for Electromagnetic Engineering (Wiley, 2005). His current research interests are in computational electromagnetics and antennas designs, particularly in time domain numerical techniques for the study of microwave devices and scattering applications.

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The Resonance Mode Theory for Exterior Problems of Electrodynamics and Its Application to Discrete Antenna Modeling in a Frequency Range Igor P. Kovalyov and Dmitry M. Ponomarev, Member, IEEE

Abstract—The problem under consideration here is that of building a versatile discrete antenna model in a frequency range. The eigenfunction (mode) expansion or singularity expansion method (SEM) is used in working out the problem. A detailed substantiation of SEM is provided and an expression determining the dependence of mode amplitudes on exterior source fields is derived. A simple example is provided in support of the theory. The SEM is used in inquiring into the frequency dependence of antenna-generalized matrix entries. It is intimated that all of the matrix entries contain resonance factors and factors that are Hankel transforms of radial coordinate functions. A sampling theorem, which is a generalized extension of the classic sampling theorem, is used for the Hankel transform. The application of the generalized sampling theorem completes the process of building a versatile discrete antenna model in a frequency range.

Combination of vectors

Index Terms—Antenna-generalized scattering matrix, complex resonance frequencies, sampling theorem for the Hankel transform, spatial and frequency characteristics of antenna, spherical waves.

Strength of electric field of the exterior sources in free space.

and

.

Mode complex amplitude of the antenna excited by exterior field. Submatrix of the antenna-generalized scattering matrix. Entry of the

submatrix.

Auxiliary functions by means of which the mode amplitude of the antenna is determined when the latter is excited by a spherical wave. Vector of electric-field strength.

Strength of the scattering electric field. Vector of electric-field strength at the center of the th line element of the wire antenna.

NOMENCLATURE

Strength vector of the electric field caused by spherical wave radiation.

Vector of complex amplitudes of scattering field spherical waves. Complex amplitude of spherical wave of the antenna radiated or scattering field. Vector of complex amplitudes of exterior field spherical waves. Complex amplitude of exterior field spherical wave.

Strength vector of the exterior spherical wave electric field. Mode electric-field strength vector. Auxiliary function determined by the Cartesian component of the field or mode current. Complex resonance frequency.

Frequency functions used in determining entries of the generalized scattering matrix.

Auxiliary function, through which the amplitude of mode-radiated spherical wave is determined.

Prameter of discrete antenna model.

Vector of magnetic-field strength.

Combination of vectors

Strength of magnetic field of exterior sources in free space.

and

.

Strength of scattering magnetic field. Manuscript received November 10, 2010; revised April 14, 2011; accepted April 14, 2011. Date of publication August 12, 2011; date of current version November 02, 2011. I. P. Kovalyov is with the Nizhny Novgorod State University of Technology, Nizhny Novgorod 603126, Russia. He is also with MERA NN, Nizhny Novgorod 603126, Russia (e-mail: [email protected]). D. M. Ponomarev is with MERA NN, Nizhny Novgorod 603126, Russia (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2011.2164205 0018-926X/$26.00 © 2011 IEEE

Hankel spherical functions of the second kind. Spherical wave radiation magnetic-field strength vector. Spherical wave exterior magnetic-field strength vector.

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Mode magnetic-field strength vector. Current at the antenna’s input. Complex amplitude of current in the line. Mode current in the line segment. mode current at the th input. Imaginary unit. Vector of currents in wire antenna elements. Density of perfect-conductor antenna mode surface current. Cartesian components of mode current concentration vector. Bessel function of order . Volume density of the electric (magnetic) equivalent current. >

Volume density of the electric (magnetic) equivalent current of exterior sources. Volume density of induced electric (magnetic) equivalent current. Spherical Bessel function of the first kind. Wave number.

entry of the

submatrix.

Antenna-generalized scattering matrix. surface of the

radius sphere.

Surface perpendicular to the electric field between the antenna’s input terminals. Submatrix of the antenna-generalized scattering matrix. entry of the

submatrix.

Voltage at the antenna’s input. Vector of complex amplitudes of the feeders’ backward waves. Complex amplitude of voltage across the line conductors. Vector of complex amplitudes of the feeders’ forward waves. mode voltage at the th input. Mode voltage in the line segment. Vector of voltages across the wire antenna elements. Region of space in which the antenna is located. line impedance within the line segment of the length .

Complex wave number corresponding to the complex resonance frequency.

Line impedance within the line segment of the length .

Length of dipole.

Unit vector of the

Line along the electric field from one antenna input terminal to the other.

Matrix inverse to the mutual impedance matrix.

Line in the feeder cross section perpendicular to the electric field.

Unit vector of the

Mode norm. Unit vector of normal to surface. Legendre associated function.

Line impedance of the th feeder. axis.

axis.

Matrix of mutual impedance between wire antenna components. unit vector of the

axis.

Radius of remote sphere.

Subscript identifying axes of the Cartesian coordinate system ( or ).

Submatrix of the antenna-generalized scattering matrix.

Coefficients of exterior field mode expansion.

entry of the

submatrix.

Vector in 3-D space. Spherical coordinates. Distance from the origin to the farthest antenna point. region boundary. Submatrix of the antenna-generalized scattering matrix.

Vector stretching from the start to the end of the th linear element of the wire antenna. Kronecker delta. Delta function. Relative electric permittivity of antenna material. Permittivity constant.

KOVALYOV AND PONOMAREV: RESONANCE MODE THEORY FOR EXTERIOR PROBLEMS OF ELECTRODYNAMICS

Spherical coordinates. Sampling functions used for the frequency representation of functions. Coefficient of reflection from ends of the line segment. Vacuum characteristic impedance. Generalized eigenvalue, a complex and factor by which the parameters of the medium should be multiplied to obtain continuous harmonic oscillation at the frequency . Relative magnetic permeability of the antenna material. Vacuum permeability. th zero of the function

.

Spherical coordinates. Specific conductivity of antenna material. Frequency. Complex resonance frequency.

I. INTRODUCTION

T

HE problem considered in this paper is representation of antenna spatial characteristics in a frequency range. A solution of the problem is paramount for the antenna technology, as it would allow a compact form of presenting antenna characteristics. Of course, antenna characteristics may be determined for every frequency within the range of interest, as is usually done in classical monographs. However, this representation of spatial characteristics is cumbersome and inconvenient. An inquiry into the structure of spatial and frequency characteristics would allow their simpler representation. The possibility of compact presentation of antenna characteristics in a frequency range has been demonstrated in [1]. This research revealed that spatial and frequency characteristics of an antenna can be determined by a small number of parameters. However, the model presented in [1] should be viewed as an opportune heuristic finding, since the paper does not offer its rigorous enough substantiation. In this study, we suggest a more rigorous approach to modeling spatial and frequency characteristics of antennas. Briefly, the general outline of the material in this paper and main ideas presented in it is as follows. To characterize an antenna at a fixed frequency, we use the generalized scattering matrix technique, finding it a more comprehensive characterization tool than the directional pattern. Section II comprises a brief survey of spherical waves and the antenna-generalized scattering matrix, a detailed study of which is available in [2].

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While describing frequency dependencies, we use the method of field expansion in eigenfunctions of the problem. This technique is also referred to in the literature as the singularity expansion method (SEM) [3] or method of presenting the solution as a sum of modes. Notwithstanding the widespread employment of SEM in attacking exterior problems of the electrodynamics [4]–[6], the equations that are used in the process have one inherent fault. They lack the relation between mode amplitudes and fields from exterior sources given in an explicit form. In [4]–[6], the authors ignore such a relationship, and this entails the necessity to introduce additional summands, which have nothing to do with the resonance modes of the object under scrutiny. In [3], the mode-exterior-field relationship is an implicit relation written as an inverse operator. The absence of an explicit correlation between mode amplitudes and exterior source fields encumbers the application of SEM in solving practical problems. Section III gives a detailed elucidation of SEM, modes, and the application of fields singularity expansion to exterior problems of the electrodynamics. In Section III, in contrast to the traditional view of mode as harmonic oscillation with exponentially decreasing amplitude, we treat mode as a continuous harmonic oscillation that occurs in the system, if the medium inside the scatterer becomes active. This approach allows us to easily prove the orthogonality of modes and find the problem solution as a sum of them. This technique is similar to the method of generalized eigenvalues considered in [7]. However, the method discussed in [7] is applicable in case of a single fixed frequency only. The method that we suggest in our study may be regarded as a special variant of the generalized eigenvalues method designed for exterior problems in a frequency range. An introduction of modes reduces the analysis of frequency characteristics to two problems. First, it is necessary to determine the frequency dependence of amplitudes of modes excited by exterior sources. The second task is to find the interrelation between the modes as well as fields scattered by the antenna and signal at its outputs. These problems are considered in Sections IV–VI. Section IV determines the frequency dependence of mode amplitudes for the antenna excited by a spherical wave. Section V finds the frequency dependence of amplitudes of spherical waves radiated by some antenna mode. Section VI presents the frequency dependence of entries of the antenna-generalized scattering matrix. It becomes apparent that all of the frequency dependencies are sums of summands, each of which comprises frequency functions of two types. These are the resonance functions characteristic of SEM, on the one part, and similar-type frequency functions, connected with the Hankel transform of some radial coordinate functions on the other. A generalized variant of the classic sampling theorem [8] is used for representation of similar functions in Section VII. A presentation of functions of the same type through the application of this theorem completes the process of building a discrete antenna model in a frequency range. Section VIII contains the conclusion. As a rule, papers have a separate section dedicated to computational modeling that substantiates the theory. We took the

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components in the spherical coordinate system are computed using the following equations:

(2a)

(2b)

(2c) Fig. 1. Diagrammatic sketch of antenna system and waves.

(2d)

liberty of departing from this tradition and included computation examples in support of the theory in Sections III–VII. This study is conceptually close to our previous work [9], but presents new important results. First and foremost, it provides a justification of SEM applications to exterior problems of the electrodynamics. In addition, the applicability of the results obtained in this paper is broader; they hold true not for electromagnetic-wave scatterers only, but for antenna systems as well.

(2e)

II. ANTENNA-GENERALIZED SCATTERING MATRIX AND ITS SUBMATRICES In this section, we dwell briefly on the generalized scattering matrix that characterizes an antenna or a scatterer at a fixed frequency. A more elaborate discussion of the scattering matrix is available in [2]. Consider the diagrammatic sketch of an -port antenna system presented in Fig. 1. The exterior fields that the antenna system is exposed to, the fields scattered by the antenna, and its radiation fields are determined according to the uniqueness theorem by fields on some closed surface enveloping the antenna. For such an enveloping exterior, it is expedient to use the simplest unbounded surface, an radius sphere as depicted in Fig. 1. We use spherical waves to represent fields outside the sphere. Expressions for the electric and magnetic fields in terms of spherical wave complex amplitudes are written according to [2] and [9]–[14] as follows: (1a)

(2f) The following designations are used in (2): stand for spherical coordinates of the observation point; is the wave number; denotes free space impedance; are components of the electric field of the E-type spherical waves; are components of the magnetic field of E-type spherical waves; stands for the Hankel spherical function of the second kind; in (2) denotes the associated Legendre function. The expression for H-type waves can be written as follows:

(3) Equations (1) represent fields as sums of spherical waves. coefficients represent complex amplitudes of E-type The spherical waves for which the radial component of the magnetic . denotes amplitudes of field is equal to zero H-type waves, for which the radial component of the electric . We use the vector to reprefield is equal to zero sent the totality of spherical wave complex amplitudes in (1). Let us agree to place E-type wave amplitudes in the beginning of the vector , followed by H-type wave amplitudes. Therefore, the vector in detail can be written as (4)

(1b) and in (1) denote complex amplitudes of E- and and field H-type spherical waves, respectively. The

to denote the length of vector . We use In addition, when dealing with vectors, it is more convenient to switch from double subscript marking of spherical waves to one-letter indexing .

KOVALYOV AND PONOMAREV: RESONANCE MODE THEORY FOR EXTERIOR PROBLEMS OF ELECTRODYNAMICS

Expressions (2) satisfy the radiation condition and represent in Fig. 1. the antenna radiation or scattering field when To represent the exterior field, in which the antenna may be, spherical Bessel (2) must be modified. Specifically, the function should be used in them to substitute for the spherical Hankel function. Similarly, for the introduction of the vector, let us introduce an -length vector , the components of which are complex amplitudes of spherical waves of the exterior field, to which the antenna system in Fig. 1 is exand to denote the exterior electric field posed. We use of spherical waves. Components of this field are determined by is substituted for in them. (2), provided that and that comWe also consider the Q-length vectors bine direct and backward feeder waves. The totality of complex amplitudes of the waves, to which the antenna is exposed, is and that of the scattering waves is desigsymbolized by . Therefore nated as (5) and are equal to . The lengths of the vectors As the antenna system in Fig. 1 is assumed to be linear, the and satisfy the matrix equality vectors (6) matrix comprised in (6) is referred to as an antenna The system scattering matrix or a generalized scattering matrix. It is this matrix that allows the discrete representation of antenna characteristics at a fixed frequency. The division (5) of the vecand in two subvectors is conducive to presenting tors generalized matrix as an array of four submatrices the (7) Let us explain the physical meaning of the submatrix entries in (7). The submatrix of the size determines the relationship between the waves of the antenna system feeders. The submatrix represents the scattering matrix of the antenna system, if considered as a microwave circuit disregarding elecarbitrary tromagnetic fields produced by the antenna. The entry of the submatrix is equal to the backward-wave complex amplitude of the th feeder with the th feeder forward wave being a unit wave. Or

(8) Note that the diagonal entry of the submatrix is the wave reflection coefficient for the th antenna input. characterizes the radiThe submatrix of the size entry of the submatrix is equal ating antenna system. The to the ratio of the complex amplitude of the th spherical wave of the antenna radiation field to the amplitude of the forward wave of the th feeder producing this radiation (9)

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determines the characThe submatrix of the size arbitrary teristics of the receiving antenna. To calculate the entry of this submatrix, the antenna system should be placed in entry the exterior field of the th spherical unit wave. The is determined by the complex amplitude of the th feeder backward wave (10) The fourth submatrix of the generalized scattering matrix (7) characterizes the antenna as a scatterer of electromagarbitrary entry of this submatrix is equal netic waves. The to the complex amplitude of the th spherical wave of the scattering field, provided that the antenna is located in the exterior field of the th spherical unit wave (11) It is worth noting that the generalized scattering matrix provides a more comprehensive characterization means than the antenna radiation pattern. According to [2], the radiation pattern is determined solely by the submatrix of the antenna’s generalized scattering matrix. This is the reason why we adopted the generalized scattering matrix as a principal tool of inquiring into antenna spatial characteristics in a frequency range. It might be well to point out that a scatterer of electromagnetic waves may be regarded as a special case of an antenna system devoid of -generalfeeders. For this reason, in case of a scatterer, the ized scattering matrix coincides with the matrix , the entries of which are determined by (11). III. MODES AND FIELD SINGULARITY EXPANSION FOR EXTERIOR PROBLEMS OF ELECTRODYNAMICS In the analysis of frequency dependence of an arbitrary entry -generalized scattering matrix, we resort to repreof the sentation of the solution as a sum of eigenfunctions, otherwise called the singularity expansion method. The method of expansion in eigenoscillations (modes) is widely applied to interior and exterior problems of the electrodynamics. One can find a diversity of variants of this method and a multitude of references in [4]–[6]. Although a general and rigorous enough substantiation of the mode expansion method for exterior problems involves great difficulties and is hard to come across in the literature, these inherent difficulties can be easily fathomed if we consider the wave factor that determines the dependence of the wave field on time and the spatial value (12) If the mode is characterized, as is the convention, with the complex frequency, the frequency’s imaginary part must be taken as positive. Then, the oscillation amplitude (12) decreases with time. However, the same frequency appears in the wave , and oscillation increases number expression ad infinitum with the growth of . This fact makes the application of SEM difficult and complicates its theoretical substantiation for exterior problems of the electrodynamics.

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here is the wave number. The wave number may be complex valued. represents free-space impedance. and stand for the dielectric and magnetic permeability of the medium within . In (13), the exterior sources are considered to be located in domain 1. With the sources located in subdomain and should be moved to 2, the summands the second pair of (13). In the problem illustrated in Fig. 2, meeting the radiation conditions should be a requirement where frequency and

(14) The approximate equality means that it holds for the terms as the distance grows. Finally, the tangential decreasing as field components should be equal at the boundary at

Fig. 2. Antenna or scatterer in the field from exterior sources.

Reference [15] comprises such substantiation for open highcavities only. The theory for open high- cavities differs but little from the classical excitation theory for cavity resonators. In a general way, with an arbitrary , the theory needs a significant modification. This section is devoted to a detailed discussion of SEM and the approach, which enables a way around difficulties in its application.

(15)

B. Modes and Eigenvalues To solve the problem in question, let us use the solutions for the following homogeneous problem:

A. Problem Statement Let us define the problem of electromagnetic-wave scattering (radiation). Field sources are considered known; these are the exterior current and the magnetic current. There is a scatterer or antenna, which is a limited size body positioned within a subdomain (Fig. 2). The body boundary is , and parameters of the medium labeled . The within are generally coordinate-dependent. The medium outside the volume is free space with the parameters and . , produced The challenge now is to determine the field by the sources in the presence of the body . Two cases will be considered when attacking the problem. In the first case, the exterior sources are located within the volume in subdomain 2 as illustrated in Fig. 2. In the second case, the sources are located without in domain 1. The exterior fields and born by the and sources in free space are also assumed known in the second case. The scattering problem involves solving the Maxwell equation which, for domain 1 and subdomain 2 in Fig. 2, is written as

(13)

(16) where here is a certain fixed wave number value. The complex parameter, for which nonzero solutions of set (16) exist, will be referred to as an eigenvalue. The solutions will be called eigenfield or mode. Let us arrange for meeting the radiation and boundary conditions requirements

(17) at

(18)

Note that the eigenvalue in set (16) is not frequency but rather the variable that determines the parameters of the medium within the volume. It is this interpretation of eigenvalue that remedies the exponential increase of fields with the growth of and provides for getting over the difficulties mentioned in the beginning of this section. This approach represents one of the versions of the generalized eigenmodes method discussed in [7]. and parameters of the medium vary In our version, the in concord. The concerted variation of the medium parameters multiplier to eigenfrequencies allows transition from the in final equations.

KOVALYOV AND PONOMAREV: RESONANCE MODE THEORY FOR EXTERIOR PROBLEMS OF ELECTRODYNAMICS

Let us prove that modes corresponding to different eigenvalues are orthogonal. To do it, we write a set of equations for the mode

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domain 1. An integration of (22) over , a replacement of the volume divergence integral with the surface one and allowing for (25) gives

(26)

if

The desired mode orthogonality condition follows from (26) and

(19) (27) For each of the two domains, we combine the equations of sets (16) and (19). Specifically, we multiply the equations of (16) by and by and the equations of set (19) by and by . Adding the results and using the vector equality (20) we obtain (21)

(22) We integrate (21) with respect to the volume enclosed between the sphere (Fig. 2) and the surface. The radius is believed to be capable of an unlimited increase. Substituting and surface integrals for the volume divergence integral, we can write

The domain identifying superscript 2 has been omitted in (27). The integration over the volume is evident from writing the integral (27). In the equations that follow, we also omit domain indication wherever possible. It is pertinent to note that the integral (27) is calculated over a limited domain, the parameters of which differ from the parameters of free space. Therefore, there is no convergence issue in this case, which always arises when you deal with improper integrals. This is an important virtue of the generalized method of eigenoscillations. permeability is not included in the orthogNote that the onality condition (27). However, the mode field is permeability , dependent since it is comprised in the field (16). With the integral (27) may be distinct from zero. Let us introduce the designation (28) In what follows, the 1. normalized, and

mode fields are assumed

C. Solving the Scattering Problem

(23)

The mode fields in domain 2 are considered known. Let us use them in solving the inhomogeneous Maxwell’s equations. To begin with, assume that exterior sources are located in domain 2 in Fig. 2. The set of (13) for domain 2 then takes on the following form:

Let us evaluate the integrand in the right part of (23) by using the radiation condition. A substitution of (17) into it gives (29)

(24) The approximate equality (24) means that there are no sumin its right part. It may comprise summands decreasing as or faster. For this reason, the integral mands decreasing as in the right part of (23) tends to zero with the increasing radius of the sphere in Fig. 2, and since the left part is independent of , the integrals in (23) are identically equal to zero

(25) In writing (25), the condition (18) has been used, and fields within domain 2 have been substituted for the fields on within

The mode fields

,

fields can be represented as a sum of

(30) in (30) denotes the frequency-dependent mode amplitude. Mode fields with negative and positive frequencies are summed up here; to be more exact, there are frequencies and frequencies in the sums. A restriction on the summation of positive frequencies only would result in differing amplitude multipliers in the first and the second sum of (30). Formally, the number of summands

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in (30) is infinite, since the number of modes in a distributed system is infinitely large. However, in calculating numerical computations, it is essential to take into account only the modes the resonance frequencies of which happen to be within or in the close vicinity to the frequency range of interest. As for the modes, the resonance curve tails of which turn out to be negligibly small in the frequency range in question can be safely disregarded. Substituting (30) in (29) and taking into account (19), we obtain

and are assumed free-space sources known. Exterior fields satisfy Maxwell’s equations

(34) With sources located outside , it is reasonable to change representing the full to determining the scattering field field as the sum of the scattering field and the field of exterior sources

(31a) (31b) and (31b) by , add We then multiply (31a) by them, and integrate over . Taking into account the mode orthogonality condition (27) and designation (28), we obtain an expression for mode amplitude

(35) Subtracting set (34) from (13), we obtain equations for the scattering field

(32) It is apparent from (32) that the mode amplitude becomes . We shall use to denote it, and infinitely large with to symbolize the corresponding complex wave number. The and taking into account mode normalization introduction of recast expression (32) in the following form:

(36) and where duced equivalent sources

in (36) stand for newly intro-

(37)

(33) Superficially, (33) differs but little from (32), though this turns out to be a very important distinction in practice. Equation arbitrary frequency and the unknown (32) includes the complex complex parameter. Equation (33) comprises resonance frequencies that can be found experimentally. modes and the value Let us prove that the in (33). The fact that the latter statement are independent of holds good is obvious from the second pair of equations of set and if (16). These remain valid with the previous stays unchanged as varies. It is important to note that the solution of the first pair of equations of set (16) varies with varying . That is, the mode fields within the scatterer remain unchanged, though those outside the scatterer change. However, (33) includes fields within the scatterer and, therefore, the mode amplitude is independent of . So, if exterior sources are located in domain 2, then mode amplitudes are determined by (33). Let us now turn to solving the scattering problem, assuming that the exterior sources are located in domain 1 as depicted in Fig. 2. In this case, and exterior fields produced by the

As the exterior field satisfies the radiation condition and the field’s tangential components are continuous on the surface, the scattering field also meets the radiation condition (14) and boundary conditions (15). It is obvious from (36) that a transition from the entire field to the scattering field leads to a set of equations, in which the and equivalent sources are located in domain 2. Therefore, it is possible to represent the scattering field as a sum of modes and use the known solution (33) to determine mode amplitudes

(38) and Mode amplitudes are determined by (33), if the exterior sources in it are replaced with the equivalent and

(39)

KOVALYOV AND PONOMAREV: RESONANCE MODE THEORY FOR EXTERIOR PROBLEMS OF ELECTRODYNAMICS

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It is pertinent to stress that (39) determines mode amplitudes of the scattering field. For representation of the entire field in , let us find an expansion for the exterior field

(40) coefficients, we multiply the first equality of set To find (40) by , and the second by . We then add the results and integrate over . In view of the orthogonality condition (27) and mode normalization, we obtain an equation for the coefficients

Fig. 3. The 1-D system.

can calculate the field outside the scatterer, when fields in domain 2 are known. To do it, let us rewrite set (13) as follows:

(41) A substitution of (38) and (40) in (35) brings (35) into the form of (30). In doing so, the mode amplitude equation takes the form (44) Set (44) shows that the field in the scattering problem can be presented as a sum of fields produced in free space by currents of exterior currents two types. These are the induced currents. As is evident from and (44), induced currents are determined by the expressions

(42)

So the field inside the scatterer is presented as a sum of modes (30); mode amplitudes are determined by (33) or (42) as the location of sources requires. A reference to (30) shows that the introduction of modes allows one to present the frequency-spatial dependencies of fields as a sum of two multiplier products. The first of the multipliers is frequency dependent only, and the second is solely dependent on the spatial value. The representation of fields as a sum of modes (30) is sometimes believed valid only for vortex fields, the rotation of which is nonzero [15]. In potential fields are added such a case, the to the sum (30). For the potential fields

(43) In fulfilling (43), the fields are determined by gradients of scalar functions that are potentials and, for this reason, are called potential fields. We do not discrimi0 0) nate vortex and potential fields, but assume that 0, the solution of (16) in domain enters into eigenvalues. If 2 describes potential fields. Therefore, they turn out included in sums (30). We emphasize that (42) determines the amplitudes of modes that represent the field within the scatterer. Let us see how one

(45) That is, the problem of calculating the field outside the scatterer reduces to the calculation of the radiation field of free-space-induced currents. Induced currents are determined by the domain 2 full field according to (45). This is what differentiates the induced currents from equivalent currents introduced for the determining mode amplitudes. The latter currents, as is evident exterior from (37), are determined by the field. D. Simple Example Illustrating Theory Consider a simple 1-D system as an example illustrating the outlined aforementioned theory. Examine the long line depicted in Fig. 3. The line is believed to be filled with a dielectric along the full length of its segment. Owing to this, the wave impedance of the segment differs from the impedance of the rest of the line. Presenting the problem solution as a sum of modes is proposed. Though the system in Fig. 3 is not an antenna system, we, nevertheless, have selected it to illustrate theory. In this example, all parameters (mode frequencies, mode currents and voltages, and their amplitudes) are readily computable analytically. The solution to the stated problem is also easily obtainable through the use of well-known long-line equations. The exact fit of the results yielded by different methods attests to the verity

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of the theory. Other illustrative examples more closely related to the antenna technology will be presented in the sections that follow. The following equations hold for the line depicted in Fig. 3:

To elucidate the physical nature of , let us determine how the amplitude of the forward wave varies along the -length segment

(52)

(46) where in (46) denotes the complex amplitudes is the wave of voltage and current in the line, and number. Making a comparison to a 3-D model, it may be novoltage corresponds to the electiced that the current corresponds to the tric-field strength, and the magnetic-field intensity. The coordinate corresponds to the spatial vector. The counterpart for the medium impedance in the 3-D model in this case is the impedance of the line. The voltage source and the current differ by the factor. This correlation corsource responds to the incidence of a forward unitary wave onto a 1-D scatterer. To solve the defined 1-D scattering problem, we use the solutions for the respective homogeneous problem

Equality (52) demonstrates that as the wave travels along the . That is, the line is line, its amplitude increases by the factor filled with an active medium, which compensates for the amplitude’s decrease during wave reflection from the boundaries. The phase variation along the segment occurs in multiples. With multia double trip taken into account, the phase varies in ples. By this means, (52) ensures the balance of amplitudes and phases and the existence of continuous oscillation in the system in Fig. 3. The mode orthogonality condition, which is the analog of general relationship (27), in terms of the 1-D system takes on the form (53) A substitution of (50) in (53) allows one to confirm the orthogonality of different modes. Finally, let us calculate

(47) The fulfillment of conditions at the boundaries of the segment is a requisite for solving set (47)

(54) A substitution of (50) in (54) gives (55)

(48)

Let us get down to solving (46), presenting it as a sum of modes Condition (48) solely requires the existence of waves receding from the scatterer. With 0, there is a backward wave 1, only a forward traveling wave exists. only and with of set (47) will be referred to as the The solution natural oscillation or mode, and as the eigenvalue or complex , resonance frequency. Let us find explicit expressions for , , and . We will use to denote reflectance when 0 and l.

(56) Substituting (56) in (46) and in view of (47), we arrive at

(57) (49) Since there is a forward and a backward wave within the segment and their ratio is equal to , and may be written as

We then multiply the first equation of set (57) by and the second by , and add and integrate over from 0 to . In terms of (53) and (54), we obtain an expression for the amplitude of a mode

(58) (50) Expressions (49), (50), and (55) enable simplification of (58) A substitution of (50) in condition (48) gives

values (51)

(59)

KOVALYOV AND PONOMAREV: RESONANCE MODE THEORY FOR EXTERIOR PROBLEMS OF ELECTRODYNAMICS

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A substitution of (59) in (56) provides a problem solution in the form of a series (64)

(60) It is easy to obtain explicit expressions for voltage and current in the problem under consideration. Using equations for computing long lines, we obtain

(61) and , viewed in (61) as the functions of the complex variable, have their singularities at poles. Therefore, they can be presented as sums of common fractions [16]. Setting the denominator (61) equal to zero determines the poles. The expression for them coincides with (51). The representation of functions (61) as common fractions coincides with (60). This example illustrates that the presentation of the solution as a sum of modes from a mathematical standpoint is the partial-fraction expansion of spatial functions of the complex variable (complex frequency). Of course, the transition from (61) to series (60) does not offer any particular benefit in this simple example. Though in tridimensional problems, where the derivation of analytical expressions is impossible, the ascertainment of the structure of partial-fraction expansion is very important. Let us use this problem to illustrate the possibility of series transformation. This opportunity exists when the solution for 0, zero frequency, that is the static solution, is known. With from (46), it is easy to obtain (62) Physically, (62) means that 1-V constant voltage persists at all points of the line. At the same time, the current flows in (60) can be modified through the line. (63) Equation (63) allows one to present the series of (60) as two sums with second sums being independent of and providing the known static solution (62). Consequently, (60) takes on the form

A numerical analysis shows that series (64) converges faster than (60). Thus, the isolation of static fields may prove useful in solving scattering problems. factor in the system depicted in Fig. 3 is The mode and impedances. If the determined by the ratio of the impedance difference is fivefold and greater, no less than 5% accuracy can be ensured by taking into account solely one mode of the resonance frequency which is in close vicinity to the frequency range of interest. Simultaneously, all modes of the resonance frequencies, which are within the considered range, should be taken into account of course. If the ratio decreases, the mode diminishes too, and it is necessary to increase the number of summands in (64), in order to retain acceptable accuracy. So the considered 1-D problem confirms the obtained general relations. In addition, it conveniently illustrates that writing a solution as a sum of modes is connected with presenting spatial functions as sums of common fractions. E. Perfectly Conducting Scatterers The computational relations obtained in Sections IV–VI are readily illustrated by the analysis of perfect conductor antennas. Assume 1, 1, for conducting bodies, including those perfectly conducting. Then, introducing mode current (65) we can write (42) and (45) in the following form:

(66) (67) Since conductivity does not enter into relations (66) and (67), they hold true whatever the conductivity is. We will use them while analyzing antennas comprising perfect conductors . Expressions (66) and (67) may be obtained with greater rigor for a perfectly conducting scatterer. To this end, it is enough to assume that the scatterer is coated with a thin film of a magnetodielectric. Such a film will have a negligible effect on the electromagnetic field. Going again through but slightly modified expressions (16)–(27), one will be able to ascertain that the previous modes orthogonality condition (27) still holds true. in this case is the thin film that envelops the scatterer, and the volume integral of (27) turns into the scatterer’s surface integral. Since the and parameters of the coating take on arbitrary values, each of the two summands in (27) is equal to zero.

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Therefore, the orthogonality condition for the perfectly conducting scatterer can be written in one of the following forms:

If the spherical wave is an E-type wave, the expression for has the following form:

(68)

(70)

In writing (68), it was taken into account that the normal components of the electric field and tangential components of the magnetic field are distinct from zero on the surface of the perfectly conducting scatterer. The tangential magnetic field is replaceable with the mode current. The use of surface mode currents in solving the inhomogeneous problem leads to (66), in which the surface integral should be substituted to replace the volume integral. The method of solving exterior problems explained in this section can be viewed as a special variant of the widespread method of moments (MoM). The use of eigenfunctions for basis functions leads to a set of MoM algebraic equations breaking up into individual independent equations. This allows writing solutions to the equations which results in an explicit expression connecting the mode amplitude with fields of exterior sources. The obtained solution is general enough in nature; it holds true for a variety of sources and objects of variegated shapes. The presented proof is trivially extended to become applicable to an aggregation of several objects and to multilayer objects. The important condition is that the object’s spatial dimensions are finite. The explained technique is unsuitable for tackling such problems as radiation from a slit in an infinite screen and propagation out of the mouth of a semi-infinite waveguide. In closing this section, let us point out that the introduction of modes divides the scattering (radiation) problem in two phases. The first phase consists in computing mode amplitudes. The second phase involves the calculation of the radiation field of induced currents. The induced currents radiation field in free space determines the scattering field (radiation field).

When the antenna is excited by a spherical wave of the H differs from (70) in that the type, the expression for and functions swap positions in it. That is

(71) and expressions:

in (70) and (71) stand for the following

(72) IV. EXCITATION OF MODES BY SPHERICAL WAVES. FREQUENCY DEPENDENCE OF MODE AMPLITUDE In this section, we consider the problem of frequency dependence of amplitudes of modes excited by spherical waves. The spherical wave source is located outside the antenna system, naturally, and to calculate the mode amplitude, it is necessary exterior field in (42) is to use (42). The the field of the spherical wave. It is determined by (2), where the Hankel spherical function should be replaced with Bessel spherical function. Such a replacement and a the substitution of (2) in (42) give an expression for the mode amplitude, which after carrying out the rearrangements described in [9] may be written as follows:

(69)

(73) functions with different superscripts, which The appear in (70)–(73) are determined by identical expressions. This being the case, we will detail one of them and indicate what modifications should be done in the expression to write other functions

(74)

KOVALYOV AND PONOMAREV: RESONANCE MODE THEORY FOR EXTERIOR PROBLEMS OF ELECTRODYNAMICS

( or ) in (74) stands for Cartesian compomode electric field. To compute nents of the in (74), it is necessary to replace with . For the calin (74), should be omitted. An exculation of pression for may be derived from (74) by substituting for and for . To obtain an expression , it is necessary to substitute for for and omit . The chief value of the obtained equations is that they reduce the problem of studying antenna frequency characteristics to examination the equitype functions (74). The mode amplitude in case of a perfect conductor antenna is determined by (66). The integral transformation in (66) is similar to the integral transformation in (42) and leads to (69). function in such a case is determined by the The following expression: for for

waves waves (75)

in (75) is determined by (74) through the mode current constituents

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(78) in (76) and carry out integration. Integration with respect to and is needless, as these coordinates have no effect on the current. Having performed the integration, we obtain

(79) 1 if l and The Kronecker delta in (79) is equal to 0 when l. Further, the functions should be computed. We restrict ourselves to calculating the scattering matrix entries determined by H-type spherical waves. So a substitution of (79) in (73) yields

(80) Since our research interests lie in frequency characteristics of the antenna, we omit frequency-independent multipliers in this and subsequent equations. In view of that, we rewrite (80) as follows: (81)

. (76) Expression (69) provides a solution for the problem considered in this section. It determines the frequency dependence of the mode amplitude during antenna excitation by a spherical wave. Let us make a comparison of the theoretical results obtained in this section with the results of a numerical analysis. We consider a circular-loop antenna and calculate the frequency dependence of mode amplitudes in it. A radius circular loop made of a thin wire is assumed to be located in the horizontal plane . The loop has one input positioned on the axis with 0. Also, given a wire antenna, it is more expedient to deal with current rather than current concentration . The current in the loop has an azimuthal component only, and current modes and . The single are determined by the expressions eigenvalue corresponds to both of these modes. However, with the spherical wave field being dependent on the azimuthal , one may assume that the angle coordinate of the form dependence of the mode current will take the form

Expression (81) shows that in a circular loop placed in the field of a spherical wave there comes to existence only one mode, the number of which coincides with the azimuthal number of the spherical wave. A substitution of (81) in (69) defines the frequency dependence of the mode amplitude for for

.

(82)

The results of analytical calculations performed by using (82) have been compared with the results of the numerical analysis. The numerical calculations have been performed by using the method described in [17]. According to this method, the wire matrix of mutual loop is divided in little segments, and the impedance between the segments is computed. The vector for current in the basic segments of the antenna is calculated by using the equation (83)

(77)

where in (83) is the matrix inverse to the mutual constituents of the vector are impedance matrix. The defined by the scalar product of two 3-D vectors

The Cartesian constituents of the mode current are computed as follows:

(84)

(78) To calculate the functions that define the entries of the generalized scattering matrix, it is necessary to substitute

stands for the intensity vector of the spherical wave where denotes electric field in the middle of the th segment, and the vector directed from the beginning of the th segment toward its end. Many versatile antenna analysis programs presently exist . However, there are two good reasons why such “make-it-yourself” method as suggested here turns out more expedient in

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in (86) denotes the electric field determined by (2) and (3), provided that the sign of in them changes to the opposite and is substituted for the function. Equation (86) determines the amplitude of the E-type wave and the H-type wave. While calculating the E-type wave amplitude, it is necessary to substitute the E-type wave electric field in (86) and the electric field of the H-type wave when calculating the amplitude of the H-type wave. magnetic current element is The contribution of the defined by the expression similar to (86) (87)

solving the problem in question. First, it enables antenna analysis for spherical waves, which is not always possible with versatile programs. Second, it allows for real and complex frequencies. Among other things, complex resonance frequencies have been calculated based on the condition of the mutual impedance matrix determinant going to zero

in (87) denotes the magnetic field that is deteris reversed mined by (2) and (3), provided that the sign of and is replaced with . To calculate the amplitude of the radiated spherical wave, it is necessary to integrate (86) and (87) over the antenna volume. During integration and substitution of induced curand rent expressions (45), inner products are obtained, which are expressed in Secand functions. The use tion IV through the of these functions allows writing the following expression for spherical wave amplitudes connected with the mode:

(85)

(88)

Fig. 4. Mode excitation by the spherical wave with indices dependence of the mode amplitude.

n; m. Frequency

Let us present the computed complex resonance frequency values for the circular loop under consideration. The loop circle 50 mm) and radius is assumed to be equal to 50 mm the wire radius is equal to 1 mm. Calculation by (85) yields complex resonance frequency values the following , , and . The obtained values were used in calculations using (82). Fig. 4 depicts the calculated curves. The chart illustrates the dependence of the mode amplitude on the frequency . The solid lines represent the theoretical dependence curves, calculated by (82). The symbol indicates the computed values. The results of the numerical analysis differ but little from the theoretical plots. By this means, the numeric calculations confirm the verity of the obtained design equations.

The function is similar to the and is calculated by using the following equation:

function

(89) if the radiated wave is of the E-type. In case of an H-type wave, and swap places

V. SPHERICAL WAVE EMISSION BY MODES. FREQUENCY DEPENDENCE OF SPHERICAL WAVE AMPLITUDE In this section, we examine the frequency dependence of the amplitude of a mode-radiated spherical wave. The radiation field is produced by induced currents connected with the mode field by relation (45). In case of a perfectly conducting antenna, the induced current is defined by (67). To solve the stated problem, we exploit the results obtained in [18], a study that considers the amplitudes of spherical waves radiated by elementary sources (i.e., electric and magnetic dipoles). amplitude of the spherical wave, produced by the The can be written as follows: electric current element

(86)

(90) For a perfectly conducting antenna, while integrating the contribution of the mode current to the spherical wave amplitude, the inner product expressed through (75) is obtained (91) Expressions (88)–(91) provide a solution to the problem stated in this section. These equations define the relationship

KOVALYOV AND PONOMAREV: RESONANCE MODE THEORY FOR EXTERIOR PROBLEMS OF ELECTRODYNAMICS

Fig. 5. Frequency dependence of mode-radiated spherical wave.

between the mode and the amplitude of the spherical wave radiated by the antenna. To confirm the obtained theoretical results, let us consider the same circular-loop antenna covered in Section IV. For the anfunction is independent of tenna in question, the and is determined by (81). Substituting (81) in (91), we obtain a theoretical frequency dependence for the amplitude of the spherical wave produced by the current mode for for

,l .

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1) the frequency dependence of the current-mode amplitude when the antenna is excited by the feeder forward wave; 2) the frequency dependence of the current-mode amplitude when the antenna is excited by the exterior field of a spherical wave; 3) the interrelation between the feeder backward wave amplitude and the amplitude of the current mode; 4) the frequency dependence of the amplitude of the moderadiated spherical wave. After solving the four problems stated brfore, it would be easy to obtain expressions for the entries of submatrices , , , and the -generalized scattering matrix. With problems 2 and 4 already solved in Sections IV and V, let us now turn to problems 1 and 3. Problem 1. Antenna Excitation by the Feeder Wave: When solving problem 1, the source exciting the antenna is assumed to be the end of the TEM line, along which the unit wave propagates. The source is assumed to have small dimensions and located in the spatial domain, the parameters of which are some. In such a case, (33) should be used thing other than and to compute the mode amplitude. The integrals in (33) can be written in a simpler form. While transforming the first of them, we write the volume integral as transverse coordinates and the integral the integral over the longitudinal coordinate over the (93)

(92)

The plot of function (92) is depicted in Fig. 5. Unlike the graph illustrated in Fig. 4, this plot does not have resonance spikes since the mode amplitude is assumed constant when computing it. The results of a numerical calculus, denoted in Fig. 5, prove the verity of the obtained by the symbol equation.

volume current density inWe take into account that the tegral over the transverse coordinates yields the current flowing integral over the longitudinal coordiin the feeder. The feeder voltage produced by nate gives the amplitude of the the mode. By this means (94)

VI. FREQUENCY DEPENDENCE OF GENERALIZED SCATTERING MATRIX ENTRIES The present section describes an inquiry into the frequency dependence of antenna-generalized scattering matrix entries. The application of SEM in the study involves the introduction of interim variables (i.e., modes). The introduction of modes divides the problem of frequency dependence finding into two steps. The first step consists in calculating the frequency dependence of amplitudes of modes produced by an exterior influence. The second step includes finding the frequency dependence of output values on modes. As is apparent from the physical meaning of entries of the generalized scattering matrix, the exterior influence is either the feeder forward wave or an exposure to the exterior electromagnetic field of a spherical wave. The output value (response) is either the feeder backward wave or the complex amplitude of the electromagnetic-field spherical wave radiated or scattered by the antenna. To find the frequency dependence of generalized scattering matrix entries, the following four factors need to be established:

In calculating the second integral of (33), it is assumed that the exterior magnetic current is the surface current and is connected with the exterior electric field of the line wave through the relation (95) The second summand in (33) then takes on the form (96)

in (96) represents the cross-section of the line, and denotes the cross-section normal. For further transformations, we use a local system of coordinates connected with the antenna input terminals which is illustrated in Fig. 6 As is evident from Fig. 6

(97)

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ideal voltage source, the mode amplitude can be computed by using the equation (103)

Fig. 6. System of coordinates connected with the electric field at the feeder end.

A substitution of (97) in (96) and a division of the integral into a longitudinal electric-field integral and a transverse integral result in the following expression:

Equations (102) and (103) provide a solution for problem 1 (i.e., they determine the frequency dependence of the mode amplitude in the antennas excited by the sources connected to them). Problem 3. Interrelation of Mode and Output in a Receiving Antenna: Problem 3 consists of finding every mode’s contribution to the antenna’s output. Assume that in addition to the mode field, the mode current is uniquely related to it and flowing through the input terminals is also known. The product of this current by the wave impedance of the th feeder amplitude of the feeder backward wave, which is gives the exactly what output is

(98) (104) in (98) is the Taking into account that the integral over wave voltage by the end of the line, and the integral over is the mode current , we obtain (99)

The integral in (33) can be brought to a yet simpler form, if we take into consideration that voltage and mode current are interrelated through the wave impedance of the line and express wave voltage and current through the reflection coefficient from the antenna’s input (100) Taking into account (94), (99), and (100), we can write (101)

A substitution of (101) in (33) shows that with the antenna exited by the feeder wave, the mode amplitude can be calculated by using the equation (102) In writing (102), we allowed for the possibility that the antenna may have several inputs by adding the input number subto and . Expression (102) provides script a solution to problem 1, that is, it determines the frequency dependence of mode amplitude in an antenna excited by the feeder unitary wave. In actual practice, it is often assumed that the antenna is excited by the ideal unitary amplitude voltage source. In such a case, only the second inner product is left enclosed in curly brackets in (33). The said inner product is expressed by (99) where 1 is set. Therefore, for the antenna excited by the

If the antenna when radiating is considered as being excited by an ideal voltage source, then it is logical to assume that in the receiving antenna, its output terminals are short-circuited. The -mode current flowing antenna’s output in this case is the through its output terminals. Having solved the four problems stated in the beginning of this section, let us write frequency-dependence expressions for the entries of different generalized scattering matrix submatrices. The combination of the solutions for problems 1 and 3 represented by (102) and (104) defines the entries of the submatrix . The combination of the solutions for problems 2 and 3 represented by (69) and (104) provides the entries of the submatrix . The combination of the solutions for problems 1 and 4 represented by (88) and (102) defines the submatrix . Finally, the combination of the solutions for problems 2 and 4 represented by (69) and (88) provides the entries of the submatrix of the generalized scattering matrix. By these means, the entries of the generalized scattering matrix can be obtained for an antenna . It is comprising a material with the parameters believed that the antenna has feeders. The numbers of equations pertaining to this antenna are identified by the letter “a.” The equations with the letter “b” in the number pertain to the perfect conductor antenna. The antenna is believed to have voltage sources connected to its input terminals when radiating. When receiving, the antenna’s output consists of currents flowing in the antenna’s short-circuited terminals. When writing entries of the submatrices, we use 0 to distinleaving the guish the subscripts of input effects subscripts of output values unchanged

(105a)

(105b)

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(106a)

(106b)

(107a)

(107b)

(108a)

(108b) The results of computations by (105b)–(108b) were compared with the results of a numerical analysis. A dipole made of a thin wire with negligible loss was considered. The dipole of the length was positioned on the axis with the input at the point of origin. The fields radiated and scattered by the so-positioned dipole are independent of the azimuthal coordinate. Therefore, the subscript is zero for all nonzero entries of the generalized scattering matrix. A numerical analysis confirms that the dipole-mode currents are readily approximated by harmonic functions (109) Equation (109) describes modes with symmetric current distribution. The excitation of modes with asymmetric distribution was not considered at all. The complex resonance frequen0.275 m are cies of modes (109) when the length , , and . These frequencies were used in computations by (105b)–(108b). The results of the computations are illustrated in Fig. 7. In case of a single-input antenna, the submatrix comprises one entry, which is equal to the ratio of the current complex amplitude at the antenna input to the amplitude of the input voltage that is the input admittance. The plot illustrating the frequency dependence of the dipole admittance is depicted in Fig. 7(a). Fig. 7(b)–7(d) illustrates the frequency dependence of entries absolute values of the , , and submatrices. The solid lines represent the results of theoretical estimates; the symbols “ ”

Fig. 7. Frequency dependence of absolute values of the dipole-generalized submatrix entry. (c) scattering matrix entries. (a) Input admittance. (b) submatrix entry. (d) submatrix entry.

S

R

T

are used to indicate the results of numerical computations. In conclusion of this section, let us emphasize an important merit of the obtained generalized scattering matrix entry fre-

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quency-dependence equations (105)–(108). Apart from resonance multipliers, these expressions include only equitype fre(74). An examination of these funcquency functions tions is the subject matter of the next section. VII. SAMPLING THEOREM FOR THE HANKEL TRANSFORM AND DISCRETE ANTENNA MODELS It was demonstrated in Section VI that the frequency dependence of all antenna-generalized scattering matrix entries is deand similar termined by the resonance multipliers functions (74) or (76). Let us consider one of these functions

(110) The volume integral in (110) is written in spherical coordinates. is the distance from the point of origin to the antenna’s denotes one of the Cartesian confarthest point. of the th mode current. stituents Let us introduce the following denotation:

(111) superscripts are of little significance in this section, The and we omit them for brevity’s sake. Equation (110) in view of designation (111) takes on the form (112)

Fig. 8. Sampling plots for Bessel function if n

= 2.

There is a simple explanation to the fact that entries of the generalized scattering matrix comprise functions that are the Hankel transform of the radial coordinate function. The frequency dependence of spherical wave fields as is apparent from Bessel spherical functions with (2) is determined by the functions appear different subscripts. This is why the in the kernel of integral transformation that links frequency despatial value functions and a transform pendencies and the kernel is the Hankel transform. with the (112) functions, let us For discrete representation of the use the sampling theorem considered in [8] and [20], and being a generalization of the classic sampling theorem. This theorem is rarely used for discrete presentation of signals, but harnessing it for the purposes of our problem seems natural. Assuming the notation convenient for propagation and scattering problems, function of (112) as which is used in [9], we can write the

spherical Bessel Let us make a transition from the function to the Bessel function of the half-integer subscript

(116)

(113) in (116) is the th zero of the Bessel spherical function Then, we can write (112) as follows: (117) (114) The Hankel Compare (114) with the Hankel transform. The function is defined by the following equatransform of the tion [19]: (115)

A comparison of (114) and (115) reveals that the function is the Hankel transform of the function.

sampling functions are determined by

(118)

The graphs of sampling functions plotted using (118) for 4, 5, 6 and 1.5 are depicted in Fig. 8. It is evident from the plot that with , the value of the function is 1. These function values are marked in the plot by the symbols . As increases, the function maximum shifts to the right along the abscissa axis. Fig. 9

2;

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by (70), (71), and by using (89) and (90). Further, the entries of the antenna’s generalized scattering matrix are calculated by using (105a)–(108a). VIII. CONCLUSION

Fig. 9. Recovery of the function a

(k ) from four values with p = 3, n = 2.

illustrates the function recovery from its discrete values in accordance with (116). We considered the same dipole covered in Section VI. The solid line in Fig. 9 represents the continuous function computed by using (110). During the computation, we used (109) for current and assumed that 3, 2, 0. To recover the continuous function, we used four discrete values, represented in the plot by symbols . The plot of the recovered function computed by using (116) is represented in Fig. 9 by the dotted curve. It is apparent from Fig. 9 that the recovery inaccuracy is minute despite the small number of used discrete samples. The offered example illustrates the feasibility of recovering a continuous function from its discrete values in accordance with the generalized sampling theorem. The application of the generalized sampling theorem completes the process of discrete antenna modelling. The obtained model is versatile and holds good for any antenna of limited size made of materials with frequency-independent and containing perfect conductors. Let us explain the use of computation equations that allows transition from discrete parameters to the generalized scattering matrix of the antenna. For an antenna comprising perfect conductors only, the discrete complex resonance frequencies (or model includes the complex wavenumbers), sets of complex numbers, and complex mode currents that flow through the antenna incontinuous function is computed by disputs. The crete values using (116). The functions in it are determined by (118). Then, the functions are calculated , and , which are defined by (72) and (73). by using (75), Finally, the entries of the generalized scattering matrix for the frequencies of interest are calculated by using (105b)–(108b). parameters For the antenna, in which the of the antenna material are dependent on space coordinates, the range of discrete parameters is vaster. , , , , and should be determined instead of . These sets enable the recovery of the functions , , , , and by means of (116). The functions help to compute

In conclusion, let us sum up the main findings of this study. One of its principal achievements is a theoretical substantiation of the applicability of the field eigenfunction expansion method, also known as the SEM, to exterior problems of the electrodynamics. The mode in the perspective of SEM is conventionally regarded as a free-oscillation field. We demonstrate that it is more expedient to view the mode as the electromagnetic field in the scatterer (antenna) with the understanding that the medium within the scatterer is activated, and the scatterer turns into an undamped oscillator. Such definition of modes facilitates the proof of their orthogonality and is conducive to presenting the field as a sum of modes. As a result, computational equations are derived that provide a tool for solving a variety of wave propagation and scattering problems. The parameters of radiating or may be dependent on spatial values; scattering objects wave sources are arbitrary. The suggested theory is used to study antenna space-frequency characteristics. The undertaken inquiry results in expressions for the frequency dependence of entries of the antenna-generalized scattering matrix. The obtained dependence expressions comprise resonance multipliers and frequency functions that are Hankel transforms of various spatial value functions. A generalized sampling theorem differing from the classic sampling theorem is used in discrete representation of the latter functions. This theorem is extremely rarely used in the representation of signals, though its application seems a natural next step in dealing with propagation and scattering problems. We hope that by harnessing the generalized sampling theorem to solve a variety of antenna engineering problems will mark the beginning of discrete electrodynamics. The performed studies result in the accomplishment of the goal that is the creation of a versatile discrete antenna model. (or ) complex resonance The model parameters include frequencies and complex numbers . the latter being disfunctions connected with crete sampling values of the Hankel transform of the spatial value function. We believe that the suggested technique will prove instrumental in frequency range measurements for antenna characterization. It also enables a compact form of presenting antenna spatial and frequency characteristics that may be beneficial in economizing the memory size required for such data storage. ACKNOWLEDGMENT The authors would like to thank Y. B. Akimov, the technical writer of MERA Networks, for his valuable advice on the manuscript. REFERENCES [1] R. J. Allard and D. H. Werner, “The model-based parameter estimation of antenna radiation pattern using windowed interpolation and spherical harmonics,” IEEE Trans. Antennas Propag., vol. 51, no. 8, pp. 1891–1906, Aug. 2003. [2] J. E. Hansen, Near-Field Antenna Measurements, ser. IEE Electromagnetic Waves Ser. 26. London, U.K.: Peregrinus, 1998.

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[3] Transient Electromagnetic Fields, L. B. Felsen, Ed. Berlin, Germany: Springer-Verlag, 1976. [4] E. K. Miller, “Model-based parameter estimation in electromagnetics, Part I. Background and theoretical development,” IEEE Antennas Propag. Mag., vol. 40, no. 1, pp. 42–52, Feb. 1998. [5] E. K. Miller, “Model-based parameter estimation in electromagnetics, Part II. Applications to EM observables,” IEEE Antennas Propag. Mag., vol. 40, no. 2, pp. 51–65, Apr. 1998. [6] E. K. Miller, “Model-based parameter estimation in electromagnetics, Part III. Applications to EM integral equations,” IEEE Antennas Propag. Mag., vol. 40, no. 3, pp. 49–66, Jun. 1998. [7] N. N. Voitovich, B. Z. Katsenelenbaum, and A. N. Sivov, Obobshchenyi metod sobstvennykh kolebaniy v teorii difraktsii. S dopolneniyem Agranovicha M.S. Spektralnyie svoistva zadach difraktsii. Moscow, Russia: Nauka, 1977. [8] H. P. Krumer, “A generalized sampling theorem,” J. Math. Phys., vol. 38, pp. 68–72, 1959. [9] I. P. Kovalyov and D. M. Ponomarev, “Discrete models of electromagnetic wave scatterers in frequency range,” IEEE Trans. Antennas Propag., vol. 58, no. 6, pp. 2046–2059, Jun. 2010. [10] J. D. Jackson, Classical Electrodynamics, 3rd ed. New York: Wiley, 1998. [11] J. A. Stratton, Electromagnetic Theory. New York: McGraw-Hill, 1941. [12] R. F. Harrington, Time-Harmonic Electromagnetic Fields. New York: McGraw-Hill, 1961. [13] P. A. Angot, Complements de mathemaiques al’usare des ingenieurs de l’elektronique et des telecommunications. Paris, France: Editions de la Revue d’Optique, 1957. [14] I. P. Kovalyov, “SDMA for multi-path wireless channels,” in Limiting Characteristics and Stochastic Models. New York: Springer, 2004. [15] L. A. Vainshtein, Electromagnitnyie Volny. Moscow, Russia: Radio i svyaz, 1988. [16] G. A. Korn and T. M. Korn, Mathematical Handbook. New York: McGraw-Hill, 1968.

[17] R. F. Harrington, Field Computation by Moment Methods. New York: Macmillan, 1968. [18] V. I. Okhmatovski and A. C. Cangellaris, “Efficient calculation of the electromagnetic dyadic green’s function in spherical layered media,” IEEE Trans. Antennas Propag., vol. 51, no. 12, pp. 3209–3220, Dec. 2003. [19] Tables of Integral Transformations. New York: McGraw-Hill, 1954, vol. II, Based, in part, on notes left by H. Bateman and staff of the Bateman manuscript project. Director A. Erdelyi. [20] A. J. Jerri, “The Shannon sampling theorem—Its various extensions and applications, a tutorial review,” Proc. IEEE, vol. 65, no. 11, pp. 1565–1596, Nov. 1977.

Igor P. Kovalyov received the Ph.D. degree in engineering science from Nizhny Novgorod University of Technology, Nizhny Novgorod, Russia, in 1969. Currently, he is Associate Professor of the Chair “Theory of circuits and telecommunications” at the University and is a Researcher with MERA NN, Nizhny Novgorod, Russia. His current research interests include multiple-input multiple-output antenna systems and communication channel modeling. He is the author of five monographs, the latest two of which are devoted to high-speed communications systems.

Dmitry M. Ponomarev (M’05) received the M.A. degree in telecommunications engineering and the Ph.D. degree in engineering science from the University of Technology, Nizhny Novgorod, Russia, in 1974 and 1979, respectively. Before starting his entrepreneurial career in 1989, he was Professor at the University of Technology, Nizhny Novgorod, Russia, where he organized a suite of specialized training courses in a variety of front-end IT spheres. His academic experience includes serving as a Research Engineer in the field of antenna measurement. He is the author of many publications while holding 12 patents.

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Compact and Multiband Dielectric Resonator Antenna With Pattern Diversity for Multistandard Mobile Handheld Devices Laure Huitema, Majed Koubeissi, Moctar Mouhamadou, Eric Arnaud, Cyril Decroze, and Thierry Monediere

Abstract—A compact multiband antenna system using a dielectric resonator antenna (DRA) is presented in this paper. Designed to be integrated in a tablet, it is not only heavily miniaturized ( at 800 MHz), but also able to cover three frequency bands for different wireless applications (DVB-H, WiFi and WiMAX). Since a reconfigurable radiation pattern can result in an improved quality and reliability of wireless links, two DRAs have been integrated to implement this feature on the three frequency bands. These improvements are demonstrated by the presentation of the correlation coefficient and the effective diversity gain, both measured in a reverberation chamber. The experimental measurements are in very good agreement with the simulated ones. Good overall performances are obtained, and the requirements of all three applications are perfectly met. Moreover, the antenna operative frequencies are independent of the ground plane dimensions, making this system a very good candidate for all kinds of mobile devices. Index Terms—Dielectric resonator antenna (DRA), Digital Video Broadcasting-Handheld (DVB-H), miniature antenna, multiband antenna, pattern diversity, WiFi, WiMAX.

I. INTRODUCTION

A

NTENNA design for mobile communications is often problematic by the necessity to implement multiple applications on the same small terminal. Dielectric resonators using high-permittivity materials were originally developed for microwave circuits, such as filters or oscillators as tuning element [1]. Then, in 1983, Long et al. introduced using the dielectric resonator as an antenna [2] by exciting different modes using multiple feeding mechanisms. In the last decade, dielectric resonator antennas (DRAs) [3]–[5] have been widely used for their multiple advantages: compact size, light weight, low cost, high radiation efficiency, and feeding simplicity. Nowadays, DRAs appear as a viable alternative to more conventional low-gain elements, such as dipoles, monopoles, microstrip patches, and 3-D planar inverted-F antennas (PIFAs) [6]. The most integrated antennas currently developed for portable wireless systems have a planar structure based on microstrip patches or PIFAs [7]–[9]. These antennas present

Manuscript received July 22, 2010; revised March 29, 2011; accepted April 04, 2011. Date of publication August 12, 2011; date of current version November 02, 2011. This work was supported in part by the French Research Agency project NAOMI. The authors are with XLIM Laboratory, OSA Department, Faculté des Sciences et Techniques, 87060 Limoges, France (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2164183

a low efficiency, especially when small, because of metallic losses. DRAs do not suffer from such losses, which makes them a good alternative for these more conventional antennas. Much attention has been given to DRA miniaturization [10], [11] in order to integrate them inside mobile handheld devices. Other studies concentrated on the enhancement of the impedance bandwidth [12]–[17], sometimes to the detriment of the size. In the last decade, the huge demand for mobile and portable communication systems has led to an increased need for more compact antenna designs. This aspect is even more critical when several wireless technologies have to be integrated on the same mobile wireless communicator. All the new services and the increased user density are driving the antenna design toward multiband operation. Recently, many studies have been devoted to multiband antennas [18]–[22], some of them dealing with DRAs [23]–[25]. A dielectric resonator indeed supports more than one resonant mode at two close frequencies, which allows them to meet the requirements of different applications with a unique device. Some studies furthermore use both the dielectric resonator and the feeding mechanism as radiator elements [24]–[26]. This explains why the DRAs present a major advantage for multistandard devices when compared to other kinds of antennas. This current study has been performed for the NAOMI project, supported by the French ANR (National Research Agency). The objective was the integration of a small antenna on a multiband mobile handheld device, working on the nine channels of the Digital Video Broadcasting-Handheld (DVB-H) band from 790 to 862 MHz, the WiFi band at 2.4 GHz, and the WiMAX band at 3.5 GHz. Additionally, the antenna radiation pattern had to be reconfigurable in order to improve the quality and the reliability of the wireless links. To obtain pattern diversity, it has been decided to integrate two orthogonally aligned antennas in the allocated space of at 800 MHz), on a 230 130-mm 30 41 mm ( ground plane. Each DRA will therefore have to be very compact to be able to fit in such a limited area and will operate around 850 MHz, 2.4 GHz, and 3.5 GHz, thus covering the nine channels of the DVB-H band, the WiMAX band, and the WiFi band. In the first section, only one radiating element is described. Firstly, a modal analysis [27] of the dielectric resonator is presented, with the influence of the essential parameters [28]. This part focuses on the DRA design and the choice of the dielectric permittivity. The Finite Integration Temporal method and the eigenmode solver of CST Microwave Studio were used to

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Fig. 1. (a) Top view of the defined printed circuit board (PCB) card. (b) Allocated size for the antenna system. (c) Bottom view of the defined PCB card.

Fig. 2. (a) Dimensions and properties of one resonator. (b) E-field distribution mm plane. of the first natural mode for the resonator in (a) in the

carry out this work. The next part will focus on the design of the global structure. We will develop that the antenna operative frequencies are independent of the ground plane dimensions. All of these parts will include both measurements and simulations results. Finally, the prototype characterization in a reverberation chamber will be detailed, and the benefit of pattern diversity will be demonstrated. This last part will also present the effective diversity gain and the correlation coefficient on all frequency bands. II. ANTENNA DESIGN AND PARAMETRIC STUDIES A. DRA Design and Modal Analysis This section focuses on the obtaining of three resonant frequencies for a single dielectric resonator. First, the resonator has to be integrated in a handheld receiver, which means that it will be placed on a FR4 substrate . Second, the allocated space for the antenna system must not exceed mm as shown in Fig. 1. The dimensions and dielectric permittivity of each resonator need to be chosen according to these constraints to ensure the integration of the antenna in the final device. As a result, the dimensions of one resonator were chosen to be mm , with a very high dielectric constant of 37. The resulting geometry is shown in Fig. 2(a). The first natural mode of this resonator is the [Fig. 2(b)], which resonates at 3.99 GHz. This resonance frequency remains too high for the intended applications. It has, however, been shown [14] that this resonance frequency depends on the metallization of the DRA’s faces. It is therefore necessary to envision the feeding mechanism of the resonator before performing the modal analysis. Indeed, if the antenna is not fed by proximity coupling, the excitation cannot be ignored during the modal analysis. Furthermore, it can be used to adjust the resonance frequency of the resonator, especially in the case of an electrically small DRA. In this paper, the feeding will indeed play a preeminent role. The chosen excitation is a line printed on the FR4 substrate and positioned under the dielectric resonator as shown in Fig. 3. The E-field distribution of the first mode (Fig. 3) is completely

Fig. 3. Design of the DRA fed by the printed line.

different from the one obtained without this line. Indeed, as stated before, the E-field and H-field distributions inside the DRA depend on the boundary conditions on its faces. The feeding line introduces partially perfect electric conducting conditions, which disturb the field distribution inside the resonator when compared to the mode. As a result, the resonance frequency of each mode will vary in accordance with the length and width of the feeding line (defined in Fig. 3). Considering the previous study, the resonator design method is as follows. First of all, in spite of the high dielectric permittivity, there is no mode around 800 MHz. In order to allow the antenna to operate on the nine channels of the DVB-H band, the printed line will be designed to resonate around 800 MHz. It will therefore behave like a printed monopole loaded by a dielectric. The length of this line, which will from now on be referred to as the “monopole,” will be set to obtain the first resonance frequency around 800 MHz. This being done, the second and third band will be covered by the resonances of the dielectric resonator disturbed by the presence of the monopole. As previously explained, the length, width, and shape of this monopole entails a modification of the DRA resonance frequencies. The shape and width of the monopole will therefore be optimized to obtain the desired resonance frequencies, while its length remains set by the first resonance. An important point concerns the radiation -factor. The high dielectric permittivity involves a high -factor as shown by the following relation [5]: (1) With such an , the radiation -factor is also important, making it difficult to obtain a wide impedance bandwidth for a given mode. Therefore, to have a suitable impedance bandwidth, the antenna will have to be matched between two peaks of the real part of input impedance. This way, the resonances will not have to be close to the operating bands. The goal being to match the DRA over the WiFi and WiMAX bands, the monopole (its shape and width) has to be optimized to obtain resonance frequencies around 2, 2.8, and 4 GHz. A modal analysis has been performed to show the variations of the resonance frequencies of the first three modes, according to the shape and width of the monopole. Fig. 4 shows the resonance frequency of the first mode according to the monopole geometry. With the first resonance frequency set to match the antenna at 2.4 GHz and the length of the monopole set to have the first resonance at 800 MHz, the graph of the modal analysis

HUITEMA et al.: COMPACT AND MULTIBAND DRA WITH PATTERN DIVERSITY

Fig. 4. Resonance frequency of the first dielectric resonator mode according to the width and the length of the monopole defined Fig. 3.

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Fig. 6. Reflection coefficient for one resonator.

Fig. 7. Final design of the antenna structure.

Fig. 5. Input impedance of the fed dielectric resonator. TABLE I VALUES OF RESONANCE FREQUENCY FOR THE FIRST THREE MODES

allows an easy determination of the monopole width, which is 1 mm. The same studies have been performed for different shapes of the monopole. All these studies allowed the shape, length, and width of the monopole to be set, which led to the final design of the dielectric resonator. Table I shows the values of the resonance frequencies for the first three modes inside the resonator, which were obtained through the modal analysis. B. Electromagnetic Study of the DRA In order to validate the previous modal study, the dielectric resonator, placed in the area dedicated to the antenna (Fig. 1) and fed by a 50- discrete port, has been simulated with the FIT method using CST Microwave Studio. It must be noticed that a discrete port is modeled by a lumped element, consisting of a current source with a 50- inner impedance that excites and absorbs power. Fig. 5 shows the simulated input impedance of the dielectric resonator with its feed. The resonance frequencies are in agreement with the modal analysis. The radiation -factor is important, and the input impedance variations confirm that the antenna matching is easier between two resonances.

The reflection coefficient is shown in Fig. 6. The antenna is matched on all of the desired bands, i.e., the nine channels of DVB-H going from 790 to 862 MHz, the WiFi band at 2.4 GHz, and the WiMAX band at 3.5 GHz. It can be noted that the matching over the first band is obtained due to the resonance of the monopole. The dielectric resonator must now be integrated in its context, i.e., on a 230 130-mm ground plane, chosen to correspond to a standard DVB-H handheld receiver, as was specified in the project presentation (Fig. 1). As explained before, another specification was to obtain a reconfigurable radiation pattern. Thus, two instances of the previously studied resonator are orthogonally integrated on the ground plane. This structure will be the subject of the following section. III. FINAL STRUCTURE AND ANTENNA PERFORMANCES A. Final Structure Based on the previous parametric study, the final structure has been designed as shown in Fig. 7. In order to obtain pattern diversity, two dielectric resonators are orthogonally disposed in a 30 41-mm area, both fed by a printed line acting as a monopole. Each line is fed by a 50- coaxial cable. They are studied on a 230 130-mm ground plane, as defined by the specifications. It will be shown in a following section that the antenna matching is not affected by the ground plane dimensions. B. Manufactured Prototype In order to ascertain the performances of this antenna, a prototype has been fabricated as shown in Fig. 8. The resonators are manufactured with a ceramic material with a dielectric permittivity of 37 and a loss tangent on the 0.5–10-GHz band. During the simulation and measurements, each resonator has been excited by a printed line fed by a coaxial cable.

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Fig. 8. Photograph of the antenna.

During the manufacturing process of the antenna, a special care has to be given to minimize the air gap between the excitation and the resonator. Indeed, an air gap results in a lower effective dielectric constant, which entails both a decrease in the -factor and a shift of the resonance frequencies [5]. This air gap has a larger impact on the dielectric resonator modes than on the resonances of the monopole. In the case of this prototype, the resonators have been pressed onto the PCB to avoid this air gap. Therefore, Section III-C focuses on the measured and simulated results without air gap. C. Measured and Simulated Performances of the Antenna This section deals with the comparison between the simulated and measured results. The first ones have been obtained using the transient solver of CST Microwave Studio, while the measurements have been performed inside an anechoic chamber. 1) -Parameters, Realized Gain and Total Efficiency: The different parameters are presented in Fig. 9, specifically the in Fig. 9(a), the in Fig. 9(b) and the in Fig. 9(c). The measurements and simulations are in very good agreement. Moreover, the antenna is matched over all the desired bands and for both inputs. Fig. 10 shows the measured and simulated realized gains, along with the total efficiency computed for the first input. As the realized gain takes into account antenna, cable and connector losses, it is defined as .A higher coupling will therefore result in a lower realized gain. The 0-dB reference of the realized gain is an isotropic antenna, which is a perfect omnidirectional radiator. Moreover, the total efficiency is defined as the ratio of radiated to stimulated power of the antenna or the ratio of the realized gain defined previously to the directivity of the antenna. These definitions have been used during the simulations and measurements processes. For the first band (nine channels of the DVB-H band), a good agreement appears between measurement and simulation. Furthermore, the measured value is always higher than the requirements, despite the strong coupling [Fig. 9(b)] between the two inputs. For the WiFi and the WiMAX bands [Fig. 10(b) and (c), respectively], the realized gain remains higher than 0 dB, thus largely satisfying the required specifications for these two applications. It must be stressed that the results are completely similar for the second input. 2) Radiation Patterns: The radiation patterns have been measured inside an anechoic chamber. Table II shows the radiation patterns (total radiated field ) in the plane

Fig. 9. (a) Measured and simulated parameter. (b) Measured and simulated parameter. (c) Measured and simulated parameter.

(the one containing the ground plane), at 830 MHz, 2.4 GHz, and 3.5 GHz and for both inputs. During the radiation pattern measurement for the first input, the second one was fitted with a 50- load and vice versa. Table III shows the 3-D simulated radiation patterns for both inputs at 830 MHz, 2.4 GHz, and 3.5 GHz. It can be seen that the radiation pattern at a given frequency will depend on the excited port. While promising, this result is not sufficient to conclude that the radiation pattern is reconfigured. This requires the characterization of the whole system in a reverberation chamber in order to determine the correlation coefficient. The experimental results are presented in Section IV. D. Effect of the Ground Plane Dimensions The study presented hitherto was conducted in order to fulfill the project requirements, i.e., the integration of the antenna structure in a tablet. This section is dedicated to the integration of the antenna structure in a 90 50 mm area, which is the standard dimension of a mobile handset. The -parameters obtained with a 230 130-mm and a 90 50-mm ground plane are compared in Fig. 11. This figure

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TABLE II SIMULATED (SOLID LINE) AND MEASURED (DASHED LINE) RADIATION PATTERNS AT 830 MHz, 2.4 GHz, AND 3.5 GHz IN THE -PLANE

TABLE III 3-D RADIATION PATTERNS AT 830 MHz, 2.4 GHz, AND 3.5 GHz FOR THE TWO INPUTS

Fig. 10. (a) Measured and simulated maximum realized gain and total efficiency for the nine channels of DVB-H versus the required realized gain for the DVB-H application. (b) Measured and simulated maximum realized gain and total efficiency for the WiFi band. (c) Measured and simulated maximum realized gain and total efficiency for the WiMAX band.

underlines that the matching bands are similar for both ground plane dimensions, and that the antenna remains matched over all three bands in both cases. Indeed, the resonance frequencies of the monopole and the DRA remain the same and are independent of the ground plane. It is important, however, to verify that the applications requirements are met at the most critical frequencies, i.e., on the first matched band. As shown in Fig. 12, the realized gain for the two ports is modified by the ground plane reduction. The directivity is indeed 1 dB lower in the case of the 90 50 mm PCB card. The resulting realized gain is further decreased by a higher antenna coupling in the case of a smaller ground plane. The realized gain nevertheless remains higher than the DVB-H requirements [Fig. 10(a)], allowing this antenna structure to be integrated in all kind of mobile handsets.

Next, the antenna structure has been characterized in a reverberation chamber in order to obtain its diversity performances in an isotropic propagation channel. This will be the subject of Section IV.

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Fig. 11. and parameters for a 230 90 50-mm ground plane dimensions.

130-mm

and a

Fig. 13. (a) Envelope correlation coefficient for the nine channels of DVB-H. (b) Envelope correlation coefficient for the WiFi band. (c) Envelope correlation coefficient for the WiMAX band.

Fig. 12. Realized gain for the two ports for the nine channels of DVB-H versus the required realized gain for the DVB-H application.

IV. DIVERSITY PERFORMANCES MEASUREMENTS The performances of wireless communication systems can be greatly improved by using antenna diversity techniques. It is well known that the antenna diversity performances on compact handheld terminals are limited, especially when the allocated area for the antennas is small. Indeed, the resulting high mutual coupling directly impacts the performances [28], [29]. It has been shown previously that there is a high mutual coupling in the nine channels of the DVB-H band, while it is lower on the WiFi and WiMAX bands. The parameters presented to evaluate the diversity performances are the correlation coefficient [30], [31] and the effective diversity gain (EDG) [32]. The definition of the diversity gain and the measurement method in a mode-stirred reverberation chamber are presented in [33]. A. Envelope Correlation Coefficient The correlation between the signals received by the two antennas can be evaluated through the envelope correlation coefficient. The latter can therefore be used to check whether the radiation pattern is actually reconfigured. The envelope correlation coefficient can be determined using the far fields [(2)] or the -parameters [(3)]. Its value varies between 0 and 1, with a higher value corresponding to more similar signals. The necessary condition for diversity is an envelope correlation coefficient lower than 0.5 (2)

Fig. 14. Measurement process in the mode-stirred reverberation chamber.

(3) Fig. 13 shows the envelope correlation coefficient calculated with the two previously given equations for the three frequency bands. The one deduced from the reverberation chamber measurements at 835 MHz, 2.4 GHz, and 3.5 GHz is also presented. It remains lower than 0.5 with a very good agreement between the results. Thus, the radiation patterns are reconfigured for the nine channels of the DVB-H band, the WiFi band, and the WiMAX band. B. Effective Diversity Gain The effective diversity gain is evaluated in a mode-stirred reverberation chamber. The schematic of the measurement process is shown in Fig. 14. A reverberation chamber is a metallic cavity that supports a large number of resonant cavity modes. The statistically uniform field distribution can be obtained by mechanically steering these modes inside the chamber [34], [35]. As shown in Fig. 14, a fixed horn antenna excites the chamber where the antenna under test is positioned on a rotating arm. To

HUITEMA et al.: COMPACT AND MULTIBAND DRA WITH PATTERN DIVERSITY

Fig. 15. Effective diversity gain at 835 MHz.

obtain a statistically uniform channel, the stirrer and the antenna under test are rotated around the -axis. The cumulative distribution function (CDF) is calculated from the signal-to-noise ratio (SNR) measurements. The CDF curves at 835 MHz are plotted in Fig. 15. In abscissa, the threshold SNR is normalized by the SNR of the reference antenna. A half-wavelength dipole antenna, also measured in the reverberation chamber, has been used to provide this reference. Three different dipoles have been used in order to evaluate the effective diversity gain, each one being matched at the measured frequency and presenting a good total efficiency. It is obvious that the diversity techniques enhance the reception quality, while the effective diversity gain at a 1% probability is higher than 0 dB. This has been determined at three frequencies with the following results: around 5 dB at 835 MHz, and around 11 dB at 2.4 and 3.5 GHz for both Maximum Ratio Combining (MRC) and Equal Gain Combining (EGC). Nevertheless, the EDG is lower at 835 MHz due to a higher mutual coupling on the nine channels of the DVB-H band. V. CONCLUSION This study started with modal analyses, which allowed the shape and dimensions of the antenna’s excitation to be defined with a dual objective. Indeed, this line had first to behave like a monopole and cover the nine channels of the DVB-H band (going from 790 to 862 MHz). Second, it had to excite the dielectric resonator and set its resonance frequencies so as to match the antenna on the WiFi and WiMAX bands. After performing these preliminary studies, two instances of the conceived dielectric resonator have been orthogonally integrated on a 230 130-mm ground plane, which is consistent with a tablet. Finally, the antenna system, which only occupies a 30 41 mm area, is matched on the three desired bands, i.e., the nine channels of the DVB-H band, the WiFi band, and the WiMAX band. A similar study has been carried out for new ground plane dimensions standing for a “classical” mobile handset. The antenna system remains matched on the three desired bands, with a slight decrease in the total efficiency and the realized gain, which nevertheless remain higher than the requirements of the DVB-H application. Thus, the presented antenna system can be integrated in all mobile handsets.

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Two DRAs have been used in a diversity antenna design. Indeed, the envelope correlation coefficient has been presented for the three matched bands. This coefficient underlines the reconfiguration of the radiation pattern. Finally, the effective diversity gain has been characterized in a mode-stirred reverberation chamber, and the results clearly demonstrate the interest of integrating two resonators. This paper has presented the design, the realization, and the measurement of a compact and multiband DRA with pattern diversity. Indeed, the allocated dimensions for the antenna structure (composed of two radiating elements) do not exceed 30 41 mm ( at 800 MHz). Three matching bands have been obtained, allowing this antenna to be used for all DVB-H, WiFi, and WiMAX applications. The quality and reliability of the wireless link have been enhanced using the pattern diversity. This system is therefore among the rare DRAs able to cover these three bands while being capable of pattern diversity. REFERENCES [1] R. D. Richtmyer, “Dielectric resonator,” J. Appl. Phys., vol. 10, pp. 391–398, Jun. 1939. [2] S. A. Long, M. W. McAllister, and L. C. Shen, “The resonant dielectric cavity antenna,” IEEE Trans. Antennas Propag., vol. AP-31, no. 3, pp. 406–412, Mar. 1983. [3] A. Petosa, A. Ittipiboon, Y. M. M. Antar, and D. Roscoe, “Recent advances in dielectric resonator antenna technology,” IEEE Antennas Propag. Mag., vol. 40, no. 3, pp. 35–48, Jun. 1998. [4] M. Saed and R. Yadla, “Microstrip-fed low profile and compact dielectric resonator antenna,” Prog. Electromagn. Res., vol. 56, pp. 151–162, 2006. [5] K. M. Luk and K. W. Leung, Dielectric Resonator Antennas. Baldock, U.K.: Research Studies Press, 2002. [6] M. Manteghi and Y. Rahmat-Samii, “A novel miniaturized wideband PIFA for MIMO applications,” Microw. Opt. Technol. Lett., vol. 49, no. 3, pp. 724–731, Mar. 2007. [7] C. W. Ling, C. Y. Lee, C. Y. Tang, and S. J. Chung, “Analysis and application of an on-package planar inverted-F antenna,” IEEE Trans. Antennas Propag., vol. 55, no. 6, pp. 1774–1780, Jun. 2007. [8] M. J. Ammann and L. E. Doyle, “A loaded inverted-F antenna for mobile handset,” Microw. Opt. Technol. Lett., vol. 28, pp. 226–228, 2001. [9] M. Ali and G. J. Hayes, “Analysis of integrated inverted-F antennas for bluetooth application,” in Proc. IEEE-APS Conf. Antennas Propag. Wireless Commun., Waltham, MA, 2000, pp. 21–24. [10] Y. Gao, B. L. Ooi, W. B. Ewe, and A. P. Popov, “A compact wideband hybrid dielectric resonator antenna,” IEEE Microw. Wireless Compon. Lett., vol. 16, no. 4, pp. 227–229, Apr. 2006. [11] K. A. A. Wei Huang, “Use of electric and magnetic conductors to reduce the DRA size,” in Proc. iWAT, Small Smart Antennas Metamater. Appl., 2007, pp. 143–146. [12] R. Chair, A. A. Kishk, and K. F. Lee, “Wideband stair-shaped dielectric resonator antennas,” Microw., Antennas Propag., vol. 1, no. 2, pp. 299–305, Apr. 2007. [13] R. Chair, A. A. Kishk, and K. F. Lee, “Low profile wideband embedded dielectric resonator antenna,” Microw., Antennas Propag., vol. 1, no. 2, pp. 294–298, Apr. 2007. [14] L. Huitema, M. Koubeissi, C. Decroze, and T. Monediere, “Ultrawideband dielectric resonator antenna for DVB-H and GSM applications,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 1021–1027, 2009. [15] T. H. Chang and J. F. Kiang, “Broadband dielectric resonator antenna with metal coating,” IEEE Trans. Antennas Propag., vol. 55, no. 5, pp. 1254–1259, May 2007. [16] J. M. Ide, S. P. Kingsley, S. G. O’Keefe, and S. A. Saario, “A novel wide band antenna for WLAN applications,” in Proc. IEEE Antennas Propag. Soc. Int. Symp., Jul. 3–8, 2005, vol. 4A, pp. 243–246. [17] K. S. Ryu and A. A. Kishk, “Ultrawideband dielectric resonator antenna with broadside patterns mounted on a vertical ground plane edge,” IEEE Trans. Antennas Propag., vol. 58, no. 4, pp. 1047–1053, Apr.. 2010.

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[18] Y.-Y. Wang and S.-J. Chung, “A new dual-band antenna for WLAN applications,” in Proc. IEEE AP-S Int. Symp., Jun. 20–25, 2004, vol. 3, pp. 2611–2614. [19] J. Y. Jan and L. C. Tseng, “Small planar monopole antenna with a shorted parasitic inverted-L wire for wireless communications in the 2.4-, 5.2-, and 5.8-GHz bands,” IEEE Trans. Antennas Propag., vol. 52, no. 7, pp. 1903–1905, Jul. 2004. [20] Z. D. Liu and P. S. Hall, “Dual-band antenna for handheld portable telephones,” Electron. Lett., vol. 32, pp. 609–610, 1996. [21] J. Y. Jan and L. C. Tseng, “Planar monopole antennas for 2.4/5.2 GHz dual-band application,” in IEEE-APS Int. Symp. Dig., Columbus, OH, 2003, pp. 158–161. [22] L. Huitema, M. Koubeissi, C. Decroze, and T. Monediere, “Compact and multiband dielectric resonator antenna with reconfigurable radiation pattern,” in Proc. 4th EuCAP, Apr. 12–16, 2010, pp. 1–4. [23] K. Hady, A. A. Kishk, and D. Kajfez, “Dual-band compact DRA with circular and monopole-like linear polarizations as a concept for GPS and WLAN applications,” IEEE Trans. Antennas Propag., vol. 57, no. 9, pp. 2591–2598, Sep. 2009. [24] Q. Rao, T. A. Denidni, A. R. Sebak, and R. H. Johnston, “Compact independent dual-band hybrid resonator antenna with multifunctional beams,” IEEE Antennas Wireless Propag. Lett., vol. 5, pp. 239–242, 2006. [25] M. Rotaru and J. K. Sykulski, “Numerical investigation on compact multimode dielectric resonator antennas of very high permittivity,” Sci., Meas. Technol., vol. 3, no. 3, pp. 217–228, May 2009. [26] Q. Rao, T. A. Denidni, and A. R. Sebak, “A hybrid resonator antenna suitable for wireless communication applications at 1.9 and 2.45 GHz,” IEEE Antennas Wireless Propag. Lett., vol. 4, pp. 341–343, 2005. [27] M. Lapierre, Y. M. M. Antar, A. Ittipiboon, and A. Petosa, “Ultra wideband monopole/dielectric resonator antenna,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 1, pp. 7–9, Jan. 2005. [28] R. A. Kranenburg and S. A. Long, “Microstrip transmission line excitation of dielectric resonator antennas,” Electron. Lett., vol. 24, no. 18, pp. 1156–1157, Sep. 1, 1988. [29] C. A. Tounou, C. Decroze, D. Carsenat, T. Monediere, and B. Jecko, “Diversity antennas efficiencies enhancement,” in Proc. IEEE APS Int. Symp., Honolulu, Jun. 2007, pp. 1064–1067. [30] G. A. Mavridis, J. N. Sahalos, and M. T. Chryssomalis, “Spatial diversity two branch for wireless devices,” Electron. Lett., vol. 42, pp. 266–268, 2006. [31] M. B. Knudsen and G. F. Pedersen, “Spherical outdoor to indoor power spectrum model at the mobile terminal,” IEEE J. Sel. Areas Commun., vol. 20, no. 6, pp. 1156–1169, Aug. 2002. [32] S. Blanch, J. Romeu, and I. Corbella, “Exact representation of antenna system diversity performance from input parameter description,” Electron. Lett., vol. 39, no. 9, pp. 705–707, May 1, 2003. [33] P. S. Kildal, K. Rosengren, J. Byun, and J. Lee, “Definition of effective diversity gain and how measure it in a reverberation chamber,” Microw. Opt. Technol. Lett., vol. 34, pp. 56–59, 2002. [34] R. G. Vaughan and J. B. Andersen, “Antenna diversity in mobile communications,” IEEE Trans. Veh. Technol., vol. VT-36, no. 4, pp. 149–172, Nov. 1987. [35] T. Bolin, A. Derneryd, G. Kristensson, V. Plicanic, and Z. Ying, “Twoantenna receive diversity performance in indoor environment,” Electron. Lett., vol. 41, no. 22, pp. 1205–1206, Oct. 27, 2005. [36] M. Mouhamadou, C. A. Tounou, C. Decroze, D. Carsenat, and Monediere, “Active measurements of antenna diversity performances using a specific test-bed, in several environments,” Int. J. RF Microw. Comput.-Aided Eng., pp. 264–271, May 2010. Laure Huitema received the M.S degree in telecommunications high frequencies and optics from the University of Limoges, Limoges, France, in 2008, and is currently pursuing the Ph.D. degree in wave and associated systems at the XLIM Research Institute, University of Limoges. Her research interests include dielectric resonator antennas, multiband antennas, and also active antennas. Ms. Huitema was the recipient of the best student paper award at the 2010 IEEE International Workshop on Antenna Technology (iWAT 2010) and the best student paper award at the 2010 Journées de Charactérisation Microondes et Matériaux (JCMM 2010).

Majed Koubeissi received the B.S. degree in telecommunication and networks from Lebanese University, Saida, Lebanon, in 2001, the degree of High Frequency Electronics Engineering from the ENSIL School, Limoges, France, in 2004, and the Ph.D. degree in telecommunication from the XLIM Research Laboratory, University of Limoges, Limoges, France, in 2007. Since 2007, he has been an R&D Engineer with the Wave and Associated Systems Department, XLIM Research Laboratory. His research interests include phased arrays, beamforming networks, multiband antennas, DRAs, and circular polarized antennas.

Moctar Mouhamadou received the M.S degree in telecommunications high frequencies and optics and Ph.D. degree in electronics and telecommunications engineering from the University of Limoges, France, in 2004 and 2007, respectively. He is currently an Associate Professor with the OSA (Wireless communications and Effect of EM Wave) Department, XLIM Research Institute, University of Limoges. His current interests include smart antennas, MIMO and wireless communications systems, propagation models and channel characterization, and numerical optimization methods in electromagnetism.

Eric Arnaud received the Diplôme D’Etudes Supérieures Specialisées (DESS) and Ph.D. degrees in electronic and telecommunication from the University of Limoges, Limoges, France, in 1994 and 2010, respectively. His Ph.D. studies focused on circularly polarized EBG antenna. From 1996 to 2001, he was with the Microwave responsible of Free-Electron Laser (L.U.R.E Laboratory), Orsay, France. Since 2001, he has been in charge of the XLIM Laboratory antenna test range, University of Limoges, and he participated in several research projects related to design, development, and characterization of antennas. His research interests are mainly in the fields of circularly polarized EBG antenna and realization of antennas through ink-jetting of conductive inks on RF substrates.

Cyril Decroze received the Ph.D. degree in telecommunications engineering from the University of Limoges, Limoges, France, in 2002. He is currently an Associate Professor with the Wave and Associated Systems (OSA) Department, XLIM Research Institute, University of Limoges, France. Since 2006, he has been in charge of the wireless systems activity in the XLIM-OSA Department. His field of research concerns antennas design, propagation channel, MIMO systems, and digital wireless communications.

Thierry Monediere received the Ph.D. degree in electronics from the IRCOM Laboratory, University of Limoges, Limoges, France, in 1990. He is a Professor with the University of Limoges and develops his research activities in the XLIM Laboratory, UMR CNRS/University of Limoges. He works on multifunction antennas, EBG antennas, and also active antennas.

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A Novel Multiband Planar Antenna for GSM/UMTS/LTE/Zigbee/RFID Mobile Devices Ting Zhang, RongLin Li, Senior Member, IEEE, GuiPing Jin, Member, IEEE, Gang Wei, Senior Member, IEEE, and Manos M. Tentzeris, Fellow, IEEE

Abstract—A new multiband planar antenna with a compact size is designed and developed for mobile devices. The proposed antenna consists of a two-strip monopole and a meandered strip antenna which occupy a compact area of only 15 mm 42 mm. This planar antenna has a bandwidth of 42% at the 900 MHz band and 53% at the 1900-MHz band. The wide bandwidth at the low frequency is attributed to the mutual coupling of an S-shaped strip and an inverted-F strip which are separately printed on the two sides of a thin substrate, forming a two-strip monopole configuration. The bandwidth at the high frequency is enhanced by inserting a meandered strip which improves the impedance matching for the high-frequency band. The experimental results verify the simulations. The featured broad bandwidths over two frequency bands and the miniaturized size of the proposed antenna make it very promising for applications in wireless communication and wireless sensing devices. Index Terms—Broadband antenna, mobile devices, multiband antenna, planar antenna, wireless communications.

I. INTRODUCTION

M

ULTIBAND internal antennas have become a necessity for the state-of-the-art multifunction “smart phones” and wireless sensor modules for the mobile devices. Such internal antennas are generally required to be capable of covering the frequency bands of GSM- 850/900/1800/1900 and UMTS (824–894/890–960/1710–1880/1850–1990/1920–2170 MHz). In addition, the ever increasing implementation of 4G devices further increases the bandwidth requirement in order to cover the LTE2300 (2305–2400 MHz) and LTE2500 (2500–2690 MHz) bands. The integration of global-operability RFID readers and RFID-enabled wireless sensors Zigbee-based controllers in state-of-the-art “smart-phones” necessitate the antenna operability over the additional bands 860–956 MHz and 2.2–2.3 GHz bands. It should be noted that the size of the multiband internal antennas inside mobile devices has to be as small as possible and preferably below the size of a credit

Manuscript received November 11, 2010; revised March 06, 2011; accepted April 14, 2011. Date of publication August 12, 2011; date of current version November 02, 2011. This work was supported in part by the National Nation Science Foundation of China (60871061), in part by the Guangdong Province Natural Science Foundation (8151064101000085), and in part by the Specialized Research Fund for the Doctoral Program of Higher Education (20080561). T. Zhang, R. L. Li, G. P. Jin, and G. Wei are with the School of Electronic and Information Engineering, South China University of Technology, Guangzhou 510641, China (e-mail: [email protected]). M. M. Tentzeris is with the School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA 30308 USA. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2164201

card. Recently, many antennas have been designed to satisfy such stringent requirements. Nevertheless most of them with a miniaturized size fail to cover the required entire frequency bands, especially at the lower frequency band due to the narrower bandwidth [1]–[6]. Antennas with sufficient bandwidths typically require a considerable antenna size or thickness, which usually makes them difficult to integrate within mobile devices or portable wireless modules [7]–[10]. In this paper, we propose a novel multiband internal antenna with the size of 15 mm 42 mm and a thickness of only 0.5 mm. This antenna is capable of generating two wide operating bands that effectively cover the GSM/UMTS/LTE/Zigbee/RFID operations in mobile devices and wireless sensors, which includes the GSM850 (824–894), GSM900 (890–960), GSM1800 (1710–1880), GSM1900 (1850–1990), UMTS (1920–2170), LTE2300 (2305–2400), and LTE2500 (2500–2690 MHz) bands. It is well known that electromagnetic coupling and two-strip configurations are two very effective methods for increasing the bandwidth of a compact antenna structure [11]–[14]. To achieve the wide bandwidth in the lower frequencies, we use two printed strips with appropriately optimized electromagnetic coupling. An additional shorter meander branch is added for the operation around 2.1 GHz. The mutual coupling among the three strips significantly enhances the bandwidth around the higher frequency band, without affecting the performance in the lower frequency band. There is no shorting via involved in the antenna structure, something that facilitates the fabrication and integration of such an antenna topology in a fully photolithographic technology. The configuration and performance of the multiband antenna is described in Section II. A parametric study is presented in Section III and experimental results are given in Section IV. II. ANTENNA CONFIGURATION AND PERFORMANCE The configuration of the presented multiband antenna is illustrated in Fig. 1. The design of the antenna is based on a TLY-5 substrate, which has a dielectric constant of and a thickness of mm. The proposed antenna consists of a two-strip monopole and a meandered strip. The two-strip monopole includes an S-shaped strip and an inverted-F strip. The inverted-F and the meandered strips are printed on the front side of the substrate and are fed by a 50- microstrip line while the S-shaped strip is etched on the backside of the substrate and terminated at a ground plane. The upper section of the inverted-F strip is printed inside the area surrounded by the upper section of the S-shaped strip, while the lower section of the inverted-F strip overlaps with the lower section of the S-shaped strip, forming a two-strip line. The width (wf) of the 50- feed line is 1.5

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Fig. 2. Comparison of the return loss between the two-strip monopole for the 900-MHz band and the meandered strip for the 1900-MHz band (the geometric parameters used are listed in Table I). Fig. 1. Geometry of the multiband planar antenna which consists of a two-strip monopole for the 900-MHz band and a meandered strip for the 1900-MHz band.

mm while the width (wts) of the two-strip lines is 1.2 mm. The meander strip is connected to the feed line through a narrower microstrip line with a width (wf2) of 1mm. The width of the strip line of each branch is optimized by simulation in order to achieve better impedance matching over the desired frequency ranges. The height (h) of the two-strip line is the same as the total height of the antenna which is equal to 15 mm. The multiband antenna was simulated using Ansoft simulation software HFSS v11. The first step of the design involved the optimization of the branches that control the lower frequency band. To achieve a wide bandwidth which can cover the 824–960 MHz frequency band, we use a driven inverted-F strip with a total length of 80 mm and a coupled S-shaped strip with a total length of 120 mm. The length of the two strips is designed to make them resonate around the lower-band frequencies of 800 and 900 MHz, respectively. The distance (dst) between the two strips is carefully chosen to optimize the mutual electromagnetic coupling for a good impedance matching over the whole band. More design details are discussed in the Section III. It can be seen from Fig. 2 that the two-strip antenna generates two dual-resonant modes around 1.2 and 2.4 GHz. The dual-resonant modes at lower frequency provide a wide operating bandwidth for the 824–960 MHz band. However, the dual-resonant modes around 2.4 GHz cannot cover the whole 1.7–2.7 GHz band. The dual-resonant modes can be tuned by the length of the inverted-F branch and the S-shaped strip. As their lengths increase, the cutoff frequencies of both the first-order mode and the higher order mode generated by them shift down. In order to enhance the bandwidth around the higher frequencies, we add a meander strip to generate an additional resonance at around 2.1 GHz. The total length of the meandered strip is about 40 mm which is approximately a quarter wavelength at 2.1 GHz. To make use of the available space, the horizontal part of the meander line is extended to the unfilled region below

the branch of the two-strip line. The center resonant frequency can be adjusted by tuning the length (lv) of the horizontal part. The combination of the additional mode of the meander line at 2.1 GHz with the higher-order modes of the two-strip line effectively forms a wide bandwidth that can easily cover the 1.7–2.7 GHz frequency band. On the other hand, the meander line has almost negligible effects on the lower-frequency bandwidth around 900 MHz, allowing for an easy geometrical optimization to cover specific operation bands depending on the choice of the respective communication and sensing applications. To enable the multi-band antenna with a good impedance matching and a sufficiently wide bandwidth around the 900-MHz and 2-GHz bands, the geometric parameters of the antenna listed in Table I need to be optimized. The optimization design was carried out with the help of numerous simulations. The optimized values for the geometric parameters are listed in Table I. In the rest of the paper, all geometric parameters assume the values in this table unless they are given specifically. Fig. 6 shows the comparison between the measured and simulated results for return loss of the optimized multiband antenna. It is found that the simulated bandwidths (for VSWR 2.5) around the 900-MHz and 2-GHz frequency bands are 36% (0.8 GHz–1.16 GHz) and 50% (1.70 GHz–2.83 GHz), respectively. III. PARAMETRIC STUDY The dual-resonant modes generated by the two-strip monopole are very critical to the realization of the broadband feature of the antenna. As illustrated in [14], the two-strip monopole can be represented by two equivalent circuits electromagnetically coupled with each other. The frequency behavior of the two-strip monopole can be manipulated by tuning the resonances of its two branches and the electromagnetic coupling between them in order to achieve a broad bandwidth impedance matching. Through numerous HFSS simulations we found that the distance (dst) between the two strips has an significant effect on the mutual coupling. Fig. 3 shows the effect of the

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TABLE I OPTIMIZED VALUES FOR THE GEOMETRIC PARAMETERS OF THE MULTIBAND PLANAR ANTENNA

Fig. 5. Photographs of the multiband planar antenna with the optimized geometric parameters. (a) Front view. (b) Back view.

Fig. 3. Effect of the distance (dst) between the s-strip and invert-F strip on the return loss of the multiband planar antenna, where dst was changed from dst = 0:7 mm to dst = 1:2 mm. The optimized value for dst is 0.7 mm.



Fig. 6. Comparison between the measured and simulated result for return loss of the multiband planar antenna.

Fig. 4. Effect of the length (lv) of meandered strip on the return loss of the multiband planar antenna, where lv was changed from lv = 6 mm to lv = 8 mm. The optimized value for lv is 7.5 mm.



distance (dst) on the impedance matching when dst varies from 1.2–0.7 mm. As the distance between the S-shaped strip and invert-F strip decreases, the mutual coupling is enhanced and

the impedance matching around 900 and 2200 MHz become better. Thus, by selecting a proper length for dst, the return loss over the whole operating band can be improved. The impedance matching over the high frequency band can be tuned independently by lv. Fig. 4 shows the simulated return loss for the proposed antenna as a function of lv for values between 6–8 mm. Although little difference is observed for the lower band, the frequency range for the upper band varies with lv. As the total length of the meandered strip increases (lv increases), the resonant frequency shifts down, thus enhancing the impedance matching around the 1. 7 GHz and deteriorating the impedance match around 2.1 and 2.6 GHz. The size of the ground plane also affects the performance of the antenna. The length (lg) of the ground plane has a more obvious impact on the lower band. When the length decreases, the impedance matching over the lower frequency band deteriorates. However, the impedance matching over the high frequency becomes better when the length increases to 120 mm or

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Fig. 7. Radiation patterns of the multiband planar antenna. (a) At 900 MHz. (b) at 1900 MHz.

decreases to 40 mm. The width (wg) of the ground plane has little effect on the antenna performance. As the width increases, the cutoff frequencies of the lower and higher bands shift down a little bit. IV. EXPERIMENTAL RESULTS To verify the performance of the multiband planar antenna, a prototype was fabricated and measured. The antenna was mm TLY-5 substrate with 0.2 oz copper fabricated on a on both sides. Two photographs of the antenna prototype are displayed in Fig. 5 showing the front view and the back view

of the planar antenna. For the purpose of measurement, the antenna is connected to a coaxial cable in the middle section of the ground plane. The measured return loss (RL) is presented in Fig. 6. It is clearly seen that two wide operating bandwidths are obtained. The lower frequency bandwidth, defined by a VSWR of 2.5:1, is 425 MHz (42%) and covers the GSM band (824–960 MHz). The bandwidth for a VSWR of 2:1 is found to be 818 MHz–1190 MHz, which is also wide enough to cover the GSM band (824–960 MHz). On the other hand, the upper band has a bandwidth as large as 1275 MHz (53%) (VSWR2.5:1) and covers the GSM1800 (1710–1880 MHz),

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cover the lower frequency range of 818–1190 MHz for the GSM850/GSM900 and the higher frequency range of 1710–3000 MHz for the GSM1800 (1710–1880 MHz), GSM1900 (1850–1990 MHz), UMTS (1920–2170 MHz), LTE2300 (2305–2400 MHz), and LTE2500 (2500–2690 MHz) bands. The multiband planar antenna is suitable for applications as an internal antenna for wireless mobile and sensors devices. ACKNOWLEDGMENT The authors would like to thank the Speed Communication Technology Corporation, Ltd., for its help in radiation pattern measurement. REFERENCES

Fig. 8. Comparison between the measured and simulated results for peak gain of the multiband planar antenna and the measured antenna efficiency. (a) In lower band. (b) in higher band.

GSM1900 (1850–1990 MHz), UMTS (1920–2170 MHz), LTE2300 (2305–2400 MHz), and LTE2500 (2500–2690 MHz) bands. The radiation patterns of the proposed antenna at the center frequencies for lower and higher bands are plotted in Fig. 7. At 900 MHz, the radiation pattern with almost omplane) is nidirectional radiation in the azimuth plane ( observed. At 1900 MHz, some variations for the radiation pattern appear due to the higher modes. The measured and simulated values for the peak gain at the frequency bands of (824–960 MHz) and (1710–2690 MHz) are exhibited in Fig. 8, featuring good agreement between the measured and simulated results. The measured antenna efficiency is also depicted in Fig. 8. It can be seen that the antenna efficiency is higher than 50% in both of the lower band and the higher band. V. CONCLUSION A novel multiband planar antenna is proposed. The proposed planar antenna has two wide frequency bands which

[1] Y.-W. Chi and K.-L. Wong, “Quarter-wavelength printed loop antenna with an internal printed matching circuit for GSM/DCS/PCS/UMTS operation in the mobile phone,” IEEE Trans. Antennas Propag., vol. 57, no. 9, pp. 2541–2547, Sep. 2009. [2] Y.-W. Chi and K.-L. Wong, “Internal compact dual-band printed loop antenna for mobile phone application,” IEEE Trans. Antennas Propag., vol. 55, no. 5, pp. 1457–1462, May 2007. [3] S. Hong, W. Kim, and H. Park, “Design of an internal multi resonant monopole antenna for GSM900/DCS1800/US-PCS/S-DMB operation,” IEEE Trans. Antennas Propag., vol. 56, no. 5, pp. 1437–1443, May 2008. [4] R. A. Bhatti and S. O. Park, “Hepta-band internal antenna for personal communication handsets,” IEEE Trans. Antennas Propag., vol. 55, no. 12, pp. 3398–3403, Dec. 2007. [5] K.-L. Wong, Y.-C. Lin, and T.-C. Tseng, “Thin internal GSM/DCS patch antenna for a portable mobile terminal,” IEEE Trans. Antennas Propag., vol. 54, no. 1, pp. 238–242, Jan. 2006. [6] R. A. Bhatti, Y.-T. Im, and S.-O. Park, “Compact PIFA for mobile terminals supporting multiple cellular and non-cellular standards,” IEEE Trans. Antennas Propag., vol. 57, no. 9, pp. 2534–2540, Sep. 2009. [7] S.-Y. Lin, “Multiband folded planar monopole antenna for mobile handset,” IEEE Trans. Antennas Propag., vol. 52, no. 7, pp. 1790–1794, Jul. 2004. [8] C.-C. Lin, H.-C. Tung, H.-T. Chen, and K.-L. Wong, “A folded metalplate monopole antenna for multiband operation of a PDA phone,” Microw. Opt. Technol. Lett., vol. 39, no. 2, pp. 135–138, Oct. 2003. [9] Z. Du, K. Gong, and J. S. Fu, “A novel compact wide-band planar antenna for mobile handsets,” IEEE Trans. Antennas Propag., vol. 54, no. 2, pp. 613–619, Feb. 2006. [10] T. K. Nguyen, B. Kim, H. Choo, and I. Park, “Multiband dual spiral stripline-loaded monopole antenna,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 57–59, 2009. [11] R. L. Li, B. Pan, J. Laskar, and M. M. Tentzeris, “A Compact broadband planar antenna for GPS, DCS-1800, IMT-2000, and WLAN applications,” IEEE Antennas Wireless Propag. Lett., vol. 6, pp. 25–27, 2007. [12] R. L. Li, B. Pan, J. Laskar, and M. M. Tentzeris, “A novel low-profile broadband dual-frequency planar antenna for wireless handsets,” IEEE Trans. Antennas Propag., vol. 56, no. 4, pp. 1155–1162, Apr. 2008. [13] H. D. Foltz, J. S. Mclean, and G. Crook, “Disk-loaded monopoles with parallel strip elements,” IEEE Trans. Antennas Propagat., vol. 44, no. 5, pp. 672–676, May 1996. [14] J.-H. Jung and I. Park, “Electromagnetically coupled small broadband monopole antenna,” IEEE Antennas Wireless Propag. Lett., vol. 2, no. 1, pp. 349–351, 2003. Ting Zhang was born in Hunan, China, in 1986. She received the B.S. degree in electronic science and technology from Yunnan University, Kunming, China, in 2004. She is currently working toward the M.S. degree in telecommunication and information systems at South China University of Technology, Guangzhou. Her current research interests include small antennas and wideband antennas for mobile terminals.

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RongLin Li (M’02–SM’03) received the B.S. degree in electrical engineering from Xi’an Jiaotong University, Xi’an, China, in 1983, and the M.S. and Ph.D. degrees in electrical engineering from Chongqing University, Chongqing, China, in 1990 and 1994, respectively. From 1983 to 1987, he was an Assistant Electrical Engineer with the Yunnan Electric Power Research Institute. From 1994 to 1996, he was a Postdoctoral Research Fellow with Zhejiang University, China. In 1997, he visited Hosei University, Japan, as an HIF (Hosei International Fund) Research Fellow. In 1998, he became a Professor in Zhejiang University. In 1999, he visited the University of Utah as a Research Associate. In 2000, he was a Research Fellow at the Queen’s University of Belfast, U.K. Since 2001, he has been a Research Scientist with the Georgia Institute of Technology, Atlanta. He is now an Endowed Professor with the South China University of Technology, Guangzhou. He has published more than 100 papers in refereed Journals and Conference Proceedings, and three book chapters. Dr. Li is a member of the IEEE International Compumag Society. He currently serves as an Editor of the ETRI Journal and a reviewer for a number of international journals, including the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, IEEE MICROWAVE AND WIRELESS COMPONENTS LETTERS, IET Microwave, Antennas and Propagation, Progress in Electromagnetic Research, Journal of Electromagnetic Waves and Applications, and International Journal of Wireless Personal Communications. He was a member of the Technical Program Committee for IEEE-IMS 2008–2011 Symposia and a session chair for several IEEE-APS Symposia. He was the recipient of the 2009 Georgia Tech-ECE Research Spotlight Award. His current research interests include new design techniques for antennas in mobile and satellite communication systems, phased arrays and smart antennas for radar applications, wireless sensors and RFID technology, electromagnetics, and information theory.

GuiPing Jin (M’07) received the B.S. degree in optoelectronic techniques from Northwest University, Xi’an, China, in 1999 and the Ph.D. degree in physical electronics from Xi’an Institute of optics and precision mechanics of Chinese Academy of Sciences (CAS), Xi’an, China, in 2004. Since 2004, she has been a Teacher at the School of Electronic and Information Engineering, South China University of Technology, Guangzhou, . Her latest research interests include optical control switching, modeling of antennas and microwave devices, and photoelectric detection.

Gang Wei (M’92–SM’09) was born in January 1963. He received the B.S. degrees from Tsinghua University, Beijing, China, and the M.S. and Ph.D. degrees from South China University of Technology (SCUT), Guangzhou, China, in 1984, 1987, and 1990, respectively. He was a Visiting Scholar at the University of Southern California, Los Angeles, from June 1997 to June 1998. He is currently a Professor with the School of Electronic and Information Engineering, SCUT. He is a Committee Member of the National Natural Science Foundation of China. His research interests are digital signal processing and communications.

Manos M. Tentzeris (M’98–SM’03–F’09) received the Diploma degree (magna cum laude) in electrical and computer engineering from the National Technical University of Athens, Athens, Greece, and the M.S. and Ph.D. degrees in electrical engineering and computer science from the University of Michigan, Ann Arbor. He is currently a Professor with School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta. He has published more than 370 papers in refereed journals and conference proceedings, five books and 19 book chapters. He has helped develop academic programs in highly integrated/multilayer packaging for RF and wireless applications using ceramic and organic flexible materials, paper-based RFIDs and sensors, biosensors, wearable electronics, inkjet-printed electronics, “Green” electronics and power scavenging, nanotechnology applications in RF, microwave MEMs, SOP-integrated (UWB, multiband, mmW, conformal) antennas, and adaptive numerical electromagnetics (FDTD, MultiResolution Algorithms) and heads the ATHENA research group (20 researchers). He served as the Georgia Electronic Design Center Associate Director for RFID/Sensors research from 2006–2010 and as the Georgia Tech NSF-Packaging Research Center Associate Director for RF Research and the RF Alliance Leader from 2003–2006. Dr. Tentzeris was a Visiting Professor with the Technical University of Munich, Germany for the summer of 2002, a Visiting Professor with GTRI-Ireland in Athlone, Ireland for the summer of 2009 and a Visiting Professor with LAAS-CNRS in Toulouse, France for the summer of 2010. He has given more than 100 invited talks to various universities and companies all over the world. Dr. Tentzeris was the recipient/corecipient of the 2010 IEEE Antennas and Propagation Society Piergiorgio L. E. Uslenghi Letters Prize Paper Award, the 2010 Georgia Tech Senior Faculty Outstanding Undergraduate Research Mentor Award, the 2009 IEEE TRANSACTIONS ON COMPONENTS AND PACKAGING TECHNOLOGIES Best Paper Award, the 2009 E. T. S.Walton Award from the Irish Science Foundation, the 2007 IEEE APS Symposium Best Student Paper Award, the 2007 IEEE IMS Third Best Student Paper Award, the 2007 ISAP 2007 Poster Presentation Award, the 2006 IEEE MTT Outstanding Young Engineer Award, the 2006 Asian–Pacific Microwave Conference Award, the 2004 IEEE TRANSACTIONS ON ADVANCED PACKAGING Commendable Paper Award, the 2003 NASA Godfrey “Art” Anzic Collaborative Distinguished Publication Award, the 2003 IBC International Educator of the Year Award, the 2003 IEEE CPMT Outstanding Young Engineer Award, the 2002 International Conference on Microwave and Millimeter-Wave Technology Best Paper Award (Beijing, China), the 2002 Georgia Tech-ECE Outstanding Junior Faculty Award, the 2001 ACES Conference Best Paper Award and the 2000 NSF CAREER Award and the 1997 Best Paper Award of the International Hybrid Microelectronics and Packaging Society. He was the TPC Chair for IEEE IMS 2008 Symposium and the Chair of the 2005 IEEE CEM-TD Workshop and he is the Vice-Chair of the RF Technical Committee (TC16) of the IEEE CPMT Society. He is the founder and chair of the RFID Technical Committee (TC24) of the IEEE MTT Society and the Secretary/Treasurer of the IEEE C-RFID. He is an Associate Editor of IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, IEEE TRANSACTIONS ON ADVANCED PACKAGING, and International Journal on Antennas and Propagation. He is a member of URSI-Commission D, a member of the MTT-15 committee, an Associate Member of EuMA, a Fellow of the Electromagnetic Academy, and a member of the Technical Chamber of Greece. Prof. Tentzeris is one of the IEEE MTT-S Distinguished Microwave Lecturers from 2010–2012.

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Internal Coupled-Fed Dual-Loop Antenna Integrated With a USB Connector for WWAN/LTE Mobile Handset Fang-Hsien Chu, Student Member, IEEE, and Kin-Lu Wong, Fellow, IEEE

Abstract—A coupled-fed dual-loop antenna capable of providing eight-band WWAN/LTE operation and suitable to integrate with a USB connector in the mobile handset is presented. The antenna integrates with a protruded ground, which is extended from the main ground plane of the mobile handset to accommodate a USB connector functioning as a data port of the handset. To consider the presence of the integrated protruded ground, the antenna uses two separate shorted strips and a T-shape monopole encircled therein as a coupling feed and a radiator as well. The shorted strips are short-circuited through a common shorting strip to the protruded ground and coupled-fed by the T-shape monopole to generate two separate quarter-wavelength loop resonant modes to form a wide lower band to cover the LTE700/GSM850/900 operation (704–960 MHz). An additional higher-order loop resonant mode is also generated to combine with two wideband resonant modes contributed by the T-shape monopole to form a very wide upper band of larger than 1 GHz to cover the GSM1800/1900/UMTS/LTE2300/2500 operation (1710–2690 MHz). Details of the proposed antenna are presented. For the SAR (specific absorption rate) requirement in practical mobile handsets to meet the limit of 1.6 W/kg for 1-g human tissue, the SAR values of the antenna are also analyzed. Index Terms—Coupled-fed loop antennas, handset antennas, LTE antennas, mobile antennas, USB connector, WWAN antennas.

I. INTRODUCTION EVERAL internal mobile handset antennas capable of covering eight-band WWAN/LTE operation in the 704–960 and 1710–2690 MHz bands have recently been demonstrated [1]–[8]. In order to provide wideband operation and low SAR (specific absorption rate) values to meet the limit of 1.6 W/kg for 1-g head tissue [9], these WWAN/LTE handset antennas are generally disposed on the no-ground portion at the entire bottom edge of the system circuit board [8], [10]–[13]. In this case, the integration of such an antenna with nearby electronic elements such as a USB (universal series bus) connector [14]–[16], which is usually mounted at the bottom edge of the handset and used as a data port for external devices, becomes a challenging problem. This is because the presence of the USB connector, which is a conducting object, is generally not considered in the antenna design. Hence, when the USB connector

S

Manuscript received January 05, 2011; revised March 21, 2011; accepted April 30, 2011. Date of publication August 12, 2011; date of current version November 02, 2011. F.-H. Chu and K.-L. Wong are with the Department of Electrical Engineering, National Sun Yat-sen University, Kaohsiung 80424, Taiwan (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2164220

is placed very close to the antenna, some undesired coupling will usually occur to cause degrading effects on the impedance matching of the antenna, thereby greatly decreasing the bandwidth of the antenna. Many traditional internal WWAN handset antennas [17]–[23] also have a similar problem, which causes a limitation in achieving compact integration of the internal antenna with associated electronic elements inside the handset. To overcome the problem, we present in this paper a promising coupled-fed dual-loop handset antenna to intemm grate with a USB connector (typical size for mini USB connector [14]) and cover eight-band WWAN/LTE operation which includes the LTE700/ GSM850/900 bands (704–787/824–894/880–960 MHz) and the GSM1800/1900/UMTS/LTE2300/2500 bands (1710–1880/1850–1990/1920–2170/2300–2400/2500–2690 MHz). The antenna is mounted above a protruded ground extended from the main ground plane of the handset, while there is a USB connector mounted on the protruded ground. The antenna is further short-circuited to the protruded ground, and the presence of the USB connector on the protruded ground is included in the antenna design. Details of the proposed antenna are described in the paper. With the presence of the protruded ground below the antenna, the design considerations for achieving two wide operating bands to cover the desired 704–960 and 1710–2690 MHz bands are addressed. A parametric study of the major parameters of the antenna is also conducted. The obtained results including the antenna’s radiation characteristics and its SAR values for 1-g head tissue are also presented. II. PROPOSED ANTENNA Fig. 1(a) and (b) shows the geometry of the proposed coupled-fed dual-loop mobile handset antenna integrated with a USB connector. Detailed dimensions of the antenna’s metal pattern are shown in Fig. 1(c). Note that a USB connector is mounted on the protruded ground of size 10 10 mm extended from the main ground plane of size 55 105 mm . The protruded ground and main ground plane are both printed on the back surface of a 0.8-mm thick FR4 substrate of size 55 115 mm , relative permittivity 4.4 and loss tangent 0.02 in the study, which is treated as the main circuit board of a practical smartphone. The selected dimensions of the main circuit board are reasonable for practical smartphones. The antenna is centered above the protruded ground. A meandered shorting strip of length 17.8 mm and width 0.5 mm ) short-circuits the antenna to the protruded ground. (section The short-circuiting integrates the USB connector mounted on the protruded ground with the antenna. Further, by using a meandered shorting strip, additional inductance is expected to be

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Fig. 1. (a) Geometry of the proposed coupled-fed dual-loop mobile handset antenna integrated with a USB connector. (b) A USB connector mounted on the protruded ground. (c) Dimensions of the antenna’s metal pattern.

contributed to the antenna’s input impedance. This causes shifting of the excited resonant modes to lower frequencies, which is helpful in achieving decreased size of the internal antenna for eight-band WWAN/LTE operation. To achieve two wide operating bands for the desired WWAN/LTE operation, the antenna is composed of two separate shorted strips and a T-shape monopole encircled therein as a coupling feed and a radiator as well. The two shorted strips (strip 1 and 2) are of the same dimensions and are short-circuited to the protruded ground through the meandered shorting strip. Each arm (arm 1/arm 2) of the T-shape monopole couples to the open end of each shorted strip (strip 1/strip 2) through a coupling gap of 0.5 mm. By tuning the length and of the two arms (one is 23.5 mm and the other is 15.5 mm), two separate quarter-wavelength loop resonant modes can be excited to form a wide lower band to cover the desired 704–960 MHz band. With the successful excitation of the quarter-wavelength loop resonant mode [11], [24], which is different from the traditional internal loop handset antennas operated at the half-wavelength resonant mode as the fundamental or lowest-frequency mode [25]–[31]. This leads to the size reduction of the internal loop handset antennas for the WWAN/LTE operation. mm in the proposed design) By adjusting the width ( of the open-end sections of the two shorted strips, improved impedance matching of the excited resonant modes, especially the quarter-wavelength loop resonant modes, of the antenna can also be obtained. Further, an additional higher-order loop resonant mode is generated to combine with two wideband resonant modes contributed by the T-shape monopole [32] to form a very wide upper band of larger than 1 GHz to cover the desired 1710–2690 MHz.

Note that in order to enhance the operating bandwidth, there mm) in the two shorted strips. are widened portions (width The widened portions and the meandered shorting strips are disposed on a 0.8-mm thick FR4 substrate of size 8 55 mm and placed orthogonal to the edge of the main circuit board. Strip 1 and 2 and T-shape monopole, except a portion (length 8 mm and width 5 mm) of the central arm as the antenna’s feeding strip made by a 0.2-mm thick copper plate, are disposed on a 0.8-mm thick FR4 substrate of size 10 55 mm and placed parallel to the protruded ground. The front end (point A) of the feeding strip is the antenna’s feeding point, which is connected to a 50microstrip feedline of length 20 mm printed on the front surface of the main circuit board. In the experiment for testing the fabricated antenna (see photos in different views shown in Fig. 2), the microstrip feedline is further connected through a via-hole to a 50- SMA connector mounted on the back surface of the main circuit board. III. RESULTS AND DISCUSSION Based on the dimensions given in Fig. 1, the antenna was fabricated as shown in Fig. 2. Fig. 3 shows the measured and simulated return loss for the antenna. The simulated results are obtained using HFSS (high frequency structure simulator) version 12 [33], and agreement between the measured data and simulated results is seen. Two wide operating bands have been obtained for the antenna. The lower band is formed by two resonant modes which are contributed by the two coupled-fed loop resonant paths provided by the antenna. Based on the bandwidth definition of 3:1 VSWR (6-dB return loss) [34]–[36], which is widely used as the design specification of the internal

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Fig. 2. Photos of the fabricated antenna in different views. (a) Front view. (b) Front view seeing the complete T-shape monopole. (c) Back view seeing the USB connector on the protruded ground. (d) Front view seeing the widened portion and USB connector.

Fig. 3. Measured and simulated return loss for the proposed antenna.

WWAN/LTE handset antenna, the lower band covers the desired 704–960 MHz band. The upper band is formed by three resonant modes and shows a bandwidth of larger than 1 GHz to cover the desired 1710–2690 MHz band. Fig. 4 shows the measured three-dimensional (3-D) total-power and two-dimensional (2-D) radiation patterns at typical frequencies. Note that the antenna is mounted at the bottom of the main circuit board. The 2-D radiation patterns are shown in three principal planes, and the normalization for the field strengths in the three planes is the same. At 740 and 925 MHz, dipole-like radiation patterns are observed, and omnidirectional radiation in the azimuthal plane ( – plane) is seen. For higher frequencies at 1795, 1920 and 2350 MHz, relatively large variations in the radiation patterns are seen. The obtained radiation patterns are similar to those of many reported internal WWAN handset antennas [3], [4]. Fig. 5 shows

Fig. 4. Measured three-dimensional (3-D) total-power and two-dimensional (2-D) radiation patterns for the proposed antenna.

the measured antenna efficiency (mismatching loss included) for the proposed antenna. Results are measured in a far-field anechoic chamber. The measured radiation efficiency is about

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Fig. 5. Measured antenna efficiency (mismatching loss included) for the proposed antenna.

Fig. 7. Simulated return loss for the proposed antenna and the case with the coupled-fed loop on the left-hand side (arm 1 and shorted strip 1) only.

Fig. 6. Simulated return loss for the proposed antenna and the case with the T-shape monopole only.

63–83% for the lower band and about 70–91% for the upper band, which are good for practical mobile phone applications. In the following, details of the operating principle of the antenna and the excited resonant modes are discussed in Figs. 6 and 7. A parametric study of the major design dimensions is analyzed in Figs. 8–11. Fig. 6 shows the simulated return loss for the proposed antenna and the case with the T-shape monopole only. It is seen that without the two shorted strips, the lower band cannot be excited. On the other hand, a wide operating band formed by two resonant modes (the first one at about 2.15 GHz contributed by arm 1 and the second one at about 2.7 GHz contributed by arm 2) is obtained for the case with the T-shape monopole only. This indicates that the T-shape monopole is also an efficient radiator in the proposed antenna. The simulated return loss for the proposed antenna and the case with the coupled-fed loop on the left-hand side (arm 1 and shorted strip 1) only is also shown in Fig. 7. It is seen that only a resonant mode is excited at about 800 MHz and is far from covering the desired lower band. An additional resonant mode at about 2.7 GHz, which is a higher-order mode contributed by shorted strip 1, is excited to enhance the bandwidth of the antenna’s upper band. It is also noted that when arm 2 is not present, the second resonant mode in the antenna’s upper band cannot be excited. A parametric study of the major design dimensions of the antenna is also analyzed. Fig. 8(a) shows the simulated return loss as a function of the length of arm 1. Other dimensions are the same as in Fig. 1. Results for the length varied from 19.5 to 23.5 mm are presented. It is seen that the first resonant mode in the antenna’s upper band shifts to lower frequencies with

Fig. 8. Simulated return loss as a function of (a) the arm length p and (b) the arm length q of the T-shape monopole. Other dimensions are the same as in Fig. 1.

an increase in the length . This behavior is reasonable since this resonant mode is mainly contributed by arm 1. In addition, since arm 1 affects the coupling between the T-shape monopole and the shorted strips, some variations in the impedance matching of the antenna’s lower band are seen. The results for the length of arm 2 varied from 13.5 to 17.5 mm are presented in Fig. 8(b). Large effects on the second resonant mode in the antenna’s upper band, which is mainly contributed by arm 2, are seen. Similarly, since arm 2 affects the coupling between the T-shape monopole and the shorted strips, some variations in the impedance matching of other excited resonant modes can be seen. Also, larger effects of the length of arm 2 than the length of arm 1 on the frequency shifts of their respective

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Fig. 9. Simulated return loss for the proposed antenna and the case with a simple linear shorting strip of length 8.8 mm. Other dimensions are the same as in Fig. 1.

Fig. 10. Simulated return loss as a function of the open-end width k of the shorted strips. Other dimensions are the same as in Fig. 1.

Fig. 11. Simulated return loss as a function of the width t of the widened portions. Other dimensions are the same as in Fig. 1.

contributed resonant modes are probably because arm 2 has a shorter length than arm 1, and hence the same length variation will cause larger frequency shift seen in Fig. 8(b). In addition, since arm 2 is shorter than arm 1, larger coupling variations for a same length variation can be expected (see related discussions in Fig. 10), which is also expected to result in larger frequency shift seen in Fig. 8(b). The results shown in Fig. 8(a) and (b) confirm that the first two resonant modes in the upper band are contributed by the T-shape monopole and can be adjusted by tuning the length and of arm 1 and 2. Fig. 9 shows the simulated return loss for the proposed antenna and the case with a simple linear shorting strip of length 8.8 mm. Shifting of the antenna’s lower band to lower frequencies is achieved by using the meandered shorting strip to replace the simple shorting strip. This is mainly because the meandered shorting strip can lead to a longer loop resonant path, hence resulting in the lowering of the excited loop resonant modes.

Fig. 12. Simulated surface current distributions on the major metal pattern of the antenna.

Effects of the open-end section of the shorted strips are also studied. Results of the simulated return loss of the open-end width varied from 1 to 3 mm are shown in Fig. 10. Since the open-end width can adjust the coupling between the T-shape monopole and the shorted strips, large effects on the loop resonant modes are seen. Also, large effects on the second mode in the upper band, which is mainly contributed by arm 2, are seen. This behavior is largely because arm 2 has a shorter length than arm 1, and hence the coupling variations caused by the variations in the width will be larger for arm 1. Thus, the variations in the width will lead to large effects on the resonant mode contributed by arm 2. Effects of the widened portions in the shorted strips are also analyzed. Fig. 11 shows the simulated return loss for the width of the widened portions varied from 4 to 8 mm. With increasing width , enhanced bandwidth of the lower band is obtained. An additional higher-order loop resonant mode at about 2.7 GHz contributed by shorted strip 1 can also be excited with a widened or 8 mm, which enhances the bandwidth of the portion of upper band. Simulated surface current distributions on the major metal pattern of the antenna are presented in Fig. 12. From the surface current distribution at 740 MHz, it is seen that the coupled-fed loop on the left-hand side formed by arm 1 and shorted strip 1 is excited at its fundamental mode and there is no current null along the resonant path. Similar results are seen at 1000 MHz for the coupled-fed loop on the right-hand side formed by arm 2 and shorted strip 2. The results confirm that the two separate quarter-wavelength resonant modes [37] are excited to form a

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USB connector disposed on the protruded ground. Further, two coupled-fed loop resonant paths capable of generating quarter-wavelength modes have been provided by the antenna. The proposed design makes the antenna not only capable of WWAN/LTE operation but also suitable to integrate nearby associated elements such as a USB connector at the bottom edge as the data port of the handset. Good far-field radiation characteristics for frequencies over the eight operating bands have also been observed. Acceptable near-field emission of the antenna with its SAR values for 1-g head tissue well less than 1.6 W/kg has been obtained. From the results shown in this study, the proposed antenna is promising for practical applications in a smartphone for eight-band WWAN/LTE operation. REFERENCES

Fig. 13. SAR simulation model and the simulated SAR values for 1-g tissue for the proposed antenna.

wide lower band for the antenna. While at 1940 and 2350 MHz, strong surface current distributions on arm 1 and arm 2 are seen, which agrees with the discussion in Fig. 8. At 2700 MHz, the surface current distribution agrees with the observation in Fig. 7, in which the resonant mode at about 2700 MHz is related to a higher-order resonant mode of the coupled-fed loop on the left-hand side. In addition, for practical applications, the SAR values of the antenna should be less than 1.6 W/kg for 1-g head tissue [9]. To analyze the SAR results, Fig. 13 shows the SAR simulation model and the simulated SAR values for 1-g head tissue for the antenna. In the simulation model, the typical parameters , of the head phantom liquid are at 900 MHz, and , at 1800 MHz. The simulated results are obtained using the SPEAG simulation software SEMCAD X version 14 [38]. The main circuit board with the antenna mounted at the bottom is placed close to the head phantom with a 5-mm distance to simulate the thickness of the handset housing. At each testing frequency (central frequencies of the eight operating bands), the SAR values are tested using input power of 24 dBm for the GSM850/900 (859, 925 MHz) system and 21 dBm for the GSM1800/1900/ UMTS (1795, 1920, 2045 MHz) and LTE700/2300/2500 (740, 2350, 2595 MHz) systems [39]–[41]. The return loss given in the table shows the impedance matching level at the testing frequency. The obtained SAR values are well below the SAR limit of 1.6 W/kg, indicating that the antenna is promising for practical handset applications. IV. CONCLUSION A promising internal eight-band WWAN/LTE antenna integrated with a USB connector at the bottom edge of the mobile handset has been proposed. The antenna is mounted above a protruded ground extended from the main ground plane of the handset, which allows the antenna to integrate with a

[1] F. H. Chu and K. L. Wong, “Simple planar printed strip monopole with a closely-coupled parasitic shorted strip for eight-band LTE/GSM/ UMTS mobile phone,” IEEE Trans. Antennas Propag., vol. 58, no. 10, pp. 3426–3431, Oct. 2010. [2] C. T. Lee and K. L. Wong, “Planar monopole with a coupling feed and an inductive shorting strip for LTE/GSM/UMTS operation in the mobile phone,” IEEE Trans. Antennas Propag., vol. 58, no. 7, pp. 2479–2483, Jul. 2010. [3] S. C. Chen and K. L. Wong, “Small-size 11-band LTE/WWAN/WLAN internal mobile phone antenna,” Microwave Opt. Technol. Lett., vol. 52, pp. 2603–2608, Nov. 2010. [4] K. L. Wong, W. Y. Chen, C. Y. Wu, and W. Y. Li, “Small-size internal eight-band LTE/WWAN mobile phone antenna with internal distributed LC matching circuit,” Microwave Opt. Technol. Lett., vol. 52, pp. 2244–2250, Oct. 2010. [5] K. L. Wong, M. F. Tu, C. Y. Wu, and W. Y. Li, “Small-size coupled-fed printed PIFA for internal eight-band LTE/GSM/UMTS mobile phone antenna,” Microwave Opt. Technol. Lett., vol. 52, pp. 2123–2128, Sep. 2010. [6] S. C. Chen and K. L. Wong, “Bandwidth enhancement of coupled-fed on-board printed PIFA using bypass radiating strip for eight-band LTE/ GSM/UMTS slim mobile phone,” Microwave Opt. Technol. Lett., vol. 52, pp. 2059–2065, Sep. 2010. [7] K. L. Wong and W. Y. Chen, “Small-size printed loop-type antenna integrated with two stacked coupled-fed shorted strip monopoles for eight-band LTE/GSM/UMTS operation in the mobile phone,” Microwave Opt. Technol. Lett., vol. 52, pp. 1471–1476, Jul. 2010. [8] C. W. Chiu, C. H. Chang, and Y. J. Chi, “A meandered loop antenna for LTE/WWAN operations in a smart phone,” Progr. Electromagn. Rese. C, vol. 16, pp. 147–160, 2010. [9] Safety Levels With Respect to Human Exposure to Radio-Frequency Electromagnetic Field, 3 kHz to 300 GHz, ANSI/IEEE standard C95.1, 1999. [10] C. H. Li, E. Ofli, N. Chavannes, and N. Kuster, “Effects of hand phantom on mobile phone antenna performance,” IEEE Trans. Antennas Propag., vol. 57, no. 9, pp. 2763–2770, Sep. 2009. [11] Y. W. Chi and K. L. Wong, “Quarter-wavelength printed loop antenna with an internal printed matching circuit for GSM/DCS/PCS/UMTS operation in the mobile phone,” IEEE Trans. Antennas Propag., vol. 57, no. 9, pp. 2541–2547, Sep. 2009. [12] C. H. Chang and K. L. Wong, “Printed =8-PIFA for penta-band WWAN operation in the mobile phone,” IEEE Trans. Antennas Propag., vol. 57, no. 5, pp. 1373–1381, May 2009. [13] K. L. Wong and S. C. Chen, “Printed single-strip monopole using a chip inductor for penta-band WWAN operation in the mobile phone,” IEEE Trans. Antennas Propag., vol. 58, no. 3, pp. 1011–1014, Mar. 2010. [14] Wikipedia, the Free Encyclopedia Universal Serial Bus [Online]. Available: http://en.wikipedia.org/wiki/Universal_Serial_Bus [15] K. L. Wong, Y. W. Chang, C. Y. Wu, and W. Y. Li, “A small-size penta-band WWAN antenna integrated with USB connector for mobile phone applications,” in Proc. 2010 Int. Conf. Appl. Electromagn. Student Innovation Competition Awards, Taipei, Taiwan, Aug. 11–13, 2010. [16] K. L. Wong and C. H. Chang, “On-board small-size printed monopole antenna integrated with USB connector for penta-band WWAN mobile phone,” Microwave Opt. Technol. Lett., vol. 52, pp. 2523–2527, Nov. 2010.

CHU AND WONG: INTERNAL COUPLED-FED DUAL-LOOP ANTENNA

[17] C. L. Liu, Y. F. Lin, C. M. Liang, S. C. Pan, and H. M. Chen, “Miniature internal penta-band monopole antenna for mobile phones,” IEEE Trans. Antennas Propag., vol. 58, no. 3, pp. 1008–1011, Mar. 2010. [18] H. Rhyu, J. Byun, F. J. Harackiewicz, M. J. Park, K. Jung, D. Kim, N. Kim, T. Kim, and B. Lee, “Multi-band hybrid antenna for ultrathin mobile phone applications,” Electron. Lett., vol. 45, pp. 7730774–7730774, Jul. 2009. [19] C. W. Chiu and Y. J. Chi, “Planar hexa-band inverted-F antenna for portable device applications,” IEEE Antennas Wireless Propag. Lett., vol. 8, no. 3, pp. 1099–1102, Mar. 2009. [20] R. A. Bhatti, Y. T. Im, J. H. Choi, T. D. Manh, and S. O. Park, “Ultrathin planar inverted-F antenna for multistandard handsets,” Microwave Opt. Technol. Lett., vol. 50, pp. 2894–2897, Nov. 2008. [21] S. Hong, W. Kim, H. Park, S. Kahng, and J. Choi, “Design of an internal multiresonant monopole antenna for GSM900/DCS1800/USPCS/S-DMB operation,” IEEE Trans. Antennas Propag., vol. 56, no. 5, pp. 1437–1443, May 2008. [22] S. Y. Lin, “Multiband folded planar monopole antenna for mobile handset,” IEEE Trans. Antennas Propag., vol. 52, pp. 1790–1794, Jul. 2004. [23] K. L. Wong, Planar Antennas for Wireless Communications. New York: Wiley, 2003. [24] Y. W. Chi and K. L. Wong, “Very-small-size printed loop antenna for GSM/DCS/PCS/UMTS operation in the mobile phone,” Microwave Opt. Technol. Lett., vol. 51, pp. 184–192, Jan. 2009. [25] Y. W. Chi and K. L. Wong, “Internal compact dual-band printed loop antenna for mobile phone application,” IEEE Trans. Antennas Propag., vol. 55, no. 5, pp. 1457–1462, May 2007. [26] Y. W. Chi and K. L. Wong, “Compact multiband folded loop chip antenna for small-size mobile phone,” IEEE Trans. Antennas Propag., vol. 56, pp. 3797–3803, Dec. 2008. [27] Y. W. Chi and K. L. Wong, “Half-wavelength loop strip fed by a printed monopole for penta-band mobile phone antenna,” Microwave Opt. Technol. Lett., vol. 50, pp. 2549–2554, Oct. 2008. [28] Y. W. Chi and K. L. Wong, “Very-small-size folded loop antenna with a band-stop matching circuit for WWAN operation in the mobile phone,” Microwave Opt. Technol. Lett., vol. 51, pp. 808–814, Mar. 2009. [29] K. L. Wong and C. H. Huang, “Printed loop antenna with a perpendicular feed for penta-band mobile phone application,” IEEE Trans. Antennas Propag., vol. 56, pp. 2138–2141, Jul. 2008. [30] B. K. Yu, B. Jung, H. J. Lee, F. J. Harackiewwicz, and B. Lee, “A folded and bent internal loop antenna for GSM/DCS/PCS operation of mobile handset applications,” Microwave Opt. Technol. Lett., vol. 48, pp. 463–467, Mar. 2006. [31] B. Jung, H. Rhyu, Y. J. Lee, F. J. Harackiewwicz, M. J. Park, and B. Lee, “Internal folded loop antenna with tuning notches for GSM/ GPS/DCS/PCS mobile handset applications,” Microwave Opt. Technol. Lett., vol. 48, pp. 1501–1504, Aug. 2006. [32] Y. L. Kuo and K. L. Wong, “Printed double-T monopole antenna for 2.4/5.2 GHz dual-band WLAN operations,” IEEE Trans. Antennas Propag., vol. 51, pp. 2187–2192, Sep. 2003. [33] Ansoft Corporation HFSS [Online]. Available: http://www.ansoft.com/products/hf/hfss/ [34] H. W. Hsieh, Y. C. Lee, K. K. Tiong, and J. S. Sun, “Design of a multiband antenna for mobile handset operations,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 200–203, 2009. [35] X. Zhang and A. Zhao, “Bandwidth enhancement of multiband handset antennas by opening a slot on mobile handset,” Microwave Opt. Technol. Lett., vol. 51, pp. 1702–1706, Jul. 2009. [36] C. L. Liu, Y. F. Lin, C. M. Liang, S. C. Pan, and H. M. Chen, “Miniature internal penta-band monopole antenna for mobile phones,” IEEE Trans. Antennas Propag., vol. 58, no. 3, pp. 1008–1011, Mar. 2010. [37] K. L. Wong, W. Y. Chen, and T. W. Kang, “On-board printed coupled-fed loop antenna in close proximity to the surrounding ground plane for penta-band WWAN mobile phone,” IEEE Trans. Antennas Propag., vol. 59, no. 3, pp. 751–757, Mar. 2011. [38] SEMCAD Schmid & Partner Engineering AG (SPEAG) [Online]. Available: http://www.semcad.com [39] “Technical Specification Group GSM/EDGE Radio Access Network; Radio Transmission and Reception (Release 9),” 3rd Generation Partnership Project (3GPP), 3GPP, TS 45.005 V9.5.0 2010.

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[40] “Technical Specification Group Radio Access Network; Terminal conformance specification; Radio transmission and reception (FDD) (Release 6),” 3rd Generation Partnership Project (3GPP), 3GPP, TS 34.121 V6.3.0 2005. [41] “Technical Specification Group Radio Access Network; Evolved Universal Terrestrial Radio Access (E-UTRA); User Equipment (UE) radio transmission and reception (Release 9),” 3rd Generation Partnership Project (3GPP), 3GPP, TS 36.101 V9.0.0 2009. Fang-Hsien Chu (S’11) was born in Kaohsiung, Taiwan, in 1983. He received the B.S. degree in electronic communication engineering from National Kaohsiung Marine University, Kaohsiung, Taiwan, in 2005, and the M.S. degree in photonics and communications from National Kaohsiung University of Applied Sciences, Kaohsiung, Taiwan, in 2007. He is currently working toward the Ph.D. degree at National Sun Yat-Sen University, Kaohsiung, Taiwan. His main research interests are in planar antennas for wireless communications, especially for the planar antennas for mobile devices applications, and also in microwave and RF circuit design. Mr. Chu was awarded the second prize at the National Mobile Handset Antenna Design Competition in Taiwan in 2008.

Kin-Lu Wong (M’91–SM’97–F’07) received the B.S. degree in electrical engineering from National Taiwan University, Taipei, Taiwan, and the M.S. and Ph.D. degrees in electrical engineering from Texas Tech University, Lubbock, TX, in 1981, 1984, and 1986, respectively. From 1986 to 1987, he was a visiting scientist with Max-Planck-Institute for Plasma Physics in Munich, Germany. Since 1987 he has been with the Department of Electrical Engineering, National Sun Yat-Sen University (NSYSU), Kaohsiung, Taiwan, where he became a Professor in 1991. From 1998 to 1999, he was a Visiting Scholar with the ElectroScience Laboratory, The Ohio State University, Columbus, OH. In 2005, he was elected to be Sun Yat-sen Chair Professor of NSYSU. He also served as Chairman of the Electrical Engineering Department from 1994 to 1997, Dean of the Office of Research Affairs from 2005 to 2008, and now as Vice President for Academic Affairs, NSYSU. He has published more than 500 refereed journal papers and 250 conference articles and has personally supervised 50 graduated Ph.Ds. He also holds over 100 patents, including U.S., Taiwan, China, EU patents, and has many patents pending. He is the author of Design of Nonplanar Microstrip Antennas and Transmission Lines (Wiley, 1999), Compact and Broadband Microstrip Antennas (Wiley, 2002), and Planar Antennas for Wireless Communications (Wiley, 2003). Dr. Wong is an IEEE Fellow and received the Outstanding Research Award three times from National Science Council of Taiwan in 1995, 2000 and 2002, and was elevated to be a Distinguished Research Fellow of National Science Council in 2005. He also received the Outstanding Research Award from NSYSU in 1995, the ISI Citation Classic Award for a published paper highly cited in the field in 2001, the Outstanding Electrical Engineer Professor Award from Institute of Electrical Engineers of Taiwan in 2003, and the Outstanding Engineering Professor Award from Institute of Engineers of Taiwan in 2004. In 2008, the research achievements of handheld wireless communication devices antenna design of NSYSU Antenna Lab that he led was selected to be one of the top 50 scientific achievements of National Science Council of Taiwan in past 50 years (1959–2009). He was awarded the 2010 Outstanding Research Award of Pan Wen Yuan Foundation and selected as top 100 honor of Taiwan by Global Views Monthly in August 2010 for his contribution in mobile communication antenna researches. He was also awarded the Best Paper Award (APMC Prize) in 2008 Asia-Pacific Microwave Conference held in Hong Kong, China. His graduate students were the winners of Best Student Paper Awards in 2008 APMC, 2009 ISAP, and 2010 ISAP (International Symposium on Antennas and Propagation). His graduate students also won the first prize of 2007 and 2009 Taiwan National Mobile Handset Antenna Design Competition.

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Experimental Analysis of a Wideband Pattern Diversity Antenna With Compact Reconfigurable CPW-to-Slotline Transition Feed Yue Li, Zhijun Zhang, Senior Member, IEEE, Jianfeng Zheng, Zhenghe Feng, Senior Member, IEEE, and Magdy F. Iskander, Fellow, IEEE

Abstract—A wideband antenna with a reconfigurable coplanar waveguide (CPW)-to-slotline transition feed is proposed for pattern diversity applications. The feed provides three modes -CPW feed, left slotline (LS) feed and right slotline (RS) feed- without extra matching structures. Changes between modes are controlled by only two p-I-n diodes. Features of the proposed switchable feed include compact size and simple bias circuit. The equivalent transmission line model is used in the analysis of the proposed design. A prototype of the proposed antenna is fabricated, tested, and the obtained results including reflection coefficient, radiation patterns and gains, are present. A measurement of channel capacity is carried out to prove the benefit of pattern diversity when using the proposed antenna in both line-of-sight (LOS) and non-line-of-sight (NLOS) communication scenarios. Index Terms—Antenna feed, channel capacity, pattern diversity, reconfigurable antennas, wideband antennas.

I. INTRODUCTION ITH the rapid progress in developing advanced wireless communication systems, the advantages of using reconfigurable antenna patterns have been recognized and widely adopted in many designs. Reconfigurable antenna patterns provide pattern diversity that could be used to provide dynamic radiation coverage and mitigate multi-path fading. The diversity and increased directional gains of pattern reconfigurable antennas also improves coverage and increase the channel capacity, especially in the multiple antennas system [1]–[3]. Among the recent designs of such antenna systems is the research work published in [4]–[13]. One method to achieve reconfigurable pattern is to adjust the structure of the radiating

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Manuscript received January 23, 2011; revised March 21, 2011; accepted May 01, 2011. Date of publication August 12, 2011; date of current version November 02, 2011. This work was supported by the National Basic Research Program of China under Contract 2010CB327402, in part by the National High Technology Research and Development Program of China (863 Program) under Contract 2009AA011503, the National Science and Technology Major Project of the Ministry of Science and Technology of China 2010ZX03007-001-01 and Qualcomm Inc. Y. Li, Z. Zhang, J. Zheng, and Z. Feng are with the State Key Laboratory on Microwave and Digital Communications, Tsinghua National Laboratory for Information Science and Technology, Department of Electronic Engineering, Tsinghua University, Beijing 100084, China (e-mail: [email protected]). M. F. Iskander is with the Hawaii Center for Advanced Communications (HCAC), University of Hawaii at Manoa, Honolulu, HI 96822 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2164224

element, including the antenna shape [4], [5], shorting sections [6] and parasitic elements [7], [8], dynamically. Another reconfigurable pattern solution may be achieved through the selection of the radiating elements [9]–[13]. In this case, radiating elements in different directions are electronically selected by switchable mechanism to achieve the desired directive beams [9]–[11]. In [12], [13], a reconfigurable CPW-to-slotline transition is proposed and a compact antenna feed, supporting both the CPW and slotline feed is described. Such a transition presents an effective solution to the feeding of different radiating elements in a relatively compact dimension. Such a feed approach has been widely studied and applied in different configurations [12]–[16]. For example, in the design described in [14], the CPW feed was converted to slotline feed by adding a 180 phase shifter. Another method to design CPW-to-slotline transition is to short transcircuit one of the two slots of the CPW, and add a former structure to avoid reflections from the shorted end and provide good impedance matching [12], [13], [15]. In [13], the matching slot was used as a radiating element, while a function similar to a transformer was realized in [16] by using a CPW series stub printed at the center conductor of the CPW. All the designs reported in [12]–[16] required extra structures for mode phase shifter [14] and matching convergence, including structures [12], [13], [15], which occupy considerable space in the feed network. In this paper, a compact switchable CPW-to-slotline transition without any extra structures is proposed and can be treated as an improvement from the design reported in [17]. The proposed CPW-to-slotline transition provides three feed modes: CPW feed, LS feed and RS feed, and is utilized to feed a wideband Vivaldi notched monopole, which is studied in [12]. In this case, the reconfigurable pattern is realized by switching the feed modes in the working frequency range from 4–6 GHz. Compared to the antenna discussed in [12], smaller dimensions of feed structure are realized. Only 2 p-i-n diodes are used in the proposed design, which is less than the 4 p-i-n diodes used in [12]. As a result, the bias circuit is simpler and the parasitic parameters as well as the insertion loss introduced by p-i-n diodes are all reduced in the proposed design. A prototype of the proposed antenna is simulated and fabricated. The reflection coefficients, radiation patterns and gains of three feed modes are measured. In order to confirm the benefits of the pattern diversity in multiple antennas systems, the channel capacity of a 2 2 antenna array is measured in a typical indoor environment. Compared to the standard omni-directional dipoles, the improvement

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of spectral efficiency is achieved by using the proposed antenna in both LOS and NLOS communication scenarios. II. ANTENNA DESIGN PRINCIPLE A. Antenna Configuration The configuration of the proposed antenna is shown in Fig. 1(a). As it may be seen, it is composed of an elliptical topped monopole, two Vivaldi notched slots and a typical CPW feed with two p-i-n diodes. The antenna is printed on the both sides Teflon substrate, whose relative permittivity of a is 2.65 and thickness is 1.5 mm. The CPW is connected to the microstrip at the back side through vias. A 0.2 mm wide slit is etched on the ground at the front side for DC isolation. Three curves are used to define the shape of antenna, fitted to the coordinates in Fig. 1(a): none Curve 1:

(1) where none Curve 2 [18]:

, and

.

(2) . Values of these pawhere rameters are chosen after optimization. Curve 3 and curve 2 are symmetrical along X axis. B. Compact CPW-Slot Transition In order to achieve reconfigurable patterns, a switchable CPW-to-slotline transition with 2 p-i-n diodes is used as shown in Fig. 1(b). This feed structure is able to switch from CPW feed to slotline feed by controlling the bias voltage of p-i-n diodes. The working configurations of the two p-i-n diodes (PIN 1 and PIN 2) are listed in Table I. When both p-i-n diodes are in the state of OFF, the elliptical topped monopole is fed through a typical CPW and a nearly omni-directional radiation pattern is achieved in XZ plane. When PIN 1 is OFF and PIN 2 is ON, the right slotline is shorted. The left Vivaldi notched slot is fed through the left slotline of the CPW, and a unidirectional radiation pattern is formed along the -X axis. In the same way, when PIN 1 is ON and PIN 2 is OFF, a unidirectional beam axis is achieved in the right Vivaldi notched slot along the through the Right Slot (RS) feed. As a result, the reconfigurable patterns are realized by switching the modes in the CPW with two p-i-n diodes. The proposed CPW-to-slotline transitions are designed in a compact size to reduce the overall dimensions of the antenna. An equivalent transmission line model is used to explain the feed transition and the p-i-n diode is expressed as perfect conductor for ON state and open circuit for the OFF state. Fig. 2(a) is tuned to match shows the typical CPW feed, the length of the radiation resistance of monopole from at the feed port. When a slotline on either side of the CPW is

Fig. 1. Geometry and configuration of the proposed antenna. (a) Front view. (b) Detailed view of feed structure.

TABLE I WORKING CONFIGURATION OF PIN 1 AND PIN 2

shorted by p-i-n diode, the feed diagram and equivalent transmission line model of RS feed are depicted in Fig. 2(b). The right slotline is used to feed the Vivaldi notched slot, and the shorted left slotline works as a matching branch. Some related approaches are given in [13] but the proposed method is significantly different as we don’t use any extra matching structures. The shorted branch which is less than a quarter of wavelength serves as a shunt inductance and its value is determined by its length. As an improvement from the matching discuss in [16], the locations of p-i-n diodes are not fixed, as shown in Fig. 3. Therefore, the value of shunt inductance can be tuned for a better matching. As a result and by optimizing the length of and , the radiation resistance

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Fig. 2. Feed diagram and equivalent transmission line model. (a) CPW feed (b) RS feed.

will be well matched to . What is more important is the fact that we only use two p-i-n diodes to control the transition instead of 4 as described in [12]. Therefore, smaller dimension of antenna is realized by using compact feed and simple bias circuit. To help evaluate the developed antenna one may make parametric study such as those described in [19], and conduct experimental validation as described in Sections III and IV of this paper. The values of parameters are optimized by using the software Ansoft High Frequency Structure Simulator (HFSS). The optimized values are listed in Table II.

Fig. 3. Matching strategy of RS feed. (a) Feed diagram. (b) Equivalent transmission line model.

TABLE II DETAILED DIMENSIONS OF THE PROPOSED ANTENNA

III. ANTENNA FABRICATION AND EXPERIMENTAL RESULTS In order to validate the design of the compact switchable CPW-to-slotline transition, a prototype of the proposed antenna with bias circuit is built and tested, as shown in Fig. 4. The selected p-i-n diode is Agilent HPND-4038 beam lead PIN diode, with acceptable performance in a wide 1–10 GHz bandwidth. When the p-i-n diode is forward-biased, it can be treated as a series resistance. The insertion loss introduced by p-i-n diodes is approximately 0.3 dB at its typical bias current of 5–10 mA. That is to say, the efficiency decreases 0.3 dB by using p-i-n diodes. When the p-i-n diode is reverse-biased, on the other hand, it is replaced by a series capacitance of approximately 0.06 pF, which will shift the working frequency of the antenna but with less insertion loss. As a result, the insertion loss mainly comes from the p-i-n diodes at ON state for CPW mode. Clearly, a reduced number of p-i-n diodes will reduce the insertion loss and improve the performance of the systems. The detailed bias configuration of p-i-n diodes is shown in Fig. 5. Specifically, Fig. 5(a) shows the 3-D view of bias circuit. The slit is etched on the front side to isolate the bias voltage of two p-i-n diodes. are soldered over the slit for RF short. Several capacitances The radius of the vias, connecting the front and back sides, is 0.3 mm. In Fig. 5(b), the complete circuit diagram is illustrated.

Fig. 4. Fabrication of the proposed antenna.

Vcontr.1, Vcontr.2 and Vcontr.3 use 3.3 V bias voltages to control the states of the two p-i-n diodes. Another capacitance is used between Vcontr.1 and the ground, in order to short the RF . Therefore, the signal leaked from the choking inductance cable of Vcontr.1 has little effect to the antenna performance. The bias circuit of p-i-n diodes is on the back side. The bias reis ; with the bias current is 7.7 mA. The RF sistance is 10 nH. The RF signal shorting cachoking inductance pacitances are all 470 pF, and the DC block capacitances

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Fig. 7. Simulated and measured reflection coefficients of slotline feed.

TABLE III MEASURED NORMALIZED RADIATION PATTERN IN XZ PLANE

Fig. 5. Bias configuration of p-i-n diodes. (a) 3-D view. (b) Back view.

Fig. 6. Simulated and measured reflection coefficients of CPW feed.

( and ) are 20 pF each. All the measurements were taken using an Agilent E5071B network analyzer. The simulated and measured reflection coefficients of CPW feed, LS and RS feeds are shown in Figs. 6 and 7. The difference between simulated and measured results is introduced by the parasitic parameter and loss of the p-i-n diodes bias circuit. The measured bandwidths are 2.02–6.49 GHz, 3.47–8.03 GHz and 3.53–8.05 GHz for CPW feed, LS feed and RS feed, respectively. The overlap band from 3.53 GHz to 6.49 GHz is treated as the operation frequency for reconfigurable patterns. The measured radiation pattern in XZ and XY planes for CPW feed, LS feed and RS feed at 4, 5, 6 GHz are listed in Tables III and IV. The results are normalized by the maximum value of each mode at each frequency point. For the CPW feed,

a nearly omni-directional radiation pattern is achieved in XZ plane and a doughnut shape in XY plane. For the LS or RS axis, feed, a unidirectional beam is achieved along -X or with acceptable front-to-back ratio better than 9.5 dB. The different patterns of radiation are able to be switched dynamically according to the environment, proving the pattern diversity. The measured gains of CPW, LS and RS feed are illustrated in Fig. 8. The maximum value in the XY and XZ plane is selected as the gain of each mode. For the CPW feed, an average gain in the desired frequency range is 2.92 dBi. For the LS and RS feed, the average gains in the 4–6 GHz band are 4.29 dBi and 4.32 dBi. The improved gain is mainly contributed to the directivity of the slotline feed mode, and the diversity gain is achieved by switching the patterns.

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TABLE IV MEASURED NORMALIZED RADIATION PATTERN IN XY PLANE

Fig. 9. Indoor environment for channel capacity measurement.

Fig. 8. Measured gains of antenna through different feed.

IV. CHANNEL CAPACITY MEASUREMENT A. Measurement Setup In this section, we describe the experimental procedure we used to test and validated advantages of using the developed antenna system in improving channel capacity in an indoor propagation environment. To this end, the channel capacity of a 2 2 multiple antenna system is measured. The antenna array consists of two proposed reconfigurable antennas at receive end and the reference two-dipole array at transmit end. The elements of the reference two-dipole array are arranged perpendicular to XZ plane along X axis. Each port of the two wire dipoles has a bandwidth of 3.9–5.9 GHz with reflection coefficient better than , and mutual coupling between the two ports is lower than over the frequency band which is achieved by tuning the distance between two elements. Also, the isolation between . two proposed antennas is lower than

The measurement system consists of an Agilent E5071B network analyzer, which has 4 ports for simultaneous measurement, transmit antennas, receive antennas, RF switches, a computer and RF cables. The transmit antennas and the receive antennas are connected to the ports of the network analyzer, respectively. The computer controls the measurement procedure and records the measured channel responses. The measurement was carried out in a typical indoor environment in the Weiqing building of Tsinghua University, shown in Fig. 9. The framework of the room is reinforced concrete, the walls are mainly built by brick and plaster, and the ceiling is made with plaster plates with aluminium alloy framework. The heights of desk partition and wood cabinet are 1.4 m and 2.1 m. The transmit antenna array is fixed in the middle of room (TX). The receive antenna array is arranged in several typical locales which are noted as RX1-4 in Fig. 9. Here, the scenarios when the receive antenna array is arranged in RX2 and RX3 are LOS, while that is NLOS when the receive antenna array is arranged in RX1 and RX4. In the measured, the antennas used are fixed at the height of 0.8 m. The measured data was taken in the frequency range of 4–6 GHz, with a step of 10 MHz. A total number of 201 data points/ results are obtained in a typical sampling. Three configurations (CPW, LS and RS) of each element of the receiver array were switched together manually and the strongest receive signal was selected for statistics. Considering the small-scale fading effect, 5 5 grid locations for each RX position were arranged. As a result, a total number of results were measured for statistics in LOS and NLOS scenario respectively. In a real scenario, the three modes of the proposed antenna can be electrically controlled by a chip depending on the strength of receiving signal. In order to validate the effect of the proposed antennas for the systems, the channel responses of the system with another

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TABLE V AVERAGE AND 95% OUTAGE CHANNEL CAPACITY (BIT/S/HZ)

are achieved in LOS and NLOS scenarios, and 2.51 bit/s/Hz and 3.75 bit/s/Hz improvement for 95% outage capacities. In the NLOS scenario, the received signal is mainly contributed from reflection and diffraction of the concrete walls and the desk partitions, arriving at the direction of endfire. The diversity gain in the endfire increases the channel capacity. However, the path loss of NLOS is higher than that of LOS, and the transmitting power should be enhanced to ensure the performance of the system. Considering the insertion loss introduced from non-ideal p-i-n diodes, better performance of the proposed antenna can be achieved by using high quality switches, such as micro-electro-mechanical systems (MEMS) type switches with less insertion loss and smaller parasitic parameters. V. CONCLUSION

Fig. 10. CCDFs of channel capacity. (a) LOS scenario. (b) NLOS scenario.

two reference dipoles used as the receive antennas instead of the pattern reconfigurable antennas in the same measurement arrangements are measured and recorded for comparison. In the measurement, a 2 2 channel matrix is obtained. The channel capacity is calculated by formula (3).

(3) where and are the numbers of receiver and transmitter anis the normalized H by the received power in the reftennas. erence dipole system, and means the Hermitian transpose. is an identity matrix, and SNR is the signal-tonoise ratio. We selected the SNR when the average channel capacity is 5 bit/s/Hz in a 1 1 reference dipole system in LOS or NLOS scenario. B. Channel Capacity Results Fig. 10 shows the measured Complementary Cumulative Distribution Function (CCDF) of channel capacity in LOS and NLOS scenarios. The results consist of the channel capacity information of 2 2 multiple antenna system using the proposed pattern reconfigurable antennas, compared with 1 1 and 2 2 systems using reference dipoles. As listed in Table V, 2.28 bit/s/Hz and 4.13 bit/s/Hz of the average capacity improvement

A compact switchable CPW-to-slotline transition feed without extra matching structures is proposed in this paper to design a wideband reconfigurable system. CPW, LS and RS feed modes are provided to feed an elliptical topped monopole with a pair of Vivaldi notched slots for reconfigurable patterns. A nearly omni-directional pattern is achieved by feeding the monopole through CPW feed, and two endfire patterns are achieved by feeding the Vivaldi notched slots through slotline feed. An equivalent transmission line model is used to analyze the feed structure. The feed modes are controlled by only two p-i-n diodes. A prototype of the proposed antenna is fabricated and tested to prove the adequacy of the feed design. Specifically, a wide bandwidth of 3.53–6.49 GHz is achieved for reconfigurable pattern with the reflection coefficient lower . The radiation patterns of each feed mode are than measured to demonstrate the successful achievement of the pattern diversity. The average gain improvement in the direction of endfire is better than 1.37 dB in the operation band. To prove the benefit of diversity gain, the channel capacity of a 2 2 multiple antenna system using the proposed antennas is measured in an indoor propagation environment. Compared with reference wire dipoles in the same measurement, the average and 95% outage capacities are both improved by using the proposed antennas, especially in a NLOS scenario. REFERENCES [1] D. Piazza, P. Mookiah, M. D’Amico, and K. R. Dandekar, “Experimental analysis of pattern and polarization reconfigurable circular patch antennas for MIMO systems,” IEEE Trans. Veh. Technol., vol. 59, no. 5, pp. 2352–2363, Jun. 2010. [2] M. Sanchez-Fernandez, E. Rajo-Iglesias, O. Quevedo-Teruel, and M. L. Pablo-González, “Spectral efficiency in MIMO systems using space and pattern diversities under compactness constraints,” IEEE Trans. Veh. Technol., vol. 57, pp. 1637–1645, 2008. [3] J. D. Boerman and J. T. Bernhard, “Performance study of pattern reconfigurable antennas in MIMO communication systems,” IEEE Trans. Antennas Propag., vol. 56, no. 1, pp. 231–236, Jun. 2008.

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[4] G. H. Huff and J. T. Bernhard, “Integration of packaged RF MEMS switches with radiation pattern reconfigurable square spiral microstrip antennas,” IEEE Trans. Antennas Propag., vol. 54, no. 2, pp. 464–469, Feb. 2006. [5] P. Deo, A. Mehta, D. Mirshekar-Syahkal, and H. Nakano, “An HISbased spiral antenna for pattern reconfigurable applications,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 196–199, 2009. [6] S.-H. Chen, J.-S. Row, and K.-L. Wong, “Reconfigurable square-ring patch antenna with pattern diversity,” IEEE Trans. Antennas Propag., vol. 55, no. 2, pp. 492–475, Feb. 2007. [7] X.-S. Yang, B.-Z. Wang, W. Wu, and S. Xiao, “Yagi patch antenna with dual-band and pattern reconfigurable characteristics,” IEEE Antennas Wireless Propag. Lett., vol. 6, pp. 168–171, 2007. [8] S. Zhang, G. H. Huff, J. Feng, and J. T. Bernhard, “A pattern reconfigurable microstrip parasitic array,” IEEE Trans. Antennas Propag., vol. 52, no. 10, pp. 2773–2776, Oct. 2004. [9] J. Sarrazin, Y. Mahé, S. Avrillon, and S. Toutain, “Pattern reconfigurable cubic antenna,” IEEE Trans. Antennas Propag., vol. 57, no. 2, pp. 310–317, Feb. 2009. [10] A. C. K. Mak, C. R. Rowell, and R. D. Murch, “Low cost reconfigurable landstorfer planar antenna array,” IEEE Trans. Antennas Propag., vol. 57, no. 10, pp. 3051–3061, Oct. 2009. [11] I.-Y. Tarn and S.-J. Chung, “A novel pattern diversity reflector antenna using reconfigurable frequency selective reflectors,” IEEE Trans. Antennas Propag., vol. 57, no. 10, pp. 3035–3042, Oct. 2009. [12] S.-J. Wu and T.-G. Ma, “A wideband slotted bow-tie antenna with reconfigurable CPW-to-Slotline transition for pattern diversity,” IEEE Trans. Antennas Propag., vol. 56, no. 2, pp. 327–334, Feb. 2008. [13] H. Kim, D. Chung, D. E. Anagnostou, and J. Papapolymerou, “Hardwired design of ultra-wideband reconfigurable MEMS antenna,” in Proc. IEEE 18th Ann. Int. Symp on Personal, Indoor and Mobile Communications, PIMRC, Sep. 3–7, 2007. [14] K.-P. Ma, Y. Qian, and T. Itoh, “Analysis and applications of a new CPW-slotline transition,” IEEE Trans. Microwave Theory Tech, vol. 47, pp. 426–432, Apr. 1999. [15] Y.-S. Lin and C. H. Chen, “Design and modeling of twin-spiral coplanar-waveguide-to-slotline transitions,” IEEE Trans. Microwave Theory Tech, vol. 48, pp. 463–466, Mar. 2000. [16] K. Hettak, N. Dib, A. Sheta, A. Omar, G. Y. Delisle, M. Stubbs, and S. Toutain, “New miniature broadband CPW-to-slotline transitions,” IEEE Trans. Microwave Theory Tech, vol. 48, pp. 138–146, Jan. 2000. [17] Y. Li, Z. Zhang, W. Chen, and Z. Feng, “Polarization reconfigurable slot antenna with a novel compact CPW-to-Slotline transition for WLAN application,” IEEE Antennas Wireless Propag. Lett, vol. 9, pp. 252–255, 2010. [18] J. Shin and D. H. Schaubert, “A parameter study of stripline-fed Vivaldi notch-antenna arrays,” IEEE Trans. Antennas Propag, vol. 47, pp. 879–886, May 1999. [19] Y. Li, Z. Zhang, Z. Feng, M. Iskander, and R. Li, “A wideband pattern reconfigurable antenna with compact switchable feed structure,” in Proc. Int. Conf. on Microwave and Millimeter Wave Technology, Chengdu, China, May 2010, pp. 1–4.

Yue Li was born in Shenyang, Liaoning Province, China, in 1984. He received the B.S. degree in telecommunication engineering from Zhejiang University, Zhejiang, China, in 2007. He is currently working toward Ph.D. degree in electrical engineering from Tsinghua University, Beijing, China. His current research interests include antenna design and theory, particularly in reconfigurable antennas, electrically small antennas and antenna in package.

Zhijun Zhang (M’00–SM’04) received the B.S. and M.S. degrees from the University of Electronic Science and Technology of China, in 1992 and 1995 respectively, and the Ph.D. from Tsinghua University, China, in 1999. In 1999, he was a Postdoctoral Fellow with the Department of Electrical Engineering, University of Utah, where he was appointed a Research Assistant Professor in 2001. In May 2002, he was an Assistant Researcher with the University of Hawaii at Manoa, Honolulu. In November 2002, he joined Amphenol

T&M Antennas, Vernon Hills, IL, as a Senior Staff Antenna Development Engineer and was then promoted to the position of Antenna Engineer Manager. In 2004, he joined Nokia Inc., San Diego, CA, as a Senior Antenna Design Engineer. In 2006, he joined Apple Inc., Cupertino, CA, as a Senior Antenna Design Engineer and was then promoted to the position of Principal Antenna Engineer. Since August 2007, he has been has been a Professor in the Department of Electronic Engineering, Tsinghua University, Beijing, China.

Jianfeng Zheng received the B.S. and Ph.D. degrees from Tsinghua University, Beijing, China, in 2002 and 2009, respectively. He is currently an Assistant Researcher with the State Key Laboratory on Microwave and Digital Communications, Tsinghua University. His current research interests include spatial temporal signal processing, MIMO channel measurements and antenna arrays for MIMO communications.

Zhenghe Feng (SM’85) received the B.S. degree in radio and electronics from Tsinghua University, Beijing, China, in 1970. Since 1970, he has been with Tsinghua University, as an Assistant, Lecture, Associate Professor, and Full Professor. His main research areas include numerical techniques and computational electromagnetics, RF and microwave circuits and antenna, wireless communications, smart antenna, and spatial temporal signal processing.

Magdy F. Iskander (F’91) is the Director of the Hawaii Center for Advanced Communications (HCAC), College of Engineering, University of Hawaii at Manoa, Honolulu, Hawaii (http://hcac.hawaii.edu). He is also a Co-director of the NSF Industry/University joint Cooperative Research Center between the University of Hawaii and four other universities in the US. From 1997-99 he was a Program Director at the National Science Foundation, where he formulated and directed a “Wireless Information Technology” Initiative in the Engineering Directorate. He spent sabbaticals and other short leaves at Polytechnic University of New York; Ecole Superieure D’Electricite, France; UCLA; Harvey Mudd College; Tokyo Institute of Technology; Polytechnic University of Catalunya, Spain; University of Nice-Sophia Antipolis, and Tsinghua University, China. He authored a textbook Electromagnetic Fields and Waves (Prentice Hall, 1992; and Waveland Press, 2001); edited the CAEME Software Books, Vol. I, 1991, and Vol. II, 1994; and edited four other books on Microwave Processing of Materials, all published by the Materials Research Society, 1990-1996. He has published over 200 papers in technical journals, holds eight patents, and has made numerous presentations in International conferences. He is the founding editor of the journal Computer Applications in Engineering Education (CAE), published by Wiley. His research focus is on antenna design and propagation modeling for wireless communications and radar systems, and in computational electromagnetics. Dr. Iskander received the 2010 University of Hawaii Board of Regents’ Medal for Excellence in Teaching, the 2010 Northrop Grumman Excellence in Teaching Award, the 2011 Hi Chang Chai Outstanding Teaching Award, and the University of Utah Distinguished Teaching Award in 2000. He also received the 1985 Curtis W. McGraw ASEE National Research Award, 1991 ASEE George Westinghouse National Education Award, 1992 Richard R. Stoddard Award from the IEEE EMC Society. He was a member of the 1999 WTEC panel on “Wireless Information Technology-Europe and Japan,” and chaired two International Technology Institute panels on “Asian Telecommunication Technology” sponsored by DoD in 2001 and 2003. He spent sabbaticals and other short leaves at Polytechnic University of New York; Ecole Superieure D’Electricite, France; UCLA; Harvey Mudd College; Tokyo Institute of Technology; Polytechnic University of Catalunya, Spain; University of Nice-Sophia Antipolis, and Tsinghua University, China.

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Near- and Far-Field Models for Scattering Analysis of Buildings in Wireless Communications Y. B. Ouattara, S. Mostarshedi, E. Richalot, J. Wiart, Senior Member, IEEE, and O. Picon

Abstract—This paper presents an efficient building modeling for urban propagation prediction. By dividing the whole computation area into near and far zones, two different models are adopted for buildings. As demonstrated for a building facade, a precise description of the heterogeneities is necessary for an observation point situated in the building near field, whereas in the far-field area, a good accuracy is maintained by homogenizing the facade electric properties. Therefore, in the far area, a homogenized reflection coefficient is taken for each facade, and a radar cross section (RCS) is calculated for one or a set of far buildings, taking the multiple reflections into account. In a near area, the simulator can apply the classical ray-tracing code using the detailed facade description while taking account of far building groups through their RCS. The distance of homogenization and RCS validity is obtained using a reference method based on Green’s functions. Index Terms—Multiple reflection, propagation prediction model, radar cross section (RCS), urban environment.

I. INTRODUCTION ITH the increasing need of accurate radio propagation prediction in urban environment and with the availability of the detailed numerical site maps, the use of site-specific tools has been multiplied in current urban environment studies. These tools are classically based on high-frequency ray-tracing or ray-launching methods. As for a complex urban scene, the high number of rays to consider sorely increases the computation time, and the efficiency of these methods would degrade rapidly. The use of acceleration techniques in such a case is thus essential. One classic way to achieve acceleration is to simplify the simulated scene; the other possible way is to accelerate the calculation of the intersection test [1]. In the scene simplification category, the main effort is focused on the reduction of environment objects such as edges and planes. The three-dimensionality of the objects is thus often neglected. Two-dimensional ray-tracing algorithms [2] give reasonably accurate results when the average building height is much larger than both the transmitter and the receiver levels; otherwise the propagation would not be purely lateral. A frequent debatable case would be the coexistence of different

W

Manuscript received July 13, 2010; revised March 08, 2011 accepted March 18, 2011. Date of publication August 12, 2011; date of current version November 02, 2011. This work was supported by the French project ANR OP2H. Y. B. Ouattara, S. Mostarshedi, E. Richalot, and O. Picon are with the ESYCOM Laboratory, Université Paris-Est Marne-la-Vallée, 77454 Marne-la-Vallée cedex 2, France (e-mail: [email protected]; [email protected]; [email protected]; odile.picon@ univ-mlv.fr). J. Wiart is with the Whist Lab, Common Laboratory of Institute Telecom and Orange Labs, 92130 Issy-les-Moulineaux cedex 9, France (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2011.2164182

building heights for rooftop antennas. Another approach consists in reducing the number of involved objects by neglecting far buildings [3]. In the ray-path search-based methods, all possible paths between transmitter and receiver are examined, and ray tracing is then carried out only on the actual existing rays. Efforts have been done to accelerate the path-finding step with efficient visibility graphs in 2-D scenarios [4] and 3-D cases [5]. Hybrid approaches with a 2-D ray-path search plus 3-D ray tracing have been also reported [6]. For the objects far away from the calculation area, visibility graphs generation can be accelerated by splitting the whole area into smaller zones by bounding boxes [7]. The same objective of computation time reduction is followed in this paper. As the computational complexity considerably increases with the scene dimensions, we propose a specific treatment of the far buildings to avoid the determination of the related propagation paths. Far buildings have received special treatments in literature not for their reflection/diffraction effects, but concerning diffuse scattering [8]. Diffuse scattering is certainly an important phenomenon for channel dispersion calculation and wideband characterization of urban radio propagation [9]. This paper proposes a method to handle the effect at the receiving point of a set of far buildings, regarding the multiple reflection effects between them. With the help of a reference method based on Green’s functions [10], we will show in Section II that the same precision is not required in the treatment of buildings situated in the vicinity of the receiver or in the far-field area of it. Via statistical studies, we demonstrate the influence of the distribution of inhomogeneities (such as windows) on the reflected electric field in different diffraction zones of the building. Consequently, an appropriate region for a simplified homogenized model is recognized. In Section III, we propose a hybrid approach that consists of combining ray techniques with the description of the far-field areas by the radar cross section (RCS) of one or a set of buildings. A simple summing of the electric fields resulting from the propagation in both near and far areas leads then to the field at the receiver position. II. VARIABILITY OF BUILDINGS IN URBAN ENVIRONMENT A convenient rapid analysis of wave propagation in a complex and variable environment such as urban area requires a simplified model with an adapted calculation method. Since the most important objects in this environment are buildings, it is necessary for the simulators to limit the complexity of building facades and find a solution to deal with the variability of them. In this part, we consider only one of the important parameters of facades, which is the distribution of architectural details. Through parametric and statistical observations with an accurate method, we find out in which zone the influence of this parameter on the prediction of reflected electromagnetic fields is negligible. Once

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the observation point is in this appropriate region, the simulator can simplify the model and speed up the computation by neglecting the information about the distribution of the architectural details and by adopting a simpler calculation method. A. Method of Green’s Functions The method of Green’s functions, presented in [10], is based on the induction equivalence principle and opens the possibility of calculating accurately the reflected field by the buildings in different areas: from a few meters to the far field of buildings. In this method, a plane wave illuminating the surface of an object can be replaced, with respect to the equivalence principle, by electric and magnetic currents at the interface between the air and the object. The total radiation of these fictive currents gives the reflected field from the object. In order to calculate the radiation of surface currents, we use the Green’s functions associated with the interface between two semi-infinite media that have the advantage of being without singularities. The choice of semi-infinite Green’s functions in the case of a concrete–glass building can be justified based on the fact that the main material is concrete: Given the wall thickness, the loss presented by the concrete at working frequencies (about 1 GHz and beyond) is large enough to consider the concrete part as a semi-infinite medium. However, for windows with thin or multilayered glass parts, another solution allows to use the same type of singularity-free Green’s functions by introducing an equivalent permittivity. This equivalent medium gives mathematically the same reflected electric field (but not the same transmitted field) as the main structure for a given wave polarization and incidence angle. As we will see in the next example, the artificial equivalent permittivity does not necessarily represent a lossy physical medium, but depending on the thickness of the layers and the working frequency, can correspond to an amplifying medium; this equivalent medium is just an artifice of calculation in order to replace the single multilayered windows by a semi-infinite medium [11]. The convolution integral of Green’s functions with the surface currents gives the reflected field at each point in space

(1) et are semi-infinite dyadic Green’s functions where corresponding to electric and magnetic currents , respectively. It is important to note that the Green’s tensors are analytical and contain the information on the complex permittivity of the materials and the observation angle. The incidence angle is included in the expression of the equivalent currents. It is worth to underline here the basic difference between the method of Green’s functions and physical optics as the other classic asymptotic method based on the equivalent currents. Physical optics employs the free-space Green’s functions and the Fresnel reflection coefficient. This coefficient contains the information about the complex permittivity of the reflecting object as well as the incidence angle, but it does not account for the observation angle. Thus, the Green’s functions method includes the presence of the second medium and gives information in all directions of observation (not only in specular direction), independently from

Fig. 1. Three different concrete–glass building facades: (a) generic with 2 1.5 m windows; (b) randomly generated with 2 2 m windows; (c) randomly generated with 1 1 m windows.

the incident angle. However, except for some special cases, this difference remains transparent to the results [10]. The method of Green’s functions provides results with good accuracy in the vicinity of buildings in different diffraction areas. Besides knowing that the integral limits carry the information about the form of the reflecting object, we can study the influence of the size and the distribution of windows on the total reflected field from a building facade. Through statistical studies, we will quantify their influence and recognize the diffraction zones where the architectural details can be neglected. B. Distribution of Windows—Diffraction Zones Consider the generic concrete–glass building facade (12 12 m ) shown in Fig. 1(a). The main part is modeled by a concrete with for as time-harmonic variation of electromagnetic fields. The windows (2 1.5 m ) are single-glazed with a glass permittivity of and a thickness of 10 mm. Consider also the two randomly created building facades (12 12 m ) in Fig. 1(b) and (c) with the same material properties as the building in Fig. 1(a) and the same percentage of glass (33.33%) distributed differently on the surface. The three buildings are illuminated by a plane wave at 900 MHz incident in -plane. In order to observe the influence of windows distribution on the reflected electric field, we propose two different approaches. First, we use the method of Green’s functions with building facades described in details as presented in Fig. 1. Then, we use a homogenization technique in order to obtain a homogeneous facade without any architectural details. The reflected electric fields obtained by both approaches are compared over a range of distances from the building facades in order to find out in which region the accurate information on the position of windows is necessary. In the first approach, the facade heterogeneity is precisely modeled. The concrete permittivity is inserted directly into the expressions of Green’s functions while, given the incidence angle and the polarization, the single-glazed windows are replaced by a semi-infinite equivalent medium. For a normal incidence, the permittivity of the equivalent medium is equal to . As mentioned above, this complex permittivity representing an amplifying equivalent medium is only equivalent to the window for the reflection problem. The transmitted wave through the single-glazed window can obviously not be calculated by this permittivity. This value is highly sensitive to the thickness of the layer and the incidence angle. In the second approach, the structures are homogenized with respect to the surface percentage covered by each material. In

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Fig. 2. Reflected electric field by building facades in Fig. 1 and by the common homogeneous facade as a function of the observation distance.

Fig. 4. Reflected electric field from the homogeneous facade (solid line) and detailed randomly created samples with (a) 2 2 m windows (b) 1 1 m windows. Fig. 3. Different standard diffraction zones.

this way, the reflection coefficient of the homogenized surface is given for the three facades of Fig. 1 by (2) is the Fresnel reflection coefficient of each material and represents the surface proportion of glass. As the three buildings have the same material percentage and dimensions, the homogenized building facade is identical in the three cases. The reflected electric field is then calculated using Green’s functions with the equivalent permittivity corresponding to . For a TE polarized plane wave V/m in normal incidence, the three curves obtained by the first approach and the single curve obtained by the second approach are compared in Fig. 2. The results show important differences in the near zone of buildings, whereas they approach each other closely in the far zones. In order to quantify the distances in near and far zones, we resume the well-known diffraction zones in Fig. 3 [12]–[15]. In this figure, D represents the largest dimension of the building, and the free-space wavelength. For our problem, with the facade dimensions of 12 12 m and the working frequency of 900 MHz, the Rayleigh zone stops at 216 m from the buildings while the classic far-field zone starts at 864 m. Referring to the reflected electric fields in Fig. 2, we observe that a good agreement between the results obtained by detailed structures and that of the homogenized one appears before the classical far-field zone. We note that the homogenized solution presents less than 10% error compared to the other results at any distance where

beyond the Rayleigh zone (r 216 m). To reinforce this hypothesis, we perform statistical studies on the size and distribution of windows on the building facades. We create randomly 5000 samples of building facades (12 12 m ) of each type: with 2 2 m and 1 1 m windows. For a normal incidence, the reflected electric field from these buildings is observed in three different zones: at the end of the Rayleigh zone (216 m), at the beginning of the far-field zone (864 m), and at a medium distance between these two positions (410 m). The results are presented in Fig. 4 using boxplots along the homogenized solution. The coefficient of variation (CV) is defined by the ratio of the standard deviation to the mean and is a measure of dispersion. The boxplots with their respective CV show that for both facade types the reflected electric field is less sensitive to the distribution of windows in far zones. Among these three distances, the maximum error between the homogenized and all accurate solutions corresponds to the beginning of the Rayleigh region and occurs with the largest windows (15%), while the mean error for the same region is about 3%. For all observation distances, the field dispersion and the errors are less important in the case of small windows. We can conclude that the distribution of architectural details on building faces is not an influential parameter in far diffraction zones. In the context of urban field prediction, we consider that the error presented by neglecting the details is acceptable by the end of Rayleigh zone. This error then decreases as the distance increases. In order to predict the propagation of electromagnetic waves in urban environment, we can thus divide the diffraction zone of the buildings into two zones: the near zone (Rayleigh zone) where an accurate description of the facade is mandatory and the far zone (beyond Rayleigh zone) where a homogenized

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keep the information on the field polarization, the incident field is decomposed on the spherical coordinates associated to the incidence direction, and the - and -components are treated separately

(3) Through this paper, an time dependence is assumed is defined as the ratio of and suppressed. The bistatic RCS scattered power density from the facade at the receiver position ( , ) to the normalized incident power density. In 3-D, it is expressed as follows: Fig. 5. Bistatic radar configuration of an inclined building facade.

(4) reflection coefficient would be sufficient. In Section III, we propose a fast and simple method based on the RCS calculation, which is appropriate for the far zone, and we use the method of Green’s functions as a reference to confirm the validity domain of the RCS method. III. EFFICIENT MODELING OF BUILDINGS IN FAR ZONES As shown in Section II, the influence of heterogeneities distribution on building facades is not significant when the observation is made outside the Rayleigh zone and consequently in far-field zone. This allows a simplification of the scene by homogenizing the facades, so that asymptotic methods can be used to estimate their scattered far field, with a reduced computation time. The RCS, used for target recognition in radar applications, constitutes a compact representation of the scattering properties of an object or a set of objects, and the associated high frequency calculation techniques have been proven to be accurate for simple or complex targets [16]–[18]. Our proposed method, based on the RCS approach, is a fast analytical method to estimate the scattering by one or a set of buildings located in the far-field region of the observation zone, which takes into account several reflections between the facades. The total scattered electric field at the receiver position is rebuilt from the individual RCS related to single or multiple reflections. To establish the validity and limitations of this RCS approach, we compare our analytical results to a commercial simulator based on the finite element method (FEM) as well as with the Green’s functions method.

or relative to the unit vector represents The index or the polarization of the incident field, oriented along vector. In the same way, index ( or ) is relative to the observed component of the scattered electric field , and the associated vector is chosen as or unit vector of the spherical coordinate system defined at the receiving point. Thus, four components are necessary to comRCS pletely characterize the far-field scattering for any incident field polarization. is calculated by The electric field scattered by the facade the vector form of Kirchhoff–Huygens’ integral [19]–[21]: The unknown electric and magnetic currents on the facade boundaries are obtained using physical optics approximations. The is evalcontribution of these surface currents to radiated field uated over the illuminated regions of the surface, and the observations are made in backward region where the surface radiation is dominant compared to the edge diffraction. The far-zone expression of the scattered field is calculated [22], and the bistatic RCS from (4) are given as follows:

(5) (6) where and directions

are functions of the incidence and receiving

A. Bistatic RCS of a Homogeneous Building Facade As already seen in Section II, the properties of the facades situated in the far-field zone can be homogenized in accordance with the constituting materials proportions. Thus, buildings are modeled as blocks with four homogeneous rectangular facades of different permittivities, dimensions, and positions. To begin with, we will investigate the electromagnetic scattering by a faand illuminated by a plane wave cade of size of wave vector (Fig. 5). For better legibility, the following expressions are given in the local coordinate system as, with the sursociated to the scattering surface face barycenter and -axis normal to the surface. Their transposition to any coordinate system will be discussed later. To

(7) (8) As Fresnel’s reflection coefficients on the homogenized facade are given for a TE or TM polarized incident plane wave, the polarization direction of the electric incident field has to be decomposed on TE and TM components. The unit vectors and indicate the incident electric field direction respectively in TE and TM polarizations. is an integral function on the planar facade depending on the surface reflection properties. In the case of a homogeneous rectangular facade, this integral function becomes analyt-

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Fig. 6. Scattered E-field as a function of observation distance obtained by the RCS approach and the method of Green’s functions (GF) for various windows distributions and sizes.

ical and can be written as a product of two cardinal sine functions (9) with (10) (11) Whereas in the case of a single facade, the dimensions of the rectangular illuminated surface , , and correspond to the dimensions of the whole facade ( and ), it is not always the case for a set of facades as we will see later. Thus, only the illuminated surface contributes to the RCS. plane In particular, for an incidence direction in , for a TM incident field and for a TE incident field. By introducing (9) into (5) and (6), the bistatic radar cross sections in the horizontal plane are given by

(12)

(13) . with This analytical form of the RCS avoids the calculus of the surface integral and reduces strongly the computation time, but is only valid in the far-field zone of the facade. In order to determine the validity domain of our RCS approach applied to building facades, we consider the three concrete–glass building facades of Fig. 1, illuminated as previously. The equivalent reflection coefficient of the homogenized facade in the RCS approach is taken to be the same as in Section II, for a normally V/m at incident plane wave polarized along -axis 900 MHz. The field obtained using the RCS approach is compared in Fig. 6 to the ones obtained using the Green’s functions,

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Fig. 7. Scattering mechanism for two inclined buildings.

for varying observation distances. We observe a good agreement between the four curves beyond the Rayleigh distance m, with an error of less than 10%. Thus, the accuracy of RCS approach for the prediction of the scattered field from inhomogeneous building facades is quite acceptable beyond the Rayleigh distance and increases with the observation distance. B. Scattering by a Set of Homogeneous Facades We will now show that electromagnetic wave propagation in the presence of several buildings can also be predicted by applying the RCS approach described in the previous section. In this case, multiple reflections can occur between the facades. In our model, only first- and second-order reflections are considered, and higher-order reflections are supposed to be sufficiently attenuated to have no significant influence. For the sake of clarity, our algorithm will be explained in the case of two buildings, but it is also valid for a higher number of buildings as we will see with the simulated examples. The two buildings , ilare modeled as boxes of dimensions and wave luminated by a plane wave of amplitude (Fig. 7). Whereas a local coordinate system vector was used in the case of the single facade, a global coordinate is chosen while evaluating the scattering by system a set of buildings. The phase origin of the incident field is chosen here at its origin . As for the single facade, perpendicular and parallel polarization components are treated separately. We note that the decomposition of the incident field on TE and TM comis dependent on the orientation of ponents the facades and can thus be different for two differently oriented buildings. Therefore, the field decomposition on TE and TM components has to be performed while treating each reflection. The first step of our method consists in determining the building sectors contributing to first- and second-order reflections. The associated RCS are calculated in a second step. Finally, the received field is rebuilt from the contributions of all the reflections.

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1) Buildings Sectors and RCS Calculation: As only the illuminated facade sectors contribute to the reflection phenomena, the illuminated areas are first determined. In the case of Fig. 7, four facades among eight are illuminated . If the incident field on a facade is partially or totally screened by another building, this screening effect is taken into account while evaluating the illuminated surface. The field reflected from these illuminated areas can reach the receiver or be intercepted by an obstacle. In this second case, a second reflection can occur. (respectively ), the reflected In case of the facade field is not screened by any obstacles, so the RCS is directly cal(respectively culated using formulas (5)–(11) with ) and (respectively ), with a local coordinate system centered at the barycenter of the facade. facade. Depending on the buildLet us now consider ings positions and on the incidence direction, the field reflected by this facade in the specular direction can either be intercepted by facade or not. To determine the facade sectors contributing to single or double reflection, the width and the height of the reflected wavefront, and , are taken into acbe the projection of on the reflected count [4], [23]. Let . ray coming from is higher than the one of , then • If -coordinate of is compared to . If , is only a single-reflection contributor. The corresponding RCS is calculated using (5) and (6), and the surface . Both single and double reflections . In this case, the single-reoccur when is evaluated with flection RCS on and the double reflection RCS on then on to for radiating area . and the are the projections of and along the reflected path and . is the minimal height respectively on and . between is lower than the one of , no single • If -coordinate of . A double reflection is observed for then onto if -coordinate of reflection occurs on is larger than the one of . then on to as The potential double reflection on toward observation positions well as the single reflection on are not considered of -coordinate smaller than the one of in the studied examples. The RCS calculation of a double-reflection mechanism is performed using the combination of geometrical optics (GO) techniques with the closed-form expression of Kirchhoff–Huygens’ integral. The reflected field from the first facade that is incident to the second facade is computed in the sense of geometrical optics: The reflected field is considered to be planar, and the reflection specular. The Kirchhoff–Huygens’ integral is then used on the second facade to obtain the radiation of the surface currents induced by the GO field. The use of GO reflected field to illuminate the second building facade that is located in the near-field region of the first one is an approximation in order to reduce the complexity of the analysis. The GO+PO approach to solve multiple scattering problems has been used by Knott [24] for the backscattering of perfect conducting dihedral corner reflectors. As in our building configuration, both planar surfaces constituting the dihedral reflector are in the near field

of each other. Griesser et al. in [25] compared Knott’s method that combines GO and PO to the use of PO integral for the first and second reflections and showed that both approaches are in agreement for the backscattering mechanism, whereas GO+PO method fails in forward region. In our paper, the observations are limited to backward zone, and the buildings are in dielectric material. As an example, we will focus on the double-reflection mechthen on to (Fig. 7). The RCS compoanism from facade nent related to this reflection is expressed as follows:

(14) is the Fresnel’s reflection coefficient and correwhere sponds to the first reflection on the facade . is the polarization of the GO reflected field from . , , , , and are related to the radiating building sector (7)–(11). As will be developed in the following section, the phase delay relative to the phase origin of the incident and observed fields, as well as the reflection coefficient phase, are taken into account while calculating the total RCS or received field. facade is similar The treatment of the reflection on to that of facade for all incidence angles, with potential single- and double-reflection phenomena. We can notice that the number of single and double reflections depends on the incidence angle. 2) Total RCS and Reflected Field: Once all the single and double reflections on both buildings with their associated RCS values are determined, the global RCS of the set of two buildings can be calculated by summing their contributions at the receiving point. In the presented examples, we restrict the analysis . In to horizontal incident and observation planes this case, the reflections induce no field depolarization, and TE and TM fields are treated separately. In our building configurations, corresponds to the number of building sectors that contributes either to a single or a double reflection. The RCS expres, sion of each radiating sector is denoted by centered previously given in the local system at sector barycenter. This RCS can easily be expressed in the global coordinate system through a transformation of incidence and observation angles with the help of Euler’s formulas [18]. In the global coordinate system, the total RCS of the set of buildings is then obtained using the relative phase method [26] (15) Three phase terms appear in this expression. denotes the phase delay of the incident field at the • of the radiating building sector with respect barycenter to the phase reference at the origin of the global coordinate system (see Fig. 7). corresponds to the sum of the in• ) or TM (for ) trinsic phase of the TE (for reflection coefficients of the successive building sectors interfering in a single or a double reflection mechanism. • is due to the local to global coordinate system transposition for the radiated field.

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The field received at the observation point is then built in a similar way than the total RCS from the analytical RCS. It can be written as

(16) and are respectively the distance from the radiating where and the reference to the receiver. We resurface center mind that this expression is only valid beyond the Rayleigh distance from the buildings. As an example, the phase terms are calculated in the case reflection mechanism: is equal to of , and the reflection coefficient phase is the phase sum of the TE or of TM reflection coefficients corresponding to facade of Building 2. The phase term Building 1 and facade corresponds to the phase delay in observation.

Fig. 8. Electric field scattered from a block of two large buildings: ; ; Single reflection on the lateral facade ; Single reflec; Double reflection on the facade then on to tion on the facade .

C. Validation of RCS Approach on Two Inclined Buildings A validation of our RCS approach was presented in [27] in the case of two three-dimensional blocks. For validation purpose using reference results issued from FEM simulations, a scaled model was studied with blocks of dimensions m . Indeed, the commercial simulator High-Frequency Structural Simulator (HFSS) used as a reference is not suitable for modeling large structures. A very good agreement between both approaches was obtained. We will now validate our results for realistic building dimensions, but due to memory limitations while using the FEM software, a two-dimensional problem is considered (by taking an infinite building height). The buildings parameters referring to Fig. 7 are the following: m, m, m, . The buildings are placed symmetrically to the global coordinate system center with m m and m m . The facade , walls are of dry concrete material (dielectric constant S/m) and have an infinite thickness. The conductivity block of two buildings is illuminated by a TE plane wave in at 900 MHz. The electric field in normal incidence far observation zone is computed from the RCS components and compared to the FEM simulations in Fig. 8. The radiating pattern, being symmetric, is restricted to the positive elevation angles. The predominant peaks are the following: single reflecand , single reflections tions from the lateral facades and , and double reflections between on the facades these facades. Although the facades are in the near zone of each other, the GO+PO approach to compute double-reflection field matches well with FEM simulations; all the predominant peaks locations, widths, and relative amplitudes are well evaluated by the RCS approach. D. Simulation Results for a Set of Three Buildings In this section, simulations are performed for a set of three buildings illuminated by a plane wave at 900 MHz (Fig. 9). For

Fig. 9. Top view of a prototype block of buildings composed of three buildings. Incident and observation angles are presented in their positive orientation.

validation purposes, buildings of reduced dimensions are first considered, as HFSS software cannot be applied to large structures. In a second time, results obtained with our RCS approach will be given in a realistic case. The facade walls are composed of dry concrete whose characteristics are given in the previous section. 1) Validation With a Scaled Model: Three small blocks are , chosen with the following dimensions: , , . The facades and are inclined by and from -axis, and and by and . A distance of along and along separates two adjacent buildings. The scenario is illu(or ), minated in the normal direction . The scattered and the incident field is TE-polarized m for an elfield is observed on a circle of diameter evation angle (Fig. 9). For this incidence , , , direction, eight facades are illuminated: , , , , and (bold lines in Fig. 9). However, the specular and facades are oriented beyond the range reflection on

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Fig. 11. Global RCS for the scaled model in the elevation plane Fig. 10. Double- and single-reflection RCS components of the scaled model: , .

of observation angles so that their contribution to the backward scattering will be very low. Therefore, the radiation of these two surfaces is neglected in our approach although our method permits to take it into account. Six RCS components are introduced by the illumination of these three buildings. Fig. 10 shows the RCS components related to the single and double reflections. corresponds to the single reflection on facade . Due to and the observation distance, the related the orientation of is defined in the global coordinate system for analytical . The maximum of this the elevation angle . curve corresponds to the specular direction around We note that the specular reflection direction is not rigorously from the global coordiequal to 90 because of the offset of , nate system. Due to the symmetry of the scene, the curve facade, is symmetric to corresponding to the reflection on . results in RCS, calculated The single reflection on and . The radiated field is partially with screened by the presence of Building 1 and Building 2, so that curve is restricted to . The maximal . reflection occurs in the specular direction given by is given by RCS. Symmetrically, the reflection on then on to as well as on Double reflections occur on then on to . and represent the associated RCS. These curves are maximal around 0 . Since the radiation facade is partially screened by , is restricted to of . is the symmetric of . The recombination of these six RCS components using (15) leads to the global bistatic RCS of the specific building scenario. Fig. 11 shows the global RCS curve of the three buildings scenario for a normal incidence illumination. Due to its symmetry, we only plotted the positive part of elevation angles of this curve: The RCS amplitudes and scattering angles are kept un. changed for the single reflection RCS The combination of both double-reflection contributions and leads to several lobes distributed around . The comparison to the global RCS given by HFSS shows a very good concordance of both approaches and validates our method in this more complex scenario. Fig. 12 representing the

Fig. 12. Total reflected field for the scaled model.

total reflected field obtained with both methods confirms this result. 2) Study of a Realistic Scene: Once our RCS approach is validated, the same building configuration is now studied at 900 MHz with realistic buildings dimensions: m, m, m, m, , , , , and m. In order to bring out the influence of the incidence direction, : , , three different cases are studied for . The global RCS of our prototype block in Fig. 9 is and at the observarepresented in Fig. 13 in the azimuth plane m. tion distance As expected, for a normal incidence, the same reflection phenomena as with the scaled scene are observed, and the angular positions of the peaks are unchanged. However, while comparing the global RCS for the three incidences (Fig. 13) to the one obtained with the scaled model, one observes that the lobe widths are thinner and higher in the realistic case because of the increase of the radiating surfaces. These lobes are the result of the following: , , , and for — either isolated reflections: , , , and for , and for ; — or the sum of single reflections RCS: for and ;

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Fig. 13. Influence of three incidence directions on the global RCS of the prototype block of buildings (Fig. 9) where facades have realistic sizes. TABLE I AMPLITUDES AND DIRECTIONS OF THE GLOBAL RCS PEAKS FOR THE REALISTIC SCENE AND VARIOUS INCIDENCE ANGLES

— or double-reflection RCS: respec, , . tively for Table I indicates the angular directions and the amplitudes of the maximal RCS values for each incidence direction. As it can be seen in Table I, the angular positions of all the RCS peaks change with the incidence direction, so it is difficult to predict their appearance if the illumination conditions as well as the positions and inclinations of the buildings are not precisely known. However, some observations can be made on the peak amplitudes. The maximum RCS values are in the ranges of 48–60 dBsm for single reflections and of 40–50 dBsm for the double reflections. The single-reflection RCS values are consistent with the nominal RCS of buildings given in [26], but it is difficult to find examples of building databases that take higher-order reflections into account in the literature. This study has proven that our RCS approach is capable of characterizing the scattering properties of a block of buildings by taking account of simple and double reflections between their facades. For the last studied structure, the simulation time for each incidence is about 3–6 s using MATLAB and on a Pentium 4 at 3 GHz with 2 GB of RAM. The rapidity of this technique could permit the construction of an RCS database of buildings blocks. The electric field computation at the receiver point using a ray-tracing software may be accelerated by the use of these stored RCS data. IV. CONCLUSION An accelerating method for urban radio propagation simulators is proposed in this paper. It is based on the homogenization of the facades dielectric properties. The validity distance of this approach has been estimated to be beyond the Rayleigh zone

of the scattering facade. For closer buildings, the precise description of the facade heterogeneity becomes necessary, and a suitable method based on Green’s functions has been used. In order to model in a rapid and simple way the contribution of far buildings, it has been shown how scattered fields from building facades are predictable with simple analytic RCS calculation of walls. The accuracy of the field predictions by our proposed RCS approach is verified with numerical methods. Single and multiple scatterings are taken into account by combining geometrical optics and RCS formulas. The RCS approach is a model to analyze the global behavior of sets of buildings including multiple reflections of the radiowaves between building facades. Restrictions on the method applicability are as follows. — Facades are of homogenous planar rectangular shapes or composed of rectangular parts with dimensions larger than a few free-space wavelengths. — Incident wave is locally plane (the transmitter is far from the building facade). — Rayleigh distance from the receiver to any facade has to be held. — Edge diffraction is not taken into account. Except in some very particular cases, these restrictions are compatible with urban propagation modeling. The rapidity of the approach with its good precision makes it appropriate to be implemented in a propagation simulator. In this way, the simulator can be accelerated for treating the far buildings. The proposed method able to characterize rapidly the scattering by a set of buildings in a compact form could also be of interest outside the urban coverage prediction, i.e., for geolocation or radar applications. REFERENCES [1] M. F. Iskander and Z. Yun, “Propagation prediction models for wireless communication systems,” IEEE Trans. Antennas Propag., vol. 50, no. 3, pp. 662–673, Mar. 2002. [2] K. Rizk, J. Wagen, and F. Gardiol, “Two-dimensional ray-tracing modeling for propagation prediction in microcellular environments,” IEEE Trans. Veh. Technol., vol. 46, no. 2, pp. 508–517, May 1997. [3] M. Neuland and T. Kürner, “Investigation of acceleration methods for radio propagation models,” presented at the IEEE Antennas Propag. Symp., USNC/URSI Nat. Radio Sci. Meeting, San Diego, CA, Jul. 2008. [4] P. Combeau, L. Aveneau, R. Vauzelle, and Y. Pousset, “Efficient 2D ray-tracing method for narrow and wide-band channel characterization in microcellular configuration,” Inst. Elect. Eng. Proc., Microw. Antennas Propag., vol. 153, pp. 502–509, Dec. 2006. [5] R. Hoppe, P. Wertz, F. M. Landstorfer, and G. Wolfle, “Advanced ray-optical wave propagation modelling for urban and indoor scenarios including wideband properties,” Eur. Trans. Telecommun., vol. 14, pp. 61–69, Jan.–Feb. 2003.

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[6] G. E. Athanasiadou, A. R. Nix, and J. P. McGeehan, “A microcellular ray-tracing propagation model and evaluation of its narrow-band and wide-band predictions,” IEEE J. Sel. Areas Commun., vol. 18, no. 3, pp. 322–335, Mar. 2000. [7] F. A. Agelet, A. Formella, J. M. H. Rábanos, F. I. de Vicente, and F. P. Fontán, “Efficient ray-tracing acceleration techniques for radio propagation modeling,” IEEE Trans. Veh. Technol., vol. 49, no. 6, pp. 2089–2104, Nov. 2000. [8] V. Degli-Esposti and H. L. Bertoni, “Evaluation of the role of diffuse scattering in urban microcellular propagation,” in Proc. IEEE VTC’, Amsterdam, The Netherlands, Sep. 19–22, 1999, pp. 1392–1396. [9] V. Degli-Esposti, D. Guiducci, A. de’ Marsi, P. Azzi, and F. Fuschini, “An advanced field prediction model including diffuse scattering,” IEEE Trans. Antennas Propag., vol. 52, no. 7, pp. 1717–1728, Jul. 2004. [10] S. Mostarshedi, E. Richalot, J.-M. Laheurte, M. F. Wong, J. Wiart, and O. Picon, “Fast and accurate calculation of scattered EM fields from building faces using Green’s functions of semi-infinite medium,” Microw., Antennas Propag., vol. 4, pp. 72–82, Jan. 2010. [11] S. Mostarshedi, E. Richalot, and O. Picon, “Semi-infinite reflection model of a multilayered dielectric through equivalent permittivity calculation,” Microw. Opt. Technol. Lett., vol. 51, pp. 290–294, Feb. 2009. [12] “IEEE standard definitions of terms for antennas,” IEEE Trans. Antennas Propag., vol. AP-17, no. 3, pp. 262–269, May 1969. [13] A. Yaghjian, “An overview of near-field antenna measurements,” IEEE Trans. Antennas Propag., vol. AP-34, no. 1, pp. 30–45, Jan. 1986. [14] S. Laybros, P. F. Combes, and H. J. Mametsa, “The ‘very-near-field’ region of equiphase radiating apertures,” IEEE Antennas Propag. Mag., vol. 47, no. 4, pp. 50–66, Aug. 2005. [15] S. Laybros and P. F. Combes, “On radiating-zone boundaries of short , and dipoles,” IEEE Antennas Propag. Mag., vol. 46, no. 5, pp. 53–64, Oct. 2004. [16] N. N. Youssef, “Radar cross section of complex targets,” Proc. IEEE, vol. 77, no. 5, pp. 722–734, May 1989. [17] J. Baldauf and S.-W. Lee, “High frequency scattering from trihedral corner reflectors and other benchmark targets: SBR versus experiment,” IEEE Trans. Antennas Propag., vol. 39, no. 9, pp. 1345–1351, Sep. 1991. [18] J. A. Jackson, B. D. Rigling, and R. L. Moses, “Parametric scattering models for bistatic synthetic aperture radar,” in Proc. IEEE Radar Conf., Rome, Italy, May 26–30, 2008, DOI: 10.1109/RADAR.2008. 4720819. [19] R. F. Harrington, Time-Harmonic Electromagnetic Fields. Piscatawa, NJ: IEEE Press, 2001. [20] C. A. Balanis, Advanced Engineering Electromagnetics. New York, USA: Wiley, 1989. [21] J. A. Kong, Electromagnetic Wave Theory. Cambridge, MA: EMW, 1998. [22] P. Pongsilamanee and H. L. Bertoni, “Specular and nonspecular scattering from building facades,” IEEE Trans. Antennas Propag., vol. 52, no. 7, pp. 1879–1889, Jul. 2004. [23] D. A. McNamara, C. W. I. Pistorius, and J. A. G. Malherbe, Introduction to the Uniform Geometrical theory of Diffraction. Boston, MA: Artech House, 1990. [24] E. F. Knott, “RCS reduction of dihedral corners,” IEEE Trans. Antennas Propag., vol. AP-25, no. 3, pp. 406–409, May 1977. [25] T. Griesser and C. A. Balanis, “Backscatter analysis of dihedral corner reflectors using physical optics and the physical theory of diffraction,” IEEE Trans. Antennas Propag., vol. AP-35, no. 10, pp. 1137–1147, Oct. 1987. [26] M. W. Long, Radar Reflectivity of Land and Sea, 3rd ed. Boston, MA: Artech House, 2001. [27] Y. B. Ouattara, E. Richalot, O. Picon, and J. Wiart, “Field prediction in urban environment using radar approach,” Microw. Opt. Technol. Lett., vol. 53, pp. 257–261, Feb. 2011. Y. B. Ouattara received the M.Sc. and Ph.D degrees in telecommunications and electronics from Université Paris-Est Marne-la-Vallée, Marne-la-Vallée, France, in 2007 and 2010, respectively. He was a Graduate Teaching/Research Assistant with Université Paris-Est Marne-la-Vallée from 2010 to 2011, and is currently preparing a post-doctorate year on RFID benchmarking at the university’s ESYCOM Laboratory. His research interests include computational electromagnetics, radiowave propagation, and antenna design.

S. Mostarshedi received the B.Sc. degree in electrical engineering from Sharif University of Technology, Tehran, Iran, in 1999, the M.Sc. degree in electrical engineering from Tehran Polytechnic, Tehran, Iran, in 2003, and the Ph.D. degree from Université Paris-Est Marne-la-Vallée, Marne-la-Vallée, France, in 2008. She worked as a Telecommunications Engineer with Siemens, Iran, before preparing her Ph.D. degree. She is currently with Université Paris-Est Marne-la-Vallée, where she works as an Assistant Professor and is a member of the ESYCOM Laboratory. Her research interests include wave propagation and computational electromagnetics.

E. Richalot received the Diploma and Ph.D. degree in electronics engineering from the Ecole Nationale Supérieure d’Electronique,d’Electrotechnique, d’Informatique,et d’Hydraulique de Toulouse, Toulouse, France, in 1995 and 1998, respectively. Since 1998, she has been with the University of Marne-la-Vallée, Champs-sur-Marne, France. Her current research interests include modeling techniques, millimeter-wave transmission lines, reflectarrays, and electromagnetic compatibility.

J. Wiart (M’96–SM’02) received the Eng. degree from the Ecole Nationale Supérieure des Telecommunication, Paris, France, in 1992, and the Ph.D. degree in physics from the ENST and P&M Curie University, Paris, France, in 1995. He has been with Orange Labs since 1992, when he joined the Research Center of France Telecom, Issy les Moulineaux, France. Since 1994, he has been working on the interaction of radio waves with the human body and heading the Orange Labs R&D unit WAVE. Since 2008, he has been the head of the Whist Lab (http://whist.institut-telecom.fr/), common laboratory of Orange Labs and Institut Telecom. He has published more than 50 papers in peer review journal. His research interests include electromagnetic compatibility, bioelectromagnetics, antenna measurements, computational electromagnetics, and statistics. Dr. Wiart has been an SEE Fellow since 2008. He is the Chairperson of URSI France. He is the French representative in COST BM0704 and is chairing the CENELEC TC 211 Working Group in charge of mobile and base-station standards.

O. Picon received the Agregation de Physique, prepared at the Ecole Normale Supérieure de Fontenay aux Roses, Lyon, France, in 1976, the Doctor Degree in external geophysics from the University of Orsay, Orsay, France, in 1980, and the Doctor in Physics Degree in microwave CAD from the University of Rennes, Rennes, France, in 1988. She was a Teacher from 1976 to 1982. Then, she was a Research Engineer with the Space and Radioelectric Transmission Division, Centre National d’Etudes des Télécommunications (CNET), Paris, France, from 1982 to 1991. Since 1991, she has been a Professor of electrical engineering, first at the Paris7-University, Paris, France, then at the University of Paris-Est Marne-la-Vallée, Marne-la-Vallée, France, where she is heading the ESYCOM Laboratory. She has published more than 150 technical papers in books, journals and conferences. Her research work deals with electromagnetic theory, numerical methods for solving field problems, and design of millimeter-wave passive devices.

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Modeling Propagation in Multifloor Buildings Using the FDTD Method Andrew C. M. Austin, Member, IEEE, Michael J. Neve, Member, IEEE, and Gerard B. Rowe, Member, IEEE

Abstract—A three-dimensional parallel implementation of the finite-difference time-domain (FDTD) method has been used to identify and isolate the dominant propagation mechanisms in a multistorey building at 1.0 GHz. A novel method to visualize energy flow by computing streamlines of the Poynting vector has been developed and used to determine the dominant propagation mechanisms within the building. It is found that the propagation mechanisms depend on the level of internal clutter modeled. Including metallic and lossy dielectric clutter in the environment increases attenuation on some propagation paths, thereby altering the dominant mechanisms observed. This causes increases in the sector-averaged path loss and changes the distance-dependency exponents across a floor from 2.2 to 2.7. The clutter also reduces Rician -factors across the floor. Directly comparing sector-averaged path loss from the FDTD simulations with experimental measurements shows an RMS error of 14.4 dB when clutter is ignored. However, this is reduced to 10.5 dB when the clutter is included, suggesting that the effects of clutter should not be neglected when modeling propagation indoors. Index Terms—Finite-difference methods, indoor radio communication, modeling, numerical analysis.

I. INTRODUCTION

T

HE INCREASING demand for wireless communication services has necessitated the reuse of frequency spectrum. Frequency reuse causes cochannel interference, which is detrimental to system performance, reducing the coverage area, reliability, throughput and the number of users that can be supported [1]. Characterizing and mitigating cochannel interference in unlicensed bands remains a major challenge. Indoor systems are particularly susceptible as all transceivers are usually located in close physical proximity. Accurately predicting system performance depends heavily on correctly characterizing the indoor propagation environment and a number of models to accurately and reliably predict signal strengths inside buildings have been proposed. Empirical models based on experimental measurements are often used, as the large variability in architectural styles and building materials can complicate deterministic modeling.

Manuscript received July 12, 2010; revised March 07, 2011; accepted April 07, 2011. Date of publication August 12, 2011; date of current version November 02, 2011. A. C. M. Austin was with the Department of Electrical and Computer Engineering, The University of Auckland, Auckland 1001, New Zealand. He is now with the University of Toronto, Toronto, ON M5S 1A1, Canada (e-mail: [email protected]). M. J. Neve and G. B. Rowe are with the Department of Electrical and Computer Engineering, The University of Auckland, Auckland 1001, New Zealand. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2164181

Empirical models typically use an exponential distance dependency to predict local means as a function of distance from the transmitter [2], [3]. Shadowing and fading are accounted for by including statistical variation around the local mean prediction [1]. However, empirically based models cannot explain the physical observations and are thus hard to generalize. For example, many of the terms and parameters in these models lack an electromagnetic basis and vary considerably between buildings [1], [2]. Consequently, the applicability of empirical models in buildings where measurements were not taken remains a concern. When used in practice, empirical models have been found to result in pessimistic estimates of system performance [1]. More accurate findings have been reported when the empirical models were complemented with physical factors, such as correlated shadowing [1]. Site-specific ray-tracing methods—such as Geometrical Optics (GO) and the Uniform Theory of Diffraction (UTD)—have also been widely applied to model propagation within buildings [4]. However, ray methods must be applied to the indoor propagation problem with caution, as many of the assumptions and approximations used in their derivation are not valid for typical indoor environments. For example, structural corners made from lossy dielectric materials (such as concrete) are frequently encountered in indoor environments, however dielectric wedge diffraction is known to be a non-ray-optical process [5]. Correctly predicting the diffracted fields is important, as in some circumstances (e.g., deeply shadowed regions) the received power is dominated by diffracted components [6]. The relatively compact size of the indoor propagation problem and advances in computational technology are allowing the application of grid-based numerical techniques, such as the finite-difference time-domain (FDTD) method. Unlike ray-based methods, the FDTD technique does not make a priori assumptions about the propagation processes. Due to the high computational requirements of the FDTD method, previous studies have limited analysis to propagation on two-dimensional or horizontal “slices” through the geometry [6]–[11]. However, two-dimensional results on a horizontal slice may fail to capture propagation mechanisms caused by interactions with the floor or ceiling and cannot be directly verified against experimental measurements. In practical indoor wireless communication systems, frequency channels are often reused between floors in a building, so characterizing interfloor propagation is important to predict the levels of cochannel interference [1]. It is important to note that, unlike the single-floor case, no single two-dimensional slice through a multifloor geometry can correctly account for all possible propagation paths. For example, many buildings have concrete shafts containing elevators and stairwells, and propagation to adjacent floors around (or through) such shafts

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can only be thoroughly examined three-dimensionally. For this reason, we regard a three-dimensional characterization of fundamental importance. Although a three-dimensional characterization of the propagating fields using the FDTD method is useful, it is the identification and isolation of the dominant propagation mechanisms that is arguably more important—especially for system planners [12]. Specifically, the FDTD simulations can be used to determine which components are important to predict the sector-averaged mean, and which can be reliably ignored. One of the original contributions of this paper is the identification (via a rigorous FDTD analysis) of the dominant propagation mechanisms, which could be incorporated into the development of accurate yet efficient models, suitable for use by system planners on a day-to-day basis. In particular, this paper focuses on providing an independent deterministic validation of the Seidel model [2] and is underpinned with experimental measurements. (This is consistent with the philosophy adopted by Walfisch and Bertoni—in their characterization of macro-cellular systems—who were able to deterministically explain (using Fresnel–Kirchhoff diffraction theory) the distance dependency of received power [13], which until then only had an experimental basis [14].) A further contribution of this paper is an assessment of the impact of clutter in the environment (such as office furniture) on the dominant propagation mechanisms. Interestingly, most existing applications of time-domain methods to model propagation within buildings have assumed these buildings to be empty [6]–[11], [15]–[17] (though [11], [16], and [17] did assess the impact of different wall types). However, actual office buildings contain varying amounts of furniture and other metallic and dielectric clutter. In this paper, comparisons are made between a basic FDTD model (with a similar level of detail to models in the existing literature) and a more detailed model that includes some furniture and similar objects. Unlike many previous FDTD characterizations of the indoor radio channel, the findings reported in this paper are validated against independent experimental measurements of the path-loss and fading distributions. A description of the FDTD models is presented in Section II. Section III describes a method of visualizing the energy flow by tracing streamlines through the Poynting vector. Section IV presents the simulation results for propagation to the same and adjacent floors. Section V proposes models for the sector-averaged path loss, while Section VI focuses on the statistical distributions characterizing multipath fading. Also considered in Sections IV–VI are the effects of increasing the level of detail in the simulation models and comparisons against experimental measurements. Section VII briefly summarizes the findings.

interior geometries, hereafter referred to as basic and detailed. The 0.20-m-thick concrete floors, 1-cm-thick exterior glass, and the hollow concrete services shaft are common to both models. The basic geometry adds internal walls (modeled as 4-cm solid slabs of drywall) creating a corridor around the shaft and dividing the remaining space into nine offices. The detailed geometry models the internal walls as two 1-cm sheets of drywall (separated by a 4-cm air gap) attached to wooden frames with studs spaced 1.5 m apart. Against each wall, metal bookcases extend floor–ceiling—these contain books, modeled as 0.20-m-thick wooden slabs. Doors into the offices and shaft are modeled as wood and extend slightly into the corridor; the elevator doors are inset and modeled as metal. Metal reinforcing bars (2 cm square) are embedded in the concrete floors on a 1-m grid, and metal window frames (2 cm square) are spaced 1.5 m apart on the external glass windows. Two flights of concrete stairs are also included in the central shaft. A single -field component is excited with a modulated Gaussian pulse, given by with parameters: GHz, ns and . This produces a pulse with a 150-MHz 3-dB bandwidth centered around 1.0 GHz. The cm is used to time-step is 18.3 ps. A square lattice size of minimize numerical dispersion [18, pp. 110–128]. The FDTD simulation domain is m and is surrounded by a 12-cell-thick convolutional perfectly matched layer (CPML) [18, pp. 294–310], resulting in approximately 3 billion mesh cells. Solving this problem using a single processor is not currently feasible, and accordingly, the lattice is subdivided and allocated to multiple processors. Field values on the boundaries are exchanged every time-step using an implementation of the Message Passing Interface [19]. On a 64-node computer cluster (using Intel Xeon 2.66-GHz processors), these problems require approximately 180 GB of memory and take 48 h to solve to steady-state (15 000 time-steps). The steady-state electric and magnetic field magnitude and phase were extracted by multiplying the time-series with a 1.0-GHz cissoid. To compare the FDTD results against experimental measurements, the steady-state fields were converted to path loss (in dB). The radiation pattern from a single component is isotropic in the azimuth plane and proportional to in the elevation plane (i.e., similar to a short dipole antenna). The steady-state fields (in the radial direction) can be converted to path loss by normalizing the values to the Friis equation. III. VISUALIZING ENERGY FLOW The time-averaged Poynting vector is given by

II. PROPAGATION MODELING WITH THE FDTD METHOD The building under investigation is the Engineering Tower at The University of Auckland, Auckland, New Zealand. This is a typical eight-floor 1960s concrete slab building with a services shaft (containing elevators and stairwell) in the center. In this paper, three floors have been considered. The nominal values for the material properties used in FDTD simulation models are the following: Concrete: , mS/m; Glass: , mS/m; Drywall: , mS/m; Wood: , mS/m; and metal: , S/m. The effects of internal detail/clutter are examined by considering two

W/m

(1)

where and are the steady-state vector electric and magnetic fields, respectively, and denotes the complex conjugate. At each point in the field, the Poynting vector indicates the direction and magnitude of energy flow. Streamlines are projected through this space by applying principles developed in fluid dynamics for studying steady flows [20]. (A similar analysis using Poynting vector streamlines to visualize energy flow escaping backwards from a pyramidal horn antenna was reported in [21].)

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The local tangent to a streamline is the vector representing energy flow at that point, and in three-dimensions the differential equation governing a streamline is given by (2) where is the position, is the parameter along the streamline, is the Poynting vector at . Starting from an approand priate initial position, , the streamline is computed by numerically solving (2) using forward differences (tracing the direction of the physical propagation of the Poynting vector out of the lattice) or backward differences (tracing the Poynting vector back to the transmitting antenna). The forward difference expression is given by (3) A step size (where is the lattice cell size) was found to be a good tradeoff between computational efficiency and accuracy. In the case where does not lie on the down-sampled FDTD lattice, it is interpolated using values from adjacent cells. Linear interpolation was found to provide an adequate result. It should be noted that a single initial position results in one streamline that may not be representative of the dominant propagation mechanism in a region. To assist in the visualization of the net energy flow in a region of space, 100 points in a “cloud” around the specified initial position are typically seeded. IV. PROPAGATION MECHANISMS A. Propagation on the Same Floor Fig. 1(a) and (b) plots the path loss on horizontal slices through the basic and detailed internal geometries. The slices are positioned 1.50 m from the floor, in the plane of the transmitting antenna (located at ). Fig. 2(a) and (b) shows streamlines (I–IV) of energy flow, calculated using (3) for both internal geometries. Initial points were selected, such that streamlines I and III are shadowed by the central shaft, and II and IV are separated from the transmitter by soft partitions. The central services shaft is observed to significantly shadow waves propagating across the floor when the transmitters are diagonally positioned. Paths penetrating through the shaft are highly attenuated by the thick lossy concrete walls, and consequently, signals received in the shadowed regions are dominated by paths propagating around the shaft. Energy reaching the shadowed regions in the basic geometry is observed to penetrate through the soft partitioned offices and reflects off the exterior glass windows. This is supported in both Figs. 1(a) and 2(a), where strong specular reflections from the glass are visible (e.g., streamlines I and II). The presence of strong reflected paths agrees well with previous two-dimensional FDTD simulations of empty buildings [6], [7], [10]. The problem is nominally symmetric, and reflections from both sides of the building contribute equal amounts of power; the resulting sector-averaged path loss in the shadowed regions is approximately 65 dB. Comparing Fig. 1(a) to 1(b) and Fig. 2(a) to 2(b) shows a distinct change in the propagation mechanisms, namely strong reflected paths from the windows and drywall are no longer visible. When shelves and books are included against the internal

Fig. 1. FDTD simulated path loss (at 1.0 GHz) on a horizontal slice through the first floor for (a) basic internal geometry and (b) detailed internal geometry. The location of the transmitter is indicated by , with the floor plan of the building superimposed.

walls, the reflected paths are attenuated to such an extent that (in this case) diffraction around the corners of the concrete shaft is observed to dominate propagation into the shadow regions. Paths involving diffraction exist in the basic geometry, but contribute a small proportion of the total received power. The inclusion of metal window frames also perturbs specular reflection from the glass. The sector averaged path loss recorded in shadowed regions is up to 15 dB higher than the basic geometry. The attenuation introduced by a single layer of clutter in the environment only slightly reduces the received power. However, the accumulation of many such effects has the potential to cast significant radio shadows and may result in other propagation mechanisms dominating. Clutter in the detailed geometry is also observed to introduce strong multipath components when propagating through the walls, e.g., streamlines II and IV in Fig. 2(a) and (b). This behavior also alters the fading distributions and is discussed in further detail in Section VI. For both the basic and detailed internal geometries, corridors have been modeled as largely clutter-free. The sides of the corridor can be considered electrically smooth, and thus the corridor has potential to act as an overmoded waveguide. This mechanism has previously been observed experimentally for relatively long ( 30 m) corridors [22]. In our simulations,

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Fig. 2. Streamlines of energy flow through the single-floor environment: (a) basic geometry and (b) detailed geometry. Seed points are marked with . Four seed points (I–IV) are identified for each case.

Fig. 3. FDTD simulated path loss (detailed geometry at 1.0 GHz) for (a) one floor separating the transmitter and receiver and (b) two floors separation. The location of the transmitter is indicated by .

the sector average path loss is observed to increase from 30 to 47 dB when moving 1.5–8.5 m away from the antenna, along the corridor. These values are between 3–5 dB lower than expected for free space and are attributed to the reflections from the walls, ceiling, and floor. At longer distances, the angle of incidence becomes increasingly glancing, and it is likely true waveguide modes may be formed.

tenuation introduced by each floor is not constant and varies depending on the location of the receiver. Fig. 4(a) shows the path loss on a vertical “slice” through the three-floor detailed geometry; the location of the slice is indicated by in Fig. 3(a). The radiation pattern of the short dipole antenna (located at ) causes greater path loss in regions above the antenna. The lower path loss around point “A” can be attributed to reflection and scattering from the metal elevator doors. The metal rebar is observed to introduce local scattering, supporting the findings of [23], which showed greater multipath is present when the rebar embedded in the concrete is included in the analysis. However, the dominant propagation path—penetration through the concrete—remains largely unchanged. Fig. 4(b) shows streamline visualizations of the Poynting vector for four seed points (indicated by and centered on the vertical slice) located one and two floors above the transmitter. These streamlines largely follow line-of-sight (LOS) paths (though refraction is also visible at the air-concrete and concrete-air interfaces), indicating the dominant propagation mechanism (in this region) is penetration through the floors. As more floors separate the transceivers, alternative propagation paths involving the lift-shafts and stairwells may con-

B. Propagation to Adjacent Floors Fig. 3 shows the path loss on horizontal slices (a) one floor and (b) two floors above the transmitter for the detailed geometry (similar to Fig. 1, the slices are positioned 1.5 m above each floor). Comparing the distribution of path loss one and two floors above the transmitter, Fig. 3(a) and (b), with the same floor case, Fig. 1(b), shows many similarities. In particular, the radio shadow cast by the shaft remains a dominant feature of the indoor environment, and propagation into the shadowed regions remains governed by diffraction at the corners of the shaft. Similar observations can be made for the basic geometry. Clutter in the environment is also observed to introduce strong local shadowing and multipath. These results suggest that many of the mechanisms identified in the same floor case still dominate propagation to adjacent floors. However, it is noted that the at-

AUSTIN et al.: MODELING PROPAGATION IN MULTIFLOOR BUILDINGS USING FDTD METHOD

Fig. 4. (a) Vertical slice through the three-floor detailed geometry [along indicated in Fig. 3(a)]. (b) Poynting vector streamlines on the vertical slice for seed points located on the adjacent floors. (c) Streamline visualizations of propagation paths (from floor 1 to 3) that travel through the shaft.

tribute significant amounts of power, and may explain the variations in attenuation across each floor. Results indicate these paths are not dominant for a single-floor separation. However, as indicated in Fig. 4(c), propagation into the lift-shaft can provide a comparable level of power two floors above the transmitter. (Streamline visualizations of the Poynting vector show the net energy flow, and for both paths to be present, they have to contribute roughly equal power.) As the walls of the shaft are thicker than the floors, these paths are only visible after two floor penetrations. If the geometry was extended to four floors, paths propagating through the shaft would be expected to dominate the received signal. It should also be noted that, depending on the environment, propagation mechanisms external to the building perimeter may dominate the received power on other floors. Diffraction at a floor/window edge [24] and reflections from surrounding buildings [15] are two such examples. V. PATH-LOSS PREDICTION MODELS Fig. 5 shows scatter plots of sector-averaged path loss (in dB) versus the transmitter-receiver separation distance. Only -field points from material-free regions are included in the sector average. Higher path losses are observed in the detailed

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Fig. 5. Scatter plots of path loss versus distance for (a) basic and (b) detailed internal geometries. The solid lines represent a Seidel model [2] fitted to the simulated data.

geometry, particularly on longer paths. Based on the results presented in Section IV, the change in propagation mechanism from reflection at the glass windows to diffraction at the concrete corner is responsible for the increased path loss. Also evident in Fig. 5 are a number of sectors (in both geometries) with distance dependency exponents . These sectors occur in the corridors and are caused by strong reflections from the walls, ceiling, and floor. Similar observations have been made in [7]. Similar to [2], models in the form are used to relate the average path loss with the transmitter–receiver separation distance dB (4) where is the distance dependency exponent for data colis the path lected on the same floor as the transmitter, loss at reference distance m, is the transmitter-receiver separation distance, is the floor attenuation is the encountered propagating through the th floor, and number of floor separations considered. The parameters and are found by fitting (4) to the data via linear regression, m dB. given

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TABLE I PARAMETERS FOR THE SEIDEL MODEL [2]

Table I shows the least-squares best-fit distance dependency exponents, floor attenuation factors, and RMS error between (4) and the FDTD and experimental data. The data collected across the floor has been divided into two regions, lit and shadowed, based on the position of the receiver relative to the transmitter and the concrete services shaft. When all regions are considered, the same floor distance dependency exponent increases for the detailed model due to increased attenuation through the clutter. The increase in RMS error for the detailed geometry also indicates scattering off the clutter introduces greater variability around (4). It is also observed that for both geometries and the experimental measurements. If the only propagation path was through the floors, and if the floors were identical, . However, the observed decrease is between 1–3 dB. In other buildings, a greater decrease in the FAF is observed (e.g., dB and dB [2]), and this behavior is difficult to explain experimentally [2]. As shown in Fig. 4(b), penetration through the floors is the dominant propagation mechanism in the lit regions (for both internal geometries), and consequently . In the , which is close to free space basic geometry, . The streamline visualizations of the Poynting vector presented in Section IV-A show the dominant propagation path in lit regions is largely LOS. The soft partitions do not perturb the propagating waves or introduce appreciable attenuation. However, in the detailed geometry , which tends to indicate clutter in the environment introduces additional attenuation. This is also supported by examining streamlines II and IV in Fig. 2(b), which show the energy tends to propagate on non-LOS paths in the lit regions. The streamlines presented in Fig. 4(c) indicate other (lower-loss) propagation paths may dominate in the shadowed region. It is thought the presence of such paths is (partly) responsible for the decrease in FAF between one and two floor separations. It is noted that for both geometries and experimental measurements. A. Comparisons to Experimental Measurements To confirm the findings made with the FDTD method, experimental measurements were conducted at 1.8 GHz over two floors of the Engineering building. Similar to [1], the transmitter carrier frequencies were spaced 400 kHz apart, allowing the power received from transmitters located on adjacent floors to be measured in a single sweep. Identical, vertically orientated,

Fig. 6. Scatter plot comparing sector-averaged measurements of the path loss with values obtained from FDTD simulations in the same locations. Both basic and detailed internal geometries have been considered, and in both cases the FDTD method generally underestimates the sector-averaged path loss.

discone antennas were used in both transmitter and receiver and located approximately 1.6 m from the floor. The radiation pattern of the discone antennas is isotropic in the azimuth plane, and the gain was calculated from test measurements in an anechoic chamber. Fifty-two measurements were made across the floor, and the receiving antenna was rotated over a 1-m-diameter locus to average out the effects of multipath fading. The voltage envelope was also recorded and used to determine the fading distributions. As the relevant material properties do not change significantly over the 1.0–1.8 GHz frequency range [3], a direct comparison between the path loss for 1.0 GHz FDTD simulations and 1.8 GHz experimental measurements only needs to account for the increased free-space loss (5.1 dB). Fig. 6 shows a pointwise comparison between the average path loss recorded for the 104 experimental data points and sector-averaged FDTD simulations of the path loss made at the same locations. It is noted that the FDTD simulations underestimate the path loss for many sectors. This underestimation cannot be accounted for in the frequency difference and is largest in regions furthest from the transmitter (and on highly cluttered paths), particularly paths shadowed by the shaft or passing through multiple partitions. Adding internal clutter improves the prediction accuracy, as shown in Fig. 6, as the RMS error between the measurements and simulation is 14.4 dB for the basic geometry; this is reduced to 10.5 dB when the clutter is included. This result is significant, as it suggested that to correctly predict the path loss, clutter present in the indoor propagation environment must be considered when applying full-wave electromagnetic methods. Although further environmental details/clutter could be added to improve the accuracy of the FDTD predictions, the random nature of the clutter (e.g., size, position, and properties) complicates a fully deterministic characterization of the indoor radio channel. VI. FADING DISTRIBUTIONS Rayleigh and Rician distributions are frequently used to characterize multipath fading for indoor environments [1]. When a specular component is stronger than the scattered

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Fig. 8. Streamline visualizations of the Poynting vector traced from seed points (located in the shaded region in Fig. 7) to the transmitting antenna for (a) basic and (b) detailed internal geometries. Fig. 7. Cumulative distribution functions of the received power envelope (in dB) normalized to the path loss for both internal geometries and experimental -field data were measurements compared to exact Rician distributions. The recorded in the shaded regions on the same floor as the transmitter (located at ).

components—for example, on an LOS path—the probability density function (PDF) of the signal envelope follows a Rician distribution, given by (5) where is the power of the dominant component, is the is the zeroth order Bessel funcmean scattered power, and tion of the first kind. The Rician -factor is defined as the ratio of specular power to scattered power, , and is determined from the FDTD simulated data using [25]

(6) where is the average square amplitude, is the avis the th-order Bessel function of erage amplitude, and the first kind. In the case where no single component dominates , the PDF of the signal envelope follows a Rayleigh distribution, given by (7) where is the mean power. Fig. 7 shows cumulative distribution functions (CDFs) of received signal envelope (in dB) normalized to the mean path loss for both internal geometries and experimental measurements. Also shown in Fig. 7 is a floor plan; the data was collected in the shaded sectors—located within an office, approximately 5 m from the transmitter. Rician distributions are fitted to these data sets using (6). In the basic geometry is 2.6—this indicates a strong dominant component exists—whereas, in the detailed geometry and measurements , suggesting more energy is being scattered in this case. The experimental data

set consists of 400 points, whereas the simulated data set has points. Consequently, there is a greater variability between the experimental and theoretical CDFs, particularly at lower signal powers. The differences in Rician -factor for the basic and detailed geometries can be explained by examining streamline visualizations of the Poynting vector. Fig. 8 shows streamlines for the (a) basic and (b) detailed geometries, reaching the same sectors considered in Fig. 7. In the basic geometry, it is observed that the energy is traveling on the LOS path with some attenuation when penetrating through the soft partitions. Accordingly, is greater than zero. The same streamline in the detailed geometry shows the LOS path is blocked by bookshelves. The increased attenuation allows additional scattered paths (of similar magnitude to the attenuated-LOS path) to exist, and accordingly, the PDF of the signal envelop follows a Rayleigh distribution. Similar phenomena are observed in other locations shadowed by clutter. VII. CONCLUSION A three-dimensional parallel FDTD algorithm has been used to identify and isolate the dominant propagation mechanisms in a multistorey building. A simplified model based on the dominant mechanisms has been compared against experimental measurements of the path loss. Streamline projections through the Poynting vector show that the dominant propagation mechanisms can change significantly when metallic and lossy dielectric clutter is included (the clutter also reduces Rician -factors across the floor). The change in propagation mechanisms results in a lower RMS error when the FDTD simulation results are compared with measurements. ACKNOWLEDGMENT The authors wish to thank the reviewers for their valuable and useful comments; Y. Yang for her assistance with the experimental measurements; Y. Halytskyy and the Centre for eResearch at The University of Auckland for facilitating access to the BeSTGRID Auckland Cluster; and the New Zealand Tertiary Education Commission for providing a scholarship to A. C. M. Austin.

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REFERENCES [1] K. S. Butterworth, K. W. Sowerby, and A. G. Williamson, “Base station placement for in-building mobile communication systems to yield high capacity and efficiency,” IEEE Trans. Commun., vol. 48, no. 4, pp. 658–669, Apr. 2000. [2] S. Y. Seidel and T. S. Rappaport, “914 MHz path loss prediction models for indoor wireless communications in multifloored buildings,” IEEE Trans. Antennas Propag., vol. 40, no. 2, pp. 207–217, Feb. 1992. [3] “Propagation data and prediction methods for the planning of indoor radiocommunication systems and radio local area networks in the frequency range 900 MHz to 100 GHz,” Recom. ITU-R P.1238-6, 2009. [4] S. Loredo, L. Valle, and R. P. Torres, “Accuracy analysis of GO/UTD radio-channel modeling in indoor scenarios at 1.8 and 2.5 GHz,” IEEE Antennas Propag. Mag., vol. 43, no. 5, pp. 37–51, Oct. 2001. [5] H. El-Sallabi and P. Vainikainen, “Improvements to diffraction coefficient for non-perfectly conducting wedges,” IEEE Trans. Antennas Propag., vol. 53, no. 9, pp. 3105–3109, Sep. 2005. [6] E. C. K. Lai, M. J. Neve, and A. G. Williamson, “Identification of dominant propagation mechanisms around corners in a single-floor office building,” in Proc. IEEE APS/URSI Int. Symp., 2008, pp. 424–427. [7] A. Alighanbari and C. D. Sarris, “Rigorous and efficient time-domain modeling of electromagnetic wave propagation and fading statistics in indoor wireless channels,” IEEE Trans. Antennas Propag., vol. 55, no. 8, pp. 2373–2381, Aug. 2007. [8] A. Alighanbari and C. D. Sarris, “Parallel time-domain full-wave analysis and system-level modeling of ultrawideband indoor communication systems,” IEEE Trans. Antennas Propag., vol. 57, no. 1, pp. 231–240, Jan. 2009. [9] T. M. Schäfer and W. Wiesbeck, “Simulation of radiowave propagation in hospitals based on FDTD and ray-optical methods,” IEEE Trans. Antennas Propag., vol. 53, no. 8, pp. 2381–2388, Aug. 2005. [10] T. T. Zygiridis, E. P. Kosmidou, K. P. Prokopidis, N. V. Kantartzis, C. S. Antonopoulos, K. I. Petras, and T. D. Tsiboukis, “Numerical modeling of an indoor wireless environment for the performance evaluation of WLAN systems,” IEEE Trans. Magn., vol. 42, no. 4, pp. 839–842, Apr. 2006. [11] Z. Yun, M. F. Iskander, and Z. Zhang, “Complex-wall effect on propagation characteristics and MIMO capacities for an indoor wireless communication environment,” IEEE Trans. Antennas Propag., vol. 52, no. 4, pp. 914–922, Apr. 2004. [12] M. J. Neve, K. W. Sowerby, A. G. Williamson, G. B. Rowe, J. C. Batchelor, and E. A. Parker, “Physical layer engineering for indoor wireless systems in the twenty-first century,” in Proc. Loughborough Antennas Propag. Conf., 2010, pp. 72–78. [13] J. Walfisch and H. L. Bertoni, “A theoretical model of UHF propagation in urban environments,” IEEE Trans. Antennas Propag., vol. 36, no. 12, pp. 1788–1796, Dec. 1988. [14] K. Allsebrook and J. D. Parsons, “Mobile radio propagation in British cities at frequencies in the VHF and UHF bands,” IEEE Trans. Veh. Technol., vol. VT-26, no. 4, pp. 313–323, Nov. 1977. [15] A. C. M. Austin, M. J. Neve, G. B. Rowe, and R. J. Pirkl, “Modeling the effects of nearby buildings on inter-floor radio-wave propagation,” IEEE Trans. Antennas Propag., vol. 57, no. 7, pp. 2155–2161, Jul. 2009. [16] M. Thiel and K. Sarabandi, “3D-wave propagation analysis of indoor wireless channels utilizing hybrid methods,” IEEE Trans. Antennas Propag., vol. 57, no. 5, pp. 1539–1546, May 2009. [17] M. L. Stowell, B. J. Fasenfest, and D. A. White, “Investigation of radar propagation in buildings: A 10-billion element cartesian-mesh FETD simulation,” IEEE Trans. Antennas Propag., vol. 56, no. 8, pp. 2241–2250, Aug. 2008. [18] A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. Boston, MA: Artech House, 2005. [19] “MPICH2: High-performance and widely portable MPI,” Mar. 2010 [Online]. Available: http://www.mcs.anl.gov/research/projects/ mpich2/ [20] H. Landstorfer, H. Liska, H. Meinke, and B. Müller, “Energiestr ömung in elektromagnetischen wellenfeldern,” (in German) Nachrichtentechnische Zeitschrift, vol. 25, no. 5, pp. 225–231, 1972, Oct. 1974, English translation: “Energy flow in electromagnetic wave fields,” NASA Technical Translation, NASA TT F-15,955. [21] J. F. Nye, G. Hygate, and W. Laing, “Energy streamlines: A way of understanding how horns radiate backwards,” IEEE Trans. Antennas Propag., vol. 42, no. 9, pp. 1250–1256, Sep. 1994.

[22] D. Porrat and D. C. Cox, “UHF propagation in indoor hallways,” IEEE Trans. Wireless Commun., vol. 3, no. 4, pp. 1188–1198, Jul. 2004. [23] R. A. Dalke, C. L. Holloway, P. McKenna, M. Johansson, and A. S. Ali, “Effects of reinforced concrete structures on RF communications,” IEEE Trans. Electromagn. Compat., vol. 42, no. 4, pp. 486–496, Nov. 2000. [24] W. Honcharenko, H. L. Bertoni, and J. Dailing, “Mechanisms governing propagation between different floors in buildings,” IEEE Trans. Antennas Propag., vol. 41, no. 6, pp. 787–790, Jun. 1993. [25] F. van der Wijk, A. Kegel, and R. Prasad, “Assessment of a pico-cellular system using propagation measurements at 1.9 GHz for indoor wireless communications,” IEEE Trans. Veh. Technol., vol. 44, no. 1, pp. 155–162, Feb. 1995. Andrew C. M. Austin (M’11) received the B.E. (Hons.) degree and has completed the requirements for the Ph.D. degree in electrical and electronic engineering from the University of Auckland, Auckland, New Zealand, in 2007 and 2011, respectively. He is currently a Research Fellow with the University of Toronto, Toronto, ON, Canada. His research interests are in the areas of radiowave propagation, mobile communications, and computational electromagnetics. Mr. Austin was awarded a New Zealand Tertiary Education Commission Bright Futures Top Achiever Doctoral Scholarship in 2007. Michael J. Neve (M’91) received the B.E. (Hons.) and Ph.D. degrees in electrical and electronic engineering from the University of Auckland, Auckland, New Zealand, in 1988 and 1993, respectively. From May 1993 to May 1994, he was a Leverhulme Visiting Fellow with the University of Birmingham, Birmingham, U.K. During this time, he was involved with radiowave propagation research using scaled environmental models. From May 1994 to May 1996, he was a New Zealand Science and Technology Postdoctoral Fellow within the Department of Electrical and Electronic Engineering, University of Auckland, where he was a part-time Lecturer/Senior Research Engineer from May 1996 to December 2000 and is currently a Senior Lecturer with the Department of Electrical and Computer Engineering. In 2004, he was a Visiting Scientist with the CSIRO ICT Centre, Sydney, Australia. His present research interests include radiowave propagation modeling in cellular/microcellular/indoor environments, the interaction of electromagnetic fields with man-made structures, cellular system performance optimization, and antennas. Dr. Neve was jointly awarded a 1992/1993 Electronics Letters Premium for two publications resulting from his doctoral research. Gerard B. Rowe (M’84) received the B.E., M.E., and Ph.D. degrees in electrical and electronic engineering from the University of Auckland, Auckland, New Zealand, in 1978, 1980, and 1984, respectively. He joined the Department of Electrical and Computer Engineering, University of Auckland, in 1984, where he is currently a Senior Lecturer. He is a member of the Department’s Radio Systems Group and his (disciplinary) research interests lie in the areas of radio systems, electromagnetics, and bioelectromagnetics. Over the last 20 years, he has taught at all levels and has developed a particular interest in curriculum and course design. Currently, his educational research activity is concentrated on the secondary-to-tertiary transition and on the development of course concept inventories. Dr. Rowe is a member of the IET, the Institution of Professional Engineers of New Zealand (IPENZ), ASEE, STLHE, and AaeE. He was the joint recipient of the 1993 Electronics Letter Premium Award for the papers “Assessment of GTD for Mobile Radio Propagation Prediction” and “Estimation of Cellular Mobile Radio Planning Parameters Using a GTD-based Model,” which he coauthored. He has received numerous teaching awards from his institution. In 2004, he was awarded a (National) Tertiary Teaching Excellence Award in the Sustained Excellence in Teaching category, and in 2005 he received the Australasian Association for Engineering Education award for excellence in Engineering Education in the Teaching and Learning category.

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Analysis and Modeling on co- and Cross-Polarized Urban Radio Propagation for Dual-Polarized MIMO Wireless Systems Vittorio Degli-Esposti, Member, IEEE, Veli-Matti Kolmonen, Enrico M. Vitucci, Member, IEEE, and Pertti Vainikainen, Member, IEEE

Abstract—Cross-polarization coupling is an important radio propagation characteristic in dual-polarized multiple-input multiple output (MIMO) systems. Still, few studies analyze the polarimetric properties of the radio channel in relation to the actual propagation conditions and processes taking place in urban environment. The topic is studied in the present paper with the aid of dual-polarized MIMO measurements and ray tracing simulations. Several scenarios are considered, and the impact of the different propagation characteristics (LOS, NLOS, link-distance, presence of diffuse-scattering, angular distribution of the signal, etc.) on cross-polarization coupling is analyzed. Generally, a fairly high degree of coupling is observed due to multipath propagation and especially to diffuse scattering. Surprisingly, it does not appear to depend on link distance. Index Terms—Electromagnetic scattering, measurements, multiple-input multiple output (MIMO) systems, polarization, ray tracing, urban propagation.

I. INTRODUCTION ULTIPLE-INPUT multiple-output (MIMO) transmission techniques have been proposed in the last decade which can exploit the multipath nature of the radio link to ensure reliable, high-speed wireless communication in urban environment [1]. The adoption of dual polarized antennas at one or both ends of the link is a very attractive solution, since it allows in theory a doubling in the number of input/outputs of the MIMO link with a less than proportional increase of the antenna-array size. The actual performance of dual-polarized MIMO systems however, is strongly dependent on the characteristics of the propagation channel. Two paradigmatic, opposite cases can be identified to analyze the potential capacity increase of dual-polarized MIMO schemes with respect to conventional, SISO schemes.

M

Manuscript received February 04, 2011; revised March 18, 2011; accepted April 14, 2011. Date of publication August 12, 2011; date of current version November 02, 2011. This work was supported in part by the NEWCOM++ European Network of Excellence and by SMARAD Centre of Excellence, Aalto University, Finland. V. Degli-Esposti and E. M. Vitucci are with the Dipartimento di Elettronica, Informatica e Sistemistica, Alma Mater Studiorum—Università di Bologna, (DEIS), IT-40136 Bologna, Italy (e-mail [email protected]; [email protected]). V.-M. Kolmonen and P. Vainikainen are with the Department of Radio Science and Engineering, Aalto University of Technology, FI-00076, Espoo, Finland (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2164226

Case 1) The propagation channel is characterized by a low degree of cross-polarization (X-pol) coupling so that the two co-polarized (co-pol) communication branches (corresponding to the two polarization states, see below) are virtually decoupled. This is the case of free-space propagation or propagation in “poor”-scattering environments. In this case multiplexing over the two channel branches corresponding to the two polarization states seems the natural way to go to increase transmission capacity. Case 2) The propagation channel is characterized by a high degree of cross-polarization coupling and low correlation between the two X-pol branches. This case corresponds to very rich-scattering environments such as very cluttered indoor environments. In this case, a good capacity/diversity gain might be achieved exploiting the low correlation between different polarizations at both link ends. Real-life cases probably lie between Cases 1) and 2) and simple interpretations of dual-polarized MIMO performance are no longer straightforward. Moreover, recent studies highlighted the key role of other parameters such as SNR conditions and fading statistics (e.g., Rice versus Rayleigh, correlations, etc.,) to determine the actual capacity potential of dual-polarized MIMO schemes [2]. There is however a need for a thorough analysis of co-pol and X-pol propagation in urban environment to provide useful suggestions on the optimal transmission technique and thus help assess the potential of dual-polarized MIMO in such environment. The topic has been studied only by few authors so far [2]–[6]. In [7], a preliminary study on the polarization characteristics of urban radio propagation is reported, where the same advanced ray tracing (RT) model used here [8], which includes a diffuse scattering model, is validated and tuned using measurement data. The diffuse scattering model embedded into the RT program is based on a semi-statistical approach in which diffuse power at the generic surface element must satisfy a power-balance constraint and a given scattering pattern which corresponds to an angular power density function [9], [10]. In the present work, the analysis is carried further-on using measurements and multiple RT runs, aimed at determining the actual degree of X-pol coupling in urban environment and highlighting the impact of the different propagation mechanisms (LOS/NLOS propagation, street-corner effect, diffuse scattering, etc.) and of link distance. The same MIMO antenna set up and measurement scenario as in [7] is adopted.

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Fig. 1. Cartesian and spherical reference systems. Fig. 2. Coupling due to high elevation of arrival.

In Section II, a brief analysis of the main propagation mechanisms taking place in urban environment and their impact on X-pol coupling is done, and the main parameters, such as crosspolarization discrimination (XPD), are defined. In Sections III and IV the measurement set up and the RT-simulation tool used to analyze dual polarized MIMO propagation are described, respectively. The measurements used in this work were performed using a 5.3-GHz wideband MIMO channel sounder with dual-polarized planar and semi-spherical antenna arrays [11]. The multiple-antenna arrangement is used here to perform power-averages over the different propagation branches and to derive “in one-shot” spatial averages of the significant parameters. Finally, Sections V and IV report on the most significant results and then conclusions are drawn in Section VI. Results show that typical XPD values fall between 8 and 10 dBs depending on the scenario, and are surprisingly independent of link distance. RT simulations also show that X-pol coupling is largely due to diffuse scattering since XPD is overestimated when only coherent interactions (specular reflection and wedge diffraction) are considered. II. URBAN PROPAGATION AND POLARIZATION In this section, we refer for simplicity to a single radio link, between one transmitting antenna element (Tx) and one receiving antenna element (Rx). For the sake of the analysis reported in this paper, the polarization state of the propagating field is defined with respect to an earth-related, global Cartesian reference system where the vertical axis ( -axis) corresponds to the zenith and the horizontal axes ( - and -axes) are parallel to the ground (azimuth plane) (see Fig. 1). This reference system seems appropriate to the urban environment were most obstacles (e.g., building walls and edges) are either vertical or horizontal. We assume therefore to decompose the field into two orthogonal, linear polarization components: vertical polarization (V-pol) and horizontal polarization (H-pol), the latter being any linear polarization in the azimuth plane (Fig. 2, horizontal arrows arranged in circle). An alternative polarization definition, which has not been adopted as the reference here, is related to a spherical reference system centered on the Tx or Rx antenna locations (Fig. 1). In this case, the field can be decomposed into two orthogonal, linear polarization components in the directions ( -pol, being the versor of the -coordinate) and ( -pol, being the versor of the -coordinate, also corresponding to H-pol).

Of course, any physically realizable Tx antenna can only emit a far-field with transverse components with respect to the radial, propagation direction. Therefore no antenna can produce a V-pol field in a 3-D space. A nominally “vertically polarized” Tx antenna located in the origin will emit an -pol field, which corresponds to a V-pol field only over the azimuth plane. For such reason, V-pol is often confused with -pol. polarization definition might It must be pointed out that a be considered more appropriate from a transmitter-side point of polarization discrimiview. At the Rx antenna however, a nation through measurements would require direction-of-arrival (DOA) discrimination capabilities as both and are DOAdependent. Therefore, a phased array at the Rx and heavy superresolution post-processing of the received signals would be required, which might alter its characteristics. In fact such postprocessing, although feasible with the measurement setup used here, can only resolve and identify a limited number or rays, not including a significant part of the diffuse multipath component (DMC) which, although weak in relative terms, might play a key role in X-pol propagation. Therefore we refer to the V/H polarization definition, and to the original signals received at each antenna port, without any post-processing. Let us consider now a dual-polarized 2 2 MIMO radio link where the Tx antenna has two inputs and the Rx antenna has two outputs nominally corresponding to the two polarization states. In this work, we tag as V (H) a signal transmitted from the V (H) port of the Tx antenna or a signal received from the V (H) port of the Rx antenna. For example the received power is by convention the power received in the V-polarization port of the Rx antenna when the Tx antenna is fed through its H-polarization port. Considering the propagation processes that can change polarization and produce X-pol coupling, three major mechanisms, titled here below as a), b), and c), can be identified. a) Coupling due to the antennas. Beside obvious coupling due to the reciprocal inclination, antennas can indeed produce coupling due to their non-idealities, which can be measured through the cross-polarization isolation parameter (XPI) [2]. For example, a V-pol incoming wave can produce signal at both the V- and H-port of the Rx antenna because of its limited XPI. Other X-pol coupling mechanisms are due to multipath propagation: b) coupling due to elevation spread; c) coupling due to interactions with obstacles.

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X-pol coupling is expected to be primarily due to roof-top diffraction and diffuse scattering, and only to the latter if roof-top diffraction is negligible, as in typical microcellular environments. Also ground-reflected and line-of-sight (LOS) rays can contribute to X-pol coupling as long as their elevation angle is nonzero at the Rx. The degree of X-pol coupling of the radio channel between the generic couple of Tx/Rx antennas can be expressed using cross-polarization discrimination ratios (XPDs), i.e., the ratios between co- and cross- polarized components of the signal power [2] Fig. 3. Polarization rotation at roof-edge diffraction.

Mechanism b) is due to the actual mapping of the field components at the Rx into the H/V directions which can produce X-pol coupling for purely geometrical reasons. For example, when a ray arrives at the Rx from a generic direction, then even if the field is emitted directly by a V-pol Tx antenna, and is thus –polarized, it includes a H-polarized component due to the non zero elevation angle (Fig. 2). It is worth noticing that a high elevation of arrival might also indirectly generate additional X-pol coupling because of the low XPI of the circular Rx array (see Section III) away from the azimuth plane. Mechanism c) is not as simple to explain. However, considering for instance an horizontal, rain-gutter edge of a building, an H-polarized, quasi-vertical ray impinging on the wedge from close to the building facade is seen from the wedge as hard-polarized and diffracted accordingly, i.e., keeping the hard polarization into the diffracted ray, as stated by the Geometrical Theory of Diffraction [12] (see Fig. 3). If we consider an horizontal diffracted ray (among those belonging to the Keller’s cone [12]) then the diffracted field is V-polarized with respect to the global reference system, with an evident polarization rotation with respect to the impinging ray. With simple Geometrical Optics considerations it is easy to verify however that such polarization rotations do not take place if diffractions and reflections occur on vertical walls and wedges, which is the most common case in urban microcellular propagation. In microcells, most ray propagate on the lateral plane [13], which is nearly horizontal, and interactions only take place on vertical walls and wedges, therefore little X-pol coupling is in theory expected in this case from interactions with obstacles. If walls are rough or irregular, diffused (or distributed) scattering can take place [10], but diffuse-scattered rays do not necessarily belong to the lateral plane as they do not have to satisfy geometrical optics rules. Therefore, diffused rays can produce X-pol coupling due to mechanism b). Moreover, diffuse scattering interactions do not necessarily keep the polarization state as reflections and diffractions on vertical obstacle. Such potential polarization rotation property of diffuse scattering is modeled in the following through the parameter (see Section IV). Summing up, all mechanisms a)–c) combine to determine the actual polarimetric behavior of the radio channel. Besides the contribution due to the antennas, which can be quite relevant,

[dB]

(1)

[dB]

(2)

[dB]

(3)

[dB]

(4) [dB]

(5)

were the power symbols have been already defined in this section. Notice that the superscript Rx(Tx) indicates that the polarization of the Rx(Tx) is kept fixed. Expression (5) defines an overall X-pol discrimination ratio. Notice also that the higher the XPD, the lower the degree of X-pol coupling of the channel. In the narrowband case, the radio channel between the generic couple of Tx/Rx antennas can be represented by a 2 2 polarized channel matrix (6) Therefore, definitions (1)–(3) can also be expressed through the matrix elements. For example XPD can also be expressed as [dB]

(7)

XPD ratios include the effect of both the antenna and the propagation link. The XPI on the contrary only takes into account the effect of the antennas. Further, cross-polarization ratios (XPRs) can be defined which only take into account the effect of propagation [2]. III. THE MEASUREMENT CAMPAIGN Measurements have been carried out by means of a wideband MIMO radio channel sounder with antenna arrays of dual-polarized antennas at both ends developed at the Aalto University of Technology [11]. In the transmitter, the carrier at 5.3 GHz is modulated by a pseudo-noise sequence with chip-rate of 60 MHz. Tx power is 37 dBm. The transmitting antenna is a planar array of 16 dual polarized 45 -slanted antenna elements. Tx elements outputs were properly post-processed to emulate right angled patches and therefore rematch each element’s port with V or H polarization. The receiving antenna is a hemispherical array of 21 dual-polarized

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Fig. 4. Antenna arrays used in the measurements.

Fig. 7. Measurement routes in central Helsinki.

Fig. 5. Co-polar radiation pattern ( ement of Rx array.

E

component) of a single V-polarized el-

all measurement routes, the Rx was at a height of 1.6 m above ground level. The Tx was at a height of 10 m for routes 1–7 (microcellular cases), and at 40 m for route 9 (macrocellular case). Although the setup allows power-delay profiles to be extracted, in the present work only narrowband power values for each Rx/Tx pair and for each polarization combination have been considered. Narrowband channel coefficients were calculated from wideband channel impulse responses after noise removal through integration over time. The instantaneous and cross-polarized received power levels for cocomponents of the incident fields were calculated was obtained from the measured matrices. For instance, by an incoherent power sum over all vertically polarized feed pairs of the Tx and Rx arrays. IV. THE RAY TRACING TOOL AND THE DIFFUSE SCATTERING MODEL

Fig. 6. XPI of a single V-polarized element of Rx hemispherical array.

elements [6] of which only the antenna elements close to the horizontal plane (two tiers closest to the sphere’s equator, for a total of ten dual-polarized elements) have been considered in this study, to keep horizontal and vertical polarizations decoupled. A photo of the Tx and Rx antennas is shown in Fig. 4. Figs. 5 and 6 show the radiation characteristics of one element of the Rx spherical array. As shown in [11] the XPI is above 20 dB near the azimuth plane (see Fig. 6). Thus, the RX antenna should not influence much polarization for near-horizontal rays. The receiver was moved along several urban micro- and macro- cellular routes in the center of Helsinki (see Fig. 7). In

Simulation has been performed adopting the RT model de, which scribed in [8] with a number of interactions yields the best tradeoff between low computation time and good prediction accuracy. An accurate 3-D urban database has been used to represent the urban scenario under study. Typical values have been adopted for the material characteristics [14], [15] . In particular, for external brick walls of all buildings the following complex relative permittivity has been used: (8) with

and

.

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The polarimetric 3-D far field radiation patterns of all vertical and horizontal antenna elements of both radio terminals, previously measured in anechoic chamber, have been input to the RT simulator. A peculiar feature of the adopted RT model is the embedded diffuse scattering model based on the “Effective Roughness” approach [9], [10]. A directive scattering lobe centered around the direction of specular reflection (see (5) and 11 in [10]) has . In theory, such scattering been chosen here, with model allows to estimate the power scattered by each wall element dS, but for practical reasons the number of scattering points must be limited to reduce computation time. The simplest choice is to assume a single scattering point located in the center of each wall. Such “concentrated” scattering model however is too rough to yield accurate results, especially when the receiver is in a street canyon scenario, surrounded by “close” walls. A better approach consists in using a “distributed” scattering model: in the present work, each wall of the scenario has m“tiles,” each one with a scattering been subdivided into source in its barycenter: this way scattered rays are more distributed over different elevation angles, with a more realistic impact on X-pol coupling. An alternative, analytical diffuse scattering method has been proposed [16] but not implemented yet in the present work. In [7]–[10], diffuse scattering was assumed totally incoherent and every scattered ray is assumed to carry 50% of the power on each one of the two orthogonal polarization states. Here diffuse scattering is still assumed incoherent but a proper paramis inserted to establish the amount of power transeter ferred into the orthogonal polarization state after a scattering interaction. For instance the field of a linearly polarized ray after a scattering interaction, can be expressed as

(9) is the polarization vector of the incident field, and where is the polarization vector orthogonal to it, while and can be considered random phases. The scattered field strength is calculated according to the model described in [10]. If the incident field is linearly polarized, then the “scattering XPD” can be defined as the ratio between co-pol and x-pol components of the scattered field

(10) Once the polarization properties of all rays have been defined, the X-pol parameters (i.e., XPDs) are extracted from RT output in the very same manner as from measured data.

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Fig. 8. Measurement route 2 (LOS to NLOS street crossing): impact of diffuse scattering on predicted X-pol power.

V. RESULTS A. Comparison Between Measurement and Simulation In all the reported cases the polarization at the Rx has been estimates are derived. kept fixed, and therefore only The header (Rx) will be omitted throughout the rest of this paper. First of all, a preliminary investigation has been carried out to determine the optimum values of the diffuse scattering pain particular. Fig. 8 shows the tuning rameters, and of of the diffuse scattering model with respect to measurements in terms of co- and X-pol power values along route 2. It is evident that diffuse scattering (alias DMC) is necessary to achieve a good match between measurement and simulation, especially in the X-pol case, as computed power values are strongly underestimated without scattering in all but the LOS co-pol case, and does not help because the X-pol increasing the value of power values appear to saturate after two or three interactions. value appears to be 0.01, i.e., only However, the best 1% of the power is transferred by the diffuse scattering interactions into the orthogonal polarization. Simulation results with are close to those with and therefore not shown in Fig. 8 for legibility. Note however that despite value the DMC contribution is very important the low for X-pol coupling as explained further on in this section. The following four graphs refer to routes 1-2-3 all together, thus getting a single NLOS-LOS-NLOS long route. The optimum DMC model parameters have been adopted in all results. From Figs. 9 and 10 we can infer that the two polarizations do not show a symmetrical behavior as the LOS/NLOS tranthan for the other cases. The overall sitions are steeper for agreement between simulation and measurement is very good. According to our investigations the cited asymmetry might be partly due to the impact of the Brewster’s angle phenomenon, which only affect the H-polarization for vertical-wall reflections, causing a partial fading of H-polarized rays in the vicinity of street-crossings where the incidence angle of some dominant rays is close to the Brewster’s angle.

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Fig. 9. Routes 1-2-3 (NLOS-LOS-NLOS case): Measured versus simulated (with DMC) co-pol and x-pol powers (V-polarized Rx elements).

Fig. 11. Routes 1-2-3: Measured versus simulated (with and without scattering) XPD .

Fig. 10. Routes 1-2-3 (NLOS-LOS-NLOS case): Measured versus simulated (with DMC) co-pol and x-pol powers (H-polarized Rx elements).

Fig. 12. Routes 1-2-3: Measured versus simulated (with and without scattering) XPD .

Simulated and measured and curves are reported in Figs. 11 and 12, respectively: it can be observed that XPD values are higher in LOS with respect to NLOS. The gap . is more evident for Apart from that gap the graphs have a noisy, random shape and the RT-simulated curve without DMC shows an even more peaky behavior and a slightly higher mean XPD, i.e., a lower X-pol coupling degree. Therefore, the DMC contributes to increase X-pol coupling, as expected (see Section II). According to our investigations the reason why the DMC is value it that the DMC roimportant despite the low tates polarization at the receiver because its contributions are spread over a wide range of elevation angles. This is not true for reflected and diffracted rays, at least in microcellular environment, as they tend to remain nearly horizontal (see Section II). In other words, what we call diffuse scattering keeps the original polarization state at the interaction except for a marginal effect , but coupling mostly take place at the Rx antenna through mechanism b) (see Section II). As a proof, we computed the RMS elevation spread (ES) along route 2 for both the direction of arrival (DOA) and di-

Fig. 13. Route 2—RMS Elevation Spread.

rection of departure (DOD). It is evident that the ES is greater with diffuse scattering, as shown in Fig. 13. Is should be pointed our however that buildings walls in the considered environment are quite smooth, without balconies

DEGLI-ESPOSTI et al.: ANALYSIS AND MODELING ON CO- AND CROSS-POLARIZED URBAN RADIO PROPAGATION

Fig. 14. Route 2 (LOS to NLOS street crossing): simulated XPD using ideal =2 dipoles.

 XPR

Fig. 15. Routes 4-5 (NLOS-LOS-NLOS case): Measured versus simulated (with DMC) co-pol and x-pol powers (V-polarized Rx elements).

and major indentations: probably a higher value might be more appropriate in other urban environments. A relevant contribution to X-pol coupling also seems to come from the Rx-antenna limited XPI. In fact, the result of a simulation over route 2 with ideal, half-wavelength dipoles (orthogonal to the street axis) shows that by removing the effect of the antennas the influence of the DMC on X-pol coupling is much more evident, with an XPD gap between the two cases of about 15 dBs (Fig. 14). Notice that what we call XPD is actually XPR here as the effect of the antennas is almost negligible. Power-value curves for routes 4–5 (another NLOS-LOSNLOS case) are shown in Figs. 15 and 16. The behavior is quite similar as for routes 1-2-3, with slightly steeper transitions . for Power value curves with V-pol at the Rx are reported also for route 6, a fully LOS street canyon case in Fig. 17. While power values are decaying along the route, XPD values are not as the and . and decaying trend is the same for both show a very similar trend (graph not shown). XPD values are also nearly constants along routes 4–5. XPD curves for routes 4–5 and 6 are not shown here for brevity. However, XPD cumulative distributions are reported in the following sub-paragraph.

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Fig. 16. Routes 4-5 (NLOS-LOS-NLOS case): Measured versus simulated (with DMC) co-pol and x-pol powers (H-polarized Rx elements).

Fig. 17. Route 6 (LOS): Measured versus simulated (with DMC) co-pol and x-pol powers (V-polarized Rx elements).

B. Analysis of the Impact of Propagation Characteristics on the Results In this paragraph XPD values are rearranged on the base of the LOS or NLOS propagation conditions and reported in form of cumulatives in Figs. 18–21 for all considered routes. From the four graphs it appears that the XPD values are slightly lower in NLOS, as expected, and that RT simulation with DMC shows a better agreement with measurements, also as expected. In particular, RT simulation without DMC shows an abnormally high dispersion of XPD values in all cases. Tables I–IV show the main x-pol parameters in the different cases: LOS/NLOS/macrocellular (i.e., route 9), for the different routes and for different enabled propagation mechanisms in the RT simulations. It can be observed that: 1) the XPD is quite constant over the different routes, including the macrocellular case, and its mean value is and 3 dBs for slightly greater (1 dB gap for ) in LOS w.r.t. NLOS; 2) except isolated XPD drops due to Brewster’s angles with reflections-only which bring down the simulated values

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XPD

cumulatives in LOS.

Fig. 21.

XPD

cumulatives in NLOS.

TABLE I MEASURED VERSUS SIMULATED MEANXPD ON SINGLE MICROCELLULAR ROUTES WITH DIFFERENT COMBINATIONS OF MECHANISMS

Fig. 19.

XPD

cumulatives in LOS. TABLE II MEASURED VERSUS SIMULATED MEANXPD IN MICRO LOS SCENARIOS WITH DIFFERENT COMBINATIONS OF MECHANISMS

Fig. 20.

XPD

cumulatives in NLOS.

close to the measurement ones, realistic values of XPD can only be obtained including diffuse scattering into RT-simulations. Otherwise XPD is almost always overestimated, and the increase in the number of successive interactions does not help much. In order to investigate the distance-dependence of X-pol coupling the XPD values for all considered routes are reported versus the Tx-Rx link distance in Fig. 22. No clear distance

dependence is evident, and this is a quite surprising result as “multipath richness” and therefore X-pol coupling is often thought to increase with distance. To better analyze the XPD-distance trend we carried out a RT simulation in an ideal case: a 3-km-long street canyon with ideal half-wavelength dipoles at the same height as for measurements in routes 1 to 7. The dipoles are always orthogonal to the street axis. In this case a slight increase in XPD is evident (see Fig. 23), probably due to the decreasing inclination of major rays along the route (such inclination is maximum at the beginning due to the non-equal terminal height and to mechanism b), Section II), which produces a decreasing contribution to x-pol coupling.

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TABLE III MEASURED VERSUS SIMULATED MEANXPD IN MICRO NLOS SCENARIOS WITH DIFFERENT COMBINATIONS OF MECHANISMS

TABLE IV MEASURED VERSUS SIMULATED MEANXPD IN MACRO SCENARIO WITH DIFFERENT COMBINATIONS OF MECHANISMS Fig. 23. Ideal street canyon: estimated XPD

 XPR using ideal =2 dipoles.

Fig. 24. Ideal street canyon: estimated XPD using real arrays.

Fig. 22. XPD in micro and macro cellular routes as a function of distance.

Using the real antenna arrays (see Section III) the average XPD is constant throughout the route (Fig. 24). It only slightly increases if diffuse scattering is disabled, thus suggesting that while the inclination of major rays decreases with distance, the DMC component increases with distance and the two phenomena most likely compensate each other. VI. CONCLUSION In summary, results show that typical XPD values fall between 8 and 10 dBs in NLOS or LOS cases, respectively, and are surprisingly independent of other environment characteristics and of link distance.

RT simulation including diffuse scattering seems to predict in a realistic way the polarimetric behaviour of the radio channel even with a small number of interactions. RT simulations show that X-pol coupling is partly due to the DMC propagation component as XPD is overestimated when only coherent interactions (specular reflections and wedge diffractions) are considered. RT simulations also show that the polarization coupling capability of diffuse scattering is mainly due to the wider elevation-spread of the DMC. This fact seems to confirm the findings reported in [17] where diffuse scattering was identified like a distributed form of canonical interactions such as reflections and diffractions from building periodic structures such as windows, columns, indentations, and decorations. Further investigations and results will have to consider other micro- and macro- cellular scenarios and other parameters which are important to assess the potential of dual polarized MIMO systems, such as fading statistics and correlations. ACKNOWLEDGMENT The authors would like to thank BLOM KARTTA Oy, Finland, for providing the Helsinki urban database.

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REFERENCES [1] A. J. Paulraj, D. A. Gore, R. U. Nabar, and H. Bölcskei, “An overview of MIMO communications—A key to gigabit wireless,” Proc. IEEE, vol. 92, no. 2, pp. 198–218, Feb. 2004. [2] C. Oestges, B. Clerckx, M. Guillaud, and M. Debbah, “Dual-polarized wireless communications: From propagation models to system performance evaluation,” IEEE Trans. Wireless Commun., vol. 7, no. 10, pp. 4019–4031, Oct. 2008. [3] H. Kuboyama, Y. Tanaka, K. Sato, K. Fujimoto, and K. Hirasawa, “Experimental results with mobile antennas having cross-polarization components in urban and rural areas,” IEEE Trans. Veh. Technol., vol. 39, no. 2, pp. 150–160, May 1990. [4] H. M. El-Sallabi, “Polarization consideration in characterizing radio wave propagation in urban microcellular channels,” in Proc. PIMRC’00, Sep. 18–21, 2000, vol. 1, pp. 411–415. [5] J. Perez, J. Ibanez, L. Vielva, and I. Santamaria, “Capacity estimation of polarization-diversity MIMO systems in urban microcellular environments,” in Proc. PIMRC’04, Sep. 5–8, 2004, vol. 4, pp. 2730–2734. [6] L. Vuokko, A. Kainulainen, and P. Vainikainen, “Polarization behavior in different urban radio environments at 5.3 GHz,” in Proc. 8th Int. Symp. Wireless Personal Multimedia Commun. (WPMC’05), Aalborg, Denmark, Sep. 18–22, 2005, pp. 416–420. [7] M. V. Enrico, K. Veli-Matti, D.-E. Vittorio, and V. Pertti, “Analysis of radio propagation in co- and cross-polarization in urban environment,” in Proc. ISSSTA’08, Bologna, Italy, Aug. 25–28, 2008. [8] V. Degli-Esposti, D. Guiducci, A. de’Marsi, P. Azzi, and F. Fuschini, “An advanced field prediction model including diffuse scattering,” IEEE Trans. Antennas Propag., vol. 52, no. 7, pp. 1717–1728, Jul. 2004. [9] V. Degli-Esposti, “A diffuse scattering model for urban propagation prediction,” IEEE Trans. Antennas Propag., vol. 49, no. 7, pp. 1111–1113, Jul. 2001. [10] V. Degli-Esposti, F. Fuschini, E. M. Vitucci, and G. Falciasecca, “Modelling of scattering from buildings,” IEEE Trans. Antennas Propag., vol. 55, no. 1, pp. 143–153, Jan. 2007. [11] V.-M. Kolmonen, J. Kivinen, L. Vuokko, and P. Vainikainen, “5.3 GHz MIMO radio channel sounder,” IEEE Trans. Instrum. Meas., vol. 55, no. 4, pp. 1263–1269, Aug. 2006. [12] C. A. Balanis, Advanced Engineering Electromagnetics. New York: Wiley, 1989. [13] H. L. Bertoni, Radio Propagation for Modern Wireless Systems. Upper Saddle River, NJ: Prentice-Hall, 2000. [14] O. Landron, M. J. Feuerstein, and T. S. Rappaport, “A comparison of theoretical and empirical reflection coefficients for typical exterior wall surfaces in a mobile radio environment,” IEEE Trans. Antennas Propag., vol. 44, no. 3, pp. 341–351, Mar. 1996. [15] D. Peña, R. Feick, H. D. Hristov, and W. Grote, “Measurement and modeling of propagation losses in brick and concrete walls for the 900-MHz band,” IEEE Trans. Antennas Propag., vol. 51, no. 1, pp. 31–39, Jan. 2003. [16] V. Degli-Esposti, F. Fuschini, and E. Vitucci, “A fast model for distributed scattering from buildings,” in Eur. Conf. Antennas Propag. (EUCAP’09), Berlin, Germany, Mar. 23–27, 2009. [17] P. Pongsilamanee and H. L. Bertoni, “Specular and non-specular scattering from building facades,” IEEE Trans. Antennas Propag., vol. 52, no. 7, pp. 1879–1889, Jul. 2004.

Vittorio Degli-Esposti (M’94) received the “Laurea” degree (with honors) and the Ph.D. degree in electronic engineering from the University of Bologna, Bologna, Italy, in 1989 and in 1994, respectively. From 1989 to 1990, he was with Siemens Telecomunicazioni, Milan, Italy, in the Microwave Communications Group. Since November 1994, he has been with the Dipartimento di Elettronica, Informatica e Sistemistica (DEIS), University of Bologna, where he is now Associate Professor and teaches courses on electromagnetics and radio propagation. He was

a Visiting Researcher in 1998 at Polytechnic University, Brooklyn, New York, and in 2006 he held a visiting faculty position at the Helsinki University of Technology, now Aalto University, Finland, where he taught a course on deterministic propagation modeling and ray tracing. He is author or coauthor of more than 70 peer-reviewed technical papers in the fields of applied electromagnetics, radio propagation, and wireless systems. He participated in a number of European projects including the European Cooperation Actions COST 231, 259, 273, and 2100 and the European Networks of Excellence NEWCOM and NEWCOM++. He also serves as reviewer for a number of international journals including several IEEE Transactions. He chaired and organized sessions and served in the Technical Program Committees at several International Conferences, and was Vice-Chair of EuCAP2010 and EuCAP 2011.

Veli-Matti Kolmonen received the M.Sc. degree in technology from the Helsinki University of Technology, Espoo, Finland, in 2004 and the D.Sc. degree in technology from Aalto University School of Science and Technology, Espoo, in 2010. Since 2003, he has been with the Department of Radio Science and Engineering, Aalto University of Technology, as a Research Assistant, Researcher, and currently as a Postdoctoral Researcher. His current research interests include radio channel measurements and modeling.

Enrico M. Vitucci (S’04–M’08) was born in Rimini, Italy, in 1977. He graduated in Telecommunication Engineering and received the Ph.D. degree in electrical engineering and computer science from the University of Bologna, Bologna, Italy, in March 2003 and May 2007, respectively. Currently, he is a Postdoctoral Fellow at the University of Bologna. His research interests are in mobile radio propagation, ray tracing models, diffuse scattering models, and MIMO channel modelling. He participated in the European Cooperation Projects COST 273 and COST 2100 and in the European Networks of Excellence NEWCOM and NEWCOM++. He also serves as reviewer for a number of international journals including several IEEE Transactions.

Pertti Vainikainen (M’91) received the M.S., Licentiate of Science, and Doctor of Science degrees in technology from the Helsinki University of Technology (TKK), Espoo, Finland, in 1982, 1989, and 1991, respectively. He was an Acting Professor of radio engineering from 1992 to 1993, has been an Associate Professor of radio engineering since 1993, and has been a Professor of radio engineering since 1998, all with the Radio Laboratory (since 2008 the Department of Radio Science and Engineering), TKK (since 2010 Aalto University School of Science and Technology, Aalto, Finland). He was the Director of the Institute of Radio Communications (IRC), TKK, in 1993–1997 and a Visiting Professor in 2000 with Aalborg University, Aalborg, Denmark, and in 2006 with the University of Nice, Nice, France. He is the author or coauthor of six books or book chapters and about 340 refereed international journal or conference publications. He is the holder of 11 patents. His main fields of interest are antennas and propagation in radio communications and industrial measurement applications of radio waves.

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Propagation Parameter Estimation, Modeling and Measurements for Ultrawideband MIMO Radar Jussi Salmi, Member, IEEE, and Andreas F. Molisch, Fellow, IEEE

Abstract—Ultrawideband (UWB) radar is a promising method for reliable remote monitoring of vital signs. The use of multiple antennas at transmitter and receiver (MIMO) allows not only improved reliability, but also better accuracy in localization and tracking of humans and their various types of movement. This paper describes an experimental demonstration of localizing a test subject and tracking his breathing under ideal conditions. The UWB MIMO channel, which includes the test subject as well as other objects, is modeled as a superposition of multipath components (MPCs). From the measured data one can extract the parameters of the MPCs, including their directions and delays, which allows localization of the test subject as well as tracking the breathing motion. Since the breathing pattern of the test subject induces delay variations of the diaphragm-reflected MPC that are much smaller than the Fourier resolution limits, the high-resolution RIMAX algorithm (iterative maximum-likelihood estimation scheme) is employed together with a path detection scheme for determining and tracking the MPC parameters. Furthermore, it is illustrated that with a wideband array model, the requirements for antenna spacing are not as limited as for conventional narrowband array processing. Through controlled experiments with a vector network analyzer and a virtual antenna array observing both an artificial “breathing” object as well as a human subject, it is shown that one can accurately estimate the small scale movement from human respiratory activity. This is achieved both for line-of-sight between transmitter, receiver, and objects, as well as for non-line-of sight. Index Terms—Array signal processing, multipath channels, parameter estimation, position measurement, propagation measurements, ultrawideband antennas, ultrawideband radar, virtual antenna array.

I. INTRODUCTION

A

S LIFE expectancy increases, the range of diseases that can be treated expands and tasks like care of the elderly are becoming increasingly important. While medical advancements have dramatically improved the quality of care, the cost of providing care has also significantly increased. Dramatic reduction in costs can be achieved by exploiting information techManuscript received December 09, 2010; revised March 15, 2011; accepted April 07, 2011. Date of publication August 12, 2011; date of current version November 02, 2011. Part of this work was presented at the Asilomar Conference on Signals, Systems, and Computers 2010. This research was supported in part by the Academy of Finland. The work of J. Salmi was supported by the Finnish Technology Promotion Foundation (TES) and the Walter Ahlström Foundation. J. Salmi was with the Department of Electrical Engineering, University of Southern California, Los Angeles, CA. He is now with the Aalto University School of Electrical Engineering, SMARAD CoE, Espoo, Finland (e-mail: jussi. [email protected]). A. F. Molisch is with the Department of Electrical Engineering, University of Southern California, Los Angeles, CA 90089 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2164214

nology, new sensing mechanisms, and automated data collection. A particularly promising area is the continuous or quasicontinuous sensing of vital health signs, i.e., parameters that are indicative of a person’s health, such as breathing activity. While body-mounted sensors can provide these data with great reliability, there are many situations where the inconvenience of having to constantly wear such sensors renders it impractical. Therefore, remote monitoring is highly desirable. Various methods for remote breathing monitoring have been studied at least since 1975, when [1] demonstrated the observation of phase shifts of microwave chirp signals reflected off the chest of a rabbit. Relatively narrowband Doppler radars (see, e.g., [2], [3] and references therein) are by far the most popular method, as they require relatively small hardware effort. However, their drawback is that they do not allow the localization of the target; this also leads to increased difficulty in separating multiple breathing objects and suppressing interference. Further improvement can be achieved by using multiple antenna elements at transmitter and/or receiver. For the conventional Doppler radar case, [4] and [5] demonstrated the ability to resolve multiple sources. Further performance improvements can be achieved by ultrawideband (UWB) radar. UWB provides high delay resolution, and thus has the ability to determine the range to either active or passive reflectors with high accuracy [6], [7]. This allows to more accurately localize the breathing subject [8] and track the small movements of the diaphragm during breathing. A number of recent papers have provided system designs and/or experimental results, such as [9]–[12]. These papers use a single transmit and single receive antenna, usually a horn antenna that is pointed towards the target. This is useful in controlled scenarios where the location of the test subject is known. In this paper we investigate a scenario were the location of the subject is not predetermined, but finding it is one of the goals of the measurement. This can be achieved by combining UWB signalling (providing high range resolution) with the directional resolution capability of multiple antenna elements. The underlying theoretical framework formulates a wideband antenna array model and employs the double-directional radio propagation channel model [13] to characterize the UWB channel as a superposition of plane waves (multipath components, MPCs). The characteristics of the MPCs form the basis for the subject localization and the tracking of the breathing. We first establish the model in a general form, applicable to arbitrary antenna array configurations and supporting also polarization aspects. We then give an example of a reduced model, applicable to the employed measurement setup. The paper also discusses modeling of wideband antenna arrays, along with a description of a virtual array model. The virtual

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array is similar to synthetic aperture radar (SAR), employed recently in through-the-wall imaging, see e.g., [14], [15]. However, the virtual array differs from SAR imaging by relying on the MPC model, which describes the radio channel as plane waves observed at a single point in space, namely the array phase center. This also limits the total size of the array, as the MPC should originate in the far field of the array. The processing of the measurement data requires the decomposition of the observed channel response into a superposition of MPCs. For this purpose, we formulate a Maximum Likelihood (ML) based path detection framework, and jointly estimate the MPCs using a modification of the RIMAX [16] algorithm. This approach allows to overcome the Fourier limit of the delay resolution, which in our case is about 5 cm and thus insufficient for identification of the movement from breathing. The multi-antenna, UWB impulse responses are measured by a vector network analyzer (VNA) combined with a virtual antenna array. This approach does not provide real-time capabilities, and would have to be replaced by a faster excitation combined with real or switched antenna arrays for an actual deployment. However, it is eminently suitable for a proof-of-concept, and shows advantages due to better calibration of the underlying signals. For the same reason, this approach for measuring MIMO-UWB channels is in general use for channel sounding [17], although real-time implementations exist as well [18]. We show experimental results, both line-of-sight (LOS) and non-line-of-sight (NLOS), from anechoic chamber measurement for detecting small scale movement of both a passive, artificial target, as well as a human holding his breath with empty and full lungs, respectively. This information, together with the estimates of the absolute delays and angles of the corresponding MPCs, can be further utilized for passive localization and surveillance of humans or animals. Part of this work was presented in [19], where the discussion was limited to LOS measurement of an artificial object. The paper is structured as follows. Section II introduces the system model, including virtual antenna array model and the double directional MPC model. Section III describes the experiments conducted in an anechoic chamber. Section IV introduces the parameter estimation framework, that is applicable to the performed measurements. Section V illustrates the results from the measurements and concluding remarks are given in Section VI. The following notational conventions are used throughout the paper. • Boldface upper case letters (Roman or Greek ) denote matrices and lower case denote column vectors. Calligraphic uppercase letters denote higher dimensional tensors. • The vector denotes the th column of a matrix and the scalar denotes the th element of a vector . denote constants, and • Non-boldface upper case letters lower case denote scalar variables. • Superscripts , and denote matrix transpose, and Hermitian (complex conjugate) transpose, respectively. • Operator denotes Khatri-Rao (column-wise Kronecker) product.

Fig. 1. MIMO channel modeling concept. (a) The channel tensor is defined as the transfer function between all the Tx/Rx antenna port pairs. (b) The parameters of the double directional propagation path model are defined w.r.t. two points in space, namely the array phase centers.

• Symbol denotes an estimate of the vector . • Operation stacks all the elements of the input tensor into a column vector. • Operation transforms a given entity, typically a vector, into an -dimensional tensor, see also [20]. • Operation permutes the order of tensor dimensions, e.g.,

, see also [20]. II. SYSTEM MODEL

The considered system is based on measuring and characterizing the UWB radio propagation channel between two antenna arrays, one transmitting and the other receiving . Measurements provide the transfer function between all Tx/Rx antenna port pairs, as illustrated in Fig. 1(a). However, the double-directional propagation channel model, illustrated in Fig. 1(b), is defined between two points in space, namely the array phase centers. Hence, the complete channel model is comprised of two main components: i) The mapping of the MPCs onto the antenna ports, i.e., the antenna array model, and ii) the double-directional radio wave propagation model. In the following description it is further assumed that the MPCs of interest interact in the far field of the antenna arrays, i.e., the individual paths can be modeled as plane waves. A. Antenna Array Model The antenna array model describes the angular and frequency dependency of the array transfer function. The transfer function of a Tx/Rx antenna array with ports, sampled at fixed frequency points, angles, and for the (horizontal) or (vertical) orthogonal polarization component for a far field plane wave signal, is defined as a three dimensional complex-valued tensor (1)

SALMI AND MOLISCH: PROPAGATION PARAMETER ESTIMATION, MODELING AND MEASUREMENTS FOR ULTRAWIDEBAND MIMO RADAR

The two vectors

and contain angles for azimuth and elevation, respectively. The antenna array description (1) can be obtained, e.g., through a calibration measurement, where the whole antenna array is rotated w.r.t. a defined reference point, and the frequency response of each antenna element is measured for a sufficient number of angles. Interpolation to an arbitrary angle can be achieved, e.g., through a Fourier series representation, such as Effective Aperture Distribution Function (EADF) [21], [22]. The EADF also provides efficient means to obtain the derivatives, required by the estimation algorithm described in Section IV. For the experiments in this paper, a virtual antenna array is employed. A virtual array is formed by moving a single antenna such that its different positions form an array. This method not only reduces the hardware requirements, but also has the added benefit of being free from mutual coupling effects; on the downside, the channel to be measured by the array has to stay static for the duration that is required to move the antenna to all its positions. The model (1) for the virtual array can be constructed by first obtaining the angular-frequency response for the single antenna , defined at the origin of the coordinate system. The response of the th virtual antenna element, having a position vector , and a rotation , is given by

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considered frequency independent. However, for more complex propagation mechanisms, a frequency dependent path weight model may be required [17]; this is not taken into account in the remainder of the paper. Possible effects of neglecting frequency-dependent propagation would be the occurrence of “ghost components”. The wideband path model (3) differs from the well know narrowband model, see e.g., [23], in that the antenna array responses are frequency dependent. Hence, the response of a single path is no longer an outer product (or Kronecker product) of the steering vectors of the individual sampling apertures. The superposition of the MPCs (3) can be expressed as

(4)

(2) where is the unit vector in the direction , . Note that the frequency dependency in (2) results both from the directional frequency response of a single antenna, as well as the phase evolution at different frequencies according to the antenna’s position in the array. B. Double Directional Propagation Path Model A realization of the multidimensional UWB MIMO channel tensor is defined as the frequency domain transfer function of the channel between each MIMO antenna pair measured at the antenna feeds, see Fig. 1. The propagation between the two antenna arrays is described as a superposition of discrete MPCs . The model for a single path is illustrated in Fig. 1(b) and is given by

(3) denotes Khatri-Rao (column-wise Kronecker) product, is a matrix slice of (1) is for a single direction (that of path ), and the frequency response (phase shifts) resulting from the delay . The variable denotes the complex path weight of the polarization component. Note that for free space propagation, as well as reflections from materials such as large metallic surfaces, the phase shift and attenuation from the propagation channel (excluding the antennas) can be where

where

and contain the re-

sponses for paths, and , where denotes the number of polarization coefficients (here ). III. EXPERIMENTAL SETUP The experiment aims at a reproducible measurement of a UWB channel in an environment with a breathing human. The measurement was performed using a Vector Network Analyzer (VNA) and a virtual antenna array, i.e., a single antenna was moved using a high precision positioner. The VNA was calibrated to the antenna feed points, and a stepped frequency sweep was conducted for 801 points on the 2–8 GHz frequency range, see Table I. These measurements are time consuming, and can not be used for real time measurement of a human subject. They serve only as a proof-of-principle; real-time measurements could be performed with real arrays, excited by short large-bandwidth signals, such as chirps or impulses. Two types of measurements were performed. In the first ones, an object that reflects similar amount of RF radiation as a human was positioned in a controlled manner, imitating the movement of the chest while breathing. Another measurement was performed with a human subject, who was holding his breath during inhale/exhale periods for the duration of each VNA frequency sweep at each antenna position. Before each measurement, a reference measurement was taken without the object of interest inside the chamber to facilitate background removal in the postprocessing of the data, discussed in Section IV.

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antenna spacing, using the planar monopole antenna at 2.2–7.3 GHz. The ambiguity function is defined as (5) is defined as in (1), and reduces to a where narrowband steering vector for a single frequency. Fig. 3 compares the ambiguity of the wideband model to the narrowband ones at lowest and highest frequency. It can be observed that the wideband array steering vectors do not suffer from angular ambiguity, or grating lobes, which occur for narrowband steering vectors should the antenna spacing exceed , see e.g., the plot at 7.3 GHz where . This stems from the fact that the grating lobes for different frequencies occur at different angles. Hence their influence is effectively reduced using the wideband modeling. Furthermore, the angular resolution, i.e., the width of the main lobe is much narrower for the wideband case than the lowest frequency alone. It should be mentioned that one can achieve acceptable wideband ambiguity properties even with significantly smaller number of antennas. However, such analysis is outside the scope of this paper. For discussion on ultrawideband arrays, see e.g., [26], [27]. B. Measurement With Artificial “Breathing” Object

Fig. 2. (a) Antennas used in the measurements, along with the magnitudes (in dB) of their (V-polarized) angular frequency responses for (b) UWB horn antenna, and (c) UWB planar monopole antenna.

A. UWB Antennas and the Virtual Array The antennas used in the measurements are custom built UWB antennas [24], [25], shown in Fig. 2(a). The frequency response of the antennas as a function of azimuth angle (elevation fixed to horizontal plane) are obtained by a calibration measurement, and the magnitudes in the range 0.05–16 GHz are shown in Fig. 2(b) and (c). The color scale can be interpreted directly as magnitude (attenuation in dB) of the complex transfer function of the antenna for a far field plane wave. The planar monopole antenna1 is preferred over the UWB horn [25] due to its omnidirectional beampattern for situations where the location of the object is not known a priori. However, the frequency range where the omnidirectionality applies is limited to approximately 2.2–7.3 GHz. In the measurements, different combinations of these antennas were used based on their suitability and availability. Fig. 3 illustrates the ambiguity function of a virtual 10-element uniform linear antenna array (ULA-10) with 1The planar monopole antenna follows the design of solution A, listed in [24, Table I].

The measurement setup is summarized in Table I, and illustrated in Fig. 4. Two linear positioners, one for the uniform linear antenna array (ULA) and one for the breathing object, were placed inside the anechoic chamber at UltRa Lab [28]. The positions of the Rx antenna and the object, as well as the frequency sweep of the VNA, were controlled by a Labview script on a PC. Fig. 4(a) shows the approximate layout of the anechoic chamber. Two static scatterers (metallic poles) were placed in the chamber to create (controlled) multipath propagation. The Rx array positioner was sequentially placed in each corner of the chamber, and each measurement was repeated in non-line-of-sight (NLOS) conditions by blocking the line-of-sight (LOS) between the Tx and the object using a metallic bookshelf. LOS measurements were conducted using UWB horn antennas, whereas NLOS measurements were performed with UWB planar antennas. In this paper results from Rx1 position only, corresponding to Fig. 4, are reported. The path lengths and angles based on the measured center points of the interacting objects are listed in Table II. Before the actual measurement, a test measurement was performed in order to find a suitable object to be used in the measurement to replace a human. Fig. 5 illustrates the test results by comparing the signal level reflected from different objects. Based on this test, a setup with two basketballs covered by aluminum foil was chosen as the “breathing” object. C. Measurement With a Human Subject The measurement with a human subject was limited to one Rx array location (same 10-element ULA), and two “object” positions, namely “inhale” and “exhale”. The test person was sitting inside the chamber in a comfortable chair, see Fig. 6. At each virtual antenna array position, two measurements were taken, one for inhale and one for exhale. The person indicated

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Fig. 3. Ambiguity function of a ULA-10 UWB (2.2–7.3 GHz) antenna array model compared to narrowband model at 2.2 GHZ and 7.3 GHz. TABLE II POTENTIAL SIGNAL ROUTES FOR RX1 POSITION IN THE ANECHOIC CHAMBER, SEE FIG. 4(a). S1 AND S2 REFER TO THE SCATTERERS, AND B (BALL) DENOTES THE OBJECT

Fig. 5. Comparison of measured magnitude of the impulse response at 1.6–3 GHz from different types of objects: a human, one or two aluminium foil covered basketballs, and a small dummy. Two balls have the closest resemblance to that of human in terms of reflected signal level.

Fig. 4. (a) Measurement setup inside the anechoic chamber. (b) The breathing object and two scattering poles. (c) View from the opposite door.

his hold of breath, either with empty lungs, or lungs full of air, using a thumb operated light switch. Otherwise, he was immobile during the approximately 10 minute measurement. The measurement was repeated in NLOS conditions, while blocking the direct path between Tx and the person using a metallic bookshelf. In addition, each measurement was repeated so that the person had an aluminium foil wrapped around the chest. This facilitates the identification of the specific MPC, which corresponds to the reflection from the chest.

Fig. 6. Chair on which the person was sitting inside the chamber. Holding the breath was indicated by a light switch, attached on the armrest.

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IV. PARAMETER ESTIMATION A. Reduced Measurement Model For brevity, we describe the estimation methods w.r.t. a reduced measurement model, namely the one supporting the measurement setup described in Section III. The difference to the full model (4) is in that a (virtual) -polarized2 antenna array was employed at Rx only, whereas a single, -polarized antenna was used as a transmitter. In addition, the employed virtual array was a uniform linear array, and hence the angular modeling is limited to the azimuthal plane. It is assumed that these assumptions do not significantly deteriorate the system performance in the studied anechoic chamber environment, however, this assumption does not hold in general, see e.g., [22] for a discussion. With these assumptions, the model for a single MPC reduces from (3) to

either create new MPCs, or block or attenuate existing ones. Hence, it is assumed that , where denotes the MPCs resulting from the interaction with the object. From (10), the delays and azimuth angles along with corresponding path weights are estimated for a number of paths. C. Path Detection The initialization of the parameter estimation is performed using a successive cancellation type of grid search, where the detection of paths is based on single path maximum likelihood (ML) criterion. The approach is similar to [29], generalized for the wideband antenna array model (2). Typically, this initialization is performed for a larger number of paths at the first snapshot, and for a few new paths at each snapshot. The objective of the ML is to maximize the likelihood function (11)

(6) and the superposition of

Consequently, taking the logarithm of (11) and maximizing it w.r.t yields

paths can be written as

(7) where rameter vector for as

paths and

. The paparameters can be defined

(12) As the path weights are linear in (7), their best linear unbiased estimate can be expressed as (13)

(8) where the complex path weight is reparameterized into two real valued parameters and as . A single snapshot of the measured channel realization is modeled as (9) where is complex, circular symmetric white Gaussian measurement noise.

where some manipulation

in (7). Inserting (13) in (12) yields after

(14) Evaluating (14) for a single path, and assuming the non-polarimetric wideband path model (6), the ML criterion reduces to (15)

B. Background Removal As we are only interested in estimating the influence of moving a specific object in the environment, we perform the following background removal from the measured channel frequency responses. Before the actual measurement, a reference channel is measured for all antenna positions, with the object or person of interest not being inside the chamber. The MPC parameter estimation is then conducted for a data set obtained by (10) where the reference channel is subtracted from the measured channel frequency response . Ideally, all remaining, significant components in the channel should result from interaction with the scattering object; note that this interaction can 2The term V-polarized is used to highlight that the polarization alignment of the antennas was close to vertical.

. A new path eswhere timate is obtained as the set of parameters maximizing (15), and then solving for (13). The influence of this path can be subtracted from the data as , and the procedure may be repeated to detect more paths. It is also possible to optimize the already obtained path estimates jointly between detections, using the algorithm outlined in Section V. Equation (15) can also be interpreted as a matched filter, or a generalized beamformer, and the objective is to find the parameters that maximize the correlation with the measurement. Note that by setting , (15) can also be interpreted as the multidimensional power profile of the propagation channel. This has been used for the illustrations in Section V. An example of a computationally efficient implementation of a grid search over the parameters and for evaluating (15) is outlined in the Appendix.

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D. Parameter Optimization The initialization of the path estimates is based on a suboptimal strategy of successive detection and cancellation using of the ML criterion for a single-path only (15). Therefore, it is necessary to perform joint optimization of the estimates. It can be shown that finding the joint ML estimate of the parameters in (8) equals (16) No closed form solution exist for this nonlinear least squares optimization problem, so we resort to an iterative optimization technique, namely the Levenberg-Marquardt method [30], which is also used for a narrowband signal model in the RIMAX algorithm [16]. As the algorithm is gradient-based, it requires the derivatives of the data model (7) w.r.t. the estimated parameters (8). Let us define the Jacobian matrix as , the Score function , and the Fisher Information Matrix as

(17)

(18)

(19) denotes the log-likelihood function. The estiwhere mates are then optimized by iterating

(20) where denotes element-wise matrix product. The step size tuning parameter can be decreased if the iteration improved the fit (16), otherwise should be increased, see [16] for discussion. Convergence can be evaluated, e.g., by testing if is smaller than 10% of the Cramér-Rao bound, obtained from the diagonal values of in (19). After convergence, it is possible to either detect more paths from the residual , or use the current estimates as the initial values for the next snapshot of data. V. RESULTS A. Results With the Object—Los Case This section describes results from a measurement, where the breathing object (two aluminum foil covered basketballs) was moved to eight positions in the range , see Fig. 4. Fig. 7 shows the Power-Angular-Delay Profile (PADP) of the measurement, evaluated using (15) with and averaged over the eight measured ball positions. Some residual signal can still be observed after the reference channel cancelation

Fig. 7. (a) Power-Angular-Delay Profile (PADP) of the measurement averaged over the eight snapshots (ball positions). (b) PADP of , see (10), i.e., after background removal. (c) PADP zoomed to the area of interest. White x-marks indicate estimated path components (36 in total). (d) PADP at the direct reflection for the eight snapshots.

(10), especially at the LOS between the antennas around path lengths of less than 2 m, as well as for Scatterer #1 after 4 m path length. Fig. 7(c) shows the PADP zoomed at the paths of interest. Also some of the probable signal routes are identified. The white crosses drawn over the PADP denote the locations of estimated MPCs. When comparing the path lengths from Table II, it should be noted that the basketball has a radius of about 12 cm, and the scattering poles about 3 cm, which shorten the actual path length compared to measured object center points. In addition, a separate calibration was performed to determine the effect of the signal delay of the pair of UWB horn antennas to the path length estimate. This calibration resulted in an increase by 11 cm compared to the physical distance between the antennas. We limit the discussion to evaluating the strongest MPC resulting from the direct reflection from the object. Fig. 7(d) shows the corresponding PADPs for the eight individual snapshots. The small white crosses denote the position of the ML estimate of the strongest MPC. It can be observed that the sinusoidal movement of the ball is well captured by the path estimate. Comparing the estimate of the path length in Fig. 7(d) to the signal routes in Table II, it can be seen that it matches well with the direct path (route C) with , after compensating (twice the radius) for the reflection on the surface of the basketball, and adding due to electric length of the antennas.

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Fig. 8. (a) Breathing object displacement, the estimated path length difference of the strongest path , and the mapping of the path weight phase difference , with ) into equivalent distance . (b) Estimation error standard deviation for both the delay and the phase (using mapped on the path length. The estimates are obtained from the inverse of (19), see [16] for discussion.

Fig. 8 illustrates the evolution of the estimated path length difference (or differential delay, ) as well as the path weight phase difference of the strongest MPC along with the known relative object displacement . It can be observed that the relative delays and phases capture well the sinusoidal trend in the true displacement of the object. Note that although the axes are in the same scale (in mm), the influence of the object displacement on the path length depends on the projection of the displacement direction w.r.t. the angle to both the transmitter and the receiver, i.e., , where denotes the speed of light. Note also that the obtained path length resolution is much higher than , assumed e.g., in [10]. In fact, the estimation error standard deviations in Fig. 8(b), obtained as the square root of the diagonal values of the inverse of the Fisher Information Matrix (19), indicate that the estimation error can be expected to be in the order of 1 mm. B. Results With the Object—NLOS Case In the NLOS measurement, a metallic bookshelf was placed between the Tx antenna and the breathing object. Also, the omnidirectional UWB planar antennas, see Fig. 2, were employed instead of the UWB horns. The pair of UWB planar antennas was determined to have an electric length equivalent to 16 cm increase in the MPC path length. Fig. 9 shows the zoomed PADP of the NLOS measurement, comparable to Fig. 7(c) and (d). The strongest component can be identified to be the one corresponding to route A in Table II, with , while the measured path length for route A after corrections (diameters of the pole, 0.06 m, and the ball, 0.24 m, electric length of the antenna pair, 0.16 m) should have been . The 5 cm difference may result, e.g., from tilting of the Scatterer #1 pole and other uncertainties. However, it can be observed that the movement of the object is very well captured by the path delay estimate, shown as the white cross at the maximum of the PADP in Fig. 9(b). It should be noted, although not explicitly shown here, that both for LOS and NLOS cases it is possible to identify several other MPCS whose parameters capture the sinusoidal trend of the object movement. On the other hand, there are also many MPCs that do not reveal such information, either due to poor angle of reflection w.r.t. the direction of movement, or the

Fig. 9. PADPs for the NLOS case. White x-marks indicate estimated path components. (a) PADP of the overall area of interest averaged over the eight snapshots. (b) PADP at the strongest component (route A in Table II).

fact that those paths are residual errors from the background removal. Nevertheless, whenever such a trend is observed, it could be used in identifying the MPCs interacting with the target, and the path parameters could be further utilized for target localization. C. Results With a Human Target Fig. 10 shows the PADP of the measurement with the human target before (Fig. 10(a)) and after (Fig. 10(b)) background removal. The measurement setup was improved (compared to measurements in Sections V-A and V-B) by stabilizing the cable while the positioner was moving. This results in smaller residual error after background removal in Fig. 10(b) compared to Fig. 7(b). Fig. 10(c) and (d) show a comparison of the PADPs and MPC estimates in two separate measurements, where the only difference is that in Fig. 10(d), an aluminium foil was wrapped around the chest of the test subject. This resulted in a stronger reflection, which helps to identify the sources of the peaks in the PADPs, as indicated in the figures. Fig. 10(e) illustrates the estimated MPCs in the range of chest and head. It can be observed that, in addition to a path length change of about 3 cm (corresponding to about 1.5 cm chest displacement), even a slight angular shift during inhale and exhale periods is captured in the MPC estimate (P3), whereas the estimate from the head reflection MPC (P1) remains more or less constant.

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Fig. 11. PADP of the human target measurement in NLOS conditions. (a) PADP zoomed into the area of interest. (b) Estimated propagation paths in the range of chest and head for the two snapshots. It can be observed that the chest estimate of the inhale snapshot converged to another maximum, and a new path was detected at the chest position in the exhale snapshot.

model along with sequential estimation technique to track the MPCs over time [29]. VI. CONCLUSIONS

Fig. 10. PADPs of the measured channel for the human target. (a) and (b) are averaged over the two (inhale and exhale) snapshots and have the same color scale. (a) is before, and (b) is after background removal. (c) shows the PADP zoomed into the area of interest. (d) is the same as (c), but from a repeated measurement with aluminium foil around the chest. (e) illustrates the estimated propagation paths in the range of chest and head for the two snapshots.

Fig. 11 shows the PADP and MPC estimates from the human measurement in NLOS, while a metallic bookshelf was blocking the LOS between Tx and the test subject. It can be observed that the estimated MPC (P2, denoted by green x-mark) of the reflection from the chest converged to another, weaker, maximum at the second snapshot, and a new path (P21) was detected for the chest location. The path length difference of the old to the new estimate is 4 cm, corresponding to a chest displacement of about 2 cm, whereas the head component remains at a constant path length . It should be noted that in a real system, the sampling rate would be much higher than two samples per breath cycle; hence, it would be possible to employ a state-space

UWB signalling combined with MIMO processing enables a radar having very high delay resolution as well as angular information. This paper presented a study on UWB propagation for estimating small scale movement such as human respiratory activity. A model based on a superposition of MPCs, along with a wideband antenna array model was employed. We showed that the wideband antenna array model does not suffer from ambiguities, which are present for narrowband signal models, even if the antenna spacing is not strictly less than half a wavelength. Measurements were performed under ideal conditions in an anechoic chamber with a VNA and a virtual antenna array, using both an artificial “breathing object” as well as a human subject. After removal of the static background, the MPCs resulting from the interaction with the test objects were estimated and analyzed. The displacement of the artificial object, as well as the human chest while breathing, were captured non-intrusively from about 5 m distance. Based on the results, it may be possible to develop a system where the breathing rate of a person within a room could be determined. The information of angles and delays corresponding to the propagation paths showing such movement, possibly also from NLOS paths, and potentially observed with multiple receiver units, could be used to determine and track the position of the person(s) in a room. However, additional effort is required to study the performance of the envisioned system in realistic SNR conditions obtainable with a practical UWB setup. Further research should also be put on

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determining a good trade-off between the signal bandwidth and the number of antennas.

3) The matrix of detection values, or the power profile of the channel, whose elements are equivalent to (15), is obtained by

APPENDIX WIDEBAND DATA TRANSFORMATION

(28)

Wideband Data Transformation (WBDT) is a computationally efficient solution for evaluating the single path, single polarization ML criterion (15), e.g., for path detection or power profile estimation purposes. The idea is to employ the wideband array responses (steering vectors) of the antenna array(s) to transform the data from the samples collected at the antenna feeds into angular domain, and in the end employing the frequency domain steering vectors to transform into delay domain. This can be viewed as transformation from the MIMO channel tensor observed at the antenna ports, see Fig. 1(a), into a sampling grid defined by the geometrical parameters, see Fig. 1(b). For simplicity, we limit the discussion here to the reduced data model (7), i.e., Rx array only. The approach proceeds as follows, let us have a measurement (or residual) vector (10) in matrix form as (21) as well as the wideband response functions (22) (23) delay detection points, and Rx for frequency domain with array with angular detection points, respectively. Furthermore, let as define the following transformation matrices and as (24) (25)

where tensor

is the (23) and

matrix slice of the is the column of

(22). the matrix The data tensor is then transformed (filtered) from the measurement data to the parameter domain as follows. 1) First, is transformed from Rx array domain to Rx angular domain to obtain as

(26) denotes an vector of all ones (used where here to sum over antenna elements). 2) Then, is transformed from frequency domain to delay domain as (27)

where

denotes the element-wise product. ACKNOWLEDGMENT

The authors would like to acknowledge the efforts of Dr. X. Yang, S. Sangodoyin, and J. Shen in helping to conduct the measurements, as well as thank Dr. S.H. Chang for helping to set up the experiments, and Prof. H. Hashemi for many fruitful discussions. Dr. T. Lewis is also acknowledged for designing and manufacturing the UWB horn antennas, and L. Baumgartel for manufacturing the UWB planar antennas. The first author would like to thank the Academy of Finland, the Finnish Technology Promotion Foundation, and the Walther Ahlström Foundation for their financial support. REFERENCES [1] J. Lin, “Noninvasive microwave measurement of respiration,” Proc. IEEE, vol. 63, no. 10, pp. 1530–1530, 1975. [2] J. Silvious and D. Tahmoush, “UHF measurement of breathing and heartbeat at a distance,” in IEEE Radio and Wireless Symp. (RWS), 2010, pp. 567–570. [3] C. Li, J. Ling, J. Li, and J. Lin, “Accurate Doppler radar noncontact vital sign detection using the RELAX algorithm,” IEEE Trans. Instrum. Meas., vol. 59, no. 3, pp. 687–695, 2010. [4] D. Smardzija, O. Boric-Lubecke, A. Host-Madsen, V. Lubecke, I. Sizer, T. A. Droitcour, and G. Kovacs, “Applications of MIMO techniques to sensing of cardiopulmonary activity,” in Proc. IEEE/ACES Int. Conf. Wireless Commun. Appl. Comput. Electromagn., 2005, pp. 618–621. [5] O. Boric-Lubecke, V. Lubecke, A. Host-Madsen, D. Samardzija, and K. Cheung, “Doppler radar sensing of multiple subjects in single and multiple antenna systems,” in Proc. 7th Int. Conf. Telecommunications in Modern Satellite, Cable and Broadcasting Services, 2005, vol. 1, pp. 7–11. [6] S. Sahinoglu, S. Gezici, and I. Guvenc, Ultra-WIDEBAND Positioning Systems: Theoretical Limits, Ranging Algorithms, and Protocols. Cambridge, U.K.: Cambridge Univ. Press, 2008. [7] J. Sachs, “Ultra-wideband sensing: The road to new radar applications,” presented at the 11th Int. Radar Symp. (IRS), Jun. 2010. [8] E. Cianca and B. Gupta, “FM-UWB for communications and radar in medical applications,” Wireless Personal Commun., vol. 51, no. 4, pp. 793–809, 2009. [9] G. Ossberger, T. Buchegger, E. Schimback, A. Stelzer, and R. Weigel, “Non-invasive respiratory movement detection and monitoring of hidden humans using ultra wideband pulse radar,” in Proc. Int. Workshop Ultra Wideband Systems, 2004, pp. 395–399. [10] E. Conti, A. Filippi, and S. Tomasin, “On the modulation of ultra wide band pulse radar signal by target vital signs,” presented at the Int. Symp. Bioelectronics and Bioinformatics, ISBB 2009, Melbourne, Australia, Dec. 2009. [11] S. H. Chang, R. Sharan, M. Wolf, N. Mitsumoto, and J. Burdick, “UWB radar-based human target tracking,” presented at the IEEE Radar Conf. 2009, May 2009. [12] K. Ohta, K. Ono, I. Matsunami, and A. Kajiwara, “Wireless motion sensor using ultra-wideband impulse-radio,” in Proc. IEEE Radio and Wireless Symp. (RWS), 2010, pp. 13–16. [13] M. Steinbauer, A. Molisch, and E. Bonek, “The double-directional radio channel,” IEEE Antennas Propagat. Mag., vol. 43, no. 4, pp. 51–63, Aug. 2001. [14] X. Liu, H. Leung, and G. Lampropoulos, “Effects of non-uniform motion in through-the-wall SAR imaging,” IEEE Trans. Antennas Propagat., vol. 57, no. 11, pp. 3539–3548, Nov. 2009.

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[15] M. Dehmollaian, M. Thiel, and K. Sarabandi, “Through-the-wall imaging using differential SAR,” IEEE Trans. Geosci. Remote Sens., vol. 47, no. 5, pp. 1289–1296, May 2009. [16] A. Richter, “Estimation of radio channel parameters: Models and algorithms,” Ph.D. dissertation, Technischen Universität Ilmenau, Ilmenau, Germany, May 2005, www.db-thueringen.de. [17] A. F. Molisch, “Ultra-wide-band propagation channels,” Proc. IEEE, vol. 97, no. 2, pp. 353–371, Feb. 2009. [18] R. Zetik, J. Sachs, and R. Thoma, “UWB short-range radar sensing— The architecture of a baseband, pseudo-noise UWB radar sensor,” IEEE Instrumen. Meas. Mag., vol. 10, no. 2, pp. 39–45, Apr. 2007. [19] J. Salmi, S. Sangodoyin, and A. F. Molisch, “High resolution parameter estimation for ultra-wideband MIMO radar,” presented at the 44th Asilomar Conf. Signals, Systems, and Computers, Pacific Grove, CA, Nov. 2010. [20] Mathworks. Matlab. [Online]. Available: http://www.mathworks.com [21] M. Landmann, A. Richter, and R. Thomä, “DOA resolution limits in MIMO channel sounding,” in Proc. Int. Symp. Antennas and Propagation and USNC/URSI National Radio Science Meeting, Monterey, CA, Jun. 2004, pp. 1708–1711. [22] M. Landmann, “Limitations of experimental channel characterization,” Ph.D. dissertation, Technischen Universität Ilmenau, Ilmenau, Germany, Mar. 2008, www.db-thueringen.de. [23] J. Salmi, “Contributions to measurement-based dynamic MIMO channel modeling and propagation parameter estimation,” Ph.D. dissertation, Dept. Signal Process. Acoust., Univ. Technology, Espoo, Finland, Aug. 2009 [Online]. Available: http://lib.tkk.fi/Diss/2009/ isbn9789522480194/ [24] X.-S. Yang, K. T. Ng, S. H. Yeung, and K. F. Man, “Jumping genes multiobjective optimization scheme for planar monopole ultrawideband antenna,” IEEE Trans. Antennas Propagat., vol. 56, no. 12, pp. 3659–3666, 2008. [25] T. Lewis, “An ultrawideband digital signal design with power spectral density constraints,” Ph.D. dissertation, Dept. Electr. Eng., Univ. Southern California, Los Angeles, CA, Aug. 2010. [26] F. Anderson, W. Christensen, L. Fullerton, and B. Kortegaard, “Ultrawideband beamforming in sparse arrays,” IEE Proc. H, vol. 138, no. 4, pp. 342–346, Aug. 1991. [27] S. Ries and T. Kaiser, “Ultra wideband impulse beamforming: It is a different world,” Signal Process., vol. 86, no. 9, pp. 2198–2207, 2006. [28] Ultra-Wideband Radio Laboratory, UltRa Lab, Univ. Southern California, Los Angeles, CA [Online]. Available: http://ultra.usc.edu/ [29] J. Salmi, A. Richter, and V. Koivunen, “Detection and tracking of MIMO propagation path parameters using state-space approach,” IEEE Trans. Signal Process., vol. 57, no. 4, pp. 1538–1550, Apr. 2009. [30] D. Marquardt, “An algorithm for least-squares estimation of nonlinear parameters,” SIAM J. Appl. Math., vol. 11, pp. 431–441, 1963.

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Jussi Salmi (S’05–M’09) was born in Finland in 1981. He received the M.Sc. and D.Sc. degrees, both with honors, from Helsinki University of Technology (HUT), Finland, in 2005 and 2009, respectively. In 2009–2010 he worked as a Postdoctoral Research Associate in the Department of Electrical Engineering, University of Southern California, Los Angeles, CA. Currently, he works as a Postdoctoral researcher in the Department of Signal Processing and Acoustics, Aalto University School of Electrical Engineering (former HUT), Finland. His current research interests include RF-based vital sign detection, UWB MIMO radar, indoor positioning, measurement based MIMO channel modeling, and parameter estimation. He is the author of over 30 research papers in international journals and conferences. Dr. Salmi received the Best Student Paper Award in EUSIPCO’06, and co-authored a paper receiving the Best Paper Award in Propagation in EuCAP’06.

Andreas F. Molisch (S’89–M’95–SM’00–F’05) received the Dipl. Ing., Dr. techn., and habilitation degrees from the Technical University Vienna, Austria, in 1990, 1994, and 1999, respectively. From 1991 to 2000, he was with the TU Vienna, becoming an Associate Professor there in 1999. From 2000 to 2002, he was with the Wireless Systems Research Department at AT&T (Bell) Laboratories Research, Middletown, NJ. From 2002 to 2008, he was with Mitsubishi Electric Research Labs, Cambridge, MA, most recently as Distinguished Member of Technical Staff and Chief Wireless Standards Architect. Concurrently, he was also Professor and Chairholder for radio systems at Lund University, Sweden. Since 2009, he has been Professor of electrical engineering at the University of Southern California, Los Angeles, CA, where he heads the Wireless Devices and Systems (WiDeS) group. He has done research in the areas of SAW filters, radiative transfer in atomic vapors, atomic line filters, smart antennas, and wideband systems. His current research interests are measurement and modeling of mobile radio channels, UWB, cooperative communications, and MIMO systems. He has authored, co-authored, or edited four books (among them the textbook Wireless Communications, Wiley-IEEE Press), 11 book chapters, some 130 journal papers, and numerous conference contributions, as well as more than 70 patents and 60 standards contributions. Dr. Molisch is Area Editor for Antennas and Propagation of the IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS and has been co-editor of special issues of several journals. He has been General Chair, TPC Chair, or Track/Symposium Chair of numerous international conferences. He was chairman of the COST 273 working group on MIMO channels, the IEEE 802.15.4a channel model standardization group, Commission C (signals and systems) of URSI (International Union of Radio Scientists, 2005–2008), and the Radio Communications Committee of the IEEE Communications Society (2009–2010). He has received numerous awards, most recently the James Evans Avant-Garde Award of the IEEE VT Society and the Donald Fink Award of the IEEE. Dr. Molisch is a Fellow of the IEEE, Fellow of the IET, and an IEEE Distinguished Lecturer.

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Estimation of Wall Parameters From Time-Delay-Only Through-Wall Radar Measurements Pavel Protiva, Student Member, IEEE, Jan Mrkvica, Member, IEEE, and Jan Macháˇc, Senior Member, IEEE

Abstract—Knowledge of the basic characteristics of the wall is especially important for the proper function of through-wall radars (TWR). In this paper, we introduce an estimation method that can provide the unknown wall parameters in real time. Our algorithm is based on a modified version of common midpoint processing, a technique adopted from geophysics. Using this method, it is possible to estimate the thickness and the permittivity of the wall from time-delay-only measurements performed at several different transceiver-receiver separations. Time-delay estimation (TDE) of echoes backscattered from the wall at each antenna separation is performed entirely in the frequency domain by a subspace super-resolution method. Our TDE approach is capable of resolving overlapping echoes reflected from the wall even in the case of only one snapshot of the measured data, suboptimal calibration possibilities, imperfect deconvolution of the measurement system, and limited radar bandwidth. The performance of our estimation algorithm is verified by simulations, and we also present the results for two real brick walls measured by using the time-domain technique. The estimated wall parameters can be utilized for improving the accuracy of real-time locating and tracking of moving humans by TWR. Index Terms—Super-resolution, through-wall imaging, time-delay estimation, ultra-wideband radar, wall parameter estimation.

I. INTRODUCTION

NE OF the main problems of through-wall radar (TWR) imaging is the distortion due to signal propagation through walls, which can lead to ambiguities in localizing targets buried behind the wall. Several algorithms have been developed to compensate for these effects in the case of unknown wall parameters. These techniques usually involve complicated measurement procedures, such as using different array structures [1] or multiple standoff distances [2], [3]. Autofocussing approaches use computationally intensive iterative optimization schemes [4]. The other authors retrieve the unknown wall parameters by using an inverse scattering

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Manuscript received November 10, 2010; revised February 09, 2011; accepted April 23, 2011. Date of publication August 18, 2011; date of current version November 02, 2011. This work was supported in part by the Ministry of Defence of the Czech Republic under Project 080187090 “LOCALIZATION System detection and localization of protected moving and stationary object trespasser,” in part by the Czech Grant Agency under Project 102/09/0314, and in part by the Czech Technical University in Prague under Project SGS10/271/ OHK3/3T/13. P. Protiva and J. Macháˇc are with the Czech Technical University in Prague, Faculty of Electrical Engineering, 16627 Prague 6, Czech Republic (e-mail: [email protected]). J. Mrkvica is with the RETIA, Pardubice 530 02, Czech Republic. Digital Object Identifier 10.1109/TAP.2011.2164206

algorithm, which minimizes the mean square error between the measured and the predicted reflection coefficient calculated from a forward model [5]–[8]. Inverse approaches require precise measurement with a high dynamic range and a time-consuming calibration procedure. This is feasible in laboratory conditions with the use of a vector network analyzer (VNA), but it is not appropriate for practical TWR applications (e.g., during rescue operations or hostage crises). In general, portable TWRs [9]–[12] are lightweight and limited-size devices with compromise system parameters and computational power insufficient for implementing any of the algorithms mentioned before. These systems are usually based on the back projection imaging method [13]. Back projection still works if the influence of the wall is not taken into account. Unfortunately, the target images can be blurred and displaced from their true locations, which degrades the accuracy of the radar. If the permittivity and the thickness of the wall are known, the back projection can easily be modified to compensate for the refraction inside the wall [14]. Therefore, it is desirable to develop a method that can provide estimates of unknown wall parameters for the imaging algorithm. And even if not used to compensate the propagation effects, the estimation of the width of the wall can provide the operator with valuable additional information about the monitored scene. The permittivity of a wall is commonly estimated from the amplitude of the echoes backscattered from the first interface of the wall [15], [16]. However, this approach requires precise calibration similar to inverse methods. Furthermore, it is appropriate only for uniform homogeneous walls. Even a thin layer of plaster on the front side of the wall can lead to incorrect estimation of the overall effective permittivity of the wall base material. If the permittivity is known, the width of the wall can be calculated from the time difference of arrival between the first two received echoes. Measuring the time delays of echoes is more accurate than measuring their amplitude. Measuring the time delays also imposes fewer requirements regarding the performance of the measurement equipment and the measurement procedure. Since the radar range resolution is proportional to the signal bandwidth [9], ultrawideband (UWB) technology is especially advantageous for TWRs. However, the bandwidth of the TWR is considerably limited to approximately 1 to 3 GHz [17], on the one hand, by the attenuation of the building materials at higher frequencies and, on the other hand, by the minimum operating frequency of the antennas. Bandlimited UWB signals are capable of detecting the positions of targets, but they fail to resolve overlapping echoes reflected from the front and rear sides of the wall situated close to each other. At short

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PROTIVA et al.: ESTIMATION OF WALL PARAMETERS FROM TIME-DELAY-ONLY TWR MEASUREMENTS

standoff ranges, these signals further combine with direct antenna coupling. Time-delay estimation (TDE) of overlapping echoes requires advanced signal-processing methods. For example, the time resolution can be improved by deconvoluting the system response. However, deconvolution is an ill-posed problem, which requires high system bandwidth and a high dynamic range. When these requirements are not fulfilled, deconvolution is a rather problematic operation in terms of selecting an appropriate regularization parameter or a suitable filter [18], and can provide misleading results. Subspace superresolution approaches, such as multiple signal classification (MUSIC), Min-Norm, or the estimation of signal parameters via rotational invariance techniques (ESPRIT), are popular in antenna array signal processing [19]. Although these methods were originally developed for spectral and spatial analysis, they can also be used for time delay estimation. Subspace methods have been applied for improving the time resolution of narrowband data measured by the vector network analyzer (VNA) [20], [21], and wideband ground penetrating radar (GPR) data [22]. It has been demonstrated that subspace super-resolution methods can provide better resolution than conventional TDE, even if a considerably smaller processing bandwidth is available. In this paper, we present a method for wall parameter estimation that uses signals measured by ultrawideband TWR. As mentioned before, UWB technology is well appropriate for measuring the time delay. On the other hand, measuring the permittivity is a tough task. In our method, we avoid this drawback by estimating the wall thickness and permittivity using time-delay-only measurements of echoes backscattered from the wall. In order to maintain good resolution capability even in the case of real TWR with limited bandwidth, the TDE method that we use to find these delays is based on subspace super-resolution processing. It is evident that special preprocessing steps have to be performed on the UWB signal prior to the application of this frequency-domain method. We describe these steps in Section II. The estimation of the wall parameters from the measured data is based on common midpoint processing (CMP). CMP is a standard approach in geophysics, where it is used for measuring the thickness of a subsurface layer by processing the received seismic echoes [23]. CMP has also been applied to estimate the thickness of a pavement measured by GPR [24], [25]. In Section III, we apply CMP to estimated time delays measured at several different transceiver-receiver separations. Unlike traditional CMP, known from geophysics, and GPR, we use a modified propagation model that assumes a nonzero standoff distance from the wall. Experimental verification results will be presented in Section IV. We show that our method provides reliable results, even when real conditions and realistic system parameters of the radar are considered. Unlike other methods described in the literature, our approach requires neither prior measurement of the wall nor special calibration of the radar. The unknown parameters of the wall are estimated in real time from the early-time section of the same signal that contains the information about the targets. It is therefore suitable for implementation in a portable TWR that can be applied even during time-critical emergency situations.

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II. TIME-DELAY ESTIMATION OF UWB SIGNALS USING A FREQUENCY-DOMAIN SUPER-RESOLUTION ALGORITHM A. Signal Model Let us consider a scene composed of a wall illuminated by a TWR. The signal backscattered from a single-layered homogeneous wall is a sequence of time-shifted and attenuated copies of the transmitted pulse. The first two echoes, which correspond to the reflections from the front and rear side of the wall, are dominant. Real building materials are mostly inhomogeneous. However, if the size of the inhomogeneities is smaller than the pulse resolution, we can model the wall as a homogeneous layer characterized by effective permittivity [26]. The signal backscattered from such a wall is similar to the case of the ideal homogeneous wall, and the assumption also holds that reflections from the front and rear side of the wall predominate. We further assume that the material dispersion is negligible and the effective permittivity is constant in the frequency band of interest. The wall can be either lossless or weakly lossy. According to , the propagation velocity is [27], for loss tangent dominantly determined by the real part of the material permittivity, and the dependence on the loss tangent is negligible. The presence of reasonable losses within the material will therefore have a negligible impact on the time delay measurement. The received echo reflected from the front interface of the wall can then be modeled by a frequency-domain transmission equation (1) where is the voltage at the terminal of the receiving antenna; constitutes the crosstalk between the transmitting and reaccounts for the reciceiving antenna of a bistatic radar; procity principle [28]; is the free-space propagation constant; is the distance between the wall and the antennas of and are the normalized transfer functions the radar; of the transmitting and receiving antenna [29], respectively; is the reflection coefficient of the air-wall interface; and is the excitation voltage at the connector of the transmitting antenna. If the reflections from the front and the rear side of the wall are considered, the received voltage can be written by using total propagation delays encountered by the pulse as it travels from the transmitter to the th interface and back to the receiver

(2) where is the amplitude of the th echo. includes the attenuation due to reflection and transmission at the interface, and due to free-space propagation. According to the assumptions stated is assumed to be frequency independent in the frebefore, quency band of interest. The subsequent products of multiple refractions inside the wall are inherently lower in amplitude and can be neglected. Subspace-based TDE assumes that the radio channel has frequency-independent transfer characteristics [20]. It is therefore inappropriate to apply the TDE method directly to data measured in a UWB channel, which is inherently frequency dependent. In laboratory conditions, wideband frequency-domain

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data can be corrected using VNA calibration procedures. In the case of time-domain measurement, it is convenient to take into account the specific shape of the radar pulse. Unfortunately, the reference radar pulse has to be acquired a priori as the signal backscattered from a metallic plate positioned over the wall under test [22]. In order to avoid this calibration step, we only perform a partial deconvolution of the measurement system. Since the TDE will be performed entirely in the frequency domain by a subspace super-resolution method, the deconvolution is straightforward. The subspace methods require a considerably smaller bandwidth than conventional TDE methods, and the deconvolution reduces to a simple frequency-domain division of limited-bandwidth data. is suppressed by subtracLet us assume that the crosstalk tion from the received signal, and the influence of the antennas and the excitation pulse are both cancelled by deconvolution. The remaining complex exponentials form a frequency-domain transfer function. Obviously, real signals include errors due to measurement uncertainties, the influence of imperfect calibration, and due to the nonideal parameters of real walls. These effects can be represented by additive Gaussian white noise with zero mean and variance . The transfer function is then given by (3) at where stands for the additive noise. By sampling equally spaced frequencies, we obtain the discrete transfer function (4)

denotes the ensemble average, superscript denotes where is the signal correlation the Hermitian transpose, matrix, and is the identity matrix. Since snapshots of measured data are available, the correlation matrix is estimated as (7) With the eigendecomposition of the correlation matrix, we obtain the signal and the noise subspace. Sinces we are interested only in the first two strongest reflections, the rank of the signal subspace is fixed to two. Finally, either of the two subspaces can be used by a classic subspace super-resolution method, such as MUSIC, Min-Norm, or ESPRIT, to estimate the time delays [19]. Unfortunately, portable TWRs, in most cases, provide only one snapshot of the measured data. This snapshot is formed by time-averaging multiple measured waveforms, which is performed in order to improve the noise floor. Conventional estimation of the signal correlation matrix by (7) is therefore not applicable. However, even if only one snapshot is available, an estimate of the correlation matrix can be obtained using the spatial smoothing technique [21], [32]. This algorithm divides the overlapping segments (subarvector of measured data into rays) of length in such a way that (8) where and . Then, the correlation matrix is estimated by averaging the subarray correlation matrices as (9)

where matrix form

. Equation (4) can also be written in the

(5) where

In order to save computational resources, we apply an alternative approach to find the eigendecomposition of the data correlation matrix. The estimate of the correlation matrix can also be written [19] as (10) where

is a

data matrix formed as follows: (11)

and superscript denotes the matrix transpose operator. This signal model is consistent with the data form used for TDE by spectral methods [20]–[22], [30], [31].

Now, we apply the singular value decomposition (SVD) [19] to the data matrix . SVD enables us to find the eigendecomposition directly from the data matrix without needing to construct the correlation matrix.

B. Estimation of the Data Correlation Matrix

C. Improved Spatial Smoothing

Subspace super-resolution methods are based on the eigendecomposition of the data correlation matrix. Using (5), we can define the data correlation matrix as

A specific problem in time delay estimation of multipath signals originating from the same transmitter is the strong correlation between the received echoes. In the full correlation (coherent) case, the signal correlation matrix becomes singular, and subspace super-resolution methods do not work properly.

(6)

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Similarly, if only one snapshot of the measured data is available for estimating the data correlation matrix, is a deterministic vector and, therefore, the rank of decreases to one. Fortunately, the conventional spatial smoothing technique that we have used previously to estimate the correlation matrix from one snapshot of measured data also has a decorrelating effect [20], [21]. This effect can be further improved by applying the forward–backward averaging technique, which forms the basis of modified spatial smoothing [32], [33]. This method, originally introduced for estimating the direction of arrival (DOA) of coherent sources, has also gained popularity also in the field of time delay estimation [20]–[22]. However, the performance of modified spatial smoothing degrades rapidly as the correlated echoes become closely spaced. This situation frequently occurs in the case of signals backscattered from a wall. Several methods have been introduced to overcome this problem [34], [35]. In our algorithm, we apply an improved spatial smoothing method [36]. It has been shown to provide good results, even if the processed signal has the form of closely spaced coherent echoes. The improved spatial smoothing is based on forward-backward averaging, but it utilizes more information in the received data than is used by modified spatial smoothing. The data correlation matrix of the improved spatial smoothing is given as (12) where

is the exchange matrix [20], and the superscript denotes the complex conjugate. In order to simplify the algorithm by using SVD, we introduce an averaged data matrix, which is created by modifying the data matrix (11) according to the improved spatial smoothing scheme averaged data matrix is formed as (12). The (13) D. Subspace Super-Resolution Method After applying SVD to , any conventional subspace super-resolution method such as MUSIC, Min-Norm, or ESPRIT can be applied to estimate the time delays of the backscattered echoes. We prefer ESPRIT [19], [30], which is an attractive choice in terms of computational speed. ESPRIT also provides closed-form estimates of the time delays and, therefore, avoids the maximum search algorithm. A comparison of the TDE performance of polynomial versions of MUSIC, Min-Norm (root MUSIC and root Min-Norm), and ESPRIT will be published elsewhere [37]. E. Performance Test Results By using a 3-D scene modeled in the CST Microwave Studio electromagnetic simulator, we established the ability of the proposed TDE algorithm to resolve overlapping UWB pulses

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backscattered from walls of different thicknesses. To characterize the resolution capability of the algorithm, we adopted product [22], where is the bandwidth of the signal the measured at the output of the receiving antenna and denotes the time difference of arrival between the echo reflected from the front and rear side of the wall, respectively. Unlike [22], we selected the 10-dB power level to assess the bandwidth, since it is more widely used in the field of UWB systems. A TEM-horn-like antenna with a 10-dB frequency range from 1.6 to 6 GHz was directed normally to the surface of a singlelayered homogeneous wall. The antenna was placed at a distance of 0.5 m from the wall and only one antenna was employed for transmitting and for receiving. This monostatic radar configuration simplifies the simulation. The antenna was excited by a UWB waveform with a central frequency of 1.5-GHz and 1.2-GHz bandwidth and, hence, the excitation pulse is distorted and bandlimited as it passes through the frequency dispersive 1 antennas. This leads to a reduced bandwidth of about GHz at the terminal of the receiving antenna. Consequently, the in our specific case is numerically equal to the product in nanoseconds. time difference of arrival Fifteen independent simulations were performed by varying product inthe layer thickness in such a way that the creased from 0.05 to 10. Since the antenna standoff distance is small, the signal reflected from the wall overlaps with the signal reflected by the antenna structure itself. Therefore, a reference simulation with the antenna placed in a free-space environment is required to obtain the antenna response. By subtracting the reference signal from the measured data, the reflection inside the antenna can then be effectively eliminated. The time-domain waveform of the received signal was recorded at each thickness. Sampled frequency-domain data with 10-MHz resolution were obtained by the chirp-Z transform. Next, the excitation pulse and the characteristics of the antennas were deconvolved from the received data. The antenna transfer function was retrieved from the simulated transmission coefficient between two identical antennas at boresight [29]. Deconvolution suppresses any frequency dependence introduced by the excitation and by the antennas. Consequently, the resulting frequency spectrum flattens and the 10-dB bandwidth becomes wider. Since the performance of the TDE algorithms increases with the increase in the processed bandwidth, it is convenient to select a frequency range as wide as possible for subsequent processing. Unfortunately, the dynamic range of the time-domain technique is considerably limited, which can lead to the instability of the deconvolution at these frequencies, where the power level is low. At high frequencies, it is due to the decreasing spectral energy of the excitation with increasing frequency and to the attenuation of walls. Instability can also occur at very low frequencies, where the antennas show poor efficiency and the radiated spectral density decreases. Therefore, we have selected a region from the deconvoluted data that is sufficiently wide but, at the same time, is safe from instabilities. Actually, 200 frequency samples within a frequency range from 500 MHz to 2.5 GHz are utilized to form the averaged data matrix using (13). Finally, the time delays are estimated by applying SVD and the total least squares (TLS) version of ESPRIT, with the rank of

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1

Fig. 1. Relative error of the estimated time difference of arrival versus B  . Note that for the specific combination of the excitation pulse and the antennas that we selected for the simulation, B  is numerically equal to the estimated  in nanoseconds.

1

1

the signal subspace fixed to two. Thus, the algorithm provides two estimated time delays, which correspond to the reflection from the front and rear side of the wall, respectively. The time is determined by subtracting these dedifference of arrival was set in such a lays. The number of smoothing subarrays way that the smoothing quotient is equal to 0.5. This value was determined empirically as a good compromise between the resolution capability and the stability of the algorithm. Fig. 1 shows the relative error of the estimated time difference . The error is defined as of arrival versus Error

(14)

is the estimated time difference of arrival and is the true value calculated by using the actual and relative permittivity . Although the wall thickness curve in Fig. 1 does not represent a statistic, the possibilities of our method are apparent. The error decreases with increasing 0.5, which covers the large layer thickness and for majority of practical situations, the echoes can be resolved with less than about 2, a relative error 5%. Note that for pulse overlap occurs and the time difference between echoes is unlikely to be correctly estimated by conventional methods. The whole TDE algorithm, including all preprocessing steps and TLS-ESPRIT implemented in Matlab, takes only 0.2 s to provide results by using conventional PC hardware. where

III. ESTIMATION OF THE WALL PARAMETERS USING TIME-DELAY-ONLY MEASUREMENTS A. Theoretical Background Consider a bistatic TWR with a transmitting antenna and a receiving antenna separated by distance . The radar is positioned at a standoff distance from the wall. We assume for simplicity that the wall is composed of a single homogeneous dielectric layer with thickness and relative permittivity . The influence of possible surface roughness was not considered. The geometry of the problem is shown in Fig. 2. In this configuration, the received signal consists of three dominant multipath components: 1) a direct-coupling between

Fig. 2. Geometry of the problem.

the antennas, 2) a signal reflected from the air-wall interface, and 3) a signal that is refracted at the first air-wall interface, reflected back from the rear side of the wall, and again refracted at the first wall-air interface. From Fig. 2, the time delay of the latter is given by (15) and represents the position of the refraction where point P. The refraction at the air-wall interface is governed by Snell’s law, and we can write (16) where and are the angle of incidence and the angle of refraction, respectively. The basic idea of common midpoint processing (CMP) consists in measuring the time delay of signals reflected from a planar surface using different transceiver-receiver separation. In theory, if we measure the propagation time at two antenna separations, we obtain two equations that can be solved for two and . In practice, the estimated time delays are unknowns biased due to the systematic error of TWR, and due to the inhomogeneities of real walls. The wall parameters are then unlikely to be estimated using only two measurements. In order to minimize the influence of these errors, we apply an iterative optimization algorithm to a set of time delays measured at different antenna separations. Then, we estimate the unknown wall parameters by minimizing a suitable objective function, which is defined as the mean square error between the measured time and the prediction provided by a forward model delay (17) is the antenna separation corresponding to the th where measurement. The forward model is formed by (15) and (16). In order to determine the unknown position of refraction point P, the for-

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Fig. 3. CST Microwave Studio model of the porous brick. Transmitting and receiving UWB dipoles of the bistatic TWR are positioned at a 25-cm standoff distance from the 17.5-cm-thick wall.

ward model also includes a simple and robust numerical approach [14]. This method iteratively searches the value of so that equality (16) is true. Various optimization techniques may be used to minimize . We have applied the Nelder–Mead simplex algorithm. Since only two unknown parameters are considered, the objective function converges rapidly and the optimized results are available in about 1 s.

Fig. 4. Simulated signals reflected from the porous brick wall. The crosstalk between antennas was suppressed by subtracting the free-space reference from the received signal.

B. Numerical Simulations To verify the performance of our wall parameter estimation method, the algorithm was applied to simulated time-domain data obtained by the CST Microwave Studio. A UWB antenna was fixed at a standoff distance 25 cm from the wall, and it was excited by a Gaussian pulse with a full width at half maximum (FWHM) of 160 ps. The receiving antenna was positioned at the same standoff distance from the wall, and the received waveforms were gathered at each antenna separation, which was varied from 10 to 50 cm, in 41 equidistant steps of 1 cm. In order to save the computational power required by the simulation, we use simple UWB dipole antennas with a 10-dB frequency range 1.1–2.9 GHz. The wall under test was modeled as an ideal single-layered homogeneous plate. Several simulations with wall permittivity ranging from 2 to 10 cm and 10 to 50 cm in thickness were accomplished with excellent results. Instead, we present the simulation result for a more realistic scenario with a wall composed clay blocks [38]. of Wienerberger POROTHERM 17.5 The layout of the simulation is shown in Fig. 3. The permittivity of the base material of these porous bricks was set to 4, and the cavities are filled with air. Fig. 4 shows the time-domain response of the wall recorded at three different antenna separations. The crosstalk between the antennas has been already removed by subtracting the reference signal simulated with the antennas placed in a free space from the signal backscattered from the wall. In this particular case, the echoes backscattered from the front and rear sides of the wall are well resolvable, and the use of a subspace-based TDE method is therefore not implicitly necessary. However, it has

Fig. 5. B-Scan of the received data and the results of applying the TDE algorithm. Relative power in decibels is plotted. Markers show the time delays estimated for each antenna separation.

been included in our algorithm in order to maintain universality for a wide range of excitation signals and wall parameters. A 2-D plot of the received data is shown in Fig. 5. In the plot, the signal envelope is represented by a grayscale on the plane of antenna separation versus time delay. This graphical representation of the received data is similar to that used in GPR data processing; therefore, we call it the B-Scan. In addition, for better visibility of the rear wall interface, time-varying gain [27] was also applied to the raw received data. This method compensates for the losses of spherical spreading and of attenuation by the wall. It is evident from the B-Scan that the signature of the rear side of the wall is distorted due to local inhomogeneities inside the wall. The subsequent signal processing applied to each of the gathered waveforms, which includes transformation to the frequency domain, deconvolution, selecting the frequency band to be processed, and, finally, the TDE, is identical to that which we describe in Section II-E. Since the rank of the signal subspace was still set to two, TLS-ESPRIT provides two estimates of the time delay. Since the reflections from the front and rear sides of the

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Fig. 6. Estimated thickness of the wall d versus number of measurements N entering the optimization. The correct value is marked with the dashed line.

Fig. 8. Time-domain measurement system and a porous brick wall (A) composed of POROTHERM 44 P+D porous bricks 440 mm in thickness.

IV. EXPERIMENT A. Experimental Setup and Procedure

Fig. 7. Estimated permittivity of the wall versus N .

wall predominate in the received signal, the longer delay corresponds to the rear side. These time delays estimated from each of 41 measurements are marked in Fig. 5. Finally, we obtained the wall parameters from the measurements by minimizing (17). In order to study the influence of the complex structure of the wall on the accuracy of the estimation method, we executed the optimization repetitively with taken into account by a different number of measurements (17). For each ranging from 2 to 41, we selected measurements in such a way that the corresponding antenna separations span the range of 10–50 cm approximately equally distributed. The estimated thickness and permittivity of the wall versus are shown in Figs. 6 and 7, respectively. The accuracy of the estimation rises with the increasing number of measurements entering the optimization. If only measurements at a few antenna separations are considered, the errors in this sparse data set lead to suboptimal optimization results. However, as can be 5) already seen from Figs. 6 and 7, five measurements provide good estimates for the application in TWR imaging. In other words, a similar result can be achieved by using a linear array consisting of one transmitting antenna and five receiving antennas with 10-cm spacing.

In this section, we test the presented method with experimental data collected using a time-domain technique. The measurement system shown in Fig. 8 consists of a UWB shortpulse generator [39] and an Agilent 86100C sampling oscilloscope. Two identical UWB antennas with a 10-dB bandwidth 1.5–6 GHz were vertically polarized and configured in a bistatic arrangement. The TEM-horn-based antennas that we use are primarily designed for locating moving targets in the far-field, and their radiating patterns are thus not exactly omnidirectional. However, the time shift imposed by the angular dependence of the transfer function of the antennas is negligible, and it was not considered. The transmitting antenna was excited by a Gaussian pulse with an FWHM of 160 ps and about 3-GHz bandwidth [39]. The evaluated walls include: 1) a porous brick wall comclay blocks 440 posed of Wienerberger POROTHERM 44 mm in thickness (Fig. 8), and 2) a porous brick wall composed blocks 175 mm in of Wienerberger POROTHERM 17.5 thickness [38]. The antenna standoff distance should be carefully selected. If the standoff distance is too large, the received signal will be distorted by the diffraction effects from the edges of the wall samples. If the distance is too small, the wall will be located in the near field of the antenna, which can introduce an undesired time shift in the measurement [40]. The 25-cm standoff distance was selected as a tradeoff between these two effects. The transmitting antenna was fixed, while the receiving antenna was moved horizontally to a desired position using an automatic GPR scanning system (Fig. 8). We are well aware that this approach is not appropriate for real-time TWRs. For practical use, the scanning system can be replaced by an array of multiplexed receiving antennas or by individual receiver modules.

PROTIVA et al.: ESTIMATION OF WALL PARAMETERS FROM TIME-DELAY-ONLY TWR MEASUREMENTS

Fig. 9. B-Scan of the data measured on wall (A). The markers show the estimated position of the back side of the wall.

A two-step procedure is required to measure the walls. The first step is to measure the signal reflected from the wall, and the second step is to obtain the reference signal by placing the system in an open space environment. Then, crosstalk between the antennas is effectively eliminated by subtracting the reference signal from the received data. Note that the free-space calibration step has to be performed for each antenna separation individually. The antenna separations were varied from 10 to 50 cm in 41 equidistant steps. Although different antennas were used in our experiment, the configuration of the measurement has been held consistent with the simulation. Each measurement at a given antenna separation consists in recording 64 received waveforms, which are averaged to form a single snapshot of the received data. The subsequent processing is identical to that which we applied to the simulated data in Sections II-E and III-B. The excitation pulse and the normalized transfer function of the antennas, which are required by the deconvolution step of our algorithm, were measured in laboratory conditions and stored in the memory.

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Fig. 10. B-Scan of the data measured on wall (B) and the results of applying the TDE algorithm.

Fig. 11. Estimated thickness of wall (A) (crosses) and wall (B) (circles) versus . The correct values are marked with the dashed and dash-dotted lines for wall (A) and (B), respectively.

N

B. Measurement Results The measurement results and the estimated wall parameters are presented in accordance with the simulation described in Section III-B. Fig. 9 shows the B-Scan of wall sample (A). The result of our estimation algorithm depends primarily on the quality of the signature corresponding to the rear side of the wall. In Fig. 9, this signature is evidently distorted due to scattering effects inside the wall. Fig. 10 shows the B-Scan obtained by measuring wall sample (B). In this case, the signature of the rear side is clearly visible, yet the smoothness of the marked line formed by the estimated time delays is not optimal. Fig. 11 shows the estimated thickness of the two wall samples (A) and (B) versus . The results are averaged over 10 independent scans performed by the measurement system at 10 different vertical positions of the antennas. These positions were selected with minimum separation 5 cm. In Fig. 11, the mean value is plotted using markers, and the error bars are two stanin length. Again, each scan consists of dard deviation units measuring 41 different antenna separations, from which we then

Fig. 12. Estimated permittivity of wall (A) versus

N.

select measurements to be optimized. The estimated permittivities versus of wall (A) and wall (B) are shown in Figs. 12 and 13, respectively. We assume that the porous brick wall can be treated as homogeneous at TWR frequencies up to about 3 GHz. However, the nonuniform distribution of inner air cavities different in size (see Fig. 3) can lead to local fluctuations and an angular dependence of the effective permittivity. These effects can be observed in the

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Fig. 13. Estimated permittivity of wall (B) versus

N.

case of the B-Scan of wall (A), shown in Fig. 9. The signature of the rear side of the wall is evidently distorted due to scattering effects inside the wall. However, TDE of the echoes backscattered from the rear side is still possible and the optimization also yields a thickness close to the correct value. In addition, ripples in the signature of the front side of the wall (see Fig. 10) may lead to speculation that the reflected signal consists of more than two dominant components, and the longer delay estimated by our algorithm will therefore not correspond to the rear interface. However, the rippled character is caused rather by the transmitted pulse shape than by the structure of the wall. This effect is later mitigated by the deconvolution and, consequently, the TDE method becomes insensitive to these details seen in the raw data. The permittivity of the porous brick wall is spatially dependent and, therefore, difficult to verify by an alternative measurement method. For a simple comparison, the permittivity can be retrieved directly from the measured data. If the antennas are 10 cm), the refraction on the close to each other (e.g., way between the antennas and the front and rear side of the wall can be neglected. Then, the permittivity is obtained from the and from the known estimated time difference of arrivals thickness as (18) are 4.894 ns and 1.842 ns for wall samples The estimated (A) and (B), respectively. Then, the corresponding permittivities (2.78 and 2.49) are approximately equal to the estimated values shown in Figs. 12 and 13. The reliability of the algorithm increases with increasing . Unfortunately, antenna arrays of portable TWRs are limited from the point of view of number of antennas and aperture size. However, according to the results in Figs. 11–13, the use of more than 10 sensors is not reasonable. Ten receiving antennas can easily be configured in a planar array with a maximum aperture size of about 0.5 m 0.5 m. V. DISCUSSION In this section, we summarize the advantages of using the method presented here. Unlike other wall parameters estimation methods described in the literature, our approach shows

some unique properties, which were imposed by the requirements of real TWR applications. First, our method is based on time-delay-only measurement. This approach is new in the field of TWR imaging, and its advantage is that we eliminate the need to measure the reflection coefficient of the wall. Estimating the permittivity from the reflection coefficient [5]–[8], [15], [16] is an ill-posed problem requiring a high dynamic range and precise measurement procedure, which is not feasible by conventional TWRs. Second, our method does not require any timeconsuming calibration, which is usually performed prior to the measurement [5], [22] and has to be repeated after the TWR unit is moved to another position against the wall. In our case, only the excitation pulse, the characteristics of the antennas, and the free-space calibration data are used in the algorithm. These calibration data can be acquired in the laboratory and stored in the memory of a through-wall device. The wall compensation methods described in theoretical studies [1]–[3] also do not require calibration of the TWR. However, they consist of two measurement steps. In the first step, the unknown wall parameters are estimated by moving the radar to different locations and imaging a cooperative target. Then, the radar is fixed against the wall and it can be used to localize moving targets. Our method estimates the wall parameters from the signal that contains the information about the targets. It can be executed simultaneously with a real-time through-wall imaging algorithm, and prior measurement of the wall is not necessary. Finally, our paper has also provided experimental results that prove our method to be sufficiently robust in the case of real conditions. It is evident that the best results can be obtained when measuring uniform homogeneous walls. However, we have shown that the algorithm can deal with real walls composed of porous bricks. In this case, the measurement errors due to local inhomogeneities inside these bricks are mitigated by applying an optimization method on data measured at several antenna separations. Obviously, our approach is not suitable for all possible practical scenarios (e.g., when measuring multilayered or strongly inhomogeneous wall structures). In these cases, the identification of the time delays that correspond to the rear side of the wall will not be as straightforward as described and will most likely require assistance from the radar operator. VI. CONCLUSION In this paper, we have developed an algorithm capable of estimating unknown wall parameters by using an ultrawideband through-wall radar. The permittivity and the thickness of the wall are estimated by a simple optimization approach that requires the time delays of the echoes backscattered from the rear side of the wall to be measured at several different antenna separations. At each separation, the time delay is estimated by a subspace super-resolution method. We have described the essential preprocessing steps to be performed prior to the application of subspace processing. This includes estimating the signal correlation matrix from one snapshot of the measured data and improved spatial smoothing, which enhances the resolution of closely spaced echoes backscattered from the front and rear side of the wall. We have shown that by using simulated data that our time-delay estimation algorithm is capable of resolving overlapping pulses, though the processing bandwidth is small. Finally,

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the robustness of the complete wall parameter estimation algorithm was tested by simulation, and we also present the measurement results of two real walls composed of porous bricks. The results prove that our algorithm can provide reliable results although it was originally developed for the case of an ideal homogeneous wall under test. The estimation algorithm implemented in Matlab takes no more than a few seconds to provide the results, and we emphasize that no special calibration of the radar is required immediately prior to the measurement of the wall. The algorithm can therefore improve the performance of real-time through-wall radars employed in emergency or law enforcement operations.

REFERENCES [1] G. Wang, M. G. Amin, and Y. Zhang, “New approach for target locations in the presence of wall ambiguities,” IEEE Trans. Aerosp. Electron. Syst., vol. 42, no. 1, pp. 301–315, Jan. 2006. [2] F. Ahmad and M. G. Amin, “A noncoherent approach to radar localization through unknown walls,” in Proc. IEEE Conf. Radar, 2006, pp. 583–589. [3] H. Wang, Z. Zhou, and L. Kong, “Wall parameters estimation for moving target localization with through-the-wall radar,” in Proc. Microw. Millimeter Wave Technol., 2007, pp. 1–4. [4] F. Ahmad, M. G. Amin, and G. Mandapati, “Autofocusing of through-the-wall radar imagery under unknown wall characteristics,” IEEE Trans. Image Process., vol. 16, no. 7, pp. 1785–1795, Jul. 2007. [5] J. Zhang, M. Nakhkash, and Y. Huang, “Electromagnetic imaging of layered building materials,” Measure. Sci. Technol., vol. 12, pp. 1147–1152, 2001. [6] M. Dehmollaian and K. Sarabandi, “Refocusing through building walls using synthetic aperture radar,” IEEE Trans. Geosci. Remote Sens., vol. 46, no. 6, pp. 1589–1599, Jun. 2008. [7] R. Solimene, F. Soldovieri, G. Prisco, and R. Pierri, “Three-dimensional through-wall imaging under ambiguous wall parameters,” IEEE Trans. Geosci. Remote Sens., vol. 47, no. 5, pp. 1310–1317, May 2009. [8] P. Chang, R. Burkholder, and J. Volakis, “Adaptive CLEAN with target refocusing for through-wall image improvement,” IEEE Trans. Antennas Propag., vol. 58, no. 1, pp. 155–162, Jan. 2010. [9] P. Withington, H. Fluhler, and S. Nag, “Enhancing homeland security with advanced UWB sensors,” IEEE Microw. Mag., vol. 4, no. 3, pp. 51–58, Sep. 2003. [10] A. Beeri and R. Daisy, “High-resolution through-wall imaging,” in Proc. Soc. Photo-Optical Instrum. Eng. Conf. Ser., 2006, vol. 6201, pp. 1–6. [11] O. Sisma, A. Gaugue, C. Liebe, and J.-M. Ogier, “UWB radar: Vision through a wall,” Telecommun. Syst., vol. 38, pp. 53–59, 2008. [12] RETIA, a.s., ReTWis. 2009. [Online]. Available: http://www. lokalizacni-systemy.cz/en/retwisen [13] S. Foo and S. Kashyap, “Cross-correlated back projection for UWB radar imaging,” in Proc. IEEE Antennas Propag. Soc. Int. Symp., 2004, vol. 2, pp. 1275–1278. [14] E. K. Hung, W. Chamma, and S. M. Gauthier, “A second study of uwb receiver array beamformer output images of objects in a room,” Defence R&D Canada—Ottawa, 2003, Tech. Rep. [15] H. Khatri and C. Le, “Estimation of electromagnetic parameters and thickness of a wall using synthetic aperture radar,” in Proc. Soc. PhotoOptical Instrum. Eng. Conf. Ser., 2007, vol. 6547, pp. 1–9. [16] M. M. Nikolic, A. Nehorai, and A. R. Djordjevic, “Estimating moving targets behind reinforced walls using radar,” IEEE Trans. Antennas Propag., vol. 57, no. 11, pp. 3530–3538, Nov. 2009. [17] M. Farwell, J. Ross, R. Luttrell, D. Cohen, W. Chin, and T. Dogaru, “Sense through the wall system development and design considerations,” J. Franklin Inst., vol. 345, pp. 570–591, 2008. [18] T. G. Savelyev and M. Sato, “Comparative analysis of UWB deconvolution and feature-extraction algorithms for GPR landmine detection,” in Proc. Soc. Photo-Optical Instrum. Eng. Conf. Ser., 2004, vol. 5415, pp. 1008–1018. [19] H. L. V. Trees, Optimum Array Processing. Hoboken, NJ: Wiley, 2002.

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[20] H. Yamada, M. Ohmiya, Y. Ogawa, and K. Itoh, “Superresolution techniques for time-domain measurements with a network analyzer,” IEEE Trans. Antennas Propag., vol. 39, no. 2, pp. 177–183, Feb. 1991. [21] X. Li and K. Pahlavan, “Super-resolution TOA estimation with diversity for indoor geolocation,” IEEE Trans. Wireless Commun., vol. 3, no. 1, pp. 224–234, Jan. 2004. [22] C. L. Bastard, V. Baltazart, Y. Wang, and J. Saillard, “Thin-pavement thickness estimation using GPR with high-resolution and superresolution methods,” IEEE Trans. Geosci. Remote Sens., vol. 45, no. 8, pp. 2511–2519, Aug. 2007. [23] E. Fisher, G. A. McMechan, A. P. Annan, and S. W. Cosway, “Examples of reverse-time migration of single-channel, ground-penetrating radar profiles,” Geophysics, vol. 57, no. 4, pp. 577–586, 1992. [24] C. Liu, J. Li, X. Gan, H. Xing, and X. Chen, “New model for estimating the thickness and permittivity of subsurface layers from GPR data,” Proc. Inst. Elect. Eng. Radar, Sonar Navig., vol. 149, no. 6, pp. 315–319, 2002. [25] C.-P. Kao, J. Li, Y. Wang, H. Xing, and C. R. Liu, “Measurement of layer thickness and permittivity using a new multilayer model from GPR data,” IEEE Trans. Geosci. Remote Sens., vol. 45, no. 8, pp. 2463–2470, Aug. 2007. [26] U. Spagnolini, “Permittivity measurements of multilayered media with monostatic pulse radar,” IEEE Trans. Geosci. Remote Sens., vol. 35, no. 2, pp. 454–463, Mar. 1997. [27] Ground Penetrating Radar, D. Daniels, Ed., 2nd ed. London, U.K.: Inst. Elect. Eng., 2004. [28] UWB Communication Systems: A Comprehensive Overview, M.-G. D. Benedetto, T. Kaiser, A. F. Molisch, I. Oppermann, C. Politano, and D. Porcino, Eds. New York: Hindawi Publishing Corporation, 2006. [29] J. McLean, R. Sutton, A. Medina, H. Foltz, and J. Li, “The experimental characterization of UWB antennas via frequency-domain measurements,” IEEE Antennas Propag. Mag., vol. 49, no. 6, pp. 192–202, Dec. 2007. [30] H. Saarnisaari, “TLS-ESPRIT in a time delay estimation,” in Proc. IEEE 47th Vehic. Technol. Conf., 1997, vol. 3, pp. 1619–1623. [31] H. Farrokhi, “TOA estimation using MUSIC super-resolution techniques for an indoor audible chirp ranging system,” in Proc. IEEE Int. Conf. Signal Process. Commun., 2007, pp. 987–990. [32] J. Evans, J. Johnson, and D. Sun, Application of Advanced Signal Processing Techniques to Angle of Arrival Estimation in ATC Navigation and Surveillance Systems. Technical Report 582. Cambridge, MA: Mass. Inst. Technol., Lincoln Lab., 1982. [33] R. T. Williams, S. Prasad, A. K. Mahalanabis, and L. H. Sibul, “An improved spatial smoothing technique for bearing estimation in a multipath environment,” IEEE Trans. Acoust., Speech, Signal Process., vol. 36, no. 4, pp. 425–432, Apr. 1988. [34] W. Du and R. L. Kirlin, “Improved spatial smoothing techniques for DOA estimation of coherent signals,” IEEE Signal Process. Lett., vol. 39, no. 5, pp. 1208–1210, May 1991. [35] D. Grenier and E. Bosse, “Decorrelation performance of DEESE and spatial smoothing techniques for direction-of-arrival problems,” IEEE Trans. Signal Process., vol. 44, no. 6, pp. 1579–1584, Jun. 1996. [36] M. Dong, S. Zhang, X. Wu, and H. Zhang, “A high resolution spatial smoothing algorithm,” in Proc. Int Microw., Antenna, Propag. EMC Technol. Wireless Commun. Symp., 2007, pp. 1031–1034. [37] P. Protiva, J. Mrkvica, and J. Macháˇc, “Time delay estimation of UWB radar signals backscattered from a wall,” Microw. Opt. Technol. Let., vol. 53, no. 6, pp. 1444–1450, 2011. [38] A. G. Wienerberger, POROTHERM brick & ceiling system. 2010. [Online]. Available: http://www.wienerberger.com [39] P. Protiva, J. Mrkvica, and J. Macháˇc, “Universal generator of ultrawideband pulses,” Radioengineering, vol. 17, no. 4, pp. 74–78, 2008. [40] R. Yelf, “Where is true time zero?,” in Proc. 10th Int. Conf. Ground Penetrating Radar, 2004, pp. 279–282. Pavel Protiva (S’07) received the M.S. degree in radioelectronics from the Faculty of Electrical Engineering, Czech Technical University, Prague, Czech Republic, in 2007, where he is currently pursuing the Ph.D. degree in electromagnetic field. He is an author or coauthor of many scientific papers in journals and conference proceedings. His research interest is the propagation of ultrawideband signals, through-wall radar imaging, and ground penetrating radar.

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Jan Mrkvica (M’00) received the M.S. and Ph.D. degrees in radioelectronics from the Czech Technical University, Prague, Czech Republic, in 2001 and 2004, respectively. Since 2004, he has been an Ultrawideband Technology Development Engineer with RETIA, a.s., Pardubice, Czech Republic.

Jan Macháˇc (M’99–SM’01) received the M.S. and Dr.Sc. degrees from the Czech Technical University, Prague (CTU FEE), in 1977 and 1996, respectively, and the CSc. degree (Ph.D. equivalent) from the Czechoslovak Academy of Sciences, Prague, in 1982. He was appointed a Professor at CTU FEE in 2009. He is an author or coauthor of many scientific papers in journals and conference proceedings, and several textbooks. His main research interests are the modeling of waves on planar transmission lines used in microwave and millimeter-wave techniques, leaky waves, leaky wave and planar antennas, metamaterials, propagation of electromagnetic waves in periodic structures, planar microwave filters, ultrawideband fields, and propagation.

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Accuracy Evaluation of Ultrawideband Time Domain Systems for Microwave Imaging Xuezhi Zeng, Student Member, IEEE, Andreas Fhager, Member, IEEE, Mikael Persson, Member, IEEE, Peter Linner, Life Senior Member, IEEE, and Herbert Zirath, Fellow, IEEE

Abstract—We perform a theoretical analysis of the measurement accuracy of ultrawideband time domain systems. The theory is tested on a specific ultrawideband system and the analytical estimates of measurement uncertainty are in good agreements with those obtained by means of simulations. The influence of the antennas and propagation effects on the measurement accuracy of time domain near field microwave imaging systems is discussed. As an interesting application, the required measurement accuracy for a breast cancer detection system is estimated by studying the effect of noise on the image reconstructions. The results suggest that the effects of measurement errors on the reconstructed images are small when the amplitude uncertainty and phase uncertainty of measured data are less than 1.5 dB and 15 degrees, respectively. Index Terms—Biomedical imaging, measurement errors, microwave imaging, random noise, time domain measurements.

I. INTRODUCTION

A

S a potential imaging modality for biomedical applications, active microwave imaging has attracted considerable interest in the past few decades [1]–[4]. With active microwave imaging, biological tissues are classified based on their differences in dielectric properties. Several studies have shown that the variation is mainly due to different water content [5]–[8]. There are two main approaches to active microwave imaging: tomography and radar based imaging. Microwave tomography is a classic approach which leads to solving an inverse scattering problem. In a tomography system, an antenna array is used to transmit microwave signals into an object-under-test and receive scattered fields. By iteratively comparing the measured data with numerically calculated data, the dielectric properties of the object under test can be quantitatively reconstructed. Microwave tomography reconstructions based on mono-frequency, multiple frequency or time domain data have been numerically and experimentally studied [9]–[14]. It has been shown that highly stable and high-resolution reconstructions

Manuscript received February 17, 2010; revised April 21, 2011; accepted May 01, 2011. Date of publication August 12, 2011; date of current version November 02, 2011. This work was supported in part by the Swedish Agency for Innovation Systems within the Chalmers Antenna Systems VINN Excellence Centre and in part by the Swedish Foundation for Strategic Research within the Strategic Research Center Charmant. X. Zeng, A. Fhager, and M. Persson are with the Department of Signals and Systems, Chalmers University of Technology, 412-96 Gothenburg, Sweden. P. Linner and H. Zirath are with the Microwave Electronics Laboratory, Chalmers University of Technology, 412-96 Gothenburg, Sweden. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2164174

can be achieved by the use of ultrawideband (UWB) data [13], [14]. In contrast to microwave tomography, radar based imaging avoids complex image reconstruction algorithms and has been extensively investigated for biomedical applications in the last few years, e.g, breast cancer detection [15]–[19]. In this approach, a UWB signal is used to illuminate an object, and the reflected signal is measured at numerous locations. The amplitude and time arrival information of the reflected signals is utilized to identify the presence and location of significant scatters. As a result, a qualitative image of the object under investigation is obtained. Measurements for UWB microwave imaging are generally carried out either in the frequency domain with the help of a vector network analyzer (VNA) [13], [17], [18], or in the time domain using a sampling oscilloscope [12]. With the VNA, scattering parameters at a number of discrete frequencies are measured and then utilized to synthesize time domain signals. The sampling oscilloscope uses an equivalent time sampling technique which constructs a UWB signal based on measurements over several repetitive wave cycles. In contrast to the equivalent-time system, a time domain system capable of real-time data acquisition [19] was recently applied to UWB microwave imaging and with this system, the scattered signal is acquired from a single-shot measurement. In comparison with frequency domain systems, time domain systems have the advantage of fast acquisition of UWB data, which makes them more attractive for UWB applications, such as medical imaging, see-through-wall imaging radar [20], and ground-penetrating radar [21]. However, time domain systems have lower signal-to-noise-ratios (SNRs), which may produce distortion to the images. This was noted in [19] for tumor detection in breast phantoms. In order to design a suitable time domain system for UWB microwave imaging, we need quantify the required measurement accuracy and investigate the factors that affect the measurement accuracy. This paper is devoted to these issues and focuses on medical applications using the tomographic approach. In this paper, we derive analytical estimates of the measurement uncertainty of time domain systems. In order to validate the analysis, we investigate a specific time domain system and compare the estimated uncertainties with simulated results. Furthermore, we discuss the effects of antennas and propagation on the measurement accuracy of time domain imaging systems. The degree of measurement accuracy required for a microwave imaging system is dependent on the specific application. As an example, the effect of noise on the reconstruction

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A. Random Error Analysis The measurement of a time domain signal is affected by three different random error sources: thermal noise, quantization noise and time jitter [22]. The thermal noise and time jitter are nearly Gaussian and zero-mean random variables. The quantization noise is characterized by the least significant bit (LSB) and is uniformly distributed from 1/2 LSB to 1/2 LSB. , then the th sample of the We assume a noise free signal real measured signal, , can be expressed as follows:

(1) Fig. 1. Block diagram of a UWB time domain microwave imaging system.

quality of a high contrast breast model is numerically studied in order to estimate the required measurement accuracy.

where is the target time of the th sample, is the time jitter, is the thermal noise and is the quantization noise. The variability of the time domain measurements is characterized by the standard deviation of several measurements carried out under repetitive conditions [24]:

II. UWB TIME DOMAIN SYSTEM FOR MICROWAVE IMAGING Fig. 1 shows the block diagram of a UWB time domain microwave imaging system. It consists of an impulse generator, a data acquisition module, an antenna array, a switching matrix and a personal computer (PC). With this system, a UWB signal generated by the impulse generator is transmitted into an object under test and the scattered signal is received. This is performed for all the possible combinations of transmitting-receiving antenna pairs with the help of the switching matrix. The acquired signals are measured by means of the data acquisition module and the whole measurement is automated by using the PC. Antennas in the microwave imaging system typically work in a range from hundred megahertz to several gigahertz and the choice of the frequency range is a tradeoff between the spatial resolution and penetration depth in materials of interest. Directional antennas are preferable in some cases in order to concentrate the radiated power in the imaging region. In addition to the performance of the antennas, the tomographic approach imposes constraints on the properties of the antennas. The antennas need to be easily and accurately modeled in a electromagnetic computational solver. The antennas should also have small size in order to be configured in an antenna array. Monopoles are commonly used due to its simple structure [2], [3], [13]. It has been shown that when placed in a lossy medium, the bandwidth of a monopole antenna increases significantly with the associated resistive loading. Our investigation is however general and not restricted to any specific type of antenna.

(2) is the number of measurements, is the fast Here Fourier transform (FFT) of the acquired measurement of index . is the average of the FFT of all measurements. and are either amplitude (in volts when ) or phase (in degrees when ). and represent the uncertainties As a result, of spectral amplitudes and phases respectively. Defining , the amplitude uncertainty is often expressed as a relative value in terms of SNR:

(3)

B. Analytical Estimates of Measurement Uncertainties The amplitude and phase uncertainties defined by (2) and (3) can be estimated analytically based on the analysis presented below. It is usually assumed that, after averaging, time jitter acts as a low pass filter [25], [26]. Therefore, we can write:

(4) where is the frequency spectrum of the noise free signal and is the Fourier transform of the jitter’s probability density function. The latter can be expressed as [26]:

III. MEASUREMENT UNCERTAINTY The measurements are subject to errors that can be mainly classified into deterministic errors and random errors. Deterministic errors can be compensated for by using various calibration techniques [22], [23]. We therefore only study the effects of random errors in the rest of the paper.

(5) where is the standard deviation of the time jitter in seconds. Therefore, the distorting effects of the jitter may be removed by means of deconvolution [25], [26].

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If the thermal noise and quantization noise are assumed to be white, noise will have a flat spectrum in the frequency domain. , can then The standard deviation of spectral amplitude, be obtained from the following equation:

(6) Here [27] and are the quantization noise power and the thermal noise power, respectively. is the samis the effective noise bandwidth and is pling rate, the number of averages in the measurements. Therefore, given a specified UWB time domain system, the SNR of a measurement can be estimated, based on (4)to (6). Then the amplitude uncertainty can be obtained from (3). The , is proportional to the relative error of phase uncertainty, the spectral amplitude and can be roughly estimated from the SNR [28]:

(7) This equation gives the limit of the achievable phase measurement accuracy. C. Comparison Between Simulated Results and Analytical Estimates Simulations can now be performed in order to validate the analysis above. In the simulations, we model the effects of random errors on a time domain measurement according to (1) and then calculate the amplitude and phase uncertainties from the simulated data. The obtained results are compared with the uncertainties estimated analytically by (4)–(7). The specification data of an impulse generator in our lab [29] and a specific real time oscilloscope with bandwidth of 20 GHz and sampling rate of 80 Gsamples/s [30] are used for the simulations. The output voltage of the generator, , has a Gaussian shape with a full width half maximum duration of 70 picoseconds (ps). The time domain waveform and the normalized amplitude spectrum of the output are given in Fig. 2. The amplitude spectrum has a 3 dB bandwidth of around 4.5 GHz, which covers the frequency range of interest. We disregard the antennas and propagation effects and assume that the received signals have the same waveform as the . The received signals are then “measured” impulse signal in the simulations with the time domain system and the measurement uncertainties are obtained from these simulations. We take the maximum input of the oscilloscope as an example. The maximum input (the peak-to-peak amplitude) of the and the corresponding thermal noise oscilloscope is RMS. We assume that the noise is bandlimited is . to the bandwidth of the oscilloscope, that is The LSB of the quantization is given by , is the resolution of the oscilloscope. The where quantization noise is frequency limited to half of the sampling . The time jitter is RMS, frequency which is contributed by both the impulse generator and the os-

Fig. 2. The output signal of the impulse generator: (a) time domain waveform and (b) amplitude spectrum.

cilloscope. The thermal noise and time jitter are modeled as normally distributed random processes, and the modeled quantization noise has an uniform distribution. Fig. 3 shows the amplitude and phase uncertainties obtained from the simulations in comparison with the theoretical estiand , mates when the number of averages respectively. The data is presented in a frequency range from 0.5 GHz to 4.5 GHz. The measurement uncertainties of the simurepetitive lated measurement data are obtained from waveforms. The simulated results agree well with the theoretical estimates, which validates the analytical analysis. The results represent the highest accuracy can be achieved by using the investigated system for the specific simulated measurement.

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array configuration and the electric properties of the imaging medium. If we denote the number of antenna elements by and indicate the transmitting antenna with index 1, then the signal , received by the antenna element with index , can be expressed as an inverse Fourier transform (IFFT): (8) Here is the frequency spectrum of the generator output . comprises the frequency response of the transmitting and receiving antennas and the propagation effects:

(9) is the number of multi-path signals received by the where and represent antenna element with index , the amplitude and phase transfer functions respectively. The assumption we made about the antennas and propagation effects , , and in the simulations corresponds to . Here both and are constants. When dealing with far field problems, an analytical exprescan be obtained, in which antennas are described sion of by means of their impulse response [32], [33]. However, we works on a near-field imaging approach and the complicated propagation environment makes it impractical to express in an analytical way. Instead, numerical simulations can be used to estimate for a specific measurement configuration, from which the received signals can be obtained according to (8). Then by using the analytical estimates, we can evaluate the measurement uncertainties of time domain measurements. With the analytical analysis, we can also design a suitable system with given measurement accuracy. Fig. 3. Measurement uncertainties of (a) the spectral amplitudes and (b) the phases of a test signal.

D. Antennas and Propagation Effects The antennas and propagation effects were neglected in the above simulation. In practical microwave imaging measurements, the antennas and signal propagation play important roles in determining the characteristic of the received signals. First of all, an antenna works as a spatio-temporal filter, which makes the radiated signal distorted from the excitation signal. The same effect is present on the receiving side [31]. Furthermore, the signals received by different antennas vary in strength and the variation is dependent on the radiation pattern of the antennas, the size of the antenna array and the electrical properties of the imaging medium. Besides, different frequency components suffer different levels of attenuation and have different phase velocities (dispersion) in the propagation. In addition, due to multiple scattering effects, the signals acquired by the receiving antennas are superpositions of multi-path signals which have different strengths and arrival time. Therefore, the received signals in the microwave imaging system are dependent on the antenna performance, the antenna

IV. EFFECTS OF NOISE ON IMAGE RECONSTRUCTIONS Noisy measurements may degrade the image reconstruction quality. As a specific example, in this section we investigate the influence of noise on the reconstructions of a 2-D breast model by means of numerical simulations. Breast cancer detection is one of the most researched applications of microwave imaging. Recently, it has been reported that the contrast in dielectric properties between the malignant tumor and normal breast tissue varies considerably depending on the compositions of the breast [8]. Our previous study showed that the sensitivity of image reconstructions to the measurement error is dependent on the dielectric contrast of the imaged object [34]. From the image reconstruction point of view, it is more challenging to handle high contrast objects, we therefore consider a high contrast breast model in this work. Fig. 4 shows the breast model and the antenna array configuration used in the simulations. The model consists of a 2 mm skin layer, a circular healthy tissue with radius of 48 mm and two tumors with radius of 5 mm. The dielectric properties are assumed to be frequency independent and the permittivities and conductivities for skin, healthy tissue, and tumor are: , [7]; , [8];

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Fig. 5. Relative permittivity profiles reconstructed from noise-free time domain data: (a) first step reconstruction and (b) final reconstruction.

Fig. 4. Breast model and antenna array configuration used in the numerical studies.

, [8]. Sixteen antenna elements are equally spaced on a 19 cm diameter circle and immersed in and . a matching liquid with The measured scattering data was numerically generated by using a finite difference time domain (FDTD) program and totally 16 15 data sets were obtained. In the forward simulation, the grid cell size was 1 mm and the antennas were modeled as hard sources, which radiate electromagnetic energy by setting electric fields in FDTD grids. In numerical studies, the real antenna modeling gives the same quality reconstructions as those obtained based on hard source models as long as the antennas have the required bandwidths. The obtained data were then utilized to reconstruct the dielectric properties by using a nonlinear time domain inversion algorithm. A detailed description of the algorithm can be found in [13], [14]. It has been shown that permittivity reconstructions have higher quality than conductivity [13], [14], [34], therefore, we only present the permittivity reconstructions here. A. Reconstructions From Noise-Free Data We first reconstruct the image from noise-free data. In the reconstruction, the grid cell size was 2 mm in order to avoid an inverse crime, which happens when the same meshes are used in the forward and inverse simulations, resulting in an unrealistically good reconstruction due to the canceling of numerical errors. The permittivity reconstructions are shown in Fig. 5 with a reconstruction region of 120 mm 120 mm, which is the region surrounded by the dashed rectangular loop in Fig. 4. Fig. 5(a) gives the first step reconstruction using a spectral content with center frequency 1.5 GHz and bandwidth 1.5 GHz. The objects are found on the correct positions and with dimensions that correspond well to the original model. However, the skin is not well resolved and the reconstructed permittivities of both the skin and tumors are much lower than the real values. The reconstruction was then proceeded using three more steps, where this

reconstruction was used as an initial guess. In these three steps of reconstruction, both the center frequency and bandwidth of the used spectral contents were 2 GHz, 2.5 GHz, and 3 GHz respectively. Fig. 5(b) presents the final reconstruction. Compared with the initial reconstruction in Fig. 5(a), it is seen that a higher spatial resolution is achieved and the permittivities of the tumors are well consistent with those in the model. The skin is better resolved than the first step reconstruction, but the reconstructed permittivity is still lower than the real value. This is because the thickness of the skin is comparable to the FDTD cell size. B. Reconstructions From Noisy Data Using the same reconstruction settings as for the noise-free reconstruction, we now add different levels of measurement noise to the scattering data. Both the amplitude and phase noise are modeled as zero-mean normal distribution random processes. In order to quantitatively assess the influence of the noise on the reconstructions, a relative reconstruction error is defined:

(10)

where and are the permittivity profiles reconstructed from noisy and noise free data. is the 2-D reconstruction region. Fig. 6 shows the relative reconstruction errors when different levels of amplitude and phase errors are taken into account. The horizontal axis is the phase uncertainty. The dot-dashed line, dashed line and solid line are the average relative reconstruction errors of five repetitive simulations when the amplitude uncertainty is 1.5 dB, 2.0 dB and 2.2 dB respectively. The vertical lines represent the variations of relative reconstruction errors of the five simulations. It is shown that when the amplitude uncertainty is 1.5 dB, the relative reconstruction error increases with the phase uncertainty and the result varies slightly between individual runs. As the amplitude error increases, the variation of the relative reconstruction error becomes larger. Fig. 7 presents the reconstructed images with a relative reconstruction error around 0.4%, 0.9%, 4% and 12%. They are obtained for amplitude errors and phase errors of (1.5 dB, 15 degrees), (1.5 dB, 25 degrees), (2 dB, 10 degrees) and (2.2 dB, 15

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Furthermore, the required measurement accuracy for breast cancer detection has been studied numerically. It is found out that the image distortion is acceptable when the amplitude uncertainty is less than 1.5 dB and the phase uncertainty is less than 15 degrees. Although the investigation focuses on medical applications of microwave imaging, the analysis can also be applied to other UWB time domain systems. The study has concentrated on the effects of random errors. As mentioned previously, deterministic errors, although can not be fully compensated for, can be made small by effective calibrations. Another type of error need to be considered in time domain systems is the aliasing error due to inadequate sampling rate. A relevant study carried out by us indicates that the aliasing error is negligible in the frequency range of interest if the time interval resolution is higher than 10 ps. Future work includes the study of other biological tissue models and the design of a suitable time domain microwave imaging system. Fig. 6. Relative reconstruction errors when the amplitude and phase errors are taken into account. The dot-dashed line, dashed line and solid line are the average relative reconstruction errors of five repetitive simulations when the amplitude uncertainty is 1.5 dB, 2.0 dB and 2.2 dB respectively. The vertical lines represent the variation ranges of the relative reconstruction errors of the five simulations.

ACKNOWLEDGMENT The computations in this paper were performed on C3SE computing resources. REFERENCES

Fig. 7. Reconstructed relative permittivity profiles of the breast model with a relative reconstruction error of (a) 0.4%, (b) 0.9%, (c) 4% and (d) 12%. The corresponding amplitude and phase errors are (a) 1.5 dB, 15 degrees, (b) 1.5 dB, 25 degrees, (c) 2 dB, 10 degrees and (d) 2.2 dB, 15 degrees.

degrees), respectively. In comparison with the noise free reconstruction in Fig. 5(b), we can see that a relative reconstruction error less than 0.4% indicates a negligible impact on the reconstruction quality. V. CONCLUSION An analytical analysis has been developed to quantitatively predict the measurement uncertainty of UWB time domain systems due to random errors. The simulations of a specific time domain system confirm the validity of the developed theory.

[1] C. Pichot, L. Jofre, G. Peronnet, and J. C. Bolomey, “Active microwave imaging of inhomogeneous bodies,” IEEE Trans. Antennas Propag., vol. AP-33, no. 4, pp. 416–425, 1985. [2] P. M. Meaney, M. W. Fanning, D. Li, S. P. Poplack, and K. D. Paulsen, “A clinical prototype for active microwave imaging of the breast,” IEEE Trans. Microwave Theory Tech., vol. 48, no. 11, pp. 1841–1853, 2000. [3] P. M. Meaney, K. D. Paulsen, and J. T. Chang, “Near-field microwave imaging of biologically-based materials using a monopole transceiver system,” IEEE Trans. Antennas Propag., vol. 46, no. 1, pp. 31–45, 1998. [4] E. C. Fear, S. C. Hagness, P. M. Meaney, M. Okoniewski, and M. A. Stuchly, “Enhancing breast tumor detection with near-field imaging,” Microwave Mag., vol. 3, no. 1, pp. 48–56, 2002. [5] T. S. England and N. A. Sharples, “Dielectric properties of the human body in the microwave region of the spectrum,” Nature, vol. 163, pp. 487–488, 1949. [6] A. J. Surowiec, S. S. Stuchly, J. R. Barr, and A. Swarup, “Dielectric properties of breast carcinoma and the surrounding tissues,” IEEE Trans. Biomed. Eng., vol. 35, no. 4, pp. 257–263, 1988. [7] S. Gabriely, R. W. Lau, and C. Gabriel, “The dielectric properties of biological tissues: II. Measurements in the frequency range 10 Hz to 20 GHz,” Phys. Med. Biol., vol. 41, no. 11, pp. 2251–2269, 1996. [8] M. Lazebnik, D. Popovic, L. McCartney, and C. B. Watkins et al., “A large scale study of the ultra wideband microwave dielectric properties of normal, benign and malignant breast tissues obtained from cancer surgeries,” Phys. Med. Biol., vol. 52, no. 20, pp. 6093–6115, 2007. [9] S. Y. Semenov, A. E. Bulyshev, A. E. Souvorov, and A. G. Nazarov et al., “Three-dimensional microwave tomography: Experimental imaging of phantoms and biological objects,” IEEE Trans. Microwave Theory Tech., vol. 48, no. 6, pp. 1071–1074, 2000. [10] W. C. Chew and J. H. Lin, “A frequency-hopping approach for microwave imaging of large inhomogeneous bodies,” IEEE Microwave Guided Wave Lett., vol. 5, no. 12, pp. 439–441, 1995. [11] Q. Fang, P. M. Meaney, and K. D. Paulsen, “Microwave image reconstruction of tissue property dispersion characteristics utilizing multiple-frequency information,” IEEE Trans. Microwave Theory Tech., vol. 52, no. 8, pp. 1866–1875, 2004. [12] F. C. Chen and W. C. Chew, “Time domain ultra-wideband microwave imaging radar system,” J. Electromagn. Waves Appl., vol. 17, no. 2, pp. 313–331, 2003.

ZENG et al.: ACCURACY EVALUATION OF UWB TIME DOMAIN SYSTEMS FOR MICROWAVE IMAGING

[13] A. Fhager, P. Hashemzadeh, and M. Persson, “Reconstruction quality and spectral content of an electromagnetic time-domain inversion algorithm,” IEEE Trans. Biomed. Eng., vol. 53, no. 8, pp. 1594–1604, 2006. [14] J. E. Johnson, T. Takenaka, and T. Tanaka, “Two-dimensional time-domain inverse scattering for quantitative analysis of breast composition,” IEEE Trans. Biomed. Eng., vol. 55, no. 8, pp. 1941–1945, 2008. [15] E. C. Fear and M. A. Stuchly, “Microwave detection of breast cancer,” IEEE Trans. Microwave Theory Tech., vol. 48, no. 11, pp. 1854–1863, 2000. [16] E. J. Bond, L. Xu, S. C. Hagness, and B. D. Van Veen, “Microwave imaging via space-time beamforming for early detection of breast cancer,” IEEE Trans. Antennas Propag., vol. 51, no. 8, pp. 1690–1705, 2003. [17] M. Klemm, I. J. Craddock, J. A. Leendertz, A. Preece, and R. Benjamin, “Radar-based breast cancer detection using a hemispherical antenna array-experimental results,” IEEE Trans. Antennas Propag., vol. 57, no. 6, pp. 1692–1704, 2009. [18] S. M. Salvador and G. Vecchi, “Experimental tests of microwave breast cancer detection on phantoms,” IEEE Trans. Antennas Propag., vol. 57, no. 6, pp. 1705–1712, 2009. [19] J. C. Y. Lai, C. B. Soh, E. Gunawan, and K. S. Low, “UWB microwave imaging for breast cancer detection-Experiments with heterogeneous breast phantoms,” Progr. Electromagn. Res., vol. 16, pp. 19–29, 2011. [20] Y. Yang, C. Zhang, and A. E. Fathy, “Development and implementation of ultra-wideband see-through-wall imaging system based on sampling oscilloscope,” IEEE Antennas Wireless Propag. Lett., vol. 7, pp. 465–468, 2008. [21] A. G. Yarovoy, T. G. Savelyev, P. J. Aubry, P. E. Lys, and L. P. Ligthart, “UWB array-based sensor for near field imaging,” IEEE Trans. Microwave Theory Tech., vol. 55, no. 6, pp. 1288–1295, 2007. [22] W. L. Gans, “Calibration and error analysis of a picosecond pulse waveform measurement system at NBS,” Proce. IEEE, vol. 74, no. 1, pp. 86–90, 1986. [23] W. L. Gans, “Dynamic calibration of waveform recorders and oscilloscopes using pulse standards,” IEEE Trans. Instrum. Meas., vol. 39, no. 6, pp. 952–957, 1990. [24] B. N. Taylor and C. E. Kuyatt, “Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results,” NIST Tech. Note 12971994 ed. . [25] W. L. Gans, “The measurement and deconvolution of time jitter in equivalent-time waveform samplers,” IEEE Trans. Instrum. Meas., vol. IM-32, no. 1, pp. 126–133, 1983. [26] J. R. Andrews, “Removing Jitter From Picosecond Pulse Measurements,” Application Note AN-23, Picosecond Pulse Labs. Boulder, CO, 2009. [27] B. Widrow and I. Kollar, Quantization Noise: Roundoff Error in Digital Computation, Signal Processing, Control, and Communications. Cambridge, UK: Cambridge Univ. Press, 2008. [28] G. W. Stimson, Introduction to Airborne Radar. New York: SciTech Publishing, 1998. [29] “Model 3500D Impulse Generator Instruction Manual,” Picosecond Pulse Lab., Boulder, CO. [30] Agilent Technologies Infiniium 90000 X-Series Oscilloscopes (Datasheet) [Online]. Available: http://cp.literature.agilent.com/ litweb/pdf/5990-5271EN.pdf [31] B. Allen, M. Dohler, E. Okon, W. Malik, A. Brown, and D. Edwards, Ultra-Wideband Antennas and Propagation for Communications, Radar and Imaging. London, UK: Wiley, 2006. [32] A. Shlivinski, E. Heyman, and R. Kastner, “Antenna characterization in the time domain,” IEEE Trans. Antennas Propag., vol. 45, no. 7, pp. 1140–1149, 1997. [33] B. Scheers, M. Acheroy, and A. V. Vorst, “Time-domain simulation and characterisation of TEM horns using a normalised impulse response,” IEE Proc.-Microw. Antennas Propag., vol. 147, no. 6, pp. 463–468, 2000. [34] X. Zeng, A. Fhager, and M. Persson, “Study on the sensitivity of image reconstruction to the measurement uncertainty in microwave tomography,” presented at the 4th Eur. Conf. on Antennas and Propagation, Barcelona, Spain, 2010.

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Xuezhi Zeng (S’09) was born in 1980 in China. She received the M.Sc. degree in electrical engineering from Jiangsu University, Zhenjiang, China, in 2006. She is currently working towards the Ph.D. degree at Chalmers University of Technology, Göteborg, Sweden. Her research interests include electromagnetic imaging, antenna design, microwave measurement and system design.

Andreas Fhager (M’07) was born in 1976 in Sweden. He received the M.Sc. degree in engineering physics, the Licentiate of Technology degree and the Ph.D. degree, both in microwave imaging, from Chalmers University of Technology, Göteborg, Sweden, in 2001, 2004, and 2006, respectively. Currently, he is working as an Assistant Professor at Chalmers. His research interests include electromagnetic imaging methods for breast cancer detection and other biomedical applications of microwaves. Mikael Persson (M’10) received the M.Sc. and Ph.D. degrees from Chalmers University of Technology, Göteborg, Sweden, in 1982 and 1987, respectively. In 2000, he became a Professor of electromagnetics and in 2006 a professor in biomedical electromagnetics at the Department of Signal and Systems, Chalmers University of Technology. He is presently the Head of the Division of Signal Processing and Biomedical Engineering and the Director for the regional research and development platform MedTech West. At present these activities involve approximately 50 researchers. His main research interests include electromagnetic diagnostics and treatment. He is author/coauthor of more than 200 refereed journal and conference papers. Peter Linner (S’69–M’74–SM’87–LSM’10) received the M.Sc. and Ph.D. degrees from Chalmers University of Technology, Gothenburg, Sweden, in 1969 and 1974, respectively. In 1969, he became a Teaching Assistant in mathematics and telecommunications at Chalmers University of Technology. In 1973, he joined the research and teaching staff of the Division of Network Theory at the same university with research interests in the areas of network theory, microwave engineering, and computer-aided design methods. In 1974 he moved to the MI-division, Ericsson Telephone Company, Mölndal, Sweden, where he was a systems engineer and project leader in several military radar projects. He returned to Chalmers University of Technology as a Researcher in the areas of microwave array antenna systems. Since 1981 he has been an Associate Professor in telecommunications. For part of 1992 he spent a period at University of Bochum, Bochum, Germany as a Guest Researcher. His current interest is in the application of computer-aided network methods, microwave circuit technology with emphasis on filters, matching, modeling, and lumped-element methods. Herbert Zirath (M’80–F’11) received the M.Sc. and Ph.D. degrees from Chalmers University of Technology, Göteborg, Sweden, in 1980 and 1986, respectively. Since 1996, he is a Professor in high speed electronics at the Department of Microtechnology and Nanoscience, MC2, Chalmers University of Technology. He became the Head of the Microwave Electronics Laboratory during 2001. At present he is leading a group of approximately 50 researchers in the area of high frequency semiconductor devices and circuits. His main research interests include foundry related MMIC designs for millimeterwave applications based on both III-V and silicon devices, SiC and GaN based transistors and circuits for high power applications, device modeling including noise and large-signal models for FET and bipolar devices, and InP-HEMT devices and circuits. He is working part-time at Ericsson AB as a microwave circuit expert. He is author/coauthor of more than 300 refereed journal/conference papers, and holds 4 patents. Dr. Zirath is an elected IEEE Fellow.

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MultiEXCELL: A New Rain Field Model for Propagation Applications Lorenzo Luini and Carlo Capsoni

Abstract—MultiEXCELL, a new rainfall model oriented to the analysis of radio propagation impairments which was developed on the basis of a comprehensive rain field database collected by the weather radar sited in Spino d’Adda (Italy), is presented. Single rain cells are modeled by an analytical exponential profile which best represents real single-peaked rain structures. The rain cells’ probability of occurrence is analytically derived from the local rainfall statistics. The spatial features of the rain field at midand large-scale are investigated through their natural aggregative process. The clusters (aggregates) of cells are studied in terms of distance between individual cells, number of cells per aggregate, and distance between aggregates. Finally, the fractional area covered by rain, which the rainfall spatial correlation strongly depends on, is derived from radar data through the comparison with the same quantity provided by global long-term numerical weather products. The MultiEXCELL procedure for the generation of spatially correlated synthetic rain fields is duly presented and the model’s accurateness is preliminary assessed against the available radar dataset. Although MultiEXCELL is mainly oriented to propagation-related applications, its cellular approach may reveal useful also in hydrology, for the prediction/management of water resources, and in meteorology, for the nowcasting of the temporal evolution of rain structures. Index Terms—Radio propagation, rainfall effects.

I. INTRODUCTION

I

N the next future wireless telecommunication systems will operate at high frequencies, typically in the Ka band (18 to 30 GHz) and, possibly, above (Q/V bands, approximately 40/50 GHz). This choice is prevalently dictated by the always increasing demand of bandwidth from the users and by the congestion of the lower frequency bands. These systems are expected to provide real-time multimedia services, and consequently, to be reliable and guarantee the desired system availability. As well known, radio links operating at such frequencies have to cope with strong attenuation phenomena due to the atmospheric constituents, among which rain certainly plays the prevailing role [1]. In this scenario, high signal fades can no longer be overcome by simply increasing the static power margin, but require the application of smart strategies, known as Propagation Impairment Mitigation Techniques (PIMTs) [2].

Manuscript received March 04, 2010; revised March 03, 2011; accepted April 23, 2011. Date of publication August 12, 2011; date of current version November 02, 2011. The authors are with the Dipartimento di Elettronica e Informazione, Politecnico di Milano, Piazza Leonardo da Vinci, 32, 20133, Milano, Italy (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2011.2164175

A wide class of PIMTs relies on the fact that fade-producing factors, markedly rain, are unevenly distributed in space. The use of multiple receiving stations located at proper distance permits to take advantage of the spatial variability of rain, so that a separation of a few tens of kilometers strongly reduces the probability that both stations simultaneously undergo an outage (site diversity technique) [3], [4]. Analogously, in satellite communications, where wide areas are of concern (i.e., Broadcasting or Multimedia services), the irregular spatial distribution of rain makes possible and convenient to selectively distribute the limited extra power available on board the satellite towards subregions where the need of power is greater owing to adverse weather conditions [5]. Accordingly, the design of a modern telecommunication system requires the evaluation of the advantages (in terms of outage probability) deriving from the implementation of one of these techniques. To this aim, experimental campaigns are of great usefulness [6], not only because they allow to directly evaluate the PIMT performance, but also because they provide the reference data against which other methods and/or prediction models should be validated [7]. As an alternative, rain fields derived by weather radars may be used for the simulation of those PIMTs that take advantage of the temporal and spatial variability of rain. Where available, radar data reflects the peculiarities of the local climatology and topography [8]. Unfortunately, reliable radar data are not diffusely available worldwide, which has fostered the development of models of the rain fields whose spatial and temporal properties resemble those of the actual rain fields. In the literature, several models have been proposed so far, though very few of them are specifically oriented to propagation applications. Meteorological models [9] provide the most physically based estimation of the rainfall process, but imply a complex implementation and extremely time consuming calculations. Stochastic models, such as [10], simulate complete rain fields which preserve the local rainfall statistics, are spatially correlated, evolve in time and are also subject to overall advection [11]. Despite these advantages, models of this kind generate a multiplicity of rain fields whose parameters are extracted from the same rain rate (lognormal) distribution and are characterized by the same rainfall spatial correlation structure. Fractal models, which exploit the scale-invariance and self-similarity properties of the rain fields, synthesize rain maps from few parameters (e.g., the Hurst exponent and the lacunarity of the field [12]), which, however, are hardly retrievable from local data. Statistical models [13], which rely on the combination of multiple random processes, each one governing a specific feature of the field (such as the time of arrival and the duration of

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LUINI AND CAPSONI: MULTIEXCELL: A NEW RAIN FIELD MODEL FOR PROPAGATION APPLICATIONS

the storms), are based on several parameters (11 in [13], as an example): they are flexible and globally applicable, but their calibration typically requires rain gauge time series or even radar images, i.e., databases not easily retrievable worldwide. Finally, cellular models generate rain fields by means of an ensemble of synthetic rain cells. The most advanced models pertaining to this class, EXCELL [14] and HYCELL [15], present the advantage of a simple calibration on a global basis: they both allow, in principle, to derive the rain cells’ probability of occurrence from the . The site specific 1-minute integrated rainfall statistics EXCELL model is particularly oriented to propagation applications, thanks to its simple (yet effective) mathematical formulation, but it is inherently a small-scale model, as it makes use of isolated rain cells. This drawback is overcome by HYCELL, which, instead, properly combines multiple cells to reproduce rain fields over wide areas (linear size of hundreds of kilometers), while maintaining the correct large-scale spatial correlation. However, the derivation of the rain cells’ probability of ocis not straightforward, mainly due currence from the local to the combined Gaussian-exponential rain cell profile: although marginally more accurate than the simple exponential one, this feature implies a much more complex analytical formulation. This work presents MultiEXCELL, a new propagation-oriented rainfall model that aims to merge the advantages of EXCELL and HYCELL: preserve (and enhance) the simple and manageable mathematical expressions of the former and generate complete rain fields like the latter. In fact, MultiEXCELL relies on a simple rain cell profile that reflects the small-scale km ) spatial distribution of rain (approximately up to at best, a characteristic that has been inherited from EXCELL. The modeling of the large-scale (hundreds of kilometers) spatial properties of the field is achieved by combining several rain cells on the map, such that their interdistance reproduces the one observed in real rain fields. The remainder of the paper is organized as follows. Section II firstly introduces the radar derived rain field database used for the development of MultiEXCELL. Afterwards, the modeling of the single rain cell is presented by selecting the analytical profile which best represents real single-peaked rain structures and, then, by defining the rain cells’ probability of occurrence through an analytical relationship that permits to adapt the population of synthetic cells to the local rainfall statistics. Section III deals with the spatial features of the rainfall process at midand large-scale derived from the natural rain cells’ aggregative process. The clusters (aggregates) of cells are studied in terms of distance between the individual cells, number of cells per aggregate and distance between aggregates. Finally, the fractional area covered by rain, which the rainfall spatial correlation strongly depends on, is derived from radar data and compared with the results obtainable from global long-term numerical weather products (namely, the ERA-40 database). The MultiEXCELL procedure for the generation of spatially correlated synthetic rain fields is duly described in Section IV. The model is preliminary validated in Section V against the available radar data by assessing its accuracy in reproducing both the first-order (rainfall cumulative distribution) and the second-order (rain cells’ distribution and rain rate spatial correlation) statistics. Finally, Section VI draws some conclusions.

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II. RAIN CELL MODELING The rain cell is the fundamental “brick” on which MultiEXCELL is founded. In the literature, the most commonly accepted definition of rain cell is “the continuous area inside which the ” [16] and this rain rate is higher than a given threshold definition is assumed throughout the work. Before describing the rain cell modeling activity, this section introduces the radar derived rainfall databases used to develop the MultiEXCELL model. A. Radar Derived Rainfall Databases The experimental station of Spino d’Adda, located a few kilometers East of Milan (latitude 45.4 N, longitude 9.5 E, altitude 84 m a.m.s.l), Italy, is equipped with an S-band Doppler weather radar, that allowed the collection of extensive databases of rain fields, whose main characteristics are detailed below. NPC Database: The NPC (Nastri Pioggia Cartesianizzata) database is the result of a long-term measurement campaign undertaken from 1988 to 1992, which led to collect several rain events, each of them including radar snapshots of the rain field in the Padana Valley. Each radar image is a pseudo-CAPPI (Constant Altitude Plane Position Indicator) at 1.5 km above the ground, resulting from the composition of three circular radar scans at elevation angles of 3 , 5 and 7 . The maximum operational range of the radar considered in constructing the NPC database is 40 km, in order to avoid the inclusion of clutter values originating from the numerous mountains surrounding Spino d’Adda. The minimum rain rate value has been conservatively set to 0.5 mm/h, although the system could allow the detection of much lower rain rates at 40 km of distance. mm m has been converted into The radar reflectivity rain rate (mm/h) using the well-established relationship (1) whose coefficients 200 and 1.6 have been derived assuming the Marshall-Palmer drop size distribution, proven to be representative of the type of precipitation occurring in the Padana Valley [17]. Finally, CAPPI data have been remapped from their origkm Cartesian grid, therefore inal polar format to a consisting of 160 160 pixels. Consecutive images have been collected approximately each 77 seconds, the time necessary to perform the three complete radar scans. In order to select reliable rainfall data useful for rain modeling, radar snapshots have been carefully checked and problematic images have been removed, i.e., those affected by anomalous propagation effects, presence of the bright band, possible interference coming from other radio sources. The resulting NPC radar database consists of 21958 radar pictures (81 rain events collected in different months of the year and in different years), for a total of about 465 hours of observation with rain, which, as shown in [18], are fully representative of the rainfall process occurring in the Padana Valley. LR Database: In order to extend the limited coverage area of the NPC database, and, therefore, to investigate also the largescale spatial distribution of the rain field, the Spino d’Adda weather radar has been operated also at lower elevation angles

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(1.5 , 2.5 and 4.5 ), which allowed to obtain pseudo-CAPPI images with 150-km radius (hereinafter these rainfall maps will be named “Long-Range” (LR) maps). However, this goal can be achieved at the expense of a reduced quality of the snapshots, which are affected by clutter that reduces the observing area and sometimes corrupts the rain patterns. The resulting LR database consists of 3437 maps of ground rain collected during several rain events occurred in the years 1998, 1999, 2000 and 2006. As for the NPC database, the pixel km , whereas the dimension of the Cartesian grid is temporal resolution is approximately 15 minutes. B. Optimum Rain Cell Profile The rain cell modeling implies the choice of an analytical profile that describes in a simple way the spatial distribution of the rain intensity inside the single cell. On one side, the most adequate profile should preserve at best the main characteristics of the real rain cell (such as its average rain rate and its area), rather than mimic exactly its shape. On the other side, it should be mathematically manageable so as to make it applicable in practice: for instance, it is necessary to easily derive the rain cells’ probability of occurrence from local meteorological information. To this aim, the following simple rain cell profile, already employed in the EXCELL model [14], is adopted here:

specifically, the area were selected, as in [14]

and the mean rain rate (3)

(4) In (4), indicates the incomplete gamma function. The left side of (3) and (4) is calculated numerically from radar data, whereas the expressions on the right side are valid for synthetic cells whose profile is defined by (2). The most suitable profile was chosen by comparing additional descriptors of measured and synthetic rain cells, namely, the peak rain value , the root mean square of the as measured by the radar and the cell dynamic , respectively rain intensity defined as (5) where cell center and

indicates the maximum distance from the km (area of the radar pixel)

(2) In (2), (mm/h) is the rain rate, (km) is the distance from (mm/h) is located, the cell center, where the peak rain rate . whereas (km) is the equivalent radius for which Actually, (2) identifies multiple profiles, depending on , the , and respectively define shape factor: Gaussian, exponential and hyper-exponential cell profiles. First, the model parameterization has been revised, considering a much larger and accurate database now available. The 21958 NPC radar images have been processed to isolate rain from 1 to 10 mm/h. Afcells at different rain thresholds terwards, cells have been classified as corrupted (C) or uncorrupted (UC), depending on whether they are affected by (even one pixel of) clutter and/or their perimeter touches the limits of the observation area: in both cases, the incomplete observation of the rain cell would lead to a possibly biased modeling and, consequently, corrupted rain cells have been discarded, as well as those cells whose dimension cannot be resolved by the radar. is 40 km, since the 3 dB When the distance from the radar , the minimum resolvable cell size is beam width is 2 dB km. As a consequence, the minimum dB km km km . The rain cell area was set to mm/h is the lowest threshold analysis has shown that value for which cells are mostly single peaked and, therefore, for which analytical models as the ones proposed in (2) appear adequate. The number of UC cells is 20379. , and A synthetic rain cell is completely identified by . For any given cell, after selecting , 2 or 3, and are determined by imposing two conditions. We chose to involve integral quantities, as they are definitely more reliable and meaningful than point values if derived by radar images;

(6) (7) Again, (5), (6) and (7) report the expressions of each quantity as it is calculated from radar data (left side) and for the synrather than is considered in thetic cells (right side). the test because the latter is associated to a null area: in fact, as compenalready noticed in [14], “overestimated” values of sate for the spatial filtering on the km grid element performed by the radar. The spatial filtering has a negligible effect on Gaussian cells, whereas it strongly affects the peak rain rate both of exponential and hyper-exponential cells. For the testing, the following error figure is selected: (8) where and indicate the value of the descriptor under test, respectively calculated from the synthetic and from the measured rain cells. Table I lists the average (E) and root mean square (RMS) , and values of the error figure relative to , respectively named , and : all the 20379 uncorrupted rain cells at 5 mm/h have been considered. Results clearly indicate that the exponential profile reproduces at best the three selected descriptors. Nevertheless, results also suggest that a shape factor between 1 and 2, i.e., a mixed Gaussian-exponential synthetic cell, would probably allow an

LUINI AND CAPSONI: MULTIEXCELL: A NEW RAIN FIELD MODEL FOR PROPAGATION APPLICATIONS

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TABLE I MEAN (E) AND ROOT MEAN SQUARE (RMS) VALUES (%) OF " ," AND " , RELATIVE TO THE THREE RAIN CELL PROFILES (20379 UNCORRUPTED CELLS)

even more accurate modeling. This solution would be in some respect similar to the one proposed for the HYCELL model [15]. Unfortunately, the choice of the optimum noninteger value would strongly increase the mathematical complexity of the model and prevent the straightforward analytical derivation of the site dependent rain cells’ probability of occurrence (as shown later in Section II-C). As a further motivation for the choice of a purely exponential profile, the work in [19] deserves to be cited. The authors analyzed precipitation fields from the TOGA-COARE and GATE experiments and found that, on the average, the exponential profile is the most adequate to model rain cells in tropical climates as well. C. Rain Cells’ Probability of Occurrence The MultiEXCELL model assumes that the exponential profile in (2) has a global validity, as also suggested by [14] and [19]: what changes from site to site is the rain cells’ probability of occurrence, so that, for instance, in tropical areas, convective cells prevail over stratiform ones, whereas in temperate sites, their relative occurrence is more balanced. This section presents the derivation of a closed-form analytical expression that allows to easily adapt the synthetic rain cells’ probability of occurrence, , to the local input rainfall statistics. As a preliminary step towards this aim, and values extracted from the NPC database are used to com, . pute the distribution of conditioned to The probability density function (PDF) of is illustrated in Fig. 1, based on which the lognormal law appears to be an adequate fitting distribution. However, such data are inevitably truncated due to the minimum cell size resolvable by the radar, which (in devising an analytical model) forces to propose a probability distribution for , assumed to be valid also where data are not available. The general expression of the lognormal probability density function (PDF) is (9) where and are the mean and standard deviation of the variable’s logarithm, that is normally distributed. The Maximum Likelihood Estimation (MLE) of the distribution parameters provides the PDF plotted in Fig. 1 (solid line), for which and . The choice of the lognormal distribution to fit the PDF of implies that the number of rain cells reduces to zero when the cell’s dimension tends to zero. This conclusion actually differs

Fig. 1. Distribution of  and maximum likelihood estimation (MLE) lognormal PDF.

was modfrom the findings presented in [14], where eled by an exponential distribution: this different result is mainly ascribable to the reduced number of rain cells (6215) collected at that time and to the limited radar data sensitivity [14], which, in turn, permitted to identify only rain cells associated to values greater than 1 km (refer to Fig. 6 in [14]), i.e., to only partially observe the full distribution of . In the literature, several authors have investigated the distri, defined as bution of the isosuperficial rain cell diameter the diameter of the equivalent circular cell with area equal to that of the actual one. Studies published in the past, such as [20] and [21], carried out on the basis of ground radar data collected in different parts of the world, have shown that the distribution follows the exponential law (a key assumption function of of the HYCELL model, as well [15]) (10) where is measured in km, is the threshold in mm/h, and are the two parameters of the distribution. It whereas is worth pointing out that the results shown in Fig. 1 are not in contradiction with those findings, as, in fact, the diameter of the , depends not only on synthetic cells at a given threshold, , but also on , through the following expression (for an exponential cell): (11) As a result, whilst follows a lognormal distribution, for cells at 5 mm/h actually turns out to be exponentially distributed in accordance with the findings reported in the literature. conditioned to , , has The distribution of . Specifically, eight been calculated for different classes of (ranging from 5 to 250 mm/h) have classes on log scale for been chosen as a good compromise between the number of cells per class and a sufficiently fine quantization. Afterwards, several fitting distributions have been tested (exponential, gamma, inverse Gaussian, Rayleigh, Weibull, etc ) and the lognormal provided the most accurate modeling of , whichever class. The modeling accuracy of the MLE lognormal the

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Fig. 2. Comparison between P (

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j

R

) and

P ( jR

), for the most (left side) and the least (right side) numerous classes.

Fig. 3. Trend of (R ) and  (R ) as a function of the average R

value of each class.

distribution has been quantified by defining the following estimation error: (12) and indicate the values of the where MLE lognormal cumulative distribution function (CDF) and of the empirical CDF, respectively. Fig. 2 shows the comparison and , on the left for the most between numerous class (8205 cells) and on the right for the least populated one (231 cells). For all classes, the RMS of is between 11.5% and 20%. Fig. 3 shows the trend of the lognormal parameters and as a function of the average value of each class. As is clear from the figure, both trends can be accurately fitted by the following expressions (dashed lines): (13) (14) which, together with (9), give the complete description of the conditioned to . The key instatistical distribution of formation delivered by Fig. 3, as well as by (13) and (14), is that low intensity cells tend to be of larger extent mm/h with respect to cells carrying higher rain mm/h , which reasonably rates

reflects the rainfall characteristics in stratiform and convective processes, respectively. A key feature of any rainfall cellular model should be the ability to reproduce the local meteorological environment: this step is essential to derive a practical working tool. Accordingly, the following relationship expresses the superposition and links the of the cells’ contributions to generate the , to the local rain cells’ probability of occurrence, rainfall statistic (for clarity’s sake, although is idenkm km mm/h ) tified as a probability, its dimensions are

mm/h (15) where is the area of the and , for which the rain intensity cell with parameters is exceeded, km and km are respectively the minimum and the maximum value of deduced from the radar observations. The imposition of bounds and in (15) is not necessary from an analytical point of view, but, as a matter of fact, this constraint is what allows to practically apply MultiEXCELL: if no bounds were imposed, also rain cells with extremely small or extremely large equivalent radius (necessarily discarded in synthesizing realistic rain maps) would be required . to exactly match the input

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The probability of occurrence consists of two terms, one exand the other pressing the distribution of conditioned to , i.e., one relative to the distribution of (16) Therefore, the integration can be split into two parts, acand cording to the two integration variables

(17) The whole analytical procedure for the derivation of the final form of is omitted here for brevity (all the details can be found in Appendix A): at this stage, it is sufficient to as a point out that the inversion of (15) to obtain first requires the closed-form solution of the function of inner integral within square parentheses, which can be achieved follows the lognormal distribution. In this case, when assumes the final expression

(18) Fig. 4. Rain cells’ probability of occurrence, calculated through (18), relative to Spino d’Adda (45.4 N, 9.5 E) on the top and to Miami (25.65 N, 80:43 E) on the bottom.

0

where and (19) and

is the error function (20)

, are given by (13) and (14). whereas For the practical application of (18), following the approach of several rainfall models such as [11], [14] and [15], the input is fitted by an analytical model. In this work, the following expression is used [22]: (21) where , , and are regression coefficients that assure and (21). the best agreement between the input Fig. 4 shows the rain cells’ probability of occurrence calculated for Spino d’Adda (top) and for Miami (bottom) by means s predicted by recommendaof (18), using as input the tion ITU-R P.837-5 for the two sites (Spino d’Adda:

N and E; Miami: N and E). These results provide a hint of how the rain cells’ probability of occurrence varies depending on the local input : whilst in Spino d’Adda different types of cell have a comparable probability of occurrence, in Miami, intense and spatially limited precipitation clearly prevails, as it is typically the case for tropical convective-like climates. III. RAIN FIELD MODELING In light of the features presented so far, MultiEXCELL is inherently a small-scale model that reproduces the rainfall process by means of a population of isolated rain cells. Thanks to its analytical formulation, the model is particularly suitable and efficient for investigating the interaction between a radio link and precipitations in various scenarios [23]. However, when mid/large-scale applications are concerned (from tens to some hundreds of kilometers), such as the simulation of site/time diversity systems and of terrestrial networks, the evaluation of nongeostationary satellite links performance (which involves also long paths at low elevation angles) and the estimation of the radio interference due to hydrometeor scattering, the probability that more than one cell is simultaneously interacting with

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Fig. 5. Typical aggregative process of rain cells.

the link (system) is high. As a result, the isolated-cell model’s applicability to such scenarios becomes more and more critic with the increase of the system’s dimension and it is necessary to further extend the model. Weather radar data are a unique source of information of the rainfall spatial characteristics as they provide consecutive snapshots of the rain field and a simple visual inspection of rain maps points out that rain cells are not randomly distributed in space, but rather, they tend to aggregate to form larger structures [15], [24]. This behavior is evident in Fig. 5 that shows a typical rain field observed by the radar located at Spino d’Adda: the inner ellipse on the left identifies a single rain cell, the two small circles in the bottom right corner indicate two of the cells pertaining to the same aggregate, and, finally, the large ellipses delimit two aggregates. It appears from the figure that rain is organized on three spatial scales: the small-scale (up to approximately 20 km) which involves single rain cells; the mid-scale (roughly from 20 km to 50 km) that is related to how cells cluster together to form an aggregate, and finally, the large-scale (from 50 km to about 300 km) which is related to the mutual position of the aggregates on the map. The next subsections analyze the spatial distribution of precipitation on the radar maps. Specifically, the distance between rain cells belonging to the same aggregate (hereinafter referred to as intercellular distance or ICD) is investigated, as well as the distance existing between aggregates (hereinafter referred to as interaggregate distance or IAD). Finally, statistics about the number of rain cells within the same aggregate (hereinafter named daughters) are derived and studied. A. Analysis of the Spatial Characteristics of Rainfall Intercellular Distance: Throughout this work, an aggregate is considered as a rainy area thresholded at 1 mm/h with more than one peak rain rate exceeding 5 mm/h inside. It is worth noticing that this definition of aggregate is different from the one given for HYCELL in [15], where two structures at 1 mm/h are considered as separate aggregates if the distance between

Fig. 6. Definition of the ICD (top) and probability density function of the NNICD, derived both from the NPC and the LR databases (bottom).

their centers of mass exceeds a given value . The statistical characterization of the ICD has been given here in terms of the nearest neighbor ICD (NNICD) and the NPC dataset was employed, first. In fact, the calculation of the ICD involves the rain cell’s center of mass which depends on the rain rate values inside the cell itself and, therefore, the data quality (NPC database) has to be privileged. In the calculation of the NNICD, the reduced coverage of the NPC database is not expected to represent a limitation, since only the distance between each cell and its closest neighbor is considered. Nevertheless, in order to confirm the results and to assess the impact of the limited NPC observation area on the NNICD, in a second stage, also the LR dataset was employed. The ICD between two rain cells is defined as follows: (22) where , represent the position of the center of mass of cell and is the radius of the equivalent circular cell with equal area and center of mass. The top side of Fig. 6 shows the graphical definition of the ICD: the outer line indicates the aggregate at 1 mm/h, and the irregular shapes depicted in grey are two daughter cells, whose equivalent circular cells are indicated by the two surrounding circles. The definition of the ICD according to (22) is adequate and consistent with the rain cell modeling (refer to Section II). The use of an effective distance instead of the more customary real distance between cells allows preserving the isolation between the circular synthetic rain cells, and . whatever their dimension, i.e., whatever The probability density functions of the NNICD, based on 2474 and 2559 values for the NPC and the LR database, respectively, is presented in Fig. 6 (bottom side). Both distributions (they present negligible differences) show that nearest neighbor cells tend to be quite close one to the other (the peak of the PDF

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is for km) and also that the effective distance seldom exceeds 10 km (the maximum being approximately 37 km). It is worth noticing from the figure that also negative values of were found: they represent cells that are strongly anisotropic and tend to elongate orthogonally to the line connecting the centers of mass. In this case, in fact, the real distance between the cells is shorter than the sum of the two effective radii. Interaggregate Distance: Due to the large dimension of aggregates, only LR maps are suitable for the investigation of the IAD. As for the ICD, the quantity of interest is the nearest neighbor IAD (NNIAD), but, in this case, the real distance between the centers of mass is used (23) where, again, , identify the position of the center of mass of cell . In the calculation of , all cells, also the ones that were assumed to be only marginally corrupted at a visual inspection, were taken into account, the rationale being: 1) aggregates at 1 mm/h are more likely affected by clutter than single cells, both because they result from the clustering of cells at 5 mm/h and because the area covered by rain at 1 mm/h is much more extended than the one covered at 5 mm/h; 2) LR maps are more affected by clutter with respect to NPC maps, and the exclusion of all corrupted aggregates would greatly reduce the available structures; 3) due to their large dimensions, when considering aggregates at 1 mm/h, the presence of some clutter pixels has a reduced effect on the average parameters describing the aggregate. The probability density function of the NNIAD, based on 4366 values, is reported in Fig. 7 (top side), which shows that most of the NNIAD values lie within 20 and 120 km, being the most probable ones around 30 km. The minimum and the maxare 6 and 177 km, respectively. imum measured Number of Cells per Aggregate: The last information necessary to complete the description of the rain structures is the number of daughter cells (5 mm/h) per aggregate (1 mm/h). Again, LR maps were taken into account. The number of was calculated in each aggregate used to daughter cells derive the IAD. The PDF of the number of daughter cells is plotted in Fig. 7 (bottom side) and shows an exponential behavior: most of the aggregates consist of two daughters and very few of them have more than 25. Fractional Rainy Area: The generation of realistic rain fields requires the knowledge of the fraction of the area affected by rain. To this aim, the MultiEXCELL model relies on a method, originally presented in [25], that allows to derive the fractional rainy area, , from Numerical Weather Prediction (NWP) data usually provided worldwide on a regular latitude/longitude grid (24) km are respectively the area of the where and , map and the one affected by rain rates exceeding (mm) is the amount of rain accumulated during the period (hours), i.e., the temporal resolution of the available NWP data, (mm/h) is the yearly mean value of the rain intensity, mm/h and, is the value of the conditioned to local calculated for .

Fig. 7. Probability density function of the NNIAD (top side) and probability density function of the number of daughter cells (5 mm/h) per aggregate (1 mm/h) (bottom side).

For the practical application of (24) to derive worldwide, the ERA-40 database provided by the ECMWF (European Center for Medium-Range Weather Forecast) is a suitable choice: in fact, notwithstanding its limited spatial and temporal resolulat/lon grid, hours), ERA-40 tions (regular data represent the only statistically stable database (approximately 40 years) available nowadays to retrieve from NWP on a global basis. As for the calculation of in (24), recommendation ITU-R P.837-5 [26] can be employed, which, although with a certain degree of approximation, allows to obtain worldwide (and, hence, ). As clearly stated in [25], (24) is valid under the assumption that the rain rate statistical distribution is substantially invariant within . Consequently, the coarse temporal resolution of the ERA-40 database is the limiting aspect in the application of (24) (based on the quasi-ergodicity property of rain fields, is fairly independent of the observation area [27]). In fact, if only a portion of the 6-hour period is actually affected by rain, since values are calculated as the average accumulated ERA-40 rain, the value derived through (24) will underestimate the actual fractional area coverage. In order to obtain a more realistic distribution of , the expression in (24) becomes (25) is a correction factor that compensates for the avwhere actually affected by rain. The optimum erage portion of has been found by observing the CDFs of value of depicted in Fig. 8: the solid line refers to the fractional rainy

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A. Number of Rain Cells

Fig. 8. Distribution of  derived for Spino d’Adda from the ERA-40 database, according both to (24) and (25), compared with the distribution obtained from the NPC database.

area obtained from the NPC database described in Section II, the dashed curve is relative to the application of (24) to 10 years ERA-40 data for Spino D’Adda (years 1992–2001) and, of finally, the dotted line concerns the application of (25) to the allows to same ERA-40 data used for (24). The factor adjust ERA-40 data so as to better take into account, on the avactually affected by rain. erage, the portion of For the practical application of (25) to the MultiEXCELL is assumed to be globally valid: although it may model, have a climate dependency, its evaluation requires long-term radar data collected in several sites worldwide, unfortunately not currently available to the authors. IV. GENERATION OF SYNTHETIC RAIN FIELDS The quasi-ergodicity of the rainfall process is a widely accepted and established concept [25], [28]–[30], which, in fact, several rainfall models rely on, such as EXCELL [14], HYCELL [15], those proposed by Gremont-Filip [10], by Jeannin [11] and by Callaghan-Vilar [12]. This property implies that there is a negligible difference between the statistical results obtained from a set of radar derived rain maps or a single raingauge measured by sampled over time: as a consequence, the a raingauge at a given site permits to determine the rain cells’ probability of occurrence (first order statistic), as implicitly assumed in the derivation of (18). As for the spatial correlation of rain (second order statistics), such feature is tightly linked to the rain cells’ aggregation and spatial distribution: the generation of the rain fields through MultiEXCELL, described in detail in the next section, preserves the correct NNICD among the cells belonging to the same aggregate, whose dimension is determined according to the distribution of the number of daughter cells per aggregate (mid-scale spatial correlation of the field). The large-scale spatial component of the field is instead regulated by the radar derived NNIAD distribution. Finally, the fractional rainy area, whose distribution can be obtained from the ERA-40 database through (25) is preserved as well for the generation of realistic rain fields.

, calculated through (18), defines the approThe priate proportion among the various types of synthetic rain cells that must be reproduced. In order to generate a finite number of , it is synthetic rain maps, yet representative of the local necessary to determine also a finite number of cells to be used. could be normalized to its minimum In principle, so that the number of cells to be emvalue turns out to be ployed for each class . It follows that the total number of rain cells is , depends on , and, in through (18). Unfortunately, in most turn, on the local sites, assumes an extremely high value. As an example, for mm/h Spino d’Adda, for which km , ! However, the role of the less probable rain cells can be neglected without modeling accuracy. In other any significant loss in the to be words, it is possible to determine a proper threshold , such that, on one side, the modeling imposed on is reduced to accuracy is maintained and, on the other side, a manageable value. Thus, becomes (26) values lower than are set to zero where all discarded. and the associated rain cells has to be determined considering the following convalue; 2) generate a straints: 1) obtain a manageable sufficient number of rain maps to achieve statistical meaningfulness; 3) preserve the local rainfall statistics with a good degree of accuracy. As a rule of thumb, based on the authors’ between 15000 and 20000 (roughly correexperience, around ) is a good choice as it allow to sponding to fulfill all the abovementioned constraints: 300 to 500 maps are negligibly differs from the generated and the associated input rainfall statistics. B. Synthetic Rain Fields This section describes in detail the procedure according to which the MultiEXCELL model generates a set of synthetic rain and the rainfall fields preserving both the overall local spatial distribution. The procedure is based on a double aggregative process similar to the one devised in [15] and reflecting the physical characteristics of real rain fields observable in radar data: first, cells are clustered into aggregates and afterwards aggregates are combined to produce a complete synthetic rain map. Specifically, MultiEXCELL is founded on the following points: 1) Calculate the rain cells’ probability of occurrence from the local rainfall statistics using (18). values for each class of 2) Calculate the using . A total of 20 intervals both for (linear subdivision between 0.5 km and 14 km) and (logarithmic subdivision between 5 mm/h and mm/h) are selected, thus providing 400 different kinds

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is chosen so as to obtain cells. for 3) Derive the distribution of the fractional rainy area the site of interest through (25). According to the rain threshold of the NPC maps used in this study, the minimum rain rate of the generated synthetic maps is set to 0.5 mm/h). The lateral mm/h (therefore, in (25), dimension of the square synthetic map is chosen such that it maintains the area of the ERA-40 pixel containing km. Fithe site of interest: for Spino d’Adda, km , nally, the spatial resolution of the map is set to which, based on authors’ experience, is deemed to be a good compromise between the modeling accuracy (which clearly improves as the pixel dimension decreases) and the required machine resources (both memory and computation time obviously increase when the rain map resolution gets finer). 4) For each new rain map , determine the associated rainy area by randomly extracting a value from the distribution of the fractional rainy area obtained at step 3. The rain map is developed iteratively by adding sequentially (as specified in step 5) one new rain cell until the total fractional rainy reaches . As a result, the number of cells on area the map will depend also on . 5) For each map, the first aggregate of cells is defined by , randomly extracting the number of daughter cells, from the radar derived correspondent distribution shown in Fig. 7. The daughter cells are chosen uniformly from , whose dimension the set of available cells decreases by a unity each time a new cell is selected. The first cell is placed at a random point on the map, whereas each successive cell will be arranged such that its effective distance from the previous one equals a value randomly extracted from the NNICD distribution shown in of the new cell’s center Fig. 6. The coordinates of mass are, therefore, calculated as

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of cell. The value of

(27) where define the position of the previous cell’s and indicate the effective center of mass, whereas radius of the previous and of the new rain cell (at 5 mm/h), respectively; , the angle between the -axis and the line connecting the cells’ centers of mass, is extracted from a uniform distribution between 0 and . The construction of the aggregate is stopped either if its number of daughter or if reaches . After its complecells equals tion, the aggregate is randomly placed on the rain map. , a new aggregate is formed according to 6) If the procedure outlined in step 5: its position on the map is determined such that the distance between the centers of mass of the new and the previous aggregates (at 1 mm/h) randomly extracted from the NNIAD equals a value distribution shown in Fig. 7. The arrangement of the cells belonging to the same aggregate (step 5) deserves some additional comments. According to

Fig. 9. Redistribution of the overlapping rain rates around the structure.

Fig. 10. Example of a synthetic rain map generated by the MultiEXCELL model for Spino d’Adda (234 234 km ).

2

Fig. 11. Relative error between the P (R) derived from for Spino d’Adda radar data and the one calculated from synthetic rain fields.

how the NNICD distribution was derived, rain cells at 5 mm/h should not overlap. In some cases, however, during the generation of the synthetic rain field, cells at 5 mm/h overlap each to the other due to either negative values extracted from the NNICD assumes high values (both situations, distribution or when however, are quite rare, as clearly indicated by the NNICD and distributions). In this case, however, it is not possible to sum up the rain intensity values associated to the same area because, in this would not be properly reproduced anymore way, the local is computed under the assumption of isolated rain ( cells). The solution here adopted for this problem consists in redistributing around the structure of the aggregated cells all the overlapping rain rate values smaller than , as shown in Fig. 9,

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Fig. 12. Comparison between LR maps and synthetic maps (100-km radius area): PDFs of the nearest neighbor intercellular distance and overall intercellular distance (5 mm/h).

such that the overall trend of the rain profile still decreases exponentially as moving away from the center of the structure, but more slowly than the original rain cells’ profile. Although this solution may appear quite complicated, it can be performed numerically with a small computational load and, most importantly, it allows not only to preserve the original total ), but rainy area of the aggregate (in other words, the local also its average rain rate. Moreover, as a byproduct, this solution provides the shape of the synthetic rain structures with a more irregular and realistic trend (if compared to the purely exponential one): in fact, the core of a real rain cell tends to be more uniform than the surrounding debris. This is clearly visible in Fig. 10, which shows a typical synthetic rain map generated by the MultiEXCELL model: several clusters of rain cells can be identified. V. PRELIMINARY VALIDATION OF THE MODEL This section deals with the preliminary validation of MultiEXCELL against the Spino d’Adda NPC and LR radar databases. The synthetic rain maps generated by the model are analyzed in order to assess their ability in reproducing the local and the rainfall spatial correlation that characterizes real rain fields. Fig. 11 shows the relative error , as a function of rain rate, between the derived from NPC radar data (input to the model) and the one calculated from the MultiEXCELL synthetic maps, i.e., 420 rain images. The solid line refers to the original km , whilst the dashed area of the generated maps one was obtained by limiting the synthetic maps to a 40-km radius, i.e., the same dimension of the NPC maps. The figure indicates not only that, as expected, the synthetic maps correctly reproduce the input rainfall statistics, but also that they show the quasi-ergodicity property typical of real rain fields: the overall derived from radar rain maps is fairly independent of the observation area, as long as areas approximately larger than km and smaller than km are considered [27]. The effectiveness of the procedure that arranges rain cells on the map can be evaluated by comparing the intercellular distance distributions derived from the radar maps and from the synthetic maps. Specifically, cells at 5 mm/h have been isolated

and the distances between their centers of mass have been calculated (by applying (23)). The analysis produced the PDFs both of the nearest neighbor distance, NND (regardless of the fact that the cells belong to the same aggregate), and of the distance between each cell and all the others on the same map (named overall distance, OD). The former is mainly linked to the smalland mid-scale spatial correlation of rain, whereas the latter also concerns the large-scale component of the field. Obviously such distributions, especially the latter, depend on the observation area so that for the comparison with LR maps, the synthetic ones have been reduced to an area of 100-km radius (Fig. 12) and, analogously, to an area of 40-km radius for the comparison with NPC maps (Fig. 13). The analysis of the NND distribution, considering both figures, shows that, on average, synthetic rain cells tend to aggregate as real cells do: the most probable distance between cells at 5 mm/h is approximately 10 km. Concerning the OD distribution, the agreement between the PDFs is satisfactory for the observation area characterized by a 40-km radius, and acceptable for the larger one (100-km radius). Overall, the results confirm the effectiveness of the cell double aggregative process: the MultiEXCELL model clearly proves to arrange rain cells in a realistic way that resembles the rain cells’ organization observable in real rain fields. Several rainfall models, such as those proposed by Gremont and Filip [10] and by Jeannin [11], generate rain fields whose spatial correlation is defined a priori and tends to be reproduced in each of the simulated maps. This implies that has approximately the same trend in each map, which is obviously unrealistic. In fact, convective phenomena are usually characterized by very intense precipitations and by a steep decrease of with distance, whereas, on the contrary, stratiform events are often associated with rain rates of limited intensity and a wider spatial correlation. Differently from the abovementioned rain models, MultiEXCELL presents the advantage of generating rain maps whose spatial correlation is dependent on the types of cells that lie on the map, and, therefore, realistically differs from map to map. As s and the trend with distance an example, Fig. 14 shows the of the spatial correlation coefficient for two synthetic rain fields generated at Spino d’Adda, which can be roughly considered

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Fig. 13. Comparison between NPC maps and synthetic maps (40-km radius area): PDFs of the nearest neighbor intercellular distance and overall intercellular distance (5 mm/h).

Fig. 14. P (R)s (left side) and the trend with distance of the spatial correlation coefficient (right side) for two synthetic rain fields roughly classified as stratiform and convective on basis of the associated rain intensities.

respectively as stratiform and convective on the basis of the associated rain intensities. In accordance with real rain fields, the MultiEXCELL model produces rain maps whose spatial correlation characteristics are different, which allows a realistic simulation of the interaction between a telecommunication system and the rain fields. At this stage, the validation of MultiEXCELL was performed only against Spino d’Adda radar data and, indeed, some of the model’s design quantities, namely the NNICD, the NNIAD, the correction number of daughter cells per aggregate and the factor were obtained from local radar data. Although additional experimental data would be required to verify the global validity of MultiEXCELL, results reported in [31] considering several NEXRAD radar sites seem to indicate that the intercellular distance distribution shows quite a good stability across the different climates in the USA (cold, temperate and subtropical regimes) and, moreover, preliminary investigations of the authors (not reported here for brevity) using a similar dataset appear to confirm such a conclusion relatively to all the abovementioned design quantities. Based on such considerations, the MultiEXCELL model was tentatively applied to the prediction of various propagation quantities in different radio communication scenarios: preliminary results reported in [32] and [33], respectively regarding the estimation of the intersystem radio interference due to hydrometeor scattering and the evaluation of

the rain attenuation experienced by terrestrial light-of-sight microwave links, indicate a good agreement between the model’s predictions and the experimental measurements gathered by the ITU-R into its global DBSG3 database: such results represent an a posteriori hint of the global validity of MultiEXCELL. VI. CONCLUSION This paper presents MultiEXCELL, a new propagation-oriented rainfall model based on a cellular representation of rain. MultiEXCELL represents the local rainfall process by means of a multiplicity of synthetic isolated cells (associated with the proper probability of occurrence), and, in addition, simulates the natural aggregative process of rain cells while reproducing the rainfall spatial distribution. MultiEXCELL was inspired by EXCELL, of which it retains some basic characteristics: its applicability only requires the knowledge of the local rainfall statistics as input and its formulation consists of simple and manageable mathematical expressions. The MultiEXCELL model was developed on the basis of two extensive rainfall databases collected by the weather radar of Spino d’Adda. Real rain cells have been extracted from radar images and the analysis of their characteristics has shown that an exponential profile, on average, is the most suitable analytical model for the representation of real single-peaked rain structures. In fact, exponential synthetic cells, simply identified by

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the peak rain rate and the equivalent radius , have proven to estimate at best some rain cell descriptors, such as the peak , the root mean square of the rain intensity rain value and the cell dynamic . The synthetic rain cell diameter follows an exponential distribution, in accordance with the results reported in the literature conditioned to , by several authors. The distribution of , was found to be lognormal and the associated mean and standard deviation values are accurately fitted by simple analytical power laws. As a result, an analytical expression that allows to adapt the population of synthetic rain cells to local data has been devised. The observation of real rain fields allowed to investigate also the mid- and large-scale spatial properties of precipitation. Key information has been extracted from the radar databases about the distance between cells belonging to the same aggregates (ICD, intercellular distance), the number of daughter cells per , and the distance between aggregates (IAD, aggregate interaggregate distance), which, together with , the fractional area covered by rain, tightly linked to the rainfall spatial correlation, allow to generate synthetic rain maps as large as km . The procedure for the generation of the rain maps has been duly described. Each rain field consists of multiple aggregates whose interdistance reflects the one observed in real rain maps (IAD). The number of cells in each aggregate is regulated by distribution, whereas their interdistance comes from the the radar derived intercellular distribution (ICD). As a result of the simulation procedure, the MultiEXCELL model generates a set of rain fields (approximately 300 to 500, depending on the local rain cells’ probability of occurrence) that reproduce both the local rainfall statistics provided as input and a realistic largescale rainfall correlation structure. Specific validations of the model were carried out against the radar data collected at Spino d’Adda: the generated rain fields were found to correctly reproduce the overall local input and the rain cells’ spatial distribution proved to be very similar to the one observed in real rain fields. It is worth noticing that, differently from most of the existing rainfall models, MultiEXCELL presents the advantage of generating rain maps whose and spatial correlation depend on the type of cells that lie on the map, and, therefore, realistically differ from map to map, as it happens for stratiform and convective events. The model has been specifically developed for propagation related applications: preliminary tests against the propagation measurements gathered by the ITU-R into its global DBSG3 database indicate that MultiEXCELL is of great usefulness for the simulation of long microwave terrestrial links (and consequently of extended microwave terrestrial networks), for the evaluation of site diversity systems with large distance between the stations and for the analysis of the radio interference due to hydrometeor scattering. Moreover, the MultiEXCELL cellular approach for rain field modeling may reveal useful also for other applications, such as hydrology, for the prediction/management of water resources and the evaluation of the risks originating from extremely intense precipitation events [34], and meteorology, for rain events nowcasting [35].

APPENDIX A The rain cells’ probability of occurrence can be analytically derived from the input local rainfall statistics through (18). This section provides details about the derivation of such a mathematical expression. The first step concerns the closed-form solution of the inner integral in (17) (28) To this aim, the following substitution is applied: (29) Thus,

becomes

(30)

After adding and subtracting (30) can be written as

, the exponent

in

(31) As a result,

assumes the following form: (32)

By applying the following substitution: (33) we obtain

(34)

LUINI AND CAPSONI: MULTIEXCELL: A NEW RAIN FIELD MODEL FOR PROPAGATION APPLICATIONS

Considering the definition of the error function is the final expression of

in (20),

(35) Remembering that and using (35), (17) takes the form

(36) Exploiting the following mathematical identity: (37)

where obtain

and as a function of

, (36) can be inverted to

(38) As the last step, the calculation of the rain cells’ probability leads to of occurrence as the desired expression in (18). REFERENCES [1] L. J. Ippolito, “Propagation effects handbook for satellite systems design,” NASA Ref. Pub., vol. 1082, no. 4, 1989. [2] H. Fukuchi, P. A. Watson, and A. F. Ismail, “Proposal of novel attenuation mitigation technologies for future millimetre-wave satellite communications,” presented at the Millennium Conf. Antennas Prop., Davos, Switzerland, 2000. [3] J. Goldhirsh, B. H. Musiani, A. W. Dissanayake, and L. KuanTing, “Three-site space-diversity experiment at 20 GHz using ACTS in the Eastern United States,” Proc. IEEE, vol. 85, no. 6, pp. 970–980, Jun. 1997. [4] J. D. Kanellopoulos, T. D. Kritikos, and A. D. Panagopoulos, “Adjacent satellite interference effects on the outage performance of a dual polarized triple site diversity scheme,” IEEE Trans. Antennas Propagat., vol. 55, no. 7, pp. 2043–2055, Jul. 2007. [5] A. Paraboni, P. Gabellini, A. Martellucci, C. Capsoni, M. Buti, S. Bertorelli, N. Gatti, and P. Rinous, “Performance of a reconfigurable satellite antenna front-end as a countermeasure against tropospheric attenuation,” presented at the 13th Ka and Broadband Communications Conf., Torino, Italy, Sep. 2007. [6] Y. Karasawa and Y. Maekawa, “Ka band earth space propagation research in Japan,” Proc. IEEE, vol. 85, no. 6, pp. 821–842, Jun. 1997. [7] A. D. Panagopoulos, P.-D. M. Arapoglou, J. D. Kanellopoulos, and P. G. Cottis, “Long-term rain attenuation probability and site diversity gain prediction formulas,” IEEE Trans. Antennas Propagat., vol. 53, no. 7, pp. 2307–2313, Jul. 2005. [8] L. Luini, N. Jeannin, C. Capsoni, A. Paraboni, C. Riva, L. Castanet, and J. Lemorton, “Weather radar data for site diversity predictions and evaluation of the impact of rain field advection,” Int. J. Satellite Communication and Networking, 10.1002/sat.953. [9] [Online]. Available: http://www.ecmwf.int/Mar. 2010

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[10] B. C. Gremont and F. Filip, “Spatio-temporal rain attenuation model for application to fade mitigation techniques,” IEEE Trans. Antennas Propagat., vol. 52, no. 5, pp. 1245–1256, May 2004. [11] N. Jennin, L. Féral, H. Sauvageot, L. Castanet, J. Lemorton, and F. Lacoste, “Stochastic spatio-temporal modelling of rain attenuation for propagation studies,” presented at the EuCAP, Edinburgh, U.K., Nov. 11–16, 2007, EICC. [12] S. Callaghan and E. Vilar, “Fractal generation of rain fields: Synthetic realisation for radio communications systems,” IET Microw., Antennas, Propagat., vol. 1, no. 6, pp. 1204–1211, Dec. 2007. [13] C. Onof, E. Chandler, A. Kakou, P. Northrop, H. S. Wheater, and V. Isham, “Rainfall modeling using poisson-cluster processes: A review of developments,” Stochastic Environ. Res. Risk Assess., vol. 14, pp. 384–411, 2000. [14] C. Capsoni, F. Fedi, C. Magistroni, A. Paraboni, and A. Pawlina, “Data and theory for a new model of the horizontal structure of rain cells for propagation applications,” Radio Sci., vol. 22, no. 3, pp. 395–404, May–Jun. 1987. [15] L. Féral, H. Sauvageot, L. Castanet, J. Lemorton, F. Cornet, and K. Leconte, “Large-scale modeling of rain fields from a rain cell deterministic model,” Radio Sci., vol. 41, 2006. [16] G. Drufuca, “Radar-derived statistics on the structure of precipitation patterns,” J. Appl. Meteorol., vol. 16, pp. 1029–1035, 1977. [17] A. Pawlina and M. Binaghi, “Radar rain intensity fields at ground level: New parameters for propagation impairments prediction in temperate regions,” presented at the 7th Commission F, Triennal Symp., Ahmedabad, India, 1995, URSI. [18] C. Capsoni, M. D’Amico, and P. Locatelli, “Statistical properties of rain cells in the Padana Valley,” J. Atmos. Ocean. Technol., vol. 25, no. 12, Dec. 2008. [19] J. von Hardenberg, L. Ferraris, and A. Provenzale, “The shape of convective rain cells,” Geophys. Res. Lett., vol. 30, no. 24, 2003. [20] J. Goldhirsh and B. H. Musiani, “Dimension statistics of rain cell cores and associated rain rate isopleths derived from radar measurements in the mid-Atlantic coast of the United States,” IEEE Trans. Geosci. Remote Sens., vol. 30, no. 1, pp. 28–37, Jan. 1992. [21] H. Sauvageout, F. Mesnard, and R. S. Tenorio, “The relation between the area-average rain rate and the rain cell size distribution parameters,” J. Atmos. Sci., vol. 56, pp. 57–70. [22] C. Capsoni, L. Luini, A. Paraboni, and C. Riva, “Stratiform and convective discrimination deduced from local P (R),” IEEE Trans. Antennas Propagat., vol. 54, no. 11, pp. 3566–3569, Nov. 2006. [23] C. Capsoni, L. Luini, A. Paraboni, C. Riva, and A. Martellucci, “A new global prediction model of rain attenuation that separately accounts for stratiform and convective rain,” IEEE Trans. Antennas Propagat., vol. 57, no. 1, pp. 196–204, Jan. 2009. [24] P. Northrop, “A clustered spatial-temporal model of rainfall,” Proc. Math., Phys., Eng. Sci., vol. 454, no. 1975, pp. 1875–1888, Jul. 8, 1998. [25] E. A. B. Eltahir and R. L. Bras, “Estimation of the fractional coverage of rainfall in climate models,” J. Climate, vol. 6, pp. 639–644, 1993. [26] ITU-R Recommendation P.837-5, Characteristics of Precipitation for Propagation Modelling. Geneva, Switzerland, 2007. [27] C. Onof and H. Wheater, “Analysis of the spatial coverage of British rainfall fields,” J. Hydrol., vol. 176, pp. 97–113, 1996. [28] J. Goldhirsh, “Spatial variability of rain rate and slant path attenuation distributions at 28 GHz in the mid-Atlantic coast region of the United States,” IEEE Trans. Antennas Propagat., vol. 38, no. 10, pp. 1711–1716, Oct. 1990. [29] ESTEC/Contract n 17877/04/NL/JA, Recongurable Ka Band Antenna Front End for Active Rain Fade Compensation. [30] G. Drufuca, “Rain attenuation statistics for frequencies above 10-GHz from rain gauge observations,” J. Rech. Atmos., vol. 1–2, pp. 399–411, 1974. [31] N. Jeannin, L. Feral, H. Sauvageot, L. Castanet, and J. Lemorton, “Modeling of rain fields at large scale,” in Proc. Int. Workshop Satellite and Space Communications, Sep. 14–15, 2006, pp. 233–236, 2006. [32] C. Capsoni, L. Luini, and M. D’Amico, “The MultiEXCELL model for the prediction of the radio interference due to hydrometeor scattering,” presented at the EuCAP, Barcelona, Spain, Apr. 12–16, 2010. [33] L. Luini and C. Capsoni, “A physically based methodology for the evaluation of the rain attenuation on terrestrial radio links,” presented at the EuCAP, Barcelona, Spain, Apr. 12–16, 2010. [34] P. Willems, “A spatial rainfall generator for small spatial scales,” J. Hydrol., vol. 252, pp. 126–144, 2001. [35] M. Dixon and G. Weiner, “TITAN—Thunderstorm Identification, Tracking, Analysis, and Nowcasting—A radar-based methodology,” J. Atmos. Ocean. Technol., vol. 10, no. 6, pp. 785–797, Dec. 1993.

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Lorenzo Luini was born in Italy in 1979. He received the Laurea degree (cum laude) in telecommunication engineering and the Ph.D. degree in information technology (cum laude) from Politecnico di Milano, Milan, Italy, in 2004 and 2009, respectively. Since 2004, his research activities concern the radiowave propagation through the atmosphere, with a specific focus on rain field modeling for propagation applications. He has been involved in the COST 280 and COST IC0802 European projects and in the European Satellite Network of Excellence (SatNEx).

Carlo Capsoni graduated in electronic engineering from the Politecnico di Milano, Italy, in 1970. He joined the “Centro di Studi per le Telecomunicazioni Spaziali” (CSTS), research center of the Italian National Research Council (CNR), Politecnico di Milano, in 1970. In this position, he was in charge of the installation of the meteorological radar of the CNR sited at Spino d’Adda, and since then, he has been responsible for the radar activity. In 1979, he was actively involved in the satellite Sirio SHF propagation experiment (11–18 GHz) and later in the Olympus (12, 20, and 30 GHz) and Italsat (20, 40, and 50 GHz) satellites experiments. His scientific activity is mainly concerned with theoretical and experimental aspects of electromagnetic wave propagation at centimeter and millimeter wavelengths in the presence of atmospheric precipitation with particular emphasis on attenuation, wave depolarization, incoherent radiation, interference due to hydrometeor scatter, precipitation fade countermeasures, modeling of the radio channel, and design of advanced satellite communication systems. The scientific activity in the field of radar meteorology is mainly focused on rain cell modeling for propagation applications, the development of radar simulators, and the study of the precipitation microphysics. He is also active in free space optics theoretical and experimental activities. Since 1975, he has been teaching a course on aviation electronics at the Politecnico di Milano, where he became Full Professor of electromagnetics in 1986. Dr. Capsoni was a member of the ITU national group and was the Italian delegate for the COST projects of the European Economic Community related to propagation aspects of telecommunications (COST 205, 210). He is a member of the Italian Society of Electromagnetics (SIEm) and editor of the SIEm Magazine. He is also a member of the Coritel governing body.

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The Physical Basis of Atmospheric Depolarization in Slant Paths in the V Band: Theory, Italsat Experiment and Models Aldo Paraboni, Antonio Martellucci, Carlo Capsoni, and Carlo G. Riva

Abstract—After a review of the physical/mathematical definitions of anisotropy and canting angle, parameters currently used for describing the depolarization of electromagnetic waves due to hydrometeors, we present some specific algorithms to retrieve these parameters from measurements of the atmospheric polarization transfer matrix, based on data collected during the Italsat propagation experiment at 50 GHz in a slant path with elevation angle 37.7 . The data analysis, conducted after applying particular “ad hoc” procedures and severe editing, allowed to separate in many cases the ice- from the rain-effects, both in single events and statistically. We also present mathematical models derived from the observed behavior of the descriptive parameters, hopefully applicable also in different conditions. Index Terms—Depolarization, millimeter wave propagation, radiowave propagation, satellite communication.

I. INTRODUCTION

T

HE description of a dual polarization radiochannel using the so-called quasiphysical parameters, anisotropy and canting angle [1]–[13], offers significant advantages, compared to the classical description based exclusively on the depolarization ratio, because from it we can understand the physical mechanisms causing depolarization, extend available results to different experimental conditions (frequency, polarization, climatology) and, perhaps more importantly in telecommunication systems design, we can provide realistic radio channel modeling for simulation tools. The data, models and experimental techniques presented in this paper aim on one hand to improve the understanding of the effect of atmospheric hydrometeors (raindrops and ice crystals [4]) on depolarization in slant path radio links at V band and on the other hand are intended as the basis for the development Manuscript received November 18, 2010; revised April 28, 2011; accepted May 16, 2011. Date of publication August 12, 2011; date of current version November 02, 2011. This work was supported in part by the European Space Agency (ESA) in the framework of the ESA under Contract 17760/03/NL/JA “Characterisation and Modelling of propagation effects in 20–50 GHz band.” Sadly, A. Paraboni passed away on April 13th, 2011, during the review process of this paper. A. Paraboni (deceased) was with the Dipartimento di Elettronica e Informazione, Politecnico di Milano, Piazza L. da Vinci 32, 20133, Milano, Italy. C. Capsoni and C. G. Riva are with the Dipartimento di Elettronica e Informazione, Politecnico di Milano, Piazza L. da Vinci 32, 20133, Milano, Italy (e-mail: [email protected]). A. Martellucci is with the European Space Agency, ESTEC, TEC-EEP, Keplerlaan 1, PB 299, NL-2200 AG Noordwijk, The Netherlands (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2011.2164207

of a statistical model of the conditional distribution of atmospheric depolarization based on global radio-climatological parameters (like the rain attenuation), for the assessment of performances of dual polarization satellite communication systems [11], [14]–[17]. After reviewing in Section II the mathematical definitions of depolarization of an electromagnetic wave, in Section III we present a new methodology to retrieve the quasiphysical complex parameters anisotropy and canting angle from measurements of the atmospheric polarization transfer matrix, obtained using switched polarization signals [5], [10], [18]. In Section IV we apply this methodology to derive and analyze the quasiphysical parameters from a large set of experimental data at 50 GHz, collected at Spino d’Adda ground station in the frame of the Italsat satellite propagation experiment. We also discuss an extension of the definition of such parameters, to properly assess the effect of ice on atmospheric depolarization at V band. In Section V we present and discuss the statistical results of our analysis and propose an adaptation to V band of the current statistical relationship between XPD and rain. In Section VI we finally draw some conclusions. II. THEORETICAL BACKGROUND OF ATMOSPHERIC DEPOLARIZATION Any electric field vector describing a plane electromagnetic wave propagating along the -axis can be represented by a combination of linear and polarizations (1) where and are the scalar components (phasors) and and are the unit vectors along the , orthogonal axes on the plane perpendicular to the axis. In general, the relationship between the input and output field components propagating in a linear depolarizing medium (e.g., a path through the atmosphere containing non-spherical hydrometeors) can be expressed by means of the atmospheric polarization transfer matrix, (2) where is the ratio between the component received on the polarization when transmitting the polarization and the component transmitted in the polarization; the suffixes out and in stand for output and input waves respectively, as shown in Fig. 1. Of all the possible mathematical basis that can be adopted for representing the polarization state of an electromagnetic wave,

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In decibels (9)

III. REFERENCE MODELS OF THE ATMOSPHERIC DEPOLARIZATION Fig. 1. Diagram of the atmospheric polarization transfer matrix with linearly polarized waves.

the circular polarizations (right- and left-hand circular) are particularly useful because their use simplifies the solution to the inversion problems encountered in extracting the quasiphysical parameters from experimental data. Similarly to (2), the two orthogonal circular polarizations link output and input waves according to (3) The representation of any field with circular components is thus given by (4) where and are the scalar components ( and stand for right- and left-hand) and

At millimetre wavelengths, the polarization of an electromagnetic wave propagating through the atmosphere is affected by the nonspherical hydrometeors contained in precipitating clouds and in ice clouds. The depolarization modelling would require an accurate description of the microphysical properties of rain and ice which is difficult to attain on a global scale (in particular for the aloft particles found along slant paths to space), but some general properties of the particles dispersion can be assumed. In this Section we introduce a simplified (but often verified) reference model of the atmospheric depolarization. This model allows to estimate the characteristics of the depolarizing channel from experimental measurements through the quasiphysical parameters which are directly related to the physical properties of the medium and can be used to scale to different system configurations (frequency, elevation angle, type of polarization). We first derive, in Section A, the expression of the transfer matrix in circular polarization for a longitudinally homogeneous medium and show that it represents the basis for the retrieval of some quasiphysical parameters. Then, in Section B, we introduce the principal planes, an assumption that simplifies the modeling and has been verified by experimental measurements (see Section IV).

(5) A. The “Longitudinal Homogeneity Medium” Model are the circularly polarized unit vectors. Hence, the relationship between the scalar components in the two reference systems (1) and (2) is given by

(6) The elements of the matrix matrix by

are related to the elements of the

The Longitudinal Homogeneity Medium model assumes that the medium is composed of hydrometeors whose microphysical properties (distribution of diameters, shapes, orientation) do not vary along the propagation path [7], [9], [11]. This modeling can be easily generalized, although not done in this paper, to a non-homogeneous medium by means of a cascade of different layers (e.g., rain and ice) [1], [11]. Capsoni and Paraboni [1] demonstrated that, if the hydrometeors are rotationally symmetric (an assumption which is experimentally verified for rain drops and ice particles), the crosspolar elements of the polarization transfer matrix in linear polarization are equal for any particles size and rotational axes orientations (10)

(7) Other important parameters are the “complex depolarization ratios” defined, in the linear and circular polarization representations, by

(8)

In the following, the constraint (10) is applied by imposing that (11) where the and are the measured values, which, as shown below, are always very alike, so that the average (11) is meaningful. This assumption has also the advantage of mitigating the effect on measurements of the differences (in amplitude and

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Fig. 2. Reference system for a generic polarization ellipse.

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Fig. 3. Geometrical representation of the principal planes.

phase) of the copolar and crosspolar RF channels of the receiving system. In circular polarization the condition (11) results into

where the complex parameter , named complex canting angle, is given by (16) with (17)

(12) (18) With reference to the depolarization ratios (8), the matrix in (3) can be then reformulated as (13)

The real component gives the polarization angle, the imaginary component gives the axial ratio , according to (19)

whose eigenvectors, also called eigenpolarizations , , can be expressed, with the proper choice of a multiplicative constant, as

(14) where only the principal value of the roots, i.e., the one with non-negative real component, is considered. Notice that the eigenpolarizations, also defined as “characteristic polarizations” of the medium [19], propagate without being depolarized, but only differently attenuated and delayed and, in general, are elliptical. The eigenpolarizations can be represented by a) the angle formed by the major axis line with the axis (polarization angle in Fig. 2 or ellipse tilt angle); b) the axial ratio , i.e., the ratio between the minor and major axes of the ellipse (the sign of identifies the direction of rotation of the polarization, as shown in Fig. 2). To obtain these parameters it is necessary to renormalize the eigenpolarizations (14), so that the product of their components becomes unitary (in absolute value). This renormalization yields

(15)

The representation adopted for the polarization state implies an inherent ambiguity of in the determination of , which, however, has no practical consequence because identifies a not-oriented line. The propagation of the eigenvectors is described by the corresponding eigenvalues, which represent the propagation constants (20) where it is conventionally assumed that (21) This assumption allows to distinguish the less attenuated propagation constant (first eigenvector) from the most attenuated one (second eigenvector). As an example, in the case of the distribution of oblate raindrops of Fig. 3 with canting angle symmetrically distributed around the local vertical direction, and correspond to the local horizontal and vertical. axes As discussed in the analysis of single events, Section IV-B, this definition needs to be replaced at V band by the new “maximum depolarization effect” criterion, to account for ice depolarization. The eigenvalues (20) can be expressed in exponential form as

(22)

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where and are the path-integrated propagation constants, in excess to clear air conditions, given by

(23) It may worth reminding that the term gives the total attenuation expressed in Np, i.e., the natural logarithm of the ratio between the signal which would be present in absence of atmosphere and the actual one. Note that this attenuation is only due to rain and eventually ice melting particles. Similarly the term gives, in radians, the total excess phase delay due only to the propagation through the hydrometeors. For the purpose of deriving a general representation of eigenvectors, the following quantities are introduced (24)

As a result, under the hypothesis of a Longitudinally Homogeneous Medium, the depolarization process is determined only by two complex parameters: anisotropy, , and complex canting angle, . The complex depolarization ratios (8) and XPD (9) in circular polarization can be expressed as a function of and

(31) According to (31), and are not affected by and are different if is not zero (i.e., the eigenpolarizations are elliptical). With some algebra, we can express the transfer matrix in linear polarization as

(25) where it can be noted that the radicand in (25) equals the determinant of matrix (13). In exponential form we can write (24) as (32) (26) whose relative depolarization ratios are given by where the new complex parameter (27) is the differential of the complex propagation constants of the two eigenpolarizations (being and the differential attenuation and phase shift, respectively) and is called anisotropy, whereas the quantity (28) with (29) is the averaged path-integrated propagation constant (real component always expressed in Np and imaginary component in Rad). As discussed previously in this Section, any polarization can be described using the complex canting angle given by (16). Therefore the polarization transfer matrix in circular polarization can be expressed as a function of , and , as

(30) The matrix (30) represents the basis for calculating the and from depolarization meaquasiphysical parameters surements. is a multiplicative factor common to Note that as all elements of all elements of , the averaged path-integrated propagation constant, , does not affect the depolarization process, which is characterized by the complex depolarization ratios (8), as shown in (13).

(33) which shows how the depolarization of linear polarizations is highly sensitive to (and in particular to ). To conclude this subsection we underline four important mathematical properties (not demonstrated for brevity) a) In general, the two eigenpolarizations are given by two elliptical polarizations with the same axial ratio ( , determined by , in general not equal to zero) but with the major and minor axes of the ellipse interchanged (i.e., , refer to Fig. 2). their polarization angles differ by Therefore, the two eigenpolarizations are geometrically orthogonal, but in general they could be electrically nonorthogonal (i.e., the directions of rotation could be the same). This property should be taken into account when designing dual polarization systems with adaptive depolarization cancellation. b) The result that the XPDs of orthogonal circular polarizations can be different, see (31), provides a way for detecting the medium condition in which the eigenpolarizations are not linear and is not zero. Indeed this occurs [11] when the orientation of axes of the particles along the path exhibits a non-symmetrical statistical distribution. In slant paths this condition is associated with ice aloft, as frequently observed at V band during the Italsat experiment (see Section V). Therefore, ice can be detected along an Earth-to-Space radio link by comparing the amplitudes of the XPD of circular polarizations, hence reducing the experiment complexity compared to the case, implemented in the Italsat propagation campaign, which needs to estimate the full transfer matrix. Note that the radio link configuration, which determines the value of

PARABONI et al.: THE PHYSICAL BASIS OF ATMOSPHERIC DEPOLARIZATION IN SLANT PATHS IN THE V BAND

in (31), must provide enough sensitivity to the crosspolar signal. c) Both the anisotropy and the imaginary part of the canting reference angle are invariant to any rotation of the reference axes. On the contrary, in the case that the is rotated to align it with the local tilt angle of the reference horizontal polarization on the satellite then the real part of the canting angle rotates of the same angle but in the opposite direction without any change of the eigenpolarization. d) The averaged path-integrated propagation constant, defined in (28) is also invariant to any change of the polarization chosen as reference. B. The “Principal Planes” Model In the principal planes model, besides being longitudinally homogeneous, the medium is also assumed to be filled with hydrometeors whose axes of symmetry are all aligned along a common direction. As a result, the polarization transfer matrix is still characterized by the same properties of the longitudinally homogeneous medium model, see (10) and (12), with the addition of the following one (34) or equivalently (35) or also (36) This implies that the complex canting angle becomes real (37) In this case the two eigenpolarizations turn out to be linear (axial ratio equal to zero) and orthogonal, and the planes formed by the linear eigenpolarizations and the propagation direction are defined as “principal plane.s” This case is physically intuitive because one of the two principal planes contains the common axis of symmetry of the hydrometeors (see Fig. 3). The anisotropy can hence be calculated by using models of the microphysical and scattering properties of the hydrometeors [4]. Rain anisotropy is calculated for the case of oblate spheroidal equialigned raindrops in [9]. For ice clouds constituted by ice needles whose axes are all lying on an horizontal plane, some at 20, 40 and 50 GHz (equivalently expressed values of ) are reported in [11] for various degrees through of ice alignment of the needles. Actually, the assumption of a complete alignment of all axes of symmetry of the hydrometeors is too restrictive because it has not been always verified during measurements (see [5], for terrestrial links at Ka band, and [10] for spatial links at Ka and V bands). Nevertheless the Principal Planes model is also applicable to the general case of a symmetrical distribution of the axes of the hydrometeors along of circular polara common direction. In this case the in (31), the orientaizations becomes identical, being tion of the principal planes is given by the average direction of

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hydrometeors and the anisotropy of the medium decreases by a reduction factor proportional to the standard deviation of the hydrometers canting angles around the average direction [11]. When the principal planes exist, if we assume their orientation as reference coordinates - , then the transfer matrix in circular polarization becomes diagonal. of anisotropy on slant paths Considering that the modulus at V band is most of time lower than one, because of the microphysical properties of the hydrometeors, it follows that, by approximating in (31) the hyperbolic tangent with its argument, the XPD in circular polarization becomes as (38) whereas in linear polarization, by applying (33) with is given by (39) Equation (39) evidences also that the sensitivity of XPD of linear polarizations to the orientation of the principal planes: approaches for approaching 0 or (but in these conditions there is the maximum variation of XPD for is maxsmall fluctuations of ). On the other hand, . imum for equal to As the XPD level, for both linear and circular polarizations, of anisotropy (see (38) and is determined by the modulus (39)), it is then obvious how the same values of depolarization can be either due to rain or ice, with the difference, however, that the rain depolarization is dominated by the differential attenuation while the ice depolarization is dominated by differential phase. These conclusions suggest some general but opposite considerations for the design of either dual polarization communication systems in the Ka and Q/V bands, or of experimental radio links for the assessment of the atmospheric anisotropy. As for the design of dual polarization communication systems, circular polarization is always characterized by the worst performance in terms of crosspolar interference generated by the atmosphere. As for the experimental assessment of anisotropy (which results to be a key parameter for propagation modeling), a radio propagation experiments that adopts a single circular polarization, or alternatively a linear polarization with its local tilt angle , would permit to measure the value of with the equal to greatest sensitivity, see (38) and (39). IV. THE ITALSAT EXPERIMENT In Section IV-A we describe the main characteristics of the Italsat ground station at Spino d’Adda and the data pre-processing and filtering. Then we present some experimental measurements in terms of anisotropy, in Section IV-B, and of canting angle, in Section IV-C. A. The Italsat Propagation Campaign in Spino d’Adda The Italsat satellite transmitted at V band (49.5 GHz) two orthogonal linearly polarized signals, switched at the rate of

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TABLE I CHARACTERISTICS OF THE ITALSAT LINK AND OF THE GROUND STATION IN SPINO D’ADDA

Fig. 4. Time series of copolar CPA (top curve) and crosspolar XPD (bottom curve) signals of circular polarization at V band measured during the event of May 27 1993.

933 Hz [20], [21], using a similar approach already adopted in terrestrial [5] and in the COMSTAR [22], [23] and Olympus [24], [25] experiments. The Italsat ground station in Spino d’Adda, Italy [26], [27], measured, during each (approximate) 0.5 ms semiperiod, the output copolar and crosspolar signals. The detected samples were then time-averaged for obtaining one sample per second. Table I gives the main characteristics of the radiolink. Therefore it measured the polarization transfer matrix at 50 GHz during propagation campaign lasting many years. Numerous transfer matrix measurements were also performed in various European locations [13], [25]–[30] using the Olympus and Italsat satellites [31]–[33]. During the lifetime of Italsat we have recorded 131 events at Spino d’Adda and corrected these data as discussed in the following. A critical problem in atmospheric depolarization measurements is the removal of the depolarization bias due to the transmitting and receiving systems (i.e., satellite and ground antennas, and the respective radio front-ends). The need for this operation is evident by comparing the ground station antenna XPD (see Table I) with the level of rain XPD at V band, which is below 30 dB for most of the time. The bias can be removed by forcing the overall complex polarization transfer matrix (which includes transmitter and receiver transfer matrices) to be unitary just before and after the depolarization events [13], [24]. Hence, we remove the bias by multiplying the measured polarization transfer matrix by the matrix resulting from the linear temporal interpolation between the inverse of the matrices measured before and after the events. By choosing a right or left side matrix multiplication we can remove the effects of either the satellite or ground antenna. We adopt the latter choice, so that some spurious effects due to the satellite antenna cannot be completely corrected, but it can be, however, mitigated by imposing the conditions (10) for the longitudinally homogeneous model. We have found the results more than adequate for the modeling purposes.

Concerning the depolarization measurements, it must be noted that at V band the ratio between differential attenuation and attenuation of characteristic polarizations decreases with respect to the value at Ka band [4]; as a consequence for the same attenuation the signal level due to rain on the crosspolar channel is lower at V than at Ka band [11]. On the other hand, due to the increase of differential phase shift of ice crystals at shorter wavelengths, the signal level due to ice on the crosspolar channel increases significantly at V band without any concurrent attenuation on the copolar channel. Therefore at V band, even in presence of relatively deep rain fades there is a severe problems for detecting the crosspolar signal and in general depolarization due to ice is easier to detect that the rain depolarization. This can be understood by comparing the clear sky margin C/N of about 40 dB, given in Table I, with typical to of the copolar crosspolar levels ranging from level. It comes out that, for most of the time, even modest attenuation levels merge the crosspolar signal into a noise of comparable power. To avoid this inconvenience, the crosspolar signals must be additionally filtered so as to get an acceptable C/N ratio. To limit the reduced temporal resolution resulting from the reduced bandwidth, we have analyzed the data with adaptive filtering. When the crosspolar level was sufficiently high, the low-pass bandwidth was increased as a function of the actual XPD level (longer averaging time for smaller XPDs) so preserving a good time detail, and vice versa. To clarify these concepts, Fig. 4 shows the copolar attenuation (CPA) due to rain (upper line) and the XPD (lower line) recorded during a particular event in 1993. The polarization bias has been already removed, as is visible from the XPD range (from to compared to the ground station antenna XPD of the ). As evident, the crosspolar level is particularly order of noisy below with respect to the clear air. Fig. 5 shows the results of the adaptive filtering, based on XPD levels, for the same event of Fig. 4. The variations of the . XPD are now clearly visible even at about

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Fig. 5. Result of the adaptive XPD filtering applied the event of Fig. 4. The dashed curves identify the limits ( 60, 55, 50, 45, 40, 35, 30, 25 and 20 dB) of the XPD intervals of the bandwidth of the low-pass filters (from 0.01 to 1 Hz, in steps of 0.01 Hz).

0

0

0

0

0

0

0

0

0

Fig. 6. Time series of circular polarization CPA (top) and anisotropy in modulus (bottom) measured during an event on June 2, 1993. The vertical dotted line marks the separation between two different time intervals; the dashed horizontal line on the bottom figure represents the limits of the XPD intervals of Fig. 5 (from bottom to top: 60, 55, 50, 45, 40, 35, 30, 25 and 20 dB).

0

0 0 0 0 0 0 0 0

B. Anisotropy and the New Criterion of the Maximum Depolarization Effect Fig. 6 shows the time series of CPA and anisotropy (in modulus) recorded during an event on 2/6/1993. In this event the separation between rain- and ice-effects appears particularly evident. From the beginning of the event up to 2800 s, the excess attenuation reached a non-negligible value of 8 dB whereas the anisotropy remained fairly low (less than 0.05 Np in modulus ). This part of the corresponding to an XPD lower than event reveals that the attenuation was mainly caused by spherical (and thus non-depolarizing) hydrometers, as expected at low rain intensities or in non-precipitating liquid water clouds. The second part of the event, starting at 2800 s and lasting up to 5000 s, denotes the presence of relatively strong anisotropy that reaches a peak of approximately 0.25 Np (i.e., an XPD of about

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Fig. 7. Anisotropy in the complex plane for the same event of Fig. 6 (June 2, 1993).

Fig. 8. Scatterplot of CPA and XPD of circular polarization, during the same event of Figs. 6 and 7 (June 2, 1993).

) without any relevant attenuation. In this case the effect of non attenuating ice crystals is prevailing. The variation of anisotropy as a complex parameter can be better understood by observing the polar diagram of Fig. 7 which shows, in the complex plane, its real and imaginary parts ( and in Np and Rad, respectively) for the same event of Fig. 6. In the diagram, the points aligned on the imaginary axis refer to the second period (3000 to 5000 s, high anisotropy due to ice), whereas the ones off axis refer to the first period (0 to 2800 s, non depolarizing rainfall and/or clouds). Both variations are consistent with the theory that describes the scattering due to ice-needles and raindrops [11], [34]. Fig. 8 shows the scatterplot of CPA and XPD for the same event of Fig. 6. Now the different relationship between these two variables during the two different types of event (rain and ice) is clearly visible. Fig. 9 shows the time series of CPA and XPD of the event recorded the April 27, 1993. On the contrary to what observed in the previous example, attenuation and anisotropy appear to be fairly well correlated. This can be also seen in the scatterplot

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Fig. 9. Time series of CPA (top) of circular polarization and anisotropy in modulus (bottom) recorded during an event of April 27, 1993.

Fig. 11. Anisotropy in the complex plane during the same event of Figs. 9 and 10 (April 27, 1993). Dashed line refers to the new criterion for maximum depolarization.

Fig. 10. Scatter plot of CPA and XPD for the same event of Fig. 9 (April 27, 1993).

between CPA and XPD in Fig. 10, which shows, no significant depolarization with no attenuation. Fig. 11 shows the variation of the real and imaginary parts of anisotropy in the complex plane. This pattern is consistent with the theoretical prediction of rain anisotropy. As an example, Fig. 12 shows the real and imaginary part of rain anisotropy calculated at 50 GHz using a raindrop-size and Gamma distribution [35] function with exponent average depending on the rain rate plotted in abscissa. Fig. 12 evidences that the imaginary part becomes negative (as also observed in Fig. 11, solid straight line) for rain rate values higher than a relatively low threshold of 7 mm/h (which in Spino d’Adda is exceeded for about 0.2% of the annual time [36]). These examples show that the real and imaginary part of anisotropy can be either correlated or uncorrelated, depending on the prevailing sources of depolarization (large raindrops and/or oriented ice-needles). Yet, the analysis of many events has revealed the frequent occurrence of situations where the sign of the real and imaginary part of anisotropy are opposite.

Fig. 12. Theoretically predicted real (black dots curve) and imaginary parts (empty boxes curve) of anisotropy per km due to rain at 50 GHz (note the change of sign of imaginary anisotropy evidenced by symbols).

6

On the other hand, the definition given in (20) and (21) (which implies that the differential attenuation is always positive, Positive Differential Attenuation, PDA, criteria) gives rise to points always located in the right side of the horizontal axis of Fig. 13. However if the depolarization is due only to ice, we cannot adopt the criteria currently used to distinguish between the first and the second principal plane because the differential attenuation is almost zero. In these cases the sign of anisotropy undergoes frequent jumps caused only by noise. To avoid those experimentally observed inconsistencies, it appears appropriate to extend the initial definition of the reference plane for anisotropy (i.e., the “second principal plane” as the more attenuated one, see (20) and Fig. 3), by attributing its role either to the differential attenuation or to the differential

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Fig. 13. Change of the anisotropy when the imaginary part is negative and prevails, in modulus, on the real part by applying the MDE criterion; the PDA criterion discussed in Section III requires instead that Re(1) 0 in any case.



phase, whichever produces the maximum depolarization (criterion of the Maximum Depolarization Effect, MDE), i.e., the greater component in modulus. According to the MDE criterion, the new value of anisotropy is derived from the original one (in is always non-negative, see (20)–(27)), by using which when when

,

Fig. 14. Anisotropy and canting angle for the event of June 2, 1993 (dashed vertical line marks the separation between two different time intervals).

(40)

Fig. 13 shows, in schematic form, the changes introduced by the MDE criterion with respect to the PDA one. Some samples resulting from its application are also marked in Fig. 11 where, as shown, the range of the anisotropy argument changes from to . Samples closer to these two in the second and fourth quadrant limits (the solid line at of Fig. 13) indicate situations in which the two effects have similar magnitude but opposite sign; if this line is crossed then the first and the second principal plane exchange and the canting . The change of canting angle undergoes a discontinuity of angle must not be interpreted as a real jump of the orientation of the hydrometeors but as a situation in which the effects of raindrops and ice are in contrast, with the occasional predominance of one population of particles on the other, as discussed in [11]. In conclusion, with the MDE criterion the role of the first and second principal plane is changed whenever the differential phase prevails in modulus and is opposite to the differential attenuation. The presence of still possible jumps of the canting angle is now due to almost only physical causes rather than to noise, in contrast to the previous case. C. Event Based Analysis of the Canting Angle In the following figures the fixed bias on the canting angles due to the local tilt angle of the polarization transmitted by Italsat (see Table I) is removed, so that the canting angle 0 and correspond to the local horizontal and vertical axes. Fig. 14 shows the time series of anisotropy and canting angle, for the first of the events considered above (see from Figs. 6–8). at convenience Here we subtracted or added an angle of (without altering the orientation of the principal planes) for preserving continuity of the time series and allowing to identify physical patterns in the measurements. In Fig. 14 during the first period, 1000 to 2800 s, the canting angle evolves around or, equivalently, around zero, i.e., the average symmetry axis of the rain drops is vertical. During the

Fig. 15. Anisotropy and canting angle for the event of April 27, 1993.

second period (2800 to 4200 s) the canting angle is still zero, a value which agrees with a model that assumes randomly oriented ice needles in the horizontal plane [11]. After 4200 s, we jumps due, perhaps, to the change in the can observe some role of the first and second principal planes, as discussed in the previous Section. As for the second event previously analyzed (see from Figs. 9–11), we can observe that the canting angle, reported in Fig. 15, is close to (which is equivalent to zero), like the previous case, and a bit lower than (about vertical) for short time intervals after 20000 s. Various explanations are possible to justify this peculiar behavior: one is the presence of hydrometeors composed by melting ice, as it can happen in the brightband layer above rainfall. Another one is the presence of ice needles whose axes orientation is forced to be vertical by the static electric fields existing within surrounding clouds. Moreover we could assume the presence of large vibrating raindrops, whose contribution to attenuation may prevail when, with the oscillation, the raindrops assume the shape of prolate ellipsoids, a phenomenon observed even in terrestrial measurements of rain depolarization [5].

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Fig. 16. Polar histograms of the argument of anisotropy in the complex plane conditioned to three classes of XPD: argument the radius is proportional to the number density.

040, 030 and 020 dB. For any value of the

Fig. 17. Histogram of the canting angles; reference system rotated by the local tilt angle of about 20 deg (see Table I).

V. STATISTICAL ANALYSIS OF V BAND DEPOLARIZATION IN SPINO D’ADDA In this section we present and discuss some statistical results from the Italsat data described in Section IV-A, in terms of: a) Polar histogram of the complex anisotropy; b) Polar histogram of the real part of the canting angle; c) Histogram of the imaginary part of the canting angle; d) Statistics of XPD conditioned to attenuation. A. Polar Histograms of Anisotropy at 50 GHz Fig. 16 contains three polar histograms of the argument of the anisotropy, , conditioned to the following increasing depolarization intervals: , which corresponds to an XPD in circular a) (about 50000 samples); polarization of , which corresponds to an XPD in circular b) polarization of (about 20000 samples); , which corresponds to an XPD in circular c) (about 5000 samples). polarization of The clustering of the measured anisotropy argument around 90 in Fig. 16(a) and (b) confirms that ice effects are more frequent for low to medium anisotropy whereas, only in the case of strong anisotropy of Fig. 16(c), the samples with pronounced rain effects are approximately as many as the ones characterized by ice effects. B. Polar Histogram of the Real Part of Canting Angles Fig. 17 shows a conditional histogram of the canting angle subjected to the same conditions indicated in Fig. 16 ( , and ). Note that the Italsat data used in this section and in the next have been collected during two sub-periods separated by a long

gap of five years necessary to fix a malfunction occurred in the front-end of the ground station. When the acquisition was resumed, in the second period, the angular stabilization of the satellite was significantly compromised, leading to some instability that prevented from using canting angle data for the statistical analysis. The useful period was then limited to the first sub-period and then reduced to about 25%. We can observe that the canting angle assumes values clustered around the 0 and 90 (horizontal and vertical) with a bilateral spread around 10 for low to medium anisotropy and lower for high anisotropy. The average orientation of the hydrometeors seems then independent of anisotropy. The same cannot be said, however, for the spread around the average, which tends to decrease as anisotropy increases. These findings confirm that at larger rainfall rates the large hydrometeors present in the precipitation tend to be dynamically more stable, a behavior which was theoretically discussed in [37] and already verified in terrestrial [5] and slant paths [10] experiments. These considerations allow a physical modeling of the canting angle distribution [4], [37], [38]. C. The Imaginary Part of the Canting Angle The statistics of the imaginary part of the canting angle, which is related to the axial ratio of the generic elliptically polarized eigenpolarization, refer to (19), are plotted in Fig. 18, for the same conditions of Figs. 16 and 17 ( , and ). The analysis of this parameter shows that the value correin the circular depolarization unbalance, sponding to a refer to (31), is exceeded very frequently for low anisotropy, whereas the dispersion of the imaginary part of the canting angle around zero is much reduced for higher anisotropy, as

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Fig. 18. Statistics of the absolute value of the imaginary part of the canting angle.

Fig. 20. Cumulative distribution function of XPD in circular polarization conditioned to various CPA intervals (1–2 dB for solid line; 5–6 dB for dot line, 9–10 dB for dash line and 13–14 dB for dot-dash line), measured at V band in Spino d’Adda, in a Gaussian integral chart.

Fig. 19. Scatterplot (dots) between CPA and XPD in circular polarization measured in Spino d’Adda at V band: solid central line is the conditional mean of anisotropy, dashed lower and upper curve indicates the 10 and 90 percentile. The solid line with circles is ITU-R Rec. 618–10 model.

Fig. 18(c) shows. This confirms, once more, that the Principal Planes model for strong depolarization is valid. D. Statistical Prediction of Anisotropy XPD in Circular Polarization at V Band As evidenced in Section IV-A by the event-based analysis of, V band slant path measurements, the atmospheric depolarization is weakly correlated with rain attenuation, in particular at low attenuation levels, because of the presence of ice. Therefore, the statistical prediction of anisotropy is relatively difficult, if the only available input parameter is the rain attenuation. On the other hand, in communication applications, the rain attenuation CPA (also defined as excess attenuation over the clear air signal level) is usually the only available parameter (both experimentally and theoretically) for system design. Therefore it can be useful to discuss the general properties of a statistical model that links XPD and CPA for system applications at V band. Fig. 19 shows the scatterplot of XPD in circular polarization, , and CPA, measured at V band (gray dots). It also shows: the conditional mean value of measured XPD (central solid curve), the 10 and 90 percentiles of the measured conditioned distribution (dashed lower and upper curves) and the prediction of the current ITU-R model [40] (solid line

Fig. 21. Top graph: mean value of XPD in circular polarization conditioned to CPA, measured at V band in Spino d’Adda (circles) and from model (41) (solid line). Bottom graph: standard deviation of XPD in circular polarization conditioned to CPA, measured at V band in Spino d’Adda.

with circles, the ITU XPD model has been extended from 1% to 10% of time to account for increased rain attenuation at V band). Fig. 20 shows cumulative distribution functions of conditioned to various CPA intervals in a Gaussian chart. We see that the distributions can be well approximated by a Gaussian model, and thus defined only by the mean value, , and standard deviation . The dependence of (circles in the top graph) and the measured (bottom graph) on the CPA is shown in Fig. 21. It clearly , tends to increase with CPA. The two appears that variables can be described by the following linear relationship (solid line of top graph of Fig. 21), obtained by best fitting: (41) The relative error (averaged in the CPA range between 0 and 30 dB) between the model (41) and measurements is about 3.5%, with a standard deviation of about 7%.

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The standard deviation is relatively stable in the range 5 to 7 dB, across almost the whole range of measured attenuation (from 0 to 30 dB). This model (41) resulting from V band measurements deviates from the current XPD-CPA relationship widely adopted so far [28], [30], [37], [39]–[45] (42) where the and parameters depend on many variables such as frequency and the link elevation angle. We recall that (42) applies only to rain depolarization, and in the current ITU model [40], that is shown for comparison in Fig. 19 (solid line with circles), the effect of ice is modeled by an empirical parameter derived from measurements at lower frequencies (mostly C and Ku band) [46]. As shown in Fig. 19, it results that the ITU-R XPD prediction agrees with the measured mean values only for excess attenuation higher than 5 dB (i.e., in presence of relevant rain depolarization, which is also an indirect confirmation of the accuracy of the technique used to remove system depolarization bias, described in Section IV), but at lower attenuation values the ITU-R model tends to underestimate severely the effect of ice depolarization. The differences between what observed at V band and the “classical” model (42) is that at V band we have to consider the effect of the strong anisotropy due to ice. On the other hand excess attenuation at 50 GHz is remarkably greater than at lower frequency, even for clouds or light rains. Therefore the ice-induced XPD events, which at lower frequencies are normally concentrated in the very low attenuation range, at V band tend to occur also at greater attenuation and to be distributed rather uniformly in the attenuation range. This is clear when moving from single event analysis (see Fig. 8 as an example) to the statistical analysis of XPD represented in Fig. 19, where the clear separation between occurrence of ice and rain in a single event tend to disappear and to generate a rather uniform distribution of samples in the XPD-attenuation plane. VI. CONCLUSION We have generalized the concepts of the quasiphysical parameters, anisotropy and canting angle, and already known in the literature, and we have adapted them to better describe the full depolarization matrix obtained from the Italsat experimental data collected at 50 GHz in Spino d’Adda, Italy. We have processed the data for removing, as much as possible, the effect of the antennas depolarization and mitigate the effects of the critical signal-to-noise ratio in depolarization measurements. In the data analysis, we have introduced a new criterion, based on the Maximum Depolarization Effect, useful to identify unambiguously the principal planes, also when ice depolarization prevails on rain depolarization, which is frequent at V band. The analysis based on single events has shown that, due to the concurrent presence of ice and its relevant effect at V band, depolarization is not always fully correlated with rain attenuation. A statistical analysis of the quasiphysical parameters has confirmed the importance of ice in inducing depolarization at V

band: almost all the events with very strong anisotropy (and consequently depolarization) show that ice was the dominant factor. in circular polarization with XPD values of the order of little or no attenuation were measured relatively frequently. The real part of the canting angle tends to cluster around the vertical and horizontal directions for a model that assumes the existence of the principal planes. This means that these planes are near horizontal and vertical, with a spread of about 10 around these directions; the spread tends to decrease as anisotropy increases, while the average value tends to remain constant. The statistics of the imaginary part of the canting angle, which is zero in presence of the principal planes, has confirmed the applicability of the polarization channel model based on the existence of the principal planes in most cases of strong depolarization. To assess the probability of occurrence of anisotropy, we have proposed a model that assumes, as input variable, the rain attenuation. The presence of ice reduces significantly the correlation between rain attenuation and depolarization, but on the other hand there is practical interest on this relationship, being distribution of rain attenuation the only information widely available for system design. The analysis of XPD experimental statistics of Spino d’Adda, revealed that at V band the distributions of XPD conditioned to attenuation are nearly Gaussian with standard deviation independent of attenuation and the average value slowly increasing with it. ACKNOWLEDGMENT The authors would like to acknowledge the Agenzia Spaziale Italiana (A.S.I.) that supported Italsat propagation experiment and CSTS-CNR for data pre-processing. REFERENCES [1] C. Capsoni and A. Paraboni, “Depolarization of an electromagnetic wave travelling through a stratified aerosol of non spherical scatterers,” in Proc. AGARD-CP-107, Gausdal, Norway, Sep. 18–21, 1972, pp. 2.1–2.16. [2] C. Capsoni, D. Maggiori, E. Matricciani, and A. Paraboni, “Rain anisotropy prediction: Theory and experiment,” Radio Sci., vol. 16, no. 5, pp. 909–916, 1981. [3] R. L. Olsen, “Theory for measuring the effective polarization parameters of rain from orthogonal linearly-polarized transmissions,” Ann. Télécommunic., vol. 36, no. 7–8, pp. 471–476, 1981. [4] T. Oguchi, “Electromagnetic wave propagation and scattering in rain and other hydrometeors,” Proc. IEEE, vol. 71, no. 9, pp. 1029–1079, 1983. [5] A. Aresu, A. Martellucci, and A. Paraboni, “Experimental assessment of rain anisotropy and canting angle in horizontal path at 30 GHz,” IEEE Trans. Antennas Propag., vol. 41, no. 9, pp. 1331–1335, Sep. 1993. [6] A. Paraboni, M. Mauri, and A. Martellucci, “The physical basis of depolarisation,” in Proc. Olympus Utilization Conf., Sevilla, Spain, Apr. 20–22, 1993, pp. 573–581. [7] A. Paraboni and A. Martellucci, “Transfer characteristics of a dual polarisation radio channel from 10 to 50 GHz using quasi physical parameters: Theory and experimental results,” presented at the URSI General Assembly, Kyoto, Japan, Aug. 1993. [8] B. R. Arbesser-Rastburg and A. Paraboni, “European research on Ka-band slant-path propagation,” Proc. IEEE, vol. 85, pp. 843–852, Jun. 1997. [9] A. Martellucci and A. Paraboni, “The physical basis of depolarization due to hydrometeors, models and experimental results,” in Proc. XXVth General Assembly of the Int. Union of Radio Sci., Lille, France, Aug. –Sep. 28–5, 1996.

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[10] A. Martellucci, A. Paraboni, and M. Filipponi, “Measurements and modeling of rain and ice depolarization on spatial links in Ka and V frequency bands,” presented at the AP2000 Millennium Conf., Davos, Switzerland, Apr. 2000. [11] , A. Martellucci, Ed., “Rain and ice depolarisation,” in COST 255 Final Rep. Amsterdam, The Netherlands: ESA Publication Division, 2002, ch. 2.4. [12] Radiowave Propagation Information for Predictions for Earth-toSpace Path Communications. New York: ITU-R Handbook, ch. 9. [13] C. Capsoni, A. Paraboni, F. Fedi, and D. Maggiori, “A model-oriented approach to measure rain induced cross-polarisation,” Ann. Telecommun., vol. 36, pp. 154–159, Jan.–Feb. 1981. [14] M. Mauri and A. Paraboni, “Depolarization measurements and their use in the determination of dual polarization links performance,” Alta Frequen., vol. LVI, no. 1–2, pp. 47–55, Jan.–Apr. 1987. [15] A. D. Panagopoulos, P.-D. M. Arapoglou, and P. G. Cottis, “Satellite communications at Ku, Ka, and V bands: Propagation impairments and mitigation techniques,” IEEE Commun. Surv. Tutor., no. 3rd quarter, pp. 2–14, Oct. 2004. [16] J. D. Kanellopoulos and A. D. Panagopoulos, “Ice crystals and raindrop canting angle affecting the performance of a satellite system suffering from differential rain attenuation and cross-polarization,” Radio Sci., vol. 36, no. 5, pp. 927–940, 2001. [17] A. D. Panagopoulos and G. E. Chatzarakis, “Outage performance of dual polarized fixed wireless access channels in heavy rain climatic regions,” J. Electromagn. Waves Appl., vol. 21, no. 3, pp. 283–297, 2007. [18] A. Paraboni, C. Oestges, and A. Martellucci, “Experimental assessment of atmospheric depolarization at Ka and V band based on Olympus and Italsat propagation campaigns,” presented at the EuCAP 2006, Nice, France, Nov. 6–10, 2006. [19] P. Beckmann, The Depolarization of Electromagnetic Waves. Boulder, CO: Golden Press, 1968. [20] F. Fedi, A. Paraboni, A. Martinelli, and A. Vincenti, “The Italsat program: The propagation experiment,” Rivista Tecnica Selenia. Italsat Special Issue, vol. III, no. 4, pp. 40–59, 1990. [21] A. Sbardellati and O. Alberti, “The 20, 40 and 50 GHz propagation beacon,” Rivista Tecnica Selenia. Italsat Special Issue, vol. III, no. 4, pp. 181–188, 1990. [22] D. C. Cox, “An overview of the bell laboratories 19 and 28 GHz COMSTAR beacon propagation experiments,” Bell Syst. Tech. J., vol. 57, pp. 1231–1255, May-Jun. 1978. [23] D. C. Cox and H. W. Arnold, “Results from the 19- and 28-GHz COMSTAR satellite propagation experiments at Crawford Hill,” Proc. IEEE, vol. 70, no. 5, pp. 458–488, May 1982. [24] , F. Dintelmann, Ed., “OPEX: Reference book on depolarisation,” in ESA WPP-83. Noordwijk, The Netherlands: , 1994. [25] G. Brussaard, “The analysis of depolarization and anisotropy using the Olympus beacon,” in Proc. Olympus Utilization Conf., Sevilla, Spain, Apr. 20–22, 1993, pp. 561–565. [26] P. Vita, “Italsat ground based antenna systems,” Rivista Tecnica Selenia. Italsat Special Issue, vol. III, no. 4, pp. 313–323, 1990. [27] A. Clementi, G. Di Sanza, F. Fiorica, A. Florii, G. Olivieri, and A. Paraboni, “The design and the implementation of the Italsat propagation stations,” Rivista Tecnica Selenia. Italsat Special Issue, vol. III, no. 4, pp. 381–397, 1990. [28] R. Jacoby and F. Rücker, “Three years of crosspolar measurements at 12.5, 20 and 30 GHz with the Olympus satellite,” in Proc. Olympus Utilization Conf., Sevilla, Spain, Apr. 20–22, 1993, pp. 567–572. [29] F. Murr, “Evaluation and analysis of anisotropy, canting angle and copular unbalance in circular polarization,” in Proc. Olympus Utilization Conf., Sevilla, Spain, Apr. 20–22, 1993, pp. 583–587. [30] M. M. J. L. Van de Kamp, “Depolarization due to rain: The XPD-CPA relation,” Int. J. Satellite Commun., vol. 19, no. 3, pp. 285–301, 2001. [31] A. Paraboni, F. Barbaliscia, C. Riva, A. Martellucci, and A. Pawlina, “Results of the Italsat propagation measurements campaign at 18.7, 39.6 and 49.5 GHz,” presented at the AP2000 Millennium Conf., Davos, Switzerland, Apr. 2000. [32] A. Martellucci, F. Barbaliscia, and A. Aresu, “Measurements of attenuation and crosspolar discrimination performed using the Italsat propagation beacons,” presented at the 3rd Ka band Utilization Conf., Sorrento, Italy, Sep. 15–18, 1997. [33] A. Paraboni, C. Capsoni, and C. Riva, “Joint attenuation-depolarization statistics measured at 18.7 GHz in the Spino d’Adda station with the Italsat-F2 satellite,” in Proc. 12th Ka and Broadband Commun. Conf., Napoli, Italy, Sep. 27–29, 2006, pp. 397–402.

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[34] A. Martellucci, J. P. V. Baptista, and G. Blarzino, “Effects of ice on slant path earth-space radio communication links,” presented at the URSI General Assembly, Maastricht, The Netherlands, 2002. [35] C. W. Ulbrich, “Natural variations in the analytical form of the raindrop size distribution,” J. Climate Appl. Meteor., vol. 22, pp. 1764–1775, 1983. [36] C. Riva, “Seasonal and diurnal variations of total attenuation measured with the italsaT satellite at Spino d’Adda at 18.7, 39.6 and 49.5 GHz,” Int. J. Satellite Commun. Network., vol. 22, pp. 449–476, 2004. [37] G. Brussaard, “A meteorological model for rain-induced crosspolarisation,” IEEE Trans. Antennas Propag., vol. 24, no. 1, pp. 5–11, Jan. 1976. [38] K. V. Beard and A. R. Jameson, “Raindrop canting,” J. Atmos. Sci., vol. 40, pp. 448–454, Feb. 1983. [39] C. Amaya-Byrne, “Depolarisation due to troposphere and its impact on satellite-to-earth communications,” Ph.D. dissertation, Université Catholique de Louvain-la-Neuve, Belgium, 1995. [40] Propagation Data and Prediction Methods Required for the Design of Earth-Space Telecommunication Systems, ITU-R P.618-10, 2009. [41] W. L. Nowland, R. L. Olsen, and I. P. Shkarofsky, “Theoretical relationship between rain depolarization and attenuation,” Electron. Lett., vol. 13, no. 13, pp. 676–678, 1977. [42] D. C. Cox, “Depolarization of radio waves by atmospheric hydrometeors in earth-space paths: A review,” Radio Sci., vol. 16, no. 5, pp. 781–812, 1981. [43] T. S. Chu, “A semi-empirical formula for microwave depolarization versus rain attenuation on earth-space paths,” IEEE Trans. Commun., vol. 30, no. 12, pp. 2550–2554, 1982. [44] W. L. Stutzman and D. L. Runyon, “The relationship of rain-induced cross-polarization discrimination to attenuation for 10 to 30 GHz earthspace radio links,” IEEE Trans. Antennas Propag., vol. 32, no. 7, pp. 705–710, 1984. [45] A. W. Dissanayake, D. P. Haworth, and P. A. Watson, “Analytical model for cross-polarization on earth-space radio paths for frequency range 9–30 GHz,” Ann. Telecommun., vol. 35, no. 11–12, pp. 398–404, 1980. [46] H. Fukuchi, “Prediction of depolarization distribution on earth-space path,” Proc. IEE, vol. 137, no. 6, pp. 325–330, 1990. Aldo Paraboni (deceased: 1940–2011) received the Laurea degree in electronic engineering from Politecnico di Milano, Italy, in 1964. He began his academic career as a Teacher and Researcher at the now-called Dipartimento di Elettronica e Informazione, Politecnico di Milano, in early 1965. Since then, his activity was mainly devoted to antennas and radiopropagation, which he began teaching in 1969 and, from 1981 to 2010, he was a Full Professor. From 1969 to 2011, he participated in the design and the execution of various radiopropagation projects such as Sirio, Olympus and Italsat. In this area he gained scientific responsibility in the program activities developed on behalf of the Agenzia Spaziale Italiana (ASI). In 1998, he proposed a project for a new satellite-based telecommunication experiment at 22 GHz (DAVID—Resource Sharing Experiment), which was successively approved by the Italian Space Agency (ASI). A more recent proposal, partially derived from DAVID, consisted of a new scientific experiment foreseeing simultaneous attenuation and depolarization measurements in various sites of Europe in the Q/V band (40/50 GHz), aimed at assessing the effectiveness of propagation impairments mitigation techniques; this experiment has been endorsed by ASI and ESA (Alphasat TDP 5) and presumably will see the light of day within the next 5 years. He participated for a number of years in the activities of various national and international organizations such as COST 205, 255 and 280 projects of the European Community, the OPEX (Olympus Propagation EXperiment of ESA), the CEPIT (Coordinamento per l’Esperimento di Propagazione ITalsat of ASI). He gave support to the International Telecommunications Union (ITU-R) and to other institutions. He was the author of over 260 scientific papers published in national and international journals and conferences His more recent research activity was in the field of fade mitigation techniques based on the spatial correlation of the attenuation. Prof. Paraboni was a member of the Italian Commission at the International Radio Scientific Union (URSI). He was awarded two international prizes: the 1990 Piero Fanti International Prize (by Intelsat/Telespazio) for his activity in satellite technology innovation, and the Premio Internazionale Cristoforo Colombo for Space Communications (City of Genoa, 1995). He

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acted as external editor of several international publications, such as the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, Radio Science and the International Journal on Satellite Communications and Networking, and participated in various advanced research contracts in Space Telecommunications with the ASI and the ESA (the European Space Agency).

Antonio Martellucci received the Laurea degree in electrical engineering and the Ph.D. degree in applied electromagnetics from the University of Rome “La Sapienza,” Rome, Italy, in 1987 and 1992, respectively. In 1988, he joined Selenia Group (now part of Finmeccanica) as an Optical Engineer where he worked on the development of optical active systems. From 1989 to 2000, he worked at the Fondazione Ugo Bordoni “Radio communication Systems Division,” Rome, as a researcher on atmospheric propagation effects for terrestrial and spatial radio communication systems. During this period, he participated the Olympus and Italsat propagation experiments (through the ESA OPEX and Italian CEPIT working groups) for the measurement and modelling of the atmospheric attenuation and depolarization at Ka, Q, and V frequency bands. He also took part in European COST 210 and 255 projects and various ESA projects on rain scatter, clear air propagation modelling and climatological databases. In 2001, he joined the European Space Agency, ESA-ESTEC, The Netherlands, Directorate of Technical and Quality Management, as a Radiowave Propagation Engineer where he is currently involved in ESA Telecommunication (ARTES and Alphasat), Navigation (Galileo), Earth Observation (ENVISAT) and Science (Gaia, Bepi Colombo) Programmes. At ESA, he is currently involved in models for multimedia SatCom systems, including fade mitigation techniques, modelling and characterization of tropospheric effects for navigation systems and development of ground propagation equipment. He is author of more than 90 publications in books, international journals and conference proceedings. Dr. Martellucci was a recipient of the Young Scientist Award of XXV URSI General Assembly in 1996. He has been the general Editor of the EU COST 255 and since 2001 he was member of the COST 280 Management committee. Since 2008 he is chairman of the EU/FP7 COST Actions IC0802. He is also member of the ESA delegation at ITU-R SG3.

Carlo Capsoni graduated in Electronic Engineering from the Politecnico di Milano in 1970. In the same year joined the Centro di Studi per le Telecomunicazioni Spaziali (CSTS), research centre of the Italian National Research Council (CNR) at the Politecnico di Milano. In this position he was in charge of the installation of the meteorological radar of the CNR sited at Spino d’Adda and since then he is the scientific authority responsible for the radar activity. In 1979, he was actively involved in the satellite Sirio SHF propagation experiment (11–18 GHz) and later in the Olympus (12, 20 and 30 GHz) and Italsat (20,40 and 50 GHz) satellites experiments. His scientific activity is mainly concerned with theoretical and experimental aspects of electromagnetic wave propagation at centimeter and millimeter wavelengths in presence of atmospheric precipitation with particular emphasis on attenuation, wave depolarization, incoherent radiation, interference due to hydrometeor scatter, precipitation fade countermeasures, modeling of the radio channel and design of advanced satellite communication systems. The scientific activity in the field of radar meteorology is mainly focused on rain cell modeling for propagation applications, on the development of radar simulators and on the study of the precipitation microphysics. He is also active in free space optics theoretical and experimental activities. Since 1975 he has been teaching a course on Aviation Electronics at the Politecnico di Milano. In 1986 he became Full Professor of electromagnetics at the same University. Prof. Capsoni is a member of the URSI national group and of the Italian Society of Electromagnetics (SIEm). He is also a member of the Coritel governing body.

Carlo G. Riva was born in 1965. He received the Laurea degree in electronic engineering and the Ph.D. degree in electronic and communication engineering, both from Politecnico di Milano, Milano, Italy, in 1990 and 1995, respectively. In 1999, he joined the Electronics and Information Science Department, Politecnico di Milano, where, since 2006, he has been an Associate Professor of electromagnetic fields. He participated in the Olympus, Italsat and (the planned) Alphasat TDP 5 propagation measurement campaigns, in the COST255, COST280 and COSTIC0802 international projects on propagation and telecommunications and in the Satellite Communications Network of Excellence (SatNEx). He support the ITU-R Study Groups activities as a delegate of SG3 and focal point for related propagation studies. He is the author of more than 100 papers published in international journals or international conference proceedings. His main research activities are in the fields of atmospheric propagation of millimeter-waves, propagation impairment mitigation techniques, and satellite communication adaptive systems.

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Communications Bandwidth Enhancement of Low-Profile PEC-Backed Equiangular Spiral Antennas Incorporating Metallic Posts Mehdi Veysi and Manouchehr Kamyab

Abstract—This communication examines parasitic coupling to achieve higher bandwidth (BW) and further size reduction for low-profile PECbacked equiangular spiral antennas. The parasitic loading is realized by metallic posts placed between the antenna plane and metal reflector. The parasitically excited metallic posts can considerably improve the antenna axial ratio at low frequencies as well as make it more rigid in construction. However, in practical applications, the low-profile equiangular spirals, albeit compact, suffer from the lack of a proper planar feed. To obviate this problem, a planar spiral-shaped microstrip feed is proposed.

Fig. 1. Geometry of an ESA; a = 0:35 radian

,'

= 2:8 , r = 1:5.

Index Terms—Equiangular spiral antenna (ESA), freestanding equiangular spiral antenna (FESA).

I. INTRODUCTION For a freestanding spiral antenna, it is known that bi-directional radiation with opposite polarizations is achieved. But in most applications, a unidirectional pattern is needed to block radiation on the other side. So far, some investigations have been devoted to the realization of lowprofile unidirectional equiangular spiral antennas (ESAs) [1]–[3], all of which have used absorbing materials. However, the use of absorbing materials reduces the antenna efficiency. The low-profile spiral-mode microstrip (SMM) antenna has been also proposed in [4]. But a difficulty in this method is how to launch and maintain the desired spiral modes. In this communication, the use of parasitically excited metallic posts (not in contact with spiral arms) has been proposed that can result in further size reduction and BW enhancement. The proposed low-profile ESA is also more rigid and much less costly than its previous counterparts. On the other hand, feeding of the low-profile spiral antennas is also challenging and generally requires external matching networks. The vertical baluns reported in the literature (see, e.g., [5]) have a significantly long length, inconsistent with the requirements of the lowprofile spiral antenna geometry. To obviate the feeding problem, this communication presents a horizontal feeding technique for low-profile ESAs that is inspired by the infinite Dyson balun which can be applied to the slot spiral antennas [6]. The commercial software CST Microwave Studio is adopted for the simulations. II. SIMULATION RESULTS AND DISCUSSION A configuration of the self-complementary ESA is shown in Fig. 1. The distance of the first edge from the origin is obtained simply by using: r = r0 ea('+=2) where, a is a constant determining the flare rate of the spiral and ' takes on the values between 0 radians and 'end

Manuscript received September 29, 2010; revised March 12, 2011; accepted March 18, 2011. Date of publication August 12, 2011; date of current version November 02, 2011. The authors are with the Electrical Engineering Faculty, K. N Toosi University of Technology (KNTU), Tehran, Iran (e-mail: [email protected]). Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2164194

Fig. 2. (a) Location of the metallic posts on the PEC reflector, the red dots represent the circular holes in the ground plane, the diameters of the circular holes are 2.5 mm, (b) photograph of a simulated metallic post (all dimensions in millimeter).

radians. The truncated edge, Pt 0 Pe , is an arc of a circle of radius R centered at the origin. The first edge is rotated by =2 radians to obtain the second edge. The second arm of the antenna is also obtained by rotating the first arm by  radians. The dimensions of the ESA are labeled in Fig. 1. Making use of the fact that metallodielectric spiral antennas can radiate efficiently with a good return loss and acceptable radiation patterns [7], we design spiral antenna in printed-circuit technology, which reduces the size and the cost of the antenna and makes it more rigid in construction. Here, the substrate thickness and dielectric constant are selected as small as possible, to mitigate its unwanted effects. For a PEC-backed equiangular spiral antenna (ESAPEC ), it is known that when the antenna height is reduced, the antenna axial ratio will increase especially at low frequencies. The main reason for the increased axial ratio at low frequencies is the presence of the in-coming current flowing toward the feed point which in turn increases the cross-polarization component. In this communication, metallic posts (spacers) are used to improve the antenna axial ratio at low frequencies. The ESA is placed above a PEC reflector by using 8 metallic spacers. In contrast to the foam spacers, when the metallic spacers are employed, the antenna axial ratio will change, depending on the spacer locations. An additional advantage of the metallic spacers compared to the foam spacers is their capability to make the antenna

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Fig. 5. Frequency characteristics of FESA fed by proposed feed as compared to the FESA fed by an ideal port, (a) directivity (RHCP), (b) axial ratio. Fig. 3. Comparison of frequency characteristics of the ESA (Here, the ESAs are considered in the plane conductor form) with and without metallic posts, (a) axial ratio, (b) directivity (RHCP), h = 10 mm.

Fig. 6. Photograph of a fabricated ESA Fig. 4. Bottom view of fabricated FESA fed by parallel-plane spiral-shaped microstrip line.

more rigid in construction. A proper combination of the metallic posts can lead to an improved axial ratio, especially at the frequencies below 6 GHz. A parametric study on the optimum positions of the metallic posts was carried out. The spacer locations on the PEC reflector are represented by the red dots in Fig. 2(a). Photograph of a prototype individual metallic post is also shown in Fig. 2(b). Fig. 3 compares the frequency characteristics of the ESAPEC with and without metallic posts, for different antenna heights (measured from the reflector

prototype, h

= 10 mm.

plane to the spiral arms, hsp ). As can be seen, the axial ratio of the ESAPEC can be greatly improved by incorporating metallic posts. The frequency band, inside which the axial ratio of the ESAPEC is below 3 dB, ranges from 3 GHz to 12 GHz. The improvement in the antenna axial ratio can be attributed to the reduction of in-coming currents at low frequencies. On the other hand, since the metallic posts are not in contact with the spiral arms, they do not act as resistive loads and thus the antenna radiation efficiency does not degrade. Fig. 3(b) compares the broadside directivities of the ESAPEC with and without metallic posts. As revealed in the figure, the use of metallic posts does not have a significant effect on the antenna directivity.

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Fig. 7. Measured reflection coefficient of the proposed ESA posts.

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with metallic

Fig. 9. Measured radiation patterns of the proposed (b) 6 GHz, (c) 9 GHz.

ESA

, (a) 3 GHz,

first edge of the microstrip feed line has a spiral curve expressed as: r = r1 ea('+=2) , where r1 = 1:2 mm. The second edge can be obtained by rotating the first edge through an angle of  = 9:5 degrees. The spiral arms are printed on a substrate (RT-Duroid 5880) with a thickness of 0.78 mm and dielectric constant of 2.2, whereas the microstrip line is printed on the other side of the substrate, as shown in Fig. 4. The first arm of the ESA acts as a ground plane for the microstrip feed line. To have a good impedance matching, the width of the microstrip line is smoothly tapered along a spiral path in the same way as the spiral arm. The width of the microstrip line is 2.5 mm at the input for 50 characteristic impedance and gradually tapers down to a width of 0.2 mm at the end of the feed line. Finally, the microstrip line is directly connected to the second arm of the spiral at the end of the feed line. Fig. 5 presents the simulated results of the proposed freestanding equiangular spiral antenna (FESA). For comparison purposes, simulation results for a FESA fed by a 175 ideal differential port are also shown in the figure. Compared to the FESA fed by an ideal port, the proposed FESA has a relatively lower directivity, which is mainly due to the undesirable effects of the dielectric substrate [7]. The antenna axial ratio at  = 180 is also plotted in Fig. 5(b). The increased axial ratio at  = 180 can be mainly attributed to the feed radiation. Fig. 8. Frequency characteristics of the ESA (a) axial ratio, (b) RHCP gain.

IV. EXPERIMENTAL RESULTS

incorporating metallic posts,

III. FEED DESIGN The ESAs simulated in the previous section have been fed by an ideal differential port. However, in practical applications, design of a proper feed structure is of crucial importance. Thus, a planar spiral-shaped microstrip feed is examined for the use with the low-profile ESAs. To the authors’ best knowledge, this is the first attempt to feed the low profile ESAs with a completely planar balun. A configuration of the parallel-plane spiral-shaped microstrip feed is shown in Fig. 4. The

A prototype of the proposed low-profile ESAPEC was built to confirm the simulation results. Fig. 6 shows a photograph of the fabricated antenna, in which the distance between the antenna plane and PEC reflector is fixed at 10 mm. Fig. 7 shows the measured reflection coefficient of the proposed ESAPEC . It is observed that the proposed ESAPEC is matched well to the microstrip feed line. The frequency characteristics of the ESAPEC incorporating metallic posts are plotted in Fig. 8. As can be seen, the measured and simulated results show a reasonable agreement. Although the radiations from the feed line (resulting in an increased axial ratio at  = 180 , see Fig. 5(b)) and existence of the antenna substrate [7] have some undesirable effects on the antenna axial ratio, the axial ratio of the proposed ESAPEC is still

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below 5 dB over the frequency range of interest (3–10 GHz). The antenna gain in dBic is also shown in Fig. 8(b). The gain of the proposed antenna was measured in the anechoic chamber of the K. N. Toosi University of Technology using a standard linearly polarized reference antenna. And thus the antenna gain in dBic is generally 3 dB higher [2]. The deterioration in the broadside gain of the antenna at the higher frequencies is attributed to the fields reflected off the reflector. The measured radiation patterns of the proposed ESAPEC at two orthogonal planes are shown in Fig. 9. As can be seen, there is a difference between the antenna radiation patterns at two orthogonal planes. This difference can be mainly attributed to the presence of the PEC reflector and especially to the non-symmetrical arrangement of the balun circuit.

V. CONCLUSION The results show that when the freestanding equiangular spiral antenna is placed close to a PEC reflector, the broadside axial ratio does not remain small over the entire frequency band of interest. This is attributed to the in-coming currents flowing toward the feed point. In order to improve the axial ratio of the low-profile PEC-backed equiangular spiral antenna, the parasitically excited metallic posts are effectively incorporated between the spiral plane and metal reflector. A parallel-plane spiral-shaped microstrip feed has been also proposed for the use with the proposed low-profile equiangular spiral antenna. The width of the horizontal microstrip feed line slowly tapers down along a spiral path in an attempt to match the feeding and spiral antenna.

ACKNOWLEDGMENT The authors would like to thank M. Kaboli and M. Abootorab for assisting in the antenna measurements.

REFERENCES [1] H. Nakano, K. Kikkawa, Y. Iitsuka, and J. Yamauchi, “Equiangular spiral antenna backed by a shallow cavity with absorbing strips,” IEEE Trans. Antennas Propag., vol. 56, no. 8, pp. 2742–2747, Aug. 2008. [2] J. J. H. Wang and V. K. Tripp, “Design of multioctave spiral-mode microstrip antennas,” IEEE Trans. Antennas Propag., vol. 39, no. 3, pp. 332–335, Mar. 1991. [3] W. Fu, E. R. Lopez, W. S. T. Rowe, and K. Ghorbani, “A planar dual-arm equiangular spiral antenna,” IEEE Trans. Antennas Propag., vol. 58, no. 5, pp. 1775–1779, May 2010. [4] J. J. H. Wang, “The spiral as a traveling wave structure for broadband antenna applications,” Electromagn., pp. 323–342, Jul.–Aug. 2000. [5] P. H. Rao, M. Sreenivasan, and L. Naragani, “Dual band planar spiral feed backed by a stepped ground plane cavity for satellite bore-sight reference antenna applications,” IEEE Trans. Antennas Propag., vol. 57, pp. 3752–3756, Dec. 2009. [6] M. W. Nurnberger and J. L. Volakis, “A new planar feed for slot spiral antennas,” IEEE Trans. Antennas Propag, vol. 44, no. 1, Jan. 1996. [7] M. M. Fadden and W. R. Scott, Jr., “Analysis of the equiangular spiral antenna on a dielectric substrate,” IEEE Trans. Antennas Propag., vol. 55, no. 11, pp. 3163–3171, Nov. 2007.

Parasitic Current Reduction on Electrically Long Coaxial Cables Feeding Dipoles of a Collinear Array Anda R. Guraliuc, Andrea A. Serra, Paolo Nepa, and Giuliano Manara

Abstract—The reduction of parasitic currents induced on the outer conductor of electrically long coaxial cables feeding the dipoles of a collinear array is addressed. Firstly, an experimental model is presented, and its appropriateness has been verified through validation with simulations data. The above model has been used to show the effectiveness of a low-cost technique to suppress the RF currents induced on the coaxial cables running inside the metallic mast of a collinear dipole array. The proposed solution consists of a dielectric-loaded coaxial choke, realized as a shorted quarter-wavelength coaxial cylinder mounted on the external conductor of the coaxial cable, and filled with Polytetrafluoroethylene (PTFE) to reduce its physical length. The PTFE-loaded choke performance is shown through both numerical simulations and measurements on prototypes, in the 200–400 MHz frequency range. Finally, a numerical parametric analysis is performed to get some design criteria for more complex multiple-choke arrangements. Index Terms—Balun, choke, collinear array, collinear dipoles.

I. INTRODUCTION Collinear dipole arrays are commonly used in VHF/UHF communication systems requiring an omnidirectional radiation pattern in the azimuth plane, as for example in Air Traffic Control (ATC) systems. These antennas are made of vertically stacked cylindrical dipoles (as shown in Fig. 1(a) for a two-element array). The dipoles are independently fed, and electrically long coaxial cables lie side by side inside of a metallic tube (antenna mast) that supports the entire structure and acts as grounding for lightning protection (Fig. 1(b)). Usually, the coaxial cables are directly connected to each dipole, without any balun device due to lack of space. Fig. 1(c) shows how each coaxial cable is usually connected to a dipole: an arm of the dipole is connected to the central conductor of the coaxial cable, while the other one is welded to the outer conductor. On the inner surface of the outer conductor, there is the current IA and at the end of the coaxial cable, IA divides into IC and IB . It is well known that this happens because of the transition from an unbalanced transmission line to a balanced antenna. The IC current flowing on the external surface of the outer conductor causes radiation pattern asymmetry and input impedance detuning. However, main concerns regard its effect on the coupling between adjacent dipoles of the collinear array, which is related to currents flowing on the external conductor of the coaxial cables that lie side by side along the antenna metallic mast. Indeed, the outer conductor of each coaxial cable and the

Manuscript received November 05, 2010; revised February 14, 2011; accepted April 04, 2011. Date of publication August 12, 2011; date of current version November 02, 2011. This work was supported by Telsa srl (http://www.telsasrl.it) Bergamo-Italy, through funding and technical support. A. R. Guraliuc and A. A. Serra are with the Department of Information Engineering, University of Pisa, I-56122 Pisa, Italy (e-mail: anda.guraliuc@iet. unipi.it; [email protected]). P. Nepa and G. Manara are with the Department of Information Engineering, University of Pisa, I-56122 Pisa, Italy and also with CUBIT (Consortium Ubiquitous Technologies), Navacchio, Pisa, Italy (e-mail: p.nepa@iet. unipi.it; [email protected]). Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2164202

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Fig. 2. Measurement setup of a coaxial cable inside of a metallic tube.

also described. In Section III the effectiveness of the proposed solution is shown through both numerical and experimental data. Finally, some conclusions are drawn in Section IV. Fig. 1. (a) A two-element collinear dipole array; (b) connection between coaxial cable and dipole at the feeding point; (c) parasitic currents induced on the outer conductor of the coaxial cable that lies inside the metallic tube representing the metallic antenna mast.

metallic tube form a two-conductor transmission line that can support the propagation of TEM waves inside the antenna mast. A well-known simple (and relatively not-bulky) device to reduce the parasitic currents excited at the connection between a balanced antenna and a coaxial cable consists of a coaxial choke [1], which exhibits a high impedance path to the undesired current flowing on the outer conductor of the coaxial cable. However, it is not suitable for broadband or multi-band antennas as it is an inherently narrow band device. An alternative solution, usually adopted in commercial collinear dipoles, consists in coating the coaxial cable with several ferrite beads, which block (or mainly absorb) the currents flowing on the outer surface of the coaxial cable [2]–[4]. The choking effect depends on the magnetic permeability of the ferrites; however, commercial ferrite permeability commonly exhibits a cut-off frequency (typically around a few MHz), losing their efficiency at those VHF/UHF frequency bands usually adopted in communication systems [5]–[7]. Moreover, the inclusion of several ferrite beads (almost completely covering each coaxial cable) makes the structure cumbersome, increases the realization costs, and power dissipation on the ferrite beads reduces the antenna gain (above losses increase with frequency). The ferrite beads can also be combined with coaxial cables in a balun-configuration [2], [8], [9]. They are more effective if a large radius is considered, but this makes them bulky. Another technique to control the currents on wire structures includes the use of dielectric loading [10], or a combination of dielectric and magnetic bead elements [11]. Such a technique has been shown to be effective in controlling and shaping currents on dipole antennas. In this communication, the effectiveness of dielectric-loaded chokes in reducing parasitic currents on coaxial cables is investigated, with specific reference to its application in collinear dipole arrays. An ad-hoc model has been developed, for which both measurement data and numerical results are shown and compared. The communication is organized as follows. In Section II, an experimental model of a coaxial cable inside of a metallic tube is presented. The design criteria for the proposed dielectric-loaded choke is

II. MODEL FOR THE ANALYSIS OF RF PARASITIC CURRENTS The experimental model adopted to analyze the parasitic current induced on the outer conductor of a coaxial cable running inside of a metallic tube is shown in Fig. 2. A coaxial cable is placed inside of a metallic tube (length = 80 cm, radius = 2 cm) that also represents the ground reference. The inner conductors of two 50 coaxial cables connected at the two ports of an Agilent E5071C Vector Network Analyzer (VNA) are soldered to the external surface of the coaxial cable (at its two ends), while their outer conductors (ground reference for the VNA) are connected to the external metallic tube. The inner conductor of the coaxial cable is floating and irrelevant in this analysis. This setup forces a current to flow on the outer conductor of the coaxial cable, as it happens in the coaxial cables of a collinear dipole array. With this measurement setup the current on the external surface of the coaxial cable can be quantified through the transmittance parameter S21. When a choke is added, the achieved lower transmittance values indicate that the parasitic current excited on the cable has been attenuated. A technique to reduce the currents flowing on the external surface of the outer conductor of a coaxial cable can be realized by distributing some PTFE-loaded chokes along the cable. In our experimental setup, a PTFE-loaded choke is a quarter-wavelength cylindrical PTFE sample coated with a copper foil, as shown in Fig. 3. One end of the copper foil is electrically connected to the outer conductor of the coaxial cable (Fig. 3(b)). The PTFE allows us to reduce the length of the choke with respect to an air-filled choke. The radius of the PTFE cylinder (dielectric permittivity "r = 2:08, tg = 0:0002) was set at R = 5 mm (the radius of the external conductor of the coaxial cable is 1.79 mm). The choke length was calculated as L = PTFE =4 = =4 "r , where  is the free-space wavelength and PTFE is the wavelength in the PTFE medium. The numerical model of the experimental setup previously described has been implemented through the electromagnetic simulator CST Microwave Studio [12].

p

III. NUMERICAL AND EXPERIMENTAL RESULTS Measured data and numerical results were collected and compared. The transmittance was evaluated at different conditions: (a) without any choke, (b) with a single PTFE-loaded choke, c) with a set of PTFEloaded chokes (2 or 3). When no chokes are placed on the feeding cable

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Fig. 3. (a) Three PTFE-loaded chokes aligned along a coaxial cable in a metallic tube; one end of the copper foil is open (Section A in (b)) and the other end is electrically connected to the outer conductor of the coaxial cable (Section B in (b)).

Fig. 4. The transmittance parameter S21 evaluated at two different conditions: (a) without any choke; (b) with a single PTFE-loaded choke with length L1 = 17 cm.

the transmittance is close to zero, as expected, and it will be considered as a reference value in the following. When a PTFE-loaded choke is added, with length L1 = 17 cm calculated at 300 MHz, the choking effect on the currents flowing on the outer conductor of the coaxial cable is noticed around 300 MHz (Fig. 4). The frequency shift between the measured and the simulated traces is due to end-cable effects that were not taken into account in the numerical model, as absorbing boundary conditions have been imposed at the cable ends. A numerical parametric analysis has been performed, in order to show the effect of some key geometrical parameters. In Fig. 5, it is shown that the choking effect extends in a larger frequency bandwidth when the PTFE cylinder radius is increased. To extend the choke effect in a wider frequency bandwidth, two or more chokes can be properly located along the coaxial cable. Considering two chokes (Choke#1 and Choke#2 in Fig. 3(a), both with a radius R = 5 mm) with different lengths, L1 = 17 cm (PTFE /4@300 MHz) and L2 = 15:5 cm (PTFE /4@335 MHz) respectively, the frequency range where the transmittance considerably reduces is enlarged, as shown in Fig. 6. Two separated resonance frequencies are clearly visible. It was noticed that the distance between the chokes,

Fig. 5. Simulated S21 parameter as a function of the radius of the PTFE-loaded choke. Reference curve: coaxial cable without any choke.

D1, should be different from 25 cm (PTFE /4@300 MHz) to avoid a critical situation similar to a resonance condition, as apparent in Fig. 6. Three PTFE-loaded chokes with the same radius R = 5 mm, and lengths L1 = 17 cm (PTFE =4 @300 MHz), L2 = 15:5 cm (PTFE =4 @335 MHz) and L3 = 13:5 cm (PTFE =4 @380 MHz) respectively, are distributed along the coaxial cable (Fig. 3(a)). A further bandwidth enlargement is apparent in Fig. 7 where the distance between the chokes was set at D1 = D2 = 12:5 cm. A good agreement between measured and simulated results is still noticed. Finally, in order to show the improvement with respect to a solution that consists in adding a number of ferrite beads on the coaxial cable, in Fig. 7 the measured transmittance for a 20 cm-long NiZn ferrite bead is also considered (actually, 10 pieces of 2 cm-long ferrite bead with an external radius of 4 mm have been put together). An attenuation of 30 dB can be noticed, and the trace flatness (instead of a resonant behavior) demonstrates that ferrites exhibit a remarkable lossy effect at those frequencies, which is more apparent than the magnetically-induced blocking effect. Indeed, the magnetic permeability of common ferrite beads decreases at high frequency and becomes close to a few units at the VHF/UHF frequency bands [5]–[7], [10]; moreover, ferrite losses increase with frequency. This confirms that

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the performance of PTFE-loaded chokes distributed along the coaxial cable. It was also demonstrated that the frequency range of the blocking technique can be extended by combining PTFE-loaded chokes properly distributed along the coaxial cable. In realistic collinear dipole arrays more than one coaxial cable is present inside the antenna mast. Nevertheless, the simpler problem analyzed here (only one centered coaxial cable) has shown the effectiveness of the experimental/numerical model and the effect of some key design parameters of a PTFE-loaded choke arrangement. ACKNOWLEDGMENT The authors acknowledge the support of CST for providing additional resources and technical assistance for the parallel CST Microwave Studio version.

D

Fig. 6. The transmittance parameter S21 evaluated when the distance 1 between two PTFE-loaded chokes is changed. 1 = 12 5 cm in the measurement set-up.

D

:

Fig. 7. The transmittance parameter S21 evaluated when three chokes (radius = 5 mm) are placed at a distance 1 = 2 = 12 cm. Chokes length: 1 = 17 cm ( 4 @300 MHz), 2 = 15 5 cm ( 4 @335 MHz), 4 @380 MHz). The dotted line is relevant to the case 3 = 13 5 cm ( in which the coaxial cable is covered with a 20 cm-long ferrite bead.

R L L

:





=

=

D L

D :



=

commercial VHF/UHF collinear dipoles using ferrite coatings mainly exploit ferrite losses more than their magnetic properties. Also, to reduce the parasitic currents to an acceptable level, the whole coaxial cable is usually covered with ferrite beads, so increasing antenna cost and weight (ferrite density: NiZn = 4:7 g=cm3 ). In the proposed solution, the PTFE-loaded chokes must be added as close as possible to the dipole feeding point. Moreover, when a number of coaxial cables lie side by side inside the metallic tube (as it happens in a collinear dipole array), a set of PTFE-loaded chokes must be added on each coaxial cable. IV. CONCLUSIONS A model to study parasitic currents flowing on the outer conductor of electrically long coaxial cables feeding dipoles of a collinear dipole array was presented. The good agreement between transmittance results for the prototyped structure and the simulated one indicates the model appropriateness. The above model has been used to evaluate

REFERENCES [1] D. P. Kaegebein, “Parallel Fed Collinear Antenna Array,” U.S. Patent 6,057,804, May 2000. [2] R. Leitner and J. Drake, “Antenna System With Ferrite Radiation Suppressors Mounted on Feed Line,” U.S. Patent 3,680,146, Jul. 1972. [3] O. Fujiwara, T. Ichikawa, and H. Kawada, “Effect of ferrite core attachment to coaxial cable on differential mode noise caused by braided shield current,” in Proc. Int. Symp. on Electromagn. Compat., 1997, pp. 523–526. [4] H. Arai, Measurements of Mobile Antenna Systems. Boston, the approximate kernel is expected to yield good results.

1

= 2

1

= 61

Fig. 5. Magnitude of the current at the wire center and at the wire ends obtained from the two methods for L = . The new formulation (17) yields results that converge very well at all points on the wire.

= 20

Fig. 5 shows a convergence study for the current at the wire center and near the wire ends obtained from the two methods, for = 20. The approximate kernel yields a result that slowly converges to a value that is approximately 12% too large, and which diverges near the wire is increased, as expected. The new formulation (17) yields ends as results that converge quickly at all points on the wire.

L =

N

IV. CONCLUSIONS

=

20

= 81

Fig. 4. Real part of the current for L , where = and N =a : , such that the approximate kernel is not expected to provide accurate results. The new integral equation (17) uses the exact kernel and avoids the problem of current oscillations near the wire ends arising from use of the approximate kernel.

1

= 0 25

Pocklington’s integro-differential equation for thin wires with the exact kernel has been reformulated into a pure integral equation using a second derivative formula for improper integrals. Solid wire and hollow tube models have been considered. It was shown that a simple pulsefunction point matching solution of the resulting integral equation leads to results that are stable and show good convergence, and which avoids the severe and non-physical oscillations of the current that arise from the same numerical procedure used in conjunction with the approximate kernel. APPENDIX To obtain the second derivative formula similar to (7) but for open surfaces, consider the quantity

geometry considered in Fig. 3. The last column is the percent difference between the two results. It can be seen that the Pocklington form converges faster than the Hallén form; for the Pocklington form the percent difference between the = 100 and = 5000 results is 0.06%, whereas for the Hallén form it is 0.3%. Regarding computer times, both the new Pocklington form and the Hallén form only depend on 0 0 , and thus have a Toeplitz form. Only the first row of the matrix need be filled, and thus both methods are very efficient. Computing the new Pocklington form takes approximately twice as long as the Hallén form since it involves numerical 2 ), whereas the integration of two different integrands ( and 2 Hallén form involves only . On a standard PC matrix fill time for ei= 300. ther method is under a second for Fig. 4 shows the current for = 20 and = 81 (1 = 0 25), such that the approximate kernel is not expected to provide accurate results. Indeed, using the approximate kernel non-physical oscillations in the current appear at the wire ends, as discussed in [9] (for a delta-gap generator these oscillations occur near the wire center). However, the current obtained from (17) does not exhibit these non-physical oscillations, as we would expect when using the exact kernel.

N

N

z z

K

K

N L =

@ K=@z

N

=a

:

(r) = (a; ; z) = lim0 !

s(r )g(r; r )dS 0

0

0

(21)

S 0S

S

S

where is the surface of a hollow, infinitesimally-thin tube and  is an exclusion surface, depicted in Fig. 2. The quantity (21) defines a single-layer potential with source density . It is well-known that a tangential derivative of (21) can be passed through the integral (normal ) 2, where ( ) is a point on the derivatives generate a term 6 ( surface [26], [27]). Therefore,

s

s ; z =

@ @z (r) = 0 lim0S !

= lim0 !

; z

s(r ) @z@ g(r; r )dS 0

0

0

0

0S

@s(r ) g(r; r )dS @z 0

0

0

0

S 0S

0 S 0S

r  z^s(r )g(r; r ) dS 0

0

0

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 59, NO. 11, NOVEMBER 2011

n  ^zs(r0 )g(r; r0 )dl0

= !0 0 lim

C +C

+

g

0 = 0 (r r0 ) @z

S S

0

so that we have @

(r0 ) (r r0 ) 0 0

@s

;

(22)

dS

rA = nA

(23)

dl

C

where C is the contour of the open surface S . The two-dimensional divergence theorem is usually applied to open surfaces in the plane. However, it is valid for any compact differential manifold that is equipped with an inner product at each point; the inner product at a point on a tube in R3 is simply the dot product on R3 restricted to vectors tangent to the tube at the given point. In this case, these will be unit vectors parallel to the axis of the tube pointing away from the tube at the tube z. rims, i.e., n = 6^ It can be easily shown that

(r0 ) (r r0 )^z  n 0 = 0

!0

lim



s

g

;

(24)

dl

C

where it should be noted that C comprises both the top and bottom rims of S , which are taken in opposing directions. Therefore, (22) converts to @ @z

(r) = 0 n  z^ (r0 ) (r r0 ) 0 s



g

;

dl

C

+ !0

(r0 ) (r r0 ) 0 0

@s

lim

0

g

@z

S S

;

dS

(25)

which may be differentiated again to yield @

2

@z 2

(r) = 0 n^z (r0 ) s



C

0 !0 @

g

@

lim

where we have used

@ @z

0

(r r0 ) 0 dl

;

( (r0 ) 0 (r)) (r r0 ) 0 0 0 s

s

@g

@z

S S

( (r0 ) 0 (r)) = (r0 ) 0 0 s

;

dS

@z

@s

s

@z

@z

(26)

(27)

:

Using the two-dimensional divergence theorem, the second integral of (26) may be rewritten as

( (r0 ) 0 (r)) (r r0 ) 0 = 0 (r0 ) 0 (r) 0 0 S 0S S 0S 2 0) 0 0) 0 ( r r ( r r 2 + n  z^ 02 0 (r ) 0 (r) @

s

s

@g

@z

@ g

;

;

@z

dS

@g

dS

@z

s

;

C +C

As with (24), it can be shown that

n  z^ C

(r0 ) 0 (r)

s

s

@z

s

@g

s

s

(r r0 ) = 0 0 ;

@z

dl

0 (r) = 0 (r) n  ^z (r r ) @g

s

dl:

(28)

(29)

;

@z

C

+ !0

lim

0

dS

2

 @z 2

where we use @g (r; r )=@z =@z in the first line and the @g ; product rule for derivatives to obtain the second line, and where the third line comes from the divergence theorem in two dimensions,

S

4359

0

dl

(r0 ) 0 (r)

s

s

S S

2

@ g

(r r0 ) 0 ;

@z 2

dS

(30)

the desired expression. Note that the source density should be Höldercontinuous [27] in order to pass a derivative through the surface integral. This is not an issue, since we will expand the current into functions that are differentiable in the region of the singularity. This holds even for pulse functions, since a pulse function will be centered on the singularity, and over the singularity the pulse function will be constant.

ACKNOWLEDGMENT The authors would like to thank D. Gomez, UWM Electrical Engineering, for help with this project and R. Ancel, UWM Department of Mathematics, for helpful discussions concerning the two-dimensional divergence theorem.

REFERENCES [1] T. Wu, “Introduction to Linear Antennas,” in Antenna Theory, , Ed., R. E. Collin and F. J. Zucker, Eds. New York: McGraw-Hill, 1969, ch. 8, pt. 1. [2] W. L. Stutzman and G. A. Thiele, Antenna Theory and Design. New York: Wiley, 1981. [3] C. Balanis, Advanced Engineering Electromagnetics. New York: Wiley, 1989. [4] R. Elliott, Antenna Theory & Design. New York: Wiley, 1981. [5] “NEC Manual” [Online]. Available: http://www.nec2.org/ [6] S. Schelkunoff, Advanced Antenna Theory. New York: Wiley, 1952. [7] B. Rynne, “On the well-posedness of Pocklington’s equation for a straight wire antenna and convergence of numerical solutions,” J. Electromagn. Waves Applicat., vol. 14, pp. 1489–1503, 2000. [8] G. Fikioris and T. T. Wu, “On the application of numerical methods to Hallen’s equation,” IEEE Trans. Antennas Propag., vol. 49, no. 3, pp. 383–392, 2001. [9] G. Fikioris, J. Lionas, and C. G. Lioutas, “The use of the frill generator in thin-wire integral equations,” IEEE Trans. Antennas Propag., vol. 51, no. 8, pp. 1847–1854, 2003. [10] D. S. Jones, “Note on the integral equation for a straight wire antenna,” IEE Proc. Part H: Microw. Opt. Antennas, vol. 128, no. 2, pp. 114–116, 1981. [11] B. Rynne, “The well-posedness of the integral equations for thin wire antennas,” IMA J. Appl. Math., vol. 49, pp. 35–44, 1992. [12] B. Rynne, “The well-posedness of the integral equations for thin wire antennas with distributional incidental fields,” Q. J. Mechanics Appl. Math., vol. 52, pp. 489–497, 1999. [13] G. Fikioris and A. Michalopoulou, “On the use of entire-domain basis functions in Galerkin methods applied to certain integral equations for wire antennas with the approximate kernel,” IEEE Trans. Electromag. Compat., vol. 51, pp. 409–412, 2009. [14] M. Silberstein, “Application of a generalized Leibnitz rule for calculating electromagnetic fields within continuous source regions,” Radio Sci., vol. 26, pp. 183–190, 1991. [15] W.-X. Wang, “The exact kernel for cylindrical antenna,” IEEE Trans. Antennas Propag., vol. 39, no. 4, pp. 434–435, 1991. [16] P. J. Davies, D. B. Duncan, and S. A. Funken, “Accurate and efficient algorithms for frequency domain scattering from a thin wire,” J. Comput. Phys., vol. 168, no. 1, pp. 155–183, 2001. [17] D. H. Werner, J. A. Huffman, and P. L. Werner, “Techniques for evaluating the uniform current vector potential at the isolated singularity of the cylindrical wire kernel,” IEEE Trans. Antennas Propag., vol. 42, no. 11, pp. 1549–1553, 1994. [18] D. H. Werner, “An exact formulation for the vector potential of a cylindrical antenna with uniformly distributed current and arbitrary radius,” IEEE Trans. Antennas Propag., vol. 41, no. 8, pp. 1009–1018, 1993.

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[19] D. H. Werner, P. L. Werner, and J. K. Breakall, “Some computational aspects of pocklington’s electric field integral equation for thin wires,” IEEE Trans. Antennas Propag., vol. 42, no. 4, pp. 561–563, 1994. [20] O. P. Bruno and M. C. Haslam, “Regularity theory and superalgebraic solvers for wire antenna problems,” SIAM J. Sci. Comput., vol. 29, no. 4, pp. 1375–1402, 2007. [21] A. Mohan and D. S. Weile, “Convergence properties of higher order modeling of the cylindrical wire kernel,” IEEE Trans. Antennas Propag., vol. 55, no. 5, pp. 1318–1324, 2007. [22] A. Yaghjian, “Electric dyadic Green’s functions in the source region,” IEEE Proc., vol. 68, pp. 248–263, 1980. [23] A. Ishimaru, Electromagnetic Wave Propagation, Radiation, and Scattering. Englewood Cliffs, NJ: Prentice-Hall, 1991. [24] G. W. Hanson and A. B. Yakovlev, Operator Theory for Electromagnetics. Berlin: Springer, 2001. [25] R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. II. New York: Interscience, 1953. [26] J. V. Bladel, Electromagnetic Fields, 2nd ed. Piscataway, NJ: WileyIEEE Press, 2007. [27] O. D. Kellogg, Foundations of Potential Theory. Berlin: Springer Verlag, 1929.

Accelerated Source-Sweep Analysis Using a Reduced-Order Model Approach Patrick Bradley, Conor Brennan, Marissa Condon, and Marie Mullen

Abstract—This communication is concerned with the development of a model-order reduction (MOR) approach for the acceleration of a sourcesweep analysis using the volume electric field integral equation (EFIE) formulation. In particular, we address the prohibitive computational burden associated with the repeated solution of the two-dimensional electromagnetic wave scattering problem for source-sweep analysis. The method described within is a variant of the Krylov subspace approach to MOR, that captures at an early stage of the iteration the essential features of the original system. As such these approaches are capable of creating very accurate low-order models. Numerical examples are provided that demonstrate the speed-up achieved by utilizing these MOR approaches when compared against a method of moments (MoM) solution accelerated by use of the fast Fourier transform (FFT).

Typically, the relevant integral equation (IE) is discretized using the MoM and results in a series of dense linear equations. The computational burden associated with the repeated solution of the full-wave scattering problem at each source location is severe, especially for large scatterers. Different strategies are used to accelerate the solutions of these linear systems. Considerable progress has been made in incorporating sparsification or acceleration techniques [4] and preconditioners into iterative methods which permit expedited solutions of the scattering problem. The CG-FFT solution in particular is often applied in situations where the unknowns are arranged on a regular grid. An alternative approach is to develop approximate solutions to expedite the solution of EM scattering problems. Several approximations of the integral equation formulation are discussed in literature, including the Born approximations [5] and the family of Krylov subspace model order reduction techniques [6]. Although the Born approximation has been shown to efficiently simulate the EM response of dielectric bodies, these techniques are restricted to problems of relatively low frequencies and low contrast [5]. Krylov subspace approaches such as the Arnoldi algorithm [7]–[11] can produce very accurate low-order models since the essential features of the original system are captured at an early stage of the iteration. A set of vectors that span the Krylov subspace are used to construct the reduced order matrix model. By imposing an orthogonality relation among the vectors, linear independence can be maintained1 and so high-order approximations can be constructed. In this work, we modify the Arnoldi MOR procedure, introduced in [7], to efficiently perform scattering computations over a wide range of source locations for objects of varying inhomogeneity. We consider a two-dimensional dielectric object characterized by permittivity z (r) and permeability  for a TM configuration. A time dependence of exp(j!t) is assumed and suppressed in what follows. The corresponding integral equation can be expressed in terms of the unknown scattered field Ezs (r) and total field Ez (r) [13] | s Ez (r) = 4

(2)

H0

kb jr 0 r

0

j

0

0

O(r )Ez (r )dv

0

(1)

V

where O(r0 ) is the object function at point r0 given by

Index Terms—Electromagnetic propagation, method of moments, projection algorithms. 0

2

0

2

O(r ) = k (r ) 0 kb :

I. INTRODUCTION The solution of electromagnetic wave scattering problems, from inhomogeneous bodies of arbitrary shape, is of fundamental importance in numerous fields such as geoscience exploration [1] and medical imaging [2]. For such problems, it is common to require the repeated solution of the electromagnetic wave scattering problem for a variety of source locations and types. This is of particular importance in the reconstruction of unknown material parameters in inverse problems and as such is a critical step in the optimization process [3].

Manuscript received November 26, 2010; revised January 24, 2011; accepted April 04, 2011. Date of publication August 12, 2011; date of current version November 02, 2011. The authors are with the RF Modelling and Simulation Group, The RINCE Institute, School of Electronic Engineering, Dublin City University, Ireland (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TAP.2011.2164210

(2)

The background wave number is given by kb while k(r0 ) is the wave (2) number at a point in the scatterer. H0 is the zero order Hankel function of the second kind. Using m pulse basis functions and Dirac-Delta testing functions [13], (1) is discretized by employing the MoM,2 which results in the following matrix equation

(I + GA)x = b

(3)

where b is the incident field vector at the center of each basis domain, I is an m 2 m identity matrix and G is an m 2 m matrix containing coupling information between the basis functions. A is an m 2 m 1Note that due to finite precision computation loss of orthogonality between the computed vectors can occur in practical applications [6], [8], [12]. 2Note that other basis and testing functions are possible without affecting the applicability of what follows.

0018-926X/$26.00 © 2011 IEEE

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directions of interest. In theory, wn (line 9 Table I) will vanish if u1 is a linear combination of q eigenvectors of G. In the absence of superior information we choose the initial incident field as the start vector u1 . Indeed it has been shown experientially that a random vector is a reasonable choice [8], [14]. In the context of a source sweep analysis, our choice of start vector has no bearing on the accuracy of the approximation at other source locations. We do not discuss practical error controls in this communication but instead direct the reader to [6], [8], [14], where the topic is presented in detail. Suffice to say, that these techniques are applicable in what follows.

TABLE I ARNOLDI—MODIFIED GRAM-SCHMIDT ALGORITHM WITH RE-ORTHOGONALIZATION (MGSR)

III. METHODOLOGY In a source sweep analysis, the computation of the scattered fields from an inhomogeneous body requires independently solving x

= (I + GA)01b

(8)

for each step in source location. The Arnoldi algorithm produces a ROM by iteratively computing the Hessenberg reduction Hq

diagonal matrix whose diagonal elements contain the contrast at the center of each basis domain



mm =

( m ) 0 1: b

 r 

x

(4)

 xq =

Kq (G; u1 ) = spanfu1 ; Gu1 ; 1 1 1 ; Gq01 u1 g

(5)

for G generated by the vector u1 . This algorithm generates a Hessenberg reduction Hq

= UHq GUq

(6)

where Hq is an upper Hessenberg matrix [8]. Note that the subscript q is used to denote a q 2 q matrix, where q  m, the order of the MoM matrix. The columns of Uq

= [u1 ; u2 ; 1 1 1 ; uq ]

(7)

are derived iteratively using the Arnoldi process in Table I [8]. un are termed the Arnoldi vectors and they define an orthonormal basis for the Krylov subspace Kq (G; u1 ). The Arnoldi procedure can be essentially viewed as a modified Gram-Schmidt process for building an orthogonal basis for the Krylov space Kq (G; u1 ). The unit vectors un are mutually orthogonal and have the property that the columns of the generated Uq matrix span the Krylov subspace Kq . Critically, the Arnoldi iteration can be stopped part-way, leaving a partial reduction to Hessenberg form that is exploited to provide a reduced order model (ROM) for (3). It should be noted that the choice of start vector u1 is critical in the early extraction of eigenvalue information of G. As such, one should attempt to construct a start vector that is dominant in the eigenvector

q

n=1

un n

= Uq aq

(10)

where aq = [1 2 1 1 1 q ]T is a vector of expansion coefficients for the Arnoldi basis vectors un that span the Krylov subspace. The residual rq that corresponds to this approximation is introduced as

II. THE ARNOLDI ITERATION The Arnoldi algorithm is an orthogonal projection method that iteratively builds an orthonormal basis for the Krylov subspace Kq [8]

(9)

After q steps of the Arnoldi algorithm, an approximation xq , to x, can be made in terms of the q basis vectors

0

We note that for large scatterers (3) cannot be solved by direct matrix inversion.

= UqH GUq :

rq

= b 0 (Im + GA)xq :

(11)

To find the optimal approximate solution, xq is constrained to ensure that xq minimizes the residual rq . Specifically, the residual vector is constrained to be orthogonal to the q linearly independent vectors uq . This is known as the orthogonal residual property, or a Galerkin condition rq

? Kq or

UqH rq

= 0:

(12)

The residual rq is minimized when the residual vector is orthogonal to the space Kq . This requires substituting (10) into (11) rq

= b 0 (I + GA)Uq aq

(13)

and performing a Galerkin test, to give UqH rq

= UqH (b 0 (I + GA)Uq aq ) = UqH b 0 Iq + UqH GAUq aq  UqH b 0 Iq + UqH GUq UqH AUq aq = UqH b 0 (Iq + Hq A~ q2q )aq

(14) (15) (16) (17)

where

~ = UqH AUq :

Aq

(18)

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As a result of setting

aq

= (Iq + Hq A~ q )01 UHq b

(19)

the residual has been minimized as

~ q )(Iq + Hq A~ q )01 UqH b =0 UqH rq = UqH b 0(I + Hq A

(20)

yielding the following ROM for the total field

x  xq

= Uq (Iq + Hq A~ q )01UqH b

(21)

:

The key advantage is that once the Krylov matrix Uq has been gener~ q can be rapidly created and stored, ated and stored the matrix Iq +Hq A along with its inverse permitting the solution for multiple right hand side vectors, even in the case of large scatterers as typically q  m. It should be noted that the step from (15) to (16) is, in general, approximate. It is exact only if the range R(Uq ) of Uq is an invariant subspace of A. However, due to the independence of the columns of Uq imposed by the re-orthogonalization process, this step can be shown to be a very reasonable approximation. As prescribed in [8], if the columns of Uq are independent and the norm of the residual matrix

R = AUq 0 Uq Sq

Fig. 1. Case Study I: Monostatic RCS from an homogeneous circle with constant contrast and varying source location.

(22)

has been minimized for some choice of Sq , then the columns of Uq define an approximate subspace. The selection of Sq = UqH AUq = ~ q results in the norm of the residual being minimized A

min kAUq 0 Uq Sq k2 =

I 0 Uq UqH AUq

2

:

(23)

Thus, (16) becomes a valid approximation with the property that, as ! m, a better approximation is procured.

q

IV. RESULTS In this section, the monostatic radar cross section (RCS) is calculated from various geometries for a variety of source locations with a fixed contrast profile. The numerical performance of the reduced order model, generated using the Arnoldi algorithm, is compared against a MOM solution using an FFT-accelerated solver. A. Case Study I We initially consider a homogeneous cylinder embedded in free space. It is centered at the origin with radius r = 0:6 and discretized using m = 370 cells. The structure is illuminated by a transverse magnetic (TMz ) wave emanating from a line source located at (10 cos, 10 sin) where  varies in the range 0: 2 at increments of 8 . The assumed frequency is f = 300 MHz and the cylinder contrast is fixed at  = 1:5 (r = 2:5). Fig. 1 depicts the monostatic RCS obtained from the MoM and the modified Gram-Schmidt algorithm with re-orthogonalization (MGSR) technique presented in this communication, for q = 30, representing a 92% reduction in system size. The MGSR technique achieves an RCS average error (AE) of 0.12 dB over the entire source range, while yielding a RCS maximum error (ME) of 0.72 dB. B. Case Study II We now consider an inhomogeneous layered scatterer with square cross section centered at the origin, with side length l = 3 embedded

Fig. 2. Case Study II: Monostatic RCS from an inhomogeneous square with constant contrast and varying source location.

in free space. We assume the square to be composed of four equally sized slices each with width wi = 0:75 and height hi = 3. The number of basis functions used is m = 3030 with fixed contrast values of 1 = 4, 2 = 3, 3 = 2 and 4 = 1:1. The monostatic RCS is again computed for the same range of line source locations as before. The MGSR technique achieves an impressive reduction in system size while yielding an acceptable AE over the entire source range. This is highlighted in Fig. 2, which compares RCS obtained from the MoM and the MGSR technique for q = 455 representing a 85% reduction in system size. A complete CPU time analysis associated with the solution of the RCS for the FFT-accelerated MoM, and MGSR for ns = 45 samples is given in Table II. Within this table tu is representative of the time3 taken to generate the Krylov Uq matrix. Similarly, ti refers to ~ q ; tg refers to the time taken to the time taken to generate the initial A generate the FFT component of G, and tb refers to the time needed to generate b. ts is the average time taken to solve the monostatic RCS at each source location using CGNE or CGNE-FFT accordingly. ni = number of iterations taken by the solver in order to reach the tolerance 1005 . tt is the total time taken to generate and solve case 3All times discussed in the communication are equal to CPU time in seconds.

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TABLE II CPU TIME ANALYSIS FOR CASE STUDY II

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approximation over a wide source range. In addition we have demonstrated the computational saving achieved by using MOR techniques in the solution of scattering problems, when compared to techniques which are based on solving the MoM system using FFT-accelerated iterative solvers.

REFERENCES

study problem and pr = ROM size reduction expressed in %. These simulations were run on a 3.00 GHz Xeon CPU processor with 3.00 GB of RAM at 2.99 GHz and the MoM solution is solved using the conjugate gradient normal equation method accelerated with the fast Fourier transform (CGNE-FFT). The CGNE-FFT can reduce the cost of matrix vector multiplications from (m2 ) operations per iteration to (m log2 m) operations. It is evident from Table II, that the MGSR algorithm can significantly decrease the computational expense associated with the direct solution for each source location in a source sweep analysis. The main computational cost of this approach is incurred in generating the Krylov matrix q and the initial contrast profile matrix ~ q . Note that these computations can also be accelerated using the FFT and, once generated, the q and ~ q matrices can be reused in subsequent simulations involving independent source locations. From Table II, it is clear that considerable savings in CPU time can be achieved in a source sweep analysis by utilizing the MGSR method. For a 85% reduction in system size (q = 455) using the MGSR method, the associated AE and ME’s are 0.26 and 1.46 dB respectively over the entire source range. While errors of ME = 2:73 dB and AE = 0:38 dB are observed for a 90% reduction. It is also shown in Table II, that for 45 source locations (ns = 45) the ROM obtained using the MGSR technique is 21.3 times faster in producing a solution for the RCS, than the CGNE-FFT method applied to the MoM matrix system. Note that the CGNE method was used to compute the unknown field in (21) rather than direct inversion, in order to generate a conservative comparison.

O

O

U U

A

A

V. CONCLUSION We have presented a new method for accelerating source sweep analysis based on an extension of the Arnoldi MOR approach. Notably, we have shown that the Arnoldi algorithm can produce accurate low-order approximations for a relatively low computational cost. Using case studies, we demonstrated that the Arnoldi technique can produce a significant reduction in the system size while still resulting in an accurate

[1] T. J. Cui, W. C. Chew, and W. Hong, “New approximate formulations for em scattering by dielectric objects,” IEEE Trans. Antennas Propag., vol. 52, no. 3, pp. 684–692, Mar. 2004. [2] Q. H. Liu, Z. Q. Zhang, T. Wang, J. Bryan, L. A. W. Ybarra, and G. A. Nolte, “Active microwave imaging. I. 2-d forward and inverse scattering methods,” IEEE Trans. Microwave Theory Tech., vol. 50, no. 1, pp. 123–133, Jan. 2002. [3] T. Hohage, “Fast numerical solution of the electromagnetic medium scattering problem and applications to the inverse problem,” J. Comput. Phys., vol. 214, pp. 224–238, May 2006. [4] W. C. Chew, E. Michielssen, J. M. Song, and J. M. Jin, Fast and Efficient Algorithms in Computational Electromagnetics. Norwood, MA: Artech House, 2001. [5] G. Gao and C. Torres-Verdin, “High-order generalized extended born approximation for electromagnetic scattering,” IEEE Trans. Antennas Propag., vol. 54, no. 4, pp. 1243–1256, 2006. [6] E. J. Grimme, “Krylov projection methods,” Ph.D. dissertation, Coordinated-Science Laboratory, University of Illinois at Urbana-Champaign, Urbana-Champaign, IL, 1997. [7] N. Budko and R. Remis, “Electromagnetic inversion using a reducedorder three-dimensional homogeneous model,” Inverse Problems, vol. 20, no. 6, pp. S17–S26, 2004. [8] G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd ed. Baltimore, MD: Johns Hopkins Univ. Press, 1996. [9] O. Farle and R. Dyczij-Edlinger, “Numerically stable moment matching for linear systems parameterized by polynomials in multiple variables with applications to finite element models of microwave structures,” IEEE Trans. Antennas Propag., vol. 58, no. 11, pp. 3675–3684, 2010. [10] M. Zaslavsky and V. Druskin, “Solution of time-convolutionary Maxwell’s equations using parameter-dependent Krylov subspace reduction,” J. Comput. Phys., vol. 229, pp. 4831–4839, June 2010. [11] D. Weile, E. Michielssen, and K. Gallivan, “Reduced-order modeling of multiscreen frequency-selective surfaces using Krylov-based rational interpolation,” IEEE Trans. Antennas Propag., vol. 49, no. 5, pp. 801–813, May 2001. [12] K. Gallivan, E. Grimme, and P. Van Dooren, “A rational Lanczos algorithm for model reduction,” Numer. Alg., vol. 12, no. 1–2, pp. 33–63, 1996. [13] A. F. Peterson, S. L. Ray, and R. Mittra, Computational Methods for Electromagnetics, ser. IEEE Press Series on Electromagnetic Wave Theory, 1st ed. Piscataway, NJ: Wiley-IEEE Press, 1997. [14] J. Demmel, J. Dongarra, A. Ruhe, and H. van der Vorst, Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide, Z. Bai, Ed. Philadelphia, PA: Society for Industrial and Applied Mathematics, 2000.

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Extended Mie Theory for a Gyrotropic-Coated Conducting Sphere: An Analytical Approach You-Lin Geng and Cheng-Wei Qiu

Abstract—Based on the extended Mie theory with Fourier transform, the electromagnetic field in homogeneous gyrotropic media, for the first time, can be analytically obtained in spectral domain in terms of spherical eigne-vectors with their associated coefficients. The coefficients of electromagnetic fields in the gyrotropic shell and the isotropic host medium are thus exactly solved in a recursive manner. Our analytical extended Mie scattering theory have been numerically validated. Using this analytical approach, results of the scattering by the general gyrotropic coated conducting sphere are obtained. Index Terms—Coated structures, electromagnetic scattering, gyrotropic materials. Fig. 1. The geometry of a generalized gyrotropic-coated conducting sphere.

I. INTRODUCTION One century ago, the electromagnetic scattering theory by a homogeneous isotropic sphere illuminated by a plane wave has been established by Lorenz and Mie [1], [2], respectively. Since then, many scholars extended and developed the Mie theory [3]–[6]. Recently, the scattering characteristics of anisotropic media have aroused increasing interest owing to their wide applications in optical signal processing, radar cross section controlling, radar absorber synthesis, and microwave device fabrication. Much work has been done on the interaction between a plane wave and an anisotropic medium [7]–[16]. Many numerical and analytical methods have been developed to study this problem. For instance, the finite-difference time-domain (FDTD) method [7], [8], the moment method [9], the integral equation method [10], mode expansion method [11], [12], and hybrid finite element-boundary integral-multilevel fast multipole algorithm (FE-BI-MLFMA)[13]. Subsequently, the spherical vector wave functions (SVWFs) expansions along with the Fourier transform were employed to describe the plane wave scattering by a uniaxial anisotropic sphere and a plasma anisotropic sphere [14], [15] based on the plane wave expansion in terms of SVWFs in isotropic medium [17]. However, analytical scattering theory for an optically gyrotropiccoated conducting sphere or even just a single gyrotropic sphere, in which both permittivity and permeability are general gyrotropic tensors, has not been successfully reported. In this connection, this communication proposes an extended Mie theory based on Fourier transform, and tackles this existing problem. Electromagnetic fields in a gyrotropic anisotropic medium can be expanded in terms of SVWFs in gyrotropic anisotropic medium. Applying the boundary conditions at

Manuscript received October 08, 2010; revised February 27, 2011; accepted April 04, 2011. Date of publication August 12, 2011; date of current version November 02, 2011. This work was supported in part by Grant No. 60971047 of the National Natural Science Foundation of China (NSFC) and in part by Grant No. Y1080730 of the Natural Science Foundation of Zhejiang Province of China. Y. L. Geng is with the Institute of Antenna and Microwaves, Hangzhou Dianzi University, Xiasha, Hangzhou, Zhejiang 310018, China. C. W. Qiu is with Department of Electrical and Computer Engineering, National University of Singapore, Singapore 117576, Singapore (e-mail: [email protected]). Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2164195

the interface between the free space and the gyrotropic shell and at another interface between the gyrotropic shell and the PEC core, those unknown coefficients associated with eigenvectors in the gyrotropic shell as well as those in free space can be obtained analytically. Then, radar cross sections of a gyrotropic-coated conducting sphere can be readily derived. The theoretical analysis and numerical results show that the present method can be reduced to these of a homogeneous gyrotropic sphere when the radius of the conducting sphere approaches to zero [16]. This analytical solution to electromagnetic scattering by a gyrotropic-coated conducting sphere can be used to characterize the scattering of objects in microwaves as well as in optics, and also be helpful to understand wireless communication channels and radio wave propagation mechanisms. II. FORMULATION OF THE PROBLEM Consider a geometry in Fig. 1 which shows a cross section of a gyrotropic-coated conducting sphere (the outer and inner radii of the gyrotropic shell are a1 and a2 , respectively). Three distinct regions are thus divided, namely, region 0 for the free space, region 1 for the gyrotropic spherical shell, and region 2 for the conducting sphere. This composite structure is illuminated by a plane electromagnetic wave (which is assumed to have an electric-field amplitude equal to unity along the x-direction, propagating along the z -direction). In the following analysis, the time dependence of exp(0i!t) is assumed but suppressed throughout the treatment. The permittivity and permeability of the generalized gyrotropic material in the shell are characterized by

=

1 i2 0

0i2 1 0

0 0

3

;

=

1 i2 0

0i2 1 0

0 0

3

:

(1)

The wave equation in such a source-free and unbounded gyrotropic anisotropic medium can be written as

r 2 [01 1 r 2 E ] 0 !2  1 E = 0

(2)

where E denotes the electric field. Using Fourier transform and the expansion of plane wave in terms of SVWFs in isotropic media [17], one can obtain the electromagnetic

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fields (by the subscript 1 ) in the gyrotropic shell (a2 basis of the [16]:

 r  a1 ) on the

and

Es E1

2

2

l=1 q =1 mnn

(l) Fmn q

Hs

 e l) (l) Aemnq (k )M (mn (r ; kq ) + Bmnq (k )N mn (r ; kq )

0 e (l) +Cmnq (k )Lmn (r ; kq )

H1

2

2

2P

m n (cos

= l=1 q =1 mnn

2

s (3) AsmnM (3) mn (r ; k0 ) + Bmn N mn (r ; k0 )

=

k )kq2 sin k dk

=

k0 i!0

(3a)

(l) Fmn q

l) h (l) Ahmnq (k )M (mn (r ; kq ) + Bmnq (k )N mn (r ; kq )

0

2P

(l)

 Lmn (r; kq )

+C

h mnq ( k ) m n (cos

2

k )kq sin k dk

(3b)

where n0 and n are summed up both from 0 to +1 while m is summed up from 0n to n; r denotes a position vector in the spherical coordinates; the angle k = tan01 [ (kx2 + ky2 )=kz ]; the unknown coef(l) ficients, Fmn q , are to be determined using the boundary conditions; p p (k ), Cmnq (k ) (where p = e or h) and and finally Apmnq (k ), Bmnq kq are all functions of k and they have been derived in [16] as shown l) l) (l) in the Appendix. The vectors, M (mn , N (mn , and Lmn , denote SVWFs in isotropic media which are defined as follows [14], [17]

(l) l) M (mn =zn (kr ) im

Pnm (cos) dP m (cos ) 0 n  eim sin  d

(4a)

(l) 1 d rzn (kr ) zn(l) (kr) m im N mn =n(n +1) Pn (cos )e r + kr kr dr m Pnm (cos ) dPn (cos ) im  + im  e (4b) 1 d sin 

s (3) AsmnN (3) mn (r ; k0 ) + Bmn M mn (r ; k0 )

axmn

=

in+1 2n2(nn+1 +1) ; m = 1 n+1 2n+1 m = 01 i 2 ;

(7a)

x bmn

=

in+1 2n2(nn+1 +1) ; m = 1 0in+1 2n2+1 ; m = 01

(7b)

s;l

=

1;

0;

s=l s 6= l.

(l)

dzn (kr) m zn (kr) P (cos )r + d(kr) n kr

2

P m (cos ) dPnm (cos )  + im n  d sin 

eim

T12 T22 T32 T42

T13 T23 T33 T43

T14 T24 T34 T44

1 r =a

F1(1) F2(1) F1(2) F2(2) P1

=

0 0

P4

(l)

[m;1 + m;01 ]

=

T11 T21 T31 T41

(4c)

where zn (x) (with l = 1, 2, 3 and 4) denotes an appropriate kind (1) (2) of spherical Bessel functions, jn (x), yn (x), hn (x) and hn (x), respectively. To characterize the scattering fields of the gyrotropic-coated conducting sphere, the incidence wave (designated by the superscript inc) and the scattered wave (designated by the superscript s) can be expressed in free space in terms of SVWFs as follows [17]

E inc

(7c)

s The expansion coefficients of scattered fields, Asmn and Bmn in (6) (n varies from 0 to +1 while m changes from 0n to n), are unknowns to be determined using the boundary conditions, together with the un(l) known coefficients Fmn q (where l = 1, 2 and q = 1, 2) in (3). From the (3a) and (3b), it implies that when the radius a2 of the conducting sphere is infinitely small, the electromagnetic fields in the origin is still finite, but the value of the spherical Bessel functions of the second kind becomes infinitely large in the origin. Therefore the expan(2) sion coefficients Fmn q of SVWFs in (3a) and (3b) will vanish, and in this connection, the present method in this communication can be automatically reduced to a homogeneous gyrotropic anisotropic sphere, which is the same as that of [16]. Applying continuous boundary conditions of tangential field components at the interface between the gyrotropic anisotropic shell and free space (where r = a1 ), and utilizing the orthogonality of the tangential SVWFs [15], the following matrix equation is obtained:

(l)

(l)

(6b)

mn

where k0 denotes the wave number in free space, and the expansion cox efficients of the incident wave axmn , bmn , and mn have been obtained earlier in [15] and are defined by



(l) Lmn =k

(6a)

mn

=

2

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0

P2 P3

A B

+

0

C1 C2 : (8) C3 C4

Similarly on the surface of the conducting sphere (where r = a2 ), the boundary condition requires the vanishing of the tangential components of the electric field. Hence, we have

T11 T12 T13 T14 T21 T22 T23 T24

r =a

1

In the right of (9), 0 can be regarded as a N

F1(1) F2(1) F1(2) F2(2)

=0

(9)

2 1 matrix, which is

mn

2 H

inc

x (1) axmn M (1) mn (r ; k0 ) + bmn N mn (r ; k0 )

k0 = i!0

2

(5a)

[m;1 + m;01 ]

1 1 1 ; 0]

t

(10)

N

mn

x (1) axmn N (1) mn (r ; k0 ) + bmn M mn (r ; k0 )

0 = [0; 0;

(5b)

where t denotes the transpose. In (8) and (9), Tlq is N 2 N matrix; = 1, 2 and q = 1, 2); Ci and Pi (i = 1, 2, 3, 4) are N 2 1

Fq(l) (l

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matrix; A and B are 1 2 N matrix; and N is the maximum value of n and n0 in (3), (5) and (6). They appear to be as the following 

T1q (n; n ) = Aemnq jn (kq r)Pnm (cos k ) 0

0

T2q (n; n ) =



0

0 

2 kq2 sink dk ;

(11a)

2 Pnm (cos k )kq2sink dk ;

(11b)

jn (kq r) e e Bmnq Rn(1) (kq r) + Cmnq r

T3q (n; n ) = Ahmnq jn (kq r)Pnm (cos k ) 0

0

T4q (n; n ) =



0

0 

2 kq2 sink dk

(11c)

2 Pnm (cos k )kq2sink dk ;

(11d)

jn (kq r) h h Bmnq Rn(1) (kq r) + Cmnq r

T1;2+q (n; n ) = Aemnq yn (kq r)Pnm (cos k ) 0

0

T2;2+q (n; n ) =



0

0 

2 kq2 sink dk

(11e)

2 Pnm (cos k )kq2sink dk ;

(11f)

yn (kq r) e e Bmnq Rn(2) (kq r) + Cmnq r

0

T4;2+q (n; n ) =



0

0

2 kq2 sink dk ;

(11g)

2 Pnm (cos k )kq2sink dk

(11h)

yn (kq r) h h Bmnq Rn(2) (kq r) + Cmnq r

(l) Fq(l) = Fm(l)1q ; Fm(l)2q ; 1 1 1 ; FmNq

t

t

P1 = h1(1) (k0a1 );h2(1) (k0 a1 ); 1 1 1 ;h(1) N (k0 a1 ) t P2 = R1(3) (k0 a1 );R2(3) (k0 a1 ); 1 1 1 ;RN(3) (k0 a1 ) k0 h(1) (k a );h(1) (k a ); 1 1 1 ;h(1) (k a ) t P3 = i! 0 1 0 1 0 1 1 2 N 0 k0 R(3) (k0 a1 );R(3) (k0 a1 ); 1 1 1 ;R(3) (k0 a1 ) t P4 = i! 1 2 N 0

A = [Asm1 ; Asm2 ; 1 1 1 1 1 1 ; AsmN ] s B = [Bms 1 ; Bms 2 ; 1 1 1 1 1 1 ; BmN ]

(12)

(13a) (13b) (13c) (13d)

(14a) (14b)

and

C1 (n) = [m;1 + m; 1 ]axmn jn (k0 a1 ) x C2 (n) = [m;1 + m; 1 ]bmn Rn(1) (k0 a1 ) k0 [ +  ]bx j (k a ) C3 (n) = i! m;1 m; 1 mn n 0 1 0 k0 [m;1 + m; 1 ]ax R(1) (k0 a1 ): C4 (n) = i! mn n 0

In (11)to (15), m is an arbitrary integer, and the radial function Rn(l) (x) has (l) the following expression (l) Rn (x) = (1=x)(d=dx)[xzn (x)]. From (8) to (15), it shows that there are six equations and six unknown coefficients, (l) namely, Fq (where q = 1, 2 and l = 1, 2), A and B in (12)to (14). Thus all unknown coefficients of electromagnetic fields in the gyrotropic anisotropic spherical medium can be obtained. Then the coefficients of scattered fields in free space are calculated and the radar cross section of an anisotropic gyrotropic-coated conducting sphere by a plane wave can be derived. III. NUMERICAL RESULTS

T3;2+q (n; n ) = Ahmnq yn (kq r)Pnm (cos k ) 0

Fig. 2. Radar cross sections (RCSs) versus scattering angle  (in degrees) of a non-magnetic case: Results of this communication (solid curve) and FE-BIMLFMA method (circle dot).

0

(15a)

0

(15b)

0

(15c)

0

(15d)

Since there are no reported analytical or numerical results for the scattering of a general gyrotropic-coated conducting sphere, we performed two verifications to validate our analytical method. Firstly, we compared our result for a gyroelectric case with a computational method based on the FE-BI-MLFMA [13] as shown in Fig. 2. In [13], the scattering of the gyroelectric-coated PEC sphere is solved numerically. It is a good candidate to compare with the results from our analytical methods for the more general gyrotropic-coated PEC sphere. Good agreement has been observed. Secondly, the radius of the conducting sphere is assumed to be extremely small (e.g., k0 a2 = 0:002 ) which tends to a homogeneous gyrotropic sphere [16]. The degenerated results based on the present method agree well with the direction calculation [16] as shown in Fig. 3. Both trials partially verify our theory as well as the Fortran program codes in both E - and H -planes. We also obtained some new results unavailable elsewhere in literature. Two examples are considered herein, and their radar cross sections are plotted in Figs. 4 and 5. Fig. 4 depicts RCS of a lossless gyrotropic-coated conducting sphere. The electric size of the gyrotropic-coated sphere is chosen as k0 a1 = 2:1 and k0 a2 = 2 . The maximum number n0 in (11) and (15), to ensure a good convergence, is found to be 14. To illustrate further applicability of the solution to electromagnetic scattering by an electrically large gyrotropic-coated conducting sphere (for example, in its resonance region), the RCS of a lossy gyrotropic-coated sphere of relatively larger size with k0 a1 = 4 and k0 a2 = 3:9 , are obtained and depicted in both the E -plane and the H -plane in Fig. 5. As the electric dimension of the sphere is increased, the maximum number of n0 used in (11) and (15) must be increased to 18 to achieve the convergence. It is noted that for a lossy gyrotropic-coated conducting sphere of large electrical dimension, the oscillation of RCS in the backward directions will be more suppressed, which can eventually be

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IV. CONCLUSIONS An extended Mie theory based on Fourier transform and SVWFs expansion technique is successfully has been established to model the scattering by a gyrotropic-coated conducting sphere in this communication. The solution has only one-dimensional integral which can be easily evaluated using Gauss integral [18]. The theoretical analysis and numerical results demonstrate the correctness, advantages, and uniqueness since it provides, for the first time, an analytical approach for such a generalized complex media in a coated structure. In addition, the general numerical results, including the lossless and lossy gyrotropiccoated conducting sphere in resonance region, are given. The scattering control by the gyrotropy ratio is under further investigation. Fig. 3. Radar cross sections (RCSs) versus scattering angle  (in degrees): Results of this communication (solid curve) and homogeneous gyrotropic sphere (circle dot).

APPENDIX e e The unknown coefficients Aemnq , Bmnq and Cmnq in (3a) can be e1 e2 e e1 e e2 expressed as Amnq = Amnq + Amnq , Bmnq = Bmnq + Bmnq , and e e1 e2 Cmnq = Cmnq + Cmnq . One can now obtain the expansion coefficients of E-fields in a gyrotropic anisotropic medium as follows: for p = 1 and m  0

Fig. 4. Radar cross sections (RCSs) versus scattering angle  (in degrees) in the E -plane (solid curve) and in the H -plane (dashed curve) in a lossless case.

)! 41 1 Aemnq = in 2n2(nn++11) ((nn 0+ m m)! 4 2 (n + m)(n 0 m + 1)Pnm01(cos k ) 0Pnm+1 (cos k ) (A-1a) 1 ( n 0 m )! 4 1 e1 n Bmnq = i 2n(n + 1) (n + m)! 4 2 (n + 1)(n + m)(n + m 0 1)Pnm0011 (cos k ) +(n + 1)Pnm0+11 (cos k ) + n(n 0 m + 2)(n 0 m + 1)Pnm+101 (cos k ) (A-1b) +nPnm+1+1 (cos k ) 1 ( n 0 m )! 4 e1 Cmnq = in 2kq (n + m)! 41 2 (n + m)(n + m 0 1)Pnm0011 (cos k ) + Pnm0+11 (cos k ) 0 (n 0 m + 2)(n 0 m + 1)Pnm+101 (cos k ) (A-1c) 0Pnm+1+1 (cos k ) while for p

= 1 and m > 0 m)! e1 Ae01mnq = (01)m ((nn + 0 m)! Amnq m)! e1 B0e1mnq = (01)m+1 ((nn + 0 m)! Bmnq m)! e1 C0e1mnq = (01)m+1 ((nn + 0 m)! Cmnq :

Fig. 5. Radar cross sections (RCSs) versus scattering angle  (in degrees) in the E -plane (solid curve) and in the H -plane (dashed curve) in a lossy case when the size is even larger than that in Fig. 4.

approximated by the geometrical optics limit [11] when the electrical size is further increased.

Similarly, for p

(A-2a) (A-2b) (A-2c)

= 2 and m  0, we have

1 (n 0 m)! 2 Aemnq =in+1 n2(nn++1) (n + m)! 4 2 2 24 (n + m)(n 0 m +1)Pnm01(cos k ) +Pnm+1 (cos k ) + mPnm (cos k )

(A-3a)

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)! e2 Bmnq =in+1 n(n1+1) ((nn 0+ m m)! 42 (n +1)(n + m)(n + m 0 1)P m01 2 24 n01 2 (cos k )0(n +1)Pnm0+11 (cos k ) + n(n 0 m +2)(n 0 m +1) 2Pnm+101 (cos k ) 0 nPnm+1+1 (cos k ) + [n(n 0 m +1)Pnm+1(cos k ) 0(n +1)(n + m)Pnm0011(cos k ) m)! e2 Cmnq =in+1 k1q ((nn 0 + m)! 42 (n + m)(n + m 0 1)P m01 (cos k ) 2 24 n01 m+1 0 Pn01 (cos k ) 0 (n 0 m +2) 2 (n 0 m +1)Pnm+101(cos k ) +Pnm+1+1 (cos k ) 0(2n +1)cos k Pnm (cos k )g while for p

REFERENCES

(A-3b)

(A-3c)

= 2 and m > 0 m)! e2 Ae02mnq = (01)m+1 ((nn + 0 m)! Amnq m)! e2 B0e2mnq = (01)m ((nn + 0 m)! Bmnq m)! e2 C0e2mnq = (01)m ((nn + 0 m)! Cmnq :

(A-4a) (A-4b) (A-4c)

Also, one has

41 = i(b1 a2 + b2 a1 )kq2sink cosk (A-5a) 42 = b1 b3 kq2 sin2k + b12 0 b22 kq2 cos2 k kq2 sink cosk 0 (b1 a1 + b2 a2 ) (A-5b) 2 2 2 2 2 4 = 0 b2 kq cos k + a2 + b1 kq cos k 0 a1 2 b1 kq2cos2 k + b3 kq2sin2k 0 a1 (A-5c) and

a1 = !21 ; a2 = !2 2 ; a3 = !2 3 b1 = 2 01 2 ; b2 = 2 02 2 ; b3 = 1=3: 1

2

1

2

(A-6a) (A-6b)

h h h Similarly, the coefficients Amnq , Bmnq and Cmnq in (3b) can also be obtained for the gyrotropic anisotropic medium.

ACKNOWLEDGMENT The authors are indebted to Prof. X. Q. Sheng and Dr. Z. Peng from the Beijing Institute of Technology for sending them their data.

[1] L. Lorenz, “Sur la Lumière réfléchie et réfractée par une sphère transparente,” Vidensk. Selsk. Skrifter, vol. 6, pp. 1–62, 1890. [2] G. Mie, “Beitrage zur Optik truber Medien speziell kolloidaler Metallosungen,” Ann. Phys., vol. 25, pp. 377–455, 1908. [3] A. L. Aden and M. Kerker, “Scattering of electromagnetic wave from two concentric spheres,” J. Appl. Phys., vol. 22, pp. 1242–1246, 1951. [4] J. H. Richmond, “Scattering by a ferrite-coated conducting sphere,” IEEE Trans. Antennas Propag., vol. 35, no. 1, pp. 73–79, 1987. [5] J. D. Keener, K. J. Chalut, J. W. Pyhtila, and A. Wax, “Application of Mie theory to determine the structure of spheroidal scatterers in biological materials,” Opt. Lett., vol. 32, no. 10, pp. 1326–1328, 2007. [6] B. Garcia-Camara, F. Moreno, F. Gonzalez, J. M. Saiz, and G. Videen, “Light scattering resonances in small particles with electric and magnetic properties,” J. Opt. Soc. Am. A, vol. 25, no. 2, pp. 327–334, 2008. [7] A. Taflove, “Review of the formulation and applications of the finite-difference-time-domain method for numerical modeling of electromagnetic wave interactions with arbitrary structures,” Wave Motion, vol. 10, pp. 547–582, Dec. 1988. [8] L. A. Dou and A. R. Sebak, “3D FDTD method for arbitrary anisotropic materials,” Microwave Opt. Technol. Lett., vol. 48, pp. 2083–2090, 2006. [9] R. D. Graglia, P. L. E. Uslenghi, and R. S. Zich, “Moment method with isoparametric elements for three-dimensional anisotropic scatterers,” Proc. IEEE, vol. 77, no. 5, pp. 750–760, May 1989. [10] S. N. Papadakis, N. K. Uzunoglu, and C. N. Capsalis, “Scattering of a plane wave by a general anisotropic dielectric ellipsoid,” J. Opt. Soc. Am. A, vol. 7, pp. 991–997, 1990. [11] C.-W. Qiu, S. Zouhdi, and A. Razek, “Modified spherical wave functions with anisotropy ratio: Application to the analysis of scattering by multilayered anisotropic shells,” IEEE Trans. Antennas Propag., vol. 55, no. 12, pp. 3515–3523, 2007. [12] Z. Lin and S. T. Chui, “Electromagnetic scattering by optically anisotropic magnetic particle,” Phys. Rev. E, vol. 69, p. 056614, 2004. [13] X. Q. Sheng and Z. Peng, “Analysis of scattering by large objects with off-diagonally anisotropic material using finite element-boundary integral-multilevel fast multipole algorithm,” IET Microw. Antennas Propag., vol. 4, pp. 492–500, 2010. [14] Y. L. Geng, X. B. Wu, L. W. Li, and B. R. Guan, “Mie scattering by a uniaxial anisotropic sphere,” Phys. Rev. E, vol. 70, no. 5, p. 056609, 2004. [15] Y. L. Geng, X. B. Wu, and L. W. Li, “Analysis of electromagnetic scattering by a plasma anisotropic sphere,” Radio Sci., vol. 38, no. 6, pp. 1104/1–1104/12, 2003. [16] Y. L. Geng and C. W. Qiu, Analytical Spectral-Domain Scattering Theory of a General Gyrotropic Sphere [Online]. Available: http://www.arxiv.com/abs/1102.4057 2011 [17] D. Sarkar and N. J. Halas, “General vector basis function solution of Maxwell’s equations,” Phys. Rev. E, vol. 56, pp. 1102–1112, 1997. [18] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover Publications, 1972.

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Scattering by a Multilayered Infinite Cylinder Arbitrarily Illuminated With a Shaped Beam Huayong Zhang, Yufa Sun, and Zhixiang Huang

Abstract—A method of calculating the scattered electromagnetic fields of a multilayered infinite cylinder, for arbitrary incidence of a shaped beam, is presented. A complete set of coefficients, which describes the electromagnetic fields within the different regions of a multilayered infinite cylinder, is determined by solving a system of linear equations derived from the boundary conditions. As an example, for a tightly focused Gaussian beam propagating perpendicularly to the cylinder axis of a two-layered and three-layered infinite cylinder, the three-dimensional (3D) nature of the scattering problem, greatly different from the case of an incident plane wave, is described in detail, and numerical results of the normalized differential scattering cross section are evaluated. Index Terms—Cylindrical vector wave function, multilayered infinite cylinder, shaped beam.

I. INTRODUCTION The problem of electromagnetic plane wave scattering by an infinite cylinder has been treated extensively by so many researchers. Wait [1] first presented a theoretical procedure for the description of interaction between oblique plane waves and an infinite homogeneous cylinder. In their literature, Kerker [2], van de Hulst [3], Bohren and Huffman [4] also gave the exact solution of light scattering by tilted infinite homogeneous cylinders. To make explicit the physical interpretation of different scattering modes, the Debye series expansion (DSE) has been applied to the study of the scattering properties of a multilayered infinite cylinder [5]. For the case of shaped beams, Alexopoulos and park [6], Kozaki [7], [8], and Zimmerman et al. [9] investigated the scattering of a two-dimensional (2D) Gaussian beam by an infinite cylinder. Lock employed the angular spectrum of plane waves model to study the scattering of obliquely incident focused Gaussian beams by an infinite homogeneous cylinder [10]. Gouesbet et al. developed the theory of distributions, with which to analyze the interaction of Gaussian beams with an infinite homogeneous cylinder at normal and oblique incidence [11], [12]. In [13], we have obtained the expansion of a shaped beam in terms of cylindrical vector wave functions, and then, within the framework of the generalized Lorenz-Mie theory (GLMT), which effectively describes the electromagnetic scattering of a shaped beam by a spherical particle, an analytic solution to the shaped beam scattering by an infinite homogeneous cylinder is constructed [14]. As an extension of our previous works, this communication is devoted to the description of the scattering of a shaped beam by an arbitrarily oriented multilayered infinite cylinder.

Manuscript received February 06, 2011; revised May 02, 2011; accepted May 09, 2011. Date of publication August 12, 2011; date of current version November 02, 2011. This work is supported by the Key National Natural Science Foundation of China under contract 60931002 and by the 211 Project of Anhui University. The authors are with the school of Electronics and Information Engineering, Anhui University, Hefei, Anhui 230039, China (e-mail: [email protected]). Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2164227

Ox y z Oxyz z Oxyz ; ; p

Fig. 1. The Cartesian coordinate system is parallel to the shaped , and the Cartesian coordinates of beam coordinate system are (0, 0, ). is obtained by a rigid-body rotation of in through Euler angles . A -layered infinite cylinder is natural to (The cylinder axis coincides with the axis of ).

Oxyz Ox y z Oxyz

O

Oz

Oxyz

II. FORMULATION Fig. 1 shows the geometry of the scatterer. All the notations follow Fig. 1 of [14], except that in it the infinite homogeneous cylinder is replaced by a p-layered one. We have obtained an expansion in [13] of the electromagnetic fields of an incident shaped beam in terms of the cylindrical vector wave (1) (1) functions m exp(ihz ); m exp(ihz ) with respect to the system Oxyz , in the following form

m

Ei =E0

n

1  Im;TE ( )mm + Im;TM ( )nm m 01 (1)

=

0

1

Hi =

0

iE0

(1)

exp(ihz ) sin d

1 

k ! m=01

(1)

nm + Im;TM ( )mm

Im;TE ( )

0 1

(1)

(1)

exp(ihz ) sin d

(2)

m ; In;TM m are the beam shape where  = k sin  , h = k cos  , and In;TE coefficients [13], [14]. As in [14], we take the case of a TE mode as an example and, without any loss of generality, assume Euler angles  = = 0 (case of on-axis beams with diagonal incidence) [14], [15]. By following (1), (2), the scattered electric fields as well as the electric fields within the j th cylinder can be obtained in terms of infinite series with cylindrical vector wave functions as

Es = E0

0018-926X/$26.00 © 2011 IEEE

1  m ( )mm + m ( )nm m 01 (3)

=

0

1

exp(ihz ) sin d

(3)

(3)

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Ew(j) = E0

1 m 01 =



(1) (3) m(j) ( )mm + m0(j) ( )mm

0

+mj ( )nm + m0 j ( )nm 2 exp(ihz)sin d ( )

(j = 2 1 1 1 p):

(1)

(4)

The electric fields in the innermost layer (j

Ew(1) = E0

1 m 01 =



(3)

( )

= 1) are

(1) (1) m(1) ( )mm + m(1) ( )nm

0

1 exp(ihz)sin d

(5)

where j = k n2j 0 cos2  (j = 1 1 1 1 p), nj is the refractive index of the material in the j th cylinder relative to that of free space [14], [16]. The corresponding expansions of the magnetic fields can be obtained with the following relations [14], [16]

1 r 2 E; mm exp(ihz); H = i! = k1 r 2 nm exp(ihz)nm exp(ihz) = k1 r 2 mm exp(ihz): (6) To determine the expansion coefficients m ( ); m ( ) of the scat-

tered fields, it is necessary to apply the boundary conditions, i.e., continuity of the tangential electromagnetic fields at the interface in different regions. At the boundary between the surrounding medium and the outermost layer (j = p), the boundary conditions are

Ei + Es = Ew(p) Ezi + Ezs = Ezw(p) at r = rp (7) Hi + Hs = Hw(p) Hzi + Hzs = Hzw(p) where rp is the cross-sectional radius of the pth cylinder. By virtue of the field expansions, the  and z components for the

electric fields in (7) can be expressed as

m (p ) I p dJd m;TE + m cos Jm (p )Im;TM + p p (1) 1 dHmd (p ) m ( ) + m cos Hm(1) (p ) m( ) p

m (p )  p ( ) + dHm (p ) 0 p ( ) = p dJd m dp m p  Jm (p ) p ( ) + H (p ) 0 p ( ) + m ncos m m m p np p Jm (p )Im;TM + Hm (p ) m ( ) = p Jm(p )mp ( ) + Hm (p )m0 p ( ) where p = rp , p = p rp . (1)

( )

( )

2

( )

(1)

( )

(8)

(1)

2

( )

(1)

( )

(9)

at r = rj (j = 1 1 1 1 p 0 1)

where rj is the cross-sectional radius of the j th cylinder.

Let j = j +1 rj , j = j rj , and we can have the  and z components for the electric fields in (10) as follows: (1) m (j )  (j +1) ( ) + dHm (j )  0(j +1) ( ) nj+1 j dJd m dj m j + m cos  Jm (j )m(j+1) ( ) + Hm(1) (j )m0(j+1) ( ) (1) m (j )  (j ) ( ) + dHm (j )  0(j ) ( ) = nj j dJd m dj m j + m cos  Jm (j )m(j) ( ) + Hm(1) (j )m0(j) ( ) j2 Jm (j )m(j+1) ( ) + Hm (j )m0(j+1) ( ) = j2 Jm(j )m(j) ( ) + Hm (j )m0(j) ( ) :

(11)

(12)

The  and z components for the magnetic fields in (7) and (10) can also be obtained as in writing (8), (9) and (11), (12). The interested reader can refer to [14] for more information. From the system of equations derived from the boundary conditions, an adequate number of relations between the unknown coefficients is generated, then, the unknown expansion coefficients of the scattered and internal electromagnetic fields can be determined. By substituting m ( ); m ( ) into the infinite series expansion of the scattered fields, the solution of the scattering of a shaped beam by an arbitrarily oriented multi-layered infinite cylinder can be obtained [14]. III. NUMERICAL RESULTS

The boundary conditions over the surface of each layer can be written as

Ew(j+1) = Ew(j) Ezw(j+1) = Ezw(j) Hw(j+1) = Hw(j) Hzw(j+1) = Hzw(j)

Fig. 2. Normalized differential scattering cross section k  ()=(16 ) for a two-layered infinite cylinder (kr = 12:57, 18.85, n = 1:5, 1.33, i = 1, 2) (solid line) and that for a three-layered one (kr = 12:57, 18.85, 25.13, n = 1:5, 1.4, 1.33, i = 1, 2, 3) (dotted line), all with Euler angles  = = 0, = =2 and z = 0, for incidence of the Gaussian beam (TE mode) with s = 0:15.

(10)

As described in (27) of [14], the behavior of the far-zone scattered fields will be discussed in this communication. By following the same theoretical procedure as in [14], it can be easy to know that, when considering Lm;TE =  ( 0 ) and Lm;TM = 0 as the beam waist radius w0 ! 1 for the special case of plane wave illumination, the method presented in this communication reduces to a standard Mie solution to the electromagnetic plane wave scattering by a multilayered infinite cylinder [4], [14]. We have found that, for a two-layered

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p

and three-layered infinite cylinder illuminated by a tightly focused Gaussian beam, the terms 0m ( )= sin  in the integrand of (28) in [14] are the slowly varying functions with  in the neighborhood of the 0 0 ( ) cos = sin  and m ( ) sin  in saddle point  = =2 [17], m 02 (29) and (30) are so small values (of order 10 or higher) compared with 0m ( )= sin  that they can be neglected in the calculations. Then, the asymptotic expressions of the scattered electric fields, under 1, also have the form of (35) in [14]. the condition that kr A comparison of the far-zone scattered fields for an incident plane wave with those for a tightly focused Gaussian shows that they are greatly different. The former are attenuated with increasing r due to the factor 1= r , a 2D problem, while the latter due to 1=r , demonstrating the 3D nature. We will calculate the differential scattering cross section which is defined by [14]

p

p

p



p

s () = 4r E E

2

0

2

16 = 2 k

1

2

m (01) exp(im)m (=2) :

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[10] J. A. Lock, “Scattering of a diagonally incident focused Gaussian beam by an infinitely long homogeneous circular cylinder,” J. Opt. Soc. Am. A., vol. 14, pp. 640–652, 1997. [11] G. Gouesbet, “Interaction between Gaussian beams and infinite cylinders, by using the theory of distributions,” J. Opt. (Paris), vol. 26, pp. 225–239, 1995. [12] G. Gouesbet and G. Gréhan, “Interaction between a Gaussian beam and an infinite cylinder with the use of non-separable potentials,” J. Opt. Soc. Am. A., vol. 11, pp. 3261–3273, 1994. [13] H. Y. Zhang, Y. P. Han, and G. X. Han, “Expansion of the electromagnetic fields of a shaped beam in terms of cylindrical vector wave functions,” J. Opt. Soc. Am. B., vol. 24, pp. 1383–1391, 2007. [14] H. Y. Zhang and Y. P. Han, “Scattering of shaped beam by an infinite cylinder of arbitrary orientation,” J. Opt. Soc. Am. B., vol. 25, pp. 131–135, 2008. [15] A. R. Edmonds, Angular Momentum in Quantum Mechanics. Princeton, NJ: Princeton University Press, 1957, ch. 4. [16] J. A. Stratton, Electromagnetic Theory. New York: McGraw-Hill, 1941. [17] C. T. Tai, Dyadic Green’s Functions in Electromagnetic Theory. Scranton, PA: International Textbook Company, 1971.

0

m=

01

(13)

Fig. 2 shows the normalized differential scattering cross sections

k2 ()=(16) for incidence of the Gaussian beam with s = 0:15 on a two-layered cylinder and a three-layered one.

Scattering of a Gaussian Beam by a Conducting Spheroidal Particle With Non-Confocal Dielectric Coating Huayong Zhang, Zhixiang Huang, and Yufa Sun

IV. CONCLUSION In the GLMT framework, an approach to compute the shaped beam scattering by an arbitrarily oriented multilayered infinite cylinder is given. By taking a tightly focused Gaussian beam at normal incidence as an example, the 3D nature of the scattering problem is shown. As a result, this study provides a possible analytic model for interpretation of shaped beam scattering phenomena for non-spherical particles of arbitrary orientation.

Abstract—A semi-analytic solution to the Gaussian beam scattering by a conducting spheroidal particle with non-confocal dielectric coating is obtained within the framework of the generalized Lorenz-Mie theory (GLMT). By virtue of a transformation between spheroidal and spherical vector wave functions, a theoretical procedure is developed to deal with the non-confocal boundary conditions. Numerical results of the normalized differential scattering cross section are evaluated for coated spheroidal particles. Index Terms—Conducting spheroidal particle with non-confocal dielectric coating, Gaussian beam, spheroidal vector wave function.

REFERENCES [1] J. R. Wait, “Scattering of a plane wave from a circular dielectric cylinder at oblique incidence,” Can. J. Phys., vol. 33, pp. 189–195, 1955. [2] M. Kerker, The Scattering of Light and Other Electromagnetic Radiation. New York: Academic Press, 1969. [3] H. C. Van de Hulst, Light Scattering by Small Particle. New York: John Wiley & Sons. Inc., 1981. [4] C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles. New York: Wiley, 1983. [5] R. X. Li, X. E. Han, and K. F. Ren, “Generalized Debye series expansion of electromagnetic plane wave scattering by an infinite multilayered cylinder at oblique incidence,” Phys. Rev. E., vol. 79, p. 036602(16), 2009. [6] N. G. Alexopoulos and P. K. Park, “Scattering of waves with normal amplitude distribution from cylinders,” IEEE. Trans. Antennas Propag., vol. 20, pp. 216–217, 1972. [7] S. Kozaki, “Scattering of a Gaussian beam by an inhomogeneous dielectric cylinder,” J. Opt. Soc. Am., vol. 72, pp. 1470–1474, 1982. [8] S. Kozaki, “A new expression for the scattering of a Gaussian beam by a conducting cylinder,” IEEE Trans. Antennas. Propag., vol. 30, pp. 881–887, 1982. [9] E. Zimmerman, R. Dandliker, N. Souli, and B. Krattiger, “Scattering of an off-axis Gaussian beam by a dielectric cylinder compared with a rigorous electromagnetic approach,” J. Opt. Soc. Am. A., vol. 12, pp. 398–403, 1995.

I. INTRODUCTION The scattering of an electromagnetic plane wave from a perfectly conducting spheroid bas been formulated with an exact analytic solution by Bowman et al. [1], and from a homogeneous spheroid with a semi-analytic solution by Asano and Yamamoto [2], [3]. Sebak and Sinha studied the same case, but for a conducting spheroidal object with a confocal dielectric coating at axial incidence [4]. Under some practical conditions, it is necessary to consider a shaped beam rather

Manuscript received May 28, 2010; revised February 04, 2011; accepted May 09, 2011. Date of publication August 12, 2011; date of current version November 02, 2011. This work is supported by the Key National Natural Science Foundation of China under contract 60931002 and by the 211 Project of Anhui University. The authors are with the Key Lab of Intelligent Computing and Signal Processing, Ministry of education, Anhui University, Hefei, Anhui 230039, China (e-mail: [email protected]). Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2164179

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are expanded in terms of the spheroidal vector wave functions attached to the conducting spheroid and the dielectric coating, respectively, as follows:

E

w

= E0

1 1

in mn

m=0 n=m

M

(c0 ; ; ; ) M (3) (c0; ; ; ) N (1) (c0; ; ; ) N (3) (c0; ; ; )

r (1) emn

+ + i +i

mn

r emn

mn

Ox y z

mn

Fig. 1. The Cartesian coordinate system is parallel to the Gaussian beam coordinate system , and the Cartesian coordinates of in are ( = 0 ). is obtained by a rigid-body rotation through a single Euler angles . A coated spheroidal particle is of (the major axis of the spheroid is along the axis of natural to ).

Oxyz x ;y Ox y z Oxyz

Oxyz ; z Oxyz

O



z

E

Oxyz

w

= E0

1 1 m=0 n=m

(1) in mn

r omn

M

r (1) emn

0

r omn

r omn

mn

0

(c1 ; ; ; ) r emn

mn

0

(3)

+ (1) M (3) (c1 ; ; ; ) + i (1) N (1) (c1 ; ; ; ) +i (1) N (3) (c1 ; ; ; ) mn

than the infinitely extended waves [5], [6]. Within the framework of the GLMT, the scattering of a spheroidal particle illuminated by a Gaussian beam was studied by Han and Wu, but the propagation direction of the incident beam is chosen to be parallel with the symmetry axis of the spheroid [7], [8]. In [9] and [10], we have obtained an expansion of a Gaussian beam, for any angle of incidence, as an infinite series of spheroidal vector wave functions, and then analyzed the Gaussian beam scattering by a spheroidal particle. It is motivated by [4] that this communication, based on our previous works, is devoted to the description of the Gaussian beam scattering by a conducting spheroidal particle with non-confocal dielectric coating.

r omn

(4)

0

where c = k f2 , c1 = k f1 , k = kn and n is the refractive index of the material of the dielectric coating relative to that of free space. For the sake of brevity, only the electric fields are written, and the corresponding expansions of the magnetic fields can be obtained with the following relations [12]

H = i!1 r 2 E; [M N]

mn

= k1 r 2 [N

M]

mn

II. FORMULATION Fig. 1 shows the geometry of the scatterer, which is obtained by replacing the homogeneous spheroid in Fig. 1 of [10] with a coated one. The conducting spheroidal particle and the dielectric coating are concentric but not necessarily confocal, so that the conducting and coating surfaces can be in different spheroidal coordinate systems. We denote the semifocal distance, semimajor and semiminor axes by f1 , a1 and b1 for the conducting spheroid, and by f2 , a2 and b2 for the outer surface of the dielectric coating. As in [10], the electromagnetic fields of the incident Gaussian beam, for the TE mode as an example, as well as the scattered fields can be represented by infinite series with the spheroidal vector wave functions attached to the dielectric coating, in the following forms [9]–[11]:

E = E0 i

E

s

= E0

1 1 m=0 n=m

1 1 m=0 n=m

M

in Gm n;T E

r (1) emn

+iG

m n;T M

in mn

M

r (3) emn

+i

mn

(c; ; ; )

N

r (1) omn

(c; ; ; )

(1)

With 1 and 2 as the radial coordinates of the boundary surfaces of the conducting spheroid and dielectric coating respectively, the boundary conditions on the surface  = 2 are described by.

Ei + Es = Ew ; Hi + Hs = Hw ; and on the surface 

Ew

r (3) omn

(c; ; ; )

(2)

m where c = kf2 , Gm n;T E and Gn;T M are the expansion coefficients for the incident Gaussian beam, and mn and mn for the scattered fields. To overcome the difficulty of concentric non-confocal boundary conditions, which does not arise for a conducting spheroid having a confocal layer [4], the electromagnetic fields within the dielectric coating

=E =H

w  w 

at 

= 2

(5)

= 1 by = 0;

Ew

=0

at



= 1 :

(6)

By virtue of the expansions for the incident Gaussian beam, scattered and internal fields, the above boundary conditions in (5) can be written as

(c; ; ; )

N

Ei + Es Hi + Hs

1 n=m

mn mn mn mn

mn mn

in [0]

=

1 n=m

in

0G 0G 0G 0G

0 0 0 0

Umn (c) Gm n;T M Vmn (c) (1);t (1);t Umn (c) Gm n;T E Vmn (c) (1);t (1);t Xmn (c) Gm n;T M Ymn (c) (1);t (1);t Xmn (c) Gm n;T E Ymn (c)

m n;T E m n;T M m n;T E m n;T M

(1);t

(1);t

(7)

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and the boundary conditions in (6) as

1 n=m

n

i

(1);t 1 (1);t Xmn (c1 ) Umn (c )

(1);t 1 (1);t Ymn (c1 )

(3);t 1 (3);t Xmn (c1 )

Vmn (c )

Umn (c )

2

(3);t 1 (3);t Ymn (c1 ) (1) mn (1) mn = (1)

mn (1) mn Vmn (c )

0 0

(8)

The matrix [0] in (7) is given by (9), shown at the bottom of the page. (j );t (j );t (j );t (j );t The parameters Umn , Vmn , Xmn , Ymn (j = 1 or 3 depending (j ) on the usage of the radial functions Rmn (c;  ) of the first or third kind) are given by Asano et al. in [2], and (7) and (8), given the value of m, are valid for each of t  0. Obviously, it is not sufficient to solve (7) and (8) for the unknown (1) (1) expansion coefficients mn , mn , mn , mn , mn , mn , mn , mn , (1) (1)

mn and mn . We have found that a transformation from spheroidal vector wave functions to spherical ones, all with respect to the system Oxyz , can be used to generate the other relations between them, which is of the form [13], [14]

j) Mr(mn j) Nr(mn =

(h) ;  ; ; 

c

(h) ;  ; ; 

c

1 0 m+q0n mn i dq

q=0;1

(h)

c

mr(mj) m+q nr(mj) m+q

0 0 (k R; ; )

(k R; ; )

(10)

where the superscript j takes, as already mentioned, the value 1 or 3 (j ) (j ) according to the radial functions Rmn (c;  ) and Rmn (k R) of the first ( h ) or third kind, and c is c or c1 . By substituting (10) into (3) and (4), which express the electric fields within the dielectric coating in the same spherical coordinate system, we can find that, for every q , the following formulae hold because of the orthogonality and linear indepenr(j ) r(j ) dence of m m+q (k R; ; ); m m+q (k R; ; ) (j = 1, 3, q = 0; 1; 2 . . . 1) [11], [12]

0

0

m

1 n=m;m+1

0

0

n

0

mn (c0 ) 0 dmn (c ) (1) (1) mn 0 mn qmn 1 mn dq (c )mn 0 dq (c1 )mn (1) mn 0 mn dq (c ) mn 0 dq (c1 ) mn (1) mn 0 mn dq (c )mn 0 dq (c1 )mn dq

(3);t (3) ;t Umn (c) (3) ;t Ymn (c) (3) ;t Xmn (c)

[0] =

For every value of m, we can truncate the infinite system of equations consisting of (7), (8) and (11) by setting n = m, m+1; . . . ; m+N , t = 0; 1; . . . ; N and q = 0; 1; . . . ; N , N being a suitable large number for a convergent solution, so that the total number of unknown coefficients is 10(N + 1). From the above system, an adequate number of relations between the unknown coefficients can be generated, and the standard numerical techniques may be employed to solve them [2], [4], [10]. III. NUMERICAL RESULTS Of practical interest is the behavior of the scattered wave at relatively large distances from the scatterer (far field), which can be deduced by taking the asymptotic form of s , as c ! 1, under the condition that c not equals zero. By neglecting terms of order higher than 1=r in r(3) (c;  ; ; ) and r(3) (c;  ; ; ), the asymptotic forms of the emn omn scattered electric field s can be obtained from (2). The differential scattering cross section is defined by [2], [4], [10]

M

=

0 0

:

(11)

0

(3);t (3);t Vmn (c) (3);t Xmn (c) (3) ;t Ymn (c) Umn (c)

(1);t (c0 ) 0Umn (1);t (c0 ) 0nVmn (1) 0Xmn;t (c0 ) (1);t (c0 ) 0nYmn

E

N E

0

The prime over the summation sign indicates that q takes even values when n 0 m is even and odd values when n 0 m is odd, and dqmn (c) are the spheroidal expansion coefficients. For the computation of dqmn (c), an algorithm using the recursive matrix equation is adopted in this communication [15].

Vmn (c)

Fig. 2. Normalized differential scattering cross sections  (; 0)= and  (; (=2))= for a conducting sphere with a dielectric coating (ka = 3, ka = 4, a =b = a =b = 1:0001, n = 1:414, = 0) illuminated by a plane wave.

 (; ) = 4r

2

Es 2 0

E

=

2

jT1(; )j2 + jT2 (; )j2





(12)

where

1

T (; ) =

1 1 m=0 n=m

m mn

Smn (c; cos  )

+mn

(3);t (c0 ) 0Umn (3);t (c0 ) 0nVmn (3) 0Xmn;t (c0 ) (3);t (c0 ) 0nYmn

(1);t (c0 ) 0Vmn (1);t (c0 ) 0nUmn (1) 0Ymn;t (c0 ) (1);t (c0 ) 0nXmn

sin  dSmn (c; cos  ) d

(3);t (c0 ) 0Vmn (3);t (c0 ) 0nUmn (3) 0Ymn;t (c0 ) (3);t (c0 ) 0nXmn

:

sin m

(13)

(9)

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sphere. Numerical results are in good agreement with those given by Sebak and Sinha [4]. Fig. 3 shows the normalized differential scattering cross section (; )=2 for a coated spheroidal particle illuminated by the Gaussian beam with w0 = 2. A comparison is plotted in Fig. 4 between the normalized differential scattering cross section  (; 0)=2 for a coated spheroid and that for a conducting one, all illuminated by a Gaussian beam with w0 = 2. From Fig. 4, we can see that the curve of the coated spheroid shows sharper oscillations due to interference of light diffracted with light transmitted by the particle, and that the curve of the conducting spheroid is much smoother for lack of the transmitted light.

IV. CONCLUSION

Fig. 3. Normalized differential scattering cross sections  (; 0)= and  (; (=2))= for a conducting spheroid with a dielectric coating (ka = 4, ka = 6, a =b = a =b = 2, n = 1:33, = (=6)) for incidence of a Gaussian beam with w = 2.

0

Within the framework of the GLMT, an approach to compute the scattering of a Gaussian beam by a conducting spheroid with concentric non-confocal dielectric coating is presented. The electromagnetic fields within the dielectric coating are expanded in terms of the spheroidal vector wave functions with respect to appropriate different spheroidal coordinates, and then translated into the spherical ones. As a result, this study extends electromagnetic plane wave scattering by a conducting spheroidal object with confocal dielectric coating to a more general non-confocal case for arbitrary incidence of a Gaussian beam.

REFERENCES

Fig. 4. Normalized differential scattering cross section  (; 0)= for a conducting spheroid (ka = 6, kb = 3) (dotted line) and that for a coated one (ka = 6, ka = 8, a =b = a =b = 2, n = 1:33) (solid line), all with = (=4) and illuminated by a Gaussian beam of w = 2.

0

T2 (; ) =

1 1

S (c; cos ) mmn mn sin  m=0 n=m dSmn (c; cos ) + mn d

cos m:

(14)

In this communication, the normalized differential scattering cross section  (; )=2 is thereafter evaluated in the spherical coordinate system attached to the coated spheroid. In the following calculations, the incident Gaussian beam is TE polarized, and the middle of its beam waist coincides with the center of the coated spheroid at origin O(x0 = x0 = z0 = 0) [10]. Fig. 2 shows the normalized differential scattering cross section (; )=2 for incidence of a plane wave (w0 = ) on a coated

1

[1] J. J. Bowman, T. B. A. Senior, and P. L. E. Uslenghi, Electromagnetic and Acoustic Scattering by Simple Shapes. New York: Hemisphere Publishing Corporation, 1987. [2] S. Asano and G. Yamamoto, “Light scattering by a spheroid particle,” Appl. Opt., vol. 14, pp. 29–49, 1975. [3] S. Asano, “Light scattering properties of spheroidal particles,” Appl. Opt., vol. 18, pp. 712–723, 1979. [4] A. R. Sebak and B. P. Sinha, “Scattering by a conducting spheroidal object with dielectric coating at axial incidence,” IEEE Trans. Antennas Propag., vol. 40, pp. 268–273, 1992. [5] F. Xu, K. F. Ren, G. Gouesbet, X. Cai, and G. Gréhan, “Theoretical prediction of radiation pressure force exerted on a spheroid by an arbitrarily shaped beam,” Phys. Rev. E, vol. 75, p. 026613, 2007. [6] F. Xu, J. A. Lock, G. Gouesbet, and C. Tropea, “Radiation torque exerted on a spheroid: Analytical solution,” Phy. Rev. A, vol. 78, p. 013843, 2008. [7] Y. P. Han and Z. S. Wu, “The expansion coefficients of a spheroidal particle illuminated by Gaussian beam,” IEEE Trans. Antennas Propag., vol. 49, pp. 615–620, 2001. [8] Y. P. Han and Z. S. Wu, “Scattering of a spheroidal particle illuminated by a Gaussian beam,” Appl. Opt., vol. 40, pp. 2501–2509, 2001. [9] Y. P. Han, H. Y. Zhang, and G. X. Han, “The expansion coefficients of arbitrarily shaped beam in oblique illumination,” Opt. Express, vol. 15, pp. 735–746, 2007. [10] H. Zhang and Y. Sun, “Scattering by a spheroidal particle illuminated with a Gaussian beam described by a localized beam model,” J. Opt. Soc. Am. B., vol. 27, pp. 883–887, 2010. [11] C. Flammer, Spheroidal Wave Functions. Stanford, California: Stanford University Press, 1957. [12] J. A. Stratton, Electromagnetic Theory. New York: McGraw-Hill, 1941. [13] J. Dalmas and R. Deleuil, “Translational addition theorems for prolate and ,” Q. Appl. Math., vol. spheroidal vector wave functions 44, pp. 213–222, 1986. [14] J. Dalmas and R. Deleuil, “Multiple scattering of electromagnetic waves from two prolate spheroids with perpendicular area of revolution,” Radio Sci., vol. 28, pp. 105–119, 1993. [15] L.-W. Li, M.-S. Leong, T.-S. Yeo, P.-S. Kooi, and K.-Y. Tan, “Computations of spheroidal harmonics with complex arguments: A review with an algorithm,” Phys. Rev. E, vol. 58, pp. 6792–6806, 1998.

M

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Electromagnetic Wave Scattering of a High-Order Bessel Vortex Beam by a Dielectric Sphere Farid G. Mitri

Abstract—This study investigates the arbitrary scattering of an unpolarized electromagnetic (EM) high-order Bessel vortex (helicoidal) beam (HOBVB) by a homogeneous water sphere in air. The radial components of the electric and magnetic scattering fields are expressed using partial wave series involving the beam-shape coefficients and the scattering coefficients of the sphere. The magnitude of the 3D electric and magnetic scattering directivity plots in the far-field region are evaluated using a numerical integration procedure for cases where the sphere is centered on the beam’s axis and shifted off-axially with particular emphasis on the half-conical angle of the wave number components and the order (or helicity) of the beam. Some properties of the EM scattering of an HOBVB by the water sphere are discussed. The results are of some the scattering of importance in applications involving EM HOBVBs by a spherical object. Index Terms—Dielectric sphere, electromagnetic scattering, high-order Bessel vortex beam.

I. INTRODUCTION The scattering of electromagnetic (EM) radiation by particles plays an essential role in wide-ranging applications including (but not limited to) biomedicine [1], radar sensing [2], Raman scattering [3], laser fusion [4], and free-space communications [5] to name a few. In most applications, the incident beam is composed of plane EM waves [6], so as the scattering does not depend on the beam’s parameters. However, when the incident EM radiation is in the form of a Gaussian beam [7], [8], or a zeroth-order (non-vortex) Bessel beam (ZOBB) [9], the scattering is significantly affected by the beam’s geometry. For instance, because the ZOBB has a very narrow nondiffracting central core, it has been recently used to advantage for 3D subcellular microspcopic imaging with a resolution of 0:3 m [10]. Beside these types of beams, there exist other categories whose wave front is a vortex [11]–[13]. Such a wave has an axial phase singularity and the amplitude vanishes on the axis of the beam. A particular example includes the non-diffracting high-order Bessel vortex beam (HOBVB) [14] for which the transverse field (or the shape of a “slice” of the beam perpendicular to the propagation axis) is described by the cylindrical Bessel function of the first kind Jm (1) of order m > 0. The parameter m can be any negative or positive real number and denotes the order (or topological charge) of the beam that determines its helicity (see Fig. 1 in [15]). Generally m is an integer number (i.e., m 2 , where is the set of all integers) and the beam satisfies the Helmholtz wave equation. In particle manipulation applications, it provides significant advantages in particle alignment over extended distances [16], [17]. Furthermore, a HOBVB propagates over a characteristic length without spreading

Fig. 1. Geometry of the scattering problem. A homogeneous spherical particle is centered along the axis of wave propagation of an EM high-order Bessel vortex beam. The parameter is the half-cone angle of the incident beam and  and  are the polar and azimuthal angles, respectively.

[18] and possesses the property of self reconstruction [19], [20]. However, when m becomes fractional [21], the beam does not satisfy the Helmholtz equation any longer. It possesses an intrinsic opening gap across concentric intensity rings [22]. Motivated by the features and applications of such a beam, analytical and numerical analyses are undertaken here to calculate the arbitrary scattering of an unpolarized EM HOBVB of integer order m 2 by a homogeneous water drop in air. The scattering of a HOBVB with fractional order by a sphere is outside the scope of the present study. II. METHOD Based on the studies presented in [8], [23], the radial component of the (steady-state) scattered electric and magnetic fields can be expressed, respectively, as,

rsc: ( rsc: (

) r; ; )

E

r; ; 

H

r

2

p

(p + 1)

2

0

p(1) (

kr

pq ( pq ( pq (

) )

a

ka

b

ka

)Y

(1)

)

; 

where k = k0 " 1=2 , k0 = !=c, where ! and c are the angular frequency and speed of the EM wave in vacuum, " 6= 0 is the dielectric constant of the medium, a is the radius of the sphere, r is the distance (1) to a point in space, p (1) is the spherical Riccati-Hankel function of q the first kind, Yp (; ) are the spherical harmonics, and apq (ka) and bpq (ka) are the coupled scattering/beam-shape coefficients for a water sphere in air given in a compact form by [8], [23],

pq ( pq (

) )

a

ka

b

ka

p0 (  p0 (  p ( 1 pq ( ) pq ( )

n ka

Manuscript received March 02, 2011; revised April 05, 2011; accepted April 14, 2011. Date of publication August 12, 2011; date of current version November 02, 2011. This work was supported in part by a Director’s fellowship (LDRD-X9N9, Project # 20100595PRD1) from Los Alamos National Laboratory. Disclosure: this unclassified publication, with the following reference no. LA-UR 11-10347, has been approved for unlimited public release under DUSA ENSCI. The author is with the Los Alamos National Laboratory, MPA-11, Sensors & Electrochemical Devices, Acoustics & Sensors Technology Team, MS D429, Los Alamos, NM 87545 USA (e-mail: [email protected]). Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2164228

1 p 2 p=1 q=0p

a

=

n

=

n

2

) )

n ka

A

ka

ka

ka

)0

) p(1)0 (ka) 0

n ka

B

p(

p ( ) p ( ) 1 p0 ( 



n

n ka

n ka

n

n ka

p0 (

ka

)

) p(1) (ka)

(2)

where p (1) is the spherical Riccati-Bessel function of the first kind and the prime denotes a derivative with respect to the argument, the pa is the relative refractive index of the medium of wave proprameter n

0018-926X/$26.00 © 2011 IEEE

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[24], [25], the components are derived and expressed in (4)–(9), shown at the bottom of the page. All EM fields components listed in (4)–(9) are assumed to vary in time as exp(0i!t), however, for convenience the complex exponential is dropped from all subsequent expressions. The parameter E0 = ikA0 , kz = k cos , R = x2 + y2 , is the radial distance to a point in the transverse plane (x; y ), and  = tan01 (y=x) (see Fig. 1). Therefore, the radial components of the incident electric and magnetic fields can be evaluated. Their expressions are given by [26], [27]

agation (that is generally complex), and the beam-shape coefficients Apq (ka) and Bpq (ka) are given, respectively, by [8], [23],

Apq (ka) Bpq (ka)

1

=

p (p + 1) p (ka) 2  Erinc: (a; ; ) q3 2 Hrinc: (a; ; ) Yp (; ) sin  d d

(3)

0 0

where the superscript 3 denotes Erinc: (a; ; ) ; Hrinc: (a; ; ), are

a complex conjugate, and the radial components of the incident electric and magnetic fields evaluated at the radius of the sphere. As noted in (3), the incident electric and magnetic fields need to be determined to properly solve for the scattering problem. One may use the scalar wave theory that provides fairly good results only if the size of the central spot of the beam is much larger than the wavelength [24], [25] [i.e., kR (= k sin )  k , where is the half-cone angle of the HOBVB (see Fig. 1)]. In other words, the scalar wave theory may be applicable for small half-cone angle values (typically < 10 ). However, it has been shown [24], [25] that the vector nature of EM wave propagation introduces significant corrections and should be used instead for a complete analysis of circularly symmetric beams, such as the HOBVB. Thus, the Cartesian components of the electric and magnetic fields are determined prior to the determination of their radial counterparts. After some arithmetic manipulation following the method in

Ex jm2

=

1 2

E0

Erinc: (a; ; ) Hrinc: (a; ; )

+ +

E0 xy 2 2

1

x iE0 kR 2

1

1+

Hx jm2 "01=2 = Ey jm2 Hy jm2 "

=

1 2

E0

(

)

k R

+

01=2

r=a

i k z + m)] k 0 k x + m(m01)(x0iy) 1+ Jm (kR R) 2 k (yk 0x 0k 2Rimxy) k R Jm+1 (kR R) 0 k R

k

=

r=a

sin

 cos 

sin

 sin 

cos

:

(10)

(11)

exp [ ( z

m(m01) 2+i

Ez jm2

r =a

p (`) ! sin (` 0 (p=2)) p(1) (`) ! i0(p+1) exp (i`) :

i k z + m)]

=

Exinc: Hxinc: Eyinc: Hyinc: Ezinc: Hzinc:

It is common to investigate the scattered electric and magnetic fields in the far-field region, however (1) can be used to compute the scattered fields at any distance r from the sphere (in the near-field or the farfield). In the far-field (kr 0 ! 1), the expression for the scattered field is further simplified since

exp [ ( z

Ey jm2

=

kz k

2+im

(

k R

0k R

(4)

Jm (kR R)

(5)

)

Jm+1 (kR R)

i k z m)] 2 Jm (kR R) 0 kR Jm+1 (kR R)

exp [ ( z + m(10i ) R

(6) (7)

i k z + m)] k 0 k y + m(m01)(y +ix) 1+ Jm (kR R) k R k R 2 k (xk 0y +2 imxy ) Jm+1 (kR R) 0 k R

exp [ ( z

y Hz jm2 "01=2 = 12 iE0 kR

1+

kz k

i k z m)] Jm (kR R) 0 kR Jm+1 (kR R) 2

exp [ ( z + m 1+i R

(8)

:

(9)

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In that limit, (1) is simplified, and the scattering form functions for the electric and magnetic fields can be written as [9]

f E (ka; ; ) f H (ka; ; )

1 1

=

1

p

0

p (p + 1) i (p+1) apq (ka) : (12) q p 2 bpq (ka) Yp (; )

p=1 q=0

Upon the substitution of (2)–(10) into (1) using (11), the magnitude (or phase) of the scattering form functions [i.e., (12)] can be evaluated. III. RESULTS AND DISCUSSION To illustrate the theory with some examples, a computer program is developed using MATLAB software package. Magnitude profiles for the on- and off-axial 3D scattering directivity patterns are plotted for the radial components of the scattered electric and magnetic fields. Accurate computation of the spherical Riccati-Bessel/Hankel functions and their derivatives is achieved using modified versions of the specialized math functions “besselj,” “bessely” and “besselh” within the software package. The computations are performed on a personal computer with a truncation constant largely exceeding ka to ensure proper convergence of the series. Calculation of the beam-shape coefficients requires determining the surface integrals in (3). The numerical procedure consists of sampling the integrands (i.e., the radial components of the incident electric and magnetic fields) over the sphere’s surface. The integrals are then evaluated by quadrature based on a Riemann sum in the MATLAB software package. It is important to emphasize that dense grids (sampled here at 225 uniformly distributed points over the sphere’s surface) are required in both the  and  directions to obtain proper convergence of the numerical integrations. In the simulations, the value of the parameters are E0 = " = 1 and the (complex) relative 6  = 1:33 + 5 2 10 i [28] for a index of refraction of the medium is n water sphere in air. Furthermore, inspection of (3) shows that the denominators involve the spherical Riccati-Bessel function of the first kind p (ka). A particular care needs then to be paid to the choice of ka so as to avoid the zeros of p (ka), hence, the resulting indeterminacies that could appear while evaluating the integrals. For numerical convenience here, appropriate selection of ka requires excluding those to corresponding zeros of p (ka), which are listed in Table 1 of [29]. On the other hand, another way to circumvent this problem requires evaluating the beam-shape coefficients over a “control sphere” of radius b that encloses the scatterer of radius a such that b > a. In strict sense, the beam-shape coefficients describe the beam’s characteristics in the spherical coordinate system and are defined independently of the presence of the scatterer. They are used to reconstruct the incident beam field and are not connected with the particle size. This approach, however, may be used in an isotropic medium in which diffraction in the space between the two spheres as well as dissipation are negligible so that the incident EM field components evaluated at r = b are equivalent to those evaluated at r = a. With this in mind, (3) may be rewritten as

Fig. 2. Magnitude of the form functions for a first-order Bessel vortex beam , for the scattered electric [(a)-(c)] and magnetic [(d)-(f)] farwith fields. The sphere is centered on the beam’s axis. The plots in (a) and (d) corre: , (b) and (e) to ka , respectively. , (c) and (f) to ka spond to ka

= 55 =15

=5

= 12

0

Apq (kb) Bpq (kb)

=

2

1

p (p + 1) p (kb)

2  0 0

Erinc: (a; ; ) Hrinc: (a; ; )

3

Ypq (; ) sin  d d (13)

without inconvenience. Nevertheless, one has to select an appropriate radius b for the control sphere to not coincide with any of the zeros of

Fig. 3. The same as in Fig. 2, however, the sphere is shifted off the axis of the incident beam in both x and y directions such that the offset (in arbitrary units) : : . is x; y

( ) 0 oset = (0 3; 0 1)

p (kb). In contrast, this problem appears to be resolved in the description of EM beams using the formalism of the generalized Lorenz-Mie theories (GLMTs) [30]. Nevertheless, the evaluation of the beam-shape coefficients using the GLMTs, requires tedious numerical integration procedures. Computational plots for the 3D scattering directivity patterns for the magnitude of the electric and magnetic far-field form functions of an EM first-order (m = 1) Bessel vortex beam (FOBVB) with a half-cone angle = 55 incident upon a water sphere in air are displayed in Fig. 2. The plots in (a) and (d) correspond to ka = 1:5, (b) and (e) to ka = 5, (c) and (f) to ka = 12, respectively. In this case, the homogeneous water sphere is centered on the beam’s axis. The arrows on the left-hand side of each panel indicate the direction of the incident wavesbeam. One notices the significant differences in the 3D scattering directivity patterns for different ka values. The backscattering (in the direction  = 180 ) becomes more directional as ka increases. This is not the case when the water sphere is shifted off the axis of the EM FOBVB; to illustrate this effect, the 3D scattering directivity patterns are displayed in Fig. 3. For this example, the sphere is shifted





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Fig. 4. The same as in Fig. 2, however, the sphere is centered on the axis of a second-order Bessel vortex beam. Note the vanishing of the forward and backand  . ward scattering in the directions 

=0

= 180

in both x and y directions such that the offset is (x; y ) 0 oset = (0:3; 0:1). It is obvious that the 3D scattering directivity patterns for the off-axial scattering of the electric and magnetic far-fields show significant differences from the axial patterns displayed in Fig. 2. Additional numerical computations (not shown here) have shown that the spatial distributions for the off-axial scattering take particular directivity patterns determined by the amount of the offset. Additional computations are performed to investigate the effect of varying the order m of the beam on the 3D scattering directivity patterns. Fig. 4 shows the 3D axial scattering directivity patterns for the magnitude of the electric and magnetic far-field form functions of an EM second-order (m = 2) Bessel vortex beam (SOBVB) with a halfcone angle = 55 incident upon a water sphere in air. The plots in (a) and (d) correspond to ka = 1:5, (b) and (e) to ka = 5, (c) and (f) to ka = 12, respectively. Comparison of Fig. 4 with Fig. 2 shows the significant differences in the 3D scattering directivity patterns when the order m of the beam changes; as shown Fig. 4, both the forward   ( = 0 ) and backward ( = 180 ) scattering vanish for the electric and magnetic fields when the sphere is centered on the beam’s axis. This is an interesting property for the axial EM scattering of a SOBVB by a sphere. To investigate the effect of varying the half-cone angle on the 3D scattering directivity patterns, further calculations for the off-axial [(x; y ) 0 oset = (0:3; 0:1)] far-field electric and magnetic scattering are performed. The results are displayed in Fig. 5 for a SOBVB at ka = 12, and for three half-cone angle values; = 10 [(a), (d)], = 40 [(b), (e)], and = 85 [(c), (f)], respectively. As observed from this figure, the off-axial scattering directivity patterns in the far-field are sensitive to the variations of so as the scattered fields take particular directivity patterns determined by the amount of the half-cone angle. In all the preceding examples, the water particle is considered an ideal isotropic and homogeneous sphere. However, the present analysis is still valid to investigate the scattering from other types of spheres [31]–[35] provided that their appropriate scattering coefficients are used. Furthermore, it is important to emphasize that the present theoretical analysis is suitable to investigate the on- and off-axial EM scattering for

Fig. 5. Magnitude of the far-field scattering form functions for the scattered electric [(a)–(c)] and magnetic [(d)–(f)] far-fields from a homogeneous water : : versus the axis of an sphere placed off-axially x; y for three values of the incident second-order Bessel vortex beam at ka half-cone angle . In (a), (d), , in (b), (e), , and in (c), (f), , respectively.

= 85

[( ) 0 oset = (0 3; 0 1)] = 12 = 10 = 40

any beam that satisfies the homogeneous (source-free) Helmholtz wave equation. IV. CONCLUSION In summary, a quantitative analysis for the axial and off-axial farfield scattering from a homogeneous water sphere placed in the field of an unpolarized EM HOBVB is provided. The scattering coefficients of the sphere and the 3D scattering directivity plots are evaluated using a numerical integration procedure. The calculations indicate that the scattering directivity patterns are strongly dependent upon the position of the sphere facing the incident field as well as the HOBVB parameters. In addition to providing physical insight into the on- and off-axial EM scattering phenomenon, this investigation can provide a useful test of finite element codes for the evaluation of the scattering phenomenon, which is important in related applications involving the evaluation of the radiation force and torque of EM HOBVBs.

REFERENCES [1] “SPIE Proceedings on Biomedical Applications of Light Scattering IV” 2010, vol. 7573, pp. 757301–757319. [2] J. Kasparian, M. Rodriguez, G. Mejean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y. B. Andre, A. Mysyrowicz, R. Sauerbrey, J. P. Wolf, and L. Woste, “White-light filaments for atmospheric analysis,” Science, vol. 301, no. 5629, pp. 61–64, 2003. [3] J. G. Grasselli, M. K. Snavely, and B. J. Bulkin, “Applications of Raman-spectroscopy,” Phy. Rep.-Rev. Sect. Phys. Lett., vol. 65, no. 4, pp. 231–344, 1980. [4] A. Pukhov, “Strong field interaction of laser radiation,” Rep. Progr. Phys., vol. 66, no. 1, pp. 47–101, 2003. [5] B. R. Strickland, M. J. Lavan, E. Woodbridge, and V. Chan, “Effects of fog on the bit-error rate of a free-space laser communication system,” Appl. Opt., vol. 38, no. 3, pp. 424–431, 1999. [6] J. D. Murphy, P. J. Moser, A. Nagl, and H. Uberall, “A surface-wave interpretation for the resonances of a dielectric sphere,” IEEE Trans. Antennas Propag., vol. 28, no. 6, pp. 924–927, 1980. [7] J. A. Lock, “Improved Gaussian beam-scattering algorithm,” Appl. Opt., vol. 34, no. 3, pp. 559–570, 1995. [8] J. P. Barton, D. R. Alexander, and S. A. Schaub, “Internal and near-surface electromagnetic-fields for a spherical-particle irradiated by a focused laser-beam,” J. Appl. Phys., vol. 64, no. 4, pp. 1632–1639, 1988.

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[9] F. G. Mitri, “Arbitrary scattering of an electromagnetic zero-order Bessel beam by a dielectric sphere,” Opt. Lett., vol. 36, no. 5, pp. 766–768, 2011. [10] T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nature Meth., vol. 8, no. 5, pp. 417–423, 2011. [11] M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A—Pure Appl. Opt., vol. 6, no. 2, pp. 259–268, 2004. [12] D. McGloin and K. Dholakia, “Bessel beams: Diffraction in a new light,” Contemp. Phys., vol. 46, no. 1, pp. 15–28, 2005. [13] J. C. Gutierrez-Vega and C. Lopez-Mariscal, “Nondiffracting vortex beams with continuous orbital angular momentum order dependence,” J. Opt. A-Pure Appl. Opt., vol. 10, no. 1, p. 015009, 2008. [14] C. Lopez-Mariscal and J. C. Gutierrez-Vega, “The generation of nondiffracting beams using inexpensive computer-generated holograms,” Amer. J. Phys., vol. 75, no. 1, pp. 36–42, 2007. [15] L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett., vol. 96, no. 16, p. 163905, 2006. [16] J. Arlt, T. Hitomi, and K. Dholakia, “Atom guiding along LaguerreGaussian and Bessel light beams,” Appl. Phys. B-Lasers Opt., vol. 71, no. 4, pp. 549–554, 2000. [17] J. Arlt, K. Dholakia, J. Soneson, and E. M. Wright, “Optical dipole traps and atomic waveguides based on Bessel light beams,” Phys. Rev. A, vol. 63, no. 6, p. 063602, 2001. [18] J. Durnin, J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett., vol. 58, no. 15, pp. 1499–1501, 1987. [19] V. Garces-Chavez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature, vol. 419, no. 6903, pp. 145–147, 2002. [20] F. O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with selfreconstructing beams,” Nat Photon, vol. 4, no. 11, pp. 780–785, 2010. [21] S. H. Tao, W. M. Lee, and X. C. Yuan, “Dynamic optical manipulation with a higher-order fractional Bessel beam generated from a spatial light modulator,” Opt. Lett., vol. 28, no. 20, pp. 1867–1869, 2003. [22] S. H. Tao, W. M. Lee, and X. C. Yuan, “Experimental study of holographic generation of fractional Bessel beams,” Appl. Opt., vol. 43, no. 1, pp. 122–126, 2004. [23] J. P. Barton, D. R. Alexander, and S. A. Schaub, “Theoretical determination of net-radiation force and torque for a spherical-particle illuminated by a focused laser-beam,” J. Appl. Phys., vol. 66, no. 10, pp. 4594–4602, 1989. [24] S. R. Mishra, “A vector wave analysis of a Bessel beam,” Opt. Commun., vol. 85, no. 2,3, pp. 159–161, 1991. [25] F. G. Mitri, “Vector wave analysis of an electromagnetic high-order Bessel vortex beam of fractional type alpha,” Opt. Lett., vol. 36, no. 5, pp. 606–608, 2011. [26] S. A. Schaub, J. P. Barton, and D. R. Alexander, “Simplified scattering coefficient expressions for a spherical particle located on the propagation axis of a fifth-order Gaussian beam,” Appl. Phys. Lett., vol. 55, no. 26, pp. 2709–2711, 1989. [27] S. A. Schaub, J. P. Barton, and D. R. Alexander, “Erratum: Simplified scattering coefficient expressions for a spherical particle located on the propagation axis of a fifth-order Gaussian beam (Appl. Phys. Lett. 55, 2709 (1989)),” Appl. Phys. Lett., vol. 59, no. 14, pp. 1798–1798, 1991. [28] G. M. Hale and M. R. Querry, “Optical constants of water in the 200-nm to 200- wavelength region,” Appl. Opt., vol. 12, no. 3, pp. 555–563, 1973. [29] F. G. Mitri and G. T. Silva, “Off-axial acoustic scattering of a highorder Bessel vortex beam by a rigid sphere,” Wave Motion, vol. 48, no. 5, pp. 392–400, 2011. [30] G. Gouesbet, J. A. Lock, and G. Gréhan, “Generalized Lorenz-Mie theories and description of electromagnetic arbitrary shaped beams: Localized approximations and localized beam models, a review,” J. Quant. Spectr. Rad. Transf., vol. 112, no. 1, pp. 1–27, 2011. [31] A. Brunsting and P. F. Mullaney, “Light scattering from coated spheres: Model for biological cells,” Appl. Opt., vol. 11, no. 3, pp. 675–680, 1972. [32] V. V. Varadan, A. Lakhtakia, and V. K. Varadan, “Scattering by threedimensional anisotropic scatterers,” IEEE Trans. Antennas Propag., vol. 37, no. 6, pp. 800–802, 1989.

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[33] Y. L. Geng and X. B. Wu, “A plane electromagnetic wave scattering by a ferrite sphere,” J. Electromagn. Waves Applicat., vol. 18, no. 2, pp. 161–179, 2004. [34] Q.-K. Yuan, Z.-S. Wu, and Z.-j Li, “Electromagnetic scattering for a uniaxial anisotropic sphere in an off-axis obliquely incident Gaussian beam,” J. Opt. Soc. Am. A, vol. 27, no. 6, pp. 1457–1465, 2010. [35] B. R. Johnson, “Light scattering by a multilayer sphere,” Appl. Opt., vol. 35, no. 18, pp. 3286–3296, 1996.

Absorption Loss Reduction in a Mobile Terminal With Switchable Monopole Antennas Markus Berg, Marko Sonkki, and Erkki T. Salonen

Abstract—A method for compensating part of the user effect on a mobile terminal antenna was experimentally evaluated. A dynamic selection between two separately located antennas was used to reduce absorption caused by the user’s index finger. The method was examined in the 900 MHz and 1900 MHz bands with dual-band monopole antennas. The performance of the method was evaluated in terms of antenna total efficiency, body loss, mismatch loss, and absorption loss. It was found that a 3.1 dB reduction in average body loss is achieved in the low band if feed is switched to the unloaded antenna instead of the loaded one. The corresponding reduction in the higher band is 1.0 dB. Index Terms—Active antennas, antenna measurements, body loss, mobile antennas.

I. INTRODUCTION The performance of a mobile terminal antenna strongly depends on its operating environment. An antenna that is designed and optimized to operate excellently in a certain situation may exhibit moderate or unsatisfactory performance in other situations. Typical talking and browsing modes are disadvantageous in terms of total antenna efficiency. The user’s hand and head in the close vicinity of the antenna typically both tune the antenna’s resonance frequency and absorb a significant part of the power originating from the antenna. Research interest in the area of the user effect on mobile terminal antennas has recently grown, and user-induced loss is widely investigated. It has been found that antenna performance degradation caused by the user is often remarkably high [1]–[4]. Experimental results have shown up to 26 dB body loss in the worst case situation. Body loss is comprised of mismatch loss and absorption loss caused by the user’s head and hand in the proximity of the antenna [5] and the chassis of the terminal. A comparison of the user effect between different mobile terminal antenna types has been made in typical mobile terminal frequency bands [1]. Losses in the low band (frequencies near 900 MHz) were typically higher than losses in the high band (frequencies from 1700 MHz to 2100 MHz). Measured absorption loss was higher than mismatch loss with all antenna types used, without exception.

Manuscript received May 11, 2010; revised January 05, 2011; accepted April 04, 2011. Date of publication August 12, 2011; date of current version November 02, 2011. This work was supported by the Finnish Funding Agency for Technology and Innovation, Nokia Devices, and Pulse Finland Oy. The authors are with the Centre for Wireless Communications (CWC), University of Oulu, FI-90014, Finland (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2011.2164178

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Fig. 2. Finger position and height from the antenna, front and side views.

Fig. 1. Dimensions of the dual-monopole antenna structure and location of coaxial cable.

Different user hand grips and their impact on antenna input impedance variation and total efficiency were investigated in [6] and [7]. It was found that a phone grip where the index finger is located on top of the antenna element represents a significant portion of hand grips [6]. Moreover, in [8] it was found that the index and middle fingers located on top of the antenna have a significant impact on absorbed power. Although the user effect on mobile terminal antennas has been widely investigated, not many solutions for reducing or compensating for the user effect have been proposed. The majority of solutions are related to adjustable impedance tuning that compensates for user-induced mismatch loss. One example is dynamic antenna matching circuitry [9], where a total increase in efficiency from 2 dB to 4 dB compared with the situation without re-matching was reported. Dynamic impedance matching reduces only mismatch loss, and thus the effect of the finger is compensated only partially. This paper presents an approach to compensating absorption to the user’s index finger on top of the antenna. The proposed method includes two antenna elements with a capability of dynamic switching between them. Evaluation of the user effect compensation is based on measurements of total efficiency and reflection loss. A hand phantom was used to model two different hand grips where the index finger is located on one antenna element. II. ANTENNA STRUCTURE The antenna structure includes two monopole antenna elements and a ground plane, shown in Fig. 1. The elements and the ground plane were made on an RO4003 high-frequency circuit laminate with a thickness of 0.8 mm. Only one-sided copper coating was used. The antennas were shaped to obtain the first resonance frequency at 900 MHz and the second resonance frequency at 1950 MHz. To achieve an acceptable impedance bandwidth, particularly in the lower frequency band, the antenna elements were located 8 mm above the ground plane and the copper was removed from the ground plane underneath the elements. To study the effect of the non-active antenna termination on overall performance, the antenna structure was fabricated with coaxial cables (with a length of 100 mm) connected to both antenna elements. The feed point of antenna element is shown in Fig. 1. The effect of the non-active antenna termination was studied in terms of S11 -parameter and total efficiency. During the measurements one antenna element was active at a time while the other antenna element

was terminated with coaxial cable terminations (open, short, and load). Terminations were applied to the open end of the 100 mm long coaxial cable attached to the antenna feeding point shown in Fig. 1. The open termination resulted in the highest total efficiency in the low band and sufficient efficiency in the high band. Termination with a load yielded low total efficiency in both low and high bands, since power coupled to the non-active element was absorbed to the termination. Therefore the open termination was selected as the most useful in this particular study. The selected termination corresponds to the OFF state of the reflective type of single-pole double-through (SPDT) RF switch. In practice the antenna feed is realized with a SPDT switch that connects the input RF signal to either the left or right antenna. When one antenna is active the other is open-terminated because it is disconnected from the signal feed line. III. MEASUREMENTS A. Test Setup The performance of the prototype was evaluated in terms of S11 -parameter and total efficiency measurements. A phantom hand (IXB090R) was used to model two different user hand grips. In the first grip the tip of the index finger was positioned on the left antenna element (#1), as shown in Fig. 2. The position of the finger tip was then moved onto the right element to represent the second hand grip (#2). The accurate finger tip position in relation to the antenna is shown in the same figure. The distance between the finger tip and the antenna was 3 mm during the measurements. For comparison, the antenna was measured without the hand phantom in free space. To achieve accurate loss evaluation, both the impedance and total efficiency measurements were done at the same time with one hand grip. Total efficiency and S -parameters were measured with a Satimo Starlab and a Vector Network Analyzer (VNA), respectively. B. Definition of Losses User-induced losses were calculated from the measurement data (total efficiency and S11 parameter). Radiation efficiency ("rad ) is defined as the ratio of the total power radiated by an antenna to the power accepted by the antenna from the connected transmitter. Total efficiency ("tot ) is the relationship between the radiated power and the power transmitted to the antenna connector. It includes all the losses in the measured antenna structure and the losses caused by the usage environment in this case. Total efficiency can be written as

"tot = "rad

1

0 jS11j2

;

(1)

where (1 0 jS11 j2 ) is the impedance mismatch factor. Antenna mismatch loss is defined as

Lmismatch = 010 2 log

1

0 jS11j2

:

(2)

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TABLE I AVERAGE TOTAL EFFICIENCY AND AVERAGE LOSSES AT STUDIED OPERATION SITUATIONS

Body loss (BL) is the difference between measured total efficiency without ("tot;f s ) and with the user ("tot;user ) and is defined as BL = "tot;f s

0"

tot;user :

(3)

Since the radiation efficiency does not include the mismatch loss, body loss is addition of absorption loss and user influenced change in mismatch loss. Body absorption loss (Labs ) does not comprise mismatch loss and it is defined in terms of measured radiation efficiency without ("rad;f s ) and with ("rad;user ) the user Labs = "rad;f s

0"

rad;user :

(4)

IV. USER EFFECT MEASUREMENTS The measured total antenna efficiency, body loss, mismatch loss, and absorption loss values with the studied operation situations are shown in Table I. The average loss values were calculated in the frequency bands of 860–960 MHz and 1920–2150 MHz to more accurately represent antenna performance. Bands were covered by sweeping the band with 5 MHz step. The results of both the left and right antennas are shown in separate columns. Equations (1)–(4) were used in calculations. For example, in the case of finger on active (right) element in low band, body loss value of 5.9 dB is calculated by (3). It is the difference between measured total efficiency for free space (03.8 dB) and for finger on active element (09.7 dB) cases. Mismatch loss is calculated separately for free space (1.8 dB) and for finger on active element (1.9 dB) cases by (2). Now, absorption loss (Labs ) is the difference between measured radiation efficiencies for free space and for user loaded cases. As presented in (1), total efficiency equals radiation efficiency multiplied by impedance mismatch factor (1 0 jS11 j2 ). Equation is simplified to addition when dB values are used. In Table I, the free space mismatch loss is deducted from the finger on active element case and the result (0.1 dB) is deducted from corresponding body loss value (5.9 dB) resulting absorption loss of 5.8 dB. Later on, the measurement curves are shown with a larger frequency range and thus are not directly comparable with the tabulated values. A. Total Efficiency and Body Loss The measured total efficiencies of the right antenna are shown in Fig. 3 for the different studied hand grip cases. The antenna was measured with two different grips (finger left and finger right) and in free space. The measured average total efficiency in free space was 03.8 dB in the low band and 02.2 dB in the high band. According to (3), the

Fig. 3. Measured total efficiency in free space and with different hand phantom grips when the right antenna is active.

measured total efficiency difference between the loaded antenna and the antenna in free space denotes user-induced body loss. The measurement with the finger on the active element showed average body loss of 5.9 dB in the low band and 1.4 dB in the high band. When the finger was moved onto the non-active element, average body loss was 3.1 dB in the low band and 0.8 dB in the high band. The measured body loss values follow the trend of the measurements with similar hand grips [1], [8]. The measurements prove that the highest loss is caused by the index finger on the active element. The finger on the non-active element represents less body loss. Particularly in the low band, finger location has a significant effect on body loss. The difference between the two index finger states is up to 2.8 dB, on average. The corresponding value in the high band is 0.6 dB, on average. This shows that by using dynamic antenna switching, a significant decrease in body loss can be achieved with the defined index finger positions. B. S11 Parameter and Mismatch Loss The measured free space S11 parameter of the prototype with the left or right antenna element active is shown in Fig. 4. Both antennas result in sufficient impedance matching in high band (1900 MHz) but imperfect in the low band (900 MHz). However, the effect of imperfect matching shown in Fig. 4 does not cause erroneous results from absorption loss compensation point of view. The measured S11 of the left and right elements are almost identical, thus the prototype structure is assumed to be symmetrical despite the coaxial feed cable. Mismatch loss with different hand grips and without the hand phantom was calculated using (2) and is shown in Fig. 5. The solid

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Fig. 4. Measured switch.

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S

parameters of the left and right antennas without a

Fig. 6. Measured absorption loss for both antenna elements with different index finger locations.

Mismatch loss was not significantly changed in the high band (1920 MHz—2150 MHz). Generally, the hand and finger in the close vicinity of the antenna decreased average loss 0.2 dB . . . 0.4 dB compared with the free space measurement. We observed that the level of mismatch loss caused by the hand varied more in the low band than in the high band. This variation was due to the larger frequency shift in the low band. C. Absorption Loss

Fig. 5. Measured mismatch loss of both antenna elements with different index finger locations.

black curve describes the reflection loss of the unloaded antenna, which is the right antenna in free space in this case. The other black curves denote the right fed antenna and the grey curves denote the left fed antenna. The index finger on the left antenna is shown with the dotted line and the finger on the right antenna is shown with the dashed line. In the low band (860 MHz–960 MHz), the measured average mismatch loss without the phantom hand was 1.8 dB and 1.9 dB for the right and left elements, respectively. The finger on the non-active antenna reduced average mismatch loss 0.7 dB and 0.3 dB for the left and right elements compared with the unloaded antenna. The finger on the active antenna caused a 0.1 dB . . . 1.7 dB loss increase compared with the unloaded antenna. Compared with antennas having a better impedance matching, the observed increase caused by the user can be higher. We concluded that dynamic switching can also decrease user-induced mismatch loss in the low band. With both finger positions, loss increased at the upper end of the band, whereas at the lower end of the band it was significantly decreased. This indicates a downward resonance shift in low frequencies caused by the hand. An example of resonance clearly moving out of the band is shown in [2] for an antenna with better impedance matching.

Absorption loss caused by the hand was calculated using (4). Losses with different hand grips are presented in Fig. 6 for both low and high bands. When the finger was on the active antenna, the measured average absorption loss was 5.0 dB . . . 5.8 dB in the low band and 1.8 dB . . . 2.0 dB in the high band. In the low band, absorption loss was reduced 1.2 dB . . . 2.6 dB when the non-loaded antenna element was used. The corresponding reduction in the high band was 0.7 dB . . . 0.9 dB. Absorption to the hand was significantly higher in the low band than in the high band. This is partly due to the fact that the mobile terminal chassis makes a significant contribution to radiation in the low band. The index finger on the active element necessitates higher absorption losses in both frequency bands. Dielectric and lossy human tissue in close proximity to the antenna increases absorption loss by disturbing the reactive near field. V. DISCUSSION Absorption to the user hand and particularly to the user’s index finger located on top of the antenna is high [8]. In contrast to compensation systems that decrease user-induced impedance mismatch loss [9], it is proved that also absorption loss caused by the mobile terminal user can be reduced in certain situations. The measurement results showed that with the antenna arrangement used an average body loss reduction of 3.1 dB in the low band and 1.0 dB in the high band can be achieved with the defined hand grips. We conclude that the proposed dynamically switchable antenna arrangement increases antenna performance by decreasing mainly radiation absorbed to the user’s hand and index finger. The switchable antenna system can be realized with a real SPDT switch that has low insertion loss characteristics. The typical insertion loss of a switch of this kind is 0.3 dB . . . 0.4 dB, which is much lower than the attained body loss compensation. The switch location is on the

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printed circuit board of the terminal, which allows for a straightforward design, space for the switch control lines, and a 50 transmission line. The proposed dynamic antenna switching can also be used with other types of antennas, e.g. with a planar inverted-F antenna (PIFA), which is a common antenna type in mobile terminals. The use of two antennas instead of one doubles the required antenna space in the already constricted mobile terminal. In addition, the control system increases complexity, and these both affect the cost of the antenna system. However, if the achieved decrease in the user effect is large enough, the system can reduce the power consumption of the terminal and indirectly increase talk time.

VI. CONCLUSION Two dynamically switchable dual-band monopole antennas in a mobile terminal are proposed to compensate for the loading effect of the user’s index finger. The compensation was demonstrated with edge-located elements and evaluated experimentally. Measurements proved that the antenna system can decrease average body loss 3.1 dB in the low band (860 MHz–960 MHz) and 1.0 dB in the high band (1920 MHz–2150 MHz) when user’s index finger is located on the antenna element. The antenna system compensates a significant part of the absorption loss caused by the user’s index finger in both low and high bands. Also, user-induced mismatch loss is reduced in the low band.

REFERENCES [1] P. Lindberg, A. Kaikkonen, and B. Kochali, “Body loss measurements of internal terminal antennas in talk position using real human operator,” in Proc. Int. Workshop on Ant. Tech. ,iWAT, Chiba, 2008, pp. 358–361. [2] M. Berg, M. Sonkki, and E. Salonen, “Experimental study of hand and head effects to mobile phone antenna radiation properties,” in Proc. 3rd Eur. Conf. on Antennas and Propagation, Berlin, 2009, pp. 437–440. [3] J. Toftgård, S. N. Hornsleth, and J. B. Andersen, “Effects on portable antennas of the presence of a person,” IEEE Trans. Antennas Propag., vol. 41, pp. 739–746, Jun. 1993. [4] J. Krogerus, J. Toivanen, C. Icheln, and P. Vainikainen, “Effect of the human body on total radiated power and the 3-D radiation pattern of mobile handsets,” IEEE Trans. Instrum. Meas., vol. 56, pp. 2375–2385, Dec. 2007. [5] M. A. Jensen and Y. Rahmat-Samii, “The electromagnetic interaction between biological tissue and antennas on a transceiver handset,” in Proc. AP-S 1994, Seattle, 1994, pp. 367–370. [6] T. Huang and K. R. Boyle, “User interaction studies on handset antennas,” in Proc. 2nd Eur. Conf. on Antennas and Propagation, Edinburgh, 2007, pp. 1–6. [7] M. Pelosi, O. Franek, M. B. Knudsen, M. Christensen, and G. F. Pedersen, “A grip study for talk and data modes in mobile phones,” IEEE Trans. Antennas Propag., vol. 57, pp. 856–865, Apr. 2009. [8] M. Pelosi, O. Franek, G. F. Pedersen, and M. Knudsen, “User’s impact on PIFA antenna in mobile phones,” in Proc. Vehicular Technology Conference, VTC, Barcelona, 2009, pp. 1–5. [9] P. Ramachandran, Z. D. Milosavljevic, and C. Beckman, “Adaptive matching circuitry for compensation of finger effect on handset antennas,” in Proc. 3rd Eur. Conf. on Antennas and Propagation, Berlin, 2009, pp. 801–804.

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Radiation Improvement of Printed, Shorted Monopole Antenna for USB Dongle by Integrating Choke Sleeves on the System Ground Saou-Wen Su and Tzi-Chieh Hong

Abstract—By integrating a pair of short sleeves into the system ground plane of a USB dongle, the improved radiation patterns of a printed monopole antenna in the E planes cut along the axial direction of the dongle can be attained. The monopole chosen has a symmetrical, short-circuited structure and is to operate in the 2.4 GHz (2400–2484 MHz) band for ZigBee or Wi-Fi dongle applications. When there are no sleeves in the system ground, the ground-plane length significantly affects the patterns and the peak-gain direction of the antenna radiation. With the incorporation of the sleeves set at proper location and in suitable size, the radiation becomes very similar to that of a conventional two-wire, dipole antenna. In this case, for a common USB dongle having system printed circuit board 4 ( is the free-space wavelength at 2442 MHz), the (PCB) longer than maximum field strength is no more in the direction of the ground plane or even toward the main electronic device after plugging in the dongle. Index Terms—Choke sleeves, printed monopoles, radiation-pattern control, wireless USB-dongle antennas, 2.4 GHz antennas.

I. INTRODUCTION It is commonly known that the radiation patterns of the unbalanced, quarter-wavelength antennas, for example the monopoles and the PIFAs, are largely affected by the size of the antenna ground plane, especially the ground-plane length [1]. It is mainly because the ground is considered part of the radiator and has the image currents thereon [2]. When the ground-plane length is no more than =4 ( is the wavelength of the antenna resonant frequency), the antenna and the ground form a half-wavelength, dipole structure and generate omnidirectional radiation patterns. With the ground-plane increased in length exceeding =4, the radiation patterns start to show the side lobes and the nulls in the E planes [1]. In addition, the maximum field strength is also in the direction of the ground plane, which can be a major drawback for wireless USB-dongle applications. When the USB dongle is plugged into an electronics device, the antenna radiation is expected to aim at the device, causing some electromagnetic interference. In this Communication, we present a feasible solution to the abovementioned issue for USB dongles in the 2.4 GHz (2400–2484 MHz) band and show that the maximum field strength can be redirected to lie in the H plane. For 2.4 GHz USB dongles, there comes various form factors. For example, the USB dongles for wireless keyboard and mouse accessories can be as small as a thumbnail (USB connector is not counted). The size of the Wi-Fi and 3.5 GHz internet dongles [3] is about the same as that of a thumb or even larger for Zigbee operation [4] using dongle sticks [5]. As long as the system ground plane is longer than =4 or e =4 of the 2.4 GHz band (e is wavelength on the substrate obtained by (1) in Section II and smaller than ), the

Manuscript received January 25, 2011; revised April 06, 2011; accepted April 30, 2011. Date of publication August 12, 2011; date of current version November 02, 2011. The authors are with the Network Access Strategic Business Unit, Lite-On Technology Corp., Taipei County 23585, Taiwan (e-mail: stephen.su@liteon. com). Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2164225 0018-926X/$26.00 © 2011 IEEE

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= 65:5 mm: (a) the proposed = 3 mm, g = 1 mm; (b) the

Fig. 2. Measured and simulated return loss; L design with d ,h ,g reference design.

= 36 mm = 18 mm

Fig. 1. (a) Geometry of the proposed, shorted monopole antenna with sleeves integrated in the system ground plane of a USB dongle. (b) Detailed dimensions of the monopole.

radiation will become toward the system ground with noticeable side lobes and nulls. In this design, we demonstrate that by integrating a pair of quarter-wavelength sleeves in the system ground with length of 65.5 mm (about =2 at 2442 MHz), the antenna can radiate dipole-like, omnidirectional patterns with peak antenna gain very close to the H plane direction. The integrable sleeves did not occupy much circuit board space and were proven to be accepted for the circuit layout without affecting other component placement in the products. II. ANTENNA CONFIGURATION AND DESIGN CONSIDERATION Fig. 1(a) shows the proposed design with detailed dimensions given in Fig. 1(b). The antenna here is deliberately to be of a symmetrical structure by incorporating a T-shaped monopole [6], [7] and two L-shaped, short-circuiting strips to achieve symmetrical radiation patterns in the E planes [that’s, the x 0 z and y 0 z planes in Fig. 6(a)]. The corresponding reference design with no sleeves in the system ground was also studied and modified [see red numbers in the inset of Fig. 3(b); the other dimensions are kept the same] for 2.4 GHz operation. The design is formed on the same layer of a 1.6-mm thick FR4 substrate. The monopole is placed on a clearance area of size 8 mm 2 25 mm, where no grounding layout is allowed. The system ground plane has length L and width 25 mm and includes a pair of narrow sleeves of width 2.5 mm and length h. The sleeves are short-circuited at the shorted end (at the location of d 0 h), have the open ends at the location of d, and are separated from the system ground by narrow gaps (g1 and g2 ). The general principle of the sleeves in this study is similar to that of the coaxial cable with a pair of choke sleeves (see

Fig. 3. Simulated input impedance on the Smith chart for the proposed and the reference designs.

[8, Fig. 3]) or the bazooka balun (see [9, Fig. 23-2(a)]). The sleeves can be considered an integrable, quarter- wavelength sleeve choke or bazooka balun, which suppresses the surface image currents excited by the antenna. The length h of the sleeves can be approximately determined by

h= "e

pc

4 "e fr "r + 1 "r 0 1 = + 2 2

(1)

1 1+

12d w

(2)

[10] where c is the speed of light, fr is the resonant frequency of the antenna, and "e is the effective dielectric constant [10]. Notice that "r is the dielectric constant = 4:4, d is the substrate thickness 1.6 mm, and w is the width 2.5 mm in this design. Thus, the "e is about 3.28, and the length h at 2442 MHz equals about 17 mm, which is close to the obtained value 18 mm in this design.

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Fig. 4. Surface-current distributions excited at 2442 MHz.

The location and dimensions of the sleeves are important for the surface-current distributions, the radiation patterns, and the impedance matching. For the system printed circuit board (PCB) of practical length L = 65:5 mm (about =2 of the 2.4 GHz band), it is found that by integrating the sleeves at about the first current-null region of the reference design (see Fig. 4, location (d 0 h) about 18 mm), there exists no current null above the sleeves (on portion A). The currents are trapped on the sleeves and suppressed below the sleeves (on portion C). The current- effected portion is shorter than =4 at 2442 MHz. The =4-resonant monopole and the current-effected portion form a dipole-like radiating structure, resulting in the improved radiation patterns instead of aiming the maximum field strength at the system ground. Unlike the conventional add-on choke/balun [8], [9], the proposed design shows no protruding potions beyond the PCB boundary, which makes it possible for the circuit layout to easily incorporate the sleeves. The preferred dimensions were attained by means of parametric studies with the aid of the EM simulator, Ansoft HFSS [11], and the task was laborious before reaching near optimal values. III. RESULTS AND DISCUSSION A. Simulation and Experimental Results of the Constructed Prototypes Fig. 2 shows the measured and the simulated return loss. The measured data compare favorably with the simulation results. The measured impedance matching over the 2.4 GHz band is all below 10 dB (about VSWR of 2) for both the proposed and the reference designs. Notice that the 2.5 GHz (2495–2690 MHz) WiMAX band [12] is also included for the proposed design. Fig. 3 plots the simulated input-impedance curves on the Smith chart. Although the reference-design curve is centered at the 50- point, the proposed-design curve has a small loop within the 2:1-VSWR circle, which leads to wider impedance bandwidth. The results also suggest that the sleeves can be used as an impedance-matching balun to widen the achievable bandwidth. Fig. 4 shows the surface-current distributions excited at 2442 MHz, the center frequency of the 2.4 GHz band, for the proposed and the reference designs. The corresponding 2-D radiation patterns are given in Figs. 5(a) and 5(b). For the reference design, there occur two null-current regions, one at the bottom and the other close to the top, on the ground plane. This behavior is very similar to that of the 1800-MHz monopole with a =2 system ground as taught in [1]. In this case, the

Fig. 5. Far-field, 2-D radiation patterns at 2442 MHz for (a) the proposed design and (b) the reference design studied in Fig. 2(a) and 2(b).

maximum field strength exists in the lower, half space (below x 0 y cut in this study) of the polar coordinate with reduced antenna gain at around  = 60 in the patterns for the reference [see the x 0 z and y 0 z planes in Fig. 5(b)]. This means that when the USB dongle is plugged into an electronics device, most of the antenna radiation is aiming at the device, which is not desired and may introduce some electromagnetic interference. On the contrary, the proposed design shows =2, dipole-like radiation with the omnidirectional plane (in the x 0 y plane, where maximum gain is observed) perpendicular to the short side of the system ground as seen in Fig. 5(a). This is because the integrable sleeves function as a current trap, on portion B (see Fig. 4), and suppress the surface currents on portion C, the current-effected portion (portion A) becomes smaller and much shorter than =2 at 2442 MHz. Thus, the =4-resonant monopole and the current-effected ground portion form a dipole-like radiating structure. Notice that the high current density with strong magnitude is located on the sleeves. The current vectors (not shown here for brevity) are out of phase over the narrow slit. Accordingly, the near-field electric fields are cancelled out therein, making portion B non-effective radiator for far-field radiation. Figs. 6 and 7 present the measured 3-D radiation patterns. Compared with the other in-band frequencies measured, good consistency in the patterns was first obtained. From the results, it can be confirmed that the proposed design generates dipole-like, radiation patterns with good omnidirectional radiation in the horizontal, x 0 y plane compared with the reference design. The maximum field strength was found in the

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Fig. 6. Measured 3-D radiation patterns (including the x at 2442 MHz for the proposed design studied in Fig. 5(a).

0 z and y 0 z cuts)

0 z and y 0 z cuts)

Fig. 7. Measured 3-D radiation patterns (including the x at 2442 MHz for the reference design studied in Fig. 5(b).

half space at  = 105 0 y plane for the reference. Again, this drawback, the radiation toward the system ground, to USB-dongle applications can be improved by integrating a pair of sleeves into the system ground. The peak gain measured in the 2.4 GHz band for the proposed design is found at a constant level of about 2.4 dBi with the radiation efficiency of about 73%. As for the reference design, the peak gain varies from 2.2 to 3.7 dBi with the radiation efficiency above 66%. Notice that due to higher directivity, the peak gain of the reference is larger than that of the proposed.  below the x

Fig. 8. Simulated 2-D radiation patterns at 2442 MHz as a function of various parameters of (a) location d, (b) length h, (c) gap g , and (d) ground-plane length L.

B. Parametric Effects of the Sleeves on Return Loss, Radiation Patterns, and Surface Currents Various parameters of the sleeves were analyzed. The open end of the sleeves is pertinent to the shorted end of the sleeves (that’s, the location of d 0 h). The shorted end is set at about the first current-null region in

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Fig. 9. Surface-current distributions excited at 2442 MHz for the proposed design with various locations d of the sleeves as studied in Fig. 8(a).

Fig. 11. Surface-current distributions excited at 2442 MHz for the proposed = and  as studied in Fig. 8(d). design with the ground-plane length L

=3 4

Fig. 10. Surface-current distributions excited at 2442 MHz for the proposed design with various lengths h of the sleeves as studied in Fig. 8(b).

the reference design (see Fig. 4) as an optimal case (d = 36 mm). This parameter affects the resonant modes and the input matching thereof much rather than the radiation patterns [see Fig. 8(a)] when the ground plane remains unchanged. The reasons are straightforward. When the sleeves are set at about the first current-null region of the reference, it ensures that there is no current null above the sleeves on portion A as observed in Fig. 9. Further, due to the =2 system ground used, the length d 0 h of portion A is less than =4 at 2442 MHz, which results in dipole-like radiation patterns. However, if the 3=4 ground length is utilized (that’s, L = 92 mm), a current null will appear because the length of portion A exceeds =4 at 2442 MHz. The radiation then becomes similar to that of the reference design [compare Fig. 5(b) with Fig. 8(a)]. Both the matching and the patterns are affected by the length h of the sleeves; especially the patterns [see Fig. 8(b)]. This is because the surface currents are no longer trapped on portion B and instead, distributed on the entire system ground as seen in Fig. 10, which, in turn, deteriorates the desired dipole-like, omnidirectional radiation. This is due to that the effected frequencies of the current trapping are not centered at 2442 MHz but shifted to higher (at about 3.1–3.2 GHz) and lower (at about 2–2.1 GHz) frequencies in the cases of h = 13 mm and h = 23 mm respectively. It may be assumed that both the monopole and the sleeves are required to be tuned in the same frequency range to achieve the best results. For the small gap between the sleeve open end and the system ground, the parameter g1 is introduced to fine tune

the symmetry of the patterns in the upper and the lower, half space, separated by the x 0 y cut, as seen in Fig. 8(c). The return-loss curves are not much changed. Finally, the length L of the system ground is analyzed and increased to 3=4 and  at about 2442 MHz. Between L = =2 and , the impedance bandwidth still cover the 2.4 GHz band. For the patterns in Fig. 8(d), the three cases have much in common and show dipole-like radiation. Although the ground length varies, the current-effected portion remains the same in size, and the currents thereon (portion C) are very minor (see Fig. 11). It is assumed that the radiation patterns are independent of the system ground length after the proposed design applied. IV. CONCLUSION A technique of improving the radiation patterns of a 2.4-GHz monopole for USB-dongle applications has been proposed. A pair of narrow sleeves is set at the opposite, long sides of the system PCB, in which the overall dimensions 21 mm 2 3.5 mm are required in the circuit layout. The length of the sleeves is found to correspond to about 0:26e at 2442 MHz. With the sleeves, the surface currents are trapped between the open and the shorted ends, leading to suppressed surface currents below the current-trapped portion. The influential ground portion in shaping the radiation patterns thus becomes shorter in length. The results suggest that for a USB dongle with a system PCB length less than =2 of the 2.4 GHz band, no current null are spotted between the antenna and the sleeves on the ground. This ensures that the antenna yields dipole-like radiation instead of showing the maximum field strength toward the system ground. Although the quarter-wavelength sleeves are not of wideband or multiband structure, using multi-sleeves may attain extra band as suggested in the related work [13]. The studies will be conducted in the future.

REFERENCES [1] K. L. Wong, “Planar antennas for wireless communications: Small mobile device antennas,” presented at the Short Courses of the 2010 IEEE Antennas Propag. Soc. Int. Symp., Toronto, ON, Canada.

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[2] H. Morishita, Y. Kim, and K. Fujimoto, “Design concept of antennas for small mobile terminals and the future perspective,” IEEE Antennas Propag. Mag., vol. 44, pp. 30–34, Oct. 2002. [3] P. Park and J. Choi, “Internal multiband monopole antenna for wireless-USB dongle application,” Microw. Opt. Technol. Lett., vol. 51, pp. 1786–1788, Jul. 2009. [4] ZigBee, Wikipedia the free encyclopedia [Online]. Available: http://en. wikipedia.org/wiki/ZigBee [5] “Z-101A/B/D ZigBee USB Dongle,” Netvox Technology Co., Ltd [Online]. Available: http://www.netvox.com.tw/Z-101.asp [6] S. W. Su, K. L. Wong, and H. T. Chen, “Broadband low-profile printed T-shaped monopole antenna for 5-GHz WLAN operation,” Microw. Opt. Technol. Lett., vol. 42, pp. 243–245, Aug. 2004. [7] Y. L. Kuo and K. L. Wong, “Printed double-T monopole antenna for 2.4/5.2 GHz dual-band WLAN operations,” IEEE Antennas Propag. Lett., vol. 51, pp. 2187–2192, 2003. [8] B. Drozd and W. T. Joines, “Comparison of coaxial dipole antennas for applications in the near-field regions,” Microw. J., vol. 47, pp. 160–176, May 2004. [9] J. D. Kraus and R. J. Marhefka, Antennas: For All Applications, 3rd ed. New York: McGraw-Hill, 2003, ch. 23, pp. 803–805. [10] D. M. Pozar, Microwave Engineering, 3rd ed. New York: Wiley, 2005, ch. 3, pp. 143–145. [11] Ansoft Corporation HFSS [Online]. Available: http://www.ansoft.com/ products/hf/hfss [12] S. W. Su and K. L. Wong, “Wideband antenna integrated in a system in package for WLAN/WiMAX operation in a mobile device,” Microw. Opt. Technol. Lett., vol. 48, pp. 2048–2053, Oct. 2006. [13] J. Holopainen, J. Ilvonen, O. Kivekas, R. Valkonen, C. Icheln, and P. Vainikainen, “Near-field control of handset antennas based on inverted-top wavetraps: Focus on hearing-aid compatibility,” IEEE Antennas Propag. Lett., vol. 8, pp. 592–594, 2009.

Characterization of the Body-Area Propagation Channel for Monitoring a Subject Sleeping David B. Smith, Dino Miniutti, and Leif W. Hanlen

Abstract—A dynamic characterization of the wireless body-area communication channel for monitoring a sleeping person is presented. The characterization uses measurements near the 2.4 GHz ISM band with measurements of eight adult subjects each over a period of at least 2 hours. Numerous transmit-receive pair (Tx-Rx) locations on and off the body for a typical body-area-network (BAN) are used. Three issues are addressed: 1) modeling of channel gain, 2) outage probability, and 3) outage duration. It is shown that over very large durations (far in excess of a delay requirement of 125 ms that is typical for many IEEE 802.15.6 medical BAN applications) there is not a reliable communications channel for star-topology BAN. The best case outage probability, with 0 dBm Tx power and 100 dBm Rx sensitivity, is in excess of a packet-error-rate of 10%. Following from these issues the feasibility of using alternate on-body or off-body links as relays is demonstrated. Index Terms—Body area networks, channel modeling, outage duration, outage probability, relay communication, sleep monitoring.

I. INTRODUCTION Wireless communication between a network of small sensors (and/or actuators) placed around the body inspires the need for a wireless body area network (BAN) [2]. The first substantial use for BANs [3] is likely to be health-care. Any BAN, where it is used for patient care in hospital, aged-care facilities, and home health-care, requires reliable operation for the patient in bed, sleeping. There have been various analytical and experimental studies of propagation characteristics for BAN transmission from on-the-body to another position on-the-body, [4]–[12], and for transmission of on-body to a position off-the-body [13]. More general personal area network (PAN) studies for small range transmission close to the human body exist, e.g., [14]. As yet there has been no substantial propagation characterization of the particular BAN sleeping channel, either on-body or off-body1. Here we present such a characterization, with modeling and analysis, along with a solution using relays to overcome the limitations of this channel. We use an extensive measurement campaign of various adult subjects sleeping each for a period of at least two hours. Measurements were made at the candidate narrowband BAN frequency of 2.36 GHz using small radios. Our objective was to answer: 1) What are key measures of performance for the narrowband BAN radio, on-body and off-body, sleeping channel? 2) Is a star topology sufficient for IEEE 802.15.6 BAN requirements [2], with a sleeping subject? Manuscript received November 22, 2010; revised March 22, 2011; accepted May 01, 2011. Date of publication August 12, 2011; date of current version November 02, 2011. National ICT Australia is funded by the Australian Government as represented by the Department of Broadband, Communications and the Digital Economy and the Australian Research Council through the ICT Centre of Excellence program. A portion of this work appeared in [1]. The authors are with the NICTA, Canberra Research Laboratory, 7 London Circuit Canberra, Australian Capital Territory, 2600, Australia and also with the Australian National University Canberra ACT 0200, Australia (e-mail: [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2164209 1Although

a “sleep-staging” experiment, using BAN, is reported in [15].

0018-926X/$26.00 © 2011 IEEE

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Fig. 2. Typical on-body and off-body channel gain time series for the BAN sleeping channel.

Fig. 1. Illustration of the sleeping experiment set-up. TABLE I RADIO LOCATIONS. NTB(H)-NEXT TO BED HEAD, LW-LEFT WRIST, H(F)-HIP FRONT, NTB(F)-NEXT TO BED (FOOT), RW-RIGHT WRIST, H(B)-HIP BACK, LA-LEFT ANKLE; TX/RX IMPLIES RADIO ACTS AS A TRANSMITTER AND RECEIVER, RX IMPLIES RADIO OPERATES SIMPLY AS A RECEIVER

3) Are there any typical receive radio sensitivities where there is reliable communications; what are typical outage duration periods; and what implications are there for packet error rates (PER)? 4) If star-topology, single link BAN radio communications is not effective for the sleeping channel, are relays an alternative? We demonstrate effective performance measures and show that transmit-receive (Tx-Rx) links are often in outages for periods of minutes over a range of receive sensitivities. The outages are in excess of latency requirements for many medical BAN applications [2], [3], with a packet error rate greater than 10% at a very optimistic Rx sensitivity of 0100 dBm, 100 dB below transmit power. Further, we draw attention to the possible feasibility of using other links to facilitate whole-body reliable communications. A description of the experimental set-up follows in Section II. In Section III the modeling of the BAN sleeping channel is presented including time-series and distribution of the measured received signal amplitude; followed by characterization of outage probability and outage duration of the sleeping channel. Section IV provides discussion and analysis, based on the measurement data, of the potential use of relays to overcome unreliable communications in the BAN sleeping channel. Section V provides concluding remarks. II. EXPERIMENTAL SET-UP The on-body and off-body network for each measurement set was implemented using 7 small radios with three operating as transmitters and receivers (Tx/Rx) and four operating as Rx. Radios were placed beside a bed, and on a subject in bed sleeping. The positions are given in Table I. An illustration of the set-up is given in Fig. 1.

The radios used a Chipcon CC2500 2.4 GHz low power transceiver and included a ceramic multilayer chip antenna from Phycomp, which is omnidirectional in the H-plane, had a gain of 1.2 dBi, and was operational at 2.4 GHz. Full description of the radios can be found in [16]. The radios were operated at 2.36 GHz. The CC2500 transceivers have a Rx sensitivity of 0100 dBm (with the radios tuned to a Tx power of 0 dBm or 1 mW). The received signal strength indicator (RSSI) for every Tx-Rx pair is logged every 15 ms. Twelve sets of measurements were made on eight subjects.2 Each set was for 2 hours or more; over all sets there was a total of 183 link measurements, with 85 on-body links and 98 off-body links. The same radios, and hence the same antennas, were used throughout for both on-body and off-body link measurements. This gave a total of 316 hours of on-body link data and 361 hours of off-body link data. III. MODELING OF THE BAN SLEEPING CHANNEL In this section we use the RSSI-based narrowband channel received signal amplitude statistics to evaluate system performance. Channel gain is used as a negative measure of the channel attenuation; i.e., it is the ratio of receive signal amplitude to output transmit signal amplitude in any given Tx/Rx link. A. Channel Gain: Time Series and Distribution There are two cases in which the radio does not record an RSSI value (instead it records a NaN-not a number): when the channel gain drops below the radio receive sensitivity; and when there is an incorrectly decoded packet due to a CRC failure (in this case the channel gain may be greater than 0100 dB). The general poor signal quality, of any given link of the single-link BAN sleeping channel, is illustrated by the percentage of no recorded measurements where NaNs are recorded. These are 14.9% for the off-body channel and 14.8% for the on-body channel over all measurement data. A typical measured channel gain time series for one of the subject measurement sets is shown in Fig. 2 with the on-body time series for the left-wrist to the hip (back), and the off-body time series for the hip (front) to the radio next to the bed (head). The channel gain has long periods of small variation 65 dB. In these time series there are also periods of no recorded measurement; and parts where the channel gain 2Subject weights ranged from 60 kg to 90 kg, and heights ranged from 165 cm to 190 cm. All measurements were made in the home environment of the subjects. Subjects were 1 female, 7 males, and the female and 3 of the males performed the experiment twice.

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Fig. 4. Outage probability as a function of receive sensitivity, with Tx power of 0 dBm.

made for a Gamma distribution to model “shadow fading”, for which the sleeping channel provides an extreme example. B. Outage: Probability and Duration

Fig. 3. Overall gamma PDF fit to normalized measured data for sleeping channel (a) fit to agglomerate on-body data (b) fit to agglomerate off-body data.

is just above the radio receive sensitivity where there is a high frequency of incorrectly decoded packets,3 and hence no recorded measurement. These time series indicate that the channel is stable due to long periods of little movement with subjects sleeping. The channel provides unreliable communications due to very low channel gain. For agglomerate recorded measurement data for both on-body and off-body channel we fit six standard statistical distributions to the empirical normalized channel gain data: Normal, Rayleigh, Lognormal, Weibull, Gamma and Nakagami-m. Channel gain has been divided by the root-mean-square channel gain, for each measured Tx/Rx link, to obtain the normalized channel gain. The best fits were Gamma in both on-body and off-body cases. The Gamma fits had the largest log-likelihood, given the data, based on the maximum-likelihood (ML) parameter estimates of the six statistical distributions. These fits are shown in Fig. 3, on-body in Fig. 3(a) and off-body in Fig. 3(b); with probability density function given by 1

a01

f (xja; b) = a x b 0(a)

x exp 0 b

(1)

where 0(1) is the Gamma function. For the on-body sleeping channel the shape parameter a = 1:60, and scale parameter b = 0:480; and for the off-body channel a = 3:54 and b = 0:254. In [17] an argument is 3This

contrasts with the negligible loss of packets reported in [15].

In order to properly characterize the outages of the sleeping channel we attempt to remove the effects of incorrectly decoded packets, and only to account for when the channel gain stays below a certain level (or receive sensitivity). Due to the stability and very slow time-varying nature of the channel, if there is no recorded measurement for two-orless successive periods we declare this as incorrectly decoded packet/s, and set a channel gain of the previous recorded measurement value. The empirical outage probability for the on-body and off-body channel is shown in Fig. 4. The best case outage probability is more than 10% for both on-body (13.5%) and off-body (10.9%) channels. For a BAN radio receive sensitivity of 085 dBm, with Tx power 0 dBm, the probability of outages is more than 30% in both cases. Fig. 4 illustrates that the packet error rate (PER) for a BAN radio will be at least 10% for a standard one-hop star topology with a person sleeping. Due to the very slowly-varying nature of the BAN sleeping channel, it is important to understand the length of periods of which channel is in outage with respect to particular receive sensitivities. In Fig. 5 we show the percentage of time that continuous outages of larger than x seconds (on horizontal axis) occur in all measurements. For example in Fig. 5(a) a receiver with a sensitivity of 088 dBm, or 88 dB below transmit power of 0 dBm, will experience outages of larger than 1000 seconds 5% of the time. Long (duration) outages are infrequent, but because they are long they take up a large fraction of the total time. Further, in terms of BAN latency requirements for medical applications at 88 dB below transmit power for example, outages of larger than a typical latency requirement of 125 ms [2], [3], occur more than 22% of the time. Importantly, once again, the results in Fig. 5(a) and (b) are very similar for the on-body and off-body BAN sleeping channel. IV. OVERCOMING UNRELIABLE COMMUNICATIONS THE BAN SLEEPING CHANNEL

IN

How can reliable communications be established if a star topology in the BAN sleeping channel is unreliable? One solution is the use of other links as relays, when a given signal link fails. The use of a “good” link when a given link is in outage, i.e., Channel gain is below a given value G dB, for a significant duration, requires there to be an alternate link from the same T x, T x 0 Rxalternate at

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Fig. 5. Continuous outage duration below a given Rx sensitivity, with Tx power of 0 dBm, from agglomerate data (a) agglomerate on-body continuous outage duration (b) agglomerate off-body continuous outage duration.

the same time as T x 0 Rx is in outage4. Thus we seek to find for all Rxalternate for given Tx, the time when these links had a receive signal power above G, G + 5 and G + 10 dB, during the same time as the potential outag, with the below G dB link. Inspection of Table I, in Section II, shows that in the recorded measurement sets for a given T x, and T x 0 Rx pair, there are 5 T x 0 Rxalternate ; indicating alternate measurement sets may be used that occur simultaneously with a bad T x 0 Rx link. For this preliminary evaluation a rolling median was applied to all 85 on-body and 98 off-body link measurement sets, to chunks of data in periods of 1.2 s (as significant variation occurs over a much larger timescale than this). The purpose of this “median filter”, which smooths out small scale power fluctuations, is to indicate a general trend of T x 0 Rx channel gain, effectively “summarizing” the data into portions of more significant variations. The fraction of total time for which the given T x 0 Rx link piecewise-median channel gain was below a given sensitivity G dB was then calculated. Fig. 6(a) is for the off-body channel with T x next to bed head, Fig. 6(b) is for the on-body channel with T x at the front hip. The three

0

4Rx Rx is not considered, although it is clear that this is also required to be a “good” link for a reliable relay channel.

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Fig. 6. Empirical probability of good other-link signal, channel Gain > G + f0; 5; 10g dB, while poor link signal, Gain < G dB (a) off-body multi-link

comparison, Tx next to bed head (b) on-body multi-link comparison, Tx at front hip.

curves in each of Fig. 6(a) and (b), for x = f0; 5; 10g, correspond to the alternate link for G + x, when the sensitivity of the given link is below G. When the channel gain is below 090 dB, 85% of the time there is alternate link with gain greater than G for off-body, Fig. 6(a), and 80% for on-body in Fig. 6(b). Similarly there is an alternate link greater than G + 10 dB 40% of the time for off-body, and 45% of the time for on-body in Fig. 6. There is often an alternate good signal link, of greater channel quality (often significantly greater quality), to a poor link from the same T x. Relay, multi-hop, communications may be viable for the body-area communication sleeping channel. V. CONCLUDING REMARKS The body-area propagation channel (on and off-body), for the particular case of a person sleeping, has been modeled based on extensive measurements. It has been shown that standard single-link BAN radio communications is unreliable, in terms of outage probabilities greater than 10%, and in terms of general outage durations far in excess of typical medical BAN delay requirements of 125 ms. This is for sensitivities up to 100 dB below transmit power. It has been illustrated that

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the use of alternate links in the form of multi-hop communications may be viable.

REFERENCES [1] D. Miniutti, D. Smith, L. Hanlen, A. Zhang, D. Rodda, and B. Gilbert, “Sleeping Channel Measurements for Body Area Networks,” IEEE P802.15 Working Group for Wireless Personal Area Networks (WPANs) 2009. [2] B. Zhen, M. Patel, S. Lee, and E. Won, “Body Area Network (BAN) Technical Requirements,” IEEE Standard: 15-08-0037-01-0006-IEEE802-15-6, document-v-4-0, 2008. [3] D. Lewis, “802.15.6 Call for Applications—Response Summary,” IEEE 802.15 Working Group Document IEEE 802.15-08-0407-502, 2008. [4] S. Obayashi and J. Zander, “A body-shadowing model for indoor radio communication environments,” IEEE Trans. Antennas Propag., vol. 46, no. 6, pp. 920–927, Jun. 1998. [5] A. Fort, C. Desset, P. Wambacq, and L. Biesen, “Indoor body-area channel model for narrowband communications,” IET Microw. Antennas Propag., vol. 1, no. 6, pp. 1197–1203, Dec. 2007. [6] P.-S. Hall and Y. Hao, Antennas and Propagation for Body Centric Wireless Networks. Boston, MA: Artech House, 2006. [7] T. Zasowski, G. Meyer, F. Althaus, and A. Wittneben, “Propagation effects in UWB body area networks,” in Proc. IEEE Int. Conf. on UltraWideband, Sep. 2005, pp. 16–21. [8] D. Neirynck, C. Williams, A. Nix, and M. Beach, “Wideband channel characterisation for body and personal area networks,” in Proc. 2nd Int. Workshop on Wearable and Implantable Body Sensor Networks, Apr. 2004, pp. 1–3.

[9] L. Liu, F. Keshmiri, C. Craeye, P. D. Doncker, and C. Oestges, “An analytical modeling of polarized time-variant on-body propagation channels with dynamic body scattering,” EURASIP J. Wireless Commun. Network., vol. 2011, no. Article ID 362521, pp. 1–12, 2011. [10] A. Fort, F. Keshmiri, G. Crusats, C. Craeye, and C. Oestges, “A body area propagation model derived from fundamental principles: Analytical analysis and comparison with measurements,” IEEE Trans. Antennas Propag., vol. 58, no. 2, pp. 503–514, Feb. 2010. [11] S. van Roy, C. Oestges, F. Horlin, and P. De Doncker, “A comprehensive channel model for UWB multisensor multiantenna body area networks,” IEEE Trans. Antennas Propag., vol. 58, no. 1, pp. 163–170, Jan. 2010. [12] R. D’Errico and L. Ouvry, “Time-variant BAN channel characterization,” in Proc. IEEE 20th Int. Symp. on Personal, Indoor and Mobile Radio Communications, Sep. 2009, pp. 3000–3004. [13] D. Smith, L. Hanlen, J. Zhang, D. Miniutti, D. Rodda, and B. Gilbert, “Characterization of the dynamic narrowband on-body to off-body area channel,” in Proc. IEEE Int. Conf. on Communications,, Jun. 2009, pp. 1–6. [14] J. Karedal, P. Almers, A. Johansson, F. Tufvesson, and A. Molisch, “A MIMO channel model for wireless personal area networks,” IEEE Trans. Wireless Commun., vol. 9, no. 1, pp. 245–255, Jan. 2010. [15] J. Penders et al., “Potential and challenges of body area networks for personal health,” in Proc. IEEE Annu. Int. Conf. on Engineering in Medicine and Biology, Sep. 2009, pp. 6569–6572. [16] L. Hanlen et al., “Open-source testbed for body area networks: 200 sample/sec 12 hrs continuous measurement,” in Proc. IEEE 20th Int. Symp. on Personal Indoor and Mobile Radio Communication, Sep. 2010, pp. 1–5. [17] A. Abdi and M. Kaveh, “On the utility of gamma PDF in modeling shadow fading (slow fading),” in Proc. IEEE 49th Vehicular Technology Conf., Jul. 1999, vol. 3, pp. 2308–2312.

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION Editorial Board Brenn Ellsworth, Editorial Assistant Michael A. Jensen, Editor-in-Chief e-mail: [email protected] Department of Electrical and Computer Engineering (801) 422-3903 (voice) 459 Clyde Building Brigham Young University Provo, UT 84602 e-mail: [email protected] (801) 422-5736 (voice) Senior Associate Editor Karl F. Warnick Dimitris Anagnostou Hiroyuki Arai Ozlem Aydin Civi Zhi Ning Chen Jorge R. Costa Nicolai Czink George Eleftheriades Magda El-Shenawee Lal C. Godara

Associate Editors Yang Hao Andrea Massa Stuart G. Hay Derek McNamara Sean V. Hum Andrea Neto Ramakrishna Janaswamy George W. Pan Buon Kiong Lau Athanasios Panagopoulos Jin-Fa Lee Patrik Persson Kwok Wa Leung K.V. S. Rao Duixian Liu Shanker Balasubramaniam

Satish Sharma Jamesina Simpson Mei Song Tong Jon W. Wallace Fan Yang Ali Yilmaz Zhengqing Yun Zhijun Zhang Yue Ping Zhang

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