IEEE Transactions on Antennas and Propagation [volume 58 number 6]

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JUNE 2010

VOLUME 58

NUMBER 6

IETPAK

(ISSN 0018-926X)

PAPERS

Antennas Confocal Ellipsoidal Reflector System for a Mechanically Scanned Active Terahertz Imager . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . N. Llombart, K. B. Cooper, R. J. Dengler, T. Bryllert, and P. H. Siegel Compound Diffractive Lens Consisting of Fresnel Zone Plate and Frequency Selective Screen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Y. J. Fan, B. L. Ooi, H. D. Hristov, and M. S. Leong Millimeter-Wave Half Mode Substrate Integrated Waveguide Frequency Scanning Antenna With Quadri-Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Y. J. Cheng, W. Hong, and K. Wu 60 GHz Aperture-Coupled Dielectric Resonator Antennas Fed by a Half-Mode Substrate Integrated Waveguide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Q. Lai, C. Fumeaux, W. Hong, and R. Vahldieck Mu-Zero Resonance Antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J.-H. Park, Y.-H. Ryu, and J.-H. Lee The Broadband Spiral Antenna Design Based on Hybrid Backed-Cavity . . . . . . . . . . . . . . . . . C. Liu, Y. Lu, C. Du, J. Cui, and X. Shen Resonant Frequency of a Rectangular Patch Sensor Covered With Multilayered Dielectric Structures . . . . . . . . . Y. Li and N. Bowler Study of an Ultrawideband Monopole Antenna With a Band-Notched Open-Looped Resonator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S.-J. Wu, C.-H. Kang, K.-H. Chen, and J.-H. Tarng Electrically Small Magnetic Dipole Antennas With Quality Factors Approaching the Chu Lower Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . O. S. Kim, O. Breinbjerg, and A. D. Yaghjian Simple Excitation Model for Coaxial Driven Monopole Antennas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . L. C. Trintinalia Arrays A Method for Seeking Low-Redundancy Large Linear Arrays in Aperture Synthesis Microwave Radiometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J. Dong, Q. Li, R. Jin, Y. Zhu, Q. Huang, and L. Gui Experimental Demonstration of Focal Plane Array Beamforming in a Prototype Radiotelescope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. B. Hayman, T. S. Bird, K. P. Esselle, and P. J. Hall ADS-Based Guidelines for Thinned Planar Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G. Oliveri, L. Manica, and A. Massa Deterministic Synthesis of Uniform Amplitude Sparse Arrays via New Density Taper Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . O. M. Bucci, M. D’Urso, T. Isernia, P. Angeletti, and G. Toso Pattern Synthesis of Narrowband Conformal Arrays Using Iterative Second-Order Cone Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K. M. Tsui and S. C. Chan Electromagnetics A Modified Pole-Zero Technique for the Synthesis of Waveguide Leaky-Wave Antennas Loaded With Dipole-Based FSS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. García-Vigueras, J. L. Gómez-Tornero, G. Goussetis, J. S. Gómez-Díaz, and A. Álvarez-Melcón Evaluation of Weakly Singular Integrals Via Generalized Cartesian Product Rules Based on the Double Exponential Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. G. Polimeridis and J. R. Mosig Plain Models of Very Simple Waveguide Junctions Without Any Solution for Very Rich Sets of Excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P. Fernandes and M. Raffetto

1834 1842 1848 1856 1865 1876 1883 1890 1898 1907

1913 1922 1935 1949 1959

1971 1980 1989

(Contents Continued on p. 1833)

(Contents Continued from Front Cover) Numerical Methods Source and Boundary Implementation in Vector and Scalar DGTD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J. Alvarez, L. D. Angulo, M. F. Pantoja, A. R. Bretones, and S. G. Garcia Accurate FDTD Simulation of RF Coils for MRI Using the Thin-Rod Approximation . . . D. M. Sullivan, P. Wust, and J. Nadobny

1997 2004

Propagation Propagation Over Parabolic Terrain: Asymptotics and Comparison to Data . . . . . . . D. Chizhik, L. Drabeck, and W. M. MacDonald Measurement Analysis of Amplitude Scintillation for Terrestrial Line-of-Sight Links at 42 GHz . . . . . . . . . . . . . . . . . . . . . . M. Cheffena A Study of Anomalous Propagation in Persian Gulf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. U. H. Sheikh, P. Z. Khan, and S. A. Al-Semari Modeling Radio Transmission Loss in Curved, Branched and Rough-Walled Tunnels With the ADI-PE Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R. Martelly and R. Janaswamy

2012 2021 2029

Scattering Discrete Models of Electromagnetic Wave Scatterers in a Frequency Range . . . . . . . . . . . . . . . . . . I. P. Kovalyov and D. M. Ponomarev Resonance Behavior of Radar Targets With Aperture: Example of an Open Rectangular Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J. Chauveau, N. de Beaucoudrey, and J. Saillard

2046

Wireless A Comprehensive Spatial-Temporal Channel Propagation Model for the Ultrawideband Spectrum 2–8 GHz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Gentile, S. M. López, and A. Kik A Correction to Head-Wave Fields for a Simple Planar Contrast of Permittivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W.-S. Lihh

2037

2060

2069 2078

COMMUNICATIONS

Radiation From Rectangular Waveguide-Fed Fractal Apertures . . . . . . . . . . . . . . . . . . . . . . . B. Ghosh, S. N. Sinha, and M. V. Kartikeyan Experimental Demonstration of the Extended Probe Instrument Calibration (EPIC) Technique . . . . . . . . . . . . . . . . . . . R. J. Pogorzelski A Ku-Band Planar Antenna Array for Mobile Satellite TV Reception With Linear Polarization . . . . . . . . . . . . . . . . . . . . . . . R. Azadegan Hybrid-Fractal Direct Radiating Antenna Arrays With Small Number of Elements for Satellite Communications . . . K. Siakavara Ultrawideband Antennas for Magnetic Resonance Imaging Navigator Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U. Schwarz, F. Thiel, F. Seifert, R. Stephan, and M. A. Hein Dual-Band Circularly Polarized S -Shaped Slotted Patch Antenna With a Small Frequency-Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nasimuddin, Z. N. Chen, and X. Qing A Loop Loading Technique for the Miniaturization of Non-Planar and Planar Antennas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Ghosh, SK. M. Haque, D. Mitra, and S. Ghosh Unified Definitions of Efficiencies and System Noise Temperature for Receiving Antenna Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K. F. Warnick, M. V. Ivashina, R. Maaskant, and B. Woestenburg Low-Cost High Gain Planar Antenna Array for 60-GHz Band Applications . . . . . . . . . . . . . . . . . X.-P. Chen, K. Wu, L. Han, and F. He Optimal Narrow Beam Low Sidelobe Synthesis for Arbitrary Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Fuchs and J. J. Fuchs Joint Elevation and Azimuth Direction Finding Using L-Shaped Array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J. Liang and D. Liu Array Pattern Synthesis Using Digital Phase Control by Quantized Particle Swarm Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . T. H. Ismail and Z. M. Hamici Leaky-Wave Applicators: Experimental Verification of the Effectiveness of the Single Pole-Wave Approximation for the Estimation of the Power Deposition Pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G. d’Ambrosio and M. D. Migliore Near-Field Testing System for Antennas Operating in Short Millimeter Waveband . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V. H. Avetisyan Truncation-Error Reduction in 2D Cylindrical/Spherical Near-Field Scanning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K. T. Kim Pitfalls in the Determination of Optical Cross Sections From Surface Integral Equation Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. M. Kern and O. J. F. Martin

2088 2093 2097 2102 2107 2112 2116 2121 2126 2130 2136 2142 2146 2149 2153 2158

CORRECTIONS

Corrections to “Simple and Accurate Analytical Model of Planar Grids and High-Impedance Surfaces Comprising Metal Strips or Patches” . . . . . . . . . . O. Luukkonen, C. Simovski, G. Granet, G. Goussetis, D. Lioubtchenko, A. V. Räisänen, and S. A. Tretyakov

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CALL FOR PAPERS

Joint Special Issue on Multiple-Input Multiple-Output (MIMO) Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Joint Special Issue on Ultrawideband (UWB) Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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IEEE ANTENNAS AND PROPAGATION SOCIETY All members of the IEEE are eligible for membership in the Antennas and Propagation Society and will receive on-line access to this TRANSACTIONS through IEEE Xplore upon payment of the annual Society membership fee of $24.00. Print subscriptions to this TRANSACTIONS are available to Society members for an additional fee of $36.00. For information on joining, write to the IEEE at the address below. Member copies of Transactions/Journals are for personal use only. ADMINISTRATIVE COMMITTEE M. ANDO, President R. D. NEVELS, President Elect M. W. SHIELDS, Secretary-Treasurer 2010 2011 2012 2013 P. DE MAAGT A. AKYURTLU *J. T. BERNHARD G. ELEFTHERIADES W. A. DAVIS G. MANARA H. LING P. PATHAK M. OKONIEWSKI *A. F. PETERSON Honorary Life Members: R. C. HANSEN, W. R. STONE *Past President Committee Chairs and Representatives Antenna Measurements (AMTA): S. SCHNEIDER Antennas & Wireless Propagation Letters Editor-in-Chief: G. LAZZI Applied Computational EM Society (ACES): A. F. PETERSON Awards: A. F. PETERSON Awards and Fellows: C. A. BALANIS Chapter Activities: L. C. KEMPEL CCIR: P. MCKENNA Committee on Man and Radiation: G. LAZZI Constitution and Bylaws: O. KILIC Digital Archive Editor-in-Chief: A. Q. MARTIN Distinguished Lecturers: J. C. VARDAXOGLOU Education: D. F. KELLY EAB Continuing Education: S. R. RENGARAJAN Electronic Design Automation Council: M. VOUVAKIS Electronic Publications Editor-in-Chief: S. R. BEST European Representatives: B. ARBESSER-RASTBURG Fellows Nominations Committee: J. L. VOLAKIS

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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION Is the leading international engineering journal on the general topics of electromagnetics, antennas and wave propagation. The journal is devoted to antennas, including analysis, design, development, measurement, and testing; radiation, propagation, and the interaction of electromagnetic waves with discrete and continuous media; and applications and systems pertinent to antennas, propagation, and sensing, such as applied optics, millimeter- and sub-millimeter-wave techniques, antenna signal processing and control, radio astronomy, and propagation and radiation aspects of terrestrial and space-based communication, including wireless, mobile, satellite, and telecommunications. Author contributions of relevant full length papers and shorter Communications are welcomed. See inside back cover for Editorial Board.

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Digital Object Identifier 10.1109/TAP.2010.2051741

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Confocal Ellipsoidal Reflector System for a Mechanically Scanned Active Terahertz Imager Nuria Llombart, Member, IEEE, Ken B. Cooper, Member, IEEE, Robert J. Dengler, Member, IEEE, Tomas Bryllert, and Peter H. Siegel, Fellow, IEEE

Abstract—We present the design of a reflector system that can rapidly scan and refocus a terahertz beam for high-resolution standoff imaging applications. The proposed optical system utilizes a confocal Gregorian geometry with a small mechanical rotating mirror and an axial displacement of the feed. For operation at submillimeter wavelengths and standoff ranges of many meters, the imaging targets are electrically very close to the antenna aperture. Therefore the main reflector surface must be an ellipse, instead of a parabola, in order to achieve the best imaging performance. Here we demonstrate how a simple design equivalence can be used to generalize the design of a Gregorian reflector system based on a paraboloidal main reflector to one with an ellipsoidal main reflector. The system parameters are determined by minimizing the optical path length error, and the results are validated with numerical simulations from the commercial antenna software package GRASP. The system is able to scan the beam over 0.5 m in cross-range at a 25 m standoff range with less than 1% increase of the half-power beam-width. Index Terms—Reflector antennas, scanning antennas, submillimeter-wavelength imaging, terahertz radar, THz.

I. INTRODUCTION ECENTLY attention has focused on defense and security terahertz (THz) applications because signals at these frequencies can penetrate many garments and provide moderate resolution images of the body at long standoff ranges without any exposure to ionizing radiation. One promising approach

R

Manuscript received July 01, 2009; revised November 16, 2009; accepted December 25, 2009. Date of publication March 29, 2010; date of current version June 03, 2010. This work was carried out by the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration. This work was supported under a contract to the California Institute of Technology, Division of Biology, by the Naval Explosive Ordnance Disposal Technology Division, with funding provided by the DoD Physical Security Equipment Action Group (PSEAG). N. Llombart was with the Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109 USA. She is now with the Optics Department, Universidad Complutense de Madrid, E-28040 Madrid, Spain (e-mail: nuria. [email protected]). K. B. Cooper and R. J. Dengler are with the Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109 USA. T. Bryllert was with the Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109 USA. He is now with the Physical Electronics Laboratory, Chalmers University of Technology, SE-412 96 Gothenburg, Sweden and also with Wasa Millimeter Wave AB, 423 41 Torslanda, Sweden. P. H. Siegel is with the Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109 and also with the Department of Biology, California Institute of Technology, Pasadena, CA 91125 USA (e-mail: phs@ caltech.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.2010.2046860

to THz imaging utilizes the frequency-modulated continuouswave (FMCW) radar technique, where at submillimeter wavelengths, ultra-high resolution range measurements are used to detect objects concealed under clothing. In the FMCW radar imager described in [1], a 570–600 GHz radar beam is focused onto targets at 4 m standoff range using a folded-path antenna system with a 40 cm diameter ellipsoidal main reflector. To obtain imagery with a single transceiver located at the reflector’s close focus, the entire radar platform was mechanically rotated about two axes at rates of a few degrees per second. This slow scanning speed limited the THz radar’s imaging rate to roughly 10–20 ms per pixel, or one useful frame every several minutes. Other THz imaging systems have implemented advanced scanning mechanism to steer the main beam, as for example in [2], where two off-axis continuously rotating reflectors are used achieve a fast vertical scan without any acceleration, and in [3], where a periscope-based conical scan is implemented. However these systems rely on steering the beam after the main aperture, making them difficult to be used with large diameter apertures. A goal of the next-generation THz imaging radar system is to greatly increase the imaging speed [4] as well as the standoff range distance. The system will be operating at 25 m with a 1 m antenna aperture. The cross-range field of view of the system was chosen to be 0.5 0.5 m in area to span a field of view the size of a human torso. One route to achieving fast imaging is to fabricate a camera-like imaging system, where a focal plane radar array would acquire information from several pixels simultaneously. However, this approach would require a very large development cost because THz heterodyne detector array technology is in its infancy [5], [6]. Therefore we consider here an alternative that while still relying on mechanical scanning of a single beam projected onto a target, can nonetheless achieve rapid imaging by rotating a small, lightweight secondary mirror to steer the beam. We estimate that the imaging radar’s frame rate could increase by up to two orders of magnitude in this way [4]. Moreover, our design leaves open the possibility of zooming the THz beam’s focal point throughout a swath of near field distances in order to attain high quality imagery of targets over a long span of standoff ranges. The idea of using a small rotating mirror in order to increase the imaging acquisition time has already been used in several existing active THz imagers [7]–[9]. In [7] and [8], a telecentric lens design is used to focus each target pixel into a collimated beam. The collimated beams can be steered with a small mirror towards the frontand back-end electronics. In [9] a large focusing mirror is used in combination with a small rotation mirror that will steer the beam towards different parts of the mirror before focusing to

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LLOMBART et al.: CONFOCAL ELLIPSOIDAL REFLECTOR SYSTEM FOR A MECHANICALLY SCANNED ACTIVE THz IMAGER

the target. These optical designs are very effective in terms of acquisition time. However, they pay a large cost in resolution because the main apertures are very under illuminated. The scanning THz imager’s design proposed here uses a nearfield or confocal Gregorian reflector system (CGRS) [10]–[13] consisting of two paraboloid reflectors sharing a common focus. This type of reflector system has been proposed for satellite applications because of its excellent scanning performance [13], [14], a consequence of a cancellation of the coma and astigmatism aberrations that are normally present in more conventional reflector configurations [15]. In that proposed satellite application of CGRS, a phased array is used to perform the beam scanning, while for the terrestrial THz imaging of persons addressed here, we propose using a small rotating mirror for beam steering due to the technical challenge of scaling phased array technology to submillimeter wavelengths. In [16], beam scanning was proposed to be achieved by using small rotating reflectors in the aperture-image space of two reflectors. A tertiary reflector was shaped in order to have very good beam quality while scanning by rotational movement of this reflector. In our design, the rotating mirror is illuminated by a collimated beam rather than a expanding beam as in [16]. This is important because it relaxes the tolerances on the position of the rotating mirror’s principal axes, which otherwise would be difficult to align at these high frequencies. The paper is divided as follows. Section II presents the geometry of the reflector system. Section III studies the distortions of the scanned beam over the symmetric plane of a CGRS, while Section IV extends the design to a system that focuses in the near-field using an ellipsoidal reflector. In Section V we consider the reflector’s ability to refocus at variable ranges using a displacement of the feed. Finally, Section VI discusses the possibility of simultaneous scanning in the azimuth and elevation directions using the secondary mirror. II. REFLECTOR ANTENNA GEOMETRY We consider first a CGRS based on a paraboloidal main reand , as shown flector and subreflector, with diameters in Fig. 1. These parabolas share the same focal point, which is taken as the origin of the system. The focal distances of both paraboloidal reflectors are related by the system magnification . The overall dimension of the antenna system as ) will depend on this magnification and the f-number ( in the plane. of the system. The main reflector is offset by plane, , by rotating an angle The beam is steered in the about the x-axis the flat mirror (diameter ) located at a disfrom the subreflector. The flat mirror is illuminated by tance , in order to achieve a paraboloidal feed reflector, diameter plane wave incidence over the secondary reflector. The focal and , can be addistance and angle of the feed reflector, justed to optimize the far field characteristics of the available feeds and the refocusing requirements. The THz imager described in [1] uses a silicon etalon beam splitter to duplex the transmit and receive signals. In the antenna geometry of Fig. 1, the signal duplexing can be done at the feed reflector level by placing the beam splitter between the horn and the feed reflector. The characteristics of the feed reflector can also be changed to accommodate this beam splitter without

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Fig. 1. Paraboloid CGRS geometry: (a) XZ-, (b) YZ- and (c) XY -planes.

affecting the scanning performance. Another more compact way of duplexing is to use a waveguide coupler. III. STUDY OF THE SCANNING PERFORMANCE For cm-scale 3D radar imaging of targets at standoff ranges of 25 m, diffraction-limited resolution requires the main aperture to have a diameter of about 1 m. The typical cross-range ), span needed for imagery of persons is 0.5 m (or which means that the necessary angular displacement of the rotating mirror can be determined by the beam deviation factor , where is the beam deviation to be: factor [17], and it is defined as the ratio between the main beam varies bescan angle and the subreflector scan angle. The tween 0.7 and 1 depending on the system f-number and offset. If the reflector is uniformly illuminated, this corresponds to scanhalf-power beam-widths (HPBW) ning by approximately at 670 GHz. The first order distortions associated with beam scanning can be assessed by computing the path length error, referred as in this paper, over the main aperture [15]. This error is the difference between the length of each ray and the length of the central one. As explained in [13], the CGRS has superior scanning performances because the coma and astigmatism errors associated with asymmetric path length errors cancel, leaving a quadratic path length error for the confocal geometry. Fig. 2 shows the ray picture for a plane wave incident at on a symmetric confocal system with and . The rays are only computed up to the rotating flat mirror for a 2D cut. The corresponding path length

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Fig. 2. Picture of the ray tracing of a CGRS system with F D for a plane wave incident at  and X

=1m

=0

= 1:5m, M = 4, = 0:6 .

Fig. 3. Normalized path length error of a symmetric Paraboloid CGRS considering different system parameters as a function of the main aperture dimension : . for a plane wave incident at 

=06

error is shown in Fig. 3. In this case the size of the rotating , which is probably too large for sigmirror is nificant imaging speed improvement. At a larger magnification , the rotating mirror would be correspondingly smaller ), but at a cost of a larger path length error in size ( (Fig. 3). A second tradeoff for larger magnification is that the needed for a fixed beam scanning distance rotational angle scales with , potentially presenting a greater challenge to the mechanical implementation of a fast rotation [4]. We have considered an f-number of 1.5 up to now, but this . Therefore smaller implies a quite large system since f-numbers are preferable, but Fig. 3 shows that the path length error for an f-number of 1 is substantially worse for the case. We believe that a good compromise between system size and distortion is achieved when and , and in Fig. 4 we summarize the results of a physical optics simulation of the entire antenna system using GRASP for these parameters. The system feed for this geometry was chosen to be the Picket-Potter horn [18], which is commonly used at submillimeter wavelengths. This feed is modeled as a Gaussian beam with a taper of 10 dB at 12 . Fig. 4 a and b show the far field for the center and scanned ( ) beams, respectively. The insets of the figures show the corresponding far field beam and . intensities with respect to When the mirror is in the central position, we obtain a pattern with a HPBW of 0.0305 and a spillover loss of 0.39 dB.

=0 XZ = 90 = 1 m = 10 = 1 2 m = 0 55 m = 0 31 m = 0 22 m = 14 25

=0 YZ = 0 19 m

Fig. 4. Paraboloid CGRS: Far field for  (a) and 3 (b), where  refers to the offset plane ( ) and  to the symmetrical plane ( ). The inset of the figures shows the far field uv -grids. The main geometrical parame: ,X : : ters are D ,M ,L , ,F ,L : ,F : : .  and 

= 20

With the flat mirror fully rotated to , the HPBW is hardly affected at all, only increasing by 1%. In both cases, the cross-polarization fields fall outside the shown scale. Optimization of the secondary reflector’s shape, for example by using a bifocal structure [19] or by deforming the sub-reflector surface [13], might result in even less beam distortion. However, the fields shown in Fig. 4 are more than adequate for imaging purposes, with virtually no change in the crossbeam scan. range resolution (i.e., the HPBW) over the GRASP simulations indicate that even larger scan angles are possible with minimal distortion; for a scanning distance of the HPBW only increases by 8%. IV. NEAR FIELD FOCUSING The previous section considered the confocal Gregorian system with a paraboloidal main reflector, which provides focusing in the far field. However, at a 25 m standoff distance, targets are electrically very close to the antenna (in the reactive near field) for a 670 GHz radar. This means the main reflector must be an ellipsoid to achieve diffraction-limited focusing at 25 m. To a first order approximation, the near field patterns of

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Fig. 5. Geometry of a CGRS with an ellipsoidal reflector as main aperture.

an ellipsoid will be the same as the far field patterns of a paraboloid, as presented in the previous section, as long as the main rays of the two mirror types are coincident and the ellipse’s eccentricity is not too large. The geometrical mapping of the paraboloidal main reflector to the corresponding ellipsoidal reflector is summarized in Fig. 5. The mapping consists of having and for both types of reflectors, as the same values of well as having the seconday focus of the ellipsoidal reflector at and the desired focusing distance from the reflector. The ellipsoidal surface is then defined by the major axis distance , the foci distance , and the axis tilt angle . This approximate optical equivalence of paraboloidal and ellipsoidal reflectors was used by the authors for the fist time in [24] where a compensated Gregorian system with reduced cross-polarization was designed. Its primary utility is that existing design rules for far field systems (for example, the low cross-polarization dual-reflector system [21] and general design rules [22], [23]) can be readily applied to the near field. We have thus adapted the antenna geometry developed for Figs. 1–4 to an ellipsoidal main reflector, and Fig. 6 shows the and . resulting ray picture for a CGRS with The inset shows a magnified view of the rays around the main reflector region. For near-field focusing, the beam is scanned to; see Fig. 6. The path length error calculation wards for this system, where the length of the rays is computed starting from the rotating mirror up to the second focal plane (see Fig. 6), is shown in Fig. 7 as a function of the rotating mirror’s -coordinate, which relates to the -coordinate of the main reflector as . The normalized error amplitude is seen to be less than 0.35 over the entire span, which is comparable to the results of Fig. 3. The small distortions that do arise are attributed to the system actually focusing to a slightly different distance with respect to the nominal when scanning. Fig. 7 shows that the error becomes much smaller if the second focal plane is dis(defined in Fig. 5), before placed by a distance increasing again at . A larger f-number system will have the minimum plane of distortion closer to the actual fois smaller. For the imaging application cusing distance, i.e., considered here, this focusing plane displacement is not very significant. In order to check the paraboloidal/ellipsoidal design equivalence, the ellipsoidal CGRS was simulated with GRASP, and the field values at 25 m are presented in Fig. 8. As expected from the

Fig. 6. Ray picture for an ellipsoidal CGRS with D . ,R and 

= 1m

= 25 m

= 03

F

= 1 m, M = 10,

Fig. 7. Path length error for a symmetric CGRS with an ellipsoidal main reflector, and F and M as a function of the second focal plane . distance for 

= 1m =3

= 10

design equivalence, the fields at and 3 exhibit virtually the same HPBW and spillover losses as the ones associated with the paraboloidal reflector. In fact, the beam profiles would be indistinguishable from the ones shown in Fig. 4 if they were plotted on top of one another. The HPBW of the center beam at . 25 m, corresponding to 0.0305 , is V. REFOCUSING THE IMAGING SYSTEM In this section an approach for focusing at different near-field distances is considered. In particular, we propose to achieve zooming by the mechanical displacement of a system component. As with beam steering, the alternative of phased array zooming [25], while very attractive, is prohibited by the available technology at THz frequencies. Our goal is to evaluate redeviation covering the span 12.5 m – focusing over a 37.5 m in range. A single ellipsoidal reflector has two optimal focuses defined by the ellipse parameters, and one can focus to a different secas ondary focus by axially displacing the first focal point by shown in the inset Fig. 9. This displacement results in the second to . Fig. 9 shows the ray focus of the ellipse moving from picture of an offset single reflector where the feed is displaced . This displacement will introduce a certain by path length error as shown in Fig. 10. The feed displacement is

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=22

= 0 55 m

Fig. 9. Ray picture for an ellipsoidal reflector with f : ,X : , : . The inset of the figure shows the R ,R and geometrical description of a single ellipsoidal reflector with the feed displaced ( ) to change the focusing distance towards R from the nominal one R .

= 25m

= 15m

1 = 14 2cm

1

=0 = 1 32

Fig. 8. Ellipsoidal CGRS: Near field at 25 m for  (a) and 3 (b). The inset of the figures shows the near field xy -grids. The system has the same geometrical parameters as Fig. 4 plus R , : and c . :

23 869 m

= 25 m

2 =

chosen by minimizing the path length error computed up to in each of the main planes. For offset reflectors, the optimum ) feed displacement is different in the symmetrical ( ) as can be seen Fig. 10. In the case of and offset planes ( a larger f-number, the relative difference between the optimum distances for each plane is smaller as well as the maximum path length error, resulting in higher quality beams, as can be seen in the same figure. Fig. 11 shows the near fields at 12.5 m for two offset reflectors with different f-numbers. The asymmetrical results in significant path length error of Fig. 10 for coma and beam tilting distortions. In an actual system, the feed offset will be in between the optimum distances for each plane, and it would be a compromise between the beam qualities in both planes. In the confocal geometry, the focal point of the main ellipsoidal reflector needs to be displaced in order to refocus the system. One could achieve the refocusing by translating the whole secondary optical system (i.e., secondary reflector, rotating mirror, feed reflector and feed). In such a case the beam quality will depend on the main reflector’s f-number, which has been fixed to 1.2 in previous sections in order to keep the system

Fig. 10. Path length error when refocusing at 12.5 m for several reflector system : : configurations: (1) Single reflector, SR, (F and X ) where : and : ; (2) SR, for  for  : and X : ) where : for  and (F : for  ); and (3) same CGRS of Fig. 8 ( : ).

=0 1 = 6 3 cm = 22m = 0 55 m 1 = 22 65cm = 90

= 12m = 0 55 m 1 = 7 3 cm = 90 1 = 21 5 cm =0 1 = 13 5cm

overall dimension reasonably small. This small f-number will cause large beam distortions, as shown in Fig. 11. A solution that presents better beam quality is the displacement of the feed only. The advantage here is that the f-number of the feed reflector is higher (i.e., 2.2). The phase error associated with the confocal geometry is also shown in Fig. 10 when only the feed is displaced. The feed displacement provided in the figure caption has been chosen as a compromise between the beam qualities in and 90 . This error is comparable to that of both planes, the single reflector with the same f-number. However the main reflector is under illuminated (see Figs. 10 and 12). This aperand , which ture illumination depends on the distances have been chosen to avoid blockage effects when displacing the . Because of this poorer feed for refocusing at illumination, the HPBW associated with the confocal geometry is larger than the one of the single reflector. However, the beam ) are tilting effect and side lobe level in the offset cut ( much better for the CRGS, as indicated by the smaller beam length error. Fig. 13 shows that fields, when refocusing at 37.5 m, have a comparable side lobe level and beam tilting effect than

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Fig. 13. Near fields at (a) R = 12:5 m (1 = 13:5 cm) and (b) R = 37:5 m (1 = 08 cm) of the ellipsoidal CGRS.

= 12:5 m of the geometries (1)-(2) described in 1 = 6 5 cm and 1 = 22 cm, respectively: (a)  = 0 and

Fig. 11. Near field at R : Fig. 10 with (b) 90 cuts.

Fig. 14. Near field at R CPGRS with  .

=3

Fig. 12. Ray picture for the ellipsoidal CGRS with the main reflector is under illuminated.

1 = 13:5 cm. Notice that

the ones at 12.5 m. In both cases, the cross-polarization fields fall outside the shown scale. Finally, both operations, scanning and refocusing, can be simultaneously combined. To check the quality of the fields, we have computed the near fields at the external boundaries of the deg) with focusing region at the maximum scan angle ( GRASP. The fields are shown in Fig. 14 and Table I presents a summary of the simulated parameters of the system for both

= 12:5 m and R = 37:5 m of the ellipsoidal

operations. Note that the spill over (SO) provided in the table is associated with a Gaussian beam excitation, and the SO associated with the actual feed pattern may be larger. Even when refocusing, the scanned beams are of excellent quality. On one hand, the spillover loss is below 0.8 dB over the whole scanning and focusing operation range. For example, the HPBWs and ) when scanning increase by less than 6% with re( ). From Table I, we can see that spect to the center beam ( and at the several focusing distances are different from each other. Actually, if the main aperture would have the same illumination when refocusing, i.e., the same spillover loss, the HPBW at 12.5 m and 37.5 m would be half and one and a half that at 25 m, respectively. Instead, the actual values are larger and smaller than these ideal ones because of different illumination of the main reflector. Even so, they are within a factor of two of one another and should provide comparable imaging performance at the different distances. VI. 2-AXIS SCANNING SYSTEM A flat secondary mirror that can rotate in two planes could achieve faster frame rates than a single-axis mirror by steering the radar beam in a circular or spiral pattern [26]. Therefore it is valuable to examine the beam quality in the orthogonal scanplane of Fig. 1 for the antenna configurations ning plane (the

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both operations are applied, yielding to very low beam distortions. The design was implemented by generalizing a confocal dual reflector system with a paraboloidal main reflector into a near-field focusing system with an ellipsoidal main reflector using a simple equivalence rule. The beam patterns were numerically calculated using GRASP for all extreme system specifi) and refocusing distance (50% from cations of scan angle ( the nominal). These results indicate that the THz imaging radar system will able to scan a target area of 0.5 m at 25 m stand-off with less than 3% increase of the HPBW at the nominal focusing distance, and to refocus from 12.5 m up to 37.5 m while maintaining cm-scale imaging resolution.

TABLE I SCANNING AND REFOCUSING PARAMETERS OF THE ELLIPSOIDAL CGRS

REFERENCES

Fig. 15. Near field at 25 m when scanning in the offset plane (  ).

=0

= 03

,

TABLE II PARAMETERS OF THE 2D SCANNING ELLIPSOIDAL CGRS

presented here). In order to avoid beam blockage effects, simu, of 60 cm rather lations were performed with a larger offset, than 55 cm used in Fig. 8. Fig. 15 shows the scanned beam when in this offset plane ( ). the flat mirror is rotated by As shown in Table II, which compares the beam widths and spillover loss for scanning in the two directions, there will be only a negligible impact (at most 3%) on the radar’s imaging . resolution when the beam is steered in any direction up to The asymmetry of the HPBWs when scanning towards positive is associated with the offset of the or negative angles for structure. VII. CONCLUSION The next generation of terahertz imagers will require fast scanning and will benefit from refocusing capabilities. In this paper, a reflector system design was presented that can achieve both functionalities using mechanical rotation and translation of small secondary optical elements while the large primary mirror remains stationary. The reflector geometry consists on a confocal system. Using this system, very low path length errors are obtained when either scanning or refocusing the beam, or

[1] K. B. Cooper, R. J. Dengler, N. Llombart, T. Bryllert, G. Chattopadhyay, E. Schlecht, J. Gill, C. Lee, A. Skalare, I. Mehdi, and P. H. Siegel, “Penetrating 3D imaging at 4 and 25 meter range using a submillimeter-wave radar,” IEEE Trans. Microw. Theory Tech., vol. 56, pp. 2771–2778, Dec. 2008. [2] A. H. Lettington, D. Dunn, N. E. Alexander, A. Wabby, B. N. Lyons, R. Doyle, J. Walshe, M. F. Attia, and I. Blankson, “Design and development of a high-performance passive millimeter-wave imager for aeronautical applications,” SPIE Opt. Eng., vol. 44, p. 093202, 2005. [3] E. N. Grossman, C. R. Dietlein, M. Leivo, A. Rautiainen, and A. Luukanen, “A passive, real-time, terahertz camera for security screening, using superconducting microbolometers,” in Proc. IEEE Int. Microwave Symp., Jun. 7–12, 2009, pp. 1453–1456. [4] K. B. Cooper, R. J. Dengler, N. Llombart, T. Bryllert, G. Chattopadhyay, I. Mehdi, and P. H. Siegel, “An approach for sub-second imaging of concealed weapons using terahertz (THz) radar,” Int. J. Infrared Millim. Wave, vol. 30, no. 12, pp. 1297–1307, Dec. 2009. [5] C. Groppi, C. Walker, C. Kulesa, D. Golish, J. Kloosterman, P. Pütz, S. Weinre, T. Kuiper, J. Kooi, G. Jones, J. Bardin, H. Mani, A. Lichtenberger, T. Cecil, A. Hedden, and G. Narayanan, “SuperCam: A 64 pixel heterodyne imaging spectrometer,” in Proc. SPIE, 2008, vol. 7020, pp. 702011-1–8. [6] G. Chattopadhyay, “Heterodyne arrays at submillimeter wavelengths,” presented at the XXVIIIth General Assembly of Int. Union of Radio Science, New Delhi, India, Oct. 2005. [7] T. M. Goyette, J. C. Dickinson, K. J. Linden, W. R. Neal, C. S. Joseph, W. J. Gorveatt, J. Waldman, R. Giles, and W. E. Nixon, “1.56 terahertz 2-frames per second standoff imaging,” in Proc. SPIE, Terahertz Technology and Applications, Jan. 2008, vol. 6893, Photonics West 2008. [8] J. Xu and G. C. Cho, “A real-time terahertz wave imager,” presented at the Lasers and Electro-Optic and Conf. on Quantum Electronics and Laser Science Conf. CLEO/QELS, May 4–9, 2008. [9] Q. Song, A. Redo-Sanchez, Y. Zhao, and C. Zhang, “High speed imaging with CW THz for security,” in Proc. SPIE, 2008, vol. 7160, pp. 716016–716016-8. [10] C. Dragone and M. J. Gans, “Imaging reflector arrangements to form a scanning beam using a small array,” Bell Syst.Tech. J., vol. 58, no. 2, pp. 501–515, Feb. 1979. [11] S. Morgan, “Some examples of generalized Cassegrain and Gregorian antennas,” IEEE Trans. Antennas Propag., vol. 12, no. 6, pp. 685–691, Nov. 1964. [12] W. D. Fitzgerald, Limited Electronic Scanning with an Offset-Feed Near-Field Gregorian System MIT Lincoln Lab., 1971, Tech. Rep. 486. [13] J. A. Martinez-Lorenzo, A. Garcia-Pino, B. Gonzalez-Valdes, and C. M. Rappaport, “Zooming and scanning Gregorian confocal dual reflector antennas,” IEEE Trans. Antennas Propag., vol. 56, no. 9, pp. 2910–2919, Sep. 2008. [14] Y. Imaizumi, Y. Suzuki, Y. Kawakami, and K. Araki, “A study on an onboard Ka-band phased-array-fed imaging reflector antenna,” in Proc. IEEE Antennas and Propag. Society Int. Symp., 2002, vol. 4, pp. 144–147. [15] C. Dragone, “A first-order treatment of aberrations in Cassegrain and Gregorian antennas,” IEEE Trans. Antennas Propag., vol. AP-30, no. 3, pp. 331–339, 1982. [16] P. C. Werntz, W. L. Stutzman, and K. Takamizawa, “A high-gain trireflector antenna configuration for beam scanning,” IEEE Trans. Antennas Propag., vol. 42, no. 9, pp. 1205–1214, Sep. 1994.

LLOMBART et al.: CONFOCAL ELLIPSOIDAL REFLECTOR SYSTEM FOR A MECHANICALLY SCANNED ACTIVE THz IMAGER

[17] A. W. Rudge, K. Milne, A. Olver, and P. Knight, The Handbook of Antenna Design. London, U.K.: Peter Peregrinus. [18] P. D. Potter, “A new horn antenna with suppressed sidelobes and equal beamwidths,” Microw. J., p. 71, Jun. 1963. [19] J. Rao, “Bicollimated Gregorian reflector antenna,” IEEE Trans. Antennas Propag., vol. 32, no. 2, pp. 147–154, Feb. 1984. [20] F. Ulaby, R. Moore, and A. Fung, Microwave Remote Sensing. Norwood, MA: Artech House, 1982. [21] Y. Mizugutch, M. Akagawa, and H. Yokoi, “Offset dual reflector antennas,” in Proc. IEEE Antennas and Propag. Society Int. Symp., Oct. 1976, pp. 2–5. [22] K. W. Brown and A. Prata, “A design procedure for classical offset dual reflector antennas with circular apertures,” IEEE Trans. Antennas Propag., vol. 42, no. 8, Aug. 1994. [23] W. V. T. Rusch, A. Prata, Y. Rahmat-Samii, and R. A. Shore, “Derivation and application of equivalent paraboloid for classical offset Cassegrain and Gregorian antennas,” IEEE Trans. Antennas Propag., vol. 38, no. 8, pp. 1141–1149, Aug. 1990. [24] N. Llombart, A. Skalare, and P. H. Siegel, “High-efficiency array for submillimeter-wave imaging applications,” presented at the IEEE Antennas and Propag. Soc. Inter. Symp., Jul. 5–11, 2008. [25] W. H. Carter, “On rrefocusing a radio telescope to image sources in the near field of the antenna array,” IEEE Trans. Antennas Propag., vol. 37, no. 3, pp. 314–319, Mar. 1989. [26] C. am Weg, W. von Spiegel, B. Hills, T. Loeffler, R. Henneberger, R. Zimmermann, and H. G. Roskos, “Fast active THz camera with range detection by frequency modulation,” Int. J. Infrared Millim. Wave, vol. 30, no. 12, pp. 1281–1296, Dec. 2009.

Nuria Llombart (S’06–M’07) received the Ingeniero de Telecomunicación degree and the Ph.D. from the Universidad Politécnica de Valencia, Spain, in 2002 and 2006 respectively. During her Master’s degree studies she spent one year at the Friedrich-Alexander University of Erlangen-Nuremberg, Germany, and worked at the Fraunhofer Institute for Integrated Circuits, Erlangen, Germany. From 2002 until 2007, she was with the Antenna Unit at the TNO Defence, Security and Safety Institute, The Hague, The Netherlands, working as Ph.D. student and afterwards as Researcher. From 2007 until 2009, she was a Postdoctoral Fellow at the California Institute of Technology, Pasadena, working for the SWAT group of the Jet Propulsion Laboratory. Since January 2010, she is a “Ramón y Cajal” researcher at the Universidad Complutense de Madrid, Spain. Her research interests include the analysis and design of printed array antennas, EBG structures, reflector antennas, lens antennas and submillimeter-wave components. Dr. Llombart was a co-recipient of the 2008 H. A. Wheeler Applications Prize Paper Award from the IEEE Antennas and Propagation Society.

Ken B. Cooper (M’06) received the A.B. degree in physics (summa cum laude) from Harvard College, Cambridge, MA, in 1997, and the Ph.D. degree in physics from the California Institute of Technology, Pasadena, in 2003. Since 2006, he has been a Member of the Technical Staff with the Jet Propulsion Laboratory, Pasadena, CA. His current research interests include submillimeter-wave radar, spectroscopy, and device physics.

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Robert J. Dengler (M’09) received the B.Sc. degree in electrical and computer engineering from California State Polytechnic University, Pomona, in March 1989. He began his work with Dr. Siegel at the Jet Propulsion Laboratory as an intern in 1988, developing beam pattern acquisition and analysis software. Since then he has been involved in the design and construction of submillimeter-wave receivers and components, including design and fabrication of test instrumentation for submillimeter flight mixers. His recent work is focused on THz active imaging and heterodyne spectrometers, including design and construction of a 110 dB dynamic range biosample transmission imaging system at 2.5 THz, a high resolution imaging radar system operating at 670 GHz, and an ultra high-sensitivity room-temperature 550–620 GHz absorption spectrometer.

Tomas Bryllert received the degree of M.S. degree in physics and the Ph.D. degree in semiconductor physics from Lund University, Sweden, in 2000 and 2005, respectively. In 2006, he joined the Microwave Electronics Laboratory, Chalmers University of Technology, Sweden, where his main research interest was device- and circuit-technology for terahertz frequency multipliers. During 2007–2009 he was with the Jet Propulsion Laboratory (JPL), Pasadena, CA, funded by a research fellowship from the Wallenberg foundation – working on submillimeter-wave imaging radar and terahertz time-domain imaging systems. Starting in September 2009, he joined the Physical Electronics Laboratory at Chalmers University of Technology, working on circuits and devices for millimeter wave applications. Since 2007, he is also the CEO of Wasa Millimeter Wave AB, Torslanda, Sweden, a company that develops and produces millimeter wave modules.

Peter H. Siegel (F’01) received the BA degree in Astronomy from Colgate University, Hamilton, NY, in 1976, and the MS in physics and Ph.D. degree in electrical engineering from Columbia University, New York, in 1978 and 1983, respectively. He served as an NRC Fellow at the Goddard Institute for Space Studies, NY and then as a staff member in the Electronics Development Lab, National Radio Astronomy Observatory, Charlottesville, VA, until 1987. He then moved to the Jet Propulsion Laboratory, Pasadena, CA, to work on submillimeter wave sensors for NASA space astrophysics and Earth remote sensing applications. At JPL he has been involved in four space flight missions and more than 65 research and development programs. He founded and has led for more than 15 years a large technical team, SWAT – Submillimeter Wave Advanced Technology, focused on NASA applications of terahertz technology. In 2001, he joined the staff at the California Institute of Technology, where he holds appointments as Member of Professional staff in Biology and Faculty Associate in Electrical Engineering. At Caltech he has been expanding terahertz applications into biology and medicine as well as into defense and security areas. His interests cover all areas of THz technology, techniques and applications. Dr. Siegel is an active member of the IEEE THz community and has served as Vice-Chair and Chair of MTT-4, THz Technology, as an IEEE Distinguished Lecturer and continuing member of the speaker’s bureau, as Organizer and Chair of seven special THz sessions at sequential IMS meetings, as a long term member of the TPC and Special Guest Editor for the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES and JIMT. He is also Chair of the International Organizing Committee and Founding Chair of the International Society of Infrared, Millimeter, and Terahertz Waves, the largest continuous forum devoted to THz science and technology.

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Compound Diffractive Lens Consisting of Fresnel Zone Plate and Frequency Selective Screen Yijing Fan, Ban-Leong Ooi, Senior Member, IEEE, Hristo D. Hristov, Senior Member, IEEE, and Mook-Seng Leong, Senior Member, IEEE

Abstract—We describe a new Fresnel zone plate (FZP) and frequency selective screen (FSS) compound lens consisting of a binary FZP and FSS. The FZP has eight circular zones totally, four open and four closed, and the FSS is a square array of 40 40 four-leg loaded elements cut in a thin metal sheet. The FZP-FSS lens is designed for a paraxial plane-wave illumination at the frequency of 12 GHz, with a focal length of 15 cm. The compound lens is simulated and analyzed numerically by means of a specially developed hybrid PSTD-FDTD algorithm and software. The PSTD-FDTD results are contrasted with those obtained by lens prototype measurements. As a result, some attractive focusing and spectral properties of the FZP-FSS lens compared to the same-size FZP lens have been found: a frequency filtering property enhancement, about 2 dB increase in the peak focusing intensity, and more than 4 dB reduction of the first off-axis maximum. Both lenses have roughly the same transverse angular resolution. Index Terms—Focusing, frequency selective surface (FSS), Fresnel zone plate (FZP), microwave lens.

Some FZP lenses have complex and irregular structures [4]–[6], [10]–[13], which may consist of hundreds of small apertures. In such structures, the hybrid PSTD- FDTD method with coarse PSTD grids and dense FDTD grids is a good choice in order to achieve a good computational efficiency and accuracy [14], [15]. This paper presents a novel microwave compound lens consisting of two basic components: FZP lens and frequency selective surface or screen (FSS). It was proposed and examined briefly in a recent conference paper as one of several numerical examples used for evaluation of the PSTD-FDTD algorithm for a numerical simulation of complex irregular FZP lenses [13]. Here the same algorithm is applied for more extensive analysis of the FZP-FSS lens and its components. The PSTD-FDTD results for the FZP and FZP-FSS lenses are contrasted with those obtained by prototype measurements.

I. INTRODUCTION

II. FZP-FSS LENS DESIGN

HE microwave/millimeter wave Fresnel zone plate (FZP) has been successfully used in some mountain radio-relay communications links as a passive signal repeater for a transmit distance expansion and line-of-sight alteration. Also, it has been employed as a focusing element in a variety of lens antennas, measurement setups and imaging systems. The classical microwave FZP lens consists of circular concentric metal rings that lay over the odd or even Fresnel zones. For this simple focusing structure the Kirchhoff’s diffraction integral (KDI) [1]–[6] is the classical choice for analysis. By adopting the cylindrical or linear lens symmetry the complexity of KDI calculation can be significantly reduced, and with a similar accuracy, the method will generally outperform the already established full-wave numerical methods, as for instance, the finite-difference time-domain (FDTD) method [7], [8] or the Method of Moments [9] in term of the computation time.

The FSS periodic structures have gained wide applications [16]. They can be designed to reflect or transmit electromagnetic waves with required frequency discrimination, or to act as resonant periodic filters. Adding together the FZP and FSS makes a compound diffractive lens, the FZP-FSS (Fig. 1(a)). The new lens was designed for a paraxial plane-wave illumination at a frequency (wavelength ), and for a primary focal measured from the FZP plane. length Thus, the FZP-FSS lens is acting like a receiving lens antenna with a 0 dBi isotropic feed. The FZP lens has eight Fresnel zones totally: four open and four closed, or covered by metal rings, with the -th zone radius calculated according to [4]

T

Manuscript received May 22, 2009; revised November 03, 2009; accepted December 08, 2009. Date of publication March 29, 2010; date of current version June 03, 2010. The work of H. D. Hristov was supported by the Chilean Fondecyt Project 1095012/2009. Y. J. Fan, and M. S. Leong are with Department of Electrical and Computer Engineering, National University of Singapore (NUS), Singapore 119260, Singapore (e-mail: [email protected]). B. L. Ooi (deceased) was with Department of Electrical and Computer Engineering, National University of Singapore (NUS), Singapore 119260, Singapore. H. D. Hristov is with Departamento de Electronica, Universidad Tècnia Fedrico Santa Maria (UTFSM), Valparaiso, Chile (e-mail: hristo.hristov@usm. cl). Digital Object Identifier 10.1109/TAP.2010.2046849

(1) where . In Table I are listed the FZP lens zone radii calculated by (1). The entire lens aperture diameter is . The ratio between the focal length and the FZP radius is . The FZP lens shown in Fig. 1(a) has its odd zones (1st, 3rd, 5th and 7th) open and is called a positive FZP, unlike the negative FZP, in which even zones are open. The transmission type band-pass FSS plate placed at the back side of the FZP-FSS lens is a square array of 40

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Fig. 2. Illustration of PSTD-FDTD hybrid grids for the FSS lens component.

III. PSTD-FDTD METHOD FOR FSS PLATE ANALYSIS

Fig. 1. FZP-FSS design illustrations: (a) FZP-FSS computer model, (b) geometry of the FSS four-leg cross element, (c) FZP prototype and (d) FSS prototype.

TABLE I FRESNEL ZONE PLATE RADII IN CENTIMETERS

40 four-leg loaded elements cut in a thin metal screen. The four-legged (cross) loaded element is frequently used in the FSS plates for its capacity of interacting with an arbitrary polarization. The general design guidelines for this FSS plate are given in [16]. Each FSS element was designed for a resonant frequency of 12 GHz, which resulted in the dimensions shown in the element geometry (Fig. 1(b)). The initial cross length of the four-legged element was set to . Next, the cross width and the gap between adjacent elements were optimized using computer simulations to fix the resonant frequency to 12 GHz. The FZP-FSS lens size of 40 cm was beyond the capability of the existing etching and milling machines in the NUS RF & Microwave Laboratory, where the prototype was built and experimented. Therefore, the FZP metal rings were handmade of a thin copper sheet and were glued on the front side of a Styrofoam board of thickness 4 mm (Fig. 1(c)). The FSS plate was made by four FSS square sub-plates of 20 cm in size each (Fig. 1(d)). The sub-plates were fabricated by use of the etching process on a PC board 1.2 mm thick, with a substrate permittivity equal to 4.4 and a loss tangent of about 0.01. The FSS sub-plates were combined together and glued to the back side of the FZP Styrofoam board to form the complete compound lens with a total distance between FZP and FSS equal to 6.4 mm or .

For explanation simplicity, the hybrid PSDT-FDTD algorithm is described here not for the entire FZP-FSS lens but for one of its components only: the more complex FSS structure The whole computation domain of the FSS is separated into two sub-domains and different algorithms are applied to them. An interface is setup to exchange the results between the two sub-domains. If the metal thickness is ignored, the planar FSS can be approximated as a 2D plate ABCD in XY-plane as shown in Fig. 2. This 2D plate can be meshed and updated with quasi-2D FDTD grid points, while PSTD is updated with 3D grid points. Next the PSTD-FDTD meshing is contrasted in complexity to the Kirchhoff’s diffraction layout (KDL), illustrating the corresponding initial and boundary electromagnetic conditions [1], [4]. The diffraction aperture is supposed to be cut in an infinitely thin conductive plane. Here the KDL method is applied to the FSS structure. in size square For the KDL numerical simulation, the FSS plate comprising cross-element apertures, is meshed with nodes, where for meshing size. The field is computed in the diffraction apertures only and then is transformed to the receiving focal plane, nodes. Each cross-element aperture is meshed with approximated by elements, where . For receiving planes inside the 3D region of FSS, the overall flops for one time step are calculated by: (for 3D field distribution). For the computation complexity is . If PSTD-FDTD method is employed the FSS plate is analyzed with 2D FDTD method using a fine meshing . The region around FSS is computed with 3D PSTD using a coarse meshing . For the fine meshing and the complexity is . The complexity is much lower for large . If , the overall flops for one time step are . The complexity in the case of PSTD-FDTD method can be dramatically reduced further. The number of flops required for KDL method and PSTDFDTD method with two different (meshing 1 and meshing

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TABLE II COMPLEXITY COMPARISON FOR KDL AND PSTD-FDTD IN CASE OF FSS PLATE SIMULATION

Fig. 3. Simulated field intensity at the focal point vs. the frequency of single FZP and composite FSS-FZP lenses.

2) are listed in Table II. It is evident from the table that the PSTD-FDTD meshing requires much smaller number of flops than the meshing in case of KDL. We also found that the PSTD-FDTD numerical method can provide similar accurate solution as the classical Kirchhoff’s diffraction integral (KDI), obtained analytically from the KDL approximation [1], [4]. IV. NUMERICAL AND EXPERIMENTAL RESULTS The PSTD-FDTD method is employed first to perform 3D transient analysis of the FZP lens alone. The numerical frequency response of the focusing intensity is obtained from the time domain results and is graphed in Fig. 3 by a dashed line. After adding the FSS, the frequency response of the resulted complex FZP-FSS lens (solid line) is narrowed around the design frequency of 12 GHz, and is lowered drastically for the lower frequencies around 2–6 GHz. Hence, the FZP-FSS lens band-pass filtering properties are improved significantly compared to those of the single FZP lens. The numerical optimization has shown that the proper distance between the FSS and FZP is around a quarter wavelength like in the folded or reflector-type FZP. Fig. 4 illustrates the cartographic YZ-plane field intensity distribution obtained by the PSTD-FDTD space-domain analysis for the single FZP lens (a), and composite FSS-FZP lens (b). In both lenses, the focal field areas around the primary foci show up much better transverse than axial resolution. In the case of FSS-FZP lens the focal spot is well shifted to the lens direction, which makes its effective focal length by 15–20% smaller compared to the FZP design focal length of

Fig. 4. Cartographic YZ-plane intensity distribution of (a) FZP lens, and (b) FSS-FZP lens.

15 cm. A more appropriate choice of the FSS substrate thickness or permittivity, and the distance between FZP and FSS would probably help to further diminish the focal area shift. The 3D graphs in Fig. 5 are very informative for the intensity distribution in the transverse focal plane . Obviously, the compound FZP-FSS lens greatly surpasses in focusing the same-size single FZP lens. More clear distinction between the transverse focal planes focusing of the two lenses can be made by Fig. 6, which illustrates the numerical and measured focusing intensity vs. the coordinate . A good agreement between the measured and numerical results is observed, especially for the single FZP lens. Some of the basic lens parameters, found from Fig. 5 and Fig. 6, are summarized in Table III. These are the focal point (or main lobe) intensity , the first (or maximum) side lobe intensity normalized to the lens resolution. The power resolution of a circular lens of diameter , illuminated by a paraxial plane wave, is define as the angle between the lens axis and the first minimum (zero) in the focal plane intensity distribution.

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Fig. 6. Intensity distribution for the FZP and FSS-FZP lenses (simulated and F. measured) along the transverse axis X , at the focal plane Z

=

TABLE III BASIC FOCUSING PARAMETERS OF FZP AND FSS-FZP LENSES (NUMERICAL AND MEASURED)

Fig. 5. Intensity distribution in transverse focal plane Z and (b) FZP-FSS lens.

In optics, the resolution angle mately found by [4]

= F for (a) FZP lens

(in radians) can be approxi-

(2) Where is called a lens resolution factor. For the microwave wave FZP lenses, which number of zones is relatively small (usually less than a few dozens) has a bit different value. The negative FZP has , or a somewhat better resolution, while the positive FZP has , or a slightly worse resolution. In the transverse intensity distribution curves (Fig. 6) the lens resolution angle can be expressed as , where is the distance from the FZP axis to the first intensity minimum. Because the numerical and measured values of , and , respectively, for both FZP and FZP-FSS lenses differ slightly it is preferable to calculate their mean .

From Table III is concluded that adding the FSS to the FZP lens results in about 2 dB increase in the peak focusing intensity while the first intensity sidelobe is reduced by more than 4 dB. The mean of the resolution angle for both FZP and FSS-FZP lenses is roughly the same. The computer simulations of the composite FZP-FSS structure and its components FZP and FSS have grounded the following simplified lens hypothesis. The FSS transmission function is preserved authentic only in the annular regions behind the FZP open zones. For that reason, the doublets of corresponding FZP and FSS transmissive zones are expected to produce a focusing comparable to that of the single FZP. Alternatively, the FZP metal rings greatly influence the nearby FSS resonant elements and deteriorate their transmission properties. Thus, the FSS annular regions behind the FZP metal rings become partially reflective, and jointly with the metal rings form open resonant cavities. Under the chosen initial and boundary field conditions the open-cavity multi-reflection/diffraction fields transfer constructively into the transmission volumes behind the open zones. Accordingly, the complex FZP-FSS lens surpasses in focusing intensity the same-size FZP lens. V. CONCLUSION Combining the FZP lens and the FSS structure leads to a new compound diffractive FZP-FSS lens with enhanced focusing and frequency filtering characteristics.

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A hybrid PSTD-FDTD algorithm using coarse PSTD grids and dense FDTD grids is explored for the FZP-FSS lens numerical analysis. A lens laboratory prototype is fabricated and measured. The main numerical and experimental lens characteristics agree well. As a result, some important focusing and spectral properties of the FZP-FSS lens compared to the samesize FZP lens have emerged: (i) enhancement of the lens frequency filtering property, (ii) around 2 dB increase in the peak focusing intensity and a 4 dB reduction in the maximum side lobe, and (iii) both FZP-FSS and FZP lenses have similar transverse resolutions. The future applications of the FZP-FSS lens may include microwave/mm-wave lens antennas, imaging and measurement systems.

REFERENCES [1] M. Born and E. Wolf, Principles of Optics, 2nd ed. New York: Pergamon Press, 1964. [2] , J. Ojeda-Castaneda, C. Gomez-Reino, and B. J. Thomson, Eds., Selected Papers on Zone Plates. Washington: SPIE Opt. Eng. Press, 1996. [3] D. N. Black and J. C. Wiltse, “Millimeter-wave characteristics of phase-correcting Fresnel zone plates,” IEEE Trans. Microw. Theory Tech., vol. 35, no. 12, pp. 1122–1128, 1987. [4] H. D. Hristov, Fresnel Zones in Wireless Links, Zone Plate Lenses and Antennas. Boston-London: Artech House, 2000. [5] Y. J. Guo and S. K. Barton, Fresnel Zone Antennas. Boston-London: Kluwer Academic , 2002. [6] O. V. Minin and I. V. Minin, Diffraction Optics of Millimeter Waves. Bristol-Philadelphia: Institute of Physics Publishing, 2004. [7] D. W. Prather and S. Shi, “Formulation and application of the finitedifference time-domain method for the analysis of axially symmetric diffractive optical elements,” J. Opt. Soc. Am A, vol. 16, pp. 1131–1142, 1992. [8] D. R. Reid and G. S. Smith, “A full electromagnetic analysis for the Soret and Folded zone plate antennas,” IEEE Trans. Antennas Propag., vol. 34, no. 12, pp. 3638–3646, 2006. [9] D. W. Prather and S. Shi, “Electromagnetic analysis of axially symmetric diffractive lenses with the method of moments,” J. Opt. Soc. Am A., vol. 17, no. 4, pp. 729–739, 2000. [10] A. Petosa, A. Lttipiboon, and S. Thirakoune, “Investigation on arrays of perforated dielectric Fresnel lenses,” Proc. Inst. Elect. Eng. Microw. Antennas Propag., vol. 153, no. 3, pp. 270–276, 2006. [11] M. Hajian, G. A. deVree, and L. P. Ligthart, “Electromagnetic analysis of beam-scanning antenna at millimeter-wave band based on photoconductivity using Fresnel-zone-plate technique,” IEEE Antennas Propag. Mag., vol. 45, no. 5, pp. 13–25, 2003. [12] A. Petosa, S. Thirakoune, I. V. Minin, and O. V. Minin, “Array of hexagonal Fresnel zone plate lens antennas,” Electronics Lett., vol. 42, no. 15, pp. 834–836, 2006. [13] Y. J. Fan, B. L. Ooi, M. S. Leong, X. C. Shan, A. Lu, H. D. Hristov, and R. Feick, “PSTD-FDTD analysis for complex irregular Fresnel zone plates,” presented at the 2nd EuCAP 2007, Edinburgh, U.K., Nov. 11–16, 2007, paper WePa035. [14] Y. F. Leung and C. H. Chan, “Combining the FDTD and PSTD method,” Microw. Opt. Technol. Lett., vol. 23, no. 4, pp. 249–254, 1999. [15] Q. L. Li, Y. Chen, and C. K. Li, “Hybrid PSTD-FDTD technique for scattering analysis,” Microw. Opt. Technol. Lett., vol. 34, no. 1, pp. 19–24, 2002. [16] B. A. Munk, Frequency Selective Surface Theory and Design. New York: Wiley, 2000.

Yijing Fan (S’05) received the B.Sc. degree in electronics from Peking University of China, in 2003 and the Ph.D. degree in microwave engineering from the National University of Singapore, in 2009. She joined Nanyang Technological University (NTU) in 2007. She is currently a Research Engineer in the Satellite Engineering Center of NTU and is working on X-band transmitter for a micro-satellite project. Her research interests include computational electromagnetics, RF circuits design, and wireless telecommunication.

Ban-Leong Ooi (M’91–SM’04) (deceased) was born in Taiping, Perak State, Malaysia on 17 Dec 1967. He received the B. Eng and Ph.D. degrees from the National University of Singapore (NUS), in 1992 and 1997, respectively. He was an Associate Professor of Electrical and Computer Engineering at the NUS. He was the Past Director for the Centre of RF and Microwaves. He also served as the Deputy Director for the MMIC and Packaging Laboratory and the Lab supervisor for the Microwave Laboratory in the NUS. His main research interests included active antenna, microwave semiconductor device modeling and characterization, microwave and millimeter-wave circuits design, and novel electromagnetic numerical methods. Prof. Ooi was a Member of IET and URSI, and Senior Member of IEEE. He served on the Singapore IEEE MTT/EMC/AP Chapter as Secretary (2000–2001) and Chapter’s Vice-Chairman (2002–2003). His last held position in IEEE was the Chairman of the Singapore IEEE MTT/AP and EMC Chapters. He was actively involved in organizing the 1999 Asia Pacific Microwave Conference, the 2003 Progress in Electromagnetics Research Symposium, the 2005 International Workshop on Antenna Theory, the 2006 EMC Zurich and the 2008 Asia-Pacific EMC. He was also an active member of a Consultancy team which provided EMC services to many private and government sectors. He was the recipient of the 1993 URSI XXIV General Assembly’s Young Scientist Award. He published over 140 peer-reviewed international journals and conference papers and participated as either Principal Investigator or Collaborator of over S$14.5 M research grants.

Hristo D. Hristov (SM’87) received the Ph.D. and D.Sc. degrees in wireless communications from the Technical University, Sofia, Bulgaria. Since 1965, he has been with the Technical University of Varna, Bulgaria, and currently, he is a Research Professor with the Universidad Técnica Federico Santa María, Valparaíso, Chile. His research interests include high-frequency electromagnetism, antennas, propagation, microwave/millimeter-wave devices and mobile/PCS wireless communications. He was a Researcher at the Strathclyde University of Technology (1971–1972) and Queen Mary College, London, UK (1975–1976), and at the Eindhoven University of Technology, the Netherlands (1993). He was also a short-term Visiting Professor at universities and research institutions in Russia, Japan, USA, Greece, Denmark and other countries. He is the coauthor of Microwave Cavity Antennas (Boston: Artech House, 1989) and the author of Fresnel Zones in Wireless Links, Zone Plate Lenses and Antennas (Boston-London: Artech House, 2000). Dr. Hristov was an invited lecturer at IEEE Tokyo AP-S Chapter and Sweden AP-S/MTT-S Chapter, and at the European Space Technology Center, the Netherlands. He has been a participant in several EU COST Actions on antennas for satellite and mobile communications. He was the Co-Organizer and served as a Chair of the Bulgarian IEEE Section and MTT/AP-S Chapter, and was awarded the IEEE Third Millennium Medal. He currently serves as an Associate Editor of the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION.

FAN et al.: COMPOUND DIFFRACTIVE LENS CONSISTING OF FZP AND FSS

Mook-Seng Leong (M’81–SM’92) received the B.Sc. degree in electrical engineering (first-class honors) and Ph.D. degree in microwave engineering from the University of London, London, U.K., in 1968 and 1972, respectively. Since 1989, he has been a Professor of electrical engineering with the National University of Singapore (NUS), Singapore. His main research interests include antenna analysis and design, solution of electromagnetic (EM) boundary-value problems associated with electromagnetic compatibility (EMC), semiconductor device characterization, and EM energy harvesting.

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Dr. Leong is a Fellow of the Institution of Engineering and Technology (IET), U.K. He was an Editorial Board member of IET Microwaves, Antennas and Propagation (2007–2009) and Microwave and Optical Technology Letters (2000–present). He was a Guest Editor for the December 2008 APMC’07 Special Issue of the IET Microwaves, Antennas and Propagation. He is an Associate Editor for the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION (2004–2007, 2007–2010). He was General Chairman for APMC1999, PIERS2003, and APMC2009. He is the Founding Chairman of the IEEE Microwave Theory and Techniques (MTT)/Antennas and Propagation (AP)/EMC joint Chapter, Singapore Section. He was the recipient of the MINDEF-NUS Joint Research and Development Award for his outstanding contributions to the MINDEF-NUS Research and Development program in 1996.

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Millimeter-Wave Half Mode Substrate Integrated Waveguide Frequency Scanning Antenna With Quadri-Polarization Yu Jian Cheng, Student Member, IEEE, Wei Hong, Senior Member, IEEE, and Ke Wu, Fellow, IEEE

Abstract—A millimeter-wave frequency scanning antenna, which is capable of dynamically changing the state of polarization thereby providing four modes of operation, is investigated and synthesized in the half mode substrate integrated waveguide (HMSIW) technology. This antenna is a planar passive circuit and fabricated by the low-cost PCB process. It is able to operate in either linear or circular polarization (LP or CP), depending on the requirements of its specific application. A wide angular region can be covered by 3 dB beam-widths of the continuous scanning LP beams and CP beams varying the frequency. It has good performance validated by measurements and has nearly a half reduction in size compared with the substrate integrated waveguide (SIW) version. The axial ratios of CP modes are excellent in the main beam directions within whole frequency band of interest and the isolations between each channel are good as well. Index Terms—Frequency scanning antenna, half mode substrate integrated waveguide (HMSIW), leaky-wave antenna, quadri-polarization.

I. INTRODUCTION

M

ULTIPLE-POLARIZATION antenna possesses an ability to change its polarization state dynamically depending on the requirements of its specific application. This characteristic offers great flexibility for antenna systems, because a single antenna could be used to satisfy various requirements of different systems. It provides a powerful technique, which can be called as the polarization diversity, for combating the multipath fading and increasing the channel capacity. Although many works have already been done [1]–[16], multiple-polarization antenna is still in an early stage of development. The requirements of performance, size and cost have been irreconcilable until now, especially for the millimeter-wave application [1]. Manuscript received March 20, 2009; revised June 30, 2009; accepted January 05, 2010. Date of publication March 29, 2010; date of current version June 03, 2010. This work was supported in part by the NSFC under Grant 60621002, in part by the National High-Tech Project under Grant 2007AA01Z2B4, and in part by the Scientific Research Foundation of Graduate School of Southeast University. Y. J. Cheng and W. Hong are with the State Key Laboratory of Millimeter Waves, School of information Science and Engineering, Southeast University, Nanjing 210096, China (e-mail: [email protected]). K. Wu is with the Poly-Grames Research Center, Department of Electrical Engineering, Ecole Polytechnique, University of Montreal, Montreal, QC H3V 1A2 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.2010.2046844

On the other hand, the frequency scanning technology is an efficient and economical way to realize the multibeam capability, which can generate several high-gain narrow beams to cover a solid angle without complex beamforming networks [18]–[25]. As is well known, the beam generated by the leakywave antenna can be scanned by varying the frequency. Generally speaking, the circular polarization (CP) beam seems difficult to be scanned compared with the linear polarization (LP) one. The challenge is to maintain a pure CP mode within a wide scan angular region. In our opinion, the CP frequency scanning antenna is a good solution to keep good axial ratio in the beam direction over a wide frequency band. If we combine two technologies together, it will effectively improve the signal-to-noise ratio (SNR), bit-error rate (BER), channel capacity, and power savings in a mobile link. Some useful multiple-polarization antennas have been introduced in [15]–[17], which help us to propose a novel antenna with four switchable polarizations using half mode substrate integrated waveguide (HMSIW) [26]–[28]. Specifically, it has 45 LP, LP, left-handed circular polarization (LHCP) and righthanded circular polarization (RHCP) according to the different input ports. The frequency is used as a variable to make LP and CP beams scanning. Moreover, this millimeter-wave antenna is an absolute passive printed circuit, which has lower loss, lower cost, better stability and reliability, and higher power handling capability than the active ones. II. DESIGN PROCEDURE A. Basic Principle of Operation The 3-D configuration of HMSIW multiple-polarization frequency scanning antenna is shown in Fig. 1. It is constructed by a 3 dB directional coupler and two leaky-wave antennas. Each or 45 radiating slots arleaky-wave antenna has sixteen ranged on the broad wall of the radiating HMSIWs. All of them were designed on a single Rogers5880 substrate with a relative permittivity of 2.2 and a thickness of 0.508 mm. Four input ports are divided to attach at both ends of the leaky HMSIWs. By changing the direction of propagation of the leaky wave, a pair of orthogonal LP modes and a pair of orthogonal CP modes can be generated in the opposite beam directions with a tilting angle other than broadside as shown in Fig. 2. Such a structure can not accommodate all states of polarization for each one of the scan /45 angle, but two orthogonal states of polarization, i.e., LP or L/RHCP, can be generated to cover the same angular region.

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Fig. 1. Configuration of the proposed HMSIW quadri-polarization frequency scanning antenna.

Fig. 2. Basic principle of the quadri-polarization frequency scanning antenna.

The basic principle of such a quadri-polarization frequency scanning antenna is explained as follows. When excited at port 1 or 4, the incident wave is divided into two equal portions with 90 phase difference. Therefore, an ideal RHCP or LHCP wave slots [29]. Due to the is obtained by a combination of . On the leaky wave excitation, the main beam points to other hand, when excited at port 2 or 3 by the wave propagating direction, a or 45 LP beam will be generated to the and radiates into the direction. With increasing the frequency, the beam direction is moving close to the broadside. B. HMSIW When an SIW is used only with the dominant mode, the maximal -field is at the vertical center plane along the propagation

direction, so the center plane can be considered as an equivalent magnetic wall. If an SIW is divided into two parts along this plane, each one can support a half of the field independently because of the large width-to-height ratio. To design a HMSIW device, the steps should be taken in the way similar to the design of an SIW circuit. Firstly, the width of an SIW operating at the same frequency range should be determined as described in [30]. Then, the width of HMSIW could be set to be approximately a half of the corresponding SIW structure. The substrate should be extended only a little in the horizontal direction. Comparing with the SIW structure, the original HMSIW has a smaller physical dimension, but it has higher element-to- element parasitic coupling when operated at the relative high frequency band due to the high current density along the open edges of transmission line. To overcome this drawback, we add an additional row of shunt vias (edge via) near the open edge of HMSIW as shown in Fig. 3. The distance between edge via and open edge of HMSIW, eo, should be greater than a certain value to remove the influence on the input VSWR, which can be determined by the full-wave simulation implemented by Ansoft HFSS. Fig. 4 presents the simulated input VSWR of HMSIW versus eo at different frequency, and shows that eo must be . Finally, eo is chosen greater than 0.25 mm for to be 0.5 mm to make all input VSWR less than 1.3 within the frequency band of interest. C. 90 3 dB HMSIW Directional Coupler The 3 dB directional coupler is employed to realize the 90 phase difference feeding mechanism herein. The coupling section of HMSIW-based coupler is a continuous aperture. The design process can be referenced in [26], [27]. The configuration of the proposed structure is shown in Fig. 5 and the simulated results are shown in Figs. 6–7. The reflection and coupling are alwithin the frequency band of . most below

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Fig. 3. Configurations of the SIW, original HMSIW and improved HMSIW. Fig. 6. Simulated S-parameters of the proposed HMSIW 3 dB coupler.

Fig. 4. Simulated input VSWR versus eo at different frequency. (The operation frequency increases from 32 GHz to 38 GHz following the arrow). Fig. 7. Simulated phase difference between output ports of the proposed HMSIW 3 dB directional coupler.

Fig. 5. Configuration of the proposed HMSIW 3 dB coupler (unit: mm).

It also has well-balanced outputs. The phase difference between within the interested frequency band. output ports is D. Leaky-Wave Antenna If the slot spacing of a slot array antenna is not just the half of guide wavelength and a matched termination is placed at the end, a leaky-wave array results [25]. Considering the beam direction of leaky-wave antenna changing in frequency, it is the frequency scanning antenna. The antenna array should be long enough to make the input power reduced to an acceptable level after radiating from those slots. Considering the leaky-wave antenna should be excited at both ends, we had better to design a symmetrical structure. In this design, we employ a uniformly spaced leaky-wave slot array with large bandwidth.

Fig. 8. Configuration of the proposed HMSIW leaky-wave antenna. (unit: mm).

When the frequency is increased, the beam is scanned toward the direction of propagation of the leaky wave. After the full-wave simulation and optimization, the configuration and dimension of the proposed leaky-wave antenna are shown in Fig. 8. The simulated -parameters of the proposed leaky-wave antenna are shown in Fig. 9. It shows that the residual power is down by the increase in the frequency. The non-resonant array achieves a wide band due to the phase differences between the reflections from various slots [25]. The curve of is plotted in this figure as well, which demonstrates such an antenna has good leakage efficiency. The simulated -plane radiation patterns according to different frequency are shown in

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Fig. 12. Simulated E-plane patterns of the proposed single antenna at 36 GHz. Fig. 9. Simulated S-parameters of the proposed HMSIW leaky-wave antenna.

Fig. 13. Simulated E-plane patterns of the proposed single antenna at 37 GHz. Fig. 10. Simulated E-plane patterns of the proposed single antenna at 34 GHz.

Fig. 14. Configuration of the two-element antenna array.

Fig. 11. Simulated E-plane patterns of the proposed single antenna at 35 GHz.

Figs. 10–13. The beam directions are simulated to be 26 , 21 , respectively. 17 , and 13 corresponding to E. Two-Elements Array Such a or 45 slot array antenna can generate or slot antennas are combined sym45 LP mode. If such metrically as shown in Fig. 14, a CP wave will be realized with the excitations of 90 relative phase difference. Therefore, we can use this two-element array to generate LHCP and RHCP modes. To achieve a high isolation and suppress the mutual coupling and 45 leaky-wave antennas, there between the adjacent

needs an offset distance between two leaky-wave antennas [15]. Through simulation, we found that the offset distance will not deteriorate the axial ratio of CP modes. Thus the offset distance will be selected with only one goal, i.e., to achieve the best isolation. As shown in Fig. 15, the offset distance equal to 2.5 mm is the best choice at all frequency of interest. Now, the whole antenna array consisting of two elements is simulated. The performances of those LP beams are similar to the beams generated by one leaky-wave antenna as shown in Figs. 10–13. The radiation patterns and axial ratios of RHCP plane are depicted at different frequency as shown beams in in Figs. 16–19. The detailed simulated results of such an array are listed in Table I. at III. EXPERIMENT The whole circuit was fabricated on a single-layer substrate through normal PCB process as shown in Fig. 20. As shown in Figs. 21–22, the reflections S11 and S33 are within . The isolation coefficients below

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Fig. 15. Isolation coefficients between two leaky-wave antennas versus the offset distance at different frequency.

Fig. 18. Simulated patterns and axial ratios of the two-element antenna array having RHCP mode in xoz plane at 36 GHz.

Fig. 16. Simulated patterns and axial ratios of the two-element antenna array having RHCP mode in xoz plane at 34 GHz.

Fig. 19. Simulated patterns and axial ratios of the two-element antenna array having RHCP mode in xoz plane at 37 GHz.

TABLE I SIMULATED RESULTS OF SUCH A TWO-ELEMENT ARRAY HAVING RHCP MODE

Fig. 17. Simulated patterns and axial ratios of the two-element antenna array having RHCP mode in xoz plane at 35 GHz.

are all better than 10 dB within . Compared with the simulated results of separate components, it is found that the return loss and isolation are deteriorated due to the mismatching

between the coupler and the leaky-wave antenna. An optimization for the distance between the first slot and the coupler can improve the characteristics. In other words, the offset between the two leaky-wave arrays also influence the reflection and isolation. Therefore, we can optimize the distance between the first slot and the coupler and the offset between the two leaky-wave arrays to achieve a better reflection and isolation. Such a HMSIW quadri-polarization frequency scanning antenna was characterized in an anechoic chamber using the setup

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Fig. 20. Photograph of the fabricated HMSIW quadri-polarization frequency scanning antenna.

Fig. 23. Sketch of the measurement setup used to test the HMSIW multiplepolarization frequency scanning antenna. A standard-gain LP horn antenna is used as the transmitter and measured antenna is used as the receiver.

Fig. 21. Measured S-parameters of the fabricated HMSIW quadri-polarization frequency scanning antenna excited at port 1.

Fig. 24. Measured CP radiation patterns of the fabricated quadri-polarization frequency scanning antenna in xoz plane excited at port 1 versus the frequency.

Fig. 22. Measured S-parameters of the fabricated HMSIW quadri-polarization frequency scanning antenna excited at port 3.

shown in Fig. 23. A standard gain LP horn antenna as the transmitter is rotated at a given angle by varying the polarization modes. The measured antenna as a receiver is rotated from to 180 , and the receiving power level is detected and recorded at the corresponding port. To estimate the characteristics of axial ratio, the antenna is measured several times at the different angle . Fig. 24 presents the RHCP radiation patterns plane received at port 1 when the transmitting antenna in is in the horizontal polarization. The measured results excited at port 1 are listed in Table II. It is able to cover the angular re, 0.6 ) with 3 dB beam-widths. When excited gion of ( plane with 45 LP are at port 3, the radiation patterns in also tested as shown in Fig. 25. Here, the transmitting horn has the same polarization sense with the tested antenna, i.e., 45 LP. The measured results excited at port 3 are also listed in

Fig. 25. Measured LP radiation patterns of the fabricated quadri-polarization frequency scanning antenna in xoz plane excited at port 3 versus the frequency.

Table II. It can cover the angular region of (1.3 , 34.8 ) with 3 dB beam-widths. Fig. 26 indicates the 35 GHz LP beam has a cross-polar level in the beam direction. Compared with the simulated ones, these measured patterns present higher sidelobe levels, which are related to the microstrip feed lines. The residual powers are radiated by the feed lines and spoil the antenna performance. We can use the

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TABLE II MEASURED RESULTS OF SUCH FABRICATED QUADRI-POLARIZATION FREQUENCY SCANNING ANTENNA

Fig. 26. Measured 35 GHz co-polar and cross-polar radiation patterns in xoz plane excited at port 3.

metallic shield or feed the HMSIW by waveguide to avoid the direct radiation from the feed discontinuity. IV. CONCLUSION In this paper, we propose and fabricate a millimeter-wave HMSIW frequency scanning antenna, which supports a pair of orthogonal CP senses and a pair of orthogonal LP senses. Four operation modes are well-isolated between each other. From 33 GHz to 39 GHz, it generates continuous scanning LP to cover a 33.5 angular range and CP beams to cover another 33.4 angular range with 3 dB beam-widths, while the CP beams keep good axial ratios in the beam directions. The addition of such antenna at the very front end of a link offers significantly potential to reduce the multipath fading and increase the channel capacity. REFERENCES [1] S. Gao, A. Sambell, and S. S. Zhong, “Polarization-agile antennas,” IEEE Antennas Propag. Mag., vol. 48, no. 6, pp. 28–37, Jun. 2006. [2] D. H. Schaubert, F. Farrar, A. Sindoris, and S. Hayes, “Microstrip antenna with frequency agility and polarization diversity,” IEEE Trans. Antennas Propag., vol. 29, no. 1, pp. 118–123, Jan. 1981. [3] P. Haskins and J. S. Dahele, “Varactor-diode loaded passive polarisation-agile patch antenna,” Electron. Lett., vol. 30, no. 13, pp. 1074–1075, Jun. 1994. [4] P. Haskins, P. S. Hall, and J. S. Dahele, “Polarization-agile active patch antenna,” Electron. Lett., vol. 30, no. 2, pp. 98–99, Jan. 1994. [5] M. Boti, L. Dussopt, and J. M. Laheurte, “Circularly polarized antenna with switchable polarization sense,” Electron. Lett., vol. 36, no. 18, pp. 1518–1519, Aug. 2000.

[6] F. Yang and Y. Rahmat-Samii, “A reconfigurable patch antenna using switchable slots for circular polarization diversity,” IEEE Microw. Wireless Compon. Lett., vol. 12, no. 3, pp. 96–98, Mar. 2002. [7] H. Scott and F. V. Fusco, “Polarization-agile circular wire loop antenna,” IEEE Antennas Wireless Propag. Lett., vol. 1, pp. 64–66, Dec. 2002. [8] N. J. McEwan, R. A. Abd-Alhameed, E. M. Ibrahim, P. S. Excell, and J. G. Gardiner, “A new design of horizontally polarized and dual-polarized uniplanar conical beam antennas for HIPERLAN,” IEEE Trans. Antennas Propag., vol. 51, no. 2, pp. 229–237, Feb. 2003. [9] M. Fries, M. Grani, and R. Vahldieck, “A reconfigurable slot antenna with switchable polarization,” IEEE Microw. Wireless Compon. Lett., vol. 13, no. 11, pp. 490–493, Nov. 2003. [10] Y. Sung, T. Jang, and Y. Kim, “A reconfigurable microstrip antenna for switchable polarization,” IEEE Microw. Wireless Compon. Lett., vol. 14, no. 11, pp. 534–536, Nov. 2004. [11] E. A. Soliman, M. S. Ibrahim, and A. K. Abdelmageed, “Dual-polarized omnidirectional planar slot antenna for WLAN applications,” IEEE Trans. Antennas Propag., vol. 53, no. 9, pp. 3093–3097, Sep. 2005. [12] Y. F. Wu, C. H. Wu, D. Y. Lai, and F. C. Chen, “Reconfigurable quadripolarization diversity aperture-coupled patch antenna,” IEEE Trans. Antennas Propag., vol. 55, no. 3, pp. 1009–1012, Mar. 2007. [13] S. L. S. Yang, A. A. Kishk, K. F. Lee, K. M. Luk, and H. W. Lai, “The design of microstrip patch antenna with four polarizations,” in Proc. IEEE Radio and Wireless Symp., Jan. 2008, pp. 467–470. [14] R. H. Chen and J. S. Row, “Single-fed microstrip patch antenna with switchable polarization,” IEEE Trans. Antennas Propag., vol. 56, no. 4, pp. 922–926, Apr. 2008. [15] S. Park, Y. Okajima, J. Hirokawa, and M. Ando, “A slotted post-wall waveguide array with interdigital structure for 45 linear and dual polarization,” IEEE Trans. Antennas Propag., vol. 53, no. 9, pp. 2865–2871, Sep. 2005. [16] K. Sakakibara, Y. Kimura, J. Hirokawa, and M. Ando, “A two-beam slotted leaky waveguide array for mobile reception of dual- polarization DBS,” IEEE Trans. Veh. Tech., vol. 48, no. 1, pp. 1–7, Jan. 1999. [17] Y. J. Cheng, W. Hong, K. Wu, and X. W. Zhu, “Millimeter-wave reconfigurable antenna with polarization and angle diversity,” in IEEE Radio Wireless Symp. Dig., San Diego, CA, Jan. 2009, pp. 300–303. [18] M. W. Shelley, A. G. Mason, and D. J. Brain, “A circularly polarised frequency scanning antenna for space applications,” in Proc. Inst. Elect. Eng./SEE Seminar on Spacecraft Antennas, May 9, 1994, pp. 1/1–1/6. [19] M. W. Shelley, A. G. Mason, and K. Markus, “Frequency scanning antennas for aeronautical communications,” in Proc. IEE Colloquium on Satellite Antenna Technology in the 21st Century, Jun. 1991, pp. 12/1–12/4. [20] M. Danielsen and R. Jorgensen, “Frequency scanning microstrip antennas,” IEEE Trans. Antennas Propag., vol. 27, no. 3, pp. 146–150, Mar. 1979. [21] A. Ishimaru and H. S. Tuan, “Frequency scanning antennas,” in IRE Int. Convention Record, Mar. 1961, vol. 9, pp. 101–109. [22] Y. D. Lin and T. Itoh, “Frequency-scanning antenna using the crosstieoverlay slow-wave structures as transmission lines,” IEEE Trans. Antennas Propag., vol. 39, no. 3, pp. 377–380, Mar. 1991. [23] D. Sievenpiper, J. Schaffner, J. J. Lee, and S. Livingston, “A steerable leaky-wave antenna using a tunable impedance ground plane,” IEEE Antennas Wireless Propag. Lett., vol. 1, pp. 377–380, Dec. 2002. [24] J. L. Volakis, Antenna Engineering Handbook, 4th ed. New York: McGraw-Hill, 2007. [25] R. S. Elliott, Antenna Theory and Design, revised ed. Piscataway, NJ, NJ: IEEE Press, 2003.

CHENG et al.: MILLIMETER-WAVE HALF MODE SUBSTRATE INTEGRATED WAVEGUIDE FREQUENCY

[26] B. Liu, W. Hong, Y. Q. Wang, Q. H. Lai, and K. Wu, “Half mode substrate integrated waveguide (HMSIW) 3 dB coupler,” IEEE Microw. Wireless Compon. Lett., vol. 17, no. 1, pp. 22–24, Jan. 2007. [27] Y. J. Cheng, W. Hong, and K. Wu, “Half mode substrate integrated waveguide (HMSIW) directional filter,” IEEE Microw. Wireless Compon. Lett., vol. 17, no. 7, pp. 504–506, Jul. 2007. [28] Y. Q. Wang, W. Hong, Y. D. Dong, B. Liu, H. J. Tang, J. X. Chen, X. X. Yin, and K. Wu, “Half mode substrate integrated waveguide (HMSIW) bandpass filter,” IEEE Microw. Wireless Compon. Lett., vol. 17, no. 4, pp. 265–267, Apr. 2007. [29] Z. J. Chen, W. Hong, Z. Q. Kuai, J. X. Chen, and K. Wu, “Circularly polarized slot array antenna based on substrate integrated waveguide,” in Int. Conf. on Microwave and Millimeter Wave Technology ICMMT, Nanjing, China, Apr. 2008, vol. 3, pp. 1066–1069. [30] Y. Cassivi, L. Perregrini, P. Arcioni, M. Bressan, K. Wu, and G. Conciauro, “Dispersion characteristics of substrate integrated rectangular waveguide,” IEEE Microw. Wireless Compon. Lett., vol. 12, no. 9, pp. 333–335, Sep. 2002. Yu Jian Cheng (S’08) was born in Sichuan Province, China, on April, 1983. He received the B.S. degree in electric engineering from University of electronic Science and Technology of China, in 2005. He is currently working toward the Ph.D. degree without going through the conventional Master’s degree at Southeast University, Nanjing, China. His current research interests include microwave and millimeter-wave passive circuits, antennas. Mr. Cheng served as a reviewer for the IEEE Microwave and Wireless Components Letters and IEEE/ ASME Journal of Microelectromechanical Systems.

Wei Hong (M’92–SM’07) received the B.S. degree from the University of Information Engineering, Zhengzhou, China, in 1982, and the M.S. and PhD degrees from Southeast University, Nanjing, China, in 1985 and 1988, respectively, all in radio engineering. Since 1988, he has been with the State Key Laboratory of Millimeter Waves, Southeast University, and is currently a professor of the School of Information Science and Engineering. In 1993, 1995, 1996, 1997 and 1998, he was a short-term Visiting Scholar with the University of California at Berkeley and at Santa Cruz, respectively. He has been engaged in numerical methods for electromagnetic problems, millimeter wave theory and technology, antennas, electromagnetic scattering and RF technology for mobile communications etc. He has authored and coauthored over 200 technical publications, and authored two books, Principle and Application of the Method of Lines (in Chinese, Southeast University Press, 1993) and Domain Decomposition Methods for Electromagnetic Problems (in Chinese, Science Press, 2005). He twice awarded the first-class Science and Technology Progress Prizes issued by the Ministry of Education of China in 1992 and 1994 respectively, awarded the fourth-class National Natural Science Prize in 1991 etc. Besides, he also received the Foundations for China Distinguished Young Investigators and for “Innovation Group” issued by NSF of China. Dr. Hong is a senior member of CIE, Vice-President of Microwave Society and Antenna Society of CIE, and served as the reviewer for many technical journals including the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, IET Proc.-H, Electronics Letters, etc., and now serves as an Associate Editor of the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES.

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Ke Wu (M’87–SM’92–F’01) received the B.Sc. degree (with distinction) in radio engineering from Nanjing Institute of Technology (now Southeast University), China, in 1982 and the D.E.A. and Ph.D. degrees in optics, optoelectronics, and microwave engineering (with distinction) from Institut National Polytechnique de Grenoble (INPG) and University of Grenoble, France, in 1984 and 1987, respectively. He is a Professor of electrical engineering, and Tier-I Canada Research Chair in RF and millimeter-wave engineering at Ecole Polytechnique (University of Montreal). He also holds a number of visiting (guest) and honorary professorships at various universities including the first Cheung Kong Endowed Chair Professorship at Southeast University, the first Sir Yue-Kong Pao Chair Professorship at Ningbo University, and Honorary Professorships at Nanjing University of Science and Technology and City University of Hong Kong. He has been the Director of the Poly-Grames Research Center and the Founding Director of “Centre de recherche en électronique radiofréquence” (CREER) of Quebec. He has (co)authored over 720 refereed papers, a number of books/book chapters and patents. His current research interests involve substrate integrated circuits (SICs), antenna arrays, advanced CAD and modeling techniques, and development of low-cost RF and millimeter-wave transceivers. He is also interested in the modeling and design of microwave photonic circuits and systems. He serves on the Editorial Board of the Microwave Journal, Microwave and Optical Technology Letters,, Wiley’s Encyclopedia of RF and Microwave Engineering, and . He is an Associate Editor of the International Journal of RF and Microwave Computer-Aided Engineering (RFMiCAE). Dr. Wu is a member of Electromagnetics Academy, the Sigma Xi Honorary Society, and the URSI. He has held many positions in and has served on various international committees, including Co-Chair of the Technical Program Committee (TPC) for 1997 and 2008 Asia-Pacific Microwave Conferences (APMC), General Co-Chair of 1999 and 2000 SPIE’s Int. Symposia on Terahertz and Gigahertz Electronics and Photonics, General Chair of 8th Int. Microwave and Optical Technology (ISMOT’2001), TPC Chair of 2003 IEEE Radio and Wireless Conference (RAWCON’2003), General Co-Chair of RAWCON’2004, Co-Chair of 2005 APMC Inter. Steering Committee, General Chair of 2007 URSI Int. Symp. on Signals, Systems and Electronics (ISSSE), and General Co-Chair of 2008 and 2009 Global Symposia on Millimeter-Waves, and Int. Steering Committee Chair of the 2008 Int. Conference on Microwave and Millimeter-Wave Technology. In particular, he will be General Chair of 2012 IEEE MTT-S International Microwave Symposium (IMS). He has served on Editorial or Review Boards of various technical journals, including the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, IEEE PROCEEDINGS, and the IEEE Microwave and Wireless Components Letters. He served on the Steering Committee for the 1997 joint IEEE AP-S/URSI Int. Symp. and the TPC for the IEEE MTT-S Int. Microwave Symp. He is currently Chair of the joint IEEE chapters of MTTS/APS/LEOS in Montreal. He is an elected MTT-S AdCom member for 2006-2012 and serves as Chair of the IEEE MTT-S Member and Geographic Activities (MGA) Committee. He was the recipient of a URSI Young Scientist Award, Inst. Elect. Eng. Oliver Lodge Premium Award, Asia-Pacific Microwave Prize, IEEE CCECE Best Paper Award, University Research Award “Prix Poly 1873 pour l’Excellence en Recherche” presented by the Ecole Polytechnique on the occasion of its 125th anniversary, Urgel-Archambault Prize (the highest honor) in the field of physical sciences, mathematics and engineering from ACFAS, 2004 Fessenden Medal of IEEE Canada, and 2009 Thomas W. Eadie Medal of the Royal Society of Canada (The Canadian Academy of the Sciences and Humanities). In 2002, he became the first recipient of the IEEE MTT-S Outstanding Young Engineer Award. He is Fellow of the Canadian Academy of Engineering and Fellow of the Royal Society of Canada. He is an MTT-S Distinguished Microwave Lecturer from January 2009 to December 2011.

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60 GHz Aperture-Coupled Dielectric Resonator Antennas Fed by a Half-Mode Substrate Integrated Waveguide Qinghua Lai, Graduate Student Member, IEEE, Christophe Fumeaux, Senior Member, IEEE, Wei Hong, Senior Member, IEEE, and Rüdiger Vahldieck, Fellow, IEEE

Abstract—Dielectric resonator antennas (DRAs) fed by a halfmode substrate integrated waveguide (HMSIW) are proposed and studied in this paper. The investigated antenna configuration consists of a dielectric resonator (DR) mounted on the conducting back plane of an HMSIW. Energy is coupled from the interior HMSIW to the DR through an aperture between them. Using this excitation scheme, a 60 GHz linearly polarized HMSIW-fed DRA is first designed by applying a transverse rectangular slot to feed a dielectric cylinder. This design experimentally exhibits a bandwidth of 10 dB. In particular, a gain higher than 5.5 24.2% for S11 dB and a radiation efficiency between 80% and 92% are obtained over the whole operation band, indicating that the HMSIW can be an efficient feed for DRAs operating around 60 GHz. In addition to this linearly polarized example, a DRA of circular polarization, coupled through a pair of cross slots, is also designed, presenting a measured 3-dB axial ratio bandwidth of 4.0%. Index Terms—Dielectric resonator antennas (DRAs), half-mode substrate integrated waveguide (HMSIW).

I. INTRODUCTION

S

INCE the first application of a cylindrical dielectric resonator as an antenna [1], a number of excitation schemes have been proposed for the dielectric resonator antennas (DRAs) [2]–[6]. One of the most common feeding mechanisms consists in coupling energy into the resonator through an aperture in its ground plane. This technique can be applied in conjunction with various transmission lines, including the conventional rectangular waveguide [7] and various planar waveguides such as the microstrip line [8], [9], coplanar waveguide [10], and substrate integrated waveguide [11]. One distinguished advantage of the aperture-fed DRAs is the easy realization of the desired polarization, which can directly be implemented either by selecting suitable slot shapes [8], [9], [12]–[14] or by adjusting the relative position between Manuscript received May 25, 2009; revised November 04, 2009; accepted December 16, 2009. Date of publication March 29, 2010; date of current version June 03, 2010. This work was supported in part by the Laboratory for Electromagnetic Fields and Microwave Electronics at ETH Zurich and in part by the NSFC under grant 60621002 from China. Q. Lai and W. Hong are with the State Key Lab. of Millimeter Waves, School of Information Science and Engineering, Southeast University, Nanjing 210096, China (e-mail: [email protected]). C. Fumeaux is with the School of Electrical and Electronic Engineering, The University of Adelaide, Adelaide 5005, South Australia (e-mail: [email protected]). R. Vahldieck is with the Laboratory for Electromagnetic Fields and Microwave Electronics, ETH Zürich, Zürich CH-8092, Switzerland (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2010.2046852

the slot and resonator [15]. In some cases, a wide operation bandwidth can also be conveniently achieved by merging offset resonances of the slot and dielectric resonator [16], i.e., these two resonances are placed on each side of the center frequency of interest and their combined frequency ranges determine the bandwidth of operation. Furthermore, aperture-coupled DRAs based on planar waveguides appear as a competitive antenna choice for antenna-in-package designs at mm-wave frequencies, considering their compatibility with integrated circuits and their high radiation efficiency [17]. However, some drawbacks associated with the feeding transmission line need to be taken into account when considering a DRA for a given application. For example, for the DRAs fed by a microstrip line, the overall radiation efficiency might drastically degrade with rising frequency, due to the increasing conductor and radiation losses in the feeding microstrip line. Another illustrative example is the DRAs fed by a substrate integrated waveguide (SIW), for which the large transverse size of the feeding SIW might become a barrier for some application. Despite the restrictions imposed by the SIW transverse size, it is worth noting that the SIW can be an attractive alternative to the microstrip line as a feed for DRAs operating in the millimeter wave band, when considering the high quality factor of the SIW. In an effort to create a compact and efficient feeding scheme for DRAs operating at millimeter wave frequencies, a novel excitation structure based on a half-mode substrate integrated waveguide (HMSIW) is proposed and studied throughout this work. The concept of HMSIW was first proposed in [18] and since then, a number of high-performance devices have been developed with this technique, including filters [19], [20] antennas [21], and power dividers [22]. Recently, a theoretical and experimental study has been carried out to characterize the propagation properties of the HMSIW [23]. The investigation results have demonstrated that for an HMSIW and a standard 50microstrip line fabricated on the same substrate, the HMSIW can present an attenuation constant much smaller than the microstrip line at frequencies above 40 GHz. This motivates the introduction of the HMSIW as feed for the DRAs. Sketches of the investigated antenna configuration are shown in Fig. 1. A cylindrical dielectric resonator is mounted on the conducting back plane of an HMSIW and the energy is coupled from the interior waveguide to the dielectric resonator through an aperture that is etched into the HMSIW back plane, beneath the dielectric cylinder. Fig. 1(b) shows that the HMSIW is fed by a

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end. As depicted in Fig. 1(d), a transverse rectangular slot is applied to feed the dielectric cylinder for linearly polarized radiation, while a pair of cross slots is adopted for realizing circular polarization as illustrated in Fig. 1(e). Design procedures are first described for both types of antennas in Section II. Subsequently, the measured performance of two fabricated prototypes, of linear and circular polarization respectively, is demonstrated for 60 GHz operation in Section III. The good agreement between the measured and simulated results validates the concept of the HMSIW-fed DRA as a high efficiency feed-antenna combination for millimeter-wave band applications. II. DESIGN PROCEDURES In this section, design approaches are sequentially described firstly for the linearly polarized and secondly for circularly polarized HMSIW-fed aperture-coupled DRAs. A. HMSIW-Fed DRAs of Linear Polarization

Fig. 1. Sketches of the HMSIW-fed DRAs. (a) 3D view. (b) View of the HMSIW feeding side. The substrate is shown as white and the dark gray areas are metalized. The metal posts (vias) are shown as circles. The width and length are w and l for the HMSIW, w and l for the microstrip line, and w and l for the taper impedance transition. The vias diameter is d and the spacing between them is s. (c) Side view. The dielectric cylinder has a diameter D and a height h. The glue layer thickness is h and the substrate thickness is t. (d) Top view of a linearly polarized HMSIW-fed DRA. The cylindrical dielectric resonator is shown as circular area in light gray and the slot is plotted as a rectangle in white. The slot has a width w and a length l . The slot center has a distance of z from the HMSIW shorted end and a distance of x from the HMSIW open side. (e) Top view of a circularly polarized HMSIW-fed DRA.

microstrip line through a tapered impedance transition at one end, and is shorted by a column of metallic vias at the other

HMSIW Dimensions: The first step in the design procedure is to determine the dimensions of the feeding HMSIW. For a practical antenna design, the HMSIW cutoff frequency is first appropriately selected by considering that the whole operation frequency band of the DRA should be on one hand far away from the HMSIW cutoff frequency, and on the other hand, still within the single mode operation region of the waveguide. A full design methodology for the HMSIW has been presented in [23] and is only briefly outlined here. Based on the cutoff frequency , an HMSIW effective width can be calculated using [23, Eq. (11)]. Subsequently, the substrate thickness and permittivity as well as the metallic vias diameters and spacing are properly selected to guarantee a good confinement of fields inside the HMSIW [24]. Lastly, substituting the known param, , , , and into [23, Eq. (8)–(10), (13)] yields the eters HMSIW physical width , which is the only unknown in that set of design formulas. Dielectric Resonator Dimensions: Explicit formulas are available in the literature for determining the resonant frequencies of dielectric resonators according to their shapes, dimensions, and operation modes [25]. In general, a non-unique set of dimensions can make dielectric resonators of a given shape operate at a specified resonant frequency on the same mode. Therefore, care should be taken to select an appropriate set of dimensions by considering the antenna specifications and fabrication aspects. For instance, the dielectric cylinder used mode and in our work is designed to operate on the its dimensions, i.e., its height and diameter , are calculated from [25, Eq. (5)]. A large diameter-to-height ratio is chosen for the cylindrical resonator out of consideration to obtain a broad bandwidth [26], while lowering the antenna profile and providing large space to flexibly tune the coupling slot length. Slot Location and Dimensions: As only a small amount of energy leaks from the waveguide into the uncovered substrate area (the region in white in Fig. 1(b)), the whole slot is cut inwards the HMSIW as shown in Fig. 1(d). Furthermore, in order to maximize energy coupling from the HMSIW interior to the dielectric resonator, the slot is positioned at a distance of from the HMSIW shorting end.

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The guided wavelength can be calculated according to the , where the phase constant can be deterrelation mined by [23, Eq. (12)]. in order to avoid The slot width is selected as is the resonant wavelength cross-polarized radiation, where is optimized with the aid of the in free space. The slot length electromagnetic simulation tool Ansoft HFSS. This optimization process on one hand helps achieve the impedance matching between the HMSIW and dielectric resonator, and on the other hand, enables combining the offset resonances of the slot and dielectric resonator [16] to produce a broadband antenna. Impedance Matching Network: For the sake of measurements, a microstrip line is used to feed the HMSIW. The impedance matching between the feeding microstrip line and the HMSIW can be realized through an intermediate microstrip taper, as shown in Fig. 1 with optimized dimensions and . B. HMSIW-Fed DRAs of Circular Polarization

Fig. 2. An X-slot etched on the conducting back plane of an end-shorted HMSIW. It is noted that the origin of current coordinate system is located at the corner of the HMSIW, compared to that in Fig. 1 placed at the cylinder center.

the relation of , the following equation can be derived to determine possible locations of the X-slot center (2)

For the case of circularly polarized HMSIW-fed DRAs, the design of the HMSIW, dielectric resonator, and matching network can be carried out using the same methods as for the linearly polarized HMSIW-fed DRAs. Therefore, the following design procedure concentrates firstly on the positioning of the coupling slots and secondly on the determination of their dimensions. Positioning of the X-Slot: It is known that a pair of crossed slots cut into the broad wall of a rectangular waveguide can radiate circularly polarized wave, if the X-slot is located at the points where the transverse and longitudinal magnetic mode are equal in field components of the dominant mode amplitude [27]. Considering that the dominant in an HMSIW of width is approximately half the dominant mode in a rectangular waveguide of width [22], [23], the -slot used in the circularly polarized HMSIW-fed DRAs should also be centered at the spots where two orthogonal mode have magnetic field components of the dominant an identical amplitude. Fig. 2 depicts an X-slot etched on the conducting back wall of an end-shorted HMSIW which extends to infinity along the + direction. Field equations have been derived as [23, Eq. (4)] for the dominant traveling wave in a thru-HMSIW. By reformulating those equations and adapting them to the coordinates of Fig. 2, two expressions can be obtained for the two magnetic mode in the shorted field components of the standing HMSIW as follows: (1a) (1b) in which is the angular frequency, is the permeability of substrate, is the HMSIW width, and is a real amplitude constant. Wave numbers and are in the form of [23, (3)]. A comparison of (1a) and (1b) indicates that the two magand have an inherent 90 phase difnetic components ference. Therefore, circular polarization will be radiated at the spots where the two components are equal in amplitude. With

It is noted that for a given certain value of , two values of can be obtained by adopting “ ” or “ ” in (2), respectively. But is physically valid as the X-slot is only the value of centered within the HMSIW. For illustration purpose, Fig. 3 plots a curve computed with (2), corresponding to the example of circularly polarized HMSIW-fed DRA described in the present work. The antenna is designed for ideal circular polarization at broadside at 60 , a GHz. The feeding HMSIW has a width height , and substrate permittivity . The maximum electric field in the HMSIW across section is according to [23, estimated to occur at the spot of Fig. 4]. That is to say the value of the parameter in [23, Eq. (3)] is assessed at . A mapping of Fig. 3 onto Fig. 2 shows , , and that boundaries of Fig. 3 at correspond to the shorted end, open side, and shorted side of the HMSIW, respectively. The curve is divided into segments, where the solid lines are related to the use of ‘ ’ in (2), indi, while the dash lines result from the adoption cating . Depending of “ ” in (2), representing a relation of or , either on the excitation type, i.e., circular polarization sense (i.e., clockwise or counterclockwise) can be excited by centering the X-slot on either branch of the line. It is noted that the possible positions for the two circular polarization senses alternate with a period of half a guided wavelength along -axis, coinciding with the cyclic behavior of (2). To validate (2), a model of slotted HMSIW in the configuration of Fig. 2 is analyzed using HFSS. The HMSIW and the pair of slots are identical with their counterparts in the circularly polarized HMSIW-fed DRA, whose dimensions are given in Table I. Varying the X-slot position on the conducting back wall of the HMSIW, a minimum value of axial ratio can be observed at 60 GHz at some spots where the two magnetic components . The coare considered in the equal amplitude, i.e., ordinates of the optimal locations are recorded and then marked in star symbols in Fig. 3. A fairly good agreement is gained between the calculation and simulation results, validating the

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TABLE I CONFIGURATIONS OF TWO TYPES OF HMSIW-FED DRAs

“LP” and “CP” denote “linear polarization” and “circular polarization,” respectively.

III. PERFORMANCE Fig. 3. Position of the X-slot center on the conducting back wall of the HMSIW for radiating circularly polarized wave.

proposed design (2). It is noted that due to the HMSIW small thickness, the etching of the X-slot distorts the field distribution inside the waveguide. That effect is not considered in the derivation of (2), which mainly explains the discrepancy between the calculations and simulations. Finally, it is pointed out that although an infinite number of points can be theoretically calculated from (2), only those close can be used in practice, to the middle of the HMSIW as the slot has to be entirely etched within the HMSIW. Slot Dimensions: As in the case of linearly polarized antenna, the pair of slots has a width much smaller compared to the wave. In order to excite two near-delength in free space generate orthogonal modes in the dielectric cylinder, two slots are orthogonally crossed and have identical or slightly different lengths [12], [14]. Theoretically, for a pair of identical cross slots, the center positioning obtained from (2) can guarantee a circularly polarized radiation regardless of the slot length [27], which significantly simplifies the design. However, in order to obtain sufficient coupling from the HMSIW interior to the dielectric res, onator, an optimal slot length is empirically found around is the guided wavelength calculated from the relation where , as in the previous case of linearly polarized antennas. In the real design, due to the distortion effects of the X-slot on the field distribution beneath the aperture inside the HMSIW, the individual slot length might have to be optimized slightly for achieving a desired axial ratio level as well as for ) and the merging the impedance band (where ). Therefore, the resulting axial ratio band (where optimized slots might have slightly different lengths. It is worth emphasizing that, since the polarization sense is only dependent on the slot location, i.e., either on the solid branch or dash branch, exchanging the slot lengths does not change the circular polarization sense for the present HMSIW-fed circularly polarized DRA. Up to this point, design approaches have been given for the proposed HMSIW-fed aperture-coupled DRAs of linear and circular polarizations.

To validate the concept of the HMSIW-fed aperture-coupled DRAs, both a linearly polarized antenna and a circularly polarized antenna have been designed following the procedures above and optimized with the aid of the simulation tool HFSS. Both antennas are designed to operate at around 60 GHz and their dimensions are listed in Table I. In both cases, the sub,a strate made of Rogers 5880 has a thickness , and a loss tangent of 0.001. The permittivity dielectric cylinders used for both antennas are made of Rogers and a loss tanTMM10i with a permittivity gent of 0.002. The dielectric cylinders are glued over the slot on the HMSIW conducting back plane using Araldite resin with a and an estimated loss tangent of 0.06 at permittivity 2.45 GHz [28], [29]. The glue layer labeled as “ ” in Fig. 1(c) is estimated thick. Simulated performance of the two designs has been compared to the experimental results of two corresponding fabricated prototypes. Before conducting a quantitative discussion on the comparison, it is worth pointing out that all electrical characteristics of the two Rogers dielectric materials above, i.e., Rogers 5880 and Rogers TMM10i, are specified at 10 GHz by their manufacturer [30]. However, considering the reported stability of those materials properties [31], [32], it has been empirically assumed in our design that the variation of the material properties with frequency is still within tolerance even up to 60 GHz. The results in the following section validate this assumption through a good agreement between simulations and measurements. In order to investigate the influence of glue layer on the antenna performance, the linearly polarized HMSIW-fed DRA has been numerically analyzed with glue layers of different thick, 15, and 20 , respectively. From the renesses sulting reflection coefficient curves plotted in Fig. 4, it can be seen that even a slight increase in the glue layer thickness can significantly influence the impedance matching over the whole band. Therefore, it is crucial to take the glue layer effects into account while designing DRAs operating in the millimeter-wave band. The simulated input impedance obtained from the use of thick glue layer is also depicted in the inset of Fig. 4. a 15 A lower resonance due to the slot occurs at around 46 GHz, whereas the higher offset resonance caused by the dielectric resonator cylinder is observed at around 60 GHz.

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Fig. 4. Simulated and measured reflection coefficients of the linearly polarized HMSIW-fed DRA. The simulation results are obtained with the use of different , 15, and 20  . The inset depicts the glue layer thicknesses, i.e., h simulated input impedance obtained in the case of 15  thick glue layer.

= 10

m m

The measured reflection coefficient is also shown in Fig. 4, . By conpresenting a bandwidth of 24.0% for trast, a bandwidth of 25.5% is obtained in the simulation using thick glue layer. These results confirm that offset resthe 15 onances of the slot and the dielectric resonator can be combined to achieve broadband operation. The radiation patterns obtained at three frequency points, i.e., 50 GHz, 55 GHz, and 63 GHz, are illustrated in Fig. 5. All patterns are broadside and correspond to the pattern of a horizontal magnetic dipole on the ground plane. This is expected for both DR mode and for the slot resonance. In both simthe ulations and measurements, ripples are observed in the E-plane pattern over the whole operation band, which can mainly be explained by diffraction at the edges of the finite ground plane. The level of ripples measured at the antenna broadside is about 2.2 dB at 50 GHz, 0.9 dB at 55 GHz, and 2.4 dB at 63 GHz. As for radiation patterns in the H-plane, the measured pattern has a 3 dB beamwidth of 76 at 50 GHz, of 60 at 55 GHz, and of 74 at 63 GHz. All angles above are estimated in an error of 2 . Due to the finite dimension of the ground plane, a relatively large back radiation occurs in both E and H plane patterns. Radiation leakage from the HMSIW open side causes the asymmetry in the H plane pattern. Similarly as in the analysis of reflection coefficient, the near broadside maximum gain of the linearly polarized antenna is also investigated with the aid of the simulation tool HFSS for , three different thicknesses of the glue layer, i.e., 15 , and 20 . The simulation results are plotted as a funcin tion of frequency in Fig. 6, showing that a change of 5 the glue layer thickness with an estimated loss tangent of 0.06 can lead to a shift of 0.10 0.07 dB in the maximum gain over the whole operation band. In addition, the measured gain is also

Fig. 5. Co- and cross-polarized radiation patterns of the linearly polarized HMSIW-fed DRA at (a) 50 GHz, (b) 55 GHz, and (c) 63 GHz. The measurement results are presented in solid line, while the simulation results in dash line.

illustrated in Fig. 6, exhibiting a level of above 5.5 dB in the frequency range of 49.5 to 62.5 GHz. Variations observed in the measurement data can mainly be explained by large ripples in the radiation patterns (as shown in Fig. 5) and uncertainties in the measurements. It is noted that the measured gain shown in Fig. 6 is calibrated by removing the losses in the feeding structure composed of a 90 degree bend and an MMPX connector, which are experimentally estimated at about 0.5 0.1 dB from 50 GHz to 56 GHz, about 0.6 0.1 dB from 56 GHz to 58 GHz, and about 0.9 0.1 dB above 60 GHz. The discrepancy between the measured and simulated results is mainly attributed to some experimental losses unaccounted for in the simulations. The radiation efficiency of the linearly polarized HMSIW-fed DRA has been measured using the directivity/gain method (D/G method) [17], [33], [34]. For this measurement, the gain is directly measured with a reference standard gain horn in an anechoic chamber. To determine the directivity with reasonable measurement accuracy [17], both the co- and cross-polarized 3D radiation patterns of the DRA are first sampled with an azimuth

LAI et al.: 60 GHz APERTURE-COUPLED DIELECTRIC RESONATOR ANTENNAS FED BY A HMSIW

Fig. 6. Gain vs. frequency for the linearly polarized HMSIW-fed DRA. The simulated gain is obtained with the use of different glue layer thicknesses, i.e., h , 15, and 20  . The measured data are calibrated with the experimentally estimated losses of a 90 degree bend and an MMPX connector, which are used for feeding the antenna.

= 10

m

Fig. 7. Radiation efficiency of the linearly polarized HMSIW-fed DRA. The simulated results are obtained with the use of different glue layer thicknesses, , 15, and 20  . The losses due to the 90 bend, MMPX connector, i.e., h and microstrip line together with the taper transition have been excluded from the measured radiation efficiency.

= 10

m

interval of 10 and an elevation interval of 2 . Subsequently, the directivity is calculated through a numerical integration over the sampled 3D patterns [33]. Lastly, the radiation efficiency of interest can be computed according to its definition, i.e., division of the gain by the directivity. On the side of the simulation, the radiation efficiency (labeled as “simulated” in Fig. 7) can also be accurately determined using the same procedure, in which however, the gain and 3D radiation patterns are provided by simulations. Fig. 7 depicts the radiation efficiency obtained from simulations and measurements. Firstly, a comparison of the simulated radiation efficiency obtained for different glue layer thick, 15 , and 20 , shows that the nesses, i.e., in the glue layer thickness impact of a small change on the radiation efficiency is not significant for the presently used glue. This finding is consistent with the fact that the gain is only slightly affected by the specified variations of the glue

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Fig. 8. Reflection coefficient of the circularly polarized HMSIW-fed DRA.

layer, as discussed for Fig. 6. Secondly, as the present investigation aims at characterizing the combination of the dielectric resonator and its feeding HMSIW, the losses due to the 90 degree bend, the MMPX connector, and the microstrip line together with the taper transition have been excluded from the measured radiation efficiency. The losses in the bend and connector are as given previously in the discussion of Fig. 6, while those due to the microstrip line and the taper transition are experimentally estimated about 0.3 0.1 dB over the whole operation band. Therefore, the radiation efficiency presented in Fig. 7 only includes the dielectric losses of the cylindrical resonator, as well as the dielectric and conductor losses of the feeding HMSIW. From Fig. 7, it can be seen that the measured radiation efficiency exhibits a maximum of 92% around 57 GHz and generally stays above 80% over the whole operation band, indicating that the HMSIW can be an efficient feed for DRAs operating around 60 GHz. Variations in the measured gain as shown in Fig. 6 contributes directly to significant spread in the measured radiation efficiency. The discrepancy between the measurement and simulation results mainly arises from some unaccounted losses and from uncertainties in the measurements which is considered an intrinsic drawback of the D/G method [17], [34]. In summary, the measured performance of the linearly polarized HMSIW-fed DRA demonstrates attractive characteristics in terms of bandwidth and radiation efficiency. The following discussion will now focus on the performance of the circularly polarized HMSIW-fed DRA. First, the measured reflection coefficient is presented in Fig. 8, exhibiting a . Then, Fig. 9 bandwidth of 4.5% or 2.7 GHz for depicts the axial ratio (AR) measured at the antenna boresight, demonstrating a 3-dB AR bandwidth of 4.0%, i.e., from 58.6 to 61.0 GHz, with a minimum of about 0.8 dB around 59.5 GHz. A comparison between the frequency responses in Fig. 8 and at the Fig. 9 reveals that the antenna operates with broadside over most of the operation band. Fig. 10 plots the measured major and minor axes of the polarization ellipse as a function of the elevation angle in two orthogonal planes, i.e., the yz- and xy-plane as defined in Fig. 1(a) at 59.4 GHz. The results show that the beamwidth for is 81 in the yz-plane (from to 62 ) and 99 in the xy-plane (between and 55 ).

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Fig. 9. Axial ratio at the broadside of the circularly polarized HMSIW-fed DRA.

linear or planar array. The combination of a low-loss transmission line with a highly efficient antenna strongly suggests the possibility achieving significantly higher radiation efficiency than with conventional microstrip-fed patch antennas arrays. The challenges towards realization of arrays can be described in terms of design and fabrication. On the side of the design, a relatively straightforward realization of a linear array would make use of a resonant series array DRAs fed by a common end-shorted HMSIW. This of array can be analyzed and designed using an approach similar to that for the resonant slot array fed by an HMSIW [21] or by a boxed stripline [35]. For application where frequency scanning is not desired, a corporate feed based on multi-way T-junctions [22] would need to be realized, while accommodating the larger transverse directions of the HMSIW compared to the microstrip line. This might increase the overall complexity, especially for a planar array. However, as the HMSIW can be considered as a nearly closed structure on the side of its vias, it can be intuitively expected that the mutual coupling between parallel HMSIWs will be lower than that of parallel microstrip lines in a feeding network, allowing to place them in close proximity. A detailed investigation of the mutual coupling between HMSIWs will be the subject of a future communication. On the side of the fabrication, the realization of a large number of metalized via holes in a circuit will certainly increase the complexity of the physical realization. However, in the context of the application of 3D multi-layer techniques such as Low-Temperature Co-fired Ceramic (LTCC) technology, the realization of such vias appears realistically feasible. Another practical alternative for standard planar circuit technology is to replace the row of vias in each HMSIW with a narrow longitudinal slot cut through the substrate. The inner surface of the slot needs to be metalized to realize the electric wall and to prevent the wave leakage. IV. CONCLUSION

Fig. 10. Measured major and minor axes of the polarization ellipse vs. elevation angle in the (a) yz-plane and (b) xy-plane for the circularly polarized HMSIW-fed DRA at 59.4 GHz. The coordinates are as shown in Fig. 1(a). The major axis is plotted in solid line, while the minor axis in dash line.

Up to this point, the simulated and measured performance has been demonstrated for both the linearly and circularly polarized HMSIW-DRA. With the attractive features of those single antennas, it appears desirable to assemble them in a

This paper has proposed an HMSIW-fed dielectric resonator antenna. The combination of an efficient feeding transmission line with an efficient radiator allows achieving a radiation efficiency above 80% around 60 GHz, with a maximum of 92%. The attractive performance of two HMSIW-fed DRA prototypes at 60 GHz, respectively for linear polarization and for circular polarization, has been substantiated in terms of bandwidth and radiation efficiency through simulations and experiments. The results demonstrate the feasibility of the HMSIW as an efficient feed for DRAs operating in the millimeter wave frequency band. The individual HMSIW-fed DRA can further be applied as an element for constructing HMSIW-fed series arrays or planar arrays of DRAs. ACKNOWLEDGMENT The authors thank H. Benedickter and G. Almpanis (ETH Zurich) for their helpful discussion and kind assistance in the measurement. The authors are also grateful to M. Lanz and C. Maccio (ETH Zurich) for their excellent manufacture of the devices.

LAI et al.: 60 GHz APERTURE-COUPLED DIELECTRIC RESONATOR ANTENNAS FED BY A HMSIW

REFERENCES [1] S. A. Long, M. W. McAllister, and L. C. Shen, “The resonant cylindrical dielectric cavity antenna,” IEEE Trans. Antennas Propag., vol. 31, no. 3, pp. 406–412, May 1983. [2] K. W. Leung, K. M. Luk, and K. Y. A. Lai, “Theory and experiment of a coaxial probe fed dielectric resonator antenna,” IEEE Trans. Antennas Propag., vol. 41, no. 10, pp. 1390–1398, Oct. 1993. [3] G. P. Junker, A. A. Kishk, and A. W. Glisson, “Input impedance of dielectric resonator antennas excited by a coaxial probe,” IEEE Trans. Antennas Propag., vol. 42, no. 7, pp. 960–966, Jul. 1994. [4] R. A. Karnenburg and S. A. Long, “Microstrip transmission line excitation of dielectric resonator antennas,” Electron. Lett., vol. 24, no. 18, pp. 1156–1157, Sep. 1988. [5] R. Kranenberg, S. A. Long, and J. T. Williams, “Coplanar waveguide excitation of dielectric resonator antennas,” IEEE Trans. Antennas Propag., vol. 39, no. 1, pp. 119–122, Jan. 1991. [6] M. T. Birand and R. V. Gelsthorpe, “Experimental millimetric array using dielectric radiators fed by means of dielectric waveguide,” Electron. Lett., vol. 13, no. 18, pp. 633–635, Sep. 1981. [7] K. W. Leung and K. K. So, “Rectangular waveguide excitation of dielectric resonator antenna,” IEEE Trans. Antennas Propag., vol. 51, pp. 2477–2481, Sep. 2003. [8] J. T. H. ST. Martin, Y. M. M. Antar, A. A. Kishk, A. Ittipiboon, and M. Cuhaci, “Dielectric resonator antenna using aperture coupling,” Electron. Lett., vol. 26, pp. 2015–2016, Sep. 1990. [9] K. W. Leung, K. M. Luk, K. Y. A. Lai, and D. Lin, “Theory and experiment of an aperture coupled hemispherical dielectric resonator antenna,” IEEE Trans. Antennas Propag., vol. 43, no. 9, pp. 1192–1198, Nov. 1995. [10] R. N. Simons and R. Q. Lee, “Effect of parasitic dielectric resonators on CPW/aperture coupled dielectric resonator antennas,” IEE Micorw. Antennas Propag., vol. 140, no. 5, pp. 336–338, Oct. 1993. [11] Z. C. Hao, W. Hong, A. D. Chen, J. X. Chen, and K. Wu, “SIW fed dielectric resonator antennas,” in Proc. Int. Symp. Microw. Theory Tech., San Francisco, Jun. 2006, pp. 202–205. [12] C. Y. Huang, J. Y. Wu, and K. L. Wong, “Cross slot coupled microstrip antenna and dielectric resonator antenna for circular polarization,” IEEE Trans. Antennas Propag., vol. 47, no. 4, pp. 605–609, Apr. 1999. [13] K. W. Leung, W. C. Wong, and H. K. Ng, “Circularly polarized slot coupled dielectric resonator antenna with a parasitic patch,” IEEE Antennas Wireless Propag. Lett., vol. 1, pp. 57–59, Jul. 2002. [14] G. Almpanis, C. Fumeaux, and R. Vahldieck, “Offset cross slot coupled dielectric resonator antenna for circular polarization,” IEEE Microw. Wireless Compon. Lett., vol. 16, no. 8, pp. 461–463, Aug. 2006. [15] M. B. Oliver, Y. M. M. Antar, R. K. Mongia, and A. Ittipiboon, “Circularly polarized rectangular dielectric resonator antenna,” Electron. Lett., vol. 31, no. 6, pp. 418–419, Mar. 1995. [16] A. Buerkle, K. Sarabandi, and H. Mosallaei, “Compact slot and dielectric resonator antenna with dual resonance, broadband characteristics,” IEEE Trans. Antennas Propag., vol. 53, no. 3, pp. 1020–1027, Mar. 2005. [17] Q. H. Lai, C. Fumeaux, W. Hong, and R. Vahldieck, “Comparison of the radiation efficiency for the dielectric resonator antenna and the microstrip antenna at Ka band,” IEEE Trans. Antennas Propag., vol. 56, no. 11, pp. 3589–3592, Nov. 2008. [18] W. Hong, B. Liu, Y. Wang, and Q. Lai et al., “Half mode substrate integrated waveguide: A new guided wave structure for microwave and millimeter wave application,” in Proc. Joint 31st Int. Conf Infra. Millimeter Waves/14th Int. Conf. Terahertz Electron., Shanghai, Sep. 2006, pp. 18–22. [19] Y. Wang, W. Hong, Y. Dong, and B. Liu et al., “Half mode substrate integrated waveguide (HMSIW) bandpass filter,” IEEE Microw. Wireless Compon. Lett., vol. 17, no. 4, pp. 256–267, Apr. 2007. [20] Y. Cheng, W. Hong, and K. Wu, “Half mode substrate integrated waveguide (HMSIW) directional filter,” IEEE Microw. Wireless Compon. Lett., vol. 17, no. 7, pp. 504–506, Jul. 2007. [21] Q. H. Lai, W. Hong, Z. Q. Kuai, Y. S. Zhang, and K. Wu, “Half mode substrate integrated waveguide transverse slot array antennas,” IEEE Trans. Antennas Propag., vol. 57, no. 4, pp. 1064–1072, Apr. 2009. [22] B. Liu, W. Hong, L. Tian, H. B. Zhu, W. Jiang, and K. Wu, “Half mode substrate integrated waveguide (HMSIW) multi-way power divider,” in Proc. Asia Pacific Microw. Conf., Suzhou, China, Dec. 12–15, 2006, pp. 917–920. [23] Q. H. Lai, C. Fumeaux, W. Hong, and R. Vahldieck, “Characterization of the propagation properties of the half-mode substrate integrated waveguide,” IEEE Trans. Microw. Theory Tech., vol. 57, no. 8, pp. 1996–2004, Aug. 2009.

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[24] F. Xu and K. Wu, “Guided-wave and leakage characteristics of substrate integrated waveguide,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 1, pp. 66–72, Jan. 2005. [25] R. K. Mongia and P. Bhartia, “Dielectric resonator antennas—A review and general design relations for resonant frequency and bandwidth,” Int. J. Microw. Millim.-Wave Comput.-Aided Eng., vol. 4, no. 3, pp. 230–247, Mar. 1994. [26] K. M. Luk and K. W. Leung, Dielectric Resonator Antennas. Hertfordshire, England: Research Studies Press, 2002, pp. 179–183. [27] A. J. Simmons, “Circularly polarized slot radiators,” IEEE Trans. Antennas Propag., vol. 5, no. 1, pp. 31–36, Jan. 1957. [28] Huntsman Advanced Materials American Inc., 2004, Product Data Araldite AW 106 Resin [Online]. Available: http://adhesive.leaderseal.com/download/TDS-A106-953(US).pdf [29] M. J. Akhtar, L. Feher, and M. Thumm, “Measurement of dielectric constant and loss tangent of epoxy resins using a waveguide approach,” in Proc. IEEE Antenna Propag. Symp., Albuquerque, NM, Jul. 9–14, 2006, pp. 3179–3182. [30] Rogers Corporation, 1991–2002, High Frequency Circuit Materials Product Selector Guide [Online]. Available: http://www.leiton.de/formulare/Rogers.pdf [31] G. J. Simonis, J. P. Satter, T. L. Worchesky, and R. P. Leavitt, “Characterization of near millimeter wave materials by means of non-dispersive Fourier transform spectroscopy,” Int. J. Infrared Millim. Wave, vol. 5, no. 1, pp. 57–72, May 1984. [32] V. N. Egorov, V. L. Masalov, Y. A. Nefyodov, A. F. Shevchun, and M. R. Trunin, “Measuring microwave properties of laminated dielectric substrates,” Rev. Sci. Instrum., vol. 75, no. 11, pp. 4423–4433, Nov. 2004. [33] C. A. Balanis, Antenna Theory. Hoboken, NJ: Wiley, 2005, pp. 1028–1036. [34] D. M. Pozar and B. Kaufman, “Comparison of three methods for the measurement of printed antenna efficiency,” IEEE Trans. Antennas Propag., vol. 36, no. 1, pp. 136–139, Jan. 1988. [35] R. Shavit and R. S. Elliott, “Design of transverse slot arrays fed by a boxed stripline,” IEEE Trans. Antennas Propag., vol. 31, no. 4, pp. 545–552, Jul. 1983.

Qinghua Lai (S’09) received the B.S. degree in radio engineering from University of Electronic Science and Technology, Chengdu, China, in 2004. He is currently working toward the M.Phi. and Ph.D. degrees in radio engineering at Southeast University, Nanjing, China. From July 2007 to June 2008, he was Visiting Student with the Swiss Federal Institute of Technology (ETH), Zurich, Switzerland, where he was engaged in the research on the radiation efficiency of dielectric resonator antennas and on the propagation properties of half-mode substrate integrated waveguides. From September 2008 to September 2009, he was a Academic Guest with ETH, where he was involved in the development of high-performance filter for the base station. His current research interests focus on array antennas and filters.

Christophe Fumeaux (M’03–SM’09) received the Diploma and Ph.D. degrees in physics from the ETH Zurich, Switzerland, in 1992 and 1997, respectively. From 1998 to 2000, he was a Postdoctoral Researcher with the School of Optics, University of Central Florida, Orlando. In 2000, he joined the Swiss Federal Office of Metrology, Bern, Switzerland, as a Scientific Staff Member. From 2001 to 2008, he was a Research Associate and Group Leader with the Laboratory for Electromagnetic Fields and Microwave Electronics (IFH), ETH, Zurich, Switzerland. During the Fall of 2005, he was a Visiting Scientist with the Laboratory of Sciences and Materials for Electronics, and of Automatic (LASMEA), University Blaise Pascal, Clermont-Ferrand, France. In 2008, he joined the School of Electrical and Electronic Engineering, the University of Adelaide, Australia, as an Associate Professor. His current main research interest concerns computational electromagnetics, microwave circuits and antenna engineering.

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Dr. Fumeaux was the recipient of the ETH Silver Medal of Excellence for his doctoral dissertation. From 2006 to 2008, he was the Chairman of the IEEE Swiss Joint Chapter on Microwave Theory and Techniques, Antennas and Propagation, and EMC. He was the corecipient of the outstanding paper award of the Applied Computational Electromagnetics Society (ACES) in 2004.

Wei Hong (M’92–SM’07) received the B.S. degree from the University of Information Engineering, Zhengzhou, China, in 1982, and the M.S. and Ph.D. degrees from Southeast University, Nanjing, China, in 1985 and 1988, respectively, all in radio engineering. Since 1988, he has been with the State Key Laboratory of Millimeter Waves, Southeast University, where he is currently a Professor with the School of Information Science and Engineering. In 1993 and 1995–1998, he was a short-term Visiting Scholar with the University of California at Berkeley and the University of California at Santa Cruz. He has been engaged in numerical methods for electromagnetic problems, millimeter-wave theory and technology, antennas, electromagnetic scattering, RF technology for mobile communications, etc. He has authored or coauthored over 200 technical publications. He authored Principle and Application of the Method of Lines (in Chinese) (Southeast Univ. Press, 1993) and Domain Decomposition Methods for Electromagnetic Problems (in Chinese) (Sci. Press, 2005). He has served as a reviewer for IET Microwave Antennas, and Propagation and Electronics Letters. Dr. Hong is a Senior Member of the Chinese Institute of Electronics (CIE). He is vice president of the Microwave Society and Antenna Society, CIE. He has served as a reviewer for the IEEE TRANSACTIONS ONMICROWAVE THEORY AND TECHNIQUES and the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION. He is an associate editor for the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUE. He was a two-time recipient of the 1992 and 1994 First-Class Science and Technology Progress Prizes presented by the Ministry of Education of China and the 1991 Fourth-Class National Natural Science Prize. He was also the recipient of the Foundations for China Distinguished Young Investigators and for the “Innovation Group” presented by the National Natural Science Foundation of China (NSFC).

Rüdiger Vahldieck (M’85–SM’86–F’99) received the Dipl.-Ing. and the Dr.-Ing. degrees in electrical engineering from the University of Bremen, Germany, in 1980 and 1983, respectively. He was a Postdoctoral Fellow with the University of Ottawa, Ottawa, ON, Canada, until 1986. In 1986, he joined the Department of Electrical and Computer Engineering, University of Victoria, BC, Canada, where he became a Full Professor in 1991. During Fall and Spring of 1992 to 1993 he was a Visiting Scientist at the “Ferdinand-Braun-Insitute fór Hochfrequenztechnik”, Berlin, Germany. In 1997, he was appointed Professor for electromagnetic field theory at the Swiss Federal Institute of Technology, Zurich, and became head of the Laboratory for Electromagnetic Fields and Microwave Electronics (IFH) in 2003. In 2005, he became President of the Research Foundation for Mobile Communications and was elected Head of the Department of Information Technology and Electrical Engineering (D-ITET), ETH Zurich. Since 1981 he has published more than 300 technical papers in books, journals and conferences. His research interests include computational electromagnetics in the general area of EMC and in particular for computer-aided design of microwave, millimeter wave and optoelectronic integrated circuits. Dr. Vahldieck received the Outstanding Publication Award of the Institution of Electronic and Radio Engineers in 1983, the K.J. Mitra Award of the IETE (in 1996) for the best research paper in 1995, and the ACES Outstanding Paper Award in 2004. He is the Past-President of the IEEE 2000 International Zurich Seminar on Broadband Communications (IZS’2000) and since 2003 President and General Chairman of the international Zurich Symposium on Electromagnetic Compatibility (EMC Zurich). He is a member of the editorial board of the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES. From 2000 until 2003 he served as Associate Editor for the IEEE MICROWAVE AND WIRELESS COMPONENTS LETTERS and from July 2003 until the end of 2005 as the Editor-in Chief. Since 1992, he has been on the Technical Program Committee of the IEEE International symposium, the MTT-S Technical Committee on Microwave Field Theory, and in 1999 on the TPC of the European Microwave Conference. From 1998 until 2003 he was the Chapter Chairman of the IEEE Swiss Joint Chapter on MTT, AP, and EMC.

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Mu-Zero Resonance Antenna Jae-Hyun Park, Young-Ho Ryu, and Jeong-Hae Lee, Member, IEEE

Abstract—We present mu-zero resonance (MZR) antennas that use an artificial mu-negative (MNG) transmission line (TL). The equivalent circuit for verifying the peculiarity of the MNG TL is derived and analyzed. To operate the MZR antenna properly, the antenna is fed by magnetic coupling. The analysis and design of the MZR antenna are performed according to theory and simulation based on a dispersion diagram and field distribution. The surface current distribution shows that the radiation mechanism of the MZR antenna is essentially identical to that of a small-loop antenna. Applying the novel concept of the MZR antenna, a dualband MZR antenna using two MZR antennas with different MZR frequencies is proposed. The radiation characteristics of the antenna are simulated and measured at two frequencies. The measured characteristics show agreement with the simulated results. It is confirmed that the characteristics of the MZR antenna, including the efficiency, gain, and fractional bandwidth, are suitable for a multiband antenna. Index Terms—Metamaterials, mu-negative (MNG), mu-zero resonance (MZR), small-loop antenna.

I. INTRODUCTION

T

HE novel electromagnetic properties of metamaterials [1]–[5], such as backward-wave propagation, the negative index refraction, and the infinite wavelength wave have opened up new areas of applications. Since the beginning of research on this area, the field and the number of applications of metamaterials have shown rapid growth. From a practical application standpoint, the composite right- and left-handed (CRLH) transmission line (TL) has been broadly applied to radio frequency (RF) devices because it is associated with a low level of loss and a broad LH bandwidth [6]–[9]. In addition, the CRLH TL provides an inherent parasitic right-handed property because it contains the same components utilized in a conventional TL. Therefore, the CRLH TL has the unique property of an infinite wavelength wave (zero propagation constant) at a discrete frequency of the boundary of the LH and RH bands. Various size-independent RF devices using the infinite wavelength property of the CRLH TL, such as a power

Manuscript received June 19, 2008; revised January 24, 2009; accepted January 01, 2010. Date of publication March 29, 2010; date of current version June 03, 2010. This work was supported by the Korea Science and Engineering Foundation (KOSEF) grant funded by the Korean government (MOST) (No. R01-2007-000-20495-0) . J.-H. Park is with the Department of Electronic Information and Communication Engineering, Hongik University, Seoul 121-791, Korea. Y.-H. Ryu is with the School of Electrical Engineering and Computer Science, Kyungpook National University, Daegu 702-701, Korea. J.-H. Lee is with the Department of Electronic and Electrical Engineering, Hongik University, Seoul 121-791, Korea (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.2010.2046832

divider [10], a zeroth-order resonator (ZOR) [11], and ZOR antennas [12]–[19] have been reported. Recently, a ZOR antenna based on the fundamental infinite wavelength property of a shunt inductor-loaded TL [18] or an artificial epsilon-negative (ENG) TL [19] was presented. Given that an infinite wavelength occurs at the zero permittivity of the metamaterial TL, the epsilon-zero resonance (EZR) frequency does not depend on its electrical length, which is similar to the ZOR frequency of the CRLH TL [11]–[16]. The CRLH TL has two ZOR resonances which depend on the applied boundary condition because the TL consists of a combination of series and shunt resonances. To realize ZOR antennas using the ZORs of CRLH and ENG TLs in a number of studies [11]–[19], an open-ended boundary condition was applied because the antennas are implemented by the shunt admittance of a unit cell, which determines the ZOR frequency where the effective permittivity is zero. In this paper, a mu-zero resonance (MZR) antenna which also supports an infinite wavelength wave is proposed using an artificial mu-negative (MNG) TL [see Fig. 1]. The TL has an effectively negative mu value in the rejection band due to its artificially added series capacitance, which is similar to the ENG TL [18], [19]. Therefore, the artificial MNG TL provides a MNG rejection band in the low-frequency region and a right-handed (RH) propagation band in the high-frequency region [see Fig. 2]. Therefore, at the transition point between the MNG rejection and RH propagation bands, the MNG TL has the MZR mode of zero permeability. By varying the series capacitance of the MNG TL unit cell, the MZR frequency of the resonator using the MNG TL can be controlled. To design the MZR antenna, a short-ended boundary condition must be applied to the antenna because the series resonance of the MNG TL unit cell determines the MZR frequency, which supports the infinite wavelength, while an open-boundary condition is applied to the antenna using CRLH and ENG TLs. A matching network using a magnetic coupling to operate the MZR antenna effectively is numerically verified. MZR antennas consisting of one, two, and three unit cells are investigated based on periodic structure theory, circuit modeling with an optimized parameter, full-wave simulation, and measurements. It is shown that the MZR antenna has good efficiency, gain, and fractional bandwidth characteristics. Furthermore, MZR antennas are suitable for multiband applications because they can be coupled and integrated by one feed line. A dual-band antenna using two mu-zero resonators as an example is presented. II. THEORY The artificial MNG TL supporting an infinite wavelength is discussed in Section II-A. The resonator using the TL is also analyzed in Section II-B.

0018-926X/$26.00 © 2010 IEEE

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Fig. 1. Equivalent circuit of a lossless artificial MNG TL.

A. Infinite Wavelength Property of Artificial Mu-Negative Transmission Line In general, practically realized metamaterial TLs are composed of artificial components such as via holes, gaps, stubs and other such components. Thus, the equivalent circuit of a lossless and ) artificial mu-negative (MNG) TL is realized ( by adding a series capacitance ( ) to the host TL, as shown in Fig. 1. The effective permeability and permittivity of the MNG TL are then obtained through the equation below, which is similar to CRLH and ENG TLs [19]

(1) , , and are the times-unit-length series capacHere, itance, the per-unit-length shunt capacitance, and the per-unitlength series inductance in terms of an infinitesimal component (in , F/m, and H/m), respectively. The propagation constant of the MNG TL can be obtained by applying periodic boundary conditions to the equivalent circuit of the unit cell, as follows [4] (2) where (3) is the propagation constant for Bloch waves, and is length of the unit cell. ( ), ( ), and ( ) are the series capacitance, shunt capacitance, and series inductance in terms of the real lumped component (in F and H), respectively. From (2), the dispersion curve of the MNG TL is obtained and those of the CRLH and ENG TLs are over plotted to compare with the ZOR frequencies of each TL, as shown in Fig. 2. The equivalent circuit elements used in plotting the dispersion curves are , , , and , respectively, in Fig. 1. (Here, is the shunt inductance for the CRLH and ENG TLs in the literature [18], [19]). In Fig. 2, the corresponding mu-zero ( ) and epsilon-zero ( ) frequencies are 3.56 GHz and 5.03 GHz, respectively. An artificial MNG TL, similar to the ENG TL, supports zero and positive propagation constants while the propagation constants of the CRLH TL are negative, zero, and positive values [11]–[15], as shown in Fig. 2. Therefore, the MNG TL also has an infinite wavelength property with a zero propagation constant. De-

Fig. 2. Dispersion relations of the metamaterial TLs.

pending on the boundary condition, the ZOR frequencies of each TL are selected. That is, the mu-zero resonance (MZR) is obtained with a short-ended boundary condition while the epsilon-zero resonance (EZR) is obtained with an open-ended boundary condition. Therefore, the MNG TL has a nonzero resonance frequency ( ), which is supported by the series resonance of the TL unit cell in Fig. 1. In this case, a short-ended boundary must be employed to apply the MZR to RF devices. B. Resonators Using an Artificial Mu-Negative Transmission Line The resonance modes of the MNG TL can be obtained by the following simple condition, similar to those of the CRLH and ENG TLs [11]–[19]: (4) ) and are the number of the unit cell and the where ( total length of the resonator, respectively. To calculate the theoretical MZR frequency of the MNG TL, the input impedances of conventional TLs based on the openand short-ended boundary conditions are calculated by (5) [11]

(5) By applying the conditions to the equivalent circuit in Fig. 1, (5) can be rewritten as

(6)

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Fig. 3. Schematic diagram for the short-ended resonator using an artificial three-stage MNG TL by magnetic coupling.

To obtain the MZR frequency, the short-ended boundary condition must be applied to the unit cell from (6). Thus, the MZR frequency, similar to those of the CRLH and ENG TLs, is given as (7) From (7), it is noted that the resonant frequency is independent of the total length ( ) of the resonators. Fig. 3 shows a schematic diagram of the short-ended resonator using an artificial three-stage MNG TL, which is related to the equivalent circuit of Fig. 1. The same circuit parameter values of Fig. 2 are used in the simulation. In Fig. 3, the short-ended resonator is connected to the magnetic coupled section with input and output ports for magnetic coupling in the resonator. The parameter values are arbitrarily chosen to be , , and , with mutual between the and in order inductance of to achieve a magnetic coupling of the mu-zero resonators to external ports and show that the CRLH and MNG TLs have the same ZOR frequency. To verify the related resonance modes in (4) and (7), the resonant frequencies of the resonator in Fig. 3 were simulated using a circuit simulator (Ansoft’s Designer). Fig. 4 shows the simulation results with that of the CRLH TL case for comparison. The three-stage CRLH resonator has two negative, two positive, and zero resonance modes while the three-stage MNG resonator has two positive and zero resonance modes, as shown in Fig. 4. In ) of the MNG resonator particular, the MZR frequency ( is identical to that of the CRLH resonator in spite of absence of the shunt inductance in the short-ended MNG and CRLH resonators, similar to the open-ended ENG and CRLH resonators [18], [19]. Thus, the MNG resonator has a simpler structure and is easier to fabricate than the CRLH resonator, as the shunt components of the CRLH TL, such as metallic vias, are not required. In Table I, the simulated resonance frequencies of the MNG and CRLH resonators are compared with the theoretical resonance frequencies of the resonators in Fig. 2. It is noted that each value of 0 , 60 , and 120 in Fig. 2 corresponds to the resonant modes of the zero, first, and second modes, respectively. The simulated results show good agreement with the theory, as shown in Table I. The slight difference in the resonance frequencies at each mode is caused by the boundary conditions. That is, the theoretical resonant frequencies are obtained with the perfectly short-ended boundary condition, whereas the simulated resonance frequencies are shifted by the coupling coefficients , , , and values at each mode, as shown from the in Fig. 3. Consequently, the infinite wavelength resonator using the artificial MNG TL with the short-ended boundary condition can be successfully applied to the antennas because the antenna

Fig. 4. Resonance modes of the three-stage MNG and CRLH resonators. TABLE I RESONANCE MODES OF THREE-CELL MNG AND CRLH RESONATORS

Fig. 5. Unit cell of artificial MNG TL (gap = 0:1 mm, w = 0:15 mm, and p = 3 mm).

size will be independent of the electrical length and because they are easily fabricated. III. MU-ZERO RESONANCE ANTENNA A. Realization of an Artificial MNG TL To realize the miniaturized artificial MNG TL, an interdigital capacitor (IDC) is employed, as shown in Fig. 5. A unit-cell structure has the following dimensions: gap of 0.1 mm, a finger width ( ) of 0.15 mm, and a period ( ) of 3 mm. The substrate of the MNG TL uses a Rogers RT/Duroid 5880 with a dielectric and a thickness . As mentioned constant in Section II-A, the effective permeability of the MNG TL has negative, zero, and positive values due to the series capacitance of the IDC. The TL has a MZR frequency at zero permeability, which can be controlled by the number of fingers ( ) of the IDC. Thus, the parameters of a related number of fingers were studied using a circuit and full-wave simulator.

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TABLE II EXTRACTED PARAMETERS OF MNG UNIT CELL VERSUS NUMBER OF FINGERS

Fig. 6. Mu-zero resonance antenna.

Table II lists the circuit parameters of Fig. 1, as extracted by a full-wave simulation (Ansoft’s HFSS) of a MNG unit cell versus the number of fingers. Increasing the number of fingers results in increases in the series capacitance ( ), as expected, whereas the series inductance ( ) decreases. Consequently, the MZR frequency decreases because the increment rate of the series capacitance value is larger than the decrement rate of the series inductance value. Thus, to obtain a lower MZR frequency, the number of fingers or length of the unit cell must increase. B. Antenna Design Fig. 6 shows the proposed MZR antenna using the MNG resonator with a short-ended boundary. Thus, both ends of the proposed antenna are connected to the ground by metallic via holes. The unit-cell dimensions of the antenna are identical to those discussed in Section III-A. As shown in Fig. 6, the antenna should be located in parallel with the feed line to enhance the magnetic coupling. The antenna uses an open-ended quarter wavelength feed line at the MZR frequency to obtain good impedance matching. The input impedance of the antenna consists of the input impedance of the quarter wavelength feed line and the coupling impedance to the MNG TL. To investigate the effect of the length of the feed line, the input impedance of the antenna is calculated for three different ( ), ( ), lengths of feed line: ( ). The length of the feed line has little and effect on the MZR frequency, as shown in Fig. 7, because the frequency is mainly determined by the geometrical parameters of the resonator, such as the finger width ( ), the number of fingers, gap, and the period ( ) shown in Fig. 5. In general, a reactance value must be zero for good impedance matching. However, the imaginary value is not zero when the

Fig. 7. Input impedance for three different cases (s = 2:5 mm) (a) real and (b) imaginary.

length of the feed line is not a quarter of the wavelength, as shown in Fig. 7(b). Therefore, a rather complex and difficult impedance-matching technique is required if the length of the feed line is not a quarter of the wavelength. Thus, good impedance matching can easily be accomplished using an open-ended quarter-wavelength feed line, as its reactance value at the MZR frequency is zero, as shown in Fig. 7(b). To investigate the behavior of the input impedance versus the separation ( ), a one-cell MZR antenna with an IDC of four fingers was designed and simulated. Fig. 8 shows the input impedance of the port versus the frequency with different separations. The input impedance for matching the antenna can be controlled by the separation between the antenna and the feeding line, which is related to the intensity of the magnetic coupling. That is, if the separation increases, the input impedance decreases due to weak magnetic coupling, as shown is in Fig. 8. For instance, an input impedance of obtained at a separation ( ) of 2.5 mm and frequency of 7.16 GHz, as shown in Fig. 8. Consequently, an optimized matching network can be determined according to the separation. Finally, the width of the feed line is also related to the coupling and impedance matching. If the width is narrower, magnetic coupling is stronger because the narrow feed line has a

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Fig. 9. (a) Simulated surface current of a three-cell MZR antenna at the MZR frequency and (b) equivalent-loop current.

Fig. 8. Input impedance (a) real and (b) imaginary (w : , and d : ).

7 9 mm

= 0 55 mm

= 1 mm, l =

strong inductive component. In this study, the width of the antenna was selected as 1 mm to consider the fabrication. If the width of feed line is narrower than 1 mm, the separation ( ) must be larger than 2.5 mm. C. Radiation Mechanism of MZR Antenna To predict the radiation pattern of the MZR antenna intuitively, the surface current of a three-cell MZR antenna with an IDC of four fingers at the MZR frequency was simulated by Ansoft’s HFSS. The surface current flows in opposite directions at the top and bottom in Fig. 9(a), resulting in the equivalent-loop current shown in Fig. 9(b). The surface current intensity is asymmetrical with respect to the -axis due to the asymmetrical IDC structure and asymmetrical feed structure. Under these circumstances, it is intuitively expected that the radiation patterns of the antenna can be tilted. It is important to note that the phase of the current was observed to be in-phase along the current loop. This indicates that an infinite wavelength is supported at the MZR frequency. As shown in Fig. 9(b), the current flow is identical to that of a small-loop antenna [20]. Therefore, it is expected that the radiation mechanism of the MZR antennas is identical to that of a small-loop antenna. To verify that the MZR antenna has the same radiation properties as a small-loop antenna, the radiation

Fig. 10. Simulated small-loop antenna on the ground plane.

properties of two antennas are simulated and compared in terms of their radiation patterns. Fig. 10 shows a small-loop antenna above the ground plane. It is oriented vertically with respect to the given coordinate, and a 50 lumped port is applied to the feed. Fig. 11 shows the simulated radiation patterns of a one-cell MZR antenna with an IDC of four fingers and a small-loop antenna above the ground plane. The size and resonance frequency of the two antennas are sim(the length and height of MZR ilar, specifically, antenna) at 7.16 GHz, and (the diameter of small-loop antenna) at 7.35 GHz, respectively. A parallel plate capacitor was included in a small-loop antenna to reduce its size. The radiation patterns of the two antennas are similar, as expected. A slightly asymmetrical radiation pattern in the MZR antenna was observed due to the asymmetrical IDC structure and asymmet). Additionally, a rical feed structure in the - plane ( slightly asymmetrical radiation pattern in the MZR antenna on ) occurs because the resonator is placed the - plane (

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TABLE III EXPERIMENTAL, SIMULATED, AND THEORETICAL RESULTS OF EACH MZR ANTENNA

Table II, MZR antennas of a one-cell MZR antenna using an IDC with four fingers, a one-cell MZR antenna using an IDC with six fingers, a two-cell MZR antenna using an IDC with four fingers, and a three-cell MZR antenna with IDC of four fingers were realized. The simulated and measured MZR frequencies, reflection coefficient values, fracantenna sizes, tional bandwidths, maximum gains, and radiation efficiencies of the four types of MZR antennas are summarized in Table III. The SMA connector is included in the simulation because the simulated results are compared with those of the measurements. First, the MZR frequency decreases as the number of fingers increases because the series capacitance increases with the number of fingers. The simulated and measured MZR frequencies converge to the theoretical MZR frequency as the number of unit cells increases because the effect of the metallic vias is removed. It can also be observed that the simulated and measured MZR frequencies are lower than the theoretically calcu) value lated frequencies. If inevitable, the inductance ( of the metallic vias for a short-ended boundary is included in an -stage resonator, as shown in Fig. 6, and the exact input impedance of (6) is modified as (8) Thus, the exact MZR frequency can be obtained by (8), as follows: (9) Fig. 11. Simulated radiation patterns (a) One-cell MZR antenna with IDC of four fingers and (b) small-loop antenna (radius of loop = 1:8 mm).

at one side of the feed line. The cross-polarization of the MZR antenna is higher than that of the small loop antenna due to the asymmetric structure, as shown in Fig. 11. Therefore, the MZR antenna as a planar type can be fabricated and integrated with other RF devices more easily compared to a conventional loop antenna that is not a planar type, as shown in Fig. 10. The proposed MZR antenna is also very useful as a multiband antenna. D. Experimental Results of MZR Antenna Four types of MZR antennas with the unit cells analyzed in Section III-A were fabricated: Using the unit cells listed in

), the above If the number of unit cells is infinite ( equation is identical to (7), while if the number of the unit cells decreases, the MZR frequencies from (9) are lower than the theoretical MZR frequency from (7). In Table III, the MZR frequencies that are obtained from (7) and (9) are compared. ) of two metallic In (9), the extracted inductance value ( vias is 0.38 nH. The exact MZR frequency from (9) is closer to the measured and simulated MZR frequencies than that from (7), as shown in Table III. The small difference between the MZR frequency from (9) and the measured and simulated MZR frequencies stems from the coupling effects, as noted in Section II-B. The measured and simulated MZR frequencies are in good agreement, as shown in Table III.

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TABLE IV FINGERS SIMULATED RESULTS OF THE THREE-CELL MZR ANTENNAS WITH THE DIFFERENT GROUND SIZES

Second, the total size of the MZR antenna becomes relatively large compared to that of the mu-zero resonator because the antenna employs a quarter-wavelength feed line to obtain good matching easily. For example, the total sizes of the one-cell and three-cell MZR antennas with four fingers are and , as shown in Table III, but the and resonator sizes of the MZR antennas are , respectively. Thus, smaller MZR antennas can be designed if the quarter-wavelength feed line is replaced by a small matching network. If a small matching network is used, this will affect the value of the input impedance, as shown in Fig. 7, because the magnetic coupling will be affected by the shorter line. Therefore, this has to be considered when designing the matching network. The important effects of the ground and SMA connector, which affect the properties of the antenna, are also considered. To investigate the effect of the SMA connector, three-cell MZR antennas with and without the SMA connector were fracsimulated. The simulated MZR frequencies and tional bandwidths are 7.46 GHz and 7.40 GHz and 4.9% and 2.1%, respectively. Although the MZR frequencies show good agreement, the bandwidth is rather different because the SMA connector operates as another radiator. To investigate the ground effect, the antenna properties with respect to the width and length of the ground were simulated using the three-cell MZR antenna without a SMA connector, as shown in Table IV. The simulated results are summarized in Table IV. In general, the size of the ground influences the radiation properties of an antenna and, especially, it affects the bandwidth. When the length of the ground is equal to a half wavelength (16 mm), it is discovered that the antenna has good antenna properties because the ground also radiates. For example, the fractional bandwidth and efficiency increase to 4.76% and 96% when the size of the ground is 10 mm 16 mm. On the other hand, the fractional bandwidth and efficiency decrease to 2.10% and 89% when the size of ground is 20 mm 16 mm because ground with a width of 20 mm is matches the feed line less than ground with a width of 10 mm at the MZR frequency.

Fig. 12. Three-cell MZR antenna with IDC of four fingers (a) Photograph, (b) simulated and measured S reflection coefficient values.

Thus, it cannot be stated that the fractional bandwidth and efficiency decrease with a small ground size. For instance, the fractional bandwidth and efficiency are 2.10% and 89% when the ground size of the fabricated antenna is 20 mm 16 mm. If it is decreased to 10 mm 11 mm, the fractional bandwidth and efficiency increase to 2.22% and 92%. In addition, the overall radiation patterns with different ground sizes are similar to that of the small-loop antenna, as shown in Fig. 11. The gain increases as the length of the ground increases in Table IV. The reason for the increment in the gain is that the ground plane is operated as a reflector of radiation. Thus, it is concluded that the fractional bandwidth and efficiency of the MZR antenna are not proportional to the ground size; that is, they can be larger or smaller, as shown in Table IV. Therefore, it is thought that a small MZR antenna with a small ground size of 10 mm 11 mm listed in Table IV can be designed. The radiation without the three-cell resonator was calculated to investigate the radiation of the feed line itself and/or the ground. The result shows that the ratio of the radiation power without the MZR resonator to that with the MZR resonator . However, it was found that some radiated power is comes from the ground because the length of the ground (20 . When mm 16 mm) in this case was a resonant length of , the length of the ground is changed to 21 mm, which is not the radiated power becomes even smaller ( ), indicating

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Fig. 13. Radiation patterns of the three-cell MZR antenna (n

Fig. 14. Dual-band MZR antenna (n

= 4): (a) Co-pol. (x-z plane), (b) X-pol. (x-z plane), (c) Co-pol. (y-z plane), (d) X-pol. (y-z plane).

= 4 and n = 6).

that some radiated power is due to ground resonance. In any case, the radiation is dominated by the MZR antenna. Fig. 12(a) shows a photograph of a three-cell MZR antenna using an IDC with four fingers as an example. The dimensions of the feed line for impedance matching are given by the method described in Section III-B, as shown in Fig. 12(a). Fig. 12(b) shows the simulated and measured results of the antenna including the SMA connector. The measured results are in good agreement with the simulated results. The fractional bandwidths were simulated and measured as 3.8% and 4.9%, respectively. These values are different due to the rather poor matching from the fabrication tolerance, as shown in Fig. 12(b). Fig. 13 shows the measured and simulated co-polarization and cross-polarization radiation patterns of the three-cell MZR antenna. According to coordination in Fig. 9, the co-polarization in the - plane and in the - plane corresponds to the radiated

Fig. 15. Photograph and MZR antenna.

S

reflection coefficient values of the dual-band

electric field in the -direction and in the -direction, respectively. In Fig. 13, the difference between the co-and cross-polarizations is less than that of a conventional small-loop antenna because the current path in Fig. 9 tilts with respect to the -axis. The measured maximum gain of the antenna was 3.9 dBi, whereas the simulated maximum gain of the antenna was 5.1 dBi. The simulated and measured maximum gains of cross-polarization are less than that of co-polarization by 12.5 dB and 11.7 dB, respectively. The gain difference between the

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Fig. 16. Radiation patterns of the dual-band MZR antenna (a) First MZR frequency of 6.2 GHz (i) Phi MZR frequency of 7 GHz (i) Phi = 0 (x-z plane) (ii) Phi = 90 (y -z plane).

simulation and measurement of the MZR antenna is thought to be caused by mismatching from the dimensional tolerance of the fabrication and the smaller measured value of efficiency. It was also noted that the measured asymmetrical radiation pattern is due to the asymmetrical surface current, as mentioned earlier. The measured radiation patterns of MZR antennas have nulls and back-lobes due to the influence of the measuring cable. However, the measured results agree with the simulated results overall. Finally, the fractional bandwidth and maximum gain of the antennas increase if the number of unit cells increases because the size of the in-phase current loop increases [21]. The radiation efficiencies were measured by the Wheeler cap method. To minimize the influence of the Wheeler cap on the current distribution on the MZR antenna, it is recommended that the cap radius of a spherical cap should be 1/6 [22]. However, our cap is not a sphere but a hexahedron. Thus, taking into account the hexahedral shape of the cap and the size of the MZR antenna, the size of the Wheeler cap was 40 mm in width, 32 , , mm in length, and 25 mm in height [23]. The and resonant modes of the cavity are 6 GHz, 7.07 GHz, and 7.61 GHz, respectively. For the one-cell MZR antenna with four fingers, the measured input impedances with and without the cap were 365-j5 and 74-j0.8 at 6.693 GHz and 6.635 GHz,

= 0 (x-z plane) (ii) Phi = 90 (y-z plane) (b) Second

respectively. Thus, the measured efficiency was determined to be 80% by calculating the above measured input impedance . The measured efficiencies of the four MZR antennas in Table III are . The results show that the measured efficiencies are smaller than the simulated efficiencies, as shown in Table III. These smaller measured efficiencies may explain the smaller gains in the measurement shown in Table III. Consequently, the results of the size, bandwidth, efficiency, and gain show that the proposed MZR antenna can be successfully applied to antenna applications. IV. DUAL-BAND MZR ANTENNA The proposed MZR antennas are suitable for multiband applications because two or more MZR resonators with different resonant frequencies can easily be integrated using one feed line. Fig. 14 shows a dual-band MZR antenna using mu-zero resonators with four fingers and six fingers as an example. Here, the ground size is 20 mm 17.5 mm. Two resonators are used: a one-cell resonator with six fingers and another with four fingers, corresponding to the MZR frequencies of 6.38 GHz and 7.03 GHz from (9) in Table III, respectively. For good impedance matching, the dimensions of the related feed line were chosen

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to be , , , , and , as discussed in Section III-B. The distance of is less than , as shown in Fig. 14, because the magnetic coupling intensity is different as to the position of the radiator along the feed line. That is, the magnetic fields of the end of the feed line are weak because the feed line is open-ended. Therefore, the distance is smaller than the distance so as to obtain the required magnetic coupling. reflection coeffiFig. 15 shows a photograph and the cient values of a dual-band MZR antenna. The simulated and measured reflection coefficient values show good agreement. The first and second MZR frequencies (and fractional bandwidths) are measured as frequencies of 6.2 GHz and 7 GHz (and 1.03% and 0.95%), respectively. The size of the anat the first MZR frequency and tenna is at the second MZR frequency. Fig. 16 shows the measured radiation patterns of the dualband MZR antenna at the first and second MZR frequency. The two radiation patterns at two MZR frequencies have similar broadside patterns. The measured maximum gains of the antenna are as 2.3 dBi and 3.3 dBi at 6.2 GHz and 7 GHz, and the differences in the maximum gains between cross- and co-polarization are 6.5 dB and 11.2 dB, respectively. The patterns at 6.2 GHz have significantly higher cross-polarization components compared to those at 7 GHz for both the - and the - planes because the resonator at 6.2 GHz is close to the SMA connector. The efficiencies are 83% and 84% at 6.2 GHz and 7 GHz, respectively. Consequently, the proposed dual-band MZR antenna can easily be expanded by placing another MZR resonator beside the feed. It is also expected that the radiation characteristics at each band are similar. V. CONCLUSIONS A novel antenna using the mu-zero resonance of an artificial MNG metamaterial TL is presented. The equivalent circuit for verifying the peculiarity of the MNG TL is derived and analyzed. The feed lines for impedance matching and magnetic coupling are designed by circuit and full-wave simulation. The analysis and design of the MZR antenna were accomplished by theory and simulation based on a dispersion diagram and field distribution. The analyzed surface current has the characteristics of an in-phase loop current, indicating that the radiation mechanism of an MZR antenna is in essence identical to that of an electrical small-loop antenna. MZR antennas with one, two, and three unit cells were designed. Appling the novel concept of the MZR antenna, a dual-band MZR antenna using two MZR antennas with different MZR frequencies is presented. The measured characteristics show good agreement with the simulated values. The results also show that the dual-band MZR antenna can be expanded to a multiband antenna. Consequently, the proposed MZR antennas are suitable for antenna applications that require feasible gain, bandwidth, efficiency, and multiband characteristics. In particular, the size, gain, effibandwidth of the dual-band MZR antenna ciency, and are ( ), 2.3 dBi (3.3 dBi),

83% (84%), and 1.03% (0.95%), at the first (second) MZR frequency, respectively. REFERENCES [1] V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of " and ,” Soviet Phys. Uspekhi, vol. 10, no. 4, pp. 509–514, Jan. 1968. [2] R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science, vol. 292, pp. 77–79, Apr. 2001. [3] N. Engheta and R. W. Ziolkowski, Metamaterials: Physics and Engineering Explorations. Hoboken, NJ: Wiley, 2006. [4] C. Caloz and T. Itoh, Electromagnetic Metamaterials: Transmission Line Theory and Microwave Applications. Hoboken, NJ: Wiley, 2006. [5] G. V. Eleftheriades and K. G. Balmain, Negative-Refraction Metamaterials: Fundamental Principles and Applications. Hoboken, NJ: Wiley, 2005. [6] C. Caloz and T. Itoh, “Transmission line approach of left-handed (LH) materials and microstrip implementation of an artificial LH transmission line,” IEEE Trans. Antennas Propag., vol. 52, no. 5, pp. 1159–1166, May 2007. [7] M. Gil, J. Bonache, J. Selga, J. Garcia-Garcia, and F. Martin, “Broadband resonant-type metamaterial transmission lines,” IEEE Microw. Wireless Compon. Lett., vol. 17, pp. 97–99, Feb. 2007. [8] J. H. Park, Y. H. Ryu, J. G. Lee, and J. H. Lee, “A novel via-free composite right- and left-handed transmission line using defected ground structure,” IEEE Microw. Wireless Compon. Lett., vol. 49, no. 8, pp. 1989–1993, Aug. 2007. [9] G. V. Eleftheriades, “Analysis of bandwidth and loss in negative-refractive-index transmission-line (NRI-TL) media using coupled resonators,” IEEE Microw. Wireless Compon. Lett., vol. 17, pp. 412–414, Jun. 2007. [10] M. A. Antoniades and G. V. Eleftheriades, “A broadband series power divider using zero-degree metamaterial phase-shifting lines,” IEEE Microw. Wireless Compon. Lett., vol. 15, pp. 808–810, Nov. 2005. [11] A. Sanada, C. Caloz, and T. Itoh, “Novel zeroth-order resonance in composite right/left- handed transmission line resonators,” in Proc. Asia-Pacific Microwave Conf., 2003, vol. 3, pp. 1588–1591. [12] A. Sanada, M. Kimura, I. Awai, C. Caloz, and T. Itoh, “A planar zerothorder resonator antenna using a left-handed transmission line,” in Proc. Eur. Microw. Conf., 2004, vol. 3, pp. 1341–1344. [13] J. G. Lee and J. H. Lee, “Zeroth order resonance loop antenna,” IEEE Trans. Antennas Propag., vol. 55, no. 3, pp. 994–997, Mar. 2007. [14] A. Rennings, T. Liebig, S. Abielmona, C. Caloz, and P. Waldow, “Tri-band and dual-polarized antenna based on composite right/lefthanded transmission line,” in Proc. Eur. Microw. Conf., Oct. 2007, pp. 720–723. [15] A. Rennings, T. Liebig, C. Caloz, and P. Waldow, “MIM CRLH series mode zeroth order resonant antenna (ZORA) implemented in LTCC technology,” in Proc. Asia-Pacific Microw. Conf., Dec. 2007, pp. 1–4. [16] M. A. Antoniades and G. V. Eleftheriades, “A folded-monopole model for electrically small NRI-TL metamaterial antennas,” IEEE Antennas Wireless Propag. Lett., vol. 7, pp. 425–428, 2008. [17] R. W. Ziolkowski, “An efficient, electrically small antenna designed for VHF and UHF applications,” IEEE Antennas Wireless Propag. Lett., vol. 7, pp. 217–220, 2008. [18] A. Lai, K. M. K. H. Leong, and T. Itoh, “Infinite wavelength resonant antennas with monopolar radiation pattern based on periodic structures,” IEEE Trans. Antennas Propag., vol. 55, no. 3, pp. 868–876, Mar. 2007. [19] J. H. Park, Y. H. Ryu, J. G. Lee, and J. H. Lee, “Epsilon negative zerothorder resonator antenna,” IEEE Trans. Antennas Propag., vol. 55, no. 12, pp. 3710–3712, Dec. 2007. [20] J. J. Carr, Practical Antenna Handbook. : McGraw-Hill, Press, 2001. [21] L. J. Chu, “Physical limitations on omni-directional antennas,” J. Appl. Phys., vol. 19, pp. 1163–1175, Dec. 1948. [22] D. M. Pozar and B. Kaufaman, “Comparison of three methods for the measurement of printed antenna efficiency,” IEEE Trans. Antennas Propag., vol. 36, pp. 136–139, Jan. 1988. [23] H. Choo, R. Rogers, and H. Ling, “On the Wheeler cap measurement of the efficiency of microstrip antennas,” IEEE Trans. Antennas Propag., vol. 53, no. 7, pp. 2328–2332, Jul. 2005.

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Jae-Hyun Park received the B.S. degree in electronic and electrical engineering and the M.S. degree in electronic information and communication engineering from Hongik University, Seoul, Korea, in 2005 and 2008, respectively, where he is currently working toward the Ph.D. degree. His research interests include metamaterial RF devices and wireless power transfer systems.

Young-Ho Ryu received the B.S. degree in electronic and electrical engineering from Jeju National University, Seoul, Korea, in 2003 and the M.S. and Ph.D. degrees in electrical engineering and computer science from Kyungpook National University, Daegu, Korea, in 2005 and 2009, respectively. He is currently an R&D staff member at Samsung Advanced Institute of Technology. His research interests include metamaterial RF devices and wireless power transfer systems.

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Jeong-Hae Lee (M’XX) received the B.S. and M.S. degrees in electrical engineering from Seoul National University, Korea, in 1985 and 1988, respectively, and the Ph.D. degree in electrical engineering from the University of California, Los Angeles, in 1996. From 1993 to 1996, he was a Visiting Scientist of General Atomics, San Diego, CA, where his major research was to develop the millimeter wave diagnostic system and to study the plasma wave propagation. Since 1996, he has been at Hongik University, Seoul, Korea, where he is a Professor of Department of Electronic and Electrical Engineering. His current research interests include the microwave/millimeter wave circuits, the millimeter wave diagnostic, and the metamaterial RF devices.

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The Broadband Spiral Antenna Design Based on Hybrid Backed-Cavity Chunheng Liu, Yueguang Lu, Chunlei Du, Jingbo Cui, and Ximing Shen

Abstract—Here, a hybrid backed-cavity with electromagnetic band-gap (EBG) structure and a perfect electronic conductor (PEC) is proposed for an Archimedean spiral antenna. This cavity makes the spiral antenna work operate over the 10:1 bandwidth and without the loss introduced by absorbing materials that are used conventionally to broadband spirals. Based on the artificial magnetic conductor (AMC) characteristic, an EBG is placed in the outer region of backed-cavity to improve the blind spot gain at low frequency. A PEC at the center of the structure is used to obtain high gain at high frequency. The antenna performances are improved significantly for of the low profile spiral antenna is improved significantly. A typical spiral antenna with a hybrid backed cavity is fabricated and studied experimentally. The experimental data are consistent with that of the simulation results. Index Terms—Artificial magnetic conductor (AMC), broadband, electromagnetic band-gap (EBG), hybrid cavity, spiral antenna. Fig. 1. The model of Archimedean spiral antenna.

I. INTRODUCTION

I

N conventional spiral antenna design, a metallic backedcavity placed at quarter wavelength position is commonly used to obtain unidirectional beam. The fixed physical length leads to destructive interferences between direct and reflected waves from metallic backed-cavity in some frequency region. Therefore, the broadband characteristic of spiral antenna is seriously limited. Recently, AMC characteristic of EBG [1]–[3] structure has been receiving more attention. When a normally incident plane wave illuminates EBG structure, the plane wave can be reflected in-phase in some frequency range. Therefore, EBG structure can be used as ground planes to design low profile antenna [4], [5]. In [5], a low profile spiral antenna based on EBG ground plane was presented, whose bandwidth was about 2.5:1 (from 7

Manuscript received February 02, 2009; revised May 13, 2009; accepted October 01, 2009. Date of publication January 22, 2010; date of current version June 03, 2010. C. Liu and Y. Lu are with Northern Institute of Electronic Equipment of China, Beijing 100083, China, and also with the State Key Lab of Optical Technologies for Microfabrication, Institute of Optics and Electronics, Chinese Academy of Sciences, Chengdu, Sichuan 610209, China (e-mail: [email protected]; [email protected]). C. Du is with State Key Lab of Optical Technologies for Microfabrication, Institute of Optics and Electronics, Chinese Academy of Sciences, Chengdu, Sichuan 610209, China (e-mail: [email protected]). J. Cui and X. Shen are with the Jiangnan Electronic Corporation of China, Jiaxing 372400, 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.2010.2041147

GHz to 17 GHz). However, the AMC characteristic of EBG always exhibits a multiband behavior and with narrow bandwidth, making them unsuitable for broadband antenna design [6]–[8] In order to improve the gain of UHF broadband spiral antenna, a reflective surface consisting of ferrite and conductive coating was introduced in [9]. Inspired by [9], we put forward a hybrid backed-cavity with EBG structure and PEC in the paper. Based on AMC characteristic, the EBG structure is placed in the outer region of backed-cavity to improve blind spot gain at low frequency. A PEC at the center of the cavity is used to keep the gain of the antenna unaffected at high frequency. The spiral antenna bandwidth with the hybrid backed cavity can be as high as 10:1 and without the loss introduced by absorbing materials. The effects of the key parameters such as hybrid cavity height and radius of PEC on the spiral antenna performance with hybrid backed cavity are studied. Some design principles are obtained for spiral antenna with hybrid backed-cavity. A typical spiral antenna with hybrid backed cavity is fabricated and is studied in anechoic chamber. There are good agreements between the measured and simulated results. Our study shows that the novel hybrid backed cavity broadens the operation bandwidth and, at the same time, improves the gain of spiral antenna. II. SPIRAL ANTENNA DESIGN WITH HYBRID BACKED-CAVITY In order to demonstrate our design on the spiral antenna with hybrid backed-cavity, an Archimedean spiral antenna with frequency range of 1–10 GHz is introduced for reference in the , the section. The width of spiral antenna arms , and the permittivity of substrate substrate thickness

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Fig. 4. Reflecting phase of UCPBG. Fig. 2. Gains of spiral antenna for reference.

Fig. 5. Gains of hybrid cavity and PEC cavity. Fig. 3. The structure of hybrid backed-cavity.

is 4.4 (FR4_epoxy substrate materials). The height of metallic backed cavity is 12 mm. The inner and outer radius of spiral and , rearms are designed to be spectively. The Archimedean spiral antenna is shown in Fig. 1, which is designated as our reference for the following discussion. The model of the spiral antenna is numerically simulated with Ansoft HFSS. The antenna gains are presented in the Fig. 2. In Fig. 2, the antenna gain appears to fluctuate from 4 GHz to 10 GHz due to destructive interferences between direct wave and reflected wave from metallic backed-cavity. The minimum gain of the antenna is at 7.625 GHz, which is the typical blind spot which seriously limits the operation bandwidth of spiral antenna. In order to lessen destructive interfering effect at the blind spot, a low profile hybrid backed-cavity, combining EBG structure in the outer region and PEC in the center region, is proposed. The structure of hybrid backed-cavity is shown in Fig. 3.

As was indicated in [4], [7], the reflecting phases of high impedance surface (HIS) and un-coplar photonic bandgap (UCPBG) are similar, because the metal patterns of the surface of HIS and UCPBG are the same seen from normal direction above. For the sake of easy fabrication and obtaining much broader in-phase bandwidth, the UCPBG is used as hybrid back-cavity. It is well known that the low frequency wave of spiral antenna is mainly reflected in the outer region of backed-cavity. In order to get constructive interferences at low frequency, an UCPBG is placed at the outer region of hybrid backed cavity, because UCPBG can reflect incident wave with in-phase,. Therefore, the multi-band behavior and narrow bandwidth limitation of EBG structure can be avoided. The PEC in the center region is used to keep the antenna gain unaffected at high frequency. The height h of hybrid backed-cavity is determined by the wavelength at high frequency. Therefore, the low profile spiral antenna with hybrid backed cavity can be successfully obtained. In our numerical simulations, the frequency of zero-phase reflection of UCPBG is adjusted to 5 GHz, the period of EBG

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Fig. 6. VSWR curve of hybrid cavity spiral antenna.

Fig. 7. Hybrid backed cavity height parameters analysis.

structure and width of patch . The reflecting phase is shown in the Fig. 4. and the height of hybrid cavity The radius of PEC are 30 mm and 7 mm, respectively. Other parameters of hybrid backed cavity take the same value as those of the reference spiral antenna. The antenna gain of hybrid backed-cavity and VSWR curve are shown in the Figs. 5 and 6. Fig. 5 shows that the antenna gains of the hybrid backed-cavity are greatly improved in the high frequency range, due to the use of a low profile of hybrid backed-cavity. However, the gain of hybrid backed cavity is lower than that of PEC backed-cavity at low frequency. It is apparent that the hybrid backed cavity provides a good compromise between low frequency and high frequency. Therefore, the hybrid backed cavity is more suitable for the broadband spiral antenna than PEC backed cavity. The VSWR curve of hybrid backed-cavity is indicated in the Fig. 6, which is less than 2 in the operational frequency band.

Fig. 8. Hybrid backed cavity radius parameters analysis.

Fig. 9. The photography of the manufactured antenna.

III. STUDIES ON THE PARAMETERS OF THE SPRIAL ANTENNA WITH HYBRID BACKED-CAVITY In the section, the effects of hybrid cavity height and radius of hybrid backed cavity on the spiral antenna gain of PEC are numerically analyzed. A. Hybrid Cavity Height The height of hybrid cavity plays an important role on antenna gain. In our analysis, the height of hybrid backed cavity takes 6 mm, 8 mm and 10 mm respectively. Other parameters of hybrid backed cavity hold the same value as those of the reference spiral antenna. Fig. 7 shows the antenna gains with different height of hybrid backed cavity. Fig. 7 shows that the antenna gains in low frequency band increase, as hybrid backed cavity increase. However, the antenna gains greatly decrease at high frequency due to the destructive interferences. When hybrid backed cavity takes 6 mm and 10 mm, the antenna gains appear to fluctuate, the blind

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Fig. 10. Measured VSWR for manufactured spiral antenna.

Fig. 13. The measured radiation pattern at 4 GHz.

Fig. 11. The comparison between measured gain and simulated gain.

Fig. 14. The measured radiation pattern at 6.25 GHz.

Fig. 12. The measured radiation pattern at 1.5 GHz.

spot of antenna gain emerges at 3.75 GHz and 7.5 GHz respectively. When h is 8 mm, the spiral antenna with hybrid backed cavity presents a compromising gain between the low frequency and the high frequency. Therefore, good performance of spiral antenna based on hybrid backed cavity can be achieved by optimizing the hybrid cavity height. B. Radius of PEC In order to demonstrate the effect of the PEC radius on the antenna gain, the radius of PEC of hybrid backed cavity takes 20 mm, 25 mm, 30 mm and 35 mm respectively. Other parameters of hybrid backed cavity take the same value as those of the reference spiral antenna. The Fig. 8 shows the antenna gain with different radius of PEC.

From Fig. 8, we can see that when the radius of PEC takes 20 mm and 25 mm, the gain of spiral antenna begin to fluctuate in the low frequency band. The gain blind spots emerge in the low frequency band at the 3 GHz and 5 GHz. However, the radius of PEC has a little effect on gain of spiral antenna in the high frequency band, which mainly depends on the height of hybrid backed-cavity. Therefore, by optimizing radius of PEC in the center region, we can improve the antenna gain in the low frequency band. IV. ANTENNA MANUFACTURE AND MEASUREMENT The wideband balun necessary for the balanced feeding of the spiral antenna is commonly required in backed-cavity. The broadband balun transforms the unbalanced coaxial mode into balanced two-wire transmission line mode that feeds the spiral antenna. Additionally, the balun provides impedance transformation (through a tapered paired-strip line) from the impedance of the coaxial line to the impedance of the 50 spiral antenna [10]. The spiral antenna of hybrid backed cavity with UCPBG and PEC is manufactured, whose photography is shown in Fig. 9.

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Fig. 15. The measured radiation pattern at 7.625 GHz.

Fig. 17. The measured axis ratio curve.

Fig. 16. The measured radiation pattern at 9.375 GHz.

The VSWR of spiral antenna with hybrid backed cavity is measured by network vector analyzer, which is shown in the Fig. 10. Fig. 10 shows that the measured VSWR is almost less than 2 from 1 GHz to 10 GHz. The gain of spiral antenna with hybrid backed cavity was measured in the anechoic chamber using standard gain double ridged horn antenna (model number: 3115, serial number: 5471). The measured antenna gain and simulation results are presented in the Fig. 11, which indicates that the measured gains are well consistent with those of simulation model. The measured normalized radiation patterns of typical frequencies at the 1.5 GHz, 4 GHz, 6.25 GHz, 7.625 GHz and 9.375 GHz are shown from the Figs. 12–16. Figs. 12–16 shows that all of the radiation patterns have normal shape. Additionally, the measured radiation patterns exhibit a little offset due to the feed line connecting to SMA. Based on Figs. 12–16, it can be concluded that the radiation pattern doesn’t deteriorate in the whole bandwidth from 1 GHz to 10 GHz, although the hybrid backed cavity has a low profile.

Fig. 18. The measured axis ratio at 1 GHz.

The axis ratios of the manufactured antenna are also measured in the anechoic chamber. The axis ratios at the normal direction are shown in the Fig. 17. The axis ratio pattern at the frequency 1 GHz, 5 GHz and 10 GHz are shown from Figs. 18–20. Fig. 17 shows that the axis ratio of hybrid backed cavity is less than 3 dB from 2 GHz to 10 GHz. For our manufactured spiral antenna, the outer radius of spiral arms is (wavelength at 1 GHz); therefore, the axis ratio at 1 GHz is more than 3 dB. If the outer radius of spiral arms increase, the axis ratio at 1 GHz will decrease accordingly. From Figs. 18–20, we can see that the axis ratios in low, middle and high frequency band are all having normal axis ration radiation pattern. V. CONCLUSION A novel hybrid backed cavity with the EBG structure in the outer region and PEC in the center region is put forward in the

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REFERENCES

Fig. 19. The measured axis ratio at 5 GHz.

[1] J. D. Jonannopoulos, R. D. Meade, and J. N. Winn, Photonic crystals. Princeton, NJ: Princeton Univ. Press, 1995. [2] R. Coccioli, F. R. Yang, K. ping Ma, and T. Itoh, “Aperture-Coupled patch antenna on UC-PBG substrate,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 11, pp. 2123–2130, Nov. 1999. [3] D. Sievenpiper, L. J. Zhang, R. F. Broas, N. G. Alexopoulos, and E. Yablonovitch, “High-impedance electromagnetic surfaces with a forbidden frequency band,” IEEE trans. Microw. Theory Tech., vol. 47, no. 11, pp. 2059–2074, Nov. 1999. [4] F. Yang and Y. Rahmat-Samii, “Reflection phase characterizations of the EBG ground plane for low profile wire antenna applications,” IEEE Trans. Antennas Propag., vol. 51, no. 10, pp. 2691–2703, Oct. 2003. [5] J. M. Bell and M. F. Iskander, “A low profile Archimedean spiral antenna using an EBG ground plane,” IEEE Antennas Wireless Propag. Lett., vol. 3, pp. 223–226, 2004. [6] L. Li, B. Li, H. X. Liu, and C. H. Liang, “Locally resonant cavity unit model for electromagnetic band gap structures,” IEEE Trans. Antennas Propag., vol. 54, no. 1, pp. 90–100, Jan. 2006. [7] G. Goussetis, A. P. Feresidis, and J. C. Vardaxoglou, “Tailoring the AMC and EBG characteristic of periodic metallic arrays printed on grounded dielectric substrate,” IEEE Trans. Antennas Propag., vol. 54, no. 1, pp. 82–89, Jan. 2006. [8] D. J. Kern et al., “The design synthesis of multiband artificial magnetic conductors using high impedance frequency selective surface,” IEEE Trans. Antennas Propag., vol. 53, no. 1, pp. 8–17, Oct. 2005. [9] B. A. Kramer, S. Koulouridis, C. C. Chen, and J. L. Volakis, “A novel reflective surface for UHF spiral antenna,” IEEE Antennas and Wireless Propag. Lett., vol. 5, pp. 32–34, 2006. [10] C. A. Balanis, Antenna Theory Analysis and Design. New York: Wiley, 1997.

Chunheng Liu received the M.S. degree from Tsinghua University, China, in 2002. He is currently working toward the Ph.D. degree at the State Key Lab of Optical Technologies for Microfabrication, Institute of Optics and Electronics, Chinese Academy of Sciences. Currently, he is an Engineer at the Northern Institute of Electronic Equipment of China. His research interests include computational electromagnetic, patch antenna and novel electromagnetic material (PBG, EBG).

Fig. 20. The measured axis ratio at 10 GHz.

paper. The spiral antenna with hybrid backed cavity is manufactured and measured. The measured results show that the novel hybrid backed cavity can effectively improve spiral antenna gain and broaden operating frequency band. The effects of important parameters of hybrid backed cavity are studied in detail. Some design principles have been obtained for spiral antenna design with hybrid backed-cavity. The hybrid backed-cavity opens new possibilities for the broadband spiral antenna applications.

ACKNOWLEDGMENT The authors thank reviewers for their careful and kindly comments.

Yueguang Lu received the Ph.D. degree in applied optics from Harbin Institute of Technology, China, in 1990. He is now a Professor at the Institute of Optics and Electronics, Chinese Academy of Science, and a Senior Researcher at the Northern Institute of Electronic Equipment of China. He was a visiting scholar at the Optical Science Laboratory, University College London, from October 2001 to August 2002. His primary research interests include photonic crystal, optical testing and information processing. Dr. Lu is a member of SPIE and OSA.

Chunlei Du is currently a Professor of micro-optics in the State Key Lab of Optical Technologies for Microfabrication, Chinese Academy of Sciences. Her research interests include methods, fabrication, and applications for nano/micro-optics and subwavelength optics.

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Jingbo Cui is an engineer with the Jiangnan Electronic Corporation of China, Jiaxing. His research interests is antenna design.

Ximing Shen is a Senior Researcher at the Jiangnan Electronic Corporation of China, Jiaxing. His research interests is antenna design.

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Resonant Frequency of a Rectangular Patch Sensor Covered With Multilayered Dielectric Structures Yang Li, Student Member, IEEE, and Nicola Bowler, Senior Member, IEEE

Abstract—This paper presents a simple analytical model for the resonant frequency of a rectangular patch sensor covered with multilayered dielectric structures. The accuracy of this model, while the relative permittivity of the dielectric top layers varies from 1 to 10.2, is verified with Ansoft’s high frequency structural simulator (HFSS) calculations and with experimental measurements. Comparison with HFSS simulation results show that the average error of the proposed model for three-layer test structures with various sets of relative permittivities and thicknesses is 2% or smaller. Merits of this model also include low calculation cost and mathematical simplicity. The robustness of this model for test structures with less than three layers is also validated by comparing with experimental measurement, HFSS simulation and well-established theory. This model can also be directly used to predict the resonant frequencies of rectangular patch antennas covered with multilayered dielectric superstrates. Index Terms—Microstrip antennas, nondestructive testing, permittivity, resonance, sensors.

I. INTRODUCTION

P

ATCH antennas are being used increasingly as nondestructive testing sensors in many applications, such as liquid/solid permittivity measurement, sensing of DNA hybridization, porosity measurement, and moisture content evaluation [1]–[5]. Recently, there is interest in applying this technique to permittivity measurement of multilayered dielectric structures [6], [7]. In this particular application, a patch sensor may be used to assess permittivities of particular layers in the multilayered dielectric structure, by measuring the shift of the sensor’s resonant frequency. Design and signal-inversion work for this sensor requires an accurate prediction of its resonant frequency. Many works have studied the effect of multilayered superstrates or/and substrates on the characteristics of patch antennas, including resonant frequency [8]–[25]. Bahl et al. [10] applied a variational technique in the Fourier domain to study the effective permittivity of a patch antenna covered with a dielectric layer. In the spectral domain, Losada et al. [11]–[13], Shavit [14],

Manuscript received September 14, 2009; revised December 01, 2009; accepted December 19, 2009. Date of publication March 29, 2010; date of current version June 03, 2010. Y. Li is with the Department of Electrical and Computer Engineering, and Center for Nondestructive Evaluation, Iowa State University, Ames, IA 50011 USA (e-mail: [email protected]). N. Bowler is with the Departments of Materials Science and Engineering, Electrical and Computer Engineering, and Center for Nondestructive Evaluation, Iowa State University, Ames, IA 50011 USA (e-mail: nbowler@iastate. edu). Digital Object Identifier 10.1109/TAP.2010.2046871

Wong et al. [15], as well as Fan and Lee [16] analyzed the resonance, radiation and input impedance characteristics of multilayered substrates and covered patch antennas. According to a special variant of the discrete-mode-matching method, Dreher and Ioffe [17], [18] analyzed patch antennas and feed networks embedded in multilayer structures of arbitrary shape. Using the method of moments (MoM), Pozar [19], Raffaelli et al. [20], He and Xu [21], as well as Makarov et al. [22] computed resonant characteristics of patch antennas. Further, Soliman et al. [23] applied a neural network model to the efficient filling of the coupling matrix in MoM to analyze patch antennas with arbitrary shape. Again, utilizing neural networks, Guney and Sarikaya [24] as well as Banerjee [25] analyzed resonant frequencies and design parameters of various patch antennas of regular geometries. These works achieved reasonably accurate results but, since they are based on rigorous mathematical procedures, none of them are appropriate for sensor design and inversion work due to their high computational and implementation cost. This paper presents accurate formulas to predict the resonant frequency of a rectangular patch sensor covered with multilayered dielectric structures, as shown in Fig. 1(a). Compared with the methods in [11]–[16], the present one has the advantages of low computational cost and mathematical simplicity. Additionally, these efficient formulas are capable of predicting accurate resonant frequencies that may be used in sensor design and inversion work for a wide range of sensor, substrate and top layer geometric and electrical parameters. This improvement is based on the work done by Wheeler [26], [27], Svacina [28], [29], Bernhard and Tousignant [30], as well as that of Guha and Siddiqui [31], [32]. As in these prior works, the conformal mapping approach, the concept of equivalent capacitance, and an improved cavity model are used in this work. The developments of this work include i) rectification of the filling fractions to eliminate the inconsistencies in previous work [29], ii) novel configuration of these filling fractions to represent the actual distribution of electric flux lines in the structure under study and iii) calibration of the patch length to account for surface wave losses due to the effects of the substrate and top layers. Since the application and theoretical background of this work both originate from the theory of rectangular patch antennas, this model can also be applied to predict resonant frequencies of rectangular patch antennas with multilayered dielectric superstrates directly. Section II introduces the equations for this model, including effective patch length, filling fractions, and effective permittivity of the resonant structure. The predicted resonant frequency with corresponding experimental measurements and comparative HFSS simulation results are presented in Section III for various multilayered structures.

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and

(3) In these expressions,

(4)

Fig. 1. (a) Top view and cross-sectional view of the rectangular patch sensor covered with a multilayered dielectric structure. (b) Conformally mapped equivalent parallel plate structure with region of note enclosed by the dashed line.

is given by (15). and the effective line width Bernhard and Tousignant [30] indicated that Svacina [28] ignored the case in which the superstrate of a three-layer structure does not exist. Similarly, the configuration in which each of the ( , 3, 4) shown in Fig. 1(a) does not top layers with , then ( , 3, exist, was also ignored in [29]. If 4) should equal zero when the -th top layer does not exist. However, derived by (2) does not become zero for this situation. It illustrates that the filling fractions of these top layers from [29] are overestimated in all cases. To rectify this inconsistency, the following calibrated filling fractions are introduced by setting as on the right-hand side of (2)

II. FORMULATION A. Effective Filling Fractions The application of conformal mapping to microstrip analysis was introduced by Wheeler [26], [27]. Based on Wheeler’s transformations, Svacina analyzed a three-layer microstrip [28] and derived the effective permittivity expressions for a generalized multilayered microstrip [29]. Zhong et al. [8], Bernhard and Tousignant [30], as well as Guha and Siddiqui [32] developed Svacina’s method to predict the resonant frequencies of patch antennas with multilayered superstrates or/and substrates. Their results achieved similar accuracy with the extra benefits of mathematical simplicity and low computational cost over many of the full wave analyses described in Section I. Using Svacina’s method for a generalized multilayered microstrip [29], the five-layer structure shown in Fig. 1(a) is conformally mapped onto a complex -plane with results as shown in Fig. 1(b). The filling fraction is defined as the ratio of each dielectric area to the entire area of the cross section in the -plane [27]. According to the equations for the filling fractions of this wide strip-line structure [29], the filling fractions for each of these five layers are (1)

(2)

(5) Using these calibrated filling fractions, the filling fractions in (1) and (2) are rectified as follows (6)

(7) The above new equations for filling fractions better represent the distribution of dielectric materials in Fig. 1(b). Firstly, becomes zero when the -th top layer does not exist, i.e., when . Secondly, accurately reflects the distribution of different dielectric materials in Fig. 1(b). Considering the area within which dielectric materials with relative permittivity are in series combination [within the dotted line in Fig. 1(b)], the calibrated filling fractions ( , 3, 4) are used to approximate it in Fig. 3 (to be discussed further in Section II-B). After this approximation, the proportion among the blocks with relative permittivities ( , 3, 4) is still in good consistency with that in Fig. 1(b). The rectified set of filling fractions (6), (7), and (3) are next configured between an equivalent parallel-plate capacitor structure, Fig. 3, to derive the effective permittivity of the microstrip structure, Fig. 1(a).

LI AND BOWLER: RESONANT FREQUENCY OF A RECTANGULAR PATCH SENSOR

Fig. 2. Illustration of electric flux paths for the structure shown in Fig. 1(a).

B. Effective Permittivity In one of the classic papers on this topic, Bernhard and Tousignant [30] state that “the arrangement of the new filling fractions within the equivalent parallel-plate structure is intended to represent the paths of electric field flux in the actual structure as closely as possible.” As shown in Fig. 2, the electric field flux lines are capable of following five different paths in the structure under study. Path 1 reflects the flux within the substrate of this structure. Paths 2 to 4 represent the existence of flux lines in substrate and top layers with relative permittivities , ( ,3), and ( , 3, 4), respectively. Finally, the flux lines extend to the space above these top layers by path 5. Considering the effect of these flux line paths, the configuration of the new filling fractions derived in Section II-A is shown in Fig. 3. Based on this configuration, the effective permittivity of this approximate equivalent structure is here derived as

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Fig. 3. Filling fraction configuration for derivation of effective permittivity that considers the electric flux paths illustrated in Fig. 2. The capacitive effect of electric flux path 1 is represented by the block with (S ;  ). The capacitive effect of electric flux path 2 is represented by the block containing (S ;  ) and a part of S with  . The capacitive effect of electric flux path 3 is represented by the block containing (S ;  ) and parts of both S with  and  , respectively. The capacitive effect of electric flux path 4 is represented by the block containing (S ;  ), a part of S with  and parts of both S with  and  , respectively. The capacitive effect of electric flux path 5 is represented by the blocks containing (S ;  ), (S ;  ), a part of S with  and parts of both S with  and  , respectively.

Alexopoulos and Jackson [33] showed that the surface wave due to the surface wave mode will be diminished dramatically when the relative permittivity of the superstrate is larger than that of the substrate for some superstrate-substrate thickness combinations. On the other hand, fringing field decreases as substrate relative permittivity increases [34]. Based on these two facts, Guha and Siddiqui [32] accounted for the surface wave losses in terms of fringing field. They [32] proposed an equivalent relative permittivity of the substrate, as in (11) below, in order to calculate the effective patch radius of a circular patch antenna covered with a dielectric superstrate

(8) (11)

with

(9) and

In terms of the effect of fringing field, modification of the patch length of a rectangular patch is similar to that of patch radius of a circular patch [34]. Hence, a similar approach to that presented in [32] is used here to rectify the patch length of the structure under study, Fig. 1(a), to account for surface wave losses. Based on [32, (12)], the effective patch length can be expressed as (12)

(10) This effective permittivity expression can be used to calculate the resonant frequency of a rectangular patch sensor covered with multilayered dielectric structures, Fig. 1(a), as shown in Section II-D. C. Effective Patch Length It has been indicated that the model for the three-layer structure presented in [30] does not consider surface wave losses in the case of high relative permittivity substrates and superstrates.

where represents the static fringing field effects at the edge of the patch capacitor [31]. is evaluated from [31, (9)–(14)] and as in with in those equations expressed as (11), respectively. D. Resonant Frequency Considering the effect of fringing fields at all edges of the patch, the resonant frequency of the rectangular patch sensor shown in Fig. 1(a), is given as [35]: (13)

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TABLE I RELATIVE PERMITTIVITY  AT 10 GHz, GEOMETRIC DIMENSIONS AND SOURCES OF THE TEST MATERIALS

where is the speed of light in vacuum, is the effective is the effective open-line length of the patch given by (12), length extension and represents the effective permittivity is calculated using formulas presented in [35] given by (8). with given by (14) [30] used in place of of [35], and given by (15) [30] used in place of of [35].

TABLE II COMPARISON OF THEORETICAL PREDICTED RESONANT FREQUENCIES FOR A RECTANGULAR PATCH SENSOR COVERED WITH VARIOUS THREE-LAYER DIELECTRIC STRUCTURES WITH EXPERIMENTAL AND HFSS SIMULATION RESULTS. THESE THREE-LAYER DIELECTRIC STRUCTURES ARE WITH h ) AND  , FIG. 1(a) DIFFERENT (h

0

(14)

(15) and are determined by a two-step iteraThe quantities tion procedure described in [30], starting with an approximation and . III. EXPERIMENTAL TESTS AND RESULTS A. Sensor Coaxial-line-fed half-wave patch sensors were designed and , , fabricated with parameters and [6]. Due to the fabrication process, the of each prototype sensor is slightly difsubstrate thickness ferent. For this reason, for each individual sensor utilized in these experiments is given with Tables II–V. To verify the model in Section II, a series of resonant frequency measurements were performed using an Anritsu 37347C vector network analyzer. Measured resonance is defined at the minimum return loss. The properties and sources of the test materials used in these measurements are listed in Table I. B. Sensor Covered With Three-Layer Structures Since this sensor is designed to assess the permittivity change in Fig. 1(a)) of a dielectric sandwich in the core (layer with structure, such as an aircraft radome, by measuring the shift of sensor resonant frequency [6], the calculated resonant frequencies of the sensor covered with various three-layer dielecand are compared tric structures having different with corresponding measurements and HFSS simulation results

in Table II. Measured frequencies differ from HFSS simulation results by less than 1.42% on average and 1.93% at maximum. Compared with those differences, the present model shows good accuracy, as can be seen from the last two columns in Table II. on Table III shows the effect of core thickness sensor resonant frequencies for different skin combinations and in Fig. 1(a)). The results show that for (layers with and 9.20, and , the resonant frequencies calculated here provide good accuracy. C. Sensor Covered With Less Than Three Dielectric Layers Verification of the model’s robustness for predicting resonant frequency when the sensor is covered with less than three dielectric top layers is examined in Tables IV and V. There are two ways to apply the present model to a structure with dielectric top layers. First, set and . Second, let and set . When , also set , . Our data indicates that the first method yields better accuracy when the

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TABLE III COMPARISON OF THEORETICAL PREDICTED RESONANT FREQUENCIES FOR A RECTANGULAR PATCH SENSOR COVERED WITH VARIOUS THREE-LAYER DIELECTRIC h ) FOR STRUCTURES WITH EXPERIMENTAL AND HFSS SIMULATION RESULTS. THESE THREE-LAYER DIELECTRIC STRUCTURES ARE WITH DIFFERENT (h h ),  , (h h ) AND  COMBINATIONS, FIG. 1(a) THREE SETS OF (h

0

0

0

TABLE IV COMPARISON OF THEORETICAL PREDICTED RESONANT FREQUENCIES FOR A RECTANGULAR PATCH SENSOR COVERED WITH VARIOUS TWO-LAYER DIELECTRIC h ),  , STRUCTURES WITH EXPERIMENTAL AND HFSS SIMULATION RESULTS. THESE TWO-LAYER DIELECTRIC STRUCTURES ARE WITH DIFFERENT (h h ) AND  COMBINATIONS (h

0

0

test layers have high permittivities, otherwise, the second approach gives a better accuracy. For the sensors tested here, covered with two dielectric top layers, the first method is more accu( calculated by the first method). rate while When the sensor is covered with one or no dielectric top layer, better accuracy is provided by the first method while ( computed by the first method). The first and second methods have been implemented and results shown in Tables IV and V, respectively. For a sensor with two dielectric top layers, Table IV compares the resonant frequencies calculated by the present model with corresponding HFSS simulation and measurement results. Generally, the calculated results are in good agreement with those of HFSS simulation and measurement. Since the first method was used, the third set of results (with ) shows the least errors. The computed resonant frequencies of the sensor with or without one dielectric top layer are compared with results of HFSS simulation, measurements and results of previous theory [30] in Table V. Due to implementation of the second way of dielectric applying the present model to the structure with

top layers, the errors in results of the present method given in . The method of Table V are proportional to the value of [30] shows the same trend in error. However, compared with method of [30], the present method yields significantly better since surface wave losses due to the accuracy when high relative permittivity of the top layers are considered, as explained in Section II-C. IV. CONCLUSION In this paper, a simple model for the resonant frequency of a rectangular patch sensor covered with multilayered dielectric structures is presented. The improvements of this work over prior works [29], [30] include i) rectification of the filling fractions of a conformally-mapped equivalent parallel plate structure to eliminate the inconsistencies in previous work, ii) novel configuration of the filling fractions to closely represent the paths followed by electric flux lines in the multilayered structure and iii) calibration of patch length to account for surface wave losses due to the effects of substrate and top layers. Compared with several previous works, this model has advantages of

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TABLE V COMPARISON OF THEORETICAL PREDICTED RESONANT FREQUENCIES, EXPERIMENTAL RESONANT FREQUENCIES, AND HFSS SIMULATION RESONANT FREQUENCIES FOR A RECTANGULAR PATCH SENSOR WITH AND WITHOUT VARIOUS ONE-DIELECTRIC TOP LAYERS. THESE DIELECTRIC TOP LAYERS ARE h AND  WITH DIFFERENT h

0

mathematical simplicity and low computational cost. The accuracy of this model while the relative permittivity of the dielectric top layers varies from 1 to 10.2 is verified by comparing with HFSS simulation and measurement results. Comparison with HFSS simulation results show that the average error of this model for a three-layer test structure with various relative permittivities and thicknesses is 2% or smaller. Comparing with measurement, HFSS simulation and well established theory, the application of this model for a sensor testing less than three top layers is also verified. This model can also be used to predict the resonant frequencies of rectangular patch antennas covered with multilayered dielectric superstrates directly. Future developments of the present model might include the following aspects. First, the model could be generalized to predict the resonant frequency of a rectangular patch sensor covered with more than four dielectric top layers. Second, considering that the accuracy was improved significantly by accounting for surface wave modes in the present model, the accuracy might be further improved by the consideration of leaky modes [36]. Leaky modes may be accounted for in a manner similar to the treatment of surface wave modes given here. ACKNOWLEDGMENT This material is based upon work supported by the Air Force Research Laboratory under Contract FA8650-04-C-5228 at Iowa State University’s Center for NDE. The authors thank American Standard Circuits for their accurate fabrication of the half-wave patch sensors used in the experiments. They also thank Rogers Corporation and Evonik Foams, Inc. for providing Microwave Laminates and Rohacell foam, respectively. REFERENCES [1] M. Bogosanovich, “Microstrip patch sensor for measurement of the permittivity of homogeneous dielectric materials,” IEEE Trans. Instrum. Meas., vol. 49, pp. 1144–1148, Oct. 2000. [2] C. Wichaidit, J. R. Peck, L. Zhang, R. J. Hamers, S. C. Hagness, and D. W. Van Der Weide, “Resonant slot antennas as transducers of DNA hybridization: A computational feasibility study,” in IEEE MTT-S Int. Microw. Symp. Dig., 2001, vol. 1–3, pp. 163–166.

[3] A. K. Verma, Nasimuddin, and A. S. Omar, “Microstrip resonator sensors for determination of complex permittivity of materials in sheet, liquid and paste forms,” in Proc. Inst. Elect. Eng., 2005, vol. 152, pp. 47–54. [4] A. Zucchelli, M. Chimenti, and E. Bozzi, “Application of a coaxial-fed patch to microwave nondestructive porosity measurements in low-loss dielectrics,” Progr. Electromagn. Res., vol. 5, pp. 1–14, 2008. [5] A. Cataldo, G. Monti, E. De Benedetto, G. Cannazza, and L. Tarricone, “A noninvasive resonance-based method for moisture content evaluation through microstrip antennas,” IEEE Trans. Instrum. Meas., vol. 58, pp. 1420–1426, May 2009. [6] Y. Li and N. Bowler, “Design of patch sensors for microwave nondestructive evaluation of aircraft radomes,” in Electromagnetic Nondestructive Evaluation XIV, J. S. Knopp, M. P. Blodgett, R. A. Wincheski, and N. Bowler, Eds. Amsterdam: IOS Press, 2010. [7] S. F. Barot and J. T. Bernhard, “Permittivity measurement of layered media using a microstrip test bed,” in Proc. IEEE Int. Symp. Antennas Propag., San Diego, CA, Jul. 2008, vol. 1–9, pp. 3507–3510. [8] S. Zhong, G. Liu, and G. Qasim, “Closed form expressions for resonant frequency of rectangular patch antennas with multidielectric layers,” IEEE Trans. Antennas Propag., vol. 42, pp. 1360–1363, Sep. 1994. [9] T. N. Kaifas and J. N. Sahalos, “Analysis of printed antennas mounted on a coated circular cylinder of arbitrary size,” IEEE Trans. Antennas Propag., vol. 54, pp. 2797–2807, Oct. 2006. [10] I. J. Bahl, P. Bhartia, and S. Stuchly, “Design of microstrip antennas covered with a dielectric layer,” IEEE Trans. Antennas Propag., vol. 30, pp. 314–318, Mar. 1982. [11] V. Losada, R. R. Boix, and M. Horno, “Resonant modes of circular microstrip patches in multilayered substrates,” IEEE Trans. Microw. Theory Tech., vol. 47, pp. 488–497, Apr. 1999. [12] V. Losada, R. R. Boix, and M. Horno, “Full-wave analysis of circular microstrip resonators in multilayered media containing uniaxial anisotropic dielectrics, magnetized ferrites, and chiral materials,” IEEE Trans Microw. Theory Tech., vol. 48, pp. 1057–1064, Jun. 2000. [13] V. Losada, R. R. Boix, and M. Horno, “Resonant modes of circular microstrip patches over ground planes with circular apertures in multilayered substrates containing anisotropic and ferrite materials,” IEEE Trans Microw. Theory Tech., vol. 48, pp. 1756–1762, Oct. 2000. [14] R. Shavit, “Dielectric cover effect on rectangular microstrip antenna array,” IEEE Trans. Antennas Propag., vol. 42, pp. 1180–1184, Aug. 1994. [15] K. Wong, S. Hsiao, and H. Chen, “Resonance and radiation of a superstrate-loaded spherical-circular microstrip patch antenna,” IEEE Trans. Antennas Propag., vol. 41, pp. 686–690, May 1993. [16] Z. Fan and K. F. Lee, “Input impedance of annular-ring microstrip antenna with a dielectric cover,” IEEE Trans. Antennas Propag., vol. 40, pp. 992–995, Aug. 1992. [17] A. Dreher and A. Ioffe, “Analysis of microstrip lines in multilayer structures of arbitrarily varying thickness,” IEEE Microw. Guided Wave Lett., vol. 10, pp. 52–54, Feb. 2000. [18] A. Ioffe, M. Thiel, and A. Dreher, “Analysis of microstrip patch antennas on arbitrarily shaped multilayers,” IEEE Trans. Antennas Propag., vol. 51, pp. 1929–1935, Aug. 2003.

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[19] D. M. Pozar, “Input impedance and mutual coupling of rectangular microstrip antennas,” IEEE Trans. Antennas Propag., vol. 30, pp. 1191–1196, Nov. 1982. [20] S. Raffaelli, Z. Sipus, and P. Kildal, “Analysis and measurements of conformal patch array antennas on multilayer circular cylinder,” IEEE Trans. Antennas Propag., vol. 53, pp. 1105–1113, Mar. 2005. [21] M. He and X. Xu, “Closed-form solutions for analysis of cylindrically conformal microstrip antennas with arbitrary radii,” IEEE Trans. Antennas Propag., vol. 53, pp. 518–525, Jan. 2005. [22] S. N. Makarov, S. D. Kulkarni, A. G. Marut, and L. C. Kempel, “Method of moments solution for a printed patch/slot antenna on a thin finite dielectric substrate using the volume integral equation,” IEEE Trans. Antennas Propag., vol. 54, pp. 1174–1184, Apr. 2006. [23] E. A. Soliman, M. H. Bakr, and N. K. Nikolova, “Neural networksmethod of moments (NN-MoM) for the efficient filling of the coupling matrix,” IEEE Trans. Antennas Propag., vol. 52, pp. 1521–1529, Jun. 2004. [24] K. Guney and N. Sarikaya, “A hybrid method based on combining artificial neural network and fuzzy inference system for simultaneous computation of resonant frequencies of rectangular, circular, and triangular microstrip antennas,” IEEE Trans. Antennas Propag., vol. 55, pp. 659–668, Mar. 2007. [25] B. Banerjee, “A self-organizing auto-associative network for the generalized physical design of microstrip patches,” IEEE Trans. Antennas Propag., vol. 51, pp. 1301–1306, Jun. 2003. [26] H. Wheeler, “Transmission-line properties of parallel wide strips by a conformal mapping approximation,” IEEE Trans. Microw. Theory Tech., vol. 12, pp. 280–289, Mar. 1964. [27] H. A. Wheeler, “Transmission-line properties of parallel strips separated by a dielectric sheet,” IEEE Trans. Microw. Theory Tech., vol. 13, pp. 172–185, Mar. 1965. [28] J. Svacina, “Analysis of multilayer microstrip lines by a conformal mapping method,” IEEE Trans. Microwave Theory Tech., vol. 40, pp. 769–772, Apr. 1992. [29] J. Svacina, “A simple quasi-static determination of basic parameters of multilayer microstrip and coplanar waveguide,” IEEE Microw. Guided Wave Lett., vol. 2, pp. 385–387, Oct. 1992. [30] J. T. Bernhard and C. J. Tousignant, “Resonant frequencies of rectangular microstrip antennas with flush and spaced dielectric superstrates,” IEEE Trans. Antennas Propag., vol. 47, pp. 302–308, Feb. 1999. [31] D. Guha, “Resonant frequency of circular microstrip antennas with and without air gaps,” IEEE Trans. Antennas Propag., vol. 49, pp. 55–59, Jan. 2001. [32] D. Guha and J. Y. Siddiqui, “Resonant frequency of circular microstrip antenna covered with dielectric superstrate,” IEEE Trans. Antennas Propag., vol. 51, pp. 1649–1652, Jul. 2003.

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[33] N. G. Alexopoulos and D. R. Jackson, “Fundamental superstrate (cover) effects on printed circuit antennas,” IEEE Trans. Antennas Propag., vol. 32, pp. 807–816, Aug. 1984. [34] R. Garg, P. Bhartia, I. Bahl, and A. Ittipiboon, Microstrip Antenna Design Handbook. Canton, MA: Artech House, 2001. [35] M. Kirschning, R. H. Jansen, and N. H. L. Koster, “Accurate model for open end effect of microstrip lines,” Electron. Lett., vol. 17, no. 3, pp. 123–125, Feb. 1981. [36] F. Mesa, D. R. Jackson, and M. J. Freire, “Evolution of leaky modes on printed-circuit lines,” IEEE Trans Microw. Theory Tech., vol. 50, pp. 94–104, Jan. 2002.

Yang Li (S’09) was born in Wuhan, China, on January 5, 1984. He received the B.S. degree in telecommunication engineering in 2006 and the M.S. degree in electromagnetic field and microwave technology in 2008 from Huazhong University of Science and Technology (HUST), Wuhan, China. He is currently working toward the Ph.D. degree at Iowa State University (ISU), Ames. He is also a Research Assistant with ISU’s Center for Nondestructive Evaluation (CNDE). His research interests include microwave nondestructive evaluation for composite structures/materials, theoretical electromagnetics, and remote sensing.

Nicola Bowler (M’99–SM’02) was born on December 6, 1968 in Hereford, U.K. She received the B.Sc. degree in physics from the University of Nottingham, U.K., in 1990 and the Ph.D. degree from the University of Surrey, surrey, U.K., in 1994, for theoretical work in the field of eddy-current nondestructive evaluation (NDE). She moved to the Center for NDE, Iowa State University, Ames, in 1999 where, in 2006, she was appointed Associate Professor of materials science and engineering. Her research interests include engineering the electromagnetic properties of composite materials by analysis and design, and electromagnetic NDE of dielectrics and metals—inventing new NDE techniques and improving accuracy in four-point potential drop, eddy-current, microwave and capacitive NDE.

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Study of an Ultrawideband Monopole Antenna With a Band-Notched Open-Looped Resonator Sung-Jung Wu, Cheng-Hung Kang, Keng-Hsien Chen, and Jenn-Hwan Tarng, Senior Member, IEEE

Abstract—A novel band-notched planar monopole ultrawideband (UWB) antenna is proposed. A notched band, located in the 5 GHz WLAN band, is created using a resonator at the center of a fork-shaped antenna. The resonator is composed of an open-looped resonator and two tapped lines. With the open-looped resonator, the antenna has a good band-notched performance and bandstop-filter-like response in the target band. A parametric study of the notched bandwidth is described that explored the antenna operating mechanism. Then, an equivalent circuit model illustrates the band-notched behaviors more clearly. The antenna input admittance calculated with the equivalent circuit model reasonably agrees with the HFSS simulated result. The proposed antenna also features flat gain frequency responses, small varied group delay and 15 to 35 dB gain suppression at the notched band. Accordingly, the band-notched antenna can effectively select target bands by adjusting these antenna parameters. Index Terms—Equivalent circuits, high quality factor, monopole antenna, open-looped resonator, ultrawideband (UWB) antenna.

I. INTRODUCTION

I

N 2002, the Federal Communication Commission (FCC) officially assigned an unlicensed 3.1–10.6 GHz bandwidth dBm/MHz effective isotropic radiated with less than power (EIRP) level for commercial applications of ultrawideband (UWB) systems. In this regulated signal condition, UWB technology is applied in extremely high transmission rates over a short distance, i.e., 480 Mbps data rate signal over 10 m transmission distance [1], [2]. Many studies have proposed an extreme broadband antenna for UWB radio systems [3]–[10]. Abbosh et al. discussed the performances of UWB planar monopole antennas with a circular or elliptical shape [5]. Chen et al. discussed ground plane effect on a small print UWB antenna [6]. Cheng et al. proposed a compact and low profile printed wide-slot inverted cone antenna for UWB applications [7]. Low et al. described a UWB suspended plate antenna (SPA) with enhanced impedance and radiation performance [9]. Over the inherently operating bandwidth of the UWB system, the existing bands are used by a wireless local-area network (WLAN). Therefore, the UWB antenna with a band-notched

Manuscript received January 16, 2009; revised October 01, 2009; accepted October 27, 2009. Date of publication March 29, 2010; date of current version June 03, 2010. This work was supported by the National Science Council, R.O.C., under Grant NSC 97-2219-E-009-012. The authors are with the Department of Communication Engineering, Taiwan National Chiao Tung University, Hsinchu, Taiwan 300, R.O.C. (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2010.2046839

characteristic is required to reduce the interference. Researches in some literatures produce band-rejection characteristics by cutting a slot on the antenna [11]–[14] or adding a tuning metal stub within the antenna structure [15], [16]. Several researchers have created transmission zero at the required notched bands by introducing associated resonators in the antenna. By placing the resonator in the antenna, the antenna impedance shifts to a very high or very low level and brings out impedance mismatch at the notch band. Simultaneously, the antenna at notched band is similar to the virtual-open or virtual-short circuit and is capable of not only preventing energy from transmitting to free space, but of also avoiding receiving the unwanted signal from free space. Qu et al. created a notched band by a coplanar waveguide resonant cell [17]. Zaker et al. used an H-shaped conductor-backed plane to generate band-notched effect [18]. Other resonator forms, such as folded strips, two T-shaped stubs and capacitive-load strips have also been applied for band-notching purposes [19]–[21]. Although the resonators are well accepted in band-notched antenna design, band-notched antenna performance is sometimes quite limited owing to the structure of the antenna and resonator. Meanwhile, the return loss level at the notched band is also a crucial factor for estimating gain suppression. In general, the return loss level is simply in reverse proportion to the gain suppression of the antenna. The following examples explain the limited performance. The first example is that the amount of gain suppression is around 20 dB at specific angles, but the gain [19]. The next exsuppression is only several dB at ample is that the bandwidth of the notched band on the 10 dB return loss condition is overlapped with wanted UWB operating frequency, i.e., the bandwidth of the notched band is unsatisfactory for UWB applications [19], [20]. Previous literatures have mainly focused on the band-notched UWB antenna for wide operating bandwidth and band-notched performance. The band-notched performance in these literatures could utilize three observed criteria to estimate notched band antenna performance, i.e., gain suppression, bandwidth and roll-off rate (frequency selectivity) of the notched band. These criteria strongly relate to the structure and quality factor of the resonator [18]–[20]. According to our knowledge and experiments, the resonator position at the antenna should be included in the notched-band antenna design because it is also related to notched band performance. Hence, the quality factor and resonator position can be accommodated simultaneously to improve the controlled ability of the notched band. To investigate notched band performance, the proposed antenna consists of a fork-shaped antenna, an open-looped resonator with a high quality factor and two taped-lines. The fork-

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shaped antenna is designed for wide operating range. The proposed resonator is placed at the center of the fork-shaped antenna, creating the 5-GHz notched band. Using this arrangement, this work emphasizes the resonator effects. The proposed antenna also shows good performance, such as fast roll-off rate of return loss, good gain suppression ability and narrow notched bandwidth at the notched band. Section II presents the geometry and design concept of the proposed antenna and discusses important parameters for the fork-shaped antenna and band-notched performance. Section III shows the equivalent circuit model as a simple way to estimate the notched frequency of the proposed antenna. The calculated antenna input impedance using the proposed circuit mode agrees with the full-wave simulation data. Section IV further examines the gain frequency response and group delay of the proposed antenna. Finally, Section V draws conclusions. II. ANTENNA CONFIGURATION AND PERFORMANCE A. Antenna Configuration and Performance Fig. 1 shows the geometry of the proposed antenna consisting of the fork-shaped antenna and the proposed resonator. The wide operating bandwidth of the fork-shaped antenna is mainly determined by three parameters, i.e., L , L and W . The lowest frequency is determined by (1) to (3) and the tapered profile of the antenna structure is described by (4). is the angle between the radiator and the ground plane

(1) (2) (3) (4) where L is the estimated longest current path along the outer radiating strip, approximated as a quarter of the length at the are the speed of light and the aplowest frequency. The and proximated effective dielectric constant, respectively. The performance of the fork-shaped antenna at the UWB high band is related to . Here, the UWB high band refers to the optional band from 5.85 to 10.65 GHz whereas the UWB low band represents the mandatory band from 3.1 to 5.1 GHz [1], [21]. In our experiments, should be 0.4–0.6 for better return loss level at the UWB high band. According to simulated current distributions of the planar monopole antenna, the current on the metal plate is inherently concentrated along the outer edges of the radiating plate, especially for the UWB low band. Based on this phenomenon, the resonator position at the interior of the antenna not only realizes a band-notched characteristic, but also preserves nearly the original characteristic of the antenna. The cutting triangular area, 0.5 3.5 mm 14 mm, is applied here to place the proposed resonator. To achieve the band-notched property, the proposed resonator is symmetric around its centerline and consists of one open-

Fig. 1. Configuration of the proposed antenna. (a) Top view. (b) Proposed resonator. (c) Schematic equivalent circuit model of proposed resonator.

looped resonator and two tapped lines as shown in Fig. 1(b). The open-looped resonator is realized by folding back a half wavelength straight strip to form a pair of coupled lines. The coupled lines are connected together at one end. The total length of , approximates one-quarter wavethe coupled lines, length. The current work uses the tapped-line to connect the proposed open-looped resonator and the fork-shaped antenna. The proposed resonator causes high input impedance level and impedance mismatching at the proposed antenna in the notched band. Conceptually, the proposed resonator can be represented by the schematic equivalent circuit model shown in Fig. 1(c), in which each coupled line and the strip of W can be represented

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Fig. 3. Simulated return loss of various ground plane sizes. Fig. 2. Measured and simulated return loss of proposed antenna.

by a lumped parallel lossy RLC circuit and inductive load, respectively. The tapped-line is treated as the inductive loads and capacitive loads. The capacitive coupling between two folded strips is ignored in Fig. 1(c). The resonant frequency of the proposed resonator can be readily controlled by adjusting the equivalent inductance and capacitance values. It is noted that the proposed resonator is operated in inhomogeneous media without background plane. Thus, the proposed resonator neither supports the TEM mode nor forms the microstrip line resonator. Section III discusses the analysis and the simplified equivalent circuit model. The antenna was fabricated on a 35 mm 30 mm 0.769 mm Rogers RO4350 substrate with dielectric constant and loss tangent = 0.004 at 10 GHz. The final design pamm, mm, mm, rameters are mm, mm, mm, mm, mm, mm, mm, mm, mm, mm, mm, mm, mm, mm and mm. Fig. 2 shows the simulated and measured return losses. The simulation was performed using Ansoft HFSS 9.2 while the measurement was taken by an Agilent E8362B performance network analyzer. The measured result agrees with the simulated result. The proposed resonator only slightly interferes with the return loss of the fork-shaped antenna except within the notched band. Fig. 2 also shows the simulated result of the antenna without the proposed resonator, evidencing that the desired band notched property is introduced by the proposed resonator. The notched band reveals the narrow bandwidth and the fast roll-off rate due to the high quality factor and the appropriate position of the resonator. In general, the ground plane can be treated as part of a small antenna. In this work, it is necessary to discuss the effect of the ground plane. Fig. 3 shows the simulated return loss of various ground plane sizes. Observations show that the bandwidth of the UWB low band becomes significantly wider as L changes from 12 mm to 18 mm but both the bandwidth of UWB high band and the return loss level of the notched band remain practically

unchanged. According to this phenomenon, the larger ground size is proportional to the bandwidth at the UWB low band. The antenna radiation patterns are measured in a 7.0 m 3.6 m 3.0 m anechoic chamber with an Agilent E8362B network analyzer along with NSI2000 far-field measurement software. Fig. 4 shows the measured radiation patterns in yz-, xz- and xy-planes at 4.5 and 8.5 GHz. The measured patterns agree with the simulated patterns. Referring to Fig. 4(a), the co-polarization patterns are probably omni-directional shaped. The cross-polarization level rises considerably as frequency increases. The cross-polarization level is comparable to the co-polarization level in the yz-plane. In Figs. 4(b) and (c), the co-polarization patterns are the roughly dumbbell-like shaped and the cross-polarization levels are generally much lower than co-polarization levels. The discrepancies of cross-polarization in xy-plane and yz-plane can be attributed to the interference of the coaxial cable and the absorber. B. Effect of Resonator on Notched Bands To comprehend the effect of the proposed resonator, this subsection discusses the geometric parameters of the proposed resonator along with the fork-shaped antenna. The following discussions evaluate band-notched performance by the bandwidth, roll-off rate and return loss level of the notched band. Fig. 5 shows the simulated return loss of various folded lengths of the resonator. The folded length of the open-looped resonator is the dominated element on the notched band. As shown in Fig. 1(c), the folded length of the resonator, , preliminarily determines the values of the parallel RLC circuit. In the meantime, the resonant frequency of the resonator is principally controlled by adjusting the values of the parallel RLC circuit. As the folded strip length becomes longer, the amount of capacitive load of the parallel RLC circuit increases accordingly. The center frequency of the notched band is simply in reverse proportion to the folded length of the resonator. Simultaneously, the bandwidth and return loss level retain their original value. Fig. 6 shows the simulated return loss of various positions of the tapped-line. The position of the tapped line is the feeding

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Fig. 6. Simulated return loss of various positions of the tapped-line.

Fig. 4. Measured and Simulated radiation patterns at (a) yz-plane. (b) xz-plane. (c) xy-plane. (Unit:dBi)

Fig. 5. Simulated return loss of various folded lengths of the resonator.

input of the resonator. In this case, the position of the tapped-line moves vertically while the position of the whole open-looped resonator at the antenna is fixed. In this arrangement, the center notched frequency varies from 5.24 GHz to 6.46 GHz as T and T respectively changes from 0.0 and 2.6 mm to 2.4 and 0.2 mm at the fixed T . The notched frequency shifts over a 1.2 GHz range as the tapped-line moves several millimeters. The result is different from the general filter design concept where

the position of the tapped-line cannot dominate the resonant frequency. To explain this phenomenon, the current study employed an equivalent circuit model of the resonator shown in Section III. According to the simulated results of the equivalent circuit model, the resonant frequency only shifts around 0.2 GHz as the position of the tapped line changes in same condition. This implies a certain degree of dependency between the two situations, i.e., the resonator with ground plane and the resonator without ground plane. In the former situation, the resonator supports the quasi-TEM mode. Hence, the feeding position of the resonator cannot dominate the resonant frequency. In the latter situation, resonant frequency is easily influenced by changing the resonator input because the resonator is placed in the antenna/radiator without the ground plane. Fig. 7 shows the simulated return loss according to various vertical positions of the resonator at the antenna, i.e., various values of T while T , T , W and W are fixed. Here, the proposed resonator moves along the x-direction. The center frequency and the return loss level of the notched band slightly changes as T changes from 1.8 mm to 7.8 mm, whereas the bandwidth of the notched band is significantly larger as the resonator is placed near the end of the fork-shaped antenna. In this case, since the proposed resonator structure is not changed, the unload quality factor of the proposed resonator or each component of Fig. 1(c) retains its original value. This study considers the external quality factor of the resonator when the resonator cooperates with the radiator/antenna. Fig. 7 implies that when the resonator is placed near the end of the fork-shaped antenna, the external quality factor becomes lower and the bandwidth of the notched band becomes wider simultaneously. Fig. 8 shows the simulated return loss according to horizontal positions of the resonator at the antenna, i.e., various values of W and W at fixed T , T and T . Different from Fig. 7, the proposed resonator moves along the y-direction. Referring to is not equal to L because W is not equal to Fig. 1(c), L W . When the proposed antenna becomes non-symmetrical, the notched band moves farther apart and splits into two individual dB at 5.56 GHz and notched bands whose peak values are dB at 6.48 GHz when mm and mm, and dB at 5.4 GHz and dB at 6.08 GHz when are mm and mm, respectively. Fig. 8 shows that the

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Fig. 7. Simulated return loss according to various vertical position of resonator at the antenna.

Fig. 9. (a) Exacting structure of the proposed resonator. (b) One-port lump equivalent circuit network of the proposed resonator. (c) Two-port lump equivalent circuit network of the proposed resonator. (d) Simplified equivalent circuit model of the proposed antenna.

Fig. 8. Simulated return loss according to various horizontal positions of resonator at the antenna.

proposed resonator can be treated as two sub-resonators, where each sub-resonator is formed by a tapped-line and a folded strip of the open-looped resonator. The notched frequencies of the proposed antenna depend on the structure of each sub-resonator.

III. THE EQUIVALENT CIRCUIT MODEL Conceptually, the schematic equivalent circuit model shown in Fig. 1(c) represents the proposed resonator. The inductive and capacitive loads explain the band-notched behaviors at the notched band. However, it is difficult for band-notched antenna modeling using the schematic equivalent circuit model to obtain accurate values of each component. To tackle this problem, this section proposes the simplified equivalent circuit model to explain complex resonant behaviors. This work first extracts the impedance characteristic of the proposed resonator shown in Fig. 9(a). Here, the dimension of each ground plane is 8 20 mm and the two delta sources excite at each node interface of the proposed resonator at the Ref plane. The proposed resonator in the fork-shaped antenna is actually floating. Therefore, during the extracting process shown in Fig. 9(a), the proposed resonator is placed at the RO4350 sub-

strate without ground plane to accompany the actual operating mechanism. To transfer the proposed resonator to the equivalent lumped circuit model, the procedure of the equivalent lumped circuit model shown in Fig. 9(b) and (c) can be accounted for using the filter design theory in [[22], Chap 6–8]. Using the HFSS simulation, the Z-parameters of the proposed resonator can be easily achieved and can transfer into a one-port lumped equivalent parallel resonant as shown in Fig. 9(b) with

(5) (6) (7) where FBW is the fractional bandwidth. and are frequencies as the input impedance magnitude of the one-port resonant network is respectively 0.707 times the maximum magnitude of the one-port network. is the center resonant frequency of the resonator. R is the real part of the impedance of the one-port network at the center frequency. L and C are the inductance and capacitance of the one-port network, respectively [21]. Because the resonator is symmetrical, the one-port resonant network can separate into the symmetrical two-port resonant network as shown in Fig. 9(c). The T-T’ line is a symmetrical plane.

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TABLE I EXTRACTED VALUES OF THE RESONATOR PARAMETERS AND J INVERTER

Here, we define two-port network elements, Q , L , C and R , as

(8) (9) (10) (11) Note that R is the real part of impedance of the two-port network and is 2R to keep the real part of implement of the two-port network equal to that of the one-port network. R or R in the lumped circuit model represents the amount of conductor and dielectric loss, not radiation loss. Fig. 9(c) presents the proposed resonator with two node interfaces by a two-port lumped circuit model. In the modeling process, the metal strip is regarded as the , which high impedance line with a length of is approximately one-quarter wavelength of the notched band. Here, the metal strip is between the proposed resonator and the feeding microstrip line. Hence, this metal strip behaves quite similarly to a quarter-wavelength transformer, and can be therefore modeled as a pair of J-inverters [21], [22]. The values of the J-inverter are determined by the characteristic admittance of the feeding microstrip line and the real-part admittance of the equivalent resonant circuit. G stands for a constant radiation conductance and accounts for the wideband nature of the antenna. Finally, a simplified equivalent circuit model presents the proposed antenna as shown in Fig. 9(d). The extracted values of is resonator parameters are summarized in Table I and Y the input admittance of the antenna given by

(12)

Fig. 10 shows the simulated impedance of the proposed resonator and the simulated admittance of the proposed antenna, respectively. Reasonable agreement exists between the results of the HFSS simulation and the simplified equivalent circuit model. The discrepancy between the curves mostly attributes to the simplistic modeling of the resonator and the J-inverter. The results, in terms of a simplified equivalent circuit with higher quality factor, reasonably explain the narrower bandwidth of

Fig. 10. Compared results between HFSS and the simplified equivalent circuit. (a) Simulated impedance of the resonator. (b) Simulated admittance of the proposed antenna.

frequency gain response in Section IV. Despite some inaccuracy in the simplified equivalent circuit model, the results still provide valuable information of antenna behavior. IV. MEASURED GAIN RESPONSE AND GROUP DELAY In measuring antenna gain frequency response, the EMCO 3115 double-ridge horn antenna with a constant group delay of 630 ps is used as a reference antenna for calibration. Fig. 11 illustrates the measured gain frequency responses in the yz-plane to 180 with 45 interval. at eight angles, ranged from The gain response of the proposed antenna is quite flat from 3.1 to 10.6 GHz except at the notched band. According to Fig. 11, the range of gain suppression is from 15 dB to 35 dB within these eight angles at the target notched band. Although and at , the notched band slightly shifts at it remains within the range of the target notched band. The notched frequencies are quite similar at the observation angles. The notched bandwidth is significantly narrower due to the fast roll-off rate and high level of return loss at the notched band. The distance between the referenced antenna and the proposed antenna is 3.4 m. In the UWB system, the constant group delay response is required. Fig. 12 presents the measured group delay of the proposed antenna. Except at the notched band, group

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V. CONCLUSION This paper proposes and analyzes a novel band-notched monopole ultrawideband antenna. By applying an open-looped resonator, the antenna shows a narrower bandwidth and high return loss level as well as good gain suppression ability at the desired notched band. The parameter studies of the proposed antenna provide brief guidelines for a band-notched antenna design using the similar monopole antenna and resonator. This study investigates these parameters in terms of the relationship between the fork-shaped antenna and the open-looped resonator. The simplified equivalent circuit model explains the rather complicated resonant behavior of the proposed antenna. The calculated antenna input admittance using the simplified equivalent circuit model agrees with the HFSS simulated result. Evaluations of return loss, radiation patterns, gain responses, and group delay confirm the antenna performance. These features of the proposed antenna demonstrate that the proposed antenna is suitable for UWB communicational applications and prevents interference from the WLAN system. ACKNOWLEDGMENT The authors would like to thank Wireless Communication and Electromagnetism Application Lab of the National Taiwan University of Science and Technology, Taiwan, ROC., for supporting the far field measurement REFERENCES

= 0 ; = = 045 .

Fig. 11. Measured gain response of the proposed antenna. (a)  ; ; . (b)  ; ; ;

90

= 180

= 090

= 45

= 135

= 0135

Fig. 12. Measured group delay of the proposed antenna.

delay variation over the 3.1 to 10.6 GHz is less than 130 ps with average of 718 ps as the spatial angle varies. Figs. 11 and 12 show that the proposed antenna has good time/frequency characteristics and a small pulse distortion over the UWB operated band.

[1] UWB, Forum Homepage [Online]. Available: http://www.uwbforum [2] Task Group 3a Homepage [Online]. Available: http://www.ieee802. org/15/pub/TG3a.html [3] J. S. McLean, H. Foltz, and R. Sutton, “Pattern descriptors for UWB antennas,” IEEE Trans. Antennas Propag., vol. 53, pp. 553–559, Jan. 2005. [4] S. B. T. Wang, A. M. Niknejad, and R. W. Brodersen, “Circuit modeling methodology for UWB omnidirectional small antennas,” IEEE J. Select. Areas Commun., vol. 24, no. 4, pp. 871–877, 2006. [5] A. M. Abbosh and M. E. Bialkowski, “Design of ultrawideband planar monopole antennas of circular and elliptical shape,” IEEE Trans. Antennas Propag., vol. 56, no. 1, pp. 17–23, Jan. 2008. [6] Z. N. Chen, T. S. P. See, and X. Qing, “Small printed ultrawideband antenna with reduced ground plane effect,” IEEE Trans. Antennas Propag., vol. 55, no. 2, pp. 383–388, Feb. 2007. [7] C. Shi, P. Hallbjorner, and A. Rydberg, “Printed slot planar inverted cone antenna for ultrawideband applications,” IEEE Antennas Wireless Propag. Lett., vol. 7, pp. 18–21, 2008. [8] T. G. Ma and S. K. Jeng, “Planar miniature tapered-slot-fed annular slot antennas for ultrawideband radios,” IEEE Trans. Antennas Propag., vol. 53, pp. 1194–1202, Mar. 2005. [9] X. N. Low, Z. N. Chen, and W. K. Toh, “Ultrawideband suspended plate antenna with enhance impedance and radiation performance,” IEEE Trans. Antennas Propag., vol. 56, no. 8, pp. 2490–2495, Aug. 2008. [10] C. D. Zhao, “Analysis on the properties of a coupled planar dipole UWB antenna,” IEEE Antennas Wireless Propag. Lett., vol. 3, pp. 317–320, 2004. [11] J. Qiu, Z. Du, J. Lu, and K. Gong, “A planar monopole antenna design with band-notched characteristic,” IEEE Trans. Antennas Propag., vol. 54, no. 1, pp. 288–292, Jan. 2006. [12] W. S. Lee, W. G. Lim, and J. W. Yu, “Multiple band-notched planar monopole antenna for multiband wireless systems,” IEEE Microwave Wireless Compon. Lett., vol. 15, pp. 576–578, Sept. 2005. [13] Y. J. Cho, K. H. Kim, D. H. Choi, S. S. Lee, and S. O. Park, “A miniature UWB planar monopole antenna with 5-GHz band-rejection filter and the time-domain characteristics,” IEEE Trans. Antennas Propag., vol. 54, pp. 1453–1460, May 2006.

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[14] T. Dissanayake and K. P. Esselle, “Predication of the notch frequency of slot loaded printed UWB antennas,” IEEE Trans. Antennas Propag., vol. 55, no. 11, pp. 3320–3325, Nov. 2007. [15] Y. Lin and K. J. Hung, “Compact ultrawideband rectangular aperture antenna and band-notched designs,” IEEE Trans. Antennas Propag., vol. 54, pp. 3075–3081, Nov. 2006. [16] K. H. Kim and S. O. Park, “Analysis of the small band-rejected antenna with the parasitic strip for UWB,” IEEE Trans. Antennas Propag., vol. 54, pp. 1688–1692, June 2006. [17] S. W. Qu, J. L. Li, and Q. Xue, “A band-notched ultrawideband printed monopole antenna,” IEEE Antennas Wireless Propag. Lett., vol. 5, pp. 495–498, 2006. [18] R. Zaker, C. Ghobadi, and J. Nourinia, “Novel modified UWB planar monopole antenna with variable frequency band-notch function,” IEEE Antennas Wireless Propag. Lett., vol. 7, pp. 112–114, 2008. [19] T. G. Ma and S. J. Wu, “Ultrawideband band-notched U-shape folded monopole antenna and its radiation characteristics,” in Ultra-Wideband Short Pulse Electromagnetics 8. Berlin, Germany: Springer Publisher, 2007, pp. 49–56. [20] C.-Y. Hong, C.-W. Ling, I.-Y. Tran, and S.-J. Chung, “Design of a planar ultrawideband antenna with a new band-notch structure,” IEEE Trans. Antennas Propag., vol. 55, no. 12, pp. 3391–3397, Dec. 2007. [21] T. G. Ma and S. J. Wu, “Ultrawideband band-notched folded strip monopole antenna,” IEEE Trans. Antennas Propag., vol. 55, pp. 2473–2479, Sept. 2007. [22] J. S. Hong and M. J. Lancaster, Microstrip Filter for RF/Microwave Application. New York: Wiley, 2001.

Sung-Jung Wu was born in Taipei, Taiwan, R.O.C., in 1980. He received the B.S. degree in electrical engineering from TamKang University (TKU), Taipei, in 2004, and the M.S. degree in electrical engineering from National Taiwan University of Science and Technology (NTUST), Taipei, in 2007. He is currently working toward the Ph.D. degree at National Chiao Tung University, Hsinchu, Taiwan. He worked with the Foxconn Technology Co., Ltd., Taiwan, and Sunplus Technology Co., Hsinchu, for RF circuit design in 2004–2006 and 2006–2008 respectively. His research interests include mobile antenna designs, RFID tag antenna designs, and UWB antenna designs, reconfigurable antenna design.

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Keng-Hsien Chen was born in Taipei, Taiwan, R.O.C., in 1986. He received the B.S. degree in electrical engineering from National Taiwan Ocean University, Keelung, Taiwan, in 2008. He is currently working toward the M.S. degree at National Chiao Tung University, Hsinchu, Taiwan. His current research interests include microwave antenna and circuits, reconfigurable antennas, and electromagnetic compatibility.

Cheng-Hung Kang was born in Yilan, Taiwan, R.O.C., in 1986. He received the B.S. degree in communication engineering from National Chiao Tung University, Hsinchu, R.O.C., in 2008, where he is currently working toward the M.S. degree. His research interests include antenna designs, UWB antenna and passive circuit designs.

Jenn-Hwan Tarng (S’85–M’89–SM’06) received Ph.D. degree in electrical engineering from Pennsylvania State University, University Park, in 1989. After obtaining the Ph.D. degree, he joined the Faculty of National Chiao-Tung University (NCTU), Hsin-Chu, Taiwan, R.O.C., where he now holds a position as Professor in the Department of Electrical Engineering. During 2003–2005, he was the Chairman of the Communication Engineering Department and Director of ARTS (Advanced Radio Technology and Systems) Center, NCTU, and then he was invited (on leave) as Chair Professor and Dean of Engineering College, Chung-Hua University from 2005–2007. From 2007–2009, he was also on leave and acted as the General Director of ISTC (Identification and Security Technology Center) of ITRI (Industrial Technology Research Institute), Taiwan, ROC. In the center, he led more than 150 R&D engineers to develop advanced RF ID and physical security technologies and systems to enhance associated local industries global competition. His professional interests include radio propagation modeling and measurement, frequency management, antenna design, RF IC and EMI/EMC.

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Electrically Small Magnetic Dipole Antennas With Quality Factors Approaching the Chu Lower Bound Oleksiy S. Kim, Olav Breinbjerg, Member, IEEE, and Arthur D. Yaghjian, Life Fellow, IEEE

Abstract—We investigate the quality factor for electrically small current distributions and practical antenna designs radiating the 10 magnetic dipole field. The current distributions and the antenna designs employ electric currents on a spherical surface enclosing a magneto-dielectric material that serves to reduce the internal stored energy. Closed-form expressions for the internal and external stored energies as well as for the quality are derived. The influence of the sphere radius and the factor material permeability and permittivity on the quality factor is determined and verified numerically. It is found that for a given antenna size and permittivity there is an optimum permeability that ensures the lowest possible , and this optimum permeability is inversely proportional to the square of the antenna electrical radius. When the relative permittivity is equal to 1, the optimum permeability yields the quality factor that constitutes the lower bound for a magnetic dipole antenna with a magneto-dielectric core. Furthermore, the smaller the antenna the closer its quality factor can approach the Chu lower bound. Simulated results for the 10 -mode multiarm spherical helix antenna with a that is 1.24 times the Chu lower bound magnetic core reach a for an electrical radius of 0.192.

TE

TE

Index Terms—Chu limit, electrically small antennas, magnetic dipole, quality factor, spherical modes, surface integral equation.

I. INTRODUCTION LECTRICALLY small antennas, that is, antennas that are small compared to the free-space wavelength at their frequency of operation, have been a subject of research for many years [1]–[8] and the challenges of these antennas are well known. Nevertheless, in recent years the escalation of miniaturized wireless technology has stimulated new research to develop small antennas with acceptable matching, bandwidth, and radiation efficiency [9]–[12]. In particular, it is of great interest to investigate how closely a small antenna can have its radiation quality factor approach the Chu lower bound . For an antenna that is resonant (equal stored electric and magnetic energies) and that radiates either an electric-dipole spherical mode) or a magnetic-dipole field ( field ( mode) at a frequency with free-space wave number , and with

E

Manuscript received July 16, 2009; revised October 14, 2009; accepted January 11, 2010. Date of publication March 29, 2010; date of current version June 03, 2010. This work was supported by the Danish Research Council for Technology and Production Sciences within the TopAnt project (http://www.topant. dtu.dk). O. S. Kim and O. Breinbjerg are with the Department of Electrical Engineering, Electromagnetic Systems, Technical University of Denmark, DK-2800 Kgs. Lyngby, Denmark (e-mail: [email protected]; [email protected]). A. D. Yaghjian is at Concord MA, 01742 USA. Digital Object Identifier 10.1109/TAP.2010.2046864

the smallest circumscribing sphere of radius , this bound can be expressed as [4] (1) It is sometimes stated that the Chu lower bound is based on the assumption that the internal volume of the circumscribing sphere does not store any energy—but this is not entirely correct. As pointed out by Chu [2, p. 1170] the internal volume can store the energy that may be needed to make the antenna reselectric-dipole antenna, onant. For an electrically small where the external stored electric energy dominates the external stored magnetic energy, the internal volume must store a magnetic energy to compensate the difference of the external stored energies, and vice versa for a magnetic-dipole antenna. In any case, this internal stored energy is taken into account since (1) is based on a total stored energy that is set to twice the maximum of the external stored electric and magnetic energies. Internal stored energy, beyond that needed to make the antenna resonant, will obviously increase the quality factor above the Chu lower bound. An electrically small, spherical surface eleccurrent distribution in free space that radiates the tric-dipole field will have a significant internal stored electric energy, while a similar current distribution that radiates the magnetic-dipole field will have a significant internal stored magnetic energy. Investigating a spherical coil antenna radiating the magnetic-dipole field, Wheeler [13] reported a quality factor 3.0 times the first term of the Chu lower bound (1). More recently, Thal [8] showed that the inclusion of the internal stored energy results in a quality factor for electric surface currents radiating the electric-dipole field that is 1.5 times the Chu lower bound, while the quality factor for electric surface curmagnetic-dipole field is 3.0 times the rents radiating the Chu lower bound. That is, (2a) (2b) Thal’s derivation [8] is based on his previously derived circuit equivalents for spherical vector waves [14] and is thus in line with the approach of Chu [2]. The authors of this manuscript have followed the more direct approach of Collin and Rothschild [4] to verify (2) by calculating the internal and external stored energies from spatial integrations of the electromagnetic fields [15]. These calculations show that for an electrically small, electric surface current density radiating the electric-dipole field, the internal stored electric energy is 0.5 times the external stored electric energy. Furthermore, for an

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KIM et al.: ELECTRICALLY SMALL MAGNETIC DIPOLE ANTENNAS WITH QUALITY FACTORS APPROACHING

electrically small, electric surface current density radiating the magnetic-dipole field the internal stored magnetic energy is 2.0 times the external stored magnetic energy. It is interesting to note that recent electrically small electric-dipole antennas by Best [9], [10] as well as magnetic-dipole antennas by Best [16] and Kim [17] exhibit quality factors that approach the values in (2) from above, and there is apparently no report in the literature of free-space antennas with quality factors below Thal’s lower bounds. In order to overcome Thal’s lower bounds (2) and approach the Chu lower bound (1), an antenna with vanishing internal stored energy is required. For an electrically small, spherical elecelectric surface current density radiating the tric-dipole field, it is easily seen that the internal stored electric energy will increase, not decrease, if the internal volume is filled with a dielectric material. However, for an electrically small, spherical electric surface current density radiating the magnetic-dipole field, the internal stored magnetic energy will decrease, not increase, if the internal volume is filled with a magnetic material; in fact, the internal stored magnetic energy will be inversely proportional to the permeability of the magnetic material. The latter case was pointed out already by Wheeler [13] and confirmed by Thal [8]; specifically, with denoting the relative permeability the quality factor becomes (3) It is noted, that the expression (3) holds only in the limit of vanishing electrical radius since it is necessary to avoid the cavity resonances that may result from filling a finite-sized spherical surface current density with a high-permeability material. Such resonances, with no external field, will of course make the quality factor approach infinity. Though the expression (3) constitutes a highly interesting limit, it is thus not guaranteed to apply to practical antenna designs of finite size. As will be shown in this manuscript, it does not, and for a given electrical size of the spherical current density, the lowest quality factor is not obtained with the highest permeability, because of resonances. The purpose of this work is to investigate the quality factor for electrically small current distributions and practical anmagnetic dipole field; in partenna designs radiating the can approach the Chu ticular, to determine how close the lower bound. The current distributions are spherical electric surface current densities, and the practical antennas are conducting wires wound on a spherical surface—in both cases enclosing a magnetic or magneto-dielectric spherical core to reduce the internal stored magnetic energy. For the current distribution an analytical solution is derived in terms of spherical vector wave functions that leads to closed-form expressions for the internal and external stored energies as well as the radiated power for arbitrary values of the electrical radius of the sphere and the permeability/permittivity of the core. The influence of these parameters on the quality factor can thus be investigated, and the optimal permeability for a given electrical radius can then be determined.

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Fig. 1. Ideal magnetic dipole antenna with a magneto-dielectric core.

As a practical antenna, the -mode multiarm spherical helix antenna [17] augmented with a spherical magnetic/magneto-dielectric core is investigated. A numerical solution is obtained using the surface integral equation and higher-order method of moments [18]. Some preliminary results of this work were reported previously at the ISAP 2009 conference [19], [20]. This manuscript concentrates on the optimal permeability and the lowest possible for a given antenna electrical radius. Furthermore, the influence of the core permittivity on the quality factor is investigated both analytically and numerically. II. SPHERICAL SURFACE CURRENT DENSITY ENCLOSING A MATERIAL CORE A. Single

Mode

The configuration consists of a time-harmonic, spherical, electric surface current density of radius and amplitude enclosing a magneto-dielectric core with relative permeability and permittivity , and wave number (Fig. 1). The core material is assumed to be linear, isotropic, homogeneous, lossless and dispersion-less.1 Introducing a spherical coordinate system with its origin at the center of the core, and employing phasor notation with suppressed time factor , (where is the speed of light), the surface current density can be expressed as (4) is the azimuthal unit vector. This electric surface curwhere rent density is thus azimuthally directed with a sinusoidal polar variation, and it is proportional to the electric equivalent surface current density of the Huygens-Love field equivalence principle for a -directed magnetic dipole located at the coordinate system origin. 1) Analytical Solution: The radiated field is obtained by exas a spherical vector wave pressing the internal field for of the standing wave type, the external field for as a spherical vector wave of the outward propagating type, and then ento deterforcing the appropriate boundary conditions at 1It is assumed that the frequency dispersion of the relevant bandwidth, and thus,  > 0 and "



and

"

 1 [21].

is negligible over

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mine the complex amplitudes and of the internal and external fields, respectively. Since the surface current density is azimuthally constant and has a first-order polar variation, this will also be the case for the radiated fields; thus, only the magnetic-dipole spherical vector wave with azimuthal index and polar index will be present; that is the mode. Employing the notation of Hansen [22], the interior and exterior electric and magnetic fields can be expressed as (5a) (5b) (5c) (5d) In these expressions and are the intrinsic admittances of free space and the core, respectively. Furthermore, the -functions are the power-normalized spherical vector wave functions (6a)

(6b) (6c)

(6d) and the

and

coefficients are given by (7a) (7b)

where

(8) The stored energies and the radiated power are determined through direct spatial integration of the radiated fields. In the calculation of the external stored energy the contribution of the propagating field is subtracted from that of the total field to obtain the energy density of the non-propagating field only [4], [6]. , , , and denoting the internal electric, inWith ternal magnetic, external electric, and external magnetic stored energy, respectively, it is found that (9a)

(9b) (9c) (9d) Finally, the radiated power is (10) It is noted that these expressions hold for arbitrary electrical raof the sphere, and thus allow us to investigate the reladius tionships between , relative permeability and permittivity , and the quality factor in the case of finite-sized spheres. 2) Stored Energy: First, we consider a pure magnetic core, . Fig. 2(a) shows the ratio of the total stored that is, to the total stored magnetic electric energy vs. free-space electrical radius energy in the range from 0 to 1 for four values of the relative perme, 2, 8, and 100. For vanishing electrical radius ability the magnetic energy clearly dominates the electrical energy and the current distribution is thus not resonant. As the electrical radius increases, the electrical energy becomes relatively larger and eventually equals the magnetic energy at the resonances. vs. the The complimentary dependence of the ratio for three different values of , relative permeability 0.25, and 0.5 is presented in Fig. 3(a). The total dominance of the is observed in the entire shown magnetic energy for and 0.5 the magnetic energy range of , whereas for dominates only away from the resonances. Fig. 2(b) shows the ratio of the internal stored energy to the total stored energy vs. free-space electrical radius . In the free-space , the internal energy is seen to constitute 2/3 of case, the total energy—and thus it is equal to twice the external energy—for vanishing electrical radius. As the relative permeincreases to 2, 8, and 100, the energy ratio decreases ability to 0.5, 0.2, and 0.02, respectively. However, for non-zero electrical radius the dependence of the energy ratio on permeability is different, and for the internal cavity resonances the total stored energy is actually entirely internal. It is thus seen that the smallest energy ratio is not obtained with the largest permeability for finite values of . The latter observation is also clearly illustrated in Fig. 3(b), is plotted (note the logarithmic where the ratio scale) vs. the relative permeability in the range from 1 to 500 for three different values of , 0.25, and 0.5. The ratio only decreases monotonically for the very small electrical radius , whereas for larger electrical radii the initial decrease reaches a minimum whereafter the ratio actually increases. 3) Quality Factor: The radiation quality factor is defined times the ratio of the stored electric and magnetic energy as to the radiated power per period. However, in order to establish a resonant system (equal electric and magnetic stored energy) the stored energy is set to twice the maximum of the electric and

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Fig. 2. Energy and quality factor as functions of ka (" = 1): (a) ratio of electric and magnetic stored energies; (b) ratio of internal and total stored energies; (c) quality factor (normalized by Chu lower bound).

Fig. 3. Energy and quality factor as functions of  (" = 1): (a) ratio of electric and magnetic stored energies; (b) ratio of internal and total stored energy; (c) quality factor (normalized by Chu lower bound).

magnetic energies. It is understood that the lesser of the two has been increased to equal the larger by use of, e.g., a lumped-commagnetic-dipole antenna, ponent tuning circuit. For the the magnetic energy dominates for small electrical radii but the electric energy may dominate at larger radii. Thus, the quality factor is determined from (9) and (10) as

Fig. 2(c) shows the quality factor , normalized by the Chu , vs. free-space electrical radius of the lower bound magnetic sphere for different relative permeabilities . For (non-magnetic core) the normalized is equal to 3 and it remains fairly constant for for vanishing (this agrees with Thal [8, Table I]). For , 8, and 100 the normalized equals 2, 1.25, and 1.02, respectively, for in agreement with (3). This is also demonstrated in small is plotted as a function of Fig. 3(c), where the ratio for different values of the free-space electrical radius .

(11)

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Fig. 4. (a) Values of  , for which the relative error between the exact Q (11) and expression (3) becomes larger than 1%, as a function of ka with " as a parameter.

In the shown range of the relative permeability, the curve for very closely follows the the smallest sphere with expression (3). the resonance behavior of the fields in For larger values of the core manifests itself in the fact that the curves start diverging from (3). In the following this divergence is investigated for a general magneto-dielectric core, that is, the relative permittivity is allowed to be greater than 1. Fig. 4 shows the values of permeability , for which the relative error between the exact (11) and the asymptotic relation (3) becomes larger than 1%, as a function of with the permittivity as a parameter. A and the lower the general observation is that the smaller larger is the range of in which the expression (3) holds. From Fig. 2(c) and 3(c) we can also conclude that the Chu lower bound can be approached with modest values of ; e.g., and . Furthermore, the smallest within 25% for is not necessarily obtained with the largest . For a given and permittivity there is an optimum permeability for which the ratio is minimal. These values have been determined numerically and are plotted in Fig. 5(a) as a with the core permittivity as a parameter. The function of curves in Fig. 5(a) very accurately follow the expression (12a) (12b) The minimum achievable ratios corresponding to (12a) are plotted in Fig. 5(b), from which it is seen that in order to reach the Chu lower bound a magnetic dipole antenna should have a vanishing electrical size . For finite values of , the shows the lower bound for a curve corresponding to magnetic dipole antenna with a magneto-dielectric core. B. Higher-Order Modes In the investigation of Section II-A above, it is assumed that there is only a single mode excited in the system. The situation changes in the presence of higher-order spherical modes

Fig. 5. (a) Optimal permeability  , yielding the lowest Q for a given ka and " ; (b) the corresponding lowest achievable ratio Q=Q —the lower bound for a magnetic dipole antenna with a magneto-dielectric core.

that might be excited in a practical antenna when the current distribution deviates from (4). If these modes are not suppressed, they increase the stored energy inside the magnetic core and, in the worst case of a resonance, make the diverge. In particular, the TM modes with the polar index are problematic, since in a magnetic sphere they generate the lowest frequency -mode resonance occurs at resonance. Whereas the first , the first resonance for the modes is at . Consequently, the presence of any of the modes makes the optimum found in Section II-A a poor choice. Furthermore, the range of usable core permeabilis now limited by the -mode resonance, i.e, it is ities reduced to almost half of the range for the single -mode and the minimum case. The optimum core permeability depend in each particular case on achievable for a given and modes as well as on the power ratios between the presence of other spherical modes. III. MULTIARM SPHERICAL HELIX (MSH) ANTENNA In this section we extend the theory presented in the previous Section II by numerical results obtained for a practical -mode antenna configuration with a magneto-dielectric core. We employ the magnetic dipole MSH antenna first reported in [17]. This antenna consists of multiple wire arms

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Fig. 6. The TE -mode MSH antenna with a spherical core.

twisted into two symmetric hemispherical helices and excited by a curved dipole placed at the equator of the antenna (Fig. 6). As shown in [17], where an air-core version of the MSH antenna is thoroughly investigated, the resonance frequency of this self-resonant electrically small antenna is determined by the length of the wire arms, whereas the input resistance at the resonance is nearly independently controlled by the length of the excitation dipole. By placing a material core inside the antenna we change its resonance frequency and thus the electrical size. To avoid this change we adjust the length of the arms for each value of permeability/permittivity so that the . resonance frequency is kept constant at For our investigation we have chosen a 4-arm antenna config. The length of the uration with the wire radius of excitation dipole is quantified in angular units; the half-length . The antenna is assumed to be lossless and fed is by a delta-gap voltage generator in the middle of the excitation dipole. To ensure a well-behaved numerical solution the radius of the core is always set 1 mm less than the helix radius . The is calculated from the antenna input impedance using the expression from [7] (13) where and respectively, and frequency.

are the input resistance and reactance, with being the resonance

A. Magnetic Core First, we consider two antenna configurations of radii (electrical radius ) and , respectively, with a pure magnetic core . The radii of the core are and for the first and second configuration, respectively. The quality factor normalized to the Chu lower bound is plotted in Fig. 7(a) as a function of the core permeability . It is observed steeply decreases from its initial freethat the ratio for and space value— for —to a level below 1.5 as the permeability . The rapid drop of the stored magnetic reaches only energy inside the core requires a corresponding decline of the mode (Fig. 8(a)), since this mode acts as a distributed reactive matching circuit providing the stored electric energy needed to make the antenna self-resonant. The strength of the mode is proportional to the number of turns, and the latter reduces as well (Fig. 7(b)). The input resistance also initially

Fig. 7. Numerical results for the TE -mode MSH antenna with a magnetic core (" = 1). (a) Quality factor (normalized by Chu lower bound); (b) number of turns; (c) input resistance R .

reduces to about 50–60 ohms and then remains relatively stable until the first resonance is reached (Fig. 7(c)). This resonance for and at occurs at roughly for , which in both cases correspond to , i.e., to the first -mode internal resonance in a magnetic . Thus, the mode present in spherical core the antenna radiation spectrum (Fig. 8(b)) gives rise to a spike curve. in the

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Fig. 8. Numerical results for the TE -mode MSH antenna with a magnetic core (" = 1). Radiated power of the (a) TM and (b) TM modes normalized to the total radiated power.

Having passed the first resonance, the for the configuration monotonically grows with and reaches with the -mode internal core resonance at , or . The minimum value of the is found below the resonance at for and at for . The corresponding ratios are 1.28 and 1.24. This result is consistent with the observations in Section II, i.e., decreases with the antenna the lowest achievable ratio electrical size.

B. Magneto-Dielectric Core For the core permittivity larger than unity , the , shift core resonances, which are functions of (Fig. 9(a)); and the minimum towards the lower values of values of the ratio increase. Again, the result is consistent with the theoretical prediction illustrated in Fig. 5. The higher the more electric energy is stored in the core, and, as a consequence, less number of turns is required to make the antenna self-resonant (Fig. 9(b)). The input resistance also decreases (Fig. 9(c)), which, if necessary, can be easily compensated by truncating the excitation dipole [17].

Fig. 9. Numerical results for the TE -mode MSH antenna with a magnetodielectric core. (a) Quality factor (normalized by Chu lower bound); (b) number of turns; (c) input resistance R .

IV. CONCLUSION An analytical solution has been derived for the -mode magnetic dipole field radiated by a spherical, electric surface current density enclosing a magneto-dielectric core. Through direct spatial integration of the radiated fields, this solution led to closed-form expressions for the internal and external electric and magnetic stored energies, as well as the radiated power, and thus the radiation quality factor for arbitrary values of

KIM et al.: ELECTRICALLY SMALL MAGNETIC DIPOLE ANTENNAS WITH QUALITY FACTORS APPROACHING

the electrical radius. It is shown that the internal stored energy of the core and, for reduces with increasing permeability , the radiation vanishing free-space electrical radius, quality factor approaches the Chu lower bound arbi, in agreement with the prediction of trarily closely as Wheeler [13]. For finite-sized spheres the internal resonances of the magneto-dielectric core limit the minimum value of the quality factor . However, it was seen that the Chu lower bound can be approached with even modest values of the permeability , e.g., within 25% for and . On basis of the analytical solution, for a given antenna size and permittivity the minimum value of the quality factor was determined numerically and it was found to occur for the relative permeability that satisfies the relation (12a). By substituting correinto (7)–(11) the lower bound for a magnetic sponding to dipole antenna with a magneto-dielectric core is established. The analytical results are consistent with numerical simula-mode multiarm spherical helix tions conducted for the antenna with a magneto-dielectric core. The radiation quality is obtained for the antenna with freefactor . space electrical radius Generally, the smaller the antenna the closer its radiation can approach the Chu lower bound. In addition, a magnetic core with and the absence of the parasitic modes are prerequisites for the lowest possible of a magnetic dipole antenna.

REFERENCES [1] H. A. Wheeler, “Fundamental limitations of small antennas,” Proc. IRE, vol. 35, no. 12, pp. 1479–1484, 1947. [2] L. J. Chu, “Physical limitations of omni-directional antennas,” J. Appl. Phys., vol. 19, no. 12, pp. 1163–1175, 1948. [3] R. F. Harrington, J. Research National Bureau of Standards-D, Radio Propag., vol. 64D, no. 1, pp. 1–12, 1960. [4] R. Collin and S. Rothschild, “Evaluation of antenna Q,” IEEE Trans. Antennas Propag., vol. 12, no. 1, pp. 23–27, Jan. 1964. [5] R. L. Fante, “Quality factor of general ideal antennas,” IEEE Trans. Antennas Propag., vol. 17, no. 2, pp. 151–155, Mar. 1969. [6] J. S. McLean, “A re-examination of the fundamental limits on the radiation Q of electrically small antennas,” IEEE Trans. Antennas Propag., vol. 44, no. 5, pp. 672–676, May 1996. [7] A. D. Yaghjian and S. R. Best, “Impedance, bandwidth, and Q of antennas,” IEEE Trans. Antennas Propag., vol. 53, no. 4, pp. 1298–1324, Apr. 2005. [8] H. L. Thal, “New radiation Q limits for spherical wire antennas,” IEEE Trans. Antennas Propag., vol. 54, no. 10, pp. 2757–2763, Oct. 2006. [9] S. R. Best, “The radiation properties of electrically small folded spherical helix antennas,” IEEE Trans. Antennas Propag., vol. 52, no. 4, pp. 953–960, Apr. 2004. [10] S. R. Best, “Low Q electrically small linear and elliptical polarized spherical dipole antennas,” IEEE Trans. Antennas Propag., vol. 53, no. 3, pp. 1047–1053, Mar. 2005. [11] H. R. Stuart, S. R. Best, and A. D. Yaghjian, “Limitations in relating quality factor to bandwidth in a double resonance small antenna,” IEEE Antennas Wireless Propag. Lett., vol. 6, no. 11, pp. 460–463, 2007. [12] A. Erentok and R. W. Ziolkowski, “Metamaterial-inspired efficient electrically small antennas,” IEEE Trans. Antennas Propag., vol. 56, no. 3, pp. 691–707, Mar. 2008. [13] H. A. Wheeler, “The spherical coil as an inductor, shield, or antenna,” Proc. IRE, vol. 46, no. 9, pp. 1595–1602, 1958. [14] H. L. Thal, “Exact circuit analysis of spherical waves,” IEEE Trans. Antennas Propag., vol. 26, no. 2, pp. 282–287, Mar. 1978. [15] O. Breinbjerg, “Stored Energy and Quality Factor for Spherical Dipole Antennas in Free Space,” Technical Univ. Denmark, 2009, Tech. Rep..

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[16] S. R. Best, “A low Q electrically small magnetic (TE mode) dipole,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 572–575, 2009. [17] O. S. Kim, “Low-Q electrically small spherical magnetic dipole antennas,” IEEE Trans. Antennas Propag., in press. [18] E. Jørgensen, O. S. Kim, P. Meincke, and O. Breinbjerg, “Higher order hierarchical legendre basis functions in integral equation formulations applied to complex electromagnetic problems,” in Proc. IEEE Antennas and Propagation Soc. Int. Symp., Washington DC, Jul. 2005, vol. 3A, pp. 64–67. [19] O. Breinbjerg and O. S. Kim, “Minimum Q electrically small spherical magnetic dipole antenna—Theory,” presented at the Int. Symp. Antennas Propag. (ISAP 2009), Bangkok, Thailand, Oct. 20–29, 2009. [20] O. S. Kim and O. Breinbjerg, “Minimum Q electrically small spherical magnetic dipole antenna—Practice,” presented at the Int. Symp. Antennas Propag. (ISAP 2009), Bangkok, Thailand, Oct. 20–29, 2009. [21] A. D. Yaghjian, “Power-energy & dispersion relations for diamagnetic media,” presented at the IEEE Antennas Propag. Soc. Int. Symp., Charleston, SC, Jun. 1–5, 2009. [22] J. E. Hansen, Spherical Near-Field Antenna Measurements. London, U.K.: Peter Peregrinus, 1988.

Oleksiy S. Kim received the M.S. and Ph.D. degrees from the National Technical University of Ukraine, Kiev, in 1996 and 2000, respectively, both in electrical engineering. In 2000, he joined the Antenna and Electromagnetics Group at the Technical University of Denmark (DTU). He is currently an Associate Professor with the Department of Electrical Engineering, ElectroScience Section, DTU. His research interests include computational electromagnetics, metamaterials, electrically small antennas, photonic bandgap and plasmonic structures.

Olav Breinbjerg (M’87) was born in Silkeborg, Denmark, on July 16, 1961. He received the M.Sc. and Ph.D. degrees in electrical engineering from the Technical University of Denmark (DTU), in 1987 and 1992, respectively. Since 1991 he has been on the faculty of the Department of Electrical Engineering (formerly Ørsted1DTU, Department of Electromagnetic Systems, and Electromagnetics Institute) where he is now Full Professor and Head of the Electromagnetic Systems Group including the DTU-ESA Spherical Near-Field Antenna Test Facility. He was a Visiting Scientist at Rome Laboratory, Hanscom Air Force Base, Massachusetts, in the fall of 1988 and a Fulbright Research Scholar at the University of Texas at Austin, in spring 1995. His research is generally in applied electromagnetics—and particularly in antennas, antenna measurements, computational techniques and scattering—for applications in wireless communication and sensing technologies. At present, his interests focus on metamaterials, antenna miniaturization, and spherical near-field antenna measurements. He is the author or coauthor of more than 40 journal papers, 100 conference papers, and 70 technical reports, and he has been, or is, the main supervisor of 10 Ph.D. projects. He has taught several B.Sc. and M.Sc. courses in the area of applied electromagnetic field theory on topics such as fundamental electromagnetics, analytical and computational electromagnetics, antennas, and antenna measurements at DTU, where he has also supervised more than 70 special courses and 30 M.Sc. final projects. Furthermore, he has given short courses at other European universities. He is currently the coordinating teacher at DTU for the 3rd semester course 31400 electromagnetics, and the 7–9th semester courses 31428 advanced electromagnetics, 31430 antennas, and 31435 antenna measurements in radio anechoic chambers. Prof. Breinbjerg received a U.S. Fulbright Research Award in 1995. Also, he received the 2001 AEG Elektron Foundation’s Award in recognition of his research in applied electromagnetics. Furthermore, he received the 2003 DTU Student Union’s Teacher of the Year Award for his course on electromagnetics.

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Arthur D. Yaghjian (S’68–M’69–SM’84–F’93 –LF’09) received the B.S., M.S., and Ph.D. degrees in electrical engineering from Brown University, Providence, RI, in 1964, 1966, and 1969. During the spring semester of 1967, he taught mathematics at Tougaloo College, MS. After receiving the Ph.D. degree he taught mathematics and physics for a year at Hampton University, VA, and in 1971 he joined the research staff of the Electromagnetics Division of the National Institute of Standards and Technology (NIST), Boulder, CO. He transferred in 1983 to the Electromagnetics Directorate of the Air Force Research Laboratory (AFRL), Hanscom AFB, MA, where he was employed

as a Research Scientist until 1996. In 1989, he took an eight-month leave of absence to accept a visiting professorship in the Electromagnetics Institute of the Technical University of Denmark. He presently works as an independent consultant in electromagnetics. His research in electromagnetics has led to the determination of electromagnetic fields in materials and “metamaterials,” the development of exact, numerical, and high-frequency methods for predicting and measuring the near and far fields of antennas and scatterers, the design of electrically small supergain arrays, and the reformulation of the classical equations of motion of charged particles. Dr. Yaghjian is a Life Fellow of the IEEE, has served as an Associate Editor for the IEEE and URSI, and is a member of Sigma Xi. He has received Best Paper Awards from the IEEE, NIST, and AFRL.

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Simple Excitation Model for Coaxial Driven Monopole Antennas Luiz C. Trintinalia, Member, IEEE

Abstract—A new excitation model for the numerical solution of the electric field integral equation (EFIE) applied to arbitrarily shaped monopole antennas fed by coaxial lines is presented. This model yields a stable solution for the input impedance of such antennas with very low numerical complexity and without the convergence and high parasitic capacitance problems associated with the usual delta gap excitation. Index Terms—Antenna feeds, antenna input impedance, electric field integral equation (EFIE), Galerkin method, method of moments (MoM).

I. INTRODUCTION

I

N 1995 Junker et al. [1] introduced a new delta gap source model that improved the convergence of the numerical solution of the electric field integral equation (EFIE) using thin wire theory. It was shown in that paper that the usual delta gap model suffers from convergence problems, especially for thicker cylindrical antennas. As pointed out, it was the imaginary part of the admittance that presented the worst convergence as the number of segments was increased, for the usual delta gap. The novel delta gap model then proposed, on the other hand, was shown to be very stable and presented a much faster convergence. That model used an axial directed Gaussian excitation field, with a standard deviation equal to half conductor radius. Since this field is quite smooth, a stable solution could be achieved for any number of segments used in the discretization. That model was even extended to better fit the magnetic frill model, but still for cylindrical monopole antennas [2]. The idea of this delta gap model is similar to the Popovic belt generator [3], which provided a distributed excitation field, but with a raised cosine distribution function rather than a Gaussian distribution but, still, only applicable to cylindrical monopoles, for which only the axial component of the excitation field is necessary. New three-dimensional EFIE codes, used to analyze antennas that cannot be modeled by thin wires, usually rely on the simple delta gap source to model the feed point of the radiation structure. Makarov [4], for example, presents an implementation of such a code, where the standard Rao, Wilton and Glisson [5]

Manuscript received September 23, 2009; revised November 23, 2009; accepted December 01, 2009. Date of publication March 29, 2010; date of current version June 03, 2010. This work was supported in part by the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq). The author is with the Department of Telecommunications and Control Engineering, Escola Politécnica da Universidade de São Paulo, São Paulo, SP 05508-900, Brazil (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2010.2046872

basis functions are used to compute the input impedance of arbitrarily shaped antennas. As we will show later, the simple delta gap source also presents convergence problems for these 3-D structures, therefore better excitation models are needed. For structures that are fed by a coaxial line, the magnetic-frill excitation model [6] usually presents very good results in predicting the input impedances of generic monopole antennas, provided the actual field distribution at the coaxial aperture is close to the transverse electromagnetic (TEM) field distribution, which seems to be the case for most practical problems. The only drawback of this model is that, for radiation structures that cannot be thin wire modeled, the field computation requires the evaluation of very complex expressions involving, even in simplified forms, integrals of elliptic functions [7]. Of course, more complex feeding structures demand also more complex excitation models such as the one proposed by Lockard and Butler [8], but still a great number of monopole antennas use a simple coaxial feed, flush mounted on a ground plane, for which simple excitation models should suffice. The need exists, then, for some simple excitation field, similar to the model of Junker et al. [1], which could be used for complex three-dimensional structures fed by coaxial lines. This new model must include also a radial component of the field, besides the axial one, similar to the radial component present in the magnetic frill model. Both components should decay axially and radially and some dependence on the inner and outer coaxial radii is expected. In an attempt to solve these issues, a new excitation model is here presented. This new model has very low numerical complexity and does not suffer from the convergence and high parasitic capacitance problems associated with the usual delta gap excitation, as it will be shown. It should be pointed out that neither the model proposed here nor the other simplified models corresponds to true physical excitations. The idea behind proposing these models is to obtain a simple model that can lead to accurate results for predicting the input impedance under certain restrictions (aperture of the coaxial line small compared to wavelength). This paper is organized as follows. Section II of this paper presents the derivation of this new excitation model. Section III shows some numerical examples comparing the behavior of this new model with the magnetic frill and the simple delta gap model. Concluding remarks are presented in Section IV.

II. FORMULATION For a coaxial feed, situated at plane, as shown in Fig. 1, we intend to derive an expression for an excitation field similar

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. interfaces and to a non-zero divergent of the field for Although it is possible, then, that this might lead to some parasitic capacitance associated with this model, we hope that it will have a less severe effect than the usual delta gap excitation, and the examples in the next section do show that this is the case. Therefore, the associated potential must not vary with in these regions. Since this potential must be continuous at and must be null for , we must set (3) This condition implies that

Fig. 1. Geometry of a coaxial feed on a ground plane situated at z

= 0.

(4)

to the field produced by the magnetic frill but with a much lower numerical complexity. This excitation field must have the axial component (along the axis) decaying with the distance to the feed, while incorporating also a radial variation and a radial component. For simplicity, we will admit that the expression of this field can be written as the product of two independent functions, with independent variations along and (the distance to the axis). Also, we will impose that the integral of this excitation electric field, along any path starting at the inner coaxial conductor and ending at the outer one , should be equal to 1 V (for 1 V excitation), and that its diver. With these gence should be zero in this region restrictions, the electric field should be the gradient of a scalar field, with axial symmetry, satisfying Laplace’s equation,

where is the th zero of the Bessel function of the first kind and zeroth order. Since we are looking for an excitation that is as close to the magnetic frill as possible near the coaxial aperture, . Therefore, we choose the potential must not oscillate for (5) Finally, imposing unity potential (for 1 V excitation) at , we obtain

(6) and the corresponding electric field components result

(1) and that can be decomposed as the product of one function of and one function of . Since we expect the field to decay very rapidly with the distance to the feed, and since the dimension of the feed should be much smaller than the wavelength, an expression of the field which is independent of frequency (static field) might be, in fact, a good approximation, if we want to keep it simple. The solution of (1) with these restrictions is of the form (2) where and are the zeroth order Bessel functions of the first and second kind respectively. We will choose the first term to represent our field because the first zero of is greater than the first zero of , leading to a higher value for the decaying constant , as we will see. This higher value seemed to lead to results closer to the ones with the magnetic frill model in our simulations. Because the magnetic frill field does not have radial compoor at (the plane where the coaxial nent for interface lies) we set the radial component of the new model to and . These restrictions, of be equal to zero for course, will lead to discontinuity of the electric field at these

(7) where is the first order Bessel function of the first kind. These expressions present low numerical complexity, are quite simple to implement and, unlike the usual delta gap, you do not have to specify which edges are part of the excitation. All that is required is that the antenna geometry be defined such as . the coaxial feed is at the origin, with the ground plane at Of course some additional work is necessary to compute the inner product between the testing functions and this excitation field, but this can be done by Gaussian quadrature integration which, because of the simplicity of the excitation field, has only a negligible effect on the total computation time. With this excitation model, using some standard EFIE formulation, such as the ones presented by Rao et al. [5] or by Trintinalia and Ling [9], and a Galerkin procedure, we obtain a matrix equation relating the current coefficients, in vector , with the weighted excitation field, in vector , (8)

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where is a square matrix, whose elements at the th row and th column represent the electric field produced by th basis function weighted by the th weight function (which is the same as the basis function for the Galerkin procedure), and are given by

(9) is the angular frequency, and and are the magnetic permeability and electric permittivity of the medium, is the freespace Green’s function, given by

Fig. 2. “Bow-tie” monopole antenna with 90 degree aperture angle and h = 50 cm length over an infinite ground, z = 0, fed by a 50 coaxial line with inner conductor radius equal to 1 mm, discretized in 378 triangular facets.

(10) is a vector of coefficients given by (11) is the th basis/testing function used in the expansion of the surface current on the antenna, (12) are the observation and source points, and finally, and respectively. Once the current coefficients are determined, the complex power provided by the excitation field can be calculated as Fig. 3. Input admittance of a “bow-tie” monopole antenna with 90 degree aperture angle and length h cm, as a function of h=. Simulation results with magnetic frill and the new excitation model, for 1598 triangular patches, compared against Brown and Woodward measurements.

= 50

(13) Since this power can also be written as a function of the input voltage and current, (14) we can write

(15) since we are imposing a unity input voltage. So, this simple expression gives us the input admittance of the antenna. III. RESULTS Using the EFIE/Galerkin formulation detailed in [9] with the excitation model presented in Section II, we analyzed first a “bow-tie” monopole antenna with 90 aperture angle, length cm, over an infinite ground plane, shown in Fig. 2.

The antenna is fed by a coaxial line with inner radius mm and outer radius mm. The results obtained for the magnetic frill excitation and for the new excitation model, are shown in Fig. 3, compared against Brown and Woodward [10] measurements. We can see that both excitation models present very similar results and both compare quite well with measurements, despite the fact that, in the measurements, the frequency of the feed was maintained at a constant value and the size of the antenna was changed to cover a range of antenna size to wavelength ratios. In order to verify convergence, Fig. 4 shows, for a single frequency (300 MHz) the behavior of both excitation models as a function of the number of triangular facets. Again, we see that the convergence trend is very similar for both models and good results could be obtained with about 400 facets. Still, the worst problems in terms of convergence arise for thick feeding structures in terms of wavelength, as it was already pointed out in [1]. In order to analyze these scenarios, we also simulated two structures using a coaxial feed with internal

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Fig. 4. Input admittance of a “bow-tie” monopole antenna with 90 degree apercm length at 300 MHz h= : simulated ture angle and length h with magnetic frill and the new excitation model as a function of the number of triangular patches used in the discretization, compared against Brown and Woodward measurement.

= 50

(

= 0 5)

=

Fig. 5. Input admittance of a hollow cylindrical monopole with radius a : cm radius and length h cm as a function of h=. Computed values for three different excitation models, with 818 triangular facets, and measurement values obtained by King.

11 29

= 50

diameter reaching up to 0.2 : a cylindrical monopole and a conical monopole. cm For a hollow cylindrical monopole, with radius cm, fed by a coaxial line with inner radius and length cm and outer radius cm, corresponding to a 10 characteristic impedance, results were obtained for the three models: delta gap, magnetic frill and the new model. The results obtained for a discretization with 818 triangular facets are shown in Fig. 5 together with measurements obtained by King [11].

Fig. 6. Current distribution for the new model excitation of a half wavelength monopole discretized with 818 triangular facets.

We see, again, that the results for the input conductance are all very close to the measurements, with a slightly higher discrepancy obtained for the magnetic frill at the higher frequencies. This is probably due to the fact that the expressions used to approximate the magnetic frill excitation field at these higher frequencies (diameter of the outer coaxial conductor at the highest frequency) do not hold anymore. That come as a consequence of the difficulty in numerically implement the magnetic frill model and shows the advantage of a simpler formulation. It must be pointed out that, at these higher frequencies, the influence of the higher order modes at the coaxial aperture should be very severe and even the magnetic frill model is probably not suitable to represent the real excitation field over the monopole structure. For the susceptances, on the other hand, we see a huge discrepancy for the delta gap model: the values obtained are more than twice the measured ones. The new model, on the other hand, gives accurate results over all frequencies. Figs. 6 and 7 show the current distribution obtained with the new model and the delta gap excitation, respectively. The gray scale represents the absolute value of the current density in A/m , for a 1 V excitation. We can see that, close to the feed , the current is much more intense for the delta gap A/m ) than for the new model ( A/m ). Far from ( the feed, both models give similar values for the current density. A convergence test is shown in Fig. 8, where we show only the input susceptance at 225 MHz, for all excitation models, as a function of the number of triangular facets used in the discretization. We can clearly see that the delta gap model not only converges slowly than the other excitation models, but converges to

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= 0 25

Fig. 9. Conical monopole antenna with length h : m and flare angle coaxial line with inner conductor of radius 2 cm, 45 degrees fed by a 50 discretized by 1744 triangular facets.



Fig. 7. Current distribution for the delta gap excitation of a half wavelength monopole discretized with 818 triangular facets.

=

Fig. 10. Input admittance of a conical monopole antenna with length h cm as a function of h=. Computed values for three different excitation models with 1744 triangular facets.

50

= 50

Fig. 8. Input susceptance of a cylindrical monopole with length h cm at : 225 MHz h= simulated with delta gap, magnetic frill and the new excitation model as a function of the number of triangular patches used in the discretization, compared against King’s measurement.

(

= 0 375)

an incorrect value. The new model, on the other hand presents fast convergence and good accuracy. Finally, for the conical monopole, which is an important example, since the radial component of the excitation plays an important role in this case, the same analysis was performed. The geometry analyzed is shown in Fig. 9. The hollow conical cm, flare angle and is fed monopole has length cm and outer radius by a coaxial line with inner radius cm, corresponding to a 50 characteristic impedance.

In Fig. 10 we plotted its input admittance as a function for the three excitation models using a 1744 triangular facets in the discretization of the cone surface. Once more, we observe the same behavior of the previous example where, again, we see a huge discrepancy in the susceptance for the delta gap model, whereas the new model gives results very close to the ones obtained with the magnetic frill model. Also in terms of convergence, Fig. 11 shows, for a frequency equal to 250 MHz , that this new model presents, again, faster convergence than the usual delta gap, which, again, converges to a value much higher than the one obtained with the magnetic frill model, showing that the delta gap model presents a much higher parasitic capacitance associated with it. IV. CONCLUSION A new excitation model for three-dimensional EFIE analysis of monopole antennas over a ground plane has been presented. The model depends only upon the inner and outer radius of the

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REFERENCES

=

Fig. 11. Input susceptance of a conical monopole antenna with length h cm at 250 MHz h= : simulated with delta gap, magnetic frill and the new excitation model as a function of the number of triangular patches used in the discretization.

25

(

= 0 208)

coaxial excitation, it can be used with any EFIE/MoM code for 3D antenna analysis, and its numerical implementation is much simpler than the magnetic frill model. The examples analyzed here show that this new model yields accurate solutions for the input impedance of monopole antennas over a ground plane, having a faster convergence and higher accuracy than the usual delta gap model, even for thick coaxial feeding structures. Though not tested yet, this model might also be used for monopoles mounted on finite ground planes or even dipoles, but its accuracy in these cases needs to be verified. Also, the analysis of structures fed by coaxial lines with higher input impedances should also be addressed in the future to verify the possible limitations of the present model.

[1] G. Junker, A. Kishk, and E. A. Glisson, “A novel delta gap source model for center fed cylindrical dipoles,” IEEE Trans. Antennas Propag., vol. 43, no. 5, pp. 537–540, May 1995. [2] G. P. Junker, A. W. Glisson, and E. A. A. Kishk, “A modified extended delta source model,” Microw. Opt. Technol. Lett., vol. 12, no. 3, pp. 140–144, June 1996. [3] B. Popovic, “Thin monopole antenna—Finite-size belt-generator representation of coaxial-line excitation,” Proc. Inst. Elect. Eng., vol. 120, no. 5, pp. 544–550, May 1973. [4] S. Makarov, “MoM antenna simulations, with Matlab: RWG basis functions,” IEEE Antennas Propag. Mag., vol. 43, no. 5, pp. 100–107, Oct. 2001. [5] S. Rao, D. Wilton, and E. A. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag., vol. 30, no. 3, pp. 409–418, May 1982. [6] L. Tsai, “A numerical solution for the near and far fields of an annular ring of magnetic current,” IEEE Trans. Antennas Propag., vol. 20, no. 5, pp. 569–576, Sept. 1972. [7] A. Sakitani and E. S. Egashira, “Simplified expressions for the near fields of a magnetic frill current,” IEEE Trans. Antennas Propag., vol. 34, no. 8, pp. 1059–1062, Aug. 1986. [8] M. Lockard and E. C. Butler, “Feed models for coax-driven monopole and dipole antennas,” IEEE Trans. Antennas Propag., vol. 54, no. 3, pp. 867–877, Mar. 2006. [9] L. Trintinalia and E. H. Ling, “First order triangular patch basis functions for electromagnetic scattering analysis,” J. Electromagn. Waves Applicat., vol. 15, no. 11, pp. 1521–1537, Nov. 2001. [10] G. H. E. Brown and O. M. Woodward, “Experimentally determined radiation characteristics of conical and triangular antennas,” RCA Rev., vol. 13, pp. 425–452, Dec. 1952. [11] R. King, “Measured admittances of electrically thick monopoles,” IEEE Trans. Antennas Propag., vol. 20, no. 6, pp. 763–766, Nov. 1972. Luiz C. Trintinalia (S’94–M’97) was born in São Paulo, Brazil, on June 12, 1964. He received the B.S. and M.S. degrees from Escola Politécnica da Universidade de São Paulo, Brazil, in 1987 and 1992, respectively, and the Ph.D. degree from the University of Texas at Austin, in 1996, all in electrical engineering. In 1987, he joined the faculty of the Escola Politécnica da Universidade de São Paulo where he is presently an Associate Professor in the Department of Telecommunications and Control Engineering. His main areas of interest are computational electromagnetics and electromagnetic signal processing.

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A Method for Seeking Low-Redundancy Large Linear Arrays in Aperture Synthesis Microwave Radiometers Jian Dong, Qingxia Li, Member, IEEE, Rong Jin, Yaoting Zhu, Quanliang Huang, and Liangqi Gui

Abstract—For one-dimensional aperture synthesis microwave radiometers, the optimal placement of antenna elements in a low-redundancy linear array (LRLA) is difficult when large numbers of elements are involved. In this paper, the general structure of large LRLAs is summarized first, and then a novel stochastic optimization technique, ant colony optimization (ACO), is applied to the search for low redundancy arrays. By combining the general structure with the ACO procedure, an efficient method is proposed for a rapid exploration for optimal array configurations. Numerical studies show that the method can generate various large LRLAs with lower redundancy than the previous algorithms did and the computational cost is greatly reduced. Based on the method, several analytical patterns for LRLAs are further derived, which can yield various array configurations with very low redundancy in nearly zero computation time. Both the method and the resulting configurations can be utilized to facilitate antenna array design in synthetic aperture radiometers with high spatial resolution. Index Terms—Ant colony optimization (ACO), aperture synthesis radiometer, redundancy, thinned linear arrays.

I. INTRODUCTION NTERFEROMETRIC aperture synthesis technique is an attractive means for improving spatial resolution in passive microwave remote sensing of the Earth [1]–[3]. Different spatial frequencies are sampled by cross-correlating antenna pairs with different separations, and all sampled spatial frequencies can then be inverted to estimate the original brightness distribution of a scene. The question of how to place the antenna elements in order to optimally sample the spatial frequency spectrum of a scene has received considerable attention. For one-dimensional aperture synthesis radiometers, minimum redundancy linear arrays (MRLAs) are usually the choice [1], [4], [5], which provide the largest instantaneous set of contiguous spatial samples for a given number of antennas, and therefore achieve the highest spatial resolution. Besides, in view of different influence on the performance of synthetic aperture radiometers (e.g., signal-to-noise

I

Manuscript received April 06, 2009; revised October 23, 2009; accepted December 01, 2009. Date of publication March 29, 2010; date of current version June 03, 2010. This work was supported in part by the National Science Foundation of China (NSF60705018) and in part by the National High Technology Research and Development Program of China (No. 2006AA09Z143). The authors are with the Department of Electronics and Information Engineering, Huazhong University of Science and Technology, 430074 Wuhan, 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.2010.2046846

ratio and radiometric sensitivity) [1], [6], [7] and different manufacturing difficulties in antenna engineering [8], various candidates of array configurations with similar spatial resolution are also desirable in practical applications. The problem of finding out optimum MRLAs has been first investigated as a purely number-theoretic issue of “difference basis” [9]–[12], in which each antenna of the array is assigned an integer representing its position, the problem is then translated into constructing a set of integers which generates contiguous differences from 1 up to the largest possible number. Leech [9], [10] demonstrated that for optimum MRLAs, for , where is the redundancy quantitatively defined as the number of possible pairs of antennas divided by the maximum spacing : . Optimum solutions are known for [9] and they have the lowest possible redundancy . With more elements, the number of possible array configurations rapidly becomes too large to sort through completely [5], [8]. Several earlier attempts have been made to construct large low-redundancy linear arrays (LRLAs). Bracewell [13] proposed a systematic arrangement of antennas and the value of approaches 2 for a large value of . Ishiguro [8] constructed large LRLAs from subarrays of small minimum redundancy arrays, but the redundancy of the resulting arrays would increase monotonically with the number of elements and LRLAs with the prime number of elements can not be constructed by Ishiguro’s method. With the help of powerful modern computers, some numerical algorithms, like greedy algorithm [14], tree-searching-based algorithm [15], and simulated annealing (SA) algorithm [4], were also used to search for LRLAs. However, the contradiction between solution quality and computational efficiency limits practical applications of these algorithms. Camps [16] proposed a fast technique for the direct synthesis of low-redundancy large arrays and summarized main LRLA configurations for which are the best and most extensive results known up to date. In this paper, the general structure of large LRLAs is summarized, and then an efficient method using the general structure to guide an ant-colony-optimization-based search is proposed for a rapid exploration of optimum LRLAs. Some new array configurations obtained by this method are also presented, superior to the results given by Camps [16]. Furthermore, several useful analytical patterns for LRLAs are discovered, which, for any given number of elements, can yield various array configurations with very low redundancy in nearly zero computation time.

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II. THE METHOD FOR SEEKING LOW-REDUNDANCY LARGE LINEAR ARRAYS Determining the optimal thinned linear array for a given number of elements is a constrained multidimensional nonlinear optimization problem expressed as (1) where the objective function to be maximized is the largest ; the condifference between element positions means that there are no missing straint spatial samples (i.e., no holes) within the segment [0, ]. The difficulty in solving this problem is that the inverse mapping from spatial sample sets to array configurations is analytically unknown. Usually, a numerical implementation described as follows is preferable: Given element number , an initial maximum difference is chosen to determine whether the elements can be placed at the range [0, ] so that all the required . If such a conspacings are present, i.e., figuration is found, can be increased and the procedure repeats until grows too large. The conventional numerical algorithms [4], [14], [15] explore all possible array configurations in the solution space. Since the solution space expands exponentially with the number of antennas, there is an intrinsic contradiction between solution quality (redundancy) and computation time, i.e., reducing computation time would lead to a poor solution [14], while obtaining a good solution would require large computation time [4], [15]. The method proposed in this paper uses the general structure of large LRLAs to effectively guide an optimization search executed by a novel stochastic optimization technique, ant colony optimization (ACO) [17]–[19], and consequently ensures obtaining low-redundancy large linear arrays while greatly reducing the size of the solution space, therefore greatly reducing computation time. The superiority of ACO for the LRLA optimization problem would come from the facility of inserting the general structure of LRLAs as a-priori information in the ACO search process. A. General Structure of Low-Redundancy Large Linear Arrays Ishiguro [8] has found that there are apparent regular patterns in the configurations of optimum LRLAs for a large value of , i.e., the largest spacing between successive pairs of antennas repeats many times at the central part of the array. Such LRLA patterns were presented by Ishiguro and improved by Jun [20]. Similar regular patterns have also been found by Camps [16]. The authors have found that a common general structure of large LRLAs can be further summarized as follows from all above patterns: (2) where each item in the sequence denotes the spacing between denotes that the “seed” adjacent antennas in an LRLA; (i.e., the largest spacing) repeats times, [16], and different values of would affect the redundancy of the

Fig. 1. LRLA geometry with the general structure expressed by (2).

array while still keeping all spacings; and denote the and lengths of the sub-sequences, , respectively, and are related to the seed by (3) The LRLA geometry with the general structure expressed by (2) is illustrated in Fig. 1. To better clarify it, a numerical example is given below. Beginning from the pattern , , and ), a family {1,1,4,2,9,9,3,7,3,1} ( of LRLAs, such as 11 40 (this notation in the paper means and ), 12 49, 13 58, and so an array with on, is generated by inserting antennas at the center spaced 9. From the comparison among the patterns described in [8], [16], [20], it can be further concluded that equality or a slight difference between and would lead to LRLAs, whereas a large difference between them would cause relatively high redundancy arrays. To ensure obtaining LRLAs, we restricted and in the general structure by (4) For a given antenna number and maximum difference , unoccupied positions are left for the remaining antennas because 0, 1, and must be included in any array solution that contains all differences between 1 and (that means ). The total number of possible configurations is (5) where is the number of combinations of items taken at a time. When the solution space is constrained by the general structure expressed in (2), the elements corresponding to the repeating seeds can be considered as a particular “element” in of possibilities in this the array, and then the total number case is

(6) For and , . Given and , . It is clear that the total number of tries is drastically reduced by introducing the constraint of the general structure. Since the repeating seeds are just located at the central part of the sequence and the spacings larger than the seed do not exist in , the actual number of tries is much less than the number estimated here.

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B. Ant Colony Optimization Technique Ant colony optimization (ACO) is a newly introduced stochastic optimization algorithm, which has been inspired by the way ants find the optimal path from their nest to the food source [17]. In a colony, each ant affects the path choice of the others by marking the path it follows with a chemical called pheromone, producing a positive feedback for the shortest path. By exploiting these ideas, the ACO technique has been initially developed to cope with hard combinatorial problems, such as the traveling salesman problem, and recently applied to some electromagnetic problems [21], [22]. The basic formulations of ACO are presented in the Appendix. The main salience of ACO could come from the positive feedback mechanism that it supplies and also from its simplicity. Different from SA [4], ACO is a population-based technique. This means that it will produce many optimized configurations simultaneously and allow for potential parallel implementation in modern computers. It also allows many different initial guesses for the configurations to be considered at once. Unlike genetic algorithms (GAs), ACO does not need encoding or decoding operations on chromosomes and hence simplifies the optimizations. In ACO for the LRLA optimization problem, the array configuration is represented by an -dimensional integer vector (7) where represents the th element position. Each ant constructs an -node path representing a candidate solution and its performance is quantified by , i.e., the number of holes. Fig. 2 illustrates the behavior of ants constructing paths according to the random proportional rule (17) in the Appendix. Each ant is navigated by its own experience (the heuristic information ) and the knowledge from other ants (the pheromone trail ). The shorter the ant’s path, the higher the amount of pheromone the ant deposits on the nodes of its path. This in turn leads to the fact that these nodes have a higher probability of being selected in the subsequent iterations of the algorithm. It is the positive feedback given through the pheromone update by ants that triggers the discovery of good paths. The heuristic information, dependent on the partial solution constructed so far, is related to the number of missing differences in this problem, i.e., (8) where denotes the number of holes when adding node to a partial path under construction. The inclusion of the heuristic information will guide the ant-based solution construction and obviously improve solution quality but at the cost of added computational expense. C. ACO-Based Method Exploiting General Structure of LRLAs ACO has demonstrated its effectiveness in locating the global optimum in nonlinear optimization problems. However, when large numbers of elements are involved in the LRLA problem, high computational cost is still unaffordable. This drawback

Fig. 2. Schematic representation of different paths constructed by ants. Each ant represents a candidate solution and its performance is quantified by h( ). Note that the position 0, 1, and L must be included in any path. At each construction step, each ant decides which node to visit according to the pheromone trail  and the heuristic information  on each node.

X

X

could be effectively avoided if the available a-priori information on the geometrical structure is incorporated into the stochastic search. We have developed a strategy, combining the general structure of LRLAs with an optimization search, for a rapid exploration of optimum LRLAs. ACO, not other metaheuristics, is chosen to carry out the optimization search because of the facility of inserting the general structure of LRLAs as a-priori information in the ant-based solution construction. With this combination, the convergence of the ACO procedure can be speeded up because of the reduced dimension of the solution space, whereas the global exploration capability is still maintained. From Section II-A, we know that by the repetition of the seed, larger LRLAs can be easily generated from parent arrays with a higher redundancy. The remaining problem concerns how to find such a parent redundancy array, i.e., how to determine the sub-sequences, and , corresponding to a given seed . To obtain the simplest parent redundancy array, the repetition number of the seed is usually chosen to be as few as possible. In ACO modeling shown in Fig. 3, the antenna elements related to ) can the seed (i.e., antennas with ID be taken as a particular “element” because of the fixed spacing between successive pairs of elements. The spacings between other adjacent elements in the array are limited to be less than and satisfy that the sum of them equals to . These mechanisms are aimed at constraining ACO to search array configurations consistent with the general structure of LRLAs. Also, with these mechanisms, it can be expected that the ant-based solution construction would be speeded up because of the use of a-priori geometrical information and the reduced neighborhood for the current solution component. When a parent redundancy array is obtained by the ACO search process described above, we can grow it to a series of large LRLAs in nearly zero time by inserting a seed repeatedly while keeping all possible antenna separations. In summary, the ACO-based method exploiting general structure of LRLAs is schematized as follows: • Initialization • While (termination condition not met) do

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TABLE II PARENT REDUNDANCY ARRAYS WITH n

= 15 AND L = 67=68=69

Fig. 3. Schematic representation of ant-based solution construction by exploiting the general structure of LRLAs expressed in (2). The antenna elements in the dashed block are considered as a particular “element” because of the fixed spacing r between successive pairs of elements.

TABLE I ACO PARAMETER VALUES FOR THE NUMERICAL COMPUTATION

— Ant-based solution construction (exploiting the general structure of LRLAs expressed by (2)) — Pheromone update • End while • Generating a set of LRLAs by the repetition of the seed in parent redundancy arrays reaches zero. where the termination condition means that For brevity, we will refer to this method as the constrained ACO.

Fig. 4. Minimum number of holes versus number of iterations using the standard ACO.

III. NUMERICAL VALIDATION In the following, numerical examples are presented to validate the effectiveness of the constrained ACO. The optimization is executed on an Intel Pentium 2.8 GHz CPU with a 1 GB RAM. The assumed parameters for the ACO-based procedures are listed in Table I, where , and are defined as in the Appendix; is the number of ants; is the number of antennas; MAXIT is the maximum number of iterations; is the given maximum difference. , we want to find the corresponding Given a seed parent redundancy arrays. From (3), we have . , the total number of elements in a parent redunAssuming . In the ACO search process, the spacing dancy array is between the three antennas at the central part of the array (e.g., antenna No. 7, 8, and 9) is fixed at 13. Numerical computations show that it only takes several tens of seconds to find out such parent redundancy arrays as listed in Table II. at the center Then, by repeatedly inserting the seed of the array configurations, we can easily obtain a few series of large LRLAs such as 19 119/120/121, 20 132/133/134, 145/146/147, 22 158/159/160, and so on. However, 21 it would probably take several hours to find out these LRLAs by applying only the standard ACO procedure described in Section II-B.

Fig. 5. Minimum number of holes versus number of iterations using the constrained ACO (r and l ).

= 13

=7

More quantitatively, Fig. 4 and Fig. 5 show the convergence rates of the standard ACO and the constrained ACO for a 20 133 LRLA. Table III gives the time cost for a 20 133 LRLA, by the standard ACO and the constrained ACO, respectively. From Fig. 4, Fig. 5, and Table III, it can be seen that the constrained ACO converges much more quickly than the standard ACO and the blindness at the start of the search is significantly reduced with the guide of the general structure of LRLAs. Moreover, the time required for each iteration of the constrained ACO

DONG et al.: A METHOD FOR SEEKING LOW-REDUNDANCY LARGE LAs IN APERTURE SYNTHESIS MICROWAVE RADIOMETERS

TABLE III COMPARISON OF TIME COST BETWEEN THE STANDARD ACO AND THE CONSTRAINED ACO FOR A 20 133 LRLA



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sulting arrays, it can be found that the least redundancy would usually occur when the repetition number is chosen to be approximately half the seed , i.e., too small or too large value of would result in a relatively high redundancy array. V. ANALYTICAL PATTERNS FOR LOW-REDUNDANCY LINEAR ARRAYS

Fig. 6. Achieved redundancy values, R, using our method as compared to Ruf’s method [4] and Camps’ method [16]. Also, the theoretical reference values of R are plotted, which were given in [23] but have never been achieved up to date.

is highly reduced owing to the use of a-priori geometrical information and the reduced neighborhood for the current solution component under construction. In brief, the constrained ACO can find out various large LRLAs at drastically lower computational cost than the standard ACO does, and therefore significantly alleviate the contradiction between solution quality and computation time. IV. NEW LRLAS AND COMPARISON WITH ARRAYS OBTAINED BY EXISTING METHODS To save space, Table IV only presents a small portion of new results obtained by the method described in this paper. Array configurations and redundancy are compared to the collections listed in Table I in [16]. The nomenclature used to denote an integers array of antennas is a bracketed sequence of indicating the spacing between adjacent antennas. It is more convenient to represent the structural characteristic of an array by its spacings instead of the absolute antenna means repetitions of the positions. In the table, notation interelement spacing . One significant and useful result of this method is some sets of new array configurations that perform better than previous ones [4], [8], [14], [16]. In order to better assess the effectiveness of the method, Fig. 6 shows the achieved redundancy values, , as compared to those obtained by Ruf’s method [4] and Camps’ –30. For the element method [16], respectively, for number larger than 19, our method can generate LRLAs with lower redundancy than the previous methods did. In addition, by repeatedly inserting the seed at the center of a given parent array and investigating the redundancy of the re-

One peculiarity of the ACO-based method exploiting general structure of LRLAs, quite different from other existing algorithms, is that the solution resulting from this method is not an isolated redundancy array, but a class of redundancy arrays with the same seed and sub-sequences. Further, there are inherent structural similarities among some subclasses of redundancy arrays, from which analytical regular patterns for LRLAs can be derived. For example, by comparing the three subclasses of redundancy arrays shown in Table V (for brevity, only one paradigm for each subclass is listed), an analytical pattern can be easily generalized. It is easy to verify that this analytical pattern is also true for , corresponding to LRLAs with element number . Since , any other can be expressed as , where . If w is positive (negative), the seed will be inserted into (deleted from) . the central portion of the array. Correspondingly, Similarly, other analytical patterns for LRLAs are also obtained by us. All these analytical patterns are summarized as follows (sorted by the seed mod 4). Type I 1 a) are both positive integers and satisfy (approximately half the seed) where . In and this case, . The redundancy

(9) b) where

. In this case,

and The redundancy

.

(10) c) where and The redundancy

. In this case, .

(11) Type II a) 1This pattern is also given in a more complex expression by Wichmann [24] and Pearson [25], respectively.

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TABLE IV COMPARISON OF THE NEW LRLAS OBTAINED BY THE PROPOSED METHOD WITH EXISTING LRLAS. SYMBOLS n: NUMBER OF ANTENNAS, L: MAXIMUM ARRAY SPACING, R: REDUNDANCY

= 20

= 135=136. = 4, L = 6.

The maximum difference L is not contiguous, e.g., for n , we have not found an LRLA with L Mirror image arrays are considered equivalent, e.g., [1 3 2] and [2 3 1] are the same MRLA with n

where and

. In this case, . The redundancy

b) where and

(12)

. In this case, . The redundancy

(13)

DONG et al.: A METHOD FOR SEEKING LOW-REDUNDANCY LARGE LAs IN APERTURE SYNTHESIS MICROWAVE RADIOMETERS

c) where and

. In this case, . The redundancy

TABLE V AN ANALYTICAL PATTERN FOR LRLAS (P

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= 1, 2, 3)

(14) Type III a) where and

. In this case, . The redundancy (15)

patterns for LRLAs might be derived from the solutions of this method. APPENDIX BASIC FORMULATIONS OF THE ACO TECHNIQUE

b) where and

. In this case, . The redundancy (16)

All the analytical patterns described herein have very low redundancy (within 1.5), which is lower than that in [8], [13], [16], [20], and for any given number of antennas, various array configurations can be yielded from these patterns in nearly zero computer time. In view of different mutual coupling effects, different radiometric sensitivities, and different manufacturing difficulties, these analytical patterns with similar low redundancy would be helpful to the selection of proper array configurations in engineering applications, especially applications at millimeter wave band. Besides, the distributed repetition embedded in these analytical patterns will facilitate the construction of space deployable structures. VI. CONCLUSION For one-dimensional aperture synthesis microwave radiometers, it is difficult to find optimum LRLAs when large numbers of elements are involved. Existing algorithms suffer from high redundancy or large computational load. In this paper, an efficient method, combining the general structure of large LRLAs with an ACO-based optimization search, is proposed for a rapid exploration for optimal array configurations. Numerical computations show that for a large number of elements, the method can generate various LRLAs with lower redundancy than the previous algorithms did and the computational cost is highly reduced. A second characteristic of the method is that the solution obtained is not an isolated redundancy array, but a class of redundancy arrays with the same seed and sub-sequences. Based on this characteristic, several analytical patterns for LRLAs are derived. With the help of them, various kinds of LRLAs with redundancy no more than 1.5 for any given number of antennas can be yielded in nearly zero computer time, so these analytical patterns may be helpful to the selection of proper array configurations in engineering applications. Both the method and the resulting configurations can be used in antenna array design in aperture synthesis microwave radiometers with high spatial resolution. Further, more analytical

ACO was developed by Dorigo in the early 1990s [17] and has been recently introduced into some electromagnetic problems, such as antennas [21] and microwave imaging [22]. The underlying concept of ACO was originated by imitating the collective behavior of ants choosing a path between the nest and the food source. In ACO, multiple ants iteratively sample the solution space through a loop mainly consisting of a path-constructing step and a pheromone-updating step. Step 1) Constructing the path. At each construction step, ant applies a probabilistic action choice rule, called random proportional rule, to decide which node to visit next. In particular, the probability of a given node to be chosen for a path by ant at iteration is defined as [17] (17) where is the pheromone level of node , is the heuristic is the tabu information (also called visibility) of node , and list of nodes still to be visited by ant . The parameters and are weights representing the relative influence of the pheromone trail and the heuristic information, and play an important role in the balance between the exploration of unvisited search regions and the exploitation of the search experience gathered so far. Step 2) Global update of pheromone trails. Global update of pheromone trails is performed after all ants have completed their paths, by applying a Max-Min Ant System (MMAS) mechanism [19]. Only the best ant in a given iteration is allowed to mark its path with the pheromone, and this will influence the path choice of ants in the subsequent iterations. For a given iteration , the amount of pheromone is updated by (18) is the pheromone decay rate, where pheromone increment defined by

is the

(19) where denotes the fitness of either iteration-best (usually preferable) or global-best solution.

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To avoid stagnation of the search, the range of possible pheromone trails on each node is limited to an interval . All pheromone trails are initialized to the upper bound in order to favor exploration at the start of the algorithm. As in [19], the upper bound is usually set as (20)

is the fitness of the optimal solution found so far, and where , where is a constant. the lower bound is set to Compared to conventional gradient-based optimization techniques, ACO does not require differentiation of the objective function and is hence less coupled to the solution space. It substantially increases the robustness of ACO in nonlinear, multidimensional optimizations. Moreover, as one out of a number of metaheuristics, several characteristics make ACO a unique approach: it is a constructive, population-based metaheuristic which exploits an indirect form of memory of previous performance [18]. This combination of characteristics is not found in any other metaheuristics. ACKNOWLEDGMENT

[15] K. A. Blanton and J. H. McClellan, “New search algorithm for minimum redundancy linear arrays,” in Proc. IEEE ICASSP, 1991, vol. 2, pp. 1361–1364. [16] A. Camps, A. Cardama, and D. Infantes, “Synthesis of large low-redundancy linear arrays,” IEEE Trans. Antennas Propag., vol. 49, no. 12, pp. 1881–1883, Dec. 2001. [17] M. Dorigo, V. Maniezzo, and A. Colorni, “Ant system: Optimization by a colony of cooperating agents,” IEEE Trans. Syst., Man, Cybern. B, vol. 26, no. 1, pp. 29–41, Feb. 1996. [18] M. Dorigo and T. Stutzle, Ant Colony Optimization. Cambridge, MA: MIT Press, 2004. [19] T. Stutzle and H. H. Hoos, “Max-min ant system,” Future Gener. Comp. Syst., pp. 889–914, 2000. [20] L. Jun, L. Shijie, Z. Zuyin, and H. Tiexia, “Methods for achieving linear arrays with minimum redundancy,” J. Huazhong Univ. Sci. Tech., vol. 23, no. 5, pp. 64–69, May 1995. [21] C. M. Coleman, E. J. Rothwell, and J. E. Ross, “Investigation of simulated annealing, ant-colony optimization, and genetic algorithm for self-structuring antennas,” IEEE Trans. Antennas Propag., vol. 52, no. 4, pp. 1007–1014, Apr. 2004. [22] M. Pastorino, “Stochastic optimization methods applied to microwave imaging: A review,” IEEE Trans. Antennas Propag., vol. 55, no. 3, pp. 538–548, Mar. 2007. [23] S. D. Bedrosian, “Nonuniform linear arrays: Graph-theoretic approach to minimum redundancy,” Proc. IEEE, vol. 74, no. 7, pp. 1040–1043, Jul. 1986. [24] B. Wichmann, “A note on restricted difference bases,” J. Lon. Math. Soc., vol. 38, pp. 465–466, 1963. [25] D. Pearson, S. U. Pillai, and Y. Lee, “An algorithm for near-optimal placement of sensor elements,” IEEE Trans. Inform. Theory, vol. 36, no. 6, pp. 1280–1284, Nov. 1990.

The authors would like to thank the anonymous reviewers for their very helpful comments and suggestions. REFERENCES [1] 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. [2] A. S. Milman, “Sparse-aperture microwave radiometers for Earth remote sensing,” Radio Sci., vol. 23, no. 2, pp. 193–206, 1988. [3] C. T. Swift, D. M. Le Vine, and C. S. Ruf, “Aperture synthesis concepts in microwave remote sensing of the Earth,” IEEE Trans. Microw. Theory Tech., vol. 39, no. 12, pp. 1931–1935, Dec. 1991. [4] C. S. Ruf, “Numerical annealing of low-redundancy linear arrays,” IEEE Trans. Antennas Propag., vol. 41, no. 1, pp. 85–90, Jan. 1993. [5] A. T. Moffet, “Minimum-redundancy linear arrays,” IEEE Trans. Antennas Propag., vol. AP-16, no. 2, pp. 172–175, Mar. 1968. [6] R. Butora and A. Camps, “Noise maps in aperture synthesis radiometric images due to cross-correlation of visibility noise,” Radio Sci., vol. 38, pp. 1067–1074, 2003. [7] A. Camps, I. Corbella, J. Bara, and F. A. Torres, “Radiometric sensitivity computation in aperture synthesis interferometric radiometry,” IEEE Trans. Geosci. Remote Sens., vol. 36, no. 2, pp. 680–685, Mar. 1998. [8] M. Ishiguro, “Minimum redundancy linear arrays for a large number of antennas,” Radio Sci., vol. 15, pp. 1163–1170, 1980. [9] J. Leech, “On the representation of 1; 2; . . . ; n by differences,” J. London Math. Soc., vol. 31, pp. 160–169, 1956. [10] C. B. Haselgrove and J. Leech, “Note on restricted difference bases,” J. London Math. Soc., vol. 32, pp. 228–231, 1957. [11] P. Wild, “Difference basis systems,” Discrete Math., vol. 63, pp. 81–90, 1967. [12] J. Miller, “Difference bases, three problems in additive number theory,” in Computers in Number Theory. London: Academic Press, 1971, pp. 229–322. [13] R. N. Bracewell, “Optimum spacings for radio telescopes with unfilled apertures,” Nat. Acad. Sci. Nat. Res. Council. Publ., vol. 1408, pp. 243–244, 1966. [14] Y. Lee and S. U. Pillai, “An algorithm for optimal placement of sensor elements,” in Proc. IEEE ICASSP, 1988, vol. 5, pp. 2674–2677.

Jian Dong received the B.S. degree in electrical engineering from Hunan University, Changsha, China, in 2004. He is currently working toward the Ph.D. degree at Huazhong University of Science and Technology (HUST), Wuhan, China. Since 2005, he has been working as a Graduate Research Assistant in the HUST Microwave Center. His research interests include antenna arrays, microwave remote sensing, and numerical optimization techniques for electromagnetic problems.

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, Wuhan. His research interests include microwave techniques, microwave remote sensing, antennas, and wireless communication.

Rong Jin received the B.S. and M.S. degrees in electrical engineering from Huazhong University of Science and Technology (HUST), Wuhan, China, in 2006 and 2008, respectively, where he is currently working toward the Ph.D. degree. Since 2007, he has been working as a Graduate Research Assistant in the HUST Microwave Center. His research interests include array signal processing, microwave radiometer calibration, and geophysical retrieval algorithms.

DONG et al.: A METHOD FOR SEEKING LOW-REDUNDANCY LARGE LAs IN APERTURE SYNTHESIS MICROWAVE RADIOMETERS

Yaoting Zhu received the B.S. degree in electrical engineering from Huazhong University of Science and Technology, Wuhan, China, in 1961. From 1984 to 1985, he was a Visiting Scholar at the University of Pittsburgh, Pittsburgh, PA. From 1986 to 1997, he was a Vice President of Huazhong University of Science and Technology, Wuhan, where he is presently a Professor in the Department of Electronics and Information Engineering. He has published more than 100 technical papers and reports in the areas of wireless communication, imaging processing, and microwave remote sensing. Mr. Zhu is a fellow of China Institute of Communications (CIC).

Quanliang Huang received the B.S. and M.S. degrees in applied physics and the Ph.D. degree in electrical engineering all from Huazhong University of Science and Technology (HUST), Wuhan, China, in 1994, 1997, and 2003, respectively. He is presently an Associate Professor in the Department of Electronics and Information Engineering, Huazhong University of Science and Technology. His research interests include microwave techniques, microwave remote sensing, and imaging processing.

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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, in 2005. He is presently an Associate Professor in 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.

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Experimental Demonstration of Focal Plane Array Beamforming in a Prototype Radiotelescope Douglas Brian Hayman, Member, IEEE, Trevor S. Bird, Fellow, IEEE, Karu P. Esselle, Senior Member, IEEE, and Peter J. Hall

Abstract—Focal plane arrays are being developed to provide dishes with a wide field of view for both the next generation of radiotelescopes and to retrofit existing large radiotelescopes. We describe a prototype radiotelescope, comprising a two dish interferometer with real-time digital beamformer that was built to study focal plane array systems. Two beamformer weightings were applied to the system: A normalized conjugate match and the maximum sensitivity (G/T). Both incorporate the uncorrelated noise from the receiver chains and the latter includes correlated noise from spillover and coupling in the array. A black box approach is taken where the assembled system is considered and the only accessible data is that typically available from an operational radiotelescope. This approach is particularly suitable for complex active antennas where there is insufficient knowledge of the system for beamformer weights to be set a priori. It also allows adaptation to changes such as electronic gain drift, partial failures and alterations in the environment. Index Terms—Active arrays, antenna array feeds, antenna array mutual coupling, antenna measurements, radio astronomy.

I. INTRODUCTION

T

HE primary benefit of focal plane arrays (FPAs) to radio astronomy is the increased field of view and hence survey speed. Interest has been developing in arrays with closely spaced elements, which we will refer to as dense FPAs, where the signals from the elements are combined to form beams in contrast to single feed per beam or discretely processed FPAs [1], [2]. These dense FPAs, also known as phased array feeds, can provide superior survey speed and have recently become more attractive due to technological advances. Discretely processed multibeam FPAs, where the output of each array element forms a separate beam, have been used in

Manuscript received March 04, 2009; revised February 15, 2010; accepted February 16, 2010. Date of publication March 29, 2010; date of current version June 03, 2010. D. B. Hayman is with Australia Telescope National Facility (ATNF) and Information and Communication Technologies Centre (ICT Centre), Australia’s Commonwealth Scientific and Industrial Research Organisation (CSIRO), Epping NSW 1710, Australia and also with Macquarie University, NSW 2109, Australia (e-mail: [email protected]). T. S. Bird is with the Information and Communication Technologies Centre (ICT Centre), Commonwealth Scientific and Industrial Research Organisation (CSIRO), Epping NSW 1710, Australia. K. P. Esselle is with Macquarie University, NSW 2109, Australia. P. J. Hall is with Curtin University of Technology, Western Australia 6845, 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.2010.2046843

radioastronomy for some time [3], [4]. These systems multiply the field of view available by the number of array elements but they do not provide a contiguous field of view as outlined by Johansson [5]. Veidt shows they typically require four or more interleaved pointings to fully sample the sky [6]. Conversely a dense FPA, where the elements are combined as a complex weighted sum, can fully sample a region of sky, providing contiguous coverage [6]. These feeds also have potentially much wider bandwidths and allow some correction of aberrations present in off-axis beams [6]–[8]. Dense FPAs have come within reach due to progress in a number of areas. Advances in analogue electronics allow the use of uncooled low-noise amplifiers (LNAs) and hundreds of receiver chains per antenna. Improvements in electromagnetic analysis capabilities allow the joint optimization of the array structure and the connecting electronics, especially the LNA and matching components. Finally, developments in digital electronics and devices have made digital beamforming practical, bringing precision, stability and flexibility. Early work on radioastronomy systems fitted with dense FPAs was done by the National Radio Astronomy Observatory, USA [1], with continuing collaboration with Brigham Young University [9], [10]. ASTRON made early contributions on several fronts including the array design in collaboration with others [11], [12] and beamforming [2]. They continue an active program with an FPA on one of the Westerbork radiotelescope dishes [13]–[15]. The National Research Council of Canada has also pursued this technology [16], reporting recent results of their 180 element prototype [17]. FPAs form an important part of plans for the Square Kilometre Array (SKA) [18]. CSIRO started investigating this area in the early 2000s, and is building the Australian SKA Pathfinder (ASKAP), a radiotelescope with 36 12-m dishes fitted with FPAs [19]. A prototyping test bed for ASKAP was used for our experimental work. This test bed was built to gain experience in receiver, beamformer and system design. The purpose here is to present results from the prototype radiotelescope, giving a more comprehensive account of beamforming than the early results in conference presentations [20]–[23]. The beamformer weightings used are based on formulations first applied to aperture arrays: The weighting of aperture arrays for maximum gain and G/T is detailed in [24] and [25]. The conjugate field match, with a history in horn feed design [26], has been used as the basis for FPA beamforming [7], [27], [28]. This approach assumes the array elements are point field samplers and does not usually include mutual coupling effects. By

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HAYMAN et al.: EXPERIMENTAL DEMONSTRATION OF FOCAL PLANE ARRAY BEAMFORMING IN A PROTOTYPE RADIOTELESCOPE

contrast the maximum gain or maximum directivity weightings do include these effects and have been applied to FPAs by Bird [29] and Lam et al. [30]. The maximum G/T has been applied to FPAs by Bird and Hayman [31], Brisken [32] and Hansen et al. [33] and Jeffs et al. [34] among others. The design of FPAs requires a full understanding of the system, including a detailed model of the electromagnetics of the FPA and its relationship to the reflector and the receiving system [32], [35], [36]. During development of an array, it is common to measure the scattering matrix and radiation patterns of the array separately from the receiver electronics. With knowledge of the characteristics of the electronics, the overall performance of the array can be predicted. However once the system is assembled, access to internal ports is usually impractical, particularly once the array is integrated with the reflector. Therefore, a black-box approach was adopted for this work. We consider the situation where there the system can operate in two modes. Firstly where there is access to the beamformed output and secondly where each element’s signal can be correlated against the others. The implementation used is presented here to make it clear how the weights are calculated and how they relate to common figures of merit. The next section discusses the implementation of the weighting formulation. Section III describes the prototype radiotelescope. The measurement details are provided in Section IV and they are discussed in Section V. Finally in Section VI, conclusions are drawn about the two beamformer weightings and the evaluation techniques. II. IMPLEMENTATION The standard definition of antenna gain [37] includes the dissipative losses in the antenna and does not include the match between the receiver and the antenna. When measuring an assembled active antenna such as a focal plane array, however, it is impossible to separate these effects from the electronic gain in the receiver electronics. The noise contributions of the antenna and receiver also cannot easily be separated. In the black box approach adopted (Fig. 1), it is convenient to define the reference plane for the gain and system noise as the radiation port, consequently directivity is used as the reference for G/T evaluation. Dissipative losses in the antenna are assigned to the receiver. Some other approaches that assume modelled or measured knowledge of parameters internal to the active antenna are given in [38]–[41]. The voltage at the beamformer output, with the assumption that the array system is linear, is given by

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Fig. 1. FPA signal and noise model showing n beamformed focal plane array ~ : desired incident plane-wave. ~e : embeddedelements. The parameters are: E element field-pattern. v : noise from all sources referred to the input of each element. w : adjustable complex weight.

ement; and , the adjustable complex weight.1 The subscript and unit vector correspond to the cross-polarized components and is the cross-polarized embedded field-pattern. The is the noise power constants have been chosen so that element (i.e. and , and the directivity of the ) is (2) This formulation leads the and to include pattern distortion due to mutual coupling but not coupling loss or dissipative loss. By using superposition it can be readily shown that the directivity of the array for the polarization is (3) where (4) and are column and vectors representing the array field-patterns and weights respectively. The superscript T denotes the transpose and H the Hermitian or conjugate transpose. The equivalent system temperature can be expressed as

(5) (1) where

is the wavelength; the impedance of free space; the field strength of the desired signal in the co-polarization, denoted by subscripts and with unit vector definition ; , the co-polarized embedded-element field-pattern (with final system loading, not open or shorted); , the noise from all sources referred to the input of each el-

where is the noise covariance matrix for the array. That are is the elements of (6) 1Another common notational convention uses the conjugate of the beamformer weights.

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where is the expected value and can be measured by correlating element pairs. The system temperature is the sum of noise from the antennareceiver combination and radiation from the surrounding scene: . Modelling the contributors to is described in other FPA treatments such as [34], [35], [42], [43]. Using (3) and (5) the G/T for the system is then

(7) The weights for the maximum G/T are [25], [44]

As with (8), the weights for the maximum G/T are

(12) This expression allows the maximum G/T weighting to be found . from the conveniently measured data and Initially the full covariance matrix was not available from the system and so a simpler weighting was employed. The conjugate match to the desired signal was normalized by the noise from each element, that is the diagonal of the noise covariance matrix and so the weighting terms are

(13)

(8)

The equivalent weights in terms of the field patterns and noise covariance matrix referred to the radiation port are

A. Measurable Quantities The model outlined above established the formulation for the maximum G/T weighting, however, in an assembled system there is usually insufficient information to determine accurate . We will show values of the embedded element patterns how the maximum G/T weighting can be found from data that is readily measured—the covariance matrix and the element responses to a distant point source—as used, e.g., in [34]. We will also detail a simpler weighting that does not require the covariance matrix, referred to here as the normalized conjugate match. To establish the implementation based on measurable quantities, the unknown contributions of electronic gain, scattering by the antenna, dissipative losses in the antenna, mutual coupling losses and mismatch effects are assigned to a set of complex is used to repreterms . A new weight vector sents the weights actually applied in the beamformer. Using (1) the voltage at the beamformer output is given by (9) is the active element response to a distant where is the noise voltage at the point source and element (i.e. beamformer output for unity weighting for the and , ). This modified model will be used to show that the maximum G/T weighting equivalent to (8) can be determined even though the electronic gains and the losses listed above are not directly known. The relative magnitudes and phases of the are determined from the correlation product from the interferometer pointed at a point source. The noise covariance matrix, scaled for the weight vector , is

Using the relationships between (7) can be expressed in terms of

and and between and , giving

(10) and ,

(11)

(14) The similarity between this normalized conjugate match and the maximum gain is described in Appendix B. The normalized conjugate match and the maximum G/T weightings are equivalent if the noise from each element is uncorrelated. With this black box approach, it was possible to obtain the data for calculating the weights using the completed radiotelescope system. The calculation of the maximum G/T weighting (12) requires the element responses to a distant point source and the noise covariance matrix with the antenna pointed away from any strong sources (off-source). III. INSTRUMENTATION The performance of a prototype radio telescope with various beamformer weightings has been studied. The radiotelescope comprised a two dish interferometer established at the CSIRO Radiophysics Laboratory in Sydney. It had a 90 m baseline, oriented approximately east-west (Figs. 2 and 3). The two dishes have a diameter of 14.2 m and an F/D of 0.40. The western dish was fitted with a single horn with an edge taper of 17 dB. The eastern dish was fitted with a single polarization 8 8 array of Vivaldi elements, originally used in the ASTRON THEA project [45]. The signals were transported from the FPA on coaxial cables to receivers in the pedestal. The 70 MHz intermediate frequency signals from the receiver (24 MHz wide) were then fed into the 24 input digital beamformer-correlator. Twenty one of the sixty four elements of the array were selected for beamforming. Fig. 4 shows the positions from above the dish and feed, the dominant polarization is electric field horizontal, west is in the top left corner and north the top right. The initial more central element selection was used first and is shown in Fig. 4(b). Element number seven developed a fault and so the connections were then moved to an offset selection for the latter measurements (Fig. 4(e)). This moved the beam 1.8 away from the reflector boresight. With this set of connections, element 17 developed an intermittent fault. Being at the edge of the selected area it was expected to have less impact than element seven in

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a noise signal was radiated from the vertex of the dish by a noise source connected to a small antenna. With the noise source turned on and a coupled signal from the source fed into the correlator, a reference phase was established and this was used to compensate for drift. The noise source used in this instance did not have a stabilized output level and so the off-source autocorrelation of each element was used as a measure of the magnitude drift instead. A stabilized noise source is recommended for future systems. Where an interferometer, with its higher sensitivity, is unavailable, a single dish can be used to obtain the beamformer coefficients and determine the G/T performance [17], [34], [46]. The prototype radiotelescope is described in more detail in [22]. IV. MEASUREMENTS

Fig. 2. The two dish prototype radiotelescope located in Sydney; the east antenna in the foreground is fitted with the FPA and the west antenna in the background with a single horn.

A series of investigations were conducted on the prototype radiotelescope to demonstrate beamforming and evaluation methods. For each set of measurements, there were two phases: First the data required for calculating the array weights was gathered and then tests were undertaken to evaluate the system with beamforming applied. A. Beamformer Weight Data Collection

Fig. 3. System diagram of the prototype radiotelescope. Twenty one signals are brought down from the FPA through the receiver to the digitizer. Autocorrelations and cross-correlations between any two FPA inputs were also possible.

the central element selection. The other three beamformer inputs were used for the reference (western) antenna feed and the two vertex noise sources used for calibration. The original FPA range of 700 to 1800 MHz was found to be too wide with significant distortion products present from the severe radio frequency interference (RFI) environment of suburban Sydney. An 1150 to 1750 MHz filter was fitted to each element between the first and second amplification stages, leaving a few RFI free bands available for measurements. The center frequency of 1200 MHz was chosen for the results presented in this paper. At this frequency the element spacing was 0.51 wavelengths and with a higher frequency, with element lower spacing, a single element would dominate and beamforming would have less effect. The separate receiver chains result in drift of the gain and phase for each element relative to the others. To correct for this

The noise covariance matrix (Section II) was used for determining the system noise for different sets of weights. To generate the matrix, the antenna was pointed off-source, i.e. away from any strong sources whilst retaining a similar surrounding scene of radiative noise. This was achieved by keeping the elevation approximately the same for the on-source and off-source measurements. The diagonal terms of this matrix dominated, being at least ten times the off diagonal terms. The correlation coefficients where are used for this comparison . Element pairs that were adjacent in the H-plane (e.g., elements 4 and 8, Fig. 4) had correlation terms between 6 and 10% of the diagonals. Element pairs that were adjacent in the E-plane had much lower terms. This corresponds to the degree of coupling that is expected with electric-source dominated elements [47]. and Initially the system was not equipped to measure only the normalized conjugate match weighting was used for the central element selection. In interferometer mode, both dishes were pointed at a suitable radio source and the correlations of each element against the reference (western) antenna were recorded. The galaxy M87 (Virgo A) was chosen for this purpose because it appeared as a point source for the 90 m baseline at 1200 MHz, it has a high flux density and it is available for 7 1/2 hrs a day for the location and range of motion of these antennas. B. Evaluation of the Beamformed System The weights were calculated from the noise covariance matrix and the point source response data (see (12) and (13)). The single element in the center of the 21 selected elements was used as a reference for the weighting cases (number one in Fig. 4. It is stressed that this single element over illuminates the reflector and so a purpose designed horn, such as the one on the

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Fig. 4. Magnitude of array element weights (decibels) at 1200 MHz for the weighting cases applied, showing the central and offset element selections. The E field polarization of the array is horizontal. (a) Single element—central. (b) Normalized conjugate match, 21 elements—central. (c) Single element—offset. (d) Maximum G/T, 5 elements—offset. (e) Normalized conjugate match, 21 elements—offset. (f) Maximum G/T, 21 elements—offset.

western antenna, would perform better in both noise and efficiency. Nonetheless, the single element was convenient to use as it was expected to have a similar noise contribution from the receiver electronics as the weighted array and its G/T would not change too much over time. Tests were also done with five elements to see the effect of the reduction from 21 elements. The weighting cases applied were the normalized conjugate match with 21 elements for both the central and offset element selections and the maximum G/T with five and 21 elements in the offset selection. Fig. 4 shows the magnitude of the calculated weights for the cases presented here. Radiation patterns were obtained with the system acting as an interferometer using a point source. The correlator output is the product of the voltage gains of the eastern and western dishes. The reference (western) dish tracks the source and the eastern dish changes its pointing relative to the source to make the pattern cuts. M87 was used for the central element selection. Sagittarius , being higher in the sky allowed a greater range in declination, was used for the offset element selection. Cuts were made in the E, H, 45 and 135 planes at 0.25 spacing over 8 and 10 spans for the central and offset selections respectively (Fig. 5). The 45 plane (with respect to the E-field) corresponds to cuts in declination (north-south) and the 135 plane corresponds to cuts in hour angle (east-west). These patterns were interpolated using the formulation described in Bucci [48]. The half-power beamwidth (HPBW) is about 1.7 with the variations shown in Table I. A readily accessible method was used to compare single element and beamformed G/T performance. A directivity measure was obtained from the interferometer response to M87 and a noise measure was obtained from the autocorrelation of weighted signal off-source. These measures do not represent the true directivity or system temperature as they do not include the overlap integral terms in the denominators of (3) and (5).

Their ratio however is proportional to the true G/T as shown by (7) and (11). Table I shows the G/T improvement seen for the beamformed cases over the single element cases. These measurements were repeated a number of times and the results vary by about 0.1 dB but the difference between the normalized conjugate match and the maximum G/T case remains within 0.03 dB of the figures quoted. The absolute G/T values for the offset element selection were determined by measuring the ratio of the power pointing at and away from a source of known strength. The G/T value then is found by

(15) is the Boltzmann constant, is the on-source/offwhere source power ratio and is the spectral power flux density of the source. M87 was used as the source and its strength was determined by extrapolating the data from Ott et al. [49], giving the value at 1200 MHz. The uncertainty in , 0.02 of dB, dominates the combined uncertainty of 0.7 dB ( 17%). The combined uncertainties quoted in this paper are calculated from the root sum of squares of the contributing uncertainties and represent the 95% probability interval. The system tem, perature over aperture efficiency (based on the dish area), is also presented as the figure of merit as this is a useful way of comparing feed systems:

(16) where is the dish area. Both figures of merit are shown in Table I.

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Fig. 5. Radiation pattern cuts at 1200 MHz for the weighting cases applied. (a) Single element—central. (b) Normalized conjugate match, 21 elements—central. (c) Single element—offset. (d) Maximum G/T, 5 elements—offset. (e) Normalized conjugate match, 21 elements—offset. (f) Maximum G/T, 21 elements—offset.

TABLE I BEAMFORMED PERFORMANCE AT 1200 MHz

1G=T refers to the ratio of the G/T with the weighting and the corresponding single element G/T. A measure of the system temperature can be arrived at by placing microwave absorber under the feed. A 1.8 m 1.8 m sheet of 457 mm thick absorber was used. The power ratios for , are shown in Table I; the uncerwith and without absorber, tainty is 0.02 dB. The interpretation of these results requires knowing either the receiver system temperature or the noise contribution from the surrounding scene [50]. An estimate of the former is detailed in Section V-B. V. DISCUSSION The measurements are used to interpret the behavior of the beamforming weights in the prototype radiotelescope. Characteristics of the aperture distribution are determined from the radiation patterns. An estimate of the FPA noise temperature is calculated and then used to find the spillover temperatures and aperture efficiencies from the power ratio measurements.

A. Radiation Patterns As a general observation, it is apparent from the pattern distortion that the H-plane radiation pattern cuts for the single and beamformed central element selection (Fig. 5(a) and (b)) suffered minor contamination with noise. This is probably due to interference being present at the moment of the distorted data in those cuts. The patterns in the E and 45 planes show a coma effect in the single and five-element cases for the offset element selection (Fig. 5(c) and (d)). However in the 21-element cases the effect is reduced (Fig. 5(e) and (f)). This improvement demonstrates the ability of FPAs to compensate for such off-axis aberrations [51]. To understand beamforming behaviour, it is valuable to determine the aperture field distribution generated by the feed patterns. Aperture distributions are commonly ascertained using microwave holography from an extensive 2D far-field map. This

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Fig. 6. Aperture illumination cuts at 1200 MHz, magnitude (dB). (a) Single element—central. (b) Normalized conjugate match, 21 elements—central. (c) Single element—offset. (d) Maximum G/T, 5 elements—offset. (e) Normalized conjugate match, 21 elements—offset. (f) Maximum G/T, 21 elements—offset.

approach was not available here and so the technique described in Appendix A is used. The magnitude and phase distributions from the measured cases are shown in Figs. 6 and 7 respectively. The magnitude plots show a slight narrowing of the feed pattern for the beamformed cases over the single element cases. The phase plots are flatter in the beamformed cases. This is most likely due to the phase center of the array elements being slightly above or below the focal plane (defocused) and the beamforming correcting the location of the phase center. In the central element selection the first sidelobe levels are higher for the beamformed case (Fig. 5(b)) than for the single element (Fig. 5(a)). The aperture distributions show an increase in aperture taper for the beamformed case (cf. Fig. 6(a) and (b)) and normally this reduces the sidelobe levels. However the concentration of the feed pattern under the feed blockage appears to be the factor increasing the sidelobes [52]. A tradeoff to be considered with the use of FPAs is the increased blockage. On this instrument, the feed, occupying 2.1 , blocks 1.3% of the physical aperture, corresponding to a 5% reduction in gain [52]. The struts, occupying 3.4 , block 2.1% of the physical aperture corresponding to a 4% reduction in gain.

an array with the same elements and LNAs but with a differing subsequent receiver chain produced a very similar result [53]. Rough uncertainties assigned to the values contributing to this estimate show the two dominant contributors to the total uncerare the dissipative loss in the antenna and the LNA tainty in noise. The former is assigned 95% confidence interval bounds of 0.1 and 0.2 dB, based on our experience with similar calculations and measurements, resulting in a 10 to 20 K unand the latter bounds of 9 and 18 K, based certainty in on the manufacturers data sheet and resulting in contributions uncertainty. The upper and lower of 9 and 18 K to the bounds are combined as a root sum of squares separately. The are 17 K combined 95% confidence interval bounds for (120 K) and 29 K (166 K). The relatively high combined uncertainty in reflects the incomplete nature of the information readily available for this FPA. The estimated value of the receiver equivalent noise temper, was used to give representative values of ature, the equivalent noise temperature from the surrounding scene for the offset element selection. This is given by

B. G/T Results

where is the temperature of the absorber, 299 K, and is the power ratio. These results are shown in Table I. The unis dominated by . This varies slightly for certainty in the different measurement sets with the worst case being 16 K. The uncertainty is much less for the relative values however: for the five-element maximum G/T case is 46 3K for the 21-element lower than the single element case. maximum G/T case is 58 4 K lower than the single element

The equivalent receiver noise temperature of the FPA was calculated to aid in the understanding of the G/T results. This was calculated as 137 K using estimates of component contributions, listed in Table II, obtained from both the FPA supplier and data from the modifications performed by CSIRO. Note that includes dissipative losses in the antenna but not radiation from the surrounding scene, such as spillover. Measurements on

(17)

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Fig. 7. Aperture illumination cuts at 1200 MHz, phase (deg). (a) Single element—central. (b) Normalized conjugate match, 21 elements—central. (c) Single element—offset. (d) Maximum G/T, 5 elements—offset. (e) Normalized conjugate match, 21 elements—offset. (f) Maximum G/T, 21 elements—offset.

The values for the offset element selection weightings range from 41–57% (Table I). The combined uncertainty in the efficiency is 10% in absolute terms and is dominated by the un. It is interesting to note that this calculation is certainty in relatively insensitive to the coarse estimate of the receiver noise with it contributing only 2–4% to the total. Higher efficiencies are expected with an FPA where more elements are used.

TABLE II FPA NOISE BUDGET

This is the energy picked up by each element that has been radiated from the termination presented by surrounding LNAs. It was calculated from the scattering matrix measured from the array (without the electronics) and an estimated equivalent noise temperature presented by the LNA inputs of 100 K [50]. Estimated by the FPA supplier [50]. Data sheet for the LNA [54]. Calculated by the designer [55]. Measured prior to installation.

case. The difference between the normalized conjugate match and maximum G/T for 21-elements is 10 1 K. The , and values can be combined to find the antenna aperture efficiency using the expression

(18)

C. Comparison of Weighting Cases In the measured results there is close correspondence between the normalized conjugate match and the maximum G/T weighting cases in the radiation patterns, the aperture distributions and the G/T measures. The normalized conjugate match has a slightly higher directivity (aperture efficiency) and the maximum G/T has slightly better noise performance. As the normalized conjugate match does not take spillover noise into account (assuming identical spillover for the individual elements) the similarity suggests aperture efficiency improvement and not noise reduction dominated both cases. The maximum G/T weighting reduces to the normalized conis diagjugate match when the noise covariance matrix onal. On this instrument the receiver noise temperature is rela. Where tively high, hence the dominant diagonal terms in the receiver temperature is very low, reduction of spillover is expected to be more dominant for the maximum G/T case and the two weightings would be less similar. The five-element case showed about half the G/T improvement of the 21-element case (Table I). The difference between the G/T comparison method (Section IV-B) and the on/off M87 result is accounted for by the uncertainty estimates: 2.1 0.1

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and 2.4 0.7 dB respectively for maximum G/T over single element. The normalized conjugate match and the maximum gain weightings are shown to be equivalent in certain cases as shown in Appendix B. They have the same relative phase and are equivalent if across the beamformed elements, the radiation efficiencies are equal and the noise temperatures are equal. The dominantly weighted elements are away from the edge of the array and so will have similar radiation efficiencies. Meavariation of up to 2.5 dB. surements on the FPA show a Taking the worst case of this disparity being on two dominant elements, the reduction in the gain is 0.09 dB. That is gain from the normalized conjugate match is within a tenth of a decibel of the maximum gain achievable. VI. CONCLUSION The practical implementation of beamforming an FPA in a prototype radiotelescope has been presented. A normalized conjugate match and the maximum G/T weightings have been applied and evaluated. The results were compared to a single element from the array and a similar moderate improvement in G/T was seen in both cases. With lower noise FPAs, a greater difference between the two weightings is expected. The formulation for the beamforming was presented using a black box approach based on standard beamforming methods. Scattering matrix or similar models are essential for good FPA system design and can provide good predictions of performance [15]. The black box approach shown here demonstrates what can be readily determined from measurements on an installed system alone. It accommodates factors such as any omissions in modelling, such as supporting structures, and drift in the electronic gain amplitude and phases with temperature and over time as well as component failures. The relationship between the black box formulation and a scattering matrix method was also detailed. The black box approach does not separate out the performance of the different components and so is less useful for synthesis or analyzing where the system could be improved. Beamforming schemes used in an operating radiotelescope are likely to be close to the maximum G/T methods explored here (and in other recent FPA prototypes) because of the importance of sensitivity to the field of radioastronomy. The maximum G/T does not however address other considerations such as sidelobe level, main lobe shape, pattern stability and polarization purity. These may need to be taken into account as well (as in [29], [34]) and would require high dynamic range measurements or accurate modelling of the element radiation patterns. The normalized conjugate match was shown to be useful in the system development due to its simplicity but is unlikely to be appropriate in an operational setting because it does not minimize correlated noise. The evaluation techniques allowed the antenna system to be characterized with some limitations. The radiation patterns gave an indication of the aperture distribution but more extensive pattern cuts or a holography grid would provide superior resolution and greater confidence in the interpretation of the results. A relatively sensitive technique for comparing G/T values as well as a less sensitive method for determining the absolute value of G/T were demonstrated. The uncertainty of the latter was

Fig. 8. E-plane cuts of test and regenerated apertures. The 2D insert shows the FPA and strut blockage used in the model. The parameters for the apertures are: , flat phase; B: n , flat phase, all with quadratic phase; C: n A: n 6 dB edge taper.

=1

=1

=2

high due to the limited strength of the source, M87. This uncertainty would be much lower with a stronger source such as the northern hemisphere’s Cassiopeia-A. While the noise performance of this system was modest, the beamforming and evaluation techniques are applicable to future systems. Modelling and design techniques for FPAs are rapidly improving with the efforts from a number of groups. Within CSIRO, FPA development is continuing using another test-bed at the Parkes observatory (where the RFI is much lower) [19], [36], [46]. APPENDIX A APERTURE DISTRIBUTION CALCULATION The available data consists of four patterns cuts, equally spaced in the azimuthal coordinate, , over eight or ten degree spans in the elevation coordinate, , where the coordinate system is aligned with the beam peak. This angle is interpolated as described in Section IV-B and , is interpolated using a cubic spline. The aperture distributions are then calculated by applying 2D Fourier transforms to this far-field data. A Hamming window is used to reduce ripples in the aperture distributions. The validity and limitations of this technique was tested using models of aperture distributions. These include feed and strut blockage. A parabolic taper on a pedestal was used for the test distribution with differing profiles and 6 dB edge illumination . Quadratic phase distortion of similar magnitude to the single element re. is sults, was introduced in the form the radius of the dish. Magnitude and phase cuts of the fields in three such test apertures are in Fig. 8 along with the regenerated apertures. The far-field patterns were generated from the

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The choice of reference plane results in the field patterns from the black box and scattering matrix models to be related by complex terms that are constant over the sphere. Equivalent weightings from the two models (i.e. producing the same output for the . same incident radiation) obey the identity A consequence of the field patterns from the two models being proportional to each other is their directivities are the same (20) (21) Fig. 9. Scattering matrix model of the array showing the location of the reference plane separating the array to the left and the beamformer to the right. Note the receiver match is on the array side.

test apertures (using the formulation in [56], p283). Comparison of the initial and regenerated apertures show the effect of spatial filtering and little can be determined by the regions around the blocked center of the dish or the edges. Within the annulus to 6 m however, the test apertures exhibited an RMS of phase error of 0.24 for the quadratic phase case and much less for the flat phase case and 2% and 5% RMS amplitude error for and cases respectively. the APPENDIX B NORMALIZED CONJUGATE MATCH AND MAXIMUM GAIN Our purpose here is to relate the black-box implementation (Section II) to a conventional scattering matrix representation, which is often used in synthesis. The correspondence between the models is used to demonstrate the conditions under which the normalized conjugate match weighting (black box) is equivalent to the maximum gain weighting (scattering matrix). Particularly noted is the difference in the choice of reference plane dividing the array and the beamformer. Scattering matrix analyses of arrays is well covered in the literature in a range of different forms (e.g. [57] and [58]). The scattering matrix model of the array is shown in Fig. 9. The receiver electronics (with the LNA having the major effect) is included in the model by splitting each receiver into a passive-reciprocal two-port, , representing the network element presented to the array port and an isolating-matched amplifier representing the electronic gain (Fig. 9). The gain is included in are embedded-element the weighting terms . The field-patterns for the polarization and the antenna gain (inport is . cluding the receiver match) at the in The voltage signal response to a polarized plane-wave terms of the scattering matrix model is (cf. (1))

The denominator of (20) is unity from the definition of the ((2)). Consider (21) in terms of the scattering matrix model in transmit. The denominator is the ratio of total power radiated port only excited. By comparing and power available for the and the is (20) and (21), the relationship between the

(22) and between

and

(23) is an (undetermined) phase. where The maximum gain condition, that is the maximum power delivered to the beamformer output, given the constraint that is constant, is given by the conjugate match to the signals . Inserting this into (23), from the array port, . and are constant across the array, If both the normalized-conjugate-match condition is equivalent to the maximum gain condition. ACKNOWLEDGMENT This work was facilitated by the ASKAP project and its predecessors and has only been possible through the contributions of the large team that contributed to the prototype radiotelescope [22]. ASTRON contributed both with the FPA itself and with advice during this project. In particular M. Ivashina and B. Woestenburg have provided data and advice on evaluating the FPA. The authors thank the anonymous reviewers for their comments resulting in an improved paper. D. B. Hayman thanks S. Hay, J. O’Sullivan, and M. Kesteven for their additional encouragement and advice during this project. REFERENCES

(19) where and are the weight and embedded-element field-pattern column vectors.

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[3] Multi-Feed Systems for Radio Telescopes, ser. Astron. Soc. Pac. Conf., D. T. Emerson and J. M. Payne, Eds. Tucson, AZ: , May 16–18, 1994, vol. 75. [4] L. Staveley-Smith, W. E. Wilson, T. S. Bird, M. J. Disney, R. D. Ekers, K. C. Freeman, R. F. Haynes, M. W. Sinclair, R. A. Vaile, R. L. Webster, and A. E. Wright, “The Parkes 21 CM multibeam receiver,” Publ. Astron. Soc. Aust., vol. 13, pp. 243–248, Nov. 1996. [5] J. F. Johansson, “Fundamental limits for focal-plane array efficiency,” in Multi-Feed Systems for Radio Telescopes, ser. Astron. Soc. Pac. Conf.. Tucson, AZ: , May 16–18, 1994, vol. 75, pp. 34–41 [Online]. Available: http://adsabs.harvard.edu/abs/1995ASPC…75…34J [6] B. Veidt, Focal-Plane Array Architectures: Horn Clusters vs. PhaseArray Techniques International Square Kilometre Array Steering Committee, SKA Memo 71, 2006 [Online]. Available: http://www.skatelescope.org/PDF/memos/71_Veidt.pdf [7] P. Loux and R. Martin, “Efficient aberration correction with a transverse focal plane array technique,” in Proc. IRE Int. Convention Record, 1964, vol. 12, pp. 125–131, IEEE. [8] R. Padman, “Optical fundamentals for array feeds,” Multifeed Systems for Radio Telescopes, ser. Astron. Soc. Pac. Conf., vol. 75, 1995 [Online]. Available: http://adsabs.harvard.edu/abs/1995ASPC…75….3P [9] K. F. Warnick, B. D. Jeffs, J. Landon, J. Waldron, D. Jones, A. Stemmons, J. R. Fisher, R. Norrod, and R. Bradley, “BYU/NRAO 19 element L-band focal plane array feed—Sensitivity, efficiency, and RFI mitigation,” in Eur. Conf. on Antennas and Propagation (EuCAP), Edinburgh, UK, Nov. 2007 [Online]. Available: http://wiki.gb.nrao.edu/ pub/Electronics/ResultPresentations/ArrayFeed_E uCAP_Nov07.pdf [10] K. F. Warnick, B. D. Jeffs, J. Landon, J. Waldron, D. Jones, J. R. Fisher, and R. Norrod, “Beamforming and imaging with the BYU/ NRAO l-band 19-element phased array feed,” in Proc. ANTEM Symp., Banff, AB, Feb. 15–18, 2009, pp. 1–4. [11] C. Craeye, A. B. Smolders, A. G. Tijhuis, and D. H. Schaubert, “Computation of finite array effects in the framework of the square kilometer array project,” in Proc. Inst. Elect. Eng. Int. Conf. on Antennas and Propagation (ICAP), Manchester, Apr. 17–20, 2001, vol. 1, pp. 298–301. [12] J. Simons, J. G. B. de Vaate, M. V. Ivashina, M. Zuliani, V. Natale, and N. Roddis, “Design of a focal plane array system at cryogenic temperatures,” in Proc. Eur. Conf. on Antennas and Propagation (EuCAP), Nice, Nov. 6–10, 2006, pp. 1–6. [13] M. A. W. Verheijen, T. A. Oosterloo, W. A. van Cappellen, L. Bakker, M. V. Ivashina, and J. M. van der Hulst, “Apertif, a focal plane array for the WSRT,” in The Evolution of Galaxies Through the Neutral Hydrogen Window, Aug. 2008, vol. 1035, pp. 265–271, Am. Inst. Phys. Conf. Proc. series. [14] M. V. Ivashina, M. N. 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, no. 2, pp. 373–382, Feb. 2009. [15] M. V. Ivashina, O. A. Iupikov, R. Maaskant, W. A. van Cappellen, L. Bakker, and T. Oosterloo, “Off-axis beam performance of focal plane arrays for the westerbork synthesis radio telescope—Initial results of a prototype system,” in IEEE AP-S Int. Symp. Digest, Charleston, SC, Jun. 1–5, 2009, pp. 1–4. [16] B. Veidt and P. Dewdney, “Development of a phased-array feed demonstrator for radio telescopes,” presented at the ANTEM Symp., SaintMalo, France, Jun. 2005. [17] B. Veidt, T. Burgess, R. Messing, G. Hovey, and R. Smegal, “The DRAO phased array feed demonstrator: Recent results,” in Proc. ANTEM Symp., Banff, AB, Feb. 15–18, 2009, pp. 1–4. [18] P. E. Dewdney, P. J. Hall, R. T. Schilizzi, and T. J. L. W. Lazio, “The square kilometre array,” Proc. IEEE, vol. 97, no. 8, pp. 1482–1496, Aug. 2009. [19] D. R. DeBoer, R. G. Gough, J. D. Bunton, T. J. Cornwell, R. J. Beresford, S. Johnston, I. J. Feain, A. E. Schinckel, C. A. Jackson, M. J. Kesteven, A. Chippendale, G. A. Hampson, J. D. O’Sullivan, S. G. Hay, C. E. Jacka, T. W. Sweetnam, M. C. Storey, L. Ball, and B. J. Boyle, “Australian SKA pathfinder: A high-dynamic range wide-field of view survey telescope,” Proc. IEEE, vol. 97, no. 8, pp. 1507–1521, Aug. 2009. [20] D. B. Hayman and T. Cornwell, “NTD THEA tile measurements,” presented at the 3rd Int. Focal Plane Array Workshop, Mar. 2007. [21] J. O’Sullivan, R. Gough, D. B. Hayman, A. Grancea, C. Granet, S. Hay, and J. Kot, “Recent focal plane array developments for the Australian SKA Pathfinder,” presented at the Int. Symp. on Microwave and Optical Technology (ISMOT), Dec. 2007.

[22] D. Hayman, R. Beresford, J. Bunton, C. Cantrall, T. Cornwell, A. Grancea, C. Granet, J. Joseph, M. Kesteven, J. O’Sullivan, J. Pathikulangara, T. Sweetnam, and M. Voronkov, “The NTD interferometer: A phased array feed test bed,” presented at the URSI Workshop on Applications of Radio Science, 2008 [Online]. Available: http://www.ncrs.org.au/wars/wars2008/Hayman [23] D. B. Hayman, T. S. Bird, P. Hall, and K. Esselle, “Evaluation of beamforming radioastronomy focal plane arrays,” presented at the UNSC/ URSI, 2009. [24] 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, no. 8, pp. 1033–1045, 1966. [25] S. P. Applebaum, “Adaptive arrays,” IEEE Trans. Antennas Propag., vol. 24, no. 5, pp. 585–598, Sep. 1976. [26] B. Minnett and H. Thomas, “A method of synthesizing radiation patterns with axial symmetry,” IEEE Trans. Antennas Propag., vol. 14, no. 5, pp. 654–656, Sep. 1966. [27] M. V. Ivashina, J.-G. bij de Vaate, R. Braun, and J. D. Bregman, “Focal plane arrays for large reflector antennas: First results of a demonstrator project,” presented at the SPIE Astronomical Telescopes and Instrumentation, Glasgow, Scotland, U.K., 2004. [28] M. V. Ivashina, J. Simons, and J. G. B. Vaate, “Efficiency analysis of focal plane arrays in deep dishes,” Exp. Astron., vol. 17, no. 1–3, pp. 149–162, Jun. 2004. [29] T. S. Bird, “Contoured-beam synthesis for array-fed reflector antennas by field correlation,” Proc. Inst. Elect. Eng. Microw. Opt. Antennas, vol. 129, pp. 293–298, Dec. 1982. [30] P. Lam, S.-W. Lee, D. Chang, and K. Lang, “Directivity optimization of a reflector antenna with cluster feeds: A closed-form solution,” IEEE Trans. Antennas Propag., vol. 33, no. 11, pp. 1163–1174, 1985. [31] T. S. Bird and D. B. Hayman, “Focal-plane array concepts for the Parkes radio telescope,” presented at the URSI General Assembly, Lille, France, Sep. 1996. [32] W. Brisken and C. Craeye, Focal Plane Array Beam-Forming and Spill-Over Cancellation Using Vivaldi Antennas National Radio Astronomy Observatory, EVLA Memo 69, Jan. 2004 [Online]. Available: http://www.aoc.nrao.edu/evla/geninfo/memoseries/evlamemo69.pdf [33] C. K. Hansen, K. F. Warnick, B. D. Jeffs, J. R. Fisher, and R. Bradley, “Interference mitigation using a focal plane array,” Radio Sci., vol. 40, p. RS5S16, Jun. 2005. [34] B. D. Jeffs, K. F. Warnick, J. Landon, J. Waldron, D. Jones, J. R. Fisher, and R. D. Norrod, “Signal processing for phased array feeds in radio astronomical telescopes,” IEEE J. Sel. Topics Signal Process., vol. 2, no. 5, pp. 635–646, Oct. 2008. [35] K. F. Warnick and M. A. Jensen, “Optimal noise matching for mutually coupled arrays,” IEEE Trans. Antennas Propag., vol. 55, no. 6, pp. 1726–1731, Jun. 2007. [36] S. G. Hay, J. D. O’Sullivan, J. S. Kot, C. Granet, A. Grancea, A. R. Forsyth, and D. B. Hayman, “Focal plane array development for ASKAP (Australian SKA Pathfinder),” in Proc. Eur. Conf. on Antennas and Propagation (EuCAP), Edinburgh, Nov. 2007, pp. 1–5. [37] Standard Definitions of Terms for Antennas, IEEE Std. 145-1993, Mar. 1993. [38] A. Waldman and G. J. Wooley, “Noise temperature of a phased array receiver,” Microw. J., vol. 9, pp. 89–96, Sep. 1966. [39] E. Jacobs, “A figure of merit for signal processing reflector antennas,” IEEE Trans. Antennas Propag., vol. 33, no. 1, pp. 100–101, Jan. 1985. [40] J. J. Lee, “G/T and noise figure of active array antennas,” IEEE Trans. Antennas Propag., vol. 41, no. 2, pp. 241–244, Feb. 1993. [41] K. F. Warnick and B. D. Jeffs, “Gain and aperture efficiency for a reflector antenna with an array feed,” IEEE Antennas Wireless Propag. Lett., vol. 5, no. 1, pp. 499–502, Dec. 2006. [42] S. G. Hay, “FPA modelling concepts at CSIRO,” in Focal Plane Array Workshop, Dwingeloo, The Netherlands, Jun. 2005 [Online]. Available: http://www.astron.nl/fpaworkshop2005/, Astron [43] K. F. Warnick and M. A. Jensen, “Effects of mutual coupling on interference mitigation with a focal plane array,” IEEE Trans. Antennas Propag., vol. 53, no. 8, pp. 2490–2498, Aug. 2005. [44] D. K. Cheng and F. I. Tseng, “Gain optimization for arbitrary antenna arrays,” IEEE Trans. Antennas Propag., vol. AP-13, pp. 973–974, Nov. 1965. [45] S. J. Wijnholds, A. G. Bruyn, J. D. Bregman, and J. G. B. Vaate, “Hemispheric imaging of galactic neutral Hydrogen with a phased array antenna system,” Exp. Astron., vol. 17, no. 1–3, pp. 59–64, Jun. 2004.

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[46] J. D. O’Sullivan, F. Cooray, C. Granet, R. Gough, S. Hay, D. B. Hayman, M. Kesteven, J. Kot, A. Grancea, and R. Shaw, “Phased array feed development for the Australian SKA Pathfinder,” presented at the URSI General Assembly, Sep. 2008. [47] R. C. Hansen, Phased Array Antennas, K. Chang, Ed. New York: Wiley, 1998. [48] O. M. Bucci, C. Gennarelli, and C. Savarese, “Optimal interpolation of radiated fields over a sphere,” IEEE Trans. Antennas Propag., vol. 39, no. 11, pp. 1633–1643, Nov. 1991. [49] M. Ott, A. Witzel, A. Quirrenbach, T. P. Krichbaum, K. J. Standke, C. J. Schalinski, and C. A. Hummel, “An updated list of radio flux density calibrators,” Astron. Astrophys., vol. 284, pp. 331–339, Apr. 1994. [50] E. E. M. Woestenburg, Personal Communication. 2007. [51] A. W. Rudge and M. J. Withers, “New technique for beam steering with fixed parabolic reflectors (Wide angle microwave antenna radiation beam steering with fixed parabolic reflectors, using adaptive primary feed for intercepted field spatial Fourier transformation),” Proc. Inst. Elect. Eng., vol. 118, pp. 857–863, Jul. 1971. [52] P.-S. Kildal, Foundations of Antennas: A Unified Approach. Sweden: Studentlitteratur, 2000. [53] E. E. M. Woestenburg and K. F. Dijkstra, “Noise characterization of a phased array tile,” in Proc. Eur. Microwave Conf. (EuMC), 2003, vol. 1, pp. 363–366. [54] “CGY2106TS Dual LNA Data Sheet,” Philips Semiconductors, 2000. [55] A. Grancea, Personal Communication. 2006. [56] W. L. Stutzman and G. A. Thiel, Antenna Theory and Design, 2nd ed. New York: Wiley, 1998. [57] S. Stein, “On cross coupling in multiple-beam antennas,” IEEE Trans. Antennas Propag., vol. AP-10, no. 5, pp. 548–557, Sep. 1962. [58] J. W. Wallace and M. A. Jensen, “Mutual coupling in MIMO wireless systems: A rigorous network theory analysis,” IEEE Trans Wireless Commun., vol. 3, no. 4, pp. 1317–1325, 2004. Douglas Brian Hayman (M’93) was born in Misawa, Japan, in 1964. He received the B.Sc. degree in pure mathematics and physics in 1986 and the B.E. degree (electrical, Hons.) in 1988, both from the University of Sydney, Sydney, Australia. In 1988, he joined Radio Transmission Engineering, working on FM broadcast transmitters and microwave links. From 1990 to 1992, he was with the satellite company, AUSSAT, working in earth station engineering. In December in 1992, he joined the Commonwealth Scientific and Industrial Research Organisation (CSIRO), in Sydney, Australia, where is he currently a Senior Research Engineer. He has focused on antenna metrology and led a comprehensive upgrade of CSIRO’s antenna measurement facility hardware and software. He has also designed components such as waveguide rotary joints and orthomode transducers and managed a number of antenna related projects. Most recently he has been involved in the ASKAP focal plane array work and, since 2003, he has been undertaking a part time Ph.D. in the field of focal plane array beamforming and evaluation at Macquarie University, NSW, Australia. Mr. Hayman has served as a reviewer for the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION since 2005 and for the Australian Symposium on Antennas since 2006.

Trevor S. Bird (S’71–M’76–SM’85–F’97) received the B.App.Sc., M.App.Sc., and Ph.D. degrees from the University of Melbourne, Melbourne, Australia, in 1971, 1973, and 1977, respectively. From 1976 to 1978, he was a Postdoctoral Research Fellow at Queen Mary College, University of London, London, U.K., followed by five years as a Lecturer in the Department of Electrical Engineering at James Cook University of North Queensland. During 1982 and 1983, he was a consultant at Plessey Radar, U.K., and in December 1983 he joined the Commonwealth Scientific and Industrial Research Organisation (CSIRO) in Sydney, Australia. He has held several positions with CSIRO and is currently a CSIRO Fellow and Chief Scientist in the CSIRO Information and Communication Technologies Centre. He is also an Adjunct Professor at Macquarie University, Sydney. Dr. Bird is a Fellow of the Australian Academy of Technological and Engineering Sciences, the IEEE, the Institution of Engineering and Technology

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(IET), London, U.K., and is an Honorary Fellow of the Institution of Engineers, Australia. He has published widely in the areas of electromagnetics and antennas, particularly related to waveguides, horns, reflectors, wireless and satellite communication applications, and he holds twelve patents. In 1988, 1992, 1995, and 1996 he received the John Madsen Medal of the Institution of Engineers, Australia for the best paper published annually in the Journal of Electrical and Electronic Engineering, Australia, and in 2001 he was co-recipient of the H. A. Wheeler Applications Prize Paper Award of the IEEE Antennas and Propagation and Society. He was awarded a CSIRO Medal in 1990 for the development of an Optus-B satellite spot beam antenna and again in 1998 for the multibeam antenna feed system for the Parkes radio telescope. He received an IEEE Third Millennium Medal in 2000 for outstanding contributions to the IEEE New South Wales Section. Engineering projects that he played a major role in were given awards by the Society of Satellite Professionals International (New York) in 2004, the Engineers Australia in 2001, and the Communications Research Laboratory, Japan, in 2000. In 2003 he was awarded a Centenary Medal for service to Australian society in telecommunications and also named Professional Engineer of the Year by the Sydney Division of Engineers Australia. His biography is listed in Who’s Who in Australia. He was a Distinguished Lecturer for the IEEE Antennas and Propagation Society from 1997 to 1999, Chair of the New South Wales joint AP/MTT Chapter from 1995 to 1998, and again in 2003, Chairman of the 2000 Asia Pacific Microwave Conference, Member of the New South Wales Section Committee from 1995–2005 and was Vice-Chair and Chair of the Section in 1999–2000 and 2001–2002 respectively, Associate Editor of the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION from 2001 to 2004, a member of the Administrative Committee of the IEEE Antennas and Propagation Society from 2003–2005, and a member of the College of Experts of the Australian Research Council (ARC) from 2006–2007. He has been a member of the technical committee of numerous conferences including JINA, ICAP, AP2000, IRMMW-THz and the URSI Electromagnetic Theory Symposium. Currently, he is member of the Editorial Boards of the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES and the Journal of Infrared, Millimeter and Terahertz Waves. He has been Editor-in-Chief of the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION since 2004.

Karu P. Esselle (M’92–SM’96) received the B.Sc. degree in electronic and telecommunication engineering (First Class Honors) from the University of Moratuwa, Sri Lanka, in 1983 and the M.A.Sc. and Ph.D. degrees in electrical engineering from the University of Ottawa, Ottawa, ON, Canada, in 1987 and 1990, respectively. He is a Professor in electronic engineering, Macquarie University, Sydney. He was the Immediate Past Associate Dean - Higher Degree Research and the Founding Director of Postgraduate Research Committee in the Division of Information and Communication Sciences. He held these positions 2003 - 2008 and was also a member of the Division Executive. He served in all Macquarie University HDR-related committees at the highest level. He is the Director of Electromagnetic and Antenna Engineering, and the Deputy Director of the Research Centre on Microwave and Wireless Applications (CMWA), which was recently expanded after recognised as a Concentration of Research Excellence. He has been invited to serve as an international expert/research grant assessor by several overseas nationwide research funding bodies form the Netherlands, Finland, Hong-Kong and Chile. In Australia, he has been invited to assess grant applications submitted to the nation’s most prestigious schemes such as Australian Federation Fellowships and Australian Laureate Fellowships. His industry experience includes full-time employment as Design Expert by the Hewlett Packard Laboratory, USA, and several consultancies for local and international companies, including Cisco Systems (USA), Optus Networks, Locata (USA)/QX Corporation, ResMed, FundEd and Katherine-Werke (Germany) through Peter-Maxwell Solicitors. He was an Assistant Lecturer at the University of Moratuwa, a Canadian Government laboratory Visiting Postdoctoral Fellow at Health Canada, a Visiting Professor of the University of Victoria and a Visiting Scientist of the Information and Communication Technologies Centre (ICT Centre), Commonwealth Scientific and Industrial Research Organisation (CSIRO), Sydney, Australia. He is an Editor of the International Journal of Antennas and Propagation. He has authored about 250 scientific publications, including six invited book chapters and over fifteen invited conference presentations. His current research interests include metamaterials and their microwave applications, photonic crystals and photonic band gap (PBG)/electromagnetic band gap (EBG) structures, millimeter-wave EBG MMIC devices, antennas based on EBG, periodic structures including frequency selective surfaces,

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antennas for mobile and wireless communication systems including WiFi, WiMAX, HyperLAN, and ultrawideband systems, antennas for multi-signal location and navigation systems, dielectric-resonator (DR) antennas, broadband and multi-band printed antennas, smart antenna systems, hybrid antennas, theoretical methods, lens and focal-plane-array antennas for radio astronomy, moment methods, FDTD methods for periodic structures and closed-form Green’s functions for layered structures. His research activities are posted in the web at www.elec.mq.edu.au/celane/. Prof. Esselle’s recent awards include the 2009 Vice Chancellor’s Award for Excellence in Higher Degree Research Supervision (the first such award ever offered in Macquarie University) and 2004 (Inaugural) Innovation Award for best invention disclose. Since 2002, he was involved with research grants and contracts worth about five million dollars, and his research team members attracted further grants worth about a million dollars. The CELANE, which he founded, has provided a stimulating research environment for a strong team of researchers including six postdoctoral fellows. His mentees have been awarded 6 extremely competitive postdoctoral fellowships. Nine international experts who examined the theses of his recent five Ph.D. graduates ranked them in the top 5% or 10% in the world. He has served on Technical Program Committees or International Committees for many International Conferences. He will be chairing the Technical Program Committee of APMC 2011; he was the Publicity Chair of the APMC 2000. He is the Chair of the IEEE New South Wales (NSW) MTT/AP Joint Chapter, Editor of MQEC, the past Chair of the Educational Committee of the IEEE NSW, and a member of the IEEE NSW Committee.

Peter J. Hall was born in Hobart, Tasmania, in 1957. He was received the B.Eng. degree from the Tasmanian College of Advanced Education, in 1980, and the B.Sc. (Hons.) and Ph.D. degrees from the University of Tasmania, in 1981 and 1985, respectively. His postgraduate study was in the field of radio astronomy, particularly the development of high time resolution polarimeter spectrometers. He began his professional career in 1985 with a Postdoctoral Fellowship at the Commonwealth Scientific and Industrial Research Organization

(CSIRO), Parkes radio telescope, Sydney, Australia, implementing a novel system for observing the encounter of the Giotto spacecraft with Halley’s Comet. He moved to the University of Sydney School of Electrical Engineering as a Lecturer in 1987, prior to re-joining CSIRO in 1989, this time as a Research Scientist and Group Leader responsible for on-site commissioning of the new Australia Telescope Compact Array (ATCA) at Narrabri, in northwestern New South Wales. He performed a similar role during a major upgrade of the Parkes radio telescope in the mid-1990s, the upgrade allowing the telescope to support the Galileo mission to Jupiter while observing the southern sky with a new 13-beam receiver. With a long-time interest in millimeter-wave astronomy, he was a prime instigator of, and system scientist during, a major extension of the ATCA to 100 GHz operation. In 1999 he became Program Leader for CSIRO’s Square Kilometre Array (SKA) endeavour, being responsible for developing the scientific and engineering concepts surrounding wide-field astronomy, as well as the initial Australian submission to host the SKA. With a growing role in the global SKA project, he accepted an invitation in 2003 to become the first International Project Engineer, a position he held until 2008. During this time he played a leading role in the establishment of a Reference Design for the SKA, the setting of initial specifications for the instrument, and the formulation of an international design phase now underway. In 2008 he accepted an invitation to join Curtin University of Technology in Perth as its Foundation Professor of Radio Astronomy Engineering. In this capacity he is also Co-Director of the Curtin Institute of Radio Astronomy and a Deputy Director of the International Centre for Radio Astronomy Research (ICRAR). Professor Hall is a Fellow of the Institution of Engineers (Australia) and a Member of the Institution of Engineering and Technology (IET), London, U.K.

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ADS-Based Guidelines for Thinned Planar Arrays Giacomo Oliveri, Member, IEEE, Luca Manica, Graduate Student Member, IEEE, and Andrea Massa, Member, IEEE

Abstract—We propose an analytical technique based on almost difference sets (ADSs) for thinning planar arrays with well controlled sidelobes. The method allows one to synthesize bidimensional arrangements with peak sidelobe levels (PSLs) predictable and deducible from the knowledge of the array aperture, the filling factor, and the autocorrelation function of the ADS at hand. The numerical validation, concerned with both small and very large apertures, points out that the expected PSL values are significantly below those of random arrays and comparable with those from different sets (DSs) although obtainable in a wider range of configurations. Index Terms—Almost difference sets, array antennas, planar arrays, sidelobe level control, thinned arrays.

I. INTRODUCTION

A

NTENNA arrays for radar tracking, remote sensing, biomedical imaging, satellite and ground communications have often to support three-dimensional scanning with a suitable beampattern shape in the whole angular region [1]. Towards this end, planar arrays have to be used and large apertures are necessary to provide satisfactory angular resolutions along both azimuth and elevation [1]. On the other hand, the inter-element spacing should not exceed half-wavelength to avoid the presence of grating lobes [1]. These requirements usually result in very inefficient, heavy, and expensive solutions consisting of planar geometries with several thousands close elements. In order to reduce the number of elements while keeping the radiation properties of the original structures, thinning techniques have been successfully introduced [2]. Designing thinned planar arrays is an important research topic since decades (see [2]–[8] and the references cited therein). As a matter of fact, a suitable thinning allows one to reduce the array costs, its weight, and the power consumption. However, it causes the loss of the control of the peak sidelobe level (PSL) [6] to be properly counteracted. To this end, several techniques has been proposed in order to fully exploit the advantages of thinned arrangements while minimizing their drawbacks. First attempts have been conceived to require low computational resources (see [9, Table I]), but they have provided no significant

Manuscript received June 25, 2009; revised November 13, 2009; accepted December 16, 2009. Date of publication March 29, 2010; date of current version June 03, 2010. This work was supported in part by the Italian National Project: Wireless multiplatform mimo active access networks for QoS-demanding multimedia Delivery (WORLD), under Grant 2007R989S. The authors are with the Department of Information Engineering and Computer Science, University of Trento, Povo 38050 Trento-Italy (e-mail: giacomo. [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.2010.2046858

improvements when compared with random placements [2], [9] extensively employed in practice [2]. More recently, the availability of large computational resources has justified the use of optimization techniques such as dynamic programming [10], [11], genetic algorithms [5], [7], [8], [12], simulated annealing [13]–[17], and particle swarm optimizers [18]. Thinned arrays synthesized with optimization tools turn out to be very effective [7], [8], [11], [14], even though it is not possible to a-priori estimate the expected performances for a given array aperture and thinning factor [6]. Furthermore, computational and convergence issues make the application of stochastic optimizers difficult and expensive when dealing with 1D large apertures [6] and, even more, when planar arrangements are considered. In order to overcome such drawbacks, an alternative approach for thinning large arrays has been introduced (see [4] for the linear case and [6], [9], [20]–[22] for both linear and planar cases). Such an approach relies on the exploitation of binary sequences derived from difference sets (DSs), which exhibit a two-level autocorrelation function [4], or from DSs extensions [19]–[21]. Besides their analytic nature and the arising inexpensive generation, DS-based thinned arrays have several interesting properties. They are deterministically designed and present predictable [6] and low PSLs (3 dB and 1.5 dB below random arrays for the linear case and the planar one, respectively). However, only a limited number of DS sequences exist and the whole set of aperture sizes and thinning values [6], [23] cannot be dealt with. The problem of obtaining sub-optimal sequences (in terms of autocorrelation levels) has been recently addressed in information theory and “close” sequences to DSs have been looked for. Almost difference sets (ADSs) [24]–[26] are a wide class of binary sequences with three-valued autocorrelations [24]–[26]. They represent the closest sets to DSs [24]–[26] (three-levels vs. two-levels) and can be defined in a much broader set of aperture sizes and thinning values with respect to DSs [27]. Furthermore large repositories of explicit sequences are now available (e.g., [28]). They have been determined by numerically implementing the generating rules coming from the information-theory/combinatorial-mathematics literature (e.g., [26]). As regards geometries, the sidelobe characteristics of ADS-based arrays have been analyzed in [27] and good performances have been predicted and numerically verified dealing with both small and large apertures. Because of these results, its deterministic nature, and preliminary examples of planar arrays based on a subset ADSs with peculiar power patterns features (see [19]–[21] and the references therein for more details), an ADS-based technique seems to be a good candidate for thinning planar arrangements of radiating elements and it

0018-926X/$26.00 © 2010 IEEE

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will be presented in this paper. More specifically, the objective is not to define the “optimal” thinning method, but rather to provide a simple and reliable technique which guarantees to the designer predictable performances to be taken into account during the feasibility study. Towards this end, the PSL behavior of ADS-based planar arrays will be analytically investigated and, for the first time to the best of the author’s knowledge, different bounds will be provided by considering the whole and in a general fashion the whole class of 2D ADSs. It should be pointed out that, despite the linear case [27] where the Blahut’s theorem [27] has been applied, a different mathematical analysis is here necessary. Unlike the mathematical approach in [27] and [6], the PSL bounds are then derived starting from an innovative formulation based on the 2D discrete Fourier transform and related theorems, which allows a compact and straightforward analytical formulation. The paper is organized as follows. After a short overview on ADSs (Section II), a set of suitable bounds of the PSL are analytically determined in Section III. Section IV provides a selected set of numerical results aimed at validating the obtained PSL estimators as well as comparing the ADS performances with both random techniques and state-of-the-art optimization approaches. The exploitation of directive elements is also considered in order to point out the flexibility of the ADS thinning theory. Finally, some conclusions are drawn (Section V).

D

D

D

Fig. 1. Autocorrelation functions and associated binary sequences of the ADSs , (b), (e) , and (c), (f) . in Table I: (a), (d)

II. TWO-DIMENSIONAL ALMOST DIFFERENCE SETS With reference to the 2D problem, let us define -almost difference set as a -subset a of the Abelian group of 1 order ( , and being chosen according the Kronecker decomposition theorem [29]) for which the multiset

whose 2D periodic autocorrelation function [6] ( , being its periodicity) is a three-level function [24], [26]

(2) if and is an element of the set . For descriptive purposes, let us consider the ADSs in Table I [26], [28]. The plots of and of the three-level function in correspondence ( , 2, 3) are shown in Fig. 1. with As regards the closeness of the ADS to the DS sequences, likewise 1D arrangements, the bidimensional autocorrelation function of a differs from [6] by a unity in only points [24], [26] [(2)]. Moreover, the ADS autocorrelation function still remains unaltered after cyclic shifts of the reference sequence [24], [26] since if is an ADS, then where otherwise, and

contains nonzero elements of each exactly times and the nonzero elements each exactly remaining times [26]. Therefore, an ADS satisfies the following existence condition [25], [26]: (1) where , , and . Moreover, it is worth noticing that DSs are ADSs for which or [26]. is a , then it is possible to deIf rive a two dimensional binary sequence

abelian groups G = fg = ( ) = 0 . . . ; P Q 0 1; j = 0 . . . 0 1 = 0 . . . 0 1g G with the comis g + g = (( + ) ( + ) ) 1The

symbol stands for the direct sum of the , that is j; h ; i ; ; ;P ;h ; ;Q and is equipped and , that ponent-wise operations derived from j j ; h h . and

,

(3) is still an ADS. As a consequence, starting from a , it is always possible to build different s by applying cyclic shifts to its elements.

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Concerning the ADS generation, several theorems have already been proposed in information theory and combinatorial mathematics (e.g., see [26] and references therein). Analogously, the explicit forms of many others ADSs have been determined and they are now available in [28]. Furthermore, suitable techniques/theorems for completing the whole set of theoretically admissible ADS sequences are still a work-in-progress [24]–[26] in related fields of research non-properly concerned with electromagnetics/antenna-theory and out-of-the-scope of this paper. III. ADS-BASED PLANAR ARRAYS—MATHEMATICAL FORMULATION elements located, Let us consider a planar array of according to the binary sequence , on a bidimensional lattice and wavelengths along the and of points spaced by directions, respectively. The array factor of such an elements arrangement turns out to be [1], [6]

(4) where

and . Moreover, can be also expressed in terms of the 2D discrete time Fourier transform (DTFT) of the sequence ,

(5) as follows:

D =D

Fig. 2. Plot of the normalized array factor derived from and associated j k; l j values: (a) i and (b) i .

( )

=1

=2

(6) Furthermore, by applying the sampling theorem [30] to the , function

For illustrative purposes, Fig. 2 shows the plot of the array and the samples of the DFT of factor in correspondence with the set [Fig. 2(a)] and the set [Fig. 2(b)]. As regards the peak sidelobe level (PSL), it is defined as

(7) being 2D (DFT) of the sequence

(10)

and , it results that where shifted set

and

is the array factor coming from the is the main-lobe region of extension

(8) Such a relationship states that the samples of the array factor at , are equal to the values of the DFT in of the weighting sequence (9)

(11) (see Appendix) with

.2

The notation G (0; 0) indicates the set of elements of the Abelian group G without the null element, (0,0). 2

n

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TABLE I EXAMPLES OF ADSs AND DESCRIPTIVE PARAMETERS

provide a PSL threshold as for DSs-based planar arrays [6]. Nevertheless, the following set of inequalities holds true for sufficiently large values of and (Appendix) (13) where , , , ,

Fig. 3. PSL values of the ADS-based planar arrays derived from the sequences , ; ;P , ; ;Q (a) and PSL bounds (b). Number of elements: P Q —aperture size:  .

D

= 0 ...

0 1 = 0 ... 2 = 7 2 11

01

3 25

By substituting (8) in (10),

(12) is

obtained,

since ,

being the two-dimensional sequence derived from . As it can be noticed, the PSL of an ADS-based array is a function . Unfortunately, since these of the coefficients coefficients cannot be expressed in closed-form (but their values are available when the generating ADS is known) and, unlike DSs, depends on the indexes and , it is not possible to

, and . and can be It is now worth pointing out that evaluated only once the ADS sequence is exactly known, since is required, while the knowledge of the term and can be a-priori determined the bounds starting for the knowledge of the characteristic parameters describing the ADS (i.e., , , , , and ). For a preliminary validation of such an estimate criterion, let in Table I. As exus refer to the planar array generated by pected, the PSL of the set depends on the values of and [Fig. 3(a)] and different shift values give the same , whose value lies into the range of confioptimal PSL, dence defined in (13) [Fig. 3(b)]. The multiplicity of the optimal evaluations are actually solutions indicates that less than needed to identify the optimal ADS-based planar array. This is a negligible computational cost compared to the burden required by stochastic optimization techniques to determine a thinned arrangement on the same aperture. As regards the simple steps required to design an ADS-based array they consist in: (a) seand thinning factor for lecting the desired aperture size the designed array (on the basis of the application constraints); (b) evaluating the expected PSL for the final ADS array by means of (13); (c) if the expected PSL complies with the application requirements, selecting from [27] (or other repositories) an ADS with size and thinning factor as close as possible to and ; (d) deriving the optimal array by applying cyclic shifts to and evaluating the arising PSL. IV. NUMERICAL ANALYSIS In this section, the results of a numerical assessment are described and discussed to point out potentialities and limitations of the ADS-based approach proposed as a suitable tool for predicting the performance of an effective set of planar thinned arrays. For comparison purposes, random arrangements [3], [6] are considered as reference since, likewise ADS arrays, their

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performances can be a-priori estimated. More in detail, the estimator of the normalized peak sidelobe level of planar random arrays (RND) turns out to be [3]

(14) where is the probability or confidence level that no sidelobe value. Moreover, random lattice planar arexceeds the rays (RNL), whose elements are located on a uniformly-spaced lattice of points over the aperture, exhibit the following PSL [6] (15) where is the thinning factor (such an expression, which is valid for not close to one, can be obtained by trivial extension to the 2D case of [6, Eq. (25) ]). The first numerical example deals with the analysis of the for different apertures PSL bounds (13) versus (Fig. 4). and when As expected (Section II), the upper bound of PSL tends to when and and its value, , is always below and except for and large apertures a small set of values close to . As a matter of fact, whatever the array dimension, the worst performances verify in correspondence with . Therefore, such an index value will be analyzed in the following to provide “worst-case” indications on ADS-based thinning. Fig. 5(b) shows the behaviors of the ADS bounds versus the . Since ADS are here availaperture dimension , , and are reported, as well. able [28], usually As it can be noticed, these plots confirm that overestimates the actual peak sidelobe of the ADS array and that is always well below the values exhibited by random families. For completeness, the remaining of Fig. 5 gives an indication on the estimated behavior of ADS arrays in correspon—Fig. 5(a), dence with different thinning percentages [ —Fig. 5(b), —Fig. 5(c)] for which ADSs are , denot still available. As regards the confidence range fined as (16) and shows a limited dependence on it slightly increases with ) (Fig. 6). the aperture dimension ( 4 dB in and the minimum value Moreover, verifies for as it can be analytically derived. of Concerning available ADSs with , Fig. 7 shows the (and related bounds) of the array genbehavior of the ( and ) erated from the sequence whose power pattern and elements arrangement are given in Fig. 7(b) and Fig. 7(c), respectively. Despite the small aperture , still lies in the range of values estimated by (13) [Fig. 7(a)] and it appears to be significantly below the . random estimates and comparable with the DS value at It is also interesting to notice that the reference array derived

Fig. 4. Numerical Validation. Plots of the PSL bounds of ADS-based planar arrays, the estimator of the PSL of random (RND) and random lattice (RNL) arrays, and values of the PSL of DS-based finite arrays versus  when  : and (a) P Q , (b) P Q , (c) P Q .

= 10

= 10

= 10

=05

from allows one to determine several shifted array configurations with [Fig. 3(b)] as well as multiple arrays with . Such a feature is not only limited to , but it is a common property of ADS-based arrays as confirmed by the examples in Figs. 8–10 and concerned with larger apertures. Furthermore, it should be pointed out that more than one cyclic shift of the reference ADS sequence

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1

Fig. 6. Numerical Validation. Plots of versus the array aperture, when  : and in correspondence with different thinning values  2

=05

Fig. 5. Numerical Validation. Plots of the PSL bounds of ADS-based planar arrays and the estimators of the PSL of random (RND) and random lattice (RNL) arrays versus the array aperture, P Q, when  : and (a)  : , (b)  : , (c)  : .

= 05

= 06

= 05

= 04

gives an array pattern with [Figs. 3(b), 8(a), 9(a), 10(a)]. Such considerations highlight evalthat: (a) also through an exhaustive search, less than uations are actually needed to identify the optimal ADS-based planar array; (b) a very limited number of evaluations is enough

[

P Q,

[0; 1]].

to synthesize an ADS array with a PSL value below that from random/random lattice distributions. As far as the radiation patterns are concerned, Figs. 7(b) and 8(c)–10(c) allow one to point out a further interesting property is a twoof ADS planar arrays. Unlike DSs, where valued function [6], the unequal magnitudes of the samples of (Fig. 2) lead to a non-uniform behavior of the array pattern outside the mainlobe region with some non-negligible variations of the sidelobes [see Figs. 7(b) and 8(c)–10(c)]. This can be exploited as an additional degree of freedom to be used in antenna synthesis. One efficient way to do that is to consider directive elements. As an example, let us consider the planar arrays synthewith isotropic or directive elements (e.g., sized from dipoles along the axis). Fig. 11(a) gives the PSL values for different shifts of the reference set. As it can be observed, reduces ( vs. the value of ) thanks to the use of directive elements and, more interestingly, the optimal shift for the directive , array is not equal to that with isotropic elements ( vs. , ). This is due to the following. One has to determine the shift generating the lowest lobes in the whole sidelobe region when dealing with the “isotropic” array [Fig. 11(b)]. Otherwise, the use of directive and values with lowest elements suggests to choose the sidelobes only near the mainlobe region [Fig. 11(c)] since the element factor “erases” the highest sidelobes far from the mainlobe region in the resulting antenna pattern [Fig. 11(d)]. The last section of the numerical validation is aimed at giving some indications on the performance of the ADS arrays versus those coming from state-of-the-art thinning techniques based on stochastic optimizers [5], [7], [18]. Since ADSs are not still available in correspondence with the thinning percentage of the test cases under analysis, the comparison cannot be considered fully fair, but it can be useful to suggest some guidelines for a fast and reliable choice of the most suitable synthesis procedure as well as on the achievable PSL results. Fig. 12 shows the PSL of the thinned arrays optimized with the methods in [5], [Fig. 12(a)], [7], [Fig. 12(b)], and [18],

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2

Fig. 7. Numerical Validation—Planar Array [Number of elements: P —aperture size:  ]. Plots of the PSL bounds versus  Q [P Q , : ] (a). Plot of the normalized array factor (b) generated -based array arrangement (c). from the

= 7 2 11 = 77 = 0 4805

D

3 25

[Fig. 12(c)], respectively, along with the PSL bounds derived for the corresponding ADS-based arrays (i.e., only the values and since the ADS sequences, although of theoretically existing, have not been yet determined). As it can be noticed, ADS-based arrays compare favorably in terms of PSL with global optimized designs since, even in the worst ), . case (i.e., V. CONCLUSION In this paper, ADSs have been considered for the design of thinned planar arrays. The research work is aimed at identifying the descriptive parameters of the ADS-based thinning technique as well as their effect on the array performances.

D

2 = 0 . . . 01 = 0 . . . 01 = 529 = 0 5

Fig. 8. Numerical Validation—Planar Array [Number of elements: P —aperture size:  ]. PSL values of the ADS-based arrays Q , ; ;P , ; ;Q derived from the sequences , : ]. Plot of the (a). Plots of the PSL bounds versus (b)  [P Q -based array arrangement. normalized array factor (c) generated from the

= 23223

D

11 211

D

Likewise the linear case [27], the objective of this study is to analytically define a “term of comparison” to help the array designer in identifying the synthesis approach allowing the optimal trade-off between computational resources and the achievable result in terms of PSL level. Towards this purpose, the performances of planar ADS-based arrays have been investigated and suitable bounds for the PSL have been determined

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Fig. 10. Numerical Validation—Planar Array D [Number of elements: P 2 2 Q = 199 2 199—aperture size: 99 2 99]. PSL values of the ADS-based 36 236 ,  = 0; . . . ; P 0 1 ,  = arrays derived from the sequences D D = 0 . . . 01 = 0 . . . 01 0; . . . ; Q 0 1 (a). Plots of the PSL bounds versus (b)  [P Q = 39601,  = 0:5]. Plot of the normalized array factor (c) generated from the D -based = 5329 = 0 5

D

Fig. 9. Numerical Validation—Planar Array [Number of elements: P —aperture size:  ]. PSL values of the ADS-based arrays Q , ; ;P , ; ;Q derived from the sequences (a). Plots of the PSL bounds versus (b)  [P Q , : ]. Plot of the -based array arrangement. normalized array factor (c) generated from the

array arrangement.

thanks to a new formulation based on the properties of the two-dimensional DFT. Such an analysis has been validated by means of a large set of numerical experiments also aimed at comparing the predicted ADS performances with those of random distributions or stochastically optimized arrays.

The obtained results have pointed out the following features of the ADS thinning technique: • the PSL of the synthesized pattern is a-priori known when the ADS sequence is available in an explicit form, while suitable bounds are predictable, otherwise;

= 73273

D

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Fig. 11. Numerical Validation—Non-Isotropic elements. PSL values , of the ADS-based arrays generated from the sequences  ; ;P ,  ; ;Q (a). Normalized array patterns of the arrays generated from the sequences (b) with isotropic elements, with isotropic (c) and directive elements (d).

= 0 ...

01

D

= 0 ...

01

D

D =D

• because of the three-level autocorrelation function, ADS sequences guarantee additional degrees-of-freedom (compared to the DS case) to be profitably exploited (e.g., using directive elements) for fitting the design constraints;

Fig. 12. Comparative Assessment. Plots of the PSL bounds versus the array : and for (a)  : [5], (b)  aperture, P Q, when  : [7], (c)  : [18], (d)  : [18].

= 0 48

=05 = 0 44

= 0 54

= 0 507

• unlike iterative optimization or trial-and-test random synthesis techniques, the approach determines the array con-

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figuration just through simple shifts of a reference ADS sequence; • thanks to the availability of rich repositories of ADSs also concerned with large and indexes, wide apertures (impracticable for stochastic optimizers) can be dealt with; • the use of ADS does not prevent their integration with optimization techniques, vice versa it could represent a way (to be explored in successive researches) to improve the convergence rate of iterative methods or for enabling stochastic searches in thinning large arrays by means of a suitable ADS-based initialization. In addition to these features, other main contributions of the present paper consist in the following methodological novelties: 1) an innovative and compact analytic formulation for the analysis of the PSL based on the relations between the DTFT and the DFT of ADS binary sequences that avoids the exploitation of the “infinite array” formalism [6], [27] generally more complicated when dealing with planar arrangements; 2) unlike ADS-based linear arrays [27], the exploitation of the information on to derive a more tight bound for planar geometries. Future works will be devoted to further extend the proposed analysis to other geometries and problems. Moreover, although out-of-the-scope of the present paper since not pertinent to antenna arrays but concerned with combinatorial mathematics, other advances in the research activities concerned with the explicit determination of other ADS sequences are certainly expected. APPENDIX Definition of the Mainlobe Region : Starting from (12) as for planar DS arrays [6], it can be proved that the PSL of ADSbased arrays is close to the values of the samples of the array , factor at

. By exploiting such an observation, one can obtain (17), shown at the bottom of the page, where the mainlobe region, , is defined analogously to [6] as the visible region where the first term in (17) exceeds the magnitude of the second one. As regards the first term, its magnitude is approximately equal to

and for large values of and . Moreover, the largest coefficients in the second term (i.e., and ) of (17) are bounded by

Thus, after simple manipulation, it is possible to show that extends to the region limited by the following boundary inequality (18) Derivation of , version of ,

in (13): With reference to discrete

(19) let us consider the following approximation of , shown in (20) at the bottom of the page, where the complex coefficient has been expressed in terms of its amplitude,

, and phase,

.

(17)

(20)

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It is worth pointing out that, likewise DSs, is not a-priori known as well as, unlike DSs, the term and they have to be estimated. Towards this end, by exploiting the circular correlation property of DFT [30], it is possible to state that

, the analysis is carried out As regards the phase terms as in [6] in order to give an estimate of the PSL. More specifiare deterministic quancally, although the phase terms tities, they are dealt with as independent identically distributed (i.i.d.) uniform random variables. Under this assumption, (25) can be expressed as

(21) (26)

and to obtain the following relationship

where for large

and (27)

(22) being

where . Concerning the real-valued coefficients the Parseval’s theorem [30]

, by applying

, , being i.i.d. random variables and is the cardinality of . Since the statistics of are not available in closed form, Monte Carlo simulations have been performed to provide an approximation of its mean value (28)

and noticing that

and , the following holds true (23)

is fiBy substituting (28) in (26), the upper bound nally obtained. in (13): By sampling (12) at ( Derivation of , ), , , one can obtain (29), shown at the bottom of the page. By substituting (22) in (29), and observing that

Therefore, since (30) (24) and substituting (22) in (20), we obtain (25) at the bottom of the page.

it turns out (31)

(25)

(29)

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Derivation of assume that the ADS

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in (13): With reference to (20), let us at hand is known. Thus, (32)

is now a known quantity. By substituting (32) in (20), we obtain (33), shown at the bottom of the page. , let us consider the same As regards the phase terms and let us rewrite (33) as procedure used for deriving follows:

By using (19) and expressing the complex coefficients in terms of their amplitude and phase, we can approximate (35) as follows in (36), shown at the bottom of the page. In order to provide a lower bound to such an expression, let us observe that the following equation holds true (see (37) at the bottom of the page). The two terms on the right-hand side of (37) are then treated separately. As regards the first factor, it can be observed that, when the ADS sequence is explicitly available, it is a known quantity equal to

(34)

(38)

where is successively approximated with its mean value . (28) to obtain in (13): Let us start from (17), and Derivation of consider the following approximation of , see (35) shown at the bottom of the page.

Therefore, by substituting (38) in (37), we obtain (39), shown at the top of the next page. As for the second term, an analysis similar to that carried out is still possible. However, the two sumfor deriving as in (27). Indeed, mations cannot be extended here up to

(33)

(35)

(36)

(37)

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(39)

(40)

by performing a Monte Carlo analysis, it turns out that the approximation (see (40) at the top of the page) holds true for large , which, substituted in (39), provides the lower values of . bound ACKNOWLEDGMENT A. Massa wishes to thank E. Vico for being and C. Pedrazzani for her continuous help and patience. Moreover, the authors are very grateful to M. Donelli for kindly providing some numerical results of computer simulations. Furthermore, the authors greatly appreciated the reviewing and valuable comments of the anonymous reviewers of a previously submitted version of the manuscript. REFERENCES [1] C. A. Balanis, Antenna Theory: Analysis and Design, 2nd ed. New York: Wiley, 1997. [2] Y. T. Lo, “A mathematical theory of antenna arrays with randomly spaced elements,” IEEE Trans. Antennas Propag., vol. 12, no. 3, pp. 257–268, May 1964. [3] B. Steinberg, “The peak sidelobe of the phased array having randomly located elements,” IEEE Trans. Antennas Propag., vol. 20, no. 2, pp. 129–136, Mar. 1972. [4] D. G. Leeper, “Thinned periodic antenna arrays with improved peak sidelobe level control,” U.S. Patent 4071848, Jan. 31, 1978. [5] R. L. Haupt, “Thinned arrays using genetic algorithms,” IEEE Trans. Antennas Propag., vol. 42, no. 7, pp. 993–999, Jul. 1994. [6] D. G. Leeper, “Isophoric arrays—Massively thinned phased arrays with well-controlled sidelobes,” IEEE Trans. Antennas Propag., vol. 47, no. 12, pp. 1825–1835, Dec. 1999. [7] S. Caorsi, A. Lommi, A. Massa, and M. Pastorino, “Peak sidelobe reduction with a hybrid approach based on GAs and difference sets,” IEEE Trans. Antennas Propag., vol. 52, no. 4, pp. 1116–1121, Apr. 2004. [8] R. L. Haupt and D. H. Werner, Genetic Algorithms in Electromagnetics. Hoboken, NJ: Wiley, 2007. [9] B. Steinberg, “Comparison between the peak sidelobe of the random array and algorithmically designed aperiodic arrays,” IEEE Trans. Antennas Propag., vol. 21, no. 3, pp. 366–370, May 1973. [10] M. I. Skolnik, G. Nemhauser, and J. W. Sherman, III, “Dynamic programming applied to unequally-space arrays,” IRE Trans. Antennas Propag., vol. AP-12, pp. 35–43, Jan. 1964. [11] S. Holm, B. Elgetun, and G. Dahl, “Properties of the beampattern of weight- and layout-optimized sparse arrays,” IEEE Trans. Ultrason., Ferroelectr., Freq. Control, vol. 44, no. 5, pp. 983–991, Sep. 1997. [12] T. G. Spence and D. H. Werner, “Thinning of aperiodic antenna arrays for low side-lobe levels and broadband operation using genetic algorithms,” in Proc. IEEE Antennas and Propagation Society Int. Symp., Jul. 9–14, 2006, pp. 2059–2062.

[13] C. S. Ruf, “Numerical annealing of low-redundancy linear arrays,” IEEE Trans. Antennas Propag., vol. 41, no. 1, pp. 85–90, Jan. 1993. [14] 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. [15] M. Vicente-Lozano, F. Ares, and E. Moreno, “Pencil-beam pattern synthesis with a uniformly excited multi-ring planar antenna,” IEEE Antennas Propag. Mag., vol. 42, no. 6, pp. 70–74, Dec. 2000. [16] F. Ares, J. Fondevila-Gomez, G. Franceschetti, E. Moreno-Piquero, and J. A. Rodriguez-Gonzalez, “Synthesis of very large planar arrays for prescribed footprint illumination,” IEEE Trans. Antennas Propag., vol. 56, no. 2, pp. 584–589, Feb. 2008. [17] M. Alvarez-Folgueiras, J. A. Rodriguez-Gonzalez, and F. Ares, “Lowsidelobe patterns from small, low-loss uniformly fed linear arrays illuminating parasitic dipoles,” IEEE Trans. Antennas Propag., vol. 57, no. 5, pp. 1584–1586, May 2009. [18] 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. [19] L. E. Kopilovich and L. G. Sodin, “Two-dimensional aperiodic antenna arrays with a low sidelobe level,” Proc. Inst. Elect. Eng., vol. H-138, no. 3, pp. 233–237, 1991. [20] L. E. Kopilovich and L. G. Sodin, “Synthesis of two-dimensional nonequidistant antenna arrays on the basis of difference set theory,” J. Commun. Technol. Electron., vol. 39, pp. 33–42, 1994. [21] L. E. Kopilovich and L. G. Sodin, Multielement System Design in Astronomy and Radio Science. Dordrecht/Boston/London: Kluwer Academic Publishers, Astrophysics and Space Science Library, 2001, vol. 268. [22] L. E. Kopilovich, “Square array antennas based on hadamard difference sets,” IEEE Trans. Antennas Propag., vol. 56, no. 1, pp. 263–266, Jan. 2008. [23] La Jolla Cyclic Difference Set Repository [Online]. Available: http:// www.ccrwest.org/diffsets.html [24] C. Ding, T. Helleseth, and K. Y. Lam, “Several classes of binary sequences with three-level autocorrelation,” IEEE Trans. Inf. Theory, vol. 45, no. 7, pp. 2606–2612, Nov. 1999. [25] K. T. Arasu, C. Ding, T. Helleseth, P. V. Kumar, and H. M. Martinsen, “Almost difference sets and their sequences with optimal autocorrelation,” IEEE Trans. Inf. Theory, vol. 47, no. 7, pp. 2934–2943, Nov. 2001. [26] Y. Zhang, J. G. Lei, and S. P. Zhang, “A new family of almost difference sets and some necessary conditions,” IEEE Trans. Inf. Theory, vol. 52, no. 5, pp. 2052–2061, May 2006. [27] 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. [28] ELEDIA Almost Difference Set Repository [Online]. Available: http:// www.ing.unitn.it/~eledia/html/ [29] M. I. Kargapolov and J. I. Merzljako, Fundamentals of the Theory of Groups. New York: Springer-Verlag, 1979. [30] J. G. Proakis and D. G. Manolakis, Digital Signal Processing: Principles, Algorithms, and Applications, 3rd ed. London: Prentice Hall, 1996.

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Giacomo Oliveri (M’09) received the “Laurea” degree in telecommunications engineering and the Ph.D. degree in space science and engineering from the University of Genoa, Italy, in 2005 and 2009, respectively. He is a member of the ELEDIA Research Group, University of Trento, Italy. His main research is focused on antenna arrays, electromagnetic propagation in complex environments and numerical methods for electromagnetic problems.

Luca Manica (S’09) was born in Rovereto, Italy. He received the B.S. and M.S. degrees in telecommunication engineering both from University of Trento, Italy, in 2004 and 2006, respectively, where he is currently working toward the Ph.D. degree. He is a member of the ELEDIA Research Group, University of Trento, Italy. His main interests are the synthesis of the antenna array patterns and fractal antennas.

Andrea Massa (M’03) received the “Laurea” degree in electronic engineering and the Ph.D. degree in electronics and computer science from the University of Genoa, Genoa, Italy, in 1992 and 1996, respectively. From 1997 to 1999, he was an Assistant Professor of Electromagnetic Fields at the Department of Biophysical and Electronic Engineering (University of Genoa) teaching the university course of Electromagnetic Fields 1. From 2001 to 2004, he was an Associate Professor at the University of Trento. Since 2005, he has been a Full Professor of Electromagnetic Fields at the University of Trento, where he currently teaches electromagnetic fields, inverse scattering techniques, antennas and wireless communications, and optimization techniques. At present, he is the Director of the ELEDIALab at the University of Trento and Deputy Dean of the Faculty of Engineering. His research work since 1992 has been principally on electromagnetic direct and inverse scattering, microwave imaging, optimization techniques, wave propagation in presence of nonlinear media, wireless communications and applications of electromagnetic fields to telecommunications, medicine and biology. Prof. Massa is a member of the IEEE Society, of the PIERS Technical Committee, of the Inter-University Research Center for Interactions Between Electromagnetic Fields and Biological Systems (ICEmB) and Italian representative in the general assembly of the European Microwave Association (EuMA).

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Deterministic Synthesis of Uniform Amplitude Sparse Arrays via New Density Taper Techniques Ovidio Mario Bucci, Fellow, IEEE, Michele D’Urso, Tommaso Isernia, Piero Angeletti, Member, IEEE, and Giovanni Toso, Senior Member, IEEE

Abstract—Uniform amplitude sparse arrays have recently gained a renewed interest and a number of synthesis techniques, mainly based on global optimization algorithms, have been presented. In this paper, after a discussion about the expected characteristics of such arrays, a simple deterministic approach for the case of pencil beams patterns is proposed and discussed. The approach, which outperforms previous synthesis techniques, takes inspiration from existing (not well-known) density taper procedures to develop a two stages synthesis where the first step is solved in a new closed analytical form, while the second one just requires local refinements or fast 1-D optimizations. As a consequence, the proposed approach avoids the need of multidimensional global optimization procedures and the inherent possibly prohibitive computational costs. Notably the first step, which also exhibits an improved flexibility with respect to other analytical techniques, already outperforms previous approaches in a number of cases of interest. Several numerical results confirm the effectiveness and usefulness of the proposed tools. Index Terms—Antenna synthesis, aperiodic arrays synthesis, array antennas, density taper techniques.

I. INTRODUCTION AND MOTIVATIONS HE synthesis of non uniformly spaced arrays with fixed amplitude excitations is a traditional problem in the antenna literature [1]–[19]. The problem has recently gained a renewed interest in the area of satellite applications [20], [21]. In particular, phased arrays represent a natural antenna solution for generating a multiple beams satellite coverage, but they have been rarely used because of their complexity, cost and weight. These drawbacks are mainly due to the required distributed and tapered power amplification which induces a poor efficiency in DC-to-RF power conversion. In this respect, unequally spaced

T

Manuscript received May 25, 2008; revised December 09, 2009; accepted December 09, 2009. Date of publication March 29, 2010; date of current version June 03, 2010. O. M. Bucci is with DIBET-Univ. Federico II of Napoli, I-80125 Napoli, Italy and also with IREA-CNR I-80124 Napoli, Italy (e-mail: [email protected]). M. D’Urso was with DIBET-Univ. Federico II of Napoli, I-80125 Napoli, Italy and DIMET-Univ. Mediterranea di Reggio Calabria, I-80100 Reggio Calabria, Italy. He is now with the Giugliano Research Center-Integrated System Analysis Unit-SELEX Sistemi Integrati, Giugliano di Napoli, I-80014 Napoli, Italy (e-mail: [email protected]). T. Isernia is with DIMET-Univ. Mediterranea di Reggio Calabria, I-80100 Reggio Calabria, Italy (e-mail: [email protected]). P. Angeletti and G. Toso are with the European Space Agency, Noordwijk, The Netherlands (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.2010.2046831

arrays with uniform amplitude have several interesting characteristics and may offer some potential advantages with respect to periodic arrays [20]–[22]. In fact, a uniform excitation allows all the amplifiers of the active antenna to be operated under the same optimal condition, thus gaining efficiency. Then, the density tapering of the elements’ positions provides the degrees of freedom of the array in place of the excitations, with a number of interesting consequences. First, as the beam-width is strongly dependent on the source dimension but weakly dependent on the number of elements [23], the same beam-width of a periodic array may be achieved with a reduced number of elements. This reduction implies an increase of the average spacing, which results in lowering the mutual coupling between the array elements. In addition, the lack of periodicity also allows reducing grating lobes in the radiation pattern, with improvements not only on the sidelobes level (SLL) at a single frequency, but even allowing extending the bandwidth of the antenna [2]. Obviously, a number of drawbacks exist as well, such as for instance the reduced maximum directivity obtained when decreasing the number of radiating elements [20], [23]. In this respect, it is worth to note that no synthesis procedure algorithm will be able to compensate for the reduced aperture efficiency of the array in case of large average spacing. In the synthesis of this type of arrays the designer has to deal with a different number and kind of degrees of freedom with respect to the typical case where only the excitations are the unknowns of the problem. In particular, as the radiated field depends linearly on the (possibly complex) excitations but non linearly on the (real) elements locations, not only the number of degrees of freedom is reduced, but also, and more important, the available degrees of freedom are much more difficult to control and to exploit in an optimal way. This latter circumstance has induced a large number of researchers to tackle the problem by means of several kinds of up-to date biologically inspired global optimization procedures (see [11]–[14], [17], [21], [22]). However, the computational cost of global optimization algorithms rapidly increases with the problem size [26] and this, in practice, can prevent the actual attainment of the global optimum [27], [28]. In order to give a contribution not only to the development of effective synthesis procedures for aperiodic arrays, but also to the comprehension of the limitations of these radiating structures, a new deterministic approach, resulting in very effective synthesis procedures is proposed and thoroughly discussed in this paper, considerably extending the preliminary ideas presented in [22].

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In particular, Section II is concerned with a very brief summary of existing approaches, with an emphasis on the relevant but not well known work by Doyle [5] (as reported in [6]). In Section III the new approach, using two different steps, is proposed. The first step, which exploits a “localization result” due to Doyle [5], [6], leads to a new analytical solution and allows to enlarge the flexibility and to improve performances with respect to previous analytical solutions. Notably, the proposed analytical solution, opposite to [5], [6], can also be applied to the case where the reference source is a discrete one (i.e., an array). Limitations of the analytical step, related to an underlying approximation and to a simplifying assumption, both needed in order to get a closed form solution, are also discussed. The second step is devoted to overcome as much as possible the above limitations by means of two procedures which may be used independently or in a combined way. Both these procedures result in fast 1D global optimizations and/or in suitable local multidimensional optimizations. In this way the need for large size global optimization procedures is drastically reduced or completely avoided, leading to reliable and effective synthesis procedures. Comparisons with enumerative techniques, in those cases where these latter are viable, also show that the proposed deterministic synthesis procedures are indeed capable to get the globally optimal locations of the array elements. Numerical results in Section IV fully confirm the theoretical analysis of Section III, and also permit deriving quantitative rules about the performances one can achieve in the different cases. Conclusions follow. II. SOME EXISTING PROCEDURES, THE DOYLE-SKOLNIK APPROACH, AND A USEFUL ‘LOCALIZATION’ RESULT A detailed review of the large existing literature on aperiodic arrays is outside of the scope of the paper. In this section a particular attention is placed on a number of recent contributions and to an existing analytical optimization approach whose outcomes are of interest for the new two-step approach presented in Section III. While an interesting (local) optimization approach is proposed in [18], [19], most of the recently published procedures for the synthesis of uniform amplitude sparse arrays are based on the exploitation of a number of (stochastic) physically inspired global optimization procedures [11]–[14], [17]. The main advantages of these kind of approaches seem to be their flexibility (any kind of constraint or goal can be easily dealt with, and multiobjective problems can be tackled [13]), as well as their capability to actually get the global optimum in case the number of unknowns is sufficiently small. As a matter of fact, even if the most recent contributions seem to enhance the range of cases which can be dealt with [17], one has to bear in mind that computational complexity of global optimization techniques grows exponentially with the number of unknowns [26], As a consequence, this kind of approach does not actually guarantee the achievement of the globally optimal solutions in a reasonable time when the number of antennas exceeds some threshold depending on the kind of problem and on the particular algorithm

at hand [26], [27]. Hence, it makes sense to take a step towards alternative, possibly deterministic, approaches. Useful hints in this direction are already available in a number of contributions that appeared as early as in the sixties [1]–[7], which share the common idea of using some kind of density taper of the (uniformly excited) elements to emulate the behavior of a properly chosen continuous source acting as a reference. In particular, as some kind of tapering is required on the continuous source to get low sidelobes, this results in a density taper of the array elements in a way such that the elements of the array will be more densely packed where the continuous source is higher. In particular, because of the fact that it is the only one which can be proved to be optimal in some sense, and because of its relevance to our developments, let us focus on the synthesis approach presented in [5] (as reported in [6]). Therein, one starts from the idea of approximating a given real and positive continuous aperture current density (acting as a reference) with an array of equal amplitude non-uniformly spaced elements. As the optimal synthesis of pencil beams generally implies real and positive aperture functions when the beam pointing direction is at broadside [28], the method is of interest any time one has to deal with the synthesis of pencil beams. is the ideal current distribution (given, for instance, If is the cumulative by a Taylor distribution [24], [25]), and current distribution, i.e., (1) where is the continuous source dimension normalized to the wavelength, the idea consists in matching the ideal cumulative current distribution given by (1) with the actual (staircase) cumulative distribution generated from the sparse array. To this has a unitary integral over the source end, assuming that extension, and is the number of elements of the array, the following procedure is suggested: 1) compute the ideal cumulative current distribution; into intervals each having the 2) divide the interval . This division identifies boundary same area , with , such that points (2) 3) for each interval point corresponding to

select on the ordinate axis the

(3) 4) determine the corresponding abscissa . As indicated in [5], [6], this synthesis procedure is equivalent to minimize a weighted distance between the desired far field pattern, i.e., (4)

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by the elements of the original uniformly spaced array, thus over-restricting the set of possible locations and worsening performances. As very flexible, fast and effective solutions do exist for the synthesis of uniformly spaced arrays [28], [29], this point also needs further attention. In order to avoid the above drawbacks, let us re-formulate the overall problem as the minimization of the cost functional

Fig. 1. Reference linear array, supposed to be symmetric.

(7)

and the synthesized one, (5) is the algebraic distance of the -th (isotropic) elewhereas ment, measured in wavelengths, from the center of the antenna (see Fig. 1). More precisely, the technique is and equivalent to minimize the functional (6) is a weighting function depending on wherein the spectral variable . Note that the minimized cost functional takes into account, in a weighted sense, both the visible space ). and the invisible part of the spectrum (i.e., the region The behavior and the properties of the cumulative current distribution (1) permit deriving a further result which is the one exploited in the following paragraphs. In fact, one can show [5], [6] that a necessary condition in order to minimize (6) is that one radiating element is required for each equal-area element of the . We will refer in the following original current distribution to this relevant property as to a localization process.

are the unknown locations of the wherein antennas and denotes either the or the uniform (i.e., ) norm in the generic interval where the matching between the reference pattern and the actual pattern is desired. Note the proposed approach is different in spirit from the Doyle-Skolnik one, as a fitting on the radiated field over a given interval is required herein, while a fitting of the cumulative currents is the starting rationale of [5], [6]. Also note that we will consider the case wherein the norm in (7) is the uniform one ), although a similar analysis is also valid in the (i.e., sense. As a general strategy, we have in mind a two step approach wherein an analytical solution of (7) is found under simplifying assumptions, and then the simplifying assumptions are removed and a more accurate solution eventually found. For the sake of clarity, the two step of the overall approach are discussed in the two following subsections. A. An Analytical Solution Under Simplifying Assumptions Minimization of the cost functional (7) corresponds to minimize in the interval the maximum value of the difference

III. THE PROPOSED APPROACH TO DENSITY TAPER The Doyle-Skolnik approach sets in a precise mathematical framework the density taper techniques, suggesting a solution procedure which gives back the globally optimal locations in weighted sense defined in (6). On the other side, it also the presents a number of drawbacks. First, the adopted cost functional is not necessarily the most suited for practical applications. In fact, although some control over the invisible part of the spectrum is required in principle, one is not necessarily interested there in the same identical behavior as the one pertaining to the reference solution, and a non super-directive behavior of the synthesized sparse array is already dictated by the (real and positive) reference aperture field. Second, also in view of the fact that the element factor can help in limiting the pattern close to the endfire directions, one could just be interested in realizing a fitting over a reduced angular zone. Besides, one could be interested in a uniform norm, one. rather than in a (weighted) Last, but not least, the Doyle-Skolnik approach cannot deal, in an actually globally optimal way, with the case wherein the reference source is by itself a regular (uniformly spaced) array. In fact, in that case the cumulative current distribution becomes a staircase function, so that the procedure [5], [6] would necessarily give back a subset of the positions already occupied

(8) Then, as in [5], [6], let us partition the overall source into segments such that (9)

with the normalization (10) Then, (8) can also be written as

(11)

Finally, let us perform a localization process analogous to the one in [5], [6], i.e., let us suppose that minimization of (11)

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can be performed by separately minimizing each of its addenda. Then, minimization of (11) would become (12)

, which amounts to enforce that the field for radiated from the -th radiating source is as near as possible (in the uniform norm sense) to the one radiated from the -th portion of the continuous reference source. Notwithstanding this localization cannot be demonstrated to be necessary (as in [5], [6]), splitting the original problem into sub-problems seems to be reasonable for a number of circumstances. First, this splitting is correct if the cost functional norm with a weight (and is enforced in the ) as in [5], [6]. Second, in order for the two far fields (i.e., the far fields corresponding to the reference source and to the actual one) to be equal, the corresponding near fields have to be equal as well (neglecting the invisible part of the spectrum). Then, if a near field matching is performed on the two fields, and the observation line is very close to the array aperture, the field at a generic observation point will mainly depend on the nearest discrete source and on the nearest portion of the reference source, respectively, so that transformation of the dense problem of (11) into the diagonal problem of (12) is again reasonable, at least if the spacing amongst the antennas is not too large. In view of the above considerations, minimizing (in sense) each term of (11) is likely to provide at least a sub-optimal solution to the problem of minimizing (11). Finally, as shown in the following, the assumed localization process allows improving the performances with respect to [5], [6] while preserving simplicity (and analyticity if desired). Then, let us enforce minimization of (12) for each , with . If one assumes (13) so that the exponential can be accurately approximated by the first two terms of its Taylor expansion, it is shown in the is ensured provided Appendix that minimization of each that (14)

is the bariSolution (14) has a very simple interpretation: in the interval . Obviously, it is also center of very easy to compute for both cases of a continuous reference source and of a discrete reference source (i.e., an array). Note that, under (13), solution (14) remains valid when the minimization of (7) (and henceforth of (12) under the localization as(un-weighted) sense. sumption) is meant in the Hypothesis (13) also permits the understanding of properties (and limitations) of the proposed approach. By deferring the reader to Section IV for more quantitative rules, let us note herein that the proposed analytical solution will be particularly

accurate when the product at the left hand side of (13) is sufficiently small, which depends on: • the maximum observation angle; • the extension of the different intervals in terms of wavelengths; • the desired degree of accuracy on final patterns, which also depends on and the expected SLL (the lower the sidelobes, the more the required accuracy). Therefore, for a given pattern to be achieved, the (approximated) analytical solution (14) will already guarantee very good results if the spacings in the aperiodic array are sufficiently small, and/or when the angular sector of interest is sufficiently small. Performances of solution (14) will instead degrade when spacings increase, or the angular region of interest is not sufficiently small. In this respect, it is worth noting that some degradation in performances is expected (whatever the solution procedure) when the average spacing of the array elements exceeds a given threshold given by half a wavelength. A simple explanation of such a circumstance is the fact that under the given assumption of isotropic sources, a reduction in the number of elements necessarily implies a reduced aperture efficiency, so that the pattern of the continuous reference source cannot be granted (but for a reduced angular range, see Section IV for more details). When considering patterns with lower and lower sidelobe level, in order to guarantee the required accuracy of approximations for the specific cases at hand, the left hand member of (13) has to be smaller and smaller, so that all considerations above remain valid, but for further reductions of the maximum allowable average spacing and/or of the angular region to be considered (see Section IV-C). Again, however, it has to be noted that in case of very low sidelobe level some degradation in performances is unavoidable whatever the solution procedure. In fact, when passing from a uniformly spaced array to an aperiodic one, the energy of the first grating lobe is spread over a region (first pseudo grating lobe), expressed in terms of , going from the reciprocal of the maximum spacing to the reciprocal of the minimum one (both expressed in wavelengths) [20]. In case of very low SLL, one will have a large tapering dynamic, and henceforth a large variance amongst the different distances. Therefore, the reciprocal of the larger spacing will be smaller, so that the pseudo grating lobes will more easily enter the observation region of interest. In summary, the developed analytical approach suffers from the limitations induced from (13), but its effectiveness should be evaluated with respect to the ultimate theoretical performances achievable by any kind of solution process, rather than by per se. Such an analysis is performed in Section IV, wherein some rules of thumb are also given. B. Removing the Simplifying Assumptions By following the strategy identified at the beginning of this section, the overall performances can be improved by relaxing, in a second processing step, the hypotheses assumed up to now. A first possibility to improve results with respect to the analytical solution of the previous subsection consists in removing the small angle/small interval assumption. As long as the problem

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can be transformed from minimization of (11) to minimiza, tion of (12), one separately looks for the variables , each having well defined bounds . Then, it is a rather obvious idea to tackle the problem as the global optimization of (12) (or the corresponding ones in the sense). Although this entails the missing of analytical formulas, and the need of global optimizations, this is still a very effective computational procedure. In fact, for each optimization, the unknowns are limited in values and known to lie in limited and relatively small intervals. As a consequence, the needed global optimizations are simple, up to the point they can also be performed by means of enumerative techniques. A second chance of refinement, which can be used either in conjunction or separately from the one above, amounts in removing the localization assumption. The localization process of minimizing (12) rather than (11) cannot be proved to be necessary. On the other side, it has been argued that it is a reasonable assumption, so that, at least for the case where spacings are not too large (so that a near field matching is hopefully possible, see discussion above), minimization of (12) should give back final , quite near to the global minimum of (8). So, results can be effectively adopted as a starting guess of another optimization procedure meant to minimize (8). As long as the localization process is a good ansatz, and accurate solutions are achieved in minimizing (11), local optimization procedures, such as for instance the ones in [18], [19] will be sufficient to get the globally optimal positions. Numerical investigations (see Section IV in the following) fully support such an ansatz. Note that a cost functional different from (8) can be adopted in this last step. For instance, in order to further enhance flexibility of the overall approach, one could ask to minimize the maximum level of the sidelobes rather than matching a specific reference pattern. IV. COMPARISONS WITH OTHER APPROACHES, RELEVANT CHARACTERISTICS, AND EXPECTED PERFORMANCES By the sake of clarity, the analysis is subdivided in four subsections. In the first one, the full-analytical procedure of Section III-A is compared with both the previous density taper methods [5], [6], and with recent procedures based on stochastic global optimizations procedure. Notably, it is shown that the analytical step alone already outperforms recently published synthesis techniques based on global optimization. In the second one, some relevant characteristics of the approach, such as the capability to deal with a very large number of elements, and the case where the reference source is by itself an array, are discussed. In the third one, the overall approach, including the two synthesis steps, is analyzed and compared with previous and present analytical techniques, as well as with a number of results recently published in the literature. In a low order problem, performances are also compared with the ones achieved by an ad-hoc developed and computationally expensive exhaustive technique, showing the capability of the new proposed approach to get the actually global optimum. Finally, the last part is devoted to fix some simple quantitative rules for performances of the proposed approaches, also in relationship with the expected ultimate limitations of this class of sources.

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Fig. 2. Synthesized array factors by using (17) and the Doyle method.

A. Performances of the Full-Analytical Procedure Let us consider the problem of synthesizing a 10 element linear sparse array with uniform excitations and with a desired radiation pattern corresponding to a Taylor distribution with , and [25]. The average elewhile ment spacing for the array to be synthesized is is the maximum size of the non uniform array. Now, by applying the deterministic synthesis procedure described in Section III, thus determining the array element positions according to (14), the radiation pattern reported in Fig. 2 is obtained, which corre. The corresponding element posponds to a sitions are [0.1839 0.5605 0.9712 1.4587 2.0473], given in terms of . The positions obtained by using the method in [5], [6] are [0.1833 0.5593 0.9678 1.4526 2.0719], corresponding to the ra. diation pattern reported in Fig. 2, with a As it can be seen, the proposed analytical technique outperforms the previous one, as it both allows improving the SLL (which is about 1 dB better) and better matching the reference Taylor pattern. In order to provide comparisons with global optimization techniques, let us consider the problem, only slightly different from the one above, of synthesizing a 10-element sparse array , while there wherein the outermost elements are fixed at are four element locations which need to be optimized. The while is the overall sparse average element spacing is array dimension (which will imply a slightly narrower main beam with respect to the first example in this section). Now, by applying the procedure proposed in Section III with a , reference Taylor source characterized by and [25], the radiation pattern reported in Fig. 3 is achieved, which corresponds to a . The achieved element positions are [0.2019 0.6158 1.0671 1.6026 2.2500], always given in terms of . The same problem has been considered in [13], where a particle swarm optimization based procedure has been adopted to get the final result (see Fig. 3). Notably, the proposed approach is able to get in real time (essentially) the same result of [13]. A further assessment of the capabilities of the proposed technique with respect to stochastic optimization procedures can be achieved by comparing the performances of our method (the one in Section III) with the ones reported in [14] and [11], wherein an ant colony optimization (ACO) based procedure has been

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Fig. 3. Synthesized array factors by using (17) and the method in [13]. Here, 2a = 4:5, N = 10 and the outermost elements have been fixed to 2:25.

6

adopted for optimizing sparse linear arrays with 10 and 28 radiating elements, respectively. While in the first case our analytical method achieves (in real time) essentially the same pattern, the proposed analytical procedure outperforms the one based on ACO in the second case, wherein a larger number of array elements locations has to be optimized. In fact, let us consider the synthesis problem of [11] dealing with the synthesis of a dimension. In 28-elements sparse linear array covering a this case, by using a Taylor reference source characterized by a , and [22], and an element factor in order to provide a fair comparison, , 2 dB better than the one one achieves an in [11]. The achieved performances, and in particular the improved results with respect to existing procedures, are fully coherent with the circumstance that the computational burden of global optimization procedures (and henceforth the performances for a given processing time) dramatically worsen when the number of unknowns to be optimized grows [26]–[28]. The NemirovsyYudin theorem proves in fact that such a complexity grows exponentially with the number of unknowns [26]. B. Some Relevant Characteristics of the Analytical Solution In view of the fact that purely numerical procedures may consider (at the price of a further increase in computational complexity) mutual coupling effects, an interesting question arises about the performances of the proposed analytical procedure in case some mutual coupling is present. This may be indeed the case with the above examples, where the minimum spacings is or even less. In fact, while mutual coupling in the order of is negligible in case of large spacings, it may have an effect on the results above, wherein, in several examples, the minimum spacing is severely reduced. A detailed discussion of such a point is outside of the scope of the paper, both in view of the fact that the amount of mutual coupling will change with the specific kind of radiating sources, and of the fact that the second step of the overall procedure could be implemented in such a way to take into account mutual coupling. However, in order to understand how much performances are affected from mutual coupling, all the above patterns were re-computed by means of the CST Microwave Studio commercial electromagnetic simulator. A small worsening of the SLL

Fig. 4. Synthesized array factor when N = 301 elements are used.

level (of order of 0.7 dB) was observed for the first two examdipoles were used as elementary sources). No ples (where kind of worsening was instead found when comparing the theoretical (mutual coupling free) and actual pattern of the third example (where a two element Yagi antenna was used to account for some directive element pattern). According to the above, one may conclude that the proposed analytical technique works quite nicely also in the cases of a moderate degree of mutual coupling. In case mutual coupling has a strong impact, the analytical procedure cannot provide the very best solution to the synthesis problem. However, the partial result can still be used as a starting point for the subsequent (second) step of the overall procedure, wherein mutual coupling can be exactly dealt with. Notably, such a strategy would avoid the heavy task of performing the computation of mutual couplings (for each element of the population, at each iteration) which is intrinsic to evolutionary based optimization procedures. As a further interesting point, note that, being based on analytical formulas plus (eventually) simple local or 1-D optimizations, the presented approach is perfectly capable to deal in an effective fashion with hundreds of unknowns (that is, the same number of antennas tackled in [1] for the simpler thinned array case). In order to show this capability, an example of a 301 element linear array is reported in Fig. 4, wherein a Taylor pat, tern corresponding to a continuous distribution with [25] is adopted as reference , and source. Also note that 501 elements would be required by using spaced uniform linear array. To the best of our knowledge, a no evolutionary technique would be able to manage such a large number of continuous unknowns. As a last characteristic which is worthy of consideration, we stress again that, in opposition to existing analytical techniques, the proposed analytical one can be applied in a straightforward way to the case wherein the reference pattern arises from a discrete source (f.i., a regular, uniformly spaced array). In fact, it is meant in a distriwill suffice that the reference source butional sense. As an example, Fig. 5 shows the radiation pateltern of a sparse and uniformly excited array with ementary sources whose locations have been achieved starting and from a Dolph-Chebychev array [24] with -spaced elementary sources (Fig. 5). As can be seen, the achieved radiation pattern fulfills the design requirements quite well. Note that, in both cases, an element factor has been adopted.

BUCCI et al.: DETERMINISTIC SYNTHESIS OF UNIFORM AMPLITUDE SPARSE ARRAYS VIA NEW DENSITY TAPER TECHNIQUES

Fig. 5. Synthesized array factor when Dolph-Chebychev source.

N = 55 and starting from a discrete

Fig. 6. Synthesized array factors with N

= 50.

C. Performances of the Two-Step Approaches Although the proposed analytic technique outperforms the ones in [5], [6] in a non negligible number of cases, and it is never outperformed from the elder ones, it has to be noted that many cases can be found wherein the previous technique [5], [6] and the proposed one achieves essentially the same results. Moreover, because of the underlying derivations, one is not capable to provide the best approximation to the reference pattern in whatsoever angular region (or for more demanding SLL) in case of largely spaced arrays. Some performance improvement can be achieved by the procedures suggested in Section III-B. As a first example, let us consider the case of a 50-element linear sparse array with uniform excitations and with a desired radiation pattern fixed according to the one radiated by a Taylor , and [25] distribution with (see Fig. 6). To determine the unknown locations, let us first use (14). The achieved radiation pattern is reported in Fig. 6. The average distance between elements is now equal to . Then, let us move to the first possibility to perform the second step as discussed in Section III-B. In particular, by assuming that an accurate matching between the actual and reference pattern is , let us minimize (12) by using required for . The achieved radiation pattern is reported in Fig. 6 (solid-yellow line). As can be seen, the simple effective idea of performing a global optimization on each interval works quite well, as a reduction of 2.5 dB over the sidelobes and a better matching of the reference field (in the considered spectral variable range) is achieved with respect to (14). Note that the relatively large average spacing between elements (and the

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Fig. 7. Synthesized array factors by using the two-step approach of Section III-B, the stochastic procedure of [12], and the global optimal solution.

reduced variance between locations) allows using non-isotropic antennas in the synthesized array, which would improve final performances. In order to understand the kind of improvements one can achieve by means of the second possibility discussed in Section III-B, we have compared the proposed synthesis approach with the results in [10] and [12], which are respectively based on a deterministic synthesis procedure based on Legendre series expansions, and on the exploitation of a Differential Evolution algorithm respectively. In both cases, the FMINIMAX procedure available in MATLAB has been used in the second step (i.e., after the analytical one) for minimizing the maximum level of the field in the sidelobes region. In particular, in order to compare with [10], a linear symisotropic elements has been conmetric sparse array of and sidered, and a Taylor pattern with [25] has been used as reference pattern. Then, the achieved locations have been used as starting guess of a local optimization procedure. The achieved pattern, as compared to the 16.87 dB , while having exachieved in [10], exhibits a actly the same beam-width as the one in [10]. Then, in order to compare the actual performances with the ones in [12], the problem of synthesizing a symmetric linear isotropic elements (i.e., the example in array with Fig. 4 of [12]) has been considered. In the first step, a reference source corresponding to a Taylor radiation pattern with and has been used [25]. Then, by applying the above discussed local optimization procedure, we , achieve the result reported in Fig. 7, with a better than the one in [12] of more than 1 dB. Notably, the result is coincident with the result one would obtain by using an ad hoc developed exhaustive search,1 looking for the array positions able to minimize the maximum sidelobes level in the sidelobes region, once the beam-width has been fixed. It is worth noting that no similar kind of performances has been possible when starting the FMINIMAX procedure from a random (or uniform) set of locations, so that the overall approach indeed takes definite advantage from the first analytical step. 1This search, very expensive from a computational point of view, exploits the expected symmetry of the solution, assumes locations are ordered as the corresponding indices, and adopts subsequent grid fittings until the synthesis result remains essentially unchanged.

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D. Some Rules of Thumb on the Expected Performances

A relevant question concerns the expected performances of the proposed one step or two step synthesis procedures, as well as the ultimate performances one can achieve from a theoretical point of view by means of sparse arrays. As far as the expected performances of the analytical step in case of large spacings are concerned, on the basis of many examples, a practical rule for the case the required SLL is not very demanding (i.e., 20 dB) can be devised. As a matter of , i.e., fact, a product seems to guarantee the fulfillment of the given constraints. In order to get quantitative rules in case of more demanding requirements on SLL, and to get a finer understanding, we have performed a parametric study on the kind of performances one is able to achieve, and the minimum number of (isotropic) antennas which is required for a given SLL. Note a first analysis of this problem, under quite restrictive hypotheses, dates back to the 1960s [30]. The corresponding results, evaluated for the , 25 case of a reference Taylor source with dB, 30 dB, respectively, are reported in Fig. 8 and show that average spacing is enough for , while a a narrower and narrower average spacing is required for lower requirements on SLL. In particular, an average spacing of about would be required for the case , and about is instead required in case . Note that this kind of performance is representative of the results from a wider set of numerical experiments, including the case wherein Dolph-Chebychev patterns are used as a reference. While the graphs in Fig. 8 have been obtained by imposing a SLL over the entire visible range, a trade off of the kind discussed in Section IV-A (the narrower the angular region of interest, the larger the average spacing in such a way to have a constant product) is expected. By so doing, Fig. 8 allows setting some interesting rules of thumb for the expected performances also in case of reduced angular ranges. A similar analysis has also been done for the overall two step approach, and the corresponding results are shown in the same figure (square markers). As can be seen, the more refined synaverage spacing is actually thesis procedure indicates that a , 25 dB, while the enough for both cases of requires an average more demanding case of . As the number of degrees of freedom spacing of less than of radiated fields reduces with the region of interest [30], a trade off is again expected amongst the angular region to be covered and the number of required elements. More precisely, if a narrower region has to be covered, then a lower number of elements can be used. The quantitative rule for understanding how the minimum number of required elements changes with the angular region to be covered can be deduced from [28]. Notably, . the rule still concerns the product In summary, the considerations we have developed on the basis of Fig. 8 and of a number of additional numerical experiments set practical criteria on how the number of required antennas (or the average spacing) changes with the required SLL (as evaluated on the entire visible range), while the product at the left hand side of (13) and/or the theory of the degrees of

0

Fig. 8. SLL versus N for SLL = 20 dB (a), SLL = SLL = 30 dB (c). In all cases 2a = 10 and n = 10.

0

025 dB (b), and

freedom of the field establish how this number will vary when reducing the angular region of interest. Note that a similar kind of trade-off will be true whatever the synthesis procedure one wants to adopt, in the sense that the number of degrees of freedom of the field in a given angular sector sets a bound on the minimum required number of radiating elements to achieve a given pattern. Then, according to the above graphs, this number needs to be improved in case of very demanding SLL to accommodate the required large tapering on the source. In view of the examples in Section IV-A and Fig. 8, (showing that a number of antennas exactly equal to the number of degrees of freedom of the field in the adopted interval is needed in case of moderate values of SLL), and of results in IV-B, (showing that the proposed approach is capable to get the same results as the one

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coming from an exhaustive search), we are confident the ultimate performances one can achieve by means of sparse arrays are not far from the ones we have found herein.

so that, as the first and last term coincide because of (9), miniis ensured provided that (14) holds true. mization of each

V. FINAL REMARKS AND FUTURE WORK

REFERENCES

In this paper, by taking advantage of a relevant ‘localization’ result available in the literature [5], [6], a simple and effective two-step deterministic approach to the synthesis of pencil beams by means of sparse isophoric arrays, has been proposed and discussed. In particular (see Section IV-A) the first step of the overall approach, resulting in closed form solutions, outperforms previous analytical solutions, and already outperforms global optimization based procedures in a number of cases. Also, (see Section IV-B) opposite to previous analytical techniques, it can deal with the case wherein the reference source is a (regular) array, and, opposite to global optimization techniques, can deal in an effective fashion with very large arrays. Notably, by virtue of analyticity, results are obtainable almost in real-time (less than one second on a desktop PC). As shown in IV-C, the second step of the proposed procedures further improves performances (definitely better compared to the ones achieved with previous available methods) at the price of a slightly increased computational burden (few tens of seconds). In case of small spacings, such a second step could take into account mutual couplings while still taking advantage from the analytical solution of step one. To this end, some recently developed fast method, such as the one in [31], could be exploited. Some rules of thumb on expected performances, as compared to ultimate limitations of such a class of sources, has been also developed (Section IV-D). Possible future works include the extension of the overall approaches and tools to the 2D geometry in either cases of general and circularly symmetric patterns, as well as the consideration of cases where properly quantized (rather than strictly isophoric) excitations are admitted. APPENDIX One has to minimize the maximum value of

(A.1)

i.e.,

(A.2)

for . Assuming that the exponential term can be accurately approximated by the first two terms of its Taylor expansion, (A.2) becomes

[1] H. Unz, “Linear arrays with arbitrarily distributed elements,” IEEE Trans. Antennas Propag., vol. 8, pp. 222–223, 1960. [2] M. Skolnik, G. Nemhauser, and J. Sherman, III, “Dynamic programming applied to unequally spaced arrays,” IEEE Trans. Antennas Propag., vol. 12, no. 1, pp. 35–43, 1964. [3] A. Ishimaru and Y.-S. Chen, “Thinning and broadbanding antenna arrays by unequal spacings,” IEEE Trans. Antennas Propag., vol. 13, no. 1, pp. 34–42, 1965. [4] R. E. Willey, “Space tapering of linear and planar arrays,” IRE Trans. Antenna Propag., vol. 10, pp. 369–377, 1962. [5] W. Doyle, “On approximating linear array factors,” RAND Corp. Mem RM-3530-PR, 1963. [6] M. I. Skolnik, “Ch. Nonuniform arrays,” in Antenna Theory, R. E. Collin and F. Zucker, Eds. New York: McGraw-Hill, 1969, pt. I. [7] Y. T. Lo, “A study of space-tapered arrays,” IEEE Trans. Antennas Propag., vol. 14, no. 2, pp. 22–30, 1966. [8] R. F. Harrington, “Sidelobe reduction by non-uniform element spacing,” IRE Trans. Antennas Propag., vol. 9, pp. 187–192, 1961. [9] D. G. Leeper, “Isophoric arrays—Massively thinned phased arrays with well controlled sidelobes,” IEEE Trans. Antennas Propag., vol. 47, no. 12, pp. 1825–1835, 1999. [10] B. P. Kumar and G. R. Branner, “Synthesis of unequally spaced arrays using Legendre series expansion,” in Proc. IEEE Antennas Propag. Int. Symp., 1997, vol. 4, pp. 2236–2239. [11] O. Quevedo-Teruel and E. Rajo-Iglesias, “Ant colony optimization for array synthesis,” in Proc. IEEE Antennas Propag. Society Int. Symp., July 9–14, 2006, pp. 3301–3304. [12] D. G. Kurup, M. Himdi, and A. Rydberg, “Synthesis of uniform amplitude unequally spaced antenna arrays using the differential evolution algorithm,” IEEE Trans. Antennas Propag., vol. 51, pp. 2210–2217, 2003. [13] N. Jin and Y. Rahmat-Samii, “Advances in particle swarm optimization for antenna design: Real-number, binary, single-objective and multiobjective implementation,” IEEE Trans. Antennas Propag., vol. 55, no. 3, pp. 557–567, 2007. [14] E. Rajo-Iglesias and O. Quevedo-Teruel, “Linear array synthesis using an ant-colony-optimization-based algorithm,” IEEE Antennas Propag. Mag., vol. 49, no. 2, pp. 70–79, 2007. [15] T. M. Milligan, “Space-tapered circular (ring) array,” IEEE Antennas Propag. Mag., vol. 46, no. 3, pp. 70–73, 2004. [16] G. Toso, M. C. Viganó, and P. Angeletti, “Null-matching for the design of linear aperiodic arrays,” presented at the IEEE Antennas Propag. Society Int. Symp., Honolulu, HI, Jun. 10–15, 2007. [17] K. Cheng, X. Yun, Z. He, and C. Han, “Synthesis of sparse planar arrays using a modified real genetic algorithm,” IEEE Trans. Antennas Propag., vol. 55, no. 4, Apr. 2007. [18] G. Miaris and J. N. Sahalos, “The orthogonal method for the geometry synthesis of a linear antenna array,” IEEE Antennas Propag. Mag., vol. 41, no. 1, pp. 96–99, Feb. 1999. [19] S. K. Soudos, G. Miaris, and J. N. Sahalos, “On the quantized excitation and the geometry synthesis of a linear array by the orthogonal method,” IEEE Trans. Antennas Propag., vol. 49, no. 2, pp. 298–303, Feb. 2001. [20] G. Toso, C. Mangenot, and A. G. Roederer, “Sparse and thinned arrays for multiple beam satellite applications,” in Proc. 29th ESA Antenna Workshop on Multiple Beams and Reconfigurable Antennas, Apr. 2007, pp. 207–210. [21] G. Caille, Y. Cailloce, C. Guiraud, D. Auroux, T. Touya, and M. Masmousdi, “Large multibeam array antennas with reduced number of active chains,” presented at the 2nd Eur. Conf. on Antennas Propag., Edinburgh, U.K., Nov. 2007. [22] O. M. Bucci, M. D’Urso, T. Isernia, P. Angeletti, and G. Toso, “A new deterministic technique for the design of uniform amplitude sparse arrays,” presented at the 30th ESA Antenna Workshop, Noordwjik, The Netherlands, May 27–30, 2008. [23] R. J. Mailloux, Phased Array Antenna Handbook, 2nd ed. London: Artech House, 2005. [24] 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.

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[25] T. T. Taylor, “Design of line-source antennas for narrow beamwidth and low sidelobes,” IRE Trans. Antennas Propag., vol. 3, pp. 16–28, Jan. 1955. [26] A. S. Nemirovsky and D. B. Yudin, Problem Complexity and Method Efficiency in Optimization, ser. Interscience Series in Discrete Mathematics. New York: Wiley, 1983. [27] O. M. Bucci, M. D’Urso, and T. Isernia, “Some facts and challenges in array synthesis problems,” Auto. J. Contr. Meas. Electron. Comput. Commun., vol. 49, no. 1, pp. 13–20, 2008. [28] T. Isernia, P. Di Iorio, and F. Soldovieri, “An effective approach for the optimal focusing of array fields subject to arbitrary upper bounds,” IEEE Trans. Antennas Propag., vol. 50, pp. 1837–1847, 2000. [29] M. G. Andreasen, “Linear arrays with variable interelement spacings,” IRE Trans. Antennas Propag., vol. 10, pp. 137–143, 1962. [30] O. M. Bucci, C. Gennarelli, and C. Savarese, “Representation of electromagnetic fields over arbitrary surfaces by a finite and nonredundant number of samples,” IEEE Trans. Antennas Propag., vol. 46, pp. 351–359, 1998. [31] C. Craeye, “A fast impedance and pattern computation scheme for finite antenna arrays,” IEEE Trans. Antennas Propag., vol. 54, no. 10, pp. 3030–3034, 2006.

Ovidio Mario Bucci (F’93) was born in Civitaquana, Italy, on November 18, 1943. He was an Assistant Professor at the Istituto Universitario Navale of Naples, 1967–1975, then Full Professor of Electromagnetic Fields at the University of Naples. He was Director of the Department of Electronic Engineering, 1984–86 and 1989–90. Vice Rector of the University of Naples, 1994–2000. He also was Director of the Interuniversity Research Centre on Microwaves and Antennas (CIRMA), 1997–2003, and of the CNR Institute of Electromagnetic Environmental Sensing (IREA), since 2001. He has been the principal investigator or coordinator of many research programs, granted by national and international research organizations, and companies. He is the author or coauthor of more than 350 scientific papers, mainly published on international scientific journals or proceedings of international conferences. His scientific interests include scattering from loaded surfaces, reflector and array antennas, efficient representations of electromagnetic fields, near-field far-field measurement techniques, inverse problems and non invasive diagnostics, biological applications of nanoparticles and electromagnetic fields. Prof. Bucci was Past President of the National Research Group of Electromagnetism and of the MTT-AP Chapter of the Centre-South Italy Section of IEEE. He was a recipient of the International Award GUIDO DORSO for Scientific Research, 1996, Presidential Gold Medal for Science and Culture, 1998. He received the Calabria prize for scientific research, 2004. He is Fellow of the Institute of Electrical and Electronic Engineers (IEEE) since 1993, and a Member of the Associazione Elettrotecnica Italiana (AEI) and of the Academia Pontaniana.

Michele D’Urso was born in Cercola, Italy, in 1976. He received the Telecommunication Engineering Master degree (summa cum laude) and the Ph.D. degree from the University Federico II of Naples, Italy, in 2002 and 2006, respectively. After graduation, he joined the Applied Electromagnetic Group at the Department for Electronics and Telecommunication Engineering of the University Federico II in Naples, first as an Associate Researcher and then as a Ph.D. Student, from 2003 to 2005. From September 2004 to January 2005, he was an intern in the Mathematics and Modeling Department of Schlumberger-Doll Research, Ridgefield, CT, under the supervision of Prof. T. Habashy and Dr. A. Abubakar. After 2006 he was an Associate Researcher at the University Federico II of Naples and then an Associate Professor at University Mediterranea of Reggio Calabria, Italy. He is now the Director of “Giugliano Research Center” of the Integrated System Analysis Unit of SELEX Sistemi Integrati. His scientific interests include forward and inverse electromagnetic scattering

methods, phase retrieval problems, radar array antenna synthesis, metamaterials synthesis and analysis. Dr. D’Urso was the recipient of the G. Barzilai Award of the Italian Electromagnetic Society (SIEM) in 2004. He has been also awarded at the European Wireless Technology Conference (EuWiT) in 2009 for the Best Young Research Paper Award. He is also the recipient of the (SELEX Sistemi Integrati) “Premio Innovazione (Innovation Award)” 2009.

Tommaso Isernia received the Laurea degree (summa cum laude) and the Ph.D. degree from the University Federico II of Napoli, Italy. After being a Researcher (from 1992 to 1998) and an Associate Professor (from 1998 to 2003) at the Federico II University, he is currently Full Professor of Electromagnetic Fields at the Università Mediterranea di Reggio Calabria,Reggio Calabria, Italy. His research interests cover inverse problems in electromagnetic with particular emphasis on phase retrieval, inverse scattering, and antenna synthesis problems. More recently, he is also engaged on effective solution procedures for forward scattering problems as applied to very large scale problems and photonic crystal devices. Prof. Isernia was the recipient of the G. Barzilai award from the Italian Electromagnetics Society (SIEM) in 1994.

Piero Angeletti (M’07) received the Laurea degree (summa cum laude) in electronic engineering from the University of Ancona, Ancona (Italy), in 1996. From 1997 to 2004, he was involved in aerospace systems engineering activities, joining in succession the following companies in Italy: Augusta Helicopters, Alenia Spazio, Elettronica, and Space Engineering. He is currently a Member with the Technical Staff of the European Space Agency (ESA), European Space Research and Technology Centre (ESTEC), Noordwijk, The Netherlands. He authored and coauthored more than 100 contributions to international conferences, peer-reviewed journals, book chapters and several patents. His current research interests are in the field of analysis, modeling, and design of telecommunication and navigation-satellite systems, payloads and antennas. Mr. Angeletti was a corecipient of the 2008 Best Paper Prize at the 30th ESA Antenna Workshop.

Giovanni Toso (S’93–M’00–SM’07) received the Laurea Degree (summa cum laude) and the Ph.D. in electrical engineering from the University of Florence, Florence, Italy, in 1992 and 1995, respectively. In 1996, he was a Visiting Scientist at the Laboratoire d’Optique Electromagnétique, University of Aix-Marseille III, Marseille, France. From 1997 to 1999, he was a Postdoctoral student at the University of Florence. In 1999, he was a Visiting Scientist at University of California, Los Angeles (UCLA) and received a scholarship from Thales Alenia Space (Rome, Italy). In 2000, he was appointed as a Researcher in a Radioastronomy Observatory of the Italian National Council of Researches (CNR). Since 2000, he is with the Antenna and Submillimeter Section, European Space and Technology Centre, European Space Agency, ESA ESTEC, Noordwijk, The Netherlands. His research interests are mainly in the field of array antennas for satellite applications and, in particular, in non regular arrays, reflectarrays and multibeam antennas. He has coauthored more than 150 contributions to international conferences, peer-reviewed journals, book chapters and several patents. Dr. Toso was a corecipient of the 2008 Best Paper Prize at the 30th ESA Antenna Workshop. In 2009, he was a coeditor of the Special Issue on Active Antennas for Satellite Applications published in the International Journal of Antennas and Propagation.

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Pattern Synthesis of Narrowband Conformal Arrays Using Iterative Second-Order Cone Programming K. M. Tsui and S. C. Chan, Member, IEEE

Abstract—A new design method is proposed for the power or shaped beam pattern synthesis problem of narrowband conformal arrays, where only the magnitude response is specified. The proposed method iteratively linearizes the non-convex power pattern function to obtain a convex subproblem in the design variables, which can be solved optimally using second-order cone programming (SOCP). In addition, a wide variety of magnitude constraints such as non-convex lower bound magnitude constraints can be incorporated. An efficient technique for determining a reasonably good initial guess to the problem is also proposed to further improve the reliability of the method. Computer simulations show that the initial guesses so obtained converge to satisfactory solutions while satisfying various prescribed magnitude constraints. Design results show that the performance of the proposed method is comparable to the optimal solution previously obtained for uniform linear arrays with isotropic elements. Moreover, we show by means of examples that the proposed method is also applicable to general non-convex power pattern synthesis problems involving arbitrary array geometries, arbitrary polarization characteristics and mutual coupling effect. Index Terms—Conformal antenna arrays, mutual coupling, narrowband design, polarization, power pattern synthesis, second-order cone programming, shaped beam pattern synthesis.

I. INTRODUCTION

HE analysis and design of conformal arrays have received considerable attention because of their flexibility in attaching to arbitrary surface of vehicles and aircrafts to save space and capability of offering wide angular coverage and avoiding boresight error, etc [1]. Unlike conventional arrays with regular geometry, the element excitations or factors of conformal arrays have to be designed by optimization techniques. In the general synthesis problem of narrowband conformal arrays, the complex element excitations are optimized so that the resulting radiation pattern satisfies a desired beam shape possibly with a specified tolerance. Sometimes it is also necessary to consider the polarization characteristics and directivity of individual element, which depends on the orientation of the elements. Fortunately, the radiation pattern is a linear function of the complex element excitations. Hence if both magnitude and phase responses of the desired beam shape are specified,

T

Manuscript received July 17, 2009; revised November 11, 2009; accepted January 11, 2010. Date of publication March 29, 2010; date of current version June 03, 2010. The authors are with the Department of Electrical and Electronic Engineering, The University of Hong Kong, Pokfulam, Hong Kong (email: kmtsui@eee. hku.hk; [email protected]). Digital Object Identifier 10.1109/TAP.2010.2046865

the overall problem is convex and is closely related to conventional filter design problems [2], [3]. This has attracted much interest in using efficient convex optimization methods to solve various beam pattern synthesis problems [4]–[6]. An important advantage of convex programming is its ability to satisfy multiple objectives expressed in terms of a set of linear and convex quadratic constraints. Moreover, the optimality of the solution, if it exists, is guaranteed. Efficient pattern synthesis techniques are useful in predicting the radiation patterns and configurations of the target array during initial design stage, or further improving its performance when real measurements are available. On the other hand, the pattern specifications are very often expressed in power (or decibels) and the pattern phase is generally unimportant, say in applications such as radar and communication systems. Beam shaping, as in cosecant squared patterns, is a typical example where phase response is of no concern. This kind of problem is known as power or shaped beam pattern synthesis problem, where only magnitude response is specified [7]–[13]. Although the resulting problem is in general non-convex and more than one solution to the synthesis problem may exist, additional design freedom is available and hence the performance of the synthesized result is usually better than the convex beam pattern synthesis problem mentioned above [13]. It should be noted that the power pattern synthesis of uniform linear arrays with isotropic elements is the only exception that convex programming is applicable [7], [8]. By using the autocorrelation sequence representation of the excitations, it is possible to find an optimal power pattern in this simple case. For practical implementation, spectral factorization is further required to extract the corresponding complex excitations from the optimal power pattern. However, these approaches are not suitable for the general array geometry and directive elements. To cope with the general non-convex power pattern synthesis problem, various optimization methods have also been proposed [9]–[13]. The stochastic optimization methods, including genetic algorithm [9], particle swarm optimization [10] and simulated annealing [11], are very general and flexible frameworks, but their computational complexities are rather high because the searching methods involved are random in nature. The semidefinite programming (SDP) method in [12] focused only on nonuniform linear arrays with isotropic elements, and it relaxed the non-convex lower bound constraint and solved a series of subproblems using SDP. Whereas the iterative least-squares method in [13] employed a virtual phase response obtained from the previous iteration and solved a least-squares pattern synthesis problem successively to improve the synthesized result. However, it is

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somewhat difficult to precisely control the beam pattern and incorporate inequality constraints, such as prescribed lower bound constraints. In this paper, we propose a general design method for the power beam pattern synthesis problem using iterative second-order cone programming (SOCP). It extends the iterative methods proposed in [14] and [15] for the design of IIR and IIR frequency-response masking digital filters to the array pattern synthesis problem. Although these filter design problems are highly nonlinear, it is shown that the iterative SOCP approach is capable of finding better solution than conventional methods in digital filter design community. The basic idea of the proposed approach is to linearize the power pattern function (i.e., the magnitude square of the radiation pattern) of the conformal array in a neighborhood of the complex excitations in each iteration. By so doing, the original non-convex problem is relaxed to a series of convex SOCP subproblems with respect to the corresponding update vectors around the previous iterates. Hence, the advantages of previously mentioned convex programming can be utilized. For example, a wide variety of magnitude constraints can be incorporated easily in the problem formulation. To ensure the accuracy of the linear approximation, a norm constraint on the update vector is further imposed under the SOCP framework. The idea is similar to the conventional trust region method [16]. As the original problem is generally non-convex, a reasonably good initial guess is very important to achieve a good solution. To this end, we propose a simple way to obtain an initial guess, which satisfies all the constraints specified in the original non-convex power beam pattern synthesis problem. Since non-convex constraints are usually involved, it is not straightforward to find such initial guess. Fortunately, as mentioned earlier, if a suitable phase response is known, the original problem can be approximated as a convex programming problem. Therefore, we propose to employ a virtual phase response, which is the average phase of all the elements, so that the initial guess satisfying all the constraints can be obtained by solving a convex programming problem. In the simple case of uniform linear array with isotropic elements, design result of a cosecant squared pattern shows that the initial guess so obtained converges to a solution which is comparable to the optimal solution found using the convex optimization method in [7]. This suggests that the proposed approach with the above choice of the initial guess is reliable and efficient in obtaining a near optimal result. A more complicated example of hemispherical array with circular polarization is also given to demonstrate the flexibility and effectiveness of the proposed approach in satisfying various magnitude constraints for different polarization components. Moreover, the proposed method can be readily extended to solve pattern synthesis problems including the effect of mutual coupling. The paper is organized as follows: Section II is devoted to the general radiation pattern of conformal arrays with polarization. In Section III, the proposed iterative SOCP method and efficient method for determining the initial guess are presented. Design examples are given in Section IV. Finally, conclusion is drawn in Section V.

Fig. 1. Global and local coordinate systems.

II. RADIATION PATTERN OF CONFORMAL ARRAY Consider a conformal array of elements located arbitrarily at a carrier surface. To determine the total far-field radiation pattern produced by the conformal array at a far-field point , it is more convenient to first consider the field of individual elements in their own coordinate systems and then transform the field back to the global coordinate system. Fig. 1 of the th shows the local coordinate system element with origin at its position , and the global coordinate . The element is pointed in the direction of the system -axis. The spherical coordinate of the common far-field point in the local coordinate system are , and . Hence, the field of the th element with respect to its coordinate system is given by (2-1) where and are respectively the unit and directions; and and vectors in the are respectively the element pattern functions of and components. Note these element pattern functhe tions can either be described by a suitable model for simulation purpose, or obtained from measurements or EM simulations so as to take the actual radiation properties of real antenna systems and their mutual coupling into account [24]. Other effects such as measurement errors and deviation in radiation characteristics due to the feeding and supporting structures etc are not considered in our formulation. Regarding real examples and physical realizations of antenna elements, interested readers are referred to [25]–[32] for more details. Since each element has its own position and orientation, coordinate transformation is required so that all the elements’ comin ponents can be superimposed at the far-field point the global coordinate system [17], [18]. More precisely, and in terms of , , we want to express , ,

TSUI AND CHAN: PATTERN SYNTHESIS OF NARROWBAND CONFORMAL ARRAYS USING ITERATIVE SOCP

and , where the unit vectors and are defined similarly and except that and are respectively replaced as by and in global coordinate system. In general, the coordinate transformation from global coorto local coordinate system dinate system can be characterized by a rotation matrix and inversely by . Very often, three separate axis rotations are sufficient to fully describe the position and orientation of the element. Interested readers are referred to [17] and [18] for more details. The relation of the former coordinate transformation is given by

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is orthogonal to , then the corresponding and assume polarization components can be expressed as:

(2-8) which are also linear functions of . On the other hand, if either one of the polarization components is considered, the corresponding radiation pattern is given by:

(2-9) (2-2) Using this result,

and

Here we assume the subscripts (2-6) – (2-8) are collectively represented by for notation simplicity.

can be expressed as

(2-3) where

; and is the unit vector in

radial direction. Next, we consider the polarization components in (2-1) after coordinate transformation. The basic idea is to first transform the polarization components into the global coordinate system and then project these rotated polarization components onto the polarization components in the global coordinate system. Consequently, the field of the th element with respect to the global coordinate system can be expressed as

(2-4) Hence, according to (2-1) – (2-4), the total far-field radiation pattern of the -element array can be written as

III. POWER PATTERN SYNTHESIS A. General Problem Definition In pattern synthesis of a conformal array, a commonly considered problem is to determine the element excitation such that the radiation pattern approximates a desired beam pattern:

(3-1) and are respectively the desired magwhere , and is the region nitude and phase responses of for . of interest. Note by definition Without loss of generality, subsequent discussions are based on in (2-9). Similar idea can be directly applied to all kinds of polarization components defined in (2-6) – (2-8). A typical pattern synthesis problem is to minimize the maximum and absolute complex approximation error between . That is

(3-2) (2-5) (2-6) and

(2-7) ; ; is the complex excitation amplitude of the th element; is the wave number with is the position vector wavelength ; and and of the th element. More generally, if we denote respectively as the co- and cross-polarization unit vectors where

Alternatively, the least squares (LS) error criterion [3] and the maximization of the partial directivity measure [19] are frequently considered. Since the desired phase response is given, the error response is a linear function of the complex excitations . Consequently, all these optimization problems can be readily formulated as convex programming problem such as SOCP and SDP [4]–[6]. The main advantages of these approaches are that additional constraints can be easily incorporated, and the solution is guaranteed to be optimal if the overall problem is convex. For example, one may impose the to magnitude constraint in the form of shape the maximum value of the magnitude response according . to a given function A more challenging pattern synthesis problem is to approximate only a desired magnitude response, while keeping the de-

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sired phase unspecified. More precisely, one wishes to obtain the desired magnitude response as

(3-3) or equivalently, after squaring both sides of (3-3),

iterative SOCP filter design methods proposed in [14] and [15], because we extend these methods to deal with real-valued power pattern function with complex variables. The main advantage of the proposed approach is that a wide variety of constraints can be incorporated because the subproblem in each iteration can be solved optimally using SOCP. As an example, we consider the following constrained optimization problem

(3-4) which is a commonly encountered problem in applications such as radar and communication systems. In fact, this problem is equivalent to conventional shaped beam or power pattern synthesis problem. The concept is intuitively simple, but the design problem is unfortunately non-convex in general due to the lack of the desired phase response. In this paper, we shall focus on the following problem of power pattern synthesis based on minimax criterion

(3-5) subject to the following constraint specifying only the bound of the desired magnitude response:

(3-6) where and are respectively the lower and upper bound functions of the desired magnitude response and . Obviously, neither the approximation problem in (3-5) nor the magnitude constraint in (3-6) is convex in na. Hence, the convex ture due to the lower bound of programming methods are not directly applicable. It should be noted that there are previous attempts in solving a similar problem in (3-5) possibly with the constraint in (3-6) using convex programming [7], [8]. However, these approaches are only suitable for uniform linear array with isotropic elements (i.e., ). For more general cases considered in this paper, the synthesis problem is usually more difficult, and very often has to be solved by means of certain forms of problem approximation through relaxations and iterative methods such as the approaches in [12] and [13]. In [13], an iterative weighted least squares method was proposed to iteratively minimize the sum of squares of the approximation error, while the desired phase response is varied and updated according to previous iterates. It was found that if the desired phase response is appropriately chosen in each iteration, the final synthesized result is usually much better than the case where the desired phase response is fixed (i.e., the problem in (3-1)). A disadvantage of this method is that it is not straightforward to incorporate additional inequality constraints in the formulation. B. Problem Formulation In the subsequent subsections, we shall describe a general design method for solving the non-convex power pattern synthesis problem. The proposed method can be viewed as a variant of the

(3-7a)

(3-7b) (3-7c) , and represent three different regions where is an additional lower bound of interest, and function. Due to the lower bound constraints of , the problem in (3-7) is in general a non-convex optimization problem. Moreover, it is a very general problem, which covers the most common types of non-convex objective function and constraints related to the magnitude-only specification in the power pattern synthesis problem. For simplicity, we do not consider convex objective function and constraints because they can similarly be handled without much difficulty. Now we introduce a slack variable and reformulate (3-7) as

(3-8) are For notation simplicity, the argument variables dropped in subsequent derivations. First of all, we rewrite (2-9) more compactly in matrix form as

(3-9) where and Then

and . Let , and be respectively real and imaginary parts of . can be rewritten as

(3-10) where

, and . Further, we define the squared magni-

tude response as

(3-11)

TSUI AND CHAN: PATTERN SYNTHESIS OF NARROWBAND CONFORMAL ARRAYS USING ITERATIVE SOCP

where gate of

and denote the complex conju. Hence, the problem in (3-8) can be reformulated as

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and are respectively two sets containing and uniformly sampled points of and on the whole range of interest, the above problem can be casted as the following SOCP

where

(3-12) C. Design Methodology The proposed iterative optimization procedure starts with a reasonably good initial guess . Suppose that after iterations, we arrive at a point . For a smooth function in a vicinity of , we consider a linear approximation of as follows

(3-13) where is the gradient of with respect , and is the linear update vector such to , evaluated at that . Substituting into the problem in (3-12) and using the linear approximation in (3-13), the new can be obtained by solving . This process is resolution peated until the relative change of two successive solutions is insignificant or the maximum number of iteration is reached. Consequently, the problem in (3-12) can be approximated as

(3-14) where is a prescribed positive step size bound to ensure that the linear approximation in (3-13) is sufficiently accurate. This can also be viewed as a trust region method by restricting is always positive the norm of . Note, to ensure that for by definition, an extra constraint is imposed, yet it is active only when . Since the first three constraints in (3-14) are linear inequalities and the last constraint is a convex quadratic constraint, the overall problem is convex, which can be solved using either SOCP with discretization of and [7], or SDP without discretization of and [8]. In this paper, we only focus on the former approach. Interested readers are referred to [7] for the pros and cons of these two approaches. By defining the set

(3-15)

(3-16) can be solved optimally in each iteration. Consequently, Moreover, additional linear equality and convex inequality constraints can be easily incorporated in the above formulation. It should be noted that the algorithm presented above converges to a local solution due to the linear approximation in (3-13). Therefore, the determination of a reasonably good initial guess, which governs the quality of the solution, is important. In Section III-D, we shall introduce an efficient technique to find such an initial guess. As illustrated subsequently in a representative example to be presented in Section IV-A for the synthesis of a uniform linear array, the proposed algorithm with the initial guess so obtained is capable of finding a solution that is close to the optimal solution obtained using the method in [7]. The good convergence performance of the proposed algorithm is largely attributed to the global convergence of individual subproblem and the sufficiently small norm bound of the update vector, as suggested in [14] and [15]. The proposed algorithm might also be viewed as a SOCP-based trust region method with simplified update steps [16]. A clear advantage of the proposed method is that the step size and step direction characterized by the norm bound constraint in each convex subproblem would be efficiently and optimally handled by well developed interior-point methods, see [17], [18] and references therein. Also, a possible way to speed up the convergence is to adaptively adjust the norm bound in each iteration. Additional control mechanism is required to make a good tradeoff between the accuracy of the linearization and the convergence rate of the proposed algorithm. For simplicity, we do not further explore these issues and options and will consider them in future work. D. Initial Design There are many possible choices of the initial guesses , provided that they are feasible to the original non-convex problem. should at least satisfy all constraints, while In other words, the actual approximation error in the objective function is less critical. We propose to find such initial guess by defining a virtual phase response for a given magnitude response. This allows us to relax the original non-convex problem to a convex problem so that the constraints can be easily specified under the convex programming framework described in (3-2). Using this result, we can easily find an appropriate , even though some of the

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constraints, if not all, are in general non-convex constraints as in (3-7). As observed in the uniform linear array and other symmetric arrays with isotropic elements, the desired phase response of the main beam is always assumed to be zero. In this paper, this idea is extended to general array geometry and the virtual phase response is chosen as the average argument (phase) of all the elements

(3-17) where denotes the argument of a complex number inside the bracket. This choice is quite natural in the case of the uniform linear array (as well as other symmetry arrays) with . isotropic elements because it gives Now consider again the problem in (3-7). The non-convex constraint in (3-7b) can be rewritten as:

(3-18) where

and

are respectively defined as

(3-19) and (3-20) With the virtual phase response in (3-17), the constraint in (3-18) can be approximated by the following convex constraint

(3-21) This constraint is particularly suitable for the case when the magnitude constraint is specified in decibel scale, which is frequently encountered in power pattern synthesis problem. corresponds to a cosecant For example, suppose that specifies the maximum beam pattern (in dB), and . Then, solving the error (in dB) of deviation from problem with the constraint in (3-21) results in a beam pattern , which has a specified error of deviation from in decibel scale. Whereas for the other non-convex lower bound constraint in (3-7c), the strategy considered in [4] is employed. To ensure that this nonconvex constraint is satisfied, we impose a set of linear equality constraints to fix to values slightly larger than at several sample points of . More precisely, the lower bound constraint can be approximated using a number of linear equality constraints as

(3-22)

, for , where is a small positive constant and is a positive integer. Note it is unnecessary to impose too many equality constraints because our primary objective is merely to ensure (3-7c) is satisfied. Using the above results, the initial guess for the problem of (3-7) can be obtained by solving the following convex problem (3-23) subject to the constraints in (3-21) and (3-22). The initial guess found in this way is reasonable because the original non-convex problem in (3-7) is similar to a minimax problem after linearization. Alternatively, it is possible to use LS objective function instead of the minimax objective function in (3-23). However, computer simulations suggest that the LS initial guess generally leads to a slightly slower convergence although the resulting beam pattern is nearly identical to that obtained using the abovementioned initial guess. For simplicity, we shall stick with the initial guess obtained by solving (3-23). E. Design Procedure Without loss of generality, we particularly describe the general design procedure for the power pattern synthesis problem considered in (3-7). It is easy to modify the procedure for a variety of pattern synthesis problems with more complicated array settings, as we will illustrate in Section IV. Given number of elements , array geometry, element pattern function , and desired power pattern, the procedure can be summarized as follows: in the Step 1: Define the virtual phase response non-sidelobe region according to (3-17), derive the convex constraints from the original non-convex constraints according to (3-18) – (3-22), and formulate and solve the resulting convex optimization problem in (3-23) for an initial guess . , linearize the power patStep 2: Set iteration number according to (3-13), define tern function , and formulate and the maximum norm bound solve the convex optimization problem in (3-16) for a linear update vector . and increase Step 3: Update the solution by one. Repeat from Step 2 until the relative change of two successive solutions is insignificant or the maximum number of iteration is reached. IV. DESIGN EXAMPLES In the following examples, the maximum norm of the linear . and are update vector is chosen as and equally respectively discretized into spaced samples for an increment of 0.2 per sample over the whole range of interest. All the SOCP optimization problems were solved by the CVX Matlab Toolbox [20] on a Pentium 4 personal computer. A. Example 1: Uniform Linear Array In this example, the power pattern synthesis of a uniform linear array with isotropic elements is considered. Since the ap-

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programming problem can be solved optimally. For practical implementation, an additional step of spectral factorization is from the optimal autocorrelation coeffirequired to extract cients. Fig. 3(a) and (b) show respectively the patterns obtained after convex optimization and spectral factorization. We notice a slight difference between these two patterns because of the numerical error caused by spectral factorization, especially when is large. That is, the factorized result may not exactly assemble the optimal autocorrelation coefficients in the reverse process. Nevertheless, we only focus on the optimal pattern derived from the autocorrelation coefficients. From the discussions in Section III-C, an initial guess of the proposed algorithm is first obtained by solving the following convex problem:

Fig. 2. Uniform linear array geometry.

proach in [7] is capable of giving an optimal pattern in this simple case, a comparison with this approach will be considered below. Such optimal result provides a gold standard to assess the performance of the proposed approach in this particular problem. elements shown in Consider the uniform linear array of Fig. 2, where the array lies on the -axis and has a center of symmetry at . Let be the distance between two adjacent elements. The position vector of the th element is then ; and given by for . Since coorhence we have dinate transformation is not required for linear array geometry for isotropic elements, the radiation pattern and of the uniform linear array is simply given by

(4-1) according to (2-9). Also, it is independent of -direction. As an illustration, suppose that we want to synthesize a beam and a cosepattern having minimum sidelobe level for (in decibel scale) with maxcant squared main beam dB for , together with a imum allowable error prescribed upper bound magnitude constraint for . The problem can be formulated as

(4-2) where and . ; The array pattern with the following parameters: ; ; ; ; ; and will be considered below. For comparison purpose, we use the method in [7] (Matlab code is available in [20]) to solve the problem in (4-2). Its basic idea is to reformulate (4-2) with respect to the autocorbecomes a linear function of relation of , so that the autocorrelation coefficients. Hence, the equivalent linear

(4-3) where and are defined according to (3-19) and corresponds to in the original (3-20), respectively. Here, problem in (4-2). Note, in the second constraint, the correis equal to zero as sponding virtual phase response defined in (3-17). Fig. 3(c) and (d) show respectively the initial and proposed array patterns. We can see that the proposed pattern have a much better sidelobe attenuation than the initial pattern, because the problem in (4-3) ignores the phase information and hence more degree of design freedom is available. Also, the proposed pattern converges to a solution which is very close to the optimal pattern shown in Fig. 3(a), even though the initial guess is far away from the optimal solution. This suggests the proposed iterative method is effective in approaching the optimal solution. Regarding the design complexity of the proposed method, each iteration is solved in about 5 seconds, thanks to the efficient SOCP solver. The number of iterations and the design time may increase with the numbers of variables and constraints involved, but the design complexity is still affordable for nowadays personal computers. Next, we shall demonstrate the effectiveness of the proposed approach by considering a more difficult power pattern synthesis problem of conformal arrays with arbitrary array geometries and polarization characteristics, where no optimal solution is guaranteed in general. B. Example 2: Hemi-Spherical Array In this example, the power pattern synthesis of a hemi-spherical array with circular polarization is considered. As mentioned earlier, the convex programming methods in [7] and [8] are not directly applicable to such nonlinear array geometry and directive elements. For simulation purpose, we follow the common practice used in related studies and assume that the radiation characteristics of each element are described by certain established empirical models. The use of such model not only helps us to verify the

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Fig. 3. Synthesized power patterns obtained using (a) optimal autocorrelation coefficients, (b) complex excitations factorized from autocorrelation coefficients, (c) initial complex excitations, and (d) proposed complex excitations. MBE: Main Beam Error.

usefulness of the proposed approach more conveniently, but also allows the antenna designers to predict the radiation pattern of the target array and determine the required configurations, such as the number of antennas used and their locations, to meet a given specification before proceeding to construct the real antenna system. Once the real system has been constructed, it is also possible to use the measured element radiation patterns in the problem formulation so as to perform final tuning of the resultant beam pattern. As an illustration, the individual element pattern is assumed to be the lowest order circular patch model used in [13] as follows

and , and

. (4-5b)

In this distribution, the average element spacing is approximately given by , where is the radius of the sphere in this example). Fig. 4 shows the corresponding array ( geometry, in which the polar coordinate of each element is and there are elements. All the elements are assumed to be pointed towards the radial direction. For coordinate transformation mentioned in Section II, the required is given rotation matrix of the element located at by

otherwise otherwise (4-4) and . Following [21], an isosahedron type of element distribution is used in the hemi-spherical array and it is given by the following parameters:

where

(4-5a)

(4-6) After simple re-indexing to replace and by , the polarization components of the array in the elevation and azimuth directions according to (2-6) can be respectively expressed in term of and (2-7). Now suppose that we want to synthesize a beam pattern having the following desired properties:

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compares the results of the initial and proposed co- and cross-polarization patterns. It can be seen that the proposed patterns satisfy all the required constraints, while achieving much lower sidelobes and cross-polarization than the initial pattern. This illustrates the design flexibility and freedom of the proposed approach in handling non-convex magnitude constraints in general power pattern synthesis problems. C. Example 3: Circular-Arc Array With Mutual Coupling Fig. 4. Hemi-spherical array geometry with radius R

It has been demonstrated that the radiation pattern is significantly degraded when mutual coupling is not considered [22]. In this example, we extend the proposed pattern synthesis method to include the effect of mutual coupling. To this end, we first rewrite (3-9) as:

= .

i) Desired co-polarization component and and and

:

(4-9) .

ii) Desired cross-polarization component : to be and . minimized, In other words, the problem objective is to minimize the sidelobes and cross-polarization simultaneously. Also, the beam pattern is shaped to have a flat magnitude response and , and a prescribed for for and sidelobe attenuation of . Note that we only consider the space for (i.e., and ), because the radiation field at lower half of the sphere is negligible according to the element pattern function and array geometry we respectively defined in (4-4) and (4-5). Equivalently, the problem can be posed as:

where the interaction among the elements are characterized by complex matrix called mutual coupling matrix. a In general, the coefficients of are difficult to determine analytically, but can be calculated using numerical techniques suggested in [1]. Consequently, the proposed method can be applied in (3-9) is replaced by to handle the mutual coupling effect if and subsequent steps described in Section III are derived similarly. As a comparison, we focus on the same array settings and specifications studied in [22], wherein an 18-element ciris considered. The array cular-arc array with radius plane and only the horizontal plane (i.e., is placed on the ) is considered. The spacing of the elevation angle of . The azimuth location of the th adjacent elements is element is , and is the angular separation. Each element has a cosine isolated . To pattern, which is given by calculate the total radiation pattern, each element undergoes a coordinate transformation with the rotation matrix

(4-10) (4-7) The corresponding convex problem for determining the initial guess can be written as:

(4-8) where , and are determined according to (3-17), (3-19) and (3-20), respectively. Fig. 5

A cosecant main beam with unequal sidelobe level is synthesized according to the pre-defined pattern masks shown as dashed lines in Fig. 6(a). Using the notations defined previously, the pattern masks can be translated into the following specifications

(4-11) denotes the cosecant squared function in decibel where scale. In [22], the alternating projection method is used to find a pattern lying within the required pattern masks. The resulting

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Fig. 5. (a) and (b) Synthesized results of the initial co- and cross-polarization power patterns. (c) and (d) Synthesized results of the proposed co- and crosspolarization power patterns. Solid lines represent the power patterns in discretized -planes and dashed lines represent the prescribed pattern constraints (excluding the main beam constraint for clarity).

pattern shown in Fig. 6(a) almost fulfills the requirements, except for a portion of the pattern near the band edge. In particular, the worst case errors deviated from the desired specifications , 0.049 dB , 0.247 in (4-11) are 0.545 dB and 1.776 dB , respectively. On the dB other hand, the proposed method is employed to minimize the , while the other pattern residelobe level at quirements are left as constraints to the minimization problem. From Fig. 6(b), the sidelobe level of the proposed pattern at is found to be , while the prescribed specifications in other three regions of are precisely satisfied. Comparing with the pattern obtained using the alternating projection method, the proposed pattern has slightly better performance in every target regions of defined in (4-11). The main reason is that the alternating projection method aims at finding a suitable pattern that closely matches with the pattern masks, rather than optimizing the pattern according to the masks, as suggested in a comprehensive work for this method [23]. Therefore, the resulting pattern is in general not guaranteed to satisfy all the requirements, especially when the masks are not chosen properly. However, it is not always straightforward to find such masks because we do not have prior knowledge on the ultimate performance of the array at hand. For example, if the required is increased to , one sidelobe level at in a trial and error has to define the masks at manner. For the proposed method, such intervention is not re-

quired. Fig. 7(a) shows another optimized power pattern obtained by setting the maximum sidelobe level at to . We can see that the sidelobe level at is now minimized to about . Of course, if this minimal in the value is employed as the upper mask at alternating projection method, a reasonably good pattern that almost satisfies the masks can still be obtained. Nevertheless, the proposed method is able to provide more accurate results than the alternating projection method in most cases. We now illustrate the effect of mutual coupling by repeating the synthesis problem without taking the mutual coupling into account for the desired pattern where the maximum sidelobe at . The degradation of level is set to the resulting pattern due to mutual coupling is demonstrated in Fig. 7(b). We can see that it significantly deviates from the desired specifications as the mutual coupling is not compensated. This suggests that the proposed method provides an attractive alternative to conventional methods for solving the general power pattern synthesis problem of conformal arrays with the effect of the mutual coupling. Table I summarizes the dynamic range ratios for all array excitations designed in Examples 1, 2 and 3. Those obtained using the proposed method are all reasonable. V. CONCLUSIONS An iterative SOCP method for the power pattern synthesis of narrowband conformal arrays is presented. The non-convex

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Fig. 6. Synthesized power patterns including mutual coupling effect, obtained using (a) the alternating projection method, and (b) the proposed method. Dashed lines show the required pattern masks.

Fig. 7. (a) Synthesized power pattern including mutual coupling effect. (b) Synthesized power pattern excluding mutual coupling effect. Dashed lines show the required pattern masks.

TABLE I DYNAMIC RANGE RATIOS IN EXAMPLES 1, 2 AND 3.

arbitrary array geometries and polarization characteristics, and mutual coupling effect. The proposed method can also be extended to handle the model errors, say due to inexact radiation characteristics and location of antenna elements. By treating the unknown errors as random variables with appropriate bounds specified by the antenna designers, the corresponding uncertainties can be taken into account in the design procedure to reduce the sensitivity of the designed pattern at the expense of slightly degraded performance. This is usually referred to as the robust beamformer design problem. Due to page limitation, we will report this interesting problem in future work. ACKNOWLEDGMENT

power pattern synthesis problem is solved via a sequence of linear approximations in which each subproblem is solved using SOCP. Design results show that the proposed method is an attractive alternative to traditional design methods in tackling a wide range of pattern synthesis problems with various types of magnitude constraints, because of its generality and flexibility. In particular, the proposed method is able to design a power pattern that is close to the optimal one in the case of uniform linear array with isotropic elements, and is also applicable to more complicated power pattern synthesis problems involving

The authors would like to thank the anonymous reviewers for their useful comments and suggestions, which greatly improved the quality of this paper. REFERENCES [1] L. Josefsson and P. Persson, Conformal Array Antenna Theory and Design. Piscataway–Hoboken, NJ: IEEE Press–Wiley, 2006. [2] Y. Liu and Z. Lin, “On the applications of the frequency-response masking technique in array beamforming,” Circ., Syst. Signal Processing, Special Issue on Computationally Efficient Digital Filters: Design Techniques and applications, vol. 25, no. 2, pp. 201–224, 2006.

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[3] K. F. C. Yiu, X. Yang, S. Nordholm, and K. L. Teo, “Near-field broadband beamformer design via multidimensional semi-infinite-linear programming technique,” IEEE Trans. Speech Audio Process., vol. 11, no. 6, pp. 725–732, Nov. 2003. [4] H. Lebret and S. Boyd, “Antenna array pattern synthesis via convex optimization,” IEEE Trans. Signal Processing, vol. 45, pp. 526–532, Mar. 1997. [5] D. P. Scholnik and J. O. Coleman, “Formulating wideband array-pattern optimization,” in Proc. IEEE Int. Conf. Phased Array Syst. Technol., May 2000, pp. 489–492. [6] J. O. Coleman, D. P. Scholnik, and P. E. Cahill, “Synthesis of a polarization-controlled pattern for a wideband array by solving a secondorder cone program,” in Proc. IEEE Int. Symp. Antennas and Propagation Society, Jul. 2005, vol. 28, pp. 437–440. [7] S. P. Wu, S. Boyd, and L. Vandenberghe, “FIR filter design via spectral factorization and convex optimization,” in Applied and Computational Control, Signals and Circuits, B. Datta, Ed. New York: Birkhauser, 1998, vol. 1, pp. 215–245. [8] T. Davidson, T. Luo, and J. Sturm, “Linear matrix inequality formulation of spectral mask constraints with applications to FIR filter design,” IEEE Trans. Signal Processing, vol. 50, pp. 2702–2715, 2002. [9] F. J. A. Pena, J. A. Rodriguez-Gonzalez, E. Villanueva-Lopez, and S. R. Rengarajan, “Genetic algorithms in the design and implementation of antenna array patterns,” IEEE Trans. Antennas Propag., vol. 47, no. 3, pp. 506–510, Mar. 1999. [10] D. W. Boeringer and D. H. Werner, “Particle swarm optimization versus genetic algorithms for phased array synthesis,” IEEE Trans. Antennas Propag., vol. 52, no. 3, pp. 771–779, Mar. 2004. [11] J. A. Ferreira and F. Ares, “Pattern synthesis of conformal arrays by the simulated annealing technique,” Electron. Lett., vol. 33, no. 14, pp. 1187–1189, 1997. [12] F. Wang, V. Balakrishnan, P. Y. Zhou, J. J. Chen, R. Yang, and C. Frank, “Optimal array pattern synthesis using semidefinite programming,” IEEE Trans. Signal Process., vol. 51, no. 5, pp. 1172–1183, May 2003. [13] L. I. Vaskelainen, “Iterative least-sauares synthesis methods for conformal array antennas with optimizaed polarization and frequency properties,” IEEE Trans. Antennas Propag., vol. 45, no. 7, pp. 1179–1185, Jul. 1997. [14] W. S. Lu and T. Hinamoto, “Optimal design of IIR digital filters with robust stability using cone-quadratic-programming updates,” IEEE Trans. Signal Processing, vol. 51, pp. 1581–1592, Jun. 2003. [15] W. S. Lu and T. Hinamoto, “Optimal design of IIR frequency- response-masking filters using second-order cone programming,” IEEE Trans. Circuits Syst. I, vol. 50, pp. 1402–1412, Nov. 2003. [16] J. Nocedal and S. J. Wright, Numerical Optimization. Berlin: Springer, 1999, Series in Operations Research. [17] T. Milligan, “More applications of Euler rotation angles,” IEEE Antennas Propag. Mag., vol. 41, no. 4, pp. 78–83, Aug. 1999. [18] T. A. Milligan, Modern Antenna Array, 2nd ed. Piscataway–Hoboken, NJ: IEEE Press-Wiley, 2005. [19] R. S. Elliott, Antenna Theory and Design. Englewood Cliffs, NJ: Prentice Hall, 1981. [20] M. Grant, S. Boyd, and Y. Ye, “CVX Version 1.1. Matlab Software for Disciplined Convex Programming,” 2007 [Online]. Available: www. stanford.edu/~boyd/cvx/ [21] D. L. Sengupta, T. M. smith, and R. W. Larson, “Radiation characteristics of a spherical array of circularly polarized elements,” IEEE Trans. Antennas Propag., vol. 16, no. 1, pp. 2–7, Jan. 1968. [22] L. Josefsson and P. Persson, “Conformal array synthesis including mutual coupling,” Electron. Lett., vol. 35, no. 8, pp. 625–627, 1999. [23] O. M. Bucci, G. D’Eila, G. Mazzarella, and G. Panariello, “Antenna pattern synthesis: A new general approach,” Proc. IEEE, vol. 82, no. 3, pp. 358–371, 1994. [24] N. Kojima, K. Hariu, and I. Chiba, “Low sidelobe pattern synthesis using projection method with mutual coupling compensation,” in Proc. IEEE Int. Symp. on Phased Array Systems and Technology, 2003, pp. 559–564.

[25] H. Steyskal, “Pattern synthesis for a conformal wing array,” in Proc. IEEE Aerospace Conf., 2002, vol. 2, pp. 2-819–2-824. [26] D. A. Wingert and B. M. Howard, “Potential impact of smart electromagnetic antennas on aircraft performance and design,” in Proc. NATO Workshop on Smart Electromagnetic Antenna Structures, Brussels, Nov. 1996, pp. 1–10. [27] M. Kanno, T. Hashimura, T. Katada, M. Sato, K. Fukutani, and A. Suzuki, “Digital beam forming for conformal active array antenna,” in Proc. IEEE Int. Symp. on Phased Array Systems and Technology, Oct. 1996, pp. 37–40. [28] S. W. Schneider, C. Bozada, R. Dettmer, and J. Tenbarge, “Enabling technologies for future structurally integrated conformal apertures,” in IEEE AP-S Int. Symp. Digest, Boston, Jul. 2001, pp. 330–333. [29] M. A. Hopkins, J. M. Tuss, A. J. Lockyer, K. Alt, R. Kinslow, and J. N. Kudva, “Smart skin conformal load-bearing antenna and other smart structures developments,” in Proc. American Institute of Aeronautics and Astronautics (AIAA), Structures, Structural Dynamics and Materials Conf., 1997, vol. 1, pp. 521–530. [30] E. Vourch, C. Caille, M. J. Martin, J. R. Mosig, A. Martin, and P. O. Iversen, “Conformal array antenna for LEO observation platforms,” in IEEE AP-S Int. Symp. Digest, 1998, pp. 20–23. [31] G. Caille, E. Vourch, M. J. Martin, J. R. Mosig, and A. Martin Polegre, “Conformal array antenna for observation platforms in low earth orbit,” IEEE Antennas Propag. Mag., vol. 44, no. 3, pp. 103–104, Jun. 2002. [32] P. Knott, “Design and experimental results of a spherical antenna array for a conformal array demonstrator,” in Int. ITG Conf. Antennas – INICA, Mar. 2007, pp. 120–123.

K. M. Tsui received the B.Eng., M.Phil., and Ph.D. degrees in electrical and electronic engineering from the University of Hong Kong, in 2001, 2004 and 2008, respectively. He is currently working as a Postdoctoral Fellow in the Department of Electrical and Electronic Engineering at the University of Hong Kong. His main research interests are in array signal processing, highspeed AD converter architecture, biomedical signal processing, digital signal processing, multirate filter bank and wavelet design, and digital filter design, realization and application.

S. C. Chan (S’87–M’92) received the B.Sc. (Eng) and Ph.D. degrees from the University of Hong Kong, in 1986 and 1992, respectively. He joined City Polytechnic of Hong Kong in 1990 as an Assistant Lecturer and later as a University Lecturer. Since 1994, he has been with the Department of Electrical and Electronic Engineering, University of Hong Kong, and is now a Professor. He was a Visiting Researcher in Microsoft Corporation, Redmond, USA, Microsoft, Beijing, China, University of Texas at Arlington and Nanyang Technological University, Singapore. His research interests include fast transform algorithms, filter design and realization, multirate and biomedical signal processing, communications and array signal processing, high-speed AD converter architecture, bioinformatics and image-based rendering. Dr. Chan is currently a member of the Digital Signal Processing Technical Committee of the IEEE Circuits and Systems Society and an Associate Editor of the IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I and Journal of Signal Processing Systems. He was Chairman of the IEEE Hong Kong Chapter of Signal Processing 2000–2002 and an Organizing Committee Member of the IEEE ICASSP 2003 and ICIP 2010.

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A Modified Pole-Zero Technique for the Synthesis of Waveguide Leaky-Wave Antennas Loaded With Dipole-Based FSS María García-Vigueras, Student Member, IEEE, José Luis Gómez-Tornero, Member, IEEE, George Goussetis, Member, IEEE, Juan Sebastian Gómez-Díaz, Student Member, IEEE, and Alejandro Álvarez-Melcón, Senior Member, IEEE

Abstract—An extension of the pole-zero matching method proposed by Stefano Maci et al. for the analysis of electromagnetic bandgap (EBG) structures composed by lossless dipole-based frequency selective surfaces (FSS) printed on stratified dielectric media, is presented in this paper. With this novel expansion, the dipoles length appears as a variable in the analytical dispersion equation. Thus, modal dispersion curves as a function of the dipoles length can be easily obtained with the only restriction of single Floquet mode propagation. These geometry-dispersion curves are essential for the efficient analysis and design of practical EBG structures, such as waveguides loaded with artificial magnetic conductors (AMC) for miniaturization, or leaky-wave antennas (LWA) using partially reflective surfaces (PRS). These two practical examples are examined in this paper. Results are compared with full-wave 2D and 3D simulations showing excellent agreement, thus validating the proposed technique and illustrating its utility for practical designs. Index Terms—Artificial magnetic conductors (AMC), electromagnetic bandgap structures (EBG), frequency selective surfaces (FSS), leaky-wave antennas (LWA), partially reflective surfaces (PRS), periodic surfaces, transmission line networks.

I. INTRODUCTION

P

ASSIVE surfaces consisting of doubly periodic arrangements of planar metallic scatterers have recently been receiving increased attention [1]–[4]. Initially employed as frequency selective surfaces [5], more recently they have been used as high impedance surfaces [6]. Since then, a large variety of geometries have been employed in the past few years to realize other electromagnetic band gap structures [6]–[8], artifiManuscript received June 15, 2009; revised November 19, 2009; accepted December 05, 2009. Date of publication March 29, 2010; date of current version June 03, 2010. This work was supported in part by the Spanish National project TEC2007-67630-C03-02/TCM, in part by the Regional Seneca project 02972/PI/05, and in part by Regional Scholarship PMPDI-UPCT-2009. M. García-Vigueras, J. L. Gómez-Tornero, J. S. Gómez-Díaz, and A. Álvarez-Melcón are with the Department of Communication and Information Technologies, Technical University of Cartagena, Cartagena 30202 Spain (e-mail: [email protected]; [email protected]; [email protected]; [email protected]). G. Goussetis was with the School of Engineering and Physical Sciences, ˙ He is now with the Institute Heriot-Watt University, Edinburgh EH14 4AS, U.K. of Electronics Communications and Information Technology, Queen’s University Belfast, Belfast BT7 1NN, Northern Ireland (e-mail: [email protected]. uk). 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.2010.2046856

cial magnetic conductors [7], [9] as well as left-handed metamaterials [10]. In this context, there is an increased tendency to describe their interaction with electromagnetic waves by equivalent homogenized parameters, such as generalized sheet transition conditions [11], averaged boundary conditions [12] or equivalent impedance/ admittance [13]–[16]. A large number of techniques have been reported for the analysis of scattering characteristics of plane waves incident on such periodic surfaces [11]–[16]. Analytical expressions have been derived that can produce the equivalent far-field surface impedance for specific element geometries; an excellent summary of the state-ofthe-art can be found in [15], and a more recent contribution in [16]. Employed in equivalent circuits, the equivalent impedance can predict reflection characteristics and dispersion diagrams [6], [13]–[16]. However their application is limited to those geometries for which analytical solutions have been obtained. Numerical techniques on the other hand can be used to directly derive the reflection and/or dispersion properties of general periodic arrays. Among those, the integral equation and its subsequent solution using the method of moments (MoM) is a popular due to its fast and efficient features. Recently a pole-zero method was proposed by Maci et al. [13], which employing full-wave results and Foster’s theorem, can derive the equivalent impedance of periodic surfaces. Despite the extensive literature available for the analysis of periodic structures, there is relatively limited published material on the inverse problem of synthesis. Assuming that the value of the equivalent impedance required by a periodic surface is known, the synthesis problem aims to derive the geometrical characteristics of the array unit cell. Analytical techniques, such as those reported in [14]–[16], can be employed for this purpose. However, e.g., for the case of a dipole array, the dependence of the equivalent impedance on the dipole length has not been accurately derived. Brute force simulations have been recently reported [17], but this is computationally expensive. Yet, this synthesis procedure is essential for the efficient design of components employing such periodic surfaces. Recently a class of hybrid waveguide-planar leaky wave antenna topology has been proposed that is capable of controlling the fan-beam pointing direction using a simple geometrical configuration [18]. By employing a periodic arrangement of dipoles, it has been demonstrated that it is possible to produce forward and backward scanning leaky wave antennas (LWA) [18]. Operating in the usual mode, radiation in that case

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substrate. In this case, the transverse propagation constant is (Fig. 1). The analogous situation is presented plane, the TM polarization will when the incidence is in the interact with the dipoles, and the waves in the structure will propagate through the axis with a constant In analogy with [13], the analytical expression of the equivalent admittance that we employ in this work is given by:

Fig. 1. Doubly periodic dipole array over a dielectric substrate under arbitrary plane wave incidence.

was imposed by the asymmetric positioning of the metallic patches or slots in respect to the horizontal dimension of the waveguide. As a result of this, independent control of the phase mode could constant and the radiation rate of the leakymode, and loaded by a be achieved [18]. Operating in the periodic array of dipoles at the open end, this topology can also produce leaky-wave antennas [19], [20] Although some work has been undertaken on the dispersion characterization of this type of LWA [20], a rigorous design would require a technique to synthesize the FSS. In this paper we propose a modified pole-zero technique [13] for doubly periodic arrays of dipoles that relates the equivalent sheet impedance with the dipole length, hence allowing to perform synthesis. Following validation with an example of a closed waveguide loaded with such an array, we proceed to demonstrate the design of hybrid planar-waveguide LWA that operate in the mode. The topology is simple and compatible with standard PCB and milling fabrication techniques. It is demonstrated that with this antenna a large range of leakage rates can be achieved by properly designing the FSS dipole length. Based on a simple transverse equivalent network and the derived expressions for the equivalent sheet impedance, the design of this type of antennas can be efficiently achieved. II. MODIFIED POLE-ZERO METHOD The pole zero technique in [13] provides an efficient method to extract the equivalent sheet impedance of doubly periodic arrays and its variation with frequency from a limited number of full wave simulations. Here we employ a similar concept to extract the impedance of doubly periodic dipole arrays and its variation with the dipole length. The schematic of the structure under consideration is shown in Fig. 1. This structure can be analyzed under an arbitrary direction of incidence at an elevation angle and azimuth (see Fig. 1). in Fig. 1), at low freAssuming thin metallic dipoles ( quencies they will only interact with the -component of the in( cident electric field. When the plane of incidence is in Fig. 1) TE polarized waves will be affected by the dipoles, while the TM will only experience a metal-backed dielectric

(1) Depending on the incidence, are used in (1) for the TE /TM respectively. If is such that TE and TM polarization are coupled, the admittance can be described as a matrix changing the problem into calculating a TE admittance and a TM one, following [13]. In this way, not only dipole-based FSS can be analyzed but also rectangular strips or square patches. A simple comparison of the above expression with (4) and on in (1) is (5) in [13] reveals that the dependence of assumed similar to that with frequency in [13]. This similarity between the dependence with frequency and comes from the fact that the dipoles are resonant in the direction of their length, and therefore both variables are expected to have a similar in. Thus, the task of obtaining an fluence on the behavior of expression for the equivalent admittance as a function of can then be reduced to extracting a set of poles and zeros (which are assumed to vary slowly with the wavenumber [13]). This rational fitting can be carried out by any of the standard procedures available in the literature; in our case we have used a least squares scheme. Using a full-wave analysis tool (for instance the MoM employed in [7]), we compute the reflection phase experienced by an incident plane wave on the FSS printed on a conductor-backed dielectric substrate. In particular, we ob) with the dipole tain the variation of the reflection phase ( length, , for different angles of incidence (the admittance can be readily obtained from these values). Next, we will illustrate our proposal by means of an example , , involving an FSS with dimensions , , and (Fig. 1). We will characterize this structure under TE incidence in the plane (the index TE will be omitted in the following expressions for convenience). In this case, the propagation of the waves in the structure will be in the axis with wavenumber . The same approach can be used for the TM case without losing validity. Fig. 2 shows the full-wave values of the reflection phase for the FSS under consideration varying the length , at angles of incidence 0 , 27 , 54 , 81 , and frequency 20 GHz. For comparison purposes, the values of our rational fitting approach are also shown in this figure. The lengths for which the FSS appears as a short circuit, and the reflection phase takes the value of 180 , should appear as real poles in the admittance. On the other hand, the values of where the reflection phase is equal to that obtained by the same structure in the absence of the periodic array should correspond to real zeros in (1). In this latter case, the FSS appears transparent to incident plane waves. The full-wave results in Fig. 2 , which is explicitly acshow the presence of a zero at counted for by the factor in the numerator of (1). The location of these poles and zeros are marked in Fig. 2. Other zeros and

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Fig. 4. Dipole FSS under consideration for TE incidence. The metallic walls are shown to demonstrate the equivalence between a 2D FSS structure and a 1D FSS inside a parallel plate waveguide.

Fig. 2. Phase of the reflection coefficient of the FSS ( ) of Fig. 1 varying L at 20 GHz. Incidence in the plane yz ( in Fig. 1) under TE polarization. : ,Q : ,a : and " : . ,D Dimensions in mm: P

=0 = 1 5 = 0 5 = 11

= 1 13

r=22

= = 1 5 mm

Fig. 5. a) Rectangular waveguide loaded with a dipole-based FSS-AMC (H ,D : : ,L ,P : a , , Q : ). b) Transverse equivalent network (TEN) of the structure.

= 11 mm = 1 13 mm = 0 5 mm

Fig. 3. Zeros and poles found in the rational fitting and its spline interpolation with the angle of incidence.

poles should also appear in the rational fitting of (1) in order to have a good numerical matching. Specifically, the rational fitting shown in the figure has been computed with just two poles and two zeros, which has provided a maximum relative error with the use of 10 full-wave simulations (per angle of of incidence). A similar good numerical performance has been obtained for other cases. In analogy to what happens with the frequency dependence can vary rapidly with , discussed in [13], although and , vary the values of the poles and zeros, slowly with respect to the incidence angle. It allows these poles/zeros can be interpolated and/or extrapolated via a low order polynomial (in our case we have performed a spline interpolation). This aspect is well illustrated in Fig. 3, where it is shown the poles and zeros obtained for the example under and , are study. This figure shows that the two zeros, complex conjugate, as expected (for other cases the poles can , placed be both real). It can also be observed that the pole

= 22

= 9 mm

around 6 mm, corresponds to the physically meaningful pole expected from the discussions on Fig. 2. This modified pole-zero method will be employed in next sections in order to model a periodic FSS located inside a parallelplate waveguide. In order to study propagation of TE modes, these structures will be excited by TE-polarized plane waves assuming the setup shown in Fig. 4 (plane-wave incidence in the plane). Due to symmetry, the proposed FSS-loaded waveguide in Fig. 4 is equivalent to the two dimensional free-standing FSS in Fig. 1, with the period in the -dimension being equal to the width “ ” of the parallel-plate guide. III. FSS LOADED WAVEGUIDE The inclusion of FSS and/or AMC inside waveguides has attracted much interest in the last decade [6]–[17], [20], [21]. Interesting properties can be achieved, such as quasiTEM propagation [20], [21], generation of bandgaps [6]–[8], size reduction [9], [10], or dispersion compensation [17]. In this section, the structure shown in Fig. 5(a) will be analyzed using the simple Transverse Equivalent Network (TEN) illustrated in Fig. 5(b). As Fig. 5(a) shows, an empty rectangular waveguide (width “ ” and height “ ”) is loaded with an AMC, which consists of a ”, width “ ” periodic dipole-based-FSS (dipoles length “ and periodicity “ ”) over a cavity backed dielectric substrate of thickness “ ” and relative permittivity “ ”.

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To analyze this structure, the following transverse resonance equation (TRE) associated to the TEN shown in Fig. 5(b) must be solved

(2) represents the equivalent admittance of the dipolewhere FSS, while and are the input admittances of the two sections of short-circuited transmission lines (the first of length “ ” and the second of length “ ” and dielectric constant “ ”, see Fig. 5(b)). The two main variables involved in the TRE ) and the unknown (2) are the analysis frequency ( longitudinal propagation constant (see axes in Fig. 5), which must be numerically solved to obtain the dispersion curves with frequency. However, we have explicitly added the length of the ) as a variable in , since we are also AMC dipoles ( interested in obtaining dispersion curves as a function of this design variable for a given fixed frequency of design. We will start by performing a frequency dispersion analysis of the structure. Using the pole-zero technique presented by Maci in [13], the AMC with a fixed geometry can be analyti. cally modeled with frequency by the admittance Introducing this expression in the TEN (Fig. 5(b)) and solving the associated TRE (2), the frequency dispersion curves shown in Fig. 6 are obtained. In particular, Fig. 6 shows the normalized ) curves derived from the TEN propagation constant ( (continuous lines), which are compared to those obtained with a full-wave EBG modal analysis technique based on the method of moments [22] (MoM, dashed line), and with full-wave modal results obtained from commercial Finite Element Method [23] (FEM, circles). Different frequency-dispersion curves have been obtained for four different values of the AMC dipoles length ( , 6, 7 and 8 mm). Excellent agreement is observed between the three techniques for all values of in all the fast-wave frequency range ( , where the values for the angles of incidence given by are real). Good agreement is observed even in the surface-wave ), validating the effectiveness of the freregion ( quency-pole-zero approach presented in [13]. However, this technique does not allow for a direct derivation of the dispersion with the length of the FSS dipoles. As a result, if we affects the performance of the AMC want to study how loaded waveguide, the pole-zero technique must be repeated for , as it has been done to obtain the every single value of curves shown in Fig. 6. This is not as efficient as the technique described in the previous section. Following the modified pole-zero method proposed in Section II we can directly obtain dispersion curves varying the length of the dipoles in the AMC for a fixed frequency. In paris derived ticular, the equivalent admittance following (1), and is then introduced in the TEN of Fig. 5(b) . Fig. 7(a) shows in order to solve (2) for any value of ) for the perturbed the normalized phase constant ( mode of the dipole-FSS-AMC-waveguide of Fig. 5(a), varying from 2 mm to 6 mm at the design frequency of 15 GHz. As expected, the longitudinal phase constant of the perturbed mode is enlarged as increases from 2 mm to 6 mm, producing the aforementioned miniaturization effect

Fig. 6. Frequency dispersion curves for different lengths of the AMC dipoles.

Fig. 7. Dispersion curve for the structure of Fig. 1 varying L

at 15 GHz.

TABLE I COMPARISON OF COMPUTATION TIME PER DISPERSION POINT

[9], [10]. The results obtained with the proposed approach are very accurate, since they match those derived from different full-wave techniques such as MoM [22] and FEM [23], while being much more efficient in terms of computational cost, as will be described later (Table I). Moreover, physical insight can be easily extracted from the TEN. For instance, Fig. 7(b) shows the reflection phase of the , Fig. 5(b)) seen by the perturbed mode when AMC ( at 15 GHz. As it can be seen in Fig. 7(b), the varying FSS is almost transparent for dipoles lengths below 4 mm, prearound 120 due to the dielectric slab senting a value of and . When the dipoles of thickness length increases, the resonance of the AMC appears, obtaining ) for , magnetic-wall boundaries ( which creates an effective waveguide of double height. Beyond becomes negative, allowing for higher miniathis value, turization, until the modes gets into the surface-wave regime.

GARCÍA-VIGUERAS et al.: A MODIFIED POLE-ZERO TECHNIQUE FOR THE SYNTHESIS OF WAVEGUIDE LWAs

Fig. 8. Dispersion curve showing the effective waveguide height , at 15 GHz. function of L

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 =2 as a

This profile reduction process can be equivalently appreciated by studying the wavelength of the propagating mode in the transverse plane ( , Fig. 8). In a waveguide with no FSS, this wavelength is approximately twice the height of ). This occurs when the FSS in our the cavity ( , see Fig. 8). structure is almost transparent ( increases, the FSS perturbs the propagating mode, As increasing its , which is equivalent to dealing with an unpermode in a bigger waveguide. Particularly, when turbed we can see that , which means we are working with an effective waveguide of double height due to the magnetic boundary condition presented by the AMC. In this situation, we have achieved a reduction of the resonant more, we can obtain further cavity to half. Increasing effective cavity heights, as it is shown in Fig. 8. This process can be illustrated by plotting the transverse elecmode in the structure, as tric field lines of the perturbed shown in Fig. 9. Again, very good agreement is observed between the fields obtained with the simple TEN model (Fig. 9(a)), and the full-wave fields obtained using a full-wave MoM technique [22] (shown in Fig. 9(b)). However, it must be noticed that the TEN is very simple and it only represents a single Floquet-mode, losing all the visual information given by evanescent higher-order Floquet-modes in the proximity of the printed dipoles. Nevertheless, the boundary condition of the AMC is perfectly depicted from the simple TEN model, showing trans, AMC boundaries at parency at , and surface-wave regime at . IV. ANALYSIS AND DESIGN OF A PRS LWA FSS can also be used to synthesize partially reflective surfaces (PRS) for Fabry-Perot resonant antennas [15]. These type of structures increase the directivity of low-gain radiators (such as dipoles, patch antennas or slot apertures), thanks to the excitation of a leaky-wave, which illuminates a large radiating surface [25]. Dipole-based FSS have been extensively used to conceive PRS leaky-wave antennas (LWA) in 2D configuration [1]–[4]. 2D-PRS-LWA normally radiate at broadside, although conical beam-scanning has also been proposed [3]. On the other hand, 1D-LWA are also of much interest, since they are able to

Fig. 9. Transverse electric field inside the AMC-loaded waveguide, obtained (a) from TEN (b) from MoM [22].

Fig. 10. (a) 1D Fabry-Perot leaky-wave antenna formed by a parallel-plate waveguide loaded with a dipole-based FSS acting as a PRS. (b) Transverse equivalent network of the structure. (a = H = 11 mm, S = 5 mm, D = 1:13 mm,  = 2:2, L = 10 mm, P = 1:5 mm, Q = 0:5 mm).

synthesize high-directive fan-scanned-beams in a simple way [26]. Some works have been published about the ability to control the radiation pattern of a 1D-LWA by using printed-circuits which perturb a dielectric-filled parallel-plate waveguide (PPW) [22], [26], [28]. Here we present a 1D-LWA formed by a hollow cavity-backed PPW, which is loaded by a dipole-based FSS that acts as a PRS to conceive a 1D Fabry-Perot resonator antenna. The scheme of the structure and the main dimensions are shown in Fig. 10, together with its transverse equivalent network (TEN). Leaky-modes are characterized by a complex propagation constant along the longitudinal direction of the waveguide ( axis in Fig. 10(a))

(3) stands for the propagation or phase constant, and where is the attenuation rate, due to the radiation or leakage induced by the leaky-wave [26]. To analyze and design a LWA, the dispersion curves of the constituent leaky-mode are of much help determines the pointing or [4], [18], [26]–[28]. Particularly,

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radiating angle of the LWA in the elevation plane ( plane in , which is approximately expressed as [26] Fig. 10(a)),

(4) and the leakage rate is related to the radiation efficiency of , , which in the lossless case is [26]: the LWA of length (5) The main beamwidth is determined by the length of the antenna and the aperture illumination, which for a uniform LWA (exponential illumination) can be expressed as [26]: (6) In many cases, 90% radiation efficiency LWAs are desired, in these cases, (5)–(6) provide the following approximate expression for the antenna beamwidth

(7) The admittance of the dipole-PRS-FSS can be analytically expressed as a function of frequency and the unknown leakypropagation constant, , following (2) of Maci’s pole-zero procan then be introduced cedure [13]. This function in the TEN (Fig. 10(b)), and the associated TRE can be numerically solved to obtain the unknown complex (3) in the desired frequency band. The pole-zero matching procedure [13] should be performed for every single desired value of the PRS-FSS (see Fig. 10(a)). Fig. 11 shows the family dipoles length, of frequency dispersion curves obtained in this way. The results obtained with the TEN are plotted together with those given by a full-wave analysis tool for leaky-modes based on MoM [18], [22]. Curves from 3D full-wave simulations using FEM commercial software [23] are also plotted. Very good agreement is observed between these three techniques for all fre, for both the pointing quencies and different values of , Fig. 11(a)), and the normalized leakage angle curves ( , Fig. 11(b)). rate curves ( The PRS-FSS must be properly engineered to design the LWA (4), we [1]–[4]. Particularly, for a fixed pointing angle which will determine can synthesize a certain leakage rate the radiation efficiency (5) and directivity (6) of the LWA [26]. In our case, we choose a pointing angle of 30 for a design strongly frequency of 15 GHz. As shown in Fig. 11(b), affects the radiation rate of the leaky-mode due to the fact that controls the transparency of the FSS [1], [4], [11], [12]. However, the pointing angle is also affected by the FSS, and this must be taken into account for an accurate design. For this purpose geometry-dispersion curves at the frequency of design should be obtained. This will be next efficiently achieved. Following the process described in Section II, an analytical expression for the admittance of the FSS as a function of and , , can be derived using (1). In this way the TRE can be solved for the unknown leaky-mode complex

Fig. 11. Frequency dispersion curves for the 1D-PRS-LWA in Fig. 1 for dif) (a) Pointing angle (b) Normalized ferent lengths of the FSS dipoles (L leakage rate.

, as a function of , for a fixed design frequency. Fig. 12 shows the results obtained at 15 GHz for the PRS-LWA of from 4 mm to 10 mm. Again, the Fig. 10, sweeping results obtained by this procedure are compared to full-wave results obtained with 2D-MoM [18], [22] and 3D-FEM [23], showing very good agreement. This analysis is very efficient, since it allows direct derivation of dispersion curves with the . As it can be seen in geometrical variable of interest, Fig. 12(b), the leakage rate increases as is lowered. This is a simple procedure to control the radiation efficiency (5) and directivity (6) of the LWA [1], [4], [18], [28]. The TEN of Fig. 10(b) allows to obtain physical insight in this phenomenon, analyzing the reflection coefficient (modulus and phase) seen by the TE-leaky-wave incident on the PRS ( in Fig. 10(b)). This reflection coefficient can be analytically de, once the disrived from the expression of persion curves are obtained using simple microwave relations. These results are shown in Fig. 13. As it can be seen in Fig. 13, decreases, the PRS-FSS becomes more transparent as for lower values of which explains the increase of in Fig. 12(b). This can also be easily checked by inspecting the fields of the TE-leaky-mode inside the PRS-loaded cavity. The plots of the fields are shown in Fig. 14, and they are compared to those obtained with full-wave MoM [18], [22]. Excellent agreement is observed, showing how the transparency of the PRS in. Also, the fields plots along the creases for lower values of plane of the LWA (see reference axis in Fig. 10(a)), obtained with 3D-FEM [23] are shown in Fig. 15 for different values of

GARCÍA-VIGUERAS et al.: A MODIFIED POLE-ZERO TECHNIQUE FOR THE SYNTHESIS OF WAVEGUIDE LWAs

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Fig. 14. Transverse electric field of the leaky-mode inside the PRS cavity, obtained (a) from TEN (b) from MoM [22].

Fig. 12. Dispersion curves for the 1D-PRS-LWA in Fig. 1 with the length of ), at 15 GHz (a) Pointing angle (b) Normalized leakage the FSS dipoles (L rate.

Fig. 15. Leaky-wave field patterns obtained with 3D-FEM for different L

Fig. 16. Radiation pattern of the designed PRS-LWA (L 15 GHz). Fig. 13. Module and phase of the reflection coefficient of the FSS-PRS  at 15 GHz as a function of the length of the dipoles, L .

, confirming this phenomenon (see how more energy is is decreased in Fig. 15). radiated as has also However, this control of the leakage rate with some effects in the radiating angle of the LWA. The variation with observed in Fig. 12(a) is due to the deof ) with pendence of the reflection phase of the PRS-FSS ( , shown in Fig. 13. Particularly, increases as decreases, making decrease for lower values of . This effect must be taken into account for an accurate design of the LWA [1]–[4].

.

= 8 mm, f =

Finally, Fig. 16 shows the radiation pattern of the designed and with an anPRS-LWA at 15 GHz, with . Excellent agreement tenna length of is observed between the far-field pattern calculated by Fourier transforming the leaky-wave aperture field obtained from the simple TEN, and the results given by 3D-FEM analysis of the LWA [16]. Table I compares computation times to analyse each point of the dispersion curves obtained in this paper. Computation times with the proposed technique based on TEN analysis are much lower than those needed with full-wave tools. A distinction has been made when dealing with complex leaky-modes, since the observed efficiency is even better in this case, due to the fact that

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the search of complex modes is more cumbersome. These computation times are obtained using an Intel Centrino Duo T2400 processor, working at 1.83 GHz with 2 GB RAM. Regarding the comparison of our results with full-wave simulations, a better agreement with MoM has generally been observed. This was expected, since MoM full-wave data were used for the rational fitting of the FSS. V. CONCLUSION A modification of the pole-zero matching technique developed by Maci et al. has been presented in this work. The proposed expansion allows to obtain an analytical expression of the equivalent admittance of dipole-based FSS, as a function of unknown propagation constant and the length of the resonant dipoles. In this way, dispersion curves as a function of the dipoles length can be efficiently obtained for the first time, without any geometrical restriction. This technique is valid for any dimension of the dipoles, and also for square patches, apertures and slots. The only restriction is that the simple transverse equivalent network is valid if higher-order Floquet-modes are below cutoff in the transverse direction. The novel technique has been illustrated by obtaining useful parametric curves for the design of practical structures based on printed-circuit FSS, such as a miniaturized AMC-loaded waveguide, and a FabryPerot resonant leaky-wave antenna. Results have been validated by comparing with more costly full-wave techniques based on MoM and FEM. Excellent agreement has been obtained for propagating modes and also for radiating leaky-waves, thus confirming the efficiency and versatility of the proposed approach. REFERENCES [1] A. P. Feresidis and J. C. Vardaxoglou, “High gain planar antenna using optimised partially reflective surfaces,” IEE Proc. Microw., Antennas and Propag., vol. 148, no. 6, pp. 345–350, Dec. 2001. [2] S. Maci, R. Magliacani, and A. Cucini, “Leaky-wave antennas realized by using artificial surfaces,” in IEEE AP-S Int. Symp. Dig., Columbus, OH, Jun. 23–27, 2003, pp. 1099–1102. [3] T. Zhao, D. R. Jackson, J. T. Williams, H.-Y. D. Yang, and A. A. Oliner, “2-D periodic leaky-wave antennas-Part I: Metal patch design,” IEEE Trans. Antennas Propag., vol. 53, no. 11, pp. 3505–3514, Nov. 2005. [4] P. Kosmas, A. P. Feresidis, and G. Goussetis, “Periodic FDTD analysis of a 2-D leaky-wave planar antenna based on dipole frequency selective surfaces,” IEEE Trans. Antennas Propag., vol. 55, no. 7, pp. 2006–2012, Jul. 2007. [5] G. V. Trentini, “Partially reflective sheet arrays,” IRE Trans. Antennas Propag., vol. AP-4, pp. 666–671, 1956. [6] D. Sievenpiper, L. Zhang, F. J. Broas, N. G. Alexopulos, and E. Yablonovitch, “High-impedance electromagnetic surfaces with a forbidden frequency band,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 11, pp. 2059–2074, Nov. 1999. [7] G. Goussetis, A. P. Feresidis, and J. C. Vardaxoglou, “Tailoring the AMC and EBG characteristics of periodic metallic arrays printed on grounded dielectric substrate,” IEEE Trans. Antennas Propag., vol. 54, no. 1, pp. 82–89, Jan. 2006. [8] G. Goussetis, A. P. Feresidis, and P. Kosmas, “Efficient analysis, design and filter applications of EBG waveguide with periodic resonant loads,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 11, pp. 3885–3892, Nov. 2006. [9] G. Goussetis, A. P. Feresidis, and R. Cheung, “Quality factor assessment of subwavelength cavities at FIR frequencies,” Journal of Optics A, vol. 9, pp. s355–s360, Aug. 2007. [10] M. Caiazzo, S. Maci, and N. Engheta, “A metamaterial surface for compact cavity resonators,” IEEE Antennas Wireless Propag. Lett., vol. 3, pp. 261–264, 2004.

[11] J. C. Vardaxoglou, Frequency Selective Surfaces Analysis and Design. London, U.K.: Taunton Research Studies, 1997. [12] B. Munk, Frequency Selective Surfaces: Theory and Design. New York: Wiley, 2000. [13] S. Maci, M. Caiazzo, A. Cucini, and M. Casaletti, “A pole-zero matching method for EBG surfaces composed of a dipole FSS printed on a grounded dielectric slab,” IEEE Trans. Antennas Propag., vol. 53, no. 1, pp. 70–81, Jan. 2005. [14] F. Medina, F. Mesa, and R. Marqués, “Extraordinary transmission through arrays of electrically small holes from a circuit theory perspective,” IEEE Trans. Microw. Theory Tech., vol. 56, no. 12, pp. 3108–3120, Dec. 2008. [15] S. A. Tretyakov, Analytical Modeling in Applied Electromagnetics. Norwood, MA: Artech House, 2003. [16] O. Luukkonen, C. Simovski, G. Granet, G. Goussetis, D. Lioubtchenko, A. V. Räisänen, and S. A. Tretyakov, “Simple and accurate analytical model of planar grids and high-impedance surfaces comprising metal strips or patches,” IEEE Trans. Antennas Propag., vol. 56, no. 6, pp. 1624–1632, Jun. 2008. [17] G. Goussetis, N. Uzunoglou, J.-L. Gomez-Tornero, B. Gimeno, and V. E. Boria, “An E-plane EBG waveguide for dispersion compensated transmission of short pulses,” presented at the IEEE AP-S Int. Symp., Honolulu, Jun. 9–15, 2007. [18] J. L. Gómez, F. D. Quesada, and A. A. Melcón, “Analysis and design of periodic leaky-wave antennas for the millimeter waveband in hybrid waveguide-planar technology,” IEEE Trans. Antennas Propag., vol. 53, pp. 2834–2842, Sep. 2005. [19] M. Guglielmi and G. Boccalone, “A novel theory for dielectric-inset waveguide leaky-wave antennas,” IEEE Trans. Antennas Propag., vol. 39, pp. 497–504, Apr. 1991. [20] R. D. Seager and J. C. Vardaxoglou, “Characterisation of leaky wave antennas constructed from solid rectangular waveguides with a dipole frequency selective surface sidewall,” presented at the Inst. Elect. Ent. National Conf. on Antennas and Propagation, York, Mar.-Apr. 31–1, 1999. [21] F.-R. Yang, K.-P. Ma, Y. Qian, and T. Itoh, “A novel TEM waveguide using uniplanar compact photonic-bandgap (UC-PBG) structure,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 11, pp. 2092–2098, Nov. 1999. [22] J. L. Gómez, D. Cañete, F. Quesada, J. Pascual, and A. A. Melcón, “P.A.M.E.L.A: A useful tool for the study of leaky-wave modes in strip-loaded open dielectric waveguides,” IEEE Antennas Propag. Mag., vol. 48, no. 4, pp. 54–72, Aug. 2006. [23] High Frequency Structure Simulator, 11 ed. Ansoft Corporation. [24] G. V. Trentini, “Partially reflective sheet arrays,” IRE Trans. Antennas Propag., vol. AP-4, pp. 666–671, 1956. [25] D. R. Jackson and A. A. Oliner, “A leaky-wave analysis of the highgain printed antenna configuration,” IEEE Trans. Antennas Propag., vol. 36, no. 7, pp. 905–910, Jul. 1988. [26] A. A. Oliner, “Leaky-wave antennas,” in Antenna Engineering Handbook, R. C. Hansen, Ed., 3rd ed. New York: McGraw-Hill, 1993, ch. 10. [27] M. Guglielmi and G. Boccalone, “A novel theory for dielectric-inset waveguide leaky-wave antennas,” IEEE Trans. Antennas Propag., vol. 39, no. 4, pp. 497–504, Apr. 1991. [28] 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 Propag., vol. 54, pp. 1–9, Nov. 2006.

María García-Vigueras (S’09) was born in Murcia, Spain, in 1984. She received the Telecommunications Engineer degree from the Technical University of Cartagena (UPCT), Spain, in 2007, where she is currently working towards the Ph.D. degree. In 2008, she joined the Department of Communication and Information Technologies, UPCT, as a Research Assistant. From October 2009 to February 2010, she hwas a visiting Ph.D. student in the University of Seville, Spain. Her research interests focus on the development of equivalent circuits to characterize periodic surfaces, with application to the analysis and design of leaky-wave antennas.

GARCÍA-VIGUERAS et al.: A MODIFIED POLE-ZERO TECHNIQUE FOR THE SYNTHESIS OF WAVEGUIDE LWAs

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 Ph.D. degree in telecommunication engineering (laurea cum laude) 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 de 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. Dr. Gómez-Tornero received the national award from the foundation EPSONIbérica to the best Ph.D. project in the field of Technology of Information and Communications (TIC) in July 2004. 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.

George Goussetis (S’99–M’02) received the Electrical and Computer Engineering degree from the National Technical University of Athens, Athens, Greece, in 1998, the B.Sc. degree in physics (first class honors) from University College London (UCL), London, U.K., and the Ph.D. degree from the University of Westminster, Westminster, U.K. In 1998, he joined the Space Engineering, Rome, Italy, as a Junior RF Engineer and in 1999 the Wireless Communications Research Group, University of Westminster, as a Research Assistant. Between 2002 and 2006, he was a Senior Research Fellow at Loughborough University, U.K. Between 2006 and 2009, he was a Lecturer (Assistant Professor) with the School of Engineering and Physical Sciences, Heriot-Watt University, Edinburgh, U.K. He joined the Institute of Electronics Communications and Information Technology, Queen’s University Belfast, Ireland, in September 2009 as a Reader (Associate Professor). He has authored or coauthored over 100 peer-reviewed journals and conference papers. His research interests include the modeling and design of microwave filters, frequency-selective surfaces and EBG structures, microwave heating as well numerical techniques for electromagnetics. Dr. Goussetis received the Onassis foundation scholarship in 2001. In October 2006 he was awarded a five-year research fellowship by the Royal Academy of Engineering, U.K.

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Juan Sebastian Gómez-Diaz (S’07) was born in Ontur (Albacete), Spain, in 1983. He received the Telecommunications Engineer degree (with honors) from the Technical University of Cartagena (UPCT), Spain, in 2006, where he is currently working towards the Ph.D. degree. In 2007, he joined the Telecommunication and Electromagnetic group (GEAT), UPCT, as a Research Assistant. From November 2007 to October 2008, he was at Poly-Grames, École Polytechnique de Montréal as a visiting Ph.D. student, where he was involved in the impulse-regime analysis of linear and non-linear metamaterial-based devices and antennas. From September 2009 to March 2010, he was a visiting Ph.D. student at the Fraunhofer Institute for High Frequency Physics and Radar Techniques (FHR), working on the modal analysis of leaky-wave antennas. His current scientific interests also include IE and numerical methods and their application to the analysis and design of microwave circuits and antennas.

Alejandro Alvarez-Melcon (M’99–SM’07) was born in Madrid, Spain, in 1965. He received the Telecommunications Engineer degree from the Technical University of Madrid (UPM), Madrid, Spain, in 1991, and the Ph.D. degree in electrical engineering from the Swiss Federal Institute of Technology, Lausanne, Switzerland, in 1998. In 1988, he joined the Signal, Systems and Radiocommunications Department, UPM, as a Research Student, where he was involved in the design, testing, and measurement of broadband spiral antennas for electromagnetic measurements support (EMS) equipment. From 1991 to 1993, he was with the Radio Frequency Systems Division, European Space Agency (ESA)/European Space Research and Technology Centre (ESTEC), Noordwijk, The Netherlands, where he was involved in the development of analytical and numerical tools for the study of waveguide discontinuities, planar transmission lines, and microwave filters. From 1993 to 1995, he was with the Space Division, Industry Alcatel Espacio, Madrid, Spain, and also with the ESA, where he collaborated on several ESA/ESTEC contracts. From 1995 to 1999, he was with the Swiss Federal Institute of Technology, École Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland, where he was involved in the field of microstrip antennas and printed circuits for space applications. In 2000, he joined the Technical University of Cartagena (UPCT), Cartagena, Spain, where he currently develops his teaching and research activities. Dr. Alvarez-Melcón was the recipient of the Journée Internationales de Nice Sur les Antennes (JINA) Best Paper Award for the best contribution to the JINA’98 International Symposium on Antennas, and the Colegio Oficial de Ingenieros de Telecomunicación (COIT/AEIT) Award for the best doctoral thesis in basic information and communication technologies.

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Evaluation of Weakly Singular Integrals Via Generalized Cartesian Product Rules Based on the Double Exponential Formula Athanasios G. Polimeridis and Juan R. Mosig, Fellow, IEEE

Abstract—Various weakly singular integrals over triangular and quadrangular domains, arising in the mixed potential integral equation formulations, are computed with the help of novel generalized Cartesian product rules. The proposed integration schemes utilize the so-called double exponential quadrature rule, originally developed for the integration of functions with singularities at the endpoints of the associated integration interval. The final formulas can easily be incorporated in the context of singularity subtraction, singularity cancellation and fully-numerical methods, often used for the evaluation of multidimensional singular integrals. The performed numerical experiments clearly reveal the superior overall performance of the proposed method over the existing numerical integration methods. Index Terms—Double exponential quadrature rule, generalized Cartesian product rules, method of moments, mixed potential integral equations, weakly singular integrals.

I. INTRODUCTION

M

IXED potential integral equation formulations have been extensively used over the last years for solving a wide variety of practical electromagnetic radiation and scattering problems [1]–[3]. A typical numerical solution of the mixed potential integral equations using the Rao-Wilton-Glisson basis functions [1] and a Galerkin method of moments approach [2], calls for the computation of 4-D integrals over surface subdomains (triangles, rectangles or general polygons), according to the selected discretization scheme. The aforementioned multidimensional integrals can be classified as smooth, near-singular and weakly singular (though integrable) integrals, depending on the behavior of the integrand, which is also strongly related to the proximity of the discretized subdomains. To be more specific, the weakly singular cases arise when the two surface elements coincide (coincident integration), share a common edge (edge adjacent integration), or share a common vertex (vertex adjacent integration). Correspondingly, the near-singular integrals arise when the outer and inner elements don’t share any common points but their distance is very small, while the smooth integrals cover all the other possible cases. Manuscript received July 17, 2009; revised November 24, 2009; accepted January 11, 2010. Date of current version March 29, 2010; date of current version June 03, 2010. The authors are with the Laboratory of Electromagnetics and Acoustics (LEMA), Ecole Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland (e-mail: [email protected]; juan.mosig@epfl. ch). Digital Object Identifier 10.1109/TAP.2010.2046866

Since the very first applications of the mixed potential integral equation formulations, the weakly singular integrals over triangular domains appeared to be the most challenging of all, as the discretization into triangular elements is considered to be the most flexible one, being able to model surfaces of arbitrary shape. In general, weakly singular integrals are treated by using mainly the singularity subtraction method [4]–[9] or the singularity cancellation method [10]–[12]. Despite their widespread usage, both singularity subtraction and singularity cancellation methods fail to meet the requirements for an accurate and efficient numerical integration of weakly singular integrals. On the other hand, some very promising new methods have appeared in the literature that seem to outperform the traditional techniques. In [13], for instance, a method originated in the context of mechanics, which utilizes a series of coordinate transformations followed by an appropriate Duffy transform [14] is presented. Similarly, a direct approach for the evaluation of hyper-singular static surface integrals has been introduced in [15]. Moreover, the later direct evaluation method was generalized by the first author for the case of the weakly singular integrals over coincident triangular elements [16]. The only drawback of these methods is their limited applicability together with their highly analytical complexity. Hence, it is quite difficult to generalize them in order to deal with different types of integrands and/or integration domains. The present work is motivated mostly by our recent results presented in [16], where it is obvious that in both singularity subtraction and singularity cancellation methods the accuracy as well as the efficiency are limited by the remaining numerical integrations of smooth functions (without blow up singularities). Moreover, in some recent publications [11], [12], [17]–[19] it was clearly demonstrated that in many cases the near-singular potential integration (the inner 2-D integration) is more challenging than the singular one. Therefore, a reasonable resolution in the never ending quest for a machine precision general-purpose code, not only for 4-D weakly singular integrals but also for 4-D hyper-singular and general near-singular integrals, could be given by the combination of semi-analytical formulations together with appropriate cubature rules [20]. In this manuscript, we present generalized Cartesian product rules based on the double exponential quadrature formulas. These formulas are well known in the mathematical literature since their introduction in the mid-seventies by Takahasi and Mori [21]–[24]. In a recent paper, they have been compared with other numerical alternatives and hailed as one of the most promising high precision quadrature schemes [25]. However,

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POLIMERIDIS AND MOSIG: EVALUATION OF WEAKLY SINGULAR INTEGRALS VIA GENERALIZED CARTESIAN PRODUCT RULES

they remain mostly unknown and unused by the computational electromagnetics community. In this paper, we first briefly recall for the sake of completeness and internal coherence the main steps of the technique, following the excellent summary given in [25]. Then, the double exponential formulas are integrated with classic methods for the computation of weakly singular and near-singular 2-D and 4-D integrals. The resulting overall scheme is compared with the most representative alternatives based on Gauss formulas, showing a significant improvement in almost all of the most challenging cases. II. THE DOUBLE EXPONENTIAL QUADRATURE RULE Standard interpolatory quadrature rules like Newton-Cotes and Gauss formulas can normally be used for integrands that are regular at the endpoints of the integration interval. On the other hand, although Gauss-Jacobi formulas have been widely used for integrands with infinite derivatives or integrable singularities at the endpoints, the type of singularity that can be treated by such formulas is quite limited. Coming to fill this gap, the double exponential quadrature rule is not based on ad hoc transformations or very specific weight functions, but on an appropriate variable transformation which results in general purpose quadrature formulas so robust and efficient that deserve a prominent place in standard mathematical subroutine libraries. As it is well known, the trapezoidal rule with an equal mesh size gives highly accurate results for analytic functions over . In fact, it was proved in [26] that for an integral of the trapezoidal rule with an an analytic function over equal mesh size is asymptotically optimal among formulas with the same density of sampling points. The optimality of the trapezoidal formula turned out to play a crucial role in the process of the discovery by Takahasi and Mori [21] of the double exponential formula. We now briefly outline the main steps leading to the development of a practical implementation of the double exponential quadrature formula. Additional details can be found in [25].

Hence, the original integral is given by

(4) which is solved with the help of the trapezoidal formula

(5) The remaining point is now the optimal selection of the . The main idea, based on the change of variables Euler-Maclaurin formula [27], is to select a transformation such as all its derivatives tend exponentially to zero for or large values (positive and negative) of . Then, even if its derivatives have an integrable algebraic singularity at one or both endpoints, they will disappear within the smooth and at infinity. fast convergence of the new integrand In these cases, according to the Euler-Maclaurin argument, the quadrature error should decrease faster than any power of . As described in [25], based on the above reasoning, Takahasi and Mori came out with a very interesting variable transformation (6) (7) as . Due to this double exponential decay, (6) is called the double exponential (DE) transformation. An alternate frequently used name, based again on (6), is the “tanh-sinh” tech, we get the DE formula, nique. If we truncate (5) at

(8)

A. Double Exponential Transformation Without loss of generality, we will confine ourselves to the following integral over :

(1) Since Takahasi and Mori had already proved the optimality of , it was quite natural that the trapezoidal formula over they focused on a variable transformation which maps the orig[21], i.e., inal interval of integration onto

(2) which we assume is analytic over

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and satisfies

(3)

where is the number of the quadrature rule’s abscissas. The key feature of the DE formula’s superior performance compared to other quadrature rules designed to handle endpoint singularities lies in the fact that via the DE transformation one can approach the endpoint singularity as close as one wants, because the DE rule has an infinite number of points in the neighborhood of the endpoints [22]. In cases where the integrand has a blow up singularity at an endpoint, this scheme permits one to sum terms with abscissas very close to the endpoints until the rapidly decreasing weights overwhelm the large function values [25]. B. On the Implementation of the Double Exponential Quadrature Rule Although it is very easy to compute the weights and abscissas of the DE quadrature rule, we encounter two significant problems in the actual coding process. According to [24], the problems arising from the careless coding may be one of the reasons that prevented the spread of the DE formula. More specifically,

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TABLE I NUMBER OF INTEGRATION POINTS ( ) FOR THE DE RULE IN TERMS OF THE ASSOCIATED LEVEL OF THE QUADRATURE RULE ( )

N

M

the first and most severe problem is the loss of significant digits: has a singularity at the endpoint like , where If is a small positive constant, we often encounter a large error due to the loss of significant digits for very close to . One way to overcome the aforementioned problem is to compute and and in addition to the acstore the values tual weights and abscissas. Of course, this modification requires the a priori identification of those binomials in the integrand, reducing the overall generality of the DE scheme. The second problem is the numerical underflow and overflow which arises in the denominator of the weights in (8). More specifically, the constraints that are imposed due to the double precision format lead, after some algebraic manipulation, to the following (safe) choice in the construction of the DE quadrature rule:

(9)

C. Final Double Exponential Formulas Taking into account the aforementioned constraints we proceed to the construction of the DE formula, according to the following parametrization, which is slightly different from the one adopted in the numerical experiments presented in [25] (aiming at 400 significant digits)

(10) is the so-called level of the quadrature rule. This where choice fulfils the constraint (9) to avoid numerical underflow and overflow. Besides the parametrization, one could follow different strategies in the final algorithms. The most suitable candidate for a general purpose integration scheme, though, is based on truncating the series of weights and abscissas at for which the following inequality holds: the point , where eps stands for the machine precision in double precision arithmetic, ensuring that and are never equal to zero. The final truncated number of points is , as shown in Table I, for each level of the equal to rule. Another possible variant could be derived without the aforementioned truncation together with the pre computation of and H. In that case, the number of integration points is given , almost double compared to by our preferred choice. This formula guaranties much higher accuracy in some specific problems, but its problem oriented nature is limiting dramatically the repertoire of possible applications and, hence, will be excluded from our study.

D. Extension to Generalized Cartesian Product Rules As described above, DE quadrature rule does not integrate exactly any polynomial and therefore it doesn’t belong to the family of the interpolatory quadrature formulas like NewtonCotes, Gauss, Radau, Lobatto and many others. Nevertheless, it turns out to be highly accurate for the integration of analytic functions with algebraic or/and logarithmic singularities at the endpoints of the integration interval, as will be shown in the sample test integrals following in the next section. This is exactly the reason that engages us in the quest for the construction of multidimensional integration schemes based on sophisticated quadrature formulas like DE rule, which are suitable for the treatment of functions with boundary weakly singular behavior. A first attempt for the generalization of the DE quadrature rule in 2-D and 3-D integration formulas was given in [28]. More specifically, a progressive strategy was utilized where the integration interval is divided into subdomains and, after a specific search pattern, the optimal abscissas and weights are found. The main drawback, though, of this method is its problem oriented nature, since the progressive search has to be performed for each new integrand. On the other hand, another research group followed a different approach while trying to find optimal cubatures for weakly singular, strongly (or Cauchy) singular and hyper-singular multidimensional integrals [29], [30]. In [29], a combination of Gauss product rules and DE formulas for the integration of 2-D singular integrals, arising in the context of computational mechanics, was implemented. Later, the same team in [30] focused on various 4-D singular integrals. The 4-D integrals are treated in the following way: First, an analytical integration is performed for the inner 2-D integral. Then, the type of singularity of the aforementioned analytical results as a function of the outer variables of integration is studied, in order to give some indications about the integration formulas needed for the remaining outer 2-D integral. In the end, again, generalized 2-D Cartesian product rules for the integration of at most boundary weakly singular functions are proposed based on a combination of 1-D Gauss product rules and DE quadrature rule. As a general comment, we could add that a hybrid scheme with analytical 2-D inner integrations and numerical 2-D outer integrations, as in [30], clearly reduces the computation cost and increases the accuracy. Furthermore, the study of the analytical results as function of the outer variables helps significantly in the suitable choice of the basic quadrature rules used for the construction of optimal multidimensional integration formulas. Of course, DE quadrature rule stands as one of the most appealing choices due to its unmatched precision in integrating functions with endpoint algebraic or/and logarithmic singularities. In the computational electromagnetics community there are generally two main families of methods for the solution of weakly singular integrals arising in the mixed potential integral equation formulations. Starting with the singularity subtraction method, it is well known that the integrand of the final 4-D integral may be non-singular, but with infinite derivatives. Also, the subtracted function can only be evaluated for specific

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geometries, leading to limited applicability. On the other hand, in the singularity cancellation method only the computation of the potential (inner) integrals to machine precision has been studied, and most of the authors imply that this is enough in order to assure highly accurate results for the original 4-D integrals. Unfortunately, this is not the case, since a simple study reveals that the integrand of the outer integral may still have infinite derivatives at the endpoints of the integration interval. In this manuscript, following the basic philosophy in [30], we utilize a DE rule as the main building block in the generalized Cartesian product rules for the solution of the aforementioned shortcomings. III. NUMERICAL RESULTS In this section, various numerical results will be presented in order to illustrate the worthiness of incorporating the DE quadrature rule in the construction of generalized Cartesian product rules for the solution of multidimensional weakly singular integrals arising in the mixed potential integral equation formulations. After some test 1-D and 2-D problems, we will show the results of some representative numerical experiments including different approaches, like singularity cancellation and singularity subtraction methods. A. Sample 1-D and 2-D Test Integrals In the beginning, it would be intuitive to give some results for a selection of 1-D test integrals found in [25]. The importance of such numerical experiments is twofold: first we validate the DE quadrature rule, as proposed in Section II, and, second, we observe its limitations for various types of endpoint singularities. The selected groups of test integrals (keeping the same numbering as in [25]) are listed below: • 5: continuous function on finite interval, but with an infinite derivative at an endpoint; • 7, 8, 10: functions on finite intervals with an integrable singularity at an endpoint.

(11) A straightforward application of the DE formulas as well as the Gauss-Legendre formulas leads to the relative error presented in Fig. 1, in terms of the associated level of the quadrature rule. For a fair comparison, we keep obviously the same number of integration points in both Gauss and DE formulas. Clearly, for the case of the test integral #5, function with infinite derivative at an endpoint, the DE formulas converge to the exact solution, while Gauss based formulas give very poor results. The same behavior is also observed for the test integral #8. For the other two test integrals #7 and #10, functions with an integrable singularity at an endpoint, the DE formulas cannot converge to the exact solution, but in any case their performance is by far superior to the performance of the Gauss rules.

Fig. 1. Relative error in calculating the test integrals in (11). (a) Test integral #5. (b) Test integral #7. (c) Test integral #8. (d) Test integral #10.

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The reference solution of the integral (absolute error less than ) is directly copied from [12], (14) In Fig. 3 the relative error in calculating the real part (singular portion) and the imaginary part (nonsingular portion) of the weakly singular potential integral (13) using the DE quadrature rule as well as basic Gauss formulas in 2-D generalized Cartesian product rules, is shown. The level of cubature is equal to the level of the 1-D quadrature rules , i.e., . By a simple inspection of the figures, it is easy to come to the conclusion that DE cubatures converge to the numerically exact solution for the most challenging case (the singular real part). As it could be expected, there is no interest in using DE rules for non-singular functions (the imaginary part) where Gauss-Legendre is optimally suited. Fig. 2. Relative error in calculating the test integral (12).

C. 4-D Weakly Singular Integrals Over Triangles via Singularity Cancellation

Finally, we choose a 2-D test integral #2 with endpoint singularities found in [28]:

Next, we deal with the far more interesting 4-D weakly singular integral arising in the mixed potential integral equation formulations solved via a Galerkin triangular discretization together with the linear Rao-Wilton-Glisson basis functions. More specifically, the contribution due to the scalar potential is given by

(12) (15) is the level of the 1-D quadrature rules where is also called level of the cubature) used in the construction ( of the 2-D Cartesian product rule. Hence, each cubature utilizes the same quadrature rule for each of the dimensions. As shown clearly in Fig. 2, again, DE based cubatures succeed in giving highly accurate results compared to the Gauss based cubatures. To account for the incidental presence of error propagation effects, we cautiously assume a result to be numerically exact if , as suggested in [12]. its relative error is lower than B. Weakly Singular Potential Integrals Moving to some more challenging problems, we choose the weakly singular potential integral

(13)

is the unit triangle defined by the following where and vertices: . Following the basic idea of the singularity cancellation method, we get

(16) where the outer integration over the triangle is numerically computed using a generalized Cartesian product rule. On the other hand, the inner or potential integral is reduced to a 1-D smooth integral via an appropriate coordinate transformation and a further analytical evaluation. The remaining 1-D inner integral is computed via a simple quadrature rule. The reference solution of the integral (absolute error less than ) is directly copied from [13],

where

is the unit triangle defined by the following vertices: is the dis. The observation point is given by tance function and . The singularity lies in the integration interval, for which case is very hard to find efficient cubatures. Hence, we split the integral into three subintegrals, isolating the singularity only at one vertex of the new triangles. Next, we proceed to the fully numerical integration of all three 2-D integrals via generalized Cartesian product rules based on Gauss-Legendre and DE formulas.

(17) In Figs. 4 and 5 the relative error in calculating the real part (singular portion) of the 4-D weakly singular integral (15) using the singularity cancellation method together with a generalized Cartesian quadrature rule based on the DE quadrature rule and the Gauss-Legendre formulas, is shown. More specifically, for the results in Fig. 4 we used a standard Gauss-Legendre quadrature rule for the remaining 1-D integral, while for the results in

POLIMERIDIS AND MOSIG: EVALUATION OF WEAKLY SINGULAR INTEGRALS VIA GENERALIZED CARTESIAN PRODUCT RULES

Fig. 3. Relative error in calculating the weakly singular potential integral I in (13) using a combination of triangle splitting and 2-D generalized Cartesian product rules based on Gauss-Legendre and DE quadrature rules. (a) Real part. (b) Imaginary part.

Fig. 4. Relative error in calculating the real part of the 4-D weakly singular integral I in (15) via singularity cancellation and 3-D generalized Cartesian product rules based on Gauss-Legendre and DE quadrature rules. For the inner 1-D integral a Gauss-Legendre rule is utilized. (a) Level of inner integration rule: M . (b) Level of inner integration rule: M .

=4

Fig. 5 we utilized the DE quadrature rule, again for the inner 1-D integration. The level of the outer cubature is equal to the level of each quadrature rule employed in the construction of . A simple comparison the 2-D formulas, i.e., of the aforementioned figures leads to the safe conclusion that only the 3-D generalized Cartesian product rule which is solely can give numerically based on DE quadrature rules exact results, as is clearly depicted in Fig. 5(b). The performance of the scheme that is presented in this manuscript is of paramount importance, since it is the first time that the singularity cancellation method can produce results close to the machine precision with a reasonable number of integration points. A major factor that affects crucially the accuracy of the

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=5

3-D numerical integration seems to be the behavior of the inner integral as a function of outer’s integration variables. Generally, the outer integration hasn’t been thoroughly studied in previous publications, where it was believed that machine precision results of the inner integral would physically lead to highly accurate results also for the final 4-D weakly singular integral by a straightforward implementation of Gaussian cubatures. One could easily jump to the conclusion that although the singularity is canceled, numerical shortcomings are encountered in the integration of the remaining function due to the non smooth behavior of its higher order derivatives.

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The key feature of this method is the analytical evaluation of the potential (inner) integral,

(19) as explained in [4] and [5]. The second and final step is the numerical computation of the remaining 2-D integral,

(20)

Fig. 5. Relative error in calculating the real part of the 4-D weakly singular integral I in (15) via singularity cancellation and 3-D generalized Cartesian product rules based on Gauss-Legendre and DE quadrature rules. For the inner 1-D integral a DE rule is utilized. (a) Level of inner integration rule: M . (b) Level of inner integration rule: M .

=5

=4

D. 4-D Weakly Singular Integrals Over Quadrangles via Semi-Analytical Method As a final test, we shall present sample numerical results in the context of the 4-D weakly singular integrals over quadrangular domains for the static kernel. More specifically, the 4-D integral to be solved is written as follows:

(18) where and are general quadrilaterals and is the distance function. The most common approach for treating the aforementioned integral is the so-called semi-analytical method.

Without loss of generality, we confine ourselves to the case of square domains (with sides equal to 2). Fig. 6 depicts the relative error in calculating the 4-D weakly singular integral (18) utilizing the analytical expressions provided in [4] together with a 2-D generalized Cartesian product rule based on the DE quadrature rule and the Gauss-Legendre formulas. More specifically, in Fig. 6(a), (b) and (c)–(d), we examine respectively the self-term case (coinciding squares), the orthogonal case (orthogonal squares sharing one edge) and the parallel case (parallel squares separated by distance ). The reference results are derived with the help of the complete analytical formulas provided in [9]. As with the previous examples, generalized Cartesian product rules based on the DE formula outperform the standard GaussLegendre cubatures for the solution of the most challenging cases, i.e., the self-term case (Fig. 6(a)) and the parallel case when the distance between the elements is very small (Fig. 6(c)). Obviously, for non-singular cases Gauss-Legendre still remains the optimal solution. Trying to give a fair explanation for the poor results of the Gauss-Legendre cubatures or the superior performance of the proposed DE based schemes, we analyze the behavior of the analytically evaluated results of the inner 2-D integral in terms of the observation point, i.e., the point given by the outer cubature in the final computation of the 4-D integral. For example, the potential integral for the observation points and , where , due to the source square with the following vertices: and , is presented in Fig. 7. Clearly, in the second case (encountered in the evaluation of the self-term), the potential has infinite derivatives as the observation point passes from one edge of the source square. The aforementioned behaviors, together with the overall discussion in previous sections, come to elucidate the actual causes for the performance of the integration schemes used in our numerical experiments. IV. CONCLUSION Novel generalized Cartesian product rules are presented for the computation of various multidimensional weakly singular integrals, arising in mixed potential integral equation formulations. The proposed formulas utilize the double exponential quadrature rule, ideally suited for the integration of functions with endpoint singularities. Due to the use of such non interpolatory quadrature rules, the algorithms presented in this manuscript can lead, together with common singularity subtraction

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Fig. 7. Behavior of the analytically derived potential integral (20) for a ; ; ;r ; ; ; square with the following vertices: r r ; ; ; ; and r and different observation points ; ; c and r c; ; , respectively. r

= (1 1 0) = (01 0 )

= (01 01 0) = (01 1 0) = ( 0 0)

= (1 01 0)

and cancellation techniques, to unmatched accuracy. The superior performance of the proposed method is verified in comparison with standard interpolatory cubatures through a series of representative numerical experiments. Moreover, in many cases the accuracy of the integration via double exponential based formulas is close to the machine precision, working on typical double precision arithmetic. Finally, the detailed analysis presented herein forms the backbone for the treatment of a plethora of cumbersome integrals, like strongly singular and hyper-singular integrals arising in various integral equation formulations, always in the context of computational electromagnetics. REFERENCES

Fig. 6. Relative error in calculating the 4-D weakly singular integral (18) (with k ) utilizing the analytical expressions provided in [4] together with a 2-D generalized Cartesian product rule based on DE and Gauss-Legendre quadrature rules. (a) Self-term. (b) Orthogonal cells. (c) Parallel cells d : . (d) Parallel cells d : .

=1

( = 0 01)

( = 0 00001)

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[9] S. López-Peña and J. R. Mosig, “Analytical evaluation of the quadruple static potential integrals on rectangular domains to solve 3-D electromagnetic problems,” IEEE Trans. Magn., vol. 54, no. 3, pp. 1320–1323, Mar. 2009. [10] L. Rossi and P. J. Cullen, “On the fully numerical evaluation of the linear-shape function times the 3-D Greens function on a plane triangle,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 4, pp. 398–402, Apr. 1999. [11] M. A. Khayat and D. R. Wilton, “Numerical evaluation of singular and near-singular potential integrals,” IEEE Trans. Antennas Propag., vol. 53, no. 10, pp. 3180–3190, Oct. 2005. [12] R. D. Graglia and G. Lombardi, “Machine precision evaluation of singular and nearly singular potential integrals by use of Gauss quadrature formulas for rational functions,” IEEE Trans. Antennas Propag., vol. 56, no. 4, pp. 981–998, Apr. 2008. [13] D. J. Taylor, “Accurate and efficient numerical integration of weakly singular integrals in Galerkin EFIE solutions,” IEEE Trans. Antennas Propag., vol. 51, no. 7, pp. 1630–1637, July 2003. [14] M. G. Duffy, “Quadrature over a pyramid or cube of integrands with a singularity at a vertex,” SIAM, J. Numer. Anal., vol. 19, no. 6, pp. 1260–1262, 1982. [15] L. J. Gray, A. Salvadori, A. V. Phan, and A. Mantic, “Direct evaluation of hypersingular Galerkin surface integrals. II,” Electronic Journal of Boundary Elements, vol. 4, no. 3, pp. 105–130, 2006. [16] A. G. Polimeridis and T. V. Yioultsis, “On the direct evaluation of weakly singular integrals in Galerkin mixed potential integral equation formulations,” IEEE Trans. Antennas Propag., vol. 56, no. 9, pp. 3011–3019, Sep. 2008. [17] Ismatullah and T. F. Eibert, “Adaptive singularity cancellation for efficient treatment of near-singular and near-hypersingular integrals in surface integral equation formulations,” IEEE Trans. Antennas Propag., vol. 56, no. I, pp. 274–278, Jan. 2008. [18] M. A. Khayat, D. R. Wilton, and P. W. Fink, “An improved transformation and optimized sampling scheme for the numerical evaluation of singular and near-singular potentials,” IEEE Antennas Wireless Propag. Lett., vol. 7, pp. 377–380, 2008. [19] P. W. Fink, D. R. Wilton, and M. A. Khayat, “Simple and efficient numerical evaluation of near-hypersingular integrals,” IEEE Antennas Wireless Propag. Lett., vol. 7, pp. 469–472, 2008. [20] Z. Wang, J. Volakis, K. Saitou, and K. Kurabayashi, “Comparison of semi-analytical formulations and Gaussian-quadrature rules for quasi-static double-surface potential integrals,” IEEE Antennas Propag. Mag., vol. 45, no. 6, pp. 96–102, Dec. 2003. [21] H. Takahasi and M. Mori, “Double exponential formulas for numerical integration,” Publ. RIMS, Kyoto Univ., no. 9, pp. 721–741, 1974. [22] M. Mori, “Quadrature formulas obtained by variable transformation and the DE-rule,” J. Comput. Appl. Math., no. 112, pp. 119–130, 1985. [23] M. Mori and M. Sugihara, “The double exponential transformation in numerical analysis,” J. Comput. Appl. Math., no. 127, pp. 287–296, 2001. [24] M. Mori, “The discovery of the double exponential transformation and its developments,” Publ. RIMS, Kyoto Univ., no. 41, pp. 897–935, 2005. [25] D. H. Bailey, K. Jeyabalan, and X. S. Li, “A comparison of three highprecision quadrature schemes,” Exp. Math., vol. 3, no. 14, pp. 317–329, 2005.

[26] H. Takahasi and M. Mori, “Error estimation in the numerical integration of analytic functions,” Rep. Comput. Centre Univ. Tokyo, no. 3, pp. 41–108, 1970. [27] A. R. Krommer and C. W. Ueberhuber, Computational integration. Philadelphia, PA: SIAM, 1998. [28] V. U. Aihie and G. A. Evans, “A comparison of the error function and the tanh transformation as progressive rules for double and triple singular integrals,” J. Comput. Appl. Math., no. 30, pp. 145–154, 1990. [29] A. Aimi, M. Diligenti, and G. Monegato, “Numerical integration schemes for the BEM solution of hypersingular integral equations,” Int. J. Numer. Methods Eng., vol. 45, pp. 1807–1830, 1999. [30] A. Aimi and M. Diligenti, “Hypersingular kernel integration in 3D Galerkin boundary element method,” J. Comput. Appl. Math., no. 138, pp. 51–72, 2002.

Athanasios G. Polimeridis was born in Thessaloniki, Hellas, in 1980. He received the Diploma degree in electrical engineering and the Ph.D. degree from the Department of Electrical and Computer Engineering, Aristotle University of Thessaloniki (AUTh), Thessaloniki, Hellas, in 2003 and 2008, respectively. Since October 2008, he has been a Postdoctoral Research Fellow with the Laboratory of Electromagnetics and Acoustics, Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland. His research interests include computational electromagnetics, with emphasis on the development and implementation of integral-equation based algorithms.

Juan R. Mosig (S’76–M’87–SM’94–F’99) was born in Cádiz, Spain. He received the Electrical Engineer degree from the Universidad Politécnica de Madrid, Madrid, Spain, in 1973, and the Ph.D. degree from the Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland, in 1983. In 1976, he joined the Laboratory of Electromagnetics and Acoustics, EPFL. Since 1991, he has been a Professor with EPFL, and since 2000, he has been the Head of the Laboratory of Electromagnetics and Acoustics (LEMA), EPFL. In 1984, he was a Visiting Research Associate with the Rochester Institute of Technology, Rochester, NY, and Syracuse University, Syracuse, NY. He has also held scientific appointments with the University of Rennes, Rennes, France, the University of Nice, Nice, France, the Technical University of Denmark, Lyngby, Denmark, and the University of Colorado at Boulder. He is currently the Chairman of the EPFL Space Center and is responsible for many Swiss research projects for the European Space Agency (ESA). He has authored five book chapters on microstrip antennas and circuits and over 100 reviewed papers. His research interests include EM theory, numerical methods, and planar antennas. Dr. Mosig has been a member of the Swiss Federal Commission for Space Applications. He is currently a member of the Board of the Applied Computational Electromagnetics Society (ACES), the chairman of the European COST Project on Antennas ASSIST (2007-2011), and a founding member and acting chair of the European Association and the European Conference on Antennas and Propagation (EurAAP and EuCAP).

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Plain Models of Very Simple Waveguide Junctions Without Any Solution for Very Rich Sets of Excitations Paolo Fernandes and Mirco Raffetto

Abstract—Almost trivial waveguide junctions involving standard media and metamaterials modelled by effective constitutive parameters are investigated. It is shown that, when some dielectric configurations are present, no solution can be found for these models, for some excitations on the ports which are very regular, are not at all pathological and allow simple modal expansions. The set of excitations for which no solution exists is very rich and contains excitations almost indistinguishable from those for which the solution exists. This lack of solution does not originate, as usual in electromagnetics, from the excitation of a resonance of an ideal cavity. It rather arises from a mechanism similar to the one that causes ill posedness of inverse problems. What is new and unexpected is to find this kind of ill posedness in a direct problem. The well known modal technique is exploited heavily but, quite unusually, to prove the non-existence of solutions rather than to find them. Finally, the importance of results on the a priori well posedness of models is pointed out. Index Terms—Electromagnetic boundary value problems, electromagnetic simulators, electromagnetic theory, guided waves and wave-guiding structures, ill posedness, metamaterials.

I. INTRODUCTION

I

N any branch of the science we deal invariably with simplified models of the “real world”, rather than with the “real world” itself (whatever this more philosophical than scientific term may mean!). After all, even physical laws are nothing but particularly well established models accepted by the whole scientific community. A model always catches the features of the phenomena under consideration that are essential in the context where the model itself is to be used, while neglecting finer details. This approach is both unavoidable and fruitful and most of the progress of the science has been based on it. Familiar examples of that in electromagnetics are many models of radiation phenomena and microwave circuits usually considered in applications, numerical simulations and education [1], [2]. In many scientific communities, good properties of models are often taken for granted. In particular, most of them are im-

Manuscript received January 26, 2009; revised October 09, 2009; accepted December 16, 2009. Date of publication March 29, 2010; date of current version June 03, 2010. P. Fernandes is with the Istituto di Matematica Applicata e Tecnologie Informatiche del Consiglio Nazionale delle Ricerche, Via De Marini 6, I-16149 Genoa, Italy (e-mail: [email protected]). M. Raffetto is with the Department of Biophysical and Electronic Engineering, University of Genoa, I-16145, Genoa, Italy (e-mail: raffetto@dibe. unige.it). Digital Object Identifier 10.1109/TAP.2010.2046840

plicitly supposed to be “well posed problems”, that is to say, to have a unique solution, which continuously depends on the data [3], [4]. It is sometimes argued that, since a physical system driven by a specific excitation will behave in a specific way, its model should automatically have one and only one solution for any allowed excitation. This is not a good argument, however. Since any model neglects something, it cannot be taken for granted that it behaves like the corresponding physical system, not even approximately. As a matter of fact, as we will see later on, very reasonable simplifying assumptions may lead, on occasion, to unsolvable problems. On the other hand, it is now recognized that there exist sensible models, whose physical interpretation and mathematical solution are somewhat more delicate, that are inherently ill posed [5]. Hence, a priori investigations about well posedness of models are important and whenever facing an ill posed model, we should ask ourselves whether ill posedness is inherent to the physical situation or rather arises from some flaw in modelling. In this paper, we will deal with steady state time harmonic electromagnetic boundary value problems. Hence, we are modelling conceptually simple physical situations that should lead to well posed problems. In fact, problems in this class have been supposed for long time to be well posed except when a cavity resonator is driven at any of its resonant frequencies and the unrealistic hypothesis of total absence of losses is assumed [4], [6]. In this case, if the excitation is in some way coupled to the field of any mode which resonates at that frequency, then no solution exists, while, if it does not, then the solution is not unique [4]. However, quite unexpectedly, it has been recently shown that the solution of some steady state time harmonic electromagnetic boundary value problems involving metamaterials modelled by effective constitutive parameters may fail to depend continuously on the data [7], [8]. In this paper, we will show a fact that is even more unexpected for the engineering community, even though, from a mathematical point of view it is intimately linked to the lack of continuous dependence of the solution on the data. As a matter of fact, we will exhibit steady state time harmonic boundary value problems that model very simple waveguide junctions which involve metamaterials, have losses on part of the boundary and have no solution for a very rich set of non-pathological excitations on the ports. It is remarkable that ill posedness, even in its strongest form of absence of solution, may arise in modelling very simple, non-pathological electromagnetic situations under seemingly reasonable assumptions. The lack of solution for the considered models is even more surprising for microwave engineers since the usual techniques based on mode completeness and modal expansion [4], [9] are almost invariably believed to

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be sufficient to readily handle simple waveguide discontinuity problems, as the ones here considered, when uniqueness is ensured by the presence of impedance boundary conditions with a real and positive impedance on the two ports. It is also remarkable that any excitation for which the solution does exist can be perturbed by arbitrarily small quantities in such a way that the solution no longer exists. This is the reason why the excitations for which the considered models are unsolvable give rise to very rich sets. It can be useful to point out, even though this aspect will not be developed any further, that the same conclusion could be drawn also for cavity resonators, as those considered in [8], with losses in part of the media involved and driven by non-pathological current density distributions. It is also worth noticing that the mechanism by which this kind of ill posedness arises is similar to the well known one that makes ill posed inverse problems, when the expansion of the solution of the corresponding direct problem contains infinite many evanescent waves. In that case, however, the direct problem, which models the physical system under consideration, is well posed and ill posedness merely concerns the process by which we obtain information about a source from the knowledge of the field it generates. Our case unexpectedly shows that a similar mechanism can lead to ill posedness of direct problems, in the strongest form of absence of solution for a rich set of admissible data. In particular, this happens when and because infinite many surface waves can be excited by the sources. No doubt, these direct problems which fail to have any solution for many non-pathological excitations are not good models of the physical systems they are intended to represent. Moreover, if a numerical simulation is based on one of them, the solution algorithm either fails or, even worse, gives an output anyhow, whose unreliability may pass unnoticed. We stress that this unfavourable situation arises for frequently considered configurations involving metamaterials and cannot be improved by acting on the numerical solver because we are facing ill posedness, not just ill conditioning. Hence, a time harmonic solver cannot reliably produce a waveguide solution when some combinations of ordinary media and metamaterials are present. These considerations provide other theoretical reasons for a deeper understanding of the recently pointed out difficulties of electromagnetic simulators in solving problems that involve metamaterials [7], [8], [10], [11]. The above considerations makes the well posedness issue a practical rather than academic one. Hence, well posedness of models should be assessed rather than taken for granted and having conditions, even just sufficient, ensuring well posedness of classes of electromagnetic models would be very valuable [12]–[15]. The paper is organized as follows. In Section II the plain models of almost trivial waveguide junctions are defined. Section III develops an alternative formulation of the same simple direct problems which is crucial for obtaining the results of interest on the non-existence of the solution. In Section IV the details of the constructions of the non-pathological excitations for which no solution can be found are provided. In the same section this lack of solutions is pointed out together with some considerations on the behaviours of the fields involved. The details for obtaining some technical results exploited in Section IV are reported in two appendices.

II. DEFINITION OF THE PLAIN MODELS In this section we define the plain models of almost trivial waveguide junctions which will be shown to have very peculiar properties, in addition to the features already pointed out in [7] and [8]. For our purposes it is sufficient [7], [8] to consider junctions in rectangular waveguides [2] but one should note that for many other geometries the same results do apply. Thus, in any case, where the electhe open, bounded and connected subset of tromagnetic boundary value problems of interest will be posed is

(1) where and . For future reference, let us define, for any

(2) The waveguide walls, , are assumed to be made up of a perfect electric conductor, according to the most simple models of waveguide junctions considered in the microwave community. By retaining the same spirit, on the two ports at and of the waveguide junctions denoted by and , respectively, we will consider the usual impedance boundary conditions matched for the propagation of the dominant mode. These boundary conditions will present forcing terms, as is usually done to excite one or more modes in the waveguide [16]. In a great number of cases [17]–[21], metamaterials are modelled by using homogeneous media with sharp interfaces at their boundaries. For this reason, just piecewise homogeneous media will be considered in our plain models. According to this approach, in order to simplify as much as possible the analytical details of our analysis, we consider exactly the same dielectric configuration analyzed in [7], but it is important to note that the same conclusion can be drawn if we considered the material configuration investigated in [8]. Thus, two linear, stationary, isotropic and lossless media are assumed to be present in the the waveguide junctions. In particular, we denote by and constitutive parameters of the media in the subdomain , with , 2. These subdomains, together with most of the details of the models considered, are shown in Fig. 1. The interface be. tween the two media is denoted by With the above indicated conventions and notation, for the usual frequency values allowing only the propagation of the fundamental mode, the waveguide junctions considered are modelled with the following plain formulations: Problem 1: Given and , , 2, find such that , , 2, satisfying in in on on on on

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FERNANDES AND RAFFETTO: PLAIN MODELS OF VERY SIMPLE WAVEGUIDE JUNCTIONS WITHOUT ANY SOLUTION

Fig. 1. Geometries and dielectric configurations of the waveguide junctions considered in this work.

where is the outward unit vector normal to the boundary, is the modal admittance of the fundamental mode in the medium , 2, occupying ,

(4) and

(5) III. ALTERNATIVE FORMULATION OF PROBLEM 1 FOR FIELDS DEFINED IN DISCONNECTED SUBREGIONS OF In order to obtain the main outcome of this paper some auxiliary results are necessary. For this reason, let us introduce an alternative formulation of Problem 1 for fields defined in the dis, not including the interface , under connected region the assumption that the media are lossless and source free in the excluded region (notice that the second assumption is already present in Problem 1). , and Problem 2: Given , , 2, find , defined in , such that for any satisfying

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impedance boundary conditions one then deduces on , , 2. Now the considerations reported on p. 135 of [22] above the statement of Theorem 34 allow us to conclude in . that Moreover, one can easily check that for any solution of Problem 1 the restrictions define a solution of Problem 2. As a matter of fact, since and satisfy equations , , and we immediately obtain that , , and are satisfied by and equations . is trivially satisfied by and, then, by their restrictions since it represents a balance of active power for a source free region containing a lossless media. It could be useful of Problem 2 to note that, on the contrary, any solution (the Cartedoes not necessarily belong to , that is the sian product space of pairs of fields in ) and, therefore, is not necessarily deduced from restrictions of a solution of . Problem 1 since this solution must belong to Finally, we now show that if Problem 2 has a solution then Problem 1 does not admit any solution. As a matter of fact, suppose that it could exist a of Problem 1. By using one of the previous solution is a statements we would deduce that solution of Problem 2. Thus we could consider and obtaining a solution of the homogeneous version of Problem 2. By the uniqueness result obtained above . we could deduce But these equalities are not possible since they would imply , against the hypothesis. Thus, by contradiction, we obtain the result of interest. IV. COMPUTATION OF APPARENTLY NON-PATHOLOGICAL EXCITATIONS THAT GIVE NO SOLUTION As pointed out in [7], [8], it is easy to verify that if the two media are complementary [23] (that is and ) the following electric and magnetic fields of a mode, , satisfy Maxwell equations, the continuity of the tangential components at the interface and the boundary conditions on the waveguide walls

in in on on

in

(7)

in (6) , for any Lipschitz continuous surfaces , 2, cutting the waveguide, where is a unit vector normal to pointing outward from . It is possible to prove that the only solution of the homoge) is neous version of Problem 2 (i.e., with in . As a matter of fact, on the one hand, in is a solution of problem 2. On the other hand, as shown satisfy the homogeneous version in detail in [7], if and , and the of Problem 2, then, from Poynting theorem, are real and have always the same sign, we deduce fact that on , , 2. By using the two homogeneous

in

(8)

in where (9)

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is the usual propagation constant for a modes in a rectangular waveguide of width . The reader can verify that all these fields satisfy the required conditions just in the case of complementary media [7], [8]. When the frequency allows only the propagation of the funcan be chosen to be a strictly positive real damental mode, [7]. As will be pointed out later on, this number for any assumption is only for convenience and could be safely dropped. is an arbitrarily defined complex constant Any coefficient and the corresponding electromagnetic field thus defined satisfies Problem 1 provided that the forcing terms on the two ports are given by

These series converge very quickly due to the exponential decay of the terms of the series with increasing . Our choice of the coefficients, in particular, allows the computation of in closed form. As shown in Appendix A, one can deduce

(15) The other forcing term, on the contrary, is not found in closed form due to the presence of the second and fourth factor in the terms of the series. The series converges, however, since

(16) (10)

(11) These electromagnetic fields and the excitation fields on the ports can be superimposed to determine new solutions corresponding to new data, by linearity. This idea can be extended to consider series of electromagnetic fields and series of excitation fields on the ports. Microwave engineers are familiar to carrying out this so-called modal expansion of the solution of waveguide discontinuity problems [2], [4]. However, in this case, the presence of infinite many surface waves (modes with an exponential growth as one moves away from the ports) can play a crucial role in determining the convergence of the series can be defined to show some and the arbitrary coefficients unusual consequences. In particular, by exploiting the exponential growth of the infinite many surface wave solutions consid), we can define coefficients which guarered (for antee convergence of the excitation fields and of the electromagnetic fields in regions adjacent to the ports but making the series for the electromagnetic field to diverge near the interface of the complementary media. giving unusual As an example of sequence of coefficients consequences, let us choose, for all

thus

and ; moreover for large values of . It converges very quickly owing to the indicated exponential factor and, for this reason, can be approximated very well. , the series of On the contrary, for the chosen coefficient the fields converge only in a part of the whole domain . In particular,

in (17) in and

(12) where , , and consider the series of the forcing terms on the two ports

in (18)

in

(13) Owing to (16) we have:

(19) and (14)

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FERNANDES AND RAFFETTO: PLAIN MODELS OF VERY SIMPLE WAVEGUIDE JUNCTIONS WITHOUT ANY SOLUTION

Fig. 2. Behavior of the forcing terms f (x) on 0 and f (x) on 0 when all coefficients V are defined by (12) with a = 0:02 m, b = a=2, f = 10 GHz, z = a and s = z =2. f is available in closed form whereas the behavior of f has been calculated by considering 30 terms in the series, which are sufficient, with the indicated data, to obtain very accurate results, stable up to the first five significant digits.

0

j j

so that all series indicated converge uniformly for , . for any both the above series converge not uniformly and For both the above series do not converge, so that this for procedure does not identify a solution of Problem 1. In the regions of uniform convergence the series identify two and , which can be derived term by term. Thus, fields, , and we deduce since any surface wave satisfies that the two fields and satisfy , and . is also satisfied term by term and then by the corresponding se, and . Finally, is trivially verified since the ries , and , the fields converge uniformly in regions containing integrals can be done term by term and is satisfied by any surface wave mode making up the series since they do not propis a solution of Problem agate any active power. Thus, 2. It is not a solution of Problem 1, however, since, as shown in Appendix II

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j

j

Fig. 3. Behavior of E (x = a=2; z ) when all coefficients V are defined by (12) with a = 0:02 m, b = a=2, f = 10 GHz, z = a and s = z =2. The field has been calculated by considering 120 terms in the series. More terms of the series are necessary in this case since the convergence becomes slower as one moves away from the ports.

0

E

j 0

j j

j

Fig. 4. Behavior of E (x; z = const) in different sections of the waveguide, when all coefficients V are defined by (12) with a = 0:02 m, b = a=2, f = 10 GHz, z = a and s = z =2. The field has been calculated by considering 120 terms in the series.

j j

E

(21) and the results of Section II apply. so that In particular, Problem 1 does not admit any solution. It is important to notice that the same conclusions can be drawn for infinite many other sequences of coefficients defining infinite many other regular and apparently non-pathological excitation fields on the ports. Since a finite number of terms of a series does not affect its convergence or divergence, one can notice that the hypothesis of considering frequency values allowing only the propagation of the fundamental mode is not essential and has been introduced just to consider the usual situation of interest in practice. In order to complete this analysis, it can be useful to show and and of the behaviours of the forcing terms the field , for the sequence of coefficients considered. In parand ticular, in Fig. 2 we show the regular behavior of , when , , and all coare defined by (12) with , , efficients , and , and in Fig. 3 we plot the

magnitude of the y-component of along the waveguide axis, under the same conditions. The phase of the field is not reported since it is constant along the axis (being a series of waves having a real propagation constant in a lossless waveguide; see also (17)). Finally, in Fig. 4 the behavior of the magnitude of on different sections is the y-component of shown. It is possible to notice that near the ports the term with is dominant but as the distance from the ports becomes larger and larger the behavior of the field changes and the peak value increases in a very significant way. As a final remark of this section, it could be interesting to notice that the above excitation terms could be superimposed to a much bigger excitation of the fundamental mode. Fig. 5 shows the behaviors of the magnitude of these excitation terms when a unit amplitude fundamental mode is assumed to be excited just on port 1 and on port 2) and is on port 1 (as and when, as before, superimposed to the fields , , and all coefficients are defined by (12) with , , ,

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Fig. 5. Behavior of jf (x)j on 0 and jf (x)j on 0 when to a unit amplitude excitation of the fundamental mode on port 1 only are superimposed the high-order surface wave modes excited by using the coefficients V defined by (12), with a = 0:02 m, b = a=2, f = 10 GHz, z = 0a and s = jz j=5. As before, f is available in closed form whereas the behavior of f has been calculated by considering 30 terms in the series, which are sufficient, with the indicated data, to obtain very accurate results, stable up to the first five significant digits.

V. CONCLUSION In this work we have studied a trivial waveguide discontinuity and have proved that, in the presence of some dielectric configurations, no solution exists for infinite many excitations on the two ports. The nonexistence of the solution is neither due to pathological nonphysical excitations nor to the excitation of resonant modes of ideal cavities but is a consequence of the fact that the models considered admit infinite many modes with anomalous characteristics. Owing to this infinite set of surface waves, the existence of the solution can always be disrupted by arbitrarily small perturbations of the excitation and the set of excitations that make unsolvable the model is very rich. This provides also a deeper understanding of the unsatisfactory behavior of some electromagnetic simulators with some problems involving metamaterials. The well known technique based on mode completeness and modal expansion is exploited to obtain the main result of the paper. Quite unexpectedly, it cannot be used to find a solution. The importance of an a priori assessment of the well posedness of models is pointed out. APPENDIX CLOSED FORM FOR THE EXCITATION TERM ON PORT 1 By using the fact that

as before but . As can be seen from the figure, in this way we obtain an excitation “almost” identical to the excitation of the fundamental mode only. Quite unexpectedly, the solution does exist for the latter excitation while it does not for the former. Moreover, by dropping an increasing number of lower-order surface waves, we can make arbitrarily small the difference between these two excitations. In other words there are infinite many “bad” excitations arbitrarily near the “good” one. Hence, electromagnetic simulators are expected to perform badly also for the excitation of the fundamental mode only. In fact, owing to roundoff errors or to the way boundary data are dealt with, the problem we are actually trying to solve in this case has a slightly perturbed excitation and, thus, is very likely to be unsolvable. The crucial point is that we have infinite many surface waves whose slope increases without bound with the order. Hence, provided a surface wave of sufficiently high order is excited, we can have an arbitrarily large error at the interby arbitrarily small perturbations of the excitations at face and . It is this hopeless situation that makes the the ports problem ill posed. If we had only a finite number of surface waves, instead, the problem would be just ill conditioned, the solution would exist for any excitation and, in principle, could be evaluated to any desired accuracy by using more significant digits in the computation, even though it may be difficult to obtain it in practice. As already pointed out, the series (12), (13), (16) and (17) can be added to excitations and fields of any solution after dropping an arbitrarily large finite number of terms. coefficients Moreover, infinite many different sequences of give series with the same properties. Hence, we have infinite many ways to obtain a “bad” excitation by perturbing a single “good” one and it is in this sense that the set of excitations for which the problem is unsolvable is very rich. Figs. 2–5 show just an example of how a “good” excitation can be made “bad” by a small perturbation, but producing many other of them would be a trivial exercise.

(22) the forcing term on port 1 given by (13) becomes

(23) The series appearing in (23) are the well known geometric series. It is known that (24) so that (25) The series in (23) converge since

so that . By using (25) we

then deduce

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LIMIT OF THE

APPENDIX NORM OF THE COMPONENT OF THE FIELD CONSIDERED IN SECTION IV

By definition we have

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converges since, on the one hand, the common factor considered in (31), (32) and (33) is dominated by the exponential terms, and, on the other hand, these exponential terms alone would and , determine a converging series since for all . These two relations and (16) imply also that such that and then (35)

(27)

an inequality that will be useful later on. Moreover, as and

so that

and observing that

(28) from (27) we easily deduce that

(36) converges as the exponential factors of its terms present an ex. ponential decay as Finally, (37)

(29) which is a numerical series. By using (18) we obtain

converges since . As before, the common factor considered in (31), (32) and (33) do not affect the character of the last two series. and the sum of the Let us now denote by series shown in (36) and (37). It is important to note that these quantities do not depend on . On the contrary, the series in (34) depends on . Now, from (33) and (35) we get

(30) In order to get two inequalities that will be exploited in that follows we note that the common factor

(38) , for

It is well known that so that (31) Then, for big values of

, we have (39) (32)

Moreover, since

, for all

and we conclude that

, we deduce (40) (33) Then

again for big values of . , from (30) we can define three converging For any series. As a matter of fact,

(41) REFERENCES

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[1] C. A. Balanis, Antenna theory: Analysis and Design. New York: Wiley, 1997.

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[2] R. E. Collin, Foundations for Microwave Engineering. New York: McGraw-Hill, 1992. [3] J. Hadamard, “Sur les problèmes aux dérivées partielles et leur signification physique,” Princeton Univ. Bulletin, pp. 49–52, 1902. [4] J. G. V. Bladel, Electromagnetic Fields, 2nd ed. New York: Wiley, 2007. [5] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory. New York: Wiley, 1998. [6] S. Caorsi and M. Raffetto, “Uniqueness of the solution of electromagnetic boundary value problems in the presence of lossy and piecewise homogeneous lossless dielectrics,” IEEE Trans. Microw. Theory Tech., vol. 46, no. 10, pp. 1353–1359, October 1998. [7] M. Raffetto, “Ill posed waveguide discontinuity problem involving metamaterials with impedance boundary conditions on the two ports,” IET Sci. Meas. Tech., vol. 1, no. 5, pp. 232–239, Sep. 2007. [8] G. Oliveri and M. Raffetto, “A warning about metamaterials for users of frequency-domain numerical simulators,” IEEE Trans. Antennas Propag., vol. 56, no. 3, pp. 792–798, Mar. 2008. [9] R. E. Collin, Field Theory of Guided Waves. New York: McGrawHill, 1960. [10] G. Cevini, G. Oliveri, and M. Raffetto, “Performances of electromagnetic finite element simulators in the presence of three-dimensional double negative scatterers,” IET Microw. Antennas Propag., vol. 1, no. 3, pp. 737–745, Jun. 2007. [11] G. Oliveri and M. Raffetto, “An assessment by a commercial software of the accuracy of electromagnetic finite element simulators in the presence of metamaterials,” COMPEL, vol. 27, no. 6, pp. 1260–1272, Nov. 2008. [12] P. Fernandes and M. Raffetto, “Existence, uniqueness and finite element approximation of the solution of time-harmonic electromagnetic boundary value problems involving metamaterials,” COMPEL, vol. 24, no. 4, pp. 1450–1469, 2005. [13] A. Alonso and M. Raffetto, “Unique solvability for electromagnetic boundary value problems in the presence of partly lossy inhomogeneous anisotropic media and mixed boundary conditions,” Math. Models Methods Appl. Sci., vol. 13, no. 4, pp. 597–611, 2003. [14] P. Monk, Finite Element Methods for Maxwell’s Equations. Oxford: Oxford Science Publications, 2003. [15] A. S. B.-B. Dhia, P. C. , Jr., and C. M. Zwölf, “Two- and three-field formulations for wave transmission between media with opposite sign dielectric constants,” J. Comput. Appl. Math., vol. 204, pp. 408–417, 2007. [16] J. Jin, The Finite Element Method in Electromagnetics. New York: Wiley, 1993.

[17] J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett., vol. 85, pp. 3966–3969, 2000. [18] C. Caloz, C.-C. Chang, and T. Itoh, “Full-wave verification of the fundamental properties of left-handed materials in waveguide configurations,” J. Appl. Phys., pp. 5483–5486, Dec. 2001. [19] R. W. Ziolkowski and A. L. 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, Oct. 2003. [20] N. Engheta, “An idea for thin subwavelength cavity resonators using metamaterials with negative permittivity and permeability,” IEEE Antennas Wireless Propag. Lett., vol. 1, no. 1, pp. 10–13, 2002. [21] M. Vogel, “Application of double-negative materials to improve an ultra-wideband TEM horn,” in Proc. Antennas and Propagation Society Int. Symp., Monterrey, CA, Jun. 2004, pp. 1776–1779. [22] C. Müller, Foundations of the Mathematical Theory of Electromagnetic Waves. Berlin: Springer-Verlag, 1969. [23] A. Alù and N. Engheta, “Guided modes in a waveguide filled with a pair of single-negative (SNG), double-negative (DNG) and/or double positive (DPS) layers,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 1, pp. 199–210, Jan. 2004.

Paolo Fernandes was born in Genoa, Italy, in 1952. He received the M.Sc. degree in physics from the University of Genoa, Italy, in 1977. He has been with the National Research Council since 1975 where currently he is a Research Director of the Institute of Applied Mathematics and Information Technologies, Department of Genoa. He was actively involved in particle accelerator physics and engineering for several years. Now his field of interest is computational electromagnetics: in particular, differential models of electromagnetic phenomena and their finite element approximation.

Mirco Raffetto was born in Genoa, Italy, in 1967. He received the M Sc. degree in electronic engineering and the Ph.D. degree in “models, methods and tools for electromagnetic and electronic systems” from the University of Genoa, Genoa, Italy, in 1990 and 1997, respectively. At present, he is an Assistant Professor of Electromagnetic Fields in the Department of Biophysical and Electronic Engineering, University of Genoa. His main research interests are on electromagnetic theory, microwave circuits, innovative materials and computational electromagnetics.

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Source and Boundary Implementation in Vector and Scalar DGTD J. Alvarez, Member, IEEE, Luis D. Angulo, M. Fernández Pantoja, Member, IEEE, A. Rubio Bretones, Senior Member, IEEE, and Salvador G. Garcia, Member, IEEE

Abstract—We summarize the boundary and source implementation for the several formulations of the discontinuous Galerkin time domain method (DGTD). Since DGTD with zeroth-order scalar basis functions using the upwind flux, coincides with the finite volume time domain (FVTD), many of the concepts developed for FVTD can be ported to DGTD in any of its different formulations (scalar/vector basis, upwind/centered flux). Numerical examples illustrate the different alternatives. Index Terms—Absorbing boundary conditions, finite difference methods, finite element methods, finite volume methods, time domain analysis.

I. INTRODUCTION

T

HE discontinuous Galerkin time domain (DGTD) method is a rapidly emerging technique in computational electromagnetics in the time domain [1]–[7], which provides an alternative to finite elements time domain (FETD), finite volume time domain (FVTD) and finite difference time domain (FDTD) methods. Like FETD, DGTD employs a variational formulation (discontinuous Galerkin) to integrate the spatial part of time-domain Maxwell’s curl equations, with a differential integration scheme non-overlapping for the time part. The space is divided into elements, in each of which the solution is expanded in a set of local scalar [8] or vector [4] basis functions of arbitrary order. The weak form of Maxwell’s curl equations are found element by element by employing a Galerkin test procedure. Unlike FETD, the solution is not enforced to be continuous at the boundaries between adjacent elements. Instead, continuous numerical fluxes are defined at the interface in order to connect the solution between them in the manner used in FVTD methods. Two common flux conditions are found in the literature: the centered flux [1], and the upwind flux [9]. The latter is the one actually employed in FVTD, and in fact, FVTD can be Manuscript received June 19, 2009; revised October 16, 2009; accepted December 01, 2009. Date of publication March 29, 2010; date of current version June 03, 2010. This work was supported in part by the European Community’s Seventh Framework Programme FP7/2007-2013, under Grant 205294 (HIRF SE project), in part by the Spanish National Projects TEC2007-66698-C04-02, CSD2008-00068, DEX-530000-2008-105, and in part by Junta de Andalucia Project TIC1541. J. Alvarez is with EADS-CASA, Military Air System, 28906 Getafe, Spain (e-mail: [email protected]). L. D. Angulo, M. F. Pantoja, A. R. Bretones, and S. G. Garcia are with Department of Electromagnetism and Matter Physics, University of Granada, 18071 Granada, Spain. 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.2010.2046857

regarded as a special case of DGTD with this flux, and 0th order (constant) scalar basis functions [10]. The main advantage of DGTD over FVTD is its higher order in space, while over FETD, the advantage resides in the fact that square matrices of DGTD needs only the inversion of elements (with the number of basis functions), while larger are involved in FETD. matrices In this paper, we take advantage of the resemblances of FVTD and DGTD to derive simple boundary conditions and to implement sources into DGTD for the different formulations: vector/scalar, centered/upwind flux approximation (this idea was successfully employed in [7] to derive a hybrid FDTD/DGTD algorithm). Here, we use the numerical flux to: a) incorporate wave sources directly by using the total field/scattered field formulation [11]; b) implement perfect electric/magnetic conducting (PEC/PMC) surfaces; and c) incorporate Silver-Müller Absorbing boundary conditions (SM-ABCs) [12] into DGTD. Though some of those ideas are well-known in FVTD, we think it may be useful to extend everything under a common framework for the different formulations of the DGTD method. This paper is organized as follows. In Section II we summarize the DGTD fundamentals in 3D with vector/scalar basis and the centered/upwind flux. Section III shows the implementation of boundary conditions, wave sources and SM-ABC into DGTD, and Section IV presents some results. II. DGTD FUNDAMENTALS A. Scalar Basis Formulation Let us assume Maxwell’s symmetric curl equations for linear isotropic homogeneous media in Cartesian coordinates. Now, , each let us divide the space in non-overlapping elements and enforce the inner product of each equation bounded by with a set of local continuous scalar test functions, to nullify element by element

(1) (2) With being, respectively: electric field, magnetic field, electric current density, magnetic current

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density, electric conductivity, magnetic conductivity, permittivity and permeability. After some algebra we can write (1) as

Two common choices of the numerical flux are reported in the literature i) A centered flux [8] found by averaging the solutions at both sides of the interface

(3) (7)

(4) Equations (3), (4) together with a tangential field continuity1 condition between adjacent elements leads to a FETD method cal[13]. Namely, adding the superscript to the fields at culated in the element adjacent to , the continuity on the tanof two adgential field components on the common face jacent elements requires for FETD that

ii) The upwind flux usually employed in FVTD [9] arising from the solution of the advection equations with discontinuous initial values (Riemann problem), as shown in (8) at the bottom of the page, [1] with being the intrinsic impedance of the element , and being that of the adjacent one. The semi-discrete algorithm is found by assuming that the space and time dependencies of the fields can be separated, and that the spatial part is expanded within each element in a set of basis functions equal to the set of test functions (Galerkin method)

(5) The main drawback of the resulting algorithm resides in its implicit nature, which requires the solution of large matrix equations [14]. The core idea of DGTD is to relax the continuity conditions to yield a quasi-explicit algorithm. Namely, instead of plugging (5) into (3), (4), DGTD defines numerical values of , henceforth called numerical fluxes the tangential fields on and ), which do not coincide with any of ( but depend the values of the tangential fields at any side of on them

(9) The final form of the semi-discrete algorithm at the element is

(10) (11) (6) This numerical flux is the one actually employed by any pair of adjacent elements to calculate the surface (flux) integrals in and . the RHS of (3), (4), instead of 1Let

us assume at the moment that no PEC/PMC are present. We will show later how to handle these.

where • and

are the field coefficients

(12) (13)

(8)

ALVAREZ et al.: SOURCE AND BOUNDARY IMPLEMENTATION IN VECTOR AND SCALAR DGTD

•

and

are the weak form of the source terms

1999

. They are first demagnetic basis functions fined in a standard reference element [14] as a function of the by simplex coordinates (14) (21)

(15)

•

is the mass matrix

(16)

•

is the stiffness matrix

(17)

•

nodal points in the element requiring to form a complete basis. The local basis for each element is found by computing the mapping of the transformation from the leads to the reference element to the actual one. The case classical FVTD algorithm [9]. The resulting system of ordinary differential equations in time can be solved in a number of ways: second-order leapfrog (LF) [7], 4th order Runge-Kutta [4], implicit Crank-Nicolson [10], symplectic [15], etc.

B. Vector-Basis Formulation The fundamentals of the vector formulation are similar to those of the scalar one. Now the basis and test functions are chosen to be vectorial: . The weak form of Maxwell’s equations is found by using the scalar product of the vector test-functions and the fields

are the flux matrices

(18)

where, for the centered flux

(22)

(19) and for the upwind flux

(23)

(20) on the LHS of (10)–(11) are Notice that the flux terms factors appearing only when the upwind flux is employed. A common choice for the basis functions [1], is the set 3D Lagrange interpolating th order polynomials (Legendre polynomial basis can be found in [5]) with equal set of electric and

where we already assumed the fluxes in the RHS to be the numerical ones. Comparing (3), (4) and (22), (23) we find similar flux-density integrals in their RHSs. Thus the same upwind and centered fluxes of the scalar case can be used here. For vector-basis functions the expansion (9) now becomes

(24)

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The semi-discrete algorithm is formulated by plugging (24) into (22), (23). The resulting equations are formally equal to (10), (11), now with

3) PMC conditions are reciprocal of PEC

(34) (25) (26) (27)

(28) (29)

Let us note that for the upwind flux, both for PEC and and . PMC, we must also assume 4) Regarding the ABCs, the straightest ones are the so-called first-order Silver-Müller (SM-ABC) [12] which are based on considering that outside the computation domain, the fields propagate as plane waves normally to the interface . For the upwind flux, this is directly implemented since it is equivalent to assuming that there is no contribution to the flux from outside the region of solution, only remaining in (6)

(30)

(35) (31) (32)

A common election of the basis functions is the hierarchical high-order vector-basis functions, widely used in finite elements methods [4], [16], which present some implementation advantages in order to reduce computation and memory requirements. Namely, since only the edge- and face-basis functions associhave non-zero tangential components, ated with the face function the flux matrices are sparse. Furthermore, the space is separated in the gradient space and the rota[16], and only the functions belonging to tional space , leading the rotational space have non-zero to be also sparse. Finally, the and matrices do not depend on geometrical information (this does not apply to ) , only needing one storage when an and can be shared by all explicit time integration scheme is used.

SM-ABC for the upwind flux provide a reflection coeffidB for normal incidence, this rapidly decient of up to grading when the angle of incidence increases [17]. For the centered flux, SM-ABC conditions can also be employed [19], but, in this paper, we have implemented instead perfectly matched layer (PML ) ABCs [5], [20]. 5) Incident-wave conditions can also be generated in a straightforward way. Let us consider that, inside a total-field zone (TFZ), a known wave is propagating, while outside it (scattered-field zone (SFZ)) the field is denote the wave fields at each point of null. If the TFZ/SFZ interface (Fig. 1), the flux across the face in the TFZ (with this face lying on the of an element TFZ/SFZ interface) needs to be modified according to

(36) and if

III. BOUNDARY CONDITIONS The flux conditions which serve to connect adjacent fields, also serve to implement boundary conditions. 1) The interface of two elements with different and is handled in an indirect manner in the DGTD formulation, thanks to taking the same tangential components of the and in the flux integrals for fields two adjacent elements. re2) PEC boundary conditions on a face of an element quire the setting of the tangential electric field employed in the flux integrals to be null, and the tangential magnetic field to be continuous

(33)

is in the scattered field zone

(37) This technique can also be applied in a reverse way to incorporate the fields created by other sources (Hertzian dipoles, wire antennas, etc.). Let us assume that the sources are inside the SFZ, while the TFZ is outside. If we know the fields on the SFZ/TFZ interface, we can use them as incident fields in (36), (37), to get null fields inside SFZ and the original ones in the TFZ. This form of the Huygen’s principle was successfully employed in [18] in a hybrid method implementation.

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Fig. 1. Total field/scattered field decomposition.

IV. RESULTS We have implemented 3D codes, both with the scalar and vector-basis functions, and with the upwind and centered numerical fluxes, incorporating sources and ABCs (SM-ABC and PML). Second-order accurate centered differences (LF2) as well as fourth-order Runge-Kutta (RK4) schemes, have been used for the time integration. The study of the stability (and dispersion) of the resulting schemes will not be addressed here, and we have limited ourselves to derive heuristic estimations [1], [19] for the maximum time steps, yielding stable schemes in each case. To validate the ideas presented in this paper, three simple examples are shown. Exhaustive side-by-side comparisons of the accuracy of scalar- and vector-DGTD are beyond the scope of this work, and they are left to a forthcoming publication.

Fig. 2. Scattered field error as a function of the minimum space resolution: ()=(h). Vector centered-flux approximation. LF2.

A. Plane-Wave Generation In order to test the TFZ/SFZ formulation, we have generated a known field inside the total field zone, and we have measured the field that escapes to the scattered field zone due to numerical errors. The TF zone consists of a 1 m-side cube where a plane, and polarized along wave is traveling with . The time variation is chosen to be a sine modulated by a Gaussian pulse

(38) Fig. 2 shows the normalized field in a scattered-field point near the TFZ/SFZ interface (Fig. 1) as a function of the minimum space resolution (minimum wavelength normalized to the maximum edge length), for different orders of the basis functions (1, 2, 3), using a hierarchal vector-basis DGTD, with the centered flux, and with a LF2-scheme in time. As in FDTD [21], the field that escapes from the TF zone, due to dispersion errors, decreases with the space resolution. B. ABCs To test the ABC performance, we have measured the enm), with a ergy decay with time in a cubic region ( Hertzian dipole at its center with the time variation given by

Fig. 3. Energy decay with time in the simulation region. A centered dipole fed with a 300 MHz continuous wave modulated by a Gaussian pulse ( = 2 ns) is placed at its center. Scalar upwind-flux formulation. SM-ABCs. RK4.

(38). Fig. 3 shows results for the SM-ABCs, placed in a sphere m) that is concentric with the dipole, found with ( , upscalar basis (Lagrange polynomials) of orders wind flux approximation, and with RK4-scheme in time. In this case the absorption of SM-ABCs is especially efficient because of the spherical nature of the waves impinging on the truncation boundary, which satisfies the SM-ABCs principle. The Hertzian dipole illumination has been incorporated into DGTD using the reverse TFZ/SFZ formulation described above. Finally, let us show in Fig. 4 results of the RCS in the E-plane of a 1 m radius PEC sphere in a UPML-truncated region. We compare results found by FDTD (under conditions similar to those of DGTD) and Mie series solution, with DGTD results found with the centered flux approximation together with hierarand (with and chal vector-basis of orders being the th order gradient and rotational spaces), and with

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It bears noting from Figs. 4 and 5 that to achieve the accu(less than 1 dBsm almost everyracy of DGTD with where), we needed to employ FDTD resolutions over 90 cells/ , requiring the solution of over unknowns, while DGTD unknowns2. only needs V. CONCLUSION In this paper, we have described the implementation of boundary conditions and total field/scattered field zone separation into the DGTD method. We have made use of the concept of numerical flux to generate them, taking advantage of the similarities between FVTD and DGTD. A common framework to incorporate them into the different formulations of DGTD (vector/scalar basis, centered/upwind flux) has been described. These concepts have been numerically tested in canonical examples, and validated in RCS prediction, with extremely accurate results. ACKNOWLEDGMENT Fig. 4. Bi-static RCS in the E-plane of a 1 m radius PEC sphere at 300 MHz illuminated with a 300 MHz continuous wave modulated by a Gaussian pulse ns). DGTD results. Vector centered-flux approximation. UPML, LF2. ( White sphere: PEC (1 m. radius). Innermost crown: TF zone (1.17 m. ext. radius). Middle-portion crown: Maxwellian zone (1.34 m ext. radius). Outermost crown: UPML (1.75 m ext. radius).

=2

Fig. 5. Bi-static RCS in the E-plane of a 1 m radius PEC sphere at 300 MHz illuminated with a 300 MHz continuous wave modulated by a Gaussian pulse ( ns). FDTD results. UPML.

=2

a LF2-scheme in time. We have used quadratic curvilinear tetrahedra (not detailed in the inset of Fig. 4) to further remove discretization errors. The sphere is illuminated with the TFZ/SFZ formulation described in this paper, by a plane wave with time variation (38). The UPML for the DGTD method is chosen to be a spherical crown (see inset of Fig. 4), with parabolic condB. For ductivity, and a theoretical reflection coefficient of the FDTD method a parallelepiped crown with similar characteristics is chosen.

The authors wish to thank SPEAG for providing them with the SEMCAD X software employed in the FDTD simulations of this paper. REFERENCES [1] J. S. Hesthaven and T. Warburton, “Nodal high-order methods on unstructured grids—I. Time-domain solution of Maxwell’s equations,” J. Comput. Phys., vol. 181, pp. 186–211, 2002. [2] T. Xiao and Q. H. Liu, “Three-dimensional unstructured-grid discontinuous Galerkin method for Maxwell’s equations with well-posed perfectly matched layer,” Microw. Opt. Technol. Lett., vol. 46, no. 5, pp. 459–463, Sep. 2005. [3] M. H. Chen, B. Cockburn, and F. Reitich, “High-order RKDG methods for computational electromagnetics,” J. Sci. Comput., vol. 22–23, pp. 205–226, Jun. 2005. [4] S. D. Gedney, C. Luo, B. Guernsey, J. A. Roden, R. Crawford, and J. A. Miller, “The discontinuous Galerkin finite element time domain method (DGFETD): A high order, globally-explicit method for parallel computation,” in Proc. IEEE Int. Symp. on Electromagnetic Compatibility, Jul. 2007, pp. 1–3. [5] T. Lu and P. Z. W. Cai, “Discontinuous Galerkin methods for dispersive and lossy Maxwell’s equations and PML boundary conditions,” J. Comput. Phys., no. 200, pp. 549–580, 2004. [6] L. Pebernet, X. Ferrieres, S. Pernet, B. L. Michielsen, F. Rogier, and P. Degond, “Discontinuous Galerkin method applied to electromagnetic compatibility problems: Introduction of thin wire and thin resistive material models,” IET Sci. Meas. Technol., vol. 2, no. 6, pp. 395–401, Nov. 2008. [7] S. G. García, M. F. Pantoja, C. M. de Jong van Coevorden, A. R. Bretones, and R. G. Martín, “A new hybrid DGTD/FDTD method in 2-D,” IEEE Microw. Wireless Compon. Lett., vol. 18, no. 12, pp. 764–766, Dec. 2008. [8] M. Bernacki, L. Fezoui, S. Lanteri, and S. Piperno, “Parallel discontinuous Galerkin unstructured mesh solvers for the calculation of three-dimensional wave propagation problems,” Appl. Math. Model., vol. 30, no. 8, pp. 744–763, Aug. 2006. [9] A. H. Mohammadian, V. Shankar, and W. F. Hall, “Computation of electromagnetic scattering and radiation using a time-domain finitevolume discretization procedure,” Comput. Phys. Comm., vol. 68, pp. 175–196, 199.

! Calculation speed = 14 1 10 ; Memory = 1104 Mb. DGTD (G ; R ) ! Calculation speed = 126 1 10 ; Memory = 300 Mb. ; Memory = 60 DGTD (G ; R ) ! Calculation speed = 728 1 10 2Computational

requirements in a 1.66 Ghz Core 2 Duo T5500: FDTD

Mb. Calculation speed is given in terms of physical time normalized to the CPU time (e.g., a calculation speed of implies that 1 s is needed by the CPU to simulate a physical time of 1 ps).

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[10] A. Catella, V. Dolean, and S. Lanteri, “An unconditionally stable discontinuous Galerkin method for solving the 2-D time-domain maxwell equations on unstructured triangular meshes,” IEEE Trans. Magn., vol. 44, no. 6, pp. 1250–1253, Jun. 2008. [11] A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method. Boston, MA: Artech House, 1995. [12] J.-C. Nédelec, Acoustic and Electromagnetic Equations. Heidelberg, Berlin: Springer-Verlag, 2001. [13] J.-F. Lee, R. Lee, and A. Cangellaris, “Time-domain finite-element methods,” IEEE Trans Antennas Propag., vol. 45, no. 3, pp. 430–442, Mar. 1997. [14] P. P. Silvester and R. L. Ferrari, Finite Elements for Electrical Engineers, 2nd ed. Cambridge, U.K.: Cambridge Univ. Press, 1990. [15] T. Lau, E. Gjonaj, and T. Weiland, “Investigation of higher order symplectic time integration methods for discontinuous Galerkin methods with a centered flux,” in Proc. Workshop on Computational Electromagnetics in Time-Domain, Oct. 2007, pp. 1–4. [16] J. P. Webb, “Hierarchical vector basis functions of arbitrary order for triangular and tetrahedral finite elements,” IEEE Trans. Antennas Propag., vol. 47, no. 8, pp. 1244–1253, 1999. [17] K. Sankaran, “Accurate domain truncation techniques for time-domain conformal methods,” Ph.D. dissertation, ETH Zurich, Switzerland, 2007. [18] A. R. Bretones, R. Mittra, and R. G. Martín, “A hybrid technique combining the method of moments in the time domain and FDTD,” IEEE Microw. Guided Wave Lett., vol. 8, no. 8, pp. 281–283, Aug. 1998. [19] L. Fezoui, S. Lanteri, S. Lohrengel, and S. Piperno, “Convergence and stability of a discontinuous Galerkin time-domain method for the 3D heterogeneous Maxwell equations on unstructured meshes,” ESAIM: Math. Model. Numer. Anal., vol. 39, no. 6, pp. 1149–1176, Jun. 2005. [20] S. D. Gedney, C. Luo, J. A. Roden, R. D. Crawford, B. Guernsey, J. A. Miller, and E. W. Lucas, “A discontinuous Galerkin finite element time domain method with PML,” in Proc. IEEE Antennas and Propagation Society Int. Symp., July. 5–11, 2008, pp. 1–4. [21] A. Taflove and S. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. Boston, MA: Artech House, 2005.

Mario Fernández Pantoja (M’00) received the B.Sc., M.Sc., and Ph.D. degrees in electrical engineering from the University of Granada, Granada, Spain, in 1996, 1998 and 2001, respectively. Between 1997 to 2001, he was an Assistant Professor at the University of Jaen, Jaen, Spain. In 2001, he joined the University of Granada where, in 2004, he was appointed Associate Professor. He has been a Guest Researcher at the Dipartimento Ingenieria dell’Informazione, University of Pisa, Italy, the Computational Mechanics Group, Imperial College, the Antenna and Electromagnetics Group, Denmark Technical University, and Pennsylvania State University. His research is mainly focused on the areas of interaction of electromagnetic waves with structures in time domain, and optimization methods applied to antenna design.

Jesus Alvarez (M’07) was born in Leon, Spain. He received the B.Sc. degree in electrical engineering from the University of Cantabria, Spain, in 2001, and the M.Sc. degree from the University Carlos III of Madrid, Spain, in 2008. In 2006, he joined the EADS CASA, Military Air Systems, Spain. His current research interests include computational electrodynamics in time domain, method of moments and fast algorithms for integral equations in frequency domain and computational electromagnetic applied to electromagnetic compatibility, antenna and RADAR cross section problems.

Salvador G. Garcia (M’93) received the M.S. and Ph.D. degrees (with extraordinary honors) in physics from the University of Granada, Granada, Spain, in 1989 and 1994, respectively. In 1999, he joined the Department of Electromagnetism and Matter Physics, University of Granada, as an Assistant Professor (with tenure). He has published over 40 refereed journal articles and book chapters, and over 80 conference papers and technical reports, and participated in several national and international projects with public and private funding. He has received grants to stay as a Visiting Scholar at the University of Duisburg (1997), the Institute of Mobile and Satellite Communication Techniques (1998), the University of Wisconsin-Madison (2001), and the University of Kentucky (2005). His current research interests include computational electrodynamics in the time domain, microwave imaging and sensing (GPR), bioelectromagnetics, and antenna design.

Luis Diaz Angulo was born in Basque Country, Spain, in 1985. He received the B.Sc. and M.Sc. degrees in physics from the University of Granada, Spain, in 2005 and 2007, respectively. From 2006 to 2007, he spent one year at the University of Manchester, U.K., on an Erasmus Mundus Fellowship. Since 2007, he has been with the Department of Electromagnetism and Matter Physics, the University of Granada, Granada, Spain, where he is working toward the Ph.D. degree. He has worked in time domain numerical methods applied to electromagnetism, specially discontinuous Galerkin time domain methods. Other interests are applications of numerical methods in terahertz technologies, GPR imaging, and bioelectromagnetics.

Amelia Rubio Bretones (SM’08) was born in Granada, Spain. She received the Ph.D. degree in physics (cum laude) from the University of Granada, Granada, Spain, in 1988. Since 1985, she has been employed at the Department of Electromagnetism, University of Granada, first as an Assistant Professor, and then in 1989 as an Associate Professor, and finally, since 2000 as a Full Professor. On several occasions, she was a Visiting Scientist at the Delft University of Technology, Eindhoven University of Technology, and at The Pennsylvania State University. Her research interest is mainly in the field of numerical techniques for applied electromagnetics with an emphasis on time-domain techniques such as finite difference time domain, the application of the method of moments in the time domain for antenna and scattering problems, and hybrid techniques.

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Accurate FDTD Simulation of RF Coils for MRI Using the Thin-Rod Approximation Dennis M. Sullivan, Senior Member, IEEE, Peter Wust, and Jacek Nadobny, Senior Member, IEEE

Abstract—The finite-difference time-domain (FDTD) method is being used in the design and analysis of highly resonant radio frequency (RF) coils often used for magnetic resonance imaging (MRI). It is shown that by using the thin-rod approximation (TRA) for accurate wire radius calculation, the FDTD method can model the inductance of the metal wires for different radii at radio frequencies, as well as the capacitance of discrete elements. FDTD can also model human bodies in the near field of the loop in order to determine the influence of the body on the loop, as well as the heating patterns produced in the body by the electromagnetic radiation. Index Terms—Coils, dosimetry, finite-difference time-domain (FDTD) methods, MRI.

I. INTRODUCTION AGNETIC resonance imaging, or MRI, is a valuable diagnostic tool in modern medicine [1]. Among the key elements of the MRI instrumentation are highly resonant radio frequency (RF) coils, often designed as wire loops, which consist of conducting wires and capacitors. The resonant frequency at which a given coil is to be tuned depends on the inductance of the wire loop itself and on the capacitors situated in the loop. The loop inductance depends on the geometry of the loop as well as the cross-section of the wire. The design can also be influenced by a human body in the near field of the coil. Furthermore, the body can be heated by the RF radiation of the coil. Electromagnetic simulation is being used to study both the design of RF coils and the heating due to RF energy [2]. This paper describes the use of the finite-difference time-domain (FDTD) method [3]–[6] in simulating the RF coils. One of the main advantages of using FDTD is that it can also model human bodies in the near field of the loop in order to determine the influence of the body on the loop, as well as the heating patterns produced in the body by the electromagnetic radiation. The simulation of an RF coil centers on the resonance of the coil and must include the current flowing through the loop. Since FDTD models only the and fields, special measures must be taken. Because the radius of the wires of the loop is important, we employ the thin rod approximation (TRA) [7] for increased accuracy.

M

Fig. 1. The LC resonant loop that is simulated in this paper.

This paper will focus on the FDTD simulation of the LC coil illustrated in Fig. 1. This type of loop is often used in MRI surface coils. Furthermore, in large MRI body coils these LC loops can be grouped together in a configuration known as a “bird cage” to produce a large homogeneous magnetic field. The paper is organized as follows: Section II describes the FDTD simulation of a metal loop without capacitors. Using an analytic method for comparison, it is shown that FDTD using the TRA can accurately determine the inductance corresponding to wires of different diameters. Section III describes how capacitors are added using the method of lumped elements and how the loop is tuned to resonate at a specific frequency. The loss due to radiation is also calculated. Section IV describes the simulation of a loop in proximity to a human body. This is important for two reasons. First, it is shown that the presence of a body in close proximity to the loop affects the resonance characteristics of the loop. Second, it is important to be able to determine the heating characteristics in the body due to the close proximity of the loop. Section V summarizes the results. II. THE BASIC FDTD METHOD We begin with the Maxwell equations: (2.1a) (2.1b) One of the most straight-forward FDTD formulations leads to the following -directed equations in three-dimensions [5]:

Manuscript received September 03, 2009; revised November 30, 2009; accepted December 08, 2009. Date of publication April 05, 2010; date of current version June 03, 2010. D. M. Sullivan is with the Department of Electrical and Computer Engineering, University of Idaho, Moscow, ID, 83844-1023 (e-mail: dennis@ee. uidaho.edu). P. Wust and J. Nadobny are with the Clinic for Radiation Oncology, ChariteCampus Virchow-Klinikum, Humboldt University, 13353 Berlin, Germany. 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.2010.2047370 0018-926X/$26.00 © 2010 IEEE

(2.2a)

SULLIVAN et al.: ACCURATE FDTD SIMULATION OF RF COILS FOR MRI USING THE THIN-ROD APPROXIMATION

Fig. 2. Current through a wire is simulated by the surrounding H fields. Fig. 3. The calculation of proximity to a wire.

H

2005

must be modified by the TRA if it is in close

(2.2b) where Fig. 4. Diagram of the rectangular metal loop to be simulated.

(2.2c) The values and are the relative dielectric constant and conis the dielectric ductivity, respectively, at each of the cells. constant of free space, is the magnetic permeability, and is the speed of light in a vacuum. The time step is determined by

where is the cell size. Different materials are simulated and at different points in the in FDTD by specifying the problem space. Metal is specified by setting ca and cb equal to zero, which insures that the corresponding -field at that point will be zero. The and fields are in fact split to accommodate the formulation of the perfectly matched layer (PML) that surrounds the problem space in all simulations [8]. However, that is not germane to the discussion in this paper. All of the above is covered extensively in the literature, so the details will not be given here. fields. Therefore, the curFDTD only models the and rent flowing through a wire must be modeled indirectly through Ampere’s law:

In the standard FDTD, the calculation of a current moving through a wire in the direction becomes (2.3) as illustrated in Fig. 2.

Unfortunately, this simple method does not allow for different wire diameters. In order to be able to simulate wires of different thicknesses, we employ the thin-rod approximation [7]. This requires the modification of the surrounding fields, as illustrated if Fig. 3. in Fig. 3 must be The calculation of modified in the following way:

(2.4) The factor

is calculated by the thin-rod approximation (2.5)

where , the radius of the wire, is given as a fraction of the cell size. The details are in [7] and will not be repeated here. Voltage in FDTD is simulated by the field of a cell times the length of the cell (2.6) In an LC loop, the capacitance is provided by individual capacitors, but the inductance is provided by the wire. Therefore, in order to evaluate the capability of FDTD to correctly model the inductance of the wire, we begin with the simulation of just a square wire loop without capacitors, as shown in Fig. 4.

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Fig. 6. Simulation of an inductive loop. The top plot is the Gaussian pulse that is used as the excitation. The bottom plot is the time domain current as calculated by (2.3). After a certain amount of ringing, the current settles at 0.444 mA.

loop is 80 by 80 cells. The problem space in the direction perpendicular to the plane is 40 cells. This allows room for a 7 cell in Fig. 4, is a PML on each boundary. The input to the loop, narrow Gaussian pulse. Fig. 5 shows the fields in the plane of the loop after 175, 245, 305 and 350 time steps. Each time is one of the fields perpendicular to step is 0.0167 ns. Since the wire, it may be regarded as an indication of the current flow. The inductance is determined through the standard relationship between current and voltage at an inductor, (2.7) The current is calculated by (2.3). The integral of the input voltage is very easily calculated in FDTD by (2.8) can be calculated after a sufficient number of time steps

by

(2.9) As an example we simulate the 40 cm square loop having circular wires that are 0.1 mm in diameter. Equation (2.5) then gives (2.10)

H

Fig. 5. The field in the plane of the loop at various times after the input voltage has been applied at (a) 175 time steps; (b) at 245 time steps; (c) at 305 timed steps; (d) at 350 time steps.

The FDTD cells are 1 cm cubed. The 40 cm square loop then requires 40 cells on a side. The problem space in the plane of the

The results are shown in Fig. 6. The ringing that appears in the plot of the current is simply due to the time that it takes the pulse to circle around the loop. Notice that it was necessary to let the program run long enough for the transient response of the current to die out. After , and 6000 time steps, . Fortunately, there is an analytic so from (2.9),

SULLIVAN et al.: ACCURATE FDTD SIMULATION OF RF COILS FOR MRI USING THE THIN-ROD APPROXIMATION

2007

Fig. 7. Comparison of the FDTD calculated value of inductance vs. the analytic values of (2.11).

formula available to check he accuracy of this calculation [9]. For a one-turn square loop with sides of length and wire of diameter , the inductance is

(2.11) For the given problem, . Fig. 7 is a plot comparing the FDTD calculated values of inductance for various wire diameters as well as the analytic calculation using (2.11). The wire diameters vary between and 10 mm. In all cases, the FDTD value is within 3 percent of the analytic value. III. SIMULATION OF A RESONANT LC LOOP We now move on to the simulation of a resonant LC loop, as illustrated in Fig. 1. The first task is to add the simulation of the four capacitors. The addition of lumped elements to an FDTD simulation is not new, and there are many different approaches [10]–[13]. A very straight-forward approach will be used here. A capacitor is added to an FDTD simulation by assuming that one cell in the FDTD grid is a parallel plate capacitor filled with a material with a dielectric constant . The capacitance is given by (3.1) Therefore, to specify a capacitance the relative dielectric constant to

at an FDTD cell, we set

and calculate the values of ca and cb in (2.2) accordingly. In actuality, Fig. 1 is an LC circuit with four capacitors in series. is desired, must be Therefore, if a total capacitance of . set to Recall that the goal was to simulate a loop that was resonant at one frequency, say . We know the inductance of the loop for a given wire radius, at least before we added the capacitors. So a good place to begin is

(3.2)

Fig. 8. The impulse response of an LC coil with L = 2:63 mH and C = 0:516 pf . The top is the Gaussian input voltage; the middle plot is the time domain current; and the bottom plot is the Fourier transform of the current showing resonance at 125 MHz.

We start with a wire diameter of 0.1 mm which has an induc. If a resonant frequency of tance of 2.63 is desired, then a capacitance of is required. . This In fact, the resulting resonance is at about should not be too surprising. We did nothing to allow for the fact that out of the 40 cells that make up each arm of the loop, one has been converted to a capacitor. This is bound to have an effect on our original calculation of inductance. Furthermore, our estimation of the capacitance given in (3.2) must be regarded as a “first order” approximation. After a little trial and error, it results in a resonant frequency was found that field at four different of 125 MHz, as shown in Fig. 8. The times during the simulation is shown in Fig. 9. The last parameter to be calculated is the resistance of the loop. This is necessary to calculate the quality factor of the loop. This will be determined simply by observing the attenuation of the current in the loop, as shown in Fig. 10. The amplito over 0.114 . tude decreases from 2.45 Therefore, the attenuation constant can be determined by

In a series RLC circuit [14]

From this, the

of the circuit is determined by

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Fig. 10. (a) The response to the LC loop to an impulse. (b) A close-up of the response between 0.102 s and 0.245 s that shows that the amplitude is attenuated from to 2.45 10 A to 1.65 10 A.

2

2

Fig. 11. The H field in the plane of a loop set to resonate at 125 MHz. The plots on the left show the two-dimensional fields in the plane of the loop. The plots on the right are the one-dimensional cross sections through the middle. Note that the H field is relatively flat in the middle.

Fig. 12. An LC loop of the type described in Section III is placed 5 cm from materials representing a human body. The characteristics of the fat layer are " = 5 and  = 0:05 and those of the muscle are " = 30 and  = 0:3.

IV. SIMULATION OF AN LC LOOP WITH A BODY IN THE NEAR FIELD

H

Fig. 9. The field in the LC loop resulting from the impulse input at (a) 175 time steps; (b) 230 time steps; (c) 300 time steps; (d) 550 time steps.

Now that the parameters to achieve a desired resonant frequency have been found, the impulse input can be replaced with a sinusoid, which would be the normal mode of operafields at two different times tion. Fig. 11 shows the resulting during the cycle. Note that the field in the middle goes up and down uniformly at the resonant frequency.

Next we simulate the situation shown in Fig. 12, i.e., an LC loop of the type described in the previous section with a dielectric slab in close proximity. The slab has the dielectric properties to roughly model a human body. The body is simulated by a one-centimeter layer of fat, with a dielectric constant of 5, and conductivity of 0.05, and muscle tissue, with a dielectric constant of 30 and a conductivity of 0.3. We repeat the simulation of the previous section with a square loop having wires of a 0.1 mm diameter and discrete capacitors with a total value of 0.51 pF. The results are shown in Fig. 13. The results are similar to Fig. 8, except that the resonant frequency has been shifted from 125 MHz to 130 MHz due to the

SULLIVAN et al.: ACCURATE FDTD SIMULATION OF RF COILS FOR MRI USING THE THIN-ROD APPROXIMATION

2009

Fig. 14. (a) The time domain response of the LC circuit in Fig. 12; (b) A close-up of the time-domain response. The amplitude drops from 0.78 to 0.2 in the time interval between 0.099 s and 0.199 s.

Fig. 13. The impulse response of the LC circuit in close proximity to a body, as illustrated inFig. 12. The body has caused the resonant frequency to shift to 130 MHz.

presence of the body in the near field. We then repeat the simulation to determine the resistance of the loop, as shown in Fig. 14. We see that the amplitude drops from 0.78 to 0.2 over a period of 0.1 . Now the resistance can be calculated from

In a series RLC circuit Fig. 15. The SAR on the surface of the body under the coil. The top is a mesh plot and below is a contour diagram.

As expected, the resistance is substantially higher due to the body in the near field that is absorbing energy. From this, the of the circuit is determined by

Fig. 16. Contour diagram of the SAR distribution under the middle of the loop. The maximum energy deposition is at the fat-muscle interface.

which is lower than the of the LC circuit in free space. One of the key reasons for using FDTD simulation in the design of the LC coils is to be able to evaluate the effect of the EM radiation from the coil on the body being scanned by the MRI apparatus. The quantity most widely used in EM dosimetry is the specific absorption rate (SAR) given by [15], [16]

(4.1)

This is considered to be the rate at which EM energy is being . Fig. 15 shows the SAR on the surabsorbed at a point face of the body. It is not surprising that the maximum energy absorption takes place directly under the wires of the coil. Something else of interest is the SAR deposition going into the body, as shown in Fig. 16. Although there will be some energy absorption leading to heating as much as four or five centimeters deep, the “hot spots” tend to occur at the fat-muscle

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Fig. 17. A low pass bird cage coil. These coils can produce a very homogeneous magnetic field within the coil.

interface. This has been observed both in laboratory measurements [17] and in clinical hyperthermia cancer treatments [18]. Much more realistic models of the human body could be used, if necessary [19]. V. DISCUSSION The LC coil of Fig. 1 is not a realistic coil for use with MRI. It was chosen as a simple example to illustrate the capabilities of the simulation. More elaborate coils could be modeled with more elaborate simulation techniques. One relatively straight forward extension of the coil of Fig. 1 is the birdcage coil, illustrated in Fig. 17. The bird cage coils are perhaps the most popular type of coil because they can produce a very homogeneous magnetic field over a large volume within the coil [1]. In summary, it has been shown that the FDTD method is capable of accurately simulating the parameters of an LC coil for MRI. It can also account for the presence of a human body in the near field and the energy deposition within the body due to the coil. REFERENCES [1] J. Juanming, Electromagnetic Analysis and Design in Magnetic Resonance Imaging. Boca Raton, FL: CRC Press, 1999. [2] D. M. Sullivan and J. Nadobny, “FDTD simulation of RF coils for MRI,” presented at the IEEE Int. Symp. APS, Charleston, SC, Jun. 1–5, 2009. [3] K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag., vol. 14, pp. 303–307, Mar. 1966. [4] K. S. Kunz and R. J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics. Boca Raton, FL: CRC Press, 1993. [5] D. M. Sullivan, Electromagnetic Simulation Using the FDTD Method. Piscataway, NJ: IEEE Press, 2001. [6] A. Taflove and S. C. Hagness, Computational Electromagnetics: The Finite-Difference Time-Domain Method, 3rd ed. Norwood, MA: Artech House, 2005. [7] J. Nadobny, R. Pontalti, D. M. Sullivan, W. Wlodarczk, A. Vaccari, P. Deuflhard, and P. Wust, “A thin-rod approximation for the improved modeling of bare and insulated cylindrical antennas using the FDTD method,” IEEE Trans. Antennas Propag., vol. 51, pp. 1780–1796, Aug. 2003. [8] J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comp. Phys., vol. 114, pp. 185–200, 1994. [9] F. E. Terman, Radio Engineers’ Handbook, 1st ed. London, U.K.: McGraw-Hill, 1950.

[10] W. Sui, D. A. Christiansen, and D. H. Durney, “Extending the two-dimensional FDTD method to hybrid electromagnetic systems with active and passive lumped elelments,” IEEE Trans. Microw. Theory Tech., vol. 40, pp. 724–730, Apr. 1992. [11] C. A. Thomas, M. E. Jones, M. J. Piket-May, A. Taflove, and E. Harrington, “The use of SPICE lumped circuits as sub-grid models for FDTD analysis,” IEEE Microw. Guided Wave Lett., vol. 4, pp. 141–143, May 1994. [12] J. A. Pereda, F. Alimenti, P. Mezzanotte, L. Roselli, and R. Sorrentino, “A new algorithm for the incorporation of arbitrary linear lumped networks into FDTD simulators,” IEEE Trans. Microw. Theory Tech., vol. 47, pp. 943–949, Jun. 1999. [13] J. Lee, J. Lee, and H. Jung, “Linear lumped loads in the FDTD method using piecewise linear recursive convolution method,” IEEE Mircrow. Wireless Compon. Lett., vol. 16, pp. 158–160, Apr. 2006. [14] C. Paul, Fundamentals of Electric Circuit Analysis. New York: Wiley, 2001. [15] D. M. Sullivan, D. T. Borup, and O. Gandhi, “Use of the finite-different time-domain method in calculating EM absorption in human tissues,” IEEE Trans. Biomed. Eng., vol. 34, pp. 204–211, Feb. 1987. [16] D. M. Sullivan, O. P. Gandhi, and A. Taflove, “Use of the finite-difference time-domain method in calculating EM absorption in man models,” IEEE Trans. Biomed. Eng., vol. 35, pp. 179–186, Mar. 1988. [17] D. M. Sullivan, “Three-dimensional computer simulation in deep regional hyperthermia using the FDTD method,” IEEE Trans. Microw. Theory Tech., vol. 38, pp. 204–211, 1990. [18] R. Ben-Yosef and D. M. Sullivan, “Peripheral neuropathy and myonecrosis following hyperthermia and radiation therapy for recurrent prostatic cancer: correlation damage with predicted SAR pattern,” Int. J. Hyperthermia, vol. 8, pp. 173–185, 1992. [19] B. J. James and D. M. Sullivan, “Direct use of CT scans for hyperthermia treatment planning,” IEEE Trans. Biomed. Eng., vol. 39, pp. 845–851, Feb. 1992.

Dennis M. Sullivan (M’88–SM’96) received the Ph.D. degree in electrical engineering from the University of Utah, Salt Lake City, in 1987. From 1987 to 1992, he worked at Stanford University School of Medicine, where he developed a treatment planning system for hyperthermia cancer therapy. Since 1992, he has been with the Department of Electrical Engineering, University of Idaho, Moscow, where he is presently an Associate Professor. His research interests include electromagnetic and quantum simulation. He is the author of the book Electromagnetic Simulation Using the FDTD Method (Wiley-IEEE, 2000). Dr. Sullivan won the R. W. P. King Award for the Best Paper by a Young Investigator in 1998.

Peter Wust was born in Berlin, Germany in 1953. He received the Diploma in Physics (Dipl.-Phys.) and the M.D. degree (Dr. Med.) from the Free University of Berlin, in 1978 and 1983, respectively. He received the Board Certification of Radiation Oncology in 1990. He was a Coordinator of the Collaborative Research Project SFB 273, DFG, on Hyperthermia (1994–2002). Since 2000, he has been a Professor of radiation oncology. He is currently with the Department of Radiation Oncology, Charité Universitätsmedizin Berlin-Corporate Medical School, Freie Universität Berlin and the Humboldt-Universität zu Berlin, Campus Virchow-Klinikum, Berlin. His current research include biomedical thermal effects, design, controls and applications to biomedical engineering, in particular, in the area of MR-guided hyperthermia cancer therapy treatments.

SULLIVAN et al.: ACCURATE FDTD SIMULATION OF RF COILS FOR MRI USING THE THIN-ROD APPROXIMATION

Jacek Nadobny (M’98–SM’06) received the Ph.D. degree in electrical engineering from the Technical University, Berlin, Germany, in 1993. His dissertation was focused on the development of a new formulation for the volume-surface integral equation for electrically inhomogeneous media on tetrahedral grids. He is a Research Engineer in the Department of Radiation Oncology, Charité Universitätsmedizin Berlin-Corporate Medical School, Freie Universität Berlin and the Humboldt-Universität zu Berlin,

2011

Campus Virchow-Klinikum, Berlin. His research activities include design and modeling of antenna arrays and magnetic resonance (MR) coils and development of electromagnetic and thermal simulation methods for biomedical applications, in particular, for the MR-guided hyperthermia cancer therapy applications and for the safety requirements for the use of MR tomographs.

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Propagation Over Parabolic Terrain: Asymptotics and Comparison to Data Dmitry Chizhik, Senior Member, IEEE, Lawrence Drabeck, and W. Michael MacDonald, Member, IEEE

Abstract—Analysis of radio propagation over varying, clutter-covered terrain was carried out aiming at prediction of power received by a terminal immersed in clutter, with the transmitter placed above clutter. The need for such prediction arises, for example, in planning and assessing coverage and interference in radio communications. Following a general formulation of the problem, particular solutions were found when the terrain has constant curvature. Asymptotic evaluation yielded compact expressions both for parabolic valleys and ridges. In both cases, ray-optical term dominated for short ranges, while a single mode dominated at large ranges. Strong focusing was found to occur in valleys, while ridges produced strong blockage beyond the “horizon”. The resulting procedure for predicting pathloss over varying terrain is therefore to apply the formulae using the terrain curvature extracted from terrain files. In comparison to measured power across a valley, mean errors of less than 1 dB were found, a marked improvement over standard terrain-unaware models that produce a mean error of 30 dB. Index Terms—Propagation, terrain factors.

I. INTRODUCTION LANNING and performance assessment of radio communications often requires prediction of received power of both desired and interfering signals. A widely occurring arrangement is that of a transmitter (e.g. cellular base station) placed somewhat above terrestrial clutter and a terminal receiver immersed in clutter, such as buildings or trees. While prediction of exact received power requires unreasonably detailed knowledge of the environment as well as models of exceptionally high fidelity, it is often of interest to predict average received power that may be expected based on relatively crude information, such as terrain height variation and clutter height. In the cellular industry a widespread practice is to employ empirical models of path loss such as [2]–[5]. These models were obtained through a reduction of measured path loss data and specify a linear relationship between pathloss in dB and the logarithm of transmitter-receiver separation. Resulting predictions are usually supplemented by adjusting the model parameters (i.e. slope and intercept) by fitting to locally measured data. Such practice leads to additional expense and delay of collecting

P

Manuscript received June 12, 2009; revised December 09, 2009; accepted December 10, 2009. Date of publication March 29, 2010; date of current version June 03, 2010. The authors are with Bell Laboratories, Alcatel-Lucent, Holmdel, NJ 07733 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.2010.2046855

data, but has the virtue of allowing for empirical adjustments due to unmodeled effects such as terrain and clutter variations. Models based on first principles offer the promise of predictions without additional measurements. Examples of such models are the Walfisch-Bertoni model [6], derived for equal height buildings on flat terrain, and with buildings modeled as multiple absorbing half-screens. This was extended to buildings as screens on variable terrain [7]. Whitteker [8] has applied marching Huygens principle evaluation between elevated terrain points and included reflection from intervening terrain. This is a very general approach and is similar to parabolic equation methods [9], both involving iterative numerical computation as the field is “marched” in range. Blaunstein, et al. [10] represents propagation loss as a consequence of probabilistic visibility representation and interaction with multiple random scatterers. Blaunstein and Andersen [10] provide a extensive analysis of propagation over obstacles in rural areas, with each tree acting as a random phase-amplitude screen. Piazzi and Bertoni [12] extended [6] to variable terrain, exploring numerically the effect of clutter covered variable terrain with buildings represented as half-screens. Numerical solutions were found to be amenable to ray-optical interpretations. Barrios [21], Barrios et al. [22], Dockery and Kuttler, [23], Donohue and Kuttler [24] have applied the numerically efficient split-step/Fourier algorithm to generally varying terrain, allowing accounting both for terrain variation as well as atmospheric refraction at large ranges with no significant clutter. In [1] local scattering around the mobile was treated in the case of flat terrain covered by constant height clutter. Present work is an extension of [1] aimed at allowing terrain height variation, particularly in the case of parabolic terrain, both in the case of a valley and a ridge. The goal is to derive relatively simple expressions for pathloss for this special case that wireless system planners would find useful and easy to use. The problem of solving for a field due to a source above a concave boundary, subject to the Dirichlet boundary condition has been treated by Felsen, et al. in [13] and [14]. It was found that the solution may be expressed asymptotically as a hybrid mix of rays and modes. In this work, it is of interest to find the received power for a terminal in terrestrial clutter. It was found in [1] that the key quantity of interest is the derivative of the Green’s function at the clutter surface. In this work, the variable boundary problem is addressed through transforming the wave equation into a parabolic equation, which is then solved asymptotically to arrive at a hybrid ray-mode mixture. It is found that the solution has a simple interpretation of a ray optical contribution and only a single mode, other modes being negligible, both in the case of a

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CHIZHIK et al.: PROPAGATION OVER PARABOLIC TERRAIN: ASYMPTOTICS AND COMPARISON TO DATA

2013

height of the base above clutter be , and recognizing that for , it was found that most cases of interest, (3) Considering now the varying cluttered terrain boundary, the in air above the clutter due to a point source at field satisfies the Helmholtz equation (4) Fig. 1. Propagation over variable terrain.

concave and a convex boundary. The predictions are compared both to full modal sum solution as well as to measurements collected in variable terrain. In Section II the problem of calculating fields in air over a generally rough dielectric surface is cast as a parabolic equation with a variable index of refraction. In Section III this equation is solved as a sum of modes for constant terrain curvature case (parabolic valley or ridge). In Section IV the sum of modes is found to be well approximated by a ray optical term and a single mode contribution, while Section V presents comparisons to measured power over a river valley.

The top of the surface may be viewed as an inhomogeneous varies as a function of range . It dielectric whose height may be noted, however, that for a large range of material properties and for both polarizations, plane wave incidence at small grazing angles results in nearly perfect reflection, with a reflection coefficient of nearly 1. The interaction with a surface is then approximated here as the Dirichlet boundary condition (5) The problem may be transformed into a simpler problem by substituting (6) into (4) to get

II. PROPAGATION IN AIR OVER A GENERALLY VARYING DIELECTRIC SURFACE

(7)

This work addresses prediction of average power received at a terminal beneath terrestrial clutter, such as trees and buildings, from a transmitter antenna placed at height above local clutter, where the terrain between the transmitter and the receiver is generally varying, illustrated in Fig. 1. In [1] it was found that the received power is related to by the transmitted power (1)

Assuming nearly grazing propagation, , (7) becomes the “parabolic equation” (8) or a Schrödinger’s equation with two spatial dimensions and , and range playing the role of time. A transformation proposed by Beillis and Tappert [8] allows further simplification. Letting (9)

where the mean square of the vertical derivative of the Green’s function above clutter, , may be interpreted as the factor accounting for propagation over the clutter-covered varying terrain and the wavelength is related to the . The local scattering factor wavenumber by (2) for the case of a terminal in the middle of a street of width . The other quantities are local clutter height and mobile height above local ground , shown in Fig. 1. Similar expressions have also been found [1] for flat terrain covered by vegetation, as opposed to buildings. For flat terrain with uniform height clutter, treated in [1], image theory has been used to determine . Letting the

and substituting (10)

into (8), results in

(11) where the prime notation on is omitted for notational clarity. The boundary condition (5) is transformed by (9) to a flat surface boundary condition: (12)

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The balance of the work addresses particular solutions of (11) with (12) as the boundary condition.

The received power (1) depends on the absolute value of the -derivative of the Green’s function at the surface. As may be observed from (6) and (10)

III. CONSTANT TERRAIN CURVATURE A. Sum of Modes Solution Of particular interest is the case of constant curvature ter, corresponding to a parabolic valley rain, or a parabolic ridge . Equation (11) subject to (12) may be solved [16], [17] by introducing a horizontal Green’s func, describing the field at due to a point tion source at , and a vertical Green’s function due to a point source at . The two Green’s functions satisfy correspondingly

(20) where the last equality follows from the boundary condition depends on (12). The -derivative of (17), where only , is, then, using (19),

(13) and (21) (14) where where the separation constant may be interpreted as a square of the vertical component of the wavevector. The solution to (11) may be expressed as (15) The contour integral in (15) encompasses the poles of the vertical Green’s function . The coordinates of the source (base antenna) in the horizontal plane have been set to in (15) for convenience. Equation (13) may be solved using Fourier transforms to yield

(22)

is defined for later convenience. The form of the vertical mode function depends on the sign of the terrain curvature, as described in the following sections. B. Valley In the case of a valley, vertical mode solutions satisfying the source-free version of (14) are Airy functions [18]

(16) The solution to (15) may be expressed as a sum over residues at the poles of the integrand in (15):

(23) where the effective waveguide width (24)

(17)

recognized as a sum over modes where is the vertical mode, satisfying the source-free vermode function of the sion of (14) and the boundary condition (12), with the corresponding Wronskian determinant [18]: (18)

depends on the curvature and the characteristic vertical spatial frequency is determined from boundary condition to be approximately [18] defined by (25)

C. Ridge In the case of a ridge, the vertical mode solutions satisfying the source-free version of (14) are Airy functions with complex arguments [18]

and (19)

(26)

CHIZHIK et al.: PROPAGATION OVER PARABOLIC TERRAIN: ASYMPTOTICS AND COMPARISON TO DATA

where depends on the curvature and is determined the characteristic vertical spatial frequency to approximately [18] from boundary condition satisfy

2015

in comparison to the rapid fluctuation of the numerator with respect to . Using (25) for the characteristic spatial frequencies , the asymptotic behavior of vertical mode function (32) may be seen to be

(27)

IV. ASYMPTOTIC EVALUATION (33) A. Valley While summation over modes in (21) may be carried out directly, the number of terms that are needed is often large, particularly for short ranges and/or small terrain curvatures. It is therefore of interest to derive an asymptotic expression that would provide approximate yet accurate results. This may be obtained has the following by recognizing that the Airy function asymptotic representation [18], [19] for large values of (28)

The denominator of (22) used in (31) depends on the behavior and may thus be evaluated using of the Airy function near the second approximation in (32) to be

(34) Substituting 1st expression in (33), (34) into (30) and (31), leads to

The summation over the modes (21) may now be split into two sums, one for each of the regions in (28): (29) where terms for

are summed in (35) (30) and similarly using 2nd expression in (33)

and terms for

are summed in (36) (31)

, with the subscript indicating the domwhere inant “whispering gallery” mode and mode functions (23) are used. Now using the asymptotic form (28), one finds that for

(32) Where the expression in the numerator was expanded to first order in the Taylor series for . In the last approximation in (32) the -dependence of the denominator is neglected

At short ranges, the terms in the sum (35) vary slowly with , which mode index , with the exception of the factor leads to near cancellation of the successive terms. The resulting “telescoping” series thus has only the first and the last terms in the sum that offer significant contribution. The last term corresponds to the characteristic spatial frequency given by . The first term, corresponding to the lowest order mode, may be seen in (35) to be exponentially smaller than the last term, and is neglected. At larger ranges, the phase differences become important, between modes due to the term making it necessary to include more modes. From (35) it may be seen that the dominant contribution comes from modes with characteristic vertical spatial frequencies in the neighborhood of . The dominant mode contribution is (37)

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where (38) and (39) The sum in (36) is now examined. The difference in the square of the spatial frequencies of neighboring modes may be deduced from (25) as (40) . This may be used to approxileading to mate the sum into an integral over a continuum of characteristic spatial frequencies:

Fig. 2. Path gain as a function of range for various models over flat terrain and a parabolic valley. 2 GHz, Transmitter height 20 m, Clutter height 9 m, receiver height 2 m, terrain curvature 2 10 m

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The last step is justified through a numerical comparison of (43) with the complete modal sum solution (17). Finally, the quantity of interest for evaluating the average received signal power (1) is

(41) The integral in (41) may be approximated by representing the as a sum of two complex exponentials and using stationary phase techniques to evaluate the integral asymptotically, resulting in

(42) where the Heavyside step function is introduced to indicate that in the stationary phase approximation, the integral conis tributes only when the stationary phase point within the limits of the integral (41). The signal arriving at the receiver may be thought of as consisting of two parts: the “whispering gallery mode”, represented by the first term , significant at large ranges, and the ordinary optical term , corresponding to the superposition of the direct and reflected from the top of the clutter paths. The ordinary optical term is significant at shorter ranges, and, within the approximations used here, corresponds to the case of the flat terrain used in (3). More precise evaluation of (41) may be obtained from higher order asymptotic evaluations of the integral, which would remove the abrupt transition indicated by the step function. It has been found that a simple approximation may be obtained by recognizing that each of the two terms in (42) dominates in a different range of source-receiver separation , with a switch occurring around It is decided here in an ad hoc manner to remove the step function , resulting in

(43)

(44) has been set to zero under where the cross term the assumption that the relative phase between the whispering gallery mode field and the ordinary optical contribution will depend significantly on the precise shape of the top of the clutter boundary and is assumed to be uniformly distributed. Using (44), the received power in (1) the case of the valley may therefore be expressed as

(45) where

is defined in (22) with the mode function as in (23) and using (38). Also, defined in (2) for areas with buildings and by (29) in [1] for dense vegetation. In Fig. 2, the asymptotic result (45) is compared to the exact sum over modes (1), (21), (23), (25). Also plotted for comparison are flat terrain predictions [1] as well as the widely used Walfisch-Bertoni model [6], derived for propagation over flat urban terrain with equal height buildings, modeled as absorbing half-screens. Okumura-Hata model [2]–[4], gives similar results to the flat-terrain and Walfisch-Bertoni models, as discussed in [1]. It may be observed that while all models give similar predictions at shorter ranges, guiding by the valley results in signals that are over 20 dB stronger at 10 km. The exact sum of modes solution to propagation over a valley produces “beating” between modes, resulting in some oscillation in path gain as a function of range. Nevertheless, the asymptotic formula (45) captures the exact sum of modes behavior quite well.

CHIZHIK et al.: PROPAGATION OVER PARABOLIC TERRAIN: ASYMPTOTICS AND COMPARISON TO DATA

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Fig. 3. Characteristic spatial frequencies and integration contours.

B. Ridge In the case of a ridge, the modal expansion (21) may be expressed using (16) and (26) as

Fig. 4. Path gains predicted by accepted models, flat terrain model, and ridge models. Various heights are: Transmitter 20 m, Clutter 9 m, receiver 2 m, terrain curvature 3:2 10 m

(46) The characteristic vertical spatial frequencies (27) are lodepicted in the third quadrant cated [18] on a ray of Fig. 3. The asymptotic evaluation of (46) may be carried out by separating the sum over modes into two groups, and , motivated by the change in the Airy function behavior, as described by (28)

It may be shown that the integral over the contour is exponentially smaller than other terms and is therefore neglected is the same as (41), and may therefore here. The integral over be evaluated approximately through a stationary phase technique in the same way. Each of the discrete modes in the sum in (49) decays exponentially with range, corresponding to the shedding of energy by creeping waves [20]. The lowest order mode suffers the least decay and all higher order modes are here neglected. These considerations allow the vertical derivative of the Greens function (49) to be approximated as a sum of the ordinary ray-optical component and the lowest order creeping wave

0 2

(47)

where the summand expression from (46) is not written explicitly for compactness. The second sum in (47) is now approximated by an integral over the continuous range , denoted as the contour in Fig. 3

(50) The step function in (50) indicates that, in the stationary phase approximation, the second integral in (48) contributes only when the stationary phase point is within the limits of the integral. Finally, substituting (50) in (1) results in the power received by an antenna buried in the clutter approximated as

(48)

where the modal density is introduced using (40) and following the argument leading to (41). The integrand in the contour integral in (48) is analytic in the region , allowing for the changing of the contour from to . Using the asymptotic form of the Airy function (28), this results in

(49)

(51) where

is defined in (22) with the mode function as in (26) and for mode given by (27). Also, defined in (2) for areas with houses and by (29) in [1] for dense vegetation. The step function in (51) has the effect of removing the contribution of the ordinary optical term at large , interpreted as blockage by ranges, the ridge. Path gain predicted by several models is shown in Fig. 4. The exact solution, computed from a sum of residues (46) is indicated as a thick dashed line in Fig. 4 follows the flat terrain model for short ranges, but shows increasingly larger

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Fig. 5. Terrain height across the Columbia river, Portland OR. Transmitter is marked as a white triangle, receiver locations as white stars.

Fig. 6. Actual terrain profile across the river valley and its parabolic fit (curvature of 5:2 10 m )

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loss at larger ranges. The asymptotic ridge solution (51) follows closely the exact solution at both large and small ranges, with the exception of the transition around 1 km, where the ray-optical contribution is blocked by the terrain. Higher order asymptotic evaluation of the integral over in (49) may allow further improvement in agreement, which is not pursued here. Similar conclusions have been reached in Piazzi and Bertoni [12] where a marching diffraction solution over sequences of half-screens over variable terrain were found to be well approximated by corresponding ray-optical expressions similar in form to that obtained for diffraction over smooth cylinders. V. COMPARISON WITH MEASUREMENTS Received power was measured as part of a data collection campaign carried out in Portland, Oregon. Grey scale illustration of terrain height, together with locations of transmitter and receivers is in Fig. 5. The transmitter was a 7 m high, 20 W, 120 sector antenna radiating at 1.9 GHz, placed on the north bank of the Columbia river, Fig. 5, aimed south across the river. The receiver was an omnidirectional antenna placed on a roof of a vehicle and driven on the roads on the south bank. The terrain along a typical vertical profile from the transmitter to the measurement area is shown in Fig. 6. A parabolic fit to the terrain profile provides the value of terrain curvature needed

Fig. 7. Measured (stars) and predicted pathloss as a function of range (linear scale) at 1.9 GHz. Predictions are made using a standard (terrain-unaware) model [5] and asymptotic valley prediction.

in (24). Terrain variations that deviate from the to define parabola are not treated here. Pathloss predicted using (45) as well as a standard (terrain-unaware) model [5] is compared against measurements as a function of the separation range in Fig. 7, with range plotted on a linear scale for clarity. Measured power discussed here is actually a local average of the instantaneous values of received signal power, a process that largely removes small scale variations and leaves only the slower variations of average power of interest here. Measured received power is observed to deviate significantly from predictions, with the error having a standard deviation of 14 dB, attributed to the unmodeled terrain variations, beyond the simple parabolic shape. Nevertheless, the model (45) results in a small mean error ( 1 dB). Standard model results in a mean error of 30 dB, underscoring the importance of modeling the whispering gallery modes guided by the valley. Similarly large errors would result from the use of any standard terrain-unaware model. Clearly, further modeling is required to account for general terrain variations, perhaps through numerical solutions of the parabolic wave (11). Solutions of the parabolic equation using the very efficient split step algorithm have been applied to predicting field strengths over variable terrain and variable refractivity in [21]–[24]. Model accuracy was assessed through comparison to measurements carried out mostly in areas of negligible vegetation. Clutter such as vegetation would have two primary effects on propagation: raising of the effective terrain height to that of clutter top and changing the propagation mechanism at the mobile from direct illumination to scattering [1]. At 2 GHz these effects exceed 30 dB. Extending the numerical solutions of the parabolic wave equation to include the effects of scattering into clutter would be a promising combination of proper treatment of general terrain and proper treatment of near-mobile scattering. More generally, numerical solutions offer treatment of arbitrary terrain variation while analytical methods, such as presented here, offer insight in certain canonical cases. The work presented here is aimed at the extension of flat terrain formulations [1], [2], [4], [6], some widely used, to include terrain curvature, where it is important.

CHIZHIK et al.: PROPAGATION OVER PARABOLIC TERRAIN: ASYMPTOTICS AND COMPARISON TO DATA

VI. CONCLUSION Analysis of radio propagation over varying, clutter-covered terrain was carried out, for the case when one end of the radio link, e.g. a receiver, is immersed in clutter. Asymptotic evaluation yielded compact expressions for received power for constant curvature terrain, i.e. parabolic valleys and ridges. For both cases, ray-optical term dominated for short ranges, while a single mode dominated at large ranges. Strong focusing was found to occur in valleys, while ridges produced strong blockage beyond the “horizon”. Presented closed-form expressions for pathloss require terrain curvature as a parameter. Terrain curvature may be obtained from a parabolic fit to the terrain profile available from a terrain elevation database. In the limit of zero curvature, the pathloss formulas match flat terrain predictions. In comparison to measured power across a valley, mean errors of less than 1 dB were found, a marked improvement over standard terrain-unaware models that produce a mean error of 30 dB. Still, large standard deviation of error points to a need to account for general terrain variations, beyond the parabolic case. ACKNOWLEDGMENT The authors wish to thank A. Diaz for his insightful comments that have improved and enriched the theoretical treatment.

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[13] E. Topuz, E. Niver, and L. Felsen, “Electromagnetic fields near a concave perfectly reflecting cylindrical surface,” IEEE Trans. Antennas Propag., vol. 30, no. 2, pp. 280–292, March 1982. [14] T. Ishihara and L. Felsen, “High-frequency propagation at long ranges near a concave boundary,” Radio Sci., vol. 23, no. 8, pp. 997–1012, Nov.–Dec. 1988. [15] A. Beilis and F. D. Tappert, “Coupled mode analysis of multiple rough surface scattering,” J. Acoust. Society Amer., vol. 66, 1979. [16] L. B. Felsen and N. Markuvitz, Radiation and Scattering of Waves, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, 1973. [17] R. E. Collin, Field Theory of Guided Waves. New York: IEEE Press, 1991. [18] L. M. Brekhovskikh and Y. P. Lysanov, Fundamentals of Ocean Acoustics, 2nd ed. Berlin, Germany: Springer-Verlag, 1990. [19] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. [20] A. D. Pierce, “Acoustics: An introduction to its physical principles and applications,” Acoust. Soc. Amer., 1991. [21] A. E. Barrios, “A terrain parabolic equation model for propagation in the troposphere,” IEEE Trans. Antennas Propag., vol. 42, no. 1, pp. 90–98, Jan. 1994. [22] A. E. Barrios, K. Anderson, G. Lindem, and G. , “Low altitude propagation effects—A validation study of the advanced propagation model (APM) for mobile radio applications,” IEEE Trans. Antennas Propag., vol. 54, no. 10, pp. 2869–2877, Oct. 2006. [23] D. Dockery and J. R. Kuttler, “An improved impedance-boundary algorithm for Fourier split-step solutions of the parabolic wave equation,” IEEE Trans. Antennas Propag., vol. 44, no. 12, pp. 1592–1599, Dec. 1996. [24] D. J. Donohue and J. R. Kuttler, “Propagation modeling over terrain using the parabolic wave equation,” IEEE Trans. Antennas Propag., vol. 48, no. 2, pp. 260–277, Feb. 2000.

REFERENCES [1] D. Chizhik and J. Ling, “Propagation over clutter: Physical stochastic model,” IEEE Trans. Antennas Propag., vol. 56, no. 4, pp. 1071–1077, April 2008. [2] M. Hata, “Empirical formula for propagation loss in land mobile radio services,” IEEE Trans. Veh. Tech., vol. 29, no. 3, pp. 317–325, Aug. 1980. [3] Y. Okumura, E. Ohmori, T. Kawano, and K. Fukuda, “Field strength and its variability in VHF and UHF land-mobile radio service,” Rev. Elec. Com. Lab., vol. 16, pp. 825–873, 1968. [4] COST Action 231, “Digital Mobile Radio Towards Future Generation Systems, Final Report,” European Communities, EUR 18957, 1999, technical report. [5] V. Erceg, L. J. Greenstein, S. Y. Tjandra, S. R. Parkoff, A. Gupta, B. Kulic, A. A. Julius, and R. Bianchi, “An empirically based path loss model for wireless channels in suburban environments,” IEEE J. Sel. Areas Commun., vol. 17, no. 7, pp. 1205–1211, Jul. 1999. [6] 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. [7] L. Piazzi and H. L. Bertoni, “Effect of terrain on path loss in urban environments for wireless applications,” IEEE Trans. Antennas Propag., vol. 46, pp. 1138–1147, Aug. 1998. [8] J. H. Whitteker, “Physical optics and field-strength predictions for wireless systems,” IEEE J. Sel. Areas Commun., vol. 20, no. 3, pp. 515–522, Apr. 2002. [9] M. Levy, Parabolic Equation Methods for Electromagnetic Wave Propagation. Stevenage, U.K.: IEE, 2000, p. 353. [10] N. Blaunstein, D. Katz, D. Censor, A. Freedman, I. Matityahu, and I. Gur-Arie, “Prediction of loss characteristics in built-up areas with various buildings’ overlay profiles,” IEEE Antennas Propag. Mag., vol. 43, no. 6, pp. 181–191, Dec. 2001. [11] N. Blaunstein and J. B. Andersen, Multipath Phenomena in Cellular Networks. Norwood, MA: Artech House, 2002. [12] L. Piazzi and H. L. Bertoni, “Effect of terrain on path loss in urban environments for wireless applications,” IEEE Trans. Antennas Propag., vol. 46, no. 8, pp. 1138–1147.

Dmitry Chizhik received the Ph.D. degree in electrophysics at the Polytechnic University, Brooklyn, NY. His thesis work has been in ultrasonics and non-destructive evaluation. He joined the Naval Undersea Warfare Center, New London, CT, where he did research in scattering from ocean floor, geoacoustic modeling of porous media and shallow water acoustic propagation. In 1996, he joined Bell Laboratories, Holmdel, NJ, working on radio propagation modeling and measurements, using deterministic and statistical techniques. He has worked on measurement, modeling and channel estimation of MIMO channels. The results are used both for determination of channel-imposed bounds on channel capacity, system performance, as well as for optimal antenna array design. His recent work has included system and link simulations of satellite and femto cell radio communications that included all aspects of the physical layer. His research interests are in acoustic and electromagnetic wave propagation, signal processing, communications, radar, sonar, medical imaging.

Lawrence Drabeck received the Ph.D. degree in physics from the University of California Los Angeles. He joined Bell Laboratories, Holmdel, NJ, in 1992 where his initial work was focused on RF properties and potential wireless applications of high-temperature superconductors. He has also worked on next generation radio front ends, interference modeling and smart antennas. He is now part of the Bell Labs E2E Wireless Networking Group where he works on real time network monitoring and optimization.

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W. Michael MacDonald received the B.S. degree in physics from The University of Notre Dame, New York, in 1979, and the M.S. and Ph.D. degrees in physics from the University of Illinois at Urbana-Champaign, in 1980 and 1984, respectively. He joined the research staff at AT&T Bell Laboratories, Holmdel, NJ, in 1984 and has been with Bell-Labs since then. He has worked on optical as well as wireless communications topics.

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2021

Measurement Analysis of Amplitude Scintillation for Terrestrial Line-of-Sight Links at 42 GHz Michael Cheffena

Abstract—Tropospheric scintillation is a rapid fluctuation of the signal amplitude and phase due to irregularities in the refractive index caused by atmospheric turbulence. If not considered properly, amplitude scintillation may affect the performance of a lowpower margin communication system. In this paper we study the amplitude scintillation observed on three converging terrestrial 42 GHz links. Probability density functions of amplitude scintillation for different seasons of a year are presented. In addition, the cumulative distributions of the dry and wet amplitude scintillation are shown. The measured long-term probability density functions of the three links are compared with the Moulsley-Vilar model. The results show that for small values of amplitude scintillation the Moulsley-Vilar model fits well the measured statistics, but overestimates the statistics for large values of amplitude scintillation. Good agreements are found between the Gaussian model and the measured short-term amplitude scintillation distributions. In addition, the link diversity of amplitude scintillation over combinations of two and three links are presented. Furthermore, the monthly average wet component of refractivity has been calculated using meteorological measurements, and as expected it shows a peak during the summer time. Index Terms—Amplitude scintillation, fading, propagation.

I. INTRODUCTION S the demand for high-capacity data communication increases, the use of higher frequencies may be required due to congestion in the lower frequency bands. Millimeter-wave bands offer very large bandwidths compared to lower frequencies. However, millimeter-wave signals are more sensitive to propagation degradation due to rain, scintillation and vegetation, compared to lower-frequency systems. Ionospheric scintillation arises from random fluctuations of electron density of the ionosphere. While tropospheric scintillation (which is the topic of this paper) is a rapid fluctuation of the signal amplitude and phase due to irregularities in the refractive index. These refractive index variations are caused by atmospheric turbulence (small-scale, irregular air motions characterized by winds that vary in speed and direction). In the troposphere, there are two main regions in which turbulence is likely to be strong. In the lower part of the troposphere turbulence fluctuations produce mixing of the air resulting in vertical transport processes near the Earth’s surface. In clouds, where turbulence results from

A

Manuscript received August 21, 2009; revised December 11, 2009; accepted December 12, 2009. Date of publication March 29, 2010; date of current version June 03, 2010. The author is with the University Graduate Center (UNIK), N-2027 Kjeller, Norway (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.2010.2046869

the entrainment of air (the process by which the outer edge of the cloud mixes with dry air outside of the cloud) [1]. It has been shown that turbulence in cumulus clouds (especially fair-weather clouds) induces most of the scintillation effects observed on satellite links [1]. The scintillation effects observed on terrestrial links are mainly due to turbulence in the lower part of the atmosphere [2]. The amplitude of scintillation increases with increasing signal frequency, path length, and antenna beam width (decreasing antenna size). Under turbulent tropospheric conditions scintillation effects can be both deep and fast. In [3] fast peak-to-peak variations of 20 dB were observed in a clear-sky line-of-sight (LOS) link from a mountain to a valley. Amplitude scintillation can also disturb fade mitigation technique (FMT) control loops [4]. Thus, if not considered properly, amplitude scintillation can significantly affect the performance of a communication system. A number of studies on amplitude scintillation are reported in the literature, e.g., [1], [2], [5]–[12], most of them dealing with Earth-space communication. Studies of amplitude scintillation on terrestrial links are rather limited. However, knowledge of amplitude scintillation over millimeter wavelength terrestrial LOS links is important for detailed design of such links and for the development of comprehensive time series synthesizers for FMT calibration, see e.g., [13], [14]. In this paper we present statistical analysis of amplitude scintillation on three converging terrestrial links using available measurements at 42 GHz. A description of the measurement set-up is given in Sections II. Theoretical considerations of amplitude scintillation is discussed in Section III. Measurement results and discussions are presented in Section IV. Finally, conclusions are given in Section V.

II. MEASUREMENT SET-UP Measurements of both signal and meteorological data were performed for a star-like network around Kjeller, Norway. The signal measurements were taken from five links star-like network; see Fig. 1. In this paper data from three of the links have been used, as these link’s data consist of a large portion concurrent time series measurements. The transmitter (base station) was located at Kjeller (59.58 N and 11.02 E), about 20 kilometer northeast of Oslo and had an antenna beam width of about 64 in azimuth and 10 in elevation. The receiver locations were at Ahus, Rælingen and Lillestrøm. The path lengths, elevation angles and angles measured clockwise from the north are listed

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receiver locations as well as at the base station, however rain gauges at receiver sites except Lilllestrøm gave more an indication of rainfall rather than an exact rain rate measurement. The other meteorological measurements were made at the base station located at Kjeller. The measurement was performed from the period September 1997 up to September 2001. A complete description of the measurements can be found in [15]. III. THEORETICAL CONSIDERATIONS

Fig. 1. Links in the star-like network around Kjeller, Norway [15].

TABLE I PATH LENGTH AND ANGLES

in Table I. The antenna beam width of the receiver antennas was about 1.7 . The base station transmits 100 mW power at 42 GHz with vertical polarization using QPSK modulation. At each receiver point the 42 GHz signal is converted down to intermediate frequency (1.5 MHz) in the outdoor unit, and then transmitted via coaxial cable to the indoor unit. The indoor unit is an integrated radio receiver and decoder (Philips set top box), which has been modified to make it possible to record the automatic gain control (AGC). The AGC is sampled once per second and converted to signal strength. The meteorological measurements included precipitation, air temperature, pressure, humidity, and wind force and direction. The precipitation was measured by two different methods: drop size distribution, and accumulated rain depth. The latter simply collects and measures the accumulated amount of water. Some oil and a non-freezing liquid were added to prevent evaporation and freezing of water in the buckets. All meteorological data were collected with a one-minute sampling interval, except the drop size data which were collected every 10 seconds. The precipitation and temperature information were collected at all

Atmospheric turbulence is a feature of fluid flows that occurs at very high Reynolds number value (a measure of the ratio of inertial to viscous forces), and are highly irregular and chaotic. Turbulence carries out an intensive mixing of the fluid and then a homogenization of its physical features such as temperature, humidity, etc., which are caused by random movements of fluid parcels, called turbulent eddies [1], [5], [16]. Atmospheric turbulences affects the propagation of radio waves by creating random variations of the refractive index (which are the result of temperature, humidity and pressure irregularities) that gives amplitude and phase variations of the received signal. A scintillation event can be described by the log-amplitude variation (in dB). It is the ratio of the instantaneous amplitude to the mean amplitude obtained by high-pass filtering the measured time series in dB. This technique allows scintillation events to be separated from any long-term instability of the receiver or attenuation due to hydrometeors. Under conditions of constant scintillation intensity (up to about 10 minutes), the short-term probability density function (PDF) of amplitude scintillation follows a Gaussian distribution [2], [5]. However, due to changing tropospheric conditions the scintillation intensity varies with time. Moulsley-Vilar [7] proposed a model for the long-term distribution of amplitude scintillation. In the model the scintillation log-amplitude PDF, , is considered to be conditionally Gaussian. While the scintillation intensity PDF, , is log-normal distributed , of ampli(over a period of a month). The long-term PDF, tude scintillation can then be expressed as [7]

(1) where

(2)

(3) Substituting (2) and (3) in (1) yields

(4)

CHEFFENA: MEASUREMENT ANALYSIS OF AMPLITUDE SCINTILLATION FOR TERRESTRIAL LOS LINKS AT 42 GHz

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where

and are the mean and standard deviation of , and . Analytical approximation of (4) is reported in [8]. can be exFor LOS paths, the scintillation variance pressed as [1]

(5) is the wave-number, (with and is the frequency), is the structure constant, is the path length and is the turbulence outer scale which varies between 10 and 100 meter [5]. Equation (5) is valid for point receivers only. For receivers with directional antenna, is reduced from its point receiver value by a factor defined as [9] where

Fig. 2. Average power spectral density of 57 rain events from the Ahus link.

IV. MEASUREMENT ANALYSIS (6) and are the diameter and where illumination efficiency of the antenna. The structure constant is the measure of the refractive index inhomogeneity due to turbulence, and is given by [7]

(7) is the difference between the refractive where (caused by temperature, humidity index at some time and and pressure variations), and is the wind velocity. In general, varies between to m [18]. In [10], Karasawa et al. developed a prediction model for based on link and meteorological parameters, and is defined as

(8) where is the frequency in GHz, is the elevation angle in is antenna aperture averaging factor defined in degrees, is the wet component of refractivity. can be (6), and determined from meteorological parameters such as the mean and mean relative humidity (%), given by temperature [19]

(9) where vapor pressure. and riod of a month.

is the saturated water should at least be averaged over a pe-

A. Long and Short-Term Distributions Using a one year measurement data from 1998 having the largest simultaneous available data from the three links, we studied the long-term distribution of amplitude scintillation. Spectra analyses of the measured data were first performed in order to find the cut-off frequency of a high-pass filter to be used to separate amplitude scintillation effects from attenuation due to precipitation. Simultaneous signal strength and rain rate measurements as well as visual inspection were used to identify periods of precipitation events, see [20] for more information on the event identification process. Fig. 2 shows the average power spectral density (PSD) of 57 rain events identified from the Ahus link. We can observe that on the average the theoretdB/decade slope [21] due to rain is valid up to about ical 0.01–0.03 Hz, above which amplitude scintillation and rain attenuation effects are superposed and eventually separated. Amplitude scintillation effects can be obtained by filtering the original time series using a high-pass filter. Choosing a cut-off frequency of 0.01 Hz may bring some rain power into scintillation. On the other hand, a cut-off frequency of 0.03 Hz might be a conservative choice (could reduce the power of scintillation obtained in the analysis). For a better trade-off, we used a cut-off frequency of 0.02 Hz, i.e., cut-off frequency Hz. Figs. 3(a) to (c) show the long-term probability density functions (PDFs) of combined dry (clear-sky) and wet (including periods of rain, sleet and snow) scintillation log-amplitude for Lillestrøm, Rælingen and Ahus links for different seasons i.e., summer (June–August), fall (September–November), winter (December–February) and spring (March–May). The seasonal dependency of amplitude scintillation is evident from Figs. 3(a) to (c). For all links the scintillation effects are higher during summer compared to the other seasons. This is due to increase of temperature and humidity during the summer time.

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Fig. 3. PDFs of amplitude scintillation for different seasons of the three links. Summer (June–August), fall (September–November), winter (December–February) and spring (March–May). (a) Lillestrøm link, (b) Rælingen link. (c) Ahus link.

Generally, the scintillation effects on these terrestrial links are not as severe as the ones observed in low-elevation Earth-space links. For example, for the low elevation angle Earth-space path

Fig. 4. Comparisons of the Moulsley-Vilar model and the measured long-term PDFs of amplitude scintillation of the three links. (a) Lillestrøm link, (b) Rælingen link. (c) Ahus link.

at 11 GHz reported in [8] with a path length of about 15.1 km (estimated using the rain height of the ITU-R P. 839-3 [22]), a signal enhancement (positive log-amplitude) of 2.1, 2.9, and 3.9 dB occurred for 0.1, 0.01 and 0.001 percent of the time,

CHEFFENA: MEASUREMENT ANALYSIS OF AMPLITUDE SCINTILLATION FOR TERRESTRIAL LOS LINKS AT 42 GHz

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TABLE II MOULSLEY-VILAR MODEL PARAMETERS

for the same low elevation angle Earth-space path [8], a fade depth (negative log-amplitude) of 2.3, 3.7 and 6 dB occurred for 0.1, 0.01 and 0.001 percent of the time, respectively. For the same time percent, a corresponding fade depth of 1.6, 2.2, and 2.9 for Lillestrøm, 1.3, 2.3, and 3.1 for Rælingen and 1, 1.3, and 2.2 for Ahus links are observed during the summer season. This might be explained by the fact that scintillation effects on terrestrial links are mainly due to turbulence in the lower part of the atmosphere [2] with clouds well above the path. For satellite links the scintillation effects are largely due to turbulence both in the lower part of the atmosphere and in clouds [23]. Moreover, the path through turbulence is longer for a low elevation angle satellite link than for a terrestrial link. Comparisons of the long-term PDF of the Moulsley-Vilar model given in (4) and the measured amplitude scintillation which includes both dry and wet periods for Lillestrøm, Rælingen and Ahus links are shown in Figs. 4(a) to (c). Table II shows parameters of the Moulsley-Vilar model and which are obtained by fitting the model to the measured long-term PDFs of amplitude scintillation. We can observe from Fig. 4(a) to (c) that for small log-amplitude values the Moulsley-Vilar model fits well the measured PDFs, but overestimates the statistics for large log-amplitude values. Furthermore, as discussed above under conditions of constant scintillation intensity (up to about 10 minutes), the short-term amplitude scintillation follows a Gaussian distribution [2], [5]. To verify this, Figs. 5(a) to (c) show the cumulative distributions of the Gaussian model and measured short-term amplitude scintillation for Lillestrøm, Rælingen and Ahus links where straight lines represent Gaussian distributions. We can observe from Fig. 5(a) to 5(c) that there is a good agreement between the Gaussian and the measured short-term distributions. B. Dry and Wet Amplitude Scintillation

Fig. 5. Probability plot: comparisons of the Gaussian model and measured short-term (10 minutes) amplitude scintillation of the three links. (a) Lillestrøm link, (b) Rælingen link, (c) Ahus link.

The complementary cumulative distributions (CCDFs) of enhancement (positive log-amplitude) and fade depth (negative log-amplitude) of dry (clear-sky) and wet (including periods of rain, sleet and snow) amplitude scintillation for the Lillestrøm, Rælingen and Ahus links are shown in Figs. 6(a) to (c). We can note that the effect of wet scintillation is large compared to the dry one for all three links both in case of enhancement and fade depth. This might be due to increasing atmospheric turbulence during precipitation events [24]. C. Link Diversity

respectively. While for the same time percent, a corresponding signal enhancement of 1.5, 2 and 2.8 dB for Lillestrøm (path length of 2.4 km), 1.2, 2.1 and 3.1 dB for Rælingen (path length of 4.9 km) and 1, 1.3 and 2.4 dB for Ahus (path length of 5.6 km) links are observed during the summer season. On the other hand,

Using concurrent amplitude scintillation time series of the three links, we studied the link diversity over combination of two and three links using selection combining technique. The amplitude scintillation on the diversity link is given by where for is the

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Fig. 7. PDFs of single and diversity links.

Fig. 8. Monthly measured standard deviation of amplitude scintillation for the year 1998 for Lillestrøm, Rælingen, and Ahus links.

the single link which is more or less symmetrical. We can also observe that the diversity obtained over combination of three links is more skewed to the right than the one obtained from two links. This is because the probability of getting positive log-amplitude value at a given time increases with increasing number of links. In addition, for 0.01 percent of the time a diversity gain of 0.41 and 0.49 dB were obtained from combinations of two and three links, respectively. D. Meteorological Dependence

Fig. 6. CCDFs of the dry and wet amplitude scintillation of the three links. (a) Lillestrøm link, (b) Rælingen link, (c) Ahus link.

amplitude scintillation on link . Fig. 7 shows the PDFs of amplitude scintillation over single (Lillestrøm) and diversity links. We can observe that the PDFs of the diversity links are skewed to the right (to positive log-amplitude values) comparing to the PDF of

As seen from (5) the intensity of amplitude scintillation rewhich lates directly to the refractive index structure constant in turn depends on the wind velocity and variation of the refractive index . As mentioned above the irregularities of is due to temperature, humidity and pressure variations caused by atmospheric turbulence. This suggest that amplitude scintillation is expected to exhibit diurnal, seasonal and geographical variations. Fig. 8 shows monthly measured standard deviation of amplitude scintillation for the year 1998 for Lillestrøm, Rælingen, and Ahus links. We can observe from Fig. 8 that the amplitude scintillation standard deviation is highest during

CHEFFENA: MEASUREMENT ANALYSIS OF AMPLITUDE SCINTILLATION FOR TERRESTRIAL LOS LINKS AT 42 GHz

Fig. 9. Predicted monthly average wet component of refractivity for the year 1998 using (9) with measured input metrological parameters: average temperature and relative humidity obtained from Table III.

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long-term PDFs of the three links are compared with the Moulsley-Vilar model. The results show that for small values of amplitude scintillation the Moulsley-Vilar model fits well the measured PDFs, but overestimates the statistics for large values of amplitude scintillation. Good agreements are found between the Gaussian model and the measured short-term amplitude scintillation distributions. In addition, the link diversity of amplitude scintillation over combinations of two and three links are presented. Meteorological measurements have been used to calculate the monthly average wet component of refractivity which shows a peak during the summer period, as expected. In general, the amplitude scintillation effects observed on these terrestrial links are not as severe as the ones observed in low-elevation Earth-space links. But still they have to be taken into account for detailed design of millimeter wavelength terrestrial LOS links and for the development of comprehensive time series synthesizers for FMT calibration. ACKNOWLEDGMENT

TABLE III MONTHLY AVERAGE METEOROLOGICAL DATA FOR THE YEAR 1998

The author would like to thank Telenor GBDR for providing measurement data which resulted from the EU projects CRABS and EMBRACE. The author would also like to thank Dr. T. Tjelta of Telenor GBDR for his very useful comments. REFERENCES

the summer time. In addition, Table III shows the monthly average temperature, relative humidity and wind velocity for the year 1998 measured at the base station located at Kjeller. Fig. 9 shows the corresponding monthly average wet component of recalculated using (9), with input metrological pafractivity rameters and obtained from Table III. We can observe form Fig. 9 that the wet component of refractivity has a peak in July month and is largest during the summer season which is consistent with the results found in Figs. 3(a) to (c) (with largest amplitude scintillation effects during the summer time) and Fig. 8.

V. CONCLUSION In this paper we present measurement results of amplitude scintillation observed on three converging terrestrial links at 42 GHz. The PDFs of amplitude scintillations for the different seasons of a year are presented. In addition, the cumulative distributions of the dry and wet amplitude scintillation are shown. The

[1] “Radiowave Propagation Effects on Next Generation Fixed Service Terrestrial Telcommunications Systems,” Final rep., 1996, COST 235, ESBN 92- 827-8023-6. [2] H. Vasseur and D. Vanhoenacker, “Characterization and modelling of turbulence induced scintillation on millimetre-wave line-of-sight links,” in Proc. Inst. Elect. Eng. Int. Conf. Antennas Propag., Eindhoven, Netherlands, Apr. 4–7, 1995, vol. 2, no. 407, pp. 292–295. [3] A. Shukla and R. Harrod, “High temporal-resolution channel characteristics at 40 GHz from a 30 km slant path radiowave propagation experiment,” in Proc. Inst. Elect. Eng. Proc. 11th Int. Conf. Antennas Propag., Colorado, Apr. 2001, vol. 2, no. 480, pp. 839–843. [4] A. Bolea-Alamanac, M. Bousquet, L. Castanet, and M. Van de Kamp, “Implementation of short-term prediction models in fade mitigation techniques control loops,” presented at the Joint COST 272/280 Workshop, Noordwijk, The Netherlands, May 26–28, 2003. [5] A. Ishimaru, Wave Propagation and Scattering in Random Media. New York: Academic, 1978, vol. 1–2. [6] R. S. Cole, K. Ho, and N. Mavrokoukoulakis, “The effect of outer scale of turbulence and wavelength on scintillation fading at millimeter wavelegths,” IEEE Trans. Antennas Propag., vol. 26, no. 5, pp. 712–715, Sep. 1978. [7] T. J. Moulsley and E. Vilar, “Experimental and theoretical statistics of microwave amplitude scintillations on satellite down-links,” IEEE Trans. Antennas Propag., vol. 3, pp. AP–30, 1982. [8] O. Paul and E. Vilar, “Measurement and modeling of amplitude scintillation on low-elevation Earth-space path and impact on communication systems,” IEEE Trans. Commun., vol. 34, no. 8, pp. 774–780, Aug. 1986. [9] J. Haddon and E. Vilar, “Scattering induced microwave scintillations from clear air and rain on earth space paths and the influence of antenna aperture,” IEEE Trans. Antennas Propag., vol. 34, no. 5, pp. 646–657, May 1986. [10] Y. Karasawa, M. Yamada, and J. E. Allnutt, “A new prediction method for troposheric scintillation on earth-space paths,” IEEE Trans. Antennas Propag., vol. 36, no. 11, pp. 1608–1614, 1988. [11] H. Vasseur, “Prediction of tropospheric scintillation on satellite links from radiosonde data,” IEEE Trans. Antennas Propag., vol. 47, no. 2, pp. 293–301, Feb. 1995. [12] C. Catalán and E. Vilar, “Simultaneous analysis of downlink beacon dynamics and sky brightness temperature-part II: Extraction of amplitude scintillations,” IEEE Trans. Antennas Propag., vol. 50, no. 4, pp. 535–544, 2002.

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[13] M. Cheffena, L. E. Bråten, and T. Tjelta, “Time dynamic channel model for broadband fixed wireless access systems,” presented at the 3rd Int. COST 280 Workshop, Prague, Czech Republic, Jun. 6–7, 2005. [14] M. Cheffena, L. E. Bråten, T. Tjelta, and T. Ekman, “Time dynamic channel model for broadband fixed wireless access systems,” presented at the IST Mobile and Wireless Summit, Myconos, Greece, Jun. 4–8, 2006. [15] O. Grøndalen, Ed., “Interactive broadband technology trials,” AC215 CRABS, Deliverable D4P1 Jan. 25, 1999 [Online]. Available: http:// www.telenor.no/fou/prosjekter/crabs [16] V. I. Tatarskii, Wave Propagation in a Turbulent Medium. New York: McGraw-Hill, 1967. [17] G. Brussaard and P. A. Watson, Atmosheric Modelling and Millimetre Wave Propagation. New York: Chapman & Hall, 1995, ISBN: 0-41256230-8. [18] J. M. Warnock, T. E. Vanzandt, and J. L. Green, “A statistical model to estimate mean values of parameters of turbulence in the free atmosphere,” presented at the 7th Symp. of Turbulence Diffusion, 1985. [19] The Radio Refractive Index: Its Formula and Refractivity Data. Geneva, 2003, ITU-R P.453-9. [20] M. Cheffena, L. E. Bråten, and T. Ekman, “On the space-time variations of rain attenuation,” IEEE Trans. Antennas Propag., vol. 57, no. 6, pp. 1171–1782, 2009. [21] E. Matricciani, “Physical-mathematical model of the dynamic of rain attenuation with application to power spectrum,” Electron. Lett., vol. 30, pp. 522–524, 1994.

[22] Rain Height Model for Prediction Methods. Geneva, 2001, ITU-R P.839-3. [23] K. H. Craig, Ed., “Propagation planning procedures for LMDS,” AC215 CRABS, Deliverable D3P1b Jan. 1999 [Online]. Available: http://www.telenor.no/fou/prosjekter/crabs, Available [24] E. Matricciani, M. Mauri, and C. Riva, “Relationship between scintillation and rain attenuation at 19.77 GHz,” Radio Sci., vol. 31, pp. 273–279, Mar. 1996.

Michael Cheffena was born in Asmara, Eritrea, in 1977. He received the M.Sc. degree in electronics and computer technology from University of Oslo, Norway, in 2005, and Ph.D. degree from the Norwegian University of Science and Technology (NTNU), Trondheim, Norway, in 2008. For one year he was a Visiting Researcher at the Communications Research Centre Canada (CRC) in Canada. Currently he is a Postdoctoral Fellow at the University Graduate Center (UNIK), Kjeller, Norway. His research interests include modeling and prediction of radio channels for both terrestrial and satellite links.

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A Study of Anomalous Propagation in Persian Gulf Asrar U. H. Sheikh, Fellow, IEEE, Pervez Z. Khan, and Saud A. Al-Semari, Member, IEEE

Abstract—Problem of anomalous propagation with regard to radio broadcasting in the countries surrounding the Persian Gulf is studied in this paper. The extreme conditions of humidity, temperature, water vapor and atmospheric pressure result in anomalous propagation conditions that disturbs the normal mode of radio propagation. The daily and seasonal changes in the meteorological conditions directly affect the carrier to interference plus noise ratio (CINR) in the coverage area with the consequence that the desired minimum CINR within the designated coverage area cannot be guaranteed. In this paper, we study these changes to the propagation conditions by analyzing the meteorological data over nearly three decades and develop a signal strength prediction model that integrates these conditions with the environmental features. The contour maps of refractive index and its gradient over the Persian Gulf Area are produced. To validate the developed model, signal propagation conditions were simulated and measurements made over an experimental link between Manamah in Bahrain and Dhahran in Saudi Arabia. Index Terms—Propagation, propagation measurements.

I. INTRODUCTION ADIO spectrum is a precious and non-renewable resource. International Telecommunication Union (ITU) regulates the usage of radio spectrum and to increase its utilization, frequencies are shared on spatial and/or temporal basis. The spatial frequency reuse is based on the signal attenuation rate with distance and the radio coverage is defined by the boundary within which the carrier to interference plus noise (CINR) is maintained above a minimum value. The same frequency is assigned for use to a distant station as long as it does not interfere with the other stations. Radio wave propagation loss is not only affected by the distance and the ground features but also by the prevailing atmospheric conditions like temperature, atmospheric and water vapor pressures. Variations in these parameters affect the atmospheric refractive index, which due to the land-sea interactions significantly vary particularly in the coastal areas. These variations produce abnormal conditions that result in what we call anomalous propagation. The anomalous propagation is serious when the desired carrier

R

Manuscript received April 13, 2009; revised September 29, 2009; accepted November 14, 2009. Date of publication March 01, 2010; date of current version published June 03, 2010. This work was supported in part by King Fahd University of Petroleum and Minerals (KFUPM), in part by the GCC Telecommunications Bureau, Bahrain, and in part by contracts CCCR2202 and CCCR2209. A. U. H. Sheikh and S. A. Al-Semari are with the Electrical Engineering Department, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia (e-mail: [email protected]; [email protected]). P. Z. Khan is with Center for Communications and IT Research, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia (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.2010.2044336

to noise ratio within the coverage area of a broadcasting station is disturbed by the signal originating from a distant station. The radio refractivity plays an important role in determining propagation modes, particularly for VHF, UHF, and microwave communication systems. A sufficiently high vertical gradient of the atmospheric refractive index causes the curvature of the propagation path to exceed the earth’s surface curvature thus enabling the radio waves to travel as if confined in a sort of a virtual tube. The confined space, called the duct, does not allow the signal to spread as would have been in the case of free space propagation, thereby reducing the propagation loss. The ITU publishes maps of refractive indexes and their gradients over different regions of the world. However, the maps over Gulf Co-operative Council (GCC) countries (Oman, UAE, Kuwait, Qatar, Bahrain and Saudi Arabia) are not of sufficient detail to predict the propagation loss for radio links within and between the GCC countries. The GCC Telecommunication Bureau initiated an anomalous propagation study program with two objectives. The first pertains to formulation of recommendations on a theoretical model in order to predict the impact of weather on radio propagation. The second objective is to collect propagation data at frequencies ranging from 30 MHz to 3 GHz and use the results to refine the developed model. The first objective is achieved by studying a large set of weather data and creating refractivity and refractivity gradient maps over the Gulf area and use these maps to determine the presence or absence of anomalous signal propagation conditions. The total propagation loss of a radio link is then found by integrating the empirical propagation loss models with refractivity gradient maps. Many studies on radio refractivity have been made around the world [1]–[4]. Bean and Dutton [1] reported synoptic radio climatology for the United States of America. Their studies on the design of ground-based microwave relay systems included the effects of climatology on the optimal separation between the terminals and of bending of radiowave in earth satellite links. A number of studies related to surface and upper air meteorology in particular refractivity have been reported for Dhahran, a Saudi Coastal city on the Persian Gulf and the authors found evidence of anomalous propagation [2]–[4]. Many studies have been reported on the prediction of radar coverage under anomalous conditions. Of these [5], weather radar [6], surface radar ducts [7], and statistical maritime duct estimation [8] and impact of air pollution on radar coverage [9], [10] relate to the Persian Gulf area. However, a comprehensive study on anomalous propagation applicable to radio broadcasting has not been made in the countries surrounding the Persian Gulf. Only some results are available for Dhahran where the average water vapor pressure at the sea surface has been measured to vary between 12 hPa in January to 25 hPa in September and the seasonally surface refractivity is found to vary between 320 and 355 N-units [3].

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TABLE I METEOROLOGICAL DATA USED FOR SIX STATES

Intense temperature and high humidity levels prevail in the GCC countries for about six months in a year. Under certain anomalous propagation conditions in the Gulf area, it is not uncommon to find an FM transmitter interfering at distances more than 600 km due to less propagation loss as the waves are trapped in so called ducts. To avoid the incidences of undesired levels of interference, the radio system designers must acquire tools to determine likelihood of anomalous propagation or unintentional interference. This paper addresses the efforts made towards the development of such design tool and its validation through experiments. In the next section, we present and discuss the results of our study on refractivity and its gradient. The sections that follow deal with the description of the development of a propagation model, the measurements and validation of the model, and conclusions of the study. II. REFRACTIVE INDEX AND REFRACTIVE GRADIENT MAPS IN THE GULF AREA As mentioned earlier, the radio refractivity gradient plays an important role in the appearance of anomalous propagation particularly in formation of propagation ducts. The radio refractivity -units is a function of temperature, pressure, and water vapor pressure and is defined as [11] (1) where is the atmospheric pressure (hPa), is the water vapor pressure (hPa), and is the absolute ambient temperature in Kelvin. The above expression is valid with less than 0.5% error for radio frequencies up to 100 GHz. A. Refractivity Gradient and Models When a horizontally stratified atmosphere is assumed, the gradient of radio refractivity is more important than the index itself. The refractivity gradient is a function of altitude and it is measured as the average gradient over a thickness or . The most commonly used values of are or alternatively over the over the first 100 m giving , is suitable for broadcasting first km. The first parameter, is used in ground to aircraft or stations using towers while ground to satellite links. In a simple linear model, the atmospheric refractivity is assumed to decrease linearly with height. at the sea level , the refracIf the ground refractivity is tivity , at altitude is given by

(2)

is the absolute value of the refractivity where gradient. The linear model is an approximation only in the first kilometer of the atmosphere and beyond this altitude, the refractivity decreases more slowly. A global average of is often used [11]. Other models have also been used [12], [13]. However, the real atmosphere seldom follows the reference model, which is usually based on the average behavior, therefore additional statistical measures are also used. We consider a stratified atmosphere, with layers , or displaying refractivity gradient less than with a possibility of becoming greater than positive. The chosen values differentiate between different propagation modes. B. Refractivity Data Analysis King Fahd University of Petroleum and Minerals (KFUPM) acquired the upper air meteorological data from the National Center for Atmospheric Research (NCAR) (http://www.ncar.ucar.edu), for Dhahran, Kuwait, Abu-Dhabi, Qatar, Bahrain, and Muscat, covering a period from January 1, 1973 to March 20, 2000 (approximately 27 years, see Table I for details). The refractivity analysis system (RAS) package [14], modified at KFUPM, is used in statistical analysis of the refractivity and its gradient. We analyzed the data for the six stations mentioned above at local times of 00:00 h and 12:00 h and height intervals of 50 and 100 m in two ranges 0–500 m and 500–1000 m respectively. The vertical refractivity profiles and gradients for above defined height intervals, on hourly, daily, monthly and yearly were used in determining the propagation modes supporting the atmosphere. The statistical analysis resulted in determination of percentage occurrences, refractivity index, refractivity gradients, modified refractivity profiles, and layer thickness for types of anomalous propagation such as super-refraction, sub-refraction, and ducting) and average refractivity for different times and locations. It is observed that the atmospheric pressure dominates and the average refractivity decreases with height [18]. Fig. 1 shows the daily percentage of time that surface super-refraction occurs. This percentage never exceeds 20% and Doha has the highest percentage. In Fig. 2 daily percentage occurrence of duct is shown. In this figure, Doha, Qatar have the highest percentage, which can exceed 50% of the time. The collected refractivity gradient data were analyzed for number of occurrences in a certain range of refractivity gradients and their percentage cumulative frequencies. Figs. 3 and 4 show the results of the analysis for Dhahran at 00:00 h and 12:00 h during the June to October period. From the figures, it is clear that the refractive gradient varies significantly with time. A significant change in the

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Fig. 4. Variability of refractive gradient for Dhahran at 12:00 h. Fig. 1. Daily percentage of occurrence of surface super refraction for all sites.

Fig. 2. Daily percentage of time occurrence of ducts for all stations.

Fig. 3. Variability of refractivity gradient for Dhahran at 00:00 h.

number of occurrence of a certain refractivity gradient in 12 h is indicated. It is shown that chances of formation of duct are much higher at 00:00 h as compared to noon time in the month of September. On the average August is the worst month for duct formation. No attempt was made to fit a statistical model to the time variability. This is a topic for further research. III. ANOMALOUS PROPAGATION MODEL A set of favorable meteorological conditions described earlier results in anomalous propagation conditions and formation of a duct. A propagation duct traps the signal, which due to lower

propagation loss within the duct, is able to propagate over longer distances. The propagation loss during the passage of the signal depends on factors like the length and thickness of the duct as well angles at which the signal enters into or exits from the duct. To include the effect of the duct in the model, the presence of a duct must be confirmed. For this purpose, we use refractive gradient maps over the area under consideration to determine the presence, the location, beginning, and the end of the duct in addition to its thickness. In addition to vertical variability, horizontal homogeneity of the atmosphere must be examined. The distance, over which a horizontal homogeneity is assumed, depends on the prevailing length of stratified region as well as on the terrain profile along the path. The dynamically changing atmospheric condition should be taken into consideration. However, the instantaneous changes in the weather conditions may be difficult to acquire. Generally, the worst month scenario is used to overcome this difficulty. Thus, the impact of the duct, measured in terms of a correction factor to the total propagation loss, is determined by applying the refractivity gradient to the part of the link over which the presence of the duct is confirmed. It may also be instructive to include some measure of variability of the meteorological conditions using a statistical model. The results on seasonal averages show that June, July, and August are the most critical months for the refractivity and refractivity gradient conditions that lead to duct formation. For the entire Persian Gulf area, refractivity gradient maps that use the average values of refractivity gradients are produced covering the entire region at local times of 00.00 and 12:00 h at heights of 100 and 1000 m above sea level. Locations of the six stations are shown with area map in the background. The average refractivity gradient contours drawn using the 27 years’ data at 00:00 h (local time) at a height of 100 m in June is shown in Fig. 5. The important point is to locate the refractive , which is indicative of the index gradient of presence of anomalous propagation. It is seen that the refractive contour passes over Muscat index gradient in the east and dipping towards Doha, moving West passing north of Bahrain and Dhahran and terminating nearly midway contour between Kuwait and Dammam. The indicates the presence of a duct, which is likely to impact the Kuwait links. It should be noted that these contours are based on the monthly average values of refractivity gradient. To apply

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Fig. 7. Average refractive index gradient map – July 00:00 h. Fig. 5. Average refractive index gradient map – June 00:00 h.

Fig. 8. Average refractive index gradient map – July 12:00 h. Fig. 6. Average refractivity gradient map – June 12:00 h.

to a certain time, we need to use either statistical distribution which will provide the probability of the presence of anomalous propagation or the current weather conditions, if available. The average refractivity contours for the month of June at 12:00 h local time shown in Fig. 6 are significantly different from those prevailing 12 h ago. The refractive index gradient of starts at latitude 30 and longitude 60 . The contour has tilted in more north-South direction from East contour passes over – West direction. The Dhahran. The contours are more compressed in the vicinity of Dhahran indicating a faster rate of change in refractivity gradient. Kuwait-Dhahran, Kuwait-Bahrain, Kuwait-Doha, and Dhahran-Doha links are likely to be affected. The refractivity maps for the month of July at 00:00 h local time is shown in Fig. 7. The gradient map has moved north located between Dhahran and Kuwait but closer to latter. The radio link between Bahrain and Kuwait as well as between Dhahran and Kuwait are likely to get affected. Fig. 8 shows the averaged refractivity gradient map at 12:00 conh local time for the month of July. The tour has moved east towards Dhahran and Bahrain and its shape is quite distinct from that for July 00:00 h. The Doha-Kuwait, Dhahran-Kuwait and Bahrain-Kuwait links are affected. Such a change is attributed to increase in temperature and humidity. Fig. 9 shows the averaged refractivity gradient contours drawn for 00:00 h for the month of August. The refractive

Fig. 9. Average refractive index gradient contour map – August 00:00 h.

gradient of contour is similar to the one for July at 12:00 h. The contour for August at 00:00 h are more compressed between Kuwait and Dhahran. The contour of interest starts from south going north passing West of Dhahran. The Kuwait-Dhahran, Bahrain-Doha links are likely to be affected. Fig. 10 shows the averaged refractivity gradient contour at 12:00 h for the month of August. A significant change in the contour map from that at 00:00 h is noticed. The contour has moved east and runs practically north-south passing midway between Abu Dhabi and Muscat. The Abu Dhabi-Muscat is only link likely to be affected. The correction of the propagation loss is determined by using the model given in the ITUR P.542-10 recommendation [21],

SHEIKH et al.: A STUDY OF ANOMALOUS PROPAGATION IN PERSIAN GULF

Fig. 10. Average refractive index gradient map – August 12:00 h.

which states that the signal trapped in the duct attenuates linearly with distance rather than the square of the distance. Thus the signal attenuation is much lower than the free space propagation loss. The decrease in the propagation loss may be partly offset by an increase in the loss due to signal leakage but this does not offset the lower distance related attenuation. A. Implementation of the Ducting Effect in the Propagation Models To predict the radio wave propagation loss in the Persian Gulf region consisting of desolate desert tract and coastline that has land, sea and land-sea paths, we propose a propagation model, which is derived from a number of existing models found in ITU-R 452-10 [17] and ITU-R 1546 Recommendations [18]. The analysis of propagation conditions over twelve months period indicates that in addition to the basic propagation loss estimation procedures, existence of sub-refraction, super-refraction, and ducting must be determined. However, determination of the presence of a duct and propagation loss through it independent of the selection of a basic model. In this regard, we found ITU-R 452-10 Recommendation to be the most appropriate basic model. The ducting model is derived from the following considerations. 1) Implementation of ducting component from ITU-R 452-10 which is independent of the selection of the main model. This implies that we can integrate ducting component taken from ITU-R 452-10 into the ITU-R-1546 model. variability. 2) Include the percentage time parameter % 3) Include the local refractivity gradients with an option to import user-made refractivity gradient maps for the chosen Gulf region. 4) Include the compliance of Hadamard conditions to calculate the effective earth’s radius based on imported refractivity data. The ducting part has been added as an additional option similar to the one for rain given in ITU-R P.676 [19], and for gas in ITU-R P.840 [20]. For ducting a part of ITU-R P.542-10 is used. The new ducting model is able to automatically acquire the applicable refractivity gradient by interpolating the refractivity maps. When the ducting option is selected but ducting is not found, no correction is applied to the signal strength. If the path is larger than the area covered by the imported map, the

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differential gradient path value is calculated from both the adjacent values available on the refractivity map, and the default values of the earth radius in the propagation model box. The equivalent earth radius needed in the model is automatically calculated from the refractivity map. The refractivity map is digitized, geo-coded and integrated into ICS Telecom (a radio modeling and simulation package from ATDI, France http://www. atdi.com/icstelecom.php), for each point defined by its latitude and longitude [22], [23] along the path. The developed package has an option to import the user-defined refractivity maps as complementary refractivity gradient maps for integration into the model. These models, implemented in software, are compliant with Hadamard condition that ensures that the signal retains continuity along the path. The software package divides the link into a number of sections on the basis of the presence or absence of the duct. For all sections the parameter fields are filled with imported data and calculated data from the geo-climatic data information. For each section, the package tool reads a combined data file with parameters given below. 1) The length of each section; and with longitude and latitude at height ; 2) . 3) Surface refractivity Based upon these parameters, the presence of a duct is determined and the loss due to the duct is calculated. This loss is in addition to the propagation loss estimated along the propagation path. It has been observed that the developed propagation model does affect, as expected, the field strength computation and predicts the transmission loss by taking the anomalous propagation into account. Several simulations were made for a line-of-sight link and a link with intervening obstructions. In case of line-of-sight, zero loss due to ducting component was observed but in the presence of obstruction, the loss component due to ducting did appear and it was dependent upon the transmitter parameters like transmitter frequency, transmitter and receiver antenna heights, polarization, etc. IV. MEASUREMENTS AND VALIDATION OF THE DEVELOPED MODEL To validate the developed model, a point-to-point link was established with the transmitter located at Gulf Cooperation Council Telecommunications Bureau (CGCCTB) office and the receiver at Research Institute (RI) Building at KFUPM. Initially, the transmitter used two Logperiodic directional antennas (LPDA) to cover the frequency range 80 to 3000 MHz. The receiver used two omni directional antennas. However, during testing of the link it was discovered that the link is not functional at some frequencies. Consequently, the antennas at both ends were replaced with four LPDA and two dish antennae. This functionality is provided on both sides through the manual switching device. The six antennae provide the facility to cover the specified frequency range and two types of polarizations (horizontal and vertical). The received signal is connected to a spectrum analyzer, which converts the signal strength data into a digital format. The spectrum analyzer is interfaced via a General Purpose Interface Board (GPIB) card to a personal computer where the received data is recorded and

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TABLE II COMPARISON OF PREDICTED AND MEASURED SIGNAL STRENGTH AND PROPAGATION LOSS

Fig. 12. Measured field strength for the point-to-point link (over five days). Fig. 11. Experimental setup of point-to-point link between Bahrain and Dhahran.

analyzed. In order to be able to control setting up of frequency, polarization, and sweep time, reference levels, and other required parameters from the personal computer to the receiver, a graphical user interface was developed. This interface provides full control to the receiver for the selection of frequencies, polarization, sweep time, other required parameters, and text file handling and storage facilities. The received field strength is stored in text files. A number of tests were conducted prior to the launch of the study for observing the reception of signal at the receiver. A complete setup is shown in Fig. 11. A. Field Strength Measurements A number of frequencies in VHF, UHF and microwave range were assigned for the measurement. These frequencies were used with vertical and horizontal polarization. Hence, a total of 56 different combinations of transmitted signals (28 frequencies with horizontal or vertical polarization) were monitored during this measurement study. The recording of field strength data proceeded on a 24 hour basis. For each frequency, the field strength data are collected every 15 s and is recorded for at least an hour. The data files are annotated with value of the frequency, time, and type of polarization used. For analytical purposes, another utility was written to group the data, as per requirement,

on hourly, weekly, monthly, or even yearly basis for a particular type of polarization and frequency. The field strength measurement for low, medium and high range of frequencies were recorded for a period of six months. During this period, it was observed that the signal reception was strong at the lower end of the frequencies (30–600 MHz) but became weaker for frequencies above 700 MHz and the signal almost disappeared beyond 1760 MHz. At higher frequencies the propagation losses, as expected, were higher than those at the lower frequencies. Fig. 12 shows the measured field strength over the experimental Bahrain-Dhahran Link. It has been observed that the minimum power required for the frequencies greater than 400 MHz is more than 10 W. A test conducted with a transmitted power of 12 W is plotted for horizontal and vertical polarization in the range of frequencies between 80–600 MHz and is shown in Fig. 13. B. Model Validation The developed model is tested against the experimental results obtained over a radio link established between Bahrain and Dhahran. A selection of results is given in Table II. A total of 17 frequencies were used. The Bahrain-Dhahran Link was simulated as well as measured. The ICS Telecom Software was used in simulations to test the developed model. Formation of a duct was predicted by the integrated refractive index gradient map and its loss estimated. The transmitter powers of 11 and 12 Watts were used as indicated in the table. The predicted and the

SHEIKH et al.: A STUDY OF ANOMALOUS PROPAGATION IN PERSIAN GULF

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REFERENCES

Fig. 13. Measured field strength for vertical and horizontal polarizations.

actual measured results were compared. The predicted model overestimates the received signal power. The error varies between a minimum of 0.46 dB to a maximum of 9.00 dB. The average error is 4.77 dB and its standard deviation is 2.65 dB. The median error is 5.98 dB. The main reason for the discrepancy is thought to be due to the use of monthly averaged values in the simulations rather than the prevailing value of the refractive index gradient, which were not available for use in simulations. V. CONCLUSION This paper described the findings of the research conducted into the problem of anomalous propagation in the GCC countries. The surface and upper air data sets are used to compute the refractivity indexes and their gradient over the Persian Gulf Region. The statistical analysis of the refractivity and its gradient was performed using the refractivity analysis system (RAS) package. It also provides detailed statistical analysis of sub-refraction, super-refraction, and the presence or the absence of propagation ducts for different time and locations. RAS analysis has been conducted for all specified stations for 00:00 and 12:00 h (both local). The analysis is performed on monthly, yearly, hourly, as well as a seasonal basis. Refractivity and refractivity gradient maps were constructed using the overall average values of all the locations for 00:00 h and 12:00 h separately. The refractivity maps were drawn for all locations at two altitudes (100 m and 1000 m) from the surface at the site for 00:00 h and 12:00 h respectively. Several approaches were considered toward developing a suitable radio propagation model. A hybrid Approach using empirical and analytical results was used. The analytical portion of the model uses propagation loss models derived from the radio propagation principles. The introduction of the ducting effect included in ICS Telecom package is based on the ducting part defined in ITU-R P.452-10 Recommendation. Refractivity maps are integrated in the tool that import data in ASCII format. The link design package automatically includes the effect of duct if present. The developed model is also validated by simulations and real life measurements. The model will be further validated on the radio links between the six selected stations. ACKNOWLEDGMENT The authors would like to thank the GCC Telecommunications Bureau, Bahrain for granting permission to publish the preliminary findings.

[1] B. R. Bean and E. J. Dutton, “Radio-meteorological parameters and climatology,” Telecommunications, vol. 43, no. x, pp. 427–436, 1976. [2] T. O. Halawani and S. Rehman, “Variation of surface water vapor pressure and the refractivity over the Arabian Peninsula,” Arab. J. Sci. Eng., vol. 17, no. 3, pp. 371–386, 1992. [3] T. O. Halawani and P. Z. Khan, “Subrefraction occurrence at coastal and desert sites using synoptical meteorological data,” in Proc. 9th National Radio Science Conf., Cairo, Egypt, Feb. 18–20, 1992, pp. 1–7 F3. [4] T. Husain, T. O. Halawani, S. Rehman, C. E. Schemm, L. P. Manzi, and W. Acree, “Modeling radar coverage in the planetary boundary layer under anomalous propagation conditions,” Arab. J. Sci. Eng., Theme Issue: Commun., vol. 14, no. 4, pp. 599–607, 1989. [5] S. H. Abdul-Jauwad, P. Z. Khan, and T. O. Halwani, “Prediction of radar coverage under anomalous propagation condition for a typical coastal site: A case study,” Radio Sci., vol. 26, no. xx, pp. 909–919, 1991. [6] J. Beck, B. Codina, and J. Lorento, “Forecasting weather radar propagation conditions,” Meteorol. Atmos. Phys., vol. 96, no. xx, pp. 229–243, 2007. [7] I. M. Brooks, A. K. Goroch, and D. P. Rogers, “Observations over the Persian Gulf,” J. Appl. Meteorol., vol. 38, pp. 1293–1310, 1999. [8] C. Yardim, A. K. Gerstoft, and W. S. Hodgkiss, “Statistical maritime radar duct estimation using hybrid genetic algorithm-Markov chain Monte Carlo method,” Radio Sci., vol. 42, 2007, DOI:10.1029/ 2006RS003561. [9] T. O. Halawani, P. Z. Khan, and S. Rehman, “Effect of the Kuwaiti oil field fires on AP and radar coverage,” in Proc. Air Pollution Symp., Nov. 15–17, 1993, pp. 492–498. [10] T. O. Halawani, S. Rehman, and P. Z. Khan, “Air pollution impact on anomalous propagation and radar coverage,” Arab. J. Sci. Eng., Theme Issue: Air Sea Pollution, vol. 18, no. 2, pp. 143–156, 1993. [11] ITU, Handbook on Radio Meteorology, Radio-Meteorological Parameters and Climatology Geneva, Radio Communication Bureau, 1996. [12] B. R. Bean and G. D. Thayer, “Models of the atmospheric radio refractivite index,” Proc. IRE, vol. 47, pp. 740–755, 1959. [13] A. L. Beck, “New equations for computing vapour pressure and enhancement factor,” J. Appl. Meteor., vol. 20, pp. 1527–1532, 1981. [14] J. P. Skura, The Refraction Analysis System (RAS) The Johns Hopkins Univ., Applied Phys. Lab., Maryland, 1984, Report No. 42. [15] J. P. Skura, Anomalous Propagation and Radar Coverage: Anomalous Refraction in Saudi Arabia The Johns Hopkins Univ., Appl. Phys. Lab., Maryland, 1984, Report No. STD R-1131. [16] The Radio Refractive Index: Its Formula and Refractivity Data ITU, 2005, Recommendation ITU-R P.453-6. [17] Prediction Procedure for the Evaluation of Microwave Interference Between Stations on the Surface of the Earth at Frequencies Above About 0.7 GHz ITU, 2001, Recommendation ITU-R P.452-10. [18] VHF and UHF Propagation Curves for the Frequency Range from 30 MHz to 1000 MHz Broadcasting Services ITU, 1998, Recommendation ITU-R P.1546. [19] Attenuation by Atmospheric Gases Broadcasting Services ITU, 1998, Recommendation ITU-R P.676. [20] Attenuation Due to Clouds and Fog Broadcasting Services ITU, 1998, Recommendation ITU-R P.840. [21] VHF and UHF Propagation Curves for the Frequency Range from 30 MHz to 1000 MHz Broadcasting Services ITU, 1998, Recommendation ITU-R P.542-10. [22] T. Martin, Implementation of the Propagation Model ITU-R P.370-7 in ICS-Telecom Software. Paris, France: ICS, 1999. [23] VHF and UHF Propagation Curves for the Frequency Range from 30 MHz to 1000 MHz Broadcasting Services ITU, 1995, Recommendation ITU-R P.370-7. [24] ITU, Handbook on Radiometerology Geneva, Radiocommunicaiton Bureau, ITU-R Study Group 3 (Radio Propagation), 1996. [25] P. Z. Khan, A. U. H. Sheikh, and A. Belghonaim, Technical Aspects of Radiowave Propagation Measurement Campaign in GCC Countries Center for Commun. Comput. Res., King Fahd Unive. Petroleum Minerals, 2005, Rep. PN CCCR2213. [26] Prediction Methods for the Terrestrial Land Mobile Service in the VHF and UHF Bands ITU, 1995, Recommendation ITU-R P.529-2.

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Asrar U. H. Sheikh (F’04) graduated from the University of Engineering and Technology, Lahore, Pakistan, with the first class honors degree in 1964 and the M.Sc. and Ph.D. degrees from the University of Birmingham, Birmingham, U.K., in 1966 and 1969, respectively. He held positions in Pakistan, Iran, U.K., and Libya before joining Carleton University, Ottawa, Canada, in 1981, first as an Associate Professor and later as a Professor and Associate Chairman for Graduate Studies. He was a Professor and Associate Head of the Department of Electronic and Information Engineering at the Hong Kong Polytechnic University, where he established the Wireless Information Systems Research (WISR) Centre. In March 2000, he joined King Fahd University of Petroleum and Minerals (KFUPM), Saudi Arabia, as the Bugshan/Bell Lab Chair in Telecommunications, the first chair professorship in Saudi Arabia. At KFUPM, he established Telecommunications Research Laboratory (TRL). He also established teaching laboratories in Iran and Libya. He held the position of the Rector, Foundation University Islamabad, Pakistan during his leave of absence from KFUPM last year. He is the author of a published book Wireless Communications – Theory & Techniques (Springer, 2004). He has published over 270 papers in international journals and conference proceedings. Dr. Sheikh has delivered several keynote speeches at international conferences. He also received several honors and awards including the 1984 Paul Adorian Premium from IERE (London), Teaching Achievement Awards in 1984 and 1986, the Research Achievement Award in 1994 (Carleton University), and Research Excellence Awards (College of Engineering Sciences, KFUPM) in 2005 and 2006. He has been member of Technical Program Committees of many international conferences and organized and chaired many technical sessions. He was an editor of the IEEE TRANSACTION ON WIRELESS COMMUNICATIONS (2003–2005), and was a Technical Associate Editor of the IEEE COMMUNICATION MAGAZINE (2000–2001). He is on the Editorial Board of Wireless Personal Communications, and Wireless Communications and Mobile Computing, the two highly rated international journals. He is listed in Marquis’s Who’s Who in the World and Who’s Who in Science and Engineering. He is a Fellow of the IET.

Pervez Z. Khan received the M.S. degree in electrical engineering from King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia, in 1984 and the B.S. degree in electrical engineering from Aligarh Muslim University, Aligarh, India, in 1980. He is currently working as a Research Engineer in the Center for Communications and Information Technology Research of King Fahd University of Petroleum and Minerals. His research interests include radio wave propagation, channel modeling, weather effects on wireless systems, Radio Frequency related measurements in microwave and mobile domain and radar performance assessment. Since 1986, he has been involved in applied research in the field of radiowave propagation and has conducted several projects for Saudi Arabia and GCC countries and has published in peer-journals.

Saud A. Al-Semari (M’96) received the B.S. and M.S. degrees in electrical engineering from King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia, in 1991 and 1992, respectively, and the Ph.D. degree from the University of Maryland at College Park, in 1995. He is currently a Vice Minister of the Saudi Government. He left KFUPM in late 2004. During his service at KFUPM, He served as the Director of the Communications and Computer Research Center and also the Director of the Information Technology Center. His research interests are in the field of wireless communication systems.

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Modeling Radio Transmission Loss in Curved, Branched and Rough-Walled Tunnels With the ADI-PE Method Richard Martelly, Member, IEEE, and Ramakrishna Janaswamy, Fellow, IEEE

Abstract—We discuss the use of the parabolic equation (PE) along with the alternate direction implicit (ADI) method in predicting the loss for three specialized tunnel cases: curved tunnels, branched tunnels, and rough-walled tunnels. This paper builds on previous work which discusses the use of the ADI-PE in modeling transmission loss in smooth, straight tunnels. For each specialized tunnel case, the ADI-PE formulation is presented along with necessary boundary conditions and tunnel geometry limitations. To complete the study, examples are presented where the ADI-PE numerical results for the curved and rough-walled tunnel are compared to known analytical models and experimental data, and the branched tunnel data is compared to the numerical solutions produced by HFSS. Index Terms—Alternate direction implicit (ADI), parabolic equation, radiowave propagation, tunnels.

I. INTRODUCTION HE alternate direction implicit (ADI) method coupled with the vector parabolic equation (PE) has previously been shown to model radiowave propagation in straight tunnels with smooth walls [1]. However, due to the rapid growth of telecommunication systems, different tunnel environments also need to be studied. Subway and underground road tunnels typically curve or branch out into side tunnels and have walls which are not smooth. These tunnel geometries are not always well described by analytical models and accurate numerical solutions become important. In real tunnels, it has been shown that, over a long distance, high order modes are heavily attenuated and low order modes dominate [3]. When tunnels are treated as imperfect waveguides, these fields represent waves which travel near the axis of propagation. The PE can accurately model low order modes for electrically large tunnels [1], [2]. The standard PE is an approximation of the Helmholtz to the axis of equation that assumes fields travel within

T

Manuscript received July 13, 2009; manuscript revised November 18, 2009; accepted January 11, 2010. Date of publication March 29, 2010; date of current version June 03, 2010. This work was supported by the Army Research Office under Grant ARO W911NF-04-1-0228. R. Martelly was with the Department of Electrical and Computer Engineering, University of Massachusetts, Amherst, MA 01003 USA. He is now with The MITRE Corporation, Bedford, MA 01730-1420 USA (e-mail: [email protected]; [email protected]). R. Janaswamy is with the Department of Electrical and Computer Engineering, University of Massachusetts, 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.2010.2046862

propagation. The PE lends itself to numerical discretization more easily than the Helmholtz equation but does not account for backscattered fields. Furthermore, the slow varying nature of low order modes implies the use of implicit finite difference (FDM) techniques where large discretizations along the axis of propagation are allowed. The Crank-Nicolson method is an unconditionally stable implicit FDM that has been traditionally used to solve for the vector PE. However, the Crank-Nicolson method can also become computationally intensive when dealing with fine meshes or when a large number of propagation steps is required [1]. The alternate direction implicit (ADI) technique is a modification of the Crank-Nicolson method that reduces computational labor by solving for the fields one dimension at a time [1], [6]. The truncation error introduced by the ADI modification is of the same order as the error already introduced by the Crank-Nicolson method. Previous work has shown, for modest discretizations, the ADI and Crank-Nicolson solutions are nearly identical [1]. In this paper, we use the vector PE, following the formulation of Popov [2], to solve for fields in specialized tunnel environments. For each case, we briefly discuss the ADI formulation as well as the boundary conditions used to characterize the tunnel wall. In Section II we discuss the curving tunnel and compare the ADI-PE results to known analytical approximations and published experimental data [7]. In Section III, we study the branch tunnel and compare our ADI-PE numerical results to the numerical results obtained using HFSS [13] for a smaller sample problem. Finally, in Section IV, we formulate a model for tunnels with surface roughness and compare our ADI-PE results with known, experimentally verified, analytical solutions. II. TUNNELS WITH SMOOTHLY CURVED AXIS A. Curved Tunnel Propagation Model Let us consider a tunnel with a curved axis. The geometry of a typical curved tunnel with a rectangular cross-section is shown is the in Fig. 1, where is the curved axis, or range, and range dependant radius of curvature. The vector PE was formulated by Popov and was shown to accurately model electromagnetic propagation in curved tunnels [2]. The vector PE for a tunnel with a smooth curve in the horitime dependence, zontal plane, as formulated in [2], (with where is frequency and is time) is

0018-926X/$26.00 © 2010 IEEE

(1)

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, where ence quotients are

and

. The differand and

(7) (8) Fig. 1. A typical tunnel with a curved axis.

where is the free space wave number, and is a vector function that is directly related to the transverse electric field. to the transverse electric field is given by The relationship of (2) and are the and components of the electric where field, respectively. The vector PE is formulated using asympand totic analysis, where it is assumed that (where is the wavelength), which means it is only valid for high frequency propagation and in tunnels with smooth axis of curvature. Along the tunnel wall, the impedance boundary condition is enforced and the transverse fields become coupled, as shown by (3):

(3) where and are the and components of the unit is the normalized surface normal vector at the boundary and impedance [2]. For a wall with relative permittivity and con(in S/m), we use the grazing angle approximation ductivity for surface impedance [1], [5]

, and become unity and (5) Note that when and (6) reduces to the ADI formulation for the straight axis PE shown in [1, equations (22) and (23)]. The ADI equations in (5), (6) represents a marching technique where the transverse electric field is solved step by step within the tunnel domain [2]. plane, the Starting with the known initial field at the fields of each successive plane is solved in consecutive order, at , until the field at the desired range is propagation steps of solved. B. Curved Tunnel Field Approximations Approximate analytical solutions describing fields in curved tunnels with rectangular cross sections and constant curvature radii are well known and discussed in [3], [8]. At high fre, and for large radii of curvature, , the quency, fields are best described in terms of Airy functions [3], [8]. As and components of the normal vector shown in (3), the vanishes alternatively on the vertical and horizontal walls in tunnels with rectangular cross sections. As a result, the combecome decoupled and can ponents of the vector function be solved independently. The horizontal and vertical polarizations can now be solved separately by enforcing the decoupled impedance boundary conditions on all four walls. Following the derivation shown in [2], the vertical component, , of the vector function is shown to be (keeping the same notation as [2]): (9) (10) (11)

(4) where , and , is the complex permittivity and relative conductivity, respectively. The discretizations , , and , along the , and axes are represented by respectively. A Peaceman-Rachford [6] ADI formulation of (1) can be summarized by

where and are the Airy functions of the first and second kind, is the propagation constant, , and are constant eigenvalues to be found from the boundary conditions, given by

(12) For large curvature radii, the vertically polarized fields are (5)

(13) (14)

(6)

where the integers and physically represent field variations along the and axes and specifies a possible mode. The mode

MARTELLY AND JANASWAMY: MODELING RADIO TRANSMISSION LOSS IN CURVED, BRANCHED AND ROUGH-WALLED TUNNELS

TABLE I ANALYTICAL AND NUMERICAL MAFS FOR THE CURVED TUNNEL WITH RECTANGULAR CROSS SECTION FOR 950 MHZ

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TABLE II ANALYTICAL AND NUMERICAL MAFS FOR THE CURVED TUNNEL WITH RECTANGULAR CROSS SECTION FOR 1.8 GHZ

attenuation per unit length can be found from the complex exponent, , to be [2]

(15) where

(16) . The part of (15) deand part describes small radii of curvature and the scribes large radii of curvature (or almost straight tunnels). Mahmoud and Wait [9] numerically calculated the attenuation factors for curved waveguides using more precise transcendental equations. As shown in [2, Fig. 3], the aysmptotic equation (15) is in very good agreement with Mahmoud and Wait’s numerical solutions. We will use the aysmptotic equation to validate the numerical simulations of the ADI-PE. C. Comparison of the ADI-PE to Analytical Solutions In this sub-section, we validate the ADI-PE with numerical examples. We consider a rectangular tunnel with dimensions of 8 m 4 m and walls with relative dielectric constant, , and conductivity, . We used the dominant mode of the vertically polarized field, as shown by (13) and (14), as our initial field. The discretizations used , and for the 950 MHz case were ; and the discretizations used for the 1.8 GHz , and . simulations were The mode attenuations per unit length, or mode attenuation factors (MAFs), from (15) and the ADI-PE simulations are tabulated in Tables I and II for different curvature radii. As shown in Tables I and II, the numerical MAFs closely follow the aysmptotic solutions over the range of curvature radii. Furthermore, as the curvature radii decreases, the number of reflections from the walls along the curved path increases and so does the loss. The percent error, defined by

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Fig. 2. Percent error of the MAFs of the curved waveguide for 950 MHz (dot, : . The percent error of solid) and 1.8 GHz (circle, solid) for s the MAFs for 950 MHz (dot,dashed) and 1.8 GHz (circle, dashed) for s : . The vertical lines show the region where t for 950 MHz (solid) and 1.8 GHz (dashed).

1 875

1 = 3 75

= 61

1 =

is shown in Fig. 2 as a function of , (where is the length of the diagonal of the rectangular cross section) for 950 MHz (dot, solid) and 1.8 GHz (circle, solid). The figure also shows the ) with dashed results for smaller discretizations ( curves to show that the solutions are convergent. The vertical for the 950 MHz lines in Fig. 2 show the region where (solid) and 1.8 GHz (dashed) cases, respectively. The parameter, , was selected from [2] because it is a wavelength dependent term that must be much less than unity to ensure accurate , is the Fresnel number and is one results. Furthermore, of the parameters which govern the diffraction processes in the waveguide [2]. As Fig. 2 shows, there is less than 5% error in MAFs over the entire range of considered; even when there are sharp bends and the condition, , is not satisfied. Finally, one of the important characteristics of the curved tunnel is the accumulation of the field near the concave wall at (the whispering gallery mode). Fig. 3(a) shows the dominant field as defined by (13) and (14) across the initial plane at 1.8 GHz. The whispering gallery mode feature can be seen in Fig. 3(b), where the field generated by the ADI-PE, at , has an accumulation along a range of 2000 m for the wall. D. Comparison of the ADI-PE to Ray Tracing Simulations In this section, we compare the ADI-PE simulations to the ray tracing simulations shown in Wang et al. for a curved waveguide [7]. The geometry of the waveguide (top view) is shown in Fig. 4 and is comprised of two straight sections and a curved section.

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Fig. 3. The jV at 1.8 GHz.

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j

mode at (a) s

= 0 m and at (b) s = 2000m for  = 2000m Fig. 5. Normalized received power from ray tracing (solid) and ADI-PE (dashed) along the axis of propagation for the straight tunnel.

Fig. 4. Geometry of the curved tunnel with straight sections.

The axial length of the waveguide is 400 m and the length of the curved section is 200 m with a radius of curvature of 300 m. The cross-sectional dimensions of the waveguide are 8 m 6 m. In [7], the transmitter and receivers are vertically polarized half-wave dipoles operating at a frequency of 1 GHz. The transmitter and receivers are located at the center of the waveguide at heights of 3 and 1.5 m. To establish a basis of comparison, we first look at the straight waveguide. Fig. 5 shows the normalized received power of the ray tracing simulations (solid) and the ADI-PE simulations (dashed). The ADI-PE is simulated using the far field expressions of a half-wave dipole in free space placed 30 m outside the entrance as the initial field. The field is tapered by a unit to minimize error Gaussian with standard deviation, associated with using incorrect field values at the walls [1]. The dielectric constant and conductivity of the waveguide walls are taken to be the typical values of 5 and 0.01 S/m, respectively. Simulations show that over the range of acceptable values of ), there is little change in the dielectric constants ( received power. The simulations are done with discretizations , and . As the figure of shows, the ADI-PE closely models the nulls in the received power generated from ray tracing simulations. The ADI-PE results are vertically offset so that the least squares fit line of the ADI-PE data and the experimental data (in the curved region) intersect at the start of the curved region (at 100 m). As discussed in [1] and [3], propagation in straight tunnels can be characterized by a near and far zone. In the near zone, rays propagating at large angles make significant contributions to the field and take the form of rapid oscillations. In the far zone, these rays are severely attenuated and paraxial rays are dominant. In the near zone, the PE is not accurate, but so long as the low order modes are illuminated, the PE will be accurate in the far zone. The start of the far zone is determined by the size, shape and frequency of the tunnel [3]. The far zone can be found by calculating the attenuation constant of each mode. However, the attenuation constant and amplification term for each mode may not always be found analytically. If we summed

Fig. 6. Normalized received power from ray tracing (solid) and ADI-PE (dashed) along the axis of propagation for the curved tunnel.

the equally weighted first 100 modes for the rectangular waveguides using the propagation constant expressions defined by [3, Eqs. (54) and (55)], we can see that the region before 500 m is in the near zone. In the range considered here, the contributions of the high order modes account for the discrepancies in the ADI-PE and ray tracing results. These discrepancies are in the form of rapid oscillations in the experimental data along the axis of propagation. Fig. 6 shows the normalized received power for the curved waveguide. As shown in the figure, the ADI-PE closely tracks the received power of the ray tracing simulations within the curved section. E. Comparison of the ADI-PE to Experimental Data In this section, we compare the ADI-PE to experimental data for the Lin-sen subway tunnel, shown in [7, Fig. 6]. The Lin-sen subway is comprised of two curved sections with radii of curvature of 455.68 m and 354.74 m, respectively, separated by a straight section. The straight entrance and exit sections are not considered here because they do not lie within the same horizontal plane as the rest of the tunnel. The tunnel cross-section is approximately rectangular with dimensions of 6 m 8 m. The transmitter is located outside the tunnel and is a vertically polarized Yagi-Uda antenna operating at 942 MHz. The receiver is placed off center at a height of 1.85 m above the ground. Fig. 7 shows the received power from measurements (solid) and the ADI-PE simulations. As before, the dielectric constant and conductivity are assumed to be the typical values of 5 and 0.01 S/m. In this case, the field entering the horizontal section of the subway is unknown and represents a possible source of error.

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Fig. 8. The incident and reflected rays entering the branch tunnel when (a) and (b) .

9

HE

Fig. 7. Experimental data (solid) and ADI results with the (long-dashed), (short-dashed) mode and the half wave dipole (dot-solid) as the initial field for the Lin-sen subway tunnel.

HE

However, we may choose a low order mode as our initial field because, at large distances, only low order modes dominate [1]. Simulations done for various initial fields show that the mode of the field (long-dashed), as defined by (13) and (14), produces solutions that best fits the experimental data. , and . The discretizations are As Fig. 7 shows for this initial field, even with our limitations, there is still good agreement between the ADI-PE solutions and the measured results. The ADI-PE models the nulls and overall trend of the received power. The figure also shows the field inmode (short-dashed) and the tensity when the far field expressions for the half-wave dipole (dot-line) are used as initial field. What this shows is that the field intensity is very sensitive to the initial field. This is because we are dealing with a short curved section (approximately 50 m), and high order modes make significant contributions in to the straight section. As in the previous example, we are not operating in the far zone and high order modes make significant contributions. The high order modes are represented by the rapid fluctuations in the measured data. The PE approximation does not accurately model these modes and it is a source of error. These results were obtained in a matter of minutes using a typical PC (1 GB RAM). By comparison, a typical ray tracing code would require another simulator to generate the initial field and must track each reflection and diffraction for each ray and can become computationally intensive. III. BRANCH TUNNELS A. Branch Tunnel Model Let us now consider the case of a straight tunnel that branches into a side tunnel. A typical branch tunnel geometry is shown in Fig. 8(a) and (b), where the main tunnel axis is shown as a solid bold line and the branch tunnel axis is shown as a long dashed line. The branch angle, , is the angle between the axes of the straight and branch tunnels. Fig. 8(a) and (b) show the incident and reflected rays as it enters the branch. The short dashed line marks the input plane of the branch tunnel. The grazing

9>

angle, , and the angle of the ray entering the branch, , are also shown. In order for the PE approximation to be valid, the branch angle must be small enough. More precisely, the branch angle must be less than 30 to ensure reflected rays entering the . Also, considering the branch are within our PE limit of rays diffracted from the corners of the junction, we can arrive . Diffracted at a much more stringent requirement of rays entering the main tunnel at angles greater than 15 will be weak when compared to reflected rays and will experience severe attenuation after the tunnel junction. As in the previous section, we solve the vector PE shown . The slope of the branching wall is in (1) with modeled using a staircase approximation (see Fig. 9(a)) and the impedance boundary condition is enforced on all four walls as outlined in [1] and [2]. The fields along the planes marking the entrance of the main tunnel (line C) and the branch tunnel (line B) are solved simultaneously and then used as the initial fields for the two separate diverging tunnels. B. Comparison of ADI-PE Results to HFSS Simulations In this section we validate the ADI-PE branch model with a numerical example. We simulate a 0.9 m 1.0 m rectangular tunnel with a branch angle of 15 and operating at a frequency of 900 MHz. The initial field is a unit strength Gaussian field source in the far zone. The source, with standard deviation of 0.75 , is centered 5 m before the tunnel junction (only the region near the tunnel junction is shown in Fig. 9). This means we are only using the lowest order modes as our initial field [1], [3]. The fundamental mode propagates near our PE limit at an angle of 14 with respect to the axis of the main branch. The ADI-PE simulations are done with discretizations , and (within the tunnel junction) of . The cross-sectional coordinates are indicated by and , while the axial coordinate in the main tunnel is denoted by the -axis. The -axis discretizations are made small within the tunnel junction to ensure small step sizes for the staircase approximation. Outside the junction region, discretizations along the axis of propagation can be made as large as a few wavelengths [1]. To validate our results, we compare our solutions with HFSS [13] and plot the magnitude of the field along the main and branch tunnel axes in Fig. 9(b). The HFSS simulations use radiation boundary conditions to terminate the tunnel and symmetry planes to reduce computational labor. HFSS is a full wave simulator and, unlike the ADI-PE, solves for backscattered waves as well as waves traveling in the

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Fig. 9. (a) Geometry of the branch tunnel. (b) The axial field intensity of the main tunnel from ADI-PE (solid), HFSS (dashed), the branch tunnel from ADI-PE (asterisk) and HFSS (dashed).

forward direction. The backscattering is seen as fluctuations in the axial field in Fig. 9(b) near the diverging tunnels. The dielectric constant and conductivity of the tunnel walls is 5 and 0.1 S/m, respectively. A high conductivity is chosen so there is appreciable loss in the small tunnel dimensions allowed in HFSS. As we can see from Fig. 9(b), there is strong agreement in the axial field intensity along the main and branch tunnel axes between the ADI-PE and HFSS. The figure also shows there is about a 10 dB drop when going from the main to the branch tunnel (at the point marked C in Fig. 9(b)). Although the ADI-PE is used to simulate a tunnel with a relatively 3 ), it is capable of small electrical cross-section (2.7 handling larger tunnels at higher frequencies without running into memory problems on an average (2 GB RAM) PC [1].

shown in [4] to agree with experimental data taken for coal mine tunnels for frequencies ranging from 200 – 1000 MHz. In our model, we treat small scale roughness by replacing the rough surface with a flat impedance surface that produces an equivalent specular reflection coefficient. The equivalent impedance for horizontal and vertical polarization is shown in (19) and (20), [5]

(19)

IV. ROUGH-WALLED TUNNELS A. Rough-Walled Tunnel Model So far we considered only tunnels with smooth walls, but in this section we investigate the effects of surface roughness. Surface roughness is the local variation of the tunnel wall relative to a mean surface level [4], [5]. In this study we consider random surface deviations in an otherwise smooth wall. For the purpose of numerical computations we assume the random deviations to be Gauassian distributed. A Gaussian distribution of the surface level can be characterized by a root-mean-square height deviand correlation length, [4], [5]. Smooth tunnels have ation a typical RMS height deviation of 0.01 m and rough surfaces, such as those found in coal mine tunnels, have a RMS height field due to deviation of 0.1 m [4]. The excess loss of the roughness in a rectangular tunnel is given by (18)[4]

(18) where and are the tunnel dimensions in the and axes, respectively, and is the range. The excess loss is derived by treating the rough surface as a random process and from ray tracing techniques, as outlined in [10] and [4]. Equation (18) is

(20) is the surface impedance of the smooth wall, is where the Gamma function, and is the grazing angle. When taking into account the effects of roughness, we substitute the surface impedance in (3) with the equivalent impedance of either (19) or (20). , the maximum slope Due to the PE angle limitation of angle of the rough surface and the grazing angle must satisfy the following relationship

(21) where is defined as shown in Fig. 10. As we can see from Fig. 10, the angle of the specular reflection of the incident ray, denoted by , depends on the height deviation of the roughness. The roughness angle is related to the RMS height and correlation length by

(22)

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Fig. 10. The geometry of the rough surface.

TABLE III ANALYTICAL AND NUMERICAL MAFS FOR THE RECTANGULAR TUNNEL

Fig. 11. The field intensity for the straight tunnel without roughness (solid), for the straight-curved-straight case (dashed), and for the straight-curved-straight case with roughness (dot-line).

TABLE IV ANALYTICAL AND NUMERICAL MAFS FOR THE CIRCULAR TUNNEL

B. Comparison of Numerical and Analytical Solutions We consider a rectangular 4.26 m 2.10 m tunnel and a cirmode is cular tunnel with radius of 2 m. The fundamental used as the initial field of the rectangular tunnel and the fundamode generated by a loop ring excitation is used as mental the initial field for the circular tunnel. Both tunnels operate at a frequency of 1 GHz and the dielectric constant and conductivity of the tunnel walls are taken to be 12 and 0.02 S/m, respectively. Tables III and IV summarize the mode attenuation factors (MAFs), or the loss in dB/km, of the smooth and rough tunnels with rectangular and circular cross-sections, respectively. In real tunnels, as in our simulations, the lowest order mode will determine the MAF over a long distance. We used (18) as our analytical loss factor for both the rectangular and circular tunnel. As we can see from (18), the loss due to roughness is a function of wavelength. To notice an appreciable loss at 1 GHz, we assume the walls are as rough as cave walls. Therefore, the RMS height is 0.1 m (0.33 ) and correlation length is 2.5 m (8.33 ) for both tunnels. The grazing angle of the fundamental mode of the rectangular and circular tunnels is computed using the analytical expresrepresents sions for the propagation constant, (where propagation in the positive direction), outlined in [4] and [3]. The grazing angle is obtained from plane wave theory by [12]

(23) and is found to be 4.56 and 5.25 for the rectangular and circular tunnel, respectively. The roughness angle is found to be 2.29 from (22), and (21) is satisfied. The ADI is simulated , and using discretizations of

for the rectangular tunnel and and for the circular tunnel. As we can see from Table III, the excess loss due to roughness for the rectangular tunnel is about 7 dB when using either (18) and 6.3 dB when using ADI-PE. Simulations using a unit strength Gaussian initial field (where multiple modes are allowed) with and show much less than 1% difference in MAF. Table IV shows the same close agreement in numerical and theoretical excess loss for the tunnel with circular cross-section. In this case, the excess loss due to roughness is about 16 dB. The accuracy of the results suggests that the equivalent surface impedance, along with ADI-PE, is an adequate model for determining loss due to surface roughness. For completeness, we look at the case of the curved tunnel with roughness. Fig. 11 shows the field intensity for the rectangular tunnel for the straight smooth case (solid), the curved smooth case (dashed) and the curved case with roughness (dot-line) with a radius of curvature of 1800 m. The curved section is in between two straight sections marked off by two vertical lines. The MAF of the curved section with roughness is about 37.7 dB/km. Tunnels with this type of configuration represents a topic of possible future work. C. Comparison of the ADI-PE to Experimental Data In this section, we examine a real underground tunnel with rough walls and a curving axis. The tunnel is approximately 3 m 3 m and the walls are characterized by , and with side wall roughness of [11]. The transmitter is placed at the entrance 1.71 m above the ground and the receivers are 2.73 m above the ground and operate at a frequency of 2.4 GHz. The data was recorded at every meter along the length of the tunnel. The curvature of the tunnel axis is shown in the boxed region in Fig. 12. The experimental data, shown as the solid line, was provided by the research team of Moutairou et al. [11]. The numerical simulations were done by computing the radii of curvature as a function of the tunnel axis and applying roughness for the side walls. A unit strength Gaussian with

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The limitations are dependent on the grazing angle as well as on the ratio of the RMS height to the correlation length. In this case, simulation data suggests propagation step sizes of about 1.5 – 4 . As shown in previous work, the discretizations along the and axes must be less than 1.9 to satisfy Nyquist’s theorem [1]. However, for the cases considered here, good results were obtained for discretizations less than 1 for the curved and rough-walled tunnels and less than 0.5 for the branch tunnel. ACKNOWLEDGMENT

Fig. 12. The experimental axial field intensity of the rough-curved tunnel (solid), the ADI-PE simulations (dashed). (Boxed) The axial geometry of the rough-curved tunnel.

is used as the initial field. The correlation length was selected to be 0.5 m in order to avoid a steep slope in the wall , irregularities. The discretizations are taken to be . The equivalent impedance represents averaged values within one correlation length, so is chosen to be 2 (0.25 m) (not much smaller than the correlation length). The ADI-PE results, shown as the dashed line, are shown in Fig. 12. In the simulations, a long straight section (85 m) precedes the tunnel in order to minimize the presence of higher order modes in the ADI-PE model. A sliding average with a window of 1 meter was used to match the resolution of the experimental data. As in previous cases, the overall trend is captured after the near zone region. V. CONCLUSION The ADI-PE method is used to investigate radiowave propagation in three specialized tunnel environments; curving tunnels, branch tunnels and rough-walled tunnels. We compared ADI-PE numerical examples to analytical and experimental data or to numerical data provided by HFSS. For the curved tunnel, there is good agreement between the ADI-PE and known analytical solutions for a wide range of curvature radii. For branch tunnels, even at the PE limit, there is good agreement between the ADI-PE and with commerical simulation codes such as HFSS. Also, the excess loss created by rough walls is accurately modeled using equivalent impedances. The ADI-PE method compares well with known theoretical loss factors for tunnels with either rectangular or circular cross-sections. For each case, the ADI-PE method accurately models propagation in the tunnel environment, but only when there are limitations in the tunnel geometry. For curved tunnels, the curves must be smooth and propagation step sizes of 2 – 3 is adequate for accurate results. For the branch tunnels, the branch angle must be less than 15 to ensure accurate results. Within the tunnel junction, where the staircase approximation is used, propagation step sizes of less than 1 is recommended for accurate results. In rough tunnels, the maximum RMS height deviations and correlation lengths of the rough walls are limited.

The authors wish to thank Dr. Wang from the Department of Electrical Engineering, Chin-Min Institute of Technology, Miaoli, Taiwan, R.O.C., for providing the experimental data shown in Figs. 5, 6 and 7. The authors would also like to thank Dr. Moutairou from the Underground Communications Research Laboratory/UQAT, Quebec, Canada for providing the data shown in Fig. 12. REFERENCES [1] R. Martelly and R. Janaswamy, “An ADI-PE approach for modeling radio transmission loss in tunnels,” IEEE Trans. Antennas Propag., vol. 57, pp. 1759–1770, Jun. 2009. [2] A. V. Popov and N. Y. Zhu, “Modeling radio wave propagation in tunnels with a vectorial parabolic equation,” IEEE Trans. Antennas Propag., vol. 48, pp. 1403–1412, Sep. 2000. [3] D. G. Dudley, M. Lienard, S. F. Mahmoud, and P. Degauque, “Wireless propagation in tunnels,” IEEE Antennas Propag. Mag., vol. 49, no. 2, pp. 11–26, Apr. 2007. [4] A. G. Emslie, R. L. Lagace, and P. F. Strong, “Theory of the propagation of UHF radio waves in coal mine tunnels,” IEEE Trans. Antennas Propag., vol. AP-23, pp. 192–205, Mar. 1975. [5] R. Janaswamy, Radiowave Propagation and Smart Antennas for Wireless Communications. New York: Springer, 2000, pp. 26–31. [6] J. C. Strikwerda, Finite Difference Schemes and Partial Differential Equations, 2nd ed. Philadelphia, PA: SIAM, 2004. [7] T. Wang, “Simulations and measurements of wave propagations in curved road tunnels for signals from GSM base stations,” IEEE Trans. Antennas Propag., vol. 54, pp. 2577–2584, Sep. 2006. [8] S. F. Mahmoud, “Modal propagation of high frequency electromagnetic waves in straight and curved tunnels within the earth,” J. Electromagn Waves Applicat., vol. 19, no. 12, pp. 1611–1627, May 2005. [9] S. F. Mahmoud and J. R. Wait, “Guided electromagnetic waves in a curved rectangular mine tunnel,” Radio Sci., vol. 9, no. 5, pp. 567–572, May 1974. [10] P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces. New York: MacMillan, 1963, pp. 72–81. [11] M. Moutairou, G. Y. Delisle, H. Aniss, and M. Misson, “Wireless mesh networks performance assessment for confined areas deployment,” Int. J. Comput. Sci. Netw. Security, vol. 8, no. 8, pp. 12–23, Aug. 2008. [12] R. F. Harrington, Time-Harmonic Electromagnetic Fields. New York: Wily-Interscience, 2001. [13] User’s Guide: High Frequency Structure Simulator (HFSS) V. 9.2 Ansoft Corporation, 2003, Ansoft Documentation. Richard Martelly (M’04) was born in Brooklyn, NY, on November 2, 1977. He received the M.S. degree in electrical engineering from Polytechnic University, Brooklyn, in 2004, the B.E. degree in engineering physics from Stevens Institute of Technology, Hoboken, NJ, in 2000, and the Ph.D. degree in electrical engineering from the University of Massachusetts, Amherst, in 2010. From September 2004 to November 2010, he worked in the Antenna and Propagation Laboratory in the Department of Electrical and Computer Engineering, University of Massachusetts. He is currently with The MITRE Corporation, Bedford, MA.

MARTELLY AND JANASWAMY: MODELING RADIO TRANSMISSION LOSS IN CURVED, BRANCHED AND ROUGH-WALLED TUNNELS

Ramakrishna Janaswamy (F’03) received the Ph.D. degree in electrical engineering from the University of Massachusetts, Amherst, in 1986. From August 1986 to May 1987, he was an Assistant Professor of electrical engineering at Wilkes University, Wilkes Barre, PA. From August 1987-August 2001 he was on the faculty of the Department of Electrical and Computer Engineering, Naval Postgraduate School, Monterey, CA. In September 2001, he joined the Department of Electrical and Computer Engineering, University of Massachusetts, Amherst, where he is a currently a Professor. He is the author of the book Radiowave Propagation and Smart Antennas for Wireless Communications, (Kluwer Academic Publishers, November 2000) and a contributing

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author in Handbook of Antennas in Wireless Communications (CRC Press, August 2001) and Encyclopedia of RF and Microwave Engineering (Wiley, 2005). His research interests include deterministic and stochastic radio wave propagation modeling, analytical and computational electromagnetics, antenna theory and design, and wireless communications. Prof. Janaswamy is a Fellow of IEEE. He was the recipient of the R. W. P. King Prize Paper Award of the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION in 1995. For his services to the IEEE Monterey Bay Subsection, he received the IEEE 3rd Millennium Medal from the Santa Clara Valley Section in 2000. He is an elected member of U.S. National Committee of International Union of Radio Science, Commissions B and F. He served as an Associate Editor of Radio Science from January 1999-January 2004 and as an Associate Editor of the IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY.

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Discrete Models of Electromagnetic Wave Scatterers in a Frequency Range Igor P. Kovalyov and Dmitry M. Ponomarev, Member, IEEE

Abstract—We investigate the spatial and frequency characteristics of electromagnetic fields scattered by some object. Frequency functions that allow the expression of all elements of the object scatter matrix are introduced. The sampling theorems that are proved for the introduced functions make it possible to characterize the scatterer by a set of discrete parameters. The application of the proved theorems provides for the creation of a versatile scatterer discrete model that determines the dependence of scattered fields both on spatial coordinates and frequency. A random scatterer is considered and the conditions under which the discrete parameters are statistically independent normalized random variables are stated. The article presents the results of a random scatterer modeling. Index Terms—Complex resonance frequencies, current modes, discrete model of a wave scatterer, random scatterer, sampling theorems, scatter matrix, scatterer, spatial and frequency characteristics, spherical waves.

I. INTRODUCTION

T

HIS study inquires into the correlation of spatial and frequency characteristics of electromagnetic fields scattered by some object. The solution to this problem is essential for scatterer, antenna, and channel modeling. At present, experimental research on wireless channels, among them indoor channels, poses a serious engineering challenge. A wireless channel study involves measurements done at numerous points of space [1], [2]. There is good reason to believe that finding a relation between spatial and frequency characteristics may facilitate the experimental research on wireless channels, thanks to a decrease in the number of spatial measurement points and the replacement of measurement by frequency scanning. The method of eigenfunction expansion, otherwise known as the singularity expansion method (SEM) [3], [4], is often used in research on the frequency dependence of fields. The three-part work [4] contains reference to abundant literature on the theory and practical application of SEM in electrodynamics. It might be well to point out, however, that SEM is more often used to demonstrate frequency dependence rather than spatial. An interesting model for antenna characterization in a frequency range has been suggested in [5]. This model clearly demonstrates that

Manuscript received October 08, 2008; revised September 07, 2009; accepted December 22, 2009. Date of publication March 29, 2010; date of current version June 03, 2010. I. P. Kovalyov is with the Nizhny Novgorod State Technical University, Nizhny Novgorod 603126, Russia and 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.2010.2046833

the spatial-frequency characteristics of antennas may be determined by a relatively small number of parameters. However, the model described in [5] can be regarded as a fortunate heuristic find only, because the article does not offer strong enough substantiation for it. In this paper, we suggest a stricter model for representation of electromagnetic fields in a frequency range. The remainder of the paper is organized as follows. Section II cites some general postulates of electrodynamics that form the basis of the present work. Specifically, it focuses on spherical waves and the scatter matrix built based on them. The singularity expansion equations presented in Section II are applied to demonstrate the frequency dependence of the scatter matrix elements. Section III is devoted to studying the frequency dependence of the scatter matrix elements. It presents frequency functions and shows that all matrix elements can be expressed in terms of the above functions. Section IV presents a functions, which proof of sampling theorems for the are conducive to the scatterer discrete model. Section V discusses a random spherical scatterer. It is proved that the discrete parameters characteristic of the scatterer are a set of independent normal random variables. This section also includes the results of modeling a random scatterer. Section VI is a conclusion. Appendix I provides brief information about the Hankel transform, as well as proof of the sampling theorem for the functions the Hankel transform, which is nonzero on some interval. Appendix II shows the nomenclature. II. SCATTER MATRIX AND GENERAL EQUATIONS FOR COMPUTING ITS ELEMENTS Consider some limited-size scattering object placed in an incident field. The coordinate dependence of the scattered field can be regarded as known if the field on the closed surface that envelops the scatterer is known. It is reasonable to select the simplest closed surface, which is a sphere, for the enclosing shape (Fig. 1). , let us use the funcTo represent scattered fields with tions called vector spherical wave functions (spherical waves) [6]–[11]. The field of spherical waves can be expressed through a scalar spherical wave function with the help of the following equations [6]: (1a) (1b) where is the strength of the E-type wave electric denotes the field (the transverse magnetic wave), and strength of the H-type wave electric field (the transverse electric stands for the wavenumber. wave), while

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The scattered field is actually a sum of spherical waves: (6a) (6b)

Fig. 1. Scatterer in electromagnetic field.

With time dependence of the fields believed to be scalar spherical wave function can be expressed as

, the

(2) in (2) is the associated Legendre function. With it is calculated by the equation (3a)

To calculate expression

with

, one can use the

The components of the spherical wave magnetic field and are not used in the equations that follow, therefore presenting equations for them seems unnecessary. We and coefficients as complex ampliwill refer to the tudes of E- and H-type spherical waves, respectively. Equations (5) satisfy the radiation condition and represent the in Fig. 1.) To present the field scattered by some object ( incident field, in which the scatterer is located, these equations Bessel spherical should be modified by substitution of the function in lieu of the Hankel spherical function in (5). and to denote spherical wave incident We will use fields. It is known that series (5) is convergent when fields are continuous position functions [7]. Therefore, the scattered field representation in terms of spherical waves holds good, provided the enclosing sphere in Fig. 1 is not in contact with any of the scatterer’s points or edges. and to denote the vectors that represent Let us use the incident and scattered fields, respectively (Fig. 1). Because the relationship between the scattered and the incident field is linear, true is the matrix equality

(3b) (7) represents the second-kind Hankel spherical function, correlated with the first-kind Bessel spherical function second-kind Bessel spherical function by the and the relationship (4) Equation (1) in terms of (2) allows us to write the expression for components of the spherical wave electric field as

(5a)

(5b)

The matrix in (7) is the so-called scatter matrix. It is the that ensures discrete representation of the object-scattering properties at a fixed frequency. It is worth mentioning that the said scatter matrix is a special case of the antenna scatter matrix described in [9]. Let us agree to place complex amplitudes of E-type waves ahead of H-type wave complex amplitudes when writing the and vectors. With that understanding, the vector is given by

Let us denote vector’s length as . In writing the matrix, it would be more convenient to use a single common subscript rather than the two-element subscript for components. Equation (7) demonstrates the physics of the scatter matrix arbitrary element

(5c) (8) (5d) (5e) (5f)

element equals the ratio of the comBy this means the plex amplitude of the -th spherical harmonic of the scattered field to the complex amplitude of the -th spherical harmonic of the incident field.

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While considering the frequency dependence of an arbitrary matrix element, it is reasonable to represent the result as a sum of eigenfunctions, the technique also known as the singularity expansion method (SEM). We first give a brief overview of SEM [12]. Suppose that the problem of determining a field scattered by some object is defined as an inhomogeneous integral equation written as (9) where denotes a linear integral operator, stands for the specis the specified function (source), is ified parameter, the sought function, and denotes a point in an abstract multidimensional space. For a singular expansion, the eigenvalues and eigenfunctions , representing solutions of a homogeneous integral equation, are used (10) According to [12], the singular expansion of the solution of inhomogeneous equation (9) can be written as

(11)

The scalar product in (11) is determined by the expression

(12)

The integration domain in (12) is determined by the source, and the superscript * indicates the complex conjugate value. Let us specify what form SEM calculation equations (11) and (12) take in the scattering problem. The unknown function is believed to represent the scattering object current . It may be current concentration, the vector of surface current density for an ideally conductive object, or the current-division function for a scatterer made of a multitude of thin wires. The eigenvalue is the complex resonance frequency or . It is also convenient to use the complex wavenumber instead of . The eigenfunctions are current modes , current distribution in the volume or over the scatterer’s surface, corresponding to the complex resonance frequency. The current modes are assumed to be normalized. The source is the field of the -th incident spherical wave. In this manner, in terms of wave scattering by an object placed in the field of the -th spherical wave, SEM equation (11), which represents the expansion of current in eigenfunctions, can be written as

The scalar product (12) denoted in (13) by form

takes on the

(14) Equations (13) and (14) are the equations from [3], provided the scatterer is in the field of a unitary spherical wave. Scalar product (14) is referred to as a coupling coefficient in [3]. Expression (14), as is evident from a dimensionality analysis, needs the inclusion of some multiplicative constant. The presence of this constant is not essential for the discussion that follows. We will assign it to the norm of mode current, which is believed unitary when this constant is taken into account. It is evident from (13) that the introduction of current modes makes it possible to represent frequency spatial characteristics as the sum of products of two factors—one factor being solely frequency-dependent, the other spatial value-dependent only. III. FREQUENCY DEPENDENCE OF SCATTER MATRIX ELEMENTS element As seen in the previous section, to calculate the of the scatter matrix, it is necessary to place the scatterer in the field of the -th unitary spherical wave. The complex amplitude of the -th spherical wave of the scattered field will determine element of the scatter matrix. Under SEM, the scattered the field can be determined through current modes . There, it is fore, in order to find the frequency dependence of necessary 1) to determine the frequency dependence of current mode amplitudes when the scatterer is excited by the field of the -th spherical wave, and 2) to find out the frequency dependence of the -th spherical wave complex amplitude produced by the current mode. Let us begin with the first task: finding the frequency dependence of the amplitude of the current mode excited by the -th spherical wave. We use (13) and (14) to find the dependence. As is evident from (13), the frequency dependence of the current mode amplitude is determined by two multiplier factors. The first factor is independent of the excitement type and is governed by the proximity of the excitation field frequency to the eigenfrequency of the mode. This is a resonance factor (15) depends on the incident field strucThe second factor ture. This factor can be determined with greater precision by projecting the incident field onto the current mode and is cal, it is necessary to subculated by (14). To calculate stitute (5) in (14) and carry out the integration. We write the scalar product as

(13) (16)

KOVALYOV AND PONOMAREV: DISCRETE MODELS OF ELECTROMAGNETIC WAVE SCATTERERS

Integration (16) allows writing waves as follows:

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for E- and H-type (20c)

(20d) A rearrangement of (19), with the help of (20), results in (17a)

(21)

(17b)

The frequency functions introduced in (17) are very important for the discussion that follows: (18)

in (18) represents the Cartesian components of the mode current ( or ). It is worth noting that (17a) and (17b) both include one-type frequency functions (18). An inquiry into these functions is the subject matter of the next section focused on proving sampling theorems for functions (18), which is conducive to scatterer discrete models. The derivation of (17) is based on the recurrent relation of Bessel spherical functions and associated Legendre functions. Although uncomplicated, this derivation is rather cumbersome. This being so, let us consider a simplified way of deriving (17b), assuming one current component being non-zero . The integration element (16) in view of (5) can be written as

(19) To rearrange (19), we use the following equations for the associated Legendre functions [13]:

. Expression (21) is actually (17b) with Similar rearrangements with all current components available result in (17). After the first task, i.e., considering the excitation of current mode by a spherical wave, we undertake the next task of calculating the amplitude of the spherical wave produced by the current mode. The second problem statement is quite simple. With current frequency-independent current distribution (the mode) known, it is necessary to determine the frequency decomplex amplitude of the spherical wave pendence of the produced by this current mode. The quickest and most elegant way of doing it is based on the use of the reciprocity theorem or the Lorentz lemma, to be more exact. Before turning to the Lorentz lemma, let us consider the following two problems. Let the electric field in the first problem be the complex conjugate for the incident field of the -th spherical wave while the magnetic field also differs in the sign. (22) is determined by (5) if the Bessel spherThe field ical function is substituted for the Hankel spherical function and a complex conjugate is obtained from the resulting expression. The second problem is the one that we need to solve. Specifically, it is believed that the field source in the second problem mode current. The electromagnetic field in this is the problem can be represented as a sum of spherical wave fields, that is, by (5) and (6). Rewriting expression (6) making no difference between E- and H-waves, we get (23) The Lorentz lemma states that the fields in the first and the second problem satisfy the following equation:

(20a)

(24) (20b)

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Where is the surface of the radius sphere, is the ball and are sources that produce enclosed in it, and fields in the first and second problem. Consider the right part of (24) first. The second term under the integral in it vanishes as we believe that the sources that produce field are located outside the sphere. Owing to this, the the right part of (24) becomes

Expression (31) and (17) and (28) provide a solution to the problem of this section, i.e., determining the frequency dependence of scatter matrix elements. It is worth noting that the amplitude of the spherical wave produced by the current mode can be calculated in a different way, without Lorentz lemma. A comparison of so-obtained expression with (17) allows one to make certain that (29) holds true, which attests to the correctness of (17), (29), and (31) derived in this section.

(25) IV. SAMPLING THEOREMS IN ELECTROMAGNETIC WAVE SCATTERING PROBLEMS A comparison of (25) and (14) reveals that the considered integral equals . In other words, it is determined expressions that are complex conjugate to (17a) and (17b). By this means, we can write for the right part of (24): (26) To re-express the left part of (24), we substitute (22) and (23) in it and interchange the order of integration and summation. Taking into account the orthogonality of spherical waves, we write

The objective of Section IV is to build a scatterer discrete model. The section begins with an introduction of the model discrete parameters obtained from the expansion of Cartesian components of mode currents in functions that form a basis in the tridimensional space. The discrete parameters are then used to determine frequency characteristics of scatterers. While considering this problem, sampling theorems are proved for functions. A comparison of the proved theorems with the classic sampling theorem completes Section IV. Let us address expression (18) that determines the functions. In its expanded form, it is written as

(32) (27)

For the sake of compact writing of (32), let us introduce the following notation:

The integral in the right part of (27) equals twice the radiation radiation power has power of the -th spherical wave. The been calculated in [9], for example. With regard to the normalizing constants presented in [9], the expression for power is

(33)

By this means, (26), (27), and (28) provide an equation for calculating the complex amplitude of the spherical wave produced by the current mode

It may be necessary to point out that the functions are frequency-independent, although they vary with the radial and . In view of coordinate and the discrete indices expression (33), (32) for the function can be written as

(29)

(34)

A summation over all modes of the products and resonance factor (15) provides a solution to the problem considered in this section, i.e., it gives an expression for calcuelement of the lating the frequency dependence of the scatter matrix:

For discrete representation of functions, we use the Fourier-Bessel series [12]. We briefly outline the theory of Fourier-Bessel series in the perspective of the Bessel spherical functions. Consider the set of functions

(28)

(35)

(30) A substitution of (29) in (30) results in a final expression for scatter matrix arbitrary element: (31)

where

is the -th zero of the Bessel spherical function (36)

It is possible to demonstrate that functions (35) are orthogonal on the interval . To prove this, it is with the weight

KOVALYOV AND PONOMAREV: DISCRETE MODELS OF ELECTROMAGNETIC WAVE SCATTERERS

enough to calculate the integral of the product of functions (35). A calculation by equations from [13] gives

(37)

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functions are deIt should be remembered that the termined by (33) through the Cartesian components of the mode current. Taking into account (33), (41), and (44), it is possible to define a physical interpretation of the introduced values are determined by exdiscrete parameters. The pansion coefficients of the Cartesian component of the mode current in the base of the following functions: (45)

can be Therefore, the substantially arbitrary function represented by the following series, known as the Fourier-Bessel series: (38) To get an expression for coefficients, we multiply the right and and integrate left part of (38) by over . Changing the order of integration and summation in the right part and using the property (37), we get (39)

As is known from the Fourier series theory, series (38) is convergent if is sectionally continuous and has bounded variation on . Furthermore, there must exist the integral (40)

Functions (45) represent the product of three single-variable functions from three different sets. Each of the sets includes a system of functions orthogonal in their domains. The functions are orthogonal when . funcat a fixed and diftions are orthogonal with the weight . Finally, the functions ferent values of when are orthogonal with the weight at a fixed and . various values of when This assertion explains why discrete parameters have and determine the base five indices. The three indices function (45). The fourth index denotes the mode. And the or ). fifth index represents the current component ( After the introduction and physical interpretation of the discrete parameters, let us turn to the main problem of this section, which is the use of the introduced parameters for determining the scatterer’s frequency characteristics. To solve the problem, we substitute series (41) in (34) and change the order of integration and summation. Taking into account (43) and (44), we write

Assuming that functions (33) meet the necessary conditions, we represent them as series (38) (46) (41)

coefficients in accordance with (39) are calculated The by the equation

Let us use which the

to represent wavenumber functions by parameters are multiplied in (46) (47)

(42) Let us denote the integral in (42) as

The introduced notation allows us to write sum (46) in a compact form (48)

(43)

The integral in (47) is calculated by the equations presented in [13]

The variables are the parameters of the scatterer versatile model. As is evident from (42) and (43), they are related to coefficients by the following simple relationship: (44)

(49)

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A substitution of (49) in (47) produces an explicit expression for functions (50)

Equations (48) and (50) provide a solution to the problem , set in this section. Through the discrete parameters they determine the functions required for finding all scatter matrix elements by (17), (28), and (31). In addition to providing calculation tools, the derived formulas allow another physical interpretation of the discrete parameters. parameters, let To elucidate the physical meaning of us first consider in greater detail the properties of the functions used in (48) to represent the frequency dependence. First, we will study the function behavior in the neighborhood of where expression (50) has the uncertainty the point Bessel spherical of the kind 0/0. A representation of the function in the neighborhood of this point by the Taylor series gives

(51) A substitution of (51) in (50) produces a simple expression function for the (52) It should be recognized that in the neighborhood of the point, it is preferable to use approximation (52) rather than exact expression (50). Owing to inaccuracies during calculation by (50), the obtained results may differ greatly from the true value. is unity As is evident from (52), the variable when . It also follows from (36) and (50) that when . These equations demonstrate functions possess the properties of sampling that functions (53) functions are presented in Fig. 2(a) for The plots of and in Fig. 2(b) for . When plotting the curves, was assumed to be 0.3 m. The symbols on the charts mark function values at points . The charts of Fig. 2 confirm that the values of functions at the said points are 0 or 1. Property (53) allows an interesting physical interpretation of the discrete parameters. Let us substitute the value in (48), instead of taking into account (53). Thus, we get (54)

Fig. 2.

'

k

( ) sampling functions with

r

:

= 0 3. (a)

n = 3; (b) n = 6.

By this means, the discrete parameter represents the value of the function at the point. This can be also verified through a comparison of (34) and (43). The obtained expressions allow the following theorem statement. functions Theorem 1 (Sampling Theorem): The required in finding the scattered field are fully determined by the set of their discrete values at points, where is the -th zero of the Bessel spherical function, and is the distance from the origin of coordinates to the scatterer’s farthest point. Equation (48), which in view of (54), takes on the form (55) provides the proof of Theorem 1. Equations (48) and (55) may be regarded as an expansion of frequency dependence in function sethe ries. Indeed, it can be demonstrated that

KOVALYOV AND PONOMAREV: DISCRETE MODELS OF ELECTROMAGNETIC WAVE SCATTERERS

functions are mutually orthogonal on the ray with the weight . To prove the orthogonality, it is necessary to calculate the integral (56) Integral (56) can be calculated through a transition to the contour integral on a complex number plane similar to how it was done in [14]. These calculations determine the orthogonality and basis functions. the norm of

(57)

is a second-kind spherical Bessel function. where Hence, the obtained equations demonstrate that the discrete parameters, which determine the scatterer electrodynamics in a frequency range, allow double interpretation. On the one hand, as is evident from (33) and (43), they are determined by the coordinate function expansion of mode current. represent On the other hand, as is apparent from (48), coefficients of the frequency function expansion in bases. the To summarize the present section, let us compare Theorem 1 and the classic sampling theorem used for discrete representation of signals. It should be recorded first of all that Theorem 1 includes an infinite multitude of sampling theorems, and for there is a corresponding every value of set of basis functions and a specific sampling theorem. At the same time, it is safe to assert that whatever the value of is, the basis functions differ from the classic basis functions. Additionally, it is valid to say that at any the functions asymptotically approximate to the classic functions with increasing , when does not differ much from . This fact can be deduced from observing the charts in Fig. 2. It also can be proved analytically using the asymptotic expression for the Bessel spherical function. A few words about the layout of sampling points in Theorem , the sampling point determined by 1 seem pertinent. At (36) are equidistant exactly like in the regular sampling theorem. The following inequality [13] holds for the spherical Bessel function zeros: (58) Owing to this, the number of sampling points decreases in the frequency interval with increasing. That is, the increase of allows one to get by with fewer samples. By no means, however, do we see the main virtue of Theorem 1 in economical sampling. As is evident from the equations derived in this section and as the example of the next section illustrates, functions form a natural basis for representation the

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of the scatterer’s frequency characteristics. Finally, noteworthy also is the mathematical distinction between Theorem 1 and the classic sampling theorem. Theorem 1 is based on the Hankel transform [15], while the classic is based on the Fourier transform. A brief overview of the Hankel transform and a proof of the sampling theorem based on the Hankel transform are presented in Appendix 1. V. DISCRETE MODEL OF A RANDOM SPHERICAL SCATTERER In the previous sections of this paper, mode currents were believed known during the building of scatterer discrete models. The known currents were used to find discrete parameters . These parameters, together with complex resonance frequencies, fully determine the scatterer characteristics. In this section we consider a different problem. It is believed now that although modes exist, exact mode current values remain unknown and we can make probability assertions about the currents only. We emphasize that the scatterer remains the same, and mode currents are fixed functions of spatial coordinates. The currents’ random nature is accounted for by the complexity or impossibility of their calculation. The model discrete parameters in this case are represented by random values. The objective of this section is to determine the statistical properties of the random parameters. It is supposed that mode currents meet the following conditions. They are nonzero in the region , which is a sphere of the radius . It is further believed that the mean of the complex amplitude of whichever component of every mode is zero at any spatial point (59) The superscript in (59) denotes any individual current comor ). It is believed that all components of ponent ( different modes are not correlated with each other (60a) The same equation is believed to hold true for different components of the same mode (60b) Finally, it is supposed that complex amplitudes of a single-mode current component at different points of space are not correlated either (61) , expression (61) determines the variance of a With single Cartesian component of the mode current . It is thought that the variance value does not depend on spatial coordinates and equals (62) denotes the sphere volume in (62). Postulates (59) to (62) mean that random mode currents are uniformly distributed within the sphere. Simultaneously, no preference is given to

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any spatial point, any direction or current value. Equation (62) follows from the mode current normalization condition. The stated suppositions do not allow for the orthogonality of different modes known from the electrodynamics. It is impossible to take the orthogonality into account when the electrophysical parameters of the scatterer material are unknown. In the above assumptions, the statistical independence (60a) is used in lieu of the orthogonality property. Suppositions (59) to (62) allow us to determine the statistical discrete parameters. To do that, let us use properties of (43), which in view of notation (33), can be written as

(63) With the help of (63), it is easy to verify that the mean value of parameter is zero. Averaging (63) in view of (59) any readily attests to it (64) random values have Expression (63) suggests that normal distribution. Indeed, these values can be regarded as a sum of random currents at various spatial points with certain coefficients. As currents at different spatial points are not parameter is a sum of a great number correlated, the of independent random values. In these conditions, the central limit theorem of the probability theory is applicable, which is gives grounds to affirm that the random variable normally distributed. Let us prove that any two random parameters are statistically independent, that is, their correlation is zero

and (66) With determines the variance of discrete model parameters. A substitution of (37) in (66) produces an expression for calculation of the variance: (67) are Thus, it has been found that discrete parameters statistically independent normal random values. This property allows us to reason that, with assumptions (59) to (62) being parameters represent an adequate model of a true, the random spherical scatterer. In the calculation of fields produced by a random scatterer, it is necessary to know complex resonance frequencies in adparameters. Complex resonance frequencies dition to can be found experimentally through measurement in the time or frequency domain [4]. When experimentation is impractical or objectionable, complex resonance frequencies can be taken a priori as random values. With the help of complex resonance frequencies and parameters, one can calculate the elements of the scatter matrix. These elements determine the object scattering properties in terms of spherical waves. It should be recognized, however, that in actual practice, it is conventional to determine the scattered field through exposure of the object to a plane wave. For a transition to plane waves, it is necessary to write a plane wave expansion in spherical waves. A method for tackling this problem is known from the literature [6]–[11]. Spherical wave complex amplitudes for the plane electromagnetic wave are determined by the following expressions:

(65) when the equality fails to hold for at least one of the five index . As is pairs and obvious from (60a) and (60b), (65) holds true for . Therefore, let us consider (65) with and . The use of (63) in view of (62) gives (66) at the bottom of the page. It is obvious from the above expression that with , the integral over equals zero and (66) goes to zero because . When of the orthogonality of harmonic functions , (66) goes to zero with , owing to the orthogonality of associated Legendre functions [13]. Finally, and , (66) goes to zero with because of the orthogonality of Bessel spherical when functions (37).

(68a)

(68b)

(68c)

(68d)

(66)

KOVALYOV AND PONOMAREV: DISCRETE MODELS OF ELECTROMAGNETIC WAVE SCATTERERS

and in (68) denote angular coordinates of the plane represent complex ampliwave source. tudes of E-type spherical waves produced by the sources of vertically and horizontally polarized waves, respectively. stand for complex amplitudes of H-type spherical waves produced by the sources of vertical and horizontal polarization. Equations (68) complete the theoretical considerations that allow the formulation of spatial frequency characteristics of a random spherical scatterer. Let us dwell for a while on another issue. In calculations, it is important to ascertain the number of terms that we can restrict ourselves to in formally infinite series. , and by and Let us denote the maximum values of respectively. The maximum for can be estimated on the basis of the fact that Bessel spherical function values turn out to be small when the value of the index is greater than the argument. Therefore, to estimate , we can use the expression (69) where is the wavenumber corresponding to the maximum frequency of the band in question. According to [9], the recis 10 for not too large amounts. ommended value for The number of Fourier-Bessel series terms should be selected so as to allow for all the Bessel spherical function zeros in the . Since with , the zero interval of interest has the least value of all zeros, it is reasonable to estimate by (70) Of course, equality (70) provides only a speculative result for , and the value used actually should be slightly increased. Let us turn now to ascertaining the number of modes that should be taken into consideration during modeling. Naturally, it is necessary to take into account all the modes the real resonance frequencies of which are within the frequency band in question: (71) If the imaginary parts of complex resonance frequencies are small , then the corresponding resonance curves turn out to be saw-toothed. Then, it is possible to restrict oneself to values satisfying (71). In case resonances are less pronounced, the number of modes should be increased. Their number must include the modes the tails of resonance curves of which cannot be neglected in the frequency band of interest. Now we have all the calculation equations that enable modeling of a random spherical scatterer. However, the necessary equations are rather cumbersome and interspersed throughout the paper, which hampers the calculation job. To facilitate the task, let us explain the way the obtained equations can be utilized. The proposed calculation method includes the following steps: coordiStep 1. Defining Basic Data: Specify the and nates of the plane wave source, the scatterer size , the number

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of spherical waves taken into account (the maximum value for the first index of the spherical wave to be more exact), the number of Fourier-Bessel series terms, and the number of modes . and Complex Step 2. Obtaining Discrete Parameters a Resonance Frequencies: values are easily obtained from a set of independent normal complex numbers with the unit variance. Their product by the square root of variance expression (67) produces discrete parameters with specified statistical properties. Complex resonance frequencies are either taken as known or generated as random complex numbers. Functions: Interim funcStep 3. Calculation of required for subsequent computations are caltions culated by (48). eleStep 4. Calculation of Scatter Matrix Elements: ments of the scatter matrix are calculated with the help of (31). Step 5. Calculation of Scattered Wave Complex Amplitudes: vector of complex amplitudes of scattered-waves is The vector of incident wave amplitudes determined by (7). The is obtained from (68). Step 6. Scattered Field Calculation: Components of the scattered electromagnetic field are calculated by (5) and (6). The and variables in (6) are the components of the vector calculated during Step 5. The results of modeling are demonstrated in Figs. 3 and 4. It was believed during modeling that the plane wave source is and produces a vertilocated on the axis cally polarized field. The size of the spherical scattering region m; the and values equal 8 and 14, respectively; is . Mode frequencies represented by the number of modes real parts of complex resonance frequencies were believed to be random numbers uniformly distributed in the range of 0.6–1.4 GHz. Imaginary units were taken as uniformly distributed in the – /s range. Figs. 3 and 4 demonstrate plots that illustrate the angle dependence of scattered field intensity in the horizontal plane. vertical component of Figs. 3(a) and (b) refer to the the scattered field, while Fig. 4(a) and (b) refer to the horizontal component. Figs. 3(a) and 4(a) show changes in the scattered field diagrams following minor frequency changes (from 1 GHz to 1.02 GHz). It is obvious from the curves that small frequency variations correspond to minor changes in the plots. Figs. 3(b) and 4(b) illustrate more significant frequency variations (from 1 GHz to 1.2 GHz) accompanied by notable changes in the plots. Their look is determined by different modes not correlated with previous modes. The considered example of a random spherical scatterer demonstrates that the created simulation is a versatile model of the scattering object. Indeed, it allows the description of a scatterer the mode currents of which vary at random in magnitude and direction. The model parameters are statistically independent normally distributed random variables. Such statistics give reason to state that the model is devoid of surplus parameters, and none of its parameters can be expressed through any other. It might be pertinent to point out that the presented model allows the examination of both frequency and spatial coordinate

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VI. CONCLUSION

Fig. 3. Intensity of electric field vertical component in the horizontal plane. (a) Frequency change by 2%; (b) frequency change by 20%.

The objective of this paper was to build a discrete model of the electromagnetic wave scatterer that would represent the dependence of the field scattered by it both on spatial coordinates and frequency. The results attained, while solving the problem, include the following: 1. The frequency dependence ( wavenumber dependence) of scatter matrix elements has been found. The mathematical expression of the dependence includes functions of two and types. These are resonance factors functions. 2. Sampling theorems conducive to discrete scatterer modfunctions. Chareling have been proved for the acteristics of the scattered field in a frequency range are dediscrete parameters and comtermined by a set of plex resonance frequencies. 3. A study of a random spherical scatterer has been done. Conditions under which discrete parameters are statistically independent random normal complex values have been formulated. 4. A random scatterer modeling case study has been presented. The charts resulting from the modeling confirm that the model allows the observation of both spatial and frequency characteristics of the scattered field.

APPENDIX I SAMPLING THEOREM FOR HANKEL TRANSFORM The Hankel transform of order of the function is determined by the expression from [15]:

(A.1)

is the Bessel function of order with . The where inverse Hankel transform coincides with the direct one:

(A.2)

Hankel transform is equal to zero with Suppose the . Consider for as a function of and present it as a Fourier-Bessel series: Fig. 4. Intensity of electric field horizontal component in the horizontal plane. (a) 2% frequency change; (b) 20% frequency change.

dependence of various electric and magnetic field components and is equally fit for field investigation both in the far- and near-field regions. Put another way, the model reflects spatial, frequency, and polarization characteristics of the scattered field.

(A.3)

where

is the -th zero of the

Bessel function. (A.4)

KOVALYOV AND PONOMAREV: DISCRETE MODELS OF ELECTROMAGNETIC WAVE SCATTERERS

Substituting (A.3) in (A.2) and changing the order of integration and summation, we write

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function in (A.7) is not specified. To The type of the define it, let us multiply (A.3) by and integrate it over from 0 through . Thus, we get

(A.8) (A.5) A comparison of integrals in (A.5) with (A.2) allows an asserat the tion that the integrals are discrete values of points

A calculation of integrals by equations presented in [13] results function: in the following expression for the (A.9)

(A.6)

Taking into account (A.6) in (A.5) gives

(A.7)

Expression (A.7) enables formulation of a sampling theorem for the Hankel transform. Sampling Theorem: The function the Hankel only is transform of which is different from zero with completely determined by the multitude of its discrete values at , where is the -th zero of the points Bessel function.

The sampling theorem proved in this Appendix has a more general character than the theorem presented in the main text of this study. For a transition from (A.1) to (A.9) to the equations considered in the paper, one may make the index of the Bessel function as being half-integer and use an expression for the Bessel spherical function: (A.10) It appears that with integer values of the index , the sampling theorem may be used for discrete representations in bidimensional wave emission and scattering problems. APPENDIX II

NOMENCLATURE Complex amplitude of the scattered field spherical wave. Vector of complex amplitudes of scattered field spherical waves. complex amplitude of spherical wave of the -th current mode radiation field. Complex amplitude of spherical wave of the plane wave field; the first superscript letter ( or ) designates the type of spherical wave, the second superscript letter ( or ) denotes vertical or horizontal plane wave polarization. Complex amplitude of the incident field spherical wave. Vector of complex amplitudes of incident field spherical waves. Parameter of scatterer discrete model. Frequency functions determining scatter matrix elements. Complex amplitude of current mode of scatterer excited by unitary spherical wave. Electric field strength vector. Spherical wave radiation electric field intensity vector. Components of the field. Spherical wave incident electric field intensity vector. Incident field components.

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Auxiliary function determined by mode current Cartesian component. Band maximum frequency. Complex resonance frequency. Hankel transform of Magnetic field intensity vector. Spherical wave magnetic field intensity vector. Hankel first- and second-kind spherical functions. Unit imaginary number. Scatterer current concentration vector.

Permeability of free space. The -th zero of the

function.

Basis functions used in expansion of frequency functions. Solution to linear integral equation. Integral equation eigenfunctions. Sampling functions. Scalar spherical wave function. Frequency. Maximum frequency. Complex resonance frequency.

Mode current concentration vector. Cartesian components of mode current concentration vector. Bessel function of the order . First-kind Bessel spherical function. Wavenumber. Wavenumber corresponding to maximum frequency. Complex wavenumber corresponding to complex resonance frequency. Linear operator. Number of Fourier-Bessel series basis functions. Greatest value of spherical wave first index Number of modes. Spherical wave radiation power. Associated Legendre function. Spherical coordinates. Unit vector aligned with . Radius of the scatterer-enclosing sphere. Scatter matrix. Surface of the radius sphere. Scatter matrix element. Integration domain determined by the scatterer. Volume of the radius sphere. In (9)–(13) a point in abstract multidimensional space; scalar variable in all other expressions. Scalar variable. Second-kind Bessel spherical function. Index identifying a Cartesian axis ( or ). and

Permittivity (constant). Plane wave source angular coordinates.

ACKNOWLEDGMENT The authors wish to thank Y. B. Akimov, the technical writer of MERA Networks, for his valuable advice on the manuscript.

REFERENCES [1] 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. [2] V. Degli-Esposti, G. Lombardi, C. Passerini, and G. Riva, “Wide-band measurement and ray-tracing simulation of the 1900-MHz indoor propagation channel: Comparison criteria and results,” IEEE Trans. Antennas Propag., vol. 49, pp. 1101–1110, Jul. 2001. [3] Transient Electromagnetic Fields., L. B. Felsen, Ed. Berlin–New York: Springer-Verlag, 1976. [4] E. K. Miller, “Model-based parameter estimation in electromagnetics,” IEEE Antennas Propag. Mag., vol. 40, no. 1–3, Feb., Apr., Jun. 1998. [5] 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, pp. 1891–1906, Aug. 2003. [6] J. D. Jackson, Classical Electrodynamics, 3rd ed. New York: Wiley, 1998. [7] J. A. Stratton, Electromagnetic Theory. New York: McGraw-Hill, 1941. [8] R. F. Harrington, Time-Harmonic Electromagnetic Fields. New York: McGraw-Hill, 1961. [9] , J. E. Hansen, Ed., Near-Field Antenna Measurements, ser. IEE Electromagnetic Waves Series 26. London, U.K.: Peter Peregrinus Ltd., 1998. [10] P. A. Angot, Complements de mathemaiques al’usare des ingenieurs de l’elektronique et des telecommunications. Paris, 1957. [11] I. P. Kovalyov, SDMA for Multi-Path Wireless Channels. Limiting Characteristics and Stochastic Models.. Berlin, Heidelberg, New York: Springer, 2004. [12] G. A. Korn and T. M. Korn, Mathematical Handbook. New York: McGraw-Hill, 1968. [13] Handbook of Mathematical Functions, M. Abramowitz and I. A. Stegun, Eds. Washington, DC: National Bureau of Standards, 1964, Edited by. [14] O. M. Büyükdura, S. D. Goad, and R. G. Kouyoumjian, “A spherical wave representation of the dyadic Green’s function for a wedge,” IEEE Trans. Antennas Propag., vol. 44, no. 1, pp. 12–22, Jan. 1996. [15] 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.

KOVALYOV AND PONOMAREV: DISCRETE MODELS OF ELECTROMAGNETIC WAVE SCATTERERS

Igor P. Kovalyov graduated from the Nizhny Novgorod University of Technology, Russia, in 1964 and received the Ph.D. degree in 1969. He is currently an Associate Professor and holds the “Theory of Circuits and Telecommunications” Chair at the Nizhny Novgorod State Technical University, and also works as a Researcher for MERA NN, Nizhny Novgorod. He is the author of five monographs, the latest two of which are devoted to high-speed communications systems. His current research interests include MIMO antenna systems and communication channel modeling.

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Dmitry Ponomarev (M’05) received the M.A. degree in telecom engineering and the Ph.D. degree both from the Nizhny Novgorod State Technical University, Nizhny Novgorod, Russia, in 1974 and 1979, respectively. Prior to starting his entrepreneurial career in 1989, he was a Professor at the Nizhny Novgorod State Technical University 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 more than 70 publications and is the holder of 12 patents. He has been President of MERA Group since the day of its inception in 1989. He is also Chairman of the Board of Directors of MERA Networks, Inc. He has built MERA from a small software development business with a handful of employees to a global software engineering leader offering services in the forefront areas of technology. His entrepreneurial spirit is the driving force behind the creation, growth and ongoing success of MERA Networks and MERA Group. The company owes much of its leadership in the Russian software outsourcing market to his personal commitment and contribution.

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Resonance Behavior of Radar Targets With Aperture: Example of an Open Rectangular Cavity Janic Chauveau, Nicole de Beaucoudrey, and Joseph Saillard

Abstract—In the resonance region, the radar scattering response of any object can be modelled by natural poles with the formalism of the singularity expansion method. These natural poles are resonance parameters which provide useful information for the discrimination of radar targets as their general shape, characteristic dimensions and constitution. In the case of an open radar target, high-Q internal resonances and low-Q external resonances occur respectively inside the target and on its surface. Because internal resonances have a higher Q, they may have a higher total energy and can thus be used for target identification. In this paper, we choose to study the resonance behavior of a perfectly conducting rectangular cavity with a rectangular aperture. With this simple example, we intend to show how to distinguish between the two origins of these resonances: external resonances corresponding to traveling waves on the surface of the target and internal resonances corresponding to cavity waves. Indeed, this can be applied to characterize aircrafts, whose apertures (such as inlets, open ducts, airintakes, cavities etc.) contribute significantly to the overall radar cross section. Index Terms—Electromagnetic scattering, natural poles, quality factor, rectangular cavity, resonance.

I. INTRODUCTION

INCE the development of radar systems, extensive research has been conducted on the detection, characterization and identification of radar targets, from a scattered electromagnetic field. In a military context, these targets try to get stealthy to assure their security. A reduction in the radar cross section (RCS) can be provided by using composite materials which absorb electromagnetic waves in usual radar frequency bands. Consequently, studies are involved in lower frequencies. These lower frequency bands correspond to the resonance region for object dimensions of the same order as electromagnetic wavelengths. Therefore, the energy scattered by the target fluctuates significantly and resonance phenomena clearly appear in this region. Extensive studies have been performed to extract these resonances, the most important being the singularity expansion method (SEM) introduced by Baum et al. [1], [2]. For years,

S

Manuscript received October 28, 2008; revised December 04, 2009; accepted December 09, 2009. Date of publication March 29, 2010; date of current version June 03, 2010. The authors are with the Institut de Recherche en Electrotechnique et Electronique de Nantes Atlantique (IREENA), Université de Nantes, Fédération CNRS Atlanstic, Nantes, 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.2010.2046837

the SEM has been used to characterize the electromagnetic response of structures in both time and frequency domains. The SEM was inspired by observing that typical transient temporal responses of various scatterers (e.g., aircrafts, antennas) behave as a combination of exponentially damped sinusoids. Such temporal damped sinusoids correspond to complex conjugate poles called “natural” poles, in the complex frequency domain. The knowledge of these singularities is useful for the discrimination of radar targets and can be used for different purposes of recognition and identification. For example, the E-pulse technique consists in synthesizing “extinction-pulses” signals from the natural poles of an expected target, then convolving them with the measured late-time transient response of a target under test, which leads to zero responses if both targets match [3]–[5]. In fact, the mapping of these natural poles in the Cartesian complex plane behaves as an identity card, enabling one to recognize the detected target by comparison with a database of mapping of poles, created before the experiments for a set of possible targets. Moreover, the information contained in natural poles can give some indication of the general shape, nature and constitution of the illuminated target. The scattering characterization of an aircraft is significant among radar target identification problems. Indeed, jet engine inlets contribute significantly to the overall RCS of an airplane structure. More generally, apertures such as inlets, open ducts and air-intakes can be used for aircraft identification. Thus, numerous studies on scattering from open cavities are found in the literature. The reader is invited to read the paper of Anastassiu [6], which presents an extensive review of methods used to calculate the RCS of such open structures (more than 150 references). Particularly, when illuminated in a suitable frequency band, an aperture in a radar target body can give rise to high-Q internal resonances. Hence the use of natural frequencies of resonance, mainly internal ones, is a relevant basis for open target identification. In this way, Rothwell et al. [7] analytically calculated the natural frequencies of a canonical target: a hollow, perfectly conducting sphere with a circular aperture. Their study presents an interesting behavior of poles in the complex plane depending on whether the poles originate from internal or external sphere resonances. However, they conclude that the sphere is probably not a good candidate for target identification studies because of its modal degeneracy. This paper addresses a somewhat different approach: Using the RCS of a target, we intend to show how resonance parameters depend on its dimensions. For this purpose, we choose to study a perfectly conducting rectangular cavity with a rectangular aperture. On one hand, this example is a more realistic

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model of an air-intake than a spherical cavity. On the other hand, the search for poles no longer uses an analytical method: it uses a numerical one based on SEM and therefore applicable to any target, not just canonical ones. However, the SEM method extracts the whole set of poles, without separating the internal and external poles. Consequently, the main objective of this paper is to show how we can discriminate between the two origins of these poles: external poles corresponding to traveling waves on the surface of the target and internal poles corresponding to internal cavity waves. For this purpose, we first compare poles of the open rectangular cavity with those of a closed rectangular box of the same size. Next, we study trajectories of poles in the complex plane as a function of each dimension of the cavity. In Section II, we present the extraction of resonance parameters from the simulated transfer function of a target and their use in characterizing objects. In Section III, we study how resonance parameters of an open, perfect electric conductor (PEC) rectangular cavity, depend on its dimensions and we intend to distinguish between internal and external resonances.

a damping caused by radiation losses, surface losses and, evenis tually, losses inside dielectric targets. In the following, named the damping coefficient. Indeed, each pair of complex conjugate natural poles is the mathematical representation of a physical resonance. In [9], we stated that the transfer function of a resonator ‘ ’, with a natural angular frequency of resonance and a quality factor , given by

II. EXTRACTION OF RESONANCE PARAMETERS FROM THE TRANSFER FUNCTION OF A RADAR TARGET

The interest in the characterization of targets led to the development of algorithms to extract natural poles and their associated residues, by studying either the impulse response of targets in time domain (1) or the transfer function in frequency domain (2) [2]. For all examples presented in this paper, we use numerical data obtained by an electromagnetic simulation software, which is based on the method of moments [10] and gives the scattered field at different frequencies. Consequently, we choose a frequency domain method [11]–[13]. As an example, we present in Fig. 1, the modulus of the of a perfectly conducting scattered-field transfer function , in free-space, in the angular sphere of diameter (the frequency range corresponding frequency range is [240 MHz–2.52 GHz]). For efficient target characterization, it is important to define a frequency range adapted to the scatterer dimensions, such that it contains not only the fundamental angular frequency of resonance, but also further harmonics [2]. Fig. 2 shows the mapping of natural poles for the studied and ), target in one quarter of the complex plane ( because poles have a negative real part and are complex conjugate. These poles are obtained by using the Cauchy method is approximated by a ratio [11]–[13]: the transfer function of two complex polynomials (2) and the zeros of the denomiare the poles of . nator polynomial For canonical (sphere, cylinder, dipole etc.) as well as complex shape targets, poles are distributed over branches joining the fundamental pole (dominant pole: ) and the harmonic poles [9], [14]. The main advantage of using such natural poles for discrimination of targets is that only three parameters are required to define each resonance mode. Furthermore, in a homogeneous medium, the mapping of is independent of the target orientation natural poles relative to the excitation [15]. Among the whole set of possible poles, only a few contribute appreciably to the target response and are thus sufficient to characterize a radar target [16]. In general, the selected poles are those which are close to the vertical axis.

A radar target is illuminated in the far field by an incident broadband plane wave including resonant frequencies of the target. Consequently, induced resonances occur at these particular frequencies. In the frequency domain, the scattered-field is given by the ratio of the scattered field transfer function to the incident field, for each frequency. In the time domain, the scattered transient response is composed of two successive , corresponding to the early parts. First, the impulsive part, time, comes from the direct reflection of the incident wave on the object surface. In general, for a monostatic configuration in , free space, this forced part is of duration where is the greatest dimension of the target [8]. Next, for the late time , the oscillating part, , is due to the resonance phenomena of the target. The resonant behavior of the late time is characteristic of the studied target and can be used to define a method of identification. The singularity expansion method (SEM) [1], [2] provides a convenient methodology, describing the late time response of various scatterers as a finite sum of exponentially damped sinusoids (1) Conversely, the Laplace transform of (1) gives the scatteredcorresponding to the sum of pairs field transfer function of complex conjugate poles in the complex frequency plane (2) is the complex variable in the Laplace where plane. is the total number of modes of the series. For the singularity, is the residue associated with each natural ( and are complex conjugates pole and ). The imaginary part, , is the resonance anof gular frequency. The real part, , is negative, corresponding to

(3) can be expressed, in the complex plane, as (4) where the complex roots of the denominator of and , are related to and by

, (5)

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Fig. 1. Modulus of the transfer function of the conducting sphere (monostatic ). configuration;

D = 0:17 m

Fig. 3. Representation of the Q-factor as a function of sphere .

(D = 0:17 m)

!

for the conducting

eter , with a wavelength equal to the perimeter the sphere

of

(6)

H (s) of the conducting sphere  ; ! g.

Fig. 2. Mapping of natural poles extracted from of Fig. 1, in the complex plane f

(D = 0:17 m)

In [9], we propose to represent a natural pole not only in Cartesian coordinates, , but also in , where the natural angular frequency of the form , and the quality factor, , are given by (5). resonance, Indeed, this representation of natural poles in separates information better than the usual Cartesian mapping in gives some indication regarding the the complex plane: brings out the resonance bedimensions of the target while havior of the target. Moreover, is discriminatory of the aspect ratio of targets and consequently, gives some indication regarding the general shape of targets. As an example, poles of the conducting sphere, plotted in in Fig. 2, are now plotted in a Cartesian coordinates representation in Fig. 3. We now investigate each and , successively. parameter, First, we compare the natural angular frequency of resonance , to the natural anof the fundamental pole of the sphere, , of a creeping wave travgular frequency of resonance, eling on the surface of the sphere along a great circle of diam-

with , the speed of light in vacuum, and , the path travelled by the wave over the surface. , we get and, With , to be comfrom (6), (Fig. 3), while pared to (Fig. 2). We notice that nearly equals , the modulus of , and differs from , the imaginary part of . Indeed, the representation in permits to show that this external resonance belongs to the creeping wave. Harmonic resonances (2 to 5) follow the same behavior. Secondly, we examine the parameter: we can see that the conducting sphere is a weak resonant target ( for the fundamental pole ). Indeed, the sphere is a compact object: its surface is large relative to its dimensions. Consequently, there are significant losses on the surface, corresponding to a low value [17]–[19]. Moreover, this low resonant behavior can also come from the degeneracy phenomenon of the external poles, due to geometrical symmetries of the sphere [7], [17]. After this general presentation of resonance parameters, we intend to focus on the resonance behavior of radar targets with apertures. We choose the example of an open rectangular cavity. III. STUDY OF A PEC OPEN RECTANGULAR CAVITY A rectangular cavity with an aperture is a more realistic model of an air-intake than a spherical one; moreover, its resonance angular frequencies are well-known. That is why we chose this example to discriminate between the two origins of resonances: either external resonances, corresponding to traveling waves on the surface of the target, or internal resonances, corresponding to internal cavity waves. For this purpose, we first compare results of the PEC open rectangular cavity with those of a PEC closed rectangular box of the same size. Next, in order to understand the resonance behavior of the target, we study trajectories

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Fig. 4. Geometry of the open cavity (slot centered in the wall).

TABLE I CHARACTERISTIC DIMENSIONS OF THE OPEN CAVITY AND THEORETICAL NATURAL ANGULAR FREQUENCIES OF RESONANCE

H!

Fig. 5. Comparison between j ( )j of the open cavity and the closed box.

of natural poles in the complex plane as a function of each dimension of the cavity.

natural angular frequencies, given by (6), which correspond to , , and . Values of these perimeeach perimeter, ters and their corresponding fundamental natural angular frequencies are given in Table I. Moreover, for each characteristic dimension, it is possible to find further harmonic natural angular frequencies. B. Comparison With a PEC Closed Rectangular Box

A. Parameters of the Open PEC Rectangular Cavity The studied rectangular cavity is open on one side with a centered slot (Fig. 4). Its characteristic dimensions are given in Table I: height , width , depth and slot height . The configuration of the excitation is: • frequency band of investigation: [50 MHz; 685 MHz]; • excitation: electric field parallel to the vertical direction and direction of propagation perpendicular to the aperture plane ; • monostatic study; • target (inside and outside) in vacuum. The internal modes of a closed cavity have natural angular frequencies of resonance given by (7) [20] (7) and are respectively the permittivity and the where permeability of the medium inside the cavity; and with ; ; ; being excepted. For an open cavity, the presence of the slot and makes the -factor finite, slightly perturbs the resonant . corresponding to a non-zero damping factor, In the frequency band of investigation, with such dimensions and of the target and such orientations of the electric field , the slot, there are only three possible cavity modes, and , satisfying (7). Their values for the corresponding dimensions are given in Table I, with and in vacuum. The external modes of resonance are waves traveling on the outside surface of the PEC cavity. There are four fundamental

First, we propose to compare the PEC open rectangular cavity with a PEC closed rectangular box of the same dimensions without the slot, both objects being studied with the same excitation configuration. The box being closed, internal resonances cannot be excited from an outside illumination, hence only external resonances are present. Fig. 5 compares the modulus of the scattered-field transfer for the open cavity (solid line) and the PEC function closed box (dashed line). The cavity response presents narrow peaks of resonance occurring at expected resonance angular fre, , . We quencies of each cavity mode, can see another peak of resonance occurring near the natural angular frequency corresponding to the resonance of the slot . Wider peaks are also present corresponding to lower quality of resonance. Indeed, these wide peaks exist for the box response too. Figs. 6 and 7 compare the cavity and the box using both representations of resonance parameters (see Section II): the Cartesian mapping of natural poles in the complex plane and the quality factor as a function of the natural angular frerespectively. First, we exquency of resonance amine poles existing only for the open cavity, i.e., those numbered “1,” “2,” “3,” and “4.” We can see that the angular frequencies of resonance of poles of the open cavity numbered “1,” “2,” and “3” almost correspond to the theoretical angular frequen, and cies of the closed cavity modes (see Table I and Fig. 7). Indeed, these three angular frequencies of resonance coincide with the narrowest resonance peaks of the open cavity response (Fig. 5). Accordingly, these three (Fig. 6) and corpoles have very low damping coefficients (Fig. 7). The resonance anrespond to a high quality factor

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C. Variation in Dimensions of the Open Cavity

Fig. 6. Comparison of the mapping of poles f the closed box.

;

Fig. 7. Comparison of resonance parameters in f! the open cavity and the closed box.

! g of the open cavity and

;

Q g representation of

gular frequency of the pole “4” corresponds to the resonance of the slot with , as in (6). This angular frequency and a lower quality pole has a higher damping coefficient factor than poles “1,” “2” and “3.” Indeed, the peak of resonance occurring at the resonance angular frequency of the slot is wider than the previous peaks corresponding to internal resonances “1,” “2” and “3.” We can see that natural poles “5,” “6,” “7,” and “8” of the open cavity are very close to the natural poles of the box. Consequently, we can state that these four poles correspond to traveling waves on the outside surface of the perfectly conducting cavity. From (6), we can see that these natural poles depend on various perimeters of the target given in Table I. For example, the pole numbered “5” can correspond to both characteristic or . The other three poles appear to correperimeters, spond to harmonic poles, which mainly depend on dimensions and . In conclusion, with this comparison, we have seen that resonances of an open perfectly conducting rectangular cavity depend on the dimensions of the object and have two origins: external resonances corresponding to surface traveling waves and internal resonances corresponding to internal cavity waves.

In order to confirm the above statements, we propose to vary the characteristic dimensions of the cavity and observe the traand jectories of natural poles in both representations . While in the spherical case treated in [7] the diameter of the aperture is the only varying parameter, four parameters, , , and , are successively considered for the open rectangular cavity. 1) Variation of Height of the Cavity: Height varies from 0.30 m to 0.50 m by steps of 0.05 m (5 configurations) and other characteristic dimensions are the same as the initial configuration, denoted as reference (Table I). Fig. 8(a) and (b) show the evolution of resonance parameters as a function of in and representations respectively. We can see in Fig. 8(b) that the natural angular frequency, , of poles “1,” “2,” and “3,” corresponding to cavity modes, hardly varies with . This is due to the orientation of the incident electric field , which points to the direction of (Fig. 4) involving in (7). Furthermore, we can notice that these three internal resonances “1,” “2,” and “3” become more resonant increases) when increases. Next, we can see that the ( resonance angular frequency of the pole “4,” corresponding to , is also nearly constant when varies. On the the slot contrary, poles “5,” “6,” “7,” and “8” are strongly affected by the variation of . Indeed, these poles move as well horizontally as vertically in Fig. 8(a) and mainly horizontally in Fig. 8(b). Thus, the natural angular frequency of resonance, , of these poles strongly depends on the height of the cavity while their value is relatively constant. To summarize: depends on the dimension for poles numbered “5,” • “6,” “7,” and “8.” value depends on the dimension for poles numbered • “1,” “2,” and “3.” 2) Variation of Width of the Cavity: Width varies from 0.45 m to 0.65 m in steps of 0.05 m (5 configurations) and other characteristic dimensions are the same as the reference configuration. We present the results as earlier: the evolution of resonance parameters as a function of in the complex plane (Fig. 9(a)) and in the form (Fig. 9(b)). First, we can see that the change in has a strong effect on poles corresponding to cavity modes (poles “1,” “2,” and “3”). decreases when The angular frequency of resonance increases, according to (7) for and , 2, 3. Moreover, the damping coefficient of these three poles is nearly identical . Next, the for each value of , leading to a low decrease in strong decrease in the angular frequency of pole “4” when increases proves that this pole indeed corresponds to the resonance (6). The angular frequency of pole of the slot “5” varies slightly with . Finally, Fig. 9(b) shows that poles “6” and “7” are dimly affected by the variation of . Furthermore, pole “8” is not always extracted by the Cauchy method; the damping coefficient of this pole is very high and increases with , which makes this pole more difficult to extract: • depends on the dimension for poles numbered “1,” “2,” “3,” “4,” and “5;” • Low variation of for every pole.

CHAUVEAU et al.: RESONANCE BEHAVIOR OF RADAR TARGETS WITH APERTURE: EXAMPLE OF AN OPEN RECTANGULAR CAVITY

Fig. 8. Variation of the height h of the cavity: (a) f

;

! g representation (left); (b) f!

;

Q g representation (right).

Fig. 9. Variation of the width w of the cavity: (a) f

;

! g representation (left); (b) f!

;

Q g representation (right).

3) Variation of Depth of the Cavity: Depth varies from 0.75 m to 0.95 m in steps of 0.05 m (5 configurations) and other characteristic dimensions are the same as the reference configuration. We notice in Fig. 10(a) and (b) that the resonance angular frequency of all the poles, except pole “4,” decreases when the depth of the cavity increases. As for the variation of , we can of poles “1,” see in Fig. 10(a) that the damping coefficient “2,” and “3” is almost constant with . The position of the pole “4” is almost independent of because this pole corresponds to . In Fig. 10(b), we can see that the resonance of the slot only the natural angular frequency of resonance of poles varies with . Indeed, their quality factor stays almost the same when varies and consequently, this variation of does not enhance the resonance of the cavity: depends on the dimension for poles numbered “1,” • “2,” “3,” “5,” “6,” “7,” and “8;” • Low variation of for every pole. 4) Variation of Height of the Slot: The dimension varies from 0.05 m to 0.20 m in steps of 0.05 m. Results for , i.e. when the cavity has one side open (4 1 config-

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urations), are also presented. The other characteristic dimensions are the same as the reference configuration. We observe, in Fig. 11(a) and (b) respectively, the mainly horizontal varia, and the mainly tion of poles in the complex plane , as vertical variation of poles in the representation a function of the dimension . Indeed, when increases, poles “1,” “2,” and “3” corresponding to cavity modes are increasingly damped and, consequently, their quality of resonance decreases. This is due to the fact that, when increases, the power stored in the cavity is lower and dissipation losses are much higher. Next, in Fig. 11(a), we see an intriguing behavior of poles “4” and “7.” the modulus of their damping coefficient first increases with and then, suddenly decreases. Rothwell et al. [7] report similar complicated trajectories of poles for the open spherical cavity: for poles numbered “4” and “7.” • Strong variation of for poles numbered “2” and “3;” Low variation of • Strong variation of for poles numbered “1,” “2,” and “3.” 5) Overview of the Variation of Each Dimension: We first . To consider the natural angular frequency of resonance, quantify its dependence on each dimension, we compute the

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Fig. 10. Variation of the depth d of the cavity: (a) f

Fig. 11. Variation of the height s of the slot: (a) f

;

;

! g representation (left); (b) f!

! g representation (left); (b) f!

TABLE II RELATIVE VARIATION OF ! WITH DIMENSIONS OF THE CAVITY (mean(jd! =dyj=! ) IN m )

variation of with each dimension of the cavity ( , , or ), i.e., , relative to . In Table II, we give the mean for of the absolute value of this variation: each dimension of the cavity and each pole. We estimate that the natural angular frequency of resonance of a pole depends appreciably on the dimensions of the cavity that make the relative variation significant (in bold in Table II).

;

;

Q g representation (right).

Q g representation (right).

Poles “1,” “2,” and “3” are internal resonances corresponding to the first modes of the cavity. Indeed, their angular frequencies of resonance depend only on dimensions (Fig. 9) and (Fig. 10) because of the orientation of the incident electric field , which points to the direction of (Fig. 4), involving in (7). Next, pole “4” corresponds to the resonance of the slot and depends strongly on consequently, is an external resonance. the slot width (Fig. 9) and to a smaller degree, on the slot height (Fig. 11). In the same way, poles “6,” “7,” and “8” are external resonances with a similar behavior according to the variation of dimensions of the cavity: their angular frequencies of resonance are strongly affected by the variation of dimensions (Fig. 8) and (Fig. 10). Thus, we deduce that these poles depend on of the cavity. In fact, these three poles are the perimeter . Moreover, harmonic poles corresponding to the perimeter pole “7” is affected by the slot, its angular frequency of resonance slightly varying as a function of (Fig. 11). mainly depending Pole “5” is an external resonance, with (Fig. 9) and on the dimension (Fig. 8), but also on

CHAUVEAU et al.: RESONANCE BEHAVIOR OF RADAR TARGETS WITH APERTURE: EXAMPLE OF AN OPEN RECTANGULAR CAVITY

(Fig. 10). Consequently, we cannot determine which perimeter is associated with this pole. Anyway, poles “5,” “6,” “7,” and “8,” due to external traveling waves on the surface of the open cavity, are very close to those of the perfectly conducting box with the same dimensions, even if they are modified by the slot (Figs. 6 and 7). We now consider the quality of resonance. Even if the Q-factor of internal cavity modes “1,” “2,” and “3” decreases when the slot height, , increases, it always remains higher than the Q-factor of external poles “4,” “5,” “6,” “7,” and “8.” Indeed, radiating losses are much stronger for external resonances than for internal ones making the internal resonances predominant. IV. CONCLUSION AND PERSPECTIVES In this paper, we have studied the dependence of resonance parameters of an object with its characteristic dimensions. The choice of the open PEC rectangular cavity with its well-known resonance angular frequencies, as well as the comparison with the PEC box, permits us to match resonances with the corresponding characteristic dimensions. Indeed, the variation of these dimensions confirms this dependence. The new representation better separates information: gives some indications on the dimensions of the target while brings out the resonance behavior of targets and gives some indications on the general shape of targets. This study allows determining which resonances correspond to either external surface traveling waves or internal cavity waves. Indeed, internal resonances having a lower damping than external resonances, have a higher quality coefficient of resonance , and can therefore be more easily extracted. This interesting property of internal resonances can be applied for the identification of complex shape targets, such as aircrafts, by characterizing their open cavities (inlets, open ducts, air-intakes etc.). Resonance parameters can be used to recognize a detected target by comparison with a database of poles, created before experiments for a set of possible targets. For automated target recognition applications, the new representation in natural resonant angular frequency and Q-factor turns out to be very efficient as a feature set. REFERENCES [1] C. E. Baum, “The singularity expansion method,” in Transient Electromagnetic Field, L. B. Felsen, Ed. New York: Springer-Verlag, 1976, pp. 129–179. [2] C. E. Baum, E. J. Rothwell, K. M. Chen, and D. P. Nyquist, “The singularity expansion method and its application to target identification,” Proc. IEEE, vol. 79, no. 10, pp. 1481–1492, 1991. [3] E. J. Rothwell, K. M. Chen, and D. P. Nyquist, “Extraction of the natural frequencies of a radar target from a measured response using E-Pulse techniques,” IEEE Trans. Antennas Propag., vol. 35, no. 6, pp. 715–720, 1987. [4] K. M. Chen, D. P. Nyquist, E. J. Rothwell, L. L. Webb, and B. Drachman, “Radar target discrimination by convolution of radar return with extinction-pulses and single-mode extraction signals,” IEEE Trans. Antennas Propag., vol. 34, no. 7, pp. 896–904, 1986. [5] R. Toribio, P. Pouliguen, and J. Saillard, “Identification of radar targets in resonance zone: E-Pulse techniques,” Progr. Electromagn. Res., vol. PIER 43, pp. 39–58, 2003.

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[6] H. T. Anastassiu, “A review of electromagnetic scattering analysis for inlets, cavities and open ducts,” IEEE Antennas Propag. Mag., vol. 45, no. 6, pp. 27–40, 2003. [7] E. J. Rothwell and M. J. Cloud, “Natural frequencies of a conducting sphere with a circular aperture,” J. Electromagn. Waves Applicat., vol. 13, pp. 729–755, 1999. [8] E. M. Kennaugh and D. L. Moffatt, “Transient and impulse response approximations,” Proc. IEEE, vol. 53, pp. 893–901, 1965. [9] J. Chauveau, N. de Beaucoudrey, and J. Saillard, “Characterization of perfectly conducting targets in resonance domain with their quality of resonance,” Progr. Electromagn. Res., vol. PIER 74, pp. 69–84, 2007. [10] FEKO, Software of Electromagnetic Simulations [Online]. Available: http://www.feko.info [11] A. L. Cauchy, “Sur la formule de Lagrange relative à l’interpolation”. Paris: Analyse algébrique, 1821. [12] D. L. Moffatt and K. A. Shubert, “Natural resonances via rational approximants,” IEEE Trans. Antennas Propag., vol. AP-25, no. 5, pp. 657–660, 1977. [13] R. Kumaresan, “On a frequency domain analog of Prony’s method,” IEEE Trans. Acoust. Speech Signal Processing, vol. 38, no. 1, pp. 168–170, 1990. [14] C.-C. Chen, “Electromagnetic resonances of immersed dielectric spheres,” IEEE Trans. Antennas Propag., vol. 46, no. 7, pp. 1074–1083, 1998. [15] A. J. Berni, “Target identification by natural resonance estimation,” IEEE Trans. Aerosp. Electron. Syst., vol. 11, no. 2, pp. 147–154, 1975. [16] J. Chauveau, N. de Beaucoudrey, and J. Saillard, “Selection of contributing natural poles for the characterization of perfectly conducting targets in resonance region,” IEEE Trans. Antennas Propag., vol. 55, no. 9, pp. 2610–2617, 2007. [17] Y. Long, “Determination of the natural frequencies for conducting rectangular boxes,” IEEE Trans. Antennas Propag., vol. AP-42, no. 7, pp. 1016–1021, 1994. [18] P. J. Moser and H. Überall, “Complex eigenfrequencies of axisymmetic perfectly conducting bodies: Radar spectroscopy,” Proc. IEEE, vol. 71, pp. 171–172, 1983. [19] D. L. Moffatt and R. K. Mains, “Detection and discrimination of radar targets,” IEEE Trans. Antennas Propag., vol. AP-23, pp. 358–367, 1975. [20] R. F. Harrington, Time-Harmonic Electromagnetic Fields. New York: McGraw-Hill, 1961.

Janic Chauveau was born in Saint-Nazaire, France, in 1981. He received the M.S. degree in electronics and electrical engineering in 2004 and the Ph.D. degree in 2007 from the University of Nantes, Nantes, France. He is currently working in the Radar team of the IREENA Laboratory (Institut de Recherche en Electrotechnique et Electronique de Nantes Atlantique), University of Nantes. His research interests focus on detection and classification of objects in the resonance region.

Nicole de Beaucoudrey was born in 1954. She received the Ph.D. degree in physics from the University of Paris XI, Orsay, France, in 1979. She was a Researcher with the CNRS (Centre National de la Recherche Scientifique) since 1979. She was at the Institut d’Optique, Orsay, France, until 1999. She is now with IREENA (Institut de Recherche en Electrotechnique et Electronique de Nantes Atlantique), University of Nantes, Nantes, France. Her present research interests concern electromagnetic scattering and the resonance region.

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Joseph Saillard was born in Rennes, France, in 1949. He received the Ph.D. degree and the “Docteur d’Etat” in physics from the University of Rennes, France, respectively, in 1978 and 1984, respectively. From 1973 to 1988, he was employed by the University of Rennes, France, as an Assistant and then as an Assistant Professor. He worked for CELAR Ministry of Defense (1985–1987) on radar signal processing. Now he is a Professor with Polytech’Nantes, University of Nantes, France and is in charge of the Radar Research Team in IREENA Laboratory (In-

stitut de Recherche en Electrotechnique et Electronique de Nantes Atlantique). His fields of interest are the radar polarimetry, adaptive antennas and electronics systems. These activities are done in close collaboration with other public research organizations and industry. Dr. Saillard is the organizer of JIPR’90, ’92, ’95 and ’98. He is a member of the Electromagnetic Academy.

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A Comprehensive Spatial-Temporal Channel Propagation Model for the Ultrawideband Spectrum 2–8 GHz Camillo Gentile, Sofía Martínez López, and Alfred Kik

Abstract—Despite the potential for high-speed communications, stringent regulatory mandates on ultrawideband (UWB) emission have hindered its commercial success. By combining resolvable UWB multipath from different directions, multiple-input multiple-output (MIMO) technology can drastically improve link robustness or range. In fact, a plethora of algorithms and coding schemes already exist for UWB-MIMO systems, however these papers use simplistic channel models in simulation and testing. While the temporal characteristics of the UWB propagation channel have been well documented, surprisingly there currently exists but a handful of spatial-temporal models to our knowledge, and only two for bandwidths in excess of 500 MHz. This paper proposes a comprehensive spatial-temporal channel propagation model for the frequency spectrum 2–8 GHz, featuring a host of novel parameters. In order to extract the parameters, we conduct an extensive measurement campaign using a vector network analyzer coupled to a virtual circular antenna array. The campaign includes 160 experiments up to a non line-of-sight range of 35 meters in four buildings with construction material varying from sheetrock to steel. Index Terms—Multiple-input multiple-output (MIMO), uniform circular array.

I. INTRODUCTION

U

LTRAWIDEBAND (UWB) technology is characterized by a bandwidth greater than 500 MHz or exceeding 20% of the center frequency of radiation [1]. Despite the potential for high-speed communications, the FCC mask of 41.3 dBm/MHz EIRP in the spectrum 3.1–10.6 GHz translates to a maximum transmission power of 2.6 dBm. This limits applications to moderate data rates or short range. Multiple-input multiple-output (MIMO) communication systems exploit spatial diversity by combining multipath arrivals from different directions to drastically improve link robustness or

Manuscript received October 23, 2008; revised October 13, 2009; accepted December 24, 2009. Date of publication March 29, 2010; date of current version June 03, 2010. C. Gentile is with the Emerging and Wireless Networking Technologies Group, Advanced Network Technologies Division, National Institute of Standards and Technology, Gaithersburg, MD 20899 USA (e-mail: camillo. [email protected]). S. M. López was with the National Institute of Standards and Technology, Emerging and Wireless Networking Technologies Group, Gaithersburg, MD 20899 USA. She is now with Orange-France Télécom Group, Portet-sur-Garonne, 31128, France. A. Kik is with the Tokyo Institute of Technology, Tokyo 152-8550 Japan. 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.2010.2046834

range [2]. Ultrawideband lends to MIMO by enabling multipath resolution through its fine time pulses and the fact that most UWB applications are geared towards indoor environments rich in scattering provides an ideal reception scenario for MIMO implementation; in addition, the GHz center frequency relaxes the mutual-coupling requirements on the spacing between antenna array elements. For these reasons UWB and MIMO fit hand-in-hand, making the best possible use of radiated power to promote the commercial success of UWB communication systems. In fact, a plethora of algorithms and coding schemes already exist for UWB-MIMO systems, exploiting not only spatial diversity, but time and frequency diversity as well [3]–[5]. Yet these papers use simplistic channel models in simulation and testing. While the temporal characteristics of the UWB channel have been well documented in [1], [6]–[14], surprisingly there currently exists but a handful of spatial-temporal channel models to our knowledge [15]–[21], and only two for UWB with bandwidths in excess of 500 MHz [22], [23]. Most concentrate on independently characterizing a few parameters of the channel, but none furnish a comprehensive model in multiple environments which allows total reconstruction of the spatial-temporal response, analogous to the pioneering work in the UWB temporal model of Molisch et al. [1]. In order to fill this void, we propose a detailed UWB spatial-temporal model. Specifically, the main contributions of this paper are: • a frequency-dependent pathgain model: allows reconstructing the channel for any subband within –8 GHz, essential to test schemes using frequency diversity, and incorporates frequency-distance dependence previously modeled separately; • a spatial-temporal response model: introduces the distinction between spatial clusters and temporal clusters, and incorporates spatial-temporal dependence previously modeled separately; • diverse construction materials: to model typical building construction materials varying as sheetrock, plaster, cinder block, and steel rather than with building layout (i.e. office, residential typically have the same wall materials); • high dynamic range: the high dynamic range of our system allows up to 35 meters in non line-of-sight (NLOS) range to capture the effect of interaction with up to 10 walls in the direct path between the transmitter and receiver. The paper reads as follows: Section II describes the frequency and spatial diversity techniques used to measure the spatial-temporal propagation channel. The subsequent section explains the

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design and specifications of our measurement system realized through a vector network analyzer coupled to a virtual circular antenna array, and outlines our suite of measurements. The main Section IV features our proposed stochastic model characterizing the channel with parameters reported individually for eight different environments; given the wealth of accumulated data furnished through our measurement campaign, we attempt to reconcile the sometimes contradictory findings amongst other models due to limited measurements. The last section summarizes our conclusions. II. MEASURING THE SPATIAL-TEMPORAL RESPONSE A. Measuring the Temporal Response Through Frequency Diversity

Fig. 1. The uniform circular array antenna.

of the indoor propagation The temporal response channel is composed from an infinite number of multipath arrivals indexed through (1) denotes the delay of the arrival in propagating the where distance between the transmitter and receiver, and the comaccounts for both attenuation and phase plex-amplitude change due to reflection, diffraction, and other specular effects introduced by walls (and other objects) on its path. has a frequency response The temporal response (2) suggesting that the channel can be characterized through frequency diversity: we sample at rate by transmitting tones across the channel and then at the receiver. Characterizing the channel in measuring the frequency domain offers two important advantages over transmitting a UWB pulse and recording the temporal response directly: 1) it enables extracting the frequency parameter ; 2) a subband with bandwidth and center frequency can be selected a posteriori in reconstructing the channel. The discrete frequency spectrum transforms to a signal with period in the time domain [26], and so choosing allows for a maximum multipath spread of 800 ns which proves sufficient throughout all four buildings for the arrivals to subside within one period and avoid time aliasing. B. Measuring the Spatial Response Through Spatial Diversity Replacing the single antenna at the receiver with an antenna array introduces spatial diversity into the system. This enables measuring both the temporal and spatial properties of the UWB channel. We chose to implement the uniform circular array (UCA) over the uniform linear array (ULA) in light of the following two important advantages: 1) the azimuth of the UCA covers 360 in contrast to the 180 of the ULA; 2) the beam pattern of the UCA is uniform around the azimuth angle while that of the ULA broadens as the beam is steered from the boresight.

Consider the diagram in Fig. 1 of the uniform circular array. The elements of the UCA are arranged uniformly around its , . perimeter of radius , each at angle The radius determines the half-power beamwidth corresponding to [27]. Let be the frequency response of the channel between the transmitter and reference center of the receiver array. Arrival approaching from angle hits element with a delay with respect to the is a center [28], hence the element frequency response phase-shifted version of by the steering vector, or (3) is generated through The array frequency response beamforming by shifting the phase of each element frequency response back into alignment at the [28] (4) The spatial-temporal response can then be recovered through the inverse discrete Fourier transform of its array frequency response by synthesizing all the frequencies in the subband (5) where

. III. THE MEASUREMENT SYSTEM AND CAMPAIGN

A. The Measurement System Fig. 2 displays the block diagram (a) and a photograph (b) of our measurement system. The transmitter antenna is mounted on was realized virtually by a tripod while the UCA with mounting the receiver antenna on a positioning table. We sweep elements of the array by automatically repositioning the around its perimeter. At the receiver at successive angles each element , a vector network analyzer (VNA) in turn sweeps the discrete frequencies in the 2–8 GHz band. A total channel measurement, comprising the element sweep and the frequency

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Fig. 2. The measurement system using a vector network analyzer and a virtual circular antenna array. (a) Block diagram; (b) photograph.

sweep at each element, takes about 24 minutes. To eliminate disturbance due to the activity of personnel throughout the buildings and guarantee a static channel during the complete sweep, the measurements were conducted after working hours. During the frequency sweep, the VNA emits a series of tones with frequency at Port 1 and measures the relative ampliwith respect to Port 2, providing autotude and phase matic phase synchronization between the two ports. The long cable enables variable placement of the transmitter and receiver antennas from each other throughout the test area. The height of the two identical conical monopole antennas was set to 1.7 m (average human height). The preamplifier and power amplifier on the transmit branch boost the signal such that it radiates at approximately 30 dBm from the antenna. After it passes through the channel, the low-noise amplifier (LNA) on the receiver branch boosts the signal above the noise floor of Port 2 before feeding it back. -parameter of the network in Fig. 2 can be exThe pressed as follows:

field with dimensions exceeding 100 m 100 m to minimize ambient multipath to a single ground bounce which we subsequently removed by placing electromagnetic absorbers on the ground at their midpoint. Both antennas were set to a height of 1.7 m (average human height). Note in particular the following implementation considerations: • to account for the frequency-dependent loss in the long cable when operating across such a large bandwidth, we ramped up the power at Port 1 with increasing frequency to equalize the radiated power from the transmitter across the whole band; • we removed the LNA from the network in experiments with range below 10 m to protect it from overload and also avert its operation in the non-linear region; • to extend the dynamic range of our system, we exploited the configurable test set option of the VNA to reverse the signal path in the coupler of Port 2 and bypass the 12 dB loss associated with the coupler arm. The dynamic range of the propagation channel corresponds to 140 dB as computed through [30] for an IF bandwidth of 1 kHz and a SNR of 15 dB at the receiver. B. The Measurement Campaign

(6) where the antenna transfer functions and are indexed by to account for the dependence of the antenna gains on the azimuth and elevation of the departure and arrival paths. These terms are approximated to angle independent terms since the interior elevation angle of the conical radiation pattern of the antennas is less than 40 and close to uniform in that small spread, and also because we employed a technique described in [29] which spatially averages the response by rotating the antennas with respect to each other every ten degrees. The branch was measured in a closed-circuit response fashion through a full two-port calibration step. Finally, in order , we to characterize the antenna response separated the antennas by a distance of 1.5 m to avoid near-field operation. The antenna calibration was carried out on a flat open

The measurement campaign was conducted in four separate buildings on the NIST campus in Gaitherburg, Maryland, each constructed from a dominant wall material varying from sheet rock to steel. Table I summarizes the 40 experiments in each building (10 LOS and 30 NLOS), including as an element the maximum number of walls separating the transmitter and receiver. As an example, consider the floor plan of NIST North in Fig. 3: the experiments were drawn from two sets of 22 transmitter locations and 4 receiver locations (marked by the empty and solid circles respectively) to the end of achieving a uniform distribution in range in both LoS and NLoS conditions. The solid line identifies the experiment with the longest range traversing 9 walls between the transmitter and receiver. The between the ground-truth distance and ground-truth angle transmitter and receiver were calculated in each experiment by pinpointing their coordinates on site with a laser tape, and subsequently finding these values using a computer-aided design (CAD) model of each building floor plan.

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and so can equivalently be computed for each experiment through the measured element frequency response in (3). In order to generate a model for the pathgain, consider decomposing the arrival amplitude in (2)

TABLE I EXPERIMENTS CONDUCTED IN MEASUREMENT CAMPAIGN

(8) as a product of the reference amplitude valid at reference and the pathgain factor representing the distance point and frequency dependences of the amplitude. Incorporating the frequency parameter into the model in addition to the conventional attenuation coefficient [25] has been shown to improve channel reconstruction up to 40% for bandwidths in excess of 2 GHz [24]. Now by substituting (2) into (7) and expanding, the pathgain model can be written explicitly in terms of to account for the distance of each experiment as

(9) for and the The reference pathgain attenuation coefficient were extracted at the center frequency by fitting the model above to the data points of the experiments given with varying distance from (7). We actually found the breakpoint model [9] to represent the data much more accurately

Fig. 3. The building plan of NIST North.

IV. THE PROPOSED SPATIAL-TEMPORAL CHANNEL PROPAGATION MODEL

(10)

This section describes the proposed spatial-temporal channel propagation model. It is divided into two components: 1) the reference spatial-temporal response characterizes the shape of and 2) the frethe two-dimensional multipath profile quency-dependent pathgain scales its amplitude according to the distance between the transmitter and receiver and the frequency band of operation. The two corresponding subsections explain the extraction and modeling of the parameters of each component, following by a subsection that outlines in pseudocode how to implement the parameters to generate a stochastic channel response in the eight environments. A. The Frequency-Dependent Pathgain Model The frequency-dependent excess pathgain1 is defined as

augmented by the long-established parameter which quantifies the deviation between our model and the measured data and in that capacity represents the stochastic nature of the pathgain. Next the frequency parameter in (9a) was fit to the remaining data points by allowing the frequency to vary. Based on the geometric theory of diffraction, in previous work [32] we noticed that wall interactions such as transmission, reflection, and diffraction increase from the free space propagation value of zero. As the number of expected interactions increases with distance, a linear dependence of the frequency parameter on can be observed and modeled as (11)

(7) 1There are alternative definitions to ours for the pathgain as summarized in [31].

with positive slope through all environments.2 Fig. 6(a) illustrates the frequency parameter versus the distance for the experiments in the Child Care in NLOS environment and Table II lists the parameters of the pathgain model for all eight environments. 2Only Child Care in LOS exhibited a small negative slope due to lack of data where the building structure limited the longest LOS distance to only 15.3 m.

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TABLE II THE PARAMETERS OF THE PROPOSED SPATIAL-TEMPORAL CHANNEL MODEL FOR THE EIGHT ENVIRONMENTS

B. The Reference Spatial-Temporal Response Our model for the reference spatial-temporal response valid at essentially follows from (1) by augmenting in the dimension as (12) and exchanging with . In order to extract the pawas computed for each rameters of the model, –8 GHz, however by experiment through (5) for first normalizing the measured array frequency response in (4) by the pathgain factor and so replacing it in (5) with instead. Note that the parameters of the pathgain model in the previous and so were subsection were necessary to generate extracted beforehand. Once generated, the arrival data points were extracted from the responses through the CLEAN algorithm in [16]. Only those experiments for which the distance between the transmitter and receiver exceeded 7.7 m, equivaat lent to the Fraunhofer distance for our UCA with , were used to extract in compliance with our far-field assumption in beamforming; however in some scenarios there may have been scatterers present in the near-field of the UCA in violation of that assumption, resulting in a dilated shape of the corresponding arrival in the measured response. In order to mitigate these non-linear effects when isolating the arrivals, the power proved more robust than the complex amplitude to the discrepancies between the template spatial-temporal response used for deconvolution (see [33]) and the actual shape of each arrival which varied from path to path. The iterations of the CLEAN algorithm ceased when less than 20% residual power remained from the initial amount in the response. Only the most significant arrivals, as determined by a power threshold of 27 dB from the maximum peak in the response, were used to fit the model parameters in the sequel. Fig. 5(a) illustrates a synthetic spatial-temporal response generated from 20 arrivals by scaling the template response in amplitude and shifting it in time and angle, all three parameters distributed uniformly. The response was subsequently perturbed by substantial Gaussian

.3 At each arrival, note noise for an equivalent the distinct X sidelobe pattern of the template response.4 Despite the low SNR, the CLEAN algorithm reliably detected all of the arrivals: the ground-truth arrivals are marked as red solid circles while the estimated arrivals are marked as black hollow circles. Fig. 5(b) shows the residual power left after all of the arrivals are identified and removed iteratively. The reference spatial-temporal response partitions the arrivals indexed through into spatial clusters, or superclusters indexed through , and subordinate temporal clusters, or simply clusters indexed through . It reflects our measured responses composed consistently from 1) one direct supercluster arriving first from the direction of the transmitter and 2) one or more wave-guided superclusters arriving later from the door(s) (when placing the receiver in a room) or from the hallway(s) (when placing it in a hallway); the doors and hallways effectively guide the arrivals through, creating “corridors” in the response. Consider as an example the measured response in Fig. 4(a) taken in Child Care with three distinct superclusters highlighted in different colors. The partial floor plan in Fig. 4(b) shows the three corresponding paths colored accordingly and the coorof each path appears as a dot on the response. dinate The direct supercluster arrives first along the direct path and the later two along the wave-guided paths from the opposite for NLOS through the directions of the hallway. We model Poisson distribution5 as (13) for LOS.6 and set The notion of clusters harks back to the well-known phenomenon witnessed in temporal channel modeling [1], [9], [35] caused by larger scatterers in the environment which induce a 3The cumulative amount of noise exceeds that of the signal as the former is spread throughout the whole response while the letter is composed from only a few arrivals. 4The template spatial-temporal response is formulated and illustrated in [33].

P (N ; ) =  e =N !.

5

6We actually observed two superclusters in all our LOS experiments, however

the second arriving with an offset of 180 relative to the first was clearly due to the reflections off the opposite walls attributed to our testing configuration in the hallways rather than to the channel.

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Fig. 4. A measured spatial-temporal response in Child Care with three distinct superclusters. (a) The response h(t; ); (b) the partial floor plan.

Fig. 5. A synthetic spatial-temporal response generated from 20 arrivals and perturbed by Gaussian noise. (a) Before running the CLEAN algorithm; (b) after running the CLEAN algorithm.

delay with respect to the first cluster within a supercluster. Notice the two distinct clusters of each wave-guided supercluster in Fig. 4(a). 1) The Delay : The equations in (14) govern the arrival delays. The delay of the direct supercluster coincides equals the with that of the first arrival. In LOS conditions, , i.e. the time elapsed for the signal ground-truth delay to travel the distance at the speed of light . However our previous work [33] confirms that the signal travels through walls at a speed slower than in free space, incurring an additional . As illustrated in Fig. 6(b), the additional delay delay scales with according to in (14a) since the expected number of walls in the direct path increases with ground-truth delay. Based on the well-known Saleh-Valenzuela (S-V) model [35], , the delay between wave-guided superclusters depends on the randomly located doors or hallways and so obeys the exponential distribution7 in (14a); so does the delay  0

7E (

; L) = (1=L)e

.

between clusters within supercluster in (14b) between arrivals within cluster and the delay in (14c) due to randomly located larger and smaller scatterers respectively

(14) 2) The Angle : As the walls retard the delay of the direct from the groundsupercluster , they also deflect its angle through refraction and diffraction. Our previous truth angle work [33] reveals that the degree of deflection also scales with according to in (15a). Concerning the angle of the wave, our experiments confirm the guided superclusters , uniform distribution in (15a) supported by the notion that the

GENTILE et al.: A COMPREHENSIVE SPATIAL-TEMPORAL CHANNEL PROPAGATION MODEL FOR THE UWB SPECTRUM 2–8 GHz

Fig. 6. Plots of selected model parameters for the Child Care in NLOS environment. (a) Frequency parameter versus distance; (b)  (c) cluster amplitude versus cluster delay; (d) versus cluster delay.

doors and hallways could fall at any angle with respect to the in (15b) aporientation of the receiver. The cluster angle proaches from the same angle as the supercluster due to the guiding effect of the doors and hallways, and in agreement with [15]–[17] the Laplacian distribution8 models the intra-cluster , i.e. the deviation of the arrival angle from angle the cluster angle in (15c)

(15) 3) The Complex Reference Amplitude : Like in the fades exponentially versus S-V model, the cluster amplitude according to in (16a) and as illustrated the cluster delay also fades exponentially in Fig. 6(c); the arrival amplitude versus the intra-cluster delay according to in (16b). Our experiments suggest a linear dependence of on in some buildings confirmed by other researchers [1], [9]. The capparameter drawn from a Normal distribution tures the stochasticity of the amplitude, of particular use when

L

8 (

0

;  ) = (1=2 )e

.

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0

versus cluster delay;

simulating time diversity systems [5]. The arrival phase in (16c) is well-established in literature as uniformly distributed [26]

(16) also to fade exponenWe found the arrival amplitude according to tially versus the intra-cluster angle in (16b). In NLOS the walls spread the arrival amplitude in angle with each interaction on the path to the receiver. The number of expected interactions increases with the cluster delay, as modeled through justifying a linear dependence of on (16b). Fig. 6(d) illustrates this phenomenon for the Child Care in NLOS environment. However we exercise caution in generalizing this phenomenon as indeed it depends on the construction material: in NIST North, even though sheetrock walls are

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the most favorable of the four buildings in terms of signal penetration, the aluminum studs inside the walls spaced every 40 cm act as “spatial filters”, reflecting back those arrivals most deviant from the cluster angle and hence sharpening the clusters in angle with increasing cluster delay, as indicated through negative ; this dependence is less noticeable in Sound and Plant where an order of magnitude less in comparison to the we record other two buildings since the signal propagates poorly through cinder block and steel respectively, and so wave guidance defaults as the chief propagation mechanism. In the past, Spencer [15], Cramer [16], and Chong [17] have claimed spatial-temporal independence: for Spencer, these conclusions were drawn from experiments conducted in buildings with concrete and steel walls similar to Sound and Plant respectively, where we too notice scarce dependence; for Chong and Cramer, the dependence was less observable because the experiments were conducted at a maximum distance of 14 m. C. Reconstructing the Spatial-Temporal Response A stochastic spatial-temporal response can be reconstructed from our model through the following steps: 1) Select (and in turn ), , and the parameters from one of the eight environments in Table II; , , , 2) Generate the stochastic variables and ( ) of the arrivals from the reference spatial-temporal model in Section IV-B: set in (16a) and then normalize the amplitudes to satisfy (9b), keeping only those clusters and arrivals with amplitude above some threshold; –8 GHz with bandwidth 3) Choose a subband in and center frequency , and sample interval ; compute in (2) for each sample frequency from the pathgain model in Section IV-A and the generated arrivals; in (3) from for each 4) Select and compute element in the circular antenna array (note that any array shape can be used by applying the appropriate steering vector); in (4) from which yields the 5) Compute sought spatial-temporal response through (5). V. CONCLUSIONS In this paper, we have proposed a detailed spatial-temporal channel propagation model with 20 parameters for the UWB spectrum 2–8 GHz in eight different environments. The parameters were fit through an extensive measurement campaign including 160 experiments using a vector network analyzer coupled to a virtual circular antenna array. The novelty of the model captures the dependence on the signal propagation delay of the frequency parameter, the delay and angle of the first arrival, and the cluster shape. Most importantly for UWB-MIMO systems, our model discriminates between clusters arriving from the direct path along the direction of the transmitter and those guided through doors and hallways. REFERENCES [1] A. F. Molisch, K. Balakrishnan, D. Cassioli, C.-C. Chong, S. Emami, A. Fort, J. Karedal, J. Kunisch, H. Schantz, U. Schuster, and K. Siwiak, “A comprehensive model for ultrawideband propagation channels,” in Proc. IEEE Global Commun. Conf., Mar. 2005, pp. 3648–3653.

[2] H. A. Khan, W. Q. Malik, D. J. Edwards, and C. J. Stevens, “Ultra wideband multiple-input multiple-output radar,” in Proc. IEEE Radar Conf., May 2005, pp. 900–904. [3] W. P. Siriwongpairat, W. Su, M. Olfat, and K. J. R. Liu, “MultibandOFDM MIMO coding framework for UWB communications systems,” IEEE Trans. Signal Processing, vol. 54, no. 1, Jan. 2006. [4] L. Yang and G. B. Giannakis, “Analog space-time coding for multiantenna ultra-wideband transmissions,” IEEE Trans. Commun., vol. 52, no. 3, March 2004. [5] A. Sibille, “Time-domain diversity in ultra-wideband MIMO communications,” EURASIP J. Appl. Signal Processing, vol. 3, pp. 316–327, 2005. [6] Z. Irahhauten, H. Nikookar, and G. J. M. Janssen, “An overview of ultra wide band indoor channel measurements and modeling,” IEEE Microw. Wireless Compon. Lett., vol. 14, no. 8, Aug. 2004. [7] S. S. Ghassemzadeh, L. J. Greenstein, T. Sveinsson, A. Kavcic, and V. Tarokh, “UWB delay profile models for residential and commercial indoor environments,” IEEE Trans. Veh. Technol., vol. 54, no. 4, Jul. 2005. [8] D. Cassioli, A. Durantini, and W. Ciccognani, “The role of path loss on the selection of the operating band of UWB systems,” in Proc. IEEE Conf. on Personal, Indoor and Mobile Commun., Jun. 2004, pp. 3414–3418. [9] D. Cassioli, M. Z. Win, and A. F. Molisch, “The ultra-wide bandwidth indoor channel: From statistical model to simulations,” IEEE J. Sel. Areas Commun., vol. 20, no. 6, Aug. 2002. [10] S. M. Yano, “Investigating the ultra-wideband indoor wireless channel,” in Proc. IEEE Conf. on Veh. Technol., Spring, May 2002, pp. 1200–1204. [11] C. Prettie, D. Cheung, L. Rusch, and M. Ho, “Spatial correlation of UWB signals in a home environment,” in Proc. IEEE Conf. on Ultra Wideband Systems and Technologies, May 2002, pp. 65–69. [12] A. Durantini, W. Ciccognani, and D. Cassioli, “UWB propagation measurements by PN-sequence channel sounding,” in Proc. IEEE Conf. on Commun., Jun. 2004, pp. 3414–3418. [13] A. Durantini and D. Cassioli, “A multi-wall path loss model for indoor UWB propagation,” in Proc. IEEE Conf. on Veh. Technol., Spring, May 2005, pp. 30–34. [14] J. Kunisch and J. Pump, “Measurement results and modeling aspects for the UWB radio channel,” in Proc. IEEE Conf. on Ultra Wideband Systems and Technologies, May 2002, pp. 19–24. [15] Q. H. Spencer, B. D. Jeffs, M. A. Jensen, and A. L. Swindlehurst, “Modeling the statistical time and angle of arrival characteristics of an indoor multipath channel,” IEEE J. Sel. Areas Commun., vol. 18, no. 3, Mar. 2000. [16] R. J.-M. Cramer, R. A. Scholtz, and M. Z. Win, “Evaluation of an ultrawide-band propagation channel,” IEEE Trans. Antennas Propag., vol. 50, no. 5, May 2002. [17] C.-C. Chong, C.-M. Tan, D. I. Laurenson, S. McLaughlin, M. A. Beach, and A. R. Nix, “A new statistical wideband spatio-temporal channel model for 5-Ghz band WLAN systems,” IEEE J. Sel. Areas Commun., vol. 21, no. 2, Feb. 2003. [18] K. Haneda, J.-I. Takada, and T. Kobayashi, “Cluster properties investigated from a series of ultrawideband double directional propagation measurements in home environments,” IEEE Trans. Antennas Propag., vol. 54, no. 12, Dec. 2006. [19] N. Czink, X. Yin, H. Ozcelik, M. Herdin, E. Bonek, and B. H. Fleury, “Cluster characteristics in a MIMO indoor propagation environment,” IEEE Trans. Wireless Commun., vol. 6, no. 4, April 2007. [20] J. Medbo and J.-E. Berg, “Spatio-temporal channel characteristics at 5 GHz in a typical office environment,” in IEEE Veh. Technol. Conf., Fall, pp. 1256–1260. [21] K. Yu, Q. Li, D. Cheung, and C. Prettie, “On the tap and cluster angular spreads of indoor WLAN channels,” in Proc. IEEE Veh. Technol. Conf., Spring, pp. 218–222. [22] A. S. Y. Poon and M. Ho, “Indoor multiple-antenna channel characterization from 2 to 8 GHz,” presented at the IEEE Conf. on Commun., May 2003. [23] S. Venkatesh, V. Bharadwaj, and R. M. Buehrer, “A new spatial model for impulse-based ultra-wideband channels,” presented at the IEEE Veh. Technol. Conf., Fall, Sep. 2005. [24] W. Zhang, T. D. Abhayapala, and J. Zhang, “UWB spatial-frequency channel characterization,” presented at the IEEE Veh. Technol. Conf., Spring, May 2006. [25] H. Hashemi, “The indoor radio propagation channel,” Proc. IEEE, vol. 81, no. 7, pp. 943–968.

GENTILE et al.: A COMPREHENSIVE SPATIAL-TEMPORAL CHANNEL PROPAGATION MODEL FOR THE UWB SPECTRUM 2–8 GHz

[26] X. Li and K. Pahlavan, “Super-resolution TOA estimation with diversity for indoor geolocation,” IEEE Trans. Wireless Commun., vol. 3, no. 1, Jan. 2004. [27] C. A. Balanis, Antenna Theory: Analysis and Design, Second Edition. New York: Wiley, 1997. [28] T. B. Vu, “Side-lobe control in circular ring array,” IEEE Trans. Antennas Propag., vol. 41, no. 8, Aug. 1993. [29] S. Zwierzchowski and P. Jazayeri, “A systems and network analysis approach to antenna design for UWB communications,” in IEEE Antennas Propag. Society Symp., Jun. 2003, pp. 826–829. [30] J. Keignart and N. Daniele, “Subnanosecond UWB channel sounding in frequency and temporal domain,” in Proc. IEEE Conf. on Ultra Wideband Systems and Technologies, May 2002, pp. 25–30. [31] R. Vaughan and J. B. Andersen, “Channels, propagation and antennas for mobile communications,” IET Electromagnetic Wave Series 50, 2003, Appendix A. [32] C. Gentile and A. Kik, “A frequency-dependence model for the ultrawideband channel based on propagation events,” IEEE Trans. Antennas Propag., vol. 56, no. 2, pp. 2775–2780, Aug. 2008. [33] C. Gentile, A. J. Braga, and A. Kik, “A comprehensive evaluation of joint range and angle estimation in ultra-wideband location systems for indoors,” EURASIP J. Wireless Commun. Netw., 2007, id. 86031. [34] C. Gentile and A. Kik, “A comprehensive evaluation of indoor ranging using ultra-wideband technology,” EURASIP J. Wireless Commun. Netw., 2008, id. 248509. [35] A. Saleh and R. A. Valenzuela, “A statistical model for indoor mulipath propagation,” IEEE J. Sel. Areas Commun., vol. 5, pp. 128–137, Feb. 1987. Camillo Gentile received the B.S. and M.S. degrees from Drexel University, Philadelphia, PA, and the Ph.D. degree from the Pennsylvania State University, University Park, all in electrical engineering. He has been a Researcher in the Advanced Network Technologies Division, National Institute of Standards and Technology (NIST), Gaithersburg, MD, since 2001. His current interests include RF location systems, channel modeling, and smart grids. He served as the Technical Editor for the location annex of the IEEE 802.15.4a standard. In the past, he has worked in the areas of computer vision, pattern recognition, and neural networks.

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Sofía Martínez López received the Telecommunications Engineer degree from the Universidad Politécnica de Cartagena, Spain and the Ph.D. degree from the Ecole Nationale Supérieure des Télécommunications, France. She is currently working for the Orange-France Télécom Group, where she is involved in quality supervision and optimization of the 3G mobile network. Her areas of research interests include radio channel parameter estimation and wireless communication systems.

Alfred Kik was born in Rayak, Lebanon, on November 30, 1978. He received the Electrical Engineering degree from the Lebanese University, in 2001, and a Specialized Masters degree from the Ecole Nationale Supérieure des Télécommunications, Paris, France, in 2004. From 2005 to 2007, he was with the National Institute of Standards and Technology, Gaithersburg, MD, as a Guest Researcher. He was involved in evaluating localization techniques for emergency applications and in UWB indoor channel modeling. Since 2008, he is with the Tokyo Institute of Technology, Tokyo, Japan, working toward the Ph.D. degree. His main research interests are electromagnetic compatibility and materials metrology. Mr. Kik is a member of The Institute of Electronics, Information and Communication Engineers (IEICE), Japan.

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A Correction to Head-Wave Fields for a Simple Planar Contrast of Permittivity Won-seok Lihh

Abstract—When asymptotically describing the wave fields on the denser side of a planar permittivity interface, the canonical headwave fields of the tangential-magnetic (TM) set show discrepancies with the full-wave fields. This paper derives the correction fields that supplement the canonical TM head-wave fields, by taking into account the proximity between the head-wave branch point and an extraneous pole in the wavenumber domain. The validity of the correction is demonstrated in the time domain by comparing the waveforms of a corrected field and the full-wave field, for various ratios of permittivity and at various locations of observation. Consideration is also given to the case of a conducting medium. Index Terms—Approximation methods, dielectric interfaces, electromagnetic radiation, electromagnetic transient propagation, half space.

I. INTRODUCTION

A

POINT source at the interface between two half-space media induces a critical cone, which is a geometrical cone defined by the critical angle of total reflection. Two waves with different ray paths arrive at an observation point in the region between the critical cone and the interface. One wave has conical fronts, and the other has spherical fronts. The conical wave, called the head wave, is associated with a branch point in the complex plane of the wavenumber variable [1, Ch. 3 and 7], [2, Sec. 15-8], [3, Sec. 5.5]. The spherical wave is associated with a stationary-phase point. Well outside the critical cone, the head wave is little affected by the spherical wave, with the branch point lying well off the stationary-phase point. A simple asymptotic evaluation of the integrals around the branch point gives the canonical expressions of the head-wave fields. An artifice in extracting the head-wave component from a wavenumber integral is to construct an integrand having the branch-point radical as a factor, through the denominator rationalization. When the permeabilities of the two media are equal, the extraction of the tangential-electric (TE) head-wave fields involves the branch point only. On the other hand, the extraction of the tangential-magnetic (TM) ones involves not only the branch point but also an extraneous pole introduced in the course of the rationalization. Although the isolated contributions of the pole sum to zero, the pole still exerts an effect unless it lies well off the branch point. Consideration must be given to the proximity between the branch point and the pole, to better describe the TM head-wave fields.

Manuscript received March 22, 2009; revised September 17, 2009; accepted November 03, 2009. Date of publication March 29, 2010; date of current version June 03, 2010. The author is with Global Communication Technology (GCT) Research, Inc., Seoul 156-714, Korea (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2010.2046845

This situation has some similarity with the elastic vacuumsolid case of [4], where the influence of two extraneous Rayleigh poles upon a time-harmonic interfacial field was investigated for a tangential line load applied at the interface. Attention was given to the proximity of the extraneous poles to the head-wave branch point. Unlike the elastic case, only one extraneous pole enters into the present electromagnetic problem. There have been analyses for high contrast in (complex) permittivity [5], [6], [7, Ch. 4], [8], [9, Ch. 3 and 5], assuming that the source and probe depths are fairly small compared with the source-to-probe (interface-projected) distance. This paper does not confine the problem to the close vicinity of the interface or to the high contrast of permittivity, though the interest is in the far-zone fields well outside the critical cone. The author aims to develop the correction fields that supplement the canonical TM head-wave fields, starting from the wavenumber-integral representations of the far-zone fields (Section II). The canonical fields, contributed by the branch point, are extracted in Section III from the wavenumber integrals. Section V derives the correction fields for a lossless medium, based on the asymptotic approximation in Section IV for the integral involving a branch point and a pole. The validity of the correction is demonstrated in Section VI by comparisons in the time domain against the wavenumber-synthetic (full-wave) transients. The correction is also validated for a conducting medium. Dealing with the combined effect of the branch point and the stationary-phase point, in the presence of the pole, might yield more accurate description near the critical cone [10], but at the cost of simplicity and explicitness of the field expressions. This paper does not cover the fields near the critical cone. II. INTEGRAL REPRESENTATION OF TM FIELDS The physical configuration for analysis is shown in Fig. 1. plane The point current source is placed at the origin of the that coincides with the interface. The upper and lower media are and , relossless and have positive permittivities spectively. The two media have the same permeability . The and the zenithal angle azimuthal angle are such that , and , where and . The critical angle is given by (1) which defines the critical cone.

0018-926X/$26.00 © 2010 IEEE

LIHH: A CORRECTION TO HEAD-WAVE FIELDS FOR A SIMPLE PLANAR CONTRAST OF PERMITTIVITY

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For a function of the wavenumber variable , say and operate as the wavenumber syntheses

, let

(2)

and are the second-kind Hankel functions of where order 0 and 1, respectively. In the complex plane, the Sommerfeld integration path runs from the left extreme in the third quadrant to the right extreme in the first quadrant, through the origin. The six TM fields are given by [10]

(3)

(4)

Fig. 1. The head-wave problem with a point current source placed at the permittivity interface. The critical angle  defines the critical cone.

The probe detects the electric and magnetic fields below the interface. Assuming that the source is excited by a Dirac imin the time domain, the analysis is performed in the pulse domain under the Fourier transformaangular-frequency . With the letter standing for , or , tion denotes the -directional electric field genthe notation erated by the tangential source . denotes the field generated by the normal suThe notation . The perinterfacial source and . The fields of the same for the magnetic fields subinterfacial source are times the fields of the superinterfacial source. simply Excluding the -directional source incurs little loss of generality. The TE-TM decomposition in the wavenumber domain allows a grouping of the fields into 1) TE set:

2) TM set:

The first subsets, to the left of the union operators, consist of those fields containing the dominant far-zone components (of the conical as well as the spherical wave), whereas the second subsets consist of the fields without. Of the eight TM fields, this paper deals with the six fields of the first subset. and of the second subset are neglected The fields in the far zone with respect to and , respectively, of the first TE subset.

for the tangential and normal sources, respectively, where and with for . has been omitted in the right-hand sides of (3) The factor terms in (3) are neglected in the far zone and (4). The two with respect to the terms. The analysis for the head wave involves the integration in the neighborhood of the branch . By the asymptotic behavior point (5) for large , the far-zone fields are represented as (6) with the superscript TM dropped, where (7) (8) The path notation indicates the straight line from to , where is a free positive quantity (not necessarily the same each time it occurs). It helps to perform a “rationalization” making the denomiin (6) free of the branch point (as well nator as another branch point ). Then, with the radical split off, the fields in (6) can be put in the form

(9) The functions Table I, where .

and

for the six fields are listed in with

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TABLE I EXPRESSIONS OF (k ) AND

III. CANONICAL HEAD-WAVE FIELDS

(k )

TABLE II EXPRESSIONS OF (k )

is sufficiently smaller than Suppose that the angle the critical angle . Then, there is no close proximity between the branch point and the stationarythat satisfies . (The prime phase point denotes differentiation with respect to .) Hence, well outside the critical cone, the spherical wave associated with has little effect on the head wave associated with . The canonical head-wave fields are obtained by treating the in an isolated manner. For large , the branchbranch point point approximation (10)

as in [3, Sec. 4.8] and [11, Sec. 2.5.2(b)] is used to extract (11) from (9), where where (12) (15) (13) Evaluating (11) with the expressions in Table II gives the canonical head-wave (HW1) fields (14)

with for . As will be seen in Section VI, there are discrepancies between (the time-domain waveforms of) the HW1 fields and the full-wave fields. Resorting to (11) is not sufficient for describing and the head-wave fields. Note that the expressions of possess a pole at . This pole is a removable singularity, since the original expression (6) has no

LIHH: A CORRECTION TO HEAD-WAVE FIELDS FOR A SIMPLE PLANAR CONTRAST OF PERMITTIVITY

proper pole. After the integral in (9) is split for the bracketed terms, the pole is treated as a proper one in the two integrals, while its isolated contributions sum to zero. Despite this canceling, the pole still influences the second integral unless it lies well off the branch point. Consideration must be given to the proximity between the pole and the branch point for better description of the head-wave fields. In the next section, an asymptotic approximation is made for the integral having a square-root branch point and a simple pole.

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TABLE III EXPRESSIONS OF

IV. INTEGRAL WITH BRANCH POINT AND POLE IN PROXIMITY For large , consider the integral (16)

and are real , and is an analytic where indicates the straight line from function. The path notation to . The function in the exponent is real, differentiable, and strictly increasing for real between and . By the change of variable from to , the integral (16) is written as (17)

where

and

(24) (18)

with, say,

. Near

, the linear fit

respectively, where and are positive imaginaries. Rearranging (20) leads to the following asymptotic formula for the integral (16):

(19) (25) lets (17) be approximated by [12, Sec. 2.5] (20)

where (26)

where

(27) (21)

(22) (28) with being an incomplete gamma function [13, Sec. 6.5], [14, Sec. 2.1], [15, Sec. 8.3]. The error of the approximation (22) in absolute value, but is so oscillatory as exceeds to be regarded as negligible. It may be said that (22) is a special case of (10). and are given by The expressions of (23)

with (29) The terms and are the isolated contributions of the pole is due to the and the branch point, respectively. The term proximity between the branch point and the pole.

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V. CORRECTION TO HEAD-WAVE FIELDS A. Derivation of Correction Fields Reconsider the expression (9), for which the following decomposition is now employed: (30)

and where and are respectively the residues of at the pole . The functions and are the pole-free parts. The residues

and

satisfy the relation (31)

since the pole is a removable singularity of the expression . Then, (9) can be rewritten as

(32) where the integrals are taken along the path . The first integral in (32) makes little contribution

p

0

(33) is associated with neither the branch since the function point nor the pole . The second integral in (32) involves the branch point and behaves as [see (10)]

2

Fig. 2. (a) The waveform of the Gaussian pulse g^(t) = 10 (  T ) exp( t =T ) that excites the current source with T = 500 ps. (b) A snapshot 40:9 ns for the of the vertical electric field of the vertical source, taken at t case of " = " and " = 6" . The bright (dark) shades correspond to the positive (negative) values.

according to (25)



, where (38)

(34)

with

and expressed in (12) and (13), respectively. with The third integral involves the pole and behaves as (35) where (36)

(39) in (38) is an asymptotic form of the The denominator for large [13, Eq. 6.5.32]. numerator Through (32) subject to (33)–(35) and (37), the final formula for approximating (9) is obtained as

The fourth integral, involving the combination of the branch point and the pole, behaves as

(40)

(37)

where the residue relation (31) has been used. The first term in the braces, multiplied by the leading factor, is none other than

LIHH: A CORRECTION TO HEAD-WAVE FIELDS FOR A SIMPLE PLANAR CONTRAST OF PERMITTIVITY

+

Fig. 3. The waveforms of E (dashed lines) and E E " " , (b) " " , (c) " " , and (d) " " , when " s r. and 50 ns are attributed to the spherical wave at t

=3

=5

=7

=9 =

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(dotted lines) compared with the waveforms of the full-wave field (solid lines) for (a) . The respective transients around the instants 28.9, 37.3, 44.1,

= " ; r = 5 m, and  = 110

(11) that gives the HW1 fields. The second term in the braces gives the correction fields. By use of

, or a confluent hypergeometric function function by the relation [14, Sec. 2.1], [15, Sec. 8.2, 8.3, and 9.2]

,

(41)

and the expressions of in Table III for the six TM fields, the correction (HW2) fields are obtained as (44)

(42) where

(43)

The use of the error function (erf) appears in [16, Sec. 6.3 and 7.1] and [17] with an argument similar to in (39), though the results are different from that in this paper. is It can be said that the present expression inferred more accurate than the expression from [16, Eq. (7.17)]. The latter is valid when , or when the permittivity contrast is high . Note that the ratios of the HW2 to HW1 fields [(42) to (14)] can be put in the form

The corrected (HW1 plus HW2) fields tend to the canonical as . (HW1) fields for large , by the vanishing of The column matrices in (15) and (43) both imply the wave impedance . B. Variations in (38) can also be expressed in terms of The factor the complementary error function (erfc), a parabolic cylinder

(45)

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+

Fig. 4. The waveforms of E (dashed lines) and E E (dotted lines) compared with the waveforms of the full-wave field (solid lines) for (a) r : m, (b) r m, and (c) r m, when " " ;" " , and  .

= 25 = 110

which, for

= 10

= 40

=

=6

(1 0

Fig. 5. The magnitudes of (a) the ratio E =E , (b) the quantity : " (thick solid), = , and (c) the factor , plotted versus r for " " (thick dashed), " (thick dotted), " (thin solid), " (thin dashed), " ; , and ! and " (thin dotted), when " Grad/s.

9 8) 5 80

10

=

20 = 110

= 25 40 =4

, tends to (46)

and further, to elementwise. The quantity in may be approximated as from (39). , the magnitude of beClose to the interface , which corresponds to the “numerical comes distance” in [18, Sec. 32]. For the high-contrast (lossless) and near-interface case, the head-wave fields can be expressed as

(47) where (48) Use has been made of the first equality of (44). The fields in (47) agree with those of the conducting-medium case in [9, Ch. 3 and 5]. VI. COMPARISONS IN TIME DOMAIN The validity of the correction made in the previous section can be demonstrated in the time domain. The vertical ( -directional) electric field of the vertical source is taken for illustration. The upper medium is assumed to have the free-space permeability

LIHH: A CORRECTION TO HEAD-WAVE FIELDS FOR A SIMPLE PLANAR CONTRAST OF PERMITTIVITY

Fig. 6. The waveforms of E  , (b)  , (c) 

= 95

= 105

and permittivity pulse

(dashed lines) and

E

= 115 , and (d)  = 125

+E

, when "

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(dotted lines) compared with the waveforms of the full-wave field (solid lines) for (a)

= " ; " = 6" , and r = 5 m.

. The source is excited by the Gaussian (49)

ps, shown in Fig. 2(a). For this excitation, a with grayscale snapshot of the full-wave field is shown in Fig. 2(b), ns by the Cagniard-de Hoop method for calculated at the (lossless) case of . The bright (dark) shades correspond to the positive (negative) values. The field is discontinuous across the interface, and the pattern near the source is the static field. The head-wave component has negative values followed by small positive values. In the following, comparisons are made of the time-domain and the corrected field waveforms of the canonical field against the waveforms of the full-wave field. To cover the conducting-medium case, the full-wave field is numerically calculated using the wavenumber-synthetic expression of in (4) (through an efficient evaluation). For a field spectrum , the waveform is constructed by the Fourier synthesis (50) where is the spectrum of the Gaussian pulse in (49). , and Fig. 3 compares the waveforms for at m and . The larger the value of , the larger the discrepancy between the waveforms of the canonical (dashed lines) and the head-wave transients of the field (solid lines). This increasing discrepancy is full-wave field

getting closer to the branch related with the pole point . The waveforms of the corrected field (dotted lines) provide a better description. The solid-line transients around the instants 28.9, 37.3, 44.1, and 50 ns are . attributed to the spherical wave at The discrepancy between the waveforms of and decreases with increasing , as can be seen in Fig. 4 for , and 40 m, when and . The trend of the discrepancy can be glimpsed in Fig. 5(a), which plots the versus magnitude curves of the correction ratio for various values of using (45) (the second-row secondGrad/s. The larger , column element) with the farther the slope region. The larger , in addition, the more do the curves of in Fig. 5(a) resemble the curves in Fig. 5(b) and the curves of in Fig. 5(c), of as predicted in (46). The variation of waveforms with respect to the zenithal angle is shown in Fig. 6 for and m. The dis(dashed lines) and crepancy between the waveforms of (solid lines) becomes larger with increasing . In contrast, waveforms (dotted lines) agree well with the the waveforms. The dependence of on is plotted in Fig. 7 by thick lines for , and with Grad/s. These lines indicate that, well below the critical angles, the correction by the HW2 field is more sig, nificant for larger . The thin lines belong to and are in fair agreement with the corresponding thick lines well below the critical angles. The approximation is acceptable for moderate to high contrast, with approximated or not.

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=

Fig. 7. The magnitudes of the ratio E =E plotted versus  for " " (thick solid), " (thick dashed), " (thick dotted), and " (thick cen" ;r Grad/s. The points at which the tered), when " m, and ! thick lines meet the abscissa indicate the critical angles. The thin lines are the = . magnitudes of

3

5 = =5 (1 0 9 8)

7

9

=4

+ =4

Fig. 9. The waveforms of E (dashed lines) and E E (dotted lines) compared with the waveforms of the full-wave field (solid lines) for the water medium " " of (a)  : S/m and (b)  S/m, when . " " ; r : m, and 

=

+

Fig. 8. The waveforms of E (dashed lines) and E E (dotted lines) compared with the waveforms of the full-wave field (solid lines) for the conducting lower medium of  mS/m, when " " ;" " ;r m, and  as in Fig. 6(c)  .

= 115

=4 ( = 0)

=

=6

=5

The correction for the lossless (lower) medium is also valid for a conducting medium if is replaced by , where is the conductivity. In this case, is complex, and satisfies the pole and for real . Accordingly, the residue relation (31) is replaced by (51) while the first term of (37) is replaced by (52) since (26) becomes (53) The integration path is treated as . Despite these changes, the formula (40) remains the same. mS/m. As in the lossless See Fig. 8 calculated for waveform (dotted line) case in Fig. 6(c), the waveform (solid line). shows good agreement with the Good agreement can also be seen in Fig. 9 for a water medium

( = 80 ) =25 = 91

= 04

with considerable conductivity. Fig. 9(a) is for S/m and Fig. 9(b) is for S/m. The field is for m) due to observed near the interface ( the large attenuation in the impure water. The waveforms of alone (dashed lines) fail to approximate the the field waveforms. Note that, unlike in the lossless medium, the transient HW1 field is not purely impulsive but has a tail. VII. CONCLUSION It has been shown that, to properly describe the TM headwave fields, consideration needs to be given to the proximity between the branch point and the extraneous pole in the complex wavenumber plane. The correction made to the canonical fields has been validated in the time domain by comparisons with the wavenumber-synthetic waveforms. Although the correction fields have been derived for a lossless (lower) medium, they are also valid for a conducting medium if the conductivity is incorporated into the permittivity. Results have been presented for the variations of the vertical field of the vertical source, with respect to: 1) the permittivity of the lower medium, ; 2) the source-to-probe distance ; 3) the zenithal angle of observation, ; and 4) the conductivity of the lower medium, . The interfacial-source case in this paper can be immediately extended to the case of a submerged source. If the upper and lower media have different values of permeability, the present analysis for the TM fields can be applied to the TE fields. The analysis can be extended to be applicable to the elastic headwave fields (off the interface) in the vacuum-solid configuration.

LIHH: A CORRECTION TO HEAD-WAVE FIELDS FOR A SIMPLE PLANAR CONTRAST OF PERMITTIVITY

The role of the extraneous pole is different in the acoustic fluid-fluid case. It is uncommon for an acoustically denser medium to have the lower wave speed. Usually, the pole is closer to the branch point of the inhomogeneous (head) wave rather than to the branch point of the regular head wave. REFERENCES ˇ [1] V. Cervený and R. Ravindra, Theory of Seismic Head Waves. Toronto, Canada: Univ. Toronto Press, 1971. [2] A. Ishimaru, Electromagnetic Wave Propagation, Radiation, and Scattering. Englewood Cliffs, NJ: Prentice-Hall, 1991. [3] L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves. New York: IEEE Press, 1994. [4] L. Tsang, “Time-harmonic solution of the elastic head wave problem incorporating the influence of Rayleigh poles,” J. Acoust. Soc. Amer., vol. 63, pp. 1302–1309, May 1978. [5] J. R. Wait, “The electromagnetic fields of a horizontal dipole in the presence of a conducting half-space,” Can. J. Phys., vol. 39, pp. 1017–1028, July 1961. [6] K. Sivaprasad, “An asymptotic solution of dipoles in a conducting medium,” IEEE Trans. Antennas Propag., vol. 11, pp. 133–142, Mar. 1963. [7] A. Baños Jr., Dipole Radiation in the Presence of a Conducting HalfSpace. London, U.K.: Pergamon, 1966. [8] T. T. Wu and R. W. P. King, “Lateral waves: A new formula and interference patterns,” Radio Sci., vol. 17, pp. 521–531, May/Jun. 1982. [9] R. W. P. King, M. Owens, and T. T. Wu, Lateral Electromagnetic Waves. New York: Springer-Verlag, 1992. [10] W. Lihh, “Asymptotic fields in frequency and time domains generated by a point source at the horizontal interface between vertically uniaxial media,” IEEE Trans. Antennas Propag., vol. 55, pp. 2733–2745, Oct. 2007.

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[11] W. C. Chew, Waves and Fields in Inhomogeneous Media. New York: Van Nostrand Reinhold, 1990. [12] V. A. Borovikov, Uniform Stationary Phase Method. London, U.K.: Inst. Elect. Eng. Press, 1994. [13] , M. Abramowitz and I. A. Stegun, Eds., Handbook of Mathematical Functions. Mineola, NY: Dover, 1972. [14] M. A. Chaudhry and S. M. Zubair, On a Class of Incomplete Gamma Functions With Applications. Boca Raton, FL: Chapman & Hall/CRC, 2002. [15] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, A. Jeffrey and D. Zwillinger, Eds., 7th ed. Burlington, MA: Academic, 2007. [16] A. Baños Jr. and J. P. Wesley, “The Horizontal Electric Dipole in a Conducting Half-Space,” Scripps Institute of Oceanography, La Jolla, CA, Tech. Rep., Sep. 1953 [Online]. Available: http://escholarship.org/uc/ item/5d5419sv [17] K. Sivaprasad and R. W. P. King, “A study of arrays of dipoles in a semi-infinite dissipative medium,” IEEE Trans. Antennas Propag., vol. 11, pp. 240–256, May 1963. [18] A. Sommerfeld, Partial Differential Equations in Physics. New York: Academic, 1949.

Won-seok Lihh received the B.S., M.S., and Ph.D. degrees in electronics/electrical engineering from Seoul National University, Korea, in 1993, 1995, and 2000, respectively. He is currently with Global Communication Technology (GCT) Research, Inc., Seoul, Korea. His research interests include asymptotic and time-domain analysis for wave fields, electromagnetic modeling for circuit components and packages, and CMOS circuit design for RF front-ends.

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Communications H wg M H M

Radiation From Rectangular Waveguide-Fed Fractal Apertures Basudeb Ghosh, Sachendra N. Sinha, and M. V. Kartikeyan

Abstract—We investigate the properties of fractal apertures fed by a rectangular waveguide. The self-similarity and space-filling properties of fractals have been exploited to achieve multi-band radiation. Some self-affine fractal geometries, suitable for waveguide-fed apertures, have been proposed and investigated. It is shown that the scale factor of the fractal geometry can be used as a design parameter for controlling the resonant frequencies. The problem has been solved using method of moments (MoM) and the numerical results are verified through simulation on HFSS and experimental measurements.

where t ( ) is the tangential component of magnetic field in the semi-infinite rectangular waveguide due to the equivalent current , hs t (2 ) denotes the tangential component of magnetic field in thei half-space region due to a current 2 radiating in free space and t is the tangential component magnetic field due to the incident wave. = Nn=1 n n and Expanding the unknown surface current as following the Galerkin procedure, (1) can be expressed in matrix form as

The general problem of coupling between two arbitrary regions via apertures is described in [3]. In the present case, one of the regions is a rectangular waveguide and the other is free space which are coupled through fractal apertures in the waveguide cross-section. Following [3], the apertures are closed with perfect conductors and equivalent surface magnetic currents + and 0 are placed on opposite sides. Enforcement of boundary condition on the tangential component of the magnetic field across the aperture surface yields

M

M

hs i H wg t (M ) + H t (2M ) = 0H t

(1)

Manuscript received October 26, 2008; revised July 29, 2009; accepted January 01, 2010. Date of publication March 29, 2010; date of current version June 03, 2010. The authors are with the Department of Electronics and Computer Engineering, Indian Institute of Technology, Roorkee – 247667, India. (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.2010.2046835

VM

(2)

where

[Y wg ] = 0hM m ;H twg (M n )i N 2N Y hs = 0hM m;H ths(2M n)i N 2N 0! I i = M m;H ti N 21 0! V = [Vn] :

I. INTRODUCTION

II. STATEMENT AND FORMULATION OF THE PROBLEM

M

Y wg + Y hs 0!V = 0!I i

Index Terms—Aperture antennas, fractals, waveguide antennas.

In recent years, a lot of research effort has been directed towards the development of compact multiband antennas. Fractal geometries are being widely used in antenna applications due to their multiband and space-filling properties. A fairly comprehensive review of fractal antenna engineering can be found in [1]. Although a number of fractal antennas have been developed, to the authors’ knowledge, no effort has been made to investigate the multiband radiation properties of fractal apertures. Waveguide-fed aperture antennas are widely used in radar, satellites, phased arrays, and as primary feed to parabolic reflectors and have been a center of study for several decades. It is found in [2] that the input VSWR is quite high for the entire operating band of the open-ended waveguide radiator which can be improved by using resonant apertures. It is, therefore, of interest to investigate the radiation properties of waveguide-fed fractal apertures. In this communication, we present the results on multiband radiation properties of some waveguide-fed fractal apertures.

M

M H

(3) (4) (5)

N 21

(6)

same way by using Green’s function for a semi-infinite waveguide, short circuited at one end. III. NUMERICAL RESULTS Based on the formulation, we have developed a MATLAB code in order to find out the characteristics of the fractal aperture antennas. In all the cases, a WR90 waveguide of cross-sectional dimensions = 22 86 mm and = 10 16 mm has been considered. The numerical results are compared with the results from the simulations on Ansoft’s HFSS, also verified experimentally. It is to be mentioned here that for all the geometries considered here, the solution frequency of HFSS solution setup has been kept at the center of frequency sweep. A single solution set up was defined for each antenna except for Hilbert curve aperture for which two solution set ups with frequency sweeps 7 GHz-13 GHz and 13 GHz-22 GHz were defined. For each antenna, an interpolating sweep has been performed with maximum number of solutions set at 50 with error tolerance equal to 0.5%. Maximum delta S and maximum number of passes for each setup is kept at 0.01 and 25. Some numerical results are given in the following subsections:

b

:

a

:

A. Sierpinski Gasket Aperture Antenna Owing to the arbitrary shape of the apertures, Rao-Wilton-Glisson (RWG) functions are used as the weighting and basis functions. Deterhs is discussed in detail in [4]. mination of the free space matrix The waveguide matrix [ wg ] is computed in the Sierpinski gasket [5] is one of the most preferred geometry in planar antenna design. The self-similarity of Sierpinski gasket geometry is transformed into the multiband behavior and the antenna resonates at multiple frequencies. The band separation is a function of scale factor, , and the ratio of adjacent resonant frequencies can be changed by changing it. In the present work, we have considered a self-affine gasket aperture of two iterations with = 0 8 as shown in Fig. 1. Here, the vertices of the initial triangle 2), and (shown by black lines) have been chosen to be at (0, 0), ( (0 ). The variation of input reflection coefficient of gasket aperture antenna for a TE10 mode excitation is shown in Fig. 2 for two iterations. Also shown in the figure are results of simulation on Ansoft’s

Y

Y

s

;b

s :

0018-926X/$26.00 © 2010 IEEE

a;b=

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TABLE I FREQUENCY RESPONSE OF SIERPINSKI GASKET ANTENNA FOR DIFFERENT SCALE FACTORS

Fig. 1. A self-affine Sierpinski gasket aperture of 2nd iteration with s = 0:8.

Fig. 4. Gain pattern of 2nd iterated Sierpinski gasket aperture antenna with s = 0:8 in ' = 0 plane. Fig. 2. Input reflection coefficient for different iterations of Sierpinski gasket aperture antenna with s = 0:8.

Fig. 5. A 3rd iterated Hilbert curve aperture.

Fig. 3. Input reflection coefficient for a self-affine gasket aperture of 2nd iteration for different scale factors.

HFSS where a good agreement between the two is seen. It can be seen that the first resonant frequency shifts downward in the second iteration and the ratio between the resonant frequencies is 1.32, which is a little greater than the theoretical ratio 1.25. It may be noted here that for all practical fractal geometries, the frequency ratio is always greater than the theoretical value which corresponds to infinite number of iterations. It was found that the smaller aperture of length 3.66 mm (marked as 4 in Fig. 1) had no significant effect on the performance of the antenna due to its electrically small size and location. The input reflection coefficient remained the same even when this aperture was removed. In order to find the effect of scale factor on the antenna performance, we varied the scale factor of the self-affine gasket geometry. The length of triangular aperture of 1st iteration was kept constant and the dimension of second iteration apertures were calculated accordingly. In all these cases, the smaller aperture (marked as 4) was removed. The variation of input reflection coefficient of modified 2nd iterated gasket aperture with different scale factors is shown in Fig. 3. The frequency response

Fig. 6. Input reflection coefficient for different iterations of Hilbert curve aperture antenna.

of the second iteration gasket aperture antenna for different scale factors is summarized in Table I. As seen from the plot, the 1st resonance frequency shifts downwards with the increase in scale factor due to the decrease in the width of triangular aperture. It may be noted that the first resonant frequency, which is determined by aperture marked “1,” changes relatively little with the change in scale factor. The second resonant frequency can be suitably located by changing the scale factor.

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Fig. 11. Input reflection coefficient of 2nd iterated modified plus shape fractal aperture antenna ( : ).

s =07

Fig. 7. Gain patterns of 3rd iterated Hilbert curve aperture antenna at three plane. resonant frequencies in '

=0

TABLE II FREQUENCY RESPONSE OF PLUS SHAPE APERTURE ANTENNA FOR DIFFERENT SCALE FACTORS

Fig. 8. A 2nd iterated plus shape fractal aperture.

Fig. 12. Gain patterns of 2nd iterated plus fractal aperture antenna in ' plane.

Fig. 9. Input reflection coefficient of plus shape fractal aperture antenna with : .

s = 07

=0

The gain patterns of a Sierpinski gasket aperture antenna of second iteration in ' = 0 plane are shown in Fig. 4 at the two resonant frequencies. It can be seen from the patterns that the far field characteristics remain the same at both frequencies. The maximum gain of the antenna is 5.14 dB and 5.75 dB at the two resonant frequencies. In the ' = 90 plane, the antenna was found to have an omnidirectional pattern at the two frequencies. B. Hilbert Aperture Antenna

Fig. 10. Input reflection coefficient of 2nd iterated plus shape fractal aperture antenna with different scale factors.

The space filling property of Hilbert curve [5] has been used for realizing multiband miniaturized monopole antennas, since a relatively longer curve can be fitted into a smaller area. The geometry of a third iteration Hilbert curve aperture in the transverse cross-section of a waveguide is shown in Fig. 5. The dimension of the square into which the curve can be fitted is chosen to be 8 mm 2 8 mm. The variation of input reflection coefficient for different iterations of Hilbert curve aperture antenna is shown in Fig. 6. It is evident that the primary resonant frequency of the aperture antenna decreases significantly at higher-order iterations due to the increase in end-to-end length of the aperture. The

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Fig. 13. A 3rd iterated modified Devil’s staircase fractal aperture (a = 22:86 mm, b = 10:16 mm).

Fig. 14. Input reflection coefficient of 3rd iteration modified Devil’s staircase fractal aperture antenna.

Fig. 16. Simulated and measured return loss of different 2nd iteration fractal aperture antennas. (a) Hilbert curve. (b) Plus shape fractal. (c) Devil’s staircase fractal.

C. Plus Shape Fractal Aperture Antenna

Fig. 15. Gain pattern of 3rd iterated modified Devil’s staircase fractal aperture antenna in ' = 0 plane.

best return loss is obtained for the first iteration. As the order of iteration increases, the return loss deteriorates and also, the bandwidth of the antenna decreases. The ratios between successive resonant frequencies are 1.52 and 1.67, respectively. The gain patterns of 3rd iterated Hilbert aperture antenna at three resonant frequencies in ' = 0 plane are shown in Fig. 7. The nature of the gain patterns remains similar at all the resonant frequencies and the maximum gain of the antenna is around 7.23 dB at the 3rd resonance. Again, the antenna shows an omnidirectional behavior in ' = 90 plane, with a slight increase in the directivity at higher order resonant frequencies.

Plus shape fractal geometries are used in the design of multiband FSS elements [6]. Here, we have considered a self-affine fractal aperture geometry. The initial plus shape has a length 14 mm along x-axis and 8 mm along y -axis, and is placed at the center of waveguide crosssection. This geometry is scaled by 0.7 in x-direction and by 0.5 in y -direction and four such copies are placed at (a=4; b=4), (3a=4; b=4), (3a=4; 3b=4) and (a=4; 3b=4), where a 2 b is the waveguide crosssection. The geometry of plus shape fractal of 2nd iteration is shown in Fig. 8 and the variation of input reflection coefficient for first and second iterations is shown in Fig. 9. It can be seen that 1st resonant frequency shifts downwards for higher iteration. The antenna resonates at 9.75 GHz and 15.9 GHz for 2nd iteration with a frequency ratio of 1.63, as compared to the theoretical value of 1.43. In this case, the location of first resonance is controlled by the length of the initial plus along x-direction while the second resonance, which is determined by the length of smaller plus, can be placed suitably by controlling the scale factor. Fig. 10 compares the return loss of the fractal of Fig. 8

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Fig. 17. Measured normalized radiation pattern of 2nd iteration plus shape fractal antenna. (a) 1st Resonance. (b) 2nd Resonance.

with another second-iterated plus fractal with the length of initial plus as 18 mm and s = 0:6 and the results are summarized in Table II. The effect of the size of vertical arm was also investigated and it was found that reduction in the size of the vertical arm of the initial plus results in a slight increase in the resonant frequencies. This is shown in Fig. 11 for the case where the vertical arm length has been reduced to zero resulting in five rectangular apertures. Thus, an additional parameter is available to designer for fine tuning the desired resonant frequencies. The gain patterns of the 2nd iterated plus shape fractal antenna with s = 0:7 at the two resonant frequencies are shown in Fig. 12 in the ' = 0 plane. It is found that at the 2nd resonance frequency, the pattern becomes more directive and also, side lobes are generated. Again, an omnidirectional pattern was found in ' = 90 plane. D. Devil’s Staircase Fractal Aperture Antenna The generation steps of Devil’s staircase fractal can be found in [5]. An image of the original fractal geometry is taken around the base line and then rotated by 90 . The modified Devil’s staircase fractal geometry of 3rd iteration is shown in Fig. 13 and the input reflection coefficient is shown in Fig. 14. In this case, we obtain four resonances in the frequency range 7 GHz-16 GHz. The resonance at 7.46 GHz and 10.49 GHz are due to apertures of length 20 mm and 14.29 mm of 3rd iteration while the other two resonance frequencies correspond to the apertures of previous iterations. The ratios between the successive frequencies are 1.12, 1.25, and 1.21, which are close to the theoretical values 1.17, 1.20 and 1.25. The gain patterns of the Devil’s staircase antenna at the four resonant frequencies are shown in Fig. 15 in the ' = 00 plane. As seen from the plots, the radiation patterns are stable at all the resonances and the

maximum gain of the antenna is around 5.46 dB. The gain patterns in 0 ' plane are omnidirectional at all the resonant frequencies.

= 90

IV. EXPERIMENTAL VERIFICATION Several waveguide-fed fractal aperture antennas were fabricated with a thin metal sheet and the input characteristics were measured using a HP 8720B network analyzer. The measurement was done only within the operating band (8.2 GHz–12.4 GHz) of X-band rectangular waveguide for which the calibration kit was available. The fabrication of the antennas was limited up to the second iteration. The measured return loss of various 2nd iteration waveguide-fed fractal aperture antennas are shown in Fig. 16 which are in good agreement with the simulated results. Although radiation patterns have been measured at various resonant frequencies for all the antennas described here, because of space limitations, we present here only some representative results. Fig. 17 shows the radiation pattern of plus shape fractal. The measured and simulated results match very well except for a minor difference at  = 90 and  = 270 which is due to the finite size of ground plane used in the experimental antennas. It was found that the cross-polar radiation for all antennas was less than 020 dB in H-plane and less than 010 dB in E-plane. V. CONCLUSION The fractal aperture antennas fed by a rectangular waveguide offers multiband frequency response similar to the planar fractal antennas. The scale factor of the fractal geometries can be used to allocate the resonant frequencies of the aperture antennas. The gain patterns of the aperture antenna remain nearly the same at different resonant frequencies. Most of these aperture antennas offer a gain around 5 dB. The

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space filling property of the Hilbert antenna can be efficient in reducing the antenna aperture dimension, although the input VSWR value increases with higher iterations.

REFERENCES [1] D. H. Werner and S. Ganguly, “An overview of fractal antenna engineering research,” IEEE Antennas Propag. Mag., vol. 45, no. 1, pp. 38–57, Feb. 2003. [2] S. Gupta, A. Chakraborty, and B. N. Das, “Admittance of waveguide fed slot radiators,” in Proc. IEEE Int Symp. Antennas Propag., San Jose, USA, Jun. 1989, vol. 2, pp. 968–971. [3] R. F. Harrington and J. R. Mautz, “Generalized network formulation for aperture problems,” IEEE Trans. Antennas Propag., vol. 24, pp. 870–873, Nov. 1976. [4] B. Ghosh, S. N. Sinha, and M. V. Kartikeyan, “Electromagnetic transmission through fractal apertures in infinite conducting screen,” Progr. Electromagn. Rese. B, vol. 12, pp. 105–138, 2009. [5] H. O. Peitgen, H. Jurgens, and D. Saupe, Chaos and Fractals, New Frontiers in Science. New York: Springer-Verlag, 1992. [6] J. P. Gianvittorio, J. Romeu, S. Blanch, and Y. Rahamat-Samii, “Selfsimilar prefractals frequency selective surfaces for multiband and dual polarized applications,” IEEE Trans. Antennas Propag., vol. 51, pp. 3088–3096, Nov. 2003.

Experimental Demonstration of the Extended Probe Instrument Calibration (EPIC) Technique Ronald J. Pogorzelski

Abstract—A chamber calibration technique for spherical near-field antenna measurements proposed by Pogorzelski is experimentally demonstrated. The chamber was purposely degraded by introducing a metal plate situated so as to produce a strong specular reflection from the antenna under test to the chamber probe. The effects of this artifact were easily observed in the raw data. The chamber with the artifact was calibrated using an open ended waveguide calibration antenna and the resulting calibration coefficients were used to correct the raw measurement producing a result very similar to the measurement carried out in the undegraded chamber over about 30 to 35 dB of dynamic range. Index Terms—Anechoic chambers (electromagnetic), antenna measurements, near-field far-field transformation.

I. INTRODUCTION Spherical near-field scanning has for many years been a commonly applied technique for assessing the performance of antennas in a confined space. The radiated fields are measured at a set of points distributed over a sphere of modest size using a simple probe, such as an open ended waveguide (OEWG), and the resulting field data are expanded in a set of vector spherical harmonics which, of course, have Manuscript received July 17, 2009; revised November 30, 2009; accepted January 13, 2010. Date of publication April 26, 2010; date of current version June 03, 2010. The research described in this communication was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration and at Nearfield Systems, Inc. in Torrance, CA. The author is with the Jet Propulsion laboratory of the California Institute of Technology, Pasadena, CA 91109 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.2010.2048868

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known radial dependence. This expansion is then evaluated at infinite distance to obtain the far-zone radiation pattern. [1] However, when conducting measurements in a confined space, the data are often corrupted by waves reflected from nearby objects including the walls of the room. Thus, it has been common practice to eliminate to the extent possible all unnecessary objects, cover the remaining ones with radio frequency absorbing material, and line the walls of the room with radio frequency absorbing material to reduce reflections to acceptable levels. Moreover, recognizing that the OEWG does not measure exactly the electric field components at each sample point as assumed in the theory of spherical near-field data reduction but, rather, measures an integration of the field over its aperture, practitioners of such measurements have used a conversion technique called “probe correction” to obtain the needed field samples from the probe response. Via such elaborate and extensive precautions and corrections, it has been possible to obtain very good results via spherical near-field scanning. Recently, Pogorzelski proposed a method of calibrating the spherical near-field measurement system so as to remove from the data the effects of what is termed the “extended probe.” [2] The extended probe comprises the OEWG itself as well as all stationary objects in the room including the walls. This calibration technique requires that the probe be stationary and that the spherical scan be accomplished by moving the antenna under test (AUT). One first uses the spherical near-field measurement system to measure the near-field radiation of a calibration antenna of known free-space radiation pattern. This assesses the artifacts induced by the extended probe and produces a set of calibration coefficients. One then measures the AUT and corrects this measurement using the calibration coefficients. The result is the spherical expansion of the free-space pattern of the measured antenna. Since Nearfield Systems, Inc. has equipment suitable for conducting spherical near-field measurements, an experiment using this equipment was designed to demonstrate the Extended Probe Instrument Calibration (EPIC) technique. The experiment and the results are presented and discussed below. II. THE FORMULATION OF EPIC Here, for convenience, an overview of the EPIC formulation is presented. Details may be found in [2]. EPIC is based on the JensenWacker formula for the probe response, w , in terms of the spherical expansion coefficients of the probe, P , and the AUT, T , in the form

w(R; ; ; ) =

s;`;m;

` i Ts`m eim dm ( )e Ps` (R)

(1)

where the index s identifies a TE or a TM mode, ` and m are the polar and azimuthal indices of the spherical harmonics, respectively, and d is expressed as the Fourier series ` m0 dm ( ) = i

` m

=0`

e

` ` im  1m  1 m m

:

(2)

The constants, 1, may be generated recursively. Defining the calibration coefficients

Qs`m (R; ) =



i0 1m`  Ps` (R)ei

(3)

one may express the probe response in the form

w(R; ; ; ) =

0018-926X/$26.00 © 2010 IEEE

s;`;m ;m

Ts`m im 1`m m Qs`m (R; )eim  eim (4)

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that is, a two dimensional Fourier series. As pointed out in [2], the range of each of the angles,  and , is 360 degrees so that the measurement sphere is covered twice. The two spheres of data are not redundant, however. Rather, they are necessary to fully characterize the extended probe. Defining the discrete two dimensional Fourier transform of w to be K and explicitly summing on the index s, one has T1`m im `

1`m m Q1`m +

T2`m im `

1`m m Q2`m = Km m [T ] : (5)

Now, as always, because there are two sets of T 0 s to be determined, one for each polarization, it is necessary to carry out two measurements, one with the OEWG vertically polarized and one with it horizontally polarized. Denoting these two measurements with superscripts (1) and (2), we arrive at T1`m im ` m

T1`m i `

1`m m Q(1) 1`m + 1

(2) ` m m Q1`m

T2`m im `

+

m

T2`m i `

1`m m Q2(1)`m = Km(1)m [T ] (6a)

1

(2) ` m m Q2`m

= Km m [T ] : (2)

(6b) It remains to determine the calibration coefficients, Q. As mentioned above, the calibration coefficients are determined by means a known calibration antenna. Since there are two sets of Q0 to be determined, one must conduct two measurements, with the calibration antenna in two orientations, one in vertical polarization and one in horizontal polarization. Associated with each of these orientations will be a set of known spherical expansion coefficients of the free-space field of the calibration antenna. These are denoted C (1) and C (2) and the equations for determining the Q0 become

s

s

(1)

`

`

C1`m im

1`m m Q1(1)`m +

(2) C1`m im

`

+

C1`m im

1`m m Q1(2)`m +

(2) C1`m im

1

(1)

`

1

(1) ` m m Q1`m

(2) ` m m Q1`m

+

C2`m im

1`m m Q2(1)`m = Km(1)m [C (1)]

(2) C2`m im

1

(1)

`

`

(1) ` m m Q2`m

(7a)

= Km m [C ] (1)

(2)

(7b)

C2`m im

1`m m Q2(2)`m = Km(2)m [C (1)]

(2) C2`m im

1

(1)

`

`

(2) ` m m Q2`m

(8a)

= Km m [C ]: (2)

(2)

(8b) The over determined sets of simultaneous linear equations, (6)–(8) are solved via the well-known least squares method using singular value decomposition to deal with the finite accuracy of the measured data. Basically, one conducts the two calibration measurements and solves (7) and (8) for the Q’s. One then conducts a measurement of the AUT and solves (6) using the now known Q’s to obtain the T ’s representing the free-space radiation of the AUT. III. THE EXPERIMENTAL SET-UP AND PROCEDURE The experiment was carried out at X-band (8.2 GHz) in a 6.1 by 6.1 meter anechoic chamber 6.5 meters high at Nearfield Systems, Inc. The chamber walls are lined with 0.9 meter thick pyramidal absorber. Inside the chamber was an NSI-700S-75 spherical near-field positioner.

Fig. 1. Chamber arrangement for calibration and measurement.

Associated with this positioner is an NSI-RF-90001D RF System consisting of a Panther 9000 Receiver, a Model P7020 Frequency Synthesizer, and an NSI-RF-5940 Distributed Frequency Converter. The calibration antenna was chosen to be an X-band OEWG probe (WR-90). Following Pogorzelski [2], the antenna was oriented perpendicular to the plane of the azimuth and roll axes of the positioner. While ideally it should have been positioned so that its axis passed through the intersection of the azimuth and roll axes, it was, in fact, about 24 centimeters behind that intersection (more distant from the chamber probe) for mounting convenience. The chamber arrangements for both the calibration step and the measurement are diagramed in Fig. 1 as viewed from the top of the chamber. The AUT was chosen to be an SG-90 X-band standard gain horn (SGH). To produce an artifact to be removed via EPIC, a 0.61 by 0.61 meter square aluminum plate was placed in the chamber and supported so as to produce a specular reflection from the SGH into the probe at an azimuth position of the SGH of approximately 45 degrees as indicated in Fig. 1. The calibration antenna on the positioner in the chamber is shown in Fig. 2 and the SGH, the metal plate, and the near-field probe are shown in the chamber in Fig. 3. It is important to recognize that for proper calibration of the chamber, the significant modes of the calibration antenna must cover the significant modes of the AUT. This is achieved in the present case by displacing the OEWG from the intersection of the azimuth and roll axes. . The disIn the present case, the modes of the AUT extend to ` placement of the OEWG of about nine wavelengths produces a calibration antenna with significant modes up ` thus covering all the needed modes for the measurement of the AUT. (In cases where the

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Fig. 2. Calibration antenna on the positioner in the chamber.

Fig. 4. E-plane co-polar patterns.

Fig. 3. Chamber set-up showing the AUT, the chamber probe, and the metal plate.

m

AUT has significant modes with large indices, it is also necessary to take that into consideration in selecting and positioning a calibration antenna.) The calibration antenna was characterized via measurement in the chamber without the metal plate yielding the coefficients for each orientation. As will be seen, the dynamic range of this measurement determines the attainable dynamic range using EPIC. The SGH was also measured to obtain the best possible estimate of the correct ’s for later comparison. The chamber with the plate was calibrated by measurement of the calibration antenna in two orientations with solution of (7) and (8) yielding the coefficients. Again with the plate present, the SGH was measured and (6) were solved to obtain the coefficients. Finally, far-zone pattern cuts were obtained using the ’s and these were compared with similar cuts obtained using the best possible estimate of the ’s from the chamber measurement without the plate.

C

T

Q

T T

T

IV. EXPERIMENTAL RESULTS The results are presented as a series of plots. In each case, the solid line is the best measurement of the SGH in the clean chamber (without the plate) and is taken to be the true pattern. The data are collected over 360 degrees of azimuth and 360 degrees of roll resulting in two full spheres of data. Values at corresponding points on the two spheres are averaged thus partially removing the effects of chamber reflections and even-order aziumthal modes of the probe. The dashed line shows the far-field transformation of the measurement with the plate present and no averaging. Because the specular reflection from the plate is present at every roll angle, the transformed far zone pattern has the specular lobe on both sides of boresight. The crosses show the result of EPIC

processing of the data measured with the plate present and, finally, the dots indicate the difference between the crosses and the solid curve indicating the error level after EPIC processing. The solid and dashed curves and the crosses are normalized to provide directivity using the total radiated power computed by summing the power in each of the spherical modes. Before proceeding further it is important to understand in detail the manner in which the comparisons were made. As stated above, the solid curve in each of Fig. 4 through Fig. 7 is taken to be the correct pattern as it is based on measurements taken in the undegraded chamber. The error in a given result is obtained by converting the magnitude in dB to amplitude and then converting the amplitude and phase into in-phase and quadrature components; i.e., a complex amplitude. The complex amplitudes of the results to be compared are then subtracted to produce a complex error. The amplitude of this error is then converted to error magnitude in dB. The crosses in the figures represent the measured data corrected using EPIC and the error with respect to the solid curves, computed as described above, is plotted as dots in the figures. The error prior to EPIC processing is computed in this same manner and its values at several specific points are quoted in the discussion below but not shown in the figures to preserve clarity. Fig. 4 shows the far-zone co-polar pattern in the E-plane of the SGH. Focusing first on the error at 45 degrees of polar angle, the initial error level of 6.62 dB in the dashed curve has been reduced via the EPIC algorithm to the 024 57 dB level representing more than 31 dB reduction in error. The corresponding error on boresight is at the 9.95 dB level while the residual error on boresight after EPIC processing is at the 07 27 dB level for an error reduction of 17.22 dB. Generally speaking, except near boresight, the error level is below 010 dB compared to the peak directivity of 21.5 dB. This is reasonably consistent with the measurement dynamic range of about 34 dB inferred from the singular value spectrum of the singular value decomposition used in EPIC processing to be discussed further below. The dynamic range of the clean chamber measurement shown by the solid curve is at least 50 dB but the gain of the OEWG calibration antenna is about 15 dB lower than that of the SGH so the effective dynamic range of the OEWG measurement is 15 dB less than that of the SGH measurement resulting in an overall dynamic range closer to 35 dB. Because the cross-pol level is at least 40 dB lower than the co-pol level of the SGH and because the overall dynamic range of the measurement was 30 to 35 dB, the cross-pol was not accurately recoverable

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Fig. 5. E-plane cross-polar patterns.

Fig. 6. H-plane co-polar patterns.

using EPIC as shown in Fig. 5. Incidentally, the similarity in shape of the cross and co-pol patterns (solid lines) suggests that there was a slight mis-alignment of the chamber probe relative to the SGH in roll angle resulting in some polarization mixing. Fig. 6 shows the co-polar pattern in the H-plane and the error level at 45 degrees of polar angle is initially at the 5.25 dB level whereas after EPIC processing it has been reduced to the 021:27 dB level for a net reduction of 26.52 dB. On boresight the error level is initially at the 15.73 level dB but has been reduced to the 07:27 dB level for a 23.04 dB reduction. Here again the residual error level after EPIC processing is below 010 dB except near boresight. Fig. 7 shows the H-plane cross-pol patterns and again indicates that the cross-pol level was too low to be accurately recovered. Returning now to the singular value decomposition (SVD), the threshold for the singular values was set to permit a 34 dB range of singular values resulting in the matrix ranks shown in Fig. 8 for the calibration step; i.e., solving (7a) through (8b). As shown in the plot, for a maximum ` index of 64 used in the data processing, the matrix sizes range from 2 by 2 to 128 by 128 and the ranks for the 34 dB range of singular values range from full rank for the small matrices to a maximum rank of 90. (There are actually two matrices of size

Fig. 7. H-plane cross-polar patterns.

Fig. 8. Significant modes of the SVD solution.

128 by 128 and having different rank as indicated in the plot.) For the measurement step processing; i.e., solving (6a) and (6b), the 34 dB range was used but in that case the matrices ranging up to 64 by 64 were all full rank. V. CONCLUDING REMARKS The experimental results described here clearly show that the EPIC process effectively removes artifacts induced in the data by imperfections in the measurement chamber and probe over a dynamic range that is determined in part by the relative gains of the calibration antenna and the AUT. In the context of the plots presented, dynamic range is defined to be the difference between the co-pol peak (solid curve) and the peak error level (dots) that is above the correct curve. In Fig. 4 this range is 36 dB and in Fig. 6 it is 31 dB. Of critical importance is accurate characterization of the calibration antenna and this is challenging because this antenna is typically much lower in gain than the AUT resulting, in our case, in a higher level of chamber reflection induced error that limited the dynamic range of this characterization and thus the dynamic range attainable via EPIC. It is believed that this dynamic range limitation can be mitigated by measuring the calibration antenna in the undegraded chamber at closer range. This would increase the signal

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measured by the probe relative to the ambient chamber reflections and thus increase the dynamic range of the calibration antenna characterization as is needed to increase the dynamic range of the EPIC processing. ACKNOWLEDGMENT Thanks are due Mr. G. E. Hindman, President of Nearfield Systems, Inc. for providing the AUT and calibration antenna, for the use of an NSI spherical nearfield measurement system, and for his services in operating the system to acquire the data needed for this experimental validation of the EPIC algorithm. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise, does not constitute or imply its endorsement by the United States Government or the Jet Propulsion Laboratory, California Institute of Technology.

REFERENCES [1] Spherical Near-Field Antenna Measurements, J. E. Hansen, Ed. London, U.K.: Peter Peregrinus, 1985, ch. 4. [2] R. J. Pogorzelski, “Extended probe instrument calibration (EPIC) for accurate spherical near-field antenna measurements,” IEEE Trans. Antennas Propag., vol. 57, pp. 3366–3371, Oct. 2009.

A Ku-Band Planar Antenna Array for Mobile Satellite TV Reception With Linear Polarization Reza Azadegan

Abstract—We present a high gain linearly polarized Ku-band planar array for mobile satellite TV reception. In contrast with previously presented three dimensional designs, the approach presented here results in a low profile planar array with a similar performance. The elevation scan is performed electronically, whereas the azimuth scan is done mechanically using an electric motor. The incident angle of the arriving satellite signal is generally large, varying between 25 to 65 depending on the location of the receiver, thereby creating a considerable off-axis scan loss. In order to alleviate this problem, and yet maintaining a planar design, the antenna array is designed to be consisting of subarrays with a fixed scanned beam at 45 . Therefore, the array of fixed-beam subarrays needs to be scanned 20 around their peak beam, which results in a higher combined gain/directivity. The proposed antenna demonstrates the minimum measured gain of 23.1 dBi throughout the scan range (for 65 scan) with the peak gain of 26.5 dBi (for 32 scan) at 12 GHz while occupying a circular aperture of 26 cm in diameter. Index Terms—Microstrip antenna arrays, satellite communications.

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tomotive is one of these highly demanded applications requiring complicated antennas and tracking systems so that the antenna array main beam may be locked to the direction of the satellite while the vehicle follows the road trajectory. Therefore, the antenna needs to be steerable in both elevation and azimuth to meet the high gain required by the stringent link budget for satellite communications systems. Furthermore, the antenna needs to meet the strict size, low visibility, and low profile requirement for mobile or vehicular platforms which calls for the planar solutions. One of the major challenges in antenna design for geostationary satellites is the fact that the view angle of the satellite is rather low ranging from 25 to 65 wherein the gain of the antenna drops considerably compared to its peak gain/directivity at the broadside. This drawback has been typically addressed by the use of three dimensional waveguide antennas or planar antennas mounted at an angle with respect to the horizontal plane creating a built-in slanted elevation [1]–[5]. Obviously, both approaches increase the height of the overall system, as well as its cost and complexity, which may not satisfy the low visibility requirement for many mobile and vehicular platforms. In this communication, a planar array of subarrays with fixed scanned beams at 45 is used to provide a planar array which is scanned electronically in the elevation. The use of the subarray configuration reduces the scan loss due to the fact that each subarray as a new element of the array has a peak directivity at 45 rather than the broadside. The scan loss improvement is moderate compared to what one would expect in cases where non-planar structures such as lenses and reflectors are used. Steered subarray configuration further reduces the overall system cost by lowering the number of phase shifters without contributing to unacceptable grating lobes at large scan angles [6]. Finally, the use of subarrays can eliminate scan blindness completely especially when different types of subarrays are used to break the periodic symmetry of the overall structure [7], [8]. The elevation scan in H-plane is achieved by introducing progressive phase shifts between the subarrays using commercially available SiGe or BST phase shifters, whereas the azimuth tracking is obtained mechanically using an electric motor. Fig. 1 shows the topology of the antenna array which consists of two identical halves for realization of the monopulse tracking. In the realized system, each of the 16 subarrays shown in Fig. 1 has to be connected to a phase shifter and the transceiver circuitry, which are not discussed in this communication. In what follows, the array design procedure is briefly described and the full-wave simulation results are presented. In order to validate the simulated data and to verify the overall array performance, a number of prototypes with fixed beam using fixed delay microstrip feed networks are fabricated and their measured data are compared with the simulated results. II. ARRAY DESIGN METHODOLOGY

I. INTRODUCTION The increased interest in telemetry, entertainment, security, and data communication services on mobile and vehicular platforms calls for the development of compact low cost and low profile satellite antennas with a high gain and tracking capabilities. Satellite TV reception for auManuscript received October 27, 2008; revised December 14, 2009; accepted January 10, 2010. Date of publication March 29, 2010; date of current version June 03, 2010. This work was supported in part by SES AMERICOM. The author was with the Delphi Research Laboratories, Shelby Township, MI USA. He is now with the Solid-State Electronics Lab (SSEL), University of Michigan, Ann Arbor, MI 48109-2122 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.2010.2046836

The antenna array has to meet the system requirements of Ku-band satellite TV broadcast services. As a typical example, the array is designed to meet the specifications of a satellite service operating at 12 GHz (11.8 GHz–12.2 GHz) with a linear horizontal polarization (E' ). The required elevation scan angle is between 25 to 65 with a full 360 scan in azimuth, which are realized electronically and mechanically, respectively. The above elevation scan range covers most of the northern hemisphere including North America and Europe. The required gain depends on the satellite transmit power and the maximum area that can be allocated for the antenna aperture. Considering that this maximum allocable area, as determined by a vehicle manufacturer, should not be larger than a circular disk with the diameter of 25.4 cm, the minimum antenna gain corresponding to the worst case scan angle should be larger than 22.5 dBi.

0018-926X/$26.00 © 2010 IEEE

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Fig. 3. The drawing of the top stick of the subarray type 1. The stick is uniformly excited by resonant series feed with the center patch fed through aperture coupling. The patch dimensions are L1 = L2 = L3 = L6 = 308:5 mil, L4 = 303 mil, L7 = 312 mil; w1 = w2 = w3 = w4 = w5 = 304 mil, w6 = 254 mil, and w7 = 204 mil.

Fig. 1. Topology of the antenna array consisting of three types of subarrays.

Fig. 2. A close up view of one quarter of the array consisting of three types of subarrays with fixed beam tilt at 45 .

In order to obtain a circular aperture, three types of subarrays are used in the array, two of which are corner subarrays (subarray type 1, 2) with a smaller number of elements and the third type is the center subarray (subarray type 3). Each subarray consists of two one-dimensional resonant series-fed microstrip arrays (sticks) fed through aperture coupling. The microstrip patch layer is fabricated on a 0.787 mm thick RT/Duriod 5870 high frequency laminate with a permittivity of 2.2. The feed layer is fabricated on a 0.508 mm thick RO3035 substrate with a permittivity of 3.5 [9]. A. Subarray Design As mentioned earlier, in order to mitigate the gain loss due the large scan angle, without tilting the antenna structure over the horizontal plane, the use of subarrays with a fixed 45 beam scan is proposed. To obtain a beam tilt of 45 in the subarray pattern, a fixed electric delay in the corporate feed exciting the two sticks in each subarray is introduced. The elevation beam can be scanned by introducing a progressive phase shifts among the subarrays in the y -z plane as shown in Fig. 2. The use of subarray topology further reduces the number of phase shifters thereby lowering the overall system cost. Considering that subarrays are now elements of the array, one can observe that the new element pattern has a peak directivity that is tilted to 45 without physically tilting the antenna structure above the horizontal plane. The advantage of the tilted beam element/subarray pattern is that while covering the total scan range of 25 to 65 , the array is effectively scanned up to 620 away from the element/subarray pattern peak. As a result, the scan loss in the proposed planar array with a large scan angle of 65 may be limited effectively to the moderate scan loss of a typical array as if it is scanned to 20 .

Fig. 2 shows a close-up depiction of one quarter of the full array of Fig. 1 consisting of three types of subarrays. As shown, each subarray consists of two linear arrays (sticks) of microstrip patch antennas with a resonant series feed configuration. Series feed is the preferred feeding scheme for relatively large patch arrays as it reduces the insertion loss and the radiation leakage from the feed lines when compared with the corporate feed [10], [11]. The center patch of the stick is fed through an aperture coupled microstrip line on the back of the substrate. The parallel fed center element of each stick provides a symmetry that makes each stick immune to beam squint as a function of frequency [12]. The length of each patch in the stick needs to be slightly tuned to compensate for the effect of the series feed loading. The series microstrip lines connecting the patch elements of the sticks on their radiating edges load the radiating patches capacitively. Such a capacitive loading reduces the resonant frequency of each patch which requires the length of each patch element to be shortened proportional to the loading. The magnitude of the loading along with some closed form approximations are discussed elaborately in ([10] Ch.9). The width of the patch elements in the sticks can also be varied as a parameter to maintain the impedance match and uniform excitation. Fig. 3 shows the top stick of the subarray type 1 with dimensions specified. As illustrated, the length of the center patch is slightly shorter because of the aperture and series feed loadings, but the last patch elements are longer for the lack of such couplings. In order to maintain a uniform spacing between the series elements of each stick while tuning the length of individual patches, the stick elements are placed closer than half a wavelength apart, and the series resonant feed lines connecting the two consecutive elements are meandered. As a result, the distance between the phase centers of all patches of the linear stick remains intact even though the length of the patches can be slightly different due to the tuning. As the first step in implementing the array of subarrays in accordance with the above mentioned guidelines, the subarray type 1 is designed and its simulated reflection coefficient is compared with that of measurement. As Fig. 4 shows, there is a good agreement between the simulation and the measurement, and the input impedance of the patch is perfectly matched at the frequency range of operation around 12 GHz. The simulated radiation pattern of this subarray for horizontal polarization E' is shown in Fig. 5 as a function of both elevation and azimuth angles demonstrating the idea of the tilted beam subarray with 45 fixed elevation in the y -z plane (' = 690 ). The other two types of subarrays, namely, subarray type 2 and 3 are designed following the same outlined procedure. All of the subarrays are then simulated collectively within one quarter of the array enclosed by symmetry planes so that the effect of the mutual coupling among subarrays is accounted for and incorporated in the fine tuning. Fig. 4 shows the simulated input reflections from the three types of subarrays demonstrating a perfect impedance match at the frequency band of interest around 12 GHz. The simulated results also confirm that the coupling between the adjacent subarrays is less than 017 dB (not shown here). The low mutual coupling

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Fig. 6. The schematic block diagram of the feed network of the antenna array consisting of 16 subarrays. The block can be replaced by a fixed beam microstrip lines for simplifying the measurement.

Fig. 4. The simulated and measured input reflection coefficient of the subarray type 1 in the absence of other elements, and the simulated input reflection from three types of subarrays when other elements are present.

TABLE I THE REQUIRED PHASE DIFFERENCE BETWEEN EACH OF THE TWO CONSECUTIVE SUBARRAYS FOR EACH SCAN ANGLE TOGETHER WITH THE REALIZED PEAK GAIN. THE DISTANCE BETWEEN THE STICKS AND SUBARRAYS ARE 11:43 MM (450 MIL), AND 22:48 MM (900 MIL), RESPECTIVELY

the sequential lobbing is used by electronically switching the beam in predefined steps [13]. Considering that the scanning in the elevation is done electronically, and the beam width is not very narrow (approximately 7 ), the tracking can be done adequately fast. Apparently, when the two halves of the array are excited out of phase, they will radiate in phase and provide the sum pattern for normal receiving mode. C. Feed Network

Fig. 5. The normalized horizontally polarized radiation pattern of subarray type 1 as a function of both elevation and azimuth angles. The subarray demonstrates a beam tilt in elevation around 45 in y-o-z plane (' = 90 ).

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between the subarrays can also be verified by comparing the negligible differences between the return loss of the subarray 1 with and without the presence of other subarrays as shown in Fig. 4, respectively. A reasonably low mutual coupling between the subarrays simplifies the design of the feed network as well as the integration of the antenna with the transceiver circuitry. B. Beam Tracking As shown in Fig. 1, the array configuration consists of two symmetrical halves so that the monopulse beam tracking scheme can be realized in the azimuth direction [13]. The monopulse tracking is chosen for azimuth angle finding to partially mitigate the slower response time of the mechanical scan compared with that of the electronic scan. By looking at the symmetry of the array structure, it can be seen that if all input ports of subarrays are excited in phase, the antenna pattern will have a null in the broadside in the plane of symmetry (' = 690 ) with the two adjacent lobes required for monopulse tracking (difference pattern). Considering that the monopulse is a closed-loop tacking, it should be combined with the mechanical implementation of sequential lobbing (lobe switching) for initially finding the azimuth angle by quickly rotating the turning motor in small steps. For the elevation angle finding,

The block diagram of the feed network for the proposed antenna system consisting of 16 phased subarrays is shown in Fig. 6. In the actual system, the phase of the received signals from each subarray can be controlled independently using commercially available phase shifter packages. Therefore, the total number of the phase shifters is limited to a modest number of 16, which is another advantage of using subarray topology. The number of phase shifters could have been further reduced by half if the monopulse tracking for azimuth angle finding had not been used. Given that the design of the antenna array and its performance are the main focus of this communication, fabricating the antenna with standard microstrip corporate feed network with fixed delay is more conducive to accurate test and design validation. In this approach, the phase shifters of Fig. 6 are replaced by a fixed delay microstrip corporate feed network with an electrical delay corresponding to each desired scan angle. (See Table I.) As a result, a separate prototype for each scan angle needs to be fabricated. In these prototypes the antenna layer remains intact while the microstrip corporate feed layer is changed accordingly. III. ARRAY SIMULATION AND MEASUREMENT As mentioned earlier, for both sum and difference patterns, the array maintains a plane of symmetry without which the full wave simulation of the array would not be practical even with the use of 64 bit computers and extensive memory. By taking advantage of the existing symmetry, however, the array may be simulated in less than a day with the memory requirements under 16 GB. The input reflection of 8 subarrays in the half array at the presence of PEC symmetry plane, corresponding to the normal receiving mode (sum pattern), are simulated using commercially available Finite Element Method (FEM) software package [14]. The results shown in Fig. 7 demonstrate that all of the subarrays

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Fig. 7. The HFSS simulation of the input reflection from each of the 8 subarrays within the half array at the presence of the PEC symmetry plane.

Fig. 9. The comparison between the measured and simulated input reflection coefficients of the antenna array scanned to 32 elevation realized by fixed delay microstrip feed network.

Fig. 10. The simulated and measure gain of the antenna array scanned to 32 across the frequency band of operation.

Fig. 8. The photograph of the microstrip feed network creating the required phase difference at the input of each subarray to provide 32 elevation scan.

are perfectly matched over the desired frequency band around 12 GHz. The simulated 8-port s-parameters can then be used to link the simulated antenna results to the microwave feed network including the phase shifters, low noise amplifiers, and the rest of transceiver circuitry. In order to validate the antenna array performance, independently from the phase control and the microwave circuitry, three prototypes are fabricated with the delayed microstrip line feed networks providing the phase shifts required for particular beam scans at 32 , 45 , and 58 . These values have been selected to cover the desired scan range adequately, and also to provide some insight into the performance of the array as a function of the scan angle. Additionally, the elevation scan angle of 32 and 58 correspond to the incident angle from the incoming satellite signal in two of our test facilities, and therefore, the antenna could be used for evaluating in-vehicle system performance and reception quality while driving. Fig. 8 shows the feed network of the array with 32 fixed scanned beam, in which the phase shifters are substituted with fixed delay microstrip line feed network. Fig. 9 shows the input reflection of the antenna pointing to 32 using fixed delay microstrip line feed layer of Fig. 8 as obtained by measurement and full-wave simulation. The results compare fairly well except for a slight difference due to the fact that the measured data includes a Wilkinson power divider with

loading resistor, which is not included in the simulation of the half array. The input impedances of the other two prototypes demonstrate similar agreement with those obtained by simulation. The far field characteristics of these prototype antennas are measured using the near-field scanning where a very good agreement between the simulation and the measurement is observed. The comparison between the simulated and measured gain of the array with 32 scan as a function of frequency is illustrated in Fig. 10 demonstrating a reasonably constant gain over the frequency band of operation. Note that the loss of the microstrip corporate feed has been experimentally calculated (2:7 dB) and compensated for thereafter since it is not a part of the final antenna system. The simulated and measured far-field radiation pattern of these three prototype antennas are plotted in Figs. 11, 12, and 13 for the scan angles of 32 , 45 , and 58 , respectively. As illustrated in these figures, a very good agreement is observed between the measured and simulated radiation patterns particularly for the main lobe and a few adjacent side lobes. As expected, the grating lobes in the radiation patterns of Figs. 11–13 are moderate even for the largest scan angle of 58 notwithstanding the fact that the number of phase shifters is reduced to half through the use of subarray configuration. Furthermore, the gain loss due to the large angle of 58 shown in Fig. 13 is more moderate than what one would expect in a case without a subarray with fixed 45 scanned beam. To demonstrate the effect of the proposed configuration in reducing scan loss, while maintaining a planar topology, the measured realized gain of the antenna as a function of the scan angle at the center frequency is shown in Fig. 14. For comparison, the simulated realized gain of the

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Fig. 11. Measured and simulated H-plane radiation patterns of the full array scanned to 32 ; E'() @ ' = 90 .

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Fig. 14. The realized gain of the antenna array at 12 GHz for different scan angles compared with the case of broadside radiation.

IV. CONCLUSION A high gain compact planar microstrip antenna array with a linear horizontal polarization was designed and tested successfully for the reception of the Ku-band satellite TV signals. The proposed antenna is low cost, light weight, and considerably more compact than its three dimensional waveguide based and vertically upright/slant mounted counterparts. Furthermore, the proposed design demonstrates a fairly high gain and aperture efficiency for large scan angles indicating that the expected scan loss is mitigated partially as a result of proper construction of the contiguous subarrays. ACKNOWLEDGMENT The author would like to thank D. Mateychuk, D. Morris, and S. Shi from Delphi Electronics and Safety for the helpful discussions and assistance with measurements.

REFERENCES Fig. 12. Measured and simulated H-plane radiation patterns of the full array scanned to 45 ; E'( ) @ ' = 90 .

Fig. 13. Measured and simulated H-plane radiation patterns of the full array scanned to 58 ; E'( ) @ ' = 90 .

antenna array without beam scan, when no phase shift is introduced between the sticks, is also included in Fig. 14. As shown, the scan loss at 65 compared with the case with zero scan is about 4.2 dB, which is considerably better than the best expected gain loss of cos(s = 65 ) = 7:5 dB for the standard configuration without subarrays.

[1] Y. Ito and S. Yamazaki, “A mobile 12 GHz DBS television receiving system,” IEEE Trans. Broadcasting, vol. 35, no. 1, Mar. 1989. [2] J. Hirokawa, M. Ando, N. Goto, N. Takahashi, T. Ojima, and M. Uematsu, “A single-layer slotted leaky waveguide array antenna for mobile reception of direct broadcast from satellite,” IEEE Trans. Veh. Technol., vol. 44, no. 4, pp. 749–756, Nov. 1995. [3] K. Sakakibara, Y. Kimura, J. Hirokawa, M. Ando, and N. Goto, “A twobeam slotted leaky waveguide array for mobile reception of dual-polarization DBS,” IEEE Trans. Veh. Technol., vol. 48, pp. 1–6, Jan. 1999. [4] S. Yang and A. E. Fathy, “Slotted arrays for low profile mobile DBS antennas,” in IEEE AP-S Int. Symp. Dig., Jul. 2005, vol. 1B, pp. 827–830. [5] T. Watanabe, M. Ogawa, K. Nishikawa, T. Harada, E. Teramoto, and M. Morita, “Mobile antenna system for direct broadcasting satellite,” in Proc. IEEE Antennas and Propagation Soc. Int. Symp., 1996, pp. 70–73. [6] R. J. Mailloux, Phased Array Antenna Handbook. Norwood, MA: Artech House, 1994, sec. 7.3. [7] D. M. Pozar, “Scanning characteristics of infinite arrays of printed antenna subarrays,” IEEE Trans. Antennas Propag., vol. 40, pp. 666–674, Jun. 1992. [8] D. M. Pozar and D. Schaubert, “Scan blindness in infinite phased arrays of printed dipoles,” IEEE Trans. Antennas Propag., vol. 32, pp. 602–610, Jun. 1984. [9] Advanced Circuit Materials [Online]. Available: http://www.rogerscorporation.com [10] J. R. James and P. S. Hall, Handbook of Microstrip Antennas. London, 1989, Peregrinus. [11] R. Garg, P. Bhartia, I. Bahl, and A. Ittipihoon, Microstrip Antenna Design Handbook. Norwood, MA: Artech House, 2001, ch. 12. [12] J. Huang, “A parallel-series-fed microstrip array with high efficiency and low cross-pol,” Microw. Opt. Technol. Lett., vol. 5, pp. 230–233, May 1992. [13] D. D. Howard, “Tracking Radars,” in Radar Handbook, M. Skolnik, Ed., 3rd ed. New York: McGraw Hill, 2008, ch. 9. [14] HFSS, High Frequency Simulation Software vol. 11, 2008, Ansoft Corp.

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Hybrid-Fractal Direct Radiating Antenna Arrays With Small Number of Elements for Satellite Communications Katherine Siakavara

Abstract—The fractal technique is proposed as an effective procedure for the design of direct radiating arrays (DRAs) with specific operational features, capable to serve a satellite communication network. Large fractal multibeam antenna arrays were synthesized, which meet the requirements of high end of coverage (EOC) directivity, low side lobe level and suppressed level of grating lobes. Their main advantage is the small number of elements and driving points, attributes that minimize the cost and the manufacturing complexity of the system. To obtain the high performance properties of the DRAs, the fractal algorithm was used as a fundamental process and then modification of the original arrays was made by deterministic concepts, and by combining fractal arrays with different generators. So, the entire complex process, produced arrays that would be termed as “hybrid – fractal” antenna arrays. Index Terms—Fractal antennas, large antennas, multi-beam antennas, satellite antennas.

I. INTRODUCTION Contemporary satellite services offer multimedia applications which exhibit high performances in terms of data rate. These requirements are satisfied via an overlapped beam spot coverage with high maximum and end of coverage directivity (DEOC ), high Carrier to Interference (C/I) ratio and low side lobe level (SLL). Classic antennas schemes suitable for these applications are the Focal array fed reflectors (FAFRs) at which in some cases appear problems of the reflector deployment and the feed cluster/reflector alignment. In order to overcome these problems, alternative antenna configurations are investigated. Such antenna arrangements, perhaps the most promising systems, are the direct radiating arrays (DRAs), which operate as phased arrays. In order to obtain the required high directivity, the size of the antenna has to be many wavelengths. As a consequence the number of the elements is very large and if they are fed one by one, the system demands a complex feeding network that increases the cost of manufacturing. The most critical parameter which affects this cost is the number of the required control points that determines the number of the phase shifters and modules which usually are the most expensive components. Techniques to design arrays with high directivity and low side-lobes have been effectively applied in the past, using small number of elements and variable element sizes [1]. In that case large bandwidth was also obtained. Recently, several innovative architectures as the Sparse and Thinned antenna arrays have been introduced [2]–[5], their target being to fulfill the requirements of high gain and low side-lobe level, using small number of elements. They lead to the design of planar arrays with elements positioned in an optimized lattice of non equidistant nodes. This type of arrays has an additional advantage. The deterministic or stochastic algorithms to further optimize their performance, can successfully be integrated in their design procedure [6]–[8]. Manuscript received August 19, 2009; revised November 18, 2009; accepted January 01, 2010. Date of publication March 29, 2010; date of current version June 03, 2010. The author is with the Radiocommunications Lab., Department of Applied and Environmental Physics, School of Science-Faculty of Physics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece (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.2010.2046868

An alternative promising procedure for the design of a DRA configuration is the fractal technique. Fractal arrays besides their frequency-independent multi-band characteristics, have the ability to realize low side-lobes as well as the ability to lead to rapid beamforming algorithms [9]–[11]. Moreover evolutionary techniques that combine the thinning procedure with a fractal algorithm [12] or incorporating a genetic algorithm to a fractal process and producing configurations known as fractal-random arrays [13] can lead to antenna arrangements with attributes attractive to modern applications. The fractal technique is based on the idea of realization the radiation characteristics with a repeating structure in arbitrary scales. In virtue of its nature, the fractal procedure leads to a rapid growth of the antenna size, the rapid increase of populating of the elements, and the quick increment of the gain. The fractal technique has been applied for the synthesis of a DRA [6], [14]. In these works, four stages of fractal development gave an array with a very large number of elements. In spite of this fact, the configuration offered the ability to set the elements in groups and in this way a drastic reduction of driving points was obtained. In the work at hand the properties of a fractal technique were exploited to design a DRA with high performance in terms of the maximum and end of coverage directivity, low side lobe level, suppressed grating lobes and at the same time with small size and mainly a very small number of elements and control driving points. It was ascertained that a canonical fractal process can not, by itself, guarantee operational features that fulfill the prescribed technical specifications. As a first step, a complete theoretical analysis of the fractally designed arrays and also further investigation, to find out the causes of the undesired performance characteristics, were made. Then proper modification of the array via both a deterministic way and the fractal concept followed. So, the produced final arrays are hybrid fractal configurations with two different generators and are capable to meet the technical requirements. II. FORMULATION The fractal technique was used for the synthesis of direct radiating arrays with planar configurations which when operate as phased arrays can create spots with angular width 0.65 . The spots when overlapped, with an angular separation of 0.56 , can cover, on the earth, an almost circular area, with angular width 2 21:445 , whereas the central beam is at the direction  = 0 . The central frequency of operation is 20 GHz and further requirements are: 1) maximum directive gain greater than 45 dBi; 2) directivity at the and of coverage of each spot DEOC > 43 dBi; 3) Carrier to Interference ratio, C=I > 20 dB; 4) S LL < 020 dB; 5) number of elements and control points, the smallest that could be obtained; and 6) the total size of the array to be smaller than 120 2 120 . The basic scheme of a fractally designed radiating system is a generating sub-array. In particular, the entire array can be formed recursively through repetitive application of the generating sub-array under a specified scaling factor which is one of the parameters of the problem. This factor governs how large the array growths by each recursive application of the generating array. Due to the nature of the fractal algorithm the array factor of the nth stage is produced by the multiplication of all the factors of previous stages with the array factor of the nth stage. Moreover at the nth stage, the size of the array is equal to the size of the generating sub-array multiplied by the nth power of the scaling factor and the element population is the (n + 1)th power of the number of elements of generating array. It is exactly this procedure that leads to the rapid growth of both the antenna size and the population of the elements, as well as the quick increment of the gain. In spite of all these benefits, special care has to be taken for the suppression of the grating maxima that inevitably come form the gradual increase of inter-element distances of the array during the process of fractal development.

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Fig. 1. (a) The generator of the fractal array and (b) the first stage of fractal development plus eight elements (in bold).

A. Initial Array For the implementation of the proposed DRAs, a planar array (Fig. 1(a)) with nine elements of circular aperture with radius re = 4:5 , was selected as generator. The criteria of these choices were: a) nine elements would produce, at the first stage of fractal development, a satisfactory large number of 81 elements b) the, relatively, large size of the element aperture imposes a large generator radius, rg = 12 , and in the following, with a proper selection of the scaling factor, the available area for the antenna would be filled without a further fractal development to be necessary. So, the first fractal stage with scaling factor  = 3:4 gave an initial array of 81 elements, shown in Fig. 1(b) (without the elements in bold). The eight elements depicted with bold line were placed next, in order to enforce the azimuthal uniformity of the array and as a consequence the azimuthal uniformity of the radiated field. The array factor of the scheme of Fig. 1(b) is described by (1). The first term is due to the fractal configuration and is the production of the array factor of the generator by the factor which comes from the fractal development. The second term of (1) describes the array factor of the eight additional elements. N

AF1(; ) =

A1 +

An e n=2 N

1

1+

0

jkr sin  cos(  )

+

1

AF1  = Ao

1 + (N 0 1)

1

1 + (N 0 1)

1

N

+ Ao Np

0

0

Jm (kg )e

0

1

 jm(N01)(=20s) Jm (kg )e

01

+

m=

jkr sin  cos(  )

p jkr sin  cos(  )

0

jm(N 1)(=2 s)

01

m=

p

e

An e

lobes exist, due to the large distance between the fractal sub-arrays. The aperture elements are large enough, to produce fields with low level at relatively large angles  and so they could suppress the grating lobes. So, the lobes which are expected to raise problem are solely, those found next to the main lobe, especially at  ' 0:015 rad. These lobes can not be suppressed by the element’s field pattern, because they are very close to its maximum. An approximate description of (1) could help to give a deterministic solution to the problem. The generator, the fractal objects and the array of 8 elements are circular arrays of several wavelengths and supposing that A1 = A2 = 1 1 1 = AN = Ao the array factor of (1) would approximately be described as [15]

1

0

jmN (=2 s)

Jm (kp )e

(2)

n=1

where Jm is Bessel function of the first kind and mth order

n=2 N

Fig. 2. (a) The total and part- array factors of the DRA of Fig. 1(b) ((1)) and (b) plots of the approximate factors ((3)).

(1)

n=1

where N = 9, Np = 8 and rp = 50:5  is the distance from the centre, of the elements shown with bold line. The plots of each term and of the total factor are presented in Fig. 2(a). It is observed that grating

2

Q(; ) = (sin  cos  0 sin o cos o )

2

+ (sin  sin  0 sin o sin o ) p

= rp Q(; )

1=2

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g = rg Q(; ), g imum radiation and

= rg Q(; ), (o ; o ) is the direction of max-

s(; ) = atan (sin  sin  0 sin o sin o ) 1 (sin  cos  0 sin o cos o )01

:

Taking into account that the dominant term at the summations of (2) is that of the zeroth order, a further approximate description of the array factors results AF1

AF1

AF1  Ao [1 + (N 0 1)Jo (kg )] 1 + (N 0 1)Jo (kg ) AF1

+A Np Jo (kp ) : p o

(3)

Fig. 3. The first of the modified DRAs with 89 elements and quantized excitation via one step, P2=P1.

The plots of the terms of (3) are shown in Fig. 2(b). It has to be pointed out that the above equation can describe the array factor, even approximately, for very small values of  at which the exponentials of (2) are very small. As a matter of fact, the problem arises at these space areas. In accordance to Fig. 2(b), the grating lobes around  = 0:015 rad are due to the local negative minimum of the fractal factor AF1A 1 AF1B . Solely another term, with positive values in this area could reduce that negative local minimum. This term would be created by an additional array of elements incorporated to the initial one. Two solutions are proposed in the next section. B. Modified Arrays As a first solution, the central fractal object (Fig. 1(b)) was replaced by a similar generator of nine elements with proper value of radius rgL , in order the argument krgL Q(; ) of the respective factor Jo (krgL Q(; )) to make it positive if krgL Q(; ) < 0:025. The layout of the new array is depicted in Fig. 3. The value of radius rgL = 19:7  and the radii of the new large elements is reL = 7:5 . This solution is not unique. The selection was made under the critera: a) the aforementioned condition to be fulfilled and b) to use elements with large radius and, as a consequence, with high directivity in order to enhance the directivity of the entire DRA. The exact factor of the new DRA is

AF2 = Ao 1 + 2 +

N n=2 N n=1

N n=2

ejkr

Apn ejkr

+ ALo 1 +

ejkr

sin  cos(0 )

sin  cos(0 )

(

sin  cos 0 N

n=2

ejkr

+

)

Fig. 4. The approximate radiation patterns in accordance with (5) (a) uniform excitation and (b) non-uniform excitation with one step of quantization.

(

sin  cos 0

)

: (4)

The approximate form of (4), derived as in the case of the initial array, is AF2

AF2

AF2  Ao [1 + (N 0 1)Jo (kg )] (N 0 1)Jo (kg ) + AF2

AF2

+A [1 + (N 0 1)Jo (kgL )] +A Np Jo (kp ) where gL = rgL Q(; ) L o

p o

(5)

The plots of the terms of (5) are illustrated in Fig. 4(a). The array factor AF2C 1 ALo of the new generator, positioned at the center of the array, gives positive values in the desired area of  , but they are small and can not compensate for the negative values of the fractal term AF2A 1 AF2B . The cancellation can be obtained if the elements of the central generator are driven with power higher than that of the rest elements. It is clearly depicted in Fig. 4(b) in which, as an example, the results were received with current excitation of the large elements, six times that of the smaller ones. Analytical results are presented in Section III. An alternative configuration was derived by modifying the central fractal object of the initial array in a different way (Fig. 5). In this case solely the central element was missed and the new generator with the

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Fig. 5. The second of the modified DRAs, with 97 elements and quantized excitation via one step, P2=P1.

large elements was positioned in the inter-space of the initial elements. In this arrangement the radius of the large fractal generator is rgL = 21:7  and the radius of the large elements is reL = 6:5 , instead of 7.5  of the previous configuration. The criteria for this selection are similar to those of the previous configuration and the selection was made via the exact, (6), and the approximate, (7), array factors N

AF3 = Ao

2

e n=2

jkr sin  cos(0 )

N

jkr sin  cos(0 ) e

1+

+ Ao N

+ n=1 L + Ao

jkr sin  cos(0 ) e

n=2 p jkr sin  cos(0 ) + An e N

1+

jkr e

sin  cos(0 )

n=2

 Ao [(N 0 1)Jo (k

g )]

1 + (N

0 1)Jo (k

AF3

+Ao (N

(6)

AF3

AF3

AF3

+

n=2 N



g )

+

AF3

0 1)Jo(k



g )

L +Ao 1 + (N

0 1)Jo(k

L

g )

+

AF3 p + Ao Np Jo (kp ) :

(7)

III. RESULTS The operation of the initial (Fig. 1(b)) and the modified arrays (Figs. 3 and 5) were simulated using as elements apertures with uniformly distributed field as well as conical horns (TE11 mode). The results are depicted in Figs. 6–8 and were produced via the multiplication of the respective array factors ((4) and (6)) by the field of the elements. It was verified that the ratio of the excitation power of the elements, is a critical parameter for the DRAs’ performance. Fig. 6 shows the records of all the arrays as function of P2=P1, where P2 = ALo and P1 = Ao = Apo , namely the excitation currents of the respective elements. It is ascertained that the values of the indices of operation of the initial fractal array, calculated with uniform apertures, are smaller than those of the modified arrays for all P2=P1.

Fig. 6. Variation of the performance parameters of the DRAs versus the ratio P2=P1.

With concern to the modified arrays the Dmax and DEOC directivities and the SLL, get maximum values for P2=P1 between 3 and 5

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any spot, is required to be at least 20 dB lower than the DEOC of the adjacent co-channel spot, in order to ensure low level of interference. The proposed arrays fulfill this criteria, as C=I = 22:8 dB (Fig. 8). IV. CONCLUSION Hybrid fractal Direct Radiating Arrays with high performance features and suitable for satellite applications were synthesized. The main advantage of the proposed arrays is the very small number of elements and driving points which ensure a feeding network of low cost and complexity. The synthesis was based on the fractal technique which was proved to be efficient for the initial design of the arrays but not enough to achieve the specific technical requirements. An advanced theoretical analysis gave solution to the raised problems and led to modified ’hybrid-fractal’ arrays with high performance indices. The followed process is general and can potentially be applied to the design of any fractal array.

REFERENCES

Fig. 7. Patterns of the central beam, '-cuts from 0 to 90 . (a) 97 conical horns, P2=P1 = 2 . (b) 89 conical horns P2=P1 = 6.

Fig. 8. Beam patterns of two adjacent co-channel spots, 1.12 apart. Configuration of 89 conical horns.

while the C=I ratio increases continuously with P2=P1. In order all the technical requirements to be met the advisable values of P2=P1 are 6 for the array of Fig. 3, and 25=2 for the array of Fig. 5. Fig. 7 presents the '-cuts of the radiation patterns of the arrays with conical horns, for the above ratios. Fig. 8 illustrates the radiation patterns of two beams that illuminate two adjacent co-channel spots, 2 3 0:56 , apart. In most of the applications the level of the directivity pattern, inside the area of

[1] D. Shively and W. Stutzman, “Wideband planar arrays with variable element sizes,” in Antennas and Propagation Society Int. Symp. AP-S Digest, Jun. 26–30, 1989, vol. 1, pp. 154–157. [2] A. Lommi, L. A. Massa, E. Storti, and A. Trucco, “Sidelobe reduction in sparse linear arrays by genetic algorithms,” Microw. Opt. Tech. Lett., vol. 32, pp. 194–196, Feb. 2002. [3] G. Toso, C. Mangenot, and A. G. Roederer, “Sparse and thinned arrays for multiple beam satellite applications,” in Proc. 29th ESA Antenna Workshop on Multiple Beams and Reconfigurable Antennas, Apr. 2007, pp. 207–210. [4] R. L. Haupt, “Thinned arrays using genetic algorithms,” IEEE Trans. Antennas Propag., vol. 41, pp. 993–999, Feb. 2003. [5] C. Guiraud, Y. Cailloce, and G. Caille, “Reducing direct radiating array complexity by thinning and splitting into non-regular subarrays,” in Proc. 29th ESA Antenna Workshop on Multiple Beams and Reconfigurable Antennas, Apr. 2007, pp. 211–214. [6] T. Kaifas, K. Siakavara, D. Babas, G. Miaris, E. Vafiadis, and J. N. Sahalos, “On the design of direct radiating antenna arrays with reduced number of controls for satellite communications,” presented at the 1st Int. ICST Conf. on Mobile Lightweight Wireless Systems MOBILIGHT, May 18–20, 2009. [7] T. N. Kaifas and J. N. Sahalos, “On the geometry synthesis of arrays with a given excitation by the orthogonal method,” IEEE Trans. Antennas Propag., vol. 56, no. 12, pp. 3680–3688, Dec. 2008. [8] T. Kaifas, D. Babas, and J. N. Sahalos, “Planar array optimization by using orthogonal methods,” presented at the COST IC0803 RFCSET Workshop, Apr. 27–29, 2009, WG305. [9] D. H. Werner and R. Mittra, Frontiers in Electromagnetics. Piscataway, NJ: IEEE Press, 2000. [10] C. P. Baliarda and R. Pous, “Fractal design of multiband and low sidelobes array,” IEEE Trans. Antennas Propag., vol. 44, pp. 730–739, May 1996. [11] D. H. Werner, R. L. Haupt, and P. L. Werner, “Fractal antenna engineering: The theory and design of fractal antenna arrays,” IEEE Antennas Propag. Mag., vol. 41, no. 5, pp. 37–59, Oct. 1999. [12] D. H. Werner, M. A. Gingrich, and P. L. Werner, “A self-similar radiation pattern synthesis technique for reconfigurable multiband antennas,” IEEE Trans. Antennas Propag., vol. 51, pp. 1486–1498, Jul. 2003. [13] J. S. Petko and D. H. Werner, “The evolution of optimal linear polyfractal arrays using genetic algorithms,” IEEE Trans. Antennas Propag., vol. 53, pp. 3604–3615, Nov. 2005. [14] K. Siakavara, E. Vafiadis, and J. N. Sahalos, “On the design of a direct radiating array by using the fractal technique,” in Proc. 3rd Eur. Conf. on Antennas and Propagation, Berlin, Germany, Mar. 23–27, 2009, pp. 1242–1246. [15] C. A. Balanis, Antenna Theory-Analysis and Design. Hoboken, NJ: Wiley, 2005.

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Ultrawideband Antennas for Magnetic Resonance Imaging Navigator Techniques U. Schwarz, F. Thiel, F. Seifert, R. Stephan, and M. A. Hein

Abstract—Due to its high appreciation for medical diagnostics worldwide, recent developments in magnetic resonance imaging (MRI) aim at adding the capability of creating focused images of moving objects. Among the potential navigator techniques required for such an improved MRI is ultrawideband (UWB) radar. We have studied the performance of UWB antennas for biomedical imaging inside the 3-Tesla MRI system at PTB Berlin. The strong static and time variant magnetic fields give rise to severe mechanical and electrical interactions due to the induced electromagnetic forces. On the other side, the high magnetic field homogeneity required for MR scans can also be affected adversely by the presence of the UWB antennas. The requirements resulting for the design of MRI compatible antennas have been identified and implemented in terms of a novel type of double-ridged horn antennas. We describe the general design strategy and the successful experimental verification of the antenna concept as well as biomedical results achieved with a combination of both UWB and MRI systems. Index Terms—Double-ridged horn antenna, induced electromotive force, magnetic resonance imaging, radiation pattern.

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Since the potential of a combined MRI and UWB diagnostic approach could be demonstrated [1], the performance of the UWB antennas inside the MR scanner was examined in greater detail. The most obvious criterion is minimal mutual interaction between the metallized antennas and the strong static and gradient magnetic fields. In earlier experiments with conventional wideband antennas, strong mechanical interactions were observed, pointing out the need for special antenna designs [5]. Furthermore, especially for antennas with a low cut-off frequency of 1 GHz and appreciable physical size, there was a noticeable voltage level, referred to a 50 Ohm termination, that was received from the high-power radio frequency (RF) signal used in MRI. For this purpose, detailed measurements of voltages across MR-compatible antennas are presented. In addition to our previous paper [1], we describe here further details of the design strategy and antenna performance, especially in terms of a careful experimental time domain characterization of the latest MR-compatible antenna version in comparison with an unmodified double-ridged horn antenna. Furthermore, advanced results are provided for the biomedical sensing and myocardial UWB-radar operation of the new antenna under realistic MR conditions. II. FORMULATION OF THE PROBLEM

I. INTRODUCTION Magnetic resonance imaging (MRI) systems are among the most sensitive diagnostic methods in medicine for the visualization of soft tissue [1]. At present, more than ten million MRI examinations of patients are performed per year worldwide. Given such a progressive development, further improvements of this diagnostic technique are under way. However, MRI systems are not per se capable of creating focused images of moving objects like the human heart or the thorax of the patient while breathing. Rather, additional techniques like breath holding, ECG triggering, or MR navigation methods are required. Such techniques cause inconvenience for the patient, or they are even not applicable for upcoming generations of MR scanners. A novel approach which overcomes these obstacles is the use of low-power bistatic ultrawideband (UWB) radar as a contactless navigator technology for MR tomography [2]. The specific advantages of UWB sensors [3] that make them suitable for this application are high temporal and spatial resolution, determined by the frequencies of operation and the simultaneous frequency bandwidth covered, the penetration into objects, low integral power, and compatibility with established narrowband systems such as MRI [2]. While no influence on the quality of the MR scan was observed for our M-sequence UWB radar system with an integral radiation power of 4 mW [1]–[4], we could show previously that real-time UWB radar signals can be used to trigger the MR data acquisition, to successfully suppress motional artifacts [4]. Manuscript received April 21, 2009; revised September 03, 2009; accepted October 09, 2009. Date of publication March 29, 2010; date of current version June 03, 2010. This work was supported by the German Science Foundation (DFG) in the priority programme UKoLoS (SPP1202, project acronym ultraMEDIS). U. Schwarz, R. Stephan, and M. A. Hein are with the Institute for Information Technology, Ilmenau University of Technology, 98684 Ilmenau, Germany (e-mail: [email protected]; [email protected]; matthias. [email protected]). F. Thiel and F. Seifert are with the Physikalisch-Technische-Bundesanstalt (PTB), 10587 Berlin, Germany (e-mail: [email protected]; frank.seifert@ptb. de). 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.2010.2046848

The operational conditions inside a MR scanner are determined by three different types of fields [1]. First, a static magnetic field of Bstat = 1:5 to 7 T provides a reference orientation of the nuclear spins of the regions under inspection. Furthermore, gradient magnetic fields with a slope of dBgrad =dt = 50 T/s at the rising edge are switched during diagnostic measurements, to allow for spatial (depth) information of the acquired molecular information. The gradient fields induce eddy currents in the metallized sections of the antenna according to Faraday’s law of induction. In turn, these eddy currents interact with the static magnetic field by exerting a mechanical torque on the antenna structure. The torque can reach peak values of the order of 0.045 Nm for a contiguous metallized area of 20 mm 2 30 mm. This value is high enough to result in mechanical amplitudes of several millimeters, deforming or moving the antenna structure, especially in case of mechanical resonances. Furthermore, the magnetic fields of the eddy currents can lead to artifacts of the MR-image. These interactions inhibit the beneficial application of UWB navigation for magnetic resonance imaging and, therefore, must be avoided [1]. Finally, the nuclear spins are set into precession by a strong RF signal at 42.58 MHz/T [1]. The power of the RF signal lies in the kW range. We used a 3-T MR scanner with the resulting RF frequency of 127.8 MHz, which is ten times smaller than the lower cut-off frequency of the UWB antennas employed. As the frequency response of a typical antenna corresponds to a high-pass filter of first order, the stopband attenuation amounts to 20 dB per decade, indicating the risk of collecting RF power even in the presence of path-loss and shadowing. III. MR COMPATIBLE DOUBLE RIDGED HORN ANTENNA Based on our initial attempt [5], the subsequent development, described in [1] and here, aimed at improving and experimentally characterizing the microwave performance of the MR-compatible antennas under the following constraints: 1. Weakly frequency dependent radiation characteristics over the entire operational bandwidth. 2. Lower cut-off frequency around 1 GHz. 3. Minimized contiguous metallized areas. 4. Good decoupling between neighboring antenna elements for bistatic radar arrangements. As double-ridged horn (DRH) antennas are well known for their wide usable bandwidth, high directivity, low sidelobes, and low dispersion [6], we focused our studies on this type.

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TABLE I COMPARISON OF H-PLANE CHARACTERISTICS OF A CONVENTIONAL DOUBLE-RIDGED HORN ANTENNA (CONV.) AND THE MR-COMPATIBLE DOUBLE-RIDGED HORN ANTENNA (MR) AT SELECTED FREQUENCIES. DATA DERIVED FROM NUMERICAL SIMULATIONS

Fig. 1. MR-compatible double-ridged horn antenna for a lower cut-off frequency of 1.5 GHz. All electrically conductive parts were fabricated from metallized dielectrics. Further explanations are given in the text. From [1].

We note that, due to the functional principle of double-ridged horn antennas, the above conditions (1) and (3) are in conflict with each other. These antennas are typically made entirely out of plane metallic parts of high electrical conductivity  , thus suffering from the induction of eddy currents under MR-scanner conditions. Therefore, the major challenge was to modify the double-ridged horn antenna to achieve MR-compatibility, without compromising the favorable radiation properties [1]. At the same time, such a systematic incremental procedure assures that the principles of operation of double-ridged horn antennas remain valid, rendering basic theoretical considerations unnecessary. Inspired by commercial counterparts of double-ridged horn antennas, in a first step, we removed the H-plane sidewalls of the pyramidal horn, leaving just a thin wire in the plane of the aperture, as illustrated by Fig. 1. As a result, the lower cutoff frequency could be reduced from 2.6 GHz to 1.5 GHz for otherwise unchanged dimensions and operation in air ("r = 1). The reduction of the lower cutoff frequency is of great relevance for the envisaged biomedical applications due to the limited penetration of electromagnetic waves into human tissues. The comparison with a conventional double-ridged horn antenna with a similar bandwidth, according to Table I, revealed that this improvement was achieved at the expense of increased sidelobes (column entitled “0 /90 ”) and backward radiation (column “0 /180 ”), predominantly at frequencies below 3 GHz, due to the modified aperture distribution and diffraction at the edges of the open construction. Furthermore, as a result of the halved footprint of the antenna compared to the conventional version, the beamwidth increased, leading to a slightly increased crosstalk [1]. While the widened beamwidth is desirable, e.g., for the reconstruction of irregular surfaces by UWB imaging [7], the slightly increased crosstalk, as described in [1], can be easily compensated for by re-orienting the antennas relative to each other. While the crosstalk for conventional DRH antennas becomes small for an H-plane alignment, the MR-compatible versions have to be aligned along the E-plane due to the removed H-plane sidewalls and thus reduced shielding. The generally applied co-polarization of the antennas is not effected by such a measure. In a second step, the thickness of the metallization was reduced, in order to exploit the electromagnetic skin effect for a decoupling of the low-frequency eddy current paths. The metal planes, displaying an original thickness of 2 mm, were replaced by metallized dielectric boards with a metallization thickness of d = 12 m. This value corresponds to about twice the skin depth at the lowest frequency used. The high-frequency currents determining the radiation of the antenna remain essentially undisturbed while the eddy currents in the kHz range are strongly attenuated by the high sheet resistance 1=d. Altogether, the total volume of the conductor of the antenna could be reduced by a factor of 240 [1].

Fig. 2. Simulated, normalized current distribution of an unmodified doubleridged horn antenna for a frequency of 5 GHz (left-hand part), layout of the resulting MR-compatible double-ridged horn antenna structure (centre part), and current distribution of the modified double-ridged horn antenna for a frequency of 5 GHz (right-hand part). The plots are normalized to their respective maxima (color-coded: dark = high; light = low). Further explanations are given in the text.

For further optimization of the remaining metallized areas, the distribution of surface currents in the UWB frequency range was inspected by electromagnetic simulations. Typical results for the normalized surface current are illustrated by a snap shot for 5 GHz in the left-hand part of Fig. 2. The surface current is concentrated near the position of the ridge and the edges of the pyramidal frame. In accordance with expectation, the number of current loops was found to increase with frequency; in contrast, the current distribution across the backward cubical part of the antenna showed little frequency dependence. Based on these observations, a compromise was sought to reduce the plane metallization with the minimal possible distortion of the broadband current distribution. As a result, the conductor faces of the horn sections were separated into striplines, straight and elliptically shaped, separated by 1 mm, and oriented parallel to the most common current paths, with plain connections at the face edges only. The central part of Fig. 2 illustrates the resulting geometric arrangement of the slots [1]. The normalized surface current of the modified antenna is shown in the right-hand part of Fig. 2 by a snap shot for 5 GHz. The main features of the current distribution could be sustained qualitatively both on the pyramidal faces and the backward cubical part of the antenna. Differences occurred mainly for the currents oriented perpendicular to the slots. It is this minor change in current distribution which causes the modified radiation properties discussed above. The ridges themselves required special attention. A grid of holes was eventually identified as the proper solution to reduce the metallization

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Fig. 3. Measured reflection coefficient of the MR-compatible double-ridged horn antenna (lower curve) and the measured antenna gain (upper curve) versus frequency. Data from [1].

area of the ridges without disturbing the high-frequency current distribution too much. In order to reduce the maximal loop size for low-frequency currents, the outer contour of the horn section was cut and shortened by standard surface-mounted device capacitors. The impact of the sum of these modifications on the performance of the antennas was first checked by numerical simulations. As the results turned out to be promising, first laboratory versions were manufactured from metal-plated plastics [1]. The connection of the antenna ridges to the coaxial feed cable was accomplished by brass bushes integrated inside the plastic ridges. According to Fig. 1, the outer dimensions of the resulting antenna amounted to 48 mm 2 70 mm for the aperture and 67 mm of height [1]. In terms of the wavelength c at the lower cut-off frequency, amounting to 200 mm, these dimensions correspond to an electrical size between c =4 and c =2. With a weight of as little as 30 g, the antenna is extremely lightweight, which removes the need for heavy mechanical fixing inside the MR scanner. IV. CHARACTERISTICS OF THE MR COMPATIBLE DOUBLE RIDGED HORN ANTENNA Fig. 3 displays measured results for the reflection coefficient and the gain of the modified double-ridged horn antenna. A return loss above 10 dB was achieved over the frequency range from 1.5 to 12 GHz, by optimizing the shapes and mutual arrangement of the ridges. Subsequent radiation measurements in an anechoic chamber yielded the radiation patterns illustrated by Fig. 4 for two orthogonal cuts with respect to the plane of the ridges. The half-power beamwidth, indicated as the black contour line in Fig. 4, was found to vary between 30 and 50 degrees, thus covering a range suitable for the envisaged radar applications. Except for frequencies around 2 GHz, the main lobe showed little spectral variation. The corresponding frequency variation of the antenna gain is displayed in Fig. 3. These results were found in good agreement with the numerical simulations. In addition to the illustration of the radiation pattern in the frequency domain, the transient response of the antenna is essential for UWB applications, especially for high-resolution radar. Based on a Fourier-transform of the previously accurately measured frequency-domain transfer function over a wide range from 1 to 10 GHz, we compared the transient response of a conventional double-ridged horn antenna with the modified version, as shown in Figs. 5 and 6. The conventional version was designed for a similar bandwidth, resulting in geometrical dimensions of 90 mm 2 130 mm for the aperture and a height of 130 mm (all of them being about twice as long as for the MR-compatible version). As expected, the beamwidth of the conventional antenna is narrower due to the closed pyramidal shape, leading

Fig. 4. Two-dimensional representation of the measured radiation pattern of the MR-compatible double-ridged horn antenna for the E-plane (top) and the H-plane (bottom) through the main beam. The scales indicate the antenna gain in dBi. The black and white contour lines illustrate the corresponding beam widths at 3 and 10 dB below the frequency dependent maximum gain (cf., Fig. 3), respectively. Raw data from [1].

to a higher directivity. Despite the open geometry of the MR-compatible version, a low signal distortion could be sustained, as can be concluded from almost identical decay times of the transient responses of 200 ps, which are primarily limited by the measurement bandwidth. A slight angular dependence of the time responses is observed in all four diagrams of Figs. 5 and 6, which can be attributed to an offset between the impulse centers of the antennas and the center of rotation of the antenna positioning system. Fig. 6 exhibits a time shift of 0.135 ns for the main pulse at an azimuth of 90 compared to broadside, corresponding to a distance of 40 mm between the impulse center and the center of rotation. The rising time indicates an antenna position slightly in front of the rotation center, increasing the measurement distance proportional to the angle of rotation and the propagation time, respectively. Due to smaller antenna dimensions, the impulse center of the MR compatible antenna lies still a bit closer to the rotation center, represented by an almost angle independent main pulse in both panels of Fig. 5. Systematic differences between the transient responses are caused by the different architectures of the antennas, enabling multiple propagation angles and transient responses for the open coverage type.

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Fig. 5. Two-dimensional representation of the measured time domain response of the MR-compatible double-ridged horn antenna for the E-plane (top) and the H-plane (bottom) through the main beam. The scales indicate the normalized impulse response of the antenna.

Fig. 6. Two-dimensional representation of the measured time domain response of a conventional double-ridged horn antenna with a similar bandwidth as the MR-compatible design for the E-plane (top) and the H-plane (bottom) through the main beam. The scales indicate the normalized impulse response of the antenna.

The impulse center of the smaller MR compatible antenna is located about 0.26 ns, or 78 mm, behind the impulse center of the reference antenna, as concluded from a quantitative comparison of Figs. 5 and 6. This is in good agreement with the geometrical dimensions of the antennas and the different measurement setups, which included an additional 40 mm long feed cable for the MR-compatible antenna. Based on the time domain response, the angle-dependent coherence pattern of the MR-compatible antenna can be evaluated for both principal radiation planes, as depicted in Fig. 7. This parameter refers to the fidelity factor F , calculated as the peak value of the cross-correlation of the antenna impulse responses for a given setup and orientation [8]. The coherence pattern of a single antenna can be extrapolated from the fidelity factor by the use of the impulse response at boresight (bs) as a reference:

According to the time domain radiation patterns in Fig. 5, the maximum coherence of the MR-compatible antenna is found along the main beam direction, with a pronounced decay for increasing angles. For an angular span of 640 , corresponding to the 3-dB beamwidth, the correlation remains above 55%. For the extreme limits of 690 , correlation coefficients of about 20% could still be sustained. Similar results were obtained for the conventional double-ridged horn antenna as also shown in Fig. 7. The major difference is a less focused coherence pattern of the MR-compatible antenna, especially in the H-plane. While the broader pattern is caused by a smaller aperture plane and thus by less directivity, the pronounced effect in the H-plane is due to the missing sidewalls of the MR-compatible antenna.

C (;

1 ) = max 

01

h(t; ;

) 1 hbs (t +  )

hbs k22

k

dt

(1)





V. ANTENNAS INSIDE THE MR SCANNER Another set of experiments was performed with the antennas mounted inside the 3-T MR scanner, in order to specify the electrical and mechanical effects of the strong magnetic fields on the antenna [1]. As the design of complex measurement gauges was beyond the

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Fig. 7. Coherence patterns C(2; ) of the MR-compatible double-ridged horn antenna (solid curves) and of the conventional double-ridged horn antenna (dashed curve) as the mean value of both quite similar antenna planes, based on the transient responses shown in Figs. 5 and 6, respectively.

Fig. 9. The upper panel shows the respirative time response signal of a test person, measured by UWB radar and the double-ridged horn antenna described here inside the 3-T MR scanner. The inhale and exhale phases are marked. The lower panel shows the heart beat signal, taken by an identical measurement setup. The distinction of the hearts contraction and relaxation phase is given within the diagram.

Fig. 8. Time dependent voltage induced at the feed point of the MR-compatible antenna inside the MR-scanner during a test sequence. The RF-power amounted to 156 W at 127.8 MHz.

scope of this work, the mechanical forces acting on the antenna were evaluated by subjective perception of a test person holding the antenna at the place of maximum gradients inside the MR scanner while in full operation. This scenario represents the worst case. As a quantitative determination of the forces is not available by this approach, the human hand can, in principle, resolve motions in the range of 1 m [9], which is a much higher resolution than required for the UWB navigation. Slight vibrations were noted when the block capacitors were not yet mounted. This effect disappeared for the fully modified MR-compatible version, indicating the beneficial effect of the direct current blocking on the mechanical robustness. Even more relevant than the evaluation of mechanical displacements is the investigation of the forces required to move the antenna in the static magnetic field, in the absence of the pulsed gradient fields. This force can be as high as several N for conventional antenna structures [5]. In contrast, MR-compatible antennas could be moved without any noticeable difficulty. In turn, if the induction of eddy currents is minimized, image artifacts in the MR-scan will be reduced to a minimum as well. The electrical measurement of voltages induced by the RF-system of the MR-scanner in the antenna feeding point delivered an overall peak-to-peak value of 20 mV at 156 W of RF power, as illustrated by Fig. 8. This level, which is of similar magnitude as for the previous antenna version with much poorer radiation properties, can be easily

tolerated at the input stage of the UWB receiver. As can be further seen from Fig. 8, the signal of the RF system with a repetition rate of 22 Hz as well as the switched gradient magnetic fields with a period of 148 Hz could be detected, the latter including a voltage at the antenna feed point of 2 mV for the worst case. Both signals are uncorrelated to the used M-sequence, avoiding any drawbacks for the signal processing. Due to the improved antenna properties of the current design, the stop-band attenuation of the antenna below the lower cut-off frequency could be evaluated to 40 dB per decade, resulting in strongly attenuated spectral components at frequencies below the operational bandwidth as desired. VI. BIOMEDICAL RESULTS First radar measurements were performed inside PTB’s 3-T MR-scanner with the modified double-ridged horn antenna. For test purposes, the amplitude spectra and the time responses of the respiration and heart beat of a test person were monitored, processed, and compared for the different versions of MR-compatible antennas [1]. While such measurements, in principle, do not require the presence of a MR-scanner [10], they provide the essential information necessary for a UWB-navigated MR-imaging [2]. For each test, a pair of antennas was placed directly above the thorax of a volunteer in a distance of approximately 20 cm. Due to the improved signal amplitude and spectral purity reflecting the excellent impulse response of the actual antenna, it became possible to extract basic physiological information from the recorded time-variant radar signals. The upper panel of Fig. 9 illustrates a bandpass-filtered respirative signal versus time [1], with a clear distinction between the inhale and exhale phases. The considerable breathing pause after exhalation was taken into account as the separation criterion. The respiration rate was evaluated to be 0.26 Hz. While the respirative signature could be monitored with comfortable signal amplitudes due to surface reflections of the radar signal of the moving chest, the heart beat signature has to be extracted from signals occurring in the inner part of the thorax and hence, is

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much weaker. The signal levels decreased due to the propagation loss of the human body and the dielectric interface between free space and the body [2], [6]. With the actual antenna design, we were able to reconstruct the myocardial events of a volunteer, as shown in the lower panel of Fig. 9. The bandpass filtered radar signal provides the opportunity to distinguish between different heart cycles, in terms of the different contributions of the left and right ventricles to the systolic (contraction) and diastolic (relaxation) phases [9]. Eventually, the heart rate was evaluated as 66 min01 . Mapping the voltage axes of Fig. 9 to calibrated mechanical amplitudes and further combined UWB and MRI measurement campaigns are parts of current work. VII. CONCLUSION We have described the design, realization, and test of MR-compatible double-ridged horn antennas for use in a bistatic radar arrangement in strong static and dynamic magnetic and electromagnetic fields. The scattering parameters and radiation properties could be improved significantly compared to previous antenna designs. This progress enabled us to perform first valuable measurements of the respiration and heart beat of test persons in a 3-T MR scanner under full operation, underlining the great potential for medical diagnostics. Based on these introductory results, advanced studies are in progress, including tests in a 7-T MR scanner. The fusion of MRI diagnosis with UWB-radar navigation as well as the MR-compatible double-ridged horn antenna design are patented. ACKNOWLEDGMENT The authors are grateful to E. Hamatschek, M. Fritz, and U. Schmidt at TU Ilmenau for valuable technological support.

REFERENCES [1] U. Schwarz, F. Thiel, F. Seifert, R. Stephan, and M. Hein, “Ultra-wideband antennas for combined magnetic resonance imaging and UWB radar applications,” presented at the IEEE MTT-S Int. Microwave Symp., Boston, MA, Jun. 2009. [2] F. Thiel, M. Hein, J. Sachs, U. Schwarz, and F. Seifert, “Combining magnetic resonance imaging and ultrawideband radar: A new concept for multimodal biomedical imaging,” Rev. Sci. Instrum., vol. 80, no. 1, 2009. [3] J. Sachs, D. J. Daniels, Ed., “M-sequence RADAR,” Ground Penetrating Radar2nd ed. 2004, pp. 225–237, IEE Radar, Sonar, Navigation and Avionics Series 15. [4] F. Thiel, M. Hein, J. Sachs, U. Schwarz, and F. Seifert, “Physiological signatures monitored by ultra-wideband-radar validated by magnetic resonance imaging,” in Proc. IEEE Int. Conf. on Ultra-Wideband, Hannover, Germany, Sep. 2008, vol. 1, pp. 105–108. [5] U. Schwarz, F. Thiel, F. Seifert, R. Stephan, and M. Hein, “Magnetic resonance imaging compatible ultra-wideband antennas,” in 3rd Eur. Conf. on Antennas and Propagation, Berlin, Germany, Mar. 2009, pp. 1102–1105. [6] U. Schwarz, M. Helbig, J. Sachs, F. Seifert, R. Stephan, F. Thiel, and M. Hein, “Physically small and adjustable double-ridged horn antenna for biomedical UWB radar applications,” in Proc. IEEE Int. Conf. on Ultra-Wideband, Hannover, Germany, Sep. 2008, vol. 1, pp. 5–8. [7] M. Helbig et al., “Improved breast surface identification for UWB microwave imaging,” in Proc. World Congress of Medical Physics and Biomedical Engineering, Munich, Germany, 2009, vol. 25/II, pp. 853–856. [8] W. Sörgel, “Charakterisierung von Antennen für die Ultra-WidebandTechnik,” Ph.D. dissertation, University of Karlsruhe (TH), Karlsruhe, Germany, 2006. [9] R. Klinke and S. Silbernagel, Lehrbuch der Physiologie. Stuttgart, Germany: Thieme, 1996. [10] J. C. Lin, “Microwave sensing of physiological movement and volume change: A review,” in Bioelectromagnetics, 1992, vol. 13, pp. 557–565.

Dual-Band Circularly Polarized -Shaped Slotted Patch Antenna With a Small Frequency-Ratio Nasimuddin, Zhi Ning Chen, and Xianming Qing

Abstract—A dual-band single-feed circularly polarized, -shaped slotted patch antenna with a small frequency-ratio is proposed for GPS applications. An -shaped slot is cut at the centre of a square patch radiator for dual-band operation. A single microstrip feed-line is underneath the center of the coupling aperture ground-plane. The frequency-ratio of the antenna can be controlled by adjusting the -shaped slot arm lengths. The measured 10-dB return loss bandwidths for the lower and upper-bands are 16% (1.103–1.297 GHz) and 12.5% (1.444–1.636 GHz), respectively. The measured 3-dB axial-ratio (AR) bandwidth is 6.9% (1.195–1.128 GHz) for the lower-band and 0.6% (1.568–1.577 GHz) for the upper-band. The measured gain is more than 5.0 dBic over both the bands. The measured frequency-ratio is 1.28. The overall antenna size is 0 46 0 086 at 1.2 GHz. 0.46 Index Terms—Circular polarization, circularly polarized antenna, dualband antenna, GPS antenna, microstrip antenna, slotted patch, slot.

I. INTRODUCTION Recently, dual-band circularly polarized (CP) microstrip antennas (CPMAs) have received much attention in the field of wireless communications. In many dual-band applications such as global positioning system (GPS), a small frequency-ratio is required. This poses a challenge for a single-feed, single-patch, microstrip antenna structure. A single-band broadband CP can be generated from a patch antenna with an aperture-coupled feed with cross-slots using two parallel feed-lines [1]. Various types of antenna structures with different feeding network systems for dual-band CPMAs have been reported [2]–[8]. Tanaka et al. have proposed a dual-feed CPMA which combines slots and patch for dual-band operation [2]. Yang and Wong have investigated a singlelayer slit-loaded square microstrip patch antenna for dual-band CP radiation with a frequency-ratio of 1.76 [3]. A single-feed dual-band CPMA has been proposed in [4]. They have realized dual-band CP operation by cutting two arc-shaped slots close to the boundary of a circular patch radiator and protruding one of the arc-shaped slots with a narrow slit. The frequency-ratio of the dual-band antenna is 1.48. In [5], Cai et al. have proposed a ring type slot, aperture-coupled, angular-ring patch antenna with L-shaped feed-line for dual-band CP operation. The dual-band frequency-ratio of the antenna is 1.32. Bao and Ammann have been proposed a probe-feed single-layer dual-band, CPMA with a small frequency-ratio of 1.21 [6]. The antenna consists of a small circular patch surrounded by two concentric annular-rings. An unequal lateral cross-slot is loaded on the ground-plane for dual-band CP operation. The gain of the antenna at 1.224 GHz and 1.480 GHz is around 1.35 dBic and 3.5 dBic, respectively. Su and Wong have studied a dual-band CP stacked microstrip antenna using a coaxial-feed for GPS applications. The antenna comprises two stacked patches with a combination of air and dielectric layers. The gain of the antenna is less than 2.0 dBic for the lower-band and more than 4.0 dBic for the Manuscript received May 25, 2009; revised October 19, 2009; accepted December 01, 2009. Date of publication March 29, 2010; date of current version June 03, 2010. The authors are with the Institute for Infocomm Research, Singapore 138632, 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.2010.2046851

0018-926X/$26.00 © 2010 IEEE

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TABLE I DIMENSIONS OF THE OPTIMIZED ANTENNA DESIGN

Fig. 1. Proposed dual-band CPMA: (a) cross-section view, (b) slotted patch radiator, and (c) aperture-coupled feeding structure.

S -shaped

upper-band [7]. A dual-band circularly polarized stacked microstrip antenna with cross-slot and aperture-coupled feed has been proposed for GPS [8]. Three Wilkinson power combiners have been used to add the signals from the four feed-lines at the slots with equal amplitudes and 90 phase-shifts. In this communication, a dual-band single-feed single-patch CPMA with a small frequency-ratio is proposed. The antenna consists of an S -shaped slotted square patch radiator and an aperture-coupled feeding structure. Dual-band CP radiation is achieved by cutting an asymmetrical S -shaped slot from the radiating patch, without increasing the size and the thickness of the patch antenna. The antenna design and optimization is conducted with the help of commercial EM software, IE3D [9].

Fig. 2. Effect of the S on antenna parameters: (a) return loss, (b) axial-ratio at the boresight, and (c) gain at the boresight.

II. ANTENNA STRUCTURE AND DESIGN The proposed antenna configuration is shown in Fig. 1. The patch is fed through an aperture-coupled 50- microstrip feed-line under the ground-plane. The overall size of the antenna is G 2 G 2 H . The

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Fig. 4. Measured and simulated return loss.

Fig. 3. Current distributions on the (a) 1.227 GHz and (b) 1.575 GHz.

S -shaped

slotted patch radiator at Fig. 5. Measured and simulated axial-ratio at the boresight.

50- microstrip feed-line and the aperture are etched on the opposite sides of an RO4003 substrate (h1 = 1:524 mm, "r1 = 3:38 and tan 1 = 0:0027). The open end of the microstrip feed-line extends Sf from center of the aperture. The aperture size is Wa 2 La . The asymmetrical S -shaped slot acts as a perturbation of the patch to excite the two orthogonal modes with a 90 phase-shift for CP operation at the lower-band. The asymmetrical S-shaped slot itself resonates at the upper-band and generates CP radiation for the upper-band [10]. By varying the length of one of the slot arms, the operating frequency-ratio of the two operating bands can be controlled. From simulation, it is found that the dimensions of the S -shaped slot significantly affect the performance of the antenna. Based on the simulation, a procedure of the antenna design is suggested as follows. 1. Determine the initial dimensions of the square patch with an S -shaped slot according to the lower-band frequency and the antenna size constraint; 2. Optimize the aperture-coupled feeding structure to achieve good impedance matching over the operating bands; 3. Select the length of one arm of the S -shaped slot (S1 ) and adjust the length of the other arm (S3 ) to generate dual-band CP operation. Make sure that the 3-dB AR bandwidth falls totally within the 10-dB return loss bandwidth; and 4. Further optimize the antenna by changing the foam thickness and S -shaped slot parameters (S1 ; S6 , and S5 ). If the desired performance over the required frequency is not achieved at the end of Step 4, change the initial parameters in Step 1 and iterate the steps. The optimal dimensions of the antenna are tabulated in Table I. A parametric study is conducted to understand the effect of the S -shaped slot on the dual-band CP operation. The procedure adopted for study is that only one parameter is changed at a time while all other parameters are kept unchanged.

Fig. 6. Measured and simulated gain at the boresight.

Fig. 2 shows the effect of S3 on the performance of the antenna, where S3 varies from 9.5 mm to 23.0 mm. From Fig. 2(a), it is found that as S3 increases, the 10-dB return loss bandwidth and impedance matching improve at the upper-band. Fig. 2(b) illustrates the axial-ratio (AR) at the boresight with different S3 . When S3 = S1 = 23:0 mm, namely the S -shaped slot is symmetrical, the antenna generates CP radiation only at the lower-band. As S3 decreases, the antenna generates CP radiation at the upper-band with an increase in the operating frequency, whereas the operating frequency at the lower-band changes slightly. As a result, the frequency-ratio of the two operating frequency bands can be controlled by adjusting S3 . It is also found that the bandwidth at the lower-band is slightly affected by S3 , which offers more flexibility to achieve a desired frequency-ratio. The boresight gain with varying S3 is shown in Fig. 2(c). The boresight gain

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slotted patch. This implies the lower-band operation is dependent on the patch size. From Fig. 3(b), it is observed that the majority of the current distribution is around the S -shaped slot at 1.575 GHz. This suggests that the upper-band radiation is mainly from the asymmetrical S -shaped slot. III. MEASURED RESULTS AND DISCUSSIONS The optimized antenna was fabricated and measured. Fig. 4 compares the measured and simulated return loss. The measured 10-dB return loss bandwidth is 16% (1.103–1.297 GHz) for the lower-band and 12.5% (1.444–1.636 GHz) for the upper-band. Fig. 5 shows the measured and simulated AR at the boresight. The measured 3-dB AR bandwidths at the lower- and upper-bands are 6.9% (1.195–1.280 GHz) and 0.6% (1.568–1.577 GHz), respectively. Both the GPS bands are covered with less than 3-dB AR. The measured AR at 1.227 GHz and 1.575 GHz are 2.0 dB and 1.34 dB, respectively. The frequency-ratio of the measured minimum AR for dual-band is 1.28. Fig. 6 shows the measured and simulated gain at the boresight. The gain is more than 5.0 dBic with a variation of less than 0.5 dB across the 3-dB AR bandwidth for both the bands. The gain at the upper-band is around 2.0 dB below that of the lower-band because of a gain dip. However, the upper-band gain is still greater than 5.0 dBic which is suitable for GPS application [11]. Fig. 7 shows the measured radiation patterns at 1.227 GHz and 1.575 GHz in the x 0 z and y 0 z planes, respectively. The 3-dB AR beamwidth is more than 90 for the lower-band and more than 60 for the upper-band. IV. CONCLUSION A single-feed single-patch dual-band circularly polarized microstrip antenna with a small frequency-ratio has been investigated. The proposed antenna with an S -shaped slot has achieved good impedance matching, high gain and wide CP beamwidth at the GPS lower and upper-bands. The antenna has been realized for a small dual-band frequency-ratio of 1.28. The proposed single-feed single-patch S -shaped slotted patch antenna is useful for small frequency-ratio dual-band CP antenna and array designs.

REFERENCES

Fig. 7. Measured radiation patterns at (a) 1.227 GHz, and (b) 1.575 GHz.

variation with frequency does not significantly depend on the S3 for the lower-band. However, upper-band gain dip decreases with an increase in S3 . Increasing S3 shifts the operating frequency down. Note that when S3 = S1 = 23:0 mm, there is no gain dip at the upper-band. However, such a symmetrical S -shaped slotted microstrip antenna can produce CP radiation only at the lower-band. The current distributions of the antenna at 1.227 GHz and 1.575 GHz are shown in Figs. 3(a) and (b), respectively. It is found that at 1.227 GHz, the current is much stronger around the edges of the S -shaped

[1] S. D. Targonski and D. M. Pozar, “Design of wideband circularly polarized aperture-coupled microstrip antennas,” IEEE Trans. Antennas Propag., vol. 41, no. 2, pp. 214–219, 1993. [2] T. Tanaka, T. Houzen, M. Takahashi, and K. Ito, “Circularly polarized printed antenna combining slots and patch,” IEICE Trans. Commun., vol. E90-B, no. 3, pp. 62–628, 2007. [3] K. P. Yang and K. L. Wong, “Dual-band circularly-polarized square microstrip antenna,” IEEE Trans. A Antennas Propag., vol. 49, no. 3, pp. 377–382, 2001. [4] K. B. Hsieh, M. H. Chen, and K. L. Wong, “Single-feed dual-band circularly polarized microstrip antenna,” Electron. Lett., vol. 34, no. 12, pp. 1170–1171, Jun. 1998. [5] C. H. Cai, J. S. Row, and K. L. Wong, “Dual-frequency microstrip antenna for dual circular polarization,” Electron. Lett., vol. 42, no. 22, pp. 1261–1262, Oct. 2006. [6] X. L. Bao and M. J. Ammann, “Dual-frequency circularly-polarized patch antenna with compact size and small frequency ratio,” IEEE Trans. Antennas Propag., vol. 55, no. 7, pp. 2104–2107, 2007. [7] C.-M. Su and K.-L. Wong, “A dual-band GPS microstrip antenna,” Microw. Opt. Technol. Lett., vol. 33, no. 4, May 2002. [8] D. M. Pozar and S. M. Duffy, “A dual-band circularly polarized aperture-coupled stacked microstrip antenna for global positing satellite,” IEEE Trans. Antennas Propag., vol. 45, no. 11, pp. 1618–1624, 1997. [9] IE3D Version 14.0, Zeland Software Inc.. Fremont, CA, Oct. 2007. [10] S. Shi, S. Hirasawa, and Z. N. Chen, “Circularly polarized rectangular bent slot antennas backed by a rectangular cavity,” IEEE Trans. Antennas Propag., vol. 49, no. 11, pp. 1517–1524, 2001. [11] G. Z. Rafi, M. Mohajer, A. Malarky, P. Mousavi, and S. Safavi-Naeini, “Low-profile integrated microstrip antenna for GPS-DSRC application,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 44–48, 2009.

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A Loop Loading Technique for the Miniaturization of Non-Planar and Planar Antennas Bratin Ghosh, SK. Moinul Haque, Debasis Mitra, and Susmita Ghosh

Abstract—A size reduction technique of non-planar and planar antennas using a self-resonant topology is highlighted. The antennas considered are the dipole, the monopole and the slot radiator. The dipole below its resonant frequency is known to possess a capacitive reactance. It is shown that the inductive reactance of a loop can be used to match a below-resonant dipole similar to the inductive load offered by a metamaterial shell. The configuration reduces the resonant frequency of the dipole by 26.01%, causing the dipole to almost reach the electrically small limit. Despite the loaded dipole touching the electrically small limit, the antenna needs no matching networks, offers a high efficiency and exhibits a bandwidth better than the unperturbed dipole. The radiation pattern is also seen to be unaffected by the presence of the loops. The concept is also demonstrated for the size-reduction of a monopole and for a planar slot. Index Terms—Dipole, loop, miniaturized, monopole, self-resonant, slot.

I. INTRODUCTION The demand for miniaturized antennas has risen considerably to keep pace with the stringent needs of modern communication systems. It is however known that realization of compact and particularly electrically small antennas leads to degradation of both antenna bandwidth and efficiency. The term “electrically small antennas” is ascribed to a class of size-reduced antennas with a level of miniaturization such that ka  1:0, where “a” is the radius of the smallest sphere circumscribing the antenna. In the presence of an infinite and perfect electric conductor, however, ka must not be greater than 0.5 to satisfy the “electrically small” criterion [1], [2]. In addition to a decrease in bandwidth and efficiency, antennas become more difficult to match with a reduction in their electrical size, due to an increase in antenna Q [1], [3], [4]. One of the methods used in the literature to miniaturize an antenna is by folding an antenna structure [5], [6]. This technique has also been used to achieve reduction in the resonance frequency of antennas by using top-loading in [6], [7]. However, the resonant resistance of the top-loaded monopole in [6, Fig. 6] significantly decreases from 50

with decrease in resonant frequency [6]. The resistance could be increased using multiple loading arms in [6] or a primary-secondary feed combination in [7]. In addition, the folded structure leads to a potential increase in conductor loss due to the physical length of the antenna structure. Another method used for the size-reduction of antennas is by using metamaterial shells surrounding the antenna structure [8], [9]. Several metamaterial inspired designs towards the realization of electrically small antennas were also reported in [2]. Miniaturized slot antennas were also reported in [10], [11] using a coiled structure to provide an open circuited or reactive termination. In this communication, a novel topology is highlighted to achieve reduction in the antenna resonant frequency, based on the self-resonance concept. Observing that the reactance of the electrically small dipole in [8] and the electrically small loop in [9] are of opposite signs, it can Manuscript received February 24, 2009; revised September 18, 2009; accepted November 12, 2009. Date of publication March 29, 2010; date of current version June 03, 2010. The authors are with the Department of Electronics and Computer Engineering Indian Institute of Technology, Roorkee, Roorkee 247667, India (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2010.2046842

be argued that it should be possible to construct a miniaturized antenna structure where the reactance of the dipole can be tuned out by the opposite reactance of the loop, without the presence of the metamaterial shell. It is seen that the strategy can be used to reduce the resonant frequency of a half-wavelength dipole by about 26.01%. The level of miniaturization also converts the half-wavelength dipole very close to an electrically small antenna. The concept is also applied for the size reduction of a monopole and a planar slot antenna. The proposed design offers simplicity of fabrication and reduced conductor loss compared to the electric-based antenna structures in [2], which is particularly important from an efficiency perspective for an electrically small antenna. This is also important due to the loss resulting from the changing current distribution in the helix [2]. The antenna configurations also do not need additional matching networks for the dipole or the monopole and only a quarter-wave transition for affecting match in the case of the planar slot. It is also seen that despite the high degree of miniaturization, the loaded dipole and monopole antenna configurations afford a high efficiency, better bandwidths than the unloaded counterparts with almost no effect on the radiation performance compared with the unperturbed dipole or monopole. A high degree of reduction in resonant frequency with good efficiency is also obtained for the planar slot, though the level of miniaturization is lower than the nonplanar dipole or monopole antenna configurations. II. REDUCTION IN RESONANT FREQUENCY FOR LOOP-LOADED DIPOLE AND MONOPOLE ANTENNAS It is known that the input impedance of a half-wavelength dipole is capacitive below the resonance frequency. Any attempt to reduce the resonant frequency of the dipole antenna must therefore add an inductive load to the dipole in such a manner so that the capacitive impedance of the dipole is exactly cancelled at the reduced resonant frequency. The inherent wire inductance has been successfully used in the self-resonant configuration in Fig. 8, [2] to offer the inductive load on a below resonant monopole with the required inductance being synthesized using multi-turns of the wire in a helical configuration. However, considering that the reactive impedances of an electrically small dipole and loop antenna to be opposite in signs, an alternate topology to effectively load the below-resonant capacitive dipole might be to couple it to a loop element for which the free space offers an inductive load. Also, if such an antenna configuration could be driven to self-resonance by the exciting dipole, it would obviate the need to wholly surround the antenna with an inductive DNG shell or an inductive helix. This would potentially also reduce conductor loss leading to an increase in the efficiency of the antenna by avoiding a helical matching element. It is shown below that such a combined dipole-loop antenna configuration can be designed to afford a high degree of miniaturization to the half-wavelength dipole. The configuration of the antenna structure is shown in Fig. 1. A dipole antenna is loaded by two loops placed in the upper and lower arms, with the dipole extremities touching the inner edges of the loops. The inner and outer diameters of the loop are D1 = 6 mm and D2 = 12 mm respectively. The upper and lower arms of the dipole are connected to the inner and outer conductors of a coax feed respectively through a quarter-wave long sleeve balun. The gap between the upper and lower arms is G = 0:50 mm. The diameter of the dipole conductor D3 is 1 mm. The total length of the loop loaded antenna structure is L = 36:00 mm. The loop positions are selected near the extremities of the dipole arms taking into account the current distribution on the dipole. As the dipole current is maximum in the center, placing the loops near to the central region of the dipole would cause more disturbance to the dipole current in this region. As such, the loops as matching elements to the

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Fig. 1. Double-loop loaded dipole antenna.

dipole are positioned near the dipole ends to cause minimal interference to the dipole current. The current on the dipole excites the loops to induce a magnetic field in the near-field region and in the extremities of the antenna which contributes to an inductive stored energy. This compensates the near field capacitive effect generated by the electric field produced by the dipole current. The simulated and measured reflection coefficient of the structure is shown in Fig. 2. The simulations have been conducted using the HighFrequency Structure Simulator (HFSS) [12]. The dipole conductor is modeled as a copper wire considering its finite conductivity of 5:82107 Siemens/m [12]. Excellent agreement is observed between the simulated and measured results. The reflection coefficient of an ordinary dipole of length 36 mm., which is the same as the total length of the antenna structure in Fig. 1, is also shown in Fig. 2 for comparison. While the measured resonant frequency of the unloaded dipole is at 3.73 GHz, the double-loop loaded dipole (DLLD) is seen to resonate at 2.76 GHz, corresponding to a reduction in resonance frequency of 26.01% relative to the reference antenna. A single-loop loaded antenna of the same dimensions as the antenna in Fig. 1 and with only the upper loop present was also fabricated. The resonant frequency for this antenna was found to decrease by 11.26% compared to the unloaded reference antenna. The increased reduction in resonant frequency with the double loops can be explained by considering the additional inductive load offered by the two loops near the two extremities of the dipole arms. The simulated input impedance for the structure is shown in Fig. 3. As can be seen from the input reactance plot for the unperturbed dipole in Fig. 3, the additional inductance offered by the two loops compensates for the increased capacitance of the dipole at a lower resonant frequency. Next, the reduction in resonant frequency of the DLLD is examined in terms of realization of an electrically small antenna. The value of ka at the resonance frequency of the antenna is computed at 1.04, which is very close to unity for the electrically small structure without a ground plane. The quality factor of the loaded dipole QVSWR was also compared with the minimum quality factor Qmin = QChu 2 RE where QChu = (1=ka) + (1=((ka)3 )) is the quality factor predicted by the Chu-limit [1] and RE is the radiation efficiency [2]. QVSWR was obtained from the half-power matched VSWR fractional bandwidth FBWVSWR and given by QVSWR = (2=(FBWVSWR )) [2]. The values of Qmin and QVSWR were found to be 1.81 and 3.83 respectively, corresponding to a Qratio = (QVSWR )=(Qmin ) of 2.12. Thus it can be seen that the size reduction offered by the double-loop loaded antenna structure converts the half-wavelength dipole to an almost electrically small structure. This is accomplished without the in-

Fig. 2. Measured and simulated reflection coefficients of the double-loop loaded and the reference dipole antenna.

Fig. 3. Simulated input impedance and resonant frequencies of the DLLD and the reference dipole antenna.

clusion of conductor structures surrounding the whole antenna. In addition, an excellent match is achieved to the antenna near the electrically small limit without the need of any external matching networks. The resonant bandwidths of the DLLD and the reference antenna are next compared. The measured 010 dB bandwidths for the DLLD and the reference dipoles are observed to be 13.97% and 10.41% respectively from Fig. 2, corresponding to a 34.20% improvement in bandwidth of the loaded antenna compared with the normal dipole. The enhancement in bandwidth of the loaded antenna should be particularly noted as the antenna is close to the electrically small limit. This is caused by the reduction in Q of the antenna due to the loop together with the dipole taking part in the radiation. The inductive environment around the dipole in Fig. 1 is synthesized without using additional turns of a conductor, which is one of the ad-

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Fig. 5. Loop-loaded monopole antenna.

Fig. 4. Measured and simulated radiation patterns of the DLLD at resonance. (a) E-plane. (b) H-plane.

vantages of the current design in terms of reduced conductor loss. The proposed design is also very simple to fabricate. The effect of two loops relative to a single loop has been to offer the inductive effect on the dipole in a more distributed and symmetric manner, leading to a more effective reduction in resonant frequency. The radiation characteristics of the DLLD for the E (x-z plane) and H (x-y plane) planes are presented in Fig. 4. It is seen that the pattern is similar to that of a half-wavelength dipole. Thus, introduction of the loops do not have any degrading influence on the antenna performance. Excellent agreement is also seen between the simulated and the measured results. The resonant gain of the miniaturized antenna was measured at 2.3 dBi, compared to a gain of 2.8 dBi obtained in the case of the reference dipole. The efficiencies of the loaded and the reference dipoles were also determined using the Wheeler cap method described as in [13]. The measured efficiencies for the former and latter were

96.60% and 97.80% respectively, which compare well with the corresponding simulated values of 98.67% and 99.50%. Next, it is investigated if the above technique can be used for reducing the resonant frequency of a monopole above a ground plane. The antenna structure is shown in Fig. 5. The monopole is of length L = 36 mm. The inner and outer diameters D1 and D2 respectively of the loop located at the top of the monopole and the monopole wire diameter D3 is the same as in Fig. 1 for the loaded dipole. A 50

coax feed was used to feed the monopole, with a ground plane size of LG = WG = 40 mm. The reflection coefficient for the structure is shown in Fig. 6. The reference monopole is of length 36 mm, the same as that of the loaded antenna. The measured resonant frequency of the loaded monopole is at 1.53 GHz, compared to that of the reference antenna at 2.01 GHz. Thus, a 23.88% reduction in resonant frequency is obtained with the loaded monopole configuration. Looking at the resonant frequency reduction from the perspective of an electrically small antenna, the value of ka at the loaded resonant frequency of the monopole is found to be at 0.58, which is again very close to the electrically small antenna limit of 0.5 for a grounded antenna structure. The computed values for Qmin and QVSWR in this case are 1.46 and 3.03 respectively, with a Qratio of 2.08. Also, despite the high degree of miniaturization provided by the loaded monopole, an easy match can be affected to the antenna without any matching networks, as in the dipole case. In fact, an interesting feature which can be observed from the matching characteristics in Fig. 6 is the significant improvement in matching obtained for the loaded miniaturized monopole relative to the unloaded antenna. The resonant dip for the loaded monopole is measured at 033:00 dB, relative to a dip of 017:50 dB for the normal monopole antenna. Thus it is seen that the loading in this case also helps to achieve a better impedance match with the monopole. Together with the improvement in matching, a very significant extension of bandwidth is also observed in Fig. 6 for the loaded monopole configuration, relative to the reference antenna. The 010 dB bandwidths measured for the unloaded and loaded monopoles have been measured at 14.63% and 22.95% respectively, signifying a bandwidth enhancement by about 56.87% of the loaded monopole structure compared to the unloaded monopole. Also, comparing with the bandwidth broadening for the loaded dipole structure in Fig. 2, it is observed that a much higher bandwidth improvement is obtained for the loaded monopole antenna relative to the reference monopole antenna. This is

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Fig. 6. Measured and simulated reflection coefficients of the loop loaded and the reference monopole antenna.

specially very important due to the loaded monopole being very close to the electrically small limit. The E and H-plane patterns of the loaded monopole are similar to the loaded dipole in Fig. 4 and are not repeated for brevity. Also, similar to the dipole case, it is observed that the presence of the loops do not affect the radiation characteristics of the loaded monopole relative to the unloaded reference monopole. The measured gains for the loaded and unloaded monopoles were 03:4 dBi and 01:4 dBi respectively. The efficiency of the monopole was determined by surrounding the antenna with a conducting shell, including the region behind the ground plane. The efficiencies of the loaded and the unloaded monopole antennas were measured at 96.10% and 97.40% respectively, compared with the corresponding simulated values of 98.86% and 99.78%. The effect of change in ground plane size on the antenna characteristics is next investigated. The variation in the simulated reflection coefficient of the loaded monopole with the size of the ground plane is shown in Fig. 7. The maximum ground plane size of 200 mm by 200 mm is slightly over 10 by 10 at the loaded monopole resonance of 1.53 GHz. It is seen that the reflection coefficient improves significantly as the ground plane size is reduced to 40 mm by 40 mm, corresponding to almost 0 =5 by 0 =5, with almost no change in resonant frequency. With reference to the radiation characteristics, it has also been observed that both the E and H-plane cross-pol improves with reduction in the size of the ground plane of the monopole. III. REDUCTION IN RESONANT FREQUENCY FOR LOOP-LOADED SLOT ANTENNA The miniaturization of the loop-loaded dipole and monopole in Section II has been achieved by affecting self-resonance in a below resonance capacitive antenna using the inductive effect of a loop. In this section, it is investigated if a similar technique could be used to reduce the resonant frequency of a planar slot. The proposed antenna structure is shown in Fig. 8. The slot antenna is fed by a CPW line, which affords a uniplanar design, and allows us to compare the structure with the dipole antenna presented in Section II. The dielectric material Rogers RT6010LM with a dielectric constant of

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Fig. 7. Simulated reflection coefficient of the loop loaded monopole with change in ground-plane size.

Fig. 8. Double-loop loaded slot antenna.

10.20 is used as the substrate. The higher dielectric constant of the substrate makes it easy to fabricate the 50 CPW feed, due to a smaller contrast between the CPW trace width SCPW and the gap between the CPW trace and ground WCPW . The length (Ls ) and width (Ws ) of the substrate are 110 mm each with a thickness of H = 2:54 mm. The dimensions SCPW and WCPW are chosen to be 1.80 mm and 0.76 mm respectively for the CPW line to achieve a characteristic impedance of 50 , with a CPW line length of LCPW = 54:40 mm. The slot length together with the loop loading is LSL = 20 mm with a slot width of WSL = 1:20 mm. The inner and outer diameters of the loop are D1 = 3:00 mm and D2 = 7:00 mm respectively. A quarter-wave CPW matching section of length LST = 11:10 mm and width WST = 0:98 mm is used to enhance the match between the CPW line and the slot antenna. The reflection coefficient for the antenna is shown in Fig. 9. The reflection coefficient for the unperturbed linear slot of the same total length LSL = 20 mm as the antenna in Fig. 8 is also included for comparison. The simulations were done considering the finite conductivity of copper for the ground plane and a dielectric loss tangent of 0.0023 for the substrate material. The resonances for the unperturbed and the loop-loaded antenna structures are measured at 3.26 and 2.63 GHz respectively, corresponding to a reduction of 19.33% using the loaded configuration. The resonant frequency of the antenna loaded with a single loop and of total length 20 mm has also been measured at

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Fig. 10. Simulated reflection coefficient of the DLLSA with change in groundplane size. Fig. 9. Measured and simulated reflection coefficients of the DLLSA and the unperturbed slot antenna.

2.98 GHz, which is 8.59% lower than the unloaded reference antenna, and is not included for brevity. The self-resonance in the planar configuration is affected by the compensation of the capacitive reactance of the unperturbed slot below resonance by the inductance of the loop, in a similar manner as the loop-loaded dipole. Though a significant miniaturization is achieved using the planar topology, the degree of miniaturization is seen to be smaller compared to the two-loop loaded dipole configuration in Fig. 2. Also, the twoloop loaded slot antenna, though substantially smaller than the unperturbed slot antenna, corresponds to ka = 1:30, with the value of k computed from the average of the substrate and the air permittivities to take into account the loading effect of the substrate. The reduced size slot antenna also offers easy matching with the CPW input feed. The resonant input resistance for the loaded slot is 68 as found from the simulations. The dimensions LST and WST of the quarter-wave matching section have been chosen accordingly to affect the match with the 50 CPW line. In addition, the resonant 010 dB bandwidth of the double-loop loaded slot antenna (DLLSA) is measured at 3.75%, which can be compared to a measured bandwidth of 3.03% for the reference slot antenna, corresponding to an increase of around 23.76% in bandwidth for the loaded slot antenna. Also, similar to the radiation characteristics of the loop-loaded dipole and the monopole structures, the co and cross pol patterns in the E and H-planes for the loop loaded slot is observed to be similar to those of an unperturbed slot, thus showing minimal effect due to loop radiation. The measured resonant gain of the loaded slot antenna is at 01:2 dBi compared with 0.1 dBi measured for the reference slot. For determining the efficiency of the antenna using the Wheeler cap method, the slot antenna together with the substrate was completely covered by the conducting shell to prevent back-radiation. The measured efficiencies for the loaded and the reference slot antennas were at 87.70% and 88.30% respectively. These can be compared with the corresponding simulated efficiency values of 89.55% and 90.59% for the two structures from HFSS. The loss in this case is due to the finite conductivity of ground plane and substrate dielectric loss, surface-wave loss and edge currents on the ground plane. The effect on the antenna characteristics with variation in the size of the ground plane was also investigated. The initial ground plane dimensions of 110 mm by 110 mm corresponds to about 0:960 2 0:960

at the resonance frequency of the loaded slot antenna. The reflection coefficient of the antenna as the ground plane size is reduced is shown in Fig. 10. It can be observed that the matching performance improves with the reduction in ground plane size, with the dip in the reflection coefficient at resonance at 029:14 dB for Ls = Ws = 55 mm compared to 019:22 dB for Ls = Ws = 110 mm. Thus it is seen that reducing the substrate dimensions by half results in a better reflection coefficient for the loaded slot. The effect can also be compared to the loaded monopole structure discussed in Section II. Also, similar to the loaded monopole, the resonant frequency remains almost unaltered with the reduction in ground plane size. IV. CONCLUSION In this communication, miniaturization of a dipole antenna is achieved using a loop as a reactive loading element. It is seen that the reactive loading reduces the resonant frequency of the dipole antenna to bring it very close to the electrically small antenna limit without affecting the radiation pattern. Even at this level of miniaturization, the antenna structure is very easy to feed without the use of additional matching networks. The antenna also possesses a better bandwidth than the unperturbed dipole, due to the reduction in the quality factor afforded by the loop radiation. The design also offers ease of fabrication and a high efficiency, due to the reactive compensation effect being provided by the loops without using a coiled conductor structure. The technique was also used to effectively reduce the resonant frequency of a monopole over a ground plane to very near the electrically small limit. The presence of the loop for the loop loaded monopole structure was also found to improve the reflection coefficient in this case relative to the unperturbed monopole. The bandwidth of the loop loaded miniaturized monopole, near the electrically small limit, was also significantly greater than the unloaded monopole. In addition, the miniaturization of a planar slot using the loop loading technique was also investigated. Besides a reduction in resonant frequency, the loaded slot antenna offered a high efficiency, though lower than the miniaturized dipole and monopole, due to the dielectric substrate.

REFERENCES [1] J. S. McLean, “A re-examination of the fundamental limits on the radiation Q of electrically small antennas,” IEEE Trans. Antennas Propag., vol. 44, pp. 672–676, May 1996.

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[2] A. Erentok and R. W. Ziolkowski, “Metamaterial-inspired efficient electrically small antennas,” IEEE Trans. Antennas Propag., vol. 56, pp. 691–707, Mar. 2008. [3] L. J. Chu, “Physical limitations of omnidirectional antennas,” J. Appl. Phys., vol. 19, pp. 1163–1175, Dec. 1948. [4] H. A. Wheeler, “Fundamental limitations of small antennas,” in IRE Proc., Dec. 1947, vol. 35, pp. 1479–1484. [5] S. R. Best, “The radiation properties of electrically small folded spherical helix antennas,” IEEE Trans. Antennas Propag., vol. 52, pp. 953–960, Apr. 2004. [6] S. R. Best, “The performance properties of electrically small resonant multiple-arm folded wire antennas,” IEEE Antennas Propag. Mag., vol. 47, pp. 13–27, Aug. 2005. [7] S. Lim, R. L. Rogers, and H. Ling, “A tunable electrically small antenna for ground wave transmission,” IEEE Trans. Antennas Propag., vol. 54, pp. 417–421, Feb. 2006. [8] R. W. Ziolkowski and A. D. Kipple, “Application of double negative materials to increase the power radiated by electrically small antennas,” IEEE Trans. Antennas Propag., vol. 51, pp. 2626–2640, Oct. 2003. [9] B. Ghosh, S. Ghosh, and A. B. Kakade, “Investigation of gain enhancement of electrically small antennas using double-negative, single-negative and double-positive materials,” Phys. Rev. E., vol. 78, pp. 026611/ 1–026611/13, Aug. 2008. [10] K. Sarabandi and R. Azadegan, “Design of an efficient miniaturized UHF planar antenna,” IEEE Trans. Antennas Propag., vol. 51, pp. 1270–1276, June 2003. [11] R. Azadegan and K. Sarabandi, “A novel approach for miniaturization of slot antennas,” IEEE Trans. Antennas Propag., vol. 51, pp. 421–429, Mar. 2003. [12] Ansoft Corporation. Pittsburgh, PA, HFSS ver. 10.2. [13] D. M. Pozar and B. Kaufman, “Comparison of three methods for the measurement of printed antenna efficiency,” IEEE Trans. Antennas Propag., vol. 36, pp. 136–139, Jan. 1988.

Unified Definitions of Efficiencies and System Noise Temperature for Receiving Antenna Arrays

the past few years of research in this area is that understanding the interaction between antenna element mutual coupling and receiver noise is critical to optimizing system performance. To account for these important interaction effects, several methods have been developed for modeling the sensitivity of a receiver system in a direct numerical manner [1]–[4]. Such methods are powerful tools for characterizing a complex system that includes an antenna array, multi-channel receiver, and beamforming network, but are unwieldy for the purpose of design optimization of low noise phased array receivers. For array design optimization, practical figures of merit and measurement procedures are required that allow predominant factors affecting receiver sensitivity to be isolated and understood. Two approaches were recently introduced for characterizing the performance of mutually coupled antenna arrays in terms of efficiencies and equivalent noise temperatures [5], [6]. Warnick and Jeffs [5] used network theory and the electromagnetic reciprocity principle to express the efficiencies and system noise temperature of an array receiver in terms of the antenna isotropic noise response. The isotropic noise response of the array can be determined using a generalization of the Y -factor noise measurement technique, which provides a method for experimental array characterization [7]–[9]. Another approach for defining antenna figures of merit for array receivers has been given by Ivashina, Maaskant, and Woestenburg [6], using an equivalent system representation with efficiencies and noise temperature as the model parameters of the equivalent single-port antenna and equivalent amplifier. The purpose of this paper is to present definitions for antenna terms that reflect these recent results on coupled, low noise array receivers and can be implemented in practical measurement techniques. For single-port, passive antennas, these definitions reduce to existing IEEE standard antenna terms. We will show that the proposed definitions are consistent with the results in [5], [6], which demonstrates that the two approaches can be unified and places the definitions presented in this paper on a solid theoretical foundation.

Karl F. Warnick, Marianna V. Ivashina, Rob Maaskant, and Bert Woestenburg

Abstract—Two methods for defining the efficiencies and system noise temperature of a receiving antenna array have recently been developed, one based on the isotropic noise response of the array and the other on an equivalent system representation. This letter demonstrates the equivalence of the two formulations and proposes a new set of standard definitions of antenna figures of merit for beamforming arrays that accounts for the effect of interactions between antenna element mutual coupling and receiver noise on system performance. Index Terms—Amplifier noise, antenna array feeds, antenna array mutual coupling, antenna measurements.

I. INTRODUCTION A major international effort is currently underway in the radio astronomy community to develop dense phased arrays with high sensitivity, broad bandwidth, and wide field of view. One of the lessons from Manuscript received June 23, 2009; revised December 18, 2009; accepted December 22, 2009. Date of publication March 29, 2010; date of current version June 03, 2010. K. F. Warnick is with the Department of Electrical and Computer Engineering, Brigham Young University, Provo, UT 84602 USA. M. V. Ivashina, R. Maaskant, and B. Woestenburg are with the Netherlands Institute for Radio Astronomy (ASTRON), P.O. Box 2, 7990 AA Dwingeloo, The Netherlands. Digital Object Identifier 10.1109/TAP.2010.2046859

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II. DEFINITIONS The definitions given below apply to any one output of a receiving antenna system, whether a single antenna, passive array, active array with beamforming network, or an array with beams formed in digital signal processing. The system may include nonreciprocal components such as amplifiers. For array antennas, these quantities and figures of merit are beam-dependent, and their values are contingent on a given set of beamformer coefficients. Isotropic noise response. Antenna output noise power with a noiseless receiver when in an environment with brightness temperature distribution that is independent of direction and in thermal equilibrium with the antenna. Available receiver gain. The ratio of the isotropic noise response of the antenna to the available power at the terminals of any passive antenna over the same noise equivalent bandwidth and in the same isotropic noise environment. Beam equivalent available power. Antenna output power divided by the available receiver gain. Beam equivalent noise temperature. Temperature of an isotropic thermal noise environment such that the isotropic noise response is equal to the noise power at the antenna output per unit bandwidth at a specified frequency. Effective area. In a given direction, the ratio of the beam equivalent available power due to a plane wave incident on the antenna from that direction to the power flux density of the plane wave, the wave being polarization matched to the antenna.

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Radiation efficiency. The ratio of the isotropic noise response with noiseless antenna and receiver electronics to the isotropic noise response. Noise matching efficiency. The ratio of the receiver noise power, with all receiver channels and array element ports ideally noise matched, to the actual receiver noise power at the antenna output. These definitions are based on the theoretical framework developed in [5], [6], [10]. The concept of isotropic noise response was used to derive array antenna figures of merit in [5], [10], and is related to solidbeam efficiency [11]. Available receiver gain for a beamforming array was defined in [6]. Noise matching efficiency was introduced in [5] and is related to coupling efficiency [6], [12]. It can be shown that these definitions are equivalent for a reciprocal antenna to the IEEE standard for antenna terms [5]. The proof proceeds from the fundamental result that the isotropic noise response of a receiving antenna is closely related by the reciprocity principle to the total radiated power of the same antenna when excited as a transmitter.

Fig. 1. (a) Beamforming array receiver system diagram. (b) Equivalent system with reference planes indicated.

III. APPLICATION TO A BEAMFORMING ARRAY To illustrate these definitions, we will apply them to a canonical antenna elebeamforming array shown in Fig. 1(a) consisting of ments with each port connected to a low noise amplifier (LNA) and receiver. The receiver output voltages are arranged into a vector v, which includes contributions from a signal of interest, external noise and interference sources, noise due to ohmic losses in the array, and receiver noise according to

M

v = vsig + vext + vloss + vrec

:

Rv = E[vv H ] = Rsig + Rext + Rloss + Rrec

(2)

v

=

T

t iso

t iso

w

(3)

where Rt;iso = Rext;iso + Rloss and Rloss is obtained under the condition that the array is at temperature iso . Explicit formulas for the isotropic noise response are given in Section III-B and a measurement procedure is described in Section III-C. Available receiver gain. The available receiver gain is

T

G = k PT; B av rec

B

t iso

b iso

k

av sig

sig av rec

(5)

iso

noise

noise av rec

t iso

where noise noise n n ext b loss Rrec . This definition can also be applied to individual components of the system noise due to external thermal sources, antenna losses, or receiver electronics. The beam equivalent receiver noise temperature, for example, is

T = T PP ; rec

P

rec

iso

(7)

t iso

where rec = wH Rrec w. The reference plane for beam equivalent system noise temperature is indicated in Fig. 1(b). By convention, equivalent system noise powers and temperatures are referenced to the antenna ports after antenna losses, whereas external noise sources are referenced to an antenna temperature before losses (“to the sky”), so that the beam equivalent external noise temperature is

T = T PP ; (8) where P = wH R w and P ; = wH R ; w. With these definitions for beam equivalent noise temperatures, it can be shown that the single-port antenna temperature formula T =  T + (1 0  )T + T , where T is the physical temperature of the antenna, ext

ext

ext

iso

ext

ext iso

ext iso

ext iso

rad ext

sys

rad

p

rec

p

generalizes to an active beamforming array. Effective area. The beam effective area is

A = PS e

(4)

where is the system noise equivalent bandwidth and b is Boltzmann’s constant. Beam equivalent available power. The beam equivalent available power due to the signal of interest is

P = GP

sys

P

where we have assumed that the signal and noise contributions are uncorrelated. The receiver output voltage phasors are combined using a vector of complex coefficients w to produce the beam output voltage signal H out = w v . For simplicity, we assume that all elements are beamformed, although the treatment readily accomodates arrays with parasitic elements [6] or beams formed from subarrays. From the definition of the voltage correlation matrix, the time average beam output power relative to a 1 load is out = (1 2)wH Rv w. It is customary in the array signal processing literature to drop the factor of 1/2. Isotropic noise response. If the array is in an isotropic thermal noise environment with brightness temperature iso , then we label the external noise correlation matrix as Rext;iso . The beam isotropic noise response is

P ; = wH R ;

T = T PP ; = TG (6) = k BT = wH R w and R = R + R +

(1)

The receiver output voltage correlation matrix is

P

P

where sig = wH Rsig w. The relationship between beam equivalent available power and the available power at the terminals of the array elements is considered in Section III-D. Beam equivalent noise temperature. The beam equivalent system noise temperature is

av sig

(9)

sig

S

where the signal of interest is a plane wave with power flux density sig and polarization matched to the beam such that e is at a maximum. For an arbitrarily polarized incident field, partial gain and polarization efficiency can be defined according to the usual conventions. Radiation efficiency. The beam radiation efficiency is

A

 = PP ; ; = P P; +; P rad

ext iso t iso

ext iso

ext iso

loss

(10)

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where the beam output noise power due to antenna losses, Ploss = H loss , is measured under the condition that the antenna and isotropic environment are in thermal equilibrium, so that the physical temperature Tp of the antenna is Tiso . Noise matching efficiency. If the minimum equivalent noise temperature of each LNA and receiver chain is Tmin , then the beam noise matching efficiency is

w R w



= TT

min

n

rec

= TT PP ; : min

t iso

iso

rec

(11)

The receivers and array element ports are ideally matched if the optimal source reflection coefficient parameter for each LNA is equal to the active reflection coefficient at the corresponding array element port [12], [13]. In this case, Trec = Tmin and the noise matching efficiency is unity.

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where Text is the equivalent antenna noise temperature due to spillover, atmosphere, and other external sources and Tp is the physical temperature of the antenna. The sensitivity, efficiencies, and equivalent temperatures are in general dependent on the beamformer coefficients (i.e., on the beam scan angle). B. Isotropic Noise Response The definitions given in Section II rely on knowledge of the isotropic noise response of the array. Using network theory, the isotropic noise response can be related to the antenna array S -parameter matrix. Let t be a vector of the forward wave amplitudes of the noise emanating from the array element ports into matched loads with the array in thermal equilibrium with an isotropic thermal environment at temperature Tiso . The correlation matrix of the noise wave amplitudes is defined to be

a

R;

; =E

t iso f

A. Aperture Efficiency, Antenna Efficiency, and Sensitivity Using these fundamental figures of merit, derived antenna parameters such as aperture efficiency, antenna efficiency, and receiver sensitivity can be expressed. These parameters satisfy expected relationships from classical antenna theory, which indicates the consistency of the definitions given in Section II. Aperture efficiency. For large aperture-type antennas, the standard directivity is

D

std

= 4 A

= 4  D

where Dsig is the directivity in the direction of the signal of interest, the aperture efficiency is

= DD = kAb TS B PP ; iso

sig

ap

p

std

sig

sig

ant

=

A A

e

p

=

iso

p

sig

sig

(15)

A T

e

sys

=  T + (1 0  A)T + T = (m =K) rad

rad

ext

ap

rad

p

2

p

v

min

n

(16)

(19)

t

from which it follows that the isotropic noise voltage correlation matrix is

R ; = E v vH = GR ; ; GH : t iso

t

t iso f

t

(20)

The isotropic noise response is related in a similar way to the array mutual impedance matrix. From Twiss’s theorem [18].

R ; = 8kb T t iso

iso

BQRe[Z

A

Q

]QH

(21)

where is a transformation from open circuit voltages at the array element ports to the receiver output voltages [5, Eq. (3)]. The portion of the isotropic noise correlation matrix that is caused by external noise sources is [5]

R

v

; =R;

ext iso

t iso

In view of (10) and (14), it can be seen that ant = rad ap , as expected. While this definition of antenna efficiency conforms to the IEEE standard for antenna terms, other notions of efficiency are found in the literature [14]–[16]. Coupling or mismatch efficiency is sometimes included as an additional factor to yield a total efficiency (which implies that the reference plane is located to the right of the antenna/receiver junction in Fig. 1). While this concept is useful for transmitting arrays, for active receiving antennas, impedance mismatches are inextricably linked with receiver noise and it is therefore more natural to incorporate mismatch losses as an increase in equivalent receiver noise rather than as a decrease in antenna efficiency. Receiver sensitivity. In terms of the effective area and efficiencies, the receiver sensitivity or figure of merit can be expressed as [5]

(18)

A

a

(14)

kb T B P : A S P;

A

v = Ga

ext iso

where we have used (9) and (10) for the effective area and radiation efficiency and (4) and (5) for the available receiver gain and beam equivalent available signal power, respectively. Antenna efficiency. The antenna efficiency is



S

0

iso

where A is the antenna array S -parameter matrix. The forward wave amplitudes t are related to the receiver output voltages t by a linear transformation of the form

(13)

sig

rad

(17)

t

I S SH

; = 2kb T B

t iso f

2

e



R;

t

where for oblique incidence the aperture area Ap is taken to be the projected area in a plane transverse to the signal arrival direction. Using

A

t

where E[1] denotes expectation. From Bosma’s theorem, the forward thermal noise wave correlation matrix is [17]

(12)

p

2

a aH

where

t iso 0

R = I1 16kbT loss

iso

2

j 0j

B QAQH

A is a matrix of the pattern overlap integrals given by 3 Amn = 1 2 E m (r) En (r)r d

2

1

(22)

(23)

0

and E n is the radiation pattern of the nth array element with input current I0 and all other element ports open circuited. C. Measuring the Isotropic Noise Response The element pattern measurements required to implement (22) can be time consuming and impractical. By measuring the array outputs for isotropic thermal environments with two different temperatures, the Y -factor technique can be used to determine ext;iso directly. The array output voltage correlation matrix for a noise environment with brightness temperature Thot is

R

R = TT R hot

; +R

ext iso

hot

iso

loss

+R : rec

(24)

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With a second measurement at temperature Tcold , assuming that the physical antenna temperature does not change between the hot and cold measurements, straightforward manipulations lead to

R

ext;iso

= ThotT0isoTcold (Rhot 0 Rcold ):

(25)

This approach has been implemented successfully using microwave absorber as the hot source and sky as the cold source with a conducting screen to shield the antenna from thermal noise radiated by the ground [7]–[9], [19]. A similar setup was used in earlier work to measure the radiation efficiency of single-port antennas [20], [21]. D. Available Power The relationship given above between the isotropic noise correlation matrix and the array mutual resistance matrix can be used to show that the maximum beam equivalent available power is equal to the available power at the array element terminals. For a single point source, H H , where sig;oc is a vector of the open cirsig = sig;oc sig;oc cuit voltages induced by the signal of interest at the array element terminals. With (21), the beam equivalent available power due to the signal of interest is

R

Qv

v

Q

P

w

Q w

av sig

v

= w8wv Re[vZ ]ww H

H

oc sig;oc sig;oc H

A

oc

w

P = 18 vsig oc Re[ZA ]01 vsig oc : (27) For a reciprocal array (ZA = ZA ), this is the signal power that would H

LNA

Z

Z

IV. EQUIVALENCE OF RECEIVER NOISE TEMPERATURE FORMULATIONS The figures of merit defined above appear at first glance quite different from the treatment of [6], but we will now demonstrate the rigorous equivalence of the two formulations. The critical term is the beam equivalent receiver noise temperature

Trec

=

Tmin n

= Tiso

w R w w R w: H

H

rec

t;iso

R

LNA;f

b

= SA b 0 a 



:

(32)

= E aLNA aLNA = E SA b b SA 0 SA b a 0 a b SA + a a = 2k B T + SA T SA 0 SA T 0 T SA H

H 



where

H

H 







H 



H

T = 2k1 B E a a T = 2k1 B E b b T = 2k1 B E b a

H

H 

(33)

H 

(34a)



H 

(34b)



H 

b





b



H

H





b

:

(34c)

If the intrinsic noise produced by one amplifier is uncorrelated with the noise produced by the others, then  , , and are diagonal matrices.

T T

T

B. Equivalent System Representation Because we assumed that the contribution of receiver stages after the front end LNAs was negligible, rec;f = LNA;f . Using (18) and (33) in (31) leads to

R

TLNA =

(28)

This is the temperature of an isotropic thermal noise distribution, with the array in thermal equilibrium with the environment, required to produce an output thermal noise power equal to the receiver noise power at the output. The goal is to show that this expression is equivalent to the corresponding formula in [6] for a general beamforming array receiver.

(31)

f

The amplifier forward noise wave correlation matrix is

T

be delivered by the array to a conjugate matched, M -port load network H attached to the array element with mutual impedance matrix L = A terminals.

;

t;iso;f

f

a

;

;

H

H

a

(26)

oc

= Tiso wwf RRrec f wwf

in terms of the correlation matrix of forward wave amplitudes at the amplifier input ports. At this point, we will neglect noise contributed by components in the receiver chains other than the LNAs (i.e., we will assume that Trec = TLNA ). Amplifier noise can be modeled in terms of vectors of forward and reverse amplifier noise wave amplitudes 0  and  , respectively, where the negative sign on the forward wave amplitudes is by convention. The total forward amplifier noise amplitude vector is (see [22, p. 51])

v

Z

av;max sig

Trec

b

oc

. The beamformer weight vector that maximizes where oc = this quadratic form is oc = Re[ A ]01 sig;oc , and the resulting maximum beam equivalent available signal power is H

then the beam receiver noise temperature becomes

R

w T +S T S 0S T 0T S w w (I 0 S S ) w H

f

A





H

A

A

H

A

f

H



H

H

A

f

f

A

(35)

for the equivalent amplifier noise temperature. Equation (35) can be expressed as

TLNA =

M

m=1 H

j

wf;m j2 T;m

w (I S S ) w f

A

0

H

A

f

(36)

where

T;m = T;m + j0act;m j2 T ;m 0 2Re[0act;m T ;m ]

A. Receiver Noise Waves Using network theory (28) can be reformulated in terms of forward wave amplitudes into the amplifier input ports. Using (19) in (28) leads to

Trec

GRrec f G w : = Tiso ww GR t iso f G w H

;

;

H

(29)

If we define the beamformer weight vector referred to forward wave amplitudes at the amplifier inputs as

w =G w f

H

0act = w31 f

(30)

M

wf3;n SA;nm

;m

(37)

;m n=1

H

;

H

and the active reflection coefficients are defined by

T;m can be recognized as the numerator of the extended effective input noise temperature in [22, Eq. (4.50)]. Using a standard derivation from two-port noise theory (see, e.g., [13, Appendix]), it can be shown that T;m =

1 0act 0 j

2

;m j

TmLNA

(38)

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where

TmLNA = Tmin;m +

4Rn;m T0 0act;m 0opt;m 2 Z0 1 + 0opt;m 2 (1 0act;m 2 ) j

j

0

j j

j

0 j

j

(39)

is the equivalent noise temperature of the mth LNA in terms of the noise parameters Rn;m , 0opt;m , and Tmin;m . Inserting (38) in (36) leads to

TLNA

M 2 1 0act;m 2 TmLNA : = m=1 wwf ;m H H f (I SA SA ) wf j

j

0 j

j

0

(40)

Equation (40) can be placed in the form

TLNA

H

= wH (I wfSwA Sf H ) w f

0

f

A

M 1 S21 LNA 2 w 2 0act;m 2 TmLNA : f ;m 1 m=1 M 1 LNA 2 wH w f f M S21 j

2

j

j

0 j

j

(41)

j

Using the definitions of the radiation efficiency, mismatch efficiency

mis , uncorrelated noise power gain GPower Eq , and available channel gain Gav m given in [6], this becomes TLNA = Power With mis GEq

M Gav LNA m=1 m Tm : Power mis GEq

(42)

LNA = Gav rec and the definition of Tout in [6] we have

T LNA TLNA = out Gav rec

(43)

which completes the proof that (28) is equivalent to equation (17) of [6]. It follows that the equivalent system representation of [6] is consistent with the definitions given in Section II. V. CONCLUSION We have developed a framework for defining antenna figures of merit for active receiving arrays that unifies several years worth of recent work on mutually coupled, high sensitivity array receivers. This framework extends existing definitions for antenna terms to active arrays and provides standard figures of merit for characterizing array antenna performance, which will benefit ongoing research efforts in low noise receiving aperture arrays and array feeds.

REFERENCES [1] J. P. Weem and Z. Popovic´, “A method for determining noise coupling in a phased array antenna,” in IEEE MTT-S Int. Microwave Symp. Digest, May 2001, vol. 1, pp. 271–274. [2] C. Craeye, B. Parvais, and X. Dardenne, “MoM simulation of signal-tonoise patterns in infinite and finite receiving antenna arrays,” IEEE Trans. Antennas Propag., vol. 52, pp. 3245–3256, Dec. 2004. [3] K. F. Warnick and M. A. Jensen, “Effect of mutual coupling on interference mitigation with a focal plane array,” IEEE Trans. Antennas Propag., vol. 53, pp. 2490–2498, Aug. 2005. [4] R. Maaskant, E. E. M. Woestenburg, and M. J. Arts, “A generalized method of modeling the sensitivity of array antennas at system level,” presented at the Eur. Microwave Week, Amsterdam, Oct. 2004. [5] K. F. Warnick and B. D. Jeffs, “Efficiencies and system temperature for a beamforming array,” IEEE Antennas Wireless Propag. Lett., vol. 7, pp. 565–568, 2008.

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[6] M. V. Ivashina, R. Maaskant, and B. Woestenburg, “Equivalent system representation to model the beam sensitivity of receiving antenna arrays,” IEEE Antennas Wireless Propag. Lett., vol. 7, pp. 733–737, 2008. [7] E. E. M. Woestenburg and K. F. Dijkstra, “Noise characterization of a phased array tile,” in Proc. 33rd European Microwave Conf., 2003, pp. 363–366. [8] E. E. M. Woestenburg and J. C. Kuenen, “Low noise performance perspectives of wideband aperture phased arrays,” in The Square Kilometre Array: An Engineering Perspective. Berlin: Springer, 2005, pp. 89–99. [9] K. F. Warnick, B. D. Jeffs, J. Landon, J. Waldron, R. Fisher, and R. Norrod, “BYU/NRAO 19-element phased array feed modeling and experimental results,” presented at the URSI General Assembly, Chicago, IL, Aug. 8–15, 2008. [10] K. F. Warnick and B. D. Jeffs, “Gain and aperture efficiency for a reflector antenna with an array feed,” IEEE Antennas Wireless Propag. Lett., vol. 5, no. 1, pp. 499–502, 2006. [11] IEEE Standard Definitions of Terms for Antennas, IEEE Std 145-1993. [12] K. F. Warnick, B. Woestenburg, L. Belostotski, and P. Russer, “Minimizing the noise penalty due to mutual coupling for a receiving array,” IEEE Trans. Antennas Propag., vol. 57, pp. 1634–1644, Jun. 2009. [13] E. E. M. Woestenburg, “Noise matching in dense phased arrays” ASTRON, Dwingeloo, The Netherlands, Tech. Rep. RP-083, Aug. 2005. [14] H. Minnett and B. M. Thomas, “Fields in the image space of symmetrical focusing reflectors,” IEE Proc., vol. 115, pp. 1419–1430, 1968. [15] J. Robieux, “Lois generales de la liaison entre radiateurers d’onders. application uax ondes de surface et a la propagation,” Ann. Radioelectron., vol. 14, pp. 187–229, 1959. [16] M. V. Ivashina, M. N. M. Kehn, P. 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, 2009. [17] H. Bosma, “On the theory of linear noisy systems,” Philips Res. Rep., Supplement, no. 10, pp. 1–189, 1967. [18] R. Q. Twiss, “Nyquist’s and Thevenin’s theorems generalized for nonreciprocal linear networks,” J. Appl. Phys., vol. 26, pp. 599–602, May 1955. [19] K. F. Warnick, B. D. Jeffs, J. Landon, J. Waldron, D. Jones, J. R. Fisher, and R. Norrod, “Phased array antenna design and characterization for next-generation radio telescopes,” presented at the IEEE Int. Workshop on Antenna Technology, 2009. [20] J. Ashkenazy, E. Levine, and D. Treves, “Radiometric measurement of antenna efficiency,” Electron. Lett., vol. 21, pp. 111–112, Jan. 1985. [21] D. M. Pozar and B. Kaufman, “Comparison of three methods for the measurement of printed antenna efficiency,” IEEE Trans. Antennas Propag., vol. 36, pp. 136–139, Jan. 1988. [22] J. Engberg and T. Larsen, Noise Theory of Linear and Nonlinear Circuits. New York: Wiley, 1995.

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Low-Cost High Gain Planar Antenna Array for 60-GHz Band Applications Xiao-Ping Chen, Ke Wu, Liang Han, and Fanfan He

Abstract—An effective development of a class of low-cost planar antenna arrays having a high reproducibility is presented for 60-GHz band system applications. The proposed antenna arrays, based on the substrate integrated waveguide (SIW) scheme, consists of one compact SIW 12-way power divider and 12 radiating SIWs each supporting 12 radiating slots. A 50- conductor-backed coplanar waveguide (CBCPW) integrated with CBCPW-to-SIW transition is directly used as the input of the antenna array, thus allowing to accommodate other circuits or MMICs at a minimum cost. An antenna array prototype was implemented on Rogers RT/Duroid 6002 substrate with thickness of 20 mils by our standard PCB process. Measured gain is about 22 dBi with a side lobe suppression of 25 dB in the H-plane and 15 dB in the E-Plane while the bandwidth for the 10-dB return loss is 2.5 GHz.



Index Terms—Planar antenna array, power divider, substrate integrated waveguide (SIW), 60-GHz band.

I. INTRODUCTION 60-GHz band wireless applications have recently received much attention because the allocated unlicensed bandwidth of 7 GHz enables attractive gigabit-per-second applications, including high definition multimedia interface, uncompressed high definition video streaming, high-speed internet, wireless gigabit Ethernet, and close-range automotive radar sensor. One of the most important parts of such systems is the antenna since it strongly influences the overall receiver sensitivity and the link budget. With the consideration on the higher path loss and oxygen absorption of 15 dB/km around 60 GHz band, high-gain and mass-reproducible planar arrays have strongly been desired. High radiation efficiency is also important for the system cost reduction as well as the system performance enhancement [1]. To date, a vast amount of different planar antennas have been studied for millimeter-wave radio and radar applications. Although high gain operations have been demonstrated with microstrip patch antenna arrays, these configurations suffer from serious loss in the millimetrewave band; the efficiency decreases as the gain and/or frequency becomes higher even though those antenna design techniques are basically mature. It was roughly estimated that the efficiency of microstrip arrays with gain of 35 dBi would be lower than 20% in the 60 GHz band [2]. On-chip antennas also have other drawbacks. Their radiation efficiency on conductive high-permittivity silicon is poor and in spite of the short wavelength, they still occupy a non-negligible area on an MMIC chip, which is an important cost factor. The situation is even worse if arrays need to be realized to achieve necessary gain of about 15 to 20 dBi in order to bridge intended distances of up to 10 meters in a WPAN environment [3]. On the other hand, waveguide slot antenna arrays are the most attractive candidates for high-gain planar antennas, having the smallest Manuscript received July 13, 2009; revised October 09, 2009; accepted January 11, 2010. Date of publication March 29, 2010; date of current version June 03, 2010. This work was supported in part by the Natural Sciences and Engineering Research Council (NSERC) of Canada. The authors are with the Poly-Grames Research Center, Department of Electrical Engineering, Ecole Polytechnique de Montreal (University of Montreal), QC H3T 1J4, Canada (e-mail: [email protected]; [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.2010.2046861

Fig. 1. Geometric configuration of the proposed 60 GHz SIW slot antenna.

conductor loss among all the planar feeding structures [4]. However, the complicated 3D waveguide structure has prevented its use in cost-sensitive commercial applications with few exceptions of military or professional applications. A drastic reduction of manufacturing cost of the waveguide arrays to the level of microstrip counterparts has been desired for a long time. Single-layer waveguides for a mass reproducible planar array were presented in [5], [6]. All the waveguides consist of two parts, which are the top plate with slots and the bottom plate. Several types of single-layer waveguide arrays over 12-GHz and 20-GHz bands intended for high efficiency and manufacturability were developed and extended to higher frequencies up to 60-GHz. Nevertheless, costly mechanical manufacturing is still required for single-layer waveguide arrays and special transition structure should be used for the integration with other planar circuits. Substrate integrated waveguide (SIW), also called post-wall waveguide or laminated waveguide in some publications, is realized with two rows of metallised via-holes in a metal-clad dielectric substrate by standard print-circuit-board fabrication technique at low cost. The antenna based on the SIW scheme can easily be integrated with other circuits, which leads to the cost-effective subsystem. Some SIW slot antenna arrays and beam forming networks have been developed [7]–[10]. This communication extends the design of SIW antennas to 60-GHz band and a high-gain 60-GHz SIW slot antenna which can be directly integrated with other planar circuits was prototyped by our standard PCB process and experimentally demonstrated for its performance. II. DESIGN OF THE PROPOSED ANTENNA Fig. 1 shows the geometric configuration of the proposed 60 GHz SIW slot array. With consideration on the dielectric properties and temperature properties of dielectric substrate, Rogers/duroid 6002 with 0.5 oz. rolled copper foil is used in this work. Generally, a thick substrate should be used to reduce the losses in connection with the top and bottom conductors and obtain appropriate offset for the design of radiating slots. In this context, 50 conductor-backed coplanar waveguide (CBCPW) with metalized via holes on both sides for the suppression of unexpected modes should be used as the input of antenna by using a transition between CBCPW and SIW. A 12-way SIW power divider is deployed to feed 12 linear SIW slot arrays, and each of them carries 12 radiation slots etched on the broad wall of SIW. The SIW structure is terminated with a short-circuit three-quarter guided wavelength beyond the centre of the last radiation slot. In order to allocate the slots at the

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standing wave peaks and excite all the slots with the same phase condition (in-phase), the slots in a linear array are placed half a guided wavelength at the required centre frequency and the adjacent slots have the opposite offset with respect to the SIW centre line. The width of radiation slot should be much smaller than the slot length, usually between one tenth and one twentieth of slot length. This, of course, depends on the bandwidth requirements. The detailed design procedure similar to that presented in [11] is as follows. A. Parameter Extraction of Isolated Radiation Slot When a longitudinal slot in the broad wall of SIW is designed around resonance and slot offset is not very big or very small, the forward and backward wave scatterings from the slot are symmetrical in SIW and then the slot can be equivalent to shunt admittance on transmission line. According to the Stegen’s factorization [12], the equivalent shunt admittance can be given as follows

Y (x; y ) Gr G + jB = 1 = g (x)h(y ) = g (x) [h1 (y ) + jh2 (y )] G0 G0 Gr where x is the offset of slot, g (x) = Gr =G0 is the resonant conductance normalized to the conductance G0 of SIW, h(y ) = h1 (y ) + jh2 (y) = (G + jB )=Gr is the ratio of slot admittance to resonant conductance, y = l=l(x; f ) is the ratio of length to resonant length, l(x; f ) =  1 v(x)=2 = c0 1 v(x)=2f is the resonant length. In this way, the calculation of the equivalent slot admittance is reduced to the calculation of three single variable functions g (x), v (x), and h(y ). Commercial full-wave simulator package HFSS is used to extract g (x), v(x), and h(y) of the isolated longitudinal slot. In our work, slot width is 0.18 mm and SIW width a is 2.56 mm. Fig. 2(a) and (b) show g (x=a) and v (x=a) for a discrete number of relative offsets x=a in the range 0.03–0.1. Curve fitting has been applied to approximate g (x=a) and v(x=a) in a continuum which can be directly used in the design of the slot array by the classical iteration procedure [13]. For each offset, the function h(y ) as shown in Fig. 2(c) has also been extracted for y in the range 0.82–1.18. A table-look method for h(y ) is used in the design procedure of the slot array. It is clear that h1 (y ) and h2 (y ) rapidly change with the change of y around the matching point y = 1, which shows that the bandwidth of the SIW slot array is smaller than the bandwidth of conventional rectangular waveguide slot array [14]. B. Design of Antenna Array Based on the classical pattern synthesis procedure, the excitation voltage of slots can be obtained by using Taylor distribution for H-plane pattern with 25 dB first side lobe level and uniform distribution for E-plane pattern. Elliott’s method [13] is used to obtain the length and offset of each slot for a given aperture distribution by considering the internal and external mutual couplings. In this method, active input admittance Y a of each radiating SIW includes both self admittance and mutual coupling effects with the remaining slots. A set of initial values for slot lengths and offsets are assumed, and the mutual coupling between slots is estimated according to the required slot voltage distribution. An optimization routine is then used to identify a new set of slot lengths and offsets such that all the slots are resonant and the matching conditions are satisfied for each subarray. Afterwards, a new set of mutual coupling terms are evaluated again. The procedure is iterated until a convergence is reached and the final slot lengths and offsets are obtained. As for the SIW slot array, the slot offset in SIW may be very small due to the dielectric-filling and height-reduced effects of SIW. Therefore, a fine-tuning procedure may be needed to modify the slot length and slot offset obtained by using the Elliot’s method. Some practical design aspects are considered in this work, for example, slot width and

Fig. 2. (a) g (x), (b) v (x), and (c) h(y ), of isolated longitudinal slot with slot width 0.18 mm in the broad wall of SIW with width of 2.56 mm.

length due to the over-etching and rectangular-end slot will be changed to rounded-end slot in the etching process. Fig. 5 shows the simulated radiation pattern. C. Design of Feeding Network A 12-way power divider similar to that in [2] is used to feed the radiating SIWs. With consideration on the symmetry of feeding network, the feeding network shown in Fig. 3(a) consists of one CBCPW-to-SIW transition, SIW bends and five SIW T-junctions. The method presented in [15] is used to accurately design the transition

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Fig. 3. (a) Configuration, (b) simulated frequency characteristics, of feeding network.

Fig. 5. (a) Experimental setup, (b) simulated and measured radiation patterns in both E-plane and H-Plane at 60.5 GHz.

Fig. 4. Photograph and measured reflection coefficient of the proposed antenna.

between 50- CBCPW and SIW. Usually, the slots in the CBCPW and transition structure should be placed on the opposite side to reduce

the spurious radiation. In this work, the CBCPW slots are etched on the same side as the radiating slots to facilitate measurements. For the design of T-junctions, the size of coupling post-wall window is determined by the power dividing ratio while the position of metalized via hole is used to obtain good input matching. The adjacent radiating SIWs are spaced by a half guided wavelength in the feeding SIW. Therefore, the radiating SIWs are excited with alternating-phase of 180 degree by an incident travelling wave from the input port. Finally, the overall feeding network is analyzed and optimized to compensate the mutual coupling effect from adjacent discontinuities. Fig. 3(b) depicts the simulated frequency characteristics of the overall feeding network. Over the simulation frequency band, the magnitude difference of the

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Fig. 6. Gain at different frequency points for the proposed antenna.

input power between adjacent radiating SIWs is smaller than 0.5 dB, while the input reflection of the feeding network is better than 18 dB. The phase difference of the input power between adjacent radiating SIWs is within the range of 180 6 15 degree over the simulation frequency band, which may reduce the gain bandwidth of the SIW slot array antenna. III. FABRICATION AND MEASUREMENT The proposed antenna array was implemented by using linear arrays of metallized via hole having the diameter of 0.3 mm and the center-to-center pitch of 0.6 mm, which can be made with our laboratory’s standard PCB process. The photograph of the developed antenna is displayed in Fig. 4. Anritsu 37397C vector network analyzer and Anritsu Wiltron 3680 V test fixture are used to measure the reflection coefficient that is depicted in Fig. 4. The measured bandwidth for 10 dB return loss is 2.5 GHz from 59.3 GHz to 61.8 GHz. Fig. 5 shows the experimental setup for the measurement of radiation patterns, and the measured and simulated E-plane and H-plane patterns which very well agree with each other. Due to the restriction of absorbers surrounding the antenna under test, the measurement was operated in the range from 050 degree to 50 degree. The measured side lobe level is better than 15 dB in the E-plane while better than 26 dB in the H-Plane. The gain shown in Fig. 6 was calculated from the Friis transmission equation for different frequency points. The maximum gain is about 22 dBi, which corresponds to the efficiency of about 68% estimated from the gain and directivity. IV. CONCLUSION Planar antenna array based on the substrate integrated waveguide (SIW) scheme is designed and realized on a standard dielectric substrate by a low cost PCB process. Simulated and measured results show that the proposed antenna has good efficiency and side lobe level, and it can be used as a potential candidate for 60-GHz-band applications at low cost.

REFERENCES [1] S. K. Yong and C.-C. Chong, “An overview of multigigabit wireless through millimeter wave technology: Potentials and technical challenges,” EURASIP J. Wire. Comm. Netw., vol. 2007, pp. 1–10, 2007.

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[2] M. Ando and J. Hirokawa, “High-gain and high-efficiency singlelayer slotted waveguide array in 60 GHz band,” in Proc. 11th Int. Conf. Antennas Propag., Edinburgh, U.K., 1997, vol. 1, pp. 464–468. [3] Y. P. Zhang, M. Sun, and L. H. Guo, “On-chip antennas for 60-GHz radios in silicon technology,” IEEE Trans. Electron Devices, vol. 52, pp. 1664–1668, Jul. 2005. [4] S. R. Rengarajan, M. S. Zawadzki, and R. E. Hodges, “Design, analysis, and development of a large Ka-band slot array for digital beamforming application,” IEEE Trans. Antennas Propag., vol. 57, no. 10, pp. 3103–3109, Oct. 2009. [5] M. Ando and J. Hirokawa, “System integration of planar slot array antennas for mmwave wireless applications,” presented at the Proc. XXVIIIth URSI, New Delhi, India, 2005. [6] J. Hirokawa and M. Ando, “Millimeter-wave post-wall waveguide slot array antennas,” in Proc. IEEE AP-S Int. Antennas Propag. Symp., June 2007, pp. 4381–4384. [7] L. Yan, W. Hong, G. Hua, J. Chen, K. Wu, and T. J. Cui, “Simulation and experiment on SIW slot array antennas,” IEEE Microw. Wireless Comp. Lett., vol. 14, no. 9, pp. 446–448, 2004. [8] W. Hong, B. Liu, G. Q. Luo, Q. H. Lai, J. F. Xu, Z. C. Hao, F. F. He, and X. X. Yin, “Integrated microwave and millimeter wave antennas based on SIW and HMSIW technology,” in Proc. Int. Workshop on Antenna Technology: Small and Smart Antennas Metamaterials and Applications, Mar. 2007, pp. 69–72. [9] P. Chen, W. Hong, Z. Kuai, and J. Xu, “A double layer substrate integrated waveguide Blass matrix for beamforming applications,” IEEE Microw. Wireless Comp. Lett., vol. 19, no. 6, pp. 374–376, Jun. 2009. [10] S. Cheng, H. Yousef, and H. Kratz, “79 GHz slot antennas based on substrate integrated waveguides (SIW) in a flexible printed circuit board,” IEEE Trans. Antennas Propag., vol. 57, no. 1, pp. 64–71, 2009. [11] X.-P. Chen, L. Li, and K. Wu, “Multi-antenna system based on substrate integrated waveguide for Ka-band traffic-monitoring radar applications,” in Proc. 39th Eur. Microw. Conf. Symp., Roma, Italy, 2009, pp. 417–420. [12] R. J. Stegen, “Slot radiators and arrays at X-band,” IEEE Trans. Antennas Propag., vol. AP-1, pp. 62–64, Feb. 1952. [13] R. S. Elliott, “An improved design procedure for small arrays of shunt slots,” IEEE Trans. Antennas Propag., vol. AP-31, pp. 48–53, Jan. 1983. [14] R. S. Elliott, Antenna Theory and Design. Hoboken, NJ: Wiley, 2003, ch. 8. [15] X.-P. Chen and K. Wu, “Low-loss ultra-wideband transition between conductor-backed coplanar waveguide and substrate integrated waveguide,” presented at the IEEE MTT-S Int. Microw. Symp., Boston, MA, Jun. 7–12, 2009.

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Optimal Narrow Beam Low Sidelobe Synthesis for Arbitrary Arrays Benjamin Fuchs and Jean Jacques Fuchs

Abstract—The classical narrow main beam and low sidelobe synthesis problem is addressed and extended to arbitrary sidelobe envelopes in both one and two dimensional scenarios. The proposed approach allows also to handle arbitrary arrays. This extended basic synthesis problem is first rewritten as a convex optimization problem. This problem is then transformed into either a linear program or a second order cone program that can be solved efficiently by readily available software. The convex formulation is not strictly equivalent to the initial synthesis problem but arguments are given that establish that the obtained optimum is either identical or can be made as close as wanted to the desired one. Finally, numerical applications and a comparison with a classical solution are presented to both confirm the optimality of the solution and illustrate the potentialities of the proposed approach. Index Terms—Array pattern synthesis, convex optimization, linear programming, semidefinite programming.

I. INTRODUCTION Due to its many applications in communications and signal processing, array pattern synthesis has been extensively investigated over the last several decades [1], [2]. It consists in determining the weightings to be applied to an array that lead to a specified pattern. The synthesis of narrow beam and low sidelobes is a frequently encountered problem that has first been solved by Dolph [3] for uniformly spaced linear arrays. He obtained an analytical expression for the weightings that optimizes the compromise between beamwidth and sidelobe level. Since then, numerous papers have been proposed to generalize and extend Dolph’s work in order to deal with any sidelobe envelope and arbitrary arrays. An arbitrary array can have a non linear shape and may be composed of non uniformly spaced elements having arbitrary and differing radiation patterns. Analytical solutions are of course not available for such arrays. Many numerical synthesis methods have therefore been proposed. Drawing up an exhaustive list would be impossible but one can distinguish two main classes. One class consists in methods that use iteratively re-weighted least square algorithms [4]–[7]. A good review can be found in [8]. Another important class of methods is based on adaptive array algorithms [9]–[13] that adjust their patterns so as to maximize the signal to noise ratio while rejecting a set of adaptively specified interferers. In both classes, an iterative scheme is implemented. At each step, the difference between the desired pattern and the currently obtained pattern is computed and, in an ad hoc way, the weightings in the least squares criterion or the angles and powers of the interferers in the adaptive array schemes are adjusted, before computing the new “optimal” pattern. Moreover, quite often in these schemes the phase of the optimal Manuscript received July 16, 2009; revised October 28, 2009; accepted December 16, 2009. Date of publication March 29, 2010; date of current version June 03, 2010. B. Fuchs is with the IETR/University of Rennes I, 35042 Rennes Cedex, France (e-mail: [email protected]). J. J. Fuchs is with the IRISA/University of Rennes I, 35042 Rennes Cedex, France (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.2010.2046863

pattern needs to be defined and clever phase adaptation means are designed to avoid a wrong choice of phase that would preclude optimal performance. This recursive feedback scheme is fully automatized and requires no outside intervention. The procedure is iterated until a suitable pattern is obtained. According to the authors, convergence generally occurs after a few steps. As rightly pointed out in [14], [15], if the synthesis problem is not convex, there is no guarantee that the absolute optimum is reached and indeed the problems, as they are considered in most of these papers, are not convex. Moreover, while for regular arrays with isotropic sensors, optimal methods, such as Dolph’s technique, tell us what the achievable performance is, for arbitrary arrays no such guidelines pre-exist. It is then difficult to both set a priori reasonable constraints and draw a definite conclusion, regarding the achievable performance of these arrays. Few papers have fully investigated the potentialities of convex optimization in antenna pattern synthesis problems. Optimal focusing arrays in presence of arbitrary sidelobe bounds are synthesized in [16], [17]. In [14], [15], interior point methods are used to constrain the beampattern level of linear, adaptive and broadband arrays. In this communication, the classical narrow main beam and low sidelobe level synthesis problem is considered and extended to arbitrary arrays and arbitrary sidelobe envelopes. The search for the optimal corresponding weightings is formulated as a convex optimization problem and the way to transform this problem into either a linear program (LP) or a second order cone program (SOCP) [18], a recently introduced more general optimization method, is detailed. Note that iterative algorithms using SOCP have been proposed in [19] to synthesize robust array patterns. For uniform linear arrays, the Dolph-Chebyshev’s (DC) excitations are recovered and for more general cases a global optimum is similarly obtained. To simplify the exposition but without any loss of generality, planar arrays, far field synthesis and one-dimensional patterns (function of the bearing only) are considered, but extensions to higher dimensional settings are straightforward. In Section II, the synthesis problem is defined and formulated as a convex optimization problem. The translation of this problem into a LP and into a SOCP is then detailed. Numerical applications are presented in Section III to illustrate the potentialities of the proposed techniques. Conclusions are drawn in Section IV. II. PROBLEM FORMULATION AND RESOLUTION METHODS A. Array Pattern Synthesis

N

Let us consider an antenna array with elements placed at arbitrary but known locations. For the sake of generality, one further assumes that each element has a different pattern with k ( ) the complex gain of the pattern of element in the direction measured with respect to a given reference direction (“broadside”). The array works at the angular frequency and no mutual coupling between elements is considered. The N-dimensional steering vector ( ) associated with direction is then

k

!



g 

a



a()H = [g1 ()e0j! () . . . gk () e0j! () . . . 0j! . . . gN ( )e :

 

()

]

;

(1)

where H denotes the Hermitian transposition, k ( ) is the time-delay for a signal between a given reference point and the element

0018-926X/$26.00 © 2010 IEEE

k

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Fig. 1. Schematic view of the pattern synthesis problem. The magnitude of the far field pattern jf ()j must remain below a sidelobe level ( ) over an angular region S and jf ( )j is maximized at the angular position  known as direction of look.

incident from direction  . The response of the array to such a signal is then f () = a()H w (2) with w the N-dimensional vector of complex weightings. The array pattern synthesis problem addressed, amounts then to find the weighting vector w that makes the magnitude of the array response jf ()j to be below a given envelope () specified in the sidelobe regions denoted S , while being maximal in the main beam. We now translate this synthesis problem into the following convex optimization problem:

H

max