IEEE Transactions on Antennas and Propagation [volume 58 number 1]

Citation preview

JANUARY 2010

VOLUME 58

NUMBER 1

IETPAK

(ISSN 0018-926X)

Editorial .. ......... ......... ........ ......... ......... ........ ..... ..... ......... ........ ......... ......... ........ ......... T. S. Bird

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PAPERS

Antennas A Dual-Linearly-Polarized MEMS-Reconfigurable Antenna for Narrowband MIMO Communication Systems ........ .. .. ........ ......... ......... ........ ......... ......... .. A. Grau, J. Romeu, M.-J. Lee, S. Blanch, L. Jofre, and F. De Flaviis A Lumped Circuit for Wideband Impedance Matching of a Non-Resonant, Short Dipole or Monopole Antenna ....... .. .. ........ ......... ......... ........ ......... ......... ...... V. Iyer, S. N. Makarov, D. D. Harty, F. Nekoogar, and R. Ludwig A Simple Ultrawideband Planar Rectangular Printed Antenna With Band Dispensation ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ...... K. G. Thomas and M. Sreenivasan Novel Broadband Circularly Polarized Cavity-Backed Aperture Antenna With Traveling Wave Excitation .... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... ........ K.-F. Hung and Y.-C. Lin Frequency Selective Surfaces for Extended Bandwidth Backing Reflector Functions ... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ....... M. Pasian, S. Monni, A. Neto, M. Ettorre, and G. Gerini Arrays Antenna Modeling Based on a Multiple Spherical Wave Expansion Method: Application to an Antenna Array ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ..... M. Serhir, P. Besnier, and M. Drissi An External Calibration Scheme for DBF Antenna Arrays ....... ......... ........ ......... ...... H. Pawlak and A. F. Jacob Study and Design of a Differentially-Fed Tapered Slot Antenna Array .. ........ ......... ......... ........ ......... ......... .. .. ........ ......... E. de Lera Acedo, E. García, V. González-Posadas, J. L. Vázquez-Roy, R. Maaskant, and D. Segovia Metallic Wire Array as Low-Effective Index of Refraction Medium for Directive Antenna Application ........ ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... . R. Zhou, H. Zhang, and H. Xin Electromagnetics A Novel Analysis of Microstrip Structures Using the Gaussian Green’s Function Method ...... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ...... M. M. Tajdini and A. A. Shishegar Application of Kummer’s Transformation to the Efficient Computation of the 3-D Green’s Function With 1-D Periodicity ...... ......... ........ ......... ......... ........ ......... ......... ........ .. A. L. Fructos, R. R. Boix, and F. Mesa

4 18 27 35 43

51 59 68 79

88 95

(Contents Continued on p. 1)

(Contents Continued from Front Cover) Electromagnetic Scattering by an Infinite Elliptic Dielectric Cylinder With Small Eccentricity Using Perturbative Analysis ........ ......... ........ ......... ......... ........ ......... ... G. D. Tsogkas, J. A. Roumeliotis, and S. P. Savaidis Numerical Methods Improved Model-Based Parameter Estimation Approach for Accelerated Periodic Method of Moments Solutions With Application to the Analysis of Convoluted Frequency Selected Surfaces and Metamaterials . ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... ...... X. Wang and D. H. Werner Pareto Optimal Microwave Filter Design Using Multiobjective Differential Evolution . ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... . S. K. Goudos and J. N. Sahalos Scattering and Imaging A Sparsity Regularization Approach to the Electromagnetic Inverse Scattering Problem ........ ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ...... D. W. Winters, B. D. Van Veen, and S. C. Hagness Adaptive CLEAN With Target Refocusing for Through-Wall Image Improvement ...... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ........ P. C. Chang, R. J. Burkholder, and J. L. Volakis Wireless A Comprehensive Channel Model for UWB Multisensor Multiantenna Body Area Networks .. ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... .. S. van Roy, C. Oestges, F. Horlin, and P. De Doncker Path-Loss Characteristics of Urban Wireless Channels .. .. K. T. Herring, J. W. Holloway, D. H. Staelin, and D. W. Bliss A MIMO Propagation Channel Model in a Random Medium ... ......... ........ ......... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ..... A. Ishimaru, S. Jaruwatanadilok, J. A. Ritcey, and Y. Kuga Effect of Optical Loss and Antenna Separation in 2 2 MIMO Fiber-Radio Systems .. ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... . A. Kobyakov, M. Sauer, A. Ng’oma, and J. H. Winters Experimental Evaluation of MIMO Capacity and Correlation for Narrowband Body-Centric Wireless Channels ....... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... ......... .. I. Khan and P. S. Hall

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145 155

163 171 178 187 195

COMMUNICATIONS

Application of Characteristic Modes and Non-Foster Multiport Loading to the Design of Broadband Antennas ........ .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... .. K. A. Obeidat, B. D. Raines, and R. G. Rojas 77 GHz Stepped Lens With Sectorial Radiation Pattern as Primary Feed of a Lens Based CATR ....... ......... ......... .. .. ........ ....... M. Multari, J. Lanteri, J. L. Le Sonn, L. Brochier, C. Pichot, C. Migliaccio, J. L. Desvilles, and P. Feil A Single-Layer Ultrawideband Microstrip Antenna ..... ......... ......... ........ ......... ...... Q. Wu, R. Jin, and J. Geng Influence of the Finite Slot Thickness on RLSA Antenna Design ........ ........ .. A. Mazzinghi, A. Freni, and M. Albani Efficient Determination of the Poles and Residues of Spectral Domain Multilayered Green’s Functions That are Relevant in Far-Field Calculations ....... ......... ......... ........ ... A. L. Fructos, R. R. Boix, R. Rodríguez-Berral, and F. Mesa Electromagnetic Scattering From a Slotted Conducting Wedge .. ......... ........ . J. J. Kim, H. J. Eom, and K. C. Hwang The Optimal Spatially-Smoothed Source Patterns for the Pseudospectral Time-Domain Method ....... ......... .... Z. Lin Experimental Microwave Validation of Level Set Reconstruction Algorithm ... ......... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... . D. A. Woten, M. R. Hajihashemi, A. M. Hassan, and M. El-Shenawee Scattering of Electromagnetic Waves From a Rectangular Plate Using an Enhanced Stationary Phase Method Approximation . ......... ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... ......... .. .. C. G. Moschovitis, K. T. Karakatselos, E. G. Papkelis, H. T. Anastassiu, I. C. Ouranos, A. Tzoulis, and P. V. Frangos Experimental Characterization of UWB On-Body Radio Channel in Indoor Environment Considering Different Antennas ....... ......... ........ ..... A. Sani, A. Alomainy, G. Palikaras, Y. Nechayev, Y. Hao, C. Parini, and P. S. Hall

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CORRECTIONS

Corrections to “Modeling Antenna Noise Temperature Due to Rain Clouds at Microwave and Millimeter-Wave Frequencies” ... ......... ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ... F. S. Marzano

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List of Reviewers for 2009 ....... ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... ......... .

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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 R. D. NEVELS, President Elect 2011 2012 A. AKYURTLU *J. T. BERNHARD W. A. DAVIS H. LING M. OKONIEWSKI

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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.2009.2039676

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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 58, NO. 1, JANUARY 2010

Editorial Reflections on 2009

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ESEARCHERS and publishers alike eagerly await the release of citation data. Irrespective of the significant limitations of these data, they are now an integral part of applications for grants, fellowships and promotions as well as individual CVs. For publishers and authors, citations equal impact. The IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION’s Impact Factor has risen more than 160% over the past five years. For the first time, the IEEE TRANSACTIONS is ranked among the top ten electrical engineering publications, based on the latest data from Thomson ISI. These data show that the journal Impact Factor in 2008 was 2.479, compared with 1.636 in 2007. While this is pleasing, more significant is the 53% increase in the total number of citations for the Transactions from 10 375 to 15 884. The total number of papers published (about 476) has remained roughly the same for the past three years. Significantpublicationmilestoneswerepassedin2009without the world coming to an end. Pre-processing of manuscripts went paperless in May when the IEEE TRANSACTIONS migrated to an on-line system. The first issue to be completely assembled on-line was last November, Volume 57, No. 11. This transition was not without its hiccups, as some of you may have noticed. The change has been likened to a pilot migrating from a small plane to a jumbo jet without simulator training! Whilst most changes occurred smoothly, others produced unexpected results: some authors received a publication date by email without the Editorial Office being aware that a paper had been scheduled and papers were queued for publication without the usual pre-processing. The publication of some papers has been delayed, but we should pick this up with some large issues early in 2010. An important benefit of on-line processing is that papers cleared for publication are now available on IEEE Xplore about three months ahead of the printed Transactions (click “view articles”). This has required IEEE to formally specify the date of publication, which it recently determined will be the date an article is publicly available, either as an on-line pre-print via IEEE Xplore or in print, whichever comes first. Further, by January 1, 2011, articles in all IEEE TRANSACTIONS, journals, and letters will include the following dates in the footnotes: manuscript received, manuscript revised, manuscript acceptance, publication and current version. The IEEE TRANSACTIONS commenced doing this last September. The topic of pre-publications leads to a related matter. Some authors of papers under review have been requested to either display a notice indicating the paper is subject to IEEE copyright or to remove it from the web site. Authors and their employers have the right to post their IEEE-copyrighted material on their own web sites without special permission, providing the paper prominently displays a notice alerting readers to their obligations with respect to copyrighted material and that the posted work includes an IEEE copyright notice. An example of

an acceptable notice is: “This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders. All persons copying this information are expected to adhere to the terms and constraints invoked by each author’s copyright.” Please refer to http://www.ieee.org/web/ publications/rights/policies.html for further details. In another part of this issue is listed the names of reviewers for the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION over the past 12 months. Without quality reviewers and Associate Editors, the IEEE TRANSACTIONS would cease to be the leading journal in this specialization of electrical engineering. Conspicuous service of our leading reviewers was recognized with the presentation of certificates during the Antennas and Propagation Society Symposium in Charleston last June. The criteria were: quality (based on reports and scores of Editors in ScholarOne); timeliness (within 30 days); and number of reviews completed in the past 12 months (the minimum was arbitrarily set at ten). Certificates were presented to: • Jorgen Bach Andersen • Thomas Ellis • Stuart Hay • Hon Tat Hui • Michael Jensen • Tzyh-Ghuang Ma • Aldo Petosa • John Sahalos • Kin-Lu Wong • Junho Yeo It is significant that for all these reviewers the average number of days taken to review a paper was less than 20 days, the quality of reviews was typically at the highest score available in ScholarOne and all exceeded the 10 papers minimum for the year (one reviewed with more than 30). So you see some reviewers are working very hard indeed for the Transactions! At the same event, certificates were awarded to high performing Associate Editors based on similar criteria, except that I scored timeliness for each manuscript handled. The Associate Editors recognized for exemplary service were: • Kwok Leung • Duixian Liu • Robert Scharstein I hope that the practice of recognizing reviewers and Associate Editors will continue after my term concludes later this year. Finally, the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION has a new web address http://ieeeaps.org/ aps_trans/index.htm. This is on the IEEE Antennas and Propagation Society web site, the use of which should allow continuity of material from one editor to the next.

Digital Object Identifier 10.1109/TAP.2009.2039157 0018-926X/$26.00 © 2009 IEEE

TREVOR S. BIRD, Editor-in-Chief CSIRO ICT Centre Epping, NSW 1710, Australia

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 58, NO. 1, JANUARY 2010

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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, James Cook University, North Queensland, Australia. During 1982 and 1983, he was a Consultant at Plessey Radar, U.K. In December 1983, he joined CSIRO, Sydney, NSW, Australia, where he held several positions and is currently Chief Scientist in the ICT Centre and a CSIRO Fellow. He is also an Adjunct Professor at Macquarie University, Sydney, Australia. He has published widely in the areas of antennas, waveguides, electromagnetics, and satellite communication antennas, and holds 12 patents. Dr. Bird is a Fellow of the Australian Academy of Technological and Engineering Sciences, the Institution of Engineering and Technology (IET), London, U.K., and an Honorary Fellow of the Institution of Engineers, Australia. 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 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 to 2005, and was Vice-Chair and Chair of the Section, from 1999 to 2000 and 2001 to 2002, respectively. He was an 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 to 2005, and a member of the College of Experts of the Australian Research Council (ARC) from 2006 to 2007. He has been a member of the technical committee of numerous conferences including JINA, ICAP, AP2000, EuCAP, and the URSI Electromagnetic Theory Symposium. He was appointed Editor-in-Chief of the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION in 2004.

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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 58, NO. 1, JANUARY 2010

A Dual-Linearly-Polarized MEMS-Reconfigurable Antenna for Narrowband MIMO Communication Systems Alfred Grau, Jordi Romeu, Ming-Jer Lee, Sebastian Blanch, Lluís Jofre, Member, IEEE, and Franco De Flaviis

Abstract—The design and characterization is described of a compact dual-linearly-polarized reconfigurable 2-port antenna. The antenna can operate in two different selectable linear polarization bases, thus being capable of reconfiguring/rotating its polarization base from vertical/horizontal (0 90 ), to slant 45 . The antenna has been implemented on a Quartz substrate, and uses monolithically integrated micro-electromechanical (MEM) switches to select between the two aforementioned polarization bases. The antenna operates at 3.8 GHz and presents a fractional bandwidth of 1.7%. The interest of the proposed antenna is two-fold. First, in LOS scenarios, the antenna enables polarization tracking in polarization-sensitive communication schemes. Second, there are the gains of using it in a multiple-input multiple-output (MIMO) communication system employing orthogonal space-time block codes (OSTBC) to improve the diversity order/gain of the system in NLOS conditions. These benefits were verified through channel measurements conducted in LOS and NLOS propagation scenarios. Despite the simplicity of the antenna, the achievable polarization matching gains (in LOS scenarios) and diversity gains (in NLOS scenarios) are remarkable. These gains come at no expenses of introducing additional receive ports to the system (increasing the number of Radio-Frequency (RF) transceivers), rather as a result of the reconfigurable capabilities of the proposed antenna. Index Terms—Antenna diversity, dual-linearly polarized antenna, micro-electromechanical systems (MEMS), multiple-input multiple-output (MIMO) , polarization diversity, reconfigurable antenna.

I. INTRODUCTION

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EW technologies in communications such as software-defined radio (SDR) and radio-frequency (RF) switches implemented using micro-electromechanical systems (MEMS), present new challenges and opportunities for antenna design. Antennas have traditionally been assumed to have a fixed radiation pattern and polarization at the operating fre-

Manuscript received November 05, 2007; revised July 22, 2009. First published November 06, 2009; current version published January 04, 2010. This work was supported by the National Science Foundation award ECS-0424454 and Project TEC-2007-66698. A. Grau, M.-J. Lee, and F. De Flaviis are with the University of California at Irvine, Irvine, CA 92697 USA (e-mail: [email protected]; mingjerl@uci. edu; [email protected]). J. Romeu, S. Blanch, and L. Jofre are with the Universitat Politècnica de Catalunya, 08034 Barcelona, Spain (e-mail: [email protected]; blanch@tsc. upc.edu; [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.2009.2036197

quency. With the introduction of reconfigurable antennas, it is possible to dynamically change these properties. Reconfigurable antennas work based on the principle that by altering the antenna’s physical configuration, the current density on the antenna may be controlled in a desirable manner and therefore its radiation pattern/polarization/frequency can be changed. To change the antenna’s physical configuration, one can use microelectromechanical (MEM) switches or active devices such as diodes or field-effect transistors (FETs). By placing these components in strategic locations over the geometry of the antenna, the current paths can be engineered in such a way that the resultant radiation patterns of the antenna, or the operating frequencies, follow some desired requirements. There is a fundamental Wheeler-Chu-McLean limitation on the gain as well as hardware limitations when realizing electrically small antennas being simultaneously efficient and broadband. Therefore covering several frequency bands concurrently with a single antenna having enough efficiency and bandwidth is a major challenge. As a result, in the literature one can find reconfigurable antenna designs which do not cover all bands simultaneously, but provide narrower instantaneous bandwidths that are dynamically selectable, such as in [1]–[3]. However, having to excite several radiation patterns and polarization states concurrently with a single antenna may also be impractical. Therefore, several designs of pattern [2], [4]–[7] and polarization reconfigurable antennas exist in the literature. In particular for polarization reconfigurable antennas, [8], [9] present the design of single-port polarization reconfigurable antennas, using PIN diodes, with the capability to reconfigure its polarization from being left-handed to right-handed circular polarization. Other designs with similar characteristics, using PIN diodes, were proposed in [10], [11]. Also [10], [12] introduce two reconfigurable microstrip patch antennas, using PIN diodes, which are able to select between a circular and linear polarization. In [13] a polarization MEMS reconfigurable antenna operating at 26.6 GHz, based on a square shaped microstrip patch, and able to select between a circular and linear polarization from the same antenna is presented. Other polarization reconfigurable antennas using MEMS and working in the 2–6 GHz range have been proposed in [14]–[16]. These single-fed antennas are in general used to change the antenna polarization to operate in multiple communication standards. We go one step further in the design of polarization reconfigurable antennas, and we present a compact dual-linearly-polarized reconfigurable 2-port antenna. Notice that some preliminary simulations of the proposed antenna were first introduced

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GRAU et al.: A DUAL-LINEARLY-POLARIZED MEMS-RECONFIGURABLE ANTENNA FOR NARROWBAND MIMO

by the authors in [17]. In this paper, we include a complete description of the proposed antenna, we present the design guidelines and its characterization through simulated and measured data. To the knowledge of the authors, the proposed antenna is the first one of its kind being reported in the literature. Simulation and measurement results show that the antenna is capable to reconfigure/rotate its polarization base from being vertical/hor, to slant . MEM switches have been used izontal to reconfigure the radiation characteristics of the proposed antenna. The reasons are several. Traditionally diodes and field-effect transistors (FETs) have relatively larger insertion losses, especially at higher frequencies, in the order of 0.5 dB or larger. On the other hand, MEM switches offer a superior technology, in which the insertion losses are of the order of 0.1 dB [7], [18], [19]. Both technologies are similar in terms of losses at low frequencies. However, other advantages of MEMS technology over diodes include larger isolation, an almost a negligible DC power consumption, and most importantly they can be monolithically integrated within the antenna because they can be fabricated on cheap substrates such as PCB or Quartz. Thus MEMS switches are seen as a powerful technology to be employed in the design of reconfigurable antennas. In addition, many studies on reconfigurable antennas omit to evaluate the gains introduced by these antennas at system level and in real propagation scenarios. In this paper, we do put emphasis on these aspects. In particular, it was found that the interest of the proposed antenna is two-fold. First, in line-of-sight (LOS) scenarios, the antenna has polarization tracking capabilities, which are of particular interest when used with communications schemes where the polarization orientation/matching between the transmit and receiver antennas is critical (such as in phase array architectures). Secondly, in non-line-of-sight (NLOS) propagation scenarios, the antenna can be used to improve the diversity order (and array gain) of MIMO communications system employing orthogonal space-time block codes (OSTBC). This is done by using heterogeneous polarization configurations among the transmit and receive antennas [20], [21], and also taking advantage of the fact that reflection, diffraction, and scattering affect differently each polarization, thus producing signals at the receiver with uncorrelated fading statistics. These benefits are quantified analytically. Finally, channel measurements conducted in LOS and NLOS propagation scenarios are used to verify the theoretical findings. The paper is organized as follows: in Section II the proposed antenna is described and measured parameters are presented. Section III introduces the system and channel model used to describe the system level performance of the antenna. Sections IV and V presents the theoretical benefits of using the proposed antenna in LOS and NLOS environments, through a phased array scheme and a MIMO system using OSTBCs, respectively. Performance (channel) measurement results are finally presented in Section VI. Notation: Throughout this paper we use bold upper-case letters to represent matrices, bold lower-case letters to represent vectors, and , and to denote transpose, complex conjugate and Hermitian, respectively. represents where is the elevation angle with the solid angle.

5

Fig. 1. Schematic of the ORIOL antenna. Notice the two input ports and the structure of the biasing system using radial stubs. Zoom-in of the feeding systems showing the structure of the MEM switches, referred in the text by SW .

origin in the z-axis, and the x-axis. Notice that

is the azimuth angle with origin on .

II. ORIOL ANTENNA A. Description The proposed antenna is a compact dual-linearly-polarized reconfigurable 2-port antenna. Henceforth, we refer to it as the octagonal reconfigurable isolated orthogonal (ORIOL) element antenna. The ORIOL antenna consists of a single octagonal microstrip patch, as shown in Fig. 1, in which its two ports always excite two orthogonal polarizations (dual-polarization) of the radiated electric field. This is achieved by exciting the patch from two points located in perpendicular sides of the octagonal patch. We refer to these two polarizations as a polarization base. Moreover, the antenna has the capability to reconfigure/rotate its polarization base in two different radiation states, that is, or ), to slant from being vertical/horizontal ( (or ), or viceversa, where notice that we use the reference coordinate system given in Fig. 1. Each of the ways in which a particular reconfigurable antenna can radiate is defined as a radiation state. We use index to numerate them. If represents the total number of radiation states in which a reconfigurable antenna can operate, the ORIOL an. We use index tenna is such that to numerate the ports of the ORIOL antenna. A detailed schematic of the structure of the ORIOL antenna with its basic dimensions is shown in Fig. 2(a). Most of the design complexity in the ORIOL antenna resides in the feeding structure, thus we proceed to describe it. Notice that each port 50 feeding line connects to a quarter wave transformer through a high-impedance line which, after a few millimeters, splits into two high-impedance lines that connect to the octagonal patch at two adjacent sides. The purpose of these high-impedance lines is to transform the high input antenna impedance value seen at the edges of the octagonal patch into a 50 impedance value. At each location where the high-impedance quarter wave transformer splits into two lines, we have located four switches,

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Fig. 3. Representations of the simulated (using HFSS 10) current distributions associated with each antenna-port, for the two radiation states. In particular, (a) J (r ), (b) J (r ), (c) J (r ), (d) J (r ).

Fig. 2. In (a) the basic dimensions of the ORIOL antenna are being shown and (b) represents a zoom-in of the feeding system and the series MEMS-switch structure.

denoted by , as shown in Fig. 1. For a proper operation of the antenna, the activation of these switches has to guarantee that the currents do not pass trough the two split high-impedance lines simultaneously, but only through must be conone of them. I.e., the switches trolled simultaneously, to ensure that in the first radiation state they be both in the electrical of the ORIOL antenna state ON, thus exciting the patch from locations (from port 1) and (from port 2), as shown in Fig. 2(a). Logically, in this ramust be both in the diation state, the switches electrical state OFF. If we use to denote the current input port when the ORIOL distribution excited from the radiation state, then in the radiation antenna is set in the the current distributions excited on the octagonal state patch are . Simulated plots (using HFSS 10 [22]) of the aforementioned current distributions are shown in Fig. 3(a) and (b). In the second radiation state of the ORIOL

antenna , the switches must be simulin taneously in the ON state and the switches the OFF state, such that the patch is excited from locations (from port 1) and (from port 2), as shown in Fig. 2(a). In this , case, the excited current distributions are as shown in Fig. 3(c) and (d). Reconfiguring the location of the input feeding point along the edge of the antenna, at points , , , , results in reconfiguring the polarization of the ORIOL antenna. Because these points are separated by an angular distance of 45 , the two desired polarization bases can be excited. Table I summarizes the logic that relates the radiation states of the ORIOL antenna to the electrical state of the MEM switches in order to excite a particular current distribution one the ORIOL antenna. as the normalized farLet us define now port of the ORIOL antenna, field radiation pattern at the radiation state. Notice when the antenna is operated in the is the radiation pattern associated with the current that, distribution . In the first radiation state of the ORIOL , the distribution currents antenna produce the radiation patterns . The polarization of these radiation patterns in the -axis direction is and thus the polarization base referred to pre), is excited. On viously as the vertical/horizontal (or the other hand, in the second radiation state of the ORIOL an, the distribution currents protenna , which in the duce the radiation patterns -axis direction are polarized in the directions . This corresponds to the second polarization base of slant . The polarization of these radiation -axis direction and its relation to the patterns in the electrical state of the MEM switches, is summarized in Table I.

GRAU et al.: A DUAL-LINEARLY-POLARIZED MEMS-RECONFIGURABLE ANTENNA FOR NARROWBAND MIMO

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TABLE I CORRESPONDENCE TABLE BETWEEN THE ORIOL ANTENNA RADIATION STATES AND THE REQUIRED STATE OF THE MEM SWITCHES TO EXCITE A PARTICULAR POLARIZATION, RADIATION PATTERN AND CURRENT DISTRIBUTION

B. Structure of the Switches and Bias Networks The feeding and biasing structures, as well as the non-ideal isolation capabilities of switches, were previously found to have measurable effects on the polarization of the excited fields. In our case, the switches were implemented with MEMS technology. In particular, the switch used in this work is a capacitive MEM switch whose structure is based on a double-supported suspended membrane over a microstrip line, and was developed during previous research studies within our group [7], [18], [19], [23], [24]. The basic dimensions of the used MEM switch are given in Fig. 2(b). In addition, the gap of the metal bridge is 5 , the thickness of the membrane is 0.5 , and the diameter of the membrane holes is 10 . Notice that the switches are composed of two series-MEM switches in series (instead of one single series-MEM switch), as shown in Fig. 1. The reason for this is to improve signal isolation and reduce the coupling of currents on the lines that need to be disconnected, in each antenna radiation state. At 3.8 GHz, the isolation achieved with a single series-MEM switch was found to be around 16 dB [18], while with two series-MEM switches in series the isolation level was increased up to 23 dB. Given the above isolation level, and to compensate the effect of the feeding and biasing structures, the exact locations of the four input points ( , , , , shown in Fig. 2(a)) were then varied along the edge and optimized such that the two desired polarizaand slant ), could be excited. By doing tion bases ( this, the produced radiated fields have a clean polarization (in direction) that deviates from the desired one by less the than 2 . Finally, notice that the proposed antenna uses a total of eight monolithically integrated series MEM switches, which are strategically located in the microstrip feeding structure of the antenna to achieve the desired reconfigurable capabilities. The actuation voltage of the used MEMS was 30 V. Although this required voltage can be a drawback for certain applications, electrostatically activated MEMS switches are suitable for very low power consumption applications. The bias networks are used to activate or deactivate the MEM switches. These networks are composed of DC control lines and radial open stubs, which are specifically designed to assure that the RF signals do not penetrate on the bias networks, that is, to create open circuits at RF frequencies. Basically, both the radial open stubs and the microstrip transmission lines that connect the radial stubs to either the octagonal patch or the MEM switches, are quarter-wave transformers [25], which translate the open-circuit of the radial stubs to the biasing points. Fig. 2(a) shows the location of the bias networks on the surroundings of the ORIOL antenna and its basic dimensions.

Fig. 4. Picture of the ORIOL antenna fabricated on a Quartz substrate, and ). zoom in of one of the four MEM switches (SW

C. Fabrication The ORIOL antenna was fabricated on a Quartz substrate. A picture of the antenna and a zoom in of one of the four MEM is shown in Fig. 4. The thickness of the switches , with relative dielectric conQuartz substrate is , and dissipation factor . The stant fabrication process is that developed by the authors and extensively described in [19]. The metal used to pattern the microstrip . patch and the bias networks is gold with a thickness of 0.5 D. Scattering Parameters Figs. 5 and 6, show the simulated and measured scattering parameters, at the input ports of the ORIOL antenna, over a frequency spanning from 3.5–4 GHz, in the radiation states and , respectively. In both states, there exists a good isolation level between ports of about 30–32 dB and the return loss is above 15 dB in any port. As commented in Section II-B, such large isolation values allow the ORIOL antenna to radiate with a very small deviation from the desired polarization. In fact, as it will be shown in the next sections, the co-cross polarization level is around 18–30 dB. The 10 dB return loss bandwidth of the ORIOL antenna is 1.7%, and its resonant frequency is 3.82 GHz. Notice that in applications with specific bandwidth requirements, several well known techniques could be used to expand the bandwidth of this antenna, such as using smaller permittivity substrates, or changing the excitation mechanism of the patch antenna, among other. E. Radiation Patterns Fig. 7 shows the simulated and measured normalized far-field radiation patterns of the ORIOL antenna at any of its two ports, in the radiation states and , for the (x-axis) cut plane. Fig. 8 shows the same quantities for the (y-axis) cut plane. At both figures, in the radiation state [sub-figures (a)–(b)], the co-polar component for port (a) is set in the direction , and at port (b) the co-polar

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III. SYSTEM MODEL

A. System

Fig. 5. Simulation (gray lines) and measurement (black lines) of the ORIOL antenna scattering parameters in the radiation state p = 1.

In order to evaluate the performance of the ORIOL antenna at system level, we first present our system and channel model. Consider a communication system with one ORIOL antenna at the transmitter in a fixed radiation state, let us assume in the ra, such that its polarization base is always verdiation state . At the receiver, assume one ORIOL tical/horizontal antenna which can change its radiation state such that it can seor . Thus, we can define lect a polarization base of the following two possible channel propagation states (CPSs) for our system. configura• CPS 1: transmit ORIOL antenna in the configuration and receive ORIOL antenna in the tion. configura• CPS 2: transmit ORIOL antenna in the . tion and receive ORIOL antenna in the to numerate them. The ORIOL where we use index antenna arrangements for the two CPSs are illustrated in Fig. 9. Notice that in this work, we do not use reconfigurability at the transmitter. We assume also perfect knowledge of the channel matrix at the receiver only. Because the ORIOL antenna is a 2-port antenna, the proposed system model can be analyzed as transmit ports a MIMO communications system with receive ports (2 2 system). For the sequel, we use and index and to numerate the ports of the transmit and receive ORIOL antenna, respectively. as the channel matrix in the We define now propagation state, given by

(1) where denotes the channel coefficient containing the gain and phase information of the paths between the transmit port of the ORIOL antenna and the receive port of the ORIOL antenna, during CPS .

Fig. 6. Simulation (gray lines) and measurement (black lines) of the ORIOL antenna scattering parameters in the radiation state p = 2.

component is set in the direction . In the polarization state [sub-figures (c)–(d)], at port (c) the co-polar component is set in the direction , and at port (d) the co-polar component is set in the direction . The measured maximum gain is 4.9 dBi. This value is relative low for a patch antenna, but it can be explained from the fact ) is a fraction of the skin depth. that the metal thickness (0.5 Notice that the gain could be easily improved by depositing a thicker metal layer. Finally, the cross-polarization component is always below 18 dB in any port in any of the two state, for the two cut planes. The radiation patterns have been measured at the resonant frequency of 3.82 GHz.

B. Channel Model With the Ricean K-factor, , defined as the ratio of deterministic to scattered power, the channel matrix can be expanded into

(2) where the entries are in general correlated complex Gaussian random variables with zero-mean. These variables are used to describe the scattering nature of the (NLOS) propare agation channel. On the other hand, the entries of deterministic variables which describe the LOS component of can be directly computed from the transmit the channel.

GRAU et al.: A DUAL-LINEARLY-POLARIZED MEMS-RECONFIGURABLE ANTENNA FOR NARROWBAND MIMO

9

=0 =2

=1

Fig. 7. Co (solid line) and cross (dotted line) polarization components of the normalized radiation pattern for the  cut plane. In the polarization state p (a) the co-polar component is set in the direction y , and at port n (b) the co-polar component is set in the direction x. In the polarization (a), (b), at port n state p (c), (d), at port n (c) the co-polar component is set in the direction y x = , and at port n (d) the co-polar component is set in the direction y x = . Measurements are in black color, and simulations in gray color.

=1 =2 (0 + ) p2

=2 p ( + ) 2

=1

and receive antenna radiation patterns and the propagation loss factor of the LOS component [26], [27], as follows: (3) where and are the angle-of-departure (AoD) and the angle-of-arrival (AoA) of the LOS path component with respect to the local coordinate systems at the transmitter and receiver, and are the maximum gain associated with respectively, the and ports of the transmit and receive ORIOL antennas, respectively, and is the distance between the transmitter and the receiver. Notice that denotes the radiation states of the receive ORIOL antenna, in the CPS. Unless specified . otherwise, we use For the scattering component of the channel, we assume the Kronecker channel model described in [28] ((2)). This channel model has been widely used in the literature and in several IEEE standards (such as IEEE 802.11n) [29]. Contrary to [29], we do not model the power delay profile (i.e., narrowband assumption) and Doppler effects, since we aim to measure the performance gains due to the spatial diversity provided by the ORIOL antenna. IV. POLARIZATION-TRACKING USING THE ORIOL ANTENNA IN LOS ENVIRONMENTS Due to the capability of the ORIOL antenna to reconfigure its polarization base from being vertical/horizontal to slant

, it can be used to perform polarization tracking in polarization-sensitive communication schemes, such as phased array architectures. Phased array architectures have been shown to be [30], [31], the optimal transmission technique when that is in LOS conditions where the channel matrix is given by . The advantage of phased array schemes over other architectures is that only long-term channel state information consisting of the relative location of the transmitter and receiver, needs to be sent back to the transmitter. However, one drawback of phased array using dual-linearly-polarized antennas is the antenna orientation. Having an unexpected polarization misalignment between the transmit and receive antennas may cause important losses on the Signal-to-Noise Ratio (SNR) of the system. The use of circularly polarized antennas is sometimes proposed to overcome this limitation, because they do not require any alignment [8]. However, these antennas have the inconvenience that the axial ratio (which is a measure of the quality of the circularly polarized waves) tends to increase rapidly as the scanning angle from boresight increases, thus rendering serious reductions on the SNR on these angles. In addition, the axial-ratio bandwidth for microstrip antennas is normally much smaller than the 10 dB bandwidth. On the other hand, MEMS technology enables us to design linearly-polarized reconfigurable antennas that can track the polarization of the incoming waves, such as the ORIOL antenna, and thus solve the problem of antenna orientation without having to use circularly-polarized antennas.

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= 90

Fig. 8. Same representation as in Fig. 7, but for the  cut plane. Measurements are in black color, and simulations in gray color. (a) State ; (b) state at port ; (c) state at port ; (d) state at port .

n=1

p=1

n=2

p=2

n=1

p=2

n=2

Fig. 9. ORIOL antenna arrangements for the two channel propagation states of the proposed system. (a) Channel propagation state 1 Channel propagation state 2 45 degrees. : V-H to

(p = 1)

+0

For simplicity we limit our analysis to the 2 2 system described in Section III, using one transmit and one receive ORIOL antenna. However, the conclusions are valid for phased array structures with an arbitrary number of transmit and receive as the transmitted vector, given antennas. Let us define by

p = 1 at port

(p = 2): V-H to V-H. (b)

is the average transmit power from all transmit ports. The can be written as follows received spatial vector (5) is the received spatial of additive white where Gaussian noise (AWGN) vector. Using (3) and (5), the combined received signal can then be expressed as follows:

(4) where the same symbol is sent from the two ports of the is given by transmit ORIOL antenna. The variance of , where is the average energy per data symbol where at the transmitter. On the other hand,

(6) Notice that the operation in (6) consists of directly combining the signals from all the receive ports, as it is similarly done in

GRAU et al.: A DUAL-LINEARLY-POLARIZED MEMS-RECONFIGURABLE ANTENNA FOR NARROWBAND MIMO

analog phased arrays through an external power combining netand are also the maximum gain work. Assume that direction of the transmit and receive antennas, and that . Finally, using (6), the average receive SNR of the , can be written as system in the CPS , denoted by

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V. DIVERSITY GAIN ENHANCEMENT USING THE ORIOL IN NLOS ENVIRONMENTS As shown in [21], [32], in NLOS propagation scenarios, reconfigurable antennas (such as the ORIOL antenna) can be used to improve the diversity order of a MIMO system using orthogonal space-time block codes (OSTBC) [33], where OSTBCs are a kind of widely used modulation schemes designed for quasistatic flat fading channels. These systems have been shown to , be excellent schemes in NLOS propagation scenarios because they provide maximum diversity gain with very simple decoding complexity. In these systems, the instantaneous SNR , can be written as [33] at the input of the decoder,

(7) represents the noise power. To illustrate the where benefits of the ORIOL antenna, assume in this section, a third CPS, given by: CPS 3: transmit ORIOL anconfiguration and receive ORIOL tenna in the configuration. Notice that in antenna in the , the CP3, and . Therefore and no power is being delivered. On the other hand, for the other two CPSs, it holds that and , and therefore . Thus, a phased array system using reconfigurable ORIOL antennas at the receiver, will pick the radiation state of the receiving antennas that maximizes the receive SNR, , thus tracking the polarthat is ization of the transmit antennas. Notice that since the transmit and receive ORIOL antennas can be oriented randomly, other combinations of polarization bases between the transmitter and the receiver may arise, however, those will produce SNR value and . To summarize, for a non-reconfigurable within dual-polarized phased array architecture, the ratio of worst to best SNR is

(8) while for a reconfigurable dual-polarized phased array architecture using the ORIOL antennas, such ratio is

(10) , with and where we define is the average transmit power from all transmit antennas. We define the average array gain (in the CPS ) of such system as . This quantity gives us an insight on the average received signal power as a result of coherently combining the signals propagating through the channel from the transmit antennas to all the receiving antennas. Notice that OSTBC systems, in contrast to phased array architectures, are not sensitive to antenna orientation, because the SNR is proportional to the summation of the squared absolute values of the channel coefficients. The reconfigurable capabilities of the ORIOL antenna can be used to improve the array gain (and the diversity order) of such systems. A simple way to do it is by allowing the system to select, among the two CPS given in Section III, the CPS that maximizes the receive SNR [21]. We . Assume an ideal NLOS propdenote the optimal state by agation scenario and that we use the Alamouti OSTBC [34]. and , corAssume for now that the channel matrices responding to the two possible CPSs, are iid Gaussian random matrices with zero-mean and unit-variance entries. Then, for the proposed reconfigurable MIMO system using OSTBCs, the SNR is given by

(11) where decoder,

. The average SNR at the , is henceforth given as

(9) (12) which means that the percentage of power received varies in between 50% and 100%, instead of 0% to 100% in the former case. As shown above, the ORIOL antenna does not completely solve the problem of polarization mismatch in linearly polarized phases array architectures, but proves the fact that reconfigurable antenna technology can be used to solve this problem. To solve the problem completely, one would need to use a polarization reconfigurable antenna in which the polarization base can be changed at smaller angular steps than 45 , which is a feasible feature.

where it is possible to find a close-form expression for , given by

(13)

12

where coefficient of

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 58, NO. 1, JANUARY 2010

,

in our case, and is the in the expansion of

VI. MEASUREMENT RESULTS A. Measurement Setup

(14) can be found in [28] A more general expression of for the case in which the ports of a reconfigurable antenna are allowed to be reconfigured independently of each other (as it is not the case in the proposed antenna). For the sake of completeness, in [28] it is also shown that the diversity order of such a . Notice that the expression given above system would be is a simplification of that given in [28]. and , are correlated Gaussian random In general, matrices, and therefore the expression of given in (13) represents an upper-limit. In addition, the diversity order of the but smaller than [28]. system would be larger than Correlation may appear due to the characteristics of the antenna or lack of scattering richness in the propagation channel. Assuming for now an ideal NLOS environment (uniform transmit and receive power spectrums), we can estimate the transmit and receive correlation matrices from the measured radiation patterns only [28], [35]. That is, the entries of the transmit correlation matrix, as defined in [28], are given by

(15)

and the entries of the receive correlation matrix among CPSs and are computed by

(16)

and . Equation (17) where for simplicity we assume summarizes the receive envelope correlation values computed from the measured 3D normalized far-field radiation pattern. As shown, because in the proposed system using the ORIOL , then and , are correlated antenna Gaussian random matrices. However, notice that the correlation values are small enough ( 0.7) to provide significant diversity gain, as commented in [36], and as it was verified through measurements

To compute the received power (or SNR) for any of the two aforementioned architectures, one needs to collect the ampli. tude and phase information of the channel coefficients This information can be extracted using a virtual array technique (VAT) [37]. Using this method, two antennas are moved along distinct linear trajectories, virtually creating the transmit and receive arrays of a particular MIMO system. For each CPS , the coefficients are then computed by switching among the ports of the transmit and receive antennas, then sending a single tone at the operating frequency of the antennas (3.8 GHz in our case), and measuring the propagated signal with a vector network analyzer connected to both the transmit and receive antennas. In our case, the experiments were conducted within a room with several metallic objects and walls, as shown in Fig. 10. For the NLOS measurements, the two robotic arms that move the antennas, were placed in a NLOS configuration and separated by a distance of approximately 6 m. During the LOS measurements, they were placed in a LOS configuration and separated by a distance of 1.5 m. Two ORIOL antennas were used, one being installed in the transmit arm and the other in the receive arm. During the measurements, the transmit and receive and , respecORIOL antennas were moved at steps of tively. One thousand realizations of the channel matrices and were collected. The samples of the channel matrices and were sequentially obtained on a time-invariant channel. To verify the time-invariant channel assumption, the correlation among two sequentially obtained sets of samples of was measured and found to be 0.98. B. Polarization-Tracking Measurements in LOS Environments In this section the presented measurement results are conducted in a LOS environment. Fig. 11 shows the measured cumulative distribution function (CDF) of the quantity (in dB), which is proportional to the receive SNR, for the phased array architecture using one ) and transmit ORIOL antenna (with fixed radiation state one receive ORIOL antenna, as described in Section IV, in each of the CPSs. Notice that in the CPS 1, when the polarizations of the transmit and receive antennas are perfectly aligned, the receive power is maximum. In the CPS 3, when the polarization misalignment is maximum, the receive power drops about 8 dB. Notice that the receive power in the CPS 3 is not zero, which can be explained by the fact that the channel is not ideally LOS, and the radiated polarizations by the ORIOL antennas may deviate from the ideally desired ones by about 2 , as commented in Section II-B. For the same reason, the receive power in CPS 2 only drops about 1.75 dB below that of CPS 1. However, the trend of the measurements agree well with the findings presented in Section IV. C. Diversity and Capacity Gain Measurements in NLOS Environments

(17)

The performance gains of a MIMO system using OSTBCs and reconfigurable ORIOL antennas are now verified through

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TABLE II CORRESPONDENCE TABLE BETWEEN THE CPS IN CASE 1 AND THE REQUIRED STATE OF THE MEM SWITCHES TO EXCITE A PARTICULAR POLARIZATION AND RADIATION PATTERN

channel measurements in a NLOS environment. Two cases are considered: • Case 1: system described in Section III; • Case 2: system described in Section III where the receive ORIOL antenna is followed by an external switch that selects one of the two output ports according to the receive SNR. 2 MIMO Notice that while case 1 can be analyzed as a 2 system with two CPSs, case 2 can be analyzed as a 2 1 MIMO system with four CPSs. Tables II and III describe the logic that relates the CPSs, the electrical state of the MEM switches, the active port selected by the external switch (case 2 only), and the excited radiation pattern and polarization, for case 1 and 2, respectively. We first investigate the correlation properties of the measured channel. From the measured channel coefficients it is possible to estimate the transmit and receive correlation matrices. In particular for the receive correlation matrix, it is given by

(18)

for any value of

. Equations (19) and (20)

Fig. 10. Layout and dimensions of the room where the LOS and NLOS measurements were conducted.

(19)

(20)

X

Fig. 11. Measured CDF of the quantity for a phased array architecture using one transmit (with fix radiation state) and one receive ORIOL antenna, for the three CPSs described in Section IV. These measurements were conducted in a LOS environment.

summarize the measured receive envelope correlation for case 1 and case 2, respectively, computed using (18). From our meathus and , surements we observe that are correlated Gaussian random matrices. Notice that the envelope correlation values follow similar trends to those obtained from the measured radiation patterns assuming an ideal NLOS scenario (see (17)). This is in agreement with the fact that the measurements are conducted in a NLOS environment (non-ideal

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TABLE III CORRESPONDENCE TABLE BETWEEN THE CPS IN CASE 2 AND THE REQUIRED STATE OF THE MEM SWITCHES AND EXTERNAL SWITCH TO EXCITE A PARTICULAR POLARIZATION AND RADIATION PATTERN

G

Fig. 12. Measured CDF of (in dB), for case 1. Solid lines represent the in each one of the CPSs, while the dashed line represents the CDF curves of CDF curve after selecting the optimal CPS, for each measured sample. NLOS measurements.

G

but) with rich scattering objects. In both cases, from the channel measurements. 1) Diversity Gain: Fig. 12 and Fig. 13 show the measured (in dB), for cases 1 and 2, respectively. In Fig. 12 CDF of (case 1), the received power is that collected from the two ports and combined using maximal-ratio-combining (MRC). In Fig. 12 (case 2), the received power is that collected from a single port after selecting the optimal radiation state and port of the receiving ORIOL antenna. In both figures, the solid in each CPSs, while the lines represent the CDF curves of after selecting dashed line represents the CDF curves of the optimal CPS, for each measured sample. , as the ratio We define now the incremental array gain, , using the reconfigof received power, at a probability of to urable ORIOL antenna and selecting the optimal CPS the received power in any of the states. can be expressed as

(21) Notice that is in fact the diversity gain of the system as a result of using the ORIOL reconfigurable antenna. In case 1, is equal to 1.13 dB, and in case 2 it equals 3.86 dB.

G

Fig. 13. Measured CDF of (in dB), for case 2. Solid lines represent the in each one of the CPSs, while the dashed line represents the CDF curves of CDF curve after selecting the optimal CPS, for each measured sample. NLOS measurements.

G

On the other hand, the average incremental array gain, computed as

,

(22) is equal to 0.74 dB and 1.77 dB, for cases 1 and 2, respectively. These diversity gains come at no expenses of introducing additional receive ports to the system (increasing the number of radio-frequency (RF) transceivers), but rather as a result of the reconfigurable capabilities of the ORIOL antenna. Using (13), the theoretical values of the average incremental array gain asand are iid Gaussian random matrices, suming that are 1 dB and 2.5 dB, for cases 1 and 2, respectively. Notice that the measured average values are slightly below those predicted through theory, which has to do with the fact that in our case, and are correlated Gaussian the channel matrices random matrices. Also notice that the received power in case 1 is larger that in case 2. This has to do with the fact that in case 1 we are combining the received power from two ports using MRC, while in case 2 only one receiving port is available after the external switching mechanism. On the other hand, due to the fact that in case 2 not only the optimal radiation state of the ORIOL antenna is selected but also the optimal output port, the incremental array gain is logically larger than in case 1. For the sake of completeness, the above curves are finally compared to

GRAU et al.: A DUAL-LINEARLY-POLARIZED MEMS-RECONFIGURABLE ANTENNA FOR NARROWBAND MIMO

Fig. 14. Simulated bit-error rate vs. SNR curves for the case 1 of the proposed system, using the Alamouti code and binary phase-shit key (BPSK) modulation.

those of a Single-Input Single-Output (SISO) system (1 1), consisting in using only the port 1 in the transmit and receive ORIOL antennas, and not allowing the receive ORIOL antenna to reconfigure its states. In Fig. 12 and Fig. 13, each CPS guarantees a different level of received power. This can be explained by the fact that reflection, diffraction, and scattering affect differently each polarization [26]. Notice that in these figures the channel matrices are not normalized and correspond to those directly extracted from measurements. Finally, Fig. 14 shows the bit-error (BER) rate vs. SNR curves for the proposed system (case 1), using the Alamouti code [34] and Binary Phase-Shift Key (BPSK) modulation. To obtain these curves we have conducted Monte Carlo simulations using the Kronecker channel model given in Section III-B and the measured complex correlation values associated with those given in (19). Notice that the diversity order (defined as the ) of the MIMO-OSTBC slope of the curves when system using the ORIOL antenna is larger than that of a non-reconfigurable system with the same number of transmit , the and receive ports. In fact, at a BER probability of improvement on the SNR is about 2.1 dB. These curves are compared also with that of an ideal reconfigurable system in and are iid. In all the cases, the channel which matrices have been normalized according to [38]. Notice that the diversity order of the system using the ORIOL antenna is and , due to the fact that in between , and in particular it is approximately equal to 6, which is equivalent to the diversity order of a 2 3 MIMO system using OSTBCs. However, only two receive RF transceivers are needed in the system using the ORIOL antenna. Therefore, from a cost perspective, reconfigurable antennas, such as the ORIOL antenna, allow us to build cheaper RF front ends. 2) Capacity Gain: Fig. 15 and Fig. 16 show the CDF of the system capacity (in bits/s/Hz) for cases 1 and 2, respectively. The system capacity, , assuming that the transmitted power

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Fig. 15. Measured CDF of the system capacity (in bits/s/Hz) for case 1. Solid lines represent the CDF curves of the capacity in each one of the CPSs, while the dashed line represents the CDF curve after selecting the optimal CPS, for each measured sample. NLOS measurements.

is equally distributed among the transmit antennas, is defined as

(23) In both figures, the solid lines represent the CDF curves of the capacity in each one of the CPSs, while the dashed line represents the CDF curve after selecting the optimal CPS, for each measured sample. As shown in Fig. 15 and Fig. 16, the capacity gain, computed as the increase on the system capacity at a 10% probability due to the benefits introduced by using the reconfigurable ORIOL antenna, is equal to 0.44 bits/s/Hz and 1.86 bits/s/Hz, for cases 1 and 2, respectively. These capacity gains result from the increase in received power, as commented in Section VI-C-1. Finally, notice that when using (23), the channel matrices have also been normalized according to [38]. VII. CONCLUSION A compact dual-linearly-polarized reconfigurable 2-port antenna was designed, fabricated and tested. Measurements have shown that the antenna is capable of reconfiguring/rotating its , to polarization base from being vertical/horizontal . The antenna has been implemented on a Quartz subslant strate, and uses monolithically integrated micro-electromechanical (MEM) switches to select among the two aforementioned polarization bases. MEMS technology has been chosen in this work because MEM switches are able to offer small insertion loss, large isolation, almost a negligible DC power consumption, and most importantly they can be monolithically integrated within the antenna because they can be fabricated on cheap substrates such as PCB or Quartz. The return loss level seen at the antenna ports was found to be always above 15 dB, and the isolation among ports larger than 30 dB. The measured maximum gain is 4.9 dBi and the fractional bandwidth 1.7%. The system level performance of the ORIOL antenna has also been

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Fig. 16. Measured CDF of the system capacity (in bits/s/Hz) for case 2. Solid lines represent the CDF curves of the capacity in each one of the CPSs, while the dashed line represents the CDF curve after selecting the optimal CPS, for each measured sample. NLOS measurements.

investigated analytically and through measurement. In LOS scenarios, it has been shown that the ORIOL antenna has polarization tracking capabilities, which are of particular interest in polarization-sensitive applications, such as in phased arrays. In NLOS environments, when used on MIMO system employing OSTBCs, it has been probed that the proposed antenna improves the diversity gain/order of the system. As shown, despite the simplicity of the antenna, the achievable polarization matching gains (in LOS scenarios) and diversity gains (in NLOS scenarios) are remarkable. These diversity gains come at no expenses of introducing additional receive ports to the system (increasing the number of radio-frequency (RF) transceivers), but rather as a result of the reconfigurable capabilities of the ORIOL antenna. ACKNOWLEDGMENT The authors are thankful for the support of the Balsells fellowships and the California-Catalonia Engineering Innovation Program 2004–2005. REFERENCES [1] D. Anagnostou, G. Zheng, M. Chryssomallis, J. Lyke, G. Ponchak, J. Papapolymerou, and C. Christodoulou, “Design, fabrication, and measurements of an RF-MEMS-based self-similar reconfigurable antenna,” in Proc. Int. Symp. on Circuits and Syst. ISCAS’03, Feb. 2003, vol. 54, pp. 422–432. [2] S. Nikolaou, R. Bairavasubramanian, C. Lugo, I. Carrasquillo, D. Thompson, G. Ponchak, J. Papapolymerou, and M. Tentzeris, “Pattern and frequency reconfigurable annular slot antenna using PIN diodes,” IEEE Trans. Antennas Propag., vol. 54, no. 2, pp. 439–448, Feb. 2006. [3] A. Grau, M.-J. Lee, J. Romeu, H. Jafarkhani, L. Jofre, and F. De Flaviis, “A multifunctional MEMS-reconfigurable pixel antenna for narrowband MIMO communications,” presented at the IEEE Antennas Propag. Symp., 2007. [4] G. H. Huff and J. T. Bernhard, “Integration of packaged RF MEMS switches with radiation pattern reconfigurable square spiral microstrip antennas,” IEEE Trans. Antennas Propag., vol. 54, no. 2, pp. 464–469, Feb. 2006. [5] C. Jung, M.-J. Lee, G. Li, and F. De Flaviis, “Monolithic integrated re-configurable antenna with RF-MEMS switches fabricated on printed circuit board,” presented at the 32nd Ann. Conf. IEEE Ind. Electron. Society IECON, Nov. 2005.

[6] P. Panaia, C. Luxey, G. Jacquemod, R. Staraj, G. Kossiavas, L. Dussopt, F. Vacherand, and C. Billard, “EMS-based reconfigurable antennas,” in Proc. IEEE Int. Symp. on Ind. Electron., May 2004, vol. 1, pp. 175–179. [7] C. won Jung and F. De Flaviis, “RF-MEMS capacitive series switches for reconfigurable antenna application,” IEEE Trans. Antennas Propag. , submitted for publication. [8] H. Aissat, L. Cirio, M. Grzeskowiak, J.-M. Laheurte, and O. Picon, “Reconfigurable circularly polarized antenna for short-range communication systems,” IEEE Trans. Microw. Theory Tech. , vol. 54, no. 6, pp. 2856–2863, Jun. 2006. [9] N. Jin, F. Yang, and Y. Rahmat-Samii, “A novel reconfigurable patch antenna with both frequency and polarization diversities for wireless communications,” in Proc. IEEE Antennas and Propag. Society Int. Symp., Jun. 2004, vol. 2, pp. 1796–1799. [10] 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. [11] R.-H. Chen and J. S. Row, “Circular polarized antenna with switchable polarization sense,” IEEE Electron. Lett., vol. 36, pp. 1518–1519, Aug. 2000. [12] Y. Sung, T. Jang, and Y.-S. Kim, “A reconfigurable microstrip antenna for switchable polarization,” in Proc. IEEE Antennas and Propag. Society Int. Symp., Nov. 2004, vol. 14, no. 11, pp. 534–536. [13] R. Simons, D. Chun, and L. Katehi, “Polarization reconfigurable patch antenna using microelectromechanical systems (MEMS) actuators,” in Proc. Antennas and Propag. Society Int. Symp. , 2002, vol. 2, pp. 6–9. [14] B. Cetiner, H. Jafarkhani, J.-Y. Qian, H. J. Yoo, A. Grau, and F. D. Flaviis, “Multifunctional reconfigurable MEMS integrated antennas for adaptive MIMO systems,” IEEE Commun. Mag., vol. 42, no. 12, pp. 62–70, Dec. 2004. [15] B. Cetiner, E. Akay, E. Sengul, and E. Ayanoglu, “A MIMO system with multifunctional reconfigurable antennas,” IEEE Antennas Wireless Propag. Lett., vol. 5, no. 1, pp. 463–466, Dec. 2006. [16] A. R. Weily and J. Y. Guo, “An aperture coupled patch antenna system with MEMS-based reconfigurable polarization,” in Proc. Int. Symp. on Commun. and Inf. Technol., Oct. 2007, pp. 325–328. [17] A. Grau, J. Romeu, L. Jofre, and F. De Flaviis, “On the polarization diversity gain using the ORIOL antenna in fading indoor environments,” presented at the Antennas and Propag. Symp., 2005. [18] C. won Jung and F. De Flaviis, “RF-MEMS capacitive series switches of CPW and MSL configurations for reconfigurable antenna application,” in Proc. IEEE Antennas and Propag. Society Int. Symp., Jul. 2005, vol. 2, pp. 425–428. [19] C. won Jung, M. jer Lee, G. Li, and F. De Flaviis, “Reconfigurable scan-beam single-arm spiral antenna integrated with RF-MEMS switches,” IEEE Trans. Antennas Propag., vol. 54, no. 2, pp. 455–463, Feb. 2006. [20] C.-J. Ahn, Y. Kamio, S. Takahashi, and H. Harada, “Reverse link performance improvement for an wideband OFDM using alamouti coded heterogeneous polarization antennas,” in Proc. Comput. and Commun. ISCC, Jun. 2004, vol. 2, pp. 702–707. [21] A. Grau, J. Romeu, S. Blanch, L. Jofre, H. Jafarkhani, and F. D. Flaviis, “Performance enhancement of the alamouti diversity scheme using polarization-reconfigurable antennas in different fading environments,” in Proc. IEEE Antennas and Propag. Society Int. Symp., Jul. 2006, pp. 133–136. [22] “HFSS 11” Ansoft Corporation (ANSYS) [Online]. Available: www. ansoft.com [23] H.-P. Chang, J. Qian, B. Cetiner, F. D. Flaviis, M. Bachman, and G. P. Li, “Design and process considerations for fabricating RF MEMS switches on printed circuit boards,” J. Microelectromechan. Syst., vol. 14, no. 6, pp. 1311–1322, Dec. 2005. [24] J. Y. Qian, G. P. Li, and F. D. Flaviis, “A parametric model of low-loss RF MEMS capacitive switches,” in Proc. Asia-Pacific Microw. Conf., Dec. 2001, vol. 3, no. 3–6, pp. 1020–1023. [25] D. M. Pozar, Microwave Engineering, 2nd ed. New York: Wiley, 1998. [26] C. A. Balanis, Antenna Theory, Analysis and Design, 2nd ed. New York: Wiley, 1997. [27] J. P. Kermoal, L. Schumacher, K. I. Pedersen, P. E. Mogensen, and F. Frederiksen, “A stochastic MIMO radio channel model with experimental validation,” IEEE J. Sele. Areas Commun., vol. 20, no. 6, pp. 1211–1226, Aug. 2002. [28] A. Grau, H. Jafarkhani, and F. De Flaviis, “A reconfigurable multipleinput multiple-output communication system,” IEEE Trans. Wireless Commun., vol. 7, no. 5, pp. 1719–1733, May 2008.

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[29] TGn Channel Models, IEEE 802.11-03/940r4 [Online]. Available: http://www.ieee802.org/11/ May 2004 [30] F. Farrokhi, A. Lozano, G. Foschini, and R. Valenzuela, “Spectral efficiency of FDMA/TDMA wireless systems with transmit and receive antenna arrays,” IEEE Trans. Wireless Commun., vol. 1, no. 4, pp. 591–599, Oct. 2002. [31] A. Lozano, F. Farrokhi, and R. Valenzuela, “Asymptotically optimal open-loop space-time architecture adaptive to scattering conditions,” in Proc. IEEE VTS 53rd Veh. Technol. Conf. VTC Spring , 2001, vol. 1, pp. 73–77. [32] D. Piazza and K. Dandekar, “A reconfigurable antenna solution for MIMO-OFDM systems,” IEEE Electron. Lett., vol. 42, no. 8, pp. 446–447, Apr. 2006. [33] H. Jafarkhani, Space-Time Coding: Theory and Practice, 1st ed. Cambridge, U.K.: Cambridge Univ. Press, 2005. [34] S. Alamouti, “A simple transmit diversity technique for wireless communications,” IEEE J. Sel. Areas Commun., vol. 16, no. 8, pp. 1451–1458, Oct. 1998. [35] C. Waldschmidt and W. Wiesbeck, “Compact wide-band multimode antennas for MIMO and diversity,” IEEE Trans. Antennas Propag., vol. 52, no. 8, pp. 1963–1969, Aug. 2004. [36] C. Waldschmidt, J. Hagen, and W. Wiesbeck, “Influence and modelling of mutual coupling in MIMO and diversity systems,” in Proc. IEEE Antennas and Propag. Society Int. Symp., Jun. 2002, vol. 3, pp. 16–21. [37] R. J. E. , O. Fernandez, and R. P. Torres, “Empirical analysis of a 2 2 MIMO channel in outdoor-indoor scenarios for BFWA applications,” IEEE Antennas Propag. Mag., vol. 48, no. 6, pp. 57–69, Dec. 2006. [38] M. A. Jensen and J. W. Wallace, “A review of antennas and propagation for MIMO wireless communications,” IEEE Trans. Antennas Propag., vol. 25, no. 11, pp. 2810–2824, Nov. 2004.

Ming-Jer Lee received the B.S. and M.S. degrees in physics from the National Tsing Hua University, Hsin-Chu, Taiwan, R.O.C., in 1990 and 1994, respectively, and the Ph.D. degree in electric engineering from the University of California, Irvine, in 2006. From 1996 to 2001, he was the section head of the thin lm process section in Fab4 Taiwan Semiconductor Manufacturing Co. (TSMC), Hsin-Chu, Taiwan, R.O.C. He was in charge of dielectric and metal lm deposition including various CVD and sputter.

Sebastian Blanch was born in Barcelona, Spain, in 1961. He received the Ingeniero and Doctor Ingeniero degrees in Telecommunication Engineering, both from the Polytechnic University of Catalonia (UPC), Barcelona, Spain, in 1989 and 1996, respectively. In 1989, he joined the Electromagnetic and Photonics Engineering Group, Signal Theory and Communications Department, UPC, where he is currently an Associate Professor. His research interests are antenna near field measurements, antenna diagnostics,

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Alfred Grau was born in Barcelona, Spain, in 1977. He received the Telecommunications Engineering degree from the Universitat Politecnica de Catalunya (UPC), Barcelona, Spain in 2001 and the M.S. degree and Ph.D. degree in electrical engineering from the University of California at Irvine (UCI), in 2004 and 2007, respectively. He is currently working as a Scientist at Broadcom Corporation. His research interest are in the field of miniature and integrated antennas, multi-port antenna (MPA) systems, MIMO wireless communication systems, software defined antennas, reconfigurable and adaptive antennas, channel coding techniques, microelectromechanical systems (MEMS) for RF applications, and computer-aided electromagnetics.

Jordi Romeu was born in Barcelona, Spain, in 1962. He received the Ingeniero de Telecomunicacin and Doctor Ingeniero de Telecomunicacin degrees, both from the Universitat Politecnica de Catalunya (UPC), in 1986 and 1991, respectively. In 1985, he joined the Photonic and Electromagnetic Engineering Group, Signal Theory and Communications Department, UPC, where he is currently a Full Professor and is engaged in research on antenna near-field measurements, antenna diagnostics, and antenna design. He was a Visiting Scholar at the Antenna Laboratory, University of California, Los Angeles, in 1999, on a NATO Scientific Program Scholarship, and in 2004 at University of California Irvine. He holds several patents and has published 35 refereed papers in international jounals and 50 conference proceedings. Dr. Romeu was grand winner of the European IT Prize, awarded by the European Commission, for his contributions in the development of fractal antennas in 1998.

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and antenna design.

Lluís Jofre (M’78) was born in Barcelona, Spain, in 1956. He received the M.Sc. (Ing) and Ph.D. (Doctor Ing.) degrees in electrical engineering (telecommunications engineering) from the Technical University of Catalonia (UPC), Barcelona, in 1978 and 1982, respectively. From 1979 to 1980, he was a Research Assistant with the Electrophysics Group, UPC, where he worked on the analysis and near-field measurement of antenna and scatterers. From 1981 to 1982, he was with the Ecole Superieure dElectricite, Paris, France, where he was involved in microwave antenna design and imaging techniques for medical and industrial applications. In 1982, he was appointed Associate Professor with the Communications Department, Telecommunication Engineering School, UPC, where he became a Full Professor in 1989. From 1986 to 1987, he was a Visiting Fulbright Scholar at the Georgia Institute of Technology, Atlanta, working on antennas, and electromagnetic imaging and visualization. From 1989 to 1994, he served as Director of the Telecommunication Engineering School (UPC), and from 1994 to 2000, he was UPC Vice-rector for Academic Planning. His research interests include antennas, scattering, electromagnetic imaging, and wireless communications. He has published more than 100 scientific and technical papers, reports, and chapters in specialized volumes. During 2000 and 2001, he was a Visiting Professor with the Electrical and Computer Engineering Department, Henry Samueli School of Engineering, University of California, Irvine, where he focused on antennas and systems miniaturization for wireless and sensing applications.

Franco De Flaviis was born in Teramo, Italy, in 1963. He received the Laurea degree in electronics engineering from the University of Ancona, Italy, in 1990, and the M.S. degree and Ph.D. degree in electrical engineering from the University of California at Los Angeles (UCLA), in 1994 and 1997 respectively. In 1991, he was an Engineer with Alcatel working as a researcher specializing in the area of microwave mixer design. In 1992, he was a Visiting Researcher at the UCLA working on low intermodulation mixers. Currently, he is an Associate Professor in the Department of Electrical and Computer Engineering, University of California Irvine. His research interests are in the filed of computer-aided electromagnetics for high-speed digital circuits and antennas, microelectromechanical systems (MEMS) for RF applications fabricated on unconventional substrates such as printed circuit board and microwave laminates.

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A Lumped Circuit for Wideband Impedance Matching of a Non-Resonant, Short Dipole or Monopole Antenna Vishwanath Iyer, Sergey N. Makarov, Senior Member, IEEE, Daniel D. Harty, Faranak Nekoogar, and Reinhold Ludwig, Senior Member, IEEE

Abstract—A new technique is proposed for wideband impedance matching of short dipole- or monopole-like antennas in the VHF-UHF bands. Instead of constructing the network topology for every particular antenna, we propose a simple network of one fixed topology. This network is an inductive L-section cascaded with a high-pass T-section. The network includes five discrete components—three inductors and two capacitors. Although the approach is not general, the paper proves that matching with the present network is close to the theoretical limit impedance matching confirmed by Bode-Fano theory. The matching performance also approaches the performance of the Carlin’s equalizer for short dipoles and monopoles. The dipoles and monopoles may have different shape and different matching bandwidths. By using the matching circuit of fixed topology we avoid greater difficulties related to the practical realization of the Carlin’s equalizer. The key point is to minimize the antenna’s matching network complexity (and loss) so that the circuit can be designed and constructed in a straightforward manner. Index Terms—Dipole antennas, impedance matching, small antennas.

I. INTRODUCTION

F

OR modern wideband or multiband hand-held radios typically used in either civilian or military applications, it is often desired to match a non-resonant (i.e., relatively short) monopole, or dipole-like, antenna over a wide frequency band. Current impedance matching techniques involve modifying the antenna structure. Although this is quite popular, such antennas are sometimes complicated to design, build, and analyze. This paper attempts to solve the problem of wideband matching for short non-resonant dipoles/monopoles by using lumped element matching networks, which we believe would minimize the complexity. This approach could also serve as a basis for building adaptive matching networks that modify the antenna response dynamically. The standard narrowband impedance matching techniques include , , and sections of reactive lumped circuit elements, Manuscript received August 31, 2008; revised July 26, 2009. First published November 06, 2009; current version published January 04, 2010. V. Iyer, S. N. Makarov, D. D. Harty, and R. Ludwig are with the ECE Department, Worcester Polytechnic Institute, Worcester , MA 01609 USA (e-mail: [email protected]; [email protected]; [email protected]; [email protected]). F. Nekoogar is with the Lawrence Livermore National Laboratory, Livermore, CA 94551 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.2009.2036192

single- and double-stub tuning of transmission lines, and/or a hybrid approach involving a combination of both [1], [2]. Such circuits may also include transformers. However, these techniques are not applicable when the bandwidth required is 20% or higher. The notion of wideband antenna matching to a generator with a fixed generator resistance of 50 is a classical impedance matching concept. In general, wideband matching circuit design methods can be classified into two groups [3]: the analytical approach and the real frequency technique (RFT) numerical approach. The analytical approach requires a circuit approximation of the antenna’s input impedance (the load) as it uses the data in the complex frequency plane; it has been developed for simple load circuits only. A thorough treatment of the analytical approach for different types of load is available in [3]–[9]. The Carlin’s RFT gain-bandwidth optimization approach [10]–[13] is a numerical technique that does not require a load model. Although being quite versatile, this approach, along with the pure optimization step, still requires non-unique operations with rational polynomial approximations, extraction of equalizer parameters using the Darlington procedure, and a transformer to match the obtained equalizer to the fixed generator resistance of 50 [13]. We sidestep these two approaches and prefer to introduce the equalizer topology up front: the equalizer’s first section is the section with two inductors. This section is adopted from the analysis of whip monopoles [14]. This section is cascaded with a high-pass -section comprising a shunt inductor and two capacitors, intended to broaden the narrowband response of the -section. This matching network topology can be used for all dipole-like antennas. We will show that the present simple circuit yields an average band gain that is virtually identical to the gain obtained with an improved Carlin’s equalizer [13]. The paper is organized in the following way. Section II derives the Bode-Fano limit for a short dipole/monopole based on a low-frequency RC model. Section III describes the equalizer. Section IV describes the numerical method for circuit parameter extraction, analyzes the matching performance, and provides a comparison between the present circuit and other results. Section V discusses the design and performance of a monopole antenna over a large ground plane, with the corresponding equalizer network, at a matching frequency of and a fractional bandwidth of 0.46. Finally, Section VI summarizes and concludes the paper.

0018-926X/$26.00 © 2009 IEEE

IYER et al.: LUMPED CIRCUIT FOR WIDEBAND IMPEDANCE MATCHING OF A NON-RESONANT ANTENNA

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II. THEORETICAL LIMIT ON WIDEBAND IMPEDANCE MATCHING FOR A NON-RESONANT DIPOLE/MONOPOLE A. Antenna Impedance Model , can be For a wire or strip dipole, the input impedance, approximated with a high degree of accuracy [15] as

(1) In (1), is the dipole length, is the dipole radius, , and is the wavenumber. If a strip or blade [16]. We note dipole of width , is considered, then is the equivalent radius of a wire approximation to that the strip dipole. The accuracy of (1) quickly degrades above the first resonance [15]; and at the lower end, (1) is only valid when the dipole radiation resistance is positive and does not approach zero. In the frequency domain this translates to , where is the resonant frequency of an idealized dipole, is the speed of light, and is the center frequency. When a monopole over an infinite ground plane is studied, the impedance is half. B. Wideband Impedance Matching—The Reflective Equalizer The reactive matching network is shown in Fig. 1(a) [10]. The generator resistance is fixed at 50 . This network does not include transformers. Following [10], the reactive matching network is included into the Thévenin impedance of the circuit as viewed from the antenna, see Fig. 1(b). In fact, the network in Fig. 1 is not a matching network in the exact sense since it does not match the impedance exactly, even at a single frequency. Rather, it is a reflective (but lossless) equalizer which matches the impedance equally well (or equally “badly”) over the entire frequency band. The equalizer network is reflective since a portion of the power flow is always being reflected back to generator and absorbed. Following [10], we can consider the generator or transducer gain in the form

(2) The gain is the quantity to be uniformly maximized over the bandwidth, . In practice, the minimum gain over the bandwidth is usually maximized [10], [13]. Note that the transducer gain is the squared magnitude of the microwave voltage transmission coefficient. C. Bode-Fano Bandwidth Limit The Bode-Fano bandwidth limit of broadband impedance matching ([2], [4]) only requires knowledge of the antenna’s input impedance; it approximates this impedance by one of the , , or loads ([4] and [2, p. 262]). The input canonic impedance of a small- to moderate-size dipole or monopole is

Fig. 1. Transformation of the matching network: a) reactive matching network representation, b) Thévenin-equivalent circuit representation. The matching network does not include transformers.

usually very similar to a series circuit, as seen from (1). (a small antenna or an antenna operated When below the first resonance), the antenna resistance is usually a slowly-varying function of over the frequency band of interest whereas the antenna’s reactance is almost a pure capacitance. This observation is valid at a common geometry condition: . The Bode-Fano bandwidth limit for such a circuit is written in the form [2]

(3)

D. Arbitrary Fractional Bandwidth and Arbitrary Transducer Gain Let us first obtain the simple closed-form estimate for the gain bandwidth product, given a rectangular band-pass frequency window of bandwidth and centered within the window and otherwise. at , with Using the expression for gain, , in terms of power reflection , in (2) we rewrite (3) as coefficient

(4)

over the bandwidth and applying the Substituting appropriate limits for to the integral in (4) we arrive at

(5)

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Substituting the fractional bandwidth in (5) and over this bandwidth solving for the maximum gain

(6) In (6), the half-wavelength approximation for the dipole’s resonant frequency is used; and the dipole’s capacitance in the form is chosen. The last approximation follows from (1) when is at least less than one half. Fig. 2 shows the maximum realizable gain according to (6) obtained at different desired bandwidths as a function of matching frequency. The monopole’s impedance is half of the dipole’s impedance and the dipole estimate for the bandwidth is always applicable to the equivalent monopole of half length, assuming an infinite ground plane. E. Comparison With Chu’s Bandwidth Limit It is instructive to compare the above results with Chu’s antenna bandwidth limit [18] rewritten in [17] and [19] in terms of tolerable output VSWR of the antenna and the antenna , where is the radius of the enclosing sphere. We consider, for and example, a short thick dipole of total length . According to Fig. 2(c), the case , leads to a significant generator gain of and over the frequency band, which yields a return loss of 7 dB that is uniform over the operating frequency band. For this dipole example the Chu’s bandwidth limit is about 34% [17]–[19]. Note that this estimate is less optimistic than the Bode-Fano model discussed above, but it includes an uncertainty in relating the antenna -factor to the antenna’s circuit parameters [17]. III. MATCHING CIRCUIT A. L-Section Impedance Matching A small relatively-thin monopole (whip monopole) or a small dipole is frequently matched with a simple -matching doubletuning section [14]. This section is shown in Fig. 3. Ohmic , are mostly due to losses in losses of the matching circuit, the series inductor, which may be the larger one for very short might be on the order of 0.1–1.0 mH for antennas. Namely, HF and VHF antennas. In this UHF-related study, we will neglect those losses. cancels the (large) caQualitatively, the series inductor pacitance of the whip antenna whereas the shunt inductor matches the (small) resistance of the whip antenna to the generator resistance of 50 . Quantitatively, referring to Fig. 3, the analytical result for the tuning inductances has the form for , see, for example, [14]

Fig. 2. Upper limit on transducer gain obtained by using Eq. (6), for three dipoles (a)–(c) of diameter d and length l . The curves are plotted as a function of ratio of the matching frequency to the antenna’s resonant frequency, for an infinitesimally thin dipole with the same length.

B. Extension of the L-Section Matching Network (7) is the angular matching frequency. where Although the -tuning section is very versatile and can be tuned to any frequency by varying , , its bandwidth is extremely small since impedance matching, if done analytically, is carried out for a single frequency.

To increase the bandwidth of the -tuning section at some fixed values of , , we suggest to consider the matching circuit shown in Fig. 4. It is seen from Fig. 4 that we can simply add a high-pass -network with three lumped components (a shunt inductor and two series capacitors) to the -section or, ladder with equivalently, use two sections of the high-pass

IYER et al.: LUMPED CIRCUIT FOR WIDEBAND IMPEDANCE MATCHING OF A NON-RESONANT ANTENNA

21

Fig. 3. A whip-monopole L-tuning network [14] used in the present study (monopole version). Ohmic resistance of the series inductor, R , will be neglected. The matching network does not show the DC blocking capacitor in series with L .

and a given fractional A. For a given center frequency , what are the (normalized) circuit bandwidth parameters that give the required bandwidth? B. What is the gain-bandwidth product and how does it relate to the upper estimate given by (6)? IV. NUMERICAL SIMULATION RESULTS A. Task Table and the Numerical Method We will consider the dipole case and assume monopole equivalency. The set of tested antenna parameters includes Fig. 4. An extension of the L by the T -match.

L-tuning network for certain fixed values of L

,

the series inductor and investigate the bandwidth improvement. The Thévenin impedance of the equalizer, as seen from the antenna, is given by

(8) where . Thus, we introduce three new lumped circuit elements, but avoid using transformers. C. Circuit Optimization Task We assume that the antenna is to be matched over a certain centered at , and that the variation in gain, shown band in (2), does not exceed 25% over the band. If, for this given equalizer circuit, we cannot achieve such small variations at any values of the circuit parameters, the equalizer is not considered capable of wideband impedance matching over the bandwidth . We know that a low-order equalizer (the -matching section) alone is not able to provide a nearly uniform gain over a wider band. However, increasing the circuit order helps. Thus, two practical questions need to be answered.

(9) To optimize the matching circuit with 5 lumped elements we employ a direct global numerical search in the space of circuit space includes up to nodes. The parameters. The grid in vector implementation of the direct search is fast and simple, but it requires a large (64 Gbytes or higher) amount of RAM on a local machine. First, the minimum gain over the bandwidth is calculated for every set of circuit parameters [10]. The results are converted into integer form and sorted in a linear array, in descending order, using fast sorting routines on integer numbers. Then, starting with the first array element, every result is tested with regard to 25% acceptable gain variation. Among those that pass the test, the result with the highest average gain is finally retained. After the global maximum position is found on a coarse mesh, the process is repeated several times on finer meshes in the vicinity of the anticipated circuit solution. B. Realized Gain—Wideband Matching for Fig. 5 shows the realized average generator gain over the passband based on the 25% gain variation rule at different matching , 10, center frequencies. Three dipole geometries with 5 are considered and the bandwidth is fixed at . The realized values are shown by circles; the ideal upper estimate from Fig. 2 is given by solid curves. One can see that the 5-element equalizer performs rather closely to the upper theoretical limit

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, . To scale parameters to is done for other antenna lengths one needs to multiply them by the factor . Whilst the average gain itself does not significantly change when changing capacitor/inductor values, the gain uniformity may require extra attention for a thin dipole (second row in Table I). For thicker dipoles (third and fourth row of the table) one solution to the potential tolerance problem is to slightly overestimate the circuit parameters for a better tolerance. Generally, the usual uncertainty in low-cost chip capacitors and chip inductors seems to be acceptable. Table I also indicates that the equalizer for a wideband matching of the dipole does not involve very large inductors (and large capacitors) and is thus potentially low-loss. D. Comparison With the Results of Ref. [13] In [13] a similar matching problem was solved for a thin and the radius of 0.001 m. dipole of length , .A Matching is carried out for Carlin’s equalizer with an extra LC section has been considered. Fig. 6 reports the performance of our equalizer for this problem (dashed curve). The thick solid curve within the passband is the corresponding result of [13] (and copied from Fig. 6). In our case, the optimization was done based on the 25% gain variation rule. The difference between the two average band gains was found to be 5%. The circuit components for our circuit are , 0.21 , 20 pF, 0.95 , and 17 pF. Note that without 1.06 the extra LC section, the Carlin’s equalizer may lead to a considerably lower passband gain than the gain shown in Fig. 6 [13]. Without any equalizer, the performance is, as expected, far worse. The plot indicates that a 20 dB improvement is achieved at the lower edge of the band and approximately 10 dB at the upper band edge, when the equalizer is used. A significant point that should be noted here is that our equalizer does not include any ideal transformers. V. EXPERIMENTAL RESULTS A. Short Blade Monopole

Fig. 5. Realized average generator gain T over the band (circles) based on the 25% gain variation rule at different matching center frequencies and B : for three different dipoles, obtained through numerical simulation. The realized values are shown by circles; the ideal upper estimates of T from Fig. 2 are given by solid curves, which are realized by using eqn. (6).

6

=05

when the average gain over the band, , is substituted instead. For the majority of cases, the difference between and is within 30% of . The sole -section was not able to satisfy the 25% gain variation rule in all cases except the very last centerfrequency for the thickest dipole. C. Gain and Circuit Parameters—Wideband Matching for , Table I reports circuit parameters of the equalizer for three 23 cm long dipoles with , 10, 5. In every case, matching

We have constructed, and tested several short blade monopole antennas and the corresponding equalizer networks. The antenna’s first resonant frequency is in the range 550–650 MHz. The matching is to be done over a wide, lower frequency band of 250–400 MHz, with the center frequency of 325 MHz. For equal to 20 has been used in every monopole, the ratio, the experiment. Two sets of experiments were considered for the brass monopole antennas centered in the middle of a 1 1 m aluminum ground plane: i) the monopole is resonant at 650 MHz and; ii) monopole is resonant at a slightly lower frequency of 600 MHz. In addition we also consider a smaller ground plane size of 75 75 cm and investigate the matching performance for the monopole resonant around 650 MHz. The results from these experiments are reported in this section. Fig. 7(a) shows the generic monopole setup. B. Wideband Equalizer The ubiquitous FR-4 substrate has been used for the equalizer shown in Fig. 7(b). We have chosen two tunable highcomponents among the five listed in Table II, to compensate for

IYER et al.: LUMPED CIRCUIT FOR WIDEBAND IMPEDANCE MATCHING OF A NON-RESONANT ANTENNA

TABLE I CIRCUIT PARAMETERS AND GAIN TOLERANCE FOR A SHORT DIPOLE WITH THE TOTAL LENGTH l B : BASED ON THE 25% GAIN VARIATION RULE

05

6

23

= 23 cm. MATCHING IS DONE FOR

f

=f

= 0 5, :

tunable components are and , respectively. is a tunable RF inductor from Coilcraft’s series 148 with a tuning range of is a Voltronics 56 nH–86 nH, and a nominal value of 73 nH. series JR ceramic chip trimmer capacitor with a tuning range is the only of 4.5 pF–20 pF within a half turn. Additionally, leaded component, whilst the rest are all surface mount devices.

C. Gain Comparison—1

Fig. 6. Gain variation with frequency for a short dipole or for an equivalent monopole of length 0.5 m and radius of 0.001 m, by numerical simulation of associated equalizer network. The thick solid curve is the result of Ref. [13] for an identical dipole with the modified Carlin’s equalizer, which was optimized over the same passband. Vertical lines show the center frequency and the passband. Transducer gain, in the absence of a matching network, is also shown by a dashed curve following Eq. (2).

Fig. 7. Antenna and matching network: a) A 10.1 cm long and 2.3 cm wide blade monopole over a 1 1 m ground plane used as a test antenna for wideband impedance matching, b) practical realization of the wideband equalizer for the blade monopole antenna following Table II.

2

parasitic effects due to the board assembly, the finite of the discrete components and the manufacturing uncertainties. These

1 m Ground Plane

We compare the gain performance in Fig. 8, for two different modifications of the blade monopole dimensions. The first modification involves a 10.1 cm long and 2.3 cm wide blade monopole, which is resonant at 650 MHz. In Fig. 8(a) the gain achieved by the unmatched monopole antenna (thin solid curve) and the gain of the monopole antenna with the designed wideband equalizer (thick dotted curve), are shown. The average transducer gain achieved in experiment is 0.262 over the bandand width 250–400 MHz with a gain variation of 40%. The values for this particular result are 86 nH and 7.63 pF, respectively. Next, we consider a blade monopole antenna of length 10.8 cm and width 2.3 cm. This blade monopole resonates at 600 MHz. Fig. 8(b) shows the matching performance with (thick dotted curve) and without (thin solid curve) the wideband equalizer. We see that the equalizer performs rather well even under this scenario and achieves a gain of 0.259 within the bandwidth of interest. The gain variation over the band is 28.8%. In this is changed to 73 nH while the capacitance case the value of is unchanged. Here, we also notice an approximate 10 dB improvement over the unmatched antenna, provided by the equalizer at the lower edge of the band. During the experiments we noticed that a resonance may appear at lower frequencies below 200 MHz. In the case of Fig. 8(a), the average experimental gain over the band is 0.262, slightly higher than the corresponding theoretical value of 0.245. In the case of Fig. 8(b), the average experimental gain over the band is 0.259 versus the theoretical

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TABLE II PRACTICAL COMPONENT VALUES USED IN THE MONOPOLE EQUALIZER FOR l

Fig. 8. The experimental gain data (dotted curve 1) in comparison with the theoretical result (thick curve 2) for two blade monopoles with the equalizer network from Table II: a) the blade length is 10.1 cm and the width is 2.3 cm; b) the blade length is 10.8 cm and the width is 2.3 cm. The thin solid curve 3 ) without the in both plots corresponds to the gain (based on the measured equalizer network.

S

value of 0.245. We believe that the average gain difference is within the experimental uncertainty. However, for the local gain behavior, we observe larger variations. The experimental gain is higher in the middle of the band, but is lowered at the band edges. We explain these variations by the associated tuning procedure and by the inability to exactly and capacitance follow the suggested values of inductance . An important mechanism is lumped-element losses at the higher band edge. Yet another uncertainty factor is due to the relatively small size of the measurement chamber. This effect becomes apparent at low frequencies as Fig. 8 indicates. The present results are approximate and have a very significant room for improvement. for the measured antennas is provided in dB In Fig. 9 the as well as scale. For the sake of comparison, the theoretical of the unmatched antenna is also shown. the

=d

= 20 AND = 2 3 cm t

:

S

Fig. 9. The experimental data (dotted curve 1) in comparison with the theoretical result (thick curve 2) for two blade monopoles with the equalizer network from Table II: a) the blade length is 10.1 cm and the width is 2.3 cm; b) the blade length is 10.8 cm and the width is 2.3 cm. The thin solid curve 3 in both plots without the equalizer network. corresponds to the measured

S

D. Gain Comparison—75

75 cm Ground Plane

The effect of ground plane size on the matching performance is considered. The size of the ground plane for this experiment was reduced to 75 75 cm. The length of the blade monopole was retained to be 11.5 cm since the first resonance observed with this setup is approximately 650 MHz. The width of the blade monopole is 2.3 cm. The gain performance for this setup is shown in Fig. 10. The component values are the same except and values which are 60 nH and 12 pF, respecfor the tively. The average transducer gain achieved in this experiment is 0.2854 over the bandwidth 250–400 MHz with a gain variation of 37%. Even though the overall gain variation over the band is greater than 25%, the average gain is slightly higher as compared to the experiments with the larger ground plane. Again the inability to

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Section IV, Table I provides the component values and the realized transducer gain for our 5 element equalizer. Based on the experimental results for the blade monopoles with we can safely predict that a lower VSWR can be expected for ratios by using the reflective equalizer. smaller VI. CONCLUSION

Fig. 10. The experimental gain data (dotted curve 1) in comparison with the theoretical result (thick curve 2) for a blade monopole of length 11.5 cm and width 2.3 cm over a reduced ground plane size with the equalizer network from Table II: The thin solid curve 3 in this graph corresponds to the gain (based on the measured S ) without the equalizer network.

In this study we have presented, investigated, and experimentally validated a new technique of wideband impedance-matching lumped circuit for relatively short non-resonant dipoles or monopoles of different thicknesses/widths, including thick dipoles. The belief is that one simple fixed-topology network can be used to perform wideband impedance matching for a variety of dipole-like antennas, thereby simplifying the matching process. The particular circuit suggested in the present paper includes five lumped components with common manufacturing values at VHF and UHF center frequencies. It is found that the circuit’s performance deviates on average by 30% (maximally by 40%) from the theoretical impedance matching limit when and , where is antenna’s center matching frequency and is the desired fractional bandwidth. Experimental results based on wideband impedance matching of a short monopole antenna over the frequency range 250–400 MHz indicate that an average improvement of 10 dB can be expected at the lower edge of the band. The circuit is not intended to be applied to resonant dipoles/monopoles or to dipoles/monopoles above the first resonance. Its phase characteristics and the noise figure need to be optimized separately. REFERENCES

Fig. 11. The experimental S data (dotted curve 1) in comparison with the theoretical result (thick curve 2) for a blade monopole of length 11.5 cm and width 2.3 cm over a reduced ground plane size with the equalizer network from Table II. The thin solid curve 3 in this graph corresponds to the measured S without the equalizer network.

exactly follow the suggested tuning values is the reason for this large gain variation. In Fig. 11 the for this antenna is plotted in the in dB scale. The tuning procedure results in better middle of the band as compared to the edges thus resulting in higher transducer gain in that region as observed in Fig. 10. E. Discussion on The results shown in Figs. 8–11 of the transducer gain and the , indicate that if the input VSWR requirement needs to be restricted to lie below 4:1 then as discussed in Section II-E ratio for the antenna needs to be and later in Table I, the reduced, which would result in a thicker cylindrical dipole (or monopole) or alternatively a wider blade dipole (or monopole). for a similar antenna as conSpecifically, using an sidered in the experiments resulted in a higher theoretical transducer gain and a minimum theoretical VSWR of 2.6:1. By theoretical we imply the upper limit predicted by using (6). In

[1] D. F. Bowman and E. F. Kuester, “Impedance matching, broadbanding, and baluns,” in Antenna Engineering Handbook, J. L. Volakis, Ed., 4th ed. New York: Mc Graw Hill, 2007, pp. 52-1–52-31. [2] D. M. Pozar, Microwave Engineering, 3rd ed. New York: Wiley, 2005, ch. 5. [3] S. N. Yang, H. Y. Li, M. Goldberg, X. Carcelle, F. Onado, and S. M. Rowland, “Broadband impedance matching circuit design using numerical optimization techniques and field measurements,” in Proc. IEEE Int. Symp. on Power Line Commun. and Its Applications, ISPLC ’07, Mar. 26–28, 2007, pp. 425–430. [4] R. M. Fano, Theoretical Limitations on the Broadband Matching of Arbitrary Impedances Research Laboratory of Electronics, MIT, Tech. Rep. No. 41, Jan. 2, 1948, 44 pages. [5] D. C. Youla, “A new theory of broadband matching,” IEEE Trans. Circuit Theory, vol. CT-11, pp. 30–50, Mar. 1964. [6] W.-K. Chen, “Explicit formulas for the synthesis of optimum broadband impedance-matching networks,” IEEE Trans. Circuits Syst., vol. CAS-24, no. 4, pp. 157–169, Apr. 1977. [7] W.-K. Chen and K. G. Kourounis, “Explicit formulas for the synthesis of optimum broad-band impedance-matching networks II,” IEEE Trans. Circuits Syst., vol. CAS-25, no. 8, pp. 609–620, Aug. 1978. [8] W.-K. Chen, Broadband Matching: Theory and Implementations, 2nd ed. Singapore: World Scientific, 1988. [9] R. Gudipati and W.-K. Chen, “Explicit formulas for the design of broadband matching bandpass equalizers with Chebyshev response,” in Proc. IEEE Int. Symp. on Circuits and Syst., ISCAS, Seattle, WA, Apr. 28–May 3 1995, vol. 3, pp. 1644–1647. [10] H. J. Carlin, “A new approach to gain bandwidth problems,” IEEE Trans. Circuits Syst., vol. CAS-24, no. 4, pp. 170–175, Apr. 1977. [11] H. J. Carlin and B. S. Yarman, Wideband Circuit Design. Boca Raton, FL: CRC Press, 1997. [12] T. R. Cuthbert, “A real frequency technique optimizing broadband equalizer elements,” in Proc. IEEE Int. Symp. on Circuits and Syst., ISCAS, Geneva, Switzerland, May 28–31, 2000, vol. 5, pp. 401–404.

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[13] E. H. Newman, “Real frequency wide-band impedance matching with nonminimum reactance equalizers,” IEEE Trans. Antennas Propag., vol. 53, no. 11, pp. 3597–3603, Nov. 2005. [14] M. M. Weiner, Monopole Antennas. New York: Marcel Dekker, 2003, pp. 114–118. [15] C.-T. Tai and S. A. Long, “Dipoles and monopoles,” in Antenna Engineering Handbook, J. L. Volakis, Ed., 4th ed. New York: McGraw Hill, 2007, pp. 4-3–4-32. [16] C. A. Balanis, Antenna Theory. Analysis and Design, 3rd ed. New York: Wiley, 2005. [17] A. Hujanen, J. Holmberg, and J. C.-E. Sten, “Bandwidth limitation of impedance matched ideal dipoles,” IEEE Trans. Antennas Propag., vol. 53, no. 10, pp. 3236–3239, Oct. 2005. [18] L. J. Chu, “Physical limitations of omni-directional antennas,” J. Appl. Phys., vol. 19, pp. 1163–1175, 1948. [19] H. F. Pues and A. R. Van de Capelle, “An impedance matching technique for increasing the bandwidth of microstrip antennas,” IEEE Trans. Antennas Propag., vol. 37, no. 11, pp. 1345–1353, Nov. 1989.

Vishwanath Iyer received the B.E. degree in electrical engineering from University of Mumbai, India, in 2001 and the M.S. degree in electrical engineering from Utah State University, Logan, UT, in 2004. He is currently working toward the Ph.D. degree in electrical engineering at the Worcester Polytechnic Institute, Worcester, MA. His research interests include broadband antennas, electrically small antennas, antenna arrays, and FDTD based electromagnetic modeling.

Sergey N. Makarov (M’98–SM’06) received the B.S., M.S., Ph.D., and Dr. Sci. degrees from the St. Petersburg (Leningrad) State University, Russian Federation. In 1996, he joined the Department of Mathematics and Mechanics, St. Petersburg (Leningrad) State University, where he became a a Full Professor—the youngest full professor of the Faculty. In 2000, he joined the faculty of the Department of Electrical and Computer Engineering, Worcester Polytechnic Institute, Worcester, MA, where he became a Full Professor and Director of the Center for Electromagnetic Modeling and Design in 2008. His current research interests include practical antenna design, computational and analytical electromagnetics, and educational aspects of electromagnetics and wireless power transfer.

Daniel D. Harty received the B.S. degree in electrical engineering from Worcester Polytechnic Institute, Worcester, MA, in 2009, where he is working toward the M.S. degree. His research interests include design of phased antenna arrays, microwave amplifiers and oscillators, and power dividers.

Faranak Nekoogar received the B.S.E.E. and M.S.E.E. degrees in electrical engineering from San José State University, San José, CA, in 1995 and 1996, respectively, and the Ph.D. degree in applied science/electrical engineering from the University of California, Davis, in 2005. She completed her postdoctoral fellowship at Lawrence Livermore National Laboratory (LLNL), Livermore, CA, in 2006, and currently leads several research activities at LLNL in the following areas: Long range passive RFID, and ultrawideband communications and radar imaging, and UWB Software Defined Radio. Prior to joining LLNL, she served a total of 10 years in the areas of signal processing and chip design at NASA Ames Research Center and various chip design startup companies at Silicon Valley, CA. She holds 18 patents and records of inventions in UWB and RFID, and has published two technical books with Prentice Hall in the areas of chip design (ISBN: 0130338575) and UWB communications (ISBN: 0131463268). Her new book on UWB RFIDs is currently under publication with Springer Publishing.

Reinhold Ludwig (S’83–M’86–SM’90) received the M.S.E.E. degree from the University of Wuppertal, Germany, in 1983, and the Ph.D. degree from Colorado State University, Fort Collins, in 1986. Since 1996, he has been a Full Professor in the Electrical and Computer Engineering Department, Worcester Polytechnic Institute, Worcester, MA, with joint appointments in the Mechanical and Biomedical Engineering departments. In 2004, he was appointed Chief Scientific Officer of Insight Neuroimaging, a Worcester-based biotechnology company. His research interest is in the general field of electromagnetics and acoustics with focus on RF and gradient coil design for magnetic resonance imaging and nondestructive evaluation. Dr. Ludwig is a member of various professional societies, notably ISMRM, Eta Kappa Nu, Sigma Xi, and ASNT.

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A Simple Ultrawideband Planar Rectangular Printed Antenna With Band Dispensation K. George Thomas and M. Sreenivasan

Abstract—A compact planar ultrawideband (UWB) antenna with band notched characteristics is presented. Modification in the shape of radiation element and ground plane with two symmetrical bevel slots on the lower edge of the radiation element and on the upper edge of the ground plane makes the antenna different from the rectangular printed monopole. These slots improve the input impedance bandwidth and the high frequency radiation characteristics. With this design, the reflection coefficient is lower than 10 dB in the 3.1–10.6 GHz frequency range and radiation pattern is similar to dipole antenna. With the inclusion of an additional small radiation patch, a frequency-notched antenna is also designed and good out of band performance from 5.0–6.0 GHz can be achieved. Measured results confirm that the antenna is suitable for UWB applications due to its compact size and high performance. Also an approximate empirical expression to calculate the lowest resonant frequency is proposed. Index Terms—Band-notched antenna, capacitive coupling, planar monopole, ultrawideband (UWB).

I. INTRODUCTION

T

HE advances in ultrawideband (UWB) systems and applications are progressing at a prodigious rate. Many emerging microwave techniques and applications operate on the UWB frequency spectrum, using ultra short pulses on the order of nanoseconds. UWB systems have become more prominent and attracted attention since US-FCC has assigned the frequency band of 3.1–10.6 GHz in 2002. The primary objective of UWB is the possibility of achieving high data rate communication in the presence of existing wireless communication standards. The use of UWB signals in microwave imaging applications in addition to wireless communications requires suitable antennas as transducers between UWB transceivers and the propagating medium. Broadband planar monopole antennas have received considerable attention owing to their attractive merits, such as large impedance bandwidth, ease of fabrication and acceptable radiation properties [1]–[3]. Conventional planar monopole antennas need large metallic ground planes perpendicular to the radiation element, and hence are not low profile, which limits their applications in compact systems. In order to reduce the size considerManuscript received July 28, 2008; revised November 17, 2008. First published July 16, 2009; current version published January 04, 2010. The authors are with the SAMEER-Centre for Electromagnetics, CIT Campus, Taramani, Chennai-600113, India (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.2009.2036279

ably, a series of planar UWB antennas with microstrip or CPW feeding structures were proposed in [4]–[12]. Ultrawideband (UWB) transmitter can cause EM interference to nearby communication systems such as the wireless local area network (WLAN). Therefore UWB antennas with notched characteristics in the WLAN frequency bands are required and can be found in [13]–[15]. There are various methods to achieve the band-notched characteristics. The conventional methods are cutting a slot (i.e., U-shaped, arc shaped and pi-shaped slot) on the patch [16]–[20], inserting a slit on the patch [21]–[23] and embedding a quarter wavelength tuning stub within a large slot on the patch [24]. Alternate way is putting parasitic elements near the printed monopole as filters to reject the limited band [25], or embedding a pair of T-shaped stubs inside as elliptical slot, cut in the radiation patch [26]. In this paper a simple microstrip fed UWB antenna is proposed with an empirical formula to calculate the lowest resonant frequency of planar monopole/dipole configurations. Symmetrical bevel slots are formed on the radiation and ground patch to cause a wide bandwidth from 3.1–10.6 GHz for UWB applications. The notched band covering the 5 GHz WiFi band is achieved by a small rectangular patch fed by 50 transmission line. The width and length of the patch offer sufficient freedom in selecting the notched band and the approach is capable of shifting the notched frequency with steeper rise in V.S.W.R. The antenna has a compact size of 30 mm 18 mm 0.76 mm. The measured 10-dB reflection coefficient shows that the proposed antenna achieves a bandwidth ranging from 3–11 GHz with a notched band of 5–6 GHz. The proposed antenna presents omnidirectional radiation patterns across the whole operating band in the -plane. The paper is organized as follows. Section II gives a brief description of the antenna configuration. Section III presents the proposed design method and results of simulation using Ansoft HFSS. Section IV reports on experimental results and Section V concludes the findings of this paper. II. ANTENNA CONFIGURATION Fig. 1 shows the geometry of the proposed antenna. It consists of a rectangular radiation patch with symmetrical bevel slots placed on the lower side of the patch and a partially modified rectangular ground plane with symmetrical bevel slots located and play a on its upper side. These slots with dimensions significant role in achieving a broad impedance bandwidth. The cutting of slots results in steps on the lower side of the radiation patch as well as on the upper side of the ground plane. The width of the step formed is denoted as and the gap between the radiation patch and the ground plane is denoted as . A 50

0018-926X/$26.00 © 2009 IEEE

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= 30 mm = 14 4 mm = 14 4 mm = 18 mm g = 1 2 mm w = 3 5 mm h = 3 mm s = 11 mm a = 5 3 mm b = 7 mm = 15 6 mm w = 1 4 mm H = 0 76 mm Fig. 1. Configuration of the proposed band notched UWB printed antenna L ,l : : : ,l ,W , , : , : , , , , : , : : l , .

microstrip line of width 1.4 mm is connected to the radiation patch as the feed line. It can be seen from Fig. 1 that the rectangular radiation patch and the 50 line are printed on the top side of the substrate while the ground plane is printed on the bottom side of the substrate. A small rectangular patch with dimensions and , printed on the bottom side of the substrate is connected to the 50 line through via-hole to produce a notched band in the vicinity of 5.5 GHz and thus prevents the interference with WLAN systems. The antenna was implemented on an inexpensive FR4 substrate with a thickness of 0.76 mm and relative permittivity of 4.4. A prototype of the proposed band notched UWB rectangular , printed antenna with optimal design, i.e., , , , , , , , , , , , , as shown in Fig. 1, was fabricated and tested and the reflection coefficients were measured using Agilent network analyzer E8363B. Fig. 2 shows the simulated and measured reflection coefficient curves. The measurement confirms the UWB and band-rejection characteristics of the proposed antenna, as predicted in the simulation. III. ANTENNA DESIGN In this section, the antenna covering the full UWB band (3.1–10.6 GHz) is first described. Then the new band notched structure which is equivalent to series LC circuit, is investigated. The effects of changing the geometric parameters of the proposed antenna on impedance, bandwidth and radiation pattern are discussed. The proposed antenna structure is simulated using the Ansoft High Frequency Structure Simulator (HFSS) software, with lumped port excitation. A. UWB Antenna Design The UWB antenna design features a gap (slot) between the radiation patch and the ground plane which introduces a coupling

Fig. 2. Measured and simulated reflection coefficients of the proposed UWB antenna.

capacitance and plays an important role in obtaining UWB behavior. Hence the ground plane of the proposed antenna is also a part of the radiating configuration and current distribution on the ground plane affects the characteristics of the antenna. It is to be noted that the radiation patch, the gap and the ground plane form an equivalent dipole antenna with fundamental resonance, mainly determined by the length of the antenna. It is worth mentioning that closely spaced multiple resonances which are harmonics of fundamental resonance overlap, resulting in ultrawide bandwidth. The size of the gap opening defines the impedance matching and hence by placing bevel slots on the lower side of the radiation patch and on the upper side of the ground plane, impedance bandwidth is considerably enhanced. Figs. 3 and 4 show the variation of the reflection coefficients with frequency for different dimensions of the bevel slots. It is , the reflection seen that in the absence of bevels, coefficient behavior at low frequencies is identical to a narrow band dipole with current mainly distributed over the radiation patch and the ground plane. The placing of slots significantly improves the higher band impedance matching as the shaping of the lower edge of the radiation patch has a substantial effect on the impedance matching of printed monopole antennas. The reason is that the slot formed by the lower edge of the radiation patch and the ground plane with a proper dimensions can support traveling waves at higher frequencies. Hence, properly designing the dimensions of the four bevel slots on the radiation patch and the ground plane will enhance traveling wave mode radiation and improve the impedance matching at higher frequency band. However, the ground plane on the other side of the substrate cannot form a good slot with the radiation patch to fully support traveling waves and hence the antenna operates in hybrid mode of traveling and standing waves at higher frequencies. It is also seen that the fundamental resonant frequency is lowered as the bevel slot dimensions are increased. This also contributes to widening the operating bandwidth. The lowering of the resonant frequency is due to the fact that the effective gap between the radiation element and the ground plane is increased as the bevel slots are increased in size. In other words, by properly choosing the dimensions of slots, impedance bandwidth can

THOMAS AND SREENIVASAN: A SIMPLE UWB PLANAR RECTANGULAR PRINTED ANTENNA

Fig. 3. Simulated reflection coefficients for the proposed antenna of various bevel widths (w ) with a fixed value of bevel height (h ) = 3 mm. a = b = 0 mm. Other parameters are the same as given in Fig. 1.

29

where is the probe length (gap between the ground plane and the rectangular monopole) of the 50 feed line and , and are in centimeters. It may be noted that the radiation pattern of these antennas are more like that of a dipole pattern and hence planar printed monopole antenna can be considered as printed dipole of length including the length of the ground plane. It can be seen that in [4], while determining the resonant frequencies of circular discs of different diameters, the height of ground plane is kept constant. But the total length of the antenna configuration is changed with different diameter of circular disc. It is to be noted that change of the dimension of circular disc will not cause a change in resonant frequency as long as the total length of the antenna remains same. Hence it is understood that all reported printed monopoles are in effect dipoles with a fundamental resonant frequency and the height of the ground plane contributes in determining the resonant frequency. However ultra wide band microstrip-fed planar elliptical dipole antenna has been reported [10]. Further, microstrip-fed semi-elliptical dipole antenna, covering the frequency band from 3.1–10.6 GHz, has been proposed with a small size of dimensions of about one third of the wavelength, instead of half wavelength for the lower frequency [11]. It is found that the length of the antenna is about half wavelength at the lowest frequency of operation, if the dielectric constant of the material of the substrate is taken into consideration. Hence an approximate general formula is proposed to represent the fundamental resonant frequency of any planar printed radiation configuration with a ground plane. By equating the area of the planar printed configuration to that of a cylindrical wire of length

(3) Fig. 4. Simulated reflection coefficients for the proposed antenna of various bevel heights (h ) with a fixed value of bevel width (w ) = 3:5 mm. a = b = 0 mm. Other parameters are the same as given in Fig. 1.

since

be enhanced. From the simulated results in Figs. 3 and 4, it ocand . curs when

(4) (5)

B. Simple Formula for Resonant Frequency The frequency corresponding to the lower resonance of a rectangular planar monopole can be approximately calculated by equating its area to that of an equivalent cylindrical monopole antenna of same length and equivalent radius , as given below [3]

where and are the width and the length of the planar element and is the radius of the equivalent cylindrical wire. is the effective dielectric constant of the composite (air-substrate) dielectric. At fundamental resonance, the length of cylindrical dipole for real input impedance is given by [27]

(1) where is the width of the rectangular disc is given by The lower frequency

(6) where

(2)

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Thus

(7)

TABLE I EFFECT OF WIDTHS OF THE GROUND PLANE AND RADIATION PATCH ON RESONANT FREQUENCY FOR THE PROPOSED ANTENNA WITH CONSTANT STEP WIDTH s (s = 11 mm). w DENOTES THE WIDTH OF THE GROUND PLANE AND w DENOTES THE WIDTH OF THE RADIATION ELEMENT

From the above equations, the resonant frequency is given by

(8) where and are in centimeters If and denote the length of the ground plane and radiation patch respectively and is the gap between them, then can be . expressed as Similarly if and represent the radius of equivalent cylindrical dipole corresponding to the ground plane and radiation (In the case of cylinpatch, then can be expressed as and can be considered as the radius of the drical dipole, ). dipole arms and Hence

(9) and from (5)

TABLE II SIMULATED VARIATION OF RESONANT FREQUENCY WITH CHANGE IN DIELECTRIC CONSTANT OF THE SUBSTRATE MATERIAL FOR THE PROPOSED ANTENNA WITHOUT BEVEL SLOTS

(10) (11) where and are the area of the ground plane and the radiation patch respectively. , , , and are in centimeters. In order to demonstrate the effect of the width of the ground plane and radiation patch on resonant frequency, simulations were done using HFSS by keeping the step width ” constant and the results are shown in Table I. Table II shows the simulated and calculated resonant frequencies for different . It is observed that plays a role in determining the resonant frequency and as decreases the resonant frequency is lowered and the resonant frequency is increased with increasing . Table III shows the comparison between the reported measured resonant frequencies of printed monopoles of different configurations and the calculated results using the proposed formula.

the patch to determine the center frequency and bandwidth of the rejected band. The band notched performance is related to the parameters and “ ” which are the length and width of the rectangular patch. The open circuited small rectangular patch introduced on the bottom side of the substrate is shunt connected to the radiation patch through a via hole. Here, is the coupling gap between the resonator and the ground plane. The small rectangular patch acts as a resonator and introduces capacitive coupling to offer series resonance band stop function. Since the resonator has an impedance zero at its resonant frequency . the main line is effectively shorted at and thus no power is delivered to the radiation patch. It is to be noted that capacitive coupled transmission line inductor is less than quarter wavelength at the resonant frequency. The wavelength corresponding to the resonant frequency is given as

C. The Band Rejection Function for WLAN Band The UWB system, operating between 3.1–10.6 GHz causes interference to the existing wireless communication systems, for example the WLAN operating in 5.15–5.85 GHz. The band rejection filter employed in UWB RF front-ends avoids the interference but gives complications to the UWB system. To overcome this difficulty, UWB antenna with a band rejected characteristic is required. The band rejection function of the proposed antenna is achieved by printing a small rectangular patch on the bottom side of the substrate and properly tuning the dimensions of

(12) The area of the rectangular patch with length and width can be equated to that of an equivalent cylindrical monopole antenna of height and equivalent radius . The average characteristic impedance can be defined as [28]

(13)

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TABLE III COMPARISON OF RESONANT FREQUENCIES FOR VARIOUS TYPES OF PRINTED MONOPOLE ANTENNA CONFIGURATIONS

from (3) (14) from (4) (15)

It may be noted that the capacitance of a rectangular patch primarily depends on the width of the patch, rather than its length. Therefore, when the width of the rectangular patch is changed, capacitance of the resonator changes with insignificant variation , where is the in inductance. Hence it is assumed that inductance when is changed. and represent the characteristic impedance and resIf onant frequency for the changed width then

and where The inductive reactance offered by the rectangular patch with can be expressed as resistance and (16) where and is the resistance per unit length. Since the rectangular patch resonates at a length approximately equal to quarter wavelength (17) Hence (18) and

corresponds to resonant frequency for optimal where ) and width ( ) of the resonator length ( with inductance , capacitance and characteristic impedance is Similarly, when the length of the rectangular patch changed, inductance of the resonator changes with insignificant , variation in capacitance. Hence, it is assumed that where is the capacitance when is changed. and represent the characteristic impedance and resIf onant frequency for the changed length then

(23) (19)

Therefore (20) (21) where

(22)

is the resonant frequency of the patch resonator.

Figs. 5 and 6 show the variation of simulated reflection coefficients with different values of and . It is seen that as the length and width of the rectangular patch increase, the center frequency of the rejected band shifts to the lower frequencies. Tables IV and V demonstrate the effect of the length and width of the rectangular patch (resonator) on the resonant frequency (center frequency of the rejected band). Fig. 7 shows the surface current distributions at the pass-band and stop-band. At the stop band, a strong electric coupling (capacitive coupling) occurs at the bottom edge of the resonator

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TABLE V EFFECT OF LENGTH OF THE RECTANGULAR PATCH ON RESONANT FREQUENCY OF THE REJECTED BAND FOR CONSTANT PATCH WIDTH b = 7 mm

Fig. 5. Simulated reflection coefficients for the proposed antenna of various notch element length a with a fixed notch element width b = 7 mm. Other parameters are the same as given in Fig. 1.

Fig. 6. Simulated reflection coefficients for the proposed antenna of various notch element width b with a fixed notch element length a = 5:3 mm. Other parameters are the same as given in Fig. 1.

TABLE IV EFFECT OF WIDTH OF THE RECTANGULAR PATCH ON RESONANT FREQUENCY OF THE REJECTED BAND FOR CONSTANT PATCH LENGTH a = 5:3 mm

Fig. 7. Simulated current distribution of the proposed antenna (a) pass-band (at 3.5 GHz) and (b) stop-band (at 5.5 GHz).

proposed antenna is indeed controlled by the length and width of the rectangular patch. IV. MEASURED ANTENNA PERFORMANCE

patch. Also the surface currents are concentrated at the resonator and the antenna does not radiate. This is analogous to the signal drop across the - combination in a series resonant band-stop filter, at resonance. However, at the pass band, the electric coupling does not occur at the bottom edge. The resonator does not work and the antenna returns to the normal operation. It can be concluded that the center frequency of the notch band for the

Based on the design in the previous section, the proposed band-notched UWB antenna was fabricated and fed by a 50 SMA connector. The measured and simulated reflection coefficient of the proposed antenna from 3–11 GHz is shown in Fig. 2. Measured and simulated results track fairly well. Fig. 8 shows the measured radiation patterns at 3, 6, and 9 GHz, respectively. It can be seen that the patterns of the proposed antenna present omnidirectional and stable radiation characteristics in the - plane ( -plane) over the operating frequency range which are similar to the typical dipole antenna. The - plane ( -plane) patterns shown in Fig. 9. demonstrate that at 3 GHz, the pattern is approximately symmetrical. As the width of the radiator is comparable with wave length at higher

THOMAS AND SREENIVASAN: A SIMPLE UWB PLANAR RECTANGULAR PRINTED ANTENNA

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Fig. 10. Measured and simulated gain of the proposed band notched UWB antenna.

Fig. 8. Measured radiation patterns of the proposed band notched UWB anand E component). tenna in x-y plane (E

Fig. 11. Photograph of the fabricated antenna.

UWB frequency band and a gain drop of 6–7 dB occurs at 5.5 GHz. A photograph of the proposed antenna when printed on FR4 substrate is displayed in Fig. 11. V. CONCLUSION Fig. 9. Measured radiation patterns of the proposed band notched UWB anand E component). tenna in x-z plane (E

frequencies, the patterns deviate from symmetry, as can be seen at 6.0 and 9.0 GHz. Fig. 10 shows the measured and simulated gains of the realized antenna from 3–11 GHz. The figure indicates that the proposed antenna has reasonably good gain over the band of frequencies except for the notched band. Close agreement between measured and simulated results can be found. The measured antenna gain variations are less than 2 dB throughout the desired

A microstrip-fed rectangular printed antenna is proposed and implemented for UWB applications. The overall antenna size is 30 mm 18 mm 0.76 mm. The antenna, compact and simple, has minimum design parameters which have been investigated for optimal design. A frequency-notched antenna is also realized with good out of band performance from 5–6 GHz by including an additional small radiation patch. Besides, by taking the dielectric constant of the substrate into consideration, an approximate empirical formula is presented to calculate the lowest resonant frequency for the planar printed monopole/dipole antennas in general. Study and examination of the formula have shown that various printed geometric configurations reported as planar monopoles can be defined and demonstrated as planar

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printed dipoles. The present design can easily be extended to dual or triple band-notched antennas. The operating bandwidth of the proposed antenna covers the entire frequency band from 3.1–10.6 GHz.Both simulated and measured results suggest that the proposed antenna is suitable for UWB communication applications and at the same time dispenses the interference with WLAN systems. ACKNOWLEDGMENT The authors would like to thank G. Sreekumar and P. Karthikeyan for their help in the fabrication and measurement of the antenna. REFERENCES [1] N. P. Agarwall, G. Kumar, and K. P. Ray, “Wideband planar monopole antennas,” IEEE Trans Antennas Propag., vol. 46, no. 2, pp. 294–295, 1998. [2] Z. N. Chen, M. Y. W. Chia, and M. J. Ammann, “Optimization and comparison of broadband monopoles,” Proc. Inst. Elect. Eng. Microw. Antennas Propag., vol. 150, no. 6, pp. 429–435, 2003. [3] G. Kumar and K. P. Ray, Broad Band Microstrip Antennas. Boston, MA: Artech House, 2003. [4] J. Liang, C. C. Chiau, X. Chen, and C. G. Parini, “Study of a printed disc monopole antenna for UWB systems,” IEEE Trans. Antennas. Propag., vol. 53, no. 11, pp. 3500–3504, Nov. 2005. [5] C. Y. Huang and W. C. Hsia, “Planar elliptical antenna for ultra wideband application,” Electron.Lett., vol. 41, no. 41, 6, pp. 296–297, Mar. 2005. [6] X. Qing, M. Y. WChia, and X. Wu, “Wide-slot antenna for UWB applications,” in Proc. IEEE AP-S Int. Symp., Jun. 2003, vol. 1, pp. 834–837. [7] R. Chair, A. A. Kishk, and K. F. Lee, “Ultrawideband coplanar waveguide-fed rectangular slot antenna,” Antennas Wireless Prop. Lett., vol. 3, no. 1, pp. 227–229, 2004. [8] S. H. Hsu and K. Chang, “Ultra-thin CPW-fed rectangular slot antenna for UWB applications,” in Proc. IEEE AP.-S Int. Symp., Jul. 2006, pp. 2587–2590. [9] S. H. Lee, J. K. Park, and J. N. Lee, “A novel CPW-fed ultrawideband antenna design,” Microw. Opt. Tech. Lett., vol. 44, no. 5, pp. 393–396, 2005. [10] J.-P. Zhang, Y.-S. Xu, and W.-D. Wang, “Ultra-wideband microstrip-fed planar elliptical dipole antenna,” Electron. Lett., vol. 42, no. 3, pp. 144–145, 2006. [11] J.-P. Zhang, Y.-S. Xu, and W.-D. Wang, “Microstrip-fed semi-elliptical antenna for ultrawideband communications,” IEEE Trans. Antennas. Propag., vol. 56, no. 1, pp. 241–244, Jan. 2008. [12] Y. J. Ren and K. Chnag, “Ultra-wideband planar elliptical ring antenna,” Electron. Lett., vol. 42, no. 8, pp. 447–448, 2006. [13] J. M. Qiu, Z. W. Du, J. H. Lu, and K. Gong, “A band-notched UWB antenna,” Microw. Opt. Tech. Lett., vol. 45, no. 2, pp. 152–154, 2005. [14] K. H. Kim, Y. J. Cho, S. H. Hwang, and S. O. Park, “Band-notched UWB planar monopole antenna with two parasitic patches,” Electron. Lett., vol. 41, no. 14, pp. 783–785, 2006. [15] C. Y. Huang, W. C. Hsia, and J. S. Kuo, “Planar ultrawideband antenna with a band-notched characteristic,” Microw. Opt. Tech. Lett., vol. 48, no. 1, pp. 99–101, 2006. [16] J. N. Lee and J. K. Park, “Impedance characteristic of trapezoidal ultrawideband antenna with a notch function,” Microw. Opt. Tech. Lett., vol. 46, no. 5, pp. 503–506, Sep. 2005. [17] H. K. Lee, J. K. Park, and J. N. Lee, “Design of a planar half- circle shaped UWB notch antenna,” Microw. Opt. Tech. Lett., vol. 47, no. 1, pp. 9–11, Oct. 2005.

[18] K. L. Wong, Y. W. Chi, C. M. Su, and F. S. Chang, “Band-notched ultra-wideband circular-disc monopole antenna with an arc shaped slot,” Microw. Opt. Tech. Lett., vol. 45, no. 3, pp. 188–191, May 2005. [19] C. Y. Huang, W. C. Hsia, and J. S. Kuo, “Planar ultra-wideband antenna with band-notched characteristic,” Microw. pt. Tech. Lett., vol. 48, no. 1, pp. 99–101, Jan. 2006. [20] C. Y. Huang and W. C. Hsia, “Planar ultra-wideband antenna with a frequency notch characteristic,” Microw.Opt.Tech.Lett., vol. 49, no. 2, pp. 316–320, Feb. 2007. [21] H. Yoon, H. Kim, K. Chang, Y. J. Yoon, and Y. H. Kim, “A study on the UWB antenna with band-rejection characteristic,” in Proc. IEEE AP-S Int. Symp., Jun. 2004, vol. 2, pp. 1780–1783. [22] I. J. Yoon, H. Kim, K. Chang, Y. J. Yoon, and Y. H. Kim, “Ultrawideband tapered slot antenna with band-stop characteristic,” in Proc. IEEE AP-S Int. Symp., Jun. 2004, vol. 2, pp. 1784–1787. [23] S. Y. Suh, W. L. Stutzman, W. A. Davis, A. E. Waltho, K. W. Skeba, and J. L. Schiffer, “A UWB antenna with stop-band notch in the 5 GHz WLAN band,” in Proc. IEEE ACES Int.C onf., Apr. 2005, pp. 203–207. [24] Y. Gao, B. L. Ooi, and A. P. Popov, “Band-notched ultrawideband ring monopole antenna,” Microw. Opt. Tech. Lett., vol. 48, no. 1, pp. 125–126, Jan. 2006. [25] K. H. Kim, Y. J. Cho, S. H. Hwang, and S. O. Park, “Band-notched UWB planar monopole antenna with two parasitic patches,” Electron. Lett., vol. 41, no. 14, pp. 783–785, Jul. 2005. [26] C.-Y. Hong, C.-W. Ling, I.-Y. Yarn, and S.-J. Chung, “Design of a planar ultrawideband antenna with a band-notch structure,” IEEE Antennas Propag., vol. 55, no. 12, Dec. 2007. [27] C. A. Balanis, Antenna Theory: Analysis and Design. New York: Harper and Row, 1982. [28] E. C. Jordan and K. G. Balmain, Electromagnetic Waves and Radiating Systems. Englewood Park, NJ: Prentice-Hall, 1968. K. George Thomas received the M.Sc. degree in electronics from Kerala University, India, and the M.Tech. degree in microwaves and radar from Cochin University of Science and Technology (CUSAT), Cochin, India, in 1987 and 1989, respectively. In 1990, he joined the Society for Applied Microwave Electronics Engineering and Research (SAMEER), Chennai, as a Scientist, where he has been involved in antenna design and measurements. His productive work has resulted in the development of a number of broadband and high performance antennas. In 1992, he was deputed to the Georgia Tech Research Institute, Atlanta, to work on shielding effectiveness of gaskets for EM shielding applications. He has authored and coauthored many papers in refereed journals and conference proceedings. He has one Indian patent to his credit. His main research interests include broadband planar and printed monopoles, ultrawideband VHF/UHF omnidirectional antennas and high performance directional antennas. Dr. Thomas is a Life member of the Society of EMC Engineers, India (SEMCI).

M. Sreenivasan received the M.Sc. degree in electronic science and the M.Tech. degree in microwave and radar engineering from Cochin University of Science and Technology (CUSAT), Cochin, India, in 2002 and 2004, respectively. In 2004, he joined SAMEER-Centre for Electromagnetics, Chennai, India, as a Research Scientist in the Electromagnetics and Antenna Division, where he is currently working as a Scientist. His research interests include wideband and multiband printed antennas and wideband aperture antennas and metamaterials.

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Novel Broadband Circularly Polarized Cavity-Backed Aperture Antenna With Traveling Wave Excitation Kuo-Fong Hung and Yi-Cheng Lin, Member, IEEE

Abstract—A novel broadband circularly polarized aperture antenna is presented that uses traveling-wave excitation as the design concept. The antenna configuration consists of a circular radiating aperture, a backed cavity, and an equiangular tapered strip outer-fed by a microstrip transmission line. Operating with a traveling wave excitation, the proposed antenna contains inherent broadband characteristics in terms of the impedance, axial ratio, and gain performances. The presented antenna is comprehensively investigated, including the working principles, design consideration, and parametric studies. In addition, the research interests are extended to a 2 2 antenna array. Promising results from the experimental 2 2 array are achieved, including the 10-dB return loss bandwidth of 70%, the 3-dB axial ratio bandwidth of 50%, and the half-power (3-dB) gain bandwidth of 40% with a maximum gain about 11 dBi. The measured and simulated results are well complied with each other. Index Terms—Aperture antennas, broadband antennas, circularly polarized antennas, cavity-backed antennas, microstrip-line-fed antennas.

I. INTRODUCTION

W

IDEBAND circularly polarized antennas have been desirable in high-capacity wireless communications and high-resolution radar systems. The fundamental broadband researches were first reported back in the 1950s, including the frequency independent antennas [1], the equiangular spiral antennas [2], and Archimedean spiral antennas [3]. All of these antennas are classified as the traveling wave antenna types that inherently perform a very wide bandwidth in the impedance, axial ratio, and gain patterns. However, the most difficult part in designing these center-fed spiral antennas is the feeding scheme. A successful feed design incorporating the broadband balanced transition, impedance transformation, and robust structure for connection, is considered the key to implementing a high-performance cavity-backed spiral antenna [4]. Since it requires both electrical and mechanical design work, the implementation of feeding schemes for center-fed spiral antennas is very complex and expensive. On the other hand, microstrip antennas [5]–[10] featuring the simple planar design with the

Manuscript received September 26, 2008; revised June 29, 2009. First published November 06, 2009; current version published January 04, 2010. This work was supported in part by the National Science Council of Taiwan under Grants NSC-96-2219-E-002-016 and NSC-96-2752-E-002-002-PAE and in part by Excellent Research Projects of National Taiwan University under Grant 97R0062-03. The authors are with the Department of Electrical Engineering and the Graduate Institute of Communication Engineering, National Taiwan University, Taipei 10617, Taiwan. (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2009.2036187

low-cost printed circuit board (PCB) technology have been reported and widely used over the past decades. Researches on the circularly polarized (CP) microstrip antennas have been intensively investigated in various aspects, including the size reduction [8], the analysis of infinite array and feeds [9], and the systematic broadband CP design for patch antennas [10]. Microstrip/patch antennas are classified as the resonance antenna types that inherently perform a narrow bandwidth. In addition, several recent researches on CP antennas have been proposed, including the dielectric resonator antennas [11], [12], the cavity-backed slot antennas [13], [14], and the broadband parasitic loop/slot antennas [15], [16]. The cavity-backed antennas are of special interest because they can be flush mounted on the surface of a high-speed vehicle or aircraft with applications for mobile communication and radar systems. However, the cavity-backed design usually decreases the antenna bandwidth which, in turn, affects the system capacity in broadband wireless communications. Therefore, it is desirable to design a broadband, low-cost, and cavity-backed antenna for CP operation. In this paper, we present a new broadband cavity-backed CP aperture antenna with the traveling wave excitation. A single equiangular strip outer-fed by a microstrip line is employed to excite the traveling waves for the broadband CP radiation. The presented antenna is characterized by the following features: 1) the planar PCB design, suitable for the low-cost manufacturing, 2) the outer-fed design with microstrip-line, providing the simple interface with RF circuitry or array feeding network, 3) the cavity-backed design, suitable for the conformal flush-mount deployment and providing gain-enhanced unidirectional radiation patterns, and 4) the inherent broadband characterization, including the impedance, gain, and axial ratio bandwidth. The antenna structure and design principle is described in Section II. The parametric studies and design rules are investigated in Section III. The experimental verification of a single antenna element is provided in Section IV. Finally, the design and implementation of a 2 2 array prototype and experimental results are depicted in Section V, followed by a brief conclusion in Section VI. II. ANTENNA DESIGN AND PRINCIPLES A. Antenna Structure and Design Concepts Fig. 1 shows the configuration of the proposed antenna with the coordinate systems and design parameters. A circular is etched out from the ground plane aperture of diameter of a PCB. Above the aperture is a tapered strip printed on the opposite side of the PCB with one end extended to the edge

0018-926X/$26.00 © 2009 IEEE

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Fig. 1. The configuration of the proposed antenna with the coordinate systems and design parameters.

of the aperture. Through the tapered strip, the electromagnetic energy is coupled into the aperture area from the feed point, denoted by a voltage source, where a microstrip transmission line is employed in practice. The tapered strip in the aperture is specially devised with the exponentially varied width formed and , as shown in Fig. 1. by two equiangular curves and are characterized by the The equiangular curves ): following equations in polar coordinates ( (1) (2) where is the diameter of the circular aperture; and are the shrinking coefficients of curves and , respectively; . The and is the initial strip width at the feed point strip is exponentially tapered and extended toward the center of the aperture and terminated at an ending angle, denoted by . Compared to the traditional equiangular spiral antennas fed at the center with broadband balanced structures [2], the presented antenna is simply outer-fed at the aperture edge with a microstrip transmission line. This makes the presented antenna very easy to integrate with RF circuitry or feeding networks of an array in the low-cost planar PCB design. The energy is coupled into the aperture with a traveling wave propagating along the slot between the tapered strip and the aperture edge, leading to a circularly polarized radiation. Hence, the polarization sense of the presented antenna is readily determined by the direction of the traveling wave. That is, the presented antenna shown in Fig. 1. is operated in the right-hand circular polarization. In general, aperture antennas in free space radiate bidirectional patterns [17] with the maximum gain in the normal directions of the aperture plane. Bidirectional patterns are not suitable

Fig. 2. The current distribution on the tapered strip and the ground plane in time-domain. (T = 1=f ) (Geometry parameters: D ; = 18 mm, a = 0:4;  = 0:3, and H = 16 mm).

for most wireless communication systems because the polarization senses of the two beams are opposite. To generate unidirectional patterns, the aperture antenna is devised with a backed cavity or reflector. As shown in Fig. 1., a circular cavity of diand height is placed underneath the aperture and ameter connected to the ground plane of the PCB. In practice, the cavity is slightly greater than the aperture diameter diameter considering a margin for the manufacturing and assembling tolerance. B. Operating Principles—Traveling Wave Excitation As mentioned in the previous section, the presented antenna is designed with an equiangular strip based on the concept of traveling wave excitation. Fig. 2 shows the current distribution on the tapered strip and the ground plane in time-domain. It can be observed that a right-hand (counterclockwise) rotated current distribution is moving along the aperture edge. Furthermore, there are about 1.5 guided wavelengths formed over the peripheral of the aperture at the operating frequency. Simulation work in this paper is performed with a commercial FEM-based electromagnetic simulator [18]. To further investigate the traveling wave characteristics of the presented antenna, analysis on the magnitude and phase of the current distribution is conducted. Fig. 3(a) and (b) show the normalized current distributions along the aperture edge and the tapered strip, respectively. These current distributions show a constant magnitude and linear phase along the path of aperture edge and the tapered strip, implying that a traveling wave excitation is generated. It is found that the edge surface current is traveling along the path with the total phases (meaning traveling and 270 along the paths) are about 430 ( aperture edge and the tapered strip, respectively. Proper design

HUNG AND LIN: NOVEL BROADBAND CP CAVITY-BACKED APERTURE ANTENNA WITH TRAVELING WAVE EXCITATION

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Fig. 4. Input impedance, axial ratio, and gain of the proposed aperture antenna , and mm, mm). (Geometry parameters:

D ; = 18

Fig. 3. The normalized current distributions along the aperture edge and the tapered strip. (Geometry parameters: mm, , and mm). (a) The current distribution along the aperture edge, (b) The current distribution along the tapered-strip arm.

H = 16

D ; = 18

a = 0:4; = 0:3

of the tapered strip taking into account the coupling of the aperture edge is essential to CP radiation performances; the design is discussed in detail in Section III.

C. Basic Performances The presented antenna is illustrated for right-hand circular polarization (RCP) operating in X-band with PCB RO4003 substrate of dielectric constant 3.55, loss tangent 0.0027, and thickness 0.508 mm. Before entering the optimization process, a comprehensive simulation was conducted to evaluate the typical values for some basic parameters. For example, the from initial strip width is 0.35 mm; the subtraction of is estimated at 2 mm; the ground plane size is set at 40 40 mm ; the strip ending angle is selected to be . The optimization process is conducted only on the main , the geometry parameters, including the aperture diameter shrinking coefficients and , and the height of the backed cavity . Fig. 4 shows the input impedance, axial ratio, and gain of the proposed antenna with the optimized dimensions

a = 0:4; = 0:3

H = 16

f = 9:4 GHz (GeomH = 16 mm).

Fig. 5. Radiations of the proposed aperture antenna at , and etry parameters: mm,

D ; = 18

a = 0:4; = 0:3

of the main parameters indicated in the caption. The minimum axial ratio, denoted by , is found around 0.5 dB at the GHz) with a wide 3-dB axial ratio center frequency ( (AR) bandwidth about 21%. It is also observed that the input impedance is relatively constant over the entire band. These broadband characteristics are expected from the presented antenna since it is a type of traveling wave antenna that in general has the inherent wide bandwidth in AR and impedance characteristics. The gain is about 7 dBi over the bandwidth dB) with an enhancement from the backed cavity. ( The radiation patterns of the proposed antenna are shown in Fig. 5. The unidirectional patterns are obtained with a co-polarized (RCP) gain about 7 dBi at the zenith. The backward radiation is suppressed by the cavity without any absorber with the front-to-back ratio about 35 dB. From Fig. 5, the main beam direction is found shifted slightly away from the zenith. This is because of the asymmetry of the proposed antenna that consists of an aperture and the single-arm strip. Note that the phenomenon of the tilted beam is prominent for the presented antenna as the frequency increases. This phenomenon is similar to what is reported in [19] when the circumference of the satisfies , where is the aperture guided wavelength of the current.

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Fig. 6. The axial ratio versus the frequency at different eters: , and mm).

a = 0:25;  = 0:25

H = 15

D

. (Geometry param-

(AR ) D = 18  = 0:25

(f )

Fig. 7. The minimum axial ratio value and the center frequency of the presented antenna versus the shrinking coefficients at different . (Gemm, ). ometry parameters:

H

III. PARAMETRIC STUDY AND DESIGN RULES A. Parametric Study for Axial Ratio The effect of the geometric parameters on the operating frequency of the proposed antenna is first investigated. Fig. 6 shows the axial ratio versus the frequency at different aperture diam, where , and mm. It is eter prominently affects the observed that the aperture diameter operating frequency of circular polarization for the presented mode is the fundamental resoantenna. Considering the nance of lowest frequency existing in a circular waveguide [20], it provides a cue for interpreting the operational frequency of the presented antenna. For a circular waveguide, the cut-off fre-mode is written in the quency for each allowed resonant following expression:

(3) where is the p-th zero of the n-th order Bessel function; is the diameter of the circular waveguide; and and are the permeability and permittivity, respectively. Using the fundamental of the above equation in free space, the aperture mode diameter of the presented antenna can be approximated as the following expression:

(4) and are the operating frequency and the correwhere sponding wavelength in free space, respectively. From Fig. 6, the operating center frequency, where the minimum axial ratio occurs, is found at 10.8 GHz, 9.9 GHz, and 9.25 GHz selected at 16 mm , for the aperture diameter , and 20 mm , respectively. It 18 mm shows that (4) may serve as a good first-order approximation in of the presented antenna, determining the aperture diameter given an operating frequency. From extensive simulation, Fig. 7 summarizes the effects of the shrinking coefficient on the minimum axial ratio

Fig. 8. The axial ratio versus the frequency at different mm, , and mm). eters:

D = 18

a = 0:25

H = 15



. (Geometry param-

and the operating center frequency at different cavity heights . It shows that an optimal shrinking coefficient exists in the ( dB) perrange from 0.25 to 0.45 for the sufficient is imformance. It is clearly seen that the achievable proved as increases. In practice, a small cavity height is desired for the compact low profile design. From Fig. 7, the minis shown to be about 10 mm, which is imum cavity height , where is the guided wavelength of the equivalent to 0.2 increases, the circular waveguide. Fig. 7 also shows that as operating center frequency decreases with a lower bound frequency about 9 GHz, which is compared and found close to the GHz for the TE mode) of a circular cut-off frequency ( in this case. waveguide of 20-mm diameter Fig. 8 shows the effects of the parameter on the axial ratio performance. It is observed that the tapered strip width, specified by the parameter , may influence the level of without changing the operating frequency. Fig. 8 shows that an optimal of 0.4 dB is achieved as is equal to 0.3. Note that the operating center frequency is relatively fixed at 9.9 GHz for various values of . Fig. 9 shows the effects of cavity height on the axial ratio performances. It can be observed that for shallow cavity the AR purity is degraded, but the AR bandwidth is broadened. On the other hand, for a deep cavity, the optimal AR purity may be obtained with narrow AR bandwidth. In practice, the deep cavity

HUNG AND LIN: NOVEL BROADBAND CP CAVITY-BACKED APERTURE ANTENNA WITH TRAVELING WAVE EXCITATION

Fig. 9. The axial ratio versus the frequency at different eters: mm, , and ).

D = 18

a = 0:4

 = 0:3

39

H . (Geometry param-

is suitable for single antenna, and the shallow cavity is suitable for antenna array. B. Parametric Study for Input Impedance Fig. 10 shows the input impedance versus frequency for the presented antenna while varying the parameter of shrinking coefficient from 0.15 to 0.45. As shown in Fig. 10(a), a relatively stable resistance around 90 ohms was obtained for the shrinking coefficient selected at 0.25. On the other hand, Fig. 10(b) shows that the reactance is closed to zero over the operating band while choosing the shrinking coefficient at 0.35. Combining the above observations, it can be concluded that the shrinking coefficient can be selected from the range 0.15 to 0.45 for a wideband input impedance matching. Fig. 11 shows the reflection coefficients of the presented antenna versus frequency for different shrinking coefficients with the port impedance set at 90 ohms. It shows that the lower bound frequency , which is defined by the lowest frequency for the dB, decreases as the shrinking reflection coefficient below coefficient decreases, standing for a longer tapered strip. On exceeds the the other hand, the upper bound frequency scope of display. As mentioned earlier, the backed cavity and the circular aperture of the presented antenna can be considered as circular waveguides, which have inherent property of high-pass characteristic for high-order modes. In addition, the structure of the input end of the tapered strip can be considered and the aperture as a tapered slotline between the strip edge edge. The tapered slotline serves as a wideband impedance transformer and guides the wave along the aperture edge into the radiating aperture. The zero extent of the slotline width implies that the impedance matching may be extended to infinite high frequencies. As for the lower bound frequency , the total length of the tapered strip arm has to accommodate for minimum about 270 degrees of the traveling phase at reflected wave from the end of the strip arm. Thus, the length of the tapered strip arm increases as the shrinking coefficient decreases, leading to a lower frequency . Fig. 12 shows the reflection coefficients of the presented while the port antenna at different terminating angle decreases impedance is set at 90 ohms. It is clearly seen that as increases, indicating a greater strip length. Note that as

Fig. 10. The input impedance of the presented antenna while shrinking comm, efficients range from 0.15 to 0.45 (Geometry parameters: , and mm). (a) The resistance of the input impedance, (b) The reactance of the input impedance.

a  = 0:25

D = 18

H = 10

Fig. 11. The reflection coefficients of the presented antenna while shrinking coefficients range from 0.15 to 0.45 while source impedance is equal to 90 ohms (Geometry parameters: mm, , and mm).

D = 18

 = 0:25

H = 10

the strip extends to the aperture center, the total length of the varies from to strip does not change much, while . C. Design Guideline and Discussion Through the above parametric studies with respect to the axial ratio and the input impedance performances, the design guideline of the proposed antenna is drawn as follows. First, , the diameter of the aperture was estimated to be about according to the operating frequency. Next, the initial values of the main parameters are selected from the following ranges:

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Fig. 12. The reflection coefficients of the presented antenna while  range from  to 2 , while source impedance is equal to 90 ohms (Geometry parameters: D = 18 mm, a = 0:25;  = 0:25, and H = 10 mm).

Fig. 14. Measured and simulated reflection coefficients of the prototype = antenna element versus frequency, where a = 0:4;  = 0:3;  1:75 D = 15 mm, D = 20 mm, W = 30 mm, and H = 10 mm.

Fig. 15. Measured and simulated radiation patterns of the prototype antenna element at 10.5 GHz: (a)  = 0 (XZ-plane), (b)  = 90 (YZ-plane). (a)  = 0 (XZ-plane) (b)  = 90 (YZ-plane). Fig. 13. Photo and configuration of the prototype antenna element with the backed cavity.

, and . The optimal and AR bandwidth can and the be optimized by adjusting the backed cavity height strip width . For the design of single antenna, the shrinking coefficient is selected with a slightly higher value for the wide 3-dB AR bandwidth. For the antenna array design, the lower shrinking coefficient and lower are suggested for the compact and low-profile design. Although the price is the reduced 3-dB AR bandwidth and impedance characterization, these compromised performances can be improved by employing a sequential rotation scheme to the array design, as described in the following section. IV. EXPERIMENTAL VERIFICATION To verify the proposed broadband circularly polarized aperture antenna, a prototype made of RO4003 PCB and aluminum tube cavity is fabricated and tested. Fig. 13 shows the photo and configuration of the prototype of which dimensions of design parameters are: mm, mm, mm, and mm. For measurement purpose, a tapered microstrip-line of length equal to 4 mm is employed to transform the antenna impedance (90 ) to a standard impedance (50 ) at the SMA connector. Fig. 14 shows the measured and simulated reflection coefficients of the

experimental prototype. The measured 10-dB return loss bandwidth is observed better than 20%. Fig. 15 shows the measured and simulated radiation patterns of the prototype in two elevation cuts (XZ- and YZ-plane) at 10.5 GHz. Here, the measured CP gain and axial ratio are acquired by measuring the complex fields in linear V-pol and H-pol components with a vector network analyzer followed by a post processing. The co-polarized (RCP) gain of the main beam is measured at 8 dBi with a cross-polarized isolation about 18 dB at the zenith, and a front-to-back ratio better than 25 dB. Fig. 16 shows the measured and simulated frequency response of the RCP gain and axial ratio at zenith (Z-axis) of the developed element prototype. It is observed that the 3-dB axial ratio bandwidth is about 24%. The peak gain of the antenna array is measured around 8.5 dBi with the half-power (3-dB) bandwidth around 40%. The measurement and simulation results are well complied with each other in both gain and AR performances. V. ANTENNA ARRAY DEVELOPMENT Once the antenna element is fully investigated in terms of impedance and radiation characterizations, the research interests were extended to the CP array applications. In many communication systems, the gain enhancement with antenna array is required for a sufficient link budget. In this section, an experimental study of a two-by-two (2 2) array for X-band application is described.

HUNG AND LIN: NOVEL BROADBAND CP CAVITY-BACKED APERTURE ANTENNA WITH TRAVELING WAVE EXCITATION

Fig. 16. Measured and simulated gain and AR of the prototype antenna element : ; : ; : D mm, versus frequency, where a mm, W mm, and H D mm.

= 20

= 30

= 04

= 03 = 10

= 1 75

= 15

41

Fig. 17. Top view and side view of the 2 dimensions of the feeding network.

2 2 antenna array with the detail

A. Array Configuration and Feeding Networks Fig. 17 shows the configuration of the experimental 2 2 array using the presented antennas as the radiating elements. The antenna elements are arranged in two-by-two on a square PCB of 60 60 mm . The center-to-center distance of the antenna elements/cavities is 20 mm and 22 mm along the X- and Y-axes, respectively. The elements are sequentially rotated by 90 counterclockwise with respect to its immediate neighbor for RCP operation. A compact corporate feeding network is properly designed to sequentially excite each rotated element with the correspondent phase delay of 90 step. The sequential rotation technique is an effective way to improve the CP performances of antenna array [21], [22]. In this paper, the feeding network incorporates the impedance transformation from the antenna aperture (90 ) to a standard port (50 ) of SMA connector. The detail feed network dimensions are given in Fig. 17. Through extensive simulation, the optimized dimensions of geometry parameters are obtained as: mm, , and mm. The photo of the developed antenna array is shown in Fig. 18. In this prototype the antenna element and the feeding network are printed on a RO4003 PCB, and then integrated with the backed cavities in a machined copper material. In practice, a light weight aluminum tube may be considered for the single antenna case. Furthermore, at millimeter-wave frequencies, the cavity would become so small that the cavity wall may be implemented by via ring in a multi-layered PCB or LTCC substrate, making the entire antenna module very cost effective and easy to integrate with the transceiver circuitry on the same substrate. B. Experimental Results and Discussion Fig. 19 shows the measured and simulated reflection coefficients of the experimental 2 2 array. A good impedance matching is obtained, where the 10-dB return loss bandwidth is more than 70%. As mentioned earlier, the higher frequency bound exceeds the scope of display because the presented antenna has inherent broadband impedance with traveling wave excitation. It is also shown that a very good agreement between the simulation and measurement is obtained. Fig. 20 shows the measured and simulated radiation patterns of the developed 2 2 array prototype in two elevation cuts (XZ- and YZ-plane) at 10 GHz. The co-polarized (RCP) gain

Fig. 18. Photo of the developed 2

2 2 antenna array module.

Fig. 19. Measured and simulated reflection coefficients of the 2 : ;D mm,  versus frequency, where a : ; and H mm.

= 10

= 0 25

= 0 25

= 19

2 2 array = 1:25 ,

of the main beam is measured 11 dBi with a cross-polarized isolation about 20 dB at the zenith, and a front-to-back ratio better than 25 dB. The side lobe level in the YZ-plane is slightly greater than that of the XZ-plane because of the unequal spacing of antenna elements in X- and Y-axes. Fig. 21 shows the measured and simulated frequency response of RCP gain and axial ratio at zenith (Z-axis) of the developed 2 2 array. The 3-dB axial ratio bandwidth is found about 50%, which is about two times that of a single antenna. This improvement mainly results from the optimization process of the sequential rotation schemes [22]. The antenna element and the feeding networks may be tuned slightly off the center frequency for further enhancement on AR bandwidth. The peak

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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 58, NO. 1, JANUARY 2010

Fig. 20. Measured and simulated radiation patterns of the proposed antenna (XZ-plane), (b)  array at 10 GHz in two elevation cuts (a)  (YZ-plane). (a)  (XZ-plane), (b)  (YZ-plane).

=0 = 90

=0

= 90

Fig. 21. Measured and simulated CP gain and AR of the experimental 2 array versus frequency, where a : ; : ;D mm,  :  , and H mm.

1 25

= 10

= 0 25

= 0 25

= 19

22 =

gain of the antenna array is measured around 11 dBi with the half-power (3-dB) bandwidth around 40%. The measured and simulated results are well complied with each other in both gain and AR performances. VI. CONCLUSION A novel broadband microstrip-fed circularly polarized aperture antenna with traveling wave excitation is presented in this paper. An equiangular feeding strip is carefully devised to generate a traveling wave along the aperture edge for broadband purpose. The experimental single element and 2 2 array are designed and implemented. The broadband characteristics of the presented antenna array is successfully achieved, including the 10-dB return loss bandwidth better than 70%, the 3-dB axial ratio bandwidth about 50%, and the half-power (3-dB) gain bandwidth around 40% with a peak at 11 dBi. REFERENCES [1] V. H. Rumsey, “Frequency independent antennas,” IRE National Convention Record, pt. I, pp. 114–118, Mar. 1957. [2] J. D. Dyson, “The equiangular spiral antenna,” IRE Trans. Antenna Propag., vol. 7, pp. 181–187, Apr. 1959. [3] J. A. Kaiser, “The Archimedean two-wire spiral antenna,” IRE Trans. Antenna Propag., vol. 8, pp. 312–323, May 1960. [4] C. Fumeaux, D. Baumann, and R. Vahldieck, “Finite-volume time-domain analysis of a cavity-backed Archimedean spiral antenna,” IEEE Trans. Antenna Propag., vol. 54, no. 3, pp. 844–851, Mar. 2006. [5] D. M. Pozar, “Microstrip antennas,” Proc. IEEE, vol. 80, no. 1, pp. 79–91, Jan. 1992. [6] D. M. Pozar, “Considerations for millimeter-wave printed antennas,” IEEE Trans. Antennas Propag., vol. 31, no. 31, pp. 740–747, Sept. 1983.

[7] H. M. Chen and K. L. Wong, “On the circular polarization operation of annular-ring microstrip antennas,” IEEE Trans. Antennas Propag., vol. 47, no. 8, pp. 1289–1292, Aug. 1999. [8] W. S. Chen, C. K. Wu, and K. L. Wong, “Novel compact circularly polarized square microstrip antenna,” IEEE Trans. Antennas Propag., vol. 49, no. 3, pp. 340–342, Mar. 2001. [9] J. T. Aberle and D. M. Pozar, “Analysis of infinite arrays of one- and two-probe-fed circular patches,” IEEE Trans. Antennas Propag., vol. 38, no. 4, pp. 421–432, Apr. 1990. [10] K. L. Chung and A. S. Mohan, “A systematic design method to obtain broadband characteristics for singly-fed electromagnetically coupled patch antennas for circular polarization,” IEEE Trans. Antennas Propag., vol. 51, no. 12, pp. 3239–3248, Dec. 2003. [11] Y.-C. Lin and M. Tassoudji, “Adjusted Directivity Dielectric Resonator Antenna,” Patent No. 6 344 833, Feb. 5, 2002. [12] R. Chair, S. Lung, S. Yang, A. Kishk, K. F. Lee, and K. M. Luk, “Aperture fed wideband circularly polarized rectangular stair shaped dielectric resonator antenna,” IEEE Trans. Antennas Propag., vol. 54, no. 4, pp. 1350–1352, Apr. 2006. [13] S. Shi, K. Hirasawa, and Z. N. Chen, “Circularly polarized rectangularly bent slot antennas backed by a rectangular cavity,” IEEE Trans. Antennas Propag., vol. 49, no. 11, pp. 1517–1524, Nov. 2001. [14] Q. Li and Z. Shen, “An inverted microstrip-fed cavity-backed slot antenna for circular polarization,” IEEE Antennas Wireless Propag. Lett., vol. 1, pp. 190–192, 2002. [15] R. Li, G. DeJean, J. Laskar, and M. M. Tentzeris, “Investigation of circularly polarized loop antennas with a parasitic element for bandwidth enhancement,” IEEE Trans. Antennas Propag., vol. 53, no. 9, pp. 3930–3939, Dec. 2005. [16] C.-C. Chou, K.-H. Lin, and H.-L. Su, “Broadband circularly polarized cross-patch-loaded square slot antenna,” Electron. Lett., vol. 43, no. 43, pp. 485–486, Apr. 2007. [17] K.-F. Hung and Y.-C. Lin, “Simulation of single-arm fractional spiral antennas for millimeter wave applications,” in IEEE Antennas and Propag.ion Society Symposium 2006, July 2006, pp. 3697–3700. [18] Ansoft HFSS V11. [19] H. Nakano, J. Eto, Y. Okabe, and J. Yamauchi, “Tilted- and axialbeam formation by a single-arm rectangular spiral antenna with compact dielectric substrate and conducting plane,” IEEE Trans. Antennas Propag., vol. 50, no. 1, pp. 17–24, Jan. 2002. [20] R. F. Harrington, Time-Harmonic Electromagnetic Fields. New York: McGraw-Hill, 1961, pp. 204–208. [21] J. Huang, “A technique for an array to generate circular polarization using linearly polarized elements,” IEEE Trans. Antennas Propag., vol. 34, no. 9, pp. 1113–1124, Sep. 1986. [22] U. R. Kraft, “Main-beam polarization properties of four-element sequential-rotation arrays with arbitrary radiators,” IEEE Trans. Antennas Propag., vol. 44, no. 4, pp. 515–522, Apr. 1996. Kuo-Fong Hung was born in Changhua County, Taiwan, in 1975. He received the M.S. degree in electrical engineering from the Graduate Institute of Communication Engineering, National Taiwan University, Taipei, Taiwan, in 2005, where he is currently working toward the Ph.D. degree. His research interests are the design and measurement of printed antennas, handset antennas, and diversity antennas for MIMO systems.

Yi-Cheng Lin (S’92–M’98) received the B.S. degree in nuclear engineering from National Tsing-Hua University, Hsingchu, Taiwan, in 1987, the M.S. degree in electrical engineering from National Taiwan University, Taipei, Taiwan, in 1989, and the Ph.D. degree in electrical engineering from The University of Michigan at Ann Arbor, in 1997. From 1997 to 2003, he was with Qualcomm Inc., San Diego, CA, responsible for the design and development of various antennas for satellites and cellular communication systems. Since 2003, he has been a faculty member with the Department of Electrical Engineering and Graduate Institute of Communication Engineering, National Taiwan University, Taiwan, where he is now an Associate Professor. His research interests include antenna miniaturization, ultrawideband antennas, millimeter-wave antennas, and diversity antennas for MIMO systems.

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Frequency Selective Surfaces for Extended Bandwidth Backing Reflector Functions Marco Pasian, Member, IEEE, Stefania Monni, Member, IEEE, Andrea Neto, Member, IEEE, Mauro Ettorre, and Giampiero Gerini, Senior Member, IEEE

Abstract—This paper deals with the use of frequency selective bandwidth product surfaces (FSS) to increase the efficiency in wideband antenna arrays, whose efficiency is limited by the front-to-back ratio. If the backing reflector for the antenna is realized through a single metal plane solution, its location will be suitable only on a relatively limited frequency range especially if wide angle scanning is required. In order to extend the frequency range of usability, an FSS can be sandwiched between the antenna and the ground plane, providing an additional reflecting plane for an higher frequency band. The possibility to integrate in the antenna different functionalities, otherwise performed by several antennas, is also discussed in the paper. The proposed backing structure composed by the FSS and the ground plane has been designed to be used in conjunction with a wideband antenna consisting of an array of connected dipoles. A hardware demonstrator of the backing structure has also been manufactured and tested. Index Terms—Connected array, frequency selective surface, Green’s function, wideband antenna, wide-scan antenna.

I. INTRODUCTION

C

URRENT trends in the design of military ship masts foresee the integration of several functionalities on the same antenna aperture in order to satisfy the demand of an increasing number of services to be installed on board, while still responding to the requirement of reducing the radar cross-section (RCS) [1] of the ship itself. In view of this, multiband/broadband planar or quasi-planar antennas with large scanning capabilities are required. Existing solutions, such as the Vivaldi antenna [2] show good performances at the cost of low cross polarization purity, which limits the range of possible applications. Other antennas, such as the connected array without a backing reflector, as described in [3], show excellent performances in terms of operating bandwidth, but an efficiency that can be as low as 50 %, because of the poor front-to-back ratio.

Manuscript received October 20, 2008; revised June 10, 2009. First published November 06, 2009; current version published January 04, 2010 This work was supported by the TNO and by the NL Ministry of Defense in the framework of the research program Integrated Technology Mast Systems. M. Pasian is with the Department of Electronics, University of Pavia, Pavia, Italy (e-mail: [email protected]). S. Monni, A. Neto, G. Gerini are with TNO Defence, Security and Safety, Den Haag 2597 AK, The Netherlands (e-mail: stefania.monni, andrea.neto, [email protected]). M. Ettorre was with TNO Defence, Security and Safety, Den Haag 2597 AK, The Netherlands. He is now with the Groupe Antennes and Hyperfréquences, Institut d’Electronique et de Télécommunications de Rennes (IETR), UMR CNRS 6164, Université de Rennes 1, 35042 Rennes Cedex, France (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2009.2036185

To overcome this problem, the most common approach consists in introducing a backing reflector. This backing reflector should be located at a distance of a small fraction of wavelength from the antenna if maximum gain is required, as outlined in [1]. However, a widely adopted choice is to place the reflector at a quarter wavelength distance from the antenna, as for example in [4]. Although this choice still allows obtaining a good antenna matching, with only a modest degradation of the achievable gain, the functioning of the backing reflector is optimal only in a relatively small frequency band [1]. Therefore, the improvement of the front-to-back ratio comes at the expense of the antenna bandwidth. A way to tackle this problem is by introducing a frequency dependent backing reflector, obtained by combining a frequency selective surface (FSS) with a PEC ground plane. As outlined in [1], the FSS introduces an additional frequency band, resulting in a multi-band behavior of the backing reflector. Resistively loaded FSSs [5] and electronic bandgap structures (EBGs) [6] have already been proposed to obtain a wideband behavior. In [6], EBGs were designed to extend the bandwidth of a wideband connected array, which in the lower range of the operating frequency band was backed by an absorbing layer. In fact a backing metallic reflector for the lowest frequencies should have been located at a distance from the antenna plane in the order of half a meter. However, the use of resistive elements introduces about 3 dB of losses [1], [5], [6]. In this paper, the step to achieve good front-to-back ratios for a wide-angle scanning array antennas without resistive loading is attempted, taking as starting point a schematic design of a connected dipole array [7], [8]. In view of this, the paper represents the complementary side of a unique project aimed to wideband connected arrays backed by innovative ground planes. An alternative for the antenna could be offered by the arrays proposed in [1], which are very similar to connected arrays. In [1] the continuity of the electric current between array elements is obtained by means of capacitive loadings, instead of using physically connected dipoles as in [7], [8]. However, the designs proposed in [1] achieve the required wide bandwidth when one or more dielectric layers are introduced in front of the array. On the contrary, in this paper any dielectric layer is intentionally discarded, also to preserve the highest polarization purity [10]. The considered application foresees the integration of two frequency bands, one corresponding to the typical radar X-band, 8.50–10.50 GHz, and the other one corresponding to a Tactical Common Data Link (TCDL) system, 14.40–15.35 GHz. In this paper we consider the simplified case of 1D scanning in elevation up to and one linear polarization. A connected dipole array is backed by a combination of a ground plane and

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Fig. 1. 3-D view of the antenna structure with ground plane.

TABLE I GEOMETRICAL DIMENSIONS IN MM OF THE CONNECTED DIPOLE ARRAY BACKED BY A SINGLE METAL PLATE, DEPICTED IN FIG. 1 Fig. 2. Magnitude of the active reflection coefficient of the connected dipole array antenna with the ground plane. Gray strips indicate the considered operative bands.

an FSS, which are designed to behave as perfect reflectors in two different frequency ranges. In particular, for lower frequencies, where the FSS is transparent for the impinging field, the backing reflector of the array is the real ground plane, whereas in the upper frequency range the FSS behaves as perfect reflector. A prototype of the backing structure consisting of the ground plane and the FSS has been manufactured. The magnitude of the reflection coefficient of the FSS alone and the phase of the reflection coefficient of the entire backing structure have been measured. The Green’s function (GF) and the active impedance of the structure composed by the antenna and the combined backing reflector have also been derived by extending the steps outlined in [11]. In the transmission line model of the entire structure, the FSS has been described by an equivalent network based on a modal representation in terms of spacial Floquet modes. The eventual GF provides physical insights in the problem and speeds up the antenna design and the optimization of the relative distance between the antenna, the FSS and the ground plane. The paper is organized as follows. In Section II a connected dipole array backed by a conventional metallic ground plane is presented. This configuration appears not suitable to integrate the radar and TCDL bands. The solution proposed for the backing reflector, consisting of the FSS and the ground plane is presented in Section III. The measurements of a hardware demonstrator of the backing structure are shown in Section IV. The mathematical details to derive the active impedance of the complete antenna and the simulated results for the final structure are presented in Sections V and VI, respectively. II. CONNECTED DIPOLE ARRAY ANTENNA WITH BACKING GROUND PLANE The ultra-wide band antenna considered in this paper, a connected dipole antenna, and its complementary counterpart, the long-slot array antenna, have been extensively studied in [3], [4], [11] and [7], [8]. The antenna considered in this paper is shown in Fig. 1, with the parameters defined in Table I. Such an antenna is composed by connected dipoles of width , which are depicted in Fig. 1 as gray strips along the axis. The connected dipoles are excited by couples of - gap ports of width and relative separation , depicted in Fig. 1 as thin

black lines along the axis. The dimensions of the unitary cell , where is the free space waveare length at GHz. The use of - gap ports, even if introduces some minor limitations from the simulation point of view, allows for effectively analyzing the array with no need of a detailed feeding line, which is out of the scope of the present work. The feeding lines would affect the behavior of the array as well as the behavior of the FSS, as shown for example in [12]. The first problem is accurately addressed in [8], [9], while the second problem, which is largely tied to technological aspects, working frequencies and materials will be considered in a later stage when results from both connected array and backing plane developments will be consolidated. The distance between the antenna and the ground plane has been optimized at the value 8.3 mm, trying to cover the entire . Fig. 2 8.5–15.35 GHz band, for the complete scan range shows the simulated active reflection coefficient [13], [14], for different scan angles in the H-plane (y-z plane), obtained with the commercial tool Ansoft HFSS [15]. It should be noted that only one linear polarization is required for the addressed application. Therefore, in this paper the scanning behavior of the antenna and the backing reflector will be investigated only for the H-plane. A complete evaluation of the scanning behavior of wideband connected arrays for both polarizations can be found dB as threshold limit for the active rein [8]. Considering flection coefficient, it is evident from Fig. 2 that the ground plane solution fulfils the desired requirements only in the radar frequency band, presenting very poor performances in the TCDL band. III. FREQUENCY SELECTIVE SURFACE DESIGN The previous section clearly showed that the connected array with a conventional metal ground plane is not suitable to simultaneously cover the radar and the TCDL bands. As anticipated in the introduction of this paper, this limitation can be overcome by sandwiching an FSS between the array and the metal ground plane. In particular, in this case, the FSS should separate the frequency bands allocated for naval radar operation, 8.50–10.50

PASIAN et al.: FSSs FOR EXTENDED BANDWIDTH BACKING REFLECTOR FUNCTIONS

45

TABLE III GEOMETRICAL DIMENSIONS IN M OF THE CONNECTED DIPOLE ARRAY BACKED BY THE FSS AND THE GROUND PLANE, DEPICTED IN FIG. 9

Fig. 3. Magnitude and phase of the reflection coefficient of the FSS for different angles of incidence for TE-polarized plane wave incidence.

TABLE II GEOMETRICAL DIMENSIONS IN MM OF THE BACKING PLANE COMPOSED BY THE FSS AND THE GROUND PLANE DEPICTED IN FIGS. 3 AND 4

GHz and for the TCDL, 14.40–15.35 GHz, thus providing a reflection plane for the higher frequency range and being practically transparent for the lower band, where the metallic ground plane is in charge of the reflection. On top of that, both applicain elevation. tions ask for a wide-scan capability, up to In order to achieve the desired behavior the FSS should be transparent in the 8.50–10.50 GHz frequency range and reflective in the 14.40–15.35 GHz frequency range. For this purpose, a common approach is to print metallic patches, resonating at the higher frequency band, on a dielectric substrate. The adopted solution is based on single-layer structures with a foam substrate, which is assumed to be electrically equivalent to air. This choice minimizes the manufacturing effort and the losses. The severe constraints imposed by the wide angle scanning and steep roll-off required to properly separate the two bands dominated the design. The element shape selected for the FSS is the four-legged dipole. This kind of element exhibits a good performance against angular variation and, since the total element length can be kept quite small [16], it allows a very packed lattice, with a further gain in angular independence. On the other hand, the more packed the elements are the less steep the roll-off is, due to the capacitive coupling between the elements. For this reason a proper tradeoff had to be performed. The chosen final FSS element is depicted in the inset of Fig. 3 and its dimensions are given in Table II. The lattice periodicity, mm, equal to that of the connected array, allows good angular independence, also avoiding grating lobes in the considered band. The four-legged dipoles are capacitively loaded to achieve the required roll-off needed to separate the radar and the TCDL bands, while preserving the FSS resonance frequency. The magnitude and phase of the FSS reflection coefficient, simulated with the commercial software Ansoft Designer [17] are reported in Fig. 3. Note that the phase of the reflection coefficient is evaluated at the FSS plane itself. A uniform plane

Fig. 4. Phase of the reflection coefficient of the FSS backed by a ground plane for different angles of incidence for TE-polarized plane wave incidence.

wave polarized along the -axis (TE polarization), impinging at different angles on the plane, was considered as excitation, consistently with the analysis results reported in Section II for the connected array. It can be observed that the FSS does not exactly resonate at the center of the TCDL band. Because of the proximity and thus the interaction between the FSS itself and the ground plane, the FSS design had to be tuned to optimize the phase behavior of the complete backing reflector (FSS ground plane). Then, the optimal distances between the ground plane and the FSS, and between the FSS itself and the antenna plane had to be determined. In first approximation the FSS and the ground from the antenna plane should be placed at a distance of plane, calculated at central frequencies of the two considered frequency bands. In this way, a constructive interference between the direct field generated by the antenna and the one coming back from the FSS or ground plane would be obtained. On the other hand, the interaction between the ground plane and the FSS, which are very close to each other in terms of wavelength, has a significant impact on the final results. For this reason a fast full-wave method was used to accurately calculate the best distances, as described in Section V. The obtained and , (see Fig. 9) are reported in Table III values, Fig. 4 reports the phase of the reflection coefficient exhibited by the entire structure composed by the FSS and the ground plane, for scanning angles up to 45 . The phase response deof 6.5 picted in Fig. 4 is evaluated at the optimized distance mm between FSS and antenna plane. The magnitude of the reflection coefficient is not reported because it is always equal to 0 dB, due to the presence of the perfectly metallic ground plane. It can be observed that the constructive interference, which is indicated by a reflected phase equal to 0 , is roughly achieved at the centers of the two separate operating bands. Although the chosen FSS element is geometrically symmetric, the behavior is not exactly the same for the two orthogonal polarizations [16]. Therefore, for applications in

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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 58, NO. 1, JANUARY 2010

Fig. 5. Close-up view of the manufactured FSS.

When measuring the reflection phase, even small inaccuracies may translate to several degrees of errors. For this reason, a particular measurement procedure was applied to retrieve the phase of the reflection coefficient of the backing plane under test, minimizing errors due to edge effects, to the fact that the electromagnetic field generated by the horn antenna is not a uniform plane wave and due to deviations of the horn antenna phase center position with the frequency. A conventional metallic reflector, with dimension and position exactly equal to the ones of the backing plane under test, was used as reference. The phase of the reflection coefficient given by the perfect metallic plate recorded by the network analyzer can be written as

Fig. 6. Photograph of the mechanical support.

(1) which dual linear or circular polarization is required, the design of the backing reflector should be further optimized. IV. MEASUREMENTS The backing plane design was carried out considering only foam as substrate, but manufacturing constraints required the use of an extra dielectric support, as depicted in the inset of Fig. 8. The original design was tuned to account for the different equivalent dielectric constant. The FSS was printed on a 60 60 cm thin dielectric substrate from Rogers, LG4002 mm). This layer was glued on a foam ( mm) by means of an adfrom Rohacell ( hesive film from Arlon, CuClad 6250 ( mm). A metallic plate was then fixed at the bottom of the foam, finalizing the manufacturing of the complete backing structure. A close-up photograph of the FSS without the metallic plate is shown in Fig. 5. Two measurement setups were considered: one for the FSS alone, before attaching the ground plane, and the other one for the complete backing structure. Both measurements took place in an anechoic chamber using two standard wideband double-ridge horn antennas, in a quasi-monostatic configuration for broadside measurements and in a bistatic configuration for other angles of incidence. The position of the backing plane and of the FSS were controlled thanks to a mechanical construction consisting of a mast, two vertical and horizonal poles, with elevation and azimuth adjustable joints, and a square frame (Fig. 6). The mast was specifically designed for microwave measurements showing a reflecdB. The main part of the entire tion coefficient better than support consisted of a square frame where the backing plane could be precisely slid in. This frame was connected to the rest of the support by means of two vertical and two horizontal arms bent behind the frame, to reduce their contribution to the reflection. Thanks to this arrangement most of the reflection was caused by parts located at about 30 cm away from the frame. Therefore, such reflection contributions could be removed from the measurement by applying a proper time gate in the network analyzer. Absorbing material was placed around the frame to reduce edge effects and to hide the supporting structure mentioned before. The alignment between the backing plane and the horn antennas was controlled by means of laser beams and optimized by adjusting the elevation and azimuth joints.

where is the phase recorded by the horn antenna, is the distance between the horn antenna and the metallic plate, is the analytical reflection phase given by an infinity perfectly conducting plate and takes into account any other error. Then the backing structure was measured and a similar equation holds (2) where is the phase recorded by the horn antenna and is the reflection phase given by the backing structure. The error was assumed to be equal in both cases. Now it is possible to obtain the requested value , substituting (1) into (2) (3) Of course, after this procedure, it is still necessary to report the retrieved reflection phase given by the backing structure to the plane where the antenna array will be placed, at a distance mm as shown in the previous section. of A similar reference procedure was also followed to calibrate the response of the FSS alone. In this case the magnitude of the reflection coefficient was obtained considering as reference the reflection coefficient of the complete backing structure. In fact, the latter has a magnitude of the reflection coefficient equal to 0 dB at any frequency. The experimental results are shown in Figs. 7 and 8, where they are compared with the results obtained from simulations for incidence angles of 0 , 30 and 45 . The agreement between theory and measurement is very good, both for the magnitude and the phase response. In particular, Fig. 7 shows an excellent agreement close to the resonance region of the FSS, with some minor differences at lower frequencies, where the FSS is almost invisible and therefore the mechanical structure behind it may have affected the measurement. At the highest frequency range a slight shift can be observed, which is probably due to some small unpredictable variations of the features of the dielectric support during the manufacturing process. Fig. 8 shows the phase response of the complete backing plane, calculated at the antenna plane as described in the inset of Fig. 8 itself. The agreement is very good for all considered angles, despite the extremely high

PASIAN et al.: FSSs FOR EXTENDED BANDWIDTH BACKING REFLECTOR FUNCTIONS

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Fig. 9. 3-D view of the antenna structure with the FSS and the ground plane.

Fig. 7. Measured magnitude of the reflection coefficient of the FSS for different angles of incidence for TE-polarized plane wave incidence. Lines refer to HFSS results while symbols to measurement.

active impedance for a 2-D long slot connected array was found. The complete structure is shown in Fig. 9. As demonstrated in [8] the spectral representation of the input admittance of an array of dipoles with a double feed in each cell and in the presence of a ground plane as backing reflector can be expressed analytically as

(4) and are the periodicities of the array, is the where is the distance between the two feeds inside each unit cell, free-space propagation constant, is the zeroth order Bessel and are the Floquet mode function and propagation constants along and . Please refer to [11] for a complete definition of the functions used in the previous and following expressions. From the formalism described in depth in [11] the same expression can be recognized as the explicit representation of Fig. 8. Measured phase of the reflection coefficient of the FSS backed by a ground plane for different angles of incidence for TE-polarized plane wave incidence. Lines refer to HFSS results while symbols to measurement.

(5) accuracy required for such measurement and despite the rate of the phase variation in proximity of the FSS resonance. Small discrepancies at lower frequency are probably due to a differences between the actual and nominal values of the dielectric constant and of the thickness of the dielectric support. Once a sufficient knowledge of the impact of the fabrication process on the relevant parameters of the structure (i.e., dielectric constants and layer thicknesses) will be acquired, as part of a standard engineering development, the minor shifts observed in the measurements could be tuned out by slightly adjusting the FSS geometry and the relative separations between FSS, ground plane and antenna.

V. 2-D CONNECTED ARRAY: ACTIVE IMPEDANCE IN THE PRESENCE OF THE FSS The active impedance of connected dipole arrays backed by ground plane [8] and FSS can be derived by extending the steps outlined in previous works [3] and [11], where the GF and the

The electromagnetic field in the zone of space below the con, where innected dipoles is entirely represented by dicates that the GF provides the component of the electric field radiated by the component of the electric current, and stands for backing reflector. Equating (4) and (5) it is apparent that

(6) and thus, this structure’s spectral GF is analytically known. Readers are reminded that each of the spectral components of any GF represents the response to a combination of TE and TM plane waves. For instance

(7)

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Fig. 10. Generic transmission line model for a Floquet mode in presence of the connected dipole array backed with a ground plane. The transmission line model is the same for TE and TM modes once the proper definition of the characteristic line impedance is chosen. I is a unit current generator.

where and are the amplitudes of electric fields of TM and TE plane waves respectively, in the geometrical environment that the GF represents. In the case of laterally infinite layered structures the response for each plane wave is analytically known because it only depends on the dimensions, the thickness of the layers. Thus this dependency is represented by two equivalent transmission lines, TE or TM, as in Fig. 10, where the sources are unit current generators. It is simple to show that the solution of these transmission lines combined as in (7) gives rise to (6) When an FSS is inserted between the dipoles and the ground plane, the general GF representation for the input admittance in with . However in (7) is still valid by substituting is not analytically known. The FSS loads each this case of the plane waves of the spectral domain GF in ways which depend also on the transverse details of the FSS element geometry, (e.g., lengths and width of the legs of the four-legged loop element in the present case). This information is critical to the definition of the resonance frequency of the FSS. The FSS loading impact can be accounted for by equivalent impedance loads in parallel to the Floquet mode equivalent transmission lines representing the propagation in the otherwise transversally infinite layered cross sections. Several full wave techniques can be applied to evaluate these equivalent impedance loads when only is accounted for. the main Floquet mode The overall mono-mode equivalent network valid for each of the plane wave components, TE and TM, of the spectral GF also repreaccounting for the FSS is shown in Fig. 11, where sents the equivalent FSS impedance load for the main Floquet mode. As apparent from (4) and (5), the impedance load that goes in parallel to the transmission line needs to be evaluated and , only for the plane wave directions associated to and , defined by inverting the relations i.e.,

(8) (9) If not only the fundamental mode but also higher order Floquet modes contribute to the interaction between the composite ground plane and antenna that is they constitute accessible

Fig. 11. Equivalent transmission line model of the planar structure of Fig. 9 for the main FW’s mode. h and h are respectively the distance of the FSS from the connected dipole array and from the backing ground plane.

Fig. 12. Equivalent transmission line model of the planar structure of Fig. 9.

modes according the terminology of [18], the equivalent circuit becomes a multimode network. If N accessible modes are incomponent of cluded, an accurate description of the the GF is obtained by solving the equivalent network problem depicted in Fig. 12. This is not conceptually different with respect to the much simpler equivalent circuit in Fig. 11 but it is more accurate, should the distance between the ground plane and the FSS become extremely small in terms of wavelength. Note that the relevance of this multimode extension to the present case appears to be limited. On one hand, only at very low frequencies the distance between FSS and ground plane would be such that higher order modes have to be included in the field representation. On the other hand, at these frequencies the FSS would be transparent and not significantly affect the behavior of the overall structure. Once the GF (4) is known, it is possible to study the final antenna structure shown in Fig. 9, where the FSS is the one designed in Section III. The only two parameters remaining to the and can be optimized very efficiently by using designer the GF that also includes the effect of the FSS. The dimensions of the final structure are reported in Table III. VI. ARRAY PERFORMANCES Fig. 13 shows a comparison of the results concerning the active reflection coefficient for scanning on the H-plane of the antennas, obtained with Ansoft HFSS and with the method described in the previous section. Good agreement is achieved for all scan angles. For our method, only the fundamental modes were considered. After several simulations performed including a larger number of Floquet modes, it was observed that the fundamental modes were sufficient to describe the discontinuity introduced by the FSS. The active reflection coefficient depicted

PASIAN et al.: FSSs FOR EXTENDED BANDWIDTH BACKING REFLECTOR FUNCTIONS

Fig. 13. Magnitude of the active reflection coefficient of the connected dipole array antenna with the FSS and the ground plane. Lines refer to HFSS results while symbols to the in-house method.

49

Fig. 15. Radiation patterns on the H-Plane for different scan angles at the cenGHz, for the connected dipole array tral frequency of the TCDL band, f antenna backed by the ground plane and FSS. The pattern are normalized to the maximum value for broadside radiation.

= 15

and 15 respectively, for different scan angles and considering the case of a 10 10 element finite array. VII. CONCLUSION

Fig. 14. Radiation patterns on the H-Plane for different scan angles at the cen: GHz, for the connected dipole array tral frequency of the radar band, f antenna backed by the ground plane and FSS. The pattern are normalized to the maximum value for broadside radiation.

=95

in Fig. 13 was obtained normalizing to a feeding port impedance of 400 Ohm. dB is A good matching below the threshold value of achieved in the entire scan range in both the radar and TCDL bands, showing the possibility to extend the operational bands of the connected array by using the proposed backing structure. It is also evident that such a configuration for a backing reflector, which relies on an FSS on the top of standard metallic plate, always exhibits a multi-band behavior instead of a unique wideband operative frequency region. The basic reason for that is related to the fact that each backing plate composing the entire reflector (i.e., the FSS and the metallic ground plane) constructively reflects the field generated by the array (i.e., the phase of the reflection coefficient equal to 0 ) for one precise frequency, given the distance from the antenna and the backing structure. Thus, since the phase of the reflection coefficient is a continuous and periodic function and since the sign of its derivative is constant with the frequency, in order to connect the two frequencies where the reflection phase is 0 the function must exhibits region, which correspond to a disa passage through the ruptive interference. This aspect was already observed in [1]. For the sake of completeness the normalized gain pattern on the H-plane at the central frequency of the radar and TCDL bands, obtained by using Ansoft HFSS, are shown in Figs. 14

This paper has presented a method for the extension of the operating bandwidth toward higher frequencies of a wideband antenna on top of a metallic reflector. This is achieved by sandwiching a Frequency Selective Surface between the antenna and the ground plane. In particular, the design of a connected dipole array that integrates the functionalities of radar and Tactical Common Data Link, has been presented. A prototype of the backing reflector made by the FSS and the ground plane has been manufactured and successfully tested, showing very good agreement between predictions and measurement. Besides, the procedure to derive the active impedance for the complete antenna structure has been illustrated. The results have been compared with the ones obtained through commercial tools showing an excellent agreement. It may be noted that the performance achievable from the connected dipole array backed by the composite reflector are significantly dependent on the scan angle. Thus, the distances beand , should tween the reflecting planes and the antenna always be accurately tuned for optimal performance of the full angular range. To this purpose, the availability of the active Green’s function presented in Section V greatly accelerates the design optimization. ACKNOWLEDGMENT The authors wish to thank F. Nennie, P. Kuivenhoven and M. Bruijn for their crucial contributions to the design and manufacture of the mechanic support and to the measurement campaign. REFERENCES [1] B. A. Munk, Finite Antenna Arrays and FSS. New York: Wiley, 2003, ch. 6, pp. 181–213. [2] J. J. Lee, S. Livingston, and R. Koenig, “A low-profile wideband (5:1) dual-pol array,” IEEE Antennas Wireless Propag. Lett., vol. 2, pp. 46–49, Dec. 2003. [3] A. Neto and J. J. Lee, “Infinite bandwidth long slot array antenna,” IEEE Antennas Wireless Propag. Lett., vol. 3, pp. 75–78, Dec. 2005.

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[4] J. J. Lee, S. Livingston, R. Koenig, D. Nagata, and L. L. Lai, “Compact light weight UHF arrays using long slot apertures,” IEEE Trans. Antennas Propag., vol. 54, no. 7, pp. 2009–2015, Jul. 2006. [5] Y. E. Erdemli, K. Sertel, R. A. Gilbert, D. E. Wright, and J. L. Volakis, “Frequency-selective surfaces to enhance performance of broad-band reconfigurable arrays,” IEEE Trans. Antennas Propag., vol. 50, no. 12, pp. 1716–1724, Dec. 2002. [6] J. M. Bell, M. F. Iskander, and J. J. Lee, “Ultrawideband hybrid EBG/ ferrite ground plane for low-profile array antennas,” IEEE Trans. Antennas Propag., vol. 55, no. 1, pp. 4–12, Jan. 2007. [7] D. Cavallo, A. Neto, G. Gerini, and G. Toso, “On the potentials of connected slots and dipoles in the presence of a backing reflector,” presented at the 30th ESA Antenna Workshop, Noordwijk, The Netherlands, May 27–30, 2008. [8] A. Neto, D. Cavallo, G. Gerini, and G. Toso, “Scanning performances of wide band connected arrays in the presence of a backing reflector,” IEEE Trans. Antennas Propag., vol. 57, no. 10, pp. 3092–3102, Oct. 2009, part 2. [9] A. Neto, D. Cavallo, and G. Gerini, “Common mode, differential mode and baluns: The secrets,” presented at the 5th ESA Workshop on Millimetre Wave Technology and Applications and 31st ESA Antenna Workshop, Noordwijk, The Netherlands, May 18–20, 2009. [10] A. Hoorfar, K. Gupta, and D. Cahng, “Cross polarization level in radiation from microstrip dipole antenna,” IEEE Trans. Antennas Propag., vol. 36, no. 9, pp. 1197–1203, Sep. 1988. [11] A. Neto and J. J. Lee, “Ultrawideband properties of long slot arrays,” IEEE Trans. Antennas Propag., vol. 54, no. 2, pp. 534–543, Feb. 2006. [12] S. G. Hay and J. D. O’Sullivan, “Analysis of common-mode effects in a dual-polarized planar connected-array antenna,” Radio Sci., vol. 43, Dec. 2008, RS6S04. [13] R. C. Hansen, Phased Array Antennas. New York: Wiley, 1998. [14] D. M. Pozar, “The active element pattern,” IEEE Trans. Antennas Propag., vol. 42, no. 8, pp. 1176–1178, Aug. 1994. [15] Ansoft HFSS Version 10.0, 1984–2007 Ansoft Corporation. [16] B. A. Munk, Frequency Selective Surfaces: Theory and Design. New York: Wiley, 2000. [17] Ansoft Designer Version 3.0.0, 1984–2007 Ansoft Corporation. [18] S. Monni, G. Gerini, A. Neto, and A. G. Tijhuis, “Multimode equivalent networks for the design and analysis of frequency selective surfaces,” IEEE Trans. Antennas Propag., vol. 55, no. 10, Oct. 2007. Marco Pasian (S’04–M’08) was born in 1980. He received the M.S. degree (summa cum laude) in electronic engineering and the Ph.D. degree in electronics and computer science from the University of Pavia, Pavia, Italy, in 2005 and 2009, respectively. He is currently a Postdoctoral Researcher in the Department of Electronics, University of Pavia. His main research interests are periodic structures, antennas and microwave devices for space and defence applications. In 2004, he was a trainee at the European Space Agency, Darmstadt, Germany. In 2005, he was with Carlo Gavazzi Space, Milano, Italy, as a System Engineer. In 2008, he was a Guest Scientist at TNO, Defence, Security and Safety, The Hague, The Netherlands.

Stefania Monni (S’01–M’06) received the M.Sc. degree (summa cum laude) in electronic engineering from the University of Cagliari, Italy, in 1999 and the Ph.D. degree in electronic engineering from the Technical University of Eindhoven, The Netherlands, in 2005. In 1999 and 2000, she worked at the Radio Frequency System Division, European Space Research and Technology Centre (ESA-ESTEC) as an undergraduate and graduate trainee, respectively. From 2001 until 2005, she carried out her Ph.D. research at the Netherlands Organization for Applied Scientific Research (TNO), The Hague, The Netherlands, where she is currently employed. Her main research interests concern analysis and design techniques for phased array antennas and frequency selective surfaces, wide band antennas and digital beam forming for active and passive radars.

Andrea Neto (M’00) received the Laurea degree (summa cum laude) in electronic engineering from the University of Florence, Italy, in 1994 and the Ph.D. degree in electromagnetics from the University of Siena, Italy, in 2000. Part of his Ph.D. was developed at the European Space Agency Research and Technology Center, Noordwijk, The Netherlands, where he worked for the Antenna Section for over two years. In the years 2000 to 2001, he was a Postdoctoral Researcher at the California Institute of Technology, Pasadena, working for the Sub-mm wave Advanced Technology Group. Since 2002, he has been a Senior Antenna Scientist at TNO Defence, Security and Safety, The Hague, The Netherlands. His research interests are in the analysis and design of antennas, with emphasis on arrays, dielectric lens antennas, wide band antennas and EBG structures. Dr. Neto was co-recipient of the H. A. Wheeler award for the Best Applications Paper of 2008 in the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION. He presently serves as Associate Editor of the IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS.

Mauro Ettorre (M’08) was born in Tricarico, Matera, Italy, in 1979. He received the Laurea degree (summa cum laude) in telecommunication engineering and the Ph.D. degree in electromagnetics from the University of Siena, Italy, in 2004 and 2008, respectively. During his master degree studies he spent five months at the Technical University of Denmark (DTU), Lyngby, Denmark. Part of his Ph.D. has been developed at the Defence, Security and Safety Institute of the Netherlands Organization for Applied Scientific Research (TNO), The Hague, The Netherlands, and where afterwards he worked as a Researcher. Currently, he is a Postdoctoral Fellow at the Institut d’Electronique et de Télécommunications de Rennes (IETR), Université de Rennes 1, France, working for the Groupe Antennes & Hyperfréquences. His research interests include the analysis and design of leaky-wave antennas, periodic structures and electromagnetic band-gap structures. Dr. Ettorre received the Young Antenna Engineer Prize during the 30th ESA Antenna Workshop 2008, in Noordwijk, The Netherlands.

Giampiero Gerini (M’92–SM’08) received the M.Sc. degree (summa cum laude) and the Ph.D. degree in electronic engineering from the University of Ancona, Italy, in 1988 and 1992, respectively. From 1992 to 1994, he was an Assistant Professor of Electromagnetic Fields at the University of Ancona. From 1994 to 1997, he was a Research Fellow at the European Space Research and Technology Centre (ESA-ESTEC), Noordwijk, The Netherlands, where he joined the Radio Frequency System Division. Since 1997, he has been with the Netherlands Organization for Applied Scientific Research (TNO), The Hague, The Netherlands. At TNO Defence Security and Safety, he is currently Chief Senior Scientist of the Antenna Unit in the Transceiver Department. In 2007, he was appointed part-time Professor in the Faculty of Electrical Engineering of the Eindhoven University of Technology, The Netherlands, with a Chair in Novel Structures and Concepts for Advanced Antennas. His main research interests are phased arrays antennas, electromagnetic bandgap structures, frequency selective surfaces and integrated antennas at microwave, millimeter and sub-millimeter wave frequencies. The main application fields of interest are radar, space and telecommunication systems. Prof. Gerini was co-recipient of the 2008 H. A. Wheeler Applications Prize Paper Award of the IEEE Antennas and Propagation Society. In 2008, he was also co-recipient of the Best Innovative Paper Prize of the European Space Agency 30th Antenna Workshop.

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Antenna Modeling Based on a Multiple Spherical Wave Expansion Method: Application to an Antenna Array Mohammed Serhir, Philippe Besnier, Member, IEEE, and M’hamed Drissi, Senior Member, IEEE

Abstract—A method to derive an equivalent radiation source for planar antennas is presented. This method is based on spherical near-field (NF) data (measured or computed) to ascertain an equivalent set of infinitesimal dipoles placed over the main antenna aperture. These produce the same antenna radiation field, both inside and outside the minimum sphere enclosing the antenna. A spherical wave expansion (SWE) of the NF data is written in terms of infinitesimal dipoles using a transition matrix. This matrix expresses the linear relations between the transmission coefficients of the antenna and the transmission coefficients of each dipole. The antenna a priori information are used to set the spatial distribution of the equivalent dipoles. The translational and rotational addition theorems are exploited to derive the transmission coefficients of the dipoles. Once the excitation of each dipole is known, the field at any aspect angle and distance from the antenna is rapidly calculated. Computations with EM simulation data of an antenna array illustrate the reliability of the method. Index Terms—Antenna modeling, spherical wave expansion, translational and rotational addition theorems.

I. INTRODUCTION

HE antenna modeling in its operating environment is still a challenge using full-wave solutions. Indeed, the cost of such solutions grows exponentially with the size and the complexity of the problem. To cope with these difficulties, a two-steps hybrid approach is proposed. The antenna under test (AUT) is separately characterized and a simple equivalent model is defined. This comprises point sources that emulate the antenna radiation everywhere. At the same time, when large antennas (antenna array) are hardly characterized by full-wave solutions, the measurement alternative can be of great interest, especially, the spherical multi-probe near-field technique [1]. This technique enables real-time 3D complex characterization all over the sphere surrounding the antenna provided the Nyquist sampling criterion is satisfied [2], [3]. The antenna measured near field data are then used to define an equivalent model that allows the knowledge of the radiated EM field both inside and outside the minimum sphere enclosing the antenna. Thereafter, the antenna

T

Manuscript received September 28, 2007, revised November 12, 2008. First published November 10, 2009; current version published January 04, 2010. The authors are with the Institute of Electronics and Telecommunications of Rennes (I.E.T.R), I.N.S.A. de Rennes, 35043 Rennes Cedex, France (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2009.2036284

equivalent model is incorporated into an existing EM algorithm to simulate multiple interaction scenes. However, typical constraints arise, while modeling an antenna by an equivalent current/charge distribution. First, the equivalent model must reproduce the same radiated field as the antenna in near and far field regions. Secondly, the equivalent current/charge constituting the equivalent model must be consistent with the actual antenna current /charge distribution. Thirdly, we have to define a minimization technique that allows an efficient and well-conditioned identification process. In the literature, different approaches were adopted. In [4]–[9] an electric field integral equation method (EFIE) is developed to relate the measured electric field to equivalent magnetic /electric current on the antenna aperture. The EFIE is solved using the conjugate gradient method. Equivalent currents are written as linear combinations of two-dimensional pulse basis functions, which can be approximated by Hertzian dipoles distributed over any enclosing arbitrary surface. This method has been investigated for near field to far field transformation, inverse electromagnetic radiation applications, antenna diagnosis. In [10] different strategy for antenna diagnosis is proposed. It expresses the relation between the plane wave expansion and the spherical wave expansion, which permits the calculation of the aperture field accurately. In [11]–[13] evolutionary algorithms were introduced. Using genetic algorithm (GA) technique, the AUT is substituted by a set of infinitesimal dipoles. These are distributed inside the volume enclosing the antenna. The choice of infinitesimal dipole sources is justified by the simplicity of its implementation in any EM code, and the GA shows its ability to determine the type (electric or magnetic), the position, the orientation and the excitation of each dipole. The GA shows some difficulties to determine the optimal solution, when dealing with a large number of dipoles, including convergence issues and prohibitive CPU computing time, while increasing the antenna size. In [14] quantum particle swarm optimization algorithm has been used to define a set of equivalent dipoles for antenna modeling. This method is based on an expansive iterative degrees of freedom, where process, which depends on is the number of dipoles. In a previous paper [15] the authors have introduced a modunknowns, where is eling technique depending on the number of current sources. It consists in the substitution of the antenna with a set of infinitesimal dipoles distributed over the minimum sphere circumscribing the antenna. These equivalent dipoles distribution is well adapted to build up a

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well-conditioned matrix system but is not convenient to provide the near-field within the minimum sphere for planar antennas for example. In the present paper, an improvement of the analysis presented in [15] is proposed. Starting with the a priori knowledge of the antenna geometry, the equivalent dipoles are placed over the main radiating sources. Then, the spherical wave expansion (SWE) of the antenna radiated field is rewritten as the total contribution of small number of equivalent electric and magnetic dipoles, providing adequate changes of coordinate systems. Once assessed, this set of dipoles allows the knowledge of the near-field information inside the minimum sphere circumscribing the antenna and the EM field can be rapidly calculated at any distances in a straightforward manner (faster than the SWE), using the free space Green’s function. The paper is organized as follows. Section II describes the theoretical procedure of the method. Section III presents the application process of the method for an antenna array. Particular attention is paid to the determination of equivalent dipoles number and locations. Finally, concluding remarks summarizing the potential of this method are provided in Section IV. All theoretical results are expressed in the S.I. rationalized time dependence. system with II. FORMULATIONS AND THEORETICAL DEVELOPMENTS The complete formulation is given in [15]. We therefore, recall only the essential parts of this development, with a particular attention to the modifications introduced by the new arrangement of elementary source locations. Here, a current source is defined as a combination of 4 tangential uncoupled and co-localized infinitesimal dipoles: 2 electric and 2 magnetic. We attach a local coordinate system to each current source, where the origin coincides with the position of the th source. Explicitly, we intend to rewrite the spherical wave expansion by means of multiple and (SWE) expressed in . local SWE that are expressed in Our purpose is to emulate the electromagnetic field radiated placed from an antenna by a set of current sources over the antenna main radiating surface (Fig. 1). Consequently, the field inside the minimum sphere is reached and the number of equivalent sources required to emulate accurately the antenna E-field, is reduced, while preserving the whole SWE information. A. Spherical Wave Expansion In source free region outside the minimum sphere of radius circumscribing the antenna, the spherical wave expansion in spherical coordi(SWE) of the radiated electric field nates system is expressed in terms of truncated series of spherical vector wave functions [16] as

Fig. 1. Measurement sphere of radius r sources configuration.

, AUT minimum sphere and current

where, are the transmission coefficients, and are the power-normalized spherical vector wave functions. The truncation number with, depends mainly on the antenna dimensions are determined from and the operating frequency [16]. The the knowledge of the tangential components of either or on the measurement sphere. In [16], the coefficients are determined using the orthogonal properties of spherical wave functions . Once these coefficients are known, the field outside the minimum sphere is completely characterized by (1). The SWE of the field radiated by an infinitesimal -directed placed at the origin of electric dipole expressed in the spherical coordinate system as sociated with

is as-

(2) with . Expressions for and - directed dipoles either electric or magnetic ones are provided in [15]. In the case of a planar antenna, the antenna equivalent dipoles are distributed over the antenna main surface and only the tangential dipoles are considered. Therefore, each current source , , ( th) is characterized by 4 transmission coefficients , . We refer to the th current source by the row vector . B. The Problem Synthesis

(1)

Using translational and rotational addition theorems [16]–[18], we express the field radiated by the th current

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Fig. 2. 3D view of the microstripe array structure. Dielectric stratification (middle) and exploded view of the structure (right).

source in the antenna coordinate system the row vector is expressed in

. Hence, by

(3) where the coefficients , , , are developed in [15, Appendix]. Let define . The superposition of the fields radiated by all equivalent is written in as dipoles

Fig. 3. Magnitude of the antenna array transmission coefficients.

III. RESULTS (4)

Identifying

with

, and from (1) and (4) we get

(5)

Due to the orthogonal properties of , the transmission coefficients of both sides of (5) have to be equal. Let us denote and where, where the superscript “ ” expresses the matrix transpose. Equation (5) becomes

(6) Made up of columns and lines, is the transition matrix, which expresses linear relations between the and the transtransmission coefficients of current sources mission coefficients of the actual antenna. To solve (6), we use the least square (LSQR) code of MatLab to determine the unknowns composing the row vector .

A. Example of Equivalent Models of an Antenna Array In this section, we report a numerical example by simulation of a linear microstrip array of aperture-coupled patches operating at the frequency 5.82 GHz. The antenna dimensions along the -axis, are along the -axis, and along (Fig. 2). The antenna the -axis with is composed of a linear array of 4 rectangular patches printed on a dielectric support. Each patch is aperture-coupled to the microstrip line. A common ground plane separates the radiating part from the microstrip feeding network (Fig. 2). The antenna is issued from Antenna Center of Excellence benchmarking data base, where the detailed description is provided in [19]. The antenna equivalent model construction is based on near field data obtained through the electromagnetic simulation of the antenna in a 3D simulator that uses the finite elements method (FEM). We recover at the distance of , the amplitude and phase of both and over a spherical surface E-field components , ), with the angular resolution ( . Based on these data, we assess the of antenna transmission coefficients that are presented in Fig. 3. , so The SWE is truncated at transmission coefficients is considered.

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Fig. 4. The description of the equivalent current sources distribution over the antenna surface. The dashed rectangle represents the antenna equivalent surface. (right) The black points represent the equivalent current sources distributed regularly over the antenna equivalent surface respecting the criterion x y : .

1 =1 =

0 25

The antenna being planar, we have chosen the natural way to place the current sources (rectangular grid). The antenna is seen as the rectangular surface

over which, the equivalent dipoles are uniformly distributed (Fig. 4). The antenna equivalent model is associated with a and between the current sources. spacing criterion This results in a number of current sources given by where means the integer part of . Thereafter, the positions of are geometrically equivalent current sources in determined. to each curWe attach a local coordinate system rent source, we determine the corresponding transition matrix and we solve (6) to calculate the excitation of each dipole. is Then, the E-field radiated from the equivalent dipoles calculated at different spherical observation domains, near the minimum sphere and in the far field regions and compared with using the following the actual antenna radiation pattern error function:

=0

4

Fig. 5. At the plane  , (a) Near-field observed at the distance of . (b) Far-field radiation pattern. The actual field (FEM) (-marker line), the field resulted from the SWE ( -marker line), and the field radiated from the equivalent models associated with x y :  and : .

1 = 1 = 0 25

04

where

Comparisons between the antenna radiation pattern (finite elements method) and the radiation pattern of the equivalent dipoles in near and far field regions are shown in Fig. 5 for the and . As it can be seen, the figures show good agreements and the equivalent models radiation pattern fit very well with the one resulted from the spherical wave expansion method. In Fig. 6, the error function is presented as a function of the distance from the antenna. It is seen that the equivalent model corresponding to results in and . Using the criterion , the

and equivalent model results in . For a smaller spacing , the accuracy of the antenna equivalent model is improved , ) but ( the number of equivalent sources increases corresponds to 90 current sources). These equivalent models reproduce accurately the antenna radiation everywhere outside the . minimum sphere In conclusion, the spacing between the current sources is related to the accuracy of the equivalent model. Consequently, the choice of the spacing between the equivalent current sources , may be based on a targeted error budget for a given application.

SERHIR et al.: ANTENNA MODELING BASED ON A MULTIPLE SPHERICAL WAVE EXPANSION METHOD

Fig. 6. The error function RMSE versus the distance for and :  associated equivalent models.

04

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1x = 1y = 0:25

From a numerical point of view, the spacing between sources in (6). has a direct effect over the condition of the matrix the matrix is Using a large spacing corresponds to very well-conditioned ( ), but the antenna equivalent model is not ac, ). curate ( Inversely, the conditioning of the matrix deteriorates as the and is decreased ( value of the spacing for and for ), while the accuracy of the equivalent model is improved. As a matter of fact, we have to compromise between the accuracy and the conditioning of . Inside the minimum sphere, over the plane lofrom the antenna, cated at the distance the Ey-field component estimated by the 3D simulation software (Finite elements method) and the Ey-field radiated by the equivalent dipoles are presented in Fig. 7. A good agreement is observed, between the actual antenna radiation pattern and the equivalent model radiation pattern. For the near-field presented in Fig. 7, the error function (Ey component) reaches 1.42%. As presented in Fig. 8, the equivalent models ( or ) provide a good approximation of the field inside from the minimum sphere for distances greater than the antenna. Otherwise, the field in the vicinity of the antenna is mainly reactive and the information of the evanescent modes are not available at the measurement distance . Consequently, the equivalent models presents some discrepancies , while trying to reproduce the E-field in the vicinity of the . At the distance of , antenna takes the values 4.32% and 7.28% for the equivalent models associated with and respectively. The aim of the examples presented here, is to show the feasibility of the modeling method for the case of a finite open surface (equivalence principle for an open surface) and to figure out how to compromise between the accuracy and the number of equivalent sources (simplicity of the equivalent model). The flexibility of the presented modeling methodology is outlined. Exploiting the a priori information concerning the antenna geometry, we have defined an equivalent model, which reproduces accurately the antenna radiation pattern. We have established that this modeling technique is able to define different

= 0 02m (0 4 )

Fig. 7. The aperture field Ey (dB V/m) at the plane z : :  from the AUT: (a) Resulted from the FEM simulation software. (b) Produced by the equivalent model associated to x y : .

1 = 1 = 0 25

Fig. 8. The error function RMSE evaluated inside the minimum sphere versus the distance for x y :  and :  associated equivalent models.

1 = 1 = 0 25

04

kind of equivalent models where the choice of the spacing between equivalent current sources (complexity) depends on the desired accuracy, so, on the application for which the equivalent model is intended. provides a good Definitely, the criterion compromise between the accuracy, the number of equivalent current sources and for which the matrix is well-conditioned. B. The Reduction of the Number of Equivalent Model Dipoles Solving (6), we determine the row vectors for all current sources. By means expressed as of the normalized row vector

where (7)

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TABLE I EQUIVALENT MODELS AFTER THE REDUCTION PROCEDURE CASE: x y :  FOR T h , 4%

1 = 1 = 0 25

=2

Fig. 9. The description of the reduction of the number of equivalent model dipoles procedure.

we evaluate the power contribution of each dipole constituting the current source. The elements of correspond to the power contribution of each dipole in the current source radiation power . , we remove the inBy a proper choice of the threshold significant dipoles, so that the number of the dipoles constituting the equivalent model is reduced. An iterative process is carried out. First, we choose a spacing criterion between the equivalent current sources. Second, respecting a uniform distribution, we determine the spatial position of the current sources over the planar antenna main radiating surface. Third, we solve (6) taking into account all the dipoles and we evaluate the row vectors so that one can eliminate the are less than . dipoles whose corresponding terms in Finally, a new solution of (6) is performed providing the new configuration (without the insignificant dipoles). This procedure is pursued as long as non-significant dipoles are detected. The flowchart describing the complete proposed modeling procedure is outlined in Fig. 9. Results of the antenna equivalent model based on are investigated. The reduced equivalent models are preconfigurations corresponding to different thresholds sented in Table I. Then, the E-field radiated from these reduced equivalent models is compared with the actual E-field estimated by FEM at different observation distances . as function of the From Fig. 10, where we present distance, we observe that the same level of accuracy is main,4% ( tained after the reduction procedure for , ), while reducing considerably the to number of the equivalent dipoles (240 dipoles for 126 dipoles for ). Since the actual radiating currents are due to the patches, which are aperture-coupled to the microstrip line, the actual current sources depends on the geometry of the apertures and

th =

Fig. 10. (a) The error function RMSE corresponding to the thresholds and 4% as a function of the distance resulting from the reduced equivalent models associated with x y : . (b) The near field observed at the distance  radiated from the reduced equivalent models compared to the one issued from FEM (actual antenna radiation pattern).

2

2

1 = 1 = 0 25

the patches. The apertures are aligned with the x-axis, so the antenna equivalent dipoles must include -directed magnetic dipoles. On the other hand, the rectangular patches include -directed and -directed electric dipoles. As presented in Table I, after the reduction procedure, the dipoles types constituting the reduced equivalent models are: -directed, -directed electric dipoles, and -directed magnetic dipoles. These dipoles are consistent, from a qualitative point of view, with the antenna actual current sources. , For the equivalent model based on we converge toward a reduced configuration of the equivalent

SERHIR et al.: ANTENNA MODELING BASED ON A MULTIPLE SPHERICAL WAVE EXPANSION METHOD

model (using ) while fitting accurately (99%) the actual antenna radiation pattern. This threshold allows us to reproduce accurately the radiated E-field as the actual antenna using only 126 dipoles (126 coefficients). In other words, the use of a priori information concerning the antenna geometry to define the multiple-SWE can be of great interest since it permits to rewrite the transmission coefficients of the antenna in limited number of equivalent dipole excitations. The reduction procedure tends to cancel dipoles that do not contribute effectively in the radiation pattern. Contrarily to the spherical wave expansion, the minimum number of transmission coefficients is related to the antenna dimensions. At least we should consider the truncation , which leads to the use of number transmission coefficients. The SWE is a very accurate way to describe the radiated field and the accuracy of the SWE cannot be questioned. Nonetheless, the SWE is valid outside the minimum sphere and the implementation of the SWE representation of the radiated field in an electromagnetic code is not trivial. Rewriting the transmission coefficients (SWE) in terms of equivalent dipoles, enhanced with the reduction procedure, can be a good way to express the whole information contained in the spherical wave coefficients in a limited number of dipole excitations. The rearrangement of the SWE into a multiple SWE expressed in different local coordinate systems (using a priori information) is a time consuming procedure (translational and rotational addition theorems). Nevertheless, once the equivalent model is determined, the calculation of the radiation pattern in different distances from the antenna is faster than using the SWE. This is due to the fact that the number of equivalent dipoles is smaller than the number of the transmission coefficients.

IV. CONCLUSION A method for antenna modeling has been presented. This method is based on the substitution of the original antenna by a set of equivalent infinitesimal dipoles that reproduce the same radiating field. Since the equivalent model includes a set of dipoles placed over the main antenna radiating surface, the information given by these dipoles gives qualitative information concerning the origin of the radiation. Using the antenna a priori information, it has been shown that the number of infinitesimal dipoles can be significantly reduced. The use of the transmission coefficients to elaborate the antenna equivalent model is interesting. This allows a straightforward matching procedure, where the whole antenna intrinsic information are contained in a finite number of complex values. Once the antenna equivalent model is defined, the electromagnetic radiation of the antenna can be calculated at any point both inside and outside (FF region) the minimum sphere except in the reactive region of the antenna. Also, the antenna a priori information concerning the AUT geometry have been used to express the multiple spherical wave expansion, which is of great interest since it has allowed the rewrite the SWE with a limited and reduced number of dipole excitations.

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REFERENCES [1] J. C. Bolomey, B. J. Cown, G. Fine, L. Jofre, M. Mostafavi, D. Picard, J. P. Estrada, P. G. Friederich, and F. L. Cain, “Rapid near-field antenna testing via arrays of modulated scattering probes,” IEEE Trans. Antennas Propag., vol. 36, pp. 804–814, Jun. 1988. [2] A. D. Yaghjian, “An overview of near-field antenna measurements,” IEEE Trans. Antennas Propag., vol. AP-34, pp. 30–45, Jan. 1986. [3] O. M. Bucci, C. Gennarelli, and C. Savarese, “Representation of electromagnetic fields over arbitrary surfaces by a finite and non redundant number of samples,” IEEE Trans. Antennas Propag., vol. 46, pp. 351–359, Mar. 1998. [4] P. Petre and T. K. Sarkar, “Planar near-field to far-field transformation using an equivalent magnetic current approach,” IEEE Trans. Antennas Propag., vol. 40, pp. 1348–1356, Nov. 1992. [5] P. Petre and T. K. Sarkar, “Planar near-field to far field transformation using an array of dipole probes,” IEEE Trans. Antenna Propag., vol. 42, pp. 534–537, Apr. 1994. [6] T. K. Sarkar and A. Taaghol, “Near-field to near/far-field transformation for arbitrary near-field geometry utilizing an equivalent electric current and MoM,” IEEE Trans. Antenna Propag., vol. 47, pp. 566–573, Mar. 1999. [7] F. Las-Heras and T. K. Sarkar, “A direct optimization approach for source reconstruction and NF-FF transformation using amplitude-only data,” IEEE Trans. Antennas Propag., vol. 50, pp. 500–510, Apr. 2002. [8] F. Las-Heras, M. R. Pino, S. Loredo, Y. Alvarez, and T. K. Sarkar, “Evaluating near-field radiation patterns of commercial antennas,” IEEE Trans. Antennas Propag., vol. 54, pp. 2198–2207, Aug. 2006. [9] Y. Alvarez, F. Las-Heras, and M. R. Pino, “Reconstruction of equivalent currents distribution over arbitrary three-dimensional surfaces based on integral equation algorithms,” IEEE Trans. Antennas Propag., vol. 55, pp. 3460–3468, Dec. 2007. [10] C. Cappellin, A. Frandsen, and O. Breinbjerg, “On the relationship between the spherical wave expansion and the plane wave expansion for antenna diagnostics,” presented at the AMTA Europe Symp., 2006. [11] J. R. Regué, M. Ribo, J. M. Garrell, and A. Martin, “A genetic algorithm based method for source identification and far-field radiated emissions prediction from near-filed measurement for PCB characterization,” IEEE Trans. Electromagn. Compat., vol. 43, pp. 536–542, Aug. 1996. [12] J. R. Pérez and J. Basterrechea, “Antenna far-field pattern reconstruction using equivalent currents and genetic algorithms,” Microw. Opt. Technol. Lett., vol. 42, no. 1, pp. 21–25, Jul. 2004. [13] T. S. Sijher and A. A. Kishk, “Antenna modeling by infinitesimal dipoles using genetic algorithms,” in Progr. In Electromagn. Res., PIER 52, 2005, pp. 225–254. [14] S. M. Mikki and A. A. Kishk, “Theory and applications of infinitesimal dipole models for computational electromagnetics,” IEEE Trans. Antennas Propag., vol. 55, pp. 1325–1337, May 2007. [15] M. Serhir, P. Besnier, and M. Drissi, “An accurate equivalent behavioral model of antenna radiation using a mode-matching technique based on spherical near field measurements,” IEEE Trans. Antennas Propag., vol. 56, pp. 48–57, Jan. 2008. [16] J. E. Hansen, Spherical Near-Field Antenna Measurements. London, UK: Peregrinus, 1988. [17] A. R. Edmonds, Angular Momentum in Quantum Mechanics, 3rd ed. Princeton, NJ: Princeton Univ. Press, 1974. [18] J. H. Bruning and Y. T. Lo, “Multiple scattering of EM waves by spheres—Part I: Multipole expansion and ray-optical solutions,” IEEE Trans. Antennas Propag., vol. Ap-19, pp. 378–390, May 1971. [19] [Online]. Available: http://ace1.antennasvce.org/Dissemination/view/ download?id_file=43

Mohammed Serhir was born January 8, 1981, in Casablanca, Morocco. He received the diplôme d’ingénieur degree from Ecole Mohammadia d’Ingénieurs (EMI), Rabat, Morocco, in 2003 and the Ph.D. degree in electronics from the National Institute of Applied Sciences at Rennes, INSA de Rennes, France, in 2007. His research interests include spherical wave expansion technique, spherical near-field antenna measurements, and the development of numerical methods.

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Philippe Besnier received the diplôme d’ingénieur degree from Ecole Universitaire d’Ingénieurs de Lille (EUDIL), Lille, France, in 1990 and the Ph.D. degree in electronics from the University of Lille, in 1993. He was with the Laboratory of Radio Propagation and Electronics, University of Lille, as a Researcher at the Centre National de la Recherche Scientifique (CNRS), from 1994 to 1997. Since 2002, he has been with the Institute of Electronics and Telecommunications of Rennes, Rennes, France, where he is currently a Researcher at CNRS heading EMC-related activities such as EMC modeling, electromagnetic topology, reverberation chambers, and near-field probing.

M’hamed Drissi (SM’90) received the Ph.D. degree in electronics from National Institute of Applied Sciences at Rennes, INSA de Rennes, Rennes, France, in 1989 and the HDR degree from the University of Rennes 1, France, in 1997. In 1991, he joined INSA of Rennes, where he is currently a Full Professor and Director of research. His research activities deal with the electromagnetic modelling and the design of antennas and the associated circuits, the electromagnetic compatibility of complex electronic systems, and the near-field characterization. His research interests include CAD of high-speed and MMIC circuits, neuronal modeling and simulation of high-speed interconnecting, and nonlinear circuits.

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An External Calibration Scheme for DBF Antenna Arrays Holger Pawlak, Member, IEEE, and Arne F. Jacob, Fellow, IEEE Abstract—A novel calibration scheme is presented that is especially suited for complex digital beamforming (DBF) antenna arrays at millimeter-wave frequencies. Calibration data is extracted by sampling the field of each radiator at certain locations near the array by fixed probe antennas. A scalable calibration model for evaluation of the measured data is described. First tests are performed on a small passive array representing a unit cell of larger arrays. The calibration scheme is subsequently applied to and tested on a 64 element DBF transmit antenna array. Index Terms—Antenna array, calibration, digital beamforming, probe antenna.

I. INTRODUCTION UTURE mobile satellite communication systems are driven by requirements such as large bandwidth and high flexibility. Planar antenna arrays featuring digital beamforming at Ka-band frequencies show the highest potential for advanced communication terminals [1]–[4]. Specialized architectures suitable for large-scale implementation are investigated within the frame of the project SANTANA (Smart ANTennA termiNAl) [2]–[4]. The common characteristics of the various architectures are modular array assembly and split transmit/receive functions. Parallel studies cover array calibration and other system aspects [5], [6]. The high flexibility of DBF arrays derives from the ability of full digital control over all signals forming the radiation pattern of the antenna. Therefore, each radiator must be equipped with its own receiver or transmitter circuitry. For proper operation, all channels (antenna plus circuitry) of the DBF antenna must exhibit defined amplitudes and phases. Unavoidable manufacturing tolerances call for adequate calibration. Many different techniques for antenna array calibration, each with specific advantages and drawbacks, are reported in the literature [7]–[22]. However, most of them are variations of three basic calibration techniques, which are briefly described in the following. A straightforward solution is to provide internal calibration networks to couple a small part of the output signal back to a calibration receiver [7]–[10]. In the desired application, this approach suffers from three drawbacks: space requirements on

F

Manuscript received May 14, 2008; revised June 20, 2009. First published November 06, 2009; current version published January 04, 2010. This work was supported by the German Aerospace Center (DLR) on behalf of the German Federal Ministry of Economics and Technology (BMWi) under research contract 50YB0304. H. Pawlak is with OHB-System AG, Universitätsallee 27-29, 28359 Bremen, Germany (e-mail: [email protected]). A. F. Jacob is with the Institut für Hochfrequenztechnik, Technische Universität Hamburg-Harburg, 21071 Hamburg, Germany (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2009.2036195

the densely packed RF frontend, low isolation of the calibration signal, and need of synchronization with the neighboring modules. The second calibration technique makes use of external probe antennas to detect the field generated by each channel [11]–[18]. Probe antennas placed at boresight [11]–[13] yield the best results because the calibration measurements are performed in the direction of interest. However, this approach leads to a rather bulky setup. For arrays requiring a compact assembly a placement within the array [14]–[17] may seem attractive. The drawback is the degradation of the overall array pattern due to coupling effects. This can be avoided by applying MCM (mutual coupling measurements) [15], [16] which do not disturb the array. On the other hand, MCM calibration schemes are only applicable to hybrid Rx/Tx arrays and, thus, cannot be considered here. A placement of the probe antennas at the circumference of the array [18] avoids the parasitic couplings. The third group of calibration techniques relies entirely on signal-processing methods to derive calibration data by analysis of receive signals [19]–[22]. This approach is suited only for DBF receive antennas and therefore not considered because the calibration scheme presented in this paper was designed for DBF transmit antennas. An external scheme, which has been implemented before for array monitoring purposes [18], constitutes an interesting approach for the desired application. The approach presented in this paper goes beyond monitoring (i.e., continuous comparison to a reference calibration) since it is able to derive the calibrated array excitation directly from the probe antennas measurements. For this purpose, the optimum number and locations of the probe antennas are determined. This is achieved by means of a calibration model of the antenna array and corresponding array measurements. Previous work addresses probe antenna design [5] and a measurement setup for small DBF transmit arrays along with the basic data extraction scheme [6]. Further application specific requirements like a compact calibration setup and—as far as possible—an autonomous calibration (i.e., a self-calibration without a priori knowledge of the individual radiators) are considered in the design. Section II addresses the setup of the calibration scheme and simulation results. The experimental evaluation follows in the second part of the paper. Section III provides measurement data on the inner patches of a passive 4 4 array. The setup is closely related to the previous simulation. Section IV describes the measurement results on the fully assembled 8 8 DBF transmit antenna array realized within the project SANTANA. II. CALIBRATION SCHEME AND MODEL The calibration model must be specifically tailored to the array to be calibrated. Thus, the antenna architecture is briefly

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Fig. 1. Modular construction of the 8 the individual modules.

2 8 DBF antenna and array topology of

TABLE I DBF ANTENNA CHARACTERISTICS Fig. 2. Simulation model of the array module, extended ground plane, and probe antennas (left), sketches of the simplified probe antenna (top right) and the actual probe antenna used for the measurements (bottom right).

phase difference between two channels. The array is calibrated by taking the amplitude ratios and phase differences between all channels with respect to an arbitrarily chosen reference channel. B. Optimum Locations of Probe Antennas described in Section A. The external calibration scheme compares each patch in the array with an arbitrarily chosen reference patch. The general methodology is introduced in Section B. Here, exemplary antenna patterns of two patches in the array environment are analyzed to derive the optimum number and locations of the probe antennas. Detailed formulae are provided in Section C to calculate the calibration coefficients from the probe data. Full-wave simulation results of a complete array calibrated with the proposed scheme at both patch and array level are presented. A. Array Topology and Probe Antennas Fig. 1 shows the modular construction of the DBF array from identical modules along with a sketch of the array topology. Each array module consists of sixteen circularly polarized patch antennas arranged in a quadratic grid with half-wavelength spacing. Sequential rotation within a 2 2 unit cell is used to improve polarization purity. Four unit cells constitute one array module. A summary of the nominal DBF antenna characteristics is listed in Table I. More details concerning the antenna array and the individual radiators are reported in [2]. The channel-specific amplitudes and phases needed for calibration are extracted from linearly polarized measurements. The required linear probe antennas can be realized with good characteristics at millimeter-wave frequencies [5]. A printed linear antenna is used for the calibration scheme. This antenna consists of an open-ended microstrip line protruding approximately a quarter wavelength above the ground plane. In this way, the probe antennas are easily integrated into the calibration receivers that are realized in microstrip technology, too. A sketch of this probe antenna is shown in Fig. 2. A different calibration receiver evaluates each probe antenna. The unknown absolute amplitude and phase response of each receiver is eliminated by calculating the amplitude ratio and

Fig. 2 shows the FDTD simulation model. The array module and the probe antennas share a common ground plane. Simplified probe antennas (coaxial monopoles) instead of the actual probe antennas are used to reduce model complexity. This allows simulating the entire array at reasonable computational . The cost. The probe antennas are placed on the curve curve may have an arbitrary shape but for the first implementation of the calibration scheme, a circle around the array center is chosen for symmetry reasons. Its radius is 24 mm, here. The and are compared on fields of two patch antennas . As explained above, only a relative measure of the fields matters in the desired application. Thus, it is useful to introduce the on complex field ratio

(1) probe antennas are placed on at specific look angles defined with respect to the array center. Average values and are now calculated from the complex field ratios for the amplitude and phase, respectively

(2) Only equidistant arrangements of probe antennas are considered in this paper. This leads to a constant angular spacing of . The look angle of the first probe antenna is also the of a specific arrangement. Hence, the look angles zero angle are obtained from (3)

PAWLAK AND JACOB: AN EXTERNAL CALIBRATION SCHEME FOR DBF ANTENNA ARRAYS

TABLE II COMPUTED RANGE OF VARIATION

In the following, and are assessed when using one through twelve look angles. The simulation frequency is 30 GHz. The calculation is performed for the simulated fields of two inner patch antennas (no. 2 and 5 in Fig. 2). The averages and depend on the zero angle of the arrangement. The aim is to find the number of look angles for which this dependence is smallest. This is achieved by varying the zero angle from 0 to 360 in steps of 1 . In other words, the look angles are rotated around the array center with the chosen step size while keeping the angular spacing constant. The field ratios are calculated from (1) based on the simulated fields with the and are subsequently chosen resolution. The averages and obtained from (2). Finally, the standard deviation of over all zero angles is taken. This yields the range of variation over the entire circle for each number of look angles. The numerical results are displayed in Table II. Deliberately, the arrangements for seven and eleven look angles are not listed because their angular spacing is not an integer multiple of the chosen step size. As expected, the standard deviations for a single look angle are quite large. The standard deviations generally decrease for increasing number of look angles. For arrangements of four, eight, and twelve look angles the standard deviations are virtually zero. These numbers of look angles are the obvious choice for the calibration scheme since there is no angular dependence, i.e., an arbitrary zero angle can be chosen. An intuitive explanation is provided for this behavior. The active element pattern [23] is identical (only rotated) for all inner patch antennas. When applying (2) on rotated, but otherwise identical field patterns the individual terms of the sums vanish if the angular spacing of the look angles is an integer divisor of the rotation angle. The preceding determination of the range of variation is repeated for all inner patch antennas (no. 2, 5, 12, 15) showing an and The edge antennas exhibit a identical behavior of non-vanishing range of variation for all numbers of look angles. This is because the active element pattern is different from that of the inner patch antennas. A quantitative assessment is provided in the following section. C. Calibration Data Extraction and Simulation Results The probe antennas are now placed at the optimum look angles identified in the previous section. Four probe antennas on a circle with a radius of 24 mm around the array center are used.

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The probe antennas are located near the corners of the array, which is equivalent to a zero angle of 45 (see Fig. 2). The following simulation results are performed assuming all antenna elements (ports) to be terminated with a matched load. In the later application of the calibration scheme, amplitude ratios and phase differences are measured at the output port of the probe antennas. Therefore, the field-based equations given and in so far must be rearranged. First, the quantities are replaced by the simulated scattering parameters, leading . The complex transmission coeffito cients from either patch (subscripts and ) to a probe antenna (subscript ) are denoted by and , respectively. and of two patch anThe difference of the averages tennas does not directly yield the difference of their excitations, which is the quantity needed for array calibration. The reason is that the field patterns are sampled on a circle around the array center, which does not coincide with the geometrical center of each patch antenna. This is dealt with by the following procedure. A generic field pattern is taken with its origin in the geometrical center of each patch antenna. A change of origin (patch antenna center to array center) is performed to compute the field pattern on . The calculated amplitudes and phases at the look and are incorporated in angles of the probe antennas the data extraction. An identical pattern is assumed for all patch antennas. Several choices for the generic pattern are possible: an isotropic pattern (constant amplitude and phase), a generic pattern for circularly polarized patch antennas (constant amplitude and linear phase) or simulated antenna patterns (e.g., of an inner patch antenna). The isotropic model is used here for reasons of simplicity. Hence, the equations for computing the difference of become the excitations of two patch antennas

(4) The factors depend only on the geometry of the calibration setup and are computed in advance. Patch antenna no. 15 is used as reference for the data extraction. The differences between all patch antennas with respect to the reference are computed with (4) using the simulated scattering parameters of the setup shown in Fig. 2. The calculated differences constitute the . A reference excitation is deextracted array excitation and is determined. fined and the deviation between is chosen here as a uniform excitation of equal amplitude and phase which also includes the phase shifts for the sequential rotation scheme. After data extraction, the inner patch antennas (no. 2, 5, 12, dB and from the ref15) show a low deviation of erence excitation. This small residual error is attributed to the finite discretization of the structure. The deviation of the other patch antennas is larger because their active element pattern differs from that of the reference patch. The standard deviation over all patch antennas is calculated to assess the total deviation from

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Fig. 3. Simulated array gain for excitations [R] and [K ] at 30 GHz (cut-plane  = 0 ). TABLE III FREQUENCY DEPENDENCE OF THE CALIBRATION SCHEME

the reference excitation. The numerical values are 0.75 dB and 1.81 for amplitude and phase, respectively. The calibration error of the edge elements can be significantly reduced by use of simulated antenna patterns instead of the isotropic model. This approach is only feasible for small arrays because the complete antenna pattern within the array environment has to be computed and stored. This is too time-consuming for large arrays. Furthermore, after construction of the array the measurement of each antenna pattern would be required to implement this approach. Thus, the choice of the isotropic model is a result of a trade-off between calibration accuracy and simplicity offering reasonable calibration accuracy for the whole array. The radiation pattern of the array is computed for both exciand to assess the influence of the residual deviatations tions (see Fig. 3). The pattern generated by is almost idenwith a slightly smaller main lobe and a small tical to that of increase of the sidelobes. is repeated within the operating freThe calculation of quency band from 29.5 GHz to 30 GHz with a step size of 100 MHz. The standard deviation of the difference between and is computed at each frequency. The results are shown in Table III. While the amplitude deviation increases towards higher frequencies, the phase deviation decreases. However, the frequency dependence is small because of the differential extraction scheme.

Fig. 4. Photograph of the measurement setup including the movable probe antenna and the backside of the antenna array.

the measurement setup. A 4 4 array of circularly polarized patch antennas is mounted in a quadratic cutout of a circular ground plane. The diameter of the ground plane is 75.5 mm. Field patterns are measured by means of a movable probe antenna attached to a guiding slot at the edge of the ground plane. The probe antenna is fixed at the measurement positions (look angles) by means of alignment pins. The backside of the array is shown in Fig. 4 as well. Each patch has a single feed realized as a via through all three layers of the microwave substrate. The four inner patches are connected by microstrip lines to SMP coaxial connectors. It is not possible to accommodate all sixteen feeding lines on the bottom layer (shown in the photograph) without introducing parasitic coupling between the lines. In order to connect all sixteen patches a more complicated multilayer with at least two more layers and special stripline transitions at 30 GHz are required. Thus, the other twelve patches are open-circuited. The simulation in Section II has been repeated for this different load condition to check for possible degradations. It was found that the characteristics are retained also for the open-circuited patches even though the final calibration accuracy is degraded to some extent. A vector network analyzer measures the transmission from a patch to the probe antenna. The response of the four inner patch antennas is measured at 16 equidistant look angles equivalent to a step size of 22.5 . In addition to the probe antenna measurements, the transmission from the patches to a circularly polarized, open-ended waveguide antenna at boresight is measured. The obtained differences at boresight serve to determine the ref, which is later compared to the extracted erence excitation . excitation The midband frequency of the array under consideration is 28.7 GHz. Measurement data is recorded from 28.1 to 29.3 GHz. In the following, the calibration scheme is analyzed within a fixed bandwidth of 500 MHz at midband. Here, patch no. 1 is used as reference patch (see Fig. 4).

III. PASSIVE ARRAY MEASUREMENTS B. Data Extraction Using 16 Look Angles A. Measurement Setup The first series of measurements is performed on a passive array without DBF electronics. Fig. 4 shows a photograph of

The extracted excitation is calculated from (4) using all sixteen look angles of the movable probe antenna. Subsequently, both the amplitude and the phase difference with respect to the

PAWLAK AND JACOB: AN EXTERNAL CALIBRATION SCHEME FOR DBF ANTENNA ARRAYS

Fig. 5. Phase difference between the extracted excitation [ excitation at boresight [ ] based on sixteen look angles.

R

K

]

and the reference

TABLE IV DEVIATION RANGES FROM 28.45 TO 28.95 GHz

reference excitation are determined for patches 2–4. The average difference is taken as the mean of the three individual differences. This quantity is a first estimation of the average deviation of the calibration scheme. The results are analyzed separately for amplitude and phase in the following. Fig. 5 shows the calculated difference between the extracted phases and the reference phases. It is observed that the difference curves of patches 1–2 and patches 1–3 exhibit a similar frequency dependence which differs from that of patches 1–4. From 28.45 to 28.95 GHz the average deviation lies between and +9.2 . The deviation ranges of the individual difference curves are listed in Table IV. The deviation range increases outside the specified frequency band as documented in Fig. 5. The corresponding differences for the amplitude are shown in Fig. 6. The region of smallest deviation is around 29.25 GHz, which is outside the specified frequency band. Between 28.45 GHz and 28.95 GHz the difference curve of patches 1–3 exhibits an entirely positive deviation. The average deviation dB to dB. The other deviation ranges ranges from are listed in Table IV, again. The measurements show that differences at boresight can be estimated by the calibration scheme with an average deviation range given in Table IV. The measured deviation range is higher than expected from simulation. The reason for the simulated deviations being close to zero is the almost perfect symmetry of the antenna patterns in the FDTD model. The high degree of symmetry cannot be reproduced in practice at millimeter-wave frequencies. This is particularly true for the type of patch used. Here, circular polarization is achieved by excitation of two patch modes in phase quadrature. This mechanism is susceptible to tolerances introduced by the fabrication process. The lack of symmetry causes a systematic error in the calibration scheme, which affects the deviation range.

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K

Fig. 6. Amplitude difference between the extracted excitation [ erence excitation at boresight [ ] based on sixteen look angles.

R

]

and the ref-

C. Data Extraction Using 4 and 8 Look Angles It should be noted that the average deviation range determined above results from a calibration without any specific knowledge on the individual radiators. If the achieved deviation range is not sufficient for a particular application, a one-time measurement in an anechoic chamber is performed to derive additional correction factors for each radiator. These correction factors are subsequently incorporated into the calibration scheme. The simulation shows that a minimum of four look angles is sufficient for array calibration. Therefore, the number of look angles used for computation of the extracted excitation is reduced in the following. Four (two) sets with four (eight) look angles each are defined from the measured data set of sixteen look angles. The following characteristics are expected from the simulation: each set should yield the same results as for 16 look angles and the deviation between the various sets should consequently be minimal. The analysis is performed by calculating the extracted excitation for each set of look angles. The basic properties of the various sets are studied in detail for the difference curve of patches 1–3 from the previous section. Numerical values for the other difference curves are provided at the end of this section. Fig. 7 shows the computed phase difference of the four sets with four look angles. The average curve from Fig. 5 (16 look angles) is added for comparison. The deviation between the four sets is very small at midband and increases towards the band edges. The difference curves of the four sets are in good agreement with the curve from Fig. 5. Set no. 2 yields the smallest deviation from the reference. The corresponding amplitude difference is shown in Fig. 8. The difference curves of the four sets do not correlate as well as for the phase. The curves have similar shape but there is a constant offset between them. Especially set no. 1 is lower in amplitude than the other sets. On the other hand, this set exhibits the minimum deviation from the reference excitation. The analysis shows that the deviation varies from set to set when using a reduced number of look angles. This must be considered in the assessment of the overall deviation of the calibration scheme. The typical deviation is computed by averaging the deviation of each set. By virtue of this definition, the numerical

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TABLE V WORST-CASE DEVIATION RANGE FOR FOUR LOOK ANGLES FROM 28.45 TO 28.95 GHZ

TABLE VI WORST-CASE DEVIATION RANGE FOR EIGHT LOOK ANGLES FROM 28.45 TO 28.95 GHz

K

Fig. 7. Phase difference of patches 1–3 between the extracted excitation [ ] and the reference excitation at boresight [ ] based on four sets with four look angles each.

R

listed in Table IV. Therefore, the average values of sixteen look angles can be represented by either set of eight look angles. The lower than expected symmetry is also the reason for the deviation between the various sets. Therefore, an autonomous calibration suffers from an increased deviation (i.e., reduced accuracy) when using sets of four look angles. If such a calibration is required, a minimum of eight look angles should be provided. IV. DBF ARRAY MEASUREMENTS

Fig. 8. Amplitude difference of patches 1–3 between the extracted excitation [ ] and the reference excitation at boresight [ ] based on four sets with four look angles each.

K

R

values are identical to those given in Table IV. The worst-case deviation is derived from the highest and the lowest observed value of all sets. The worst-case deviation is computed within the specified frequency band. The results for all difference curves are shown in Table V. The difference curves of patches 1–2 and 1–4 exhibit lower amplitude deviations but larger phase deviations. The relatively large amplitude deviation for the first two difference curves are caused by only one set, which exhibits a large positive deviation, compared to the other ones. The same is true for the phase of the first difference curve. Eliminating this set from the computation of the worst-case scenario reduces the deviation range significantly. The corresponding values are shown in brackets. The calculation of the worst-case is repeated for two sets of eight look angles. The numerical values are shown in Table VI. The deviations are clearly smaller than in the case of four look angles. Further on, the deviations are close to the average values

The second series of measurements is performed on the 8 8 DBF array [4]. The photograph in Fig. 9 shows the assembly. Four LTCC-modules constitute the antenna array. The DBF electronics consists of the RF, IF, and digital hardware. The RF electronics relies on MMIC technology and is integrated in the modules. The IF and digital electronics are located on separate circuit boards beneath the array. Four printed linear antennas [5] serve as probe antennas for the external calibration scheme. They are connected to two calibration receivers, one probe antenna being directly integrated into each calibration receiver and the other one being attached via a semi-rigid cable. All parts of the calibration hardware are thoroughly shielded to eliminate parasitic coupling from the nearby transmitter channels. Prior to the actual calibration measurements, several preliminary tests are carried out at 29.654 GHz. These tests are performed by measuring the transmission to a circularly polarized horn antenna placed at boresight. The long-term amplitude and phase stability of the 64 channels is measured over 20 hours under laboratory conditions to determine the average system drift. The latter is taken as the standard deviation over all channels. The result is 0.21 dB and 2.2 for amplitude and phase, respectively. The calibration receivers must be operated in the linear regime to avoid additional errors caused by compression. Therefore, the channels with the highest receive power levels at each probe antenna are determined and the calibration channel gains are set appropriately. Afterwards the series of calibration measurements is carried out. Each probe antenna measures the amplitude and phase of each channel. The calibration receivers record the data.

PAWLAK AND JACOB: AN EXTERNAL CALIBRATION SCHEME FOR DBF ANTENNA ARRAYS

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2

Fig. 9. Photograph of the 30 GHz circularly polarized 8 8 DBF array (dimensions: 40 mm 40 mm) with four probe antennas at the array edges.

2

Before computing the extracted excitation, it is useful to analyze the amplitude and phase differences at the probe antennas [6]. This allows a first qualitative assessment of the calibration accuracy. The differences are obtained by evaluating (4) for each probe antenna without taking the average. As a matter of illustration, Fig. 10 depicts the differences for module 1 in phase and amplitude, respectively. Straight lines connect the measurement points for clarity. The phase curves of the individual channels exhibit a similar shape along with a vertical shift. The latter is caused by the different phases of the transmitter channels to be calibrated. The shapes of the amplitude curves do not coincide as well. However, the agreement is sufficient to proceed with the calibration procedure. Subsequently, the extracted excitation is computed for the four modules and the complete 8 8 array. An inner patch antenna of the respective (sub-)array is chosen as reference patch . The excitation is compared with a reference for calculating obtained by measuring the differences between excitation the channels at boresight. The standard deviation of the difference between and over all channels is taken to assess the overall deviation (i.e., accuracy) of the calibration scheme. Table VII displays the results. The deviations of the four modules range from 1.9 dB to 2.8 dB and 15.0 to 17.7 for amplitude and phase, respectively. The calculated deviations of the modules are in line with those obtained for the passive array measurements (cf. Table V). With 2.9 dB and 21.3 , the overall deviations of the 8 8 array are slightly larger and thus exceed the average deviation range in Table V. Additional correction factors can compensate the residual deviations [6]. After applying the correction factors, only the average system drift (0.21 dB and 2.2 ) limits the calibration accuracy. The results presented above are compared to those obtained with a conventional scheme employing a calibration network. The latter includes a 1:16 power divider and is realized on a separate layer of the LTCC antenna multilayer. Details on the

Fig. 10. Phase differences (top) and amplitude differences (bottom) of module 1 measured at the probe antennas (operating frequency: 29.654 GHz).

TABLE VII DEVIATION RANGE AT 29.654 GHz

multilayer construction and the layout of the calibration network are reported in [7]. The calibration accuracy is assessed in a similar way as for the external scheme. For each of the four modules a reference patch in the middle of the particular array is chosen. The differences in amplitude and phase are calculated with respect to the reference patch. The accuracy is averaged over the four modules because each module has its own calibration network and calibration receiver. The calculated accuracy for the 8 8 array is 4.5 dB and 23.6 for amplitude and phase, respectively. Comparison with the external calibration scheme (cf. Table VII) shows that the calibration accuracy is slightly better than that of the internal scheme with the primary advantage that no calibration hardware like the power divider network and the receiver are required on the antenna modules. During the system tests, the DBF array was exemplarily operated at seven equidistant frequencies between 29.430 and

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Fig. 11. Amplitude and phase deviations of the 8 frequencies.

2 8 array at various operating

30.102 GHz. This allowed evaluating the bandwidth performance of the calibration scheme. For this purpose, the residual deviations of the 8 8 array are calculated. The results for amplitude and phase are shown in Fig. 11. The residual deviations exhibit only slight frequency dependence. The amplitude and phase deviations lie between 2.6 to 3.1 dB and 18.9 to 23.0 , respectively. Thus, the calibration scheme shows a nearly constant accuracy over the frequency range of the DBF array. V. CONCLUSION A compact calibration scheme is presented which relies on external probe antennas placed far from boresight. A generic design procedure is described. The subsequent analysis shows that at least four probe antennas are necessary for the array topology under consideration. The array excitation is computed from probe measurement data and array geometry. Measurements performed on a passive array demonstrate that calibration is possible with reasonable accuracy. Nevertheless, the measurements show a higher range of deviation than the simulation. The residual deviations are reduced by using a larger number of probe antennas. The calibration scheme is successfully tested on a complex 64 element DBF transmit array. The measured frequency dependence is low, making the calibration scheme attractive for broadband applications. REFERENCES [1] H. Tohyama, R. Miura, M. Oodo, K. Katsuhiko, and I. Ohiomo, “SDMA experiments using a 64-element receiving DBF antenna in the Ka-band,” in Proc. IEEE Topical Conf. on Wireless Commun. Technol., Honolulu, HI, Oct. 2003, pp. 402–403. [2] H. Pawlak, A. Dreher, A. Geise, R. Gieron, S. Holzwarth, C. Hunscher, A. F. Jacob, K. Kuhlmann, O. Litschke, D. Lohmann, W. Simon, L. C. Stange, and M. Thiel, “Modular DBF terminal antennas for broadband mobile satellite communications at Ka-band,” in Proc. 28th ESA Antenna Workshop on Space Antenna Syst. and Technol., Noordwijk, The Netherlands, Jun. 2005, pp. 786–793. [3] H. Pawlak, L. C. Stange, A. Molke, A. F. Jacob, O. Litschke, S. Holzwarth, M. Thiel, A. Dreher, and C. Hunscher, “Miniaturised DBF communication modules for broadband mobile satellite access at Ka-band,” Proc. Eur. Microw. Assoc., vol. 1, no. 3, pp. 219–226, Sept. 2005.

[4] A. Geise, A. F. Jacob, K. Kuhlmann, H. Pawlak, R. Gieron, P. Siatchoua, D. Lohmann, S. Holzwarth, O. Litschke, M. V. T. Heckler, L. Greda, A. Dreher, and C. Hunscher, “Smart antenna terminals for broadband mobile satellite communications at Ka-band,” in Proc. 2nd Int. ITG Conf. on Antennas, Munich, Germany, Mar. 2007, pp. 199–204. [5] H. Pawlak, D. Nötel, and A. F. Jacob, “Compact calibration scheme for planar DBF transmit antenna arrays using circular polarization at Ka-band,” in IEEE AP-S Int. Symp. Dig., Washington, DC, Jul. 2005, pp. 752–755. [6] H. Pawlak, A. Charaspreedalarp, and A. F. Jacob, “Experimental investigation of an external calibration scheme for 30 GHz circularly polarized DBF transmit antenna arrays,” in Proc. 36th Eur. Microwave Conf., Manchester, U.K., Sep. 2006, pp. 764–767. [7] O. Litschke, W. Simon, and S. Holzwarth, “A 30 GHz highly integrated LTCC antenna element for digital beam forming arrays,” in IEEE AP-S Int. Symp. Dig., Washington, DC, Jul. 2005, pp. 297–300. [8] A. Dreher, N. Niklasch, F. Klefenz, and A. Schroth, “Antenna and receiver system with digital beamforming for satellite navigation and communications,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 7, pp. 1815–1821, Jul. 2003. [9] C. Passmann, F. Hickel, and T. Wixforth, “Investigation of a calibration concept for optimum performance of adaptive antenna systems,” in Proc. IEEE 48th Conf. on Veh. Technol., Ottawa, Canada, May 1998, pp. 577–580. [10] Y. Takeuchi, H. Hirayama, K. Fukino, T. Murayama, Y. Notsu, and A. Hayashi, “Auto calibrated distributed local loop configuration of array antenna for CDMA cellular base station,” in IEEE 6th Int. Symp. on Spread Spectrum Tech. and Applicat., Parsippany, NJ, Sep. 2000, pp. 666–670. [11] T. W. Nuteson, J. E. Stocker, J. S. Clark, D. S. Haque, and G. S. Mitchell, “Performance characterization of FPGA techniques for calibration and beamforming in smart antenna applications,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 12, pp. 3043–3051, Dec. 2002. [12] E. Lier, D. Purdy, J. Ashe, and G. Kautz, “An on-board integrated beam conditioning system for active phased array satellite antennas,” in Proc. IEEE Int. Conf. on Phased Array Syst. and Technol., Dana Point, CA, May 2000, pp. 509–512. [13] X. Zheng, T. Gao, and X. Chen, “Mid-field calibration technique of active phased array antennas,” presented at the Asia-Pacific Microwave Conf., Suzhou, China, Dec. 2005. [14] A. Agrawal and A. Jablon, “A calibration technique for active phased array antennas,” in Proc. IEEE Int. Symp. on Phased Array Syst. and Technol., Boston, MA, Oct. 2003, pp. 223–228. [15] T. Gao, Y. Gou, J. Wang, and X. Chen, “Large active phased array antenna calibration using MCM,” in IEEE AP-S Int. Symp. Dig., Boston, MA, Jul. 2001, pp. 606–609. [16] G. Hampson and A. Smolders, “A fast and accurate scheme for calibration of active phased-array antennas,” in IEEE AP-S Int. Symp. Dig., Orlando, FL, Jul. 1999, pp. 1040–1043. [17] N. Tyler, B. Allen, and A. H. Aghvami, “Calibration of smart antenna systems: Measurements and results,” IET Microw. Antennas Propag., vol. 1, no. 3, pp. 629–638, Jun. 2007. [18] M. Sarcione, J. Mulcahey, D. Schmidt, K. Chang, M. Russell, R. Enzmann, P. Rawlinson, W. Guzak, R. Howard, and M. Mitchell, “The design, development and testing of the THAAD (theater high altitude area defense) solid state phased array (formerly ground based radar),” in Proc. IEEE Int. Symp. on Phased Array Syst. and Technol., Boston, MA, Oct. 1996, pp. 260–265. [19] S. D. Silverstein, “Application of orthogonal codes to the calibration of active phased array antennas for communication satellites,” IEEE Trans. Signal Process., vol. 45, no. 1, pp. 206–218, Jan. 1997. [20] M. Oodo and R. Miura, “A remote calibration for DBF transmitting array antennas by using synchronous orthogonal codes,” in IEEE AP-S Int. Symp. Dig., Orlando, FL, Jul. 1999, pp. 1428–1431. [21] M. Viberg and A. L. Swindlehurst, “A bayesian approach to auto-calibration for parametric array signal processing,” IEEE Trans. Signal Process., vol. 42, no. 12, pp. 3495–3507, Dec. 1994. [22] W. Yao, Y. Wang, and T. Itoh, “A self-calibration antenna array system with moving apertures,” in IEEE MTT-S Int. Microwave Symp. Dig., Philadelphia, PA, Jun. 2003, pp. 1541–1544. [23] D. M. Pozar, “The active element pattern,” IEEE Trans. Antennas Propag., vol. 42, no. 8, pp. 1176–1178, Aug. 1994.

PAWLAK AND JACOB: AN EXTERNAL CALIBRATION SCHEME FOR DBF ANTENNA ARRAYS

Holger Pawlak (S’01–M’04) was born in Salzgitter, Germany, in 1976. He received the Dipl.-Ing. degree in electrical engineering from the Technische Universität Braunschweig, Braunschweig, Germany, in 2001 and the Dr.-Ing. degree from the Technische Universität Hamburg-Harburg, Hamburg, Germany, in 2008. From 2001 to 2004, he was a member of the research staff at the Institut für Hochfrequenztechnik, Technische Universität Braunschweig. In 2004, he joined the research staff at the Institut für Hochfrequenztechnik, Technische Universität Hamburg-Harburg. Since 2007, he has been a Systems Engineer at OHB-System AG, Bremen, Germany. His research interests include antenna systems and arrays.

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Arne F. Jacob (S’79–M’81–SM’02–F’09) was born in Braunschweig, Germany, in 1954. He received the Dipl.-Ing. degree in electrical engineering and the Dr.-Ing. degree from the Technische Universität Braunschweig, Braunschweig, Germany, in 1979 and 1986, respectively. From 1986 to 1988, he was a Fellow at CERN, the European Laboratory for Particle Physics, Geneva, Switzerland. He then spent three years at the Accelerator and Fusion Research Division, Lawrence Berkeley Laboratory, University of California at Berkeley. In 1990, he joined the Institut für Hochfrequenztechnik, Technische Universität Braunschweig, as a Professor. Since 2004, he has been a Professor at the Technische Universität Hamburg-Harburg, Hamburg, Germany. His current research interests include the design, packaging, and application of integrated systems and subsystems at microwave and millimeterwave frequencies, and the characterization of complex materials.

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Study and Design of a Differentially-Fed Tapered Slot Antenna Array Eloy de Lera Acedo, Student Member, IEEE, Enrique García, Member, IEEE, Vicente González-Posadas, José Luis Vázquez-Roy, Rob Maaskant, and Daniel Segovia, Member, IEEE

Abstract—The results of a parametric study and design of an ultrawideband dual-polarized array of differentially-fed tapered slot antenna elements are presented. We examine arrays of bunny-ear antennas and discuss the capabilities and limitations of differential antenna technology. As we focus on radio astronomical applications, the absence of a balancing-feed circuit not only reduces the first-stage noise contribution associated to losses in the feed, but also leads to a cost reduction. Common-modes are supported by the antenna structure when a third conductor is present, such as a ground plane. We demonstrate that anomalies may occur in the differential-mode scan impedance. Knowledge of both types of scan impedances, differential and common mode, is required to properly design differential LNAs and to achieve optimal receiver sensitivity. A compromise solution is proposed based on the partial suppression of the undesired common-mode currents through a (low loss) balancing-dissipation technique. A fully steerable design up to 45 in both principal planes is achieved. Index Terms—Antenna array feeds, mutual coupling, phased arrays, radio astronomy.

I. INTRODUCTION

T

APERED SLOT antenna (TSA) Arrays are of large interest for ultrawideband applications from the time they were introduced by Lewis et al. [1], in particular, the widely employed exponentially tapered slot antennas (Vivaldi antennas, [2]). In recent years, a growing interest has emerged from the radio astronomy community in the so-called aperture Arrays (AA) [3], which is the European concept of a versatile array antenna composed of millions of dual-polarized TSA elements. The SKA telescope [4] will comprise a number of such antenna array concepts, thereby facing one of the biggest technological challenges in radio astronomy for the 21st century. In particular, the SKA telescope will have a collecting aperture of one Manuscript received June 05, 2008; revised June 26, 2009. First published November 06, 2009; current version published January 04, 2010. This work was supported by the European Community Framework Programme 6, Square Kilometre Array Design Studies (SKADS), under contract no. 011938 http:// www.skads-eu.org. E. de Lera Acedo was with the Astronomical Centre of Yebes, National Astronomical Observatory of Spain, E-19080 Guadalajara, Spain. He is now with the Cavendish Laboratory, University of Cambridge, Cambridge CB3 0HE, U.K. (e-mail: [email protected]). E. García, J. Luis Vázquez-Roy, and D. Segovia are with the Signal Theory and Communications Department, University Carlos III of Madrid, Leganés, Madrid 28911, Spain (e-mail: [email protected]; [email protected]; [email protected]). Vicente González-Posadas is with the Universidad Politécnica de Madrid, 28040 Madrid, Spain (e-mail: [email protected]). Rob Maaskant is with the Netherlands Institute for Radio Astronomy (ASTRON), 7990 AA Dwingeloo, The Netherlands (e-mail: [email protected]) . Digital Object Identifier 10.1109/TAP.2009.2036193

square kilometer, and will cover a frequency band from 100 MHz up to 25 GHz with dual-linear polarization. The mid-frequency band, which ranges from 0.3 to 1 GHz, is intended to be covered by a dense array of millions of cheap TSA elements that . The array design represents a challenging can scan up to problem for which different technologies are being considered [5]. This paper presents the benefits, the drawbacks, and a design example of a TSA array, which will be distinct from other studies published thus far [6]–[8] namely, the antennas are differentially-fed, thereby enabling us to use differential amplifiers and differential beam-forming technology in a straightforward manner. Fig. 1 exemplifies five different types of TSA elements. Fig. 1(A) illustrates an exponentially tapered aperture fed by a microstrip line printed on a relatively expensive and potentially lossy substrate. Such an antenna-feed transition acts as a balancing mechanism (balun) for the differential TSA element, and is only required if a direct connection to a non-differential transmission line has to be realized [7]. Fig. 1(B) represents a bilateral Vivaldi element composed of three metal layers [6]. Fig. 1(C) shows a differentially-fed unilateral TSA element, however; it still requires a relatively expensive substrate. Fig. 1(D) illustrates a modified case compared to Fig. 1(A), where the metal sheet used for the exponential taper of the aperture is no longer printed on a dielectric substrate, but realized by relatively thick metallic plates that are composed of, e.g., aluminum [7]. Finally, an example of a differentially-fed TSA element, which is solely composed of thick metals, is shown in Fig. 1(E). This, so-called bunny-ear antenna [9], which does not require a (lossy) dielectric substrate or additional balancing feed board, represents an inexpensive and suitable candidate for the SKA project when aiming for differential technology. Obviously, the expensive and noisy feed board is no longer required if a low noise amplifier (LNA) is directly attached to the antenna and is realized in differential technology (coplanar strips for instance), or if the transition to single-ended technology has been realized within the LNA. An important contribution to the design of differentially-fed array antennas that has been published by the SKA community thus far is the Australian “checkerboard” array [10], meant to be used as a focal plane array comprising a stack of a printed circuit board (PCB); a foam layer; and a ground plane. The PCB consists of an array of self-complementary rectangular conducting differentially-fed pair of patches with the appearance of a checkerboard printed on an electrically-thin dielectric substrate. The array is differentially-fed at the ground plane, with two-conductor transmission lines feeding the signals be-

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DE LERA ACEDO et al.: STUDY AND DESIGN OF A DIFFERENTIALLY-FED TAPERED SLOT ANTENNA ARRAY

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Fig. 2. Portion of a differentially-fed TSA element array with a metallic ground plane. One polarization is in dark gray and the other in light gray, the ground plane is in white. This represents a part of a larger array. Fig. 1. Types of TSA elements: (A) Printed on a dielectric substrate with microstrip feed; (B) printed on a dielectric substrate with a stripline feed; (C) printed on a dielectric substrate with a differential feed; (D) one layer of thick metal with a localized microstrip feed, (E) one layer of relatively thick metals with a differential feed.

tween these points and the gaps between the corners of neighboring patches of the array [11]. The conducting surfaces of the connected-patch array are topologically equivalent to those of the tapered-slot array studied in this paper. This equivalence has been detailed in Section III. Another array concept, which has been fabricated and is currently being evaluated using integrated active receivers, is the dual-polarized array of differentially-fed tapered slot antennas with dual cavities [12]–[14]. This array has been printed on a thin and inexpensive polyester foil. In this paper, arrays of dual-polarized differentially-fed bunny-ear antennas [Fig. 1(E)] are examined. In Section II, a parametric study of an infinite-by-infinite array of bunny-ear antennas is conducted, and used to analyze the effect of different antenna geometries on the scan impedance. The antenna elements are arranged on a square grid as shown in Fig. 2. By acquiring essential knowledge from previous papers [15], [16], and by using a commercial full-wave simulator based on finite elements method, named HFSS [17], it is now possible to design these complex structures and to analyze the vitally important mutual coupling effects as these yield the wideband performance of the array. The closely spaced elements exhibit strong mutual coupling effects which are still not modeled rigorously, so that full-wave simulations and design strategies are necessary [6]–[8]. The differential scan impedance anomalies are analyzed in Section III, which arise because of destructive mutual coupling interference effects. Generally, one can distinguish between several kinds of anomalies that may appear in the scan impedance [18]–[29]. A new type of impedance resonance for arrays of differentially-fed TSA elements has been identified and is shown to be similar to the resonance effect observed in [11]. This anomaly occurs due to the presence of a third conductor nearby the differential feeding line. This could represent the back metallic plane, which is used as a reflecting surface and a virtual ground for the differential feeding lines. Without precautions, a common-mode current may then be supported by the feed structure, and even

Fig. 3. Bunny-ear element geometry. The antenna is excited by a voltage source between the coplanar strips of the differential feed line at the bottom of the antenna as indicated in the figure (bottom left).

dominate over the differential-mode current for certain frequencies and array excitations/scan angles. As a result, a strong mismatch on the differential port is observed (see Fig. 3 for the port excitation setup), leading to a surge in the radiated power, and in turn, causes a scan blindness for the differential-mode to occur. We demonstrate that this impedance mismatch, which also leads to a noise mismatch in receiving array antennas, can be improved by means of a (low loss) balancing-dissipation technique, and we therefore study the resulting array noise figure as well. In Section IV, a design example is presented of a dual-polarized array, which is steerable up to 45 in both principal planes and operates over a frequency band ranging from 300 MHz up to 1 GHz. Finally, the most relevant aspects of the study are summarized and the future work is presented. II. PARAMETRIC STUDY OF THE TSA ELEMENT The bunny-ear element in Fig. 3, inserted in an infinite-by-infinite array as in Fig. 2, is subject to a parametric analysis, i.e.,

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the most important geometrical parameters of the antenna element are analyzed independently of each other. As indicated in Fig. 3, the antenna is excited by a voltage source which is attached to the end of the differential transmission line, and is located in the same plane as the ground plane. With reference to [8], we know that for a stripline-fed bilateral Vivaldi , and the opening rate antenna, the length of the inner curve, of the inner curve, , have a predominant effect on the scan input impedance of the array. The element width defines the and -plane spacing and has a strong effect on the frequency at which the mismatch anomaly occurs. At present, no techniques are known to suppress this type of resonance, so that it inevitably fixes the upper limit of the usable bandwidth. However, to some extent, the resonance can be shifted up in frequency by a decrease of the aperture height or width of the elements. Furthermore, an increase in the ratio of the aperture height to the antenna width will produce an enlargement of the bandwidth, though a narrower scan pattern in the -plane is to be expected when the aperture height is increased. As for the presently pro, posed differential antenna, the additional aperture length, , of the outer curve are analyzed. The and opening rate, outer curve aims to realize a gradual transition from coplanar strip to slotline technology such as to achieve an ultrawideband antenna performance. Both the inner and the outer curves are described by the exponential (1) and (2). The slot and strip width have only a minor influence on the scan impedance, as opposed to the aforementioned parameters [8]; therefore these parameters are chosen to be invariant in this study. Note that two sets , and of axes are being used, one for the inner curve . another one for the outer one The inner curve is described by

(1a) (1b) (1c) The outer curve is described by

(2a) (2b) (2c) (3) (4) The difference between half the antenna width and half the is fixed to 4.5 mm for assembly purposes. aperture width The 21 mm offset between the length of the inner curve and the total height of the antenna is kept constant. The initial geometrical parameters of the array are shown in Table I. For each of the following parametric analyses, only one of the parameters is swept, while fixing the other parameters in

TABLE I INITIAL GEOMETRICAL PARAMETERS OF THE ARRAY

Fig. 4. VSWR at broadside over frequency for various antenna widths.

accordance to Table I. Furthermore, all results in this paper are obtained by exciting only one of the polarizations. A. Width of the Element The effect on the antenna input impedance, when reducing the width of the element, is visualized in Fig. 4. From this VSWR plot we conclude that the strong impedance anomaly in the upper part of the band is downshifted in frequency when the element width is enlarged, thereby limiting the usable band and giving rise to the strong mismatch visible in the VSWR (source-reference impedance is 150 ). This effect is also observed when the array is scanned in both the - and -planes. B. Length of the Inner Curve A reduction of the taper length of the inner aperture will result in a narrowing of the impedance bandwidth while its center frequency shifts towards higher frequencies. The use of longer tapers lead to smaller fluctuations of the scan impedance, and a smaller reactance as well, particularly at lower frequencies, thereby improving the lower usable frequency (results not shown). However, the penalty is that the impedance anomalies in the upper part of the band move slightly down in frequency. A similar trend is observed when the array is scanned to 45 in both principal planes. In conclusion, the bandwidth of this type of arrays can be enlarged by increasing the length-width ratio of the inner taper of the antenna element, albeit to a certain extent. C. Opening Rate of the Inner Curve (Bin) The opening rate of the inner curve of the antenna is mainly affecting the matching level in the middle of the frequency band, though it also has a relatively significant effect on the lower limit of the usable bandwidth. Larger opening rates lead to stronger fluctuations in the scan impedance around their mean values.

DE LERA ACEDO et al.: STUDY AND DESIGN OF A DIFFERENTIALLY-FED TAPERED SLOT ANTENNA ARRAY

Fig. 5. A broadside scan over frequency for various antenna lengths of the outer curve: (a) VSWR and (b) the scan input impedance.

These mean values flatten-out for larger opening rates, especially at the upper part of the frequency band, and therefore result in better VSWRs. At the lowest end of the band, the VSWR , so that it is necessary to find a increases with increasing compromise value for the opening rate of the inner curve. A similar behavior is observed when the array is scanned up to 45 in both principal planes.

D. Length of the Outer Curve Results of scan impedance studies, and the occurrence of impedance anomalies, have been reported for the conventional stripline-fed bilateral Vivaldi antennas when the array scans offbroadside [6], [8]. To the author’s best knowledge, no results have been reported for arrays of bunny-ear antennas where the effect of the outer curve on the scan impedance has been analyzed. It will be shown that several impedance effects are attributed to a specific design of the outer taper which need to be accounted for when designing and optimizing bunny-ear arrays. The tapered outer curve is an important part of the differential feeding system as its length controls the smoothness of the transition from coplanar strip to slotline technology and, in turn, influences the scan impedance of the antenna array elements. We can appreciate in Fig. 5(b) that an increase in taper length leads to an increase in the input resistance, whereas the input

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Fig. 6. A 45 E-plane scan over frequency for various antenna lengths of the outer curve: (a) VSWR and (b) the scan input impedance.

reactance has virtually not changed, except at very low frequencies. Basically, only the matching level has changed as shown in Fig. 5(a), and the bandwidth has changed accordingly. It is important to remark that for symmetric arrays of tapered slot antennas, typically only the - and -plane scans are of primary interest, since the -plane scan impedance behaves, at least to first order, as a simple average of the - and -plane scan performances [8]. Fig. 6 illustrates that when the array is scanned up to 45 in the -plane, an impedance anomaly [Anomaly 1 in Fig. 6(a)] appears within the band of interest and is strongly dependent on the length of the outer curve. It was found that the origin of this anomaly is related to the fact that the array can support a common-mode current. A more detailed explanation of its origin and suppression is given in Section III. So far, one can see how the length of the outer curve affects the position of the anomaly as it moves towards lower frequencies for larger taper heights. The impedance results are shown in Fig. 7 for the case that the array is scanned to 45 in the -plane. An impedance resonance appears in the upper part of the frequency band, thereby reducing the usable bandwidth. As in the case of Anomaly 2 in the -plane scan of Fig. 6(a), the location of this -plane scan anomaly turns out to be virtually invariant for taper heights, as opposed to the case of Anomaly 1 in the -plane scan of Fig. 6(a). Anomaly 2 in the high part of the band is due to the

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TABLE II PARAMETERS OF THE ARRAY FOR REVEALING THE SCAN IMPEDANCE ANOMALY

Fig. 8. VSWR for an

H

Fig. 7. A 45 -plane scan over frequency for various antenna lengths of the outer curve: (a) VSWR and (b) the scan input impedance.

appearance of the endfire grating lobe and is observed in the present unit cell computation because it corresponds to the frequency and angle at which a higher-order Floquet mode starts propagating. Apart from the anomalous impedance effect, the trends observed in the scan impedances are very similar to the broadside-scanned array case, i.e., the resistance can be raised in-band by increasing the length of the outer taper, whereas the reactance remains almost the same. The mid-frequency hump of the VSWR in Fig. 7(a) also increases with the height of the outer curve, and is in accordance with the increase in the input resistance. E. Opening Rate of the Outer Curve The effects on the impedance characteristics of the array are . Furthermore, variations of the only weakly dependent on opening rate of the outer curve also have a minimal effect on the . The impedance anomalies, as opposed to a variation of respective results are therefore not shown. III. SCAN IMPEDANCE ANOMALIES ASSOCIATED TO DIFFERENTIAL ARRAY TECHNOLOGY The presently proposed bunny-ear element exhibits a few -plane scan impedance anomalies that are typically found for other Vivaldi elements as well [18], [19]. Apart from these types of resonances, a number of additional impedance anomalies exist that have been identified before and that are attributed to a particular realization of the TSA elements. Among them, (i)

E -plane scanned dual-polarized array.

the impedance resonance that appears when a surface excited wave travels along the facets of a substrate (printed TSAs), which then destructively interferes with the phasing of the array; (ii) the impedance resonances that appear due to gaps between disjoint antenna elements/tiles that tend to radiate, thereby severely disrupting the impedance characteristics [20]; (iii) the impedance resonances that occur for -plane scanned bilateral TSAs, which arise due to a voltage difference between the bilateral fins of the TSAs, and that can be eliminated by plated-through vias in the elements [6]; or (iv) the impedance resonance that arises in triangular grid arrays of phase-steered linear-polarized TSA arrays [21]. The presently employed bunny-ear antennas are dielectric-free, are electrically interconnected along their outer edges, and also unilateral and dual-polarized, so that the previously mentioned impedance resonances are not observed. However, we identify here a type of impedance resonance which can specifically be attributed to differentially-excited arrays. To examine this type of scan impedance anomaly, the geometrical parameters as listed in Table II are used. The VSWR is shown in Fig. 8 for three different -plane scans. Anomalous impedance effects occur for off-broadside scan angles in the middle of the band when exciting the antennas with a differential generator. We remark that these resonances disappear in the absence of a ground plane and shift down in frequency with an increase in scan angle. It is important to note that the position of the strong impedance anomalies in the upper part of the band also shift down for increasingly larger scan angles, in both the - and -planes, though this effect is less pronounced as compared to the effect of increasing the element width, which reduces the operational bandwidth significantly. In [19], similar anomalies appear in the middle of the band for -plane scanned single-polarized arrays of bilateral TSAs, which are due to the presence of parallel plate modes in between the parallel strokes of TSA plates. This type of resonance was also found to occur for an -plane scanned single-polarized

DE LERA ACEDO et al.: STUDY AND DESIGN OF A DIFFERENTIALLY-FED TAPERED SLOT ANTENNA ARRAY

array of bunny-ears (results omitted). As shown by Wunsch and Schaubert [22], these resonances can be suppressed by placing metallic cross walls at the locations where normally cross-polarized elements are placed. In [23], one justifies the presence of an impedance anomaly that was found to occur because of the undesired radiation from the differential feed lines used to excite an array of dipole antennas. The behavior of these anomalies with frequency and scan angle are rather similar to the impedance perturbation presented in this paper. In [24] a similar impedance anomaly for a planar array of symmetrical crossed dipoles has been identified and analyzed rigorously for the fields that are incident from grazing angle (in the -plane). However, the polarization of the incident -field vector is not the same as in the present work, where the array grid has been oriented perpendicular to the -plane scans. Nonetheless, the anomalies seem to have a common origin with the one explained in this work, where we explain how currents couple to each other due to the presence of neighboring elements and how this leads to the undesired increase in the reflection coefficient. In the work of Hay and O’Sullivan [11], an anomaly was found for a finite array of connected patches with features also observed in the present differentially-fed bunny-ear arrays1. Fig. 9 shows the topological equivalence between the two structures where is the geodesic path length along the edges of the conducting surfaces between the feed points of the elements of orthogonal polarizations. In [11] it is demonstrated that the frequency at which a differential-mode resonance occurs can approximately be predicted by the geodesic path length being half a wavelength. In the present study, the geodesic path length equals half a wavelength at 615 MHz, which is a reasonable prediction since the resonances are observed around 650–700 MHz (exact value depends on the scan angle cf. Fig. 8). Another property of this resonance is that it is characterized by anti-phase common-mode currents on the feed conductors of connected orthogonally polarized elements. In the following it is shown that this also occurs for the present structure. By removing the ground plane, we retain only the two conductors (the two arms of the antenna) as well as the balanced feed lines. We observed that, for the present frequency range and for off-broadside scan angles, where the electrical symmetry is broken, only evanescent common-mode currents are supported along the differential feed lines. However, in the presence of a third conductor, such as a ground plane, a common-mode signal can be supported by the feeding structure which may lead to severe differential-mode scan impedance resonances. From Fig. 8 one can observe for what frequencies impedance anomalies occur, and how these are dependent on the scan angle. The surface current distribution on the antenna elements has been analyzed for a number of scan angles/frequencies, thereby revealing the presence of common-mode currents on the differential feed lines. It has been confirmed that both odd- and even- modes can exist on ground-plane-backed bunny-ear ar1When revising the manuscript, the author’s attention was called to a recently published paper by Hay and O’Sullivan [11] where an equivalent situation is studied albeit for a different type of structure.

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Fig. 9. Topological equivalence between the connecting surfaces of (a) the TSA elements and (b) the patches in [11]. Each drawing represents 4 half elements (2 per polarization). The geodesic path length along the conducting surfaces between the feed points of the elements of orthogonal polarizations is p.

rays, and these modes are excited differently, depending upon the frequency and scan angle. To analyze the various modes of the currents, we let a series of plane waves be incident to the open-circuited array while monitoring the induced current distributions. Fig. 10(a) shows the current distribution at 500 MHz, where one can clearly observe that the odd mode dominates over the even mode. The even-mode current distribution in Fig. 10(b) is strongly pronounced for an incident plane wave at 22.5 at a frequency of 710 MHz, which is the frequency where a scan impedance anomaly occurs on transmit for the differentially-fed array scanned to 22.5 in the -plane. Fig. 10 therefore suggests that when the differential-mode scan impedance is at resonance, a common-mode current is supported by the structure, which may be excited strongly depending upon the type of excitation source used on transmit, or incident plane-wave field applied on receive. The presence of such a common-mode current at the anomalous frequency was also observed for an array of disconnected balanced antipodal Vivaldi antennas [25]. At this moment it is not clear how differential LNAs can be optimally noise matched to both types of scan impedances, given the fact that they are strongly related to each other. Despite elements) would largely behave an SKA-like station ( as an infinite planar array [26], the common-mode anomaly in Fig. 8 could also appear at broadside, due to a slight asymmetry brought about by the finiteness of the array. An example is the case of a large array of disjoint subarrays, or a focal plane array with offset beams (non-symmetric excitations). Anomalies may also appear as observed by Schaubert, Craeye, and Boryssenko when introducing other types of asymmetries in the structure, e.g., [27], [28]. To mitigate the common-mode problem, i.e., to eliminate the resonance in the differential-mode scan impedance, it is essential to first analyze the propagation of the corresponding surface currents through the structure at resonance. Fig. 10(b) illustrates that the common-mode current is confined to the bottom region of the bunny-ear antennas instead of along the tapered slot as for the differential-mode current, and that the phase progression of the common-mode current starts at one arm of the antenna and ends in the arm of the orthogonally connected antenna. This specific problem is only present in dual-polarized arrays like the

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Fig. 11. Distribution of the surface currents for the unit cell after placing two resistors of 3 k from each arm of the antenna to the ground plane. The current is shown at the anomalous frequency of Fig. 10(b).

Fig. 10. Distribution of the surface currents for the unit cell at (a) 500 MHz and (b) at the anomaly frequency with an incident plane wave at 22.5 .

ones in Fig. 2. Note that these characteristics were also observed for an array of connected patch antennas [11]. We will present a technique to avoid these impedance anomalies while keeping the orthogonally polarized elements in place. The comprise solution is based on using two resistors per element, each of them connected between one of the feeding lines and the GND plane, thereby dissipating a small part of the common-mode currents (see Fig. 11). Even though the proposed solution may seem very noisy for a radio astronomical application, it will be shown that the noise increment is only minor and therefore not of a big concern. The use of reactive elements has not shown to yield the desired change in the scan impedance characteristics. Fig. 11 visualizes the two resistors that have been used to partly suppress the even mode current, thereby allowing the odd mode to dominate. The propagation path of the current reveals that it is sufficient to place the resistor in only one of the polarizations. However, in order to preserve the symmetry of the array we apply them to both polarizations.

A parametric study was undertaken to examine the antenna input impedance effect for various resistor values. For this purpose, the VSWR has been computed and plotted in Fig. 12 for a 45 -plane scan. Clearly, the differential VSWR increases when low-valued resistors are used since then the differential generator becomes almost short-circuited. The differential impedance match readily improves upon raising the resistor level by only tens of ohms. In fact, the even-mode currents keep sufficiently suppressed up to 3 k , while still a . good overall matching level can be realized The differential scan impedance anomaly appears for resistor values larger than that, implying that the structure can support even-mode currents with significant amplitude. The case of raising the resistor values to infinity is equivalent to the case of removing the resistors (open circuit), so that the original anomalous differential VSWR is obtained allowing both the common-mode and the differential-mode currents to propagate. The common-mode current is stronger suppressed than the differential-mode current since the resistors are seen in series for the differential-mode excitation, whereas in parallel for the common-mode excitation (factor 4 difference, for DC). The best trade-off is reached when the resonance in the differential scan impedance becomes sufficiently suppressed while the overall impedance characteristics remain almost unaffected. This optimum resistor value was found to be near 3 k . The differential VSWR for three different -plane scans is presented in Fig. 13, both for 1 k and 3 k resistors. The -plane scans are not shown, because these are only weakly dependent on the resistors. Note that for 3 k resistors, the matching level in the band remains acceptable, even for large scan angles, and the results are very similar to those obtained for a resistor value of 1 k . A thermal noise study of the proposed method is of interest when the lumped resistors are placed across the antenna conductors in front of the amplifying circuits2. 2The two loads in Fig. 11 may be realized by the input impedance of two single-ended LNAs which are used to form a differential balanced LNA, and like that, its noise temperature contribution is accounted for by the S -parameters and noise properties of the LNAs (antenna is regarded lossless).

DE LERA ACEDO et al.: STUDY AND DESIGN OF A DIFFERENTIALLY-FED TAPERED SLOT ANTENNA ARRAY

Fig. 12. VSWR for a 45

E -plane scan for R = 50 to R = 10 k .

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Fig. 14. System sensitivity per unit cell for a 45 E -plane scan for three resistor values. The unit cell area ab = 30:6 10 m .

2

Fig. 13. VSWR for an E -plane scan, for R

= 1 k and 3 k .

The antenna impedance matching degrades by increasing the resistor values beyond a certain point as the effect of the lumped resistors decreases (Fig. 12), i.e., on transmit, a lower power is delivered to these resistors because of a low power transmission factor from the differential antenna port to the resistors. Likewise, on receive, only a small part of the thermal noise that has been generated by these resistors will be observed at the antenna port, due to this low transmission factor. In the following, we will account for the power absorption and associated noise temperature contribution through the radiation efficiency of the unit cell. For receiving (phased-array) antennas, it is common to define the system sensitivity as the ratio of the effective aperture area to the total system noise temperature

(5) The above sensitivity formula, (5), is in accordance with (22) in [30], except that is here the radiation efficiency of the unit cell. It is important to mention that the sensitivity is herein referred to the input of the LNA (output of the antenna). The effective area is defined through the antenna’s available output power which is the available incident power to the unit cell area reduced by the radiation efficiency. By definition, we assume perfect polarization match and conjugate match terminated array elements so that, in the absence of grating lobes, the effective area of a unit cell is given by the numerator of (5), where is

the direction of incidence measured from the normal to the array plane and ab is the physical size of the unit cell (cf. [31]). Apart from a power loss, the sensitivity also decreases due to an increase in the total system noise temperature caused by is a decrease in the radiation efficiency. The temperature the physical antenna temperature, which equals (for non-cooled antennas) the typical ambient temperature of 290 K, and is the noise temperature of the amplifier in the active scenario (with the source impedance of the LNA equal to the scan is the noise temperature contribution of impedance), and and are set to 0 K in the sky. For generality, both order to be able to quantify the noise increase due to only the absorption losses of the antenna. Fig. 14 demonstrates how the sensitivity per unit cell area increases when the value of the resistors increase. It is remarkable to observe that the sensitivity does not exhibit a resonance as e.g., observed in the VSWR port characteristics in Fig. 12, even not for 10 k resistor values. This is due to the fact that the radiation efficiency itself is free of resonances since it only quantifies the relative dissipated power loss which may not change much when the port impedance is at resonance. On the contrary, is expected to increase as a result of a noise mismatch to the anomalous differential source/antenna impedance, but this effect is dependent on the specific LNA design which is not discussed in this paper. In addition to the sensitivity curves, the noise temperature contributions for resistor values of 1 k , 3 k and 10 k are presented in Fig. 15 for completeness. To be consistent with the noise temperature specifications used for the SKA, as well as with -factor measurements for antennas, it is common to refer the system noise temperature in front of the antenna system (the sky), while the system sensitivity is a ratio and therefore independent on the chosen reference point. The corresponding noise temperature formula has been included in Fig. 15. We conclude that a compromise is necessary, i.e., increasing the resistor value leads to a lower noise temperature (Fig. 15) but degrades the impedance match (Fig. 12). A good compromise has been found for a resistor value of 3 k , for which both a low noise temperature (Fig. 15) and a resonant-free -plane scan impedance is realized which has a good matching level (Fig. 13). Fig. 15 demonstrates that the use of 3 k resistors leads to a minor increment of the total system noise temperature, i.e., not more than

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Fig. 15. System noise temperature for a 45 values.

E -plane scan for three resistor

10 K, which is reasonable considering the 7 K ( dB loss) that has been specified for the aperture array antennas in [13]. IV. DESIGN EXAMPLE With the knowledge acquired in the parametric analysis of Section II, and by applying the impedance anomaly elimination technique proposed in Section III, a dual-polarized array from 300 MHz to 1 GHz has been designed and optimized to cover a scan range up to 45 , thereby meeting the specifications of the SKA [26]. As we aim to minimize the cost, the element width is chosen as large as possible in order to minimize the number of amplifiers per square meter, while preventing in-band grating lobes from occurring. The final geometrical parameters of the structure are shown in Table III, and the final results for several -plane and -plane scans are plotted in Fig. 16. A resistor of 3 k has been used to obtain these results. No -plane impedance anomalies are present and the VSWR is lower than 2.5 over the entire operational frequency band, as well as for all scan angles up to 45 .

Fig. 16. Differential VSWR for various scans in the (a) plane.

H plane, and (b) E

TABLE III PARAMETERS OF THE OPTIMIZED ARRAY

V. CONCLUSION AND FUTURE WORK A dual-polarized array of differentially-excited elements (bunny-ear antennas) has been analyzed through a parametric study in order to showcase the effect of various antenna element geometries on the scan impedance characteristics of an infinite-by-infinite array. In addition, an -plane differential-mode scan impedance anomaly has been identified, which is caused by an even-mode that is supported by the differentially-fed antenna array in the presence of a third conductor, such as a ground plane. It has been shown that the impedance resonance can be suppressed by a (low loss) balancing-dissipation technique using a pair of resistors. The associated reduction in system sensitivity has been analyzed and shows that the increment of the corresponding antenna noise temperature is less than 10 K for 3 k resistors at room temperature (290 K). When passive resistors are used to prevent the differentialmode scan impedance anomaly from occurring, a compromise is necessary between: (i) the reduction in effective area as well as an increase in system noise temperature due to a reduced radiation efficiency, and; (ii) the realization of a resonant-free differential antenna impedance (source impedance) for achieving an optimal noise match between the differential LNAs and the

antenna elements. The latter depends on the specific LNA design which has not been discussed in this paper. A final optimized antenna design has been presented meeting the specifications of the SKA project while keeping the potential manufacturing cost as low as possible. The operational frequency band ranges from 300 MHz up to 1 GHz, the array is dual-polarized, and steerable up to 45 . In future, we will elaborate upon the relation between the common-mode and the differential-mode scan impedances, as well as the optimal matching conditions that apply to differential antenna and LNA designs. ACKNOWLEDGMENT The authors would like to thank to the Astronomical Centre of Yebes, National Astronomical Observatory of Spain, for their comments and suggestions. They also want to thank Prof. D. H. Schaubert for his comments, suggestions and time spent in private communications. REFERENCES [1] L. R. Lewis, M. Fasset, and J. Hunt, “A broad-band stripline array element,” in IEEE Int. Symp. Antennas Propag. Dig., 1974, pp. 335–337.

DE LERA ACEDO et al.: STUDY AND DESIGN OF A DIFFERENTIALLY-FED TAPERED SLOT ANTENNA ARRAY

[2] P. J. Gibson, “The Vivaldi aerial,” in Proc. 9th Eur. Microw. Conf., Brighton, U.K., 1979, pp. 101–105. [3] A. van Ardenne, H. Butcher, J. G. Bij de Vaate, A. J. Boonstra, J. D. Bregman, B. Woestenburg, K. van der Schaaf, P. N. Wilkinson, and M. A. Garrett, “The aperture array approach for the square kilometre array,” [Online]. Available: www.skatelescope.org May 2003, White paper [4] [Online]. Available: www.skatelescope.org [5] “The European Concept for the SKA,” [Online]. Available: www.skatelescope.org Jul. 2002, The Eur. SKA Consortium [6] H. Holter, T. H. Chio, and D. H. Schaubert, “Experimental results of 144-element dual polarized endfire tapered-slot phased arrays,” IEEE Trans. Antennas Propag., vol. 48, no. 11, pp. 1707–1718, Nov. 2000. [7] R. Maaskant, M. Popova, and R. van de Brink, “Towards the design of a low cost wideband demonstrator tile for the SKA,” presented at the Eur. Conf. on Antennas and Propag., Nice, France, 2006. [8] T.-H. Chio and D. H. Schaubert, “Parameter study and design of wideband widescan dual-polarized tapered slot antenna arrays,” IEEE Trans. Antennas Propag., vol. 48, no. 6, pp. 879–886, Jun. 2000. [9] J. J. Lee and S. Livingston, “Wideband bunny-ear radiating element,” in Proc. Antennas Propag. Society Int. Symp., 28 Jun.–2 Jul. 1993, vol. 3, pp. 1604–1607. [10] S. G. Hay, J. D. O’Sullivan, J. S. Kot, C. Granet, A. Grancea, A. R. Forsyth, and D. H. Hayman, “Focal plane array development for ASKAP (Australian SKA pathfinder),” in Proc. 2nd Eur. Conf. on Antennas and Propag., Nov. 11–16, 2007, pp. 1–5. [11] S. G. Hay and J. D. O’Sullivan, “Analysis of common-mode effects in a dual polarized planar connected-array antenna,” Radio Science, RS6S04 2008. [12] R. Maaskant, “Antenne-inrichting, antenne-array, samenstel voor het assembleren van een antenne-array en een elektronische inrichting omvattende een antenne,” Dutch patent no. NL1034102. [13] J. G. Bij De Vaate, L. Bakker, E. E. M. Woestenburg, R. H. Witvers, G. W. Kant, and W. Van Capellen, “Low cost low noise phased-array feeding systems for SKA pathfinders,” presented at the ANTEM2009 Conf., Banff, Canada, Feb. 2009. [14] M. Arts, R. Maaskant, E. de Lerea Acedo, and J. G. bij de Vaate, “Broadband differentially-fed tapered slot antenna array for radio astronomy applications,” presented at the Eucap 3rd Eur. Conf. on Antennas and Propag., Berlin, Germany, Mar. 23–27, 2009. [15] D. M. Pozar and D. Schaubert, “Scan blindness in infinite phased arrays of printed dipoles,” IEEE Trans. Antennas Propag., vol. 32, no. 6, pp. 602–610, Jun. 1984. [16] J. P. R. Bayard, M. E. Cooley, and D. Schaubert, “Analysis of infinite array of printed dipoles on dielectric sheets perpendicular to a ground plane,” IEEE Trans. Antennas Propag., vol. 39, no. 12, pp. 1722–1732, Dec. 1991. [17] [Online]. Available: www.ansoft.com [18] R. C. Hansen, Phased Arrays Antennas. New York: Wiley, 2001. [19] D. Schaubert, “A class of E-plane scan blindnesses in single-polarized arrays of tapered-slot antennas with a ground plane,” IEEE Trans. Antennas Propag., vol. 44, no. 7, pp. 954–959, Jul. 1996. [20] D. H. Schaubert and A. O. Boryssenko, “Subarrays of Vivaldi antennas for very large apertures,” in Proc. 34th Eur. Microw. Conf., Amsterdam, 2004, pp. 1533–1536. [21] A. Ellgardt, “A scan blindness model for single-polarized tapered-slot arrays in triangular grids,” IEEE Trans. Antennas Propag., vol. 56, no. 9, pp. 2937–2942, Sep. 2008. [22] J. Wunsch and D. Schaubert, “Full and partial crosswalls between unit cells of endfire slotline arrays,” IEEE Trans. Antennas Propag., vol. 48, no. 6, pp. 981–986, Jun. 2000. [23] J.-P. R. Bayard, D. H. Schaubert, and M. E. Cooley, “E-plane scan performance of infinite arrays of dipoles on protruding dielectric substrates: Coplanar feed line and E-plane metallic wall effects,” IEEE Trans. Antennas Propag., vol. 41, no. 6, pp. 837–841, Jun. 1993. [24] E. L. Pelton and B. A. Munk, “Scattering from periodic arrays of crossed dipoles,” IEEE Trans. Antennas Propag., vol. AP-27, pp. 323–330, 1979. [25] D. Schaubert, S. Kasturi, M. W. Elsallal, and W. Van Cappellen, “Wide bandwidth Vivaldi antenna arrays—Some recent developments,” presented at the EuCAP 2006, Nice, France, Nov. 6-10, 2006.

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[26] R. T. Schilizzi et al., “Preliminary specifications for the square kilometre array,” SKA Memo 100 Dec. 2007 [Online]. Available: http:// www.skatelescope.org/PDF/memos/100_Memo_Schilizzi.pdf [27] M. W. Elsallal and D. H. Schaubert, “Electronically scanned arrays of dual-polarized, doubly-mirrored balanced antipodal Vivaldi antennas (DmBAVA) based on modular elements,” in Proc. IEEE Antennas and Propag. Society Int. Symp., Jul. 9–14, 2006, pp. 887–890. [28] C. Craeye, “Efficient simulation off finite wideband arrays—Reconciling finite and infinite-array approaches,” presented at the Design of wideband receiving array systems SKADS MCCT technical workshop, Dwingeloo, The Netherlands, Nov. 26–30, 2007, Course Held as ASTRON Under the SKADS Framework. [29] E. García, E. De Lera, D. Segovia, and V. González, “Elimination of scan impedance anomalies in phased arrays,” in Proc. IEEE Antennas and Propag. Society Int. Symp., San Diego, CA, Jul. 5–11, 2008, pp. 1–4. [30] M. 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. [31] C. Craeye and M. Arts, “On the receiving cross section of an antenna in infinite linear and planar arrays,” Radio Sci., vol. 39, no. RS2010, 2004.

Eloy de Lera Acedo (S’05) was born in Spain in 1982. He received the Electrical Engineering degree (best graduate of the year distinction) from Carlos III University, Madrid, Spain, in 2005, where he is currently working toward the Ph.D. degree. From 2006 to 2007, he worked at the Yebes Astronomical Center and stayed at the Netherlands Institute for Radio Astronomy (ASTRON) working in the design of antenna arrays for radio astronomy. Currently, he has a position in the Cavendish Laboratory, University of Cambridge, Cambridge, U.K. His main research interests include ultrawideband antenna arrays as well as radio astronomy receivers.

Luis Enrique García-Muñoz (M’99) received the Telecommunications Engineer degree and the Ph.D. degree in telecommunication from the Universidad Politécnica de Madrid, Madrid, Spain, in 1999 and 2003, respectively. He is currently an Associate Professor with the Department of Signal Theory and Communications, Carlos III University, Madrid, Spain. His main research interests include radio-astronomy receivers, radiotelescopes, microstrip patch antennas and arrays, as well as periodic structures applied to electromagnetics.

Vicente González-Posadas was born in Madrid, Spain, in 1968. He received the Ing. Técnico degree in radio-communication engineering from the Polytechnic University of Madrid (UPM), Madrid, Spain, in 1992, the M.S. degree in physics from the Universidad Nacional de Educación a Distancia (UNED), Madrid, Spain, in 1995, and the Ph.D. degree in telecommunication engineering from Carlos III University, Madrid, Spain, in 2001. He is currently an Assistant Professor with the Technical Telecommunication School, Departamento de Ingeniería Audiovisual y Comunicaiones, UPM. He has authored or coauthored over 60 technical conference, letter, and journal papers. His research interest are related to active antennas, microstrip antennas, CRLH lines and metamaterials, and microwave technology.

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José Luis Vázquez-Roy was born in Madrid, Spain, in 1969. He received the Ingeniero de Telecomunicación and the Ph.D. degrees from the Universidad Politécnica de Madrid, Madrid, Spain, in 1993 and 1999, respectively. In 1999, he joined the “Teoría de la Señal y Comunicaciones” Department, Universidad Carlos III de Madrid, where he is currently an Associate Professor. His research activities and interests include the development and characterization of planar antennas and circuits, the analysis of UWB antennas and the time domain computational electromagnetics.

Rob Maaskant received the M.Sc. degree (cum laude) in electrical engineering from the Eindhoven University of Technology, The Netherlands, in 2003. Since then, he has been an Antenna Research Engineer at the Netherlands Institute for Radio Astronomy (ASTRON) where his research is carried out in the framework of the Square Kilometre Array (SKA) Radio Telescope Project. He is currently working toward the Ph.D. degree in the field of numerically efficient integral-equation techniques for large finite antenna arrays. Besides, his research

interest includes the characterization and design of antenna array receiving systems.

Daniel Segovia-Vargas (M’98) was born in Madrid, Spain, in 1968. He received the Telecommunication Engineering and Ph.D. degrees from the Polytechnic University of Madrid, Madrid, Spain, in 1993 and 1998, respectively. From 1993 to 1998, he was an Assistant Professor with Valladolid University. Since 1998, he has been an Associate Professor with Carlos III University, Madrid, Spain, where he is in charge of the microwaves and antenna courses. He has authored or coauthored over 60 technical conference, letters, and journal papers. His research interests are printed antennas and active radiators and arrays and smart antennas, left-handed (LH) metamaterials, and passive circuits. He has also been member of the European Projects Cost260, Cost284, and COST IC0603.

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Metallic Wire Array as Low-Effective Index of Refraction Medium for Directive Antenna Application Rongguo Zhou, Associate Member, IEEE, Hualiang Zhang, Member, IEEE, and Hao Xin, Senior Member, IEEE

Abstract—Two-dimensional (2-D) metallic wire arrays are studied as effective media with an index of refraction less than unity ( e 1). The effective medium parameters (permittivity e , permeability e and e ) of a wire array are extracted from the finite-element simulated scattering parameters and verified through a 2-D electromagnetic band gap (EBG) structure case study. A simple design methodology for directive monopole antennas is introduced by embedding a monopole within a metallic 1 at the antenna operating frequencies. wire array with e The narrow beam effect of the monopole antenna is demonstrated in both simulation and experiment at X-band (8–12 GHz). Measured antenna properties including reflection coefficient and radiation patterns are in good agreement with simulation results. Parametric studies of the antenna system are performed. The physical principles and interpretations of the directive monopole antenna embedded in the wire array medium are also discussed. Index Terms—Electromagnetic band gap, index of refraction, metallic wire array, metamaterial, monopole antenna.

I. INTRODUCTION

N recent years, metamaterials have attracted much attention because of their special electromagnetic properties and potential applications in microwave, infrared and optical frequencies. Examples include electromagnetic/photonic band gap (EBG/PBG) structures [1], [2] and artificial media with negative permittivity [3], negative permeability [4], and both negative permittivity and permeability [5]. In addition, an or near-zero interesting class of metamaterials with low index of refraction has been investigated by several groups because of their useful applications such as compact resonators [6], zero-phase delay lines [7], wave front transformers [8], transparent coating [9], sub-wavelength tunneling [10], highly directive beams [11]–[15] and relaxed dimension tolerance in nano-scale fabrication [16].

I

Manuscript received April 17, 2008; revised April 15, 2009. First published November 10, 2009; current version published January 04, 2010. R. Zhou and H. Xin are with the Department of Electrical and Computer Engineering and the Department of Physics, University of Arizona, Tuscon, AZ 85721 USA (e-mail: [email protected]; [email protected]). H. Zhang was with the Department of Electrical and Computer Engineering, University of Arizona, Tuscon, AZ 85721 USA. He is now with the Electrical Engineering Department, University of North Texas, Denton, TX 76203-0470 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.2009.2036282

Among the above mentioned applications, several studies concerning directive antenna design using metallic wires or other EBG structures have been reported. It is shown that the radiation of a source can be confined within a small angular region by embedding the source within two combined EBG structures (one in reflection and the other in transmission) [17]–[19]. Another technique for antenna gain enhancement is to use a superstrate composed of EBG structures [20]–[22]. In addition, EBG based Fabry-Perot cavities can be used to realize directive antennas [23]–[25]. In [12]–[15], metamaterials with low refractive index were also utilized for realizing narrow-beam antennas. Metallic wire arrays are probably the simplest structures to effective medium and can be analyzed using realize an a simple plasma theory with reduced electron density [3]. In [12], [13], [15], the structures used to achieve directive radiation were based on a ground-plane backed metamaterial slab composed of rows or meshes of metallic wires with finite length, for which the effective medium parameters could not be directly estimated by the simple plasma theory. In addition, most of the previous work did not directly apply the effective medium parameters in the antenna designs. In this work, two-dimensional (2-D) metallic wire arrays terminated by two ground planes are studied as media with low effective index of refraction . A simple methodology is utilized to design directive antennas based on the effective medium parameters of a wire array that are extracted from finite-element simulation results. First, at the geometry of the wire array is optimized to achieve low desired frequencies. The extraction results confirm the low properties of the wire array at frequencies just above the theory predicted plasma frequency. A 2-D EBG structure (a square lattice made of dielectric rods) embedded in the wire medium is then investigated. It is found that the first band gap of the 2-D EBG structure shifts to a higher frequency as expected when the is replaced by the hosting free space region wire array medium. Moreover, simulated transmission response of the actual composite of the EBG and metallic wires agrees with that of the EBG embedded in a background assigned to have the extracted effective medium parameters, confirming the validity of treating the wire array as an effective medium. Directive monopole antennas are then realized by embedding a at the monopole within a metallic wire medium with antenna operating frequency. The monopole length, wire array size and height effects on the antenna properties are studied. A prototype antenna operates within the X-band is designed, fabricated and characterized. The measured antenna properties are

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Fig. 1. Top view of a 2-D square lattice of wires with radius r and periodicity a. It can also be thought of as two independent square lattices (solid and empty circles) with a periodicity of 2a, embedded within each other.

Fig. 2. Calculated effective permittivity of a 2-D wire array (a = 4 mm, r = 1 m) using plasma theory, treating the array as either a single square lattice

in good agreement with simulation results, confirming the expected narrow beam radiation and the design methodology. The narrow beam effect achieved by the monopole/wire array antenna system is examined in the contexts of Snell’s law and effective aperture size in order to gain more physical insight. This paper is outlined as the following. In Section II, metallic wire arrays as low effective index of refraction media are reviewed. The self-consistency of the plasma theory is verified and an effective medium parameter-extraction algorithm is discussed. In Section III, an example of a 2-D EBG structure hosted within a metallic wire array as an effective medium with is presented. Section IV describes the detailed design procedure, parametric study, experimental results and physical interpretations of the proposed directive antenna made of a monopole . Finally, in embedded within a wire medium with Section V, the conclusions are drawn.

permittivity can be calculated by assuming that the second lattice (represented by solid circles) is embedded in a medium with of the first lattice (represented by the effective permittivity empty circles). The overall effective permittivity is then

p

II. METALLIC WIRE ARRAY AS EFFECTIVE MEDIUM The metamaterial structure considered in this paper is 2-D square lattice of metallic wires, as shown in Fig. 1. The effective of this kind of wire arrays can be calculated permittivity using plasma theory with reduced electron density [3], as shown in (1)

(1) where is the wire array periodicity, is the wire radius, is the angular frequency, is the speed of light in free space and is the plasma frequency, at which the effective permittivity , and thus the effective index of refraction , is zero. An important assumption of (1) is that the wires are very thin so that the plasma frequency corresponds to a free space wavelength much greater than the lattice spacing and the Bragg diffraction effect can be ignored [3]. A useful metamaterial should keep its effective medium properties when used as a constituent of a composite structure. As shown in Fig. 1, the metallic wire array itself can alternatively be decomposed into two identical but independent square lattices with lattice con, embedded within each other. Thus the effective stants of

(circles) or two square lattices (squares) embedded within each other.

(2) Equations (1) and (2) are very close to each other under the thin . As wire condition, in which case an explicit example, Fig. 2 plots the calculated effective perand mittivity of a wire array with using (1) and (2). The almost exact agreement of the two methods indicates that the plasma theory is self-consistent and a metallic wire array may be used as an effective medium with special properties such as very small or near-zero permittivity. Furthermore, the near-zero permittivity of a wire array medium can be realized in any microwave frequency by adjusting the wire array spacing and the wire radius according to (1). The above analysis is simple and effective in terms of predicting the plasma frequency. However, it neglects magnetic response and loss of the wires, which may be important in practical applications. In order to extract its effective medium parameters accurately, the exact electromagnetic responses of a wire array medium are simulated using finite-element electromagnetic solvers (i.e., Ansoft’s HFSS). The finite-element model consists of five unit cells along the propagation direction ( -axis), as shown in Fig. 3. The normal-incident plane wave is polarized along the wires ( -axis). The wire array is infinite in both and directions by using appropriate perfect electric and magnetic conducting boundary conditions. The Nicolson-Ross-Weir (NRW) approach [26]–[29] is then , implemented to extract the effective medium parameters ( and ) from the simulated scattering parameters (S-parameters). As an effective homogeneous medium, the extracted effective parameters of the wire array slab should not depend on the number of unit cells, as will be shown in Section IV.

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Fig. 3. Top view of the HFSS model of a wire array structure. A normal-incident plane wave propagates in the x direction and the array is infinite in both z and y directions.

Fig. 5. Calculated real part of n of the wire array with a : for different m values.

0 25 mm

= 5 mm and r =

Fig. 4. A two-dimensional dielectric rod EBG structure embedded in a low index of refraction wire array (top view).

III. A 2-D EBG IN A LOW INDEX WIRE ARRAY HOST As it is predicted in Section II, the effective index of refraction of a 2-D metallic wire array is close to zero around its plasma frequency, which can be realized at any microwave frequency by adjusting the wire array spacing and the wire radius. In this section, a 2-D square lattice of dielectric rods embedded within a background of such wire array is investigated to validate the description of the wire array as an effectively small index of refraction medium. The top view of the dielectric rod/wire array structure is shown in Fig. 4. In this case, the host medium of the 2-D EBG is not the typical free space but a low refractive index medium. The 2-D metallic wire array used here has a periodicity and a wire radius . Following the NRW approach [26]–[29], the finite-element simulated S-parameters of the wire array model as illustrated in Fig. 3 are utilized to extract the effective parameters of the wire array. The resulting real part of the refractive index for different values ( ,where is the free space wave number, is the transmission term and is the thickness of the sample under study along the wave propagation direction) from 1 GHz to 30 GHz is plotted in Fig. 5 (beyond 30 GHz, the Bragg effect sets in and the extracted effective medium parameters are not physically meaningful). Since the length of the wire medium slab in the wave propcontaining five unit cells is agation direction, which is much smaller than the wavelength at the lowest frequency 1 GHz (300 mm), it is reasonable to asat 1 GHz. At higher frequencies, the values sume that are selected to ensure the continuity of as a function of frequency. After the values are determined, one can easily ob-

Fig. 6. Extracted real components of " and  compared with the " calculated from (1) (a ,r : ).

= 5 mm = 0 25 mm

tain the effective and of the wire array, as plotted (the real using (1). components) in Fig. 6 along with the calculated As it is shown in Fig. 6, the calculated and extracted have similar frequency trends, both going from negative values at low frequencies to positive values at high frequencies. The extracted values from the finite-element simulation have more complicated features above 30 GHz (not plotted) and do not approach 1 as the simple plasma theory predicts. The breakdown of the simple plasma picture is expected, since above 30 GHz, the wire spacing becomes greater than the half free space wavelength and the Bragg diffraction effect sets in. In addition, the extracted are not exactly equal to 1 due to the finite radius of is slightly greater than 1 below 30 GHz the wires. Instead, and smaller than 1 (can even be negative) at higher frequencies, again due to the Bragg diffraction. Moreover, for certain frequency range (i.e., 18–30 GHz), the extracted has small positive values (ranges from 0.005 to 0.34) such that is less than 1. Embedded in the wire array discussed above, the dielectric rods shown in Fig. 4 have a dielectric constant of 2.53, a periodicity of 10 mm and a radius of 2 mm. For an incident plane wave polarized along the length of the rods (see Fig. 4), frequency band gaps exist, within which no transmission is allowed. The first band gap center frequencies and bandwidths of this dielectric rod array are calculated using the MIT Photonic-Bands

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TABLE I THE 2-D DIELECTRIC RODS BAND GAP FREQUENCY AND BANDWIDTH FOR DIFFERENT BACKGROUNDS

Fig. 8. Geometrical illustration of the beam narrowing effect for a source embedded in an n < 1 medium.

IV. DIRECTIVE MONOPOLE ANTENNA EMBEDDED IN A WIRE ARRAY MEDIUM

Fig. 7. Simulated transmissions of the actual composite of the dielectric rods and the metallic wire array (squares) and the dielectric rods embedded within a uniform slab with the extracted effective medium parameters of the wire array (circles).

(MPB) package [30] and listed in Table I for backgrounds with different permittivity . It can be seen that with the background (free space) to , both permittivity decreases from the first band gap center frequency and its bandwidth increase significantly. The physical reason for the increased band gap frequency is that for a smaller background permittivity, the effective wavelength becomes longer and the electrical length of the dielectric rod spacing becomes smaller. Therefore, when the dielectric rods are embedded within a as shown in Fig. 4, it is expected that wire array with the first band gap would occur at a higher frequency although the exact frequency and bandwidth are hard to predict due to the frequency dependency of the wire array properties. To verify the effective medium parameters of the wire array, the results of two HFSS simulations are compared in Fig. 7: the curve of squares is the transmission response of the actual composite of dielectric rods and metallic wires while the curve of circles is that of the dielectric rods embedded in a uniform background assigned to have the extracted frequency dependent complex permittivity and permeability of the wire array. Two important observations can be made from Fig. 7. First, the two simulation results agree quite well for frequencies below 30 GHz, confirming the properness of treating the wire array as a homogeneous effective medium. Second, both of the transmission responses show the first band gap at around 20 GHz with a bandwidth around 6 GHz, which agrees with our expectation, indicating a smaller than 1 index of refraction of the wire medium. In addition, the two simulated results are quite different for frequencies above 30 GHz because the Bragg diffraction effect sets in and the effective medium approach is no longer valid.

A very interesting application of low-index of refraction media is highly directive antennas. In this section, a simple design methodology for narrow beam antennas is introduced by embedding a monopole antenna within a wire array medium . The narrow beam effect due to the low refractive with index of the wire medium is demonstrated in both simulation and experiment. A. Principles of Antenna Operation When a source is placed in a medium with a refractive index , as shown in Fig. 8, at the boundary between this medium and free space, assuming simple geometric optics ap, ) requires plies, the Snell’s law ( the refractive angle to be less than , indicating that the radiated beam will be refracted towards the normal direction of the interface and a beam narrowing effect can be achieved. Therefore, a directive antenna should be obtained by embedding an omni-directional antenna within a metallic wire medium with at the antenna operating frequencies. The top view of a monopole/wire array system is shown in Fig. 9, where the monopole antenna is surrounded by a finite number of wires (11 columns and 11 rows in this case). Both the wires and the monopole are along the direction. By placing two perfect conducting ground planes at the ends of the wire array, it can be considered infinitely long in the -direction. To minimize the near field interaction between the monopole and the surrounding wires, the center element of the wire array is removed and replaced by the monopole. B. A Directive Monopole/Wire Array Antenna Design Within X-Band A monopole/wire array antenna as shown in Fig. 9 is designed to achieve narrow-beam radiation within X-band (8–12 GHz). The 11-column 11-row copper wire array has a periodicity and a wire radius . The wires have a length of 50 mm and are terminated at the ends by two copper ground planes. Following the procedures described in Section II,

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to the wire array’s less than unity index of refraction. The maxplane occurs at 9.5 GHz imum gain of the antenna in the with a magnitude of 10.4 dB [see Fig. 11(b)]. The 10-dB beam width (with four-fold symmetry as expected) of the antenna at 9.5 GHz is 37 . At this frequency, the extracted refractive index of the wire medium is 0.42. Thus, Snell’s law predicts a beam width of , which is consistent with the simulated are the limits of ). In addition, 10-dB beam width ( the simulated radiation efficiency is higher than 95%. Fig. 9. Top view of a monopole embedded within a wire array that is infinite in the z-direction.

Fig. 10. The extracted n and n of a small refractive index wire array medium and r : ). for X-band applications (a

= 9 mm

= 0 25 mm

the extracted and as a function of frequency at X-band from finite-element simulations of a 5-unit cell slab are shown in Fig. 10. The extraction results show that the plasma frequency of the wire array is around 8.7 GHz, which is not too far from the calculated plasma frequency (7.02 GHz) using (1). The effective of this wire array is less than 0.75 for the entire X-band. The effective is close to zero from 8.7–12 GHz, indicating that this wire array can be thought of as a lossless medium in that frequency range. Similar wire arrays with different number of unit cells (from 2 to 6) are also simulated and their effective material parameters are extracted, yielding the same values as the 5-unit cell case. This confirms that there is no interaction between the unit cells and the wire array may be treated effectively as a standalone uniform slab of material. A coax-fed monopole with a length of 17.4 mm is placed at the center of the wire array as indicated in Fig. 9. If the wire array can be treated as an ideal low-index medium, the resonance frequency of the monopole would satisfy , where c is the speed of light, and is the extracted frequency dependent index of refraction as shown in Fig. 10, in which case it would be 9.5 GHz [see dashed line in Fig. 11(a)]. However, the simulated antenna reflection coefficient shows multiple resonances within X-band [solid line in Fig. 11(a)] which are due to the interactions between the monopole and the finite wire array. The simulated radiation patterns confirm the expected narrow beam effect due

C. Parametric Study of the Monopole Antenna System There are several important parameters that may influence the behavior of the monopole/wire array system such as the monopole length, the wire array size and the wire array height (distance between the top and bottom metal plates). Parametric studies are performed to provide insights on how these parameters affect the radiation characteristics of the antenna system. 1) Monopole Length: The resonance frequency of the monopole antenna system is expected to be if the wire array surrounding can be treated as a standalone homogeneous effective medium, in which case it should vary slower as a function of the monopole length than a linear increases with frequency. However, dependence because , there are multiple as shown in Fig. 11(a), for resonances instead of just the predicted 9.5 GHz. To evaluate the length effect, monopoles with lengths of 7.5 mm, 10.0 mm, 12.5 mm, 15.0 mm and 17.4 mm embedded in the same 11-row wire array as described in the previous 11-column section are studied. The simulated reflection coefficients are shown in Fig. 12 (for clarity, only 3 curves are plotted). It can be seen that there are multiple resonance frequencies for all of the monopole lengths due to interactions between the monopole and the finite wire array, as mentioned previously. In addition, the resonance frequencies are quite similar for different monopole lengths from 7.5 to 17.4 mm although the detailed levels of reflection coefficient are not exactly the same. Furthermore, the radiation patterns across the entire X-band are almost identical for all the monopoles with different lengths, confirming that the wire array medium determines the radiation plane always occurs at pattern. The maximum gain in the 9.5 GHz with a magnitude of about 10.4 dB. 2) Wire Array Size: As it is mentioned above, the radiation patterns are the same for different monopole lengths when embedded within a fixed wire array size. However, for different wire array sizes, the antenna radiation patterns are different due to the finite sizes of the wire array and the interactions between the wires and the monopole. The simulation results of the same monopole as described in Section IV-B embedded within 11 11, 9 9 and 7 7 square wire arrays show that the maximum gain of the monopole in the plane increases when the wire array size increases (from 5–10.4 dB), which is related to the antenna aperture size (see discussion in Section IV-E). In addition, the resonance frequencies of the antenna are close to each other for different sizes of wire arrays. 3) Wire Array Height: The wires are terminated by two ground planes to emulate an infinitely long wire medium. The impact of the wire array height (or the separation of the ground

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Fig. 11. (a) Simulated reflection coefficient of a 17.4-mm monopole in the actual wire array (solid line) and in a uniform medium with extracted effective permittivity and permeability representing the wire array (dashed line). (b) Simulated gain (in dB) of the monopole in the wire array (solid line) comparing with the and r : ; and the 11 11 wires are terminated by two case without wire array (dashed line). The wire array has the following dimensions: a conducting ground planes separated by 50 mm.

= 9 mm

Fig. 12. The simulated reflection coefficient of the monopole/wire array system for various monopole lengths.

= 0 25 mm

2

Fig. 13. A photo of the fabricated antenna prototype. The monopole and wire lengths are 17.4 mm and 50 mm, respectively.

planes) is also studied. Simulations show that for a 10-mm 11 wire array with a long monopole embedded in the 11 height of 30, 40, and 50 mm, the resonance frequencies shift up slightly (less than 1 GHz) with the decrease of the wire array height. The maximum gain of the antenna also decreases slightly (about 2 dB) as the wire array height decreases. From these observations, it can be concluded that even though the upper ground plane interacts with the monopole, the interaction is quite small when the distance between the ground planes is significantly larger than the monopole length. D. Experimental Verification To verify the theoretical prediction of the narrow beam effect, a prototype of the proposed antenna system operating within the X-band is designed, fabricated and measured. The schematic top view of the antenna is the same as that shown in Fig. 9 (a monopole embedded at the center of an 11 11 wire array) and a photo of the fabricated prototype is shown in Fig. 13. The monopole has a length of 17.4 mm and is fed by a 50 coaxial

Fig. 14. Comparison of the measured (circles) and simulated (solid line) reof the 17.4 mm monopole antenna embedded in the flection coefficient wire array.

(S )

connector. Two copper plates of size 110 mm 110 mm are used as ground planes. Copper wires are then soldered onto the ground planes. The separation between the ground planes is 50

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Fig. 15. Measured x-y plane radiation patterns (normalized to 0 dB) of the antenna system (solid line) compared with simulated radiation patterns of the monopole in the actual wire array (dashed line) and in effective medium (dashed-dotted line) at: (a) 8.5 GHz, (b) 9.5 GHz, (c) 10.3 GHz and (d) 11.3 GHz.

mm. This antenna configuration should lead to directive radiaplane instead of the omnidirectional radiation tion in the pattern of a regular monopole. The prototype antenna in Fig. 13 is characterized using an Agilent E8361A vector network analyzer in an anechoic chamber. The measured and simulated reflection coefficient agree quantitatively well, both showing multiple resonances, as plotted in Fig. 14. The measured radiation patterns of the antenna system at various resonance frequencies (8.5, 9.5, 10.3, and 11.3 GHz) are plotted in Fig. 15, together with HFSS simulated radiation patterns of the actual monopole/wire array system and the monopole embedded in an effective medium assigned to have the extracted permittivity and permeability of the wire array. It is very interesting that the two simulation results agree excellently (except some small deviations at 11.3 GHz), which verifies the legitimacy of treating the wire array as an effective medium in this case. The agreement between measurements and simulations is also quite well as shown in Fig. 15. The narrow beam effect of the antenna system is apparent in both simulations and measurements at all the frequencies. The simulated antenna gains are 7.8, 10.4, 7.4, and 5 dB at 8.5, 9.5,

10.3, and 11.3 GHz, respectively. The simulated and measured 10-dB beam width of the monopole at 9.5 GHz is 37 and 34 , respectively, being consistent with Snell’s law estimation as discussed before. The four main beams in Fig. 15 are due to the symmetry of the antenna system. If a single main beam is desired, an effective medium with anisotropic property, for instance, wire array with plane, can different periodicities in the four directions in be applied. Another way is to add metallic reflectors at the three boundaries of the wire arrays as reported in [24]. Simulation and measurement results show that with three metallic reflectors installed on three sides of the antenna system shown in Fig. 13, a single main beam can be achieved with a gain of about 6 dB higher compared to the case without the metallic reflectors, as expected. plane is significantly shielded by The radiation in the the two ground planes, as shown in Fig. 16. The ideal -shaped radiation is truncated near the -direction. However, if the plane radiation is desired, the upper ground plane may be elimplane will be ininated, in which case the radiation in the plane creased significantly while the narrow beams in the still remains although with slightly lower gain (plot not shown).

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V. CONCLUSION

Fig. 16. The simulated and measured radiation patterns (normalized to 0 dB) of the antenna in x-z plane at 9.5 GHz.

E. Discussions

As it is pointed out in Section IV-A, the narrow beam effect of the antenna system can be explained using Snell’s law by . treating the wire array as an effective medium with Our studies on the band gaps of the 2-D dielectric rods electromagnetic crystal and the radiation patterns of the monopole embedded in the wire array show that the effective medium approach is quite accurate and provides a very useful design methodology, at least when the unit cell dimension is much smaller than the operating wavelength. However, it is important to point out an approximation used here that the longitudinal component of the incident waves is omitted (reasonable approximation for the monopole antenna configuration in this work) so that the spatial dispersion of the wire medium is not accounted for [31]. Another important thing worth mentioning is that the Snell’s law estimation of the narrow beam effect is also an approximation and care should be taken when applying it. For example, and beone may expect that the beam width decreases with is zero, which is certainly not the case comes to zero when from Fig. 15. The reason is that Snell’s law assumes plane wave incidence (or far field) while the wire array has a finite size (110 mm 110 mm) which becomes electrically smaller for smaller so that the far field assumption is no longer valid. Another physical explanation of the directive radiation is to treat the antenna system as four identical aperture antennas with a size of 110 mm 50 mm. Thus, the first null beam width of the aperture antenna is the minimum when the E-field over the whole aper, ture is constant, and can be calculated as: is the aperture where is the wavelength and length. As an example, the calculated minimum first null beam width at 9.5 GHz is 33.4 , which is very close to the Snell’s law prediction and the simulated and measured 10-dB beam width. Therefore, it can be concluded that uniform excitation of the aperture is achieved by this monopole/wire array system.

In this paper, 2-D metallic wire arrays as low-index effective media are investigated for directive antenna application. The self-consistency of the plasma model describing the wire array is demonstrated. Effective medium parameters ( , and ) of 2-D wire arrays are extracted based on finite-element simulation and the results are consistent with the plasma theory prediction. An EBG structure made of 2-D dielectric rods embedded in a wire array is studied to validate the low-index property of the wire array. A simple design methodology for directive monopole antenna is then introduced by embedding at the antenna a monopole within a wire array with working frequencies. A monopole/wire array antenna (working within X-band) is designed, fabricated and characterized. Parametric studies of this antenna system show that the antenna resonance frequencies and radiation patterns are not sensitive to the monopole length , and the antenna gain increases when the wire array size increases. Experimental results of the fabricated antenna prototype agree well with simulation results, confirming the narrow beam effect of the antenna system. The design procedure of this antenna system is flexible and can be applied at all microwave frequencies by adjusting the wire array spacing and wire radius. REFERENCES [1] H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B, vol. 58, pp. R10096–R10099, Oct. 1998. [2] J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Modeling the Flow of Light. Princeton, NJ: Princeton Univ. Press, 1995. [3] J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett., vol. 76, pp. 4773–4776, Jun. 1996. [4] J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech., vol. 47, pp. 2075–2084, Nov. 1999. [5] D. R. Smith et al., “Composite medium with simultaneously negative permittivity and permeability,” Phys. Rev. Lett., vol. 84, pp. 4184–4187, May 2000. [6] A. Alù, F. Bilotti, N. Engheta, and L. Vegni, “Compact leaky wave components using metamaterial bilayers,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2005, pp. 1733–1736. [7] R. W. Ziolkowski, “Propagation in and scattering from a matched metamaterial having a zero index of refraction,” Phys. Rev. E, vol. 70, pp. 046608(1)–046608(12), Oct. 2004. [8] A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern,” Phys. Rev. B, vol. 75, pp. 155410(1)–155410(13), Apr. 2007. [9] A. Alù and N. Engheta1, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E, vol. 72, pp. 016623(1)–016623(9), Jul. 2005. [10] M. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using "-near-zero materials,” Phys. Rev. Lett., vol. 97, pp. 157403(1)–157403(4), Oct. 2006. [11] K. C. Gupta, “Narrow-beam antennas using an artificial dielectric medium with permittivity less than unity,” Electron. Lett., vol. 7, pp. 16–18, Jan. 1971. [12] I. J. Bahl and K. C. Gupta, “A leaky-wave antenna using an artificial dielectric medium,” IEEE Trans. Antennas Propag., vol. 22, pp. 119–122, Jan. 1974. [13] S. Enoch, G. Tayeb, P. Sabouroux, N. Guérin, and P. Vincent, “A metamaterial for directive emission,” Phys. Rev. Lett., vol. 89, pp. 213902(1)–213902(4), Nov. 2002. [14] S. Enoch, G. Tayeb, and B. Gralak, “The richness of the dispersion relation of electromagnetic bandgap materials,” IEEE Trans. Antennas Propag., vol. 51, pp. 2659–2666, Oct. 2003.

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[15] G. Lovat, P. Burghignoli, F. Capolino, D. R. Jackson, and D. R. Wilton, “Analysis of directive radiation from a line source in a metamaterial slab with low permittivity,” IEEE Trans. Antennas Propag., vol. 54, pp. 1017–1030, Mar. 2006. [16] H. Xin and R. Zhou, “Low-effective index of refraction medium using metallic wire array,” in IEEE AP-S Int. Symp. Dig., Jun. 2007, pp. 2530–2533. [17] G. Poilasne, J. Lenormand, P. Pouliguen, K. Mahdjoubi, C. Terret, and P. Gelin, “Theoretical study of interactions between antennas and metallic photonic bandgap materials,” Microw. Opt. Technol. Lett., vol. 15, pp. 384–389, Aug. 1997. [18] G. Poilasne, P. Pouliguen, K. Mahdjoubi, C. Terret, P. Gelin, and L. Desclos, “Influence of metallic photonic bandgap (MPBG) materials interface on dipole radiation characteristics,” Microw. Opt. Technol. Lett., vol. 18, pp. 407–410, Aug. 1998. [19] S. Enoch, B. Gralak, and G. Tayeb, “Enhanced emission with angular confinement from photonic crystals,” App. Phys. Lett., vol. 81, pp. 1588–1590, Aug. 2002. [20] M. Thévenot, C. Cheype, A. Reineix, and B. Jecko, “Directive photonic-bandgap antennas,” IEEE Trans. Microw. Theory Tech., vol. 47, pp. 2115–2122, Nov. 1999. [21] C. Cheype, C. Serier, M. Thèvenot, T. Monédière, A. Reineix, and B. Jecko, “An electromagnetic bandgap resonator antenna,” IEEE Trans. Antennas Propag., vol. 50, pp. 1285–1290, Sep. 2002. [22] P. M. T. Ikonen, E. Saenz, R. Gonzalo, and S. A. Tretyakov, “Modeling and analysis of composite antenna superstrates consisting on grids of loaded wires,” IEEE Trans. Antennas Propag., vol. 55, pp. 2692–2700, Oct. 2007. [23] T. Akalin, J. Danglot, O. Vanbésien, and D. Lippens, “A highly directive dipole antenna embedded in a Fabry-Perot type cavity,” IEEE Microw. Wireless Comp. Lett., vol. 12, pp. 48–50, Feb. 2002. [24] F. Ghanem, G. Y. Delisle, T. A. Denidni, and K. Ghanem, “A directive dual-band antenna based on metallic electromagnetic crystals,” IEEE Antennas Wireless Propag. Lett., vol. 5, pp. 384–387, Dec. 2006. [25] B. Temelkuran, E. Ozbay, J. P. Kavanaugh, G. Tuttle, and K. M. Ho, “Resonant cavity enhanced detectors embedded in photonic crystals,” App. Phys. Lett., vol. 72, pp. 2376–2378, May 1998. [26] A. M. Nicolson and G. F. Ross, “Measurement of the intrinsic properties of materials by time domain techniques,” IEEE Trans. Instrum. Meas., vol. IM-19, pp. 377–382, Nov. 1970. [27] W. B. Weir, “Automatic measurement of complex permittivity and permeability at microwave frequencies,” Proc. IEEE, vol. 62, pp. 33–36, Jan. 1974. [28] R. W. Ziolkowski, “Design, fabrication, and testing of double negative metamaterials,” IEEE Trans. Antennas Propag., vol. 51, pp. 1516–1529, Jul. 2003. [29] X. Chen, T. M. Grzegorczyk, B. Wu, J. Pacheco, Jr, and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E, vol. 70, pp. 016608(1)–016608(7), Jul. 2004. [30] S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express, vol. 8, pp. 173–190, Jan. 2001. [31] P. A. Belov, R. Marques, S. I. Maslovski, I. S. Nefedov, M. Silveirinha, C. R. Simovski, and S. A. Tretyakov, “Strong spatial dispersion in wire media in the very large wavelength limit,” Phys. Rev. B, vol. 67, pp. 113103(1)–113103(4), Mar. 2003.

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Rongguo Zhou (A’09) received the B.S. degree in physics from the University of Science and Technology of China, in 2004 and the M.S. degree in physics from University of Arizona, Tuscon, in 2006, where he is currently working toward the Ph.D. degree. His research interests are RF/microwave component designs, including amplifier design, antenna design and measurement, metamaterial design for antenna applications, etc. He also works on the algorithm and implementation of microwave signal direction of arrival estimation.

Hualiang Zhang was born in Wuhan, China. He received the bachelor degree in electrical engineering from the University of Science and Technology of China (USTC), in September 2003 and the Ph.D. degree in electronic and computer engineering from the Hong Kong University of Science and Technology (HKUST), in January 2007. At HKUST, his research interests include design and synthesis of microwave filters, MEMS technologies especially their applications to the RF passive components, and optimization techniques. From April 2007 to July 2009, he was with the University of Arizona as a Postdoctoral Research Associate, conducting research related to RF/microwave circuits, antenna design, and metamaterials based circuits. In August 2009, he joined the Electrical Engineering Department, University of North Texas, Denton, as an Assistant Professor.

Hao Xin (SM’06) received the Ph.D. degree in physics from the Massachusetts Institute of Technology (MIT), Cambridge, in 2001. He performed research studies for five years at MIT’s Physics Department and at Lincoln Laboratory, where he investigated power dependence of the surface impedance of high-Tc superconducting films and Josephson junction properties at microwave frequencies. From November 2000 to November 2003, he was a Research Scientist with the Rockwell Scientific Company, where he conducted research as Principal Manager/Principal Investigator in the area of electromagnetic band-gap surfaces, quasi-optical amplifiers, electronically scanned antenna arrays, MMIC designs using various III-V semiconductor compound devices, and random power harvesting. From 2003 to 2005, he was a Sr. Principle Multidisciplinary Engineer at Raytheon Missile Systems, Tucson, AZ. He is now an Associate Professor in the Electrical and Computer Engineering Department and the Physics Department at the University of Arizona, Tucson. He has published over 80 refereed technical papers in the areas of solid-state physics, photonic crystals, and the applications thereof in microwave and millimeter wave technologies. He has twelve patents issued and one pending in the areas of photonic crystal technologies, random power harvesting based on magnetic nano-particles, and microwave nano-devices. His current research focus is in the area of microwave, millimeter wave, and THz technologies, including solid state devices and circuits, antennas, passive circuits, and applications of new materials such as metamaterials and carbon nanotubes.

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A Novel Analysis of Microstrip Structures Using the Gaussian Green’s Function Method Mohammad Mahdi Tajdini, Student Member, IEEE, and Amir Ahmad Shishegar, Member, IEEE

Abstract—A novel closed form expression is derived for spatial Green’s functions of microstrip structures by expanding the spectral Green’s function into a Gaussian series. This innovative method is called the Gaussian Green’s function (GGF) method due to the Gaussian form in the closed form Green’s function representation. The main advantage of the GGF method lies in its precision as well as rapid convergence. To demonstrate the versatility of this method, the current distribution of a microstrip antenna is achieved via the combination of the method of moments (MoM) and GGF method. It is shown that this method can be computationally very efficient with less than 1% error compared to the numerical integration of the spectral integral. Also, the results of the GGF method have been compared to the results of the commercial full-wave software of Agilent ADS. Index Terms—Gaussian Green’s function method, Green’s function, microstrip structure.

I. INTRODUCTION

M

ICROSTRIP structures are widely utilized in printed antennas, monolithic microwave integrated circuits (MMIC’s), and high speed digital circuits. Microwave components such as filters, couplers, and power dividers can be manufactured easily by the microstrip technology while they are extremely cheaper, lighter, and more compact than traditional waveguide structures [1]. For instance, the excellent conformability of microstrip antennas and their low cost have made them exceptionally ideal for wireless local area network (WLAN) applications [2]. In the modeling and analysis of microstrip structures especially on a large scale like printed antenna arrays, whereas the full-wave methods are not useful, the method of moments (MoM) [3] is commonly used owing to its efficiency, accuracy and applicability to various structures [4]–[6]. This method can be employed successfully to solve the mixed potential integral equation (MPIE). The important advantage of the MPIE lies in its weakly singular kernel [7]. The solution of this equation in the spatial domain usually entails the acquaintance of the spatial Green’s function. For layered media, the major problem of the approach is that the matrix approximant to the MPIE requires repetitive evaluation of the spatial Green’s function. To Manuscript received August 30, 2008; revised May 20, 2009. First published November 06, 2009; current version published January 04, 2010. M. M. Tajdini was with Sharif University of Technology, Tehran 11365-8639, Iran. He is now with the Department of Electrical and Computer Engineering, Northeastern University, Boston, MA 02115 USA (e-mail: [email protected]. edu). A. A. Shishegar is with the Department of Electrical Engineering, Sharif University of Technology, Tehran 11365-8639, Iran (e-mail: [email protected]. edu). Digital Object Identifier 10.1109/TAP.2009.2036191

find this function, much attempt should be devoted to the calculation of Sommerfeld integrals, whose integrand is composed of the closed form spectral Green’s function and the Hankel or exponential function as the kernel. This type of representation of the Green’s function is time-consuming and very expensive [8]. Therefore, the numerical appraisal of the system matrix is computationally inefficient. There are various kinds of method to make the computation of the matrix elements more efficient. One of the common approaches is based on the approximation of the spectral Green’s function by a finite series of properly chosen terms that can be transformed into the spatial domain leading to a closed form representation of the function. The image method is available for microstrip structures, although it is still time-consuming to some extent [9]. The complex images (CI) method approximates the spatial Green’s function utilizing a finite number of images of an infinitesimal source radiating in a homogenous unbounded area. The amplitudes and locations of these images can be complex values [10]–[12]. This semi-analytical method approximates the spectral Green’s function by a series of complex exponential functions. This can be acquired by any appropriate method like the simple Prony method, the least square Prony method or the generalized pencil of functions (GPOF) method [13], [14]. Then, the spectral Green’s function can be transformed to the spatial domain utilizing the Sommerfeld identity [8], resulting in a closed form spatial Green’s function. Nevertheless, there are some substantial difficulties in actual implementation of the method. For instance, the CI method has no built-in convergence measure and its precision can merely be determined a periori by testing the results with those achieved by the numerical integration of the Sommerfeld integral [15]. Moreover, the critical convergence takes place in some cases of the CI method when the Prony method is used. In other words, the accuracy of the approximation of the Green’s function is sensitive to the number of images and utilizing many complex images sometimes leads to divergence of the results [11], although this problem has been solved in the newer versions of the CI method [16]. There are other difficulties in the CI method which have been pointed out in the literature [10], [17]. Another method is the steepest descent path (SDP) method which has been exploited in the calculation of the layered media Green’s functions [18]. However, this method is applicable in far field problems, e.g., in the case of scattering in large distances from the source. One of the well-known mathematical functions is the Gaussian function. This function identifies the significant electromagnetic radiation called the Gaussian beam whose electric field and intensity (irradiance) profiles are associated with the Gaussian function [19]. The Gaussian beam is a solution of the paraxial Helmholtz equation. It is also a very

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of the spectral integral. Eventually, conclusion is provided in Section V. II. THE GREEN’S FUNCTION OF AN OPEN MICROSTRIP STRUCTURE For some electromagnetic problems including microstrip structures, the Green’s function can be found analytically in a closed form in the spectral domain and the dyadic and scalar and [15], [25] can be specified spatial Green’s function using the Sommerfeld integral in the following form [8]: Fig. 1. An infinitesimal horizontal electric dipole above an open microstrip substrate.

(1a) good approximation for many laser outputs. It is important to know that the Hankel transform of the Gaussian function is another Gaussian function. Therefore, the Gaussian function is a self-dual function [20]. In other words, this function is an eigenfunction of the Hankel transform operator. These facts are used to develop a more physical alternative to the CI method. In this paper, a new closed form representation for the spatial Green’s function of the microstrip structure is introduced. It is accomplished by expanding the spectral Green’s function into a series of Gaussian functions and making the inverse Hankel transformation to obtain the spatial Green’s function in another Gaussian series. The method is called the Gaussian Green’s function (GGF) method owing to the Gaussian form in the new closed form Green’s function representation [21]. The main advantage of the GGF method lies in its precision as well as rapid convergence. If the multilayer media Green’s functions are achieved in closed form as a summation of a few Gaussian functions, the MoM matrix entries can be calculated easily in the analytical form leading to a significant decrease in the matrix filling time. The result of this method can be valid over to . The a range of distances, i.e., from nearly parameters of the Gaussian terms required for the expansion of the Green’s function can be chosen in conformity with some simple criteria. Exploiting methods like the Gabor expansion method [22], the point matching method, or the minimum least squares method for expanding the spectral Green’s function into a Gaussian series, one can have an approximation method with the built-in convergence measure [23], [24]. Furthermore, the critical convergence does not appear in the GGF method because the precision of the Green’s function approximation in this method is not as sensitive to the number of series terms as it is in the CI-Prony method. This paper is organized as follows. In Section II, the Green’s function of an open microstrip structure is considered and the CI method is briefly explained. Section III elucidates the theory of the GGF method and derives the simple closed form expression to find the Green’s function of the open microstrip structure. The effect of the surface waves and the expansion methods are also considered in this section. To evaluate the accuracy and performance of this method, the GGF method in conjunction with the MoM is employed in Section IV to acquire the current distribution of a microstrip antenna as a practical example of microstrip structures. Numerical results are given in this section and the GGF method is compared to the numerical integration

(1b) (1c) is the second kind Hankel function of zero order, In (1), stands for the -component of spectral vector potential created due to an infinitesimal -directed electric dipole, and by the stands for the spectral scalar potential created same token owing to one charge of the dipole. For an infinitesimal horizontal electric dipole located above an open microstrip substrate, as shown in Fig. 1, these spectral potentials in the air region can be given by [26]

(2a)

(2b) where

(3a) (3b)

(3c) and where

indicate the effect of the open microstrip substrate and are defined as

(4a) (4b) where is the wave number in the air region and and are the propagation constants along the -direction in the air and substrate, respectively. The Sommerfeld integral as shown in (1) can not be calculated analytically in general, except a few special cases. There

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are a great number of articles in the literature covering a very broad range of numerical methods that have been proposed so far to overcome this problem [15], [27]. For example, a successive semi-analytical method is the CI method. To apply the CI method, a proper exponential function expansion is done in the spectral domain [11]. This can be acquired by any appropriate method like the simple Prony method, the least square Prony method or the generalized pencil of functions (GPOF) method [13], [14]. Then, the spectral Green’s function is transformed to the spatial domain using the Sommerfeld identity [8], resulting in a closed form spatial Green’s function of complex images. In other words, the total spatial Green’s function is represented in a finite series of images of an infinitesimal source radiating in a homogenous unbounded area, when both the amplitudes and locations of these images can be complex values [12]. This method has limitations which have been pointed out extensively in the literature [10], [15], [17]. III. THE GAUSSIAN GREEN’S FUNCTION METHOD A typical Gaussian function can be described as (5) represents the Gaussian function’s maximum and where is the constant which represents the spreading of the Gaussian function. It can be shown that for the Gaussian function, the inverse Hankel transform relation is [20]

(6) It means that the Gaussian function is a self-dual function, i.e., the Hankel transform of the Gaussian function is another Gaussian function. Thus, the Gaussian function is an eigenfunction of the Hankel transform operator. The Gaussian Green’s function (GGF) method is derived through the expansion of the spectral Green’s function into a Gaussian series as an alternative to the exponential series in the CI method. Consider the dyadic and scalar spectral Green’s and of (2) for example. After extracting the functions source terms, they are expanded into series of Gaussian terms, i.e., (7a) (7b) when (8a) (8b) Making the inverse Hankel transformation to the spectral Green’s functions and using (6), then we can attain the dyadic

and scalar spatial Green’s functions series of Gaussian functions, i.e.,

and

by another

(9a) (9b) (9c) where the Sommerfeld identity [8] has been applied for the first exponential term. These closed form expressions are the results of applying the Gaussian Green’s function (GGF) method [21]. It is noted that the GGF method is based on the expansion of the spectral Green’s function into a Gaussian series. The coefficients of (7) can be found exploiting the Gabor expansion method [22]. One advantage of the Gabor expansion method is that in this method, completeness is a posteriori. It also removes any arbitrariness in the preference of coefficients once the number of Gaussian terms is selected. Nevertheless, in this method finding the coefficients of the Gaussian series in (7) generally necessitates a moderately intricate numerical computation of biorthogonal integrals [23]. The coefficients of Gaussian series can be found in simpler and more straightforward approach using the point matching method or the minimum least squares method [24]. Thus, the relation between (2) and (7) can be reduced to a set of linear equations which can be solved simultaneously utilizing matrix inversion. The choice of the Gaussian expansion parameters can be carried out according to some simple criteria. For example, the number of Gaussian terms may be taken in the range of 7–9 to achieve the accuracy of 1%. The alpha parameters in (7) can be chosen to obtain the least squares error. In practice, the simple point matching method can be performed and the coefficients can be adjusted by trial and error to obtain best results. The coefficients of the Gaussian series in (7) are dependent on the source and field locations and . However, in many problems including the calculation of the current distribution of the microstrip antenna considered in the succeeding section of this paper, the and are fixed relative to each other. Thus, only one expansion is necessary to approximate the spatial Green’s function. The -independent formulation of the GGF method is currently under study. The inverse Hankel transform can be carried out along the on the complex plane as shown in Fig. 2(a). The real axis on the complex plane equivalent contour of is demonstrated in Fig. 2(b). We can approximate this contour by an oblique line, , illustrated in Fig. 2(b). It is noticed that the truncation point of the approxima, should be sufficiently selected near to or far tion path, plane to afford adequate infrom the origin of the complex formation from the spectral Green’s function for the far field or near field approximation respectively. The corresponding conon the complex plane is demonstrated in Fig. 2(a) tour of which is along a deformed integration contour passing through the origin and lying in the first and third quadrants of coordinates. Any deformed integration contour can be used while no more singularity is encountered in the deformation. Owing to the notable resemblances between the CI and GGF methods, for

TAJDINI AND SHISHEGAR: NOVEL ANALYSIS OF MICROSTRIP STRUCTURES USING GAUSSIAN GREEN’S FUNCTION

surface wave pole and the residue of and are given by [26] tively.

91

at this pole, respec-

(11) According to the Cauchy’s integral formula [28]

(12) Substituting (10) in (1) and utilizing (6) and (12), we can have the closed form dyadic and scalar spatial Green’s functions as

(13a)

Fig. 2. The integration contours C and C : (a) on the complex k plane and (b) on the complex plane.

(13b)

improvement of the approximation especially in large distances, some useful techniques such as modification of the integration contour or extraction of the surface wave poles can be examined [26]. and can have poles on the real axis of the complex plane. These poles are attributed to the source of propagating stand for the surface surface waves. The poles in waves trapped by the bottom ground plane and the poles in or epitomize the surface waves trapped by the dielectric slab [11]. The effect of surface waves is usually dominant in the far field region and may constitute a major problem for accuracy of methods based on the Hankel transform such as the CI or GGF method in large distances. To overcome this problem, all the surface wave poles can be extracted from the spectral Green’s function and the spatial Green’s function can be signified as a summation of two series. To be more explicit, by extracting all the surface wave poles from the dyadic and scalar spectral Green’s functions, the results can be expanded into series of Gaussian functions, i.e.,

Thus, the closed form spatial Green’s function can be represented as the contribution of a spherical wave, a finite number of Gaussian beams, and a finite number of cylindrical waves. The surface wave expressions have been taken in [11] and [26] as a constitutive part of the approximation in all distances from the source. Nevertheless, we consider them here as a far field approximation on the air-dielectric interface. It is noticed that nonreal poles that represent the surface waves in a lossy medium are not extracted since they are damped and disappeared at far distances. Thus, their contribution can be incorporated in the Gaussian series. There are other sources of deterioration of the approximation for large distances, such as artificial branch cuts. However, as it has been demonstrated in [8] and [29], the contribution of non-physical branch points to the spatial Green’s function can not be the main source of the problem in the far field approximation. The reason is that the branch cut contribuat the interfaces, while the surface tion is asymptotically as . Therefore, the branch cut contriwave contribution is as bution decays faster than the surface wave contribution and can be involved in the Gaussian series of (13). IV. NUMERICAL RESULTS

(10a)

(10b) where and are the th surface wave pole and the residue of at this pole, respectively. Also, and are the th

In this section, a number of numerical examples are presented to show the efficiency and versatility of the GGF method. At first, the microstrip antenna of Fig. 3 as a simple but intuitive example is considered to display the accuracy of the GGF method. This patch antenna consists of a rectangular conducting strip separated from an infinite ground plane by a relatively thick substrate and is fed at the mid point of the metallization strip. The relative permittivity of the substrate , the substrate

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Fig. 3. A planar microstrip antenna on an infinite ground plane.

G = for the open microstrip structure of Fig. 1 " = 4:3.

Fig. 4. The amplitude of , and when

z = 0; h = 0:8

thickness

cm, the strip length cm, the strip width cm, the characteristic impedance of the coaxial cable , and the frequency GHz. The analysis of this structure is customarily performed using the MoM. This usually requires the acquaintance of the spatial Green’s function of the structure. Therefore, the computation of the spatial Green’s function of the open microstrip structure, as shown in Fig. 1, looks inevitable. In this case, an infinitesimal while horizontal electric dipole is located at and . For the sake of brevity, solely is demonstrated. Fig. 4 roughly shows the dyadic Green’s function as a function of and . In Fig. 5, this function is approximated by the GGF method and compared with the results of the numer. Comparison of the 7-term closed ical integration where form Green’s function achieved through the GGF method with the numerical integration makes obvious that the difference between these two results can be unobservable with less than 1% error when compared to the numerical integration. The codes for the numerical integration and the GGF method have been carried out on a 3.2 GHz personal computer. It is perceived that the CPU time recorded for the approximation of the Green’s function utilizing the GGF method can be more than 20 times faster than the numerical integration. Fig. 6 demonstrates the current distribution of the microstrip antenna of Fig. 3 computed by the combination of the MoM and GGF method which can agree well with the implementation of

Fig. 5. Comparison of the GGF method with the numerical integration: (a) real for the open microstrip structure of Fig. 1 and (b) imaginary parts of when , and .

G = z = z = 0; h = 0:8 " = 4:3

the MoM through the numerical integration of the spectral integral. According to the numerical integration of the spectral integral in conjunction with the MoM, the input impedance of the antenna can be achieved about . By combination of the MoM and GGF method, the method that is presented in this paper, the input impedance can be accomplished as that can be in good agreement. Moreover, the antenna of Fig. 3 has been analyzed by the commercial software of Agilent ADS [30] and the current distribution of the antenna has been compared to the results of the GGF method. The input impedance of the antenna can be obtained about . The simulation results are also plotted in Fig. 6. As a final remark, the microstrip structure of Fig. 1 is considered to demonstrate the validity of the GGF method in a variety of frequencies, thicknesses and permittivities. Thus, at the freGHz, GHz, and GHz quencies of the amplitude of the dyadic Green’s function is acquired via

TAJDINI AND SHISHEGAR: NOVEL ANALYSIS OF MICROSTRIP STRUCTURES USING GAUSSIAN GREEN’S FUNCTION

Fig. 6. The current distribution on the horizontal axis of the microstrip antenna : cm, w : : , and f : cm, t cm, " of Fig. 3 when l GHz obtained from the MoM via the GGF method, MoM via the numerical integration and the simulation results.

=17

= 0 08

=5

=43

=48

93

Fig. 8. Comparison of the GGF method with the numerical integration: The amplitude of G  for the open microstrip structure of Fig. 1 when z z : , (b) h : , and f : GHz for: (a) h cm, " cm, " : . and (c) h cm, "

0

=48 = 14

=83

=5

=43

= 14

= = =43

into a series of Gaussian terms. With this method, the numerical integration of the spectral integrals can be avoided entirely, leading to a considerable decrease of calculating time. The comparison of the GGF method with the numerical integration indicates the precision and efficiency of this method. Furthermore, the combination of the MoM and GGF method is suggested as an efficient and versatile method for analysis of microstrip structures. Even though only a printed antenna with one substrate layer is studied, the method of this paper can be correspondingly applied to the multilayered microstrip structures. ACKNOWLEDGMENT The authors would like to earnestly thank the reviewers for their constructive comments. Fig. 7. Comparison of the GGF method with the numerical integration: The amplitude of G  for the open microstrip structure of Fig. 1 when z z ;h cm, and " : at f : GHz, f : GHz, and f : GHz.

0 =5

=43

=24

=48

= = =96

the GGF method and illustrated in Fig. 7 which can be in good agreement with the results of the numerical integration where cm, and . Then, this microstrip GHz but structure is considered in the frequency of when the thickness of the substrate has been changed to cm and also when the permittivity of the substrate has been al. The results of the GGF method are comtered to pared to the numerical integration in Fig. 8 which can be in good agreement for all these cases. V. CONCLUSION The spectral integral solution for the calculation of spatial Green’s functions of microstrip structures is typically time-consuming and computationally inefficient. This paper introduces an innovative method to derive a closed form expression for the Green’s function by expanding the spectral Green’s function

REFERENCES [1] Y. S. Tan, X. S. Rao, L. F. Chen, C. Y. Tan, and C. K. Ong, “Simulation, fabrication and testing of a left-handed microstrip coupler,” Microw. Opt. Technol. Lett., vol. 45, no. 3, pp. 255–258, May 2005. [2] M. M. Tajdini and M. Shahabadi, “Wideband planar log-periodic antenna,” in Proc. IEEE Int. Workshop Antenna Technol., Mar. 21–23, 2007, pp. 331–334. [3] R. F. Harrington, Field Computation by Moment Methods. New York: Wiley-IEEE Press, 1993. [4] T. Onal, M. I. Aksun, and N. Kinayman, “A rigorous and efficient analysis of 3-D printed circuits: Vertical conductors arbitrarily distributed in multilayer environment,” IEEE Trans. Antennas Propag., vol. 55, no. 12, pp. 3726–3729, Apr. 2006. [5] 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, no. 4, pp. 1174–1184, Dec. 2007. [6] J. X. Wan, T. M. Xiang, and C. H. Liang, “The fast multipole algorithm for analysis of large-scale microstrip antenna arrays,” PIER 49, Progr. Electromagn. Res., pp. 239–255, 2004. [7] L. Tsang, C. J. Ong, C. C. Huang, and V. Jandhyala, “Evaluation of the Green’s function for the mixed potential integral equation (MPIE) method in the time domain for layered media,” IEEE Trans. Antennas Propag., vol. 51, no. 7, pp. 1559–1571, Jul. 2003. [8] M. E. Yavuz, M. I. Aksun, and G. Dural, “Critical study of the problems in discrete complex image method,” in Proc. IEEE Int. Symp. Electromagn. Compat., May 11–16, 2003, vol. 2, pp. 1281–1284.

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[9] D. G. Fang, J. J. Yang, and G. Y. Delisle, “Discrete image theory for horizontal electric dipoles in a multilayered medium,” Proc. Inst. Elec. Eng., vol. 135, pp. 297–303, 1988, pt. H. [10] W.-H. Tang and S. D. Gedney, “An efficient application of the discrete complex image method for quasi-3-D microwave circuits in layered media,” IEEE Trans. Microw. Theory Tech., vol. 55, no. 8, pp. 1723–1729, Aug. 2007. [11] J. J. Yang, Y. L. Chow, G. E. Howard, and D. G. Fang, “Complex images of an electric dipole in homogenous and layered dielectrics between two ground planes,” IEEE Trans. Microw. Theory Tech., vol. 40, no. 3, pp. 1358–1362, Mar. 1992. [12] Y. L. Chow, J. J. Yang, and G. E. Howard, “Complex images for electrostatic field computation in multilayered media,” IEEE Trans. Microw. Theory Tech., vol. 39, no. 7, pp. 1120–1125, Jul. 1991. [13] Y. Hua and T. K. Sarkar, “Generalized pencil-of-function method for extracting poles of an EM system from its transient response,” IEEE Trans. Antennas Propag., vol. 37, no. 2, pp. 229–234, Feb. 1989. [14] N. Hojjat, S. Safavi-Naeini, R. Faraji-Dana, and Y. L. Chow, “Fast computation of the nonsymmetrical components of the Green’s function for multilayer media using complex images,” Proc. Inst. Elect. Eng. Microw. Antennas Propag., vol. 145, no. 4, pp. 285–288, Aug. 1998. [15] K. A. Michalski and J. R. Mosig, “Multilayered media Green’s function in integral equation formulations,” IEEE Trans. Antennas Propag., vol. 45, no. 3, pp. 508–519, Mar. 1997. [16] M. I. Aksun and G. Dural, “Clarification of issues on the closed-form Green’s functions in stratified media,” IEEE Trans. Antennas Propag., vol. 53, no. 11, pp. 3644–3653, Nov. 2005. [17] M. Ayatollahi and S. Safavi-Naeini, “A new representation for the Green’s function of multilayer media based on plane wave expansion,” IEEE Trans. Antennas Propag., vol. 52, no. 6, pp. 1548–1557, Jun. 2004. [18] B. Hu and W. C. Chew, “Fast inhomogeneous plane wave algorithm for scattering from objects above the multilayered medium,” IEEE Trans. Geosci. Remote Sens., vol. 39, pp. 1028–1038, May 2001. [19] A. Yariv, Optical Electronics in Modern Communications, 5th ed. New York: Oxford Univ. Press, 1997. [20] E. Cavanagh and B. Cook, “Numerical evaluation of Hankel transforms via Gaussian-Laguerre polynomial expansions,” IEEE Trans. Acoust., Speech, Signal Process., vol. 27, no. 4, pp. 361–366, Aug. 1979. [21] M. M. Tajdini and A. A. Shishegar, “The Gaussian expansion of the Green’s function of an electric current in a parallel-plate waveguide,” in Proc. IEEE Int. RF Microw. Conf., Dec. 2–4, 2008, pp. 223–225. [22] M. J. Bastiaans, “Gabor’s expansion of a signal into Gaussian elementary signals,” Proc. IEEE, vol. 68, no. 4, pp. 538–539, Apr. 1980. [23] H.-T. Chou, P. H. Pathak, and R. J. Burkholder, “Novel Gaussian beam method for the rapid analysis of large reflector antennas,” IEEE Trans. Antennas Propag., vol. 49, no. 6, pp. 880–893, Jun. 2001. [24] H.-T. Chou and P. H. Pathak, “Use of Gaussian ray basis functions in ray tracing methods for applications to high frequency wave propagation problems,” Proc. Inst. Elect. Eng. Microw. Antennas Propag., vol. 147, no. 2, pp. 77–81, Apr. 2000.

[25] R. E. Collin, Field Theory of Guided Waves, 2nd ed. New York: McGraw-Hill, 1991. [26] Y. L. Chow, J. J. Yang, D. G. Fang, and G. E. Howard, “A closed form spatial Green’s function for the thick microstrip substrate,” IEEE Trans. Microw. Theory Tech., vol. 39, no. 3, pp. 1358–1362, Mar. 1991. [27] K. A. Michalski, “Extrapolation methods for Sommerfeld integral tails,” IEEE Trans. Antennas Propag., vol. 46, no. 10, pp. 1405–1418, Oct. 1998. [28] G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 6th ed. Burlington, MA: Elsevier Academic Press, 2005. [29] M. I. Aksun, M. E. Yavuz, and G. Dural, “Comments on the problems in DCIM,” in Proc. IEEE APS Int. Symp. USNC/CNC/URSI North Amer. Radio Sci. Meeting Conf., Jun. 22–27, 2003, pp. 673–676. [30] Agilent ADS 2002 Agilent Technologies [Online]. Available: http:// www.agilent.com Mohammad Mahdi Tajdini (S’09) received the B.Sc. and M.Sc. degrees in electrical engineering from the University of Tehran and Sharif University of Technology, Tehran, Iran, in 2006 and 2008, respectively. He is currently working toward the Ph.D. degree at Northeastern University, Boston, MA. From 2007 to 2009, he was with the Numerical Electromagnetic Laboratory, Sharif University of Technology, as a Graduate Student Researcher. He is currently a Graduate Research Assistant in the Applied EM and Optical Devices Laboratory, Northeastern University. His research interests include theoretical electromagnetics, nanoelectromagnetics, THz communications, physics and applications of metamaterials, and modern physics.

Amir Ahmad Shishegar (M’03) received the B.Sc. (with honors), M.Sc. and Ph.D. degrees in electrical engineering from the University of Tehran, Tehran, Iran. He was a Visiting Scholar with the University of Waterloo and Apollo Photonics Inc., Canada, during 1999 and 2000. He is currently an Assistant Professor in the Department of Electrical Engineering, Sharif University of Technology. His research interests and activities include analytical and computational electromagnetics, electromagnetic wave propagation modeling, analysis and design of passive optical devices and optical integrated circuits. He held different research and development positions in the field of optical communications, microwave and antennas in a number of private and governmental telecom industries and research organizations including Iran Telecommunication Research Center (ITRC). Dr. Shishegar is currently the Treasurer of the IEEE-Iran Section.

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Application of Kummer’s Transformation to the Efficient Computation of the 3-D Green’s Function With 1-D Periodicity Ana L. Fructos, Rafael R. Boix, Member, IEEE, and Francisco Mesa, Member, IEEE

Abstract—The 3-D homogeneous Green’s function with 1-D periodicity is commonly expressed as spatial and spectral infinite series that may show very slow convergence. In this work Kummer’s transformation is applied to the spatial series in order to accelerate its convergence. By retaining a sufficiently large number of asymptotic terms in the application of Kummer’s transformation, the spatial series is split into a set of series which can be accurately obtained with very low computational effort. The numerical results obtained show that, when the number of asymptotic terms retained in Kummer’s transformation is large enough, the convergence acceleration method proposed in this work is always faster than existing acceleration methods such as the spectral Kummer-Poisson’s method and Ewald’s method. Index Terms—Convergence of numerical methods, Green’s functions, periodic structures, series.

I. INTRODUCTION N the electromagnetic analysis of periodic structures via the method of moments [1], the computation of periodic Green’s functions is always required. In particular, the 3-D homogeneous Green’s function with 1-D periodicity has been used to determine the mutual row-admittances and the scan impedances of arrays of slots and dipoles that are infinite in one direction and finite in the orthogonal direction [2], [3]. This same 3-D Green’s function with 1-D periodicity has also been employed in the analysis of the scattering by frequency selective surfaces made of dipoles or slots, which are again infinite in one direction and finite in the orthogonal direction [4]–[6]. Although models of planar arrays with double periodicity that are infinite in the two directions are easier to analyze (via Floquet’s theorem) than models of arrays with single periodicity that are infinite in one direction and finite in the orthogonal direction, these latter models have the advantage of allowing the study of the edge effects appearing in real-life arrays due to truncation [4]. The 3-D homogeneous Green’s functions with 1-D periodicity can also be applied to the solution of electromagnetic

I

Manuscript received October 06, 2008; revised March 23, 2009. First published November 06, 2009; current version published January 04, 2010. This work was supported in part by the Spanish Ministerio de Educación y Ciencia and European Union FEDER funds (project #TEC2007-65376), and in part by Junta de Andalucía (project #TIC-253). A. L. Fructos and R. R. Boix are with the Microwaves Group, Department of Electronics and Electromagnetism, College of Physics, University of Seville, 41012-Seville, Spain (e-mail: [email protected]). F. Mesa is with the Microwaves Group, Department of Applied Physics 1, ETS de Ingeniería Informática, University of Seville, 41012-Seville, Spain (e-mail:[email protected]). Digital Object Identifier 10.1109/TAP.2009.2036188

problems with 1-D periodicity involving multilayered media. Since non-periodic 3-D multilayered Green’s functions can be expressed in closed form as linear combinations of spherical waves in homogeneous media by means of the discrete complex image method [7], this makes it possible to obtain 3-D multilayered Green’s functions with 1-D periodicity in terms of 3-D homogeneous Green’s functions with 1-D periodicity, as it has been demonstrated for the 2-D periodicity case in [8], [9]. The application of the method of moments to the analysis of periodic electromagnetic problems usually requires that the periodic Green’s functions are computed thousands of times. Therefore, efficient algorithms are needed for the evaluation of those Green’s functions. In particular, the 3-D homogeneous Green’s function with 1-D periodicity can be expressed in terms of either a spatial series or a spectral series that may converge very slowly under certain circumstances: negligible losses in the case of the spatial series, observation point very close to the array of point sources in the case of the spectral series, etc. For that reason, a number of analytical and numerical methods have been reported to accelerate the convergence of these series. One of the analytical methods mostly used for the computation of homogeneous periodic Green’s functions is based on the combined use of Kummer’s transformation with Poisson’s formula [10]–[13]. As commented in [12], this method is more useful when Kummer’s transformation is applied to the spectral series since this allows for working with arbitrary complex phase shifts. The efficiency of the spectral Kummer-Poisson’s method considerably increases when several asymptotic terms are retained in the application of Kummer’s transformation. This latter approach has been successfully applied to the computation of 2-D homogeneous periodic Green’s function [14], [15] and 3-D homogeneous Green’s functions with 2-D periodicity [16]. Unfortunately, in this paper we will demonstrate that the spectral Kummer-Poisson’s method with higher order asymptotic extraction is not very efficient when applied to the determination of the 3-D homogeneous Green’s function with 1-D periodicity. And this is because the method involves the evaluation of a large number of modified Bessel functions of complex arguments, which requires an important CPU time consumption. An alternative method for the fast determination of homogeneous periodic Green’s functions is Veysoglu’s transformation, which makes it possible to express these Green’s functions in terms of infinite integrals with rapidly decaying integrands [12], [17]. Valerio et al. have shown that the application of Veysoglu’s transformation to the particular computation of the 3-D periodic Green’s function is

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not very efficient [12, Fig. 4], at least when compared with the computation of the 2-D periodic Green’s function (whereas in the 2-D case the integrands show a Gaussian decay, in the 3-D case the decay is just exponential). Lomakin and Van Orden have recently developed a method which leads to expressions of the 2-D and 3-D Green’s functions with 1-D periodicity in terms of a few terms of the spatial and spectral series plus an infinite integral with Gaussian convergence [18], [19]. Even though complicated integrals have to be numerically computed in the application of this method, efficient tailored quadrature rules have been developed for the numerical evaluation of these integrals [19]. Ewald’s method is a very efficient analytical method for the computation of homogeneous periodic Green’s functions in which these functions are expressed as a combination of two series with fast Gaussian convergence [20]–[23], [12]. The main drawback of Ewald’s method is that it suffers from high-frequency breakdown, which leads to a non-optimal choice of the splitting parameter controlling the convergence of the two Gaussian series [12], [23], [24]. The perfectly matched layers (PML) method has also proven to be an efficient approach for the computation of 3-D multilayered Green’s functions with 1-D periodicity [25]. Unfortunately, the PML method requires the repeated computation of several series by means of Ewald’s method [25, Eqs. (13)–(14)], and therefore, it is clearly less efficient than Ewald’s method alone in the homogeneous case. Among the numerical acceleration methods that have been used for the efficient computation of the 3-D homogenous Green’s functions with 1-D periodicity, we should mention Shank’s transformation [11], the -algorithm [26], Chebyschev-Toeplitz algorithm [27], and Wynn’s -algorithm [28]. These latter numerical acceleration algorithms cannot only be directly applied to the original spatial and spectral series but they can also be applied to the fast converging series resulting from the application of the analytical acceleration methods [12]. We have commented above that the Kummer-Poisson’s method is not numerically efficient when applied to the spectral series involved in the computation of the 3-D homogeneous Green’s function with 1-D periodicity. Bearing this fact in mind, in this paper we propose to accelerate the computation of this periodic Green’s function by applying Kummer’s transformation to the spatial series. Also, instead of retaining one single term in the asymptotic expansion of the th term of the spatial series [10], [12], we retain several terms in order to improve the convergence of the accelerated series [14]–[16], [29]. When Kummer’s transformation is applied to the spatial series, it is well known that the resulting asymptotic series (also called “tail” series) do not depend on the coordinates of the source and observation points. Therefore, they only have to be computed once in typical method of moments applications, which leads to important CPU time savings [29]. Unlike [29], where the efficient computation of these asymptotic series was not addressed in detail, in this paper we introduce quasi-closed form expressions for the asymptotic series so that their computation does not burden the overall computation of the periodic Green’s function and extra efficiency is achieved. The algorithm resulting from the proposed novel implemen-

tation of Kummer’s transformation has been compared with other acceleration algorithms and has proven to be faster than Ewald’s method, which has been reported to be one of the most efficient algorithms for the evaluation of periodic homogeneous Green’s functions [30]. The paper is organized as follows. In Section II we describe the mathematical derivations involved in the application of the higher order Kummer’s transformation to the efficient computation of the 3-D homogeneous Green’s function with 1-D periodicity (some cumbersome mathematical details have been relegated to Appendices II and III). In Section III we show the fast convergence of the series arising from the application of the high order Kummer’s transformation, and we also compare the CPU times required by this transformation with those required by the spectral Kummer-Poisson’s method and Ewald’s method. Conclusions are summarized in Section IV. II. ANALYSIS will be asIn the following, a time dependence of the type sumed and suppressed throughout. Let us consider a one-dimensional array of point sources that are periodically located along the direction in a non-magnetic homogeneous lossy medium (see Fig. 1). The complex permittivity of the medium is given by , and the permeability is . The periodic 3-D Green’s function generated by this array of point sources can be expressed as the following spatial series: (1) where (

is the host medium complex wavenumber is the wavelength), is the period, is the phase per period imposed either by a wave emitted by the array (positive sign) or by a wave incident and are the angular spheron the array (negative sign), ical coordinates that indicate the propagation direction of these is defined as waves, and

(2) In (2) we have used the simplified notation , and , where are the coordinates of the observation point, and are the coordinates of the point sources. The th term of the as . When series (1) decays like losses are negligible , the th term decays like , and therefore, the convergence of (1) becomes very slow. A. Previous Numerical Procedures Let be the distance between the observation point of Fig. 1 and the straight line containing the point , Poisson’s formula provides the following sources. For alternative series for the computation of [12]: (3)

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provided the limit Equation (4) can still be used for is taken in the first series. The resulting expression is

Fig. 1. Linear array of point sources with 1-D periodicity along the x direction. r ndx x nd x y y z z n is the vector ; ; ; ; pointing at the sources and r xx y y z z is the vector pointing at the observation point.

+ ^ = ( + )^ + ^ + ^( = . . . 01 0 1 . . .) = ^+ ^+ ^

where , and is the zeroth-order modified Bessel function [31, p. 374]. Although the th term of the series (3) has an exponential decay of , the series converges very slowly the type when . Also, the series (3) is not defined in the case . In order to accelerate the convergence of (3), we can apply a spectral Kummer’s transformation of higher order to this series and then apply Poisson’s formula to the asymptotic remainder. In particular, the approach proposed in [16] for the computation of the 3-D homogeneous Green’s function with 2-D periodicity can be easily extended to the 1-D periodicity case treated in this paper. When this procedure is followed, the expression obtained is given by for

(5) It can be shown that the th term of the first series in (5) decays as . Also, the th term of the second series like exponentially decays like as . Since (5) does not involve the evaluation of modified Bessel functions of complex argument, it turns out that (5) is an extremely efficient when expression for the computation of . Ewald’s method [20] provides an additional approach to speed up the convergence rate of (1). When Ewald’s method is applied, the following expression is obtained for [12], [22], [23]:

(6) (4) , the coeffiwhere is chosen to be and can be obtained in terms cients as shown in Appendix I, of and in the range and (note that and are complex quantities). Although other choices of are possible as suggested in [32]), numerical (e.g., simulations have shown that the value of chosen in this paper provides a good tradeoff between the number of terms necessary to achieve convergence in the two series of (4), and therefore, provides an optimum performance in terms of CPU time consumption. It can be shown that the first series in (4) converges very fast , even when (in fact, the th term of the first when as ). The series decays like second series in (4) also converges very fast since the th term exponentially decays like as . Therefore, it seems that (4) is an efficient expression for the compu. Unfortunately, the repeated nutation of merical evaluation of the modified Bessel functions with comand , demands a large plex argument, CPU time consumption, and this considerably reduces the efficiency of (4) as will be shown in Section III.

stands for where the complementary error function [31, p. 297], is the th-order exponential integral function defined in [31, p. 228], and is the Ewald splitting parameter, which is a positive real number. The two series of (6) over the integer index exhibit extremely fast Gaussian convergence [12], [23]. The splitting parameter simultaneously controls the convergence of the two series over . Although Valerio et al. have suggested an expression for (see [12, Eqs. (41)–(42)]), in this paper we have used a simplified expression given by (7) . Although the expression of provided in [12, with Eqs. (41)–(42)] includes an additional dependence on the parameter [which is the number of terms necessary to achieve convergence in the series over in (6)], our numerical simulations have shown that this dependence on neither increases the accuracy in the computation of nor reduces the CPU time. We have found that typically ranges between 1 and 24 when the Green’s function is computed with an accuracy of eight significant digits, and the value of is strongly dependent on the and . ratios

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In the absence of losses, is a real negative number, which implies that exponential integral functions of a real negative argument have to be computed in the second series of (6). This is a problem since the exponential integral functions have a branch cut along the real negative axis. As suggested in [23], the problem is solved when small losses are deliberately becomes a introduced in the host medium of Fig. 1 so that complex number with non-zero imaginary part (in fact, we have found that a value of the loss tangent as low as suffices to eliminate the mentioned problem). B. Our Numerical Procedure Next, we propose an alternative approach for computing the , 3-D Green’s function with 1-D periodicity which is based on the application of Kummer’s transformation to (1). In particular, the evaluation of will be restricted to the . For , one can use the interval Floquet-periodicity condition given by (8) where is an integer number. When applying Kummer’s transformation to (1), in this work we propose to retain several terms in the asymptotic expansion of the th term of the series rather than just one single term. As commented in [14], [15], [29], this strategy considerably enhances the efficiency of Kummer’s method. Specifically, let the for large be given by asymptotic expansion of

(9) where the coefficients are provided in . If we apply a th order Appendix II in the range Kummer’s transformation to (1) and make use of (9), the 3-D Green’s function with 1-D periodicity can be computed by means of the following expression (see [29] for details): (10) where (11)

(12)

and (14) in (11) is the contribution to that contains its The function (for singularity in the interval ). In accordance with (9), the th terms of the series in (12) decay like as , and therefore, the convergence of these series is always very fast is chosen large enough (even when the losses of the when host medium are negligible). Finally, the asymptotic tail series, , is expressed in (13) as a linear combination of the series and . The series in (14), which are in general a function of the complex variable , can be obtained in quasi-closed form by invoking the techniques reported in [33, Appendix A.6]. The expressions used in this paper for the computation of are collected in Appendix III for . It should be noted that and do not depend on the differences , and between source and observation points coordinates. Therefore, when the Method of Moments is applied to the analysis of a , and are fixed, periodic structure for which and only have to be computed once, thus achieving considerable CPU times savings [29]. Whereas the computation of via (4) and (6) involves the numerical evaluation of special functions (zeroth order modified Bessel function, complementary error function and exponential integrals), the computation of via (10)–(14) does not involve any special functions. This is one of the reasons why (10) is competitive with respect to (4) and (6) concerning the numerical evaluation of (see Section III). The main drawback of (10) is that it cannot be applied to the computation of for arbitrary [note that the physical complex values of the phase shift complex values of introduced by a lossy host medium in Fig. 1 can indeed be handled by (10)], which is due to the fact that (10) is based on a spatial Kummer’s transformation [12, Section II.B]. However, the expressions of (4) and (6) can handle since these expressions are both arbitrary complex values of based on a spectral transformation [12]. Some researchers have not only addressed the efficient computation of scalar homogeneous periodic Green’s functions but have also addressed the efficient computation of the gradient of these Green’s functions [34], or even the efficient computation of full dyadic periodic Green’s functions [19]. Both the gradient of and the 3-D dyadic Green’s function with 1-D periodicity can be obtained by taking first and second derivatives in (1) with , and . The resulting expressions respect to all the variables contain a set of series that decay like as . All these series can be manipulated as in (10)–(14) in order to be expressed in terms of new series that , and in terms of the series decay at least like which are obtained in quasi-closed form in Appendix III. III. NUMERICAL RESULTS

(13)

In this section we will first analyze the convergence rate of the series and of (12) for different values of . Then, we

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will study the relative weight of and in the total value of . And finally, we will carry out a comparison of the CPU times employed in the computation of by means of (4), (5), (6) and (10). In all cases we will assume that the host medium since this is the worst case scenario for is lossless the convergence of the series of (1). is the th term of [see (12)], the Bearing in mind that convergence of the series and will be studied by means of the following functions of the integer index : (15) and versus for difIn Fig. 2(a) and (b) we plot ferent values of . Note that all the curves tend to straight lines . Since the plots are logarithmic, of negative slope and show a decay of this agrees with the fact that as in the lossless case. As seen in the type and is slow in the Fig. 2(a) and (b), the convergence of case (i.e., when only one term is retained in the appliterms cation of Kummer’s transformation) with at least and with an accuracy of four being required to obtain , the convergence of significant digits. However, for and becomes very fast and only terms are required to obtain the series with an accuracy of four significant digits have to be summed (in fact, in this case, only the terms and with an accuracy of eight significant up to obtain digits). When is evaluated by means of (10), the computation of the and is the most demanding part. This two infinite series statement specially holds in Method of Moments applications and have to be computed every time , where change, unlike the infinite series and and in (13) that only have to be , and computed once because they are independent of (provided , and are kept fixed). In order to estimate and in the final value of , we the relative weight of given by introduce a new quantity (16) For a given accuracy in the evaluation of , the smaller the , the lower the accuracy needed in the computavalue of and , and the faster the calculation of by tion of is plotted as a function means of (10). In Fig. 3(a) to (c) of for different values of and . Note that aldecreases and increases. In parways decreases as is smaller than for ticular, in the case and smaller than for . Roughly speaking, this means that if is to be computed with a preciand have to be comsion of eight significant digits, puted with an accuracy of at most six significant digits in the , and with an accuracy of at interval most four significant digits in the interval (acand cording to Fig. 2, this implies that the computation of will require at most the terms of (12) in the interval , and at most the terms of (12) in the interval ).

Fig. 2. Plots of f (n) [see (15)] as a function of n for the series G (a) and G (b) of (12). d = 0:5; 1x = 0:4d; R = 0:1d;  =  = =4, and tan  = 0. (a) normalized nth term of G (b) normalized nth term of G .

An overall conclusion that can be drawn from Figs. 2 and is 3 is that a double benefit is achieved when the value of increased in (10)–(13). First, the convergence of the series and of (12) becomes faster (as shown in Fig. 2). Second, and to becomes less relevant (as the contribution of shown in Fig. 3), which makes it possible to lower the accuracy and , and therefore, to required in the computation of reduce the CPU time required in the computation of . In Fig. 4(a) and (b) we plot the ratio between the CPU time required by (4) and (5) to compute with an accuracy of eight significant digits, , and that required by (10) for , . In Fig. 5(a)–(c) we plot a similar ratio between the CPU , and . As in [29, Figs. 1 and time required by (6), and 2], the time required to compute the series of (13) has not been included since these series are independent of and , in and only need to be computed once in each of the Fig. 4(a)–(b) and 5(a)–(c). In all these five figures, it can be seen that the CPU and increase as detime ratios decreases as creases. This is due to the fact that decreases since the accuracy required in the computation of and lowers as decreases, in accordance with the results shown in Fig. 3(a)–(c). Fig. 4(a) and (b) show that

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1 =01

Fig. 3. Plots of f [see (16)] as a function of x=d. d  . (a) R , (b) R : d, (c) R = , and

4

tan = 0

=0

= 0:5;  =  = = 0:2d.

the method developed in this work for the computation of is clearly faster than the spectral Kummer-Poisson’s method of regardless the value of , and it is comparable (4) for to the spectral Kummer-Poisson’s method of (5) for when . The fact that (10) is more efficient than (4) for the computation of is attributed to the large computation time demanded by the modified Bessel functions of complex argument of (4). In fact, even though the convergence of the first series of (4) improves as increases, the CPU times observed

in Fig. 4(b) are not substantially shorter than those observed in Fig. 4(a), and this is because the number of Bessel functions with complex argument required in every summation term of is three times larger than that the first series of (4) when . Fig. 5(a) and (b) show that in the cases required when and , the method proposed in this work (between is slightly faster than Ewald’s method when 1 and 2.5 times faster) but it is comparable to Ewald’s method (note that Ewald’s method is extremely efficient when in this latter case because the series in the index of (6) does not have to be computed). However, Fig. 5(c) shows that in the the method in this work is appreciably faster case than Ewald’s method (between 1 and 4 times faster), even when . In order to explain this different behavior for different , we have to consider that the splitting paramvalues of eter used in Ewald’s method [see (7)] has its optimum value for , and separates from its optimum in order to avoid cancellation errors arising value for from the sum of two large nearly-equal numbers of opposite sign in (6) [12], [23], [24]. This means that Ewald’s method has but its efficiency worsens an optimum efficiency for increases above 1, which explains why our method is as only slightly better than Ewald’s method in Fig. 5(a) and (b) and ) but it is appreciably better than ( . Ewald’s method in Fig. 5(c) In Tables I(a) and (b) we present the number of summation terms required in the series of (4), (6), and (12) when is to be computed with an accuracy of at least eight significant digits. and stand for the number of terms required in the first and second series of (4) respectively [thus, the total ], number of summation terms required in (4) is , and stand for the number of terms required in the first and second series over of (6) respectively, stands for the number of terms required in the series over of (6) [thus, the total number of summation terms required in (6) is ], and and stand for the number of terms required in the computation and respectively (i.e., the total number of terms reof if we exclude the computation quired in (10) is of ). Tables I(a) and (b) show that the total number of summation terms required for the computation of (10) (excluding ) is always smaller than those required for the computation of the series of (4) and (6). Also, whereas the total number of summation terms required in (4) and (6) does not appreciably varies (for fixed ), the total number of sumchange as decreases, mation terms required in (10) decreases as which agrees with the conclusions drawn from Fig. 3(a)–(c) and in the computaconcerning the relative weight of tion of . Finally, whereas the total number of summation terms increases (above required in (4) and (6) increases when in the case of Ewald’s method), the total number of terms required in (10) does not appreciably change as increases. Tables II(a) and (b) present a comparison among the CPU (with times required by (4), (5), (6), and (10) to compute an accuracy of both four and eight significant digits) at 1100 different points. In fact, in the computation of the variand have been scanned all through the ables

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Fig. 4. Ratio between the CPU times required by (4)–(5) and (10) to compute with an accuracy of at least eight significant digits. , and . (a) , (b) .

G  = =4

tan  = 0

P =1

P =3

M = 6; d = 0:5; =

ranges

and while keeping fixed , and , which is a calculation that may arise the values of in a typical Method of Moments application. In those cases , (5) has been used as an alternative to (4). Unlike where what has been done in Figs. 4 and 5, in Table II the CPU times computed via (10) include the CPU time required to compute and the series in (13). It should be pointed out that the CPU time required to compute these latter series was always found to be a tiny fraction of the CPU time required to evaluate at 1100 points (usually less than 0.5%), mainly because of the high computational efficiency of the quasi-closed form expressions derived for these series in Appendix III. Tables II(a) and (b) show that the CPU times required by (4) and (5) are very similar for different values of , which indicates that there is no appreciable benefit in the computation time of when the value of is increased in (4) and (5). In order to explain this, we have to consider that even though the series of (4) converge faster as increases, the number of Bessel functions of complex argument in every summation term of the first series of (4) also increases as increases, which counterbalances the CPU time reduction provided by a faster convergence

G M = 6;  =  = =4 d = 3:5

Fig. 5. Ratio between the CPU times required by (6) and (10) to compute , with an accuracy of at least eight significant digits. and . (a) , (b) , (c) .

tan = 0

d = 0:01

d = 0:5

rate. Tables II(a) and (b) show that Ewald’s method is faster . than the spectral Kummer-Poisson’s method unless Note that whereas the CPU times demanded by Ewald’s method increases above 1 (which has to do with quickly increase as the fact that the splitting parameter of (6) quickly separates ), the CPU times demanded by (10) increase with infrom at a much lower rate (in fact, whereas the CPU creasing are roughly six times larger times required by (6) for

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TABLE I NUMBER OF SUMMATION TERMS REQUIRED BY THE SERIES OF (4), (6) AND (12) FOR THE COMPUTATION OF WITH AN ACCURACY OF AT LEAST EIGHT , AND SIGNIFICANT DIGITS. , (B) . (A)

G P = 3; M = 6; Rt = 0:1d;  =  = =4 tan  = 0 d = 0:5 d = 3:5

TABLE II CPU TIMES REQUIRED FOR THE COMPUTATION OF 1100 VALUES OF BY MEANS OF (4), (5), (6), AND (10). ALL CPU TIMES ARE NORMALIZED TO AND THE MINIMUM VALUE OBTAINED IN EACH TABLE. . (A) ACCURACY OF FOUR SIGNIFICANT DIGITS, (B) ACCURACY OF EIGHT SIGNIFICANT DIGITS

tan  = 0

G  =  = =4

). This indicates that the method proposed in this for work is especially more convenient than Ewald’s method when ). the period is electrically large (i.e., for According to Tables II(a) and (b), the application of the stan] dard first order Kummer’s transformation [(10) with to the computation of is slower than the spectral KummerPoisson’s method and Ewald’s method. However, if four significant digits are required, (10) becomes faster than (4) and (5) , and faster than Ewald’s method for (and, for for an accuracy of eight significant digits, (10) is roughly faster ). Note that for and than (4), (5), and (6) for an accuracy of four significant digits, the method proposed in this paper is roughly between 2.2 and 6.3 times faster than (4) and (5), and between 1.7 and 8 times faster than Ewald’s method (and, for and an accuracy of eight significant digits, (10) is roughly between 2.8 and 4.7 times faster than (4) and (5), and between 1.3 and 5 times faster than Ewald’s method). By using , we might have achieved even shorter CPU (10) and times than those given in Tables II(a) and (b). However, the results presented in [14, Fig. 1] indicate that there is a threshold for which the CPU time reaches a minimum, and value of larger than this threshold value the CPU time begins to for grow because of the increasing number of operations demanded , and in (12) and (13). Since by the evaluation of and do not the results obtained in Table II(a) for seems to be the threshold value of appreciable differ, that minimizes the CPU times in this Table [and also seems to be close to the threshold value of in Table II(b)]. provides the best Figs. 4 and 5 show that the case scenario in the application of the approach of (4) and (5), and in the application of (6). For that reason, the CPU time comparison carried out in Tables II(a) and (b) is repeated in Tables III(a) and (b) in the particular case where . In Tables III(a) is scanned through the range and (b) -while keeping fixed the values of , and [also, the CPU times presented include the and CPU time required to compute the series in (13)]. Note that the differand (6) ences between the CPU times required by (5) and those required by (10) are smaller in Table III turns out to be still than in Table II. However, (10) and (6) when is to be comcompetitive with (5) . Thus, the CPU times required by (10) puted in the case when are comparable to those required by (5) and (6) . And the CPU times required by (5) and in the case (6) are even between 1.5 and 3 times larger than those required in the cases and . In by (10) when the case of (5) the CPU time increase with increasing has to do with the fact that the number of terms required for conincreases vergence in the first series of (5) increases when (this behavior has also been noticed in the first series of (4) when and increases). In the case of (6) the CPU time inhas to do with a non-optimum choice crease with increasing of the splitting parameter as commented above. IV. CONCLUSION

than those required for , the CPU times required by (10) for are about 1.5 times larger than those required

In this paper Kummer’s transformation has been used to accelerate the computation of the spatial infinite series for the 3-D

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TABLE III CPU TIMES REQUIRED FOR THE COMPUTATION OF 100 VALUES OF . ALL CPU TIMES BY MEANS OF (5), (6), AND (10) IN THE CASE ARE NORMALIZED TO THE MINIMUM VALUE OBTAINED IN EACH TABLE. AND . (A) ACCURACY OF FOUR SIGNIFICANT DIGITS, (B) ACCURACY OF EIGHT SIGNIFICANT DIGITS

R =0

 =  = =4

G

tan  = 0

103

. These coefficients turn out to coincide with those previously obtained by Ivanishin and Skobelev in [16] when applying the higher order spectral Kummer-Poisson’s method to the computation of the 3-D homogeneous Green’s function with 2-D periodicity. The coefficients are

(17) (18) (19) (20) (21) (22) (23) (24) (25) (26) (27) (28)

APPENDIX II homogeneous Green’s function with 1-D periodicity. The approach is restricted to the case where the values of the phase shift (real or complex) are physical. When applying Kummer’s transformation, an arbitrary number of terms has been retained in the asymptotic expansion of the th term of the spatial series, which has made it possible to split this series into two series with algebraic convergence of arbitrarily large order plus a linear combination of series for which quasi-closed form expressions have been derived. Since these latter series do not depend on the coordinates of the source and observation points, they would only have to be computed once while the Green’s function is computed thousands of times in a typical method of moments analysis, thus leading to considerable CPU time savings. The numerical results obtained have shown that when the number of asymptotic terms retained in the spatial Kummer’s method is sufficiently large, the method proposed is typically between 2.2 and 6.3 times faster than the spectral KummerPoisson’s method, and typically between 1.3 and 8 times faster than Ewald’s method, with these speed ratios being strongly dependent on the accuracy required and on the ratio between the period and the wavelength.

In this Appendix, we provide expressions for the coefficients of (9) in the range

(29) (30)

(31)

(32)

APPENDIX I In this Appendix, we provide expressions for the coefficients and of (4) and (5) in the range

(33)

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(34)







Fig. 6. The region [0 Re(w ) 2; Im(w ) 0] of the complex w -plane is shadowed. This region is split into four subregions where the functions S (w ) (1 i 6) have been evaluated.

 

where (35) (36) (42) (37) (43)

(38)

(44)

(39) (45) APPENDIX III in In accordance with the definition given for and will Section II, in general the functions have to be evaluated in the fourth quadrant of the complex -plane. Fortunately, the periodicity condition (46) (40) makes it possible to reduce the computation of to the region of the complex -plane. In Fig. 6 this region has been shadowed and split into four subregions. Quasi-closed form expressions in each of these four subregions are proof vided below. In the region of the complex -plane (subregion 1 of Fig. 6), the functions can be obtained with an accuracy of at least ten significant digits by using the following expressions (see [33, Appendix A.6]) for details related to the derivation of the expressions):

where and are particular values of the Riemann Zeta function [31, p. 807]. of the In the region is given by (31) complex -plane (Fig. 6, subregion 2), and can be obtained with an accuracy of at least ten significant digits by means of the following expressions (see [33, Appendix A.6] for details concerning the derivations):

(47) (41)

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denotes complex conjugate. Finally, the computation in the region of the complex -plane (Fig. 6, subregion 4) can be efficiently done with an accuracy of at least ten significant digits by means of (41) and the expressions

where of

(57)

(48)

are chosen in the

where the integer numbers following way:

(58) (49)

In the above equation, closest to , and

stands for the integer number , .

ACKNOWLEDGMENT The authors would like to acknowledge one anonymous reviewer for addressing the existence of [16]. Also, the authors are indebted to that reviewer for indicating the convergence rate of the first series in (5) before they were aware of the approach described in [16]. (50)

(51) where (52) (53) (54)

(55)

The computation of

in the region of the complex -plane (Fig. 6, subregion 3) can be reduced to the computain the subregions 1 and 2 of Fig. 6 tion of when the following symmetry relation is applied: (56)

REFERENCES [1] R. F. Harrington, Field Computation by Moment Methods. New York: Macmillan, 1968. [2] R. W. Scharstein, “Mutual coupling in a slotted phased array, infinite in E-plane and finite in H-plane,” IEEE Trans. Antennas Propag., vol. 38, pp. 1186–1191, Aug. 1990. [3] R. C. Hansen and D. Gammon, “A Gibbsian model for finite scanned arrays,” IEEE Trans. Antennas Propag., vol. 44, pp. 243–248, Feb. 1996. [4] J. M. Usoff and B. A. Munk, “Edge effects of truncated periodic surfaces of thin wire elements,” IEEE Trans. Antennas Propag., vol. 42, pp. 946–953, Jul. 1994. [5] J. P. Skinner, C. C. Whaley, and T. K. Chattoraj, “Scattering from finite by infinite arrays of slots in a thin conducting wedge,” IEEE Trans. Antennas Propag., vol. 43, pp. 369–375, Apr. 1995. [6] D. S. Janning and B. A. Munk, “Effects of surface waves on the currents of truncated periodic arrays,” IEEE Trans. Antennas Propag., vol. 50, pp. 1254–1265, Sep. 2002. [7] M. I. Aksun, “A robust approach for the derivation of closed—Form Green’s functions,” IEEE Trans. Microw. Theory Tech., vol. 44, pp. 651–658, May 1996. [8] R. M. Shubair and Y. L. Chow, “Efficient computation of the periodic Green’s function in layered dielectric media,” IEEE Trans. Microw. Theory Tech., vol. 41, pp. 498–502, Mar. 1993. [9] M. J. Park and S. Nam, “Efficient calculation of the Green’s function for multilayered planar periodic structures,” IEEE Trans. Antennas Propag., vol. 46, pp. 1582–1583, Oct. 1998. [10] R. Lampe, P. Klock, and P. Mayes, “Integral transforms useful for the accelerated summation of periodic, free-space Green’s functions,” IEEE Trans. Microw. Theory Tech., vol. 33, pp. 734–736, Aug. 1985. [11] S. Singh and R. Singh, “Application of transforms to accelerate the summation of periodic free-space Green’s function,” IEEE Trans. Microw. Theory Tech., vol. 38, pp. 1746–1748, Nov. 1990. [12] G. Valerio, P. Baccarelli, P. Burghignoli, and A. Galli, “Comparative analysis of acceleration techniques for 2-D and 3-D Green’s functions in periodic structures along one and two directions,” IEEE Trans. Antennas Propag., vol. 55, pp. 1630–1643, Jun. 2007. [13] S. P. Skobelev, “Comments on “Comparative analysis of acceleration techniques for 2-D and 3-D Green’s functions in periodic structures along one and two directions”,” IEEE Trans. Antennas Propag., vol. 55, pp. 3746–3747, Dec. 2007. [14] B. Baekelandt, D. de Zutter, and F. Olyslager, “Arbitrary order asymptotic aaproximation of a Green’s function series,” AEÜ Int. J. Electron. Commun., vol. 51, no. 4, pp. 224–230, 1997.

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[15] A. L. Fructos, R. R. Boix, F. Mesa, and F. Medina, “An efficient approach for the computation of 2-D Green’s functions with 1-D and 2-D periodicities in homogeneous media,” IEEE Trans. Antennas Propag., vol. 56, pp. 3733–3742, Dec. 2008. [16] M. M. Ivanishin and S. P. Skobelev, “On the efficient computation of the Green’s function for doubly periodic structures by using the Kummer’s method of higher orders,” in Proc. 12th Int. Conf. on Mathematical Methods in Electromagnetic Theory, Odessa, Ukraine, Jun. 2008, pp. 544–546. [17] M. E. Veysoglu, H. Yueh, R. Shin, and J. Kong, “Polarimetric passive remote sensing of periodic surfaces,” J. Electromagn. Waves Appl., vol. 5, no. 3, pp. 267–280, Mar. 1991. [18] V. Lomakin and D. Van Orden, “Rapidly convergent spectral representations for periodic Green functions,” presented at the URSI General Assembly, Chicago, IL, Aug. 2008. [19] D. V. Orden and V. Lomakin, “Rapidly convergent representations for 2D and 3D Green’s functions for a linear periodic array of dipole sources,” IEEE Trans. Antennas Propag., vol. 57, pp. 1973–1984, July 2009. [20] P. P. Ewald, “Die berechnung optischer und elektrostaticher gitterpotentiale,” Ann. der Physik, vol. 64, pp. 253–287, 1921. [21] K. E. Jordan, G. E. Richter, and P. Sheng, “An efficient numerical evaluation of the Green’s function for the Helmholtz operator in periodic structures,” J. Comp. Physics, vol. 63, pp. 222–235, 1986. [22] F. Capolino, D. R. Wilton, and W. A. Johnson, “Efficient computation of the 3D Green’s function with one dimensional periodicity using the Ewald method,” in Proc. IEEE APS Symp., Albuquerque, NM, Jul. 2006, pp. 2847–2850. [23] F. Capolino, D. R. Wilton, and W. A. Johnson, “Efficient computation of the 3D Green’s function for the Helmholtz operator for a linear array of point sources using the Ewald method,” J. Comp. Phys., vol. 223, pp. 250–261, Apr. 2007. [24] A. Kustepeli and A. Q. Martin, “On the splitting parameter in the Ewald method,” IEEE Microw. Guided Wave Lett., vol. 10, pp. 168–170, May 2000. [25] H. Rogier, “New series expansions for the 3-D Green’s function of multilayered media with 1-D periodicity based on perfectly matched layers,” IEEE Trans. Microw. Theory Tech., vol. 55, pp. 1730–1738, Aug. 2007. [26] S. Singh and R. Singh, “A convergence acceleration procedure for computing slowly converging series,” IEEE Trans. Microwave Theory Tech., vol. 40, pp. 168–171, Jan. 1992. [27] S. Singh and R. Singh, “On the use of Chebyshev-Toeplitz algorithm in accelerating the numerical convergence of infinite series,” IEEE Trans. Microw. Theory Tech., vol. 40, pp. 171–173, Jan. 1992. [28] N. Kinayman and M. I. Aksun, “Comparative study of acceleration techniques for integrals and series in electromagnetic problems,” Radio Sci., vol. 30, no. 6, pp. 1713–1722, Nov.–Dec. 1995. [29] G. S. Wallinga, E. J. Rothwell, K. M. Chen, and D. P. Nyquist, “Efficient computation of the two-dimensional periodic Green’s function,” IEEE Trans. Antennas Propag., vol. 47, pp. 895–897, May 1999. [30] C. M. Linton, “The Green’s function for the two-dimensional Helmholtz equation in periodic domains,” J. Eng. Math., vol. 33, pp. 377–402, May 1998.

[31] M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, 9th ed. New York: Dover , 1970. [32] P. Baccarelli, C. Di Nallo, S. Paulotto, and D. R. Jackson, “A fullwave numerical approach for modal analysis of 1-D periodic microstrip structures,” IEEE Trans. Microw. Theory Tech., vol. 54, pp. 1350–1362, Apr. 2006. [33] R. E. Collin, Field Theory of Guided Waves, 2nd ed. New York: IEEE Press, 1991. [34] F. Capolino, D. R. Wilton, and W. A. Johnson, “Efficient computation of the 2-D Green’s function for 1-D periodic structures using the Ewald method,” IEEE Trans. Antennas Propag., vol. 53, pp. 2977–2984, Sep. 2005.

Ana L. Fructos received the Licenciado degree in physics from the University of Seville, Spain, in 2005. In 2006, she joined the Electronics and Electromagnetism Department, University of Seville, where she is currently working toward the Ph.D. degree. Mrs. Fructos was the recipient of a Scholarship financed by the Junta de Andalucía.

Rafael R. Boix (M’96) received the Licenciado and Doctor degrees in physics from the University of Seville, Spain, in 1985 and 1990, respectively. Since 1986, he has been with the Electronics and Electromagnetism Department, University of Seville, where he became Associate Professor in 1994. His current research interests are focused on the numerical analysis of periodic planar electromagnetic structures with applications to the design of frequency selective surfaces and electromagnetic bandgap passive circuits.

Francisco Mesa (M’93) was born in Cádiz, Spain, on April 1965. He received the Licenciado degree in June 1989 and the Doctor degree in December 1991, both in physics, from the University of Seville, Spain. He is currently Associate Professor in the Department of Applied Physic 1 at the University of Seville, Spain. His research interest focus on electromagnetic propagation/radiation in planar structures.

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Electromagnetic Scattering by an Infinite Elliptic Dielectric Cylinder With Small Eccentricity Using Perturbative Analysis Georgios D. Tsogkas, John A. Roumeliotis, and Stylianos P. Savaidis

Abstract—The scattering of a plane electromagnetic wave by an infinite elliptic dielectric cylinder is examined using two alternative methods. In the first the electromagnetic field is expressed in terms of elliptical-cylindrical wave functions while in the second, a shape perturbation method is applied by expressing the field in terms of circular-cylindrical wave functions only and the equation of the elliptical boundary in polar coordinates. Analytical expressions are obtained for the scattered electromagnetic field and the various scattering cross-sections, when the solution is special, ized to small values of the eccentricity with the interfocal distance of the elliptic cylinder and the length of its major axis. In this case the scattered field and the scattering cross-sections expressions have the form of (2) 2 (4) 4 6 , where the expansion co(2) (4) and are given by exact, closed form expressions. efficients Both polarizations are considered for normal incidence. Numerical results are given for various values of the parameters.

= 2 (

(0) 1 +

+

+ ( )

1) 2 ()=

Index Terms—Electromagnetic scattering, elliptic dielectric cylinder, perturbative analysis, scattering cross sections.

I. INTRODUCTION

T

HE electromagnetic scattering by cylinders of various cross-sections is an old problem with many applications, that has concentrated the interest of a great number of researchers in a lot of papers, applying different methods like, for example, the boundary integral method [1]–[3], the method of moments [4], the finite element method [5]–[10] and the boundary element method [11]–[13]. A special case are the cylinders of elliptical cross-section. Among many papers examining scattering by such cylinders are the ones referring to scattering by a perfectly conducting or by an acoustically impenetrable (soft or hard) elliptic cylinder treated in [14]–[17]. Scattering by a penetrable ribbon or by a dielectric elliptic cylinder has been studied in [18], [19] while scattering by an infinite conducting strip in [20], [21]. In this paper the scattering of a plane electromagnetic -or -wave impinging normally on an infinite elliptic dielectric cylinder, is considered. The geometry of the elliptic cylinder is Manuscript received October 17, 2008; revised April 01, 2009. First published June 10, 2009; current version published January 04, 2010. This work was supported by the Program of Basic Research PEBE 2008 of NTUA. G. D. Tsogkas and J. A. Roumeliotis are with the School of Electrical and Computer Engineering National Technical University of Athens, Athens 15773, Greece (e-mail: [email protected]; [email protected]). S. P. Savaidis is with the Department of Electronics Technological Educational Institute (TEI) of Piraeus, Athens 12244, Greece (e-mail: ssavaid@teipir. gr). Digital Object Identifier 10.1109/TAP.2009.2024527

Fig. 1. The geometry of the scatterer.

shown in Fig. 1. The interfocal distance is , while and are the lengths of its major and minor semiaxes, respectively. The permittivity, the permeability and the wavenumber are , , and , , inside the cylinder (region 1) and outside it (region 2), respectively. Two different methods are used for obtaining solution. In the first the electromagnetic field is expressed in terms of ellipticalcylindrical wave functions. In the second, a shape perturbation method, the solution is implemented by expressing the field in terms of circular-cylindrical wave functions only and the equation of the elliptical boundary in polar coordinates. For small , analytical expresvalues of the eccentricity sions of the form are obtained for the scattered field and the scattering cross-secand given by tions, with the expansion coefficients exact, closed-form expressions, independently of and corresponding to a circular cylinder of radius . The main advantage of the proposed analytical solution lies in its general validity for each small , “free” of Mathieu funcand tions (even in the first method). This means that if are known, is easily evaluated for each in contrast to the numerical techniques which require repetition of the complicated evaluation of Mathieu functions, from the beginning, for each different value of , small or large. So, the presented solution is useful for a “fat” elliptic dielectric cylinder, where it is superfluous to use the general solution. Because the terms omitted in the present solution are of the order of , or higher, the restriction to small values of is not so severe as it appears firstly. Independent numerical solution of this same problem shows that the errors in the approximate analytical results presented in this paper remain low enough, even for values ). of up to 0.3 (maximum possible The -wave polarization is examined in Section II, while the -wave polarization is examined in Section III. Finally, in

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Section IV numerical results are given for various values of the parameters. II.

—WAVE POLARIZATION

and keeping in mind that the angular Mathieu for functions in the two different regions 1 and 2 are not orthogonal ) the inner products (because

A. Solution in Terms of Elliptical-Cylindrical Wave Functions The incident plane electromagnetic wave impinging normally on the -axis and the resulting scattered field have the form [22], [23] see (1) and (2) at the bottom of the page, where , are the transverse elliptical-cylindrical coordinates with respect , and are the even (odd) radial to Mathieu functions of the first and fourth kind [the superscript (2) are the is omitted for simplicity], respectively, while even (odd) angular Mathieu functions. The normalization conare given in [23]. The angle defines the direction stants . The time dependence of incidence with respect to is suppressed throughout. The field inside the dielectric cylinder is

(5)

are required, where are the expansion coefficients for the Mathieu functions [23], with and both even or odd, and is the Neumann factor, while . Solving the equation resulting from the first of (4) one obtains (6) at the bottom of the page. Substituting next from (6) into the equation resulting from the second of (4) after the same procedure one obtains the following and : infinite linear inhomogeneous set for

(7) (3) The unknown expansion coefficients , and are calculated by satisfying the boundary conditions

,

(4) . Multiplying next both memat the elliptical boundary , integrating bers of the resulting equations by

where, see (8)–(9) at the bottom of the following page, with and both even or odd. For general values of , the set (7) can be solved only numerically, by truncation, a complicated task due to the calculation of the Mathieu functions for each different , . However, for small , an analytical, closed form solution is possible. Using, instead, the eccentricity ,( , ) as well as Maclaurin series expansions into powers

(1) (2)

(6)

TSOGKAS et al.: ELECTROMAGNETIC SCATTERING BY AN INFINITE ELLIPTIC DIELECTRIC CYLINDER

of , from [24], for each function of appearing in (5), : (7)–(9), the set (7) takes the form (up to the order

109

The elliptical-cylindrical wave functions in (2) are expanded into circular-cylindrical ones by using the expansion formula [22]

(10) As it is evident from (2) the subscripts of and are always nonnegative. In the opposite case and are equal to zero and so disappear. The same is valid also for the corresponding . The set (10) separates into two distinct subsets, one with even and the other with odd. For small values of one can set, up to the order

(18) In (18) , are the polar coordinates, the cylindrical Hankel function of the second kind, with the superscript (2) omitted for simplicity. one finds from Using the asymptotic expansion for (2) and (18) the scattered far field expression (19) where

(11) (12) Exact expressions for and ( , 2, 4) are given in (A.1)–(A.6) of the Appendix. and are obtained from the solution of the set (10) by Cramer’s rule, using [25 Eqs. (27)–(29)] found by steps similar and are obtained by exto the ones in [26]. So, finally, pressions of the form

(20) is the scattering amplitude. The backscattering , the forward scattering cross-sections are [25], [27]

and the total

(13) (14)

(21)

(15)

After lengthy but straightforward calculations the following , up to the order , is obtained from (20) expansion for

(16)

(22)

(17)

where , and are given in (9)–(11), respectively, of [25] with the only difference that is replaced . here by From (22) it is found that

where

The various symbols appearing in (15)–(17) are defined in (A.39)–(A.45) of Appendix. Similarly , and are given by (15)–(17), respectively, with the superscripts simply replaced by .

(23) with

denoting the real part and

the complex conjugate.

(8)

(9)

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Substituting in the last relation of (21) from (20) is obtained and is set in a form similar to (22)

In order to satisfy the boundary conditions

(24) , and are given by (14)–(16), rewhere spectively, of [25] with the only difference that is replaced here by . Equation (24) can be set in the form

(25) A similar relation is obtained for

and

(30) at the surface of the elliptic cylinder (the form of the second condition is explained in [28]) the equation of this surface is expressed in terms of the polar coordinates and and its ex[25], [28]. pansion into powers series of is used, for

, namely (26) (31)

Certainly, radius

and .

correspond to a circular cylinder of

B. Solution in Terms of Circular-Cylindrical Wave Functions Only The second method is applied next, using circular-cylindrical wave functions only. The incident plane electromagnetic wave impinging normally on the -axis is now expressed as [22]

(27) is the cylindrical Bessel function of the first kind. where The scattered field and the field inside the dielectric cylinder are (28) (29)

where

Using (31) the expansions for , , 2, and can be found, given in [25, Eq. (23)] (with ). and Also similar expansions are obtained for , with the only difference that one more prime is added in each one of the Bessel and Hankel functions. Finally and are given the expansions for in (41) of [25]. These expansions are substituted into (27)–(29), satisfying the boundary conditions (30) and use is made of the orthogonal properties of the trigonometric functions, concluding finally to two infinite sets of linear inhomogeneous equations for the ex, and , , up to the order : pansions coefficients See (32)–(33) at the bottom of the page. As it is evident from (28) and (29) the subscripts of , and also of , are always nonnegative. In the opposite case , and , are equal to zero and so disappear. The same is valid also for the corresponding , , and . The sets (32) and (33) separate into two distinct subsets, one with even and the other with odd.

(32)

(33)

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111

For small values of , moreover to (11) and (12) valid also for the new and in (32), similar expansions are obtained for the (39) (40) (34) and in (33) which are given again from as well as for , (11), (12) and (34), respectively, with the corresponding expan, , sion coefficients primed. Exact expressions for and , , ( , 2, 4) are given in (A.46)–(A.51) of the Appendix. and as well as and can be obtained from the solution of the sets (32) and (33) by Cramer’s rule. To make possible the application of the method presented in [26], [29] one should transform these sets so that no coefficients including a zero order term in their expansion, with respect to , lie off the main diagonal in the determinant of the sets. Following exactly the same steps with the ones described in detail in [30] from (32) and (33) there results (35)–(36) at the bottom of the page, where

(37)

while

,

,

are given by the same expressions (37) with

the corresponding unprimed symbols, if substituted by and Expansions for , and have the form

and

are

, respectively. in (35) and (37) for small values of

Similar expansions are obtained for , and in (36), with the corresponding expansion coefficients primed. Exact expres, , can be immediately obtained from sions for , , and , , (37), by simply using ,( , 2, 4) from (A.46)–(A.51) of the Appendix

(41) The same is valid also for , given by the same expressions (41) if

and and

placed by and , respectively. As is evident from (41), . Also,

which are are reand so all

the coefficients including a zero order term in their expansion, with respect to , lie on the main diagonal of the determinant of the sets (35) and (36). The determinant originates from after the substitution of the column of the coefficients and of (or ) by the column of and [ (or ) takes all integer values of the same are given in [30, parity with ]. The expansions of and Eqs. (47) and (48)], where should be replaced by 0 and which will not be repeated here. Using the expansions (38)–(40), the similar ones for , and , as well as (41) and the corresponding one for , and into [30, Eqs. (47)–(49)], (and ) is finally obtained, where

(42) (38)

while, see (43)–(45) at the bottom of the following page.

(35)

(36)

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VALUES OF g FOR  =

TABLE I : ,  =

= 2 54

The various symbols appearing in (43)–(45) are the same with the corresponding ones appearing in [30, Appendix, Eqs. (A.49)–(A.56)] if we add a superscript “e” in each case, so they will be not repeated here. is given by an expression analogous to (42) Similarly (46) where , and are again given by (43)–(45), respectively, with the superscripts replaced by . Analogous results can be obtained for the evaluation of and .

= 1, k a = 

Using the asymptotic expansion for again obtained, where in this case

in (28), (19) is

(47) Equations (21)–(23) are also valid while

is given by

(48)

(43) (44)

(45)

TSOGKAS et al.: ELECTROMAGNETIC SCATTERING BY AN INFINITE ELLIPTIC DIELECTRIC CYLINDER

Fig. 2. Backscattering cross-section for  = E -wave.

= 2:54,  = = 1, k a =  ,

Fig. 3. Forward scattering cross-section for  = k a  , E -wave.

=

= 2:54,  = = 1,

Moreover (24)–(26) are valid while , and are given by [25, Eqs. (36)–(38)]. Another check for the correctness of our results, moreover to the use of the two different methods for the solution, is given by the fulfilment of the forward scattering theorem which in the present case has the form [27] (49) Its validity was verified numerically for various values of the parameters. III.

—WAVE POLARIZATION

All steps and formulas are exactly the same as in Section II, with the only difference that and are replaced by and , respectively, while remain unchanged ( , 2).

Fig. 4. Total scattering cross-section for  = E -wave.

Fig. 5. Backscattering cross-section for  =

1:4 , H -wave.

113

= 2:54,  = = 1, k a =  ,

= 2:54,  = = 1, k a =

IV. NUMERICAL RESULTS AND DISCUSSION and appearing in (25) and In Table I the values of , for the (26) are given for various values of , for - and the -wave polarization for an infinite elliptic dielec, . The results are tric cylinder with , as it is imposed by the geometry of symmetric about the scatterer. In Figs. 2–4 the scattering cross–sections are plotted, versus , for the –wave polarization, for , and . The same is done in Figs. 5–7 for the -wave po. In each figure the corresponding larization and for scattering cross-section for (circular cylinder with radius ) is also plotted. The results are symmetric about . From Figs. 2–4 as well as from Table I the higher sensitivity to the change of is evident, as compared to that of of or , in the case of -wave. It is also evident, from all figures, that the deviation of the elliptic cylinder from the corresponding

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Fig. 6. Forward scattering cross-section for  = k a :  , H -wave.

= 14

Fig. 7. Total scattering cross-section for  = :  , H -wave.

14

Fig. 8. Backscattering cross-section in dB versus k a for

=0

,  =

=

Fig. 9. Backscattering cross-section in dB versus k a for

=0

,  =

=

= 2:54,  = = 1, 2:54,  = = 1, E -wave.

= 2:54,  = = 1, k a =

circular one increases or decreases the scattering cross-sections, depending on the values of the parameters. In Figs. 8 and 9 the backscattering cross-sections in are plotted versus for and for - and -wave, respectively. The results of Figs. 2–9 were verified, to a high degree of accuracy, by comparing to independent results obtained from the numerical solution of the same problems. In this solution the Mathieu functions were computed with custom made routines. However one can also use relative routines using “Mathieu Functions Toolbox” which can be found in Mathworks website. In Figs. 10–12 the percentage errors are given for , and , respectively, resulting from the comparison of the approximate results of this paper with the exact numerical results obtained independently. In each , while . From these figures, as well figure as from many other available, it is evident that the percentage

2:54,  = = 1, H -wave.

is much lower error for approximate results up to the order than the corresponding one for approximate results up to the (especially for ) thus permitting values of order up to 0.3 (maximum possible ) with low errors, especially for and . The phenomenal decrease of the percentage errors in our approximate results, up to the order , in the range , appearing in Figs. 10–12, is due to the failure of our analytical approximation, at least for the values of the parameters used, which means that expansion up to the order , or higher, should be necessary in this case. The values of used in Figs. 2–9 keep the percentage error of these results low enough in each case. Finally, in Figs. 13 and 14 the backscattering cross-sections for the -wave and the -wave, given in [19, Figs. 2 and 6], are also obtained by both of our methods, for the same values . The of the parameters used in [19] and for same symbols and parameters used in [19] are also used here, for reasons of easy comparison. This comparison shows that our methods give good results, thus making a further check for their correctness. The only differences appear in the minima of the

TSOGKAS et al.: ELECTROMAGNETIC SCATTERING BY AN INFINITE ELLIPTIC DIELECTRIC CYLINDER

Fig. 10. Percentage error for k  versus h for  = ,k a  , E , H -wave.

= 45

=2

Fig. 11. Percentage error for k  versus h for  = ,k a  , E , H -wave.

= 45

=2

= 2:54,  = = 1,

115

4 = 0:001.

Fig. 13. Reproduction of Fig. 2 in [19] for k q =

= 2:54,  = = 1, 4 = 0:001.

Fig. 14. Reproduction of Fig. 6 in [19] for k q =

Fig. 12. Percentage error for k Q versus h for  = ,k a  , E , H -wave.

= 45

=2

= 2:54,  = = 1,

backscattering cross-section for the -wave in Fig. 14, which are lower enough than the corresponding ones in [19, Fig. 6]. and used to plot Figs. 13 and 14 can also be The same . This used immediately for any other small values of is the great advantage and the innovation of our perturbative analytical methods, as compared with many other methods, like, for example those in [18], [19] where the evaluation should be repeated, from the beginning, for each different . value of In the special case where and the results of the present work conclude to the ones of electromagnetic scattering by an infinite elliptic metallic cylinder [25]. This was verified numerically (the analytical proof is lengthy and tedious) for various values of the parameters, using the extreme and , thus constituting a ratios further check for the correctness of our solution.

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It should be noticed here that all former results were obtained by both of our methods with a difference between them less than , at least for the values of the parameters used. APPENDIX The expressions for the various symbols appearing in the text and not defined there, are given in what follows.

A.

-Wave Polarization

Solution in Terms of Elliptical-Cylindrical Wave Funcand appearing in (11) tions: The exact expressions for and (12) are shown in (A.1)–(A.6) at the bottom of the page, where, see (A.7)–(A.38) at the bottom of the page and the pages that follow.

(A.1) (A.2) (A.3) (A.4)

(A.5)

(A.6)

(A.7)

(A.8)

(A.9) (A.10) (A.11)

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(A.12) (A.13) (A.14)

(A.15) (A.16) (A.17)

(A.18) (A.19) (A.20)

(A.21) (A.22)

(A.23)

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Moreover , and respectively, by simply replacing

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are given by (A.7)–(A.9), by and

by by (A.27)–(A.29),

, while are given by (A.30)–(A.32), respectively, and

(A.24)

(A.25)

(A.26) (A.27) (A.28)

(A.29) (A.30) (A.31)

(A.32) (A.33) (A.34)

(A.35) (A.36) (A.37)

(A.38)

TSOGKAS et al.: ELECTROMAGNETIC SCATTERING BY AN INFINITE ELLIPTIC DIELECTRIC CYLINDER

by (A.33)–(A.35), by (A.36)–(A.38), respectively, by simply replacing, in each case, the Bessel functions of the first kind (and their derivatives) with the corresponding Hankel functions of the second kind (and their derivatives). The various and appearing in the former relations have the expressions given in [25, Appendix, Eqs. (A.6)–(A.12)]. The quantities inside parentheses, for both these symbols, are used instead of the superscript in [24]. The new symbols appearing in (15)–(17) are defined as in (A.39)–(A.45) at the bottom of the page. , , the superscripts in For the evaluation of (A.39)–(A.45) are simply replaced by .

119

Solution in Terms of Circular-Cylindrical Wave Functions Only: See (A.46)–(A.51) at the bottom of the page and on the following page. , , The corresponding odd expansion coefficients , , and , , necessary for the are again given by (A.46)–(A.51), respectively, evaluation of with the only difference that is replaced by and , , and by , , and , respectively. , used in this section are The integrals given in [25, Appendix, Eqs. (A31)–(A36)].

(A.39)

(A.40)

(A.41)

(A.42)

(A.43) (A.44)

(A.45)

(A.46)

(A.47)

(A.48)

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(A.49)

(A.50)

(A.51)

B.

-Wave Polarization

All formulas are exactly the same as for the -wave polarization, with the only difference that and are replaced by and , respectively, while and are remain unchanged. REFERENCES [1] D. L. Young, C. L. Chiu, and C. M. Fan, “Nonsingular boundary integral equation for two-dimensional electromagnetic scattering problems,” Microw. Opt. Technol. Lett., vol. 48, pp. 760–765, 2006. [2] M. A. Haider, S. P. Shipman, and S. Venakides, “Boundary-integral calculations of two-dimensional electromagnetic scattering in infinite photonic crystal slabs; channel defects and resonances,” SIAM J. Appl. Math., vol. 62, pp. 2129–2148, 2002. [3] S. Venakides, M. A. Haider, and V. Papanicolaou, “Boundary integral calculations of two-dimensional electromagnetic scattering by photonic crystal fabry-perot structures,” SIAM J. Appl. Math., vol. 60, pp. 1686–1706, 2000. [4] Y. Shifman, M. Friedmann, and Y. Leviatan, “Analysis of electromagnetic scattering by cylinders with edges using a hybrid moment method,” Proc. Inst. Elect. Eng. Microwave Antennas Propag., vol. 144, pp. 235–240, 1997. [5] F. Farail, A. Roger, and M. Bouthinon, “Finite-element method for electromagnetic scattering in a lossy medium,” Proc. Inst. Elect. Eng., vol. 138, pt. H, pp. 201–206, 1991. [6] P. D. Ledger et al., “The development of an hp-adaptive finite element procedure for electromagnetic scattering problems,” Finite Elem. Anal. Des., vol. 39, pp. 751–764, 2003. [7] Y. Zhu and A. C. Cangellaris, “Application of nested multigrid finite elements to two-dimensional electromagnetic scattering,” Microw. Opt. Technol. Let., vol. 30, pp. 97–101, 2001. [8] Y. Zhu and A. C. Cangellaris, “Nested multigrid vector and scalar potential finite element method for fast computation of two-dimensional electromagnetic scattering,” IEEE Trans. Antennas Propag., vol. 50, pp. 1850–1858, 2002. [9] S. Alfonzetti and G. Borzi, “Accuracy of the Robin boundary condition iteration method for the finite element solution of scattering problems,” Int. J. Numer. Model El., vol. 13, pp. 217–231, 2000. [10] M. O. Bristeau, R. Glowinski, and J. Periaux, “Wave scattering at high wave-numbers using exact controllability and finite-element methods,” IEEE Trans. Magn., vol. 31, pp. 1530–1533, 1995. [11] K. Yashiro and S. Ohkawa, “Boundary element method for electromagnetic scattering from cylinders,” IEEE Trans. Antennas Propag., vol. 33, pp. 383–390, 1985.

[12] G. F. Wang, “A hybrid wavelet expansion and boundary-element analysis of electromagnetic scattering from conducting objects,” IEEE Trans. Antennas Propag., vol. 43, pp. 170–178, 1995. [13] J. L. Yaobi, L. Nicolas, and A. Nicolas, “2D electromagnetic scattering by simple shapes-A quantification of the error due to open boundary,” IEEE Trans. Magn., vol. 29, pp. 1830–1834, 1993. [14] J. E. Burke and V. Twersky, “On scattering of waves by an elliptic cylinder and by a semielliptic protuberance on a ground plane,” J. Opt. Soc. Am., vol. 54, pp. 732–744, 1964. [15] J. E. Burke, E. J. Christensen, and S. B. Lyttle, “Scattering patterns for elliptic cylinders,” J. Opt. Soc. Am., vol. 54, pp. 1065–1066, 1964. [16] R. Barakat, “Diffraction of plane waves by an elliptic cylinder,” J. Acoust. Soc. Am., vol. 35, pp. 1990–1996, 1963. [17] K. Udagawa and Y. Miyazaki, “Diffraction of a plane wave by a perfectly conducting elliptic cylinder-a study by conformal mapping technique,” J. Inst. Elect. Commun. Eng., vol. 48, pp. 43–53, 1965. [18] C. Yeh, “The diffraction of waves by a penetrable ribbon,” J. Math. Phys., vol. 4, pp. 65–71, 1963. [19] C. Yeh, “Backscattering cross section of a dielectric elliptical cylinder,” J. Opt. Soc. Am., vol. 55, pp. 309–314, 1965. [20] E. B. Hansen, “Scalar diffraction by an infinite strip and a circular disc,” J. Math. Phys., vol. 41, pp. 229–245, 1962. [21] D. S. Jones and B. Noble, “The low-frequency scattering by a perfectly conducting strip,” in Proc. Cambridge Phil. Soc., 1961, vol. 57, pp. 364–366. [22] J. A. Stratton, Electromagnetic Theory. New York: McGraw-Hill, 1941. [23] P. M. Morse and H. Feshbach, Methods of Theoretical Physics. New York: McGraw-Hill, 1953. [24] G. C. Kokkorakis and J. A. Roumeliotis, “Power series expansions for Mathieu functions with small arguments,” Math. Comput., vol. 70, pp. 1221–1235, 2001. [25] G. D. Tsogkas, J. A. Roumeliotis, and S. P. Savaidis, “Scattering by an infinite elliptic metallic cylinder,” Electromagnetics, vol. 27, pp. 159–182, 2007. [26] J. A. Roumeliotis and J. G. Fikioris, “Scattering of plane waves from an eccentrically coated metallic sphere,” J. Franklin Inst., vol. 312, pp. 41–59, 1981. [27] J. A. Roumeliotis, H. K. Manthopoulos, and V. K. Manthopoulos, “Electromagnetic scattering from an infinite circular metallic cylinder coated by an elliptic dielectric one,” IEEE Trans. Microw. Theory Tech., vol. 41, pp. 862–869, 1993. [28] G. C. Kokkorakis and J. A. Roumeliotis, “Acoustic eigenfrequencies in concentric spheroidal-spherical cavities: Calculation by shape perturbation,” J. Sound Vibration, vol. 212, pp. 337–355, 1998.

TSOGKAS et al.: ELECTROMAGNETIC SCATTERING BY AN INFINITE ELLIPTIC DIELECTRIC CYLINDER

[29] J. A. Roumeliotis, A. B. M. S. Hossain, and J. G. Fikioris, “Cutoff wavenumbers of eccentric circular and concentric circular-elliptic metallic waveguides,” Radio Sci., vol. 15, pp. 923–937, 1980. [30] A. D. Kotsis and J. A. Roumeliotis, “Acoustic scattering by a penetrable spheroid,” Acoust. Phys., vol. 54, pp. 153–167, 2008.

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John A. Roumeliotis was born in Corinth, Greece, in 1953. He received the Electrical Engineering and the Dr. Eng. degrees from the National Technical Univercity of Athens (NTUA), Greece, in 1975 and 1979, respectively. From 1979 to 1981, he fulfilled his military service. Since 1982, he has been with the Electrical Engineering Department, NTUA, where he is now a Professor. His research interests include scattering and wave propagation and boundary value problems in electromagnetic theory and acoustics, as well as applied mathematics.

Georgios D. Tsogkas was born in Corinth, Greece, in 1974. He received the Electrical Engineering degree from the Technological Educational Institute of Chalkis, Greece, in 1997 and the Electrical Engineering degree from the National Technical University of Athens, Greece, in 2004, where he is currently working toward the Ph.D. degree. His research interests include scattering and wave propagation in elliptic cylinders.

Stylianos P. Savaidis was born in Sparta, Greece, in 1967. He received the E.E. and Ph.D. degrees from the National Technical University of Athens, Greece, in 1991 and 1997, respectively. From 1997 to 2004, he was a Telecommunication Engineer in wireless communication networks. Since 2004, he has been with the Electronics Department, Technological Educational Institute of Piraeus, where he is now an Associate Professor. His research interests include scattering and wave propagation in complex cylindrical structures and electromagnetic applications of fractal geometry.

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Improved Model-Based Parameter Estimation Approach for Accelerated Periodic Method of Moments Solutions With Application to the Analysis of Convoluted Frequency Selected Surfaces and Metamaterials Xiande Wang, Senior Member, IEEE, and Douglas H. Werner, Fellow, IEEE

Abstract—An improved “smart” interpolation approach known as model-based parameter estimation (MBPE) is applied to the wide-band interpolation of periodic method of moments (PMM) impedance matrices for normal and oblique incidence cases. Prior to interpolation, easy to calculate but hard to interpolate, phase terms are removed from the impedance matrices. An efficient spectral-domain PMM formulation is introduced for the accelerated analysis of frequency selective surface (FSS) problems with a large number of unknowns, employing a one dimensional ( log ) FFT-based method to speed up the computation of matrix-vector products within the bi-conjugate gradient (BCG) iterative solver, which is made possible by the asymmetric multilevel block-Toeplitz structure of the impedance-matrix. The MBPE interpolation algorithm provides a faster matrix fill time than the brute force method and is comparable or even faster than the 2-D FFT-based method for a large number of unknowns. It also has the advantage that it can be applied to non-uniform gridding cases. The accuracy and efficiency of the proposed techniques for large FSS problems are demonstrated by several design examples for both the normal and oblique incidence cases. We also apply this efficient analysis tool to the design of multiband single-layer FSS filters and artificial magnetic conductors (AMC) comprised of a 2-D periodic arrangement of convoluted metallic strips in the shape of a Hilbert curve. The multiband properties of the Hilbert curve FSS filters are studied for different iteration orders (i.e., different degrees of space-filling). Index Terms—Artificial magnetic conductors (AMC), fast Fourier transform (FFT), frequency selective surfaces (FSS), impedance matrix interpolation, metamaterials, model-based parameter estimation (MBPE), multiband FSS, periodic moment method (PMM), space-filling Hilbert curves, spectral-domain periodic method of moments (PMM).

I. INTRODUCTION INCE their introduction, frequency selective surfaces (FSS) have been utilized in a wide range of applications such as electromagnetic filters, radomes, and more recently

S

Manuscript received May 24, 2008; revised December 15, 2008. First published November 06, 2009; current version published January 04, 2010. This work was supported in part by the Penn State MRSEC under NSF Grant DMR0820404 and in part by ARO-MURI Award 50342-PH-MUR. X. Wang and D. H. Werner are with the Department of Electrical Engineering, The Pennsylvania State University, University Park, PA 16802 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.2009.2036196

as components of metamaterials. Spatial-domain and spectral-domain formulations of the periodic method of moments (PMM) [1]–[3] have been widely used for the analysis of FSS’s. PMM formulations only require the discretization of one unit cell in a two-dimensional periodic FSS screen and employ a layered-media Green’s function to account for the presence of the dielectric substrate and superstrate. This leads to an important advantage of the PMM; namely, it is very fast compared to other techniques (e.g., the hybrid finite element boundary integral (FEBI)-based method [4]) in the analysis of FSS structures composed of a homogenous layered-media. The most computer-intensive operations in the PMM are filling the impedance matrix and solving the linear matrix equation to obtain the current distribution on the FSS screen. Under certain circumstances of practical importance, these operations can become computationally burdensome, especially when performing frequency and angle sweeps. In order to overcome this shortcoming, impedance matrix interpolation techniques were first proposed in [5] to significantly reduce the large matrix fill times associated with certain antenna analysis problems. A modified three-sampling-point interpolation technique was later applied in [6] to more efficiently predict the response of FSS screens. A reduced-order model was also proposed to generate a broadband approximation of the reflection coefficients for multiple-screen FSS [7]. More recently, a model-based parameter estimation (MBPE) technique was used to effectively interpolate the frequency response of electromagnetic systems [8] and for wide-band interpolation of impedance matrices [9]. However, the MBPE-based technique proposed in [9] was only applied to the normal incidence case for the wideband interpolation of PMM impedance matrices related to the FSS with substrate and superstrate configurations but not to oblique incidence cases. In this paper, we extend it to oblique incidence cases by the development of a robust and efficient algorithm that can be described as a “smart” MBPE-based approach which removes problematic phase terms from the impedance matrices prior to the interpolation process. We also describe how this approach can be applied to the analysis of large FSS problems with the help of FFT acceleration techniques produced by incorporating a generalized one-dimensional (1-D) FFT-based method into the implementation of an PMM. This method has been originally proposed in [10] for

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Fig. 1. Cross-sectional view of the FSS structure with both a dielectric substrate and a superstrate.

analyzing 3-D scattering problems, to speed up computation of multilevel block-Toeplitz matrix-vector products within the bi-conjugate gradient (BCG) iterative solver. The validation and the efficiency of the proposed algorithms are demonstrated by comparisons of the CPU times required for the conventional PMM and the improved MBPE-based interpolation algorithm for several design examples. We also describe the application of the proposed acceleration techniques to the design of a polarization dependent/independent multiband single-layer FSS as well as artificial magnetic conducting (AMC) surfaces using a 2-D periodic arrangements of convoluted Hilbert-curve-shaped strips with iteration orders as high as four [16]. In addition to AMC surfaces, the MBPE-based interpolation algorithm could also be employed for the efficient analysis and design of other types of metamaterials. II. IMPROVED MBPE INTERPOLATION AND FFT-BASED ACCELERATION A detailed discussion of the spectral-domain PMM can be found in [3] for the analysis of doubly periodic FFS structures (e.g., an FSS with both a dielectric substrate and a superstrate as shown in Fig. 1 or an FSS screen backed by a dielectric substrate as shown in Fig. 2). By expanding the surface electric current within a unit cell in terms of 2-D roof-top basis functions and then employing Galerkin’s procedure to the appropriate boundary electric field integral equation, the following linear matrix equation is obtained:

Fig. 2. The Jerusalem slot FSS screen backed by a dielectric substrate with unit cell dimensions 2.5 cm 2.5 cm and a thickness of 0.02 cm.

2

convergence of the double summation. Therefore, we refer to this conventional PMM formulation as the “brute-force” calculation method. In order to reduce the large matrix fill times that can be associated with conventional PMM formulations, we employ an efficient MBPE interpolation method to estimate the impedance matrix elements following the ideas originally presented in [9], where only the normal incidence case was investigated. Here, a more robust MBPE-based interpolation technique is introduced for both normal and oblique incidence cases by removing phase terms from impedance matrix elements prior to interpolation, which are easy to calculate but difficult to interpolate. For excan be expressed as ample, an element of the submatrix follows:

(3)

(1)

where is the phase factor associated with the source edge and test edge position and the incifor the normal incidence case. The dent wave vector, and following fitting model has been found to be very effective [the same MBPE model can be used for fitting all other elements in in (1)]: the matrices

For the sake of illustration consider an expression for the elgiven as (similar expressions ements in the submatrices , and ) exist for the other submatrices

(4)

where (2) where is the spectral-domain Green’s function, and denotes the Fourier transform of the basis function. Direct calculation of (2) is very time consuming due to the relatively slow

. The parameters

, , , are defined in [3], [9] for the FSS structure with both a dielectric superstrate and substrate. The parameters , , and in (4) denote the interpolation coefficients, while represents the different combinations of the Floquet harand , denotes the total number of Floquet harmonics monics required, which is dependent upon the total number of

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cutoff frequencies that fall within the range of interest. By samat a total of frequencies points , pling (4) can be converted into the following matrix equation: (5) is called the interpolation matrix. The interpolation where can be obtained by solving the matrix (5) via coefficients LU decomposition (LUD) or by using singular value decomposition (SVD), which is more robust than LUD. Upon obtaining interpolation coefficients, the interpolated impedance matrix elements can be calculated by multiplying the results from (4) by . It is well known the phase factor, i.e., that the cutoff frequencies are dependent on the periodicity of the FSS structure and the working frequency range and are not impacted by the resonance behavior of the FSS. Hence, the fitting frequencies can be determined by the automated selection procedure described in [9]. In addition, since the interpolation is independent of , only a one time implemenmatrix . The fast tation of LUD or SVD is required to solve for computation of the impedance matrix elements required to fill in (4) and (5), is achieved through a 2-D FFT-based method [11] which can be applied at desired sampling frequencies to calculate the double summation for the appropriate impedance . sub-matrix elements of The uniform discretization of the FSS screen and the roof-top basis functions employed here in conjunction with Galerkin’s in (1) that has an asymmethod lead to an impedance matrix metric multilevel block-Toeplitz structure. An FFT-based method with a 1-D FFT implementation [10] can be applied to speed up the matrix-vector product within the BCG iterative solver. This can be classified as a truly minimal memory method because it stores only nonredundant matrix entries for the asymmetric multilevel block-Toeplitz structure memory requirements. It may be expressed as with , where and are 1-D vectors which can be automatically generated from the asymmetric multilevel block-Toeplitz maand vector , respectively. Here, and denote trix the 1-D forward FFT and the 1-D inverse FFT, respectively. A FFT-based method for detailed discussion of the a 1-D implementation can be found in [10]. III. MULTIBAND FSS AND AMC SURFACE DESIGNS WITH HILBERT-CURVE SHAPED STRIPS Conventional approaches to the design of multiband FSS typically involve using multilayer surfaces or perturbing the elements or the spacing between elements [12]. Multiband

FSS have also been designed by utilizing certain types of self-similar fractal elements [13], or by a combination of genetic algorithm (GA) and geometry-refinement techniques [14], [15]. More recently, spacing-filling curves [16] have been introduced to the microwave engineering community, e.g., space-filling Hilbert curves have been applied in the design of FSS [17], [18] and AMC surfaces [19], [20]. However, utilizing metallic strips in the shape of Hilbert space-filling curves with high iteration order (i.e., iteration orders greater than three) for application in multiband FSS and AMC surface designs have not been adequately investigated primarily due to limitations on conventional PMM modeling tools. In this paper, polarization dependent/independent multiband single-layer FSS and AMC surfaces are investigated using 2-D periodic arrangements of Hilbert-curve-shaped strips. The developed PMM solver has been utilized to efficiently analyze Hilbert-curve FSS and AMC surfaces with a large number of unknowns and higher iteration order than those considered in previous applications. IV. NUMERICAL RESULTS AND DISCUSSION A. Validation and Efficiency of the Proposed Techniques A specific FSS design example consisting of the Jerusalem slot FSS screen shown in Fig. 2 was considered to validate the accuracy and efficiency of the proposed PMM acceleration techniques. Note that nonmagnetic materials are assumed in the following simulations unless otherwise specified. For this example, as presented in Fig. 2, the FSS is assumed to be backed by a dielectric substrate with a thickness and rel, respectively. ative permittivity of 0.02 cm and The FSS structure is doubly periodic with a periodicity de. For this configuration, the fined by total number of Floquet harmonics required to cover the frequency range of 1.0–20 GHz was found to be based on the following equation, which can be used to predict the cutoff frequency for each Floquent harmonic denoted and , shown in (6) at the bottom of the page, where by and [9]. The [see (4)] fitting frequencies were locations of the determined by using the automated selection scheme described in [9], and are given by for the normal incidence case. A discretized unit cell consisting of a 32 32 grid of pixels is used in the examples unless specified otherwise. The MBPE-based interpolation method has been successfully applied to speed up the evaluation of PMM impedance matrix elements over a wide frequency band of interest as compared to the conventional PMM [9] for the normal incidence cases. However, it is well known that the impedance matrix elements are

(6)

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Fig. 3. Comparison of impedance matrix elements calculated by the conventional PMM expressed in (2) and the PMM with the improved MBPE-based interpolation technique described in (4). The plots are for an arbitrarily chosen element in [Z ] for the oblique incidence case:  = 45 and  = 1 . (a) Magnitude in dB for an element of [Z ], (b) phase angle for an element of [Z ]; (c) magnitude in dB for an element of [Z ], (d) phase angle for an element of [Z ]; (e) magnitude in dB for an element of [Z ], (f) phase angle for an element of [Z ]; (g) magnitude in dB for an element of [Z ], (h) phase angle for an element of [Z ].

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also dependent upon the incidence angles and we have examined the accuracy and efficiency of the proposed MBPE-based interpolation technique introduced in (4) for both the normal incidence and oblique incidence cases. The technique introduced here is more robust for oblique incidence cases than the method presented in [9], due primarily to the removal of a phase factor (that varies rapidly with respect to frequency) from the impedance matrix elements prior to interpolation. Fig. 3 compares the response over the frequency range of 1.0–20 GHz for impedance matrix elements (selected at random from for the FSS configuration illustrated in Fig. 2) evaluated by the conventional PMM and the improved MBPE-based approach , . Excellent for the oblique incidence case: agreement was observed between the two methods as shown , Fig. 3(c) and (d) for , in Fig. 3(a) and (b) for Fig. 3(e) and (f) for , and Fig. 3(g) and (h) for . For the normal incidence case, good agreement has also been observed as demonstrated previously in [9]. The transmission and reflection spectra of the FSS were simulated using three PMM solvers (e.g., conventional PMM, improved MBPE-based interpolation approach proposed in this paper, and the MBPE-based method introduced in [9] with no removal of phase factors) for the validation of the developed code. Very good agreement was obtained for the reflection and transmission properties of the Jerusalem slot FSS shown in Fig. 2 as demonstrated by the comparisons shown in Fig. 4(a) for the co-polarized normal incidence. For the TM-polarized oblique incidence cases, the results computed by the improved MBPE-based interpolation algorithm agreed well with those calculated via the conventional PMM. However, good agreement was not obtained for the MBPE-based technique introduced in [9] as illustrated in Fig. 4(b) and (c), respectively, for the TM-polarized oblique incidence case (the corresponding TE-polarized results are not included here). The total CPU time required to perform each PMM simulation of the FSS unit cell illustrated in Fig. 2 has been recorded in Table I for the brute-force method, the conventional 2-D FFTbased method and the improved MBPE-based interpolation algorithms. The FSS unit cell was discretized by a 64 64 and a 128 128 grid of pixels for the normal incidence and the oblique incidence cases, respectively. The corresponding CPU times required by the different methods for nonredundant matrix element filling at a single frequency point are presented in Table II for the normal incidence case. For all of the results presented here, the computations were performed on a Xeon 3.0 GHz processor machine with 8.0 GB of RAM. The BCG iterative solver with the help of the 1-D FFT-based acceleration technique and improved MBPE-based technique provided a faster solver compared to the brute-force method and resulted for the overall solution time as indiin a speed-up of cated in Table I. As the bandwidth of interest is reduced, the corresponding number of sampling points used in the MBPE approach decreased, which results in faster matrix filling (see Table II). Alternatively, the performance of the 2-D FFT-based method [11] does not depend on a particular frequency range. The advantage of the MBPE interpolation method is that it provides a faster matrix fill time than the brute-force method and can be comparable or even faster than the 2-D FFT-based

Fig. 4. Frequency response (co-polarized) calculated by three different PMM approaches for the Jerusalem slot FSS shown in Fig. 2 at different incidence and  angles. (a) Reflection and transmission coefficients for  , (b) reflection coefficients for  and  , (c) transmission and  . coefficients for 

1

= 45

= 45 =1

=1

=1

=

method, especially in the case of frequency sweeps when solving large FSS problems with many unknowns as demonstrated in Table I and II. The MBPE-based interpolation technique also has the additional advantage of being independent of the choice of basis functions employed in the PMM formulation. Hence, (4) can still be applied to the case of

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TABLE I COMPARISON OF CPU TIMES FOR THE PMM CALCULATIONS USING THE BRUTE-FORCE METHOD, THE 2-D FFT-BASED METHOD AND THE IMPROVED MBPE-BASED INTERPOLATION ALGORITHM FOR THE FSS DESIGN PRESENTED IN FIG. 2

TABLE II CPU TIMES REQUIRED FOR MATRIX FILLING VIA THE BRUTE-FORCE ALGORITHM, THE 2-D FFT-BASED METHOD AND THE IMPROVED MBPE-BASED INTERPOLATION METHOD (NORMAL INCIDENCE CASE) FOR THE FSS DESIGN SHOWN IN FIG. 2

non-uniform discretization of the FSS unit cell; however, the 2-D FFT-based method [11] cannot be applied to this type of problem. Further, the MBPE interpolation is not influenced by the overall frequency response of the FSS unlike other scattering parameter-based methods, such as three-point-sampling interpolation techniques [5], [6]. B. Application to Multiband FSS and AMC Surfaces With Hilbert Curve Shaped Strip Elements The Hilbert curve is a continuous fractal space-filling curve, which is an important member of the family of space-filling curves [16]. For a Hilbert curve with side dimension and it, the length of each line segment and the sum eration order of the lengths of all the line segment are given by [16] (7a) (7b) The geometry for a 2-D Hilbert-curve shaped strip is illustrated in Fig. 5(a) and (b) for iteration orders equal to and , respectively. From (7b), it can be seen that the total length of all the line segments becomes longer within the same is increased. size unit cell as the iteration order 1) Polarization Dependent Multiband Hilbert Curve FSS: A multiband FSS was implemented using a 2-D periodic arrangement of metallic strip elements shaped like Hilbert-curves with as shown in Fig. 5(a). Here, an iteration order equal to the FSS was assumed to be backed by a dielectric substrate . The unit cell dimenwith a relative permittivity of sions were chosen as 2.5 cm 2.5 cm with a dielectric substrate thickness of 0.02 cm. The FSS unit cell was discretized into a

Fig. 5. Hilbert space-filling curves with various iteration orders. (a) n (b) n .

=5

= 4,

64 64 grid. In this case, a total of only Floquet harmonics were required for the MBPE fitting model, where the

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fitting frequencies were selected over the range of 0.1–10 GHz . as These fitting frequencies were determined based on the automated selection scheme described in [9] with The reflection spectra computed via the three PMM solvers are presented in Fig. 6(a) and (b) (for normal incidence cases), and in Fig. 6(c) and (d) (for oblique incidence cases) for the TE and TM modes, respectively. As can be seen, the agreement between these three different PMM solvers was excellent for the normal incidence case. It should be noted that the non-symmetric FSS screen configuration results in polarization dependent multiband filters. For the oblique incidence case, the improved MBPE-based interpolation algorithm generated the correct results, but accurate results were not obtained for the MBPE-based interpolation technique introduced in [9] as illustrated in Fig. 6(c) and (d). Note that the corresponding transmission coefficients are not plotted here. Almost the same frequency response for the oblique incidence case at the lower frequency range was observed for the TE- and TM polarization cases [see Fig. 6(c) and (d)]. Next, the iteration number for the metallic strip Hilbert curve to as illusFSS structure was increased from trated in Fig. 5(b). This case required that a finer mesh consisting of a 128 128 grid be used to achieve accurate PMM simulations of the more complex Hilbert-curve-shaped metallic strip structure (corresponding computed reflection and transmission spectra for this FSS design are not presented here). For this case we also observed that the improved MBPE-based interpolation algorithm generated the correct results for oblique incidence. It is expected that the TE and TM cases will yield a different response due to the asymmetrical geometry of the Hilbert curve pattern within the unit cell [see Fig. 5(a) and (b)]. 2) Polarization Independent Multiband Hilbert Curve FSS: Multiband FSS structures that incorporate Hilbert curve elements can also be designed that exhibit a polarization independent response. One way to accomplish this is to require that each unit cell of FSS consist of a 2 2 array of Hilbert-curve shaped metallic strips as illustrated in Fig. 7(a) and (b) for iteration order num(64 64 grid) and (128 128 bers equal to grid), respectively. Note that a symmetrical unit cell geometry is constructed by successively rotating the original Hilbert curve pattern by 90 until all four quadrants are filled. The other parameters employed here are the same as used in Fig. 6. The polarization independent reflection and transmission coefficients for normal incidence versus operating frequency were also calculated via three different PMM solvers. The case, corresponding results are plotted in Fig. 8 for the and good agreement was observed between the three different PMM solvers. The corresponding calculated results for the case as shown in Fig. 7(a) are not plotted here. We note that these single-layer Hilbert-curve FSS designs exhibited a polarization independent multiband response in the 0.1–10 and (e.g., eight bands for GHz range for ). oblique incidence case, the variation of For the reflection coefficients with respect to frequency were also computed using three different PMM solvers. The corresponding results are illustrated in Fig. 9(a) and (b) for the TE- and

Fig. 6. The reflection coefficients (co-polarized) as a function of frequency at different incident angles for polarization dependent Hilbert-curve FSS with iteration order equal to n (with a 64 64 grid) as shown in Fig. 5(a). (a) TE polarization for  and  , (b) TM polarization for  and  , (c) TE-polarization for  and  , and (d) TM-polarand  . ization for 

=1

=4 2 =1 =1 =1 = 45 =1

=1

=1

TM-polarization, respectively. We observed very good agreement between the results calculated via the conventional PMM

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Fig. 7. Unit cell composed of a 2 by 2 array of metallic strips in the shape of , (b) n . Hilbert-curves with various iteration orders. (a) n

=3

=4

Fig. 9. The reflection coefficients (co-polarized) with respect to frequency for and a 128 128 grid as shown Hilbert-curve FSS with iteration order n and  . The unit cell in Fig. 7(b) for oblique incidence:  dimensions are chosen to be 2.5 cm 2.5 cm with a dielectric substrate thickness of 0.02 cm and a relative permittivity of : j : . (a) TE-polarization, (b) TM-polarization.

=4 2 = 45 =1 2 2 0 0 0 01

Fig. 8. The reflection and transmission coefficients (co-polarized) versus frequency for polarization independent Hilbert-curve FSS with iteration orders and a 128 128 grid as shown in Fig. 7(b) for normal incidence: n  and  . The unit cell dimensions are chosen to be 2.5 cm 2.5 cm. with a dielectric substrate thickness of 0.02 cm and a relative permitj : . tivity of :

=4 2 =1 =1 2 2 0 0 0 01

and improved MBPE-based interpolation algorithms as shown in Fig. 9(a) and (b), which further demonstrated the robustness and accuracy of the proposed PMM acceleration algorithm for oblique incidence cases. Note that corresponding results for the case as shown in Fig. 7(a) are not plotted here.

3) Polarization Independent Multiband Hilbert Curve AMC: Finally, we consider a polarization independent AMC surface design that follows the idea introduced in Section B.2, where the unit cell consists of a 2 2 arrangement of Hilbert-curveshaped strip elements. Two designs will be considered, one for (illustrated in Fig. 7(a) on a 64 64 grid) and the other (illustrated in Fig. 7(b) on a 128 128 grid). The for corresponding computed reflection magnitude and phase as a function of frequency are shown in Fig. 10 for the case of normal incidence. The unit cell dimension is 0.72 cm 0.72 cm with a dielectric substrate backed by a PEC ground plane and having a , rethickness and dielectric constant of 0.6 cm and spectively. The resonant frequency and bandwidth versus the itare listed in Table III. It can be seen that the first eration order resonance shifts down lower in frequency and the corresponding is increased. This bandwidth decreases as the iteration order is because, within a unit cell of fixed dimensions, the total length is of the Hilbert-curves become longer as the iteration order increased. The polarization independent Hilbert-curve AMC deand a signs are seen to exhibit a dual-band response for

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TABLE III POLARIZATION INDEPENDENT AMC SURFACE DESIGN. COMPARISON OF THE CENTER FREQUENCY AND PERCENT BANDWIDTH VERSUS THE ITERATION ORDER FOR 2 BY 2 ARRAYS OF HILBERT-CURVE ELEMENTS

reported in the literature [17], [18]. These designs possessed a large number of unknowns due to the fine meshing needed to accurately model the higher iteration orders of the embedded Hilbert-curve structures within a unit cell of fixed dimensions and were successfully evaluated by using the very effective PMM acceleration techniques presented in this paper. REFERENCES

Fig. 10. The phase angle and magnitude of the reflection coefficient versus frequency for a polarization independent Hilbert-curve AMC with different iteration orders n at normal incidence. The unit cell size is 0.72 cm 0.72 cm with a dielectric substrate backed by a PEC ground plane and having a thickness of 0.6 cm. The unit cell is discretized by a 64 64 grid and a 128 128 grid for [shown in Fig. 7(a)] and n [shown in Fig. 7(b)], respectively. n

2

=3

=4

quadruple-band response for GHz (see Fig. 10 and Table III).

2

2

over the range of 0.1–11

V. CONCLUSION The improved MBPE approach for efficient wide-band interpolation of PMM impedance matrices is proposed in this paper through the removal of a phase factor from the impedance matrix elements prior to interpolation, producing a more robust technique for oblique incidence cases than the original method introduced in [9]. The advantage of the MBPE interpolation algorithm is that it provides faster matrix fill times than the bruteforce method and can be comparable to the 2-D FFT-based method for matrix filling as well as is independent of the choice of basis functions. The 1-D FFT technique was employed to accelerate the calculation of matrix-vector products within the BCG iterative solver. The design examples presented demonstrated the accuracy and efficiency of the developed methods for both the normal incidence and oblique incidence cases. In addition, multiband single-layer FSS filters and AMC metamaterial surfaces with polarization dependent/independent performances were designed using 2-D periodic arrangements of metallic strips in the shape of space filling Hilbert-curves. We investigated the performance of designs that employed higher iteration Hilbert-curve shaped strips than has been previously

[1] B. A. Munk, Frequency Selective Surfaces: Theory and Design. New York: Wiley, 2000. [2] Frequency Selective Surface and Grid Array, T. K. Wu, Ed. New York: Wiley, 1995. [3] R. Mittra, C. H. Chan, and T. Cwik, “Techniques for analyzing frequency selective surfaces—A review,” IEEE Proc., vol. 76, no. 12, pp. 1593–1614, Dec. 1988. [4] T. F. Eibert, J. L. Volakis, D. R. Wilton, and D. R. Jackson, “Hybrid FE/BI modeling of 3-D doubly periodic structures utilizing triangular prismatic elements and an MPIE formulation accelerated by the Ewald transformation,” IEEE Trans. Antennas Propag., vol. 47, no. 5, pp. 843–850, May 1999. [5] E. H. Newman, “Generation of wide-band data from the method of moments by interpolating the impedance matrix,” IEEE Trans. Antennas Propag., vol. 36, no. 12, pp. 1820–1824, Dec. 1988. [6] A. S. Barlevy and Y. Rahmat-Samii, “Characterization of electromagnetic band-gaps composed of multiple periodic tripods with interconnecting vias: Concept, analysis, and design,” IEEE Trans. Antennas Propag., vol. 49, no. 3, pp. 343–353, Mar. 2001. [7] D. S. Weile, E. Michielssen, and K. Gallivan, “Reduced-order modeling of multiscreen frequency selective surfaces using Krylov-based rational interpolation,” IEEE Trans. Antennas Propag., vol. 49, no. 5, pp. 801–813, May 2001. [8] E. K. Miller, “Model-based parameter estimation in electromagnetics—Part I: Background and theoretical development,” IEEE Antennas Propag. Mag., vol. 40, no. 1, pp. 42–52, Feb. 1998. [9] L. Li, D. H. Werner, J. A. Bossard, and T. S. Mayer, “A model-based parameter estimation technique for wideband interpolation of periodic moment method impedance matrices with application to genetic algorithm optimization of frequency selective surfaces,” IEEE Trans. Antennas Propag., vol. 54, no. 3, pp. 908–924, Mar. 2006. [10] B. E. Barrowes, F. L. Teixeira, and J. A. Kong, “Fast algorithm for matrix-vector multiply of asymmetric multilevel block-Toeplitz matrices in 3-D scattering,” Microw. Opt. Technol. Lett., vol. 31, no. 1, pp. 28–32, 2001. [11] J. D. Vacchione, “Techniques for analyzing planar, periodic, frequency selective surface systems,” Ph.D. dissertation, Univ. Illinois at UrbanaChampaign, Urbana-Champaign, 1990. [12] T. K. Wu and S. W. Lee, “Multiband frequency selective surface with multiring patch elements,” IEEE Trans. Antennas Propag., vol. 42, no. 11, pp. 1484–1490, Nov. 1994. [13] 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. [14] J. A. Bossard, D. H. Werner, T. S. Mayer, J. A. Smith, Y. U. Tang, R. Drupp, and L. Li, “The design and fabrication of planar multiband metallodielectric frequency selective surfaces for infrared applications,” IEEE Trans. Antennas Propag., vol. 54, no. 4, pp. 1265–1276, Apr. 2006.

WANG AND WERNER: IMPROVED MBPE APPROACH FOR ACCELERATED PMM SOLUTIONS

[15] M. Ohira, H. Deguchi, M. Tsuji, and H. Shigesawa, “Multiband singlelayer frequency selective surfaces designed by combination of genetic algorithm and geometry-refinement technique,” IEEE Trans. Antennas Propag., vol. 52, no. 11, pp. 2925–2931, Nov. 2005. [16] H. Sagan, Space-Filling Curves. Berlin, Germany: Springer-Verlag, 1994. [17] E. A. Parker and A. N. A. El Sheikh, “Convoluted array elements and reduced size unit cells for frequency-selective surfaces,” Proc. Inst. Elect. Eng. H, Microw. Antennas Propag., vol. 138, no. 1, pp. 19–22, Jan. 1991. [18] E. A. Parker, A. N. A. El Sheikh, and A. C. de Lima, “Convoluted frequency-selective array elements derived from linear and crossed dipoles,” Proc. Inst. Elect. Eng. H, Microw. Antennas Propag., vol. 140, no. 5, pp. 378–380, Oct. 1993. [19] J. McVay, N. Engheta, and A. Hoorfar, “High impedance metamaterial surfaces using Hilbert-curve inclusions,” IEEE Microw. Wireless Compon. Lett., vol. 14, no. 3, pp. 130–132, Mar. 2004. [20] J. McVay, A. Hoorfar, and N. Engheta, “Peano high impedance surfaces,” Radio Sci., vol. 40, no. 6, pp. RS6S03–RS6S03, Sep. 2005.

Xiande Wang (M’03–SM’09) received the B.Sc. degree in radio physics from Lanzhou University, China, in 1989, and the M.Eng. and Ph.D. degrees in electrical engineering from Xi’an Jiaotong University, China, in 1996 and 2000, respectively. He is currently a Postdoctoral Researcher in the Department of Electrical Engineering, Pennsylvania State University. From 1989 to 2001, he worked as an Assistant Engineer, an Engineer and then as a Senior Engineer in the China Research Institute of Radiowave Propagation (CRIRP). From October 2001 to August 2006, he was a Research Scientist in the Temasek Laboratories, National University of Singapore. His research interests include numerical techniques and fast algorithms for computational electromagnetic, metamaterials, the spatial-domain Green’s function for multilayered medium, electromagnetic scattering and radiation in complex media for applications in antennas and frequency selective surfaces, scattering cross section predication for complex objects in complex environments, wave propagation and scattering of random media and rough surfaces and their applications in target detection and microwave remote sensing. Dr. Wang is a Senior Member of Chinese Institute of Electronics and a Member of the Chinese Institute of Space Science. He is a reviewer for papers in the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, IEEE Antennas and Propagation Magazine, Radio Science, IET Microwaves, Antennas and Propagation and Journal of Electromagnetic Waves and Applications.

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Douglas H. Werner (S’81–M’89–SM’94–F’05) received the B.S., M.S., and Ph.D. degrees in electrical engineering and the M.A. degree in mathematics from The Pennsylvania State University (Penn State), University Park, in 1983, 1985, 1989, and 1986, respectively. He is a Professor in the Pennsylvania State University Department of Electrical Engineering. He is the Director of the Computational Electromagnetics and Antennas Research Lab (CEARL) http://labs.ee. psu.edu/labs/dwernergroup/ as well as a member of the Communications and Space Sciences Lab (CSSL). He is also a Senior Scientist in the Computational Electromagnetics Department of the Applied Research Laboratory and a faculty member of the Materials Research Institute (MRI) at Penn State. He edited a book entitled Frontiers in Electromagnetics (Piscataway, NJ: IEEE Press, 2000). He has also contributed a chapter for the book, Electromagnetic Optimization by Genetic Algorithms (New York: Wiley Interscience, 1999) as well as for the book, Soft Computing in Communications (New York: Springer, 2004). He has coauthored Genetic Algorithms in Electromagnetics (Hoboken, NJ: Wiley/IEEE, 2007) and has completed an invited chapter on “Fractal Antennas” for the popular Antenna Engineering Handbook (New York: McGraw-Hill, 2007). He has published over 375 technical papers and proceedings articles and is the author of eight book chapters. His research interests include theoretical and computational electromagnetics with applications to antenna theory and design, phased arrays, microwave devices, wireless and personal communication systems, wearable and e-textile antennas, conformal antennas, frequency selective surfaces, electromagnetic wave interactions with complex media, metamaterials, electromagnetic bandgap materials, zero and negative index materials, fractal and knot electrodynamics, tiling theory, neural networks, genetic algorithms and particle swarm optimization. Dr. Werner is a Fellow of the IEEE, the IET, and ACES. He is a member of the American Geophysical Union (AGU), URSI Commissions B and G, the Applied Computational Electromagnetics Society (ACES), Eta Kappa Nu, Tau Beta Pi and Sigma Xi. He was presented with the 1993 Applied Computational Electromagnetics Society (ACES) Best Paper Award and was also the recipient of a 1993 International Union of Radio Science (URSI) Young Scientist Award. In 1994, he received the Pennsylvania State University Applied Research Laboratory Outstanding Publication Award. He was the recipient of a College of Engineering PSES Outstanding Research Award and Outstanding Teaching Award in March 2000 and March 2002, respectively. He was also presented with an IEEE Central Pennsylvania Section Millennium Medal. He was the coauthor of a paper published in the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION which received the 2006 R. W. P. King Award. He has also received several Letters of Commendation from the Pennsylvania State University Department of Electrical Engineering for outstanding teaching and research. In March 2009, he received the PSES Premier Research Award. He is a former Associate Editor of Radio Science, and an Editor of the IEEE Antennas and Propagation Magazine.

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Pareto Optimal Microwave Filter Design Using Multiobjective Differential Evolution Sotirios K. Goudos, Member, IEEE, and John N. Sahalos, Life Fellow, IEEE

Abstract—Microwave filters play an important role in modern wireless communications. A novel method for the design of multilayer dielectric and open loop ring resonator (OLRR) filters under constraints is presented. The proposed design method is based on generalized differential evolution (GDE3), which is a multiobjective extension of differential evolution (DE). GDE3 algorithm can be applied for global optimization to any engineering problem with an arbitrary number of objective and constraint functions. GDE3 is compared against other evolutionary multiobjective algorithms like nondominated sorting genetic algorithm-II (NSGA-II), multiobjective particle swarm optimization (MOPSO) and multiobjective particle swarm optimization with fitness sharing (MOPSO-fs) for a number of microwave filter design cases. In the multilayer dielectric filter design case a predefined database of low loss dielectric materials is used. The results indicate the advantages of this approach and the applicability of this design method. Index Terms—Dielectric filters, differential evolution (DE), generalized differential evolution (GDE), microwave filter design, multiobjective optimization (MO), open loop ring resonator (OLRR) filter, Pareto optimization, particle swarm optimization (PSO).

I. INTRODUCTION

M

ICROWAVE filters are among the important components of a modern wireless communication system. The filter synthesis and design research field is very wide. For example among many others the problem is addressed in [1]–[6]. The filter design problem is in general multiobjective. In the case of multilayer dielectric filters there are two basic design objectives. The first is the minimization of the reflection coefficient of the multilayer structure for an incident plane wave in the passband frequency zone. The second is the minimization of the transmission coefficient in the stopband frequency zone. Both reflection and transmission coefficient depend on the thickness and the electric properties of each layer. The selection of the optimal permittivity for each layer from a predefined database of commonly available materials is also an important requirement. Additional constraints can also be imposed in the above-described problem. Such design constraints require that the reflection coefficient value in the passband and stopband zones should not lie above or below a predefined level respectively. In this paper an additional constraint of desired total layer thickness is also imposed. Similar design objectives and constraints exist also for other filter types like microstrip filters [7]. Manuscript received November 29, 2008; revised May 25, 2009. First published September 11, 2009; current version published January 04, 2010. The authors are with the Radiocommunications Laboratory, Department of Physics, Aristotle University of Thessaloniki, GR-541 24 Thessaloniki, Greece (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TAP.2009.2032100

Evolutionary algorithms (EAs) like genetic algorithms (GAs) have been applied to a variety of microwave component design problems [8]–[14]. Multilayer dielectric filters have been studied in the literature using different evolutionary algorithms. In [13] a binary coded GA is used for multilayer dielectric filter synthesis. The unknown variables are the layer thickness and permittivity, which are allowed to vary. The same technique is also used in [14] for frequency selective surfaces (FSS) filter design. In [15] a single objective approach is proposed using a self-adaptive evolutionary algorithm. This is produced from the aggregation of the objective functions and a penalty term. The layer materials are chosen from a predefined materials database consisting of 15 materials. In [16] a multiobjective EA is used for the generation of the Pareto front for the constraint dielectric filter design problem. Nondominated sorting genetic algorithm-II (NSGA-II) [17] is a popular and efficient multiobjective genetic algorithm, which has been used in several engineering design problems. The major drawback of a GA approach is the difficulty in the implementation due to the algorithm inherited complexity and the required long computational time. Particle swarm optimization (PSO) [18] is an evolutionary algorithm based on the bird fly. PSO is an easy to implement algorithm with computational efficiency. PSO has been used successfully in constrained or unconstrained electromagnetic design problems [19]–[34]. Multiobjective PSO algorithms include the multiobjective particle swarm optimization (MOPSO) [35] and multiobjective particle swarm optimization with fitness sharing (MOPSO-fs) [36]. MOPSO is utilized in [37] for microwave absorber design while MOPSO-fs is applied to the filter design problem in [38] and to antenna base station design in [39]. An evolutionary algorithm that has gained popularity recently is differential evolution (DE), proposed by Price and Storn [40], [41]. Several DE variants or strategies exist. The classical DE algorithm has been applied to microwave structures [42]–[44], antenna design [45]–[49], signal optimization [50] and microwave imaging applications [51]–[54]. One of the DE advantages is the fact that very few parameters have to be adjusted in order to produce results. Several DE extensions for multiobjective optimization have been proposed so far. Generalized differential evolution (GDE3) [55] is a multiobjective DE algorithm that has outerperformed other multiobjective evolutionary algorithm for a given set of numerical problems [56], [57]. In this paper GDE3 is used for the multiobjective filter design problem. We apply GDE3 to two different filter design cases. First, we design multiband dielectric filters using a new defined database consisting of 44 commercially available materials. It

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GOUDOS AND SAHALOS: PARETO OPTIMAL MICROWAVE FILTER DESIGN USING MULTIOBJECTIVE DE

must be pointed out that in the literature [15], [16] a database of 15 materials is used. The advantage of using the new defined database is that thinner designs can be found, as it will be shown in the results. In order to evaluate the algorithm performance combined with a numerical method, we present a multiobjective design procedure for a microstrip bandpass filter. We have selected open loop ring resonator (OLRR) filters, which are comprised of two uniform microstrip lines and an open loop between them. Synthesis of OLRRs has been presented in [58], [59]. In [59] the space mapping technique is used for filter design. This is accomplished in conjunction with FEKO [60] a commercially available EM solver. FEKO is a hybrid MoM/FEM software, which we also use for the OLRR filter design. The novelty in our work lies in the fact that we apply GDE3 to the filter design problem combined with a new predefined materials database. To the best of our knowledge this the first time that GDE3 is applied to an electromagnetics design problem. We compare GDE3 against NSGA-II, MOPSO and MOPSO-fs. The Pareto fronts produced from these algorithms are compared and discussed. We therefore validate GDE3 using a real world engineering problem. The advantages of the GDE3 algorithm approach are clearly shown. This paper is organized as follows: Section II describes the problem formulation. The definition of the general multiobjective optimization problem under constraints is given in Section III. We also present the classical DE/rand/1/bin strategy and we briefly outline the GDE3 algorithm details. Section IV presents the numerical results for four distinct filter design cases. Finally the conclusion is given in Section V. II. FORMULATION Two different filter design problems are presented. The problem of multilayer dielectric filter design has attracted our attention because it has been studied in the literature using different evolutionary algorithms (both single and multiobjective) and it is therefore suitable for testing the GDE3 algorithm. This design problem can be defined using closed form expressions. The OLRR filter design is also an interesting problem that has been defined in the literature using a single objective approach [59]. The second requires the use of a numerical method. Therefore all the algorithms have to be combined with an EM solver software.

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Fig. 1. Multilayer dielectric filter structure.

low loss dielectric nonmagnetic materials , which have frequency independent permittivity in the desired frequency range. Therefore, for a M-layer design problem, the number of the unknown variables is 2M. For this type of multilayer structure, the general expression of the reflection coeffiand at the th layer for the transverse electric cients (TE) and the transverse magnetic (TM) modes is respectively found by using the recursive formula [61]

(3)

(4) TE mode TM mode

(5) (6)

where M is the number of layers, and also and are the thickness and the dielectric constant respectively of the th layer. The microwave filter design problem is defined by the minimization of the objective functions given below (7)

A. Multilayer Dielectric Filter The structure of the multilayer dielectric filter is shown in Fig. 1. The unknown variables are the thickness and the electromagnetic characteristics of each layer. These characteristics are the frequency dependent (in general) complex permittivity and permeability given by (1) (2) The terms and are respectively the free space permittivity and permeability. The filter is assumed to be composed of

(8) Moreover, the design problem is subject to the following constraints: (9) (10) (11) (12) (13)

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Fig. 3. Open loop ring resonator filter geometry.

in the stopband frequency range. Addito minimize the tionally constraints can be set for levels in both the passband and the stopband frequency range. This design problem is therefore defined by the minimization of the objective functions Fig. 2. Dielectric filter design specifications.

where and are the overall reflection coefficients (or the reflection coefficients at the first layer) of the filter structure respectively for the transverse electric (TE) and the transverse magnetic (TM) modes, the vector of layer thickis the total layer thickness of ness and dielectric constant, is the desired total layer thickness. the design found and and define correspondingly the passband In addition, and the stopband frequency ranges, while and define respectively the passband and the stopband frequencies where and define the maxconstrains must be satisfied. imum allowable reflection coefficient values in the passband frequency ranges for TE and TM modes respectively, while and accordingly are the minimum allowable reflection coefficient values in the stopband frequency ranges. The frequency ranges where constraints apply are in general different than the frequency ranges of the objective functions. They are defined narrower than those of the objective functions as in [16]. Therefore the constraint functions make the problem more strict and difficult. The constraint functions are depicted schematically in Fig. 2.

B. Open Loop Ring Resonator Filter An OLRR filter is shown in Fig. 3. The frequency response of such a filter depends on the filter dimensions and spacings between microstrip lines. The design parameters are the ones , all exshown in Fig. 3 pressed in mm. It must be pointed out that in [59] and are considered to be constant and equal to 0.4 mm. We therefore add two additional design variables to the problem. Such a filter design problem can be defined [59] by two objectives subject to two constraint functions. The first objective is to maximize the in the passband frequency range. The second objective is

(14) (15) Subject to

(16) (17) where the vector of and define correspondingly the passband filter geometry, and define respecand the stopband frequency ranges, tively the passband and the stopband frequencies where conand define the minstrains must be satisfied. Also imum and maximum allowable values in the passband and stopband frequency ranges respectively where constrains are applied. III. MULTIOBJECTIVE OPTIMIZATION WITH CONSTRAINTS The general constrained multiobjective optimization problem (MOOP) definition is [62]

(18) (19) is the vector of the objective functions, are the constraint functions, is the number of objective functions and is the number of constraint functions. In principle multiobjective optimization is different than single-objective optimization. In single-objective optimization one attempts to obtain the best solution, which is usually the global minimum or the global maximum depending on the optimization problem. In case of multiple objectives, there may not exist one solution, which is the best (global minimum or maximum) with respect to all objectives. In a typical MOOP,

GOUDOS AND SAHALOS: PARETO OPTIMAL MICROWAVE FILTER DESIGN USING MULTIOBJECTIVE DE

it is often necessary to determine a set of points that all fit a predetermined definition for an optimum. The predominant concept in defining an optimal point is that of Pareto optimality. Pareto-optimal solutions are those solutions (from the set of feasible solutions) that cannot be improved in any objective without causing degradation in at least one other objective. Therefore, the above problem can be solved in two ways. The first way is to convert it to a single-objective optimization problem. This can be accomplished by using weights for different objective functions and penalty terms for the constraint functions. This method leads to a single solution. The second way is to use Pareto optimization, which means to optimize all the objectives simultaneously giving them equal importance. If none of the objective function values can be further improved without impairing the value of at least one objective for a given solution then this solution is Pareto-optimal and belongs to the set of non-dominated solutions which is called Pareto front. The main goal is to find some points (solutions) that belong to the Pareto front. From this set of non-dominated solutions optimal filter designs that provide a suitable compromise between the objectives for the desired constraints can be realized. A multiobjective evolutionary algorithm can be used to solve this problem. Multiobjective evolutionary algorithms have gained popularity and have been used extensively over the last years in several design problems in electromagnetics. The application areas among others include microwave absorbers [63]–[65], antenna arrays [66]–[68], wire [69]–[72] and patch antennas [73]–[75]. EAs use vectors to model the possible solutions. In order to distinguish the members of the non-dominated set from the population members we refer to the first as solutions and to the second ones as vectors. The definitions of dominance relations between two vectors (or individuals of the population) are given relation between two vectors below. The weak dominance in the search space is defined as [55]

(20) while the dominance

relation is defined as

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solution. The initial population evolves in each generation with the use of three operators: mutation, crossover and selection. Depending on the form of these operators several DE variants or strategies exist in the literature [24], [25]. The most popular is the one known as DE/rand/1/bin strategy. In this strategy a is computed by mutant vector for every target vector

(22) are randomly chosen indices from the popuwhere is a mutation control parameter. After mutalation, and tion the crossover operator is applied to generate a trial vector whose coordinates are given by if if

or and

(23) , rand(j) is a number from a uniform where random distribution from the space [0,1], rn(i) a randomly and the crossover conchosen index from stant from the space [0,1]. DE uses a greedy selection operator. According to this selection scheme for minimization problems if otherwise

(24)

where , are the fitness values of the trial and the old vector respectively. Therefore the newly found replaces the old vector only when it trial vector produces a lower objective function value than the old one. Otherwise the old vector remains in the next generation. The stopping criterion for the DE is usually the generation number or the number of objective function evaluations. DE compared with PSO has been found to produce better results on numerical benchmark problems [76]. B. Generalized Differential Evolution (GDE3)

(21) The above relations can be extended to include constraint dominance [55]: constraint-dominates when any of the following conditions are true. 1) belongs to the feasible design space and is infeasible; are both infeasible but dominates in constraint 2) function space; both belong the feasible design space but domi3) nates in objective function space. A. Differential Evolution A population in DE consists of vectors , , where is the generation number. The population is initialized randomly from a uniform distribution. Each -dimensional vector represents a possible

Multiobjective DE algorithms extend the classical DE algorithm for solving MOOP. Generalized Differential Evolution (GDE3) that introduced in [55] can solve problems that have n objectives and k constraint functions. It can handle any number of objectives and any number of constraint functions including the cases (constraint satisfaction problem) and (unconstraint problem). In case of and the algorithm is the same as the original DE. The classical DE algorithm can be considered as a special case of GDE3. Therefore one could change the current DE/rand/1/bin strategy to any other exciting DE strategy or to any method that a trial vector is compared against an old vector and the better one is preserved. Recently GDE3 has outperformed other evolutionary algorithms in numerical benchmark problems [56], [57]. It has been successfully applied to the molecular sequence alignment problem [77]. To the best of the authors’ knowledge this is the first time that the GDE3 algorithm is applied to an electromagnetics design problem.

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Fig. 4. GDE3 algorithm flowchart.

GDE3 modifies the selection rule of the basic DE. In the modified selection rule the trial vector is selected to replace the old vector in the next generation if it weakly constraint-dominates the old vector. The following selection rules apply to the GDE3 algorithm. a) If both vectors (trial and old) are infeasible, then the trial vector is selected only if it weakly dominates the old vector in constraint violation space, otherwise the old vector is preserved; b) If one vector is feasible and the other is unfeasible, then the feasible vector is selected; c) If both vectors (trial and old) are feasible, then the trial is selected only if it weakly dominates the old vector in the objective function space. If the old dominates then the old vector is selected. If neither vector dominates each other in the objective function space then both vectors are selected for the next generation. Therefore the population size may increase in the next generation. To decrease the population back to the original size a sorting technique is applied. This uses the concept of crowding distance (CD), which approximates the crowdedness of a vector in its non-dominated set like NSGA-II [17]. The vectors are sorted based on non-dominance and crowdedness. The worst population members are removed and the population size is set to the original size. The basic idea is to prune a non-dominated set to have a desired number of solutions in such a way that the remaining solutions have as good diversity as possible, meaning that the spread of extreme solutions is as high as possible, and the relative distance between solutions is as equal as possible. The pruning method of NSGA-II provides good diversity in the case of two objectives, but when the number of ob-

jectives is more than two, the obtained diversity declines drastically [78]. The method used in GDE3 is based on a crowding estimation technique using nearest neighbors of solutions in Euclidean sense, and a technique for finding these nearest neighbors quickly. More details about the GDE3 pruning method can be found in [79]. Therefore the selection based on CD is improved over the original method of the NSGA-II to provide a better-distributed set of vectors. A basic difference exists between NSGA-II and GDE3 regarding the population size after a generation. In NSGA-II the . Then population size after a generation is increased to non-dominated ranking is applied and non-dominated vectors are selected. In GDE3 after a generation the population size , where , because the population size is is and the old vector are increased only when the trial feasible and do not dominate each other. Therefore non-domipopulation size, which can nated ranking is applied to thus resulting in less computational be less in general than time than NSGA-II [55]. GDE3 can be implemented in such a way that fewer function evaluations are required because not always all the objectives and the constraints have to be evaluated, e.g., by inspecting constraint evaluations (even one constraint) can be enough to determine, which vector will be selected for the next generation. But in case of feasible vectors all the function evaluations are required. The GDE3 flowchart is given in Fig. 4. The GDE3 algorithm is outlined below. individuals. Set 1) Initialize random population of ; 2) Evaluate objective function and constraint function values for every vector of the population;

GOUDOS AND SAHALOS: PARETO OPTIMAL MICROWAVE FILTER DESIGN USING MULTIOBJECTIVE DE

TABLE I PREDEFINED MATERIALS DATABASE

3) Apply the mutation and crossover operators according to ; (22) and (23) and create a trial vector 4) Evaluate objective function and constraint function values for the trial vector; 5) Apply the selection operator according to the following criterion: if otherwise

(25)

6) Set

(26) vectors. Select 7) Apply non-dominated ranking to non-dominated vectors and set ; 8) Repeat step 3 until the maximum number of generations is reached. GDE3 variations with different DE strategies can be easily created simply by using different equations for crossover and mutation than (22) and (23). More details about the GDE3 algorithm can be found in [55].

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, , values for control parameters could be , where the number of unknown variables. In case of multiobjective optimization and conflicting objectives lower values for control parameters than those in single objective optimization are used. This is due to the fact that conflicting objectives maintain diversity and restrain the search is a speed. The value of the mutation control parameter compromise between speed and robustness. It has been found is suitable for most of the problems that a value can be rel[56], [80]. The value of the crossover operator atively low as has been shown in [80]. The population size can be selected the same way as in single objective optimization or according to the desired size of the Pareto front. Therefore the control parameters chosen for GDE3 are according to [56], [80] , . For NSGA-II [17] usual values for control parameters are for crossover probability and the mutation , where is the probability for real valued variables or string length for binary-coded variables. We have therefore set for real valued variables the crossover and mutation probabilities equal to 0.9 and 0.1, respectively. For binary coded variables the mutation probability was set to 0.16. The main characteristics of the MOPSO [35] algorithm are; the repository size, the mutation operator and the grid subdivisions. The repository is the archive where the positions of the particles that represent non-dominated solutions are stored. Therefore the parameter that has to be adjusted is the repository size. This is usually set equal to the swarm size. MOPSO introduces a mutation operator that intends to produce a highly explorative behavior of the algorithm. The effect of mutation decreases as the number of iterations increase. A setting of 0.5 has been found suitable [35] after experiments for mutation probability. Similarly, a value of 30 has been found suitable [35] for the parameter of grid divisions. The above parameters are those selected for our problem for the MOPSO algorithm. MOPSO-fs also uses a repository to store all the all the nondominated solutions [36]. As in MOPSO the repository size parameter is set equal to swarm size. Another parameter that has to be set in MOPSO-fs is the sigma share value. This is set empirically after several trials to 2.0. It must be pointed out that several modifications were made to MOPSO and MOPSO-fs algorithms. Constraint handling was added to both algorithms. Furthermore in case of discrete valued variables like the material number the velocity update rules given by the binary PSO were used [81]. More details about these modifications can be found in [37], [39]. A. Multilayer Dielectric Filter

IV. NUMERICAL RESULTS All algorithms are executed 20 times. The best results are compared. All algorithms are compiled using the same compiler (Borland C++ Builder 5.0) in a PC with Intel Core 2 Duo E8500 at 3.16 GHz with 4 GB RAM running Windows XP. The selection of the control parameters for all algorithms is explained below. A empirical rule in DE states that for single objective optimization if nothing is known about the problem then the initial

GDE3, NSGA-II, MOPSO and MOPSO-fs were applied to three dielectric filter design cases; a band-pass a dual-band and a tri-band filter. In all design cases that follow the angle of in. As in previous papers we assume cidence is set to that the resolution of layer thickness is 0.001 mm (1 micron). In order to find how sensitive the results are to layer thickness, we also present the design cases by truncating the thickness to the first decimal point. These results are denoted with a “TR” in the frequency response graphs. Such a filter could be fabricated

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TABLE II DESIGN SPECIFICATIONS FOR A FOUR-LAYER BANDPASS FILTER CASE

TABLE III DESIGN PARAMETERS OF A FOUR-LAYER BANDPASS FILTER EXAMPLE CASE

Fig. 5. Frequency response of a four-layer bandpass filter example case.

using thin glue layers among different dielectrics [14]. We assume that these glue layers are very thin and therefore not taken into account as in [13]–[16]. Each layer thickness is varied between 1 mm and 10 mm. A predefined materials database consisting of 44 commercially available low loss dielectric materials is used. This is given in Table I. We assume that permittivity values remain constant over the desired frequency range. In [15], [16] the designs use a smaller materials database of 15 materials. All previous filter designs consist of seven layers. We have found that by using our database thinner designs can be realized. After several trials we have concluded that feasible four-layer band-pass and dual-band filters can be found. In a similar way feasible tri-band filter solutions that consist of five layers can be found. The above layer numbers represent the minimum layer number that can produce feasible results for the specific design cases. We have not been able to find feasible designs with fewer layers. The population and the number of iterations is 5000 size selected is for all algorithms. The first example is a four-layer band-pass filter design. For this case, the design specifications are given in Table II. For both TE and TM modes the maximum desired reflection coefficient values in the passband frequency ranges were set to 15 dB, while the minimum desired reflection coefficient values in the stopband frequency ranges were set to 5 dB. The total desired thickness is set to 10 mm. Fig. 5 presents a filter design case with 6.626 mm total thickness. The design parameters for this filter

Fig. 6. Pareto fronts for the four-layer bandpass filter design case found by all algorithms.

are given in Table III. The truncated values present good results in the case of the TM mode. The TE mode truncated results are out of the feasible design limits. The Pareto fronts found using GDE3, NSGA-II, MOPSO and MOPSO-fs algorithms are given in Fig. 6. Each point of the Pareto front represents a feasible filter design case, which fulfils all the above constraints. GDE3 algorithm has found a larger dispersion of points in the front and clearly outperforms the other algorithms. Both the multiobjective PSO algorithms perform in a similar manner. NSGA-II has found a small number of points that are non-dominated by the points found by the GDE3 algorithm. The next example is a four-layer dual-band filter design case. Table IV has the desired design specifications. The desired reflection coefficient values in the passband and stopband freand quency ranges were set to respectively. The total desired thickness is set to 30 mm. The frequency response of an example

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TABLE IV DESIGN SPECIFICATIONS FOR A FOUR-LAYER DUAL- BAND FILTER CASE

TABLE V DESIGN PARAMETERS OF A FOUR-LAYER DUAL-BAND FILTER EXAMPLE CASE

Fig. 7. Frequency response of a four-layer dual-band filter example case.

dual-band filter is shown in Fig. 7. The total thickness of this design case is 25.583 mm. For this case the truncated filter results are very close to accurate ones. The design with the truncated values remains within allowable limits in the frequency ranges where constraints apply. The layer characteristics for this design case are given in Table V. The Pareto algorithms have found different points of the Pareto front. These are shown in Fig. 8. Although all algorithms have found the same number of non-dominated solutions, it is obvious that the GDE3 results present the larger dispersion of points. The solutions found by NSGA-II are dominated by 34 points of the GDE3 algorithm. The MOPSO points are dominated by those found by GDE3 and NSGA-II and by the 42 points found by MOPSO-fs. The MOPSO-fs results for this case present the smaller dispersion of values and they are dominated by those found by GDE3. The final dielectric filter example presents a tri-band five-layer design case. This is a complex design case with eight constraint functions. The design specifications are shown in Table VI. The desired reflection coefficient values in the passband and stopband frequency ranges were and respectively. The total desired thickness constraint is set to 35 mm. Fig. 9 depicts the frequency response between 10–40 GHz of an

Fig. 8. Pareto fronts for the four-layer dual-band filter design case found by all algorithms.

example tri-band filter design case for both TE and TM modes. This tri-band filter design has a total thickness of 31.392 mm. In this case again the truncated results present a smooth frequency response according to the design constraints. The filter design parameters are shown in Table VII. The Pareto fronts found by the algorithm are depicted in Fig. 10. One may notice that the solutions found by GDE3 dominate the ones found by the other algorithms with the exception of two points found by MOPSO-fs. The points found by MOPSO-fs and MOPSO dominate in general those found by NSGA-II algorithm. The exceptions in this case are 8 points found by MOPSO-fs and 7 found by MOPSO, which are dominated by NSGA-II points.

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TABLE VI DESIGN SPECIFICATIONS FOR A FIVE-LAYER TRI-BAND FILTER CASE

TABLE VII DESIGN PARAMETERS OF A FIVE-LAYER TRI-BAND FILTER EXAMPLE CASE

Fig. 10. Pareto fronts for the five-layer tri-band filter design case found by all algorithms.

Fig. 9. Frequency response of a five-layer tri-band filter example case.

MOPSO-fs solutions seem to dominate the MOPSO ones. There are though 40 points found by MOPSO-fs, which are dominated by MOPSO points. The NSGA-II results present the largest dispersion of values although these are dominated by those found by the other algorithms. Regarding execution time GDE3 seems to be the fastest of all algorithms. MOPSO and MOPSO-fs require about the same computational time for one algorithm run. NSGA-II is the slowest of all algorithms, which takes in some cases more than twice GDE3 time. Table VIII has the total execution time for all algorithms after 5000 iterations for each of the three design cases. B. Open Loop Ring Resonator Filter The OLRR filter was modeled in FEKO. The design specifications chosen are the same as in [59]. These are given in Table IX. For each FEKO run 15 frequency sweeps are taken in the frequency range 1.5–4.5 GHz. This requires about a total time of 3.2 sec in a PC with Intel Core 2 Duo E8500 at 3.16 GHz with 4 GB RAM. In order to integrate the in-house source code of the multiobjective algorithms with FEKO, a wrapper program was created. FEKO, except of using a graphical user interface,

offers the option to run the EM solver engine from command line. It requires an input file that defines the model geometry. This input file uses a script language that allows users to define variables and control options like the frequency range, the number of frequency points and the required data in the output file. The wrapper creates a FEKO input file for each random vector created by the algorithms and runs FEKO. The output file, which in our case is defined to contain the frequency and the S-parameters is read by the wrapper and the objective and constraint functions are evaluated. A population of 20 vectors is selected for all algorithms. The total number of generations is set to 1000. All algorithms are executed 20 times. In this case execution time plays an important role therefore fewer objective function evaluations where selected than the dielectric filter case. The same control parameters that were given in the previous section are used for all algorithms. The frequency response of an example design is given in Fig. 11. A finer mesh was used in FEKO model with 300 frequency points to provide a smooth response. One may notice that the filter has low magnitude values also out of the desired stopband frequency bands. It is evident that in the frequency bands between 1.0 and 1.5 GHz and 4.5 and 5.0 GHz the value lies below 50 dB and 30 dB respectively. The design parameters for this filter are given in Table X. The Pareto fronts produced are given in Fig. 12. It is obvious that again the GDE3 algorithm outperforms the other algorithms. GDE3 solutions dominate most of the solutions found by NSGA-II. Both algorithms have found 20 points of the Pareto front. MOPSO-fs

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TABLE VIII TOTAL EXECUTION TIME FOR 5000 ITERATIONS, 100 VECTORS FOR THE MULTILAYER DIELECTRIC FILTER

TABLE X DESIGN PARAMETERS FOR THE OLRR BAND-PASS FILTER DESIGN CASE (mm)

TABLE IX DESIGN SPECIFICATIONS FOR THE OLRR BAND-PASS FILTER

Fig. 12. Pareto fronts for the OLRR bandpass filter design case found by all algorithms.

TABLE XI TOTAL EXECUTION TIME FOR 1000 ITERATIONS, 20 VECTORS FOR OLRR FILTER

Fig. 11. OLRR bandpass filter example case.

and MOPSO have found 15 and 11 points respectively. For this problem these algorithms require more iterations. Their solutions are dominated by NSGA-II and GDE3 results. The total execution time for all algorithms regarding 1000 iterations is given in Table XI. GDE3 is the fastest of all algorithms while NSGA-II is the slowest. V. CONCLUSION Multiobjective evolutionary algorithms can be used successfully for microwave multiband filter design. A novel filter design method using GDE3 algorithm has been presented. GDE3 is a new multiobjective DE algorithm, which has been compared against NSGA-II, MOPSO and MOPSO-fs. All algorithms can

be used to produce the Pareto front in different filter design cases. However, it is obvious from the previous examples that GDE3 can produce better results for the same population size and for the same number of generations. One of the GDE3 advantages is the fact that requires less computational load than NSGA-II. This is due to the fact the population size that is ranked after a generation is usually less than the one required by NSGA-II. MOPSO and MOPSO-fs are also quite efficient and may produce better results in general than NSGA-II. But they are both outperformed clearly by GDE3. The practical filter designs subject to several constraints presented here show the applicability and the efficiency of this method. All algorithms can be combined with a numerical method. The GDE3 algorithm can be easily applied to other microwave and antenna design problems and it can also be used in conjunction with an EM solver software. In our future work we plan to extend this method to other design problems.

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[66] Y. H. Lee, B. J. Cahill, S. J. Porter, and A. C. Marvin, “A novel evolutionary learning technique for multiobjective array antenna optimization,” Progr. Electromagn. Res., vol. 48, pp. 125–144, 2004. [67] D. W. Boeringer and D. H. Werner, “Bezier representations for the multiobjective optimization of conformal array amplitude weights,” IEEE Trans. Antennas Propag., vol. 54, no. 7, pp. 1964–1970, Jul. 2006. [68] J. S. Petko and D. H. Werner, “The pareto optimization of ultrawideband polyfractal arrays,” IEEE Trans. Antennas Propag., vol. 56, no. 1, pp. 97–107, 2008. [69] R. M. Ramos, R. R. Saldanha, R. H. C. Takahashi, and F. J. S. Moreira, “The real-biased multiobjective genetic algorithm and its application to the design of wire antennas,” IEEE Trans. Magn., vol. 39, no. 3 I, pp. 1329–1332, May 2003. [70] N. V. Venkatarayalu and T. Ray, “Optimum design of Yagi-Uda antennas using computational intelligence,” IEEE Trans. Antennas Propag., vol. 52, no. 7, pp. 1811–1818, Jul. 2004. [71] H. Choo, R. L. Rogers, and H. Ling, “Design of electrically small wire antennas using a Pareto genetic algorithm,” IEEE Trans. Antennas Propag., vol. 53, no. 3, pp. 1038–1046, 2005. [72] Y. Kuwahara, “Multiobjective optimization design of Yagi-Uda antenna,” IEEE Trans. Antennas Propag., vol. 53, no. 6, pp. 1984–1992, 2005. [73] C. M. De Jong Van Coevorden, S. G. Garcia, M. F. Pantoja, A. R. Bretones, and R. G. Martin, “Microstrip-patch array design using a multiobjective GA,” IEEE Antennas Wireless Propag. Lett., vol. 4, no. 1, pp. 100–103, 2005. [74] S. Koulouridis, D. Psychoudakis, and J. L. Volakis, “Multiobjective optimal antenna design based on volumetric material optimization,” IEEE Trans. Antennas Propag., vol. 55, no. 3, pp. 594–603, 2007. [75] S. H. Yeung, K. F. Man, K. M. Luk, and C. H. Chan, “A Trapeizform U-slot folded patch feed antenna design optimized with jumping genes evolutionary algorithm,” IEEE Trans. Antennas Propag., vol. 56, no. 2, pp. 571–577, Feb. 2008. [76] J. Vesterstrom and R. Thomsen, “A comparative study of differential evolution, particle swarm optimization, and evolutionary algorithms on numerical benchmark problems,” in Proc. Congress on Evol. Comput., CEC2004, Portland, OR, 2004, pp. 1980–1987. [77] S. Kukkonen, S. R. Jangam, and N. Chakraborti, “Solving the molecular sequence alignment problem with Generalized Differential Evolution 3 (GDE3),” in Proc. IEEE Symp. on Computational Intelligence in Multicriteria Decision Making, 2007, pp. 302–309. [78] K. Deb, L. Thiele, M. Laumanns, and E. Zitzler, “Scalable test problems for evolutionary multiobjective optimization,” Evol. Multiobjective Opt. 2005, pp. 105–145. [79] S. Kukkonen and K. Deb, “A fast and effective method for pruning of non-dominated solutions in many-objective problems,” Lecture Notes in Comput. Sci., vol. 4193, pp. 553–562, 2006. [80] S. Kukkonen and J. Lampinen, “An empirical study of control parameters for the Third Version of Generalized Differential Evolution (GDE3),” in Pro. IEEE Congress on Evol. Comput., CEC 2006, 2006, pp. 2002–2009. [81] J. Kennedy and R. C. Eberhart, “Discrete binary version of the particle swarm algorithm,” in Proc. IEEE Int. Conf. on Systems, Man and Cybernetics, 1997, pp. 4104–4108.

Sotirios K. Goudos (S’01–M’05) was born in Thessaloniki, Greece, in 1968. He received the B.Sc. degree in physics, the M.Sc. degree of postgraduate studies in electronics, and the Ph.D. degree in physics from Aristotle University of Thessaloniki, in 1991, 1994 and 2001, respectively, and the Master in Information Systems from the University of Macedonia, Greece, in 2005. Since 1996 he has been working in the Telecommunications Center, Aristotle University of Thessaloniki,. He has authored or coauthored more than 50 papers in peer reviewed journals and international conferences. His research interests include antenna and microwave structures design, electromagnetic compatibility of communication systems, evolutionary computation algorithms and semantic web technologies. Dr. Goudos is a member of the Greek Physics Society and the Greek Computer Society.

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John N. Sahalos (M’75–SM’84–F’06–F’16) received the B.Sc. and Ph.D. degree in physics from the Aristotle University of Thessaloniki, (AUTH), Greece, in 1967, 1974, and 1975 respectively, and both the Diploma in Civil Engineering (BCE+MCE) and the Professional Diploma of postgraduate studies in Electronic Physics from AUTH in 1975. From 1971 to 1974, he was a Teaching Assistant in the department of Physics, AUTH, and from 1974 to 1976, he was an Instructor. In 1976, he worked at the ElectroScience Laboratory, the Ohio State University, Columbus, as a Postdoctoral University Fellow. From 1977 to 1986, he was a Professor in the Electrical Engineering Department, University of Thrace, Greece, and Director of the Microwaves Laboratory. Since 1986, he has been a Professor at the School of Science, AUTH, where he is the Director of the Postgraduate Studies in Electronic Physics and the Director of the Radio-Communications Laboratory (RCL). During 1981 to 1982, he was a Visiting Professor at the Department of Electrical and Computer Engineering, University of Colorado, Boulder. During 1989 to 1990, he was a visiting Professor at the Technical University of Madrid, Spain. He is the author of three books in Greek, six book chapters and more than 300 articles published in the scientific literature. He also is the author of the book The Orthogonal Methods of Array Syn-

thesis, Theory and the ORAMA Computer Tool (Wiley, 2006). His research interests are in the areas of antennas, high frequency techniques, communications, EMC/EMI, microwaves, and biomedical engineering. Dr. Sahalos is a Professional Engineer and a Consultant to industry. In 2002 to 2004, he was on the Board of Directors of the OTE, the largest Telecommunications Company in Southeast Europe. He served, as a technical advisor, in several national and international committees, as well as, in several Mobile Communications Companies. Since 1992, he has been a member of Commissions A and E of URSI. Since 1998, he is the president of the Greek committees of URSI. He is the president of the section of Informatics, Telecommunications and Systems of the National Committee of Research and Technology. He is an honorary member of the Radioelectrology society, a member of the Greek Physical Society and of the Technical Chamber of Greece. He is the creator and leader of an EMC network with five laboratories (3 from the academy and 2 from the industry). He has been honored with a special investigation fellowship of the Ministry of Education and Science, Spain. He also has been honored from several Institutes and Organizations. He has been on the editorial board of three scientific journals. He was elected by department representatives of the Aristotle University of Thessaloniki as the Vice-Chairman of the Research Committee of AUTH for the period 2007–2010.

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A Sparsity Regularization Approach to the Electromagnetic Inverse Scattering Problem David W. Winters, Barry D. Van Veen, Fellow, IEEE, and Susan C. Hagness, Fellow, IEEE

Abstract—We investigate solving the electromagnetic inverse scattering problem using the distorted Born iterative method (DBIM) in conjunction with a variable-selection approach known as the elastic net. The elastic net applies both 1 and 2 penalties to regularize the system of linear equations that result at each iteration of the DBIM. The elastic net thus incorporates both the stabilizing effect of the 2 penalty with the sparsity encouraging effect of the 1 penalty. The DBIM with the elastic net outperforms the commonly used 2 regularizer when the unknown distribution of dielectric properties is sparse in a known set of basis functions. We consider two very different 3-D examples to demonstrate the efficacy and applicability of our approach. For both examples, we use a scalar approximation in the inverse solution. In the first example the actual distribution of dielectric properties is exactly sparse in a set of 3-D wavelets. The performances of the elastic net and 2 approaches are compared to the ideal case where it is known a priori which wavelets are involved in the true solution. The second example comes from the area of microwave imaging for breast cancer detection. For a given set of 3-D Gaussian basis functions, we show that the elastic net approach can produce a more accurate estimate of the distribution of dielectric properties (in particular, the effective conductivity) within an anatomically realistic 3-D numerical breast phantom. In contrast, the DBIM with an 2 penalty produces an estimate which suffers from multiple artifacts. Index Terms—Breast cancer, electromagnetic tomography, FDTD methods, inverse problems, microwave imaging, regularization.

I. INTRODUCTION

S

OLVING the electromagnetic inverse scattering problem involves estimating the distribution of dielectric properties within a volume based upon observations of the scattered electromagnetic field. It is well known that the inverse scattering problem is both nonlinear and ill-posed [1]. Given an infinite number of completely precise and noise-free measurements, the inverse scattering problem has a unique solution [1]. However, in the real world, measurements will be finite in number and will be limited both in terms of accuracy and precision. In addition, the dielectric properties are typically discretized in some Manuscript received October 09, 2007; revised June 08, 2009. First published November 06, 2009; current version published January 04, 2010. This work was supported in part by the Department of Defense Breast Cancer Research Program under Award W81XWH-05-1-0365, the National Institutes of Health under Grant R01 CA112398 awarded by the National Cancer Institute, and in part by the National Science Foundation under Grant BES 0201880. D. Winters is with the The MITRE Corporation, Bedford, MA 01730-1420 (e-mail: [email protected]). B. D. Van Veen and S. C. Hagness are with the Department of Electrical and Computer Engineering, University of Wisconsin, Madison, WI USA 53706 (e-mail: [email protected], [email protected]). Digital Object Identifier 10.1109/TAP.2009.2035997

manner to simplify computations on digital computers. It is for these reasons that the inverse scattering problem does not have a unique solution in practice. There are a number of methods for solving the inverse scattering problem, including conjugate gradient methods (e.g., [2]–[6]), the distorted Born iterative method (DBIM) and equivalent Gauss-Newton methods (e.g., [7]–[12]), contrast source inversion methods (e.g., [13]–[16]), and various others (e.g., [17], [18]). For reasons that will soon become apparent, the results of this paper are formulated in terms of the DBIM. The DBIM replaces the nonlinear inverse scattering problem with a series of linear approximations of the form (1) where is a vector containing the discretized estimate of the dielectric properties contrast, is an vector containing the measurements of the scattered electromagnetic field, matrix. In each iteration of the DBIM, and is an and are functions of the properties contrast estimated in the previous iteration. The final solution is given by the summation of the estimated contrast vectors from each iteration. An important point is that for typical discretization schemes and realistic , in which case (1) is an undermeasurement systems, determined set of equations. Hence, the systems of linear (1) are typically very ill-conditioned and directly applying the method of least-squares to (1) at each iteration of the DBIM results in a solution which bears little resemblance to the true distribution of dielectric properties. Regularization is necessary to stabilize the problem and to define a unique solution [1]. A common approach to regularization with the DBIM involves solving the set of linear equations at each iteration via penalized least-squares [9]. The penalty is chosen to favor a solution of a particular form, such as those that are continuous or smooth. The relative strength of the penalty is controlled by a regularization parameter. When the penalty involves the norm of , the approach isreferred toasridgeregression[19](alsoknownas Tikhonovregularization [20]).Ridge regression achieves reduced overall mean square errorthrough a bias-variance tradeoff[21]. Aproblemwith ridge regression is that every element of in the estimated solution will generally be non-zero, even if the true solution only involves a subset of the elements. Consequently the ridge regression solution may contain artifacts that are not present in the true distribution of dielectric properties in , and this can decrease the imaging accuracy. This solution strategy is referred to in this paper as DBIM-RR. Sparse approximation methods have been widely applied in the context of linear inverse problems of the form (1) (e.g., [22]–[26]). The goal is to find a solution that accounts for the

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observed data using only a small subset of the elements of . If the true distribution of dielectric properties contrast within is sparse (that is, only a small number of the elements of are non-zero), then applying sparse approximation methods at each iteration of the DBIM should result in a solution that is free of the artifacts associated with ridge regression. While the contrast is not necessarily sparse in general, we assume there exists a set of basis functions that approximate in a sparse manner. Let (2) is a -by- real-valued matrix of basis vectors, where is an -by-1 complex-valued vector of coefficients, and is the number of basis functions in the expansion. If is sparse with respect to , then a small number of the elements of are nonzero. Such approximations are common in other fields. For example, it is well known that wavelets can represent images with a small number of coefficients [27]. While some bases may lead to more sparse representations than others, one does not need to know the optimally sparse basis to reap the benefits of sparse approximation. A direct approach to the sparse approximation problem involves penalizing the number of elements of involved in the solution to (1), which requires an exhaustive search among all combinations of elements and thus is not computationally practical [25]. An indirect approach known as convex relaxation innorm of . A growing body volves instead penalizing the solution is equal to of research indicates that the minimum the sparsest solution (or very close to it) under a variety of circumstances (e.g., [24]–[26], [28], [29]). There are several nupenalized least-squares merical approaches for solving the problem, including greedy algorithms [30]–[32], and convex programming [33]–[35]. However, in our experience, applying these purely regularized methods within the DBIM leads to numerical difficulties due to the problem size, ill-conditioning, and iterative aspects of the DBIM. In this paper we investigate solving the electromagnetic inverse scattering problem using the DBIM in conjunction with a variable-selection approach known as the elastic net [21]. We denote this approach as DBIM-EN. The elastic net applies both and penalties to (1) in order to regularize each iteration of the DBIM. The elastic net thus incorporates both the stabilizing effect of ridge regression and the sparsity encouraging effect of the penalty. One advantage of the elastic net is that the solution is computable even for very small regularization parameters. We follow [21] and use a modified version of the moderately greedy algorithm known as least angle regression (LARS) [31] for implementing the elastic net at each iteration of the DBIM. LARS efficiently produces a sequence of solutions regularization paramwhich simplifies the selection of an eter. We consider two very different 3-D numerical testbeds—a cube with an interior dielectric properties distribution that is exactly represented by a linear combination of a small number of wavelets, and an anatomically realistic breast phantom with heterogeneous dielectric properties not exactly sparse in any set of basis functions. Examples given using these testbeds demonstrate the efficacy and broad applicability of our elastic net approach to the inverse scattering problem.

We note that Baussard et al. [36] proposed an inverse scattering algorithm which is capable of producing a sparse solution. However, the algorithm in [36] is presented in the context of a specific set of multiresolution spline basis functions, and sparsity is achieved via a heuristic refinement process. In this paper we propose searching for a sparse solution to the inverse scattering problem for an arbitrary set of basis functions, and we encourage sparsity via the principle of convex relaxation. Throughout this paper, electromagnetic field vectors and dyads are denoted by upper case letters with an overline (e.g., ). Position vectors are shown as lower case letters with an arrow overline (e.g., ), while all other vector quantities are indicated by lowercase boldface type (e.g., ). Matrices are ); the matrix denoted by uppercase boldface type (e.g., transpose and complex-conjugate transpose operations are represented by superscripts and respectively. The function denotes as the independent variable and as notation a parameter. II. METHODS We begin this section by introducing the DBIM [9]. The brief review is followed by a discussion of regularization and sparsity, including descriptions of ridge regression and the elastic net. We conclude this section by discussing the choice of regularization parameters. A. Distorted Born Iterative Method Suppose that data is acquired using an antenna array located outside of , the volume throughout which the dielectric properties are to be estimated. In sequence, each antenna transmits an electromagnetic signal into while the other antennas in the array act as receivers. Consider the case where the th antenna is transmitting an electromagnetic signal at angular frequency . The nonlinear integral equation that relates the continuous spatial distribution of dielectric properties within to the scattered electric field at the th receiving antenna is given by [9]

(3) In (3), is the mathematically defined scattered field, is but unknown inside ), and is the total field (known at the known incident field. The position vectors of the th transand mitting and the th receiving antennas are given by , respectively. Inside the integral, is the dyadic Green’s are function of the homogeneous background, while and the spatially-varying complex relative permittivity of the object and the spatially invariant complex relative permittivity of the background, respectively. The difference between the object and background relative permittivity is known as the contrast [9]. function, which is denoted by We solve this nonlinear problem by using a series of simplifying assumptions. Under the Born approximation [9], the integral in (3) is linearized by replacing the total electric field in

WINTERS et al.: A SPARSITY REGULARIZATION APPROACH TO THE ELECTROMAGNETIC INVERSE SCATTERING PROBLEM

the integral with the known incident field . The scalar approximation [3] assumes that only the -directed component of the incident field is non-zero and only the -directed component of the electric field is measured by the receiving antennas. In theory the scalar approximation results in a loss of information, but in practice it has been shown that it does not significantly impact imaging performance [37]. These approximations yield the following simplified integral equation:

(4) represents the – component of the Green’s

where function dyad. Equation (4) can be discretized via the Riemann sum under the assumption that all quantities are constant over volume elements (voxels) of volume . Applying this discretization scheme to the set of approximations (4) for all transmit-receive pairs results in the following set of linear equations:

(5) is an -by- matrix, where is the number of In (5), denotes the transmit-receive pairs in the antenna array and contains the dielecnumber of voxels. The -by-1 vector voxels in , while is an tric properties contrast for the -by-1 vector and has elements equal to , for . We emphadoes not lie entirely in the span of the columns of size that , due to the linear (Born) approximation. of the true Solving (5) results in a discrete approximation . A better approximation can be obdistribution of contrast tained by adding to the background and using a series of computational electromagnetics simulations to calculate the new incident electric field and inhomogeneous Green’s function based [11]. The symbol is here used to represent a vector on of all ones. The following new set of linear equations is obtained upon substituting these updated field quantities into (4):

(6) Equation (6) is solved, resulting in an estimate , and the process is repeated for multiple iterations. Full-wave computational electromagnetics simulations are conducted at every iteration in order to calculate the updated incident electric field and Green’s function. These simulations are often collectively referred to as the “forward solver.” We note that the scalar approximation described above in the context of (4) only applies to the inverse solution; that is, the z-component of the electric field and the z-z component of the Green’s function dyad are recorded from the full-wave forward solution and incorporated into the linear system of (6). contains the residual fields after the th iteraThe vector tion. Once the norm of ceases to decrease significantly from iteration to iteration, the DBIM has converged and the estimated

contrast is given by dielectric properties is thus

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. The estimated distribution of .

B. Basis Functions, Regularization and Sparsity We re-express the vector of dielectric properties contrast at any iteration of the DBIM as a linear combination of basis functions using (2). The DBIM for this basis function expansion consists of estimating the coefficients of the basis functions instead of the contrast at each voxel. The linear problem at each iteration of the DBIM becomes (7) While it is assumed that the number of measurements is less than the number of discrete samples , the number of basis functions can be any size in theory. It is assumed that is smaller than the single-precision numerical rank of , which can be considered the effective number of measurements. In the examples considered in this paper, the rank of is always around 100. Under these assumptions, (7) is an underdetermined system of equations. In addition, does not lie entirely in the due to the linear approximation inspan of the columns of volved in obtaining (4). Solving the normal equations [38] results in the least-squares solution (8) Owing to the ill-posed nature of the inverse scattering problem, is almost never full numerical rank in single precision arithmetic. Tikhonov regularization and ridge regression stabilize (8) by adding in a fraction of the identity matrix [19], [20] (9) The statistical interpretation of (9) is that the overall mean square error of is reduced through a bias-variance trade-off [21]. In the context of the ill-posed inverse scattering problem, pads the smallest eigenvalues of the addition of and thus improves numerical stability. The disadvantage of will be ridge regression is that in general every element of non-zero, even if the corresponding basis function plays no part in the true solution. This can introduce spurious artifacts in the estimated distribution of dielectric properties. We address this issue by solving (7) via the elastic net [21]. This involves replacing (7) with the following optimization problem: (10) We denote the solution to (10) as . The elastic net can be thought of as a generalization of both ridge regression and of or become sparse approximation methods. When either will tend to . When both and go to zero, very large, then becomes (8) which is not computable. When just

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goes to zero, approaches (9). As goes to zero, then (10) becomes an -based sparse approximation problem. However, we note that the nonlinear and ill-posed nature of the inverse scattering problem can create numerical difficulties for greedy and convex optimization algorithms searching for a solution . For instance, convex optimization approaches typwith ically need to be initialized with a solution that already satisfies the system of linear equations [35]. Finding such an initialization becomes difficult since (8) is not computable. Greedy algorithms that build up a solution one basis function at a time and use least-squares to determine the coefficients [31], [32] may be unstable. These difficulties can be eliminated by using a non-zero value for in (10). We follow [21] and use a modified version of the moderately greedy algorithm known as least angle regression (LARS) [31] for implementing the elastic net at each iteration of the DBIM. The modified LARS algorithm forms a sequence of solutions norm, starting with the null with monotonically increasing . This allows to be replaced by solution and ending with a new parameter which is equal to the desired norm of the solution [21]; obviously is bounded from above and below and 0 respectively. A sparse solution is obtained by by selecting an intermediate value for . An advantage of using LARS is that its greedy nature simplifies the selection of a suitable value for . Another advantage is that LARS allows a previously selected element from to be discarded if it is determined not to be suitable for the solution. This attribute makes LARS more robust than other greedy algorithms such as orthogonal matching pursuits [31]. C. Selection of Regularization Parameters We use the L-curve principle [39] to determine the regularization parameters for the DBIM-EN and DBIM-RR. The L-curve determines a suitable regularization parameter by balancing the norm of the linearized residual and the norm of the solution [39]. The L-curve has been shown to be very robust in practice [39], although it is not optimal in any sense. The norms of L-curve applied to ridge regression balances the the linearized residual and of the solution. There are two regand ularization parameters to determine for the elastic net: . We first determine by applying the L-curve to all of the basis functions at once. With this choice for , we again use the L-curve to choose by balancing the norms of the linearized residual and of the solution. We justify the use of this two step procedure based upon the empirical observation that the shape of the L-curves do not change appreciably for a wide range of values of . We choose all regularization parameters using the point on the L-curve closest to the intersection of lines that are fit to points from the initial and tail regions of the L-curve. III. EXAMPLES We apply the DBIM-EN and the DBIM-RR to data simulated using two different 3-D computational testbeds. The first testbed consists of a generic object—a simple cube—whose distribution of dielectric properties is exactly represented by a linear combination of a small number of wavelet basis functions. We compare the DBIM-EN with those obtained using the DBIM-RR for ten different distributions of dielectric properties within the

2 2

Fig. 1. The first computational testbed consists of a 6.4 6.4 6.4 cm cubic object surrounded by a 40-element antenna array of 1.4-cm-long dipoles. The dielectric properties distribution within the cube is generated by a linear combination of 13 3-D Haar wavelets.

object. The second testbed consists of an anatomically realistic breast phantom with dielectric properties that correspond to the microwave frequency range. A previously reported inverse scattering algorithm also based upon the DBIM [40] is applied to the simulated data for the purpose of generating a low-resolution estimate of the distribution of dielectric properties within the breast phantom. The DBIM-EN and DBIM-RR estimate higher-resolution details within the breast phantom using this low-resolution estimate as an initial guess. For both examples we use a fast implementation of the modified LARS algorithm available at [41]. A. Microwave Imaging of a Heterogeneous Lossy Dielectric Cube The purpose of the first testbed is to demonstrate the performance of our sparsity approach to inverse scattering for the ideal case where the distribution of dielectric properties is sparse in a known set of basis functions. We first describe the computational testbed, which is shown in Fig. 1. Then we present qualitative and quantitative results which indicate that the elastic net outperforms the ridge regression approach for this scenario. 1) Testbed: The testbed consists of a 6.4 6.4 6.4 lossy dielectric cube. The distribution of dielectric properties within the cube is an exact linear combination of 13 3-D Haar wavelets [42]. The cube is immersed in a coupling medium (rel, conductivity: ) and is ative permittivity: surrounded, as shown in Fig. 1, by a 40-element cylindrical antenna array consisting of five elliptical rings of eight electrically-small z-directed dipole antennas. The ring spacing in the -dimension is 1 cm. Each dipole antenna is modeled by two segments of 6-mm-long copper wire separated by a 2 mm gap. Physical interaction between any of the 1.4-cm-long array elements is minimized by offsetting the placement of the dipoles in each ring by 22.5 from the placements in the neighboring rings. A finite-difference time-domain (FDTD) computational electromagnetics simulation [43] is conducted to acquire microwave signals measured at all recording antennas for every transmitting antenna in the array. The spatial grid cell size in these simulations is 2 mm. In each simulation, a different antenna array element is excited with a modulated Gaussian pulse (-20 dB bandwidth: 500–3.5 GHz) applied at the feed point. The bandwidth of the radiated signal 5 cm away in the coupling medium

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0

Fig. 2. Cross sections in the x y plane (z = 2 cm) showing the results for one of the ten trials for the first computational testbed shown in Fig. 1 The true distribution of relative permittivity; (b) estimated permittivity using the DBIM in conjunction with the oracle; (c) estimated permittivity for DBIM-RR; (d) estimated permittivity for DBIM-EN. Although (a) and (b) appear identical, Table I shows that there is some error in the oracle estimate.

is 875–3.75 GHz. The -directed electric fields at the feed point of the other antennas in the array are recorded and transformed to phasors at 2.1 GHz. The distribution of dielectric properties within the cube is given by a 3-D Haar wavelet expansion consisting of a scaling basis functions). The function and 3 levels of wavelets ( scaling function and the first level of wavelets have support over the entire cube, while the wavelets from the second and third levels have support over sub-volumes of 3.2 3.2 3.2 and 1.6 1.6 1.6 , respectively. Each of the orthonormal basis functions in the expansion is normalized by the amplitude of the scaling function. We conduct a study consisting of ten trials, where in each trial the scaling function and 12 of the Haar wavelets are randomly chosen to represent the true distribution of dielectric properties within the cube. The complex coefficient for the scaling function is chosen in each trial so that the baseline dielectric propand . erties within the cube are Four wavelets are then randomly selected from each of the three levels in the Haar expansion. The coefficients for these ran, where the sign domly selected wavelets are is chosen randomly. For each of the ten trials, ranges from 7.04 to 12.96, and ranges from 0.0511 to 0.199 S/m. An cross section of the distribution of within the cube for one of the ten trials is shown in Fig. 2(a). 2) Results: We use FDTD with a spatial grid cell size of 2 mm as the forward solver for the DBIM. Note that this is the same spatial grid cell size used for the “measurement” FDTD simulations. Committing this so-called inverse crime [1] allows the field residual (the measurement error, discussed later) to be driven to zero, which is desirable from the point of view

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of being able to set an upper bound on performance. This, together with the lack of measurement noise, allows the impact of ridge regression and the elastic net to be isolated. The example in Section III-B considers the case where the inverse crime is not committed. In these ten trials we make use of some a priori information about the cubic object. We use the DBIM to optimize just over the set of the 512 Haar wavelets; this implies knowledge of the location of the cube boundary. We also assume knowledge of the baseline dielectric properties of the cube, which are used as an initial guess by the DBIM to speed up convergence of the DBIM. We consider the performances of the DBIM-EN and DBIM-RR and compare them to the ideal solution calculated with what is known as the oracle [42]. The oracle involves solving (7) by considering only the Haar wavelets which are known to be present in the true solution. Since this knowledge will never be available in practice, the oracle provides a lower ). We bound on the error in the estimated coefficients ( simulate data acquisition with FDTD and then apply the DBIM to data collected for each of the ten trials using the oracle, ridge regression, and the elastic net. The DBIM converges in six iterations for all trials and approaches. The oracle approach produces solutions which involve only 13 of the 512 Haar wavelets. In contrast, the ridge regression solutions involve all 512 Haar wavelets. The elastic net solutions involve between 132 and 208 wavelets at each iteration with a mean of 162. The results of the ten trials for all three approaches are summarized in Table I. The normalized field residual indicates how well the three approaches fit the scattered electromagnetic field data. The normalized coefficient error measures how close the estimates are to the true solution. Table I lists the mean values and standard deviations for these two metrics. All three approaches achieve similar levels of performance with regards to fitting the scattered electromagnetic field data. The slightly lower field residuals for the approaches using ridge regression and the elastic net can be explained by the extra degrees of freedom afforded those solution strategies and the ill-posed nature of the nonlinear inverse scattering problem. As expected, the oracle performs the best in terms of coefficient error since it only optimizes over the 13 Haar wavelets that are actually involved in the true solution. The mean coefficient error for the elastic net is approximately four standard deviations lower than the corresponding value for ridge regression. This indicates that the DBIM-EN results in a significant improvement over the DBIM-RR. The improved perforcross sections of mance is evident in Fig. 2, which shows estimated at 2100 MHz for one of the ten trials. B. Microwave Imaging of a Heterogeneous Breast Phantom The purpose of the second testbed is to demonstrate the performance of our sparsity approach to the inverse scattering problem for a more realistic situation where the distribution of dielectric properties is not exactly sparse in a chosen set of basis functions. We consider an example from the field of microwave breast imaging [44]–[47]. Interest in this field has been fueled by data suggesting that the dielectric properties of breast tissue at microwave frequencies [48]–[52] are sensitive to certain physiological factors of clinical interest, such as

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TABLE I A SUMMARY OF THE RESULTS FROM THE TEN TRIALS WITH THE HETEROGENEOUS LOSSY DIELECTRIC CUBE TESTBED SHOWN IN FIG. 1. THE PERFORMANCES OF THE DBIM USING THE ORACLE, RIDGE REGRESSION, AND THE ELASTIC NET ARE COMPARED. THE METRICS SHOWN ARE THE NORMALIZED FIELD RESIDUAL AFTER SIX ITERATIONS AND THE CORRESPONDING NORMALIZED COEFFICIENT ERROR

Fig. 3. The computational testbed for the second example consists of an anatomically realistic numerical breast phantom surrounded by a 40-element antenna array of 1.4-cm-long dipoles.

water content, temperature, and vascularization. Consequently microwave tomography has the potential for characterizing normal breast tissue density—an important factor in assessing a patient’s risk of breast cancer [53]—as well as detecting and monitoring malignancies—the focus of this example. We first describe the testbed containing the breast phantom shown in Fig. 3 and then present results which show that the elastic net approach again outperforms ridge regression. 1) Testbed: The numerical breast phantom shown in Fig. 3 is derived using a 3-D MRI dataset from a patient with “scattered fibroglandular” breast tissue, based on the American College of Radiology’s BI-RAD system [54]. The scattered fibroglandular breast tissue is evident in the three cross sections of Fig. 4(a), (c), and (e). These orthogonal cross sections pass through the center of a 1-cm-diameter inclusion that has been added to the phantom to represent a malignant lesion. Data is acquired using FDTD as with the testbed of Fig. 1, save for a few differences. The spatial grid cell size in this testbed is 0.5 mm; this smaller grid cell size is required to resolve the fine geometrical features of the breast phantom. The phantom testbed uses an antenna array whose elliptical rings have dimensions 9.6 12.4 cm. These array dimensions ensure that no antenna is closer than 1 cm to the surface of the breast phantom. The numerical breast phantom is created following a procedure similar to those reported in [55], [56]. The intensity of the voxels in the MRI dataset is converted to dielectric properties via a piecewise linear mapping [6], [57]. The interior of the breast phantom is segmented into three distinct regions: adipose, fibroglandular, and transition. We adopt the dielectric properties

Fig. 4. Three orthogonal cross sections of relative permittivity at 2100 MHz for the breast phantom testbed. The left column [(a),(c),(e)] shows the true distribution of relative permittivity at a spatial resolution of 0.5 mm. The right column [(b),(d),(f)] shows the low-resolution estimated relative permittivity used as the initial guess in the DBIM-EN and DBIM-RR. Note that the estimate uses a much coarser spatial resolution (2 mm).

reported in a recent large scale dielectric spectroscopy study [52] for the adipose and fibroglandular regions in the breast phantom. Lazebnik et al. [52] reported the microwave-frequency dielectric properties for three breast tissue groups (Groups 1, 2, and 3) defined by their adipose content. Samples from Group 3 were composed primarily of adipose tissue with relatively low dielectric properties, and so we assign the dielectric properties reported for Group 3 to the adipose regions of the phantom. Groups 1 and 2 were comprised of samples with smaller amounts of adipose tissue and correspondingly higher dielectric properties, with samples from Group 1 having the least amount of adipose tissue and the highest average dielectric properties [52]. We choose to assign the properties from Group 2 to the fibroglandular region of the phantom, although using the properties from Group 1 would constitute a more challenging problem. We note that the combined use of Group 2 and Group 3 properties is still much more realistic than those considered in any other previous theoretical study of microwave imaging for breast cancer detection. MRI voxel intensities in the adipose and fibroglandular reranges about the mean properties gions are mapped to assigned to each tissue type. Voxels in the transition region are mapped to the range spanning the maximum of the adipose

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TABLE II DIELECTRIC PROPERTIES (AT 2.1 GHZ) OF THE VARIOUS MEDIA PRESENT IN THE HETEROGENEOUS BREAST PHANTOM TESTBED OF FIG. 3

range to the minimum of the fibroglandular range. The 2-mmthick skin layer is modeled using the dielectric properties for dry skin [58]. The dielectric properties assigned to the spherical inclusion are adapted from [59] and are representative of malignant breast tissue properties in our frequency range of interest. Table II gives the dielectric properties used in the FDTD model at 2.1 GHz (the frequency at which the scattered signals are recorded). 2) Results: Note that in this example the spatial resolutions of the data-generation and the DBIM forward-solver FDTD simulations are not the same. The grid cell size is 0.5 mm for the data-generation FDTD simulations and 2 mm for the forward solver. This results in an inherent mismatch between the two sets of simulations and produces as much as 15% difference even when the exact same distribution of dielectric properties is simulated. The mismatch is partially corrected through the use of the calibration procedure proposed by Meaney et al. [44], but because this calibration procedure is not perfect, the inverse crime is not being committed in this example. The data-generation FDTD simulations are run in parallel on a computing cluster, while the forward solver is run on a desktop computer utilizing Acceleware hardware acceleration technology [60]. The forward solver simulations require about 30 seconds per antenna, or approximately 20 minutes per DBIM iteration. Fig. 4(b), (d) and (f) shows the low-resolution estimate of the relative permittivity distribution at 2.1 GHz that we use as the initial guess in the DBIM-RR and DBIM-EN. The initial estimate of the conductivity is similar in appearance, except the grayscale spans 0–1.9 S/m. These estimates are obtained using the low-resolution inverse scattering algorithm reported in [40]. This algorithm, which is also based upon the DBIM, estimates the properties using a smooth set of basis functions with a nominal resolution of about 1 cm. The presence of the spherical inclusion is apparent in Fig. 4, although we note that the estimated contrast of the scatterer is less than the true contrast. We investigate the feasibility of generating a higher-resolution image of the interior of the breast phantom using the DBIM-RR and DBIM-EN with a set of 3-D Gaussian basis functions, constructed as follows. A cuboidal volume ) enclosing the breast phantom is first (7.2 11.2 6.8 defined. Fifteen 1-D Gaussians are defined along each of the axes of the cuboidal volume. The standard deviations for these 1-D Gaussians are 3.6, 5.6, and 3.4 mm for the , , and axes, respectively; the 1-D Gaussians are spaced along each axis standard deviations. Three-diof the cuboidal volume by mensional Gaussian basis functions are defined using all 3375

Fig. 5. Three cross sections of the estimated relative permittivity at 2100 MHz obtained using the DBIM-RR [(a),(c),(e)] and the DBIM-EN [(b),(d),(f)]. The output of a low-resolution inverse scattering algorithm [40], shown in Fig. 4(b), (d) and (f), is used as an initial guess for both algorithms.

combinations of the 1-D Gaussians and the Kronecker product [42]. For simplicity, we only include basis functions which have at least 95% of their support within the breast phantom interior; these chosen basis functions are truncated so that they are entirely supported within the breast phantom interior. This reduces the number of 3-D Gaussian basis functions from 3375 to 737. These 3-D Gaussian basis functions are not optimal for this example in any sense; they are selected for simplicity and to demonstrate that our sparsity approach to inverse scattering can be successful even when the true distribution of dielectric properties is not exactly sparse in the chosen set of basis functions. Figs. 5 and 6 show the results from applying the DBIM-RR and DBIM-EN using these 737 3-D Gaussian basis functions and the low-resolution initial guess of Fig. 4(b),(d) and (f). The elastic net approach takes six iterations to converge, while the ridge regression approach converges after three iterations. The two sets of images of estimated relative permittivity at 2100 MHz (Fig. 5) appear similar, although we note that the contrast of the spherical inclusion is slightly higher in the elastic net estimate. The two sets of images of estimated effective conductivity (Fig. 6) are very different from each other. The elastic net estimate is sharper and has higher contrast, while the ridge regression estimate falsely indicates the presence of additional high contrast scatterers. These artifacts are suppressed in the elastic net estimate since on average only 234 of the 737 3-D Gaussian

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consistently performed better than the DBIM-RR across ten trials involving different dielectric properties distributions within the object. Second we presented reconstructions of a geometrically complex object—an anatomically realistic breast phantom—to demonstrate the feasibility of using the DBIM-EN for microwave breast imaging. We initialized the DBIM-EN as well as the DBIM-RR with a low-resolution estimate of the dielectric properties distribution within the breast phantom obtained using the inverse scattering algorithm reported in [40]. The DBIM-RR solution made use of the entire set of 737 3-D Gaussian basis functions chosen for this formulation, while the DBIM-EN solution made use of less than one third of the basis functions. The DBIM-EN produced an enhanced contrast image of a 1-cm-diameter inclusion, while DBIM-RR reconstructions falsely indicated the presence of additional scatterers. ACKNOWLEDGMENT The authors would like to thank K. Sjöstrand for permission to use the fast implementation of the modified LARS algorithm [41]. REFERENCES

Fig. 6. Three cross sections of the estimated effective conductivity at 2100 MHz obtained using the DBIM-RR [(a),(c),(e)] and the DBIM-EN [(b),(d),(f)]. The output of a low-resolution inverse scattering algorithm [40] is used as an initial guess for both algorithms.

basis functions are involved in the solution at each iteration of the DBIM-EN. IV. SUMMARY AND CONCLUSION We demonstrated the feasibility of solving the electromagnetic inverse scattering problem using a basis function formulation of the DBIM in conjunction with a variable-selection approach known as the elastic net. The elastic net applies both and penalties to the system of linear equations that result at each iteration of the DBIM. The combined and regularizations stabilize the inverse problem and promote sparsity in the solution. A more typical approach known as ridge regression only involves the penalty. We applied the DBIM-EN to data simulated using two different 3-D computational testbeds, and compared results with those obtained using the DBIM-RR. First we presented reconstructions of a geometrically simple object whose heterogeneous dieletric properties distribution is exactly represented by a linear combination of a small number of wavelet basis functions (13 out of a possible 512). The DBIM-RR solution made use of all 512 basis functions, while the DBIM-EN solution made use of a subset of the basis functions. We used a representative set of reconstruction cross-sections to illustrate that the DBIM-EN performance is qualitatively closer than the DBIM-RR to that of the ideal approach, namely the DBIM with an oracle. We also conducted quantitative comparisons and demonstrated that the DBIM-EN

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Barry D. Van Veen (S’81–M’86–SM’97–F’02) was born in Green Bay, WI. He received the B.S. degree from Michigan Technological University, Hougton, in 1983 and the Ph.D. degree from the University of Colorado-Boulder, in 1986, both in electrical engineering. He was an ONR Fellow while working on the Ph.D. degree. In spring 1987, he was with the Department of Electrical and Computer Engineering, University of Colorado-Boulder. Since August of 1987, he has been with the Department of Electrical and Computer Engineering, University of Wisconsin-Madiso, and currently holds the rank of Professor. His research interests include signal processing for sensor arrays, magneto- and electroencephalography, and biomedical applications of signal processing. He is coauthor of Signals and Systems (1st ed. 1999, 2nd ed., 2003 Wiley). Dr. Van Veen was a recipient of a 1989 Presidential Young Investigator Award from the National Science Foundation and a 1990 IEEE Signal Processing Society Paper Award. He served as an Associate Editor for the IEEE TRANSACTIONS ON SIGNAL PROCESSING and is a member of the IEEE Signal Processing Society’s Statistical Signal and Array Processing Technical Committee and the Sensor Array and Multichannel Technical Committee. He is a Fellow of the IEEE and received the Holdridge Teaching Excellence Award from the ECE Department at the University of Wisconsin in 1997.

Susan C. Hagness (S’91–M’98–SM’04–F’09) received the B.S. degree with highest honors and the Ph.D. degree in electrical engineering from Northwestern University, Evanston, IL, in 1993 and 1998, respectively. Since August 1998, she has been with the Department of Electrical and Computer Engineering, University of Wisconsin-Madison, where she currently holds the position of Philip D. Reed Professor. She is also a faculty affiliate in the Department of Biomedical Engineering. Her current research interests include microwave imaging, sensing, and thermal therapy techniques including breast cancer detection and treatment, electromagnetic inverse scattering, ultrawideband radar, dielectric spectroscopy, bioelectromagnetics, FDTD theory and applications in biology and medicine, and global modeling of carrier-field dynamics. Dr. Hagness is currently serving as the Chair of Commission K of the United States National Committee (USNC) of the International Union of Radio Science (URSI), and the Chair of the IEEE AP-S New Technologies Committee. While pursuing the Ph.D. degree, she was a National Science Foundation (NSF) Graduate Fellow and a Tau Beta Pi Spencer Fellow. She was the recipient of the Presidential Early Career Award for Scientists and Engineers presented by the White House in 2000. In 2002, she was named one of the 100 Top Young Innovators in Science and Engineering in the World by the Massachusetts Institute of Technology (MIT) Technology Review Magazine. She is also the recipient of the University of Wisconsin Emil Steiger Distinguished Teaching Award (2003), the IEEE Engineering in Medicine and Biology Society Early Career Achievement Award (2004), the URSI Isaac Koga Gold Medal (2005), the IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING Outstanding Paper Award (2007), the IEEE Education Society’s Mac E. Van Valkenburg Early Career Teaching Award (2007), and the University of Wisconsin System Alliant Energy Underkofler Excellence in Teaching Award (2009). She served as an elected member of the IEEE Antennas and Propagation Society (AP-S) Administrative Committee from 2003 to 2005 and as an Associate Editor for the IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS from 2002 to 2007.

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Adaptive CLEAN With Target Refocusing for Through-Wall Image Improvement Paul C. Chang, Student Member, IEEE, Robert J. Burkholder, Fellow, IEEE, and John L. Volakis, Fellow, IEEE

Abstract—Signal processing techniques to improve building interior images are introduced. Specifically, an adaptive CLEAN algorithm is introduced in conjunction with target refocusing techniques to de-emphasize undesired signal effects from external walls, and thus more accurately image the lesser contrast objects. Instead of using the point spread function as in the conventional CLEAN algorithm, adaptive CLEAN incorporates a delay spread model found more appropriate for subtracting sidelobes that could obscure interior objects. Refocusing also allows for correcting target distortion and displacement due to through-wall propagation. It is shown that by combining adaptive CLEAN with target refocusing techniques, we are able to reconstruct interior targets (with minimal exterior wall distortion). The introduced processing techniques are demonstrated via simulations generated with a well validated high frequency ray tracing code, and with measured data. Index Terms—Image restoration, parameter estimation, radar imaging, radar signal processing, synthetic aperture radar.

I. INTRODUCTION

T

HERE is increasing interest to employ radar signals for information gathering and imaging of selected objects within a complex environment. Of specific interest here is through-wall imaging (TWI) of building interiors [1]. Several papers have already demonstrated that near-zone radar sensing can indeed be employed for imaging beyond a single wall [2]–[6]. These papers have employed conventional synthetic aperture radar (SAR) imaging techniques where a matched filter is applied at every pixel point for image formation. However, once the transmitting signal penetrates through the first wall, free space assumptions no longer hold and propagation phenomenology plays a more important role. In addition to multi-scattering effects, signal attenuation and distortion after propagation through one or more walls becomes an issue. As a result, the object’s detectability decreases and the target scattered signals may be displaced and defocused. In addition to distortions, monostatic TWI radar data may also be dominated by scattering from exterior walls. Thus, the building interior details are often obscured by the wall signature Manuscript received October 26, 2008; revised May 29, 2009. First published November 06, 2009; current version published January 04, 2010. The authors are with the ElectroScience Laboratory, Department of Electrical and Computer Engineering, The Ohio State University, Columbus, OH 43212 USA (e-mail: [email protected]; [email protected]; [email protected]. 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.2009.2036131

and processing sidelobes. For example, Fig. 1 illustrates how building interior SAR images appear obscured and distorted behind exterior walls (generated using the simulations and the imaging function of (4) described later). Several efforts were made to address localization errors and overcome target blurring to improve through wall imaging. For example, in [7] wall effects (for a uniform dielectric slab) were compensated via a modified matched filter so that the point target response can be reconstructed more accurately using standard SAR imaging. A similar approach was carried out in [8] using a wideband beamformer imaging technique with a stationary array. However, wall characteristics are not known a priori and we still lack suitable post-processing methods to de-emphasize undesired radar returns (e.g., exterior walls). Some recent imaging efforts have already focused on target localization estimates without wall knowledge [9]–[11]. In this context, the work in [11] proposed an interactive focusing technique for a homogeneous wall to examine image quality metrics under wall uncertainty. However, although [11] did provide insights on how wall errors can affect pixel intensity, it is based on far-field and small error assumptions. In [12], a differential SAR (D-SAR) approach was applied to isolate localized interior objects by removing features that are slowly varying in cross-range. This is effective for removing all broadside wall scattering from the image, but it may also remove features of interest such as interior walls. Herewith we employ a wall-mitigation technique based on the proposed adaptive CLEAN algorithm (A-CLEAN) along with target refocusing to systematically improve building interior images. A-CLEAN [13] is a CLEAN-like algorithm [14]–[17] that removes radar returns from external walls by subtracting signals using a pre-computed delay spread model. Unlike windowing techniques that remove the entire region around the wall, A-CLEAN resolves objects in close proximity to the wall. Target refocusing can then be used to correct the false free space assumption. The proposed signal processing scheme is intended to remove the direct radar return from exterior walls and the associated sidelobe clutter while preserving shape and location of the interior details. This paper is organized as follows. Section II discusses nearzone radar imaging and its extension to the wall mitigation. Section III provides some background information to extract parameters of an unknown wall prior to introducing the A-CLEAN algorithm in Section IV. Section V presents the target refocusing approach to correct for distortions in through-wall imaging. Numerical results from simulations are presented in Section VI for validation purposes, and Section VII applies the new methods to a real through-wall experiment.

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Fig. 2. Radar scene under interrogation.

where is the estimated reflectivity density from the individual scene points, and refers to the scene pixel location of interest. may be included to minimize sideA windowing function lobe clutter. Under the assumption of free space propagation, harmonic time convention assumed and supand with an pressed, the backscattered transfer function takes the form

(3)

Fig. 1. Interior building details heavily obscured by wall distortion and strong exterior wall scattering. The thickness of the wall is 15 cm and the relative per: : . The frequency band is 0.7–1.0 GHz: (a) building mittivity is " layout with three PEC spheres placed as objects behind walls, (b) conventional SAR image in presence of wall scattering.

= 3 10j0 372

II. NEAR ZONE RADAR IMAGING AND THE IDEA OF WALL MITIGATION Near zone radar imaging of a given scene is essentially an inverse problem. From measured or other collected data, information is extracted from the scene by applying an inverse transfer function, deconvolution, or a maximum likelihood estimator. As an example, consider a general radar target scene (shown in Fig. 2) where a stationary target is interrogated by some spatially-varying sensors. The received backscattered field by the th sensor (plus noise or clutter) can be represented in the frequency domain by the integral (1) where is the transfer function, or Green’s function, or point is the spread function of the imaging operation. wave number where is the frequency and is the speed of light, and is the volume occupied by the target or scene. represents the noise and clutter terms. The “reflectivity density” is the scattering return from the location on the target. is the nth sensor location. Given the received signal at the sensors, , the imaging function is given by

(2)

which accounts for the two-way propagation, and the imaging function thus becomes

(4)

In this expression, the inner integral over generates the range profiles and can be computed via the Fast Fourier Transform (FFT) algorithm. In essence, (4) is similar to the well-known backprojection algorithm. However, the assumption of free space propagation may not be sufficient for some scenarios. The case of through-wall imaging of building interiors is one such situation. In this paper we present an approach for wall mitigation and introduce a more suitable model-based imaging function using a similar inverse operation as in (2). The pertinent scenario (where the targets are surrounded by walls) is depicted in Fig. 3. Taking into account the wall scattering effects, the received signal plus noise (or clutter) in (1) is revised to read

(5)

and are the wall reflection and transmission where coefficients, respectively. The first term in (5) is the direct wall reflection, and as in (1), the second term represents the scattering

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extracted from slab reflection and transmission properties. To do so, the slab can be illuminated by a harmonic pulse whose early return is isolated from the rest of the target returns via windowing. This preprocessed data can then be used in conjunc& ) formulae for tion with transmission and reflection ( multilayered slabs (e.g., as derived in [23]) to extract the thickness and dielectric constant of each layer. For example, the Fresnel plane wave reflection and transmission coefficients for a single-layer dielectric slab are given by Fig. 3. Radar scene in presence of walls.

due to the wall enclosed scene. As our goal is to extract , we propose the following inverse imaging function to replace (4)

(7) where,

(6) In this, the bracket terms in (6) serve to resolve direct reflection from and propagation distortions through the exterior wall. Below we present the Adaptive CLEAN and target refocusing and isolate it from wall distortions as algorithms to render much as possible. III. ELECTROMAGNETIC WALL CHARACTERIZATION As can be understood from above, it is necessary to have an accurate model of the electromagnetic (EM) reflection and transmission properties for wall mitigation purposes. However, realistic walls may be very complex multi-layered and periodic structures with unknown material properties and dimensions. This section discusses reduced-order models for simplifying the analysis, and ways to extract the parameters of these models from measured or simulated data.

(8) is the impedance of free space and is the angle from normal in the plane of incidence. This assumes a vertical slab and a horizontal plane of incidence. Basically, a minimum mean squared error (MMSE) estimator is used here to determine the effective dielectric constant and layer thicknesses for the chosen equivalent multilayered slab. A genetic algorithm has been used as well in [24]. The unknown parameters may be estimated from measured data by isolating the broadside reflection from the exterior wall as shown in . Once the parameters of are known, Fig. 3, and fitting to then may be computed using an analytic relationship such as (7).

A. Reduced-Order Model As already noted, our proposed imaging improvements are based on (a) adaptive CLEAN with subtraction of wall scatterers, and (b) refocusing of the scattering centers behind walls to suppress distortions. These post-processing techniques imply accurate modeling of scattering from and transmission through walls. In addition to measurements for extracting the transmission and reflection properties of the unknown wall slabs, rigorous analytical techniques such as Floquet mode analysis and numerical methods have been used in the past to compute EM field interactions in commonly used walls [18]–[21]. For computational efficiency, reduced-order models were presented in [22] to evaluate the reflection and transmission characteristics of periodic composite walls (made of bricks and cinderblock). Herewith, we employ a multilayered slab model found appropriate for general wall modeling over the frequency band of interest (based on the formulas from [22]). B. Estimation of Wall Parameters To realize the multilayered slab model in our algorithms, a good estimate of the wall dielectric parameters is required to be

IV. ADAPTIVE CLEAN ALGORITHM This section presents the implementation of the second bracket term in (6) via a modification of the CLEAN algorithm. The traditional CLEAN algorithm for radar imaging [14] originated in the field of astronomy [15]. Briefly, the CLEAN algorithm searches for the brightest point in a scene and assigns it a point spread function based on its intensity. Subsequently, this brightest point is removed from the scene and the next brightest spot is searched for and extracted. This process is repeated until all unwanted bright spots are removed. However, for real-world scenes, the scatterers are not isolated (viz. uncoupled) point scatterers [17]. The adaptive CLEAN algorithm (A-CLEAN) proposed here is intended to address this issue for through-wall imaging of buildings. Our approach is to use a wall spread function (rather than a point spread function) obtained via a band-limited Fourier transform of the frequency domain reflection coefficient. This wall spread function is located and extracted from the downrange profile associated with each sensor location. For conventional CLEAN, the assumed spread function (the frequency domain version) is

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constant over the band of interest, resulting in a point spread function given by (in the downrange domain),

(9)

is the center frequency where and is the bandwidth. For adaptive CLEAN, we instead have

(10)

is the analytical reflection coefficient obtained where from the multilayered slab model. To illustrate the application of A-CLEAN, we refer to Fig. 4 showing the image of a sphere very close to and behind a wall. Specifically, Fig. 4(a) shows the scanning setup with the sensor movement parallel to the wall. For this example, the frequency band was 0.7–1.0 GHz, polarization was vertical, wall thick. ness was 15 cm and the relative permittivity was Figs. 4(c) and (d) give the corresponding downrange plots when . As is seen, the sensor position was at the point spread function of the conventional CLEAN algorithm does not adequately represent the downrange profile of the wall scattering. Therefore, subtracting it will not completely remove the wall contribution. In contrast, the pre-computed model spread function provides a much better representation of the wall scattering signature. This is depicted in Fig. 4(d) and upon coherent subtraction, it should remove the wall contribution. Indeed, a comparison of the images of Fig. 4(b), (e), and (f) demonstrates that A-CLEAN isolates the sphere quite well. It is noted that the CLEAN and A-CLEAN algorithms are carried out in the downrange domain before generating the images of Figs. 4(e) and (f). In other words, the range profile for each sensor position is pre-processed before image backprojection. A brief overview of the A-CLEAN algorithm is shown at the bottom of the page. Here, is the radar phase history of is the pre-obtained reflection coefa given sensor, and ficient in the frequency domain. Also, is the shifted and weighted spread function in the downrange domain. It is chosen to fit an undesired strong peak, such as the exterior is then given in step 2.c. We wall. The CLEANed response remark that the minimization in Step 2.b.i. may be performed

Fig. 4. Near zone through-wall sensing of a sphere behind a wall. (a) Scanning setup. (b) Conventional SAR image. (c) Conventional CLEAN point spread function for the wall contribution. (d) Adaptive CLEAN pre-computed wall signature spread function. (e) Image after conventional CLEAN pre-processing. (f) Image after adaptive CLEAN pre-processing. Color scale is linear and normalized to peak intensity in each image.

locally only over the strongest peak in the downrange profile to reduce CPU time. V. MODEL-BASED TARGET REFOCUSING As already noted, wall interactions can create ambiguities in target detection and scene understanding. To overcome this issue, we apply a model-based imaging approach that accounts for phase delay and distortion due to the outermost wall [25]. From (6), the imaging algorithm should appropriately incorporate a through-wall propagation model to compensate for wall effects. This is accomplished in (6) using the inverse of the . As described in Section III, transmission coefficient may be computed once the parameters of

1. Pre-compute spread function 2. For each sensor location : a. Compute downrange profile b. For each undesired strong peak : and such that i. Find by ii. Replace for all . c. Compute

for all

in the downrange profile.

for all . is minimized. for all .

CHANG et al.: ADAPTIVE CLEAN WITH TARGET REFOCUSING FOR THROUGH-WALL IMAGE IMPROVEMENT

Fig. 5. Through-wall SAR imaging of three PEC spheres using the proposed processing techniques. (a) Radar scanning scenario. (b) Conventional image. (c) Conventional image without front wall. (d) Image after target refocusing. (e) Conventional image after A-CLEAN pre-processing. (f) Refocused image after A-CLEAN pre-processing. Color scale is in dB relative to the peak intensity of each image.

have been estimated using an analytic relationship such as (7). is that it may One problem with using the inverse of have nulls in the frequency domain which makes the inverse singular. To avoid this it has been found that using the conjugate is sufficient for refocusing the target image. phase of The alternative imaging function is given by

(11) is the phase of the transmission coefficient. where To summarize, the A-CLEAN algorithm with target refocusing proposed here has the following three steps. 1. Estimate the wall parameters by fitting a multilayer slab to the broadside reflection reflection coefficient from the exterior wall. 2. Apply the A-CLEAN algorithm to remove the wall scattering from the downrange profiles. 3. Compute the refocused image function from (6) or (11) calculated from using the transmission coefficient the wall parameters found in step 1.

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Fig. 6. Through-wall SAR imaging of three T-shaped dielectric targets using the proposed processing techniques. (a) Radar scanning scenario. (b) Conventional image. (c) Conventional image without front wall. (d) Image after target refocusing. (e) Conventional image after A-CLEAN pre-processing. (f) Refocused image after A-CLEAN pre-processing. Color scale is in dB relative to the peak intensity of each image.

VI. NUMERICAL RESULTS In this section we demonstrate the proposed adaptive CLEAN and target refocusing techniques using two example datasets from a well-validated radar prediction code, the Numerical Electromagnetics Code—Basic Scattering Code (NEC-BSC) [26]. Specifically, we consider the scenario in Fig. 5, where a monostatic radar is moving parallel to a dielectric wall with target(s) behind it. In the first example, three one-meter radius PEC spheres are used as the targets. In Fig. 6, we show a second example where three T-shaped dielectric targets are placed relabehind the wall. The 15 cm thickness and tive permittivity are applied to both the wall and the T-shaped target plates. The frequency band is 1.0–1.3 GHz with vertical polarization. Figs. 5(b) and (c) show the conventional SAR images for the three spheres in the wall’s presence, and in free space, respectively. The color scales in the images of Figs. 5 and 6 are in dB relative to the peak value of each image. As observed, the wall reduces the scattering intensity, and slightly displaces and distorts the sphere images. Fig. 5(d) shows the refocused image using the imaging function of (6) without the A-CLEAN pre-processing step. The spheres are not only back at their original locations, but their shapes have also been mostly restored as compared to Fig. 5(c). However, the wall image is

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Fig. 7. Image formation of a building enclosing smaller wall partitions and PEC spheres. Frequency band is 0.7–1 GHz. Color scale is linear and normalized to peak intensity of each image. (a) Building geometry and four scan paths. (b) Conventional image. (c) Image after A-CLEAN.

still strong relative to the spheres. But after pre-processing with the A-CLEAN algorithm, its signature is significantly reduced. This is displayed in Fig. 5(e), even though the wall signature is still visible and the spheres images are distorted (elongated) because refocusing has not been applied. On applying the refocusing algorithm together with A-CLEAN, the resulting image in Fig. 5(f) is much improved. That is, the spheres appear nearly identical to their free space image, and the wall is barely visible. In Fig. 6, a very similar trend is observed for the T-shaped dielectric targets. However, because the double-bounce returns from the dihedrals are very strong, they display higher intensity than the PEC spheres. In a third example, the simple building shown in Fig. 7 is considered for a more practical evaluation of the A-CLEAN algorithm. To do so, NEC-BSC computations were carried out from 0.7–1 GHz (vertical polarization) as the sensor moves along the four paths shown in Fig. 7(a). Again, for this example, the exterior wall is 15 cm thick with relative permittivity . Also, the interior walls are 15 cm thick and . The floor have a relative permittivity of and ceiling are ignored in this simulation to focus on just the walls. This was important as the floor and ceiling introduce additional multibounce components that could lead to shadow images. However, these effects are generally much weaker than the direct scattering terms. As seen in Fig. 7(b), the interior walls and spheres cannot be distinguished in the conventional image due to sidelobe clutter introduced by the extremely bright surrounding walls (even though a Hamming window was used to minimize sidelobe clutter for the image formation). However, after applying the proposed A-CLEAN algorithm, as shown in Fig. 7(c), the interior walls and spheres do appear much more prominently

Fig. 8. Experimental through-wall SAR imaging of a large metal sphere and trihedral. Frequency band 0.5 to 2 GHz, vertical polarization. Color scale is in dB and is the same for all three images. (a) Experimental set-up, (b) conventional image. (c) CLEANed image. (d) CLEANed and refocused image.

as compared to the conventional image. Target refocusing is purposely left out in this example to demonstrate the clutter cancellation capability of our adaptive CLEAN algorithm. When refocusing is combined with A-CLEAN, the wall images would be reduced further and the interior walls and objects would appear even stronger. VII. EXPERIMENTAL RESULTS In this section we test the A-CLEAN algorithm with target refocusing on experimental through-wall SAR imaging data conducted indoors at the Ohio State University ElectroScience Lab. Fig. 8(a) shows the room layout with a large metal sphere and a 9” trihedral as targets. The walls are cinderblock. A 2–12 GHz vertically polarized horn antenna is moved down the hallway outside the room, parallel to the wall, in 3” increments. SAR data is collected at each antenna location via S11 measurements with a network analyzer. The backscatter data is calibrated using a large metal plate placed against the wall on the side of the antenna. Two sweeps are taken with the antenna at 41” and 47” height. Figs. 8(b)–(d) shows the SAR images. The area of the images is shown in Fig. 8(a). It is noted that the room is empty in the image area, except for the two targets and the lab benches on the far side of the room. The images are generated using a frequency

CHANG et al.: ADAPTIVE CLEAN WITH TARGET REFOCUSING FOR THROUGH-WALL IMAGE IMPROVEMENT

band from 0.5 to 2 GHz, coherently summing the images from the two sweeps. Fig. 8(b) shows a conventional image computed range correction. The front using (4), but without the of the wall is the bright band at 1.1 m downrange, followed by a slightly lighter band at 1.4 m downrange, which corresponds to the backside of the wall. The sphere and trihedral are visible, but are obscured by downrange ringing. This ringing is due to the cinderblock wall and the multi-bounce interactions between the horn antenna, the outer wall, and the floor. It is noted that the sphere and trihedral would probably not be visible if they were closer to the wall. Fig. 8(c) shows the A-CLEANed image. In step 1 of Section V, the early-time downrange reflection from the wall is matched to the analytic reflection coefficient for a three-layer model of cinderblock, as described in [22]. The parameters are found via the MMSE method. The relative dielectric constant is found to be 3.8-j0.2 for the cinderblock material. The outer two layers are of this material and are 1.25” thick. The inner periodic layer of the cinderblock is 5” thick, and is replaced with an equivalent homogeneous material with dielectric constant 1.7-j0.05. However, as expected, when the A-CLEAN algorithm is applied to the experimental data, only the two bright bands corresponding to the wall were effectively reduced in the image. The later downrange ringing in the vicinity of the target caused by the cinderblock cell structure was not removed. It is possible that a more accurate periodic model of the wall could be used to reduce some of the ringing [27]. Step 3 is to refocus the image using the transmission coefficient computed from the wall parameters obtained in step 1. Only the phase of the transmission coefficient is used as in (11) to avoid singularities. Fig. 8(d) shows the resulting A-CLEANed and refocused image. There is a noticeable focusing of the two targets compared with Fig. 8(b), and a position correction to cancel the delay caused by signal propagation through the wall. The correct position is 1.7 m downrange from the front of the wall to the center of the sphere and the vertex of trihedral. (Note that the front wall appears to also be shifted forward in Fig. 8(d) because the transmission coefficient is applied to the whole image, and not just to points beyond the wall.) It is also interesting to note that the downrange ringing associated with the cinderblock wall appears to be somewhat suppressed in the refocused image. This is because the transmission coefficient in (11) tends to accentuate only the features seen through the wall, whereas the downrange ringing is a backscatter effect from the wall itself. These experimental results verify the efficacy of the A-CLEAN with refocusing approach. VIII. CONCLUSION Two signal processing approaches have been demonstrated to improve radar imagery of building interiors. Specifically, an image refocusing technique is applied with a model-based imaging function to correct for distortion, attenuation, and displacement effects of through-wall propagation. In addition, to better accentuate the weaker interior scattering features, an adaptive CLEAN algorithm was also proposed to remove strong exterior wall returns whose sidelobes may obscure the

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interior objects. When combined, these techniques have been shown to greatly improve the building’s interior image clarity. Both techniques rely on having an accurate wall model. In the simulations the wall parameters are known, but in real situations the parameters must be estimated from measurements. The experimental results of Section VII confirm that this is indeed possible, allowing us to apply the refocusing and A-CLEAN algorithms without a priori knowledge of the wall structure. REFERENCES [1] E. Baranoski, “Through wall imaging: Historical perspective and future directions,” in Proc. Int. Conf. on Acoust., Speech, and Signal Processing, Las Vegas, NV, 2008, pp. 5173–5176. [2] A. R. Hunt, “Image formation through walls using a distributed radar sensor network,” in Proc. SPIE: Laser Phys. and Photon., Spectroscopy, and Molecular Modeling V, 2005, vol. 5778, pp. 169–174. [3] L.-P. Song, C. Yu, and Q. H. Liu, “Through-wall imaging (TWI) by radar: 2-D tomographic results and analyses,” IEEE Trans. Geosci. Remote Sensing, vol. 43, no. 12, pp. 2793–2798, Dec. 2005. [4] A. Ishimaru, S. Jaruwatanadilok, and Y. Kuga, “Short pulse detection and imaging of objects behind obscuring random layers,” Waves Random Complex Media, pp. 1–15, Nov. 2005. [5] L. M. Frazier, “Surveillance through walls and other opaque materials,” in Proc. IEEE National Radar Conf. Electron. Syst., May 1996, pp. 27–31. [6] F. Soldovieri and R. Solimene, “Through-wall imaging via a linear inverse scattering algorithm,” EEE Trans. Geosci. Remote Sensing Lett., vol. 4, no. 4, pp. 513–517, Oct. 2007. [7] M. Dehmollaian and K. Sarabandi, “Refocusing through single layer building wall using synthetic aperture radar,” presented at the IEEE Int. Symp. on Antennas and Propag., 2007. [8] F. Ahmad, Y. Zhang, and M. G. Amin, “Three-dimensional wideband beamforming for imaging through a single wall,” IEEE Geosci. Remote Sensing Lett., vol. 5, no. 2, pp. 176–179, Apr. 2008. [9] G. Wang, Y. Zhang, and M. G. Amin, “A new approach for target locations in the presence of wall ambiguities,” IEEE Trans. Aerosp. Electron. Syst., vol. 42, no. 1, pp. 301–305, Jan. 2006. [10] G. Wang and M. G. Amin, “Imaging through unknown walls using different standoff distances,” IEEE Trans. Signal Process., vol. 54, no. 10, pp. 4015–4025, Oct. 2006. [11] F. Ahmad, M. G. Amin, and G. Mandapati, “Autofocusing of through-the-wall radar imagery under unknown wall characteristics,” IEEE Trans. Signal Process., vol. 16, no. 7, pp. 1785–1795, Jul. 2007. [12] M. Dehmollaian, M. Thiel, and K. Sarabandi, “Through-the-wall imaging using differential SAR,” IEEE Trans. Geosci. Remote Sensing, vol. 47, no. 5, pp. 1289–1296, May 2009. [13] P. C. Chang, R. J. Burkholder, and J. L. Volakis, “Through-wall building image improvement via signature-based CLEAN,” in IEEE Int. Symp. on Antenna Propag., San Diego, CA, Jul. 5–11, 2008. [14] J. Tsao and B. D. Steinberg, “Reduction of sidelobe and speckle artifacts in microwave imaging,” IEEE Trans. Antennas Propag., vol. 36, no. 4, pp. 543–556, Apr. 1988. [15] J. A. Hogbom, “Aperture synthesis with a non-regular distribution of interferometer baselines,” Astron. Astrophys. Supplement, vol. 15, pp. 417–426, Jun. 1974. [16] B. G. Clark, “An efficient implementation of the algorithm CLEAN,” Astron. Astrophys., vol. 89, pp. 377–378, 1980. [17] R. Bose, A. Freeman, and B. Steinberg, “Sequence CLEAN: A modified deconvolution technique for microwave images of contiguous targets,” IEEE Trans. Aerosp. Electron. Syst., vol. 38, pp. 89–97, Jan. 2002. [18] W. Honcharenko and H. L. Bertoni, “Transmission and reflection characteristics at concrete block walls in the UHF bands proposed for future PCS,” IEEE Trans. Antennas Propag., vol. 42, no. 2, pp. 232–239, Feb. 1994. [19] R. A. Dalke, C. L. Holloway, P. McKenna, M. Johansson, and A. S. Ali, “Effects of reinforced concrete structures on RF communications,” IEEE Trans. Electromagn. Compat., vol. 42, no. 4, pp. 486–496, Nov. 2000. [20] W. P. Pinello, R. Lee, and A. C. Cangellaris, “Finite element modeling of electromagnetic wave interaction with periodic dielectric structures,” IEEE Trans. Microw. Theory Tech., vol. 42, no. 12, pp. 2294–2301, 1994.

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[21] T. F. Eibert, J. L. Volakis, D. R. Wilton, and D. R. Jackson, “Hybrid FE/BI modeling of 3D doubly periodic structures utilizing triangular prismatic elements and an MPIE formulation accelerated by the Ewald transformation,” IEEE Trans. Antennas Propag., vol. 47, no. 5, pp. 843–850, May 1999. [22] C. L. Holloway, P. L. Perini, R. R. DeLyser, and K. C. Allen, “Analysis of composite walls and their effects on short-path propagation modeling,” IEEE Trans. Veh. Technol., vol. 46, no. 3, pp. 730–738, Aug. 1997. [23] J. A. Kong, Section 3.4 in Electromagnetic Wave Theory. Cambridge, MA: EMW Publishing, 2000. [24] T. Zwick, J. Haala, and W. Wiesbeck, “A genetic algorithm for the evaluation of material parameters of compound multilayered structures,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 4, pp. 1180–1187, Apr. 2002. [25] R. J. Burkholder, P. Chang, Y. Bayram, R. J. Marhefka, and J. L. Volakis, “Model-based near-field imaging of objects inside a room,” presented at the IEEE Int. Symp. on Antennas and Propagag., Honolulu, HI, 2007. [26] R. J. Marhefka, NEC-BSC Version 4.2, User’s Manual The Ohio State University, ElectroScience Laboratory, Technical Report, Jun. 2000. [27] R. J. Burkholder, R. J. Marhefka, and J. L. Volakis, “Radar imaging through cinder block walls and other periodic structures,” presented at the IEEE Antennas Propag. Symp. and USNC/URSI Nat. Radio Science Meeting, San Diego, CA, Jul. 5–12, 2008. Paul C. Chang (S’05) received the B.S. and M.S. degrees in electrical engineering from The Ohio State University, Columbus, in 2005 and 2007, respectively, where he is currently working toward the Ph.D. degree. From 2005 to 2007, he was a Graduate Research Associate at the ElectroScience Laboratory, The Ohio State University, Columbus. Since 2008, he has been with the Sensor Signal Processing Division, SET Corporation, Dayton, OH, where he is currently a Research and Software Engineer working on signal processing algorithm development. His current research interests include radar signal processing, moving target identification, spectral estimation techniques, high frequency techniques, electromagnetic modeling, SAR inverse problems and their applications.

Robert J. Burkholder (S’85–M’89–SM’97–F’05) received the B.S., M.S., and Ph.D. degrees in electrical engineering from The Ohio State University, Columbus, in 1984, 1985, and 1989, respectively. Since 1989, he has been with The Ohio State University ElectroScience Laboratory, Department of Electrical and Computer Engineering, where he currently is a Senior Research Scientist and Adjunct Professor. He has contributed extensively to the EM scattering analysis of large and complex geometries, such as jet inlets/exhausts, targets in the presence of

a rough sea surface, and urban structures. His research specialties are high-frequency asymptotic techniques and their hybrid combination with numerical techniques for solving large-scale electromagnetic radiation, propagation, and scattering problems. Dr. Burkholder is an elected Full Member of URSI, Commission B, a member of the American Geophysical Union, and a member of the Applied Computational Electromagnetics Society. He is currently serving as an Associate Editor for the IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS.

John L. Volakis (S’77–M’82–SM’89–F96) was born on May 13, 1956, in Chios, Greece and immigrated to the U.S.A. in 1973. He received the B.E. degree (summa cum laude) from Youngstown State University, Youngstown, OH, in 1978, the M.Sc. and Ph.D. degrees from the Ohio State University, Columbus, degree in 1979 and 1982, respectively. He started his career at Rockwell International (1982–84), now Boeing Phantom Works. In 1984 he was appointed Assistant Professor at the University of Michigan, Ann Arbor, MI, becoming a full Professor in 1994. He also served as the Director of the Radiation Laboratory from 1998 to 2000. Since January 2003, he is the Roy and Lois Chope Chair Professor of Engineering at the Ohio State University, Columbus, Ohio and also serves as the Director of the ElectroScience Laboratory. His primary research deals with antennas, computational methods, electromagnetic compatibility and interference, propagation, design optimization, RF materials, multi-physics engineering and bioelectromagnetics. He has published over 260 articles in major refereed journals, 450 conference papers and 20 book chapters. He coauthored the following four books: Approximate Boundary Conditions in Electromagnetics (Institution of Electrical Engineers, London, 1995), Finite Element Method for Electromagnetics (IEEE Press, New York, 1998), Frequency Domain Hybrid Finite Element Methods in Electromagnetics (Morgan & Claypool, 2006), and edited the Antenna Engineering Handbook (McGraw-Hill, 2007). Dr. Volakis was the 2004 President of the IEEE Antennas and Propagation Society and served on the AdCom of the IEEE Antennas and Propagation Society from 1995 to 1998. He also served as Associate Editor for the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION from 1988–1992, Radio Science from 1994–97, and for the IEEE Antennas and Propagation Society Magazine (1992–2006). He currently serves as an associate editor for the J. Electromagnetic Waves and Applications and the URSI Bulletin. He chaired the 1993 IEEE Antennas and Propagation Society Symposium and Radio Science Meeting in Ann Arbor, MI., and co-chaired the same Symposium in 2003 at Columbus, Ohio, USA. He was elected Fellow of the IEEE in 1996, and is a member of the URSI Commissions B and E He has also written several well-edited coursepacks on introductory and advanced numerical methods for electromagnetics, and has delivered short courses on antennas, numerical methods, and frequency selective surfaces. In 1998 he received the University of Michigan (UM) College of Engineering Research Excellence award and in 2001 he received the UM, Department of Electrical Engineering and Computer Science Service Excellence Award. He is listed by ISI among the top 250 most referenced authors. He graduated/mentored nearly 60 Ph.D. students/post-docs, and coauthored with them 11 best paper awards at conferences.

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A Comprehensive Channel Model for UWB Multisensor Multiantenna Body Area Networks Stéphane van Roy, Claude Oestges, François Horlin, and Philippe De Doncker

Abstract—Body area networks consist of a number of biological sensors communicating over the air with a central sink placed in close proximity of the human body. A promising solution is to use multisensor multiantenna ultrawideband architecture; each sensor carries one antenna, while the central sink supports an antenna array. In this paper, a complete analytical channel model has been developed for the on-body diffracted waves mechanism. It builds on the existing IEEE 802.15.4a standard channel model and offers an innovative space-time correlation model. Index Terms—Body area networks, multisensor multiantenna, spatial correlation, ultrawideband (UWB).

I. INTRODUCTION

R

ECENT research and advances in ultra low power technologies have enabled the development of body area networks (BANs). BANs connect independent sensors placed on the human body to measure physiological and contextual information. The biological data is then sent to a central body-worn device, such as a personal digital assistant (PDA), via single or multihop transmissions. Typical applications include the realtime monitoring of heart activity, blood pressure, breathing rate, or skin temperature, for the purpose of diagnostics or generating automatic calls for emergency. Eventually, the healthcare and medical treatment of a patient can be, in part, performed remotely. Considering the potential value of BANs, the IEEE 802.15.4a group has developed a low complexity, low cost, low range, and low power physical layer based on the promising ultrawideband (UWB) technology [1]. UWB systems are characterized by a signal bandwidth exceeding the lesser of 500 MHz or 20% of the central frequency. The large bandwidth combined with the low power spectral density lead to several advantages, such as low interference to and from other systems, low sensibility to fading, and accurate positioning, thanks to fine time resolution

Manuscript received July 29, 2008; revised August 06, 2009. First published November 10, 2009; current version published January 04, 2010. This work was supported by grants from the Fonds de la Recherche pour l’Industrie et l’Agriculture, Belgium, and was conducted in the framework of the WALIBI project funded by the Walloon Region, Belgium. S. van Roy is with the OPERA dpt, Université Libre de Bruxelles, B-1050 Brussels, Belgium and also with the Microwave Laboratory, Catholic University of Leuven, B-1348 Louvain-la-Neuve, Belgium (e-mail: [email protected]). C. Oestges is with the Microwave Laboratory, Catholic University of Leuven, B-1348 Louvain-la-Neuve, Belgium (e-mail: [email protected]). F. Horlin and P. De Doncker are with the OPERA dpt, Université Libre de Bruxelles, B-1050 Brussels, Belgium (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.2009.2036280

[2]. There are two main contributions of work in the channel modeling of on-body transmissions which are clearly notable. In [3], A. Fort et al. suggest a simple statistical UWB channel for BAN, whose parameters are extracted from measurement campaigns. And in [4], A. Molisch et al. perform finite-difference time-domain (FDTD) simulations for evaluating the standard IEEE 802.15.4a for BAN communications. Recently, many efforts and initiatives have focused on the development of multisensor multiple-input/multiple-output (MSMIMO) systems. In MS-MIMO systems, the central device consolidates most of the computational complexity and uses an antenna array, while each individual sensor only carries a single antenna. This architecture is expected to enhance the signal-tonoise ratio (SNR) by exploiting the array gain, or to improve the interference reduction capability of the network, via adequate beamforming techniques. Simultaneously, it is also possible to achieve a diversity or throughput gain [5]. Deploying this technology for BAN has not been addressed thus far. A relevant paper [6] only offers measurements using two body-worn dual-polar antennas to highlight channel characteristics. Moreover, to the authors’ best knowledge, the combination of MS-MIMO systems with UWB transmissions has never been reported. Such an architecture is expected to minimize the transmit power requirement and to increase the battery lifetime of the body-worn sensors. In order to properly design and to evaluate the performance of UWB MS-MIMO systems, channel models of the on-body propagation are needed. Therefore, the purpose of this paper is to give a response to the lack of literatures by proposing an empirical space-time channel model of the first waves interfering with the human body. Future works will also consider the delayed waves incoming from reflections off the surrounding environment. The paper is organized as follows: in Section II, the measurement campaign is detailed. The general channel representation is presented in Section III. Section IV details the path-loss model, as well as the tapped delay line model, including the correlation aspects. The full channel model is then summarized in Section V and validated in Section VI. II. MEASUREMENT CAMPAIGN A. Measurement Setup A Rohde & Schwarz ZVA-24 4-port vector network analyzer (VNA) was used to measure the complex frequency-domain from 3 to 7 GHz, transfer function in a 4 GHz bandwidth with a 50 MHz frequency step. The complex time-domain response was obtained by means of an inverse fast Fourier transform on the complex baseband frequency response, applying a Hamming window to reduce side lobes (to 43 dB for the

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Fig. 2. Schematic diagram of the body. Fig. 1. Illustration of Tx and Rx antennas on the volunteer.

second lobe). Omnidirectional Skycross SMT-3T010M UWB antennas were selected since their small-size and low-profile characteristics precisely match body sensors requirements [7]. Unfortunately, the closeness of the body disrising to torts their radiation pattern and their impedance, around 5 dB. A dielectric material, a cardboard, of 5 mm is thus used to separate the antenna from the body skin, resulting . It should be noted that we did not de-embed in the antenna so our model includes both the channel and the antenna effects. Using low-loss coaxial cables to interconnect all components and selecting a fine IF-bandwidth resolution of 100 Hz to enlarge dynamic range (about 120 dB) improved the analysis capability.

Fig. 3. Illustration of two correlated impulse responses i and j for a transmitter located in the front of the body.

B. Scenario Setup The measurements were carried out around the 94 cm waist of a single volunteer (1m87, 82 kg), whose body was in a standing position, arms hanging along the side (Fig. 1). The transmit antenna was placed at a distance from the middle axis of the torso (toward the left side), with ranging from 8 to 36 cm with a step of 4 cm. In order to extend existing models to the SIMO case, a two-antenna array was used for the receiver side. One in Fig. 2), and the antenna was placed on the middle axis ( second one was located at a distance (toward the right side). Six inter-spacings were investigated ( , 5.5, 7, 8.5, 10 and meant one antenna at the re11.5 cm), whereas ceiver. In order to model the propagation channel around the body, spatial realizations were extracted from 7 levels separated by 4 cm along the body height. As seen in Fig. 2, for each of these levels, the transmitter was also shifted one level below and one level above the receiving array with the purpose of increasing the number of points for statistical accuracy. A total of 21 impulse responses were then obtained for each combination of and . III. MS-MIMO REPRESENTATION Modeling an on-body system composed of one sensor and one array of antennas consists of generating a complex distaps for each crete impulse response of array element (see Fig. 3). Such a channel describes the communication between a transmitter (Tx) placed at distance and a receiver (Rx) placed at distance .

The complex discrete impulse response of the element can be expressed as a tapped-delay line

(1) and it can be also written as a channel vector by

defined

(2)

IV. EMPIRICAL MODELING Previous works have highlighted three distinct multipath and scattering mechanisms which take place when two sensors placed on the body communicate [3], [8]–[10]. • Propagation through the body. The weak penetration in the body at high frequencies enables a rejection of this mechanism. • Diffraction around the body. The wave propagates analogous to a creeping wave, whose properties are related to the body characteristics. • Reflection off of the body parts and of objects in the surrounding environment (e.g., the ground, ceiling and walls), and then back toward the body.

VAN ROY et al.: A COMPREHENSIVE CHANNEL MODEL FOR UWB MULTISENSOR MULTIANTENNA BANs

Fig. 4. Path loss versus R

= r + d, data and power law fitting.

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decay. However, the on-body channel involves waves diffracted by body parts, each carrying a part of power. In Fig. 2, when the waves travel along the body for a long distance, these delayed multipath components are not negligible as compared to the main diffracted wave, and their contribution must be taken into account in the path loss. Extending the IEEE standard to a multiantenna system requires the validity of the path-loss model to be checked when the antenna is off-centered, involving a disymmetric of the body configuration (e.g. different distances to the arms). In Fig. 4, the path loss around the body was estimated for a set of distances by integrating the power of the first 20 taps of measured impulse responses. The fading was removed by averaging the data at the same transmit distance. In all cases, an exponential law reasonably matches the data. Whatever the position of Rx, the same path-loss parameters are extracted, and the model is only dependent on the relative distance between Tx and Rx

TABLE I PARAMETERS OF THE PATH LOSS MODEL, r AND d IN cm

(3) where is the distance between the transmit and receive an), is the path loss for a reftennas (here, erence distance , and is the slope coefficient in dB/cm. All the parameters can be found in Table I. In Fig. 4, the exponential fit of the power for the first tap is also illustrated. By comparing it to the narrowband path-loss model (integrated over all taps), the steeper slope of the main diffracted wave clearly demonstrates the growing contribution of the delayed multipath components when the distance increases, as expected. B. Tapped-Delay Line Characteristics

Fig. 5. Dual-slope model describing the mean-amplitude decay (4 GHz bandwidth).

Our channel model focuses on the first 20 taps following the , arrival of the signal. During this interval a dominant direct diffracted wave from the clockwise direction is basically followed by other diffracted waves and reflections off the body parts (mainly the arms). In a few cases, when Tx is moved towards the back, the ground reflection may appear at the end of the time interval. Nevertheless, the surrounding environment effect is not taken into account in our model.

To determine the contribution of each tap, the path-loss law is multiplied to the impulse responses so that the average relative energy around the body is in unity. Similar to the IEEE model [12], we found that, whatever the position of Rx, the smallscale fading distribution of each tap reasonably matches a lognormal distribution according to the Akaike criterion,1 hence, a Gaussian distribution in dB scale. 1) Mean in the dB Domain: Fig. 5 shows in dB the mean of the th tap relative to the mean amplitude of amplitudes . A dual-slope model best describes the decay the first tap with delay

A. Path Loss

(4) This exponential decay results from interfering echoes from the body itself. Likewise, the first-slope coefficient of this model increases with both and , due to the increased numbers of multipath components at larger Tx-Rx, as already observed in Section IV-A. also reveals a differAnalyzing the first-slope coefficient ence depending on the location of the Tx around the body, due to different contributions of arms to diffuse mechanisms. In fact,

As foreseen in [11] and assumed in [1], an isolated diffracted wave propagating around the body follows an exponential

1A set of potential distributions was compared including for example Rayleigh, Gamma, Rice or Nagakami.

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C. Correlation Aspects

TABLE II STANDARD DEVIATION VALUES

for a transmitter placed in the front of the body, reasonably matches

,

(5) while for a transmitter placed in the back of the body , is best described by (6) The breakpoint tap number corresponds to the time of arrival for diffracted waves from the counter-clockwise direction. These waves intensify the multipath component mechanism and soften the receive power decay—explaining the dualslope model. Depending on the transmitter location, in the front or in the back of the body, earlier or later breakpoint times are found, respectively. However, a single breakpoint was fixed by a best-fit algorithm over all of the measurement data. The barely varies and is therefore fixed second-slope coefficient to 1.09. 2) Standard Deviation in the dB Domain: In order to improve the extracted statistics, the number of realizations was increased by grouping the transmit positions into only two scenarios, the front and the back of the body. For each inter-spacing , a total of 84 and 105 impulse responses were then allowed for the front and back of the body, respectively. of the th tap shows that a disThe standard deviation tinct behavior is encountered independently from the receiver position, however this depends on the transmit position. In fact, variations become slightly higher in the back region of the body, especially in the vicinity of the arms (side of the body)—highlighting the scattering interference pattern from this location. Furthermore, deviations are averaged in a given channel, excluding the first tap since it is affected by lower variations and stands apart from the other ones. Table II summarizes these results.

Correlation between taps states to what extent the matching with them is in terms of electromagnetic properties. Correlated taps were illustrated for on-body single-input single-output (SISO) transmissions by previous works, see [12]. This correlation is mainly due to the vicinity of the transmitter and the receiver. As detailed in [13], very short path lengths and the natural symmetry of the body are relevant factors, drawing up overlapped trajectories and leading to high correlations. Moreover, residual correlations can be also caused by any of the following effects.2 • Although a windowing is applied, the band-limited channel signal can introduce higher correlations; • Multipath components may arrive at the boundary of two adjacent taps; • The antenna characteristics, such as non-isotropic radiation patterns, can increase the correlation. between taps In this paper, the correlation coefficient of discrete impulse responses is calculated in the dB domain is the expecby 7, shown at the bottom of page, where tation operator over space. denotes the th channel tap of the discrete impulse response with each discrete impulse response being the communication channel between which a Tx is placed, depending on the scenario, in the front or in the back is defined in a of the body, and a Rx located at a distance . similar manner. As the taps are defined in delay (UWB) and spatial (MIMO) domain, two kinds of correlation matrices are involved, depending on which Rx positions are comand delay-domain spatial corpared: delay correlation . They are illustrated in Fig. 3. relation 1) Delay Correlation: Delay correlation coefficients compare the tap variations within the same impulse response. Our tests reveal that the same statistics occur for the first 15 taps but they diverge for the last ones. Depending on the transmit position, in the front or in the back of the body, lower or higher correlations are indeed obtained at the end of the impulse responses, due to the onset of a dense multipath regime or of the coherent ground reflection, respectively. Regardless, the low power of these taps enable a rejection of their particular features in the modeling. The correlation coefficients are studied in terms of adjacent taps by averaging the upper diagonals of each correla[e.g., (first adjacent), tion matrix (second adjacent), etc.]. 2In [14], [15], the on-body propagation is proved as non-dispersive in the 3–10 GHz range and the assumption of the spreading of the pulse is then clearly rejected.

(7)

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TABLE III DELAY CORRELATION PARAMETERS OF THE TWO REGIONS OF THE BODY, d IN cm

Fig. 6. Delay correlation versus d for transmitter positions in the front of the body.

Fig. 7. Delay correlation versus d for transmitter positions in the back of the body.

Fig. 6 and Fig. 7 plot these averages versus , respectively, for a transmitter placed in the front or in the back of the body. Typically, adjacent taps are characterized by a significant correlation (above 0.6) that dramatically decreases with delay and that disappears from the third adjacent tap. Comparing the figures also shows that the correlation linearly drops with in Fig. 6, whereas it is quite constant in Fig. 7. This seems to indicate that, for transmitter positions in the back of the body, an identically correlated multipath diffraction mechanism only occurs due to overlapped trajectories while relevant decorrelated reflections off the body parts are added to this diffracted waves for a transmitter placed in the front of the body, lowering the correlation. Table III summarizes the extracted parameters describing the delay correlation in a linear fit (8) for the two regions

(8)

2) Delay-Domain Spatial Correlation: The fading spatial correlation (i.e. the correlation between taps of the array elements) should be sufficiently low for a MIMO system to offer any performance enhancement. High correlations between impulse responses reduce the degrees of freedom that can deteriorate to unity in a perfectly correlated channel. Any potential diversity or spatial multiplexing is then lost and an array gain can only be expected. In a BAN context, high spatial correlation for the first tap is intuitively assumed; the antenna array is crossed by the same diffracted wave which is attenuated only by the difference of travel distance . Spatial correlation coefficients are extracted for the two regions of the body and for inter-spacing ranging from 4 to 11.5 cm. The mean and the standard deviation over inter-spacing are estimated for each coefficient. For both of the scenarios, low dispersions are generally observed. This means that the spatial correlation coefficients can be considered as independent from the distance between the array elements and that they can be thus described by their respective mean. In Fig. 8, this mean is presented for a transmitter placed in the back of the body. The y-axis is the channel tap number whereas the x-axis is the channel tap number at the at off-centered array elements. A brighter diagonal is observable, and three correlated clusters are especially identifiable. The dominant diffracted waves at the beginning of the impulse are followed by diffracted waves response . Finally, from the counter-clockwise direction the ground reflection appears . In regards to the front of the body, the mean spatial correlation matrix diverges from the previous scenario by drawing lower coefficients. This scenario only reveals the following two clusters. • The ground reflection, traveling a shorter distance, appears earlier in an absolute time reference, but later in a relative ). one (from • The main correlated diffracted wave interacts with reflections off the arms and other body parts, creating a richer multipath environment. In Fig. 9, the spatial correlation for both of the scenarios are compared over interspacing. The multipath interference causes a decrease over to when the Tx antenna is located in the front of

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drops with the freMoreover, we also verified that quency bandwidth around 5 GHz. As a matter of fact, increasing the bandwidth distinguishes more resolvable multipath components, reducing the multipath richness in each tap and also leading to higher correlations. The covariance matrices are reproduced in [16]. V. MODEL IMPLEMENTATION A. Formulation We can summarize the channel expression of the whole MS-MIMO array by a concatenated channel vector given by (10)

Fig. 8. Spatial correlation matrix for the back of the body, averaged over d.

Adapting the formulation given by [12] to the UWB MIMO channel, the generation of discrete impulse responses of taps in dB starts with a vector of uncorrelated, zero . This mean, unit variance normal variables vector is then multiplied by the square root of the covari. The power level of each tap is ance matrix subsequently adjusted by adding the appropriate mean ampliand by subtracting the path loss tude . The covariance matrix is composed that introduce of square covariance sub-matrices the standard deviations and the correlation coefficient between each tap of the element and each tap of the element . Two kinds of covariance sub-matrices are and involved depending on and : delay covariance delay-domain spatial covariance sub-matrices (11)

Fig. 9. The spatial correlation 

(12)

of the front and the back of the body.

TABLE IV SPATIAL CORRELATION PARAMETERS FOR A TRANSMITTER PLACED IN THE FRONT OF THE BODY, d IN cm

the body. This can be modeled by (9), whose parameters are detailed in Table IV. (9) However, decorrelates (below 0.3) for antennas spaced more than the usual half wavelength of the lower frequency in use (3 GHz). A possible explanation is that the scattering mainly stays sparse and non-isotropic, due to the body configuration.

.. .

..

.

.. .

(13)

In (12), denotes the Kronecker product and is the is a square matrix with the elements canonical vector; of on the diagonal. is a vector of correlated normal The resulting variables. At last, the final channel expression is obtained by converting the amplitude taps from the dB domain to the linear domain and by adding a phase with uniform distribution be. tween B. Time of Arrival Channels in the delay domain modeled above are relative to the arrival time of the first wave. It can be useful to fix an absolute delay reference. In [14], frequency-domain spatial corre-

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Fig. 10. Relative time of arrival (in tap) between two array elements spaced of a distance d.

Fig. 11. Comparison of measured and simulated RMS delay spreads for two receive antenna positions.

lation analyses highlight that diffracted waves around the body . Fig. 10 shows the avpropagate at a velocity of erage difference in time of arrival for different inter-spacings. A is extracted which agrees with [14]. velocity of (in tap number) between two arTherefore, the difference and , rival times of impulse responses and located at , is deterministic and can be computed as respectively follows:

body parts and alters the correlation, which decreased with interspacing.

(14) where

selects the nearest lower integer. VI. VALIDATION

When deriving the model, various simplifying assumptions were done. In order to evaluate the model performance, measured and simulated channels are compared in term of RMS delay spread. Fig. 11 compares measured and simulated RMS delay spreads depicted for two array elements. Some experislightly diverge from the model, but mental points these approximations are consciously assumed to simplify the model. We conclude that the model is working properly since statistics similar to those measured around the body are yielded, the mean relative difference being below 10%. VII. CONCLUSION A new analytical space-time channel model for UWB multisensor MIMO Body Area Networks is proposed. Here, each sensor is made of one antenna and transmits pulses of 4-GHz bandwidth to a multiantenna central device. The decay of tap mean amplitude with delay is best described by a dual-slope power law whose parameters depend on the transmit and receive antenna locations. Overlapped trajectories are also identified, leading to high-correlated diffraction waves for both the delay correlation and the delay-domain spatial correlation. However, spatial interference takes place due to the reflections off the

ACKNOWLEDGMENT The authors also take the opportunity to specifically thank all reviewers for a thorough review and constructive suggestions. REFERENCES [1] IEEE P802.15 Working Group, PART 15.4: Wireless Medium Access Control (MAC) and Physical Layer (PHY) Specifications for Low-Rate Wireless Personal Area Networks (LRWPANs)—IEEE P802.15.4a LAN/MAN Standards Committee of the IEEE Computer Society, 2007, Tech. Rep.. [2] D. Porcino and W. Hirt, “Ultra-wideband radio technology: Potential and challenges ahead,” IEEE Commun. Mag., vol. 41, pp. 66–77, Jul. 2003. [3] A. Fort, C. Desset, P. De Doncker, and L. Van Biesen, “An ultra wideband body area propagation channel model: From statistics to implementation,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 4, pp. 1820–1826, Jun. 2006. [4] A. F. Molisch, D. Cassioli, C. C. Chong, S. Emami, A. Fort, B. Kannan, J. Karedala, J. Kunisch, H. G. Schantz, K. Siwiak, and M. Z. Win, “A comprehensive standardized model for ultrawideband propagation channels,” IEEE Trans. Antennas Propag., vol. 54, no. 11, pp. 3151–3166, Nov. 2006. [5] K. Makaratat and S. Stavrou, “Spatial correlation technique for UWB antenna arrays,” IET Electron. Lett., vol. 42, no. 12, pp. 675–676, Jun. 2006. [6] D. Neirynck, C. Wiliams, A. Nix, and M. Beach, “Exploiting multipleinput multiple-output in the personal sphere,” IET Microw. Antennas Propag., vol. 1, no. 6, pp. 1170–1176, Dec. 2007. [7] Skycross Inc., “3.1–10 GHz Ultrawideband Antenna SMT-3TO10M,” Tech. Rep.. [8] A. Fort, J. Ryckaert, C. Desset, P. De Doncker, and L. Van Biesen, “Ultrawideband channel model for communication around the human body,” IEEE J. Sel. Areas Commun., vol. 24, no. 4, pp. 927–933, Apr. 2006. [9] A. Alomany, Y. Hao, X. Hu, C. G. Parnini, and P. S. Hall, “UWB on-body radio propagation and system modelling for wireless bodycentric networks,” Inst. Elect. Eng. Commun. , vol. 153, pp. 107–114, Feb. 2006. [10] Y. Zhao, Y. Hao, A. Alomainy, and C. Parini, “UWB on-body radio channel modeling using ray theory and subband FDTD method,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 4, pp. 1827–183, Jun. 2006. [11] J. Ryckaert, P. De Doncker, R. Meys, A. de Le Hoye, and S. D. Belmon, “Channel model for wireless communications around human body,” IET Electron. Lett., vol. 40, no. 9, pp. 543–544, April 2004.

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[12] A. F. Molisch, K. Balakrishnan, C.-C. Chong, S. Emami, A. Fort, J. Karedal, J. Kunisch, H. Schantz, U. Schuster, and K. Siwiak, “IEEE 802.15.4a Channel Model for Body Area Networks—Final Report” Tech. Rep. P802.15-04-0662-00-004a, 2006, IEEE 802.15 Working Group Document. [13] A. F. Molisch, K. Balakrishnan, C.-C. Chong, S. Emami, A. Fort, J. Karedal, J. Kunisch, H. Schantz, U. Schuster, and K. Siwiak, Channel Model for Body Area Networks Tech. Rep. P802.15-04-0486-00-004a, 2006, IEEE 802.15 Working Group Document. [14] S. van Roy, C. Oestges, F. Horlin, and P. De Doncker, “On-body propagation velocity estimation using ultra-wideband frequency-domain spatial correlation analyses,” IET Electron. Lett., vol. 43, no. 25, pp. 1405–1406, Dec. 2007. [15] S. van Roy, C. Oestges, F. Horlin, and P. De Doncker, “Ultra-wideband spatial channel characterization for body area networks,” presented at the 2nd Eur. Conf. on Antennas Propag. (EuCAP 2007), Edinburgh, U.K., Nov. 2007. [16] S. van Roy, C. Oestges, F. Horlin, and P. De Doncker, Parameters for On-Body MS-MIMO UWB Model Université Libre de Bruxelles, Brussels, Belgium, 2008, Tech. Rep.. [17] S. van Roy, C. Oestges, F. Horlin, and P. De Doncker, “Propagation modeling for UWB body area networks: Power decay and multi-sensor correlations,” presented at the 10th Int. Symp. on Spread Spectrum Tech. and Applicat. (ISSSTA 2008), Bologna, Italy, Aug. 2008. [18] S. van Roy, C. Oestges, F. Horlin, and P. De Doncker, “A spatially-correlated tapped delay line model for body area networks,” presented at the IEEE 68th Veh. Technol. Conf. (VTC2008-Fall), Calgary, Alberta, Sep. 2008.

Stéphane van Roy received the M.Sc. degree in electrical engineering from the Université Libre de Bruxelles (ULB), Brussels, Belgium, in 2006, where he is currently working toward the Ph.D. degree. In October 2006, he joined the OPERA Department, ULB, as an F.R.I.A. Researcher. Since 2007, he was also associated with the Microwave Laboratory, Université catholique de Louvain (UCL), Louvain-la-Neuve, Belgium. His research interests focus on wireless channel modeling and digital signal processing.

Claude Oestges received the M.Sc. and Ph.D. degrees in electrical engineering from the Université catholique de Louvain (UCL), Louvain-la-Neuve, Belgium, in 1996 and 2000, respectively. In January 2001, he joined the Smart Antennas Research Group (Information Systems Laboratory), Stanford University, CA, as a Postdoctoral scholar. From January 2002 to September 2005, he was associated with the Microwave Laboratory, UCL, as a Postdoctoral Fellow of the Belgian Fonds de la Recherche Scientifique (FRS). He is presently an FRS Research Associate and Assistant Professor at UCL. He is the author or coauthor of one book and more than 100 research papers and communications. Dr. Oestges was the recipient of the 1999–2000 IEE Marconi Premium Award and the 2004 IEEE Vehicular Technology Society Neal Shepherd Award.

François Horlin received the M.Sc. and Ph.D. degrees in electrical engineering from the Université catholique de Louvain (UCL), Louvain-la-Neuve, Belgium, in 1998 and 2002, respectively. In September 2002, he joined the Inter-university Micro-Electronic Center (IMEC), Leuven Belgium, where he lead a project aimed at developing 4G mobile wireless communication systems in collaboration with Samsung Korea. He was also responsible for the digital signal processing activity for wireless communications. Since January 2007, he is a fulltime Associate Professor at the Université Libre de Bruxelles (ULB), Brussels, Belgium. Dr. Horlin is the Vice-Chair of the IEEE Benelux Signal Processing Chapter.

Philippe De Doncker received the Engineering and Ph.D. degrees from the Université libre de Bruxelles (ULB), Brussels, Belgium, in 1996 and 2001, respectively. He is currently an Assistant Professor with the Université libre de Bruxelles. His research interests focus on wireless communications and electromagnetics.

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Path-Loss Characteristics of Urban Wireless Channels Keith T. Herring, Jack W. Holloway, David H. Staelin, Life Fellow, IEEE, and Daniel W. Bliss, Member, IEEE

Abstract—Wireless channel data was collected in Cambridge, Massachusetts for diverse propagation environments over distances ranging from tens of meters to several kilometers using mobile 2.4-GHz transmitters and receivers. The 20-MHz bandwidth signals from eight individually movable van-top antennas were Nyquist sampled simultaneously with 12-bit accuracy. Although path-loss variance for any given link length within single residential/urban neighborhoods was large, single streets typically exhibited path-loss, L(dB) = 10 log10 (P P ) = 10 log10 r + C, where P is the received or transmitted power, r the link-length, the street-dependent path-loss coefficient, and C the loss incurred at street intersections. Measurements yielded = 1 5+3 2 0 27 5; is the fraction of the street length having a for 2 building gap on either side. Experiments over links as short as 100 meters indicate a 10-dB advantage in estimating path loss for this model compared to optimal linear estimators based on link length alone. Measured air-to-ground links were well modeled by = 2 for the elevated LOS path, and by stochastic log-normal attenuation for the ground-level scattering environment. These models permit path-loss predictions based on readily accessible environmental parameters, and lead to efficient nodal placement strategies for full urban coverage. Index Terms—Attenuation, communication channels, data models, fading channels, microwave propagation, multipath channels, multiple-input multiple-output (MIMO) systems, propagation, statistics, urban areas.

I. INTRODUCTION

C

ONSUMER electronics and other communications systems are increasingly utilizing wireless technology. While protocols such as 802.11b (WiFi) and bluetooth have been implemented for some time, the emergence of multiple-input multiple-output (MIMO) communications [1]–[5] has increased commercial interest. Therefore it is increasingly important to understand the propagation characteristics of environments where this new technology will be deployed [6]. The development of high-performance system architectures and protocols at low-cost depends partly on the accuracy of

Manuscript received August 11, 2008; revised July 08, 2009. First published November 10, 2009; current version published January 04, 2010. This work was supported by the National Science Foundation under Grant ANI-0333902. K. T. Herring and D. H. Staelin are with the Research Laboratory of Electronics, Massachusetts Institute of Technology (MIT), Cambridge, MA 02139 USA (e-mail: [email protected]; [email protected]). J. W. Holloway was with the Research Laboratory of Electronics, Massachusetts Institute of Technology (MIT), Cambridge, MA 02139 USA. He is currently with the U.S. Marine Corps, Corpus Christi, TX 78419 USA (e-mail: [email protected]). D. W. Bliss is with the Research Laboratory of Electronics, Massachusetts Institute of Technology (MIT), Cambridge, MA 02139 USA and also with MIT Lincoln Laboratory, Lexington, MA 02420 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.2009.2036278

the channel path-loss models used for planning, for these models determine the requirements for transmitter power, node spacing, and antenna design. Deterministic path-loss models assume that enough is known about the propagation environment that electromagnetic propagation laws can be used to calculate the power attenuation. Ray tracing plus physical modeling is perhaps the most popular such approach [7]–[12] although deterministic approaches are seldom used because of the difficulty and cost of obtaining high-resolution environment descriptions. In contrast, empirical models attempt to model path-loss L as obeying a set of equations based on real channel measurements. The most widely used of these include the Stanford University Interim (SUI) [15], Hata [16], [17], and COST-231 models [18]. The equations typically take the form

(1) where the predicted path-loss L is a function of the link-length r, path-loss coefficient , and fixed-loss component C. Observed values for and C are usually calculated using regression analysis across antenna heights, operating frequency, and different macro-environment types, e.g. urban vs. suburban. In this paper we study the empirical performance of these models based on a large database of real wireless channel data obtained from diverse urban propagation environments. A mobile channel-data collection system was built that includes an eight-channel software receiver and a collection of transmitting WiFi channel sounders [19]. The software receiver synchronously samples the signals from eight individually movable antennas in the 20-MHz band centered at 2.4 GHz. Both air-toground and ground-to-ground links were measured for distances ranging from tens of meters to several kilometers throughout the city of Cambridge, MA. These data sets demonstrate that the average attenuation across frequency within a single macro-environment can vary over 50 dB, suggesting generally large rms errors for empirically based models. In contrast we observe that individual macro-environments can be partitioned into smaller sub-regions of practically relevant size that exhibit more predictable attenuation. These empirical results demonstrate that propagation models based on simple representations of the environment can reduce rms path-loss estimates within such homogeneous subregions by roughly 10 dB for links as short as 100 meters. Section II introduces the channel measurement system and estimation algorithms. Section III presents the results of the propagation measurements that underlie the propagation model, and Section IV introduces the model itself and illustrates its application to urban streets in comparison to empirical models. Section V presents additional measurements of air-to-ground

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Fig. 1. Overview of the channel-data collection system. Top-left: architecture of eight-channel software receiver; top-right: receiver mounted to van; bottomleft: receiver with case lid removed; bottom-right: transmitting antenna mounted about 1 m above a car. Numbers [1]–[6] relate the receiver block diagram to its photograph.

Fig. 2. Illustrative 10-MHz channel power spectrum near 2.422 Ghz obtained from the algorithms applied to data from one antenna as it moved to 40 locations spaced uniformly along a line 2 wavelengths long ( = 12 cm).

B. Channel Extraction and Estimation links, which are a special case of the model, and Section VI summarizes the contributions of the work.

II. INSTRUMENTATION AND SOFTWARE A. Channel-Data Collection System The Channel-Data Collection System (CDCS) is a set of instrumentation and software that collects large amounts of coherent multi-antenna wireless channel data [19]. The CDCS incorporates an eight-channel software receiver, channel sounder (WiFi transmitter), an array of eight individually movable antenna elements, software packages for system control and database management (distributed across the laptop controller, network, and onboard receiver PC), a GPS tracking system, and infrastructure for system mobility. Fig. 1 displays the major system components. The software receiver samples the 33-MHz band centered at 2.422 GHz, which includes channel 3 of the 802.11b wireless standard. The receiver synchronously samples this band at a 67-MHz sampling rate at eight configurable antenna elements for 1-msec continuous bursts. Each data burst, called a snapshot, is stored in an on-board PC until transferred to the database for post-collection analysis of the central 10 MHz of the 20-MHz wide Channel 3. The receiver includes the following functional blocks shown in Fig. 1: RF front-end, baseband sampling/control, CPU—control/storage, clock generation/distribution, and power supply. The channel sounding system is a modified ROOFNET node where ROOFNET is an experimental 802.11b/g mesh network developed at MIT CSAIL [20]. The channel sounder continually transmits 802.11b packets with arbitrary payload and duration. The transmitter is battery operated and includes a 1-Watt output amplifier for maximizing experiment range.

The Channel Extractor is a collection of algorithms implemented in MATLAB that estimates the complex channel spectra characterizing the links between the transmitter and each receiver antenna, based on the raw channel data collected by the CDCS. The magnitudes of these complex channel spectral estimates are the base data used in the analysis and experiments that follow. The Channel Extractor software incorporates five functional blocks: matched-filter construction, front-end digital band-pass filtering, detection, time-frequency correction, and channel estimation. This software produces a channel spectral estimate for each snapshot with 33-kHz resolution across only the flat central 10-MHz of the band, which simplified the analysis. Fig. 2 illustrates the received power spectrum estimated by this software for a series of snapshots taken as the transmitter translated linearly 40 times while it moved two wavelengths. The Non-Line-Of-Sight (NLOS) 100-meter links were recorded in a rich outdoor multipath environment. III. EXPERIMENTS The following measurements were carried out across the MIT campus and city of Cambridge, Massachusetts. The multi-antenna receiver was mounted on the roof of a van and the transmitter to a car; see Fig. 1. The receiver antennas were each separated by at least 2 wavelengths ( 24 cm) and all antennas elements at both link-ends were vertically polarized 8-dBi omnidirectional units constructed as thin cylinders (see Fig. 1). A. Macro-Environment Stability We first consider the performance of empirical path-loss models that predict attenuation, averaged across frequency (10 MHz) and across the 8 antennas, according to (1) using empirically fitted values for and C observed within that environment type. Fig. 3 displays a scatter plot of power received (dBm) versus link length for a single multi-street residential neighborhood. The transmitter remained in a single location while the receiver traversed neighboring streets within a radius of several city

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Fig. 3. Path-loss measurements along urban streets. Top-left image: measurements taken within a radius of several blocks within a residential neighborhood. Remaining images: measurements taken along individual streets when the transmitter was hidden around a corner.

blocks. A large variance was observed in the relation between received power and link-length as different neighborhood sections were reached. This suggests that using a single value for for a particular neighborhood, as is assumed in popular empirical models, can lead to large errors (tens of dB) in path-loss estimates. For links of these lengths ( 1 km), the numbers and types of objects between transmitter and receiver can vary significantly as a function of direction, thus leading to this result. Since many wireless applications operate over similar link-lengths, it would be desirable to improve this performance. To improve path-loss predictability, we consider partitioning individual macro-environments into smaller subregions. Specifically, returning to the residential neighborhood, the relative power-attenuation was observed as the receiver moved along single streets within the neighborhood while the stationary transmitter was concealed around a corner. The van housing the receiver was driven down each street while channel snapshots were taken at regular intervals. Fig. 3 displays the power received as a function of link-length along three representative streets. The link-length is measured as the two-dimensional euclidian distance between transmitter and receiver. Given that the transmitter was within 5 meters of the street corner in each measurement location, this distance added less than one meter to the link distance. The path-loss coefficient associated with each street is estimated as the least-squares linear regression fit to each scatter plot and displayed accordingly. Having tens of samples per street for each street, with the regression error was less than ranging between 2 and 5. In Fig. 3 we observe that the rms error in the linear path-loss estimator is reduced to a few dB when restricted to these smaller subregions. This leads to the question of how to estimate the proper for each street. B. Predicting Path-Loss From Local Environmental Parameters The analysis in the previous section suggests that path-loss characteristics are generally stationary only within physi-

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Fig. 4. Measurement environments left: representative street measurement location. Red lines indicate gaps along the street used to calculate and . Right: air-to-ground link measurements for the transmitter mounted on top of the Green Building 90 meters above street level; each blue dot corresponds to a unique receiver location. The image is approximately 2.5 km wide.

cally homogeneous neighborhoods. We therefore explore the and for level urban streets by problem of predicting modeling them as lossy waveguides for which the loss is dominated by the departure of guided rays through gaps between buildings spaced along the street. Natural physical parameters for predicting path-loss are those that characterize these gaps. Fig. 4 displays a satellite image of a representative street and its defined gaps. To test this hypothesis the streets were characterized by two alternative metrics: is the fraction of the street for which there is a gap on either one or both sides, and is the average fraction of each side of the street that is a gap. A gap is defined as an opening between buildings where there is no third building within 10 meters of that opening. Two minimum-squared-error predictors were fit to the nine street measurements of using two free parameters where the estimated is (2) The raw rms residual error in is , which increases to reflect the fact to 0.31 after correcting by a factor of that two coefficients in (2) and (3) were used to fit 9 unknowns. The metric performs less well, where: (3) and the corrected rms error is is 0.53. If an optimum linear combination of the two metrics is used, the observed 0.22 rms error for becomes 0.27 when corrected for three fit parameters. . The advantage of the metric Fig. 5 displays where is that it correctly predicts large losses if one side of the street is without buildings for a considerable distance. One rationale for these predictive models is that waveguide-like ray-tracings suggest small uniformly distributed , gaps would produce path losses of: and are constants, is the street width, and where . This result sug-

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Fig. 5. Relationship between observed path-loss coefficients () measured over nine urban streets and estimations utilizing the building-gap-driven estimator of Eqn. (2). The raw standard deviation is 0.27, and becomes 0.31 after correction by 9=7.

gests that simple measurable local environmental parameters can be used to improve empirical path-loss models while remaining generally more practical than ray-tracing methods. These experiments also characterize the fixed power loss (C), due mostly to propagation around corners. Corner loss was defined as the increase in received power when the horizontally isotropic transmitting antenna was moved a relatively minor distance from its occluded location around a corner to a line-ofsight position within the long street. For the 10 corners measured, the associated corner losses had an empirical mean and standard deviation of 40 dB and 5.5 dB, respectively. This is quite consistent with prior measurements for urban intersections [13]. Ray tracing suggests transmission around our observed urban corners was dominated by reflections from building walls rather than by diffraction at vertical discontinuities, which typically have much smaller cross-sections. Two of these ten cases exhibited corner losses approximately 10 dB higher than the average of the remaining corners. These were the only two intersections where the buildings near the corner were constructed primarily with wood (residential houses). In contrast the other intersections had buildings made mostly of brick and concrete cinder block. The dielectric constants for wood or brick suggest that 10–15 dB might be lost due to a single reflection near normal incidence (less at grazing incidence), and that the rest of the nominal 40-dB corner loss might result because the low-loss rays propagating nearly parallel to the first street will initially reflect from corner buildings on the far side of the cross-street at angles that are more nearly normal to the building surfaces; most of the residual 25–30 dB corner loss could then result from the conversion of these initial high-loss modes to lower-loss rays and modes propagating more nearly parallel to the second street. Wood shingles and siding introduce additional loss because they slope several degrees upward and deflect incident rays toward the sky. In addition, some of the observed variance in corner loss is due to the random placement of the transmitter on the left or right sides of the street, which determines the fraction of rays that can be reflected around the corner toward the receiver; this

Fig. 6. Path-loss model for a street environment that divides into homogeneous subregions characterized by single values of .

fraction might vary 2–5 dB for the observed geometries. For the two high-loss corners both the wood-siding and wrong-side-ofstreet effects were present. These random materials and geometric contributions to corner losses account for a large fraction of the observed standard deviation of 5.5 dB. The high average loss per corner also suggest that only those paths with the minimum number of corners are likely to dominate the multipath propagation routes between any two points; it requires the sum of 10,000 incoherent paths in order to compensate for the 40-dB nominal advantage of a route with one fewer corners. IV. PATH-LOSS MODEL Consider the communications link illustrated in Fig. 6, where the transmitter and receiver are separated by approximately five blocks and a single one-corner propagation path probably dominates since all alternative paths require propagation around at least two additional corners, each corner having an expected mean loss of 40 dB. The total path loss can be estimated in five steps: 1) determine the dominant physical paths; 2) partition the dominant paths into homogeneous regions characterized by single values of ; 3) estimate for each homogeneous region using ; 4) estimate the fixed losses associated with each transition between homogeneous regions; and 5) sum the losses along all dominant paths to yield the net path loss between the two nodes. One path clearly dominates the case shown in Fig. 6, but if the transmitter were moved to the middle of a block on a street in Fig. 6 running left-right, there would be many paths having two corners, and the path loss might increase by roughly 40 dB to account for the extra corner, but then decrease by perhaps paths adding 10 dB because there would be perhaps incoherently with useful strength. Partitioning the path illustrated in Fig. 6 into three regions involves identifying where the metric changes significantly, and for each region can be estimated from using (2). The single corner on the dominant path might add 40 dB to the total path loss, and all other paths can be neglected here because they each introduce two additional high-loss corners. If there were two equal paths, each having 60 dB loss, then both the total incoherent power loss and mean coherent loss would be 57 dB.

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A. Performance To compare the performance of this model to traditional empirical models we simulated street propagation using the foland corner lowing distributions for the path-loss coefficient1 loss (b) (4) (5) In addition we assume the distribution of the

estimator to be (6)

which is consistent with the performance observed in Cambridge. For a single-corner link of length d, we estimate the path-loss (PL) to be (7)

Fig. 7. Performance comparison between our path-loss model and the optimal linear (empirical) model. Our results were obtained by simulating thousands of single corner street environments utilizing (7) and statistics consistent with our measurements for both alpha and corner loss.

where (8) As discussed in Section I, traditional empirical path-loss models are generally linear predictors trained on various environments, e.g. rural, suburban, or urban. For the street environment considered here, the empirical model would use (8) to find the single that minimizes rms errors in estimated PL over all streets in this reasonably homogeneous neighborhood. This empirical process , as picyields the estimated statistics tured in Fig. 7, where the rms errors in predicting path-loss are ten times greater than for the gap-based model for link lengths as short as 100 meters. V. PATH LOSS IN AIR-TO-GROUND LINKS Another common network topology uses air-to-ground links where one end is much higher in elevation than the other, and the channel is dominated by the scattering environment local to the ground node. This section discusses how the path-loss behavior associated with this type of link fits within our model space. A. Measurements The receiver antenna array used for the air-to-ground link measurements was identical to that used in the street measurements of Section III; antenna gains were 8 dBi. The transmitting system was mounted 90 m above street level on the roof of the Green Building located at MIT; its antenna gain was 8 dBi. The look-down angle was sufficiently small that this gain was nearly constant over the sampled neighborhoods. Fig. 8 displays an aerial view of the measured links; the transmitter location is indicated by a red dot, and each blue marker corresponds to a single receiver position for which eight independent receiver channels were measured simultaneously. Diverse 1U(a,b): Uniformly distributed on [a,b]; N(;  ): Gaussian with mean  and variance  .

Fig. 8. Left: path losses of these air-to-ground links obtained with transmitter 90 m above street level. Right: distribution of the terminal loss B(dB) incurred near the ground for the tower measurements taken in Cambridge, Massachusetts. The superimposed Gaussian curve suggests this terminal loss ensemble is approximately log-normal.

neighborhoods and physical scenarios were included in the data set. Fig. 4 displays a scatter plot of link length versus power received (dBm) for each air-to-ground channel measurement. Since the regression line has a negative slope of 1.9, the power received decreases approximately as the square of the link length. This is expected since the transmitter sits well above all surrounding buildings that might affect propagation. The standard deviation from the best-fit line is 8.3 dB, as shown in Fig. 8; this scatter is attributed to terminal loss near the ground, which has a mean of 30 dB. B. Model Air-to-ground links form a special case within our path-loss model and can be generally broken into two segments: 1) a long line-of-sight (LOS) segment that links the high end of the link to the neighborhood where the low-elevation end is located, and 2) the scattering segment at the low end as illustrated in Fig. 9.

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Fig. 9. Air-to-ground links represent a special case of our path-loss model that include two stationary sub-regions corresponding to the propagation path above and below the scattering line at the ground link. While path-loss characteristics should remain stable for segment 1, the statistics of the fixed loss associated with the ground link will vary by environment.

The wave intensity in the LOS segment decays as , so the additional “terminal” loss (dB) at the low-elevation scattering end from the can be determined by subtracting total observed link loss (dB), where is the 8-dB gain of the transmitting antenna. The terminal loss for the links illustrated in Fig. 8 was observed to be Gaussian with a standard deviation from the best-fit line of 8.3 dB. This 8.3-dB uncertainty arises from scattering from trees and buildings near the ground and significantly affects link capacity. The model path loss estimate therefore is (9) where r is the LOS path length [m] and B is the additional random terminal loss which, for this Cambridge data set was (rms). This result is consistent with the log-normal fading assumption presented by others [17], [18]. Future work may suggest simple ways to improve predictions of B based on the character of the local scattering environment. VI. CONCLUSION This paper has shown that satellite images of buildings and of the gaps between them can support useful in the path-loss model: estimates of the coefficient for propagation losses along straight level streets. Two predic; the more tors for were tested over the range successful predictor depended linearly on the fraction of the street that had a gap on either one or both sides. The residual rms error in predicted values of was 0.3 for the nine streets tested. It was also shown that rms departures of measured losses along single streets from values predicted by the model were typically less than 2 dB when the best-fit value of for that street was used and when the transmitter was hidden around a corner. This model predicted the loss for all street measurements with rms errors [dB] for that were

generally more than 10 dB better than those achieved with a minimum-square-error predictor of based only on the distance and the same data set. In addition, observed corner losses in Cambridge were 40 5.5 dB, consistent with the results of others. Since these corner losses are substantially greater than urban single-street losses, one indicated method for predicting urban path-losses involves identifying those street sequences having the fewest corners, and then determining the gap metrics for those streets using aerial photography. These results also suggest that the most efficient placement of urban street-level wireless terminals is above traffic, below rooftops, and positioned out into street intersections so that only one corner reflection is usually required to reach any address within several rectangular city blocks. Similarly, air-to-ground links can be modeled as having a (above the local building/ line-of-sight component with tree scattering line), and then an additive Gaussian random component due to local scattering near the ground-level terminal for Cambridge) (empirically REFERENCES [1] J. H. Winters, “On the capacity of radio communication systems with diversity in a Rayleigh fading environment,” IEEE J. Select Areas Commun., vol. SAC-5, pp. 871–878, Jun. 1987. [2] G. J. Foschini and M. J. Gans, “On limits of wireless communications in a fading environment when using multiple antennas,” Wireless Personal Commun., vol. 6, pp. 311–335, Mar. 1998. [3] T. L. Marzetta and B. M. Hochwald, “Capacity of a mobile multipleantenna communication link in Rayleigh flat fading,” IEEE Trans. Inf. Theory, vol. 45, pp. 139–157, Jan. 1999. [4] G. G. Raleigh and J. M. Cioffi, “Spatio-temporal coding for wireless communication,” IEEE Trans. Commun., vol. 46, pp. 357–366, Mar. 1998. [5] D. W. Bliss, A. M. Chan, and N. B. Chang, “MIMO wireless communication channel phenomenology,” IEEE Trans. Antennas Propag., vol. 52, no. 8, pp. 2073–2082, Aug. 2004. [6] E. Bonek, M. Herdin, W. Weichselberger, and H. Ozcelik, “MIMO—Study propagation first!,” in Proc. 3rd IEEE Int. Symp. on Signal Proces. and Inf. Technol. (ISSPIT 2003), Dec. 2003, pp. 150–153. [7] T. Kurner, D. J. Cichon, and W. Wiesbeck, “Concepts and results for 3D digital terrain-based wave propagation models: An overview,” IEEE J. Select Areas Commun., vol. 11, pp. 1002–1012, Sep. 1993. [8] S. Y. Seidel and T. S. Rappaport, “Site-specific propagation prediction for wireless in-building personal communication system design,” in Proc. IEEE 52nd Veh. Technol. Conf., Nov. 1994, vol. 43, no. 4, pp. 879–891. [9] G. E. Athanasiadou, A. R. Nix, and J. P. McGeehan, “A microcellular ray-tracing propagation model and evaluation of its narrow-band and wide-band predictions,” IEEE J. Select Areas Commun., vol. 18, pp. 322–335, Mar. 2000. [10] N. Blaunstein, M. Toeltsch, J. Laurila, E. Bonek, D. Katz, P. Vainikainen, N. Tsouri, K. Kalliola, and H. Laitinen, “Signal power distribution in the Azimuth, elevation and time delay domains in urban environments for various elevations of base station antenna,” IEEE Trans. Antennas Propag., vol. 54, no. 10, pp. 2902–2916, Oct. 2006. [11] T. K. Sarkar, Z. Ji, K. Kim, A. Medouri, and M. Salazar-Palma, “A survey of various propagation models for mobile communication,” IEEE Trans. Antennas Propag., vol. 45, no. 3, pp. 51–82, Jun. 2003. [12] Z. Yun, Z. Zhang, and M. F. Iskander, “A ray-tracing method based on the triangular grid approach and application to propagation prediction in urban environments,” IEEE Trans. Antennas Propag., vol. 50, no. 5, pp. 750–758, May 2002. [13] J. Lee and H. L. Bertoni, “Coupling at cross, T, and L junctions in tunnels and urban street canyons,” IEEE Trans. Antennas Propag., vol. 51, no. 5, pp. 926–935, May 2003. [14] K. R. Schaubach, N. J. Davis, and T. S. Rappaport, “A ray tracing method for predicting path loss and delay spread in microcellular environments,” in Proc. IEEE 42nd Veh. Technol. Conf., May 1992, vol. 2, pp. 932–935.

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[15] V. Erceg, K. V. S. Hari, M. S. Smith, K. P. Sheikh, C. Tappenden, J. M. Costa, D. S. Baum, and C. Bushue, Channel Models for Fixed Wireless Applications IEEE 802.16 Broadband Wireless Access Working Group, IEEE tech. rep. 802.16.3c-01/29, Jan. 2001 [Online]. Available: http://www.wirelessman.org/tg3/contrib/802163c-01_29.pdf [16] Y. Okamura, E. Ohmori, and K. Fukuda, “Field strength and its variability in VHF and UHF land mobile radio service,” Rev. Elect. Commun. Lab., vol. 16, no. 9–10, pp. 825–873, 1968. [17] M. Hata, “Empirical formula for propagation loss in land mobile radio services,” IEEE Trans. Veh. Technol., vol. VT-29, no. 3, pp. 317–325, Aug. 1980. [18] Digital Mobile Radio Towards Future Generation Systems COST Action 231, European Co-operation in Mobile Radio Research, European Commission, Brussels, Belgium, Final Tech. Rep. EUR 18957, 1999, E. Damosso, L. M. Correia (eds.). [19] K. Herring, “Propagation Models for Multiple-Antenna Systems: Methodology, Measurements, and Statistics,” Ph.D. dissertation, Dept. Elect. Eng. Comp. Sci., MIT, Cambridge, MA, 2008. [20] J. Bicket, D. Aguayo, S. Biswas, and R. Morris, “Architecture and evaluation of an unplanned 802.11b mesh network,” presented at the Mobicom 2005, Aug. 2005. [21] T. S. Rappaport, Wireless Communications: Principles and Practice. Upper Saddle River, NJ: Prentice Hall, 2002. [22] D. Tse and P. Viswanath, Fundamentals of Wireless Communication. Cambridge, UK: Cambridge Univ. Press, 2005. Keith T. Herring received the B.S. degree in computer science from the University of Illinois, Urbana-Champaign, in 2003 and the S.M. and Ph.D. degrees in electrical engineering and computer science from the Massachusetts Institute of Technology (MIT), Cambridge, in 2005 and 2008, respectively. He worked in the Research Laboratory of Electronics (RLE) at MIT from 2003 to 2008, focusing on wireless propagation research. Currently he is a Postdoctoral Associate in the MIT Research Laboratory of Electronics.

Jack W. Holloway received both the S.B. degree in applied mathematics and the S.B. degree in electrical engineering in 2003 and the M.Eng. degree electrical engineering and computer science in 2004, all from the Massachusetts Institute of Technology (MIT), Cambridge. Previously, he was engaged in Ph.D. work in integrated RF circuit design at the Micro-Technology Lab, MIT. Currently, he is a Second Lieutenant in the United States Marine Corps, currently undergoing training as a Naval Aviator in Corpus Christi, TX.

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David H. Staelin (S’59–M’65–SM’75–F’7–LF’04) received the S.B., S.M., and Sc.D. degrees in electrical engineering from the Massachusetts Institute of Technology (MIT), Cambridge, in 1960, 1961, and 1965, respectively. He joined the MIT faculty in 1965, where he is currently a Professor of electrical engineering and teaches electromagnetics and signal processing. He was the Principal Investigator for the Nimbus E Microwave Spectrometer (NEMS) and Scanning Microwave Spectrometer (SCAMS) experiments on the National Aeronautics and Space Administration’s (NASA) Nimbus-5 and Nimbus-6 satellites and a Coinvestigator for the NASA Atmospheric Infrared Sounder/Advanced Microwave Sounding Unit sounding experiment on Aqua, Scanning Multichannel Microwave Radiometer experiment on Nimbus 7, and Planetary Radio Astronomy Experiment on Voyager 1 and 2. Other research has involved estimation, radio astronomy, video coding, milliarc-second optical astrometry, random process characterization, and wireless communications. He is a member of the National Polar-Precipitation Measurement Missions and the NPOESS Preparatory Program (NPP). He was an Assistant Director of the MIT Lincoln Laboratory from 1990 to 2001.

Daniel W. Bliss (M’03) received the B.S.E.E. degree in electrical engineering from Arizona State University, Tempe, in 1989, and the M.S. and Ph.D. degrees in physics from the University of California at San Diego, in 1995 and 1997, respectively. Currently, he is a staff member at MIT Lincoln Laboratory, Massachusetts Institute of Technology (MIT), Cambridge, in the Advanced Sensor Techniques group. He was employed by General Dynamics from 1989 to 1991, he designed avionics for the Atlas-Centaur launch vehicle and performed research and development of fault-tolerant avionics. As a member of the superconducting magnet group at General Dynamics from 1991 to 1993. He performed magnetic field calculations and optimization for high-energy particle-accelerator superconducting magnets. His doctoral wor,k from 1993 to 1997, was in the area of high-energy particle physics, searching for bound states of gluons, studying the two-photon production of hadronic final states, and investigating innovative techniques for lattice-gauge-theory calculations. Since 1997, he has been employed by MIT Lincoln Laboratory, where he focuses on multiantenna adaptive signal processing and performance bounds, primarily for communication systems. His current research topics include MIMO communication channel phenomenology, space-time coding, information-theoretic bounds for MIMO communication systems, algorithm development for multichannel multiuser detectors (MCMUD), and multiple-input multiple-output (MIMO) radar concepts.

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A MIMO Propagation Channel Model in a Random Medium Akira Ishimaru, Life Fellow, IEEE, Sermsak Jaruwatanadilok, Member, IEEE, James A. Ritcey, Member, IEEE, and Yasuo Kuga, Fellow, IEEE

Abstract—Multiple input-multiple output systems have received considerable attention because of their potential to achieve high channel capacity. This paper presents a study of the effects of a random scattering medium on channel capacity. Formulations are given including stochastic Green’s functions and mutual coherence functions. Transmitter and receiver characteristics are included and analytical formulation for eigenvalues and channel capacities are given in terms of the medium scattering characteristics, optical depth, frequency, number of transmitter and receiver elements, transmitting power, and noise spectral power. As an example, we show 500 m link at 60 GHz through rain. The eigenvalues and the channel capacity are calculated in terms of SNR and the rain rate representing the optical depth. It is shown that as the rain rate increases, the correlation of waves at antennas decreases and the capacity increases. However, at high rain rate, the capacity tends to decrease due to the absorption and scattering. Index Terms—Channel capacity, communications, multiple input-multiple output (MIMO) , rain, random media.

I. INTRODUCTION HERE have been a large number of studies reported on MIMO systems because they have potential for large channel capacity [1]–[9]. In recent years, there has been increasing interest in incorporating physical propagation characteristics in MIMO studies including diffraction effects and keyholes [10]–[13], [25], [26]. In this paper, we present formulations for a MIMO propagation channel in a random medium. The random medium is a random distribution of discrete scatterers such as rain drops and a randomly varying refractive index of atmospheric turbulence. The waves in such a random medium experience random amplitude and phase fluctuations which are expressed by their correlations. The second moments are called the mutual coherence functions, which are the ensemble average of the correlation of waves at two different locations. The correlation distance is called the “coherence length” and plays an important role in this study.

T

Manuscript received September 04, 2008; revised June 29, 2009. First published November 06, 2009; current version published January 04, 2010. This work was supported in part by the National Science Foundation under Grants ECS-0601394 and ECCS-0925034, in part by the Office of Naval Research under Grants N00014-07-1-0428 and N00014-07-1-0600, and in part by ARO MURI, W911NF-07-1-0287. The authors are with the Department of Electrical Engineering, University of Washington, Seattle, WA 98195 USA (e-mail: [email protected]; [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.2009.2036189

Fig. 1. M transmitters and N receivers.

This study is, therefore, on the average channel characteristics of the medium, and not on an individual deterministic scattering event. There have been many studies made on stochastic MIMO channels [14]–[20], [30]. The key contributions of this paper are the use of analytical expressions of stochastic Green’s functions and mutual coherence functions including antenna characteristics. Analytical expressions are given for eigenvalues and channel capacity in terms of the medium characteristics. A general formulation is given in Section II. Calculation of eigenvalues based on mutual coherence function is given in Section III. Section IV gives numerical examples of 500 m link at 60 GHz. Some discussions on the results are given in Section V. Appendix A and B include a detailed explanation of mutual coherence function and rain drop size distribution. II. FORMULATION OF THE PROBLEM Let us consider transmitters and receivers (Fig. 1). and so the total power Each transmitter sends power . It is assumed that the transmitted is transmitters and receivers are matched in impedance and polarization, lossless, and the mutual couplings are neglected. to receiver at is The channel gain from transmitter at channel transfer matrix , and its elements given by the , consist of the stochastic Green’s function and the antenna characteristics (1) where source at

is the stochastic Green’s function at . In free space, is reduced to

from the

(2) In general, is a random function of , and random medium characteristics. Its first moment is the coherent field,

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ISHIMARU et al.: A MIMO PROPAGATION CHANNEL MODEL IN A RANDOM MEDIUM

and the second moment is called the mutual coherence funcand are the antenna tion, as explained in Section IV. receiver and the transmitter, recharacteristics of the spectively. The output ( matrix) at the receivers and ( matrix) at the transmitters are related by the input the channel matrix and noise as

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effects, the location and the random medium, we consider the in free space which depends on the location. normalization

(8) The normalized transfer matrix

is then given by

(3) When the transmitter and the receivers are identical and located in free space and in the far field of each other, they are reduced to

(9) Note that have

is random and because of the normalization (7), we

(4) and are the antenna gain of the receiver and transwhere mitter element, is wavelength and (4) is the Friis formula for the ratio of the received power to the transmitting power beand . In this paper, we use tween the antenna element at , half-wave dipole antennas. Then, we have and

(5) matrix is stoLet us return to the case in which characterized by the mutual coherence chastic. Consider the eigenvalues where is the confunction (MCF), and be the element of the jugate transpose of . Let matrix . It is given by

(10) . where is the eigenvalues of The factor is the attenuation due to the random medium over that in free space and is equal to (free space path loss)/(medium path loss). We also note that there are eigenvalues forming independent channels. In summary, the random channel matrix can be characterized in terms , whose elements are found from the MCF of the of medium. Since the MCF is known for some random channels, it is of interest to see how this impacts MIMO capacity. depends on the receiver The signal to noise ratio (SNR) location and the random medium characteristics. To separate the location dependence and the random medium effect, we use the following SNR in free space, which depends on the location

(11)

(6) where is the mutual coherence function. Note that in free space, there is no incoherent field and the number of the eigenand the eigenvalues of and values is equal to are the same. However, for a random medium, because is of the correlation, the number of eigenvalues for and the number of eigenvalues for is . The normalization of the transfer matrix is done by using the average channel gain using Frobenius norm [2], [10]

(7) is the sum of all eigenvalues of . The where depends on the location of the receiver and the normalization random medium characteristics. In order to separate these two

where is the noise power, is the power spectral density of the noise (W/Hz) and is the bandwidth (Hz). The random medium effects are thus represented by the normalized channel matrix .

III. CHANNEL CAPACITY Next, we consider the ergodic channel capacity for a correlated MIMO channel [13]. The power sent out from each trans. When the channel is random, the channel camitter is pacity is a random variable, characterized by its mean, variance, and distribution. In the random medium we consider the distri, which are unknown, and generating them by butions of the simulation is extremely intensive. To make analytical progress, we employ an upper bound [14], [15]. The mean ergodic capacity is given by [14], [15]

(12)

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where is the noise power at each receiver, and is identity matrix. Since Monte Carlo simulations of the scattering process are infeasible and since capacity is a concave function, we can get an upper bound of the mean capacity [14] by

(13) Noting that are

and the eigenvalues of

Fig. 2. Geometry of a random medium and transmitters and receivers. r ;z ,r  ;z L .

(

, we get

(14) Using the definition of SNR in (11), we write

= 0)

=(

= )

absorption depth. The albedo is defined by coherence length is given by

=

. The

as (19) (15)

When the channel has Gaussian distributed elements, Monte Carlo studies [28] have shown that the upper bound is a good approximation to the mean capacity. For the random medium, even Monte Carlo is intractable, but the upper bound is com. An interesting putable in terms of the eigenvalues of special case, which provides the outage capacity in the special case of a 2 2 MIMO system with highly directive antennas is diagonal, selecwith large separation so that tion combining, and lognormal fading, is given by Liolis et al. [30]. IV. CALCULATION OF The mutual coherence functions in (6) in a random medium environment illustrated in Fig. 2 have been obtained. It is given by [21]–[23]. The detailed derivation of the mutual coherence function is given in Appendix A

(16) and are free space Green’s functions and in where parabolic approximation, they are given by

(17) The effects of a random medium are given by

The scatterer characteristics are represented by the phase funcgiven by tion

(20) where

and

V. NUMERICAL EXAMPLES A. Parameters Used in Calculations We calculate the scenarios of communications in the presence of rain at different rain rates. The rain rate changes the scattering characteristics of the random medium in terms of number of particles and size of the particles. The model of the relation of rain droplet particle size distribution and rain rate , , is given in [24]. We use , and . The anisotropy factor and albedo (ratio of scattering cross section over total cross section) vary according to the rain droplet size. The number of transmitter , the number of receiver elements , elements , and is the bandnoise power , we assume the thermal width. For noise spectrum density noise , where is the is the receiving cross sections of a half-wave dipole and integral over solid angle of , the Boltzman constant , and the temperature . All elements are half-wave dipole and spacing is half wavelength. The channel characteristics as functions of rain rate are given in Table I. See Appendix B. B. Eigenvalues of

(18) is the atmospheric absorption, where is the optical scattering depth, and optical depth,

is the is the

is the scattering angle.

as Functions of Rain Rate

Fig. 3 shows the eigenvalues of . Note that is dominant and the sum of all eigenvalues is . In this case, other eigenvalues are approximately the same. Note that as the rain decreases slightly, but all rate increases, the first eigenvalue other eigenvalues increase indicating increase of multiple scattering. Smallest rain rate in Fig. 3 is 1 mm/hr, and in free space, it is less than those shown in figure, but it is not zero. All higher

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TABLE I CHANNEL CHARACTERISTICS AS FUNCTIONS OF RAIN RATE

eigenvalues are close for the distance we used. However, they become more separated as the distance becomes shorter. C. Channel Capacity Fig. 4 shows the channel capacity at different total transmit, 100 mW, and 1 W). Note that the ting powers ( higher transmitting powers give higher capacity. As the rain rate increases, the channel capacity increases because of multiple scattering, but it tends to decrease at higher rain rate because of the absorption and scattering through rain. Fig. 5 shows the correlation of waves at the receiver given where the separation by . As the rain rate increases, the correlation decreases (Fig. 5) and as seen and the channel capacity increases for large in Fig. 4. This is consistent with the observation [14] that the capacity increases as the correlation coefficient decreases. Fig. 6 shows the channel capacity as a function of signal-tonoise ratio for different transmitter and receiver elements at different rain rates. It shows that the presence of moderate rain can actually improves the capacity of the channel, but the capacity decreases at higher rain rate of 100 mm/hr. The improvement is more pronounced when number of transmitter and receiver elements increases. Fig. 7 shows channel capacity as functions of rain rate at different signal-to-noise ratio. It shows an interesting property that capacity increases as the rain rate changes from light to moderate rate, but decreases at higher rain rate. This effect is more pronounced at high SNR [Fig. 7(c)]. This is due to the multiple scattering. Fig. 8 shows channel capacity as functions of dis, tance for light, moderate, and heavy rain. Note that for , the capacity tends to decrease with distance for all rain , , the capacity increases for moderate rate. For and heavy rain more than that in light rain showing the increased capacity due to multiple scattering. Reference SNR is 20 dB at distance of 500 m and the noise is calculated at 500 m and used for all distances. Fig. 9 shows channel capacity as function of frequency at 500 m. Note that the capacity dips near 60 GHz due to the peaked atmospheric absorption. This also shows increased capacity for heavy rain due to multiple scattering. It may be noted that this study is for the average channel capacity when steady rain rate is unchanged. The coherence time

HH i as a function of rain rate. (a) M = 11, N = 7, M = 2, N = 2, and (d) M = 1, N = 1.

Fig. 3. Eigenvalues of h (b) , , (c)

M =5 N =3

has been estimated [29] to be approximately 0.135 ms and 0.064 ms for 100 GHz over 5 km in 2.5 mm/rh and 25 mm/hr repectively. If the coherence time is approximately 0.1 ms for our

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Fig. 5. Normalized correlation function as a function of the rain rate.

VI. DISCUSSION An interesting conclusion is that the channel capacity increases as rain rate increases from light to moderate because of the decorrelation of waves, but tends to decrease at higher rain rate because of the rain attenuation and scattering. A similar conclusion has been reported in [28]. In order to clarify this point, a normalization is introduced such that SNR is location dependent, and the eigenvalues depend on the random medium characteristics. This leads to the sum of eigenvalues equal to where is the attenuation. The capacity in this paper is an average over an ensemble of the rain which is assumed constant over some time, and it may be noted that the present study is a first step toward a more complete space-time analysis of channel capacity in time-varying random media. Finally, our channel capacity is an upper bound on data rate, without constraints on delay, encoding, etc. in practical systems. VII. CONCLUSION

M = 11 N = 7 M = 5 N = 3 M = 2 N = 2 M=1 N =1

Fig. 4. Channel capacity as a function of optical depth when transmitted power varies. (a) , , (b) , , (c) , , and (d) , .

case, the channel is considered steady for about 200 symbols for our bandwidth of 500 kHz.

This paper presents a MIMO channel model for propagation through a random medium. Transmitters and receivers are half-wave dipoles and the propagation characteristics are given analytically using stochastic Green’s functions and mutual coherence functions, which are expressed in terms of particle characteristics, sizes, number densities, and cross sections. As an example, 60 GHz communication through rain is studied for different rain rates. The channel transfer matrix is expressed in terms of transmission power, noise power, eigenvalues, and the medium characteristics, and the channel capacity is given by SNR and eigenvalues. The increase in capacity with MIMO on fading channels can be traced to the diversity gain, as opposed to an increase in array gain. On the rainy channel, we see two effects as the rain rate increases; the channel shows increased attenuation, but the diversity order increases. The overall effect on capacity is a mixture, with attenuation eventually winning out at high rain-rates. APPENDIX A MUTUAL COHERENCE FUNCTION The general expression of the mutual coherence function for a spherical wave in parabolic approximation is given in

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Fig. 6. Channel capacity as a function of signal to noise ratio for different transmitter and receiver elements at different rain rates. (a) Rain rate = 1 mm=hr (light rain), (b) Rain rate = 16 mm=hr (moderate rain), and (c) Rain rate = 101 mm=hr (heavy rain).

Fig. 7. Channel capacity as a function of rain rate for different transmitter and receiver elements at different signal-to-noise ratio. (a) SNR = 0 dB, (b) SNR = 10 dB, and (c) SNR = 20 dB.

(20–69) page 410 of [23]. A point source is located at the origin and the mutual coherence function which is the correlation of and , is given by waves at

(A-2)

(A-1)

where , is the atmospheric attenuation, is absorption coefficient due to particles, is scattering cois the phase function of a simple particle, and efficient, . Note that (A-1) is applicable to inhomogeneous can be functions of . , medium where , ,

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Fig. 8. Channel capacity as a function of distance at 60 GHz for SNR of 20 dB at reference distance of 500 m. (a) , , and (b) , .

M=1 N=1

Fig. 9. Channel capacity as a function of frequency at the distance of 500 m and SNR of 20 dB. (a) , , and (b) , .

. The phase function differential cross section [23, p. 10 ]

is related to the

(in ), . The value where can be found by its relation with the median volume diameter (in cm) through

M=2 N=2

M=2 N=2

M=1 N=1

(B-2)

(A-3) We made further approximation to (A-2) when the phase function is approximated by Gaussian function

The relationship of the median volume diameter with the rain rate (in mm/hr) is given by

(B-3) (A-4) We can then perform integration in (A-2) following the procedure in [21], and obtain the results shown in (16)–(20).

where , and . The value of , , , and are taken from Table 2 of [24] under category of Thunder storm rain by Jones (1956). Note that the definition of the median volume diameter is given by

APPENDIX B RAIN DROP SIZE DISTRIBUTION AND RAIN PARAMETERS Rain drop size distribution are taken from [24] which is a three parameter gamma distribution

(B-4)

where (B-1)

, but in the calculation, we use .

ISHIMARU et al.: A MIMO PROPAGATION CHANNEL MODEL IN A RANDOM MEDIUM

REFERENCES [1] A. F. Molisch, Wireless Communications. Piscataway, NJ: IEEE Press, 2005. [2] A. Paulraj, R. Nabar, and D. Gore, Introduction to Space-Time Wireless Communications. Cambridge, U.K.: Cambridge Univ. Press, 2003. [3] M. A. Jensen and J. W. Wallace, “A review of antennas and propagation for MIMO wireless communications,” IEEE Trans. Antennas Propag., vol. 52, no. 11, pp. 2810–2823, Nov. 2004. [4] G. G. Raleigh and J. M. Cioffi, “Spatio-temporal coding for wireless communication,” IEEE Trans. Commun., vol. 46, no. 3, pp. 352–366, Mar. 1998. [5] N. Blaunstein and C. Christodoulou, Radio Propagation and Adaptive Antennas for Wireless Communication Links: Terrestrial, Atmospheric and Ionospheric. New York: Wiley, 2007. [6] G. J. Foschini and M. J. Gans, “On limits of wireless communications in a fading environment when using multiple antennas,” Wireless Personal Commun., vol. 6, pp. 311–335, 1998. [7] D. Gesbert, M. Shafi, D.-S. Shiu, P. J. Smith, and A. Naguib, “From theory to practice: An overview of MIMO space-time coded wireless systems,” IEEE J. Sel. Areas Commun., vol. 21, no. 3, pp. 281–302, Apr. 2003. [8] T. L. Marzetta and B. M. Hochwald, “Capacity of a mobile multipleantenna communication link in Rayleigh flat fading,” IEEE Trans. Inf. Theory, vol. 45, no. 1, pp. 139–157, Jan. 1999. [9] A. Goldsmith, Wireless Communications. Cambridge, UK: Cambridge Univ. Press, 2005. [10] P. Kyritsi and D. Chizhik, “Capacity of multiple antenna systems in free space and above perfect ground,” IEEE Commun. Lett., vol. 6, no. 8, pp. 325–327, 2002. [11] J. B. Andersen, “Array gain and capacity for known random channels with multiple element arrays at both ends,” IEEE J. Sel. Areas Commun., vol. 18, no. 11, pp. 2172–2178, Nov. 2000. [12] J. B. Andersen, “Antenna arrays in mobile communications: Gain, diversity, and channel capacity,” IEEE Antennas Propag. Mag., vol. 42, no. 2, pp. 12–16, Apr. 2000. [13] D. Chizhik, G. J. Foschini, M. J. Gans, and R. A. Valenzuela, “Keyholes, correlations, and capacities of multielement transmit and receive antennas,” IEEE Trans. Wireless Commun., vol. 1, no. 2, pp. 361–368, April 2002. [14] S. L. Loyka, “Channel capacity of MIMO architecture using the exponential correlation matrix,” IEEE Commun. Lett., vol. 5, no. 9, pp. 369–371, Sep. 2001. [15] S. L. Loyka and G. Tsoulos, “Estimating MIMO system performance using the correlation matrix approach,” IEEE Commun. Lett., vol. 6, no. 1, pp. 19–21, Jan. 2002. [16] A. L. Moustakas, H. U. Baranger, L. Balents, A. M. Sengupta, and S. H. Simon, “Communication through a diffusive medium: Coherence and capacity,” Science, vol. 287, pp. 287–290, Jan. 2000. [17] W. Weichselberger, M. Herdin, H. Ozcelik, and E. Bonek, “A stochastic MIMO channel model with joint correlation of both link ends,” IEEE Trans. Wireless Commun.s, vol. 5, no. 1, pp. 90–100, Jan. 2006. [18] J. P. Kermoal, L. Schumacher, K. I. Pedersen, P. E. Mogensen, and F. Frederiksen, “A stochastic MIMO radio channel model with experimental validation,” IEEE J. Sel. Areas Commun., vol. 20, no. 6, pp. 1221–1226, Aug. 2002. [19] A. L. Moustakas, S. H. Simon, and A. M. Sengupta, “MIMO capacity through correlated channels in the presence of correlated interferes and noise: A (not so) large N analysis,” IEEE Trans. Inf. Theory, vol. 49, no. 10, pp. 2545–2561, Oct. 2003. [20] A. M. Tulino, A. Lozano, and S. Verdu, “Impact of antenna correlation on the capacity of multiantenna channels,” IEEE Trans. Inf. Theory, vol. 51, no. 7, pp. 2491–2509, Jul. 2005. [21] A. Ishimaru, S. Jaruwatanadilok, and Y. Kuga, “Multiple scattering effects on the radar cross section (RCS) of objects in a random medium including backscattering enhancement and shower curtain effects,” Waves in Random Media, vol. 14, pp. 499–511, 2004. [22] A. Ishimaru, S. Jaruwatanadilok, and Y. Kuga, “Short pulse detection and imaging of objects behind obscuring random layers,” Waves in Random and Complex Media, vol. 16, no. 4, pp. 506–520, Nov. 2006. [23] A. Ishimaru, Wave Propagation and Scattering in Random Media. Piscataway, NJ: IEEE Press, 1997. [24] C. W. Ulbrich, “Natural variations in the analytical form of the raindrop size distribution,” J. Climate Appl. Meteorol., vol. 22, pp. 1764–1775, 1983. [25] C. Oestges and B. Clerckx, MIMO Wireless Communications: From Real-World Propagation to Space-Time Code Design. London, U.K.: Academic Press, 2007.

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[26] R. Vaughan and J. B. Anderson, Channels, Propagation and Antennas for Mobile Communications. London, U.K.: Inst. Elect. Eng., 2003. [27] S. Loyka and A. Kouki, “New compound upper bound on MIMO channel capacity,” IEEE Commun. Lett., vol. 6, no. 3, pp. 96–98, Mar. 2002. [28] F. Bohagen, P. Orten, and G. E. Oien, “Modeling and analysis of a 40 GHz MIMO system for fixed wireless access,” in IEEE Veh. Technol. Conf., Jun. 2005, vol. 3, pp. 1691–1695. [29] S. T. Hong and A. Ishimaru, “Two-frequency mutual coherence function, coherence bandwidth, and coherence time of millimeter and optical waves in rain, fog, and turbulence,” Radio Sci., vol. 11, no. 6, pp. 551–559, Jun. 1976. [30] K. P. Liolis, A. D. Panagopoulos, P. G. Cottis, and B. D. Rao, “On the applicability of MIMO principle to 10–66 GHz BFWA networks: Capacity enhancement through spatial multiplexing and interference reduction through selection diversity,” IEEE Trans. Commun., vol. 57, no. 2, pp. 530–540, Feb. 2009.

Akira Ishimaru (M’58–SM’63–F’73–LF’94) received the B.S. degree from the University of Tokyo, Tokyo, Japan, in 1951 and the Ph.D. degree in electrical engineering from the University of Washington, Seattle, in 1958. From 1951 to 1952, he was with the Electrotechnical Laboratory, Tanashi, Tokyo, and in 1956, he was with Bell Laboratories, Holmdel, NJ. In 1958, he joined the faculty of the Department of Electrical Engineering, University of Washington, where he was a Professor of electrical engineering and an Adjunct Professor of applied mathematics. He is currently Professor Emeritus there. He has also been a Visiting Associate Professor at the University of California, Berkeley. His current research includes waves in random media, remote sensing, object detection, and imaging in clutter environment, inverse problems, millimeter wave, optical propagation and scattering in the atmosphere and the terrain, rough surface scattering, and optical diffusion in tissues. He is the author of Wave Propagation and Scattering in Random Media (New York: Academic, 1978; IEEE-Oxford University Press Classic reissue, 1997) and Electromagnetic Wave Propagation, Radiation, and Scattering (Englewood Cliffs, NJ: Prentice-Hall, 1991). He was Editor (1979–1983) of Radio Science and Founding Editor of Waves in Random Media (Institute of Physics, U.K.), and Waves in Random and Complex Media (Taylor and Francis, U.K.). Dr. Ishimaru has served as a member-at-large of the U.S. National Committee (USNC) and was Chairman (1985–87) of Commission B of the USNC/International Union of Radio Science. He is a Fellow of the Optical Society of America, the Acoustical Society of America, and the Institute of Physics, U.K. He was the recipient of the 1968 IEEE Region VI Achievement Award and the IEEE Centennial Medal in 1984. He was appointed as Boeing Martin Professor in the College of Engineering in 1993. In 1995, he was awarded the Distinguished Achievement Award from the IEEE Antennas and Propagation Society. He was elected to the National Academy of Engineering in 1996. In 1998, he was awarded the Distinguished Achievement Award from the IEEE Geoscience and Remote Sensing Society. He is the recipient of the 1999 IEEE Heinrich Hertz Medal and the 1999 URSI Dellinger Gold Medal. In 2000, he received the IEEE Third Millennium Medal.

Sermsak Jaruwatanadilok (M’03) received the B.E. degree in telecommunication engineering from King Mongkut’s Institute of Technology Ladkrabang, Thailand, in 1994, the M.S. degree in electrical engineering from Texas A&M University, College Station, in 1997, and the Ph.D. degree in electrical engineering from the University of Washington, Seattle, in 2003. He is currently a Research Assistant Professor at the University of Washington, Seattle. His research interests are optical wave propagation and imaging in random medium, as well as optical and microwave remote sensing.

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James A. Ritcey (M’09) received the B.S.E. degree from Duke University, Durham, NC, the M.S.E.E. degree from Syracuse University, Syracuse, NY, and the Ph.D. degree in electrical engineering (communication theory and systems) from the University of California, San Diego, in 1985. Since 1985, he has been with the Department of Electrical Engineering, University of Washington, Seattle, where he now holds the rank of Professor and Associate Chair. From 1976 to 1981 he was with the General Electric Company, and graduated from GE’s Advanced Course in Engineering. His research interests include communications and statistical signal processing for radar and underwater acoustics. He has published extensively in these areas. Professor Ritcey served as the General Chair of the 1995 International Conference on Communications in Seattle. He also served as Technical Program Chair in the 1992 and General Chair in 1994 of the Asilomar Conference on Signals, Systems, and Computers and is currently a member of the Steering Committee.

Yasuo Kuga (F’04) received the B.S., M.S., and Ph.D. degrees from the University of Washington, Seattle, in 1977, 1979, and 1983, respectively. From 1983 to 1988, he was a Research Assistant Professor of electrical engineering at the University of Washington. From 1988 to 1991, he was an Assistant Professor of electrical engineering and computer science at The University of Michigan. Since 1991, he has been with the University of Washington, where he is currently a Professor of electrical engineering. His research interests are in the areas of microwave and millimeter-wave remote sensing, high frequency devices and materials, and optics. Dr Kuga was an Associate Editor of Radio Science (1993–1996) and the IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING (1996–2000).

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Effect of Optical Loss and Antenna Separation in 2 2 MIMO Fiber-Radio Systems Andrey Kobyakov, Michael Sauer, Member, IEEE, Anthony Ng’oma, Member, IEEE, and Jack H. Winters, Fellow, IEEE

Abstract—We study the impact of fiber-optic loss due to realistic cable deployment and optical connector loss variation and distance between the transmitter antennas in a radio-over-fiber transmission system with 2 2 MIMO. We evaluate an upper bound on the system capacity based on measured values of the error vector magnitude and the condition number of the channel matrix. For this, assuming a Rayleigh fading environment, we derive conditional probability density functions of eigenvalues of a 2 2 Wishart matrix. We compare the obtained results with another approach, which is based on IEEE 802.11 relations between the allowed relative constellation error and the achievable data rate for an individual stream. Trends predicted by both models agree very well. Our analysis shows that (i) about 1 meter spatial separation of fiber-fed transmitter antennas is optimum and results in the maximized system performance and (ii) the maximum tolerable optical power imbalance in two fiber optic links is about 6 dB. Index Terms—Channel capacity, condition number, multiple-input multiple-output (MIMO) systems, picocellular networks, radio-over-fiber (RoF), Wishart matrices.

I. INTRODUCTION

ULTIPLE-INPUT multiple-output (MIMO) technology offers increased data rate and/or improved quality of a communication link by using multiple transmitter and receiver antennas (for an overview, see, e.g., [1]–[3]). Radio-over-fiber (RoF) is another promising approach that allows for efficient remote antenna feeding over long distances with very low loss or RF signal distortion. Using RoF signal feeding, a dense (picocellular) network comprising a very high number of remote antennas controlled from a central head-end location can be designed [4]. Incorporating MIMO principles into RoF transmission technology results in further improvements in coverage and throughput. In particular, increasing the distance between the base station antennas was shown to increase the coverage area of WLAN picocells [5]. In addition, antenna separation offers extended flexibility in the network design where time-shared use of antennas from neighboring cells becomes possible. One of

M

Manuscript received September 11, 2008; revised July 22, 2009. First published November 06, 2009; current version published January 04, 2010. A. Kobyakov, M. Sauer, and A. Ng’oma are with the Science and Technology Division, Corning Incorporated, Corning, NY 14831 USA (e-mail: [email protected]). J. H. Winters is with Jack Winters Communications, Middletown, NJ 07748 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.2009.2036190

the key goals of this work is to quantify the effect of MIMO antennas separation on the system performance. Fiber links feeding physically separated antennas may have different lengths and different losses. These losses can vary by several optical dBs in practical deployments due to optical connector loss variation and different lengths of optical cables. Multiple optical connectors are used in realistic deployments due to practical infrastructure constraints. Optical connector losses may vary from essentially 0 dB up to several tenths of dBs depending on the type of connector. When using multimode fiber links, additional loss variation may be introduced even in links of the same length due to the variable bandwidth of multimode fibers [4]. It was observed in [5] that imbalanced optical inputs can lead to a decrease in the cell coverage. Hence, another goal of this work is to quantify tolerances to the optical loss of the RoF feeding of antennas. In this paper we study a 2 2 MIMO system with fiber-fed remote antennas in the downlink. To evaluate the multi-antenna channel performance we use measured error vector magnitude (EVM) values for each stream and the measured condition number (CN) of the channel matrix. The CN characterizes invertibility of the channel matrix and shows how efficiently the multiple data streams can be decomposed. Ideally, the CN should be close to unity but as we will see later, the probability of this event is very low. The CN has been shown to be a good measure for the MIMO gain [6]–[8]. Several transmission techniques for MIMO signals are based on the CN thresholding [8]–[10]. However, the CN alone does not provide an accurate estimate of the system performance. We therefore use the CN together with the EVM of each stream to evaluate an upper bound on the system capacity. We derive a corresponding expression and use it to quantify the maximum system capacity as a function of the differential optical loss between the two fiber links and the transmitter antenna separation. The following section describes the experimental setup and the measurement procedure. In Section III we describe our statistical analysis which is used to interpret experimental results in Section IV. Section V concludes the paper. II. EXPERIMENTAL SETUP AND MEASUREMENT RESULTS The experimental setup is schematically shown in Fig. 1. To generate 16 QAM MIMO orthogonal frequency-division multiplexed (OFDM) streams occupying 16.6 MHz bandwidth at 2.4 GHz we used a pair of Agilent vector signal generators (VSGs). Two vector signal analyzers (VSAs) were used at the receiver side to decouple the two streams and measure the corresponding EVM values and the CN [11].

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Fig. 1. Experimental setup used to study the impact of optical link loss and antenna separation on 2 2 MIMO system performance. LD is the laser diode, PD is the photodetector.

2

The VSA uses a basic zero-forcing approach for MIMO channel estimation. This approach utilizes the high throughput—long training fields (HT-LTF) which are embedded in the packet, (as defined in IEEE 802.11n, section 20.3.9.4.6). The HT-LTFs are transmitted simultaneously on all active antennas, one per active HT stream, with some inverted before transmission. The net result is that the receivers have sufficient information to solve for each of the individual streams on each path. Knowledge of the actual received waveform and the ideal waveform allows for straightforward conversion to the frequency domain (OFDM demodulation) and calculation of a complex difference at each of the tone frequencies. The collection of coefficients for all tones becomes the channel response for the stream and path in question, so that a separate channel estimate is created for each tone in the OFDM signal. Since the channel response has been calculated across a very short period of time, we have used the “preamble data equalization” measurement option, i.e., used the calculated value of the channel state as the starting point for a second equalizer training pass. Then, the remaining symbols in the packet are demodulated using the initial channel estimate, and additional response coefficients are calculated for them as well (i.e., difference between actual and ideal). The improved channel estimate is the average of these coefficients across all symbols in the received packet. There is a limit to the accuracy of channel estimation, and the resulting streams may be imperfectly decoupled. Then the opposing stream appears as uncorrelated noise added to the stream under test, which results is an increase in the displayed EVM. Because residual coupling increases EVM, the best test of channel estimation accuracy is to compare measured EVM with the second transmit stream ON and OFF. Tests of the 89600 VSA suggested that the EVM floor (the minimum measurable EVM) is virtually the same in both cases ( dB). Another precaution used during measurements is related to the limited dynamic range of the VSA. We therefore used RF amplifiers to keep the signal high enough to use the full range of the A/D converters. To obtain the most accurate EVM measurements, we nearly maximized out the A/D converter by backing down the range until the overload status indicator appears and then backed off about 2 dB. The separation between the transmitter antennas was varied meter steps. The receiver antennas from 0.1 to 4 meters in were placed close to each other (0.1 m separation) as it would

be typically seen in portable or mobile receiver devices. Optical loss is introduced in one of the standard single-mode fiber links to study the system tolerance to imbalanced optical fiber-radio links feeding the remote antennas. This optical loss variation would represent practical deployment scenarios. While the electrical loss can typically be controlled within a dB, the optical loss variation may be as high as several dBs and in most cases cannot be controlled. Optical-to-electrical conversion is responsible for the fact that dB of optical loss results in 2 dB electrical loss as can be inferred from the expression for the elecof a RoF link [12] trical gain

(1) , and are the laser slope efficiency, the photodewhere tector responsivity, and the optical link loss, respectively. The distance between the transmit and receive antennas was about 8 meters in a non-line of sight (NLOS) indoor environment. RF amplifiers were used for all antennas to increase the sensitivity of the measurements. The OFDM transmit signals of the VSGs (set to 0 dBm) were directly coupled into distributed feedback lasers operating at 1300 nm and biased at 40 mA. This ensured linear operation of the laser diodes (LD). The optical RF signals then were transported over standard single-mode fibers. One of the optical fiber links had a variable optical attenuator included to vary the optical loss of this fiber link relative to the other link between 0 dB and 10 dB. After fiber transmission, two remote photodetectors (PD) converted the optical RF signals into electrical RF signals. The signal strength after phodBm, according to (1) with a laser slope todetection was W/A, an optical baseline loss (without efficiency of dB, and a photodetector variable attenuation) of efficiency of A/W. The first amplification stage had 15 dB of gain, followed by a bandpass filter (1 dB loss) and a second amplifier stage of 25 dB gain. This resulted in a transmit power of 15 dBm with omnidirectional antennas (1 dBi). On the receive side before the VSAs, low-noise amplifiers with 28 dB gain and noise figure of 2.2 dB were used. (inverse of the EVM ) Fig. 2 shows mean measured in dB and the dimensionless CN, for each OFDM carrier with the differential optical loss in the two fiber links varying from 0 dB to 10 dB. Temporal averaging is performed over about minutes. The single peak in 850 realizations taken during Fig. 2 indicates the presence of two strong multipath signals. High EVM for those carriers further increases with higher optical loss [Fig. 2(c), (d)]. The EVM of the stream without introduced optical loss remains essentially constant, while the CN increases with increased differential optical loss. As was observed before, the increased CN is accompanied by noise enhancement and signal-to-noise ratio (SNR) degradation [13]. OFDM transmission can be characterized by averaging the key system parameters over all carriers [3]. An example of the time dependence of both EVMs and the CN averaged over all 56 OFDM carriers is shown in Fig. 3. Measurement conditions correspond to highly imbalanced streams (differential optical loss of 8 dB) which results in a clearly high EVM of channel

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Fig. 2. Mean SNRs in dB of both streams and mean value of the CN,  versus the OFDM carrier number for different values of optical loss in one of the fiber-fed transmitter links: (a) 0 dB, (b) 2 dB, (c) 6 dB, and (d) 10 dB. The distance between transmit antennas is 4 meters.

Fig. 4. Mean stream EVM and the CN versus differential optical loss for various distances between the transmitter antennas: (a) 0.1 m, (b) 1 m, (c) 2 m, (d) 4 m.

derive an expression for the maximum system capacity based on measured mean values of EVM and the CN. III. STATISTICS OF THE CONDITION NUMBER AND THE 2 MIMO SYSTEM CAPACITY

2

We start with the formal definition of the CN, which is the , and the minimum, , singular ratio of the maximum, values of the channel matrix . Equivalently [1], [14], this is the square root of the ratio of the maximum and the minimum and , respectively, of matrix = , eigenvalues where denotes conjugate (Hermitian) transpose, i.e.,

(2) Fig. 3. Example of temporal evolution of the key measured parameters ; , and the condition number  (middle trace, thick curve) averaged over OFDM carriers. Transmitter antenna separation is 0.1 m, differential optical loss is 8 dB.

2 where the optical loss was introduced. One can also note a . strong correlation between the CN and Temporal averaging of the traces of Fig. 3 can give a qualitative idea about impact of the differential optical loss and antenna separation. Such a measurement summary is shown in Fig. 4 where we plot the stream EVMs and the CN averaged over both time and carrier numbers. Fig. 4 clearly shows that the overall system performance degrades with increased loss beand the CN grow. Another trend is visible by cause both comparing Fig. 4(a) and (d), namely, that increased distance between transmit antennas improves the system performance and offsets the impact of optical loss. With increased antenna separation the system becomes less sensitive to the imbalance of antenna feeding. However, Fig. 4 does not provide a quantitative estimate of the system performance. In the following section we

We note that an alternatively defined CN, the so-called Demmel condition number, was also suggested as a criterion of the MIMO efficiency [6], [8]. The Demmel condition number is related to the conventional CN as [15]. In what follows we use definition (2) when we refer to the CN. The MIMO system capacity is given by [1], [14], [16]

(3) where is the number of streams ( for 2 2 MIMO) is the corresponding signal-to-noise ratio of the th and data stream. Note that (3) is the capacity with knowledge of the channel state information (CSI) at the transmitter. In many systems the transmitter does not have this knowledge (and even in systems where there is feedback of the CSI to the transmitter, the feedback has estimation errors which reduce the capacity), in which case (3) is an upper bound on capacity which becomes tighter with increasing SNR (see, e.g., [17]). Furthermore, (3) is the capacity for a given SNR, rather than the capacity averaged

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over the fading with a given average SNR, but (3) still serves as an upper bound on our system capacity. The CN does not enter (3) explicitly. Typically one does not have access to measured eigenvalues of matrix , while monitoring the temporal evolution of the channel matrix is not convenient even in the 2 2 case, when 8 real values need to be stored. We are therefore going to develop an approach which will allow us to estimate the system capacity based on channel SNRs (or equivalently EVMs) and a single, real-valued quantity—the condition number. For this, we will for a given value of the CN need statistical properties of obtained from measurements. In what follows, we derive the conditional probability density functions of the maximum, and the minimum, eigenvalues of matrix for a fixed value . We then use the obtained statistics of their ratio to calculate the system capacity from (3). In what follows, we assume a Rayleigh channel between transmitter and receiver [7], [10], [14], [16], [18]. For our setup, this assumption seems to be reasonable as indicated by the statistical properties of the measured CN which is discussed below. With the assumption of Rayleigh channel, matrix belongs to a class of Wishart matrices, whose eigenvalues have the following joint pdf [15], [16], [19]

(4) , i.e., consider ordered In what follows, we assume eigenvalues. Therefore, despite the symmetric form of the joint and , function is asympdf (4) with respect to metric because it is determined only in the first octant where . From (4) one can find the distributions of each eigenvalue [14], [18] as

Fig. 5. Conditional pdf of the minimum,  (a) [(7)] and the maximum,  (b)  = fixed. [(8)] eigenvalues for their ratio V

=

of as

From (5) we can calculate the conditional probability density if the other random variable assumes a certain value [20] where

(6) is the pdf of random variable . Dividing (5) by (6) we obtain the desired conditional pdf from where one can obtain the expected values of the minimum and the maximum eigenvalues [14]. are The above probability density functions unconditional, i.e., they do not assume a particular value of . Since we have the CN fixed, we need conditional distributions of eigenvalues to interpret our experimental data. To obtain them, we first calculate the joint pdf of a different pair and . The correof variables, namely, sponding Jacobian determinant for the variable transformation (for details see, e.g., [20]) is and we obtain

(7) Following the same steps for the maximum eigenvalue get

we

(8) which has the same functional dependence as but with parameter replaced with its reciprocal. Both conditional pdf’s are shown in Fig. 5. in (6) one can Note that by variable transformation straightforwardly obtain the pdf of the condition number

(5) where

and

.

(9)

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Fig. 6. Histogram of the measured CN. The transmitter antenna separation is 3 m. The differential optical loss is 2 dB. These parameters correspond to nearly equal mean EVMs for both data streams. The solid curve shows the pdf (9) multiplied by the integral of the bar plot for appropriate comparison.

Fig. 7. Exact [(11), thick curves] and approximate [(12), thin curves] values of the integral J (a; b) as a function of parameter b for various values of ; a =  + 1.

which was discussed in [7], [10], [15]. One can see that the ideal case of unity CN is not realizable. The expected value of is . Fig. 6 demonstrates a reasonable agreement between statistics of the measured CN and the calculated pdf based on the assumption of a Rayleigh channel. Less than 1000 events is clearly not sufficient to appropriately describe the tails of distribution (9). From the derived conditional pdf’s (7) or (8) one can obtain the expected value of capacity. For example, since , using (3) one can express the mean capacity for given , and as

can be estimated from Fig. 7. Equation (12) is a tabular integral is the (see, e.g., [22, p. 576, formula 4.352.2]) where Euler constant. As can be seen from Fig. 7, for relatively small ), makes the approximation values of the CN (e.g., very accurate. The corresponding values of the EVM should or 18% rms which is always the therefore be less than case for the best measured stream (Fig. 4). For high values of the ), the validity of the discussed approximation CN (e.g., , i.e., % rms which requires is again true for all measured data. Our measurements can thus be analyzed with the approximation (12) which upon substitution in (10) gives

which transforms upon using (7) into

Substituting values of the auxiliary parameters finally obtain

and

we

(13) (10) where , and . In (10) we have used the relation between the SNR and the EVM [21] and associated the stream having the maximum SNR with the minimum eigenvalue to maximize capacity [14]. The exact value of the integral in (10) is given in terms of hypergeometric functions and is quite cumbersome. However, the resulting expression strongly simplifies under assumption . The validity ranges of the approximation

(11) (12)

Note that the same functional dependence of the mean capacity on the corresponding EVMs and the CN is obtained if the expectations of the eigenvalues

calculated from (7) and (8) are directly substituted in (3) for and , respectively, and is approximated as . The only difference to (13) will be in the first, numerical term whose value will be 4 rather than 3.62. IV. DISCUSSION With (13) we can use measured values of the stream EVMs and the CN to estimate the system capacity for each time instant. The result of the averaging over the measurement time

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Fig. 8. The system capacity versus optical loss calculated from (13) using measured values of EVM , EVM , and  for various distances between transmit antennas. Error bars represent the measured value of standard deviation.

(5 minutes) is shown in Fig. 8 as a function of differential optical loss for several values of the distance between transmit antennas. The trend seen from this plot is the increased capacity of the system with the increased distance between two transmit antennas. The advantage is still present even for the widest separation of 4 meters compared to the standard deployment with closely placed antennas, although a moderate separation of about 1 meter seems to be optimum. As was also seen from Fig. 4, optical loss degrades the system performance. The most significant degradation occurs, however, starting from a 6 dB differential optical loss. For antenna separation of 1, 2, and 4 meters we have observed a slight increase in the calculated system capacity when increasing optical loss from 0 dB to 2 dB (Figs. 4, 8). We attribute it to two factors. First, as indicated by Fig. 5, the distriis fairly broad near the bution of the eigenvalues of matrix respective expectations. As a result, the distribution of calculated capacity also becomes broad as indicated by error bars in Fig. 8. Second, at highest operation power (0 dB loss), one of the RF amplifiers might go slightly in compression regime so that a decrease in operation power might improve its performance. Another reason might be an imperfect balancing of the two RF-fiber links due to slightly different gain of RF amplifiers, which is difficult to control. An alternative approach to calculate the system performance is to evaluate data rates for each stream corresponding to the expected throughput in an 802.11 WLAN. Relations between the relative constellation error (RCE) which is the inverse of the SNR, the EVM and the data rate can be inferred from the IEEE 802.11 standards [23], [24]. Using Table I one can calculate the corresponding data rate of each stream for a given value of EVM. The sum of the two data rates gives the system throughput. The time-averaged throughput is shown in Fig. 9 which shows the same trend as the approach based on the statistics of the CN (Fig. 8). For example, both models show that for a considered 2 2 MIMO setup the optimum distance between meter. The expected throughput the transmitter antennas is in Fig. 9 though is about 70% of that from Fig. 8, with a 20

Fig. 9. The system throughput calculated using Table I and measured values of and .

TABLE I ALLOWED RCE, CORRESPONDING EVM, AND DATA RATE VERSUS CONSTELLATION SIZE AND CODING RATE (FROM [23], [24])

MHz bandwidth. This discrepancy is due to practical implementation constraints as reflected in the IEEE 802.11n standard, from which Table I has been taken and used to calculated curves of Fig. 9. The known constraints are, for example, the protocol overhead or the maximum constellation size of QAM 64. As a final remark, we note that a more accurate description of the system capacity, which however requires more parameters such as, e.g., frequency response of matrix is possible [25]. V. CONCLUSION We have studied performance of a 2 2 MIMO RoF system. To characterize the system capacity based on our measurements we have derived the conditional probability density functions of the two eigenvalues of the 2 2 Wishart matrix. These results allow us to interpret experimental data and evaluate the system capacity as a function of the measured MIMO condition number and error vector magnitudes of the two streams. We have found that the optimum distance between the two transmit antennas is about 1 meter. The maximum tolerable optical loss in that regime is about 6 dB. Using this approach one can optimize the system parameters such as the distance between the transmitter antennas. We have compared predictions of our model with computed values of the system throughput and found that both models agree well.

KOBYAKOV et al.: EFFECT OF OPTICAL LOSS AND ANTENNA SEPARATION IN 2

ACKNOWLEDGMENT The authors gratefully acknowledge discussions with K. Voelker and B. Reed of Agilent Technologies. REFERENCES [1] D. Tse and P. Viswanath, Fundamentals of Wireless Communications. Cambridge, UK: Cambridge Univ. Press, 2005. [2] D. Gesbert, M. Shafi, D. Shiu, P. J. Smith, and A. Naguib, “From theory to practice: An overview of MIMO space-time coded wireless systems,” IEEE J. Sel. Areas Commun., vol. 21, no. 3, pp. 281–302, Apr. 2003. [3] A. Paulraj, D. A. Gore, R. U. Nabar, and H. H. Bölcskei, “An overview of MIMO communications—A key to gigabit wireless,” Proc. IEEE, vol. 92, no. 2, pp. 198–218, Feb. 2004. [4] M. Sauer, A. Kobyakov, and J. George, “Radio over fiber for picocellular network architectures,” J. Lightw. Technol., vol. 25, no. 11, pp. 3301–3320, Nov. 2007. [5] A. Kobyakov, D. Thelen, A. Chamarti, M. Sauer, and J. Winters, “MIMO radio signals over fiber in picocells for increased WLAN coverage,” presented at the Opt. Fiber Comm. Conf. (OFC), Feb. 2008, paper JWA 113. [6] N. Kita, W. Yamada, A. Sato, D. Mori, and S. Uwano, “Measured Demmel condition number for 2 2 MIMO-OFDM broadband channels,” in Proc. IEEE Veh. Technol. Conf. (VTC), May 2004, vol. 1, pp. 294–298. [7] J. Gao, O. C. Ozdural, S. H. Ardalan, and H. Liu, “Performance modeling of MIMO OFDM systems via channel analysis,” IEEE Trans. Wireless Comm., vol. 5, pp. 2358–2362, Sep. 2006. [8] R. W. Heath and A. J. Paulraj, “Switching between diversity and multiplexing in MIMO systems,” IEEE Trans. Signal Process., vol. 53, no. 6, pp. 962–968, Jun. 2005. [9] H. Artés, D. Seethaler, and F. Hlawatsch, “Efficient detection algorithms for MIMO channels: A geometrical approach to approximate ML detection,” IEEE Trans. Signal Process., vol. 51, no. 11, pp. 2808–2820, Nov. 2003. [10] J. Maurer, G. Matz, and D. Seethaler, “Low-complexity and full-diversity MIMO detection based on condition number thresholding,” in Proc. IEEE Int. Conf. Acoust., Speech and Signal Process. (ICASSP), Apr. 2007, vol. III, pp. 2442–2446. [11] MIMO Wireless LAN PHY Layer. Operation and Measurement. Agilent application note 1509. [12] C. H. Cox, Analog Optical Links. Cambridge, UK: Cambridge Univ. Press, 2004. [13] Y. Nakaya, A. Honda, I. Ida, S. Hara, and Y. Oishi, “Measured capacity evaluation of indoor office MIMO systems using receive antenna selection,” in Proc. IEEE Veh. Technol. Conf. (VTC), 2006, vol. 6, pp. 2922–2926. [14] J. B. Andersen, “Array gain and capacity for known random channels with multiple element arrays at both ends,” IEEE J. Select. Areas Commun., vol. 18, no. 11, pp. 2172–2178, Nov. 2000. [15] A. Edelman, “Egenvalues and Condition Numbers of Random Matrices” Ph.D. dissertation, MIT, Cambridge, MA, 1989 [Online]. Available: http://www-math.mit.edu/edelman/thesis/thesis.ps [16] I. E. Telatar, “Capacity of multi-antenna Gaussian channels,” Eur. Trans. Telecomm. Related Technol., vol. 10, no. 6, pp. 585–595, Nov.–Dec. 1999. [17] M. Wennström, M. Helin, A. Rydberg, and T. Öberg, “On the optimality and performance of transmit and receive space diversity in MIMO channels,” in Proc. Inst. Elect. Eng. Seminar on MIMO: Commun. Syst. From Concept to Implementations, Dec. 12, 2001, vol. I, no. 2001/175, pp. 4/1–4/6. [18] P. G. Smith, P.-H. Kuo, and L. M. Garth, “Level crossing rates for MIMO channel eigenvalues: Implications for adaptive systems,” in Proc. IEEE Int. Conf. Commun. (ICC), May 2005, pp. 2442–2446. [19] A. T. James, “Distribution of matrix variates and latent roots derived from normal samples,” Ann. Math. Stat., vol. 35, pp. 475–501, 1964. [20] A. A. Sveshnikov, Problems in Probability Theory, Mathematical Statistics and Theory of Random Functions. New York: Dover, 1968. [21] K. M. Gharaibeh, K. G. Gard, and M. B. Steer, “Accurate estimation of digital communication system metrics—SNR, EVM, and  in a nonlinear amplifier environment,” in Proc. ARFTG Microw. Meas. Conf., Dec. 2004, pp. 41–44.

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[22] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products. Corrected and Enlarged Edition. San Diego, CA: Academic Press, 1980. [23] IEEE P802.11n/D3.00 [Online]. Available: http://ieeexplore.ieee.org/ ielD/4360106/4360107/04360108.pdf?tp=&isnumber=4360107&arnumber=4360108 [24] Comments on 802.20 Draft D1.0m [Online]. Available: http://ieee802. org/secmail/doc00653.doc [25] P. Kafle, A. B. Sesay, and J. McRory, “Capacity of MIMO OFDM systems in spatially correlated indoor fading channels,” in Proc. IEEE Veh. Technol. Conf. (VTC), 2004, vol. 1, pp. 129–133.

Andrey Kobyakov received the Masters of Science degree (with distinction) in electrical and computer engineering from the Moscow Institute of Physics and Technology, Russia, in 1992 and the Dr. rer. nat. (Ph.D.) degree (magna cum laude) in optics from Friedrich-Schiller University in Jena, Germany, in 1998. Upon completion of his degree, he worked at the Photonics Group, Friedrich Schiller University in Jena, Germany, studying nonlinear properties of optical waveguide arrays. From 1999 to 2001, he worked at the School of Optics, University of Central Florida, as a Postdoctoral Fellow where he did research in nonlinear absorption and optical limiting. In 2001, he joined the Photonics Research and Test Center, Corning Incorporated, Somerset, NJ. In 2002, he joined the Science and Technology Division, Corning Incorporated, Corning, NY. His research areas include spatial solitons, nonlinear effects in optical fibers, optical transmission systems, Raman amplifiers, photonic band-gap fibers, surface plasmons and nanophotonic applications, and photovoltaics. He has also been involved in the study of radio-over-fiber transmission and wireless networks. He has authored and coauthored over 100 technical publications including 50 journal papers. Dr. Kobyakov is a member of the Optical Society of America (OSA).

Michael Sauer (M’95) received the Dr.-Ing. (Ph.D.) degree in electrical engineering from Dresden University of Technology, Germany, in 2000. He is a Research Associate in the Science and Technology Division, Corning Incorporated, Corning, NY, where he is responsible for high-speed optical networks and communication research. His interests include fiber-wireless system design, high-speed fiber-optic transmission systems, digital signal processing techniques and modulation formats for high data rate systems, signal conditioning with fiber-based components, optical network architectures, and optical packet switching. Prior to joining Corning in 2001, he was a Research Scientist at the Communications Laboratory, Dresden University of Technology. His research areas included fiber Bragg gratings, generation and fiber-optic transmission of millimeter-wave signals, and architectures of millimeter-wave communications systems. He has authored and coauthored over 70 publications in the area of fiber-optic communication. Dr. Sauer is member of the IEEE Lasers and Electro-Optics Society (IEEE LEOS) and the IEEE Communications Society (IEEE ComSoc).

Anthony Ng’oma (M’02) received the B.Eng. degree (with Merit) in electronic engineering and telecommunications and the M.Eng. degree in electrical engineering from the University of Zambia, Lusaka, Zambia, in 1995 and 1997, respectively, and the Professional Doctorate in Engineering degree (P.D.Eng.) in information and communication technology (ICT) and the Ph.D. degree in optical communications from Eindhoven University of Technology, Eindhoven, The Netherlands, in 2002 and 2005, respectively.

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From 2005 to 2007, he was a Postdoctoral Researcher in the Electro-Optic Communications Group, Eindhoven University of Technology. In 2007, he joined Corning Incorporated, Corning, NY, where he is currently a Senior Research Scientist within the Science and Technology Division. His current research interests include fiber-optic access and local area network system architectures, hybrid fiber-wireless systems, and multi-Gbps wireless systems operating at millimeter wave frequencies. His previous research topics include network architecture design for multi-standard in-building fiber-optic systems, signal processing techniques for low-cost local-area optical networks based on multimode fibers and polymer optical fiber (POF), POF-based fiber-wireless systems for the distribution of millimeter wave signals, and the use of Artificial Intelligence for the real-time control of complex dynamic systems. He has authored and coauthored more than 40 technical publications and a book chapter in the field of fiber-optic communication. Dr. Ng’oma is a member of the IEEE Photonics Society, IEEE Communications Society, and the IEEE Microwave Theory and Techniques Society.

Jack H. Winters (F’96) received the Ph.D. degree in electrical engineering from The Ohio State University, Columbus, in 1981. He was with AT&T in the research area for over 20 years where his last position was Division Manager of the Wireless Systems Research Division at AT&T Labs-Research. At AT&T he did research on wireless and optical systems, including pioneering research on MIMO and smart antennas for wireless systems, and equalization for optical systems. Since 2002, he has been consulting on wireless and optical systems, and he is currently also Chief Scientist at Motia, Inc., and RF Advisor at Eigent Technologies, LLC, which he co-founded. He has over 40 issued patents and 60 journal publications. Dr. Winters is Area Editor for Transmission Systems for the IEEE TRANSACTIONS ON COMMUNICATIONS, a former IEEE Distinguished Lecturer, and a New Jersey Inventor of the Year for 2001.

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Experimental Evaluation of MIMO Capacity and Correlation for Narrowband Body-Centric Wireless Channels Imdad Khan and Peter S. Hall, Fellow, IEEE

Abstract—Using multiple antennas at the transmitter and receiver has shown a remarkable scope for capacity increase for indoor wireless mobile and PAN communications and is here extended to on-body channels. The capacity and multiple-input multiple-output (MIMO) channel correlation analysis has been performed at 2.45 GHz frequency using planar Inverted-F antennas (PIFA) for three on-body channels. The improvement offered by 2 2 MIMO over the conventional single-input single-output link for the on-body channels has been discussed. The variation of capacity with Rician -factor is shown and the MIMO channel spatial correlation matrices are presented. The effect of correlation on the channel capacity is discussed.

K

Index Terms—Correlation, fading channels, multiple-input multiple-output (MIMO) channel capacity, on-body channels.

I. INTRODUCTION

T

HE use of multiple antennas at the transmitter and receiver is a well known technique to increase the capacity of wireless communication systems without increasing the bandwidth. The multiple-input multiple-output (MIMO) systems have been studied extensively for the mobile and personal communication links [1]–[6]. The benefits of MIMO are limited by the correlation and power imbalance among the spatial sub-channels, mutual coupling between the spatially separated antennas, and the presence of a strong direct link between the transmitter (Tx) and receiver (Rx) in the line-of-sight (LOS) transmission [2]–[4]. Body-centric wireless communication has become popular in the recent years and a variety of application areas for the body-worn devices has emerged. Most of the work has concerned the on-body channels (both Tx and Rx are mounted on the same human body) and antenna characterization [7]–[10]. The high data rate and reliable transmission between the bodyworn wireless devices and sensors such as in military applications, sports and entertainment, and patient monitoring systems, demand the use of multiple antennas for the on-body and off-body channels. Besides the throughput gain, multiple anManuscript received November 11, 2008; revised February 10, 2009. First published June 12, 2009; current version published January 04, 2010. I. Khan is with the School of Electronics, Electrical, and Computer Engineering, University of Birmingham, Birmingham B15 2TT, U.K. and also with COMSATS Institute of Information Technology, Abbottabad, Pakistan (e-mail: [email protected]). P. S. Hall is with the School of Electronics, Electrical, and Computer Engineering, University of Birmingham, Birmingham B15 2TT, U.K. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2009.2025062

tennas can provide reasonable amount of diversity gain that makes the systems more reliable. This results in less transmit power, which is the key issue in the body-worn devices. Multiple antenna study for on-body channels has been done by using multiple antennas only at the receiver side [10]–[14]. It has been shown in [14] that the fading envelope for the on-body channels is Rician distributed. The capacity increase in Rician fading channels depends on the degree of decorrelation offered by the scattering environment, i.e., the multipath richness, and also on the signal-to-noise ratio (SNR) . The body movement may change the fading distribution of the spatially separated subchannels when multiple antennas are used [8]. For a fixed transmitted power, the presence of LOS means a high SNR at the receivers, which may increase the channel capacity compared to a non-line of sight (NLOS) link with the same configuration. But on the other hand, LOS links have a low degree of scattering, which introduces a high correlation among the spatial subchannels. Thus, for a specific value of SNR i.e., variable transmitted power, LOS can decrease the channel capacity compared to NLOS link at the same level of SNR. Hence, there is a tradeoff between the effect of increased SNR or increased correlation on the channel capacity. It has been shown in [6] that at high SNR values, the reduction in capacity due to increase in correlation is overcome by the high SNR. The performance of MIMO systems for on-body applications can either be predicted through detailed simulations or by performing realistic measurements with antennas mounted on the body. The simulations using numerical body phantoms are computation intensive and become even more so with the introduction of multiple antennas. Thus MIMO measurements in a real environment by mounting multiple transmit and receive antennas on the human body, performing realistic random movements, are required to quantify the significance of MIMO for bodyworn devices. The use of MIMO for off-body link (body to far away device) has been studied but so far no significance work has been done to signify the use of multiple antennas for the body-worn devices to the best of authors’ knowledge. It may be due to the common perception that the on-body channels exhibit a strong LOS link and hence MIMO may not be useful to increase the throughput. Some preliminary MIMO measurements for on-body channels have been reported in [2] as a small section in the context of personal area networks, but a detailed analysis is still needed. This paper concerns the use of multiple antennas at the Tx and Rx points for the on-body channels by mounting 2 transmitting and 2 receiving antennas at various positions on the body, thus forming various 2 2 on-body

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channels. The experimental characterization of 2 2 MIMO channel at frequency of 2.45 GHz, for the on-body scenario with random movements in an indoor environment, is presented. The spatial channel correlation matrices are analyzed and the effect of correlation on the capacity is discussed. The channel capacity for 2 2 MIMO channel is calculated and its cumulative distribution function (CDF) is plotted. The channel capacities for 2 2 MIMO, 1 2 multiple-input single-output (MISO), 2 1 single-input multiple-output (SIMO), and 1 1 singleinput single-output (SISO) links have been compared. The relationship between the average capacity and SNR is also shown. The variation of capacity with and without the effect of pathloss is compared with the average pathloss variation. The change of average capacity with varying -factor is also presented. Three channels, which show importance in the current application areas, were selected for measurement namely the belt-wrist, belt-chest, and belt-head channels. The measurements were performed in a highly cluttered indoor environment providing rich multipath. The subject performed random movements during the measurement. The MIMO channel model with some theoretical background is described in Section II. The description of the environment and the antennas used along with the measurement setup is presented in Section III. In Section IV, the spatial correlation matrices and the channel capacity results are discussed and analyzed. Finally some important conclusions are drawn in Section V. II. MIMO CHANNEL MODEL For a narrowband, single-user MIMO channel with transmit and receive antennas, the input-output relationship between the Tx and Rx is expressed as [1]

(1) where is the transmitted vector, is the received vector, is receive additive white Gaussian noise is the channel matrix. For a (AWGN) vector, and 2 2 MIMO channel, can be written as [1]

(2) is the complex random variable representing the where channel-fading coefficients or the complex subchannel gains from transmitting antenna to receiving antenna , as shown matrix was constructed by measuring the in Fig. 1. The . The Tx and Rx actual complex subchannel transfer gains antenna arrays mounted on the human body, as shown in Fig. 1, are surrounded by local scatterers in the form of moving body matrix, and also distant scatterers in the environment. The representing the measured channel, includes the effect of mutual coupling between the antenna elements and the correlation among the subchannels, as the subchannel gains were measured at the actual antenna ports with antenna elements placed next to each other. If the channel is completely unknown at the

transmitter, i.e., channel state information (CSI) is not available at the transmitter, the channel capacity can be expressed by (3) given below, assuming transmitted power to be uniformly distributed among the transmitting antennas [1], [5]

(3) where is identity matrix and is the average is the norsignal-to-noise ratio per receive antenna. Here malized channel matrix and represents the complex conjugate . For transpose. Equation (3) is used for the case when , the term is replaced by and identity matrix is replaced by [15], [16]. The normalization is usually done by two methods. In the first method, the matrix is normalized such that at each instance or each realization [17], [18]

(4) represents the Frobenius norm. However this norwhere malization removes any power variation of the measurement path and thus the changes in the pathloss with time are not included. This normalization is used for scenario where the transmitted power compensates for the total received power variation in order to keep the average SNR per receiver antenna fixed for each realization of the channel, irrespective of path loss. This method is useful to investigate the multipath richness of the environment [17], [18]. The second method of normalization assumes a fixed transmitted power, and hence the average SNR at the receiver for each realization of the channel changes with variation in pathloss [18]. This normalization is done such that the average Frobenius norm of the matrix (averaged over all instances) is [1], [5], [17], [18]

(5) where represents the averaging over all the instances. Thus all the instances of the matrix are normalized to a single constant value and path loss changes remain intact. The normalization is done such that average received power per subchannel is 1 [5]. The normalization techniques depicted through (4) and (5) works for a 2 2 system. However, if there are more than two antennas at one side of a MIMO system, and the antennas have high mutual coupling among them, this may not fulfil the condition of unity average received power per subchannel. The current study is constrained to a maximum of 2 2 MIMO case. The capacity calculated by (3) is a random quantity and can be represented by plotting its CDF. The outage capacity can be calculated from these CDF curves at a certain probability. In an ideal channel model, the subchannels of the MIMO channel are assumed to be independent and identically distributed (iid) with Rayleigh distribution of the envelopes and hence perfectly uncorrelated. But in practical systems, and especially in LOS scenario, the subchannels are correlated. The complex signal corre-

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Fig. 1. Placement of the antennas on the body and the MIMO channel. The Rx antenna array was placed at the three positions separately for the three on-body channels. Tx antenna remained at the waist position.

Fig. 3. Measured radiation pattern as a function of ' for one of the PIFA element with the second element terminated by 50 ohms.

Fig. 2. Top view and side view of PIFA array.

lation coefficients ( ) among the subchannels can be calculated using the following expression [19]:

(6)

is the total no. of samples of received envelope. where and represent the zero-meaned complex voltage signals. The complex correlation coefficients are useful in system modeling and characterizing the channel. The spatial correlation matrix of a MIMO system gives a comprehensive view of the degree of correlation among the subchannels. The spatial correlation matrix, , for the 2 2 MIMO channel can be constructed as [20]

(7)

is the correlation coefficient between subchannel where and . The symmetry reduces the total number of significant coefficients to 6. III. MEASUREMENT PROCEDURE To characterize 2 2 narrowband MIMO channels for the on-body wireless links, measurements were performed

at frequency of 2.45 GHz ISM band using an array of two microstip-fed planar inverted-F antennas (PIFA) on 0.8 mm thick FR4 substrate, as shown in Fig. 2, at both transmitter and receiver locations. The ground plane size was the same as the substrate size, which was 45 mm 40 mm. The thickness of the radiating plate of PIFA was 1 mm and the distance between the short-circuit pin and feeding pin was 3 mm. Other dimensions of the antenna are shown in Fig. 2. The reflection coefficients . The mutual of the PIFA antenna elements were below . The coupling between the two PIFA elements was radiation efficiency of the antennas was only slightly degraded due to mutual coupling. Very little detuning was observed when the antennas were mounted on the body and the reflection coefficient of all antenna ports at the desired frequency was . The measured radiation pattern of one still below PIFA element, with the other element in the neighborhood and terminated by 50 ohms, is shown in Fig. 3 for the xy-plane (assuming the local coordinate system for antennas), which is the plane of interest for the on-body channels. Three on-body channels were selected for the measurement. For each on-body channel, the transmitting array was placed at the waist (belt) position on the left side of the body about 100 mm away from the body centre line, and the receiving array was placed, alternately, at the right side of the head, right side of the chest, and right wrist positions, thus forming three on-body channels named belt-head, belt-chest, and belt-wrist, as shown in Fig. 1. The transmitter and receiver arrays were oriented such that vector in Fig. 2 was pointing downwards for transmitting array and upwards for receiving array for all the three channels measured, assuming the subject standing straight. The distance between the body and the antennas mounted on the body was kept to about 7–10 mm including the clothing. The coaxial cables used during the measurement were firmly strapped to the body to minimize the effect of moving cables over the channel measurement.

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Measurements were performed in an indoor environment, which was a 7.5 m 9 m sized laboratory containing equipment, tables, chairs, and computers thus providing a rich multipath propagation environment. The two transmitting antennas were connected through an RF switch to a signal generator operating at 2.45 GHz. The switching time of the RF switch was 40 , which was much less than the coherence time of the channel [12]. The two receiving antennas were connected to the two ports of a vector network analyzer (VNA) calibrated in tuned receive mode with a single frequency sweep at 2.45 GHz. The calibration was done by connecting the signal generator to each port of the VNA through the cables used, to normalize the total power delivered to the transmitting antenna port to be 0 dBm. The signal generator and the VNA were synchronized by using the 10 MHz reference output signal from the signal generator. The noise floor for the measurement was at . A total of 1600 points were collected for one sweep of 12 s duration whilst doing a sequence of pseudo-random activities. Each collected sample contained amplitude and . Thus each receiving antenna phase of the channel gain, was collecting 1600 samples with alternate samples from each transmitter with a sampling time of 15 ms, giving 800 instances of one of the four spatial subchannels. Hence, 800 instances of the 2 2 MIMO channel matrix with four subchannels were constructed. A total of 6 such sweeps were carried out, with different random movements of the body for each sweep, giving a total of 4800 instances of the channel matrix. The measurements were done with different set of movements for the three channels. The activity sets for the three channels are given in Table I. Apart from the 2 2 MIMO measurements, representing 2 1, 1 2, and 1 1 measurements, with Rx and Tx antennas, were done separately and the channel capacity results are compared. IV. RESULTS A. Spatial Correlation Matrices The presence of a strong ray introduces strong correlation among the subchannels [3], [4]. The belt-chest channel is a good example of this, for which the direct ray is much stronger than the multipath components and hence the subchannels

TABLE I MOVEMENTS DONE FOR EACH CHANNEL.

are highly correlated. A high correlation among the MIMO subchannels reduces the throughput gain and thus less improvement in the channel capacity is observed [4], [20]. The complex signal correlation coefficients among the subchannels of the 2 2 on-body MIMO channels were calculated using (6), and the spatial correlation matrices, as described in (7), were constructed for each on-body channel. The spatial correlation matrices are given below for each channel. See the equation at the bottom of the page. Referring to (7), the correlation between the two transmitat receiving antenna 1, and is at receiving ting signals is antenna 2. It is clear from the matrices given above that the correlation between the two transmitting signals is high for all the cases. This may be due to the absence of local scatterers in the near vicinity of the transmitting antennas, as the transmitting array was mounted on the waist position and there was less movement of the hands and other body parts near it to cause any significant local scattering. Similarly, the correlation between the received signals at the , assuming the signal transmitted two receiving antennas is with signal transmitted from transmitting antenna 1, and is from transmitting antenna 2. These values are low for all the cases except the belt-chest channel, which is more static channel compared to the other two. Also, the LOS component is much stronger and is only occasionally shadowed by the movement of hands. The positioning of the receiving antennas at the body for other two channels is such that the antennas are surrounded by large number of local scatterers in the form of moving body

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Fig. 4. Capacity CDFs for the belt-wrist channel. Fig. 5. Capacity CDFs for the belt-head channel.

parts. These channels also involve rapid movement of the receiving antennas in the environment as well as with respect to the Tx, thus high degree of scattering results in lower correlation. This low correlation shows that the body movement and the relative movement of Tx and Rx antennas decorrelate the subchannels significantly. The same sort of result is reported in [12], [13], where the correlation between the received diversity branch signals is quite low resulting in high diversity gains. and the correlation coefficients between The coefficients the subchannels that do not share any antenna. It is clear from the matrices that the correlation among these subchannels is very low apart from belt-chest channel, in which case it is moderate. In general, correlation among the subchannels for the three on-body channels is not significantly high apart from the belt-chest channel. B. Channel Capacity MIMO channel capacities were calculated using (3) for the three on-body channels. The second technique, using (5), was used to normalize the matrix for each channel, as the transmitted power for these channels was fixed and movement of the body introduced body shadowing which had to be kept intact while calculating the capacity. This method of normalization keeps the pathloss information while calculating the capacity. The CDF plots of the channel capacity at various values of average receive SNR, , are shown in Figs. 4–6 for the belt-wrist, belt-head, and belt-chest channels, respectively, by thin lines. The figures also show the CDFs of channel capacity for other , configurations, i.e., MISO, SIMO, and SISO at which are represented by thick lines. It can be observed that the 2 2 MIMO outage capacities, at the same probability level, are almost similar for the three channels at the lower SNR values. At the higher SNR values, there is slight difference but this difference is not significant for belt-head and belt-wrist channels, whereas, the capacity for belt-chest channel is comparatively lower. This is due to the higher correlation among the subchannels of the belt-chest link. At low SNR level, the direct ray is not strong enough to produce significant difference in the correlation, and even becomes weaker while propagating along the surface of the body in the form of creeping waves [7, (p. 47)]. Thus, the three channels behave almost similar because the multipath signals are dominant. In the high SNR regime, the direct ray is much stronger and the channels with more dominant direct ray, like the belt-chest are significantly affected by the correlation among the subchannels. It can also be noted that the

Fig. 6. Capacity CDFs for the belt-chest channel.

slope of the curves for the belt-chest channels are steeper than that of the other two channels, showing less spread in the capacity. This can be explained by less variation of the pathloss for the belt-chest channel, as the antennas are fixed with respect to each other and the path length changes only in few postures. Despite the strong correlation among the sub-channels and the presence of a strong direct link due to LOS for the belt-chest channel, the improvement in capacity offered by MIMO over the same channel with SISO, MISO, and SIMO links, is noticeable. The capacity improvement may be due to a number of factors. The most dominant is the multipath richness of the environment. Although there is strong LOS, the direct ray due to the creeping wave is attenuated while propagating on the surface of the body. This fact and the presence of rich scattering environment mean that the Rician -factor is not as high as expected [13], [14]. The other reason may be due to the fact that the communication is short range and the assumption of planar wave-front may not be valid. This means that the spherical wave-front is being exploited to achieve high capacities, as explained in [2]. Thirdly, it has been shown in [6] that at higher SNR values, there is less decrease in capacity due to high correlation. By comparing the capacity of the 2 2 MIMO channel with corresponding SISO channel capacity, the throughput gain can be calculated. This gain reflects the amount of improvement ofMIMO channel, fered by MIMO over SISO. For an ideal, this gain is approximately i.e., SISO capacity is increased by times for a fixed SNR level [1]. It is clear from the figures that the throughput gain for all the on-body channels is less than 2 due to some degree of correlation between the subchannels, with lowest values for the belt-chest channel. The MISO system offers effectively no improvement in the capacity over the SISO

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Fig. 8. Capacity CDFs with Path loss normalized at  channels.

Fig. 7. Average capacity vs. SNR.

system. This may be justified by the very high correlation between the transmitting signals, as discussed above. Also, it has been shown in [21] that MISO systems are not as good as SIMO and MIMO systems in throughput gain. SIMO offers an improvement due to low correlation between the signals at the two receiving antennas. Besides the outage capacity, the average capacity for all the obtained random movements was calculated by averaging from (3) for each on-body channel. The variation of average capacity with SNR (in dB) is shown in Fig. 7 for the three channels. It is clear from the figure that for all the three channels, the average capacity increase with each 3 dB increase in SNR is less than 2 bps/Hz, compared to [1] where it has been shown that in the high SNR regime, the increase in capacity with every MIMO system 3 dB increase in SNR for independent bps/Hz. But in general, the values are not too is far away from 2 bps/Hz for belt-head and belt-wrist channels, where the correlation is comparatively low. It is also important to find out the capacity without the pathloss effect, which is only due to the multipath richness. To see the difference between the two normalization techniques (with and without the path loss) and see the multipath richness of the environment, some results are presented with the first type of normalization i.e., normalizing matrix using (4). An SNR level of 15 dB was selected. The CDF of the capacities without pathloss at SNR level of 15 dB are shown in Fig. 8. Comparison with Figs. 4– 6 shows that the capacities with and without the pathloss are almost similar at high outage. At low outage, the capacity without pathloss effect is slightly higher than the capacity calculated with pathloss included. It was also noted that the average capacity is slightly higher with this second normalization but the difference was very small. As the difference in capacity is not high, it can be concluded that the multipath richness of the environment is a significant factor in the capacity improvement. Fig. 9 shows the capacity with the two types of normalization, i.e., with and without the pathloss effect, and its comparison , for a with the average pathloss for equivalent SISO link small portion of the data for one of the on-body channels as an example. The other two channels showed the same behavior. was calculated as defined in [3], [4]

= 15 dB for the three

The capacity with pathloss included [see Fig. 9(b)] has a downward trend with increase in the average pathloss [Fig. 9(a)], which shows that the capacity relies heavily on the SNR at the receiving antennas in this case, provided that the multipath environment is not changing significantly [3]. At any particular instant, an increase in pathloss means lower received power or decrease in the receiver SNR, and hence the capacity decreases at that instance or realization. The capacity with pathloss normalized [Fig. 9(c)] shows the opposite trend. It increases with increase in pathloss or decrease in SNR at any particular realization, and vice versa, with less spread and faster variation. The SNR would decrease with obstruction of the direct ray or shadowing due to the moving body parts, which would increase the scattering and hence the multipath richness, resulting in an increase in the capacity at that instant. Similar comparison of capacity variations with the two normalizations are shown for a mobile indoor scenario in [17]. Lastly, the effect of Rician -factor on the average capacity was studied by plotting the average capacity against the Rician -factor for several portions of the measured channel data. The -factor was calculated from the measured data using the moment method presented in [22]. The -factor was calculated for each subchannel and then averaged for the four subchannels to obtain an average -factor for the 2 2 MIMO channel. As explained above, the measurement involved various movements which contained postures with perfect LOS, obstructed LOS, and varying path lengths. Thus the -factor for the different portions of the data set was varying. The estimated -factor and the corresponding calculated average capacity are plotted in Fig. 10 as a scatter plot for one channel. The solid line shows the linear fit to the data points. It is clear that the capacity has a downward trend with increasing -factor, as reported in [3], [23]. At a fixed SNR level, higher -factor means more spatial correlation and hence a decrease in capacity. The other two channels showed the same trend, but with less steep linear fit lines. The slope was less for the other two channels due to lower values of the -factor. The LOS link was weaker for those channels resulting in more dominant multipath components and this may lead to less dependence of the average capacity on the direct ray strength. V. CONCLUSION

(8)

The significance of using narrowband MIMO for body-centric wireless communication channels has been shown through

KHAN AND HALL: EXPERIMENTAL EVALUATION OF MIMO CAPACITY AND CORRELATION

201

Fig. 9. Variation of capacity with the two normalizations of the channel matrix and the average path loss (P ) for the belt-head channel at 

= 15 dB.

REFERENCES

Fig. 10. Variation of capacity with Rician K-factor for the belt-chest channel at  = 15 dB.

measurements in an indoor environment for a 2 2 MIMO link on three on-body channels with practical PIFA antennas mountable on the body at frequency of 2.45 GHz. The MIMO channels were characterized by the correlation among the subchannels and the spatial correlation matrices were derived for the three on-body channels. The analysis of the spatial correlation matrices shows that the movement of the body and the antenna in the environment produces sufficient decorrelation among the subchannels despite the presence of LOS link. The transmitting signals are highly correlated due to less local scatterers around the transmitting antennas, whereas, the received signals at the two receiving antennas have very low correlation, as the fading for the two signals is more or less independent due to the presence of scatterers in the close proximity of the receiving antennas. The channel capacity improvement with MIMO over the SISO is considerably high and capacity gains close to a maximum gain of 2 were observed for some channels. The belt-chest channel does not provide high capacity gains due to high correlation among the subchannels. As the surface wave propagating along the creeping of the body is attenuated rapidly, the multipath components play a significant role in the capacity increase. The MISO system does not provide any improvement in channel capacity due to high correlation among the transmitted signals by the virtue of the location of transmitting array. A SIMO system provides some improvement in channel capacity but less than MIMO. The average capacity shows a downward trend with increasing value of Rician -factor and different slopes of the linear fit lines for the three channels.

[1] E. Biglier, R. Calderbank, A. Constantnides, A. Goldsmith, A. Paulraj, and H. V. Poor, MIMO Wireless Communications. New York: Cambridge Univ. Press, 2007. [2] D. Neirynck, C. Williams, A. Nix, and M. Beach, “Exploiting multipleinput multiple-output in the personal sphere,” IET Microw. Antennas Propag., vol. 1, no. 6, pp. 1170–1176, Dec. 2007. [3] H. Ozcelik, M. Herdin, R. Prestros, and E. Bonek, “How MIMO capacity is linked with single element fading statistics,” in Proc. Int. Conf. on Electromagnetics in Advanced Applications, Torino, Italy, Sep. 8–12, 2003, pp. 775–778. [4] L. Garcia, N. Jalden, B. Lindmark, P. Zetterberg, and L. Haro, “Measurements of MIMO indoor channels at 1800 MHz with multiple indoor and outdoor base stations,” EURASIP J. Wireless Commun. Network., vol. 2007, no. Article ID 28073. [5] G. J. Foschini and M. J. Gans, “On limits of wireless communications in fading environment when using multiple antennas,” Wireless Personal Commun., vol. 6, pp. 311–335, Mar. 1998. [6] K. Sakaguchi, H. Y. Chua, and K. Araki, “MIMO channel capacity in an indoor line-of-sight environment,” IEICE Trans. Commun., vol. E88-B, no. 7, pp. 3010–3019, Jul. 2005. [7] P. S. Hall and Y. Hao, Antennas and Propagation for Body-Centric Wireless Communications. London, U.K.: Artech House, 2006. [8] Y. I. Nechayev and P. S. Hall, “Multipath fading of on-body propagation channels,” in Proc. IEEE Int. AP-S Symp.USNC/URSI National Radio Science Meeting, San Diego, CA, 2008, pp. 1–4. [9] K. Fujii, M. Takahashi, and K. Ito, “Electric field distributions of wearable devices using the human body as a transmission channel,” IEEE Trans. Antennas Propag., vol. 55, pp. 2080–2087, Jul. 2007. [10] A. A. Serra, P. Nepa, G. Manara, and P. S. Hall, “Diversity measurements for on-body communication systems,” IEEE Antennas Wireless Propag. Lett., vol. 6, no. 1, pp. 361–363. [11] A. A. Serra, A. Guraliuc, P. Nepa, G. Manara, and I. Khan, “Diversity gain measurements for body-centric communication systems,” Int. J. Microw. Opt. Technol., vol. 3, no. 3, pp. 283–289, Jul. 2008. [12] I. Khan and P. S. Hall, “Multiple antenna reception at 5.8 and 10 GHz for body-centric wireless communication channels,” IEEE Trans. Antennas Propag., vol. 57, pp. 248–255, Jan. 2009. [13] I. Khan, P. S. Hall, A. A. Serra, A. R. Guraliuc, and P. Nepa, “Diversity performance analysis for on-body communication channels at 2.45 GHz,” IEEE Trans. Antennas Propag., vol. 57, pp. 956–963, Apr. 2009. [14] I. Khan, Y. I. Nechayev, and P. S. Hall, “On-body diversity channel characterization,” IIEEE Trans. Antennas Propag., to be published. [15] I. Hen, “MIMO architecture for wireless communication,” Intel Technol. J., vol. 10, no. 2, pp. 157–165, May 2006. [16] B. Vucetic and J. Yuan, Space Time Coding. New York: Wiley, 2003, pp. 7–9. [17] T. Svantesson and J. Wallace, “On signal strength and multipath richness in multi-input multi-output systems,” in Proc. IEEE Int. Conf. on Commun., May 2003, vol. 4, pp. 2683–2687. [18] H. Carrasco, R. Feick, and H. Hristov, “Experimental evaluation of indoor MIMO channel capacity for compact arrays of planar inverted-F antennas,” Microw. Opt. Technol. Lett., vol. 49, no. 7, pp. 1754–1756, Jul. 2007. [19] J. S. Colburn, Y. Rahmat-Samii, M. A. Jensen, and G. J. Pottie, “Evaluation of personal communications dual-antenna handset diversity performance,” IEEE Trans. Veh. Technol., vol. 47, pp. 737–746, Aug. 1998.

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[20] R. E. Jaramillo, O. Fernandez, and R. P. Torres, “Empirical analysis of 2 2 MIMO channel in outdoor-indoor scenarios for BFWA applications,” IEEE Antennas Propag. Mag., vol. 48, no. 6, pp. 57–69, Dec. 2006. [21] J. Gong, J. F. Hayes, and M. R. Soleymani, “Comparison of capacities of the transmit antenna diversity with the receive antenna diversity in the MIMO scheme,” in Proc. IEEE CCECE, May 4–7, 2003, vol. 1, pp. 179–182. [22] L. J. Greenstein, D. G. Michelson, and V. Erceg, “Moment-method estimation of Rician -factor,” IEEE Commun. Lett., vol. 3, no. 6, pp. 175–176, Jun. 1999. [23] Z. Tang and A. S. Mohan, “Experimental investigation of indoor MIMO Ricean channel capacity,” IEEE Antennas Wireless Propag. Lett., vol. 4, pp. 55–58, 2005.

2

K

Imdad Khan received the B.Sc. and M.S. degrees in electrical engineering from NWFP University of Engineering and Technology, Peshawar, Pakistan, in 2000 and 2003, respectively, and the Ph.D. degree from the University of Birmingham, Birmingham, U.K., in September 2009. He was with NWFP University of Engineering and Technology, from 2000 until 2001. He has been with COMSATS Institute of Information Technology, Abbottabad, Pakistan, since 2001. His major field of research is diversity and MIMO for body-centric wireless communication channels. Dr. Khan’s Ph.D. studies were funded by COMSATS Institute of Information Technology.

Peter S. Hall (F’01) received the Ph.D. degree in antenna measurements from Sheffield University, Sheffield, U.K. After graduating, he spent three years with Marconi Space and Defence Systems, Stanmore, working largely on a European Communications satellite project. He then joined The Royal Military College of Science as a Senior Research Scientist, progressing to Reader in Electromagnetics. He joined The University of Birmingham, Birmingham, U.K, in 1994, where he is currently a Professor of communications engineering, Leader of the Antennas and Applied Electromagnetics Laboratory, and Head of the Devices and Systems Research Centre, Department of Electronic, Electrical and Computer Engineering. He has researched extensively in the areas of microwave antennas and associated components and antenna measurements. He has published five books, over 250 learned papers and taken various patents. Prof. Hall is a Fellow the IEEE and the Institution of Engineering and Technology (IET), London, U.K. (formerly, IEE) and a past IEEE Distinguished Lecturer. His publications have earned six IEE premium awards, including the 1990 IEE Rayleigh Book Award for the Handbook of Microstrip Antennas. He is a past Chairman of the IEE Antennas and Propagation Professional Group and past Coordinator for Premium Awards for IEE Proceedings on Microwave, Antennas and Propagation and is currently a member of the Executive Group of the IET Professional Network in Antennas and Propagation. He is a member of the IEEE AP-S Fellow Evaluation Committee. He chaired the Organizing Committee of the 1997 IEE International Conference on Antennas and Propagation and has been associated with the organization of many other international conferences. He was Honorary Editor of IEE Proceedings Part H from 1991 to 1995 and is currently on the editorial board of Microwave and Optical Tech Letters. He is a member of the Executive Board of the EC Antenna Network of Excellence.

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203

Communications Application of Characteristic Modes and Non-Foster Multiport Loading to the Design of Broadband Antennas Khaled A. Obeidat, Bryan D. Raines, and Roberto G. Rojas

Abstract—A conceptual framework is proposed to systematically design antennas with broadband impedance and pattern characteristics using multiple reactive loads. Antennas of arbitrary geometry can have their bandwidths expanded using this technique. The technique is applied to a narrow band thin wire dipole antenna to demonstrate its main features. It is shown that the loaded antenna resonates a desired current over a wide frequency band. The loads are shown to require non-Foster elements when realized. Simulations demonstrate the broadband characteristics of both the dipole input impedance and radiation pattern. Index Terms—Antennas, broadband antennas, characteristic modes, electrically small antennas (ESA), non-Foster circuits, non-Foster loading.

extend the operational bandwidth of antennas, their combination is believed to be novel. As a point of comparison, a related work [4], [5] was performed in the past by loading a dipole with negative inductors instead of negative resonators. It differs from the proposed technique in that arbitrary control over the current distribution shape was not achieved in [4], [5] because such loading can only have one unique mode of operation. Unlike the proposed scheme, this previous work is only applicable to a small subset of antenna current distributions and geometries. Therefore, although this previous work established wideband operation of a loaded dipole antenna for both input impedance and pattern metrics, it did not actually provide a general scheme for the control of an arbitrary current distribution (and therefore, pattern) on an arbitrary antenna geometry. Furthermore, unlike the proposed technique, the analytical framework of the past work depended on transmission line theory, which is not generally applicable when applied to electrically small and arbitrarily-shaped antennas.

I. INTRODUCTION Wideband antennas have been the subject of extensive research, especially over the past ten years. The goal has been to obtain an antenna design with relatively constant pattern and impedance over some desired broad frequency range. Furthermore, market pressures for miniaturizing communication devices have encouraged the use of electrically small antennas and highly integrated RF circuitry. Electrically small antennas, however, have been shown to be fundamentally limited in bandwidth by the Chu limit [1]. Traditionally, matching networks have been used at the feed points of such antennas to improve their impedance bandwidth; however, matching networks do not directly address the problem of a potentially changing radiation pattern at higher frequencies, where the antenna features become electrically large. On the other hand, the theories of frequency-independent antennas [2] and ultra-wideband antennas [3], can yield wideband antennas, but only directly apply to electrically large antennas. In general, the best realizable electrically small antenna should only require a minimally complex matching network and should maintain a particular radiation pattern over a wide frequency range. Both of these aspects are essentially dependent on the antenna current. Wideband behavior may therefore be obtained by carefully shaping the antenna current distribution over frequency through changes to the antenna geometry, loading, or material composition. This work proposes a general design methodology which effectively controls the antenna current distribution over frequency using discrete loading. As a specific example, the proposed method will be applied to a linear wire dipole loaded with various discrete loads. It will be shown that for improved wideband control in the proposed method, non-Foster loads are required. While discrete non-Foster loading [4]–[6] and characteristic mode theory [7], [8] have been separately used in the past to Manuscript received January 21, 2008; revised January 07, 2009. First published November 10, 2009; current version published January 04, 2010. The authors are with the ElectroScience Laboratory, Department of Electrical and Computer Engineering, Columbus, OH 43212 USA (e-mail: obeidatk@ece. osu.edu; [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.2009.2036281

II. DESIGN METHODOLOGY If the current distribution over the body of an antenna can be controlled over a wide frequency bandwidth, the antenna input impedance and the radiation pattern can also be controlled. In this work, we will explore the control of the current distribution over some bandwidth through careful placement of multiple lumped, discrete loads. The theory of network characteristic modes (NCM) [9] is utilized in this work to design these loads which, together with the antenna, resonate a desired current distribution over the antenna body. There are three important decisions related to the design of a wideband antenna through current distribution control using multiple discrete loads. First, a desirable current distribution over the entire frequency band must be identified according to some metric. Second, the number and location of the loads should be determined. Third, the loads must be computed and subsequently realized over the desired frequency band. The design process will be aided through insights derived from (network) characteristic mode theory. The process is summarized graphically in Fig. 1. Identifying the required current is critical to obtain a current with small magnitude variation versus frequency variation at the antenna feed (low impedance Q factor) which should facilitate the design of a wideband passive matching network, and to produce the desired radiation pattern over the desired frequency band. Before discussing load computation, a brief background on the theory of network characteristic modes is provided. Although the theory of characteristic modes has historically been applied to the method of moments (MoM) impedance matrix, it is also possible to apply it to N-port Z networks, as in [9]. This choice allows any suitable computational electromagnetic code or experimental method to determine the N-port network matrix. The theory of network characteristic modes (NCM) for an antenna with N-port can be represented by an N-port Z matrix. The N eigenmodes are computed using the following generalized eigenvalue problem at some radial frequency ! [9]

0018-926X/$26.00 © 2009 IEEE

[

Xa (!)] In (!) = n (!) [Ra (!)] In (!)

(1)

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Fig. 2. Loaded center-fed dipole antenna with multiple load circuits. Port 1 is on the extreme right, and the port numbers increase going to the left.

Foster (elements with negative frequency derivatives) in nature. For the example provided, it will be shown that the computed loads are non-Foster in nature over a wide frequency band. Non-Foster elements may be approximately realized using negative impedance converters [10]–[12]. III. EXAMPLE

Fig. 1. Design methodology flowchart.

where [Ra (!)] and [Xa (! )] are, respectively, the real and the imaginary parts of the N-port [Za (! )] open circuit impedance matrix of the antenna, In is the nth eigencurrent, and n is the corresponding nth eigenmode. The total current is therefore a weighted summation of all these modes [9] I

!

(

) =

0

V

N

I

! 3 Voc ! In ! jn !

 ( n

n=1

)

1+



(

(

)

)

 (

(2)

)

where  oc is the N-port open circuit voltage column vector of the N-port network characterized by [Za (! )], and (1)3 indicates a Hermitian transpose. It is assumed that the modes are normalized such that 3 (! ) [R (! )]  (! ) is equal to 1. a n n Equation (2) has two important components to each modal current expansion coefficient. The denominator essentially depends on the frequency-dependent eigenvalue n , which is a function solely of the antenna geometry and materials. The numerator is determined by an inner product between a frequency-dependent eigencurrent n (! ) and the applied excitation  oc (! ). The required lossless load reactance at the ith port is given by [9]

I

I

I

V

0 Id ! Xa ! Id ! i (3) where the subscript i denotes the ith port and Id is the desired equiphase XL ! (

) =

1

(

)

[

I

(

)]  (

)



current distribution. An eigencurrent n is in resonance when its corresponding eigenvalue n equals zero. The loads described by the diagonal matrix [XL (! )] are used to enforce the quantity [Xa + XL ] d to be zero. Consequently, the N-port current d is made the dominant eigencurrent at the desired frequency points. After the frequency variation of each reactive load is computed over the frequency range of interest, each load is approximated using a finite number of lumped reactive circuit elements. For the purposes of this communication, we shall allow the elements to be both Foster and non-

I

I

The proposed design technique is applicable to any antenna; however, for simplicity, a thin wire copper dipole antenna of length 1.2 m and 1 mm radius is considered here. This antenna is known to be a narrowband resonant antenna. However, using the loading scheme proposed here, it can become a wideband antenna. The goal is to operate down to at least 50 MHz (ka = 0:628, where a = 0:6 m is the radius of the smallest circumscribing sphere about the dipole) and up to as high of a frequency as possible, with a minimum operational bandwidth of 4:1. The radiation pattern must substantially resemble a TM01 mode. The method of moments (MoM) code ESP5.4 [13] was used to extract the network Z parameters of the multiport dipole antenna, as well as input impedance and radiation pattern. ESP5 automatically generates a frequency-dependent wire segmentation in order to yield reliable results over the desired frequency band. A. Identification of Desired Dipole Current Distribution Since the goal is to obtain a wideband radiation pattern that resembles the TM01 spherical mode, the desired current distribution is the first eigenmode of the unloaded dipole [14]. Furthermore, if the first eigenmode is selected to resonate over the entire desired frequency band, it produces the desired pattern shape over that band. Thus, the desired current distribution is selected to be the first eigenmode at the lowest operational frequency, i.e., 50 MHz. The actual computation of the mode using (1) will necessarily have to wait until after the ports are selected. B. Dipole Port Placement Since this eigenmode will be computed using network characteristic mode theory, we must first define the number and locations of the load points. Five ports were selected, with four ports symmetrically distributed along the dipole and one port at the middle of the antenna, which shall also serve as the feed point. The locations of the load ports (see Fig. 2) were chosen such that the majority of ports are in regions where the current distribution is high in magnitude—that is, close to the dipole center—and the remaining 2 ports are closer to the dipole ends. Before the loads are determined, it is instructive to calculate the eigenvalue spectrum of the 5-port unloaded dipole. The spectrum is depicted in Fig. 3 and, as expected, the dominant eigenvalue (1 ) resonates at f = 120 MHz. The desired current distribution is computed to be d = T T indicates transpose. [ 0:68 0:98 1:0 0:98 0:68 ] ; where [1]

I

C. Dipole Load Computation Given the desired current distribution and the set of 5-port Z parameters over the frequency band of interest, it is now straightforward to compute the reactance of each load versus frequency using (3). The results for this example have been computed for three cases: perfect loading, approximate loading, and no loading.

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Fig. 3. Eigenvalue spectrum of 5-port unloaded 1.2 m dipole.

Fig. 4. Required series reactances (as specified by end of the dipole) and 3 (at the dipole center).

[

X

]

205

X

Fig. 5. Input reactance at the feed port of the dipole antenna for the perfect loading, approximate loading, and unloaded cases.

) at ports 1 (near the Fig. 6. Values (in dB) of the first three dominant eigenvalues of the 5-port corresponds dipole antenna matrix for the perfect loading case. Note that, to an odd mode which will not contribute to the total current of a center-fed dipole.



TABLE I LOAD ELEMENT VALUES OF THE LOADED 5 PORT DIPOLE ANTENNA

In the perfect loading case, the exact reactances computed in (3) are used, implying an infinitely complex reactive network is used at each load port to “realize” the reactance frequency behavior at that port. Two of the exact reactance curves are plotted versus frequency in Fig. 4. In the approximate loading case, the exact reactance computed from (3) at each port is approximated by a finite number of lumped reactive elements. After examining the frequency behavior of the exact computed reactances, it was found that the reactance at each port may be satisfactorily approximated by a series LC circuit where both elements (L and C) have negative values (i.e., non-Foster elements), as shown in Table I. Lastly, the unloaded case describes the dipole antenna without loads. D. Impedance Analysis

X

The value of the input reactance at the feed port ( in ) with and without loading is shown in Fig. 5. It is apparent from Fig. 5 that the magnitude of the reactance of the loaded dipole is smaller than the unloaded dipole, as well as more slowly varying in frequency. This be-

havior implies that it is easier to match the loaded dipole as compared to the unloaded dipole. The proposed technique ensures a nearly resistive feed point input impedance for frequencies where higher order eigenmodes are very weakly excited (j i j  j d j, 6= ), as shown in Fig. 6. However, as the frequency becomes higher, the electrical size of the antenna increases, which tends to excite higher order modes alongside the desired mode (where d = 0). Consequently, for a fixed number and position of the ports, the total current I, as determined by (2), will not satisfy [ a + L ] I = 0 at high frequencies, even in the perfect loading case. This implies a larger reactance at the feed port at high frequencies ( 250 MHz). As expected, the perfect loaded case generally yields the best possible frequency bandwidth performance. For the approximate loading case, the reactance at the feed port is small, but non-zero, for most frequencies between 50 MHz and 200 MHz. In this example, ideal loads have a high slope (e.g., see load 1 in Fig. 4) below 50 MHz and it is difficult to accurately match this behavior with the approximate loads. Consequently, the input reactance for the approximate loading case is higher (below 50 MHz) than the perfect loading case (see Fig. 5). To further increase the bandwidth of the antenna, additional ports are needed to better control the current. The calculated Q using (96) in [15] is shown in Fig. 7. It is evident that the loaded antenna Q is significantly reduced compared to the un-





X >

X

 i d

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Fig. 7. The Q factor of the dipole antenna for the perfect loading, approximate loading, and unloaded cases. Fig. 9. The gain in dB at 400 MHz of the approximately loaded dipole antenna compared to the unloaded antenna (excluding input impedance mismatch losses).

Fig. 8. The return loss S of the unloaded dipole verses approximately loaded dipole antenna with and without the passive lumped element matching circuit (seventh order), all referenced to 50 .

loaded case up to 400 MHz, implying that the loaded antenna bandwidth has a greater potential for expansion. To further clarify the improvement in antenna Q, especially when the dipole is electrically small, it is useful to compare these results against the Chu limit. At 20 MHz, the circumscribing sphere electrical size is ka = 0:25. The Chu limit is approximately 67, the unloaded case has a Q of 1899, the approximate loading case has a Q of about 7, and the perfect loading case has a Q of only 1. The loaded antenna return loss referenced to 50 is shown in Fig. 8. The improvementinthe inputimpedancebandwidth using multipleloads is clear. For an S11 level of 07 dB, the unloaded antenna impedance bandwidth is 1.11:1 (114–126 MHz), while the approximate loading case yields a bandwidth of 2.59:1 (61–158 MHz). The antenna bandwidth may of course be enhanced further by using a passive matching circuit placed at the feed port (see insert in Fig. 8). The improved input impedance bandwidth of 7.64:1 (47–359 MHz) is shown in Fig. 8 when a lossless seventh-order passive lumped element ladder matching circuit (designed using the Real Frequency Technique [16] from 10–350 MHz) is used at the feed port of the dipole with approximate loading. Note that ka = 0:59 at 47 MHz for the 1.2 m long dipole. E. Radiation Pattern Analysis When discussing bandwidth, it is usually insufficient to solely consider antenna input impedance bandwidth, since the radiation pattern

Fig. 10. Approximately loaded dipole realized gain ( = 90 ) with matching network (MN) compared to the unloaded dipole realized gain without MN. The realized gain includes impedance mismatch losses.

over the desired frequency band is also important. Fig. 9 illustrates the gain of the unloaded and approximately loaded antenna at 400 MHz, respectively. As far as the pattern is concerned, loading the antenna can extend the desired pattern shape up to 400 MHz by suppressing higher-order modes. Between 400 and 500 MHz, the antenna pattern degrades into the third dipole mode pattern. F. Realized Gain Analysis Fig. 10 summarizes the improved bandwidth of the approximate loading case compared to the unloaded case by comparing their overall realized gains (includes impedance mismatch losses). The unloaded case omits a passive matching network (seventh-order network designed using the RFT from 10–350 MHz) since it did not significantly improve the impedance bandwidth in that frequency range. This clearly demonstrates that the loading scheme leads to a more easily matched input impedance. The approximate loading case is clearly better than the unloaded case. The unloaded dipole manifests two different even dipole modes below 400 MHz (see Fig. 3), while the loaded antenna has only one (see Fig. 6). The loaded antenna realized gain also varies far less over frequency compared to the unloaded dipole.

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207

77 GHz Stepped Lens With Sectorial Radiation Pattern as Primary Feed of a Lens Based CATR

IV. CONCLUSION In this work, a design methodology was introduced to systematically design wideband, electrically small antennas, based on the theory of characteristic modes. This scheme can be used to implement an ideal desired antenna current distribution over a wide frequency range using a finite number of loads at a finite number of ports. The reactive loads at these ports are determined such that this current distribution resonates over a wide frequency band. As an example, the method was applied to a simple wire dipole antenna which was loaded with a set of reactive elements. For this antenna, it was found that non-Foster reactive elements were required to synthesize the reactances determined by (3). Furthermore, it has been demonstrated that through the loading of a dipole antenna using non-Foster elements, the overall antenna bandwidth may be vastly improved. Both pattern and input impedance for the loaded dipole antenna were stabilized over a substantially wider frequency range compared to the unloaded dipole antenna, even without a matching network at the feed point. As expected, improved input impedance bandwidth was obtained when a passive matching network was introduced at the feed port.

M. Multari, J. Lanteri, J. L. Le Sonn, L. Brochier, Ch Pichot, C. Migliaccio, J. L. Desvilles, and P. Feil

Abstract—We describe the design, fabrication and measurements of an axisymmetric dielectric lens, featuring a sectorial radiation pattern at 77 GHz. It will be used as the primary feed of a lens-based compact antenna test range (CATR). Due to symmetry of revolution, the sectorial lens profile can be designed in one dimension by using phase only control. The phase variation is echoed on the lens depth. The resulting stepped lens is simulated using France Telecom Orange Labs SRSRD software (“in-house” software developed for dielectric axisymmetric radiating structures) and measured in an anechoic chamber at 77 GHz. Two lenses were fabricated with different materials: PVC and polyurethane, respectively. Good agreements were obtained between simulations and measurements. Less than 0.2 dB ripple in the central beam are obtained for the polyurethane lens although relatively high secondary lobes occur at 11 . Comparisons between the near field of a CATR illuminated by a small horn providing a uniform amplitude taper and the sectorial lens are conducted using numerical simulations. Results show that on-axis oscillations are reduced from 6 to 1 dB with the sectorial lens. Index Terms—Compact antenna test range (CATR), dielectric lens, phase control, primary feed, sectorial radiation pattern.

REFERENCES [1] L. Chu, “Physical limitations of omni-directional antennas,” J. Appl. Phys., vol. 19, pp. 1163–1175, 1948. [2] P. E. Mayes, “Frequency-independent antennas and broad-band derivatives thereof,” IEEE Proc., vol. 80, pp. 103–112, Jan. 1992. [3] S. Licul, J. A. N. Noronha, W. A. Davis, D. G. Sweeney, C. R. Anderson, and T. M. Bielawa, “A parametric study of time-domain characteristics of possible UWB antenna architectures,” in Proc. IEEE 58th Veh. Technol. Conf. VTC 2003-Fall, 2003, vol. 5, pp. 3110–3114. [4] J. Quirin, “A Study of high-frequency solid-state negative-impedance converters for impedance loading of dipole antennas,” Master’s thesis, Univ. Illinois, Chicago, 1970. [5] A. Poggio and P. Mayes, “Bandwidth extension for dipole antennas by conjugate reactance loading,” IEEE Trans. Antennas Propag,, pp. 544–547, Jul. 1971. [6] R. C. Hansen, “Dipole arrays with non-foster circuits,” in Proc. IEEE Int. Symp. on Phased Array Syst. and Technol., 2003, pp. 40–44. [7] E. Antonino-Daviu, M. Cabedo-Fabrés, M. Ferrando-Bataller, and A. Valero-Nogueira, “Wideband double-fed planar monopole antennas,” Electron. Lett., vol. 39, pp. 1635–1636, 2003. [8] M. Cabedo-Fabrés, A. Valero-Nogueira, E. Antonino-Daviu, and M. Ferrando-Bataller, “Modal analysis of a radiating slotted PCB for mobile handsets,” in Proc. Eur. Conf. on Antennas and Propag. EuCAP, Oct. 2006, vol. 626. [9] R. Harrington and J. Mautz, “Modal analysis of loaded n-port scatterers,” IEEE Trans. Antennas Propag., vol. AP-21, pp. 188–199, Mar. 1973. [10] S. Sussman-Fort, “Gyrator-based biquad filters and negative impedance converters for microwaves,” Int. J. Microw. Millimeter-Wave CAE, vol. 8, no. 2, pp. 86–101, 1998. [11] R. L.-R. James and T. Aberle, Active Antennas With Non-Foster Matching Networks, C. A. Balanis, Ed. San Rafael, CA: Morgan and Claypool, 2007, pt. Synthesis Lectures on Antennas. [12] H. Kim, “Design of negative impedance converters for VHF and UHF applications,” Master’s thesis, Arizona State Univ., Tempe, 2006. [13] E. Newman, “The Electromagnetic Surface Patch Code: Version 5,” The Ohio State University [Online]. Available: http://esl.eng.ohiostate.edu [14] K. Obeidat, B. D. Raines, and R. G. Rojas, “Antenna design and analysis using characteristic modes,” in Antennas and Propagation Society Int. Symp., 2007, pp. 5993–5996. [15] A. D. Yaghjian and S. R. Best, “Impedance, bandwidth and Q of antennas,” in Proc. IEEE Antennas Propag. Int. Symp., Jun. 2003, pp. 501–504. [16] H. Carlin and P. Civalleri, Wideband Circuit Design. Boca Raton, FL: CRC, 1997.

I. INTRODUCTION This communication describes an axisymmetric dielectric lens, featuring a sectorial radiation pattern at 77 GHz. The sectorial lens is aimed to be used as primary feed for a lens-based compact antenna test range (CATR). Sectorial radiation pattern antennas are widely used in wireless communications, e.g., for base-stations or WIFI terminals [1] but they have also demonstrated their capabilities for measurement system applications such as lens-based CATR [2]. The motivation for using a lens-based CATR is to measure electrically large antennas in the millimeter-wave frequency range. In the last decade, millimeter-wave radar applications have been of increasing interest [3], [4]. Radar antennas have to fulfill the requirements for highly directive antennas such as high gain and low side lobes and result in antenna sizes of several tens of wavelengths. Most of anechoic chambers are too small for obtaining far-field conditions with respect to these antennas sizes. Therefore, a compact range system offers a solution to this test problem. Classical solutions are based on single or more often on double reflectors CATR [5], [6]. But these performing solutions have some important drawbacks such as their low flexibility (they are not easily dismounted when the anechoic chamber has to be used Manuscript received October 24, 2008; revised June 26, 2009. First published November 06, 2009; current version published January 04, 2010 M. Multari and J. Lanteri are with the Laboratory of Electronics, Antennas and Telecommunications (LEAT), University of Nice-Sophia Antipolis, CNRS, Valbonne 06560, France. J. L. Le Sonn, L. Brochier, C. Pichot, and C. Migliaccio are with the Laboratory of Electronics, Antennas and Telecommunications (LEAT), University of Nice-Sophia Antipolis, CNRS, Valbonne 06560, France and also with CREMANT, University of Nice-Sophia Antipolis, CNRS, Valbonne 06560, France, and also with France Telecom Orange Labs, Cedex 15 Paris, 75505 (e-mail: [email protected]). J. L. Desvilles is with Orange Labs, Fort de la Tête de Chien, 06320 La Turbie, France. P. Feil is with the Department of Microwave Techniques, University of Ulm, Ulm D-89069, Germany (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.2009.2036130

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Fig. 1. Lens-based CATR setup.

without CATR arrangement) and high-accuracy fabrication required at mm-waves, and to some extend, the large space they are requiring, especially for double reflector arrangements [7]. An alternative solution could be the use of Holograms [8]–[10] that have proven to be relevant especially at mm and sub-millimeter waves. We have chosen an other alternative solution consisting in lens-based CATR [11] with an outgoing plane wave of 25 cm diameter for W-band measurements. Nevertheless, one has to get rid of the on-axis oscillations (Fresnel spots) due to the strong aperture scattering effects [12] that are reinforced in a lens, due to the internal reflections. A widespread solution is the use of serrations but they may not be sufficient to completely remove the Fresnel spots in the case of the lens-based CATR. For this purpose, we propose a primary feed with a sectorial beam. It has been designed to have uniform amplitude and phase on 68% of the CATR outgoing lens surface. Moreover, the taper illumination rapidly drops at the lens periphery edge. This primary feed is also a small axisymmetrical dielectric lens. Section II describes the sectorial lens design, simulations and measurements. Due to axisymmetry, the lens profile can be obtained using one-dimensional optimization. It has been carried out using phase only control. The phase variation is echoed on the lens depth. Simulations are performed using the France Telecom Orange labs “in-house” software, so-called SRSRD developed for dielectric axisymmetric radiating structures and based on one-dimensional integral equations [13] for open or closed structures. Former studies have proven this software to be suitable for mm-wave lens simulations [14]. Section III deals with the complete CATR simulations. Comparisons between the near field of a CATR illuminated by a small horn providing a uniform amplitude taper and the sectorial lens are conducted using SRSRD with a 500 mm CATR lens diameter. Finally the CATR performances are discussed. II. SECTORIAL LENS DESIGN AND MEASUREMENTS The phase only synthesis has proved to be an efficient method for designing arrays with pre-defined radiation patterns. We apply it to the lens using a quasi-optical approach. The advantage of such a technique is its simplicity of implementation and quick execution although it might be less accurate, compared to more sophisticated methods [15]. A. Sectorial Lens Synthesis and Numerical Simulations The basic sectorial antenna set-up is shown in Fig. 1 in the Primary feed of the CATR box. It consists of a primary feed (prolate horn) illuminating a dielectric lens with focal length f and diameter d. The latter is used for sectorial beam shaping. The sectorial lens design procedure is based on the quasi-optical approach described in [16]. According to the axi-symmetry of the lens, it

has been simplified to the one-dimensional case (1D). The design parameters are the incident and transmitted angles, respectively inc and t , and the electrical length 8 on the lens surface as described in [16]. inc and t are both oriented anti-clockwise. The primary feed radiation pattern is modeled in a first step by cosn in function. The main steps of the synthesis method can be summarized as follows. — The transmitted angle t is classically obtained from the energy conservation as a function of inc ; — The electrical length 8 is derived from (1) as described in [16] with the 1D simplification; — The numerical integration of (1) gives the electrical length that has to be on the lens for the desired pattern; — Assuming that the phase profile is obtained by changing the lens depth, de , we can obtain directly the depth de from 8

d8 dx

= sin t + sin 

inc

:

(1)

The phase reference in (1) is taken in the lens centre which leads to de = 0 at this point. In order to simplify the lens fabrication, it is more convenient to avoid this configuration. Therefore, the phase reference is moved in order to have a null depth at the lens edge. Considering the respective paths of the ray going to the lens edge [left term of (1)] and another one going through the lens [right term of (1)], a new equation leading to the value of de is obtained

f 2 + (d=2)2

= (f 0 de ) + x + de p"r 2

2

(2)

where y 2 [0; d=2]. Finally, de is obtained from solving (2). Before going into design considerations, let us note that the quasioptical approach leads to a stepped lens. As a consequence, a large lens will provide a large number of steps and make the lens fabrication more complex. Therefore this method is suitable for electrically small lenses. The synthesis method described above is implemented with Scilab software [17]. The desired sectorial pattern G(t ) is chosen to be uniform over 60 and to drop rapidly beyond according to (3)

G(t ) = 1 0

tan t tan 

0

15

:

(3)

According to the desired configuration of the compact range, 0 is chosen to be 5 . The design frequency is 77 GHz. The primary feed of the sectorial lens is designed to perform a prolate radiation pattern for low secondary lobes purposes [18]. The prolate horn dimensions and measurements are shown in Fig. 2(a) and (b) respectively. It is designed for an illumination at 020 dB on the edges of the lens and corresponds to a focal length to diameter ratio of 1. Applying the method described above, a sectorial lens with diameter (d) of 50 mm, [about 13 . and using 20 x-samples per wavelength for solving (2)] was designed. The total number of steps is 270. The stepped lens profile given in the Annex is obtained for polyurethane ("r = 4). The large steps correspond to phase angle close to 180 . The design led to changes in the sign of the amplitude (which is equivalent to a 180 phase shift) due to the phase reference taken from (2). In counterpart, they create shadowing areas. Fabrication errors and focal length displacements have to be studied since we are working at millimeter waves. These errors can be close to the step dimensions of the fabricated lens. Therefore simulations are conducted with SRSRD using 650 m errors randomly distributed on the lens profile. The same way, 61 mm focal length displacements are

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(a)

(b) Fig. 2. (a) Primary feed: prolate horn—dimensions in mm. (b) Primary feed measurements at 77 GHz.

Fig. 3. Fabrication and focal length errors.

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Fig. 4. Sectorial lens picture.

Fig. 5. Simulated and measured radiation pattern at 77 GHz.

dB. Frequency measurements were also conducted showing good radiation pattern stability over the frequency range from 75 to 79 GHz. The sectorial pattern is achieved within 65 . High secondary lobes occur at 611 given that the design template was given only up to the desired zeros of the diagram. In a large anechoic chamber implementation, they are not critical for our application because they will be the spillover part of the illumination of the CATR and hence will be very low compared to the focusing done by the lens of the CATR. This effect is reinforced with increasing the size of the lens CATR. Nevertheless the high lobes levels at 11 may affect the quality of the quiet zone, given that the CATR is to be built in a small anechoic chamber, especially because of the possible reflections on the walls. Hence, some absorbing material will be used around the lens. This solution has been successfully tested in a previous study with a uniform illumination on the CATR-lens [19]. III. APPLICATION TO THE CATR

investigated. Results are plotted in Fig. 3, the total oscillation does not exceed 1 dB. B. Sectorial Lens Measurements and Discussion A first lens has been fabricated using PVC. Despite a relative good agreement with simulations, a critical, for CATR application, 1 dB oscillation in the E-plane is obtained. Looking at the lens fabrication, one can guess that debris remain in the lens grooves. They are probably due to the standard milling technique used for machining PVC. In order to overcome this difficulty, a second lens was fabricated using polyurethane. The complete sectorial lens set-up is shown in Fig. 4. Fig. 5 shows the comparisons between measurements and simulations. The focal length has been fitted at 51 mm in order to suppress the ripples in the main lobe. Measurements are in very good agreement with simulations within the sectorial behavior of the radiation pattern. The remaining oscillation has been drastically reduced to 0.2

If we aim to design a CATR, one has to face the trade-off between the strong edge diffraction, that occur with uniform illumination and the decrease of the quiet zone spot caused by the reduction of the illumination taper. The sectorial illumination seems to be the ideal compromise since it provides a uniform illumination while having a rapid drop at the CATR edges. A previous experimental study, conducted in 2006 [19] on a 500 mm diameter PVC-lens based CATR, has shown that high on-axis oscillations occur in the near field distribution when a uniform illumination is applied to the CATR. This effect has been also previously demonstrated from theoretical studies in circular apertures [20]. In order to check this behavior, our lens-CATR configuration was simulated using a uniform illumination taper. For this purpose, the primary feed is a small standard circular horn providing 1 dB taper at the CATR-lens edges. The CATR-lens has a hyperbolic profile and is made of PVC. Simulations were conducted at 77 GHz.

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TABLE I ANNEX: SECTORIAL HALF-LENS DIMENSIONS FOR "

D = 500 mm, F = 2 m,  = 7 ; X = 10 dB X = 1 dB for the sectorial illumination.

Fig. 6. CATR near field, for the uniform feed,

The amplitude field distributions at different down range distances (Z) from the CATR-lens are shown in Fig. 6. The coordinate system (X, Z) is defined according to Fig. 1, where X denotes the displacement along the lens radius and Z the distance from the center of the flat border lens to the AUT. The on-axis (X = 0) oscillations between Z = 2 and Z = 2:5 m are 6 dB. In addition, the ripple in the quite zone (along X) is 4 dB. The same simulation has been conducted using the polyurethane sectorial lens and the results are shown in Fig. 7. It is obvious that the sectorial lens dramatically reduces the ripple in the quite zone. The maximum quiet zone radius is obtained at Z = 2 m and is of 170 mm that corresponds to 68% of the CATR-lens surface. Within this

= 4; x = lens radius; z = lens depth

Fig. 7. Same CATR configuration as in Fig. 6 with an sectorial illumination.

zone, the maximum amplitude oscillation is 1 dB and the phase ripple does not exceed 8. Moreover, the quiet zone depth (along Z-axis) is 0.5 m. The installation of the CATR antenna system requires a space of 4.1 m (along Z). Furthermore, the rotation of the AUT also needs some space that has been estimated to 50 cm and some margin for the CATR construction has to be taken. Therefore, the total size of the CATR should not exceed 5 m. This overall configuration is interesting as we can build a compact test range within a small anechoic chamber. IV. CONCLUSION This communication describes the design, simulation and measurement of a new sectorial lens for primary feed of a lens-based mm-wave

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CATR. The design procedure, based on the one-dimensional quasi-optical approach, despite of its simplicity and approximations that are made regarding the internal lens reflections, is quite efficient for computing the sectorial lens profile. The axisymmetry of the lens makes it easy, in principle, to fabricate. Nevertheless the choice of the dielectric material is important since it has to be compatible with the milling technique for ensuring a good surface quality of finish. Therefore, polyurethane material is preferred to PVC. Very good agreement has been obtained between measurements and simulations carried out with SRSRD. This is a key point because the lens design procedure neglects some important lens features such as the internal reflections or step shadowing. From this knowledge, we can perform quite accurate simulations using SRSRD. The above-mentioned effects were quantified and their influences on the final CATR setup were studied.

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[16] R. Leberer and W. Menzel, “A dual planar reflectarray with synthesized phase and amplitude distribution,” IEEE Trans. Antennas Propag., vol. 53, no. 11, pp. 3534–3539, Nov. 2005. [17] [Online]. Available: www.scilab.org [18] J. Lanteri, C. Migliaccio, J.-Y. Dauvignac, and Ch. Pichot, “Reflectarray using a prolate feed at 94 GHz,” in Proc. IEEE AP-S, San Diego, Jul. 5–11, 2008, pp. 1–4. [19] M. Multari, C. Migliaccio, J.-Y. Dauvignac, L. Brochier, J.-L. Le Sonn, Ch. Pichot, W. Menzel, and J.-L. Desvilles, “Investigation of low-cost compact range W-band,” in Proc. EuCAP, Nice, Nov. 6–10, 2006, pp. 1–6. [20] R. C. Rudduck and C. L. J. Chen, “New plane wave spectrum formulation for the near-field of circular apertures,” IEEE Trans. Antennas Propag., vol. 24, no. 4, pp. 438–449, Jul. 1976.

A Single-Layer Ultrawideband Microstrip Antenna

REFERENCES [1] S. Qi and K. Wu, “Leakage and resonance characteristics of radiating cylindrical dielectric structure suitable for use as a feeder for high-efficient omnidirectional/sectorial antenna,” IEEE Trans. Antennas Propag., vol. 46, no. 11, pp. 1767–1773, Sep. 1998. [2] T. Hirvonen, J. Tuovinen, and A. V. Räisänen, “Lens-type compact antenna test range at mm-waves,” in Proc. 21st Eur. Microw. Conf., Stuttgart, Germany, Oct. 1991, vol. 2, pp. 1079–1083. [3] B. D. Nguyen, C. Migliaccio, Ch. Pichot, K. Yalmamoto, and N. Yonemoto, “W-band fresnel zone plate reflector for helicopter collision avoidance radar,” IEEE Trans. Antennas Propag., vol. 55, no. 5, pp. 1452–1456, May 2007. [4] W. Mayer, M. Meilchen, W. Grabherr, P. Nüchter, and R. Gühl, “Eight channel 77 GHz front-end module with high-performance synthesized signal generator for FM-CW sensor applications,” IEEE Microw. Theory Tech., vol. 52, no. 3, pp. 993–1000, Mar. 2004. [5] E. K. Walton and J. D. Young, “The Ohio state university compact range cross-section measurement range,” IEEE Trans. Antennas Propag., vol. 32, no. 11, pp. 1218–1233, Nov. 1984. [6] C. W. I. Pistorius, G. C. Clerici, and W. D. Burnside, “A dual chamber Gregorian subreflector system for compact range applications,” IEEE Trans. Antennas Propag., vol. 37, no. 3, pp. 305–313, Mar. 1989. [7] G. Forma, D. Dubruel, J. Marti-Canales, M. Paquay, G. Crone, J. Tauber, M. Sandri, F. Villa, and I. Ristorcelli, “30-70-100-320 GHz radiation measurements for the radio frequency qualification model of the Planck satellite,” presented at the 1st Eur. Conf. on Antennas Propag., (EuCAP2006), Nice, France, Nov. 6–10, 2006, paper 349443. [8] J. Meltaus, J. Salo, E. Noponen, M. M. Salomaa, V. Viikari, A. Lönnqvist, T. Koskinen, J. Sáily, J. Hakli, J. Ala-Laurinaho, J. Mallat, and A. V. Räisänen, “Millimeter-wave beam shaping using holograms,” IEEE Microw. Theory Tech., vol. 51, no. 4, pp. 1274–1279, Apr. 2003. [9] J. Häkli, T. Koskinen, A. Lönnqvist, J. Säily, V. Viikari, J. Mallat, J. Ala-Laurinaho, J. Tuovinen, and A. V. Räisänen, “Testing of a 1.5-m reflector antenna at 322 GHz in a CATR based on a hologram,” IEEE Trans. Antennas Propag., vol. 53, no. 10, pp. 3142–3150, Oct. 2005. [10] A. Lönnqvist, T. Koskinen, J. Hakli, J. Sáily, J. Ala-Laurinaho, J. Mallat, V. Viikari, J. Tuovinen, and A. V. Räisänen, “Hologram-based compact range for submillimeter-wave antenna testing,” IEEE Trans. Antennas Propag., vol. 53, no. 10, pp. 3151–3159, Oct. 2005. [11] Menzel and B. Huder, “Compact range for millimeter-wave frequencies using a dielectric lens,” Electron. Lett., vol. 20, pp. 768–769, Sep. 1984. [12] G. Bruhat, Optique. France: Masson, 1959, 5ème edition, RO30013068. [13] A. Berthon and R. Bills, “Integral equation analysis for radiating structures of revolution,” IEEE Trans. Antennas Propag., vol. 49, no. 4, pp. 159–170, Feb. 1989. [14] C. Migliaccio, J.-Y. Dauvignac, L. Brochier, J.-L. Le Sonn, and Ch. Pichot, “W-band high gain lens antenna for metrology and radar applications,” Electron. Lett., vol. 40, no. 22, pp. 1394–1396, Oct. 28, 2004. [15] A. P. Pavacic, D. L. Del Rio, J. R. Mosig, and G. V. Eleftheriades, “Three dimensional ray tracing theory to model internal reflections in off-axis lens antennas,” IEEE Trans. Antennas Propag., vol. 54, no. 2, pp. 604–612, Feb. 2006.

Qi Wu, Ronghong Jin, and Junping Geng

Abstract—A single-layer microstrip antenna for ultrawideband (UWB) applications, in which an array of rectangular microstrip patches was arranged in the log-periodic way and proximity-coupled to the microstrip feeding line, is presented. In order to reduce the number of microstrip patches in the UWB log-periodic arrays, a large scale factor k = 1 1 was firstly reported and proved to be highly effective. Furthermore, instead of using an absorbing terminal loading, a novel loss-free compensating stub was proposed. Detailed parameters study was also presented for better understanding of the proposed antennas. The impedance bandwidth (mea2 5) of an example antenna with only 11 elements is sured VSWR from 2.26–6.85 GHz with a ratio of about 3.03:1. Both numerical and experimental results show that the proposed antenna has stable directional radiation patterns, very low-profile and low fabrication cost, which are suitable for various broadband applications. Index Terms—Directional antennas, log-periodic antennas, microstrip antennas, ultrawideband (UWB) antennas.

I. INTRODUCTION Currently, there are increasing demands for novel ultrawideband (UWB) antennas with low-profile structures and constant directional radiation patterns for both commercial and military applications [1], [2]. Unfortunately, most of the mature UWB antennas like equiangular and Archimedean spirals [3], planar monopoles [4], [5] and wide slot antennas [6], [7] have inherently bi-directional or omnidirectional radiation patterns, which were unsuitable for conformal placement on certain platforms. Cavity-backed log-periodic slot antennas [8] could be integrated compactly into various aircrafts, but they could only provide end-fire radiation patterns and have somewhat high profile. On the other hand, microstrip antennas have some attractive merits like very low-profile and broadside radiation patterns with medium gains, which have been considered as excellent conformal radiators [9] for a long time. However, a traditional single-element microstrip antenna has inherently narrow impedance bandwidth. In the 1980s, the Manuscript received January 06, 2009; revised May 10, 2009. First published July 14, 2009; current version published January 04, 2010. The authors are with the Department of Electronic Engineering, Shanghai Jiao Tong University, Shanghai 200240, China (e-mail: [email protected]. cn). 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.2009.2027728

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series-fed log-periodic microstrip antennas (LPMAs) were firstly introduced for bandwidth enhancement [10], [11]. After that, other feeding structures like =4-microstrip feeding [12], [13], direct feeding with vias [14], slot-coupled feeding [15], inset feeding [16] for LPMAs were also reported. However, all of the LPMAs mentioned above were based on the combination of two substrate layers, in which very accurate collocation between the two layers were required. A UWB traveling-wave LPMA was usually terminated by a matched load to absorb the terminal energy and prevent reflections from the end of feeding line, which could improve its impedance matching and radiation patterns considerably at the band edges. However, the absorbing loads could remarkably degrade the efficiency of LPMAs over their operating bandwidths, especially at the band edges by the order of 10 % or more [10]. It is also well known that the operating bandwidth of a LPMA could be determined by the scale factor k [10] and the number of microstrip elements n, in a simple expression as kn01 . Therefore, for the LMPA with a small scale factor, a large number of microstrip patches were required to achieve ultrawide operating bandwidth and make the antenna unacceptably large. Obviously, the number of elements in a UWB LMPA could be significantly reduced with a larger scale factor. For example, with a scale factor k as 1.05 or below like the reported LPMAs in [10]–[16], about 21 elements were needed to achieve a 2.6:1 bandwidth; however, for the LMPA with a scale factor k as 1.1, 11 microstrip elements could be enough for the same bandwidth with a size reduction of over 40%. This communication presents a single-layer proximity-coupled logperiodic microstrip antenna for UWB applications. A large scale factor as 1.1 for UWB LPMAs is firstly reported and discussed. A novel loss-free compensating stub is also proposed for the termination and bandwidth enhancement of the proposed antennas. The configuration of the proposed LPMA is described in Section II. Numerical results are presented in Section III for better understanding of this antenna, while the measured results are presented and discussed in Section IV. This communication is concluded in Section V.

Fig. 1. Structural configuration of the proposed LPMA and the coordinate system. TABLE I OPTIMIZED PARAMETERS OF THE PROPOSED LPMA WITH COMPENSATING STUB

II. ANTENNA CONFIGURATION The proposed single-layer LPMA, as illustrated in Fig. 1, was composed of a 50 –100 impedance transition, a 100- microstrip feeding line and an array of proximity-coupled rectangular microstrip patches, which were all etched on a Teflon-based substrate with a relative permittivity of 2.65 and a thickness of 3 mm. The microstrip elements were arranged in a transposed log-periodic way with the same coupling gap G. The log-periodic scale factor k was defined as

k=

Wi Li Di;i+1 = = : Wi01 Li01 Di01;i

(1)

In (1), Di;i+1 was the center distance between patch Pi+1 and Pi ; Wi and Li were respectively the width and length of the patch Pi . The proposed LPMA was terminated by a novel compensating stub with length T, instead of a matched load or open-circuit. The LPMAs were designed by a semi-empirical way like the design procedure of LPDAs [3] except some parameters of the LPMAs should be optimized with the assistance of full-wave simulator HFSS. Following discussions of the proposed LPMAs were all based on the optimized parameters in Table I. III. NUMERICAL RESULTS Coupling gap G between the microstrip patches and the microstrip feeding line could determine the power transmission efficiency of the feeding line thus should be carefully optimized. Fig. 2 shows the simulated results of the proposed LPMA at the reference plane BB0 (see

Fig. 2. Simulated reflection coefficient and antenna efficiency of the proposed antennas with compensating stubs at the reference plane BB (k = 1:1).

Fig. 1), and the influence of the coupling gap G on the impedance matching and antenna efficiency could be easily observed. The antenna with G = 0 mm was no longer a travelling-wave array and exhibited a multiresonant behavior. The antennas with G = 0:2 mm and G = 0:4 mm, which could be respectively referred as “critical coupled” and “under coupled” [17], have very similar reflection coefficient and efficiency above 2.55 GHz because the last several microstrip patches could “absorb” the residual energy which was not properly coupled to the corresponding patches due to the “under coupled” effects. So, it is not surprising to find that the “critical coupled” one was better impedance matched in the band from 2.25–2.55 GHz than the “under coupled” one because this frequency band corresponds to the last microstrip elements PN . The characteristics of compensating stubs with positive or negative T value were different: when the stub went beyond the last patch PN (known as positive T), the stub was actually an open-circuit microstrip stub and behaved like a loss-free “series capacitive load,” the stub with negative T could be considered as a loss-free “series inductive load.” The compensating effects of the stub could be clearly observed by examining the simulated input impedance of the proposed LPMAs at the reference plane BB0 as shown in Fig. 3. The input impedance of the proposed antenna without compensating (T = 0 mm) was heavily

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Fig. 4. Simulated and measured VSWR and gain of the proposed antennas with compensating and absorbing terminations (k = 1:1). Fig. 3. Simulated input impedance of the proposed antennas with compensating stubs at the reference plane BB (k = 1:1).

capacitive just after its first resonant frequency at 2.36 GHz. The capacitive components were well compensated by the “series inductive load” (T = 04 mm), and thus the impedance matching was significantly improved; but the situation got worse if the “series capacitive load” (T = 4 mm) was applied. After the first resonant frequency, the influence of different stub T was very slight, thus it could be observed that the antennas were all well impedance matched. The performance of two identical LPMAs with the compensating and absorbing loads could be also evaluated and compared. The absorbing load was assumed to be positioned beyond the last patch PN with T = 24 mm and set as a “Lumped RLC boundary” with 100-

resistance and a dimension of 2.4 2 3.0 mm2 in HFSS. Generally both of the antennas were well impedance matched in the frequency band between 2.4 and 6.8 GHz as shown in Fig. 4. In the frequency band below 2.3 GHz, the one with an absorbing load still has good VSWR for its terminal load could absorb all of the residual energy and prevent possible terminal reflections, which also significantly degrades its efficiency as shown in Table II. The radiation patterns of the LPMAs with compensating and absorbing loads were very similar in the frequency band above 3.3 GHz as shown in Fig. 5. But the patterns have some differences in the x-z plane at 2.3 GHz, in which the pattern of the one with absorbing load has narrower beamwidth and higher directivity. That difference was mainly caused by the different dealing methods of the residual energy: for the absorbing load, all of the residual energy was absorbed at the terminal, thus good radiation patterns could be observed; for the compensating case, the residual energy was totally reflected at the termination, thus some disordered modes of patch PN appeared and the patterns were also influenced. Furthermore, although the LPMA with absorbing load has higher directivity below 3.3 GHz, its gain is obviously lower than the one with compensating stub for its extremely low efficiency. In addition, the performance of the LPMAs with k = 1:05 should also be examined. The simulated impedance bandwidth of the LPMA with k = 1:05, defined by VSWR < 2:5, was from 3.35–6.95 GHz. The radiation patterns and gain of the proposed LPMA with k = 1:05 were respectively illustrated in Fig. 5 and Table II. Generally the two LPMAs with k = 1:05 and k = 1:1 have very similar radiation patterns, and the one with k = 1:05 has higher directivity, 0.65 dB on average, than the one with k = 1:1, which shows the similar trend as the log-periodic dipole array (LPDA) [18] and could be considered as the main cost of large scale factor. Besides, the trend of patch gap D1;2 was also found to comply with the best design curve for LPDAs, and

Fig. 5. Simulated radiation patterns of the proposed LPMAs with compensating and absorbing stubs: (a) x-z plane, compensating stub, k = 1:1; (b) x-z plane, absorbing stub, k = 1:1; (c) y-z plane, compensating stub, k = 1:1; (d) y-z plane, absorbing stub, k = 1:1; (e) x-z plane, compensating stub, k = 1:05; (f) y-z plane, compensating stub, k = 1:05.

thus the well-established theory of LPMAs could be an important guidance in the design of LPMAs. IV. EXPERIMENTAL RESULTS AND DISCUSSION An example antenna with k = 1:1 was fabricated based on the optimized parameters in Table I. The impedance bandwidth was measured by using an Agilent 8722ES Vector Network Analyzer (VNA) and the

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TABLE II SIMULATED DIRECTIVITY, GAIN AND EFFICIENCY OF THE PROPOSED LPMAS

as the standard antenna, and the results could be found in Fig. 4. The simulated gains agree reasonably with the measured one, and the differences between them may be caused by the numerical errors and the calibration errors of the UWB standard horn. The gain bandwidth, defined by the simulated and measured gain better than 6.5 dB, was from 2.4–6.6 GHz with a radio bandwidth 2.75:1. In its gain bandwidth, the fluctuations of simulated and measured gain were also small and evaluated to be 2.4–2.0 dB, respectively. V. CONCLUSION A single-layer log-periodic microstrip antenna for UWB applications was presented. A large scale factor k = 1:1 was firstly reported and proved to be highly effective for the purpose of size reduction. Furthermore, instead of using an absorbing terminal loading, a novel loss-free compensating stub was also proposed. The impedance bandwidth (with measured VSWR < 2:5) of the example antenna with only 11 elements is from 2.26–6.85 GHz with a ratio of about 3.03:1. Both numerical and experimental results show that the proposed antenna has stable directional radiation patterns, very low-profile and low fabrication cost, which are suitable for various broadband applications.

REFERENCES

Fig. 6. Measured radiation patterns of the proposed antenna with compensating stub (k=1.1): (a) x-z plane; (b) y-z plane.

results were shown in Fig. 4. It shows good agreement between the simulated and measured results, and the little difference between them may be caused by the soldering effect of the SMA connector and its mechanical tolerance. Its measured impedance bandwidth defined by VSWR < 2:5 is from 2.26–6.85 GHz with a ratio of about 3.03:1. The radiation patterns of the example antenna were illustrated in Fig. 6. Generally, the measured patterns agree well with the simulated one, and the patterns were reasonably stable at the broadside. The gain was also measured by using the comparison method with a corrugated horn

[1] Z. N. Chen, M. J. Ammann, X. Qing, X. H. Wu, T. S. P. See, and A. Cai, “Planar antenna,” IEEE Microw. Mag., vol. 7, no. 6, pp. 63–73, Dec. 2006. [2] J. M. Bell, M. F. Iskander, and J. J. Lee, “Ultrawideband hybrid EBG/ Ferrite ground plane for low-profile array antennas,” IEEE Trans. Antennas Propag., vol. 55, no. 1, pp. 4–12, Jan. 2007. [3] J. D. Karus and R. J. Marhefka, Antennas for All Applications. New York: McGraw-Hill, 2003. [4] N. P. Agrawall, G. Kumar, and K. P. Ray, “Wideband planar antennas,” IEEE Trans. Antennas Propag., vol. 46, no. 2, pp. 294–295, Feb. 1998. [5] M. J. Ammann and Z. N. Chen, “A wideband shorted planar monopole with bevel,” IEEE Trans. Antennas Propag., vol. 51, no. 4, pp. 901–903, Apr. 2003. [6] J. Y. Sze and K. L. Wong, “Bandwidth enhancement of a microstripline fed printed wide-slot antenna,” IEEE Trans. Antennas Propag., vol. 47, no. 7, pp. 1020–1024, July 2001. [7] J. Lao, R. H. Jin, J. P. Geng, and Q. Wu, “An ultra-wideband microstrip elliptical slot antenna excited by a circular patch,” Microw. Opt. Technol. Lett., vol. 50, no. 4, pp. 845–846, Apr. 2008. [8] A. G. Roederer, “A log-periodic cavity-backed slot array,” IEEE Trans. Antennas Propag., vol. 16, no. 6, pp. 756–758, Nov. 1973. [9] D. M. Pozar, “Microstrip antennas,” Proc. IEEE, vol. 80, no. 1, pp. 79–91, Jan. 1992. [10] P. S. Hall, “New wideband microstrip antenna using log-periodic technique,” Electron. Lett., vol. 16, no. 4, pp. 127–128, 1980. [11] P. S. Hall, “Multi-octave bandwidth log-periodic microstrip antenna array,” Proc. Inst. Elect. Eng. Microw., Antennas Propag., vol. 133, no. 2, pt. H, pp. 127–136, 1986. [12] H. Pues, J. Bogaers, R. Pieck, and A. van de Capelle, “Wideband quasi log-periodic microstrip antennas,” Proc. Inst. Elect. Eng. Microw., Antennas Propag., vol. 128, no. 3, pt. H, pp. 159–163, 1981. [13] R. Kakkar and G. Kumar, “Stagger tuned microstrip log-periodic antenna,” in IEEE AP-S Int. Symp. Digest, Jun. 1996, pp. 1262–1265. [14] H. Ozeke, S. Hayashi, N. Kikuma, and N. Inagaki, “Quasi-log-periodic microstrip antenna with closely coupled elements,” Elect. Eng., vol. 132, no. 2, pp. 58–64, 2000, in Japan. [15] H. K. Smith and P. E. Mayes, “Log-periodic array of dual-feed microstrip patch antennas,” IEEE Trans. Antennas Propag., vol. 39, no. 12, pp. 1659–1664, 1991. [16] M. K. A. Rahim, M. N. A. Karim, T. Masri, and A. Asrokin, “Comparison between straight and U shape of ultra wide band microstrip antenna using log periodic technique,” in Proc. IEEE Int. Conf. on UWB, Sep. 2007, pp. 696–699. [17] D. M. Pozar, Microwave Engineering. Reading, MA: Addison-Wesley, 1990. [18] R. Carrel, “The design of log-periodic dipole antennas,” IRE Int. Convention Rec., pt. 1, pp. 61–75, 1961.

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Influence of the Finite Slot Thickness on RLSA Antenna Design Agnese Mazzinghi, Angelo Freni, and Matteo Albani

Abstract—In this communication, we consider the effect of the thickness of the plate where slots are cut into, with reference to the design of a radial line slot array antenna. We show that the zero-thickness approximation results in a significant lack of accuracy even though the slot thickness is of the order of one-thirtieth of the wavelength. Then, we investigate and compare the accuracy and the efficiency of various approximate numerical techniques that can be conveniently employed in the full-wave analysis and optimization process of the antenna.

Fig. 1. Geometry of the problem.

Index Terms—Antenna arrays, method of moments (MoM), radial line slot array (RLSA).

Fig. 2. Equivalent transmission line for each waveguide mode of a slot of thickness `.

I. INTRODUCTION Recently, high-gain radial line slot array (RLSA) antennas are more and more used in wireless LANs, DBS reception, and collision avoidance radar systems in millimeter-wave band [1]–[3]. Several practical antenna configurations make use of slots etched in a plate whose thickness is a not negligible fraction of the wavelength. This is true not only in the millimeter-wave band but also in the microwave band when air filled RLSAs are considered. As a matter of fact, for structural requirements the upper plate, where the slots are cut into, can even have a thickness of a few millimeters. In this communication we consider the influence of the slot thickness on the RLSA design. In particular, we show that the slot thickness ` should not be neglected even though it is quite small with respect to the wavelength (i.e., ` is in the order of =40 4 =20). In fact, the single contribution of each slot in the phase error, introduced by considering it as a zero thickness slot, is quite small but the overall error becomes relevant in the case of several hundreds of slots, as in high gain RLSAs. Moreover, we investigate the accuracy and the efficiency of various approximate techniques that can be conveniently employed in the design and optimization process of the antenna, for which many full-wave simulations are required. In particular, in the present communication a method of moments (MoM) formulation [4]–[6] is considered, where on each slot the unknown magnetic current is expanded in terms of rectangular waveguide modal current basis functions.

ports. This multipole network can be characterized by two N 2 N generalized admittance matrices: [Y1 ] for Region 1, and [Y2 ] for Region 2. The N length vectors [Vi ] and [Ii ], with i = 1; 2, collect the complex amplitudes of the modal expansions of the electric and magnetic field, respectively, scattered by the magnetic currents mi and calculated at each input port of Region i. [V0 ] and [I0 ] denote the complex amplitudes of the fields induced by the sources J0 ; M0 at the interface i = 1. Propagation of each waveguide mode through each slot of thickness ` is described by an equivalent transmission line, as shown in Fig. 2. By denoting with [Zc ]; [Yc ] and [kz ] the diagonal matrices containing the modal impedances, admittances, and propagation constants of the modes, respectively, one can easily write

[V1 (`)] = f[cos(kz `)] + j [Zc sin(kz `)][Y2 ]g[V2 (`)] [A(`)][V2 (`)] = [I0 ]

(1) (2)

where

[A(`)] = [Y1 ][cos(kz `)] + [cos(kz `)][Y2 ] + j [Yc sin(kz `)] + j [Y1 ][Zc sin(kz `)][Y2 ]: (3) A. Exact Solution

II. FORMULATION Consider the generic problem, sketched in Fig. 1, in which two arbitrary Regions 1 and 2 are separated by P rectangular apertures, drilled in a thick conducting wall. Region 1 also includes a set of impressed currents J0 ; M0 (sources). The tangential electric and magnetic fields on the nth aperture are represented in terms of Nn orthogonal waveguide modes. Each region is then characterized by a multipole network having N = P n=1 Nn Manuscript received January 14, 2009; revised June 09, 2009. First published July 14, 2009; current version published January 04, 2010. A. Mazzinghi and A. Freni are with the D.E.T., University of Florence, Florence, Italy (e-mail: [email protected]; e-mail: [email protected]). M. Albani is with the D.I.I., University of Siena, Siena, Italy (e-mail: matteo. [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.2009.2027457

In the case of a direct solution of (2), the matrix [A(`)] has first to be built, and then factorized. The former operation is dominated by the matrix product [Y1 ][Zc sin(kz `)][Y2 ] that requires a computational complexity of O(N 3 ). Moreover, the complexity related to the factorization process, i.e., O(N 3 =3), has to be added. The size of the overall required storage is O(2N 2 ), since [Y1 ] and [Y2 ] are symmetric matrices. To halve the storage size, the linear system in (2) can be solved by resorting to an iterative technique, as for example the conjugate gradient (CG). In this case, one needs to store only the two symmetric matrices [Y1 ] and [Y2 ]. As a matter of fact, the matrix [A(`)] is never directly calculated but, at each step of the CG procedure, the matrix-vector product is calculated by multiplying the search/residual vectors by each term of the right hand side of (3). Hence, the overall complexity is found to be 2 O (cS 4N ), where S is the number of steps necessary to the CG procedure to reach the desired accuracy, and c is a factor depending on the specific numerical implementation of the CG algorithm (usually it is estimated as c ' 6 [7]).

0018-926X/$26.00 © 2009 IEEE

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where

B. Application of the Asymptotic Waveform Evaluation Technique To further reduce both the computational complexity and the storage requirements, the asymptotic waveform evaluation (AWE) technique [8] can be employed. The key idea is to expand the solution of (2) in Taylor series as [

1

V2 (`)] =

where each momentum [Mn ] calculated by

Mn ] = [A0 ]01

[

[

n=0

n [Mn ]`

(4)

d=d`)[V2 (`)]j`=0 can be recursively

= (

I0 ]n0 0

n

[Aq ][Mn0q ] q=1 q! 1

Aq ] =

[

q q:

for even for odd

(6) Note that the matrices [kz ]q ; [Yc kz ]q , and [Zc kz ]q are diagonal and therefore they can be calculated with a small computational effort. From a computational point of view the formulation requires the inverse of the symmetric matrix [A0 ] = [Y1 ] + [Y2 ] (that correspond to the zero thickness slot case), and some matrix-vector multiplications. Therefore, the numerical complexity is dominated by the time required for the factorization of [A0 ], that is O(N 3 =3). Furthermore, to improve the accuracy, especially when thicker slots are considered, instead of simply truncating the Taylor series expansion (4) to Nt terms, it is more convenient to represent each sth entry of the vector solution [V2 (`)]s with a rational function, according to the Padé approximation [9]

N

V2 (`)]s '

[

Mn ]s `n

[

n=0 a0;s + a1;s ` + 1 1 1 + ap;s `p ' q 1 + b1;s ` + 1 1 1 + bq;s `

(7)

with p + q = Nt . For each sth entry, the coefficients bn;s can be obtained by solving the linear system of equations

q n=1

Mp0n+i ]s bn;s = 0[Mp+i ]s ;

[

and then

an;s = [Mn ]s +

min(n;q)

i=1

Mn0i ]s bi;s

[

with

with

i = 1; 2; . . . ; q

(8)

n = 0; 1; . . . ; p: (9)

For a given Taylor series truncation order Nt , the best accuracy in the Padé expansion is achieved when p = q or p = q + 1. Since the dimension of each linear system (8) is very small the complexity remains basically unchanged, while the required storage is O(2N 2 ). C. Application of the Approximate Green’s Function (AGF) for Thick Aperture Problems The linear system (1)–(2) can also be written as follows [10]:

Y1 ] + [Gs (`)]g[V1 ] 0 [Gm (`)][V2 ] = [I0 ] Gm (`)][V1 ] + f[Y2 ] + [Gs (`)]g[V2 ] = [0]

f[

0[

(10)

(11) (12)

The form of (10) allows the extension to the present case of the theory in [11] which introduces a correcting “delta” term for the Green’s function of the thick aperture problem. Namely, the “average” (6) and “deviation” (1) values of the magnetic current amplitudes are defined as [V6 ] = (1=2)f[V1 (`)] + [V2 (`)]g and [V1 ] = (1=2)f[V1 (`)] 0 [V2 (`)]g, respectively; by proceeding similarly to [11], at the first step (s = 0) it is assumed

(5)

in which nm is the Kronecker delta function and [Aq ] = q q (d )=(d` )[A(`)]j`=0 . In the present case, the derivatives of [A(`)] can be evaluated analytically, thus yielding

q q (01) f[Y1 ][ kz ] + [kz ] [Y2 ]g q j (01) f[Yc kz ] + [Y1 ][Zc kz ]q [Y2 ]g

Gs ] = [G6 ] + [G1 ] = 0j 2[Yc ][cot(kz `)] Gm ] = [G6 ] 0 [G1 ] = 0j 2[Yc ][csc(kz `)]:

2[ 2[

V1(0)

= [0]

V6(0)

= f[

01 I0

Y1 ] + [Y2 ] + 2[G1 ]g

[

(13) (14)

]

next, to improve the estimation, one calculates iteratively (for s ; ; ) (i) V1 = f[Y2 ] + [G6 ]g 1 f[Y2 ] + [G1 ]g V6(i 1)

V6(i)

=

0

0

1 2 ...

(15)

01

Y1 ] + [Y2 ] + 2[G1 ]g

= f[

2

I0 ] 0 ([Y1 ] 0 [Y2 ]) V1(i)

[

:

(16)

As a result, to perform the first step (s = 0), the factorization of the matrix ([Y1 ] + [Y2 ] + 2[G1 ]) in (14) requires an algorithm of computational complexity O(N 3 =3) and a storage requirement O(N 2 ). The factorization of ([Y2 ] + [G6 ]) is further needed in (16) to improve the solution estimation (s > 0), thus requiring an additional computational effort of O(N 3 =3) and an additional storage requirement of O(N 2 ). Therefore the total computational effort is of order of O(2N 3 =3) and the total memory storage of the order of O(2N 2 ). III. RESULTS In all the following examples we refer to a practical circularly polarized (CP) RLSA whose layout is sketched in the inset of Fig. 4. The structure consists of a h = 3 mm air filled parallel plate waveguide (PPW) whose upper plate, where 822 slots are cut into, is made from a ` = 0:5 mm thick, stainless steel sheet. However, through numerical simulation, a parametric analysis was conducted for varying thicknesses `. All the slots have a width w = 0:8 mm, while their length ranges from 5.8 to 9.5 mm along a spiral which moves outwardly from the center. Furthermore, a 280 mm diameter metallic rim closes the PPW. The power is fed by a probe at the center of the radial waveguide launching the m; n = 0; 0 cylindrical, omnidirectional outgoing PPW mode. Fig. 3 shows the admittance (3) of a generic slot versus its length for various slot thicknesses; namely ` = 0; 0:5 mm; 1:0 mm and 1:5 mm, that approximately correspond to =1; =35; =17 and =12, respectively, at the operating frequency of 17.2 GHz. In particular, with Amn we indicate the admittance between the rectangular waveguide modal current basis functions TEm0 and TEn0 used to expand the magnetic current on each slot at the interface between Region 2 and the slot volume. It is evident that the admittance A11 of the fundamental mode TE10 is significantly affected by the slot thickness, whose variation results in a A11 relative variation larger than 20%. Concerning higher order modes, their variation presents a trend similar to that shown in Fig. 3 with reference to A13 . Namely, a significant variation is observed just for small thickness. However, further increasing the slot thickness only slightly affects the admittance because, due the strong cutoff regime, a very weak field level reaches the other side of the slot at the interface with Region 1. The main effect of the slot admittance variation with its thickness is a change in the phase-shift experienced by the

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Fig. 3. Real and imaginary part of the admittance (3) of a pair of rectangular : mm slot versus its length, waveguide modal current basis functions of a w : GHz. at f

= 17 2

=08

217

Fig. 5. Gain versus frequency for the CP-RLSA sketched in the inset of Fig. 4, when the upper plate is made of a 0.5 mm thick, stainless steel sheet. TABLE I AVERAGE AND MAXIMUM ABSOLUTE ERROR WITH RESPECT TO THE EXACT NUMERICAL SOLUTION ON THE FREQUENCY BAND 16.2–17.8 GHz

Fig. 4. Exact gain solution versus frequency for various slot thicknesses (five rectangular waveguide modal current basis functions were used to represent the field on each slot).

feeding PPW mode, when propagating below the slot. Fig. 4 shows the RLSA gain versus frequency for various values of slot thickness. When compared to the gain obtained for the zero thickness slot case, it is apparent that even a small slot thickness results in an evident shift of the maximum gain frequency (about 300 MHz for a 1.5 mm thickness) and in a deformation of the gain curve shape. The observed maximum gain frequency shift is due to the above mentioned phase-shift introduced by the slot thickness. Indeed, the collective phase coherence between all the array slots is achieved by a slightly different electrical distance between each spiral turn. Since large gain RLSAs are moderately narrowband antennas, this frequency shift should not be neglected for an accurate antenna design and optimization. Fig. 5 compares the measured gain with the simulated one when the influence of the slot thickness is evaluated with the various methods described in the previous section. The gain curve obtained by considering zero thickness slots is also shown for comparison. It is worth noting that, despite the ripple due to a non-perfectly anechoic measurements environment, measured data fairly well match the calculated ones when the slot thickness is taken into account, thus experimentally confirming the poor accuracy of the zero thickness approximation. Table I shows the average and the maximum absolute error, in the frequency range 16.20–17.80 GHz, of the approximate solutions AWE and AGF for the three thicknesses considered previously. Furthermore, the computational complexity, the solving CPU time and the storage requirements of all the considered methods are summarized in Table II.

TABLE II SUMMARY OF THE COMPUTATIONAL COMPLEXITY AND STORAGE REQUIREMENT OF THE EXAMINED METHODS. A MATRIX FILL TIME OF 1058 S HAS TO BE ADDED TO ALL THE SIMULATIONS. CONCERNING THE CG ALGORITHM 395 ITERATIONS HAS BEEN REQUIRED TO REACH THE REQUIRED . THE CPU TIME IS RELEVANT TO A INTEL(R) RESIDUAL TOLERANCE OF CORE 2 DUO CPU, [email protected] GHz, 3 GB RAM

10

Regarding the antenna return loss, it is worth noting that a CP RLSA typically exhibits very good matching. Indeed, the reflection by each slot pair is very weak because the two slots are radially spaced by a quarter of a wavelength, thus providing two counterphased reflection contributions. Additionally, the spiral arrangement of slot pairs creates an intrinsically incoherent cumulative reflection at the feeding central point. The influence of the slot thickness on the input matching is highlighted in Fig. 6, that shows the reflection coefficient as a function of the frequency for various slot thicknesses. Despite a larger slot thickness seems to improve in this specific design the matching, the reflection coefficient is excellent for all the cases and it is not critically affected by the slot thickness. Conversely, the measured antenna reflection coefficient of about 020 dB in the whole frequency band is mainly due to the SMA connector which is not included in the simulations. To allow a fair comparison, the measurements were filtered to eliminate the SMA connector reflection by using the Network Analyzer time gate option.

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Fig. 6. Reflection coefficient generated only by the rim and the slots distribution versus frequency for the CP-RLSA sketched in the inset of Fig. 4.

This provides the dashed curve in Fig. 6, which agrees satisfactory with its simulated counterpart (` = 0:5 mm).

[4] M. Albani, G. La Cono, R. Gardelli, and A. Freni, “An efficient fullwave method of moments analysis for RLSA antennas,” IEEE Trans. Antennas Propag., vol. AP-54, no. 8, pp. 2326–2336, Aug. 2006. [5] M. Albani, A. Mazzinghi, and A. Freni, “Asymptotic approximation of mutual admittance involved in MoM analysis of RLSA antennas,” IEEE Trans. Antennas Propag., vol. AP-57, no. 4, pp. 1057–1063, Apr. 2009. [6] K. Sudo, T. Oizumi, J. Hirokawa, and M. Ando, “Reduction of azimuthal amplitude ripple in the rotating-mode feed to a radial waveguide by using a crossed dog-bone slot,” IEEE Trans. Antennas Propag., vol. 55, no. 9, pp. 2618–2622, Sep. 2007. [7] A. F. Peterson, S. L. Ray, and R. Mittra, Computational Methods for Electromagnetics. New York: IEEE Press, sec. 4.11. [8] C. R. Cockrell and F. B. Beck, “Asymptotic waveform evaluation (AWE) technique for frequency domain electromagnetic analysis,” NASA Tech. Memo. 110292, Nov. 1996. [9] E. Chiprout and M. S. Nakhla, Asymptotic Waveform Evaluation. Boston: Kluwer Academic Publishers, 1994. [10] R. F. Harrington and D. T. Auckland, “Electromagnetic transmission through narrow slots in thick conducting screens,” IEEE Trans. Antennas Propag., vol. AP-28, no. 5, pp. 616–622, Sep. 1980. [11] J. R. Mosig, “Scattering by arbitrarily-shaped slots in thick conducting screens: An approximate solution,” IEEE Trans. Antennas Propag., vol. 52, no. 8, pp. 2109–2117, Aug. 2004.

IV. RESULTS DISCUSSION AND CONCLUSION In the previous section a single RLSA design example was shown. The resulting discussion, however, leads to general conclusions since an analogous behavior was found for various other RLSA designs, not reported here for the sake of brevity. Table I shows that the application of the AGF, which uses a second step to improve the solution, gives the most accurate solutions. However, it also provides the shortest time saving (Table II, last row) when used as an alternative to the exact solution with LU factorization (Table II, second row). From Table II comes to light that the fastest method is the asymptotic waveform evaluation technique (AWE) of (4). Nevertheless, from Table I it is evident that this technique gives accurate results only for the 0:5 mm = =35 thickness. For thicker slots better accuracies are provided by the application of the single step approximate Green’s function (AGF) (i.e., the application of (14)), that presents an equal computational cost but a reduced storage requirement. We can conclude that in the design process of large RLSAs it is mandatory to take into account the slots thickness, even if in the order of =40 4 =20. To introduce this effect with reduced overall computational costs, it is possible to make use of approximate methods for the first design and optimization of the antenna, keeping the exact LU factorization procedure for a final optimization step. Among those approximate methods, on the basis of the tradeoff between accuracy and computational effort and storage reduction, the AWE seems to be the most convenient technique for slot thicknesses smaller than =30. For thicker slots the more robust single step AGF formulation is, however, preferable.

REFERENCES [1] M. Ando, K. Sakurai, N. Goto, K. Arimura, and Y. Ito, “A radial line slot antenna for 12 GHz satellite TV reception,” IEEE Trans. Antennas Propag., vol. 33, no. 12, pp. 1347–1353, Dec. 1985. [2] P. W. Davis and M. E. Bialkowski, “Linearly polarized radial-line slot-array antennas with improved return-loss performance,” IEEE Antennas Propag. Mag., vol. 41, no. 1, pp. 52–61, Feb. 1999. [3] J. I. Herranz-Herruzo, A. Valero-Nogueira, and M. Ferrando-Bataller, “Optimization technique for linearly polarized radial-line slot-array antennas using the multiple sweep method of moments,” IEEE Trans. Antennas Propag., vol. 52, no. 4, pp. 1015–1023, Apr. 2004.

Efficient Determination of the Poles and Residues of Spectral Domain Multilayered Green’s Functions That are Relevant in Far-Field Calculations Ana L. Fructos, Rafael R. Boix, Raúl Rodríguez-Berral, and Francisco Mesa

Abstract—In this work the total least squares algorithm (TLSA) is applied to the determination of the proper and improper poles of spectral domain multilayered Green’s functions that are closer to the branch point, and to the determination of the residues at these poles. The introduction of an adequate transformation in the spectral domain permits that the TLSA provides accurate values of the poles and residues, regardless of the proximity of the poles to the branch point. It is shown that the poles and residues supplied by the TLSA can be used to write the far field of the spatial domain Green’s functions in terms of closed-form expressions that are reliable in a wide variety of scenarios. Index Terms—Green’s functions, nonhomogeneous media, poles and zeros, surface waves.

I. INTRODUCTION The application of the method of moments (MoM) to the solution of mixed potential integral equations (MPIE) has proven to be an efficient numerical tool for the analysis of planar circuits and antennas

Manuscript received March 06, 2009; revised May 20, 2009. First published July 14, 2009; current version published January 04, 2010. This work was supported in part by the Spanish Ministerio de Educación y Ciencia and European Union FEDER funds (project TEC2007-65376) and in part by Junta de Andalucía (project TIC-253). A. L. Fructos and R. R. Boix are with the Microwaves Group, Department of Electronics and Electromagnetism, College of Physics, University of Seville, 41012-Seville, Spain (e-mail: [email protected]; [email protected]). R. Rodríguez-Berral and F. Mesa are with the Microwaves Group, Department of Applied Physics 1, ETS de Ingeniería Informática, University of Seville, 41012-Seville, Spain (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TAP.2009.2027344

0018-926X/$26.00 © 2009 IEEE

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(see [1] and references therein). When solving the MPIE, it is necessary to compute multilayered media Green’s functions (GF) that can be expressed as Sommerfeld integrals (SI). Although the numerical computation of SI is time consuming, several researchers have introduced the discrete complex image method (DCIM) [2], [3] as a mean to derive closed-form expressions of the spatial domain GF in terms of spherical waves. Unfortunately, the results obtained with the DCIM have been shown to deteriorate sharply for distances between source and observation points that are larger than a few wavelengths [4]. In order to avoid inaccuracies, Aksun et al. suggested the DCIM should be used up to a threshold distance, and then, a combination of far-field surface waves should be used beyond this threshold distance [5]. However, it has recently been shown that there are circumstances where the far field of spatial domain multilayered GF is dominated by a residual wave (belonging to the continuous spectrum of the spectral domain GF) rather than being dominated by surface waves [6]. The expressions of far-field surface and residual waves are obtained in terms of the proper poles of the spectral domain GF as well as those improper poles which are closer to the branch point k = k0 (and their corresponding residues) [6]. Therefore, an efficient and accurate algorithm for the determination of these proper/improper poles and residues should accompany the DCIM in order to ensure the fast derivation of closed-form expressions of spatial domain multilayered GF that are reliable both in the near-field and in the far-field. Most of the methods that have been proposed so far for the determination of the poles of spectral domain multilayered GF are computationally intensive because they require repeated numerical contour integrations in the complex spectral plane [7]–[10], and because Newton-Raphson refinement iterations may be additionally needed. Recently, two methods have been proposed which obtain estimates of the poles in terms of the roots of a polynomial, and then refine the poles by means of the Newton-Raphson procedure [11], [12]. Nevertheless, the computational burden of these two methods seems to increase quickly as the number of layers increases. In the present communication, the proper poles of the spectral-domain multilayered GF as well as those improper poles that are closer to the branch point (and the residues at all these poles) are computed 1=2 [9] in the total by introducing the transformation u0 = (k2 0 k02 ) least squares algorithm (TLSA) reported in [13] (the fundamentals of the TLSA can be consulted in [14]). Since the branch point of the spectral domain GF is removed by the aforementioned transformation, the novel implementation of the TLSA makes it possible to obtain the proper and improper poles with great accuracy regardless of the proximity between each pole and the branch point. Also, the computation of the poles and residues demands a very low CPU time consumption because the TLSA has a low computational burden (it only requires the computation of the spectral domain GF for a reduced number of samples N —typically 15  N  30—, the determination of the roots of a polynomial of degree N=2, and the solution of a system of linear equations with N equations [13]), and what is more important, because no Newton-Raphson refinement of the poles is needed for the particular dimensions of multilayered media that are encountered in practice in planar circuits and antennas. The results obtained for the poles and residues have been used for the computation of the far field of spatial domain multilayered GF. The far field results have been compared with those obtained via numerical integration of SI, and both sets of results have been found to match for distances between source and observation points beyond a few wavelenghts. II. BRIEF DESCRIPTION OF THE NUMERICAL METHOD Fig. 1 shows a multilayered medium consisting of lossy layers of complex permittivity "i = "0 "ri (1 0 j tan i ) and thickness hi (i =

219

Fig. 1. Multilayered lossy medium limited by air at the upper end and by a PEC at the lower end. The source and field points are arbitrarily located inside the multilayered medium.

1; . . . ; Nla ). Following the notation in [13], let Gn (k) (n = 0, 1) represent any of the spectral domain GF for the corrected vector and scalar potentials in formulation C of Michalski and Zheng [1]. The Sommerfeld integrals providing the spatial-domain counterpart of Gn (k ) can be written as 1 1 Gn (k )H (2) (k)kn+1 dk (1) n  4 01(SIP) (x 0 x0 )2 + (y 0 y0 )2 (see Fig. 1), Hn(2) (1) is the

Gn () =

where  = Hankel function of order n and second type, and SIP is the Sommerfeld integration path going across the first and third quadrants of the proper Riemann sheet of the complex k plane (see [6, Fig. 2]). In [13] it is suggested that Gn (k ) can be approximated by means of the following pole-residue representation:

Gn (k )  Gas n (k ) +

N i=1

an;i 2 k2 0 pn;i

(2)

, and the coefficients an;i and pn;i are where Gas n (k )  Gn (k )jk to be determined by means of the TLSA described in [13] (in fact, pn;i are estimates of the poles of Gn (k ), and an;i =2pn;i are estimates of the residues of Gn (k ) at these poles). The approximation of (2) presents two difficulties. On the one hand, whereas the functions Gn (k ) have a branch point at k = k0 , the functions used to approximate Gn (k ) in (2) do not present this branch point. As a consequence of this, since the samples of k used in the determination of an;i and pn;i are all chosen in the proper Riemann sheet of the complex k -plane (see [13, Fig. 1]), the poles pn;i capture the information about the proper poles of Gn (k ) (see [13, Table I]) but cannot provide information about the relevant improper poles (this information needs an additional fitting as commented in [6, Eq. (20)]). On the other hand, owing to the branch point of Gn (k ) at k = k0 , the approximation of (2) cannot reproduce with great accuracy those proper poles of Gn (k ) that are very close to k = k0 , as pointed out in [6] (in Section III it will be shown that the residues related to those poles are obtained with even less accuracy). In order to overcome these two difficulties, in this communication we propose to substitute the pole-residue representation of (2) by a new pole-residue representation given by



Gn (u0 )  Gas n (u0 ) +

N i=1

bn;i u0 0 qn;i

(3)

where u0 = k2 0 k02 is a new complex variable, and Gn (u0 ) is a single valued function of u0 without branch points [9]. The coefficients bn;i and qn;i of (3) are determined by working with the TLSA in the complex u0 -plane in a way completely parallel to that employed in [13]

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TABLE I NORMALIZED POLES OF THE SPECTRAL-DOMAIN GREEN’S FUNCTION K k . ,z z ,N , (a) Structure of Fig. 2 of [10]. f h : ," : ,  : ,N . (b) Structure of Fig. : ,z z ,N ,h : , 3(a) of [10]. f h : ,h : ," : ," : ," : ,    ,N

( ) = 10 GHz = = 0 =1 = 0 5 cm = 4 0 tan = 0 02 = 7 = 33 72 GHz = = 0 = 3 = 1 0 mm = 1 0 mm = 1 0 mm = 2 1 = 12 5 = 9 8 tan = tan = tan = 0 = 15

Fig. 2. Elliptic path chosen in the complex k -plane when applying the total least squares algorithm to (3). The upper half ellipse (solid line) is located in the proper Riemman sheet, and the lower half ellipse (dashed line) is located in the k " where " " ; ;" . improper sheet. k

= p

= max(

...

)

2 will be estimates of the poles of G (k ), and k02 + qn;i n 

(in this case,

[qn;i=

2 ]b k02 + qn;i n;i will be estimates of the residues of Gn (k ) at these poles). The samples used for u0 in the determination of bn;i and qn;i are chosen in the k -plane along the ellipse shown in Fig. 2. Since half the ellipse is in the proper sheet (Re(u0 ) > 0) of the k -plane and half the ellipse is in the improper sheet (Re(u0 ) < 0), the poles qn;i of (3) will not only correctly capture the information of the proper poles of Gn (u0 ) but will also capture the information of those improper poles that are closer to u0 = 0 (i.e., to the branch point k = k0 in the complex k -plane). Also, since Gn (u0 ) is a function of u0 without

branch points, the approximation of (3) does not present any problems concerning the accurate determination of the poles which are very close to u0 = 0 (i.e., very close to the branch point k = k0 in the complex k -plane) as it happens with the approximation of (2). When the approximation (3) is introduced in (1), the resulting integrals cannot be obtained in closed form. Fortunately, an asymptotic analysis makes it possible to obtain the expressions of the integrals for pp large . In order to write these expressions explicitly, let us define kn;i (i =pp1; . . . ; Npp ) as theppset of proper poles of Gn (k) for which pp Re(kn;i) > 0, and let Rn;i be the residue of Gn (k ) at k = kn;i . ip Let kn;1 be the improper pole of Gn (k ) which is closest to k0 , and ip ip c let Rn; 1 be the residue of Gn (k ) at k = kn;1 . Also, let kn be the pole of Gn (k ) (proper or improper) which is closest to k0 (i.e., pp ip c c jkn 0 k0 j  jkn;i 0 k0 j (i = 1; . . . ; Npp ) and jkn 0 k0 j  jkn;1 0 k0 j), 2 c c c let qn = (knc ) 0 k02 (where Re(qn ) > 0 if kn is a proper pole, and Re(qnc ) < 0 if knc is an improper pole), let Rnc be the residue of Gn (k ) c = [kc = (kc )2 0 k2 ]Rc . Then, an asymptotic at k = knc , and let bn n n 0 n analysis similar to that of [6, Sec. II] shows that Gn () can be approximated for large  as

Gn ()  Gas n () 0 2

u

Re

N

j pp 2 i=1 Rn;i

pp kn;i 0 k0

ip Hn(2) kn; 1 u n + (jk0 ) fRW () 2

Re

pp kn;i 0

n+1

j ip 2 Rn;1

ip k0 0 kn; 1

u

n+1 0

Im

( ) = 9 GHz = = 0 = 1 = 4 0 tan = 0 02 = 8

= 0 5 cm

contribution of knc . In fact, when jknc 0 k0 j=k0  0:1, it can be shown that fRW () is approximately given by (see [6, Eq. (21)])

bc e0jk  fRW () = n p 2  2

c c 2 + q2jnk exp j (q2nk)0  0 c qnc  sgn Im j 2jk + erf qn2jk

 

p

0

p

p

p

0

p

0

(5)

where erf(1) is the error function [15] and sgn(1) is the sign function. However, when knc is not close to k0 , all the proper and improper poles estimated in (3) have equally relevant contributions to fRW () of the type shown in (5) (note that this circumstance was not considered in [6]). In fact, when jknc 0 k0 j=k0 > 0:1, fRW () can be approximated by

fRW () =

pp Hn(2) kn;i  ip kn; 1

TABLE II NORMALIZED POLES AND RESIDUES OF THE SPECTRAL-DOMAIN GREEN’S : ,z z ,N ,h , FUNCTION K k . f : , : ,N " 

jk0 e0jk  N bn;i 2 22 i=1 qn;i

(6)

where the large argument approximation of the error function [15] has been used in (5) for the derivation of (6). ip kn; 1

(4)

where u(1) is the unit step function (u(x) = 1 if x > 0 and u(x) = 0 if x < 0), and fRW () is the residual wave (see [6, Eq. (12)] for details). The Hankel functions represent surface waves. In case the pole knc is very close to the branch point k0 , fRW () is dominated by the

III. RESULTS In Tables I(a) and (b) we present results for the normalized proper poles of the spectral scalar potential GF of a lossy one-layered substrate and a lossless four-layered substrate respectively. The results obtained for the poles in [10] are compared with those obtained with the numerical code of [9], and with those obtained by means of (3). Please

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221

TABLE III CONVERGENCE OF THE NORMALIZED POLES AND RESIDUES OF THE ( ) WITH RESPECT TO THE SPECTRAL-DOMAIN GREEN’S FUNCTION = 0, = 4, USED IN (3). = 10 GHz, = NUMBER OF TERMS = 1 5 mm, = 1 0 mm, = 1 5 mm, = 1 5 mm, = 2 1, = 12 5, = 9 8, = 8 6, tan = tan = tan = 0, = 0 08. (a) Normalized poles; (b) Normalized residues tan

h

:

"



h : "

N

:

K k f z z : h : h : : " :  

N " 

:

K

Fig. 4. Magnitude of the spatial-domain Green’s function for the fourlayered medium studied in Fig. 3. Numerical integration results (solid line) are compared with those obtained via (4) and (5) (+), and (4) and and (6) ( ). = 0, = 7. = 2 5 GHz, =

f

:

z z

N



the spectral GF, for the improper pole closer to the branch point, and for the residues at all these poles. Please note that the use of N = 11 in the pole-residue representation suffices to obtain all the poles and residues with an accuracy of six significant figures. In Fig. 3 it can be observed that (4) and (5) accurately match the far-field of the vector potential GF and scalar potential GF of a four-layered substrate in a large range of values of  (roughly when k0  1). The observed good agreement is due to the fact that the pole knc is very close to the branch point for the two GFs at the operation frequency (5.6 GHz). In the case treated in Fig. 3, (4) and (6) only provide accurate results for very large  since the approximation of erf(qnc =(2jk0 ))

K

K

h

:

Fig. 3. Magnitude of the spatial-domain Green’s functions and for a four-layered medium. Numerical integration results (solid line and dashed line) ). are compared with those obtained via (4) & (5) ( +), and (4) & (6) ( = 5 6 GHz, = 0, = 4, = 1 5 mm, = 1 0 mm, = = 1 5 mm, = 1 5 mm, = 2 1, = 12 5, = 9 8, = 8 6, tan = tan = tan = 0, tan = 0 08, = 10.

f h



: :

h 

z

z :



"

N

2; h : : " : "  : N

: "

4;

:

used in (6) is very coarse when qnc = (knc )2 0 k02 is very small. However, Fig. 4 shows a case where the approximation used in (6) is accurate. Now, (4) and (5) do not correctly reproduce the far field of the vector potential GF whereas (4) and (6) accurately reproduce this 0:5. In order to explain this behavior, it should be far field for k0  considered that, at the operation frequency studied in Fig. 4 (2.5 GHz), A only has improper poles, and all these improper poles are relaKxx tively far from the branch point. Under these circumstances, all contributions of the poles of (3) to the residual wave term, fRW (), are relevant and must be considered as in (6). In (5), all contributions to fRW () are neglected except that of knc , and therefore, the results obtained with (5) are incorrect for the present case. IV. CONCLUSION

note that all sets of results coincide within at least five significant figures. It should be pointed out that whereas the methods of [10] and [9] require some iterations for the refinement of the poles encountered, the TLSA provides the poles without any refinement. Numerical simulations have shown that the refinement of the poles supplied by the TLSA is not needed provided N i=1 hi =k0 , which is a condition fulfilled by the multilayered substrates used in planar circuits and antennas in practice. In Table II we show normalized poles and residues of the spectral scalar potential GF that have been obtained by means of (2) and (3). Whereas the results obtained for the pole more distant pp and Rpp ) coincide, the refrom the branch point and its residue (k02 02 sults obtained for the pole in close proximity to the branch point and its pp residue disagree. In particular the residue R01 provided by (2) is 7% larger than that provided by (3), and this latter result has been checked to be the correct result. This indicates (3) is clearly more accurate than (2) for the estimation of a proper pole and its residue when this pole is very close to the branch point. Finally, in Tables III(a) and (b) we show how the predictions of (3) converge for the two proper poles of

In this communication, a suitable transformation in the spectral domain makes it possible the application of the TLSA to the fast and accurate computation of the proper/improper poles of spectral domain multilayered GF which are closer to the branch point, and to the computation of the residues at these poles. The poles and residues supplied by the TLSA have proven to be useful in closed-form expressions which express the far-field of spatial domain multilayered GF in terms of a combination of surface waves plus a residual wave.

REFERENCES [1] K. A. Michalski and J. R. Mosig, “Multilayered media Green’s functions in integral equation formulations,” IEEE Trans. Antennas Propag., vol. 45, pp. 508–519, Mar. 1997. [2] Y. L. Chow, J. J. Yang, D. G. Fang, and G. E. Howard, “A closed-form spatial Green’s function for the thick microstrip substrate,” IEEE Trans. Microw. Theory Tech., vol. 39, pp. 588–592, Mar. 1991. [3] M. I. Aksun, “A robust approach for the derivation of closed-form Green’s functions,” IEEE Trans. Microwave Theory Tech., vol. 44, pp. 651–658, May 1996.

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[4] N. V. Shuley, R. R. Boix, F. Medina, and M. Horno, “On the fast approximation of Green’s functions in MPIE formulations for planar layered media,” IEEE Trans. Microw. Theory Tech., vol. 50, pp. 2185–2192, Sep. 2002. [5] M. I. Aksun and G. Dural, “Clarification of issues on the closed-form Green’s functions in stratified media,” IEEE Trans. Antennas Propag., vol. 53, pp. 3644–3653, Nov. 2005. [6] F. Mesa, R. R. Boix, and F. Medina, “Closed-form expressions of multilayered planar Green’s functions that account for the continuous spectrum in the far field,” IEEE Trans. Microw. Theory Tech., vol. 56, pp. 1601–1614, Jul. 2008. [7] F. Ling and J.-M. Jin, “Discrete complex image method for Green’s functions of general multilayered media,” IEEE Microw. Guided Wave Lett., vol. 10, pp. 400–402, Oct. 2000. [8] S.-A. Teo, S.-T. Chew, and M.-S. Leong, “Error analysis of the discrete complex image method and pole extraction,” IEEE Trans. Microw. Theory Tech., vol. 51, pp. 406–413, Feb. 2003. [9] R. Rodríguez-Berral, F. Mesa, and F. Medina, “Appropriate formulation of the characteristic equation for open nonreciprocal layered waveguides with different upper and lower half-spaces,” IEEE Trans. Microw. Theory Tech., vol. 53, pp. 1613–1623, May 2005. [10] A. G. Polimeridis, T. V. Yioultsis, and T. D. Tsiboukis, “An efficient pole extraction technique for the computation of Green’s functions in stratified media using a sine transformation,” IEEE Trans. Antennas Propag., vol. 55, pp. 227–229, Jan. 2007. [11] H. Rogier and D. V. Ginste, “A fast procedure to accurately determineleaky modes in multilayered planar dielectric substrates,” IEEE Trans. Microw. Theory Tech., vol. 56, pp. 1413–1422, Jun. 2008. [12] B. Wu and L. Tsang, “Fast computation of layered medium Green’s functions of multilayered and lossy media using fast all-modes method and numerical modified steepest descent path method,” IEEE Trans. Microw. Theory Tech., vol. 56, pp. 1446–1454, Jun. 2008. [13] R. R. Boix, F. Mesa, and F. Medina, “Application of total least squares to the derivation of closed-form Green’s functions for planar layered media,” IEEE Trans. Microw. Theory Tech., vol. 55, pp. 268–280, Feb. 2007. [14] S. V. Huffel and J. Vandevalle, The Total Least Squares Problem: Computational Aspects and Analysis. Philadelphia: SIAM, 1991, vol. 9, Frontiers in Applied Mathematics. [15] M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, 9th ed. New York: Dover Publications, 1970.

Electromagnetic Scattering From a Slotted Conducting Wedge Jin J. Kim, Hyo J. Eom, and Keum Cheol Hwang

Abstract—Electromagnetic wave scattering from a slotted conducting wedge is studied. The Kontorovich-Lebedev transform and the mode-matching are utilized to obtain a convergent series solution. Computations are performed to illustrate scattering from slotted conducting wedges. Grooved conducting wedges are also treated as a special case. The mode-matching formulation provides a stable and convergent series solution. Index Terms—Kontorovich-Lebedev transform, mode matching.

I. INTRODUCTION Electromagnetic wave scattering from conducting wedges with slots, apertures, and cavities is a canonical problem in antennas and microwave engineering. Although many studies [1]–[8] were performed to better understand wedge scattering characteristics, it is still of theoretical interest to derive a rigorous solution for scattering from a slotted conducting wedge. In order to obtain such a rigorous solution, we need to solve a boundary-value problem of scattering from a slotted conducting wedge using the boundary conditions. The purpose of the communication is to solve this boundary-value problem by using the Kontorovich-Lebedev transform and the mode-matching method, which is based on eigenfunction expansions. The Kontorovich-Lebedev transform and the mode-matching method were first applied to scattering from a conducting wedge having a single groove [9]. Scattering from a multiply slotted/grooved conducting wedge is obviously an extension of the work considered in [9]. The present work addresses the problem of multiply slotted/grooved conducting wedges, which find some practical applications in antennas, radar-cross-sections, and frequency-selective surfaces. The Kontorovich-Lebedev transform and the mode-matching method will be applied to form a convergent series solution. Numerical computations are also performed to check the convergence of series formulation.

II. SLOTTED WEDGE Consider electromagnetic scattering from a slotted conducting wedge, as shown in Fig. 1. The number of slots is finite . Slots are uniformly distributed with a period . The incident field ( -component) takes the form of uniform plane wave as

N

T

E (; ) = E0 e0

0 )

ik cos(

i z

z

(1)

0 is the incident angle. where is the wave number (= 2 ) and The time-harmonic convention 0i!t is suppressed and the cylindrical

k

e

=



Manuscript received April 02, 2009; revised June 23, 2009. First published July 14, 2009; current version published January 04, 2010. This work was supported by the Ministry of Knowledge Economy (MKE), Korea, under the Information Technology Research Center (ITRC) support program supervised by the National IT Promotion Agency (NIPA) (NIPA-2009-(C1090-0902-0034)). J. J. Kim and H. J. Eom are with the Department of Electrical Engineering, KAIST, Daejeon 305-701, Korea (e-mail: [email protected]). Keum Cheol Hwang is with the Department of Electronics Engineering, Division of Information Technology, Dongguk University, Seoul 100-715, Korea. Digital Object Identifier 10.1109/TAP.2009.2027454 0018-926X/$26.00 © 2009 IEEE

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coordinates (; ; z ) are used. The scattering geometry is two-dimensional (; ) and the electric field vector has a z -component. The primary TM (transverse-magnetic to the z -axis) electric field for an unslotted conducting wedge is 4E0 Ezp (; ) =

1

0

 ( i) m=1

J

0 (k) sin(m  ) sin(m )

223

Multiplying (10) by R (k) and integrating with respect to  from q as al to bl , we obtain equations for the modal coefficients Aqm and Bm follows:

1

4E0

(2)

where J ( 1 ) is the Bessel function of the first kind, = 2 0 t , and m = m= . To represent the scattered field in region (I), we utilize a Kontorovich-Lebedev transform pair [10]

1 01 (1) (3)  H (k)f ()d 0 i1 1 f () = J (k)F ( )d (4) 2 0i1 (1) where J ( 1 ) and H ( 1 ) are the Bessel function of the first kind and

m=1 N 1

0 1

=

0

m (0i)

l sin(m  )Kn (m )

l i  l ei  I1 0 I2 + im mn ql Aqm e

q =0 m=1 N 1 + q =0 m=1

0 1

q l 0i  l e0i  I1 0 I2 0 im mn ql Bm e

(11)

F ( ) =

where ql is the Kronecker delta and

Knl ( ) = 0

the Hankel function of the first kind, respectively. Here F ( ) is a Kontorovich-Lebedev transform of f (). The scattered field in region (I) is written as

1 ~ 0 ( )e0i]d: ~ + ( )e+i + E J (k)[E 0i1 

=

I1

=

I2

=

i

1 EzI (; ) = 2

(5)

The scattered field in region (II) is

EzII (; ) =

l mn

1 m=1

Aqm ei

0

( )

0

0

q i ( ) + Bm e

R

(k)

(6)

where R (k) = J (k)N (kbq ) 0 J (kbq )N (k); bq = q b + qT and q = 0; 1; 2; . . .. The parameter m is determined by the relation R (kaq ) = 0, where aq = a + qT . Next is to form a set of q simultaneous equations for the modal coefficients Aqm and Bm by enforcing the boundary conditions at the slotted wedge surface. The total field in region (I) is a sum of Ezp (; ) and EzI (; ). The tangential electric field continuities at  = 0 and  = are given as

j

j

Ezp (; ) =0; + EzI (; ) =0; EzII (; ) =2; ; = 0;

j

aq <  < bq otherwise

(7)

1

2



J (kal )J (kbl ) 0 J (kbl )J (kal )

kb ka 0; i

J (kal )  2 0 nl 2

 01 R2

m=n m 6= n 1 2q cos( )  I ( )K l ( ) d 0i1 m n sin( )

2

( )d;

1 2q 1  Im ( )Knl ( ) d: sin(  ) 0i1

(12) (13) (14)

i

1 2

(15)

It is expedient to convert I1 and I2 into convergent series by applying residue calculus to (14) and (15). The results are in Appendix A. Although I1 is similar to I in [9], we present it here for completeness. Similarly, the tangential continuity at  = gives 4E0

=

1

m=1 N 1

m (0i)

0 1

0

l sin(m  ) cos(m )Kn (m )

l l ei  I2 0 I1 + im mn ql Aqm

q =0 m=0 N 1 + q =0 m=0

0 1

q l l e0i  I2 0 I1 0 im : mn ql Bm

(16)

q We solve (11) and (16) for Aqm and Bm by truncating the infinite series. I q by using Next we represent the field Ez (; ) in terms of Aqm and Bm residue calculus. The results are summarized in Appendix B.

Applying the Kontorovich-Lebedev transform to (7), we obtain

0 1

N 1

~ 6 ( ) = 7 E

6

q =0 m=1 N 1

0 1

q =0 m=1

Aqm ei 

0

q i  + Bm e

q q (Am + Bm )

e7i q ( ) Im 2i sin( )

q ( ) Im 2i sin( )

(8)

III. GROOVED WEDGE Consider scattering from a grooved conducting wedge in Fig. 2, which is a special case of Fig. 1. The solution to the problem in Fig. 2 can be obtained from Section II. The fields in regions (I) and (II) are given by (5) and (6) after the following substitutions: ~ + ( ) ~ 0 ( ) 0! 0ei2 E E

where q Im ( ) = 0

(1) 2 H (kaq )J (kbq )



0

H(1) (kbq )J (kaq ) : q 2 J (kaq )  2 m

0

q Bm

(9)

The tangential magnetic field continuity (H ) at  = 0 over the (l + slot is

1)th

Hp (; 0) + HI (; 0) =

HII (; 2 );

al <  < bl :

(10)

0! 0e

i2 

Aqm

(17) (18)

The boundary conditions need to be enforced at  = 0. The tangential electric field continuity at  = 0 gives ~ + ( ) = 0 E

0 1

N 1

q =0 m=1

Aqm ei( 

0 ) sin [qm (t 0 q )] Imq ( ): sin( )

(19)

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<  < 1; 0 < a = a + qT;

Fig. 1. Slotted, perfectly conducting wedge. Region (I): 0 , Region (II): 2 : period, = + = 0 1 2 . . ..

< b b qT; q

a <  < b ; <  < ; T ; ; ;

Fig. 3. Magnitude of scattered electric field at slot aperture with different mode =1 = 4 mm, = 24 mm, = 60 =1 = 60 , numbers. and = 15 GHz.

f

N

;a

b

;E



;

TABLE I CONVERGENCE OF MODAL COEFFICIENTS

A AND B . NOTE C = [A e +B e ]R (k) WHEN  = 2 AND  = 9 MM. THE OTHER PARAMETERS USED IN COMPUTATION ARE THE SAME AS

Fig. 2. Grooved, perfectly conducting wedge. Region (I): 0 <  < 1; 0 <  < , Region (II): a <  < b ; +  <  < 2; T : period, a = a + qT; b = b + qT; q = 0; 1; 2; . . ..

The

H continuity at  = 0 over the groove apertures gives 4E0 1 m (0i) sin(m 0 )K l (m ) n

m=1 N 01 1 Aqm ei  fsin [qm (t 0 q )] I1 = 2i

THOSE IN FIG. 3

TABLE II MAGNITUDE OF SCATTERED ELECTRIC FIELD AT GROOVE APERTURE. = 4 mm, = 24 mm, = 70 = 10 =1 =1 = 60 = 15 GHz, = 0 and = 14 mm

N

;a 

;f

b



; 



;E

q =0 m=1 l l m m

l + cos  (t 0 l ) mn ql : (20) When N = 1, the final expression (20) is reduced to (6) in [9]. We q

need to solve (20) for the unknown coefficient field in region (I) is given by

EzI (; )=2i

0 1

N 1

q =0 m=1

Aqm ei



Am and the scattered

sin [qm (t 0 q )] P1 (; )

(21)

P ; ) is given by (28).

where 1 (

IV. NUMERICAL RESULTS Computations were performed to check formulation accuracy and scattering characteristics. Fig. 3 shows the behavior of electric field magnitude at the slot aperture j zI ( 0)j when different mode numbers

E ;



;

q plus addiare used. Two propagation modes corresponding to real m tional evanescent modes are used. Table I illustrates the convergence rate of these coefficients. As the number of evanescent modes increases, our results tend to approach to the results of a finite element solver COMSOL Multiphysics [11]. In addition, Table II illustrates the con)j for a grooved wedge. As max [number vergence rate of j zI ( of modes retained in the truncated series (21)] increases from 1 to 5, j zI ( )j at the center of a groove aperture converges rapidly. Our experience indicates that one or two evanescent modes in addition to the propagation modes yields acceptable results. Fig. 4 illustrates the electric field magnitude at the slot aperture ( = 0 ), where one, two, and three propagation modes are used for three different frequencies 7.5, 15, and 25 GHz, respectively. Our results agree well with the COMSOL Multiphysics. Fig. 5 shows the behavior of electric field magnitude near

E ; 

m

M

E ; 





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Fig. 4. Magnitude of scattered electric field at slot aperture ( = 0 ) with different frequencies. N = 1; a = 4 mm, b = 24 mm,  = 60 ; E = 1, and  = 60 .

Fig. 5. Contour plot of total electric field near slots. N = 3; a = 2 mm, b = 12 mm, T = 12 mm,  = 60 ; E = 1;  = 60 , and f = 30 GHz. a = ). The widths of slots are chosen the same (b

0

225

Fig. 6. Contour plot of total electric field near grooves. N = 3; a = 2 mm, b = 12 mm, T = 12 mm,  = 50 ;  = 10 ;  = 30 ;  = 20 ; E = 1;  = 60 , and f = 30 GHz.

Fig. 7. 60 mm,

jE 

j

(; 0) versus  in slotted half-plane. N = 1; a = 30 mm, = 0 ; E = 1;  = 175 , and f = 10 GHz.

b

=

the slotted wedge when the incident angle is 0 = 60 . While strong standing-wave patterns appear near  = 0 , the field near the back side of a slot ( = 300 ) becomes weaker. Fig. 6 illustrates the behavior of electric field magnitude near the grooved conducting wedge with N = 3. The field near the back side of grooved surface ( = 310 ) is almost negligible due to the shadowing effect. Fig. 7 compares our theory with [12] for a slotted half plane (N = 1 and t = 0 ) when the incident angle is 0 = 175 . Four modes (two propagation and two evanescent modes) were used in computation to achieve a stable and convergent series solution. Good agreement is seen between them. Fig. 8 shows scattering from a multiply slotted half-plane (N = 3 and t = 0 ) when the incident angle is 0 = 60 . Wave patterns near the slot appear to be more periodic than the cases in Figs. 5 and 6.

V. CONCLUSION The boundary-value problem of electromagnetic TM wave scattering from a slotted conducting wedge has been rigorously solved. The Kontorovich-Lebedev transform and the mode-matching were applied to obtain convergent series representations. Computed results agreed well with other existing ones. The formulation of Kontorovich-Lebedev transform and mode-matching is very efficient for

Fig. 8. Contour plot of total electric field near multiply slotted half-plane. N = 3; a = 2 mm, b = 12 mm, T = 12 mm,  = 0 ; E = 1;  = 60 , and f = 30 GHz.

numerical computation and is expected to be useful for the study of slotted conducting wedge scattering.

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APPENDIX A SERIES REPRESENTATIONS OF INTEGRALS I1

4i 1 = 0 

AND I2

3qlmn (p ) q 2 l2 p2 0 m p2 0 n p=1 ql0 0 i cot (mq ) 3mn (mq ) mn ql ql ( ) 4i 1  2 3mn p I2 = 0 p q 2 p  2 0 m l2  p2 0 n (01) p p=1 ql0 q 0 i 3sinmn((mqm )) mn ql I1

2

p

(22)

(23)

where

0mn ( ) ql 3mn ( ) = J (ka q )J (kal )  ql 0mn ( ) = H(1) (kaq )J (kal )J (kbq )J (kbl ) 0 H(1) (kaq )J (kbl )J (kbq )J (kal ) 0 H(1) (kbq )J (kal )J (kaq )J (kbl ) + H(1) (kbq )J (kbl )J (kaq )J (kal ) ql ql0 3mn ( ) = @ 3mn( ) ql

@

(24)

(25) (26)

APPENDIX B SERIES REPRESENTATIONS OF FIELD EzI (; )

( )=

I Ez ; 

0 1

N 1

q =0 m=1

+ Bmq e0i  P1 (; ) N 01 1 (Aqm + Bmq ) P2 (; ) +

q i  Am e

(27)

q =0 m=1

where

( ) = 02i

P1 ; 

1 p=1

p

sin(p )

(kbq ) 0 B()J (kaq ) q 2

J (kaq ) p2 0 m [qm ( 0 )] 0 iC () sinsin( q m ) 1 P2 (; ) = 2i p sin(p ) 2

()

A  J

(28)

p=1

( ) (kbq ) 0 B()J (kaq ) (01)p J (kaq ) p2 0 mq 2 (mq ) + iC () sin sin(mq )

2

( ) ( )  < aq ( ) ( ) otherwise (1) H (k)J (kbq );  > bq B () = (1) J (k)H (kbq ); otherwise (1) H (k)J (kbq ) C () = 0J (k)H(1) (kbq ); aq <  < bq 0; otherwise p p = :

( )=

A 

A  J

(29)

(1) k H kaq ; (1) H k J kaq ;

J

(30)

(31)

(32) (33)

REFERENCES [1] J. W. Silvestro and C. M. Butler, “Scattering from a right interior angle wedge loaded by a slot,” Proc. IEEE Southeastcon, vol. 2, pp. 555–558, Apr. 1989. [2] J. W. Silvestro and C. M. Butler, “TE scattering from a perfectly conducting wedge loaded by a slot,” Proc. Inst. Elect. Eng. Microw. Antennas Propag., vol. 139, no. 1, pt. H, pp. 105–109, Feb. 1992. [3] J. Silvestro, “Scattering from slot near conducting wedge using hybrid method of moments/geometrical theory of diffraction: TE case,” Electron. Lett., vol. 28, no. 11, pp. 1055–1057, May 1992. [4] M. Calamia, R. Coccioli, G. Pelosi, and G. Manara, “A hybrid FEM/UTD analysis of the scattering from a cavity-backed aperture in a face of a perfectly conducting wedge,” Int. Journal for. Computation and Mathematics in Electrical and Electronic Engineering, vol. 13, pp. 229–235, 1994. [5] G. Pelosi, R. Coccioli, G. Manara, and A. Monorchio, “Scattering from a wedge with cavity backed apertures in its faces and related configurations: TE case,” Proc. Inst. Elect. Eng. Microw. Antennas Propag., vol. 142, no. 2, pp. 183–188, Apr. 1995. [6] A. Borgioli, R. Coccioli, G. Pelosi, and J. L. Volakis, “Electromagnetic scattering from a corrugated wedge,” IEEE Trans. Antennas Propag., vol. 45, no. 8, pp. 1265–1269, Aug. 1997. [7] S. Alfonzetti, G. Borzi, and N. Salerno, “An iterative solution to scattering from cavity-backed apertures in a perfectly conducting wedge,” IEEE Trans. Magn., vol. 34, no. 5, pp. 2704–2707, Sep. 1998. [8] A. Freni, “Finite element formulation for the electromagnetic analysis of a metallic wedge arbitrarily loaded and shaped near the edge,” Proc. Inst. Elect. Eng. Microw. Antennas Propag., vol. 146, no. 3, pp. 175–180, Jun. 1999. [9] K. C. Hwang, “Scattering from a grooved conducting wedge,” IEEE Trans. Antennas Propag., vol. 57, no. 8, pp. 2498–2500, Aug. 2009. [10] G. D. Maliuzhinets, “Relation between the inversion formulas for the sommerfeld integral and the formulas of Kontorovich-Lebedev,” Soviet Phys. Dokl., vol. 3, pp. 266–268, 1958. [11] [Online]. Available: http://www.comsol.com/ [12] J. L. Tsalamengas, “TM and TE diffraction by a perfectly conducting half-plane in presence of a perfectly conducting strip: Solution by exponentially converging Nystrom and Galerkin methods,” IEEE Trans. Antennas Propag., vol. 56, no. 5, pp. 1358–1367, May 2008.

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The Optimal Spatially-Smoothed Source Patterns for the Pseudospectral Time-Domain Method Zhili Lin Abstract—Spatially-smoothed sources are often utilized in the pseudospectral time-domain (PSTD) method to suppress the associated aliasing errors to levels as low as possible. In this work, the explicit conditions of the optimal source patterns for these spanning sources are presented based on the fact that the aliasing errors are mainly attributed to the high spatial-frequency parts of the time-stepped source items and the source patterns are thereafter demonstrated to be exactly corresponding to the normalized rows of the Pascal’s triangle. The excellent performance of these optimal sources is verified by the practical 1-D, 2-D and 3-D PSTD simulations and also compared with that of non-optimal sources. Index Terms—Discrete Fourier transform (DFT), pseudospectral timedomain (PSTD) method.

I. INTRODUCTION The pseudospectral time-domain (PSTD) method has been widely applied to simulate various electromagnetic and acoustic problems since its emergence at the end of last century [1], [2]. Basically, it uses discrete Fourier transform (DFT) or Chebyshev transforms to calculate the spatial derivatives of the electric and magnetic field components both arranged in the same positions in an unstaggered space lattice of unit cells. Because the spatial-differencing process in PSTD method converges with infinite order accuracy for a low sampling density of two points per shortest wavelength, it renders lower numerical phase-velocity errors than those of the finite-difference time-domain (FDTD) method and therefore allows problems of much larger electrical size to be modeled [3]. However, the DFT, implemented by the famous fast Fourier transform (FFT), has difficulty in correctly representing the Kronecker delta function. With a single-cell source applied, zigzag wiggles are arising apparently on the excited source waves, referred to as the Gibbs phenomenon [4]. To alleviate these aliasing errors, spatially smoothed sources spanning a few (4  6) cells in each coordinate direction were proposed by Liu [5], but without further details on their optimal patterns. The compact two-identical-cell source has also been investigated by Lee and Hagness [6]. However, their study is only based on the method of comparison and the specific guidelines for the optimal source patterns remain unreported in the literature. In this work, the explicit conditions for the optimal source patterns with lowest levels of aliasing errors are investigated according to the fact that the aliasing errors are mainly attributed to the high spatial-frequency components of the added source items at each time step. We further show that the magnitude distributions of these optimal source patterns are exactly corresponding to the normalized rows of Pascal’s triangle. The practical 1-D, 2-D and 3-D PSTD simulations are conducted to verify the validity and performance of these proposed sources. II. FORMULATION Supposing a TEM plane wave propagating in the free space along the

x axis with the electric field vector oriented in the z direction. The 1-D

Manuscript received April 11, 2008; revised June 26, 2009. First published November 06, 2009; current version published January 04, 2010. This work was supported in part by the program between the China Scholarship Council and the Royal Institute of Technology, Sweden. The author is with the School of Instrumentation Science and Opto-electronics Engineering, Beihang University, Beijing 100191, China (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2009.2036198

227

grid space is composed of N cells indexed by i = 0; 1; 2; . . . ; N 0 1. According to the PSTD algorithm, updating equations for the normalized electric and magnetic fields excited by the source item S n+1=2 are given by

Ezn+1=2 = Ezn + cxF 01 Kx 1 F Hyn+1=2

(1)

Ezn+1 n Hy +3=2

= Ezn+1=2 + S n+1=2 (2) (3) = Hyn+1=2 + cxF 01 Kx 1 F Ezn+1 where cx = c1t=1x with c, 1t and 1x being the speed of light in

vacuum, the time step and cell size for a specific simulation, respectively; F and F 01 are the forward and inverse DFT defined by

X (k) = x(i) =

N 01

x(i)e0j

i=0 N 01

1

N k=0

k = 0; . . . ; N 0 1

ik ;

X (k)ej ki ;

i = 0; . . . ; N 0 1

(4) (5)

and the differential factor Kx is given by

Kx (k) =

j 2k=N

0 j 2(k 0 N )=N:

k = 0; . . . ; N=2 0 1 k = N=2 k = N=2 + 1; . . . ; N 0 1:

(6)

Assume that the superimposed item in (2) is

S n+1=2 = [0; . . . ; 0; a0 ; a1 ; . . . ; am01 ; 0; . . . ; 0]sn+1=2

(7)

a0 + a1 + . . . + am01 = 1:

(8)

comprising m source cells, say, the i0 ; i1 ; . . . ; im01 th cells with normalized amplitudes satisfying

In fact, both the soft and hard source cases have been considered in (2). n+1=2 by letting sn+1=2 = That is, (7) is standing for a soft source Ss n +1 =2 by letting n +1 = 2 or standing for a hard source Sh f

sn+1=2 = f n+1=2 0

m01 k=0

Ezn+1=2 (ik )

(9)

where f n+1=2 denotes the temporal driving function f (t) at time step n + 1=2. The pattern of the source item S n+1=2 in (2) is crucial to the problem we concerned. With (7), the spatial spectrum S (k) of the pattern [a0 ; a1 ; . . . am01 ] is

S (k) = F [S n+1=2 =sn+1=2 ] =

m01 l=0

al e0j

ki

(10)

where k stands for the discrete spatial frequency and is physically equivalent to 1=(21x) for k = N=2. Based on [(1)–(3)], we can further define that the dominant aliasing errors of the magnetic and electric fields independently caused by S n+1=2 are

n+3=2 = c sn+1=2F 01 fK 1 S g H x x En+2 = cx2 sn+1=2F 01 Kx2 1 S

(11) (12)

on the cells away from the source region. From (11) and (12), we n+3=2 and note that the spatial spectrums of the aliasing errors H n +2 E are proportional to the products of Kx and S , Kx2 and S , respectively. In Fig. 1, we show the magnitude of the spatial spectrums

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Fig. 1. Magnitudes of the spatial spectrums of

K

N K a ;a ; a k N= S N=

K

and

K

with

N = 128.

of x and x2 with = 128, both possessing ample high-spatial-frequency components. Thus, to suppress the aliasing errors, the pattern [ 0 1 . . . m01 ] should be optimally designed to make ( ) as a decreasing discrete function, as rapidly as possible, in the 2 and become zero at = 2. This requires region 0   ( 2) = 0 and j j should simultaneously reach their minimum values in the high discrete frequency range near 2, such as at 2 0 1, 2 0 2 and so on. Based upon the Lagrange multiplier method, we have the following necessary conditions for the optimal source patterns

Sk S N= N=

k

N= N=

m01

(13) S (N=2) = (01)l al = 0 l=0 @LLN=20p = @LLN=20p = . . . = @LLN=20p = 0 (14) @a0 @a1 @am01 for p = 1; 2; . . . ; pmax , where the Lagrange function LN=20p is de-

fined by

LN=20p(a0 ; . . . ; am01 ; p) = S N2 0 p + pS N2 m01 = al e0j ( 0p)l l=0

+ p



l=0

with p being the Lagrange multiplier. From (15), the (14) for each can be simplified as m01

p

(15)

m equations in

aq cos 2N (q 0 l) N2 0 p =(01)l+1 p S N2 0 p q=0 (16) for l = 0; 1; . . . m 0 1. We note that the rank of the coefficient matrix of the m equations in (16) is 2, so only two among them are independent. Thus to determine a specific source pattern, the limit of p is up to pmax = (m 0 2)=2 for an even integer m or pmax = (m 0 1)=2 for an odd m, which guarantees the magnitudes of S at the discrete spatial frequencies, N=2;N=201; . . . ; N=20 pmax reach their minimum values at the same time. Mathematically, we can also demonstrate that the rank of the coefficient matrix of the equations, including (8), (13) and the equation groups (14) with = 1 2 . . . max , is , so there is only one solution that would satisfy all these equations if it does exist. In the following, we will show that the source pattern [ 0 1 . . . m01 ] given by

p

; ; ;p

m

a ;a ; a

al = 2m101 l!(m(m0011)! 0 l)!

N = 128

l

; ; ;m ; ; ;p

m

for = 0 1 . . . 0 1, being the normalized th row of Pascal’s triangle, is exactly the solution of (8), (13) and (14) with = 1 2 . . . max . Firstly, according to the mathematical properties of Pascal’s triangle

p

1 (1 0 1)m01 = m01 al (1)m010l(01)l = 0 2 m01

1 m01 2m01 (1 + 1) =

l=0 m01 l=0

al (1)m010l(01)l = 1

so the two equations (8) and (13) are fulfilled. Further with (17), after doing some calculation, we find (15) becomes

LN=20p(a0 ; . . . ; am01 ; p) = cosm01 N N2 0 p : (18) Because LN=20p is with a constant value for each p under the specific source pattern [a0 ; a1 ; . . . am01 ] given by (17), the equation group (14)

m01

(01)l al

Fig. 2. The spatial-spectrum magnitudes of several three-cell sources under different normalized source patterns for .

(17)

always holds true. Therefore (17) is the only solution of the optimal source patterns that can simultaneously minimize the source’s high spatial frequency components, and subsequently result in much lower levels of aliasing errors introduced by the added source item n+1=2 .

S

III. NUMERICAL VERIFICATION In Fig. 2, we show the spatial spectrums of some three-cell sources under different normalized patterns, the optimal pattern [1/4,1/2,1/4], one non-optimal symmetric pattern [0.23,0.54,0.23], the identical three-cell pattern [1/3,1/3,1/3] and one asymmetric pattern = 128. It is evident that the spatial spectrum [2/8,5/8,1/8], for of the optimal pattern has much lower high frequency components than those of the other three. Further, although being derived from the 1-D problem, the proposed optimal source patterns can be easily extended to the 2-D or 3-D cases because the differencing process of each field component in PSTD is always actuated referring to only one of the three Cartesian coordinates. For instance, the optimal spatially smoothed sources comprising 3 2 3 and 3 2 3 2 3 cells, have the normalized source patterns illustrated in Fig. 3. Note that the cells in each column and row retain the proposed optimal three-cell pattern. To make our proposition more convincing, the 1-D, 2-D and 3-D practical PSTD simulations are also conducted to verify the excellent performance of these optimal source patterns. For the simulations

N

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229

applied for cells in each column and row. Evidently, we find that the magnitude of aliasing errors on the electric field Ez introduced by the optimal sources are approximately four orders smaller than those by the non-optimal sources, which holds true for all the 1-D, 2-D and 3-D PSTD problems. IV. CONCLUSION Fig. 3. Normalized optimal patterns for the 2-D and 3-D sources with three cells in each column and row. The sizes of the solid dots denote their magnitudes as specified by the values nearby in the top right.

The optimal patterns of the spatially-smoothed sources for PSTD algorithms are of great important in order to suppress the introduced aliasing errors to minimum levels. In this work, we propose that the optimal pattern of a soft or hard source composed m cells is exactly corresponding to the normalized mth row of the Pascal’s triangle. These optimal patterns deduced from 1D analysis can be easily extended to their 2D and 3D counterparts. Their excellent performance is also verified by the practical 1-D, 2-D and 3-D PSTD simulations as compared to that of non-optimal ones.

REFERENCES [1] Q. H. Liu, “The PSTD algorithm: A time-domain method requiring only two cells per wavelength,” Microw. Opt. Technol. Lett., vol. 15, no. 3, pp. 158–165, Jun. 1997. [2] Q. H. Liu, “Review of PSTD methods for transient electromagnetics,” Int. J. Numer. Model., vol. 17, pp. 299–323, 2004. [3] G.-X. Fan and Q. H. Liu, “Pseudospectral time-domain algorithm applied to electromagnetic scattering from electrically large objects,” Microw. Opt. Technol. Lett., vol. 29, no. 2, pp. 123–125, Apr. 2001. [4] T. W. Korner, Fourier Analysis. Cambridge, U.K.: Cambridge Univ., 1988, pp. 62–66. [5] Q. H. Liu, “Large-scale simulations of electromagnetic and acoustic measurements using the pseudo-spectral time-domain (PSTD) algorithm,” IEEE Trans. Geosci. Remote Sensing, vol. 37, pp. 917–926, Mar. 1999. [6] T.-W. Lee and S. C. Hagness, “A compact wave source condition for the pseudospectral time-domain method,” IEEE Antenna Wireless Propag. Lett., vol. 3, pp. 253–256, 2004.

Fig. 4. The introduced aliasing errors on the electric field E by the soft and hard sources under the optimal and non-optimal patterns are shown for the first 200 time steps in the practical 1-D, 2-D and 3-D PSTD transient simulations =. with c

=14

under test, the temporal driving function f (t) is assumed to be a Gaussian derivative pulse with

f n+1=2 = sin(0:1n)e00:002(n0100) and assigned to the electric field Ez . The sources are placed at the centers of the 1-D, 2-D and 3-D grid spaces with N = 128 in each coordinate and the artificial detectors are set at the specific cells id = 120, (id ; jd ) = (120; 64), and (id ; jd ; kd ) = (120; 64; 64) for 1-D, 2-D and 3-D spaces, respectively, to record the aliasing errors for the first 200 time steps before the perceptible source waves pass the detectors in theory. Fig. 4 shows the simulation results for the aliasing errors of electric field from the normalized soft and hard sources with the optimal pattern [1/4,1/2,1/4] and the non-optimal pattern [0.249,0.502,0.249]

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Experimental Microwave Validation of Level Set Reconstruction Algorithm Douglas A. Woten, Mohammad R. Hajihashemi, Ahmed M. Hassan, and Magda El-Shenawee

Abstract—The capability of the level set method is illustrated for shape and location reconstruction using real measurement data in the range of 3–8 GHz. An experimental setup is utilized to measure the S-parameters using two Vivaldi antennas revolving around metallic pipes of different cross-sections. A two-dimensional reconstruction of the pipe’s cross section is retrieved using the level set algorithm combined with the Method of Moments and the frequency hopping technique. The results show good agreement between the reconstructed cross section and the true configuration. This indicates to the strength of the level set method and its ability to cope with the noise in real measurements.

Fig. 1. Schematic of experimental setup.

Index Terms—Image reconstruction, inverse problem, level set algorithm, measured microwave data.

Section II [11]. In this work circular, square and hexagonal metallic pipes are utilized to approximate PEC objects and the results are reported in Section V.

I. INTRODUCTION The level set algorithm has been demonstrated to be versatile in reconstructing the shape and location of irregular shaped objects of various compositions [1]–[9]. Papers [3] and [9] consider perfect electric conductors (PEC) while [4], [7], and [8] are used with dielectric materials. The majority of published results using the level set method are based on synthetic data with varying amounts of artificial noise injected into the signal. In 2000 the Institut Fresnel started a public repository of scattered microwave data that has been used for validating inversion techniques [5]–[8], [10]. The microwave measurements system at the Institut Fresnel consisted of a spherical scanning system housed in an anechoic chamber 14.2 m in length and 6.5 m in width and height respectively. The transmitting and receiving antennas were wide band horn antennas. That system was designed to operate in a frequency between 300 MHz to 26.5 GHz with angular accuracy within 0.05 [5], [6]. Ramananjaona et al. have used the Institut Fresnel experimental data for the reconstruction of metallic pipes and presented reconstruction results for the TE and TM data on circular, rectangular and U-shaped pipes [7]. Litman employed the Institut Fresnel data for the reconstruction of dielectric objects that are homogeneous by parts and of known characteristics [8]. The purpose of the current work is to augment the experimental works described above with a different microwave measurements system. The details pertaining to the version of the level set algorithm used in this work is described in [9]. The existing system uses two Vivaldi antennas rotating around the target as will be described in Manuscript received October 23, 2008; revised May 14, 2009. First published November 06, 2009; current version published January 04, 2010. This work was supported in part by Entergy Arkansas Inc., NSF Graduate Research Fellowship, NSF GK-12 Program, NSF Award Number ECS–0524042, and in part by the Arkansas Biosciences Institute (ABI). D. A. Woten is with the Microelectronics-Photonics Program (MicroEP), University of Arkansas, Fayetteville, AR 72701 USA (e-mail: dwoten@uark. edu) R. Hajihashimi, A. M. Hassan, and M. El-Shenawee are with the Electrical Engineering Department, University of Arkansas, Fayetteville, AR 72701 USA (e-mail: mhajihas, amhassan, [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.2009.2036186

II. EXPERIMENTAL SETUP The experimental setup relies on a HP 8510C Vector Network Analyzer (VNA). This 2-port VNA model allows for frequency sweeps between 45 MHz to 26.5 GHz with up to 801 points per sweep. The VNA communicates directly with a central computer through a GPIB cable. The computer provides a central control for all the various components and acts as a collection point for the generated data. An antenna is connected to each port of the VNA to allow for the measurement of the four S-parameters (S11 ; S12 ; S21 , and S22 ). Each of these antennas (Tx ; Rx ) is connected to rotating arms controlled by independent HIS DCM 8028 motion control drivers. The drivers receive input from the National Instruments Universal Motion Controller (UMC) and each control a PowerMax II hybrid stepper motor. This allows each antenna to rotate around a central location independently. A schematic of the experimental setup is shown in Fig. 1. The transmitter, Tx , and receiver, Rx , employed in this work are Vivaldi antennas [12]. Any linearly polarized broadband antenna could be used; however, the Vivaldi antennas were readily available in the laboratory and operate between 3–8 GHz. A schematic of the Vivaldi antenna along with the material parameters is shown in Fig. 2(a). Fig. 2(b) shows the S11 of each antenna. The performance of Antenna #2 is deteriorated at frequencies above 8 GHz possibly due to a mismatch between the antenna and SMA connector (not shown here). Fig. 2(c) shows the S21 and S12 of the antennas. The Vivaldi antenna has a focused antenna beam as shown in Fig. 3. The commercial software package Ansoft High Frequency Structure Simulator (HFSS) is used to simulate the Vivaldi antenna of Fig. 2. The resulting antenna pattern in the E and H -planes are shown in Fig. 3(a) and (b) respectively. The results of Fig. 3 show the Vivaldi beamwidth is approximately 80 in the H -plane and 35 in the E -plane. The direction of the main beam shifts from 125 at 3 GHz to 97 at 8 GHz. Note that the metallizations on the front and back of the antenna substrate are not symmetric as shown in Fig. 2(a). To reduce interference from the surroundings, an anechoic chamber is built to house the rotating antennas and the pipe as shown in Fig. 4. The chamber is a one meter cube with EHP-5PCL microwave pyramidal absorber. This absorber has a maximum reflection of 030 db at

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Fig. 5. The measured S

Fig. 2. (a) Geometry of Vivaldi antenna, (b) measured measured S and S .

S

and

S

at each receiver angle for 3 GHz.

and (c)

Fig. 6. An example of the data collection for a single transmitting location.

Fig. 3. The radiation pattern of the Vivaldi antennas in the (a) E -Plane and (b) H -Plane.

calculate the CAF at all frequencies and all incident and receiving angles. The FEKO package provides faster electric field calculations of large geometries when compared with the HFSS. A sample of the S21 measured data from the square pipe is shown in Fig. 5 which illustrates the variations of the S21 with the receiver angles at 3 GHz. Fig. 5 has three curves corresponding to three different transmitter angles (0 , 90 and 180 ) where the receiver angle is always measured with respect to the transmitter angle. Theoretically, the three plots should be identical; however, discrepancies are observed around the receiving angle of 300 . This could be attributed to the coupling between the two antennas since they are closest to each other at this angle (see Fig. 6). IV. RECONSTRUCTION RESULTS A. Level Set Algorithm

Fig. 4. Anechoic chamber of 1 m with two Vivaldi antennas rotating around a pipe.

frequencies above 2 GHz and is approximately 12.5 cm thick. As a result, the useable space inside the chamber is approximately 75 cm3 . III. COMPLEX ANTENNA FACTOR Inversion algorithms use the complex electric fields as the input [1]–[10] while the VNA measures the complex scattering parameters. To validate the level set algorithm the S -parameters are first converted to electric fields using a transfer function similar to that in [13]. In this work, the transfer function represents the ratio of electric field, Ex , to the S -parameter, S21 . The Ex represents the complex electric field in the x-direction at the front end of the receiving antenna and S21 represents the complex measurement at antenna 2 when antenna 1 is excited. These quantities are both obtained when no target is present in the experimental set-up [11], [13]. The transfer function is a complex number, antenna dependent, referred to as the complex antenna factor (CAF) and given by CAF = Ex =S21 . The commercial electromagnetic simulator, FEKO, is used to simulate Ex and S21 to

In this work the configuration of the pipe immersed in air is considered two-dimensional (2D) since the length of the pipe is much larger than the wavelength with uniform cross-section. The 2D perfect electric conductor (PEC) level set algorithm of [9] is used to reconstruct the shape and location of the pipe with three different cross sections. The level set algorithm assumes that the interface is represented implicitly as the zero level of a higher order function . At each time t, the interface is defined as [3]:

0(t) = f(x; y)j(x; y; t) = 0g

(1)

Once the derivative with respect to time is found, the Hamilton-Jacobi equation for tracking the motion of the interface is obtained [3], [4]

@ + F krk = 0 @t 0 = (x; y; t = 0)

(2a) (2b)

where F is the normal component of the deformation velocity. The Method of Moments is used as the forward solver and the frequency hopping technique is used from 3–8 GHz (the antenna range) in steps

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Fig. 7. Reconstruction of circular pipe (a) initial guess at 3 GHz, (b) after 2505 iterations at 3.5 GHz, and (c) after 5025 iterations at 5 GHz.

Fig. 9. Reconstruction of hexagonal pipe (a) initial guess at 3 GHz, (b) after 1600 iterations at 3.25 GHz, and (c) after 20100 iterations at 8 GHz.

Fig. 10. Cost function for the (a) circular pipe, (b) square pipe, and (c) hexagonal pipe. Fig. 8. Reconstruction of square pipe with rounded corners (a) initial guess at 3 GHz, (b) after 4005 iterations at 4.5 GHz ,and (c) after 8490 iterations at 6.8 GHz.

that range from 200–500 MHz. Additional details of this specific level set algorithm are reported in [9]. B. Measurement Configuration The front of the two antennas, marked x in Fig. 2(a), is set at a distance of 7.8 cm from the center of the chamber where the pipe is positioned. Antenna #1 is fixed at directions from 0 –360 in steps of 10 while Antenna #2 collects data at directions 90 –270 with respect to antenna #1 in steps of 10 . Fig. 6 shows the data collection scheme. The symmetry of the pipes in this work allows using less data in the algorithm as will be discussed in each case. In all results the initial guess is a circle of diameter 20 cm centered at (3, 0).

set algorithm. The reconstruction is shown in Fig. 9(a)–(c). The reconstruction is shown after 1600 iterations in Fig. 9(b). The final reconstruction has good agreement with the true shape after 20100 iterations as shown in Fig. 9(c). The cost function, which is normalized with respect to the measurement data, gives the error between the simulated evolving shape and the measured data, is shown in Fig. 10(a)–(c). Note that the frequency hopping causes the cost function to display stair-like and/or jumping behavior as the frequency is hopped to the following frequency [3], [9]. More frequencies are used for the hexagonal reconstruction (20 frequencies in steps of 250 MHz) vs. the square (15 frequencies in steps of 500 MHz then 200 MHz) and the circle (5 frequencies in steps of 500 MHz). The need for more frequencies is due to the presence of six corners in the hexagonal vs. four corners in the square and no corners in the circular cases. It is known that higher frequencies are needed to retrieve the fine details of the target’s shape.

C. Case 1 (Circular Pipe) The first case considered in this work is a pipe with a circular crosssection. The pipe is 100 in diameter (2.54 cm) and is made of aluminum. For this work the pipe is approximated as a PEC. The image reconstruction is shown in Fig. 7(a)–(c). After 2505 iterations, Fig. 7(b) shows the evolving shape which remains circular. In the 5025th iteration, Fig. 7(c) shows the reconstructed shape that converges to the true configuration. D. Case 2 (Square Pipe) The square pipe is the second case considered in this work. The pipe has slightly rounded corners, is 100 in length on each side (2.54 cm), made of steel and is approximated as a PEC for this work. Due to symmetry of the pipe, the data measured for the transmitting angle 0 –90 is repeated for the input to the level set algorithm. The reconstruction is shown in Fig. 8(a)–(c). After 4005 iterations the evolving shape has split into multiple ones seemingly unrelated to the true shape as shown in Fig. 8(b). The reconstruction achieves good agreement with the true object after 8490 iterations as shown in Fig. 8(c). The reconstructed shape is slightly smaller especially at the corners as shown in Fig. 8(c).

V. CONCLUSION In this work measurement data from our experimental system is utilized to validate the level set algorithm for 2D PEC shapes. When considering the error in the cross-section area of the reconstructed shapes, it is observed to be on the order of 5% or less for the cases examined here. The effect of variations in the number of measurement points and antenna distances is discussed in [9]. The reconstruction requires approximately 4–7 CPU hours, depending on the geometry, on a single 64-bit AMD Opteron 246 processor running at 2 GHz. Additional higher frequencies are needed to reconstruct the fine details of complex shapes and thus increase the computational time requirements. Current work in parallelizing the code for the level set algorithm has resulted in a speedup of 100X employing 256 processors on the San Diego Supercomputer Center’s DataStar [14]. This speedup allows for reconstruction results in 2.5–4.5 minutes compared to 4–7 hours. Ongoing research by the authors considers 3D objects [15]. ACKNOWLEDGMENT

E. Case 3 (Hexagonal Pipe) The final case considered in this work is the hexagonal pipe. The pipe is 100 in diameter (2.54 cm), made of aluminum and approximated as a PEC object similar to Case 1. Due to symmetry the data measured for the transmitting angle 0 –60 is repeated for the input to the level

The authors would like to thank Mr. Kegege for his help in this work.

REFERENCES [1] J. A. Sethian, Level Set Methods and Fast Marching Methods. bridge, U.K.: Cambridge Univ. Press, 1999.

Cam-

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[2] S. J. Osher and R. P. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces. New York: Springer-Verlag, 2003. [3] R. Ferraye, J. Y. Dauvignac, and C. Pichot, “An inverse scattering method based on contour deformations by means of a level set method using frequency hopping technique,” IEEE Trans. Antennas Propag., vol. 51, no. 5, pp. 1100–1113, May 2003. [4] O. Dorn and D. Lessselier, “Level set methods for inverse scattering,” Inverse Problems, vol. 22, no. 4, pp. R67–R131, Aug. 2006. [5] K. Belkebir, S. Bonnard, F. Pezin, P. Sabouroux, and M. Saillard, “Validation of 2D inverse scattering algorithms from multi-frequency experimental data,” J. Electromagn. Waves Applicat., vol. 14, pp. 1637–1667, 2000. [6] J. Geffrin, P. Sabouroux, and C. Eyraud, “Free space experimental scattering database continuation: experimental set-up and measurement precision,” Inverse Problems, vol. 25, no. 6, pp. S117–S130, Dec. 2005. [7] C. Ramananjaona, M. Lambert, and D. Lesselier, “Shape inversion from TM and TE real data by controlled evolution of level sets,” Inverse Problems, vol. 17, no. 6, pp. 1585–1595, Dec. 2001. [8] A. Litman, “Reconstruction by level sets of n-ary scattering obstacles,” Inverse Problems, vol. 21, no. 6, pp. S131–S152, Dec. 2005. [9] M. R. Hajihashemi and M. El-Shenawee, “Using the level set method for microwave applications,” IEEE Antennas Wireless Propag. Lett., vol. 7, pp. 92–96, 2008. [10] C. Gilmore, P. Mojabi, and J. LoVetri, “Comparison of the distorted born iterative and multiplicative-regularized contrast source inversion methods: The 2D TM case,” in Proc. Annu. Rev. of Progress in Appl. Comput. Electromagn., Niagara Falls, Canada, Mar. 30–Apr. 4 2008, pp. 122–127. [11] D. A. Woten, O. Kegege, R. Hajihashimi, A. Hassan, and M. El-Shenawee, “Microwave detection using real measurement data,” in Proc. Annu. Rev. of Progress in Appl. Comput. Electromagn., Niagara Falls, Canada, Mar. 30–Apr. 4 2008, pp. 650–655. [12] O. Kegege, “Ultra-Wide Band Radar for Near-Surface and Buried Target Detection,” M.S. thesis, Univ. Texas Pan American, Edinburg, TX, 2006. [13] S. Ishigami, H. Iida, and T. Iwasaki, “Measurements of complex antenna factor by the near-field 3-antenna method,” IEEE Trans. Electromag. Compat., vol. 38, no. 3, pp. 424–432, Aug. 1996. [14] M. R. Hajihashemi and M. El-Shenawee, “MPI parallelization of the level-set reconstruction algorithm,” presented at the IEEE Int. Symp. on Antennas Propag./URSI Nat. Radio Science Meeting, Charleston, SC, Jun. 1–5, 2009. [15] M. R. Hajihashemi and M. El-Shenawee, “Three-dimensional level set algorithm for shape reconstruction of conducting objects,” presented at the IEEE Int. Symp. on Antennas Propag./URSI Nat. Radio Science Meeting, Charleston, SC, Jun. 1–5, 2009.

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Scattering of Electromagnetic Waves From a Rectangular Plate Using an Enhanced Stationary Phase Method Approximation Charalampos G. Moschovitis, Konstantinos T. Karakatselos, Efstratios G. Papkelis, Hristos T. Anastassiu, Iakovos Ch. Ouranos, Andreas Tzoulis, and Panayiotis V. Frangos

Abstract—A time-efficient high frequency analytical model for the calculation of the scattered field from a perfect electric conductor (PEC) plate is presented here, which is based on the physical optics (PO) approximation and the stationary phase method (SPM). Using the SPM analysis for the three-dimensional (3D) scattering problem under consideration, the scattered electric field is calculated analytically. It follows that the analytical formula proposed here yields an accurate and fast algorithm for the calculation of the scattered electromagnetic (EM) field, which can be used trustfully in a variety of radio propagation problems. The accuracy of the proposed analytical method is checked through a straightforward numerical integration over the PO currents, as well as through Finite Element Boundary Integral full-wave exact solution. Comparison results are given in the far field, Fresnel zone and the near field area. Index Terms—Physical optics approximation, radio coverage, scattered field calculation, stationary phase method.

I. INTRODUCTION Considering radio propagation in urban outdoor areas [1], analytical EM methods for scattering and diffraction from building walls may be preferable to empirical models or actual radio propagation experiments. For such problems analytical EM methods are usually found to be more accurate than empirical methods, even though in most cases they are slower in terms of actual calculation time. In this communication we analyze the implementation of an analytical model based on an enhanced version of the stationary phase method (SPM) approximation for the calculation of the vector potential A, and subsequently of the electric field E , in a three-dimensional (3D) scattering electromagnetic problem. The proposed method derives from the extension of the corresponding problem in two dimensions to the three-dimensional scattering problem examined here. It is found that the proposed 3D analytical method is much faster than the standard approach [1], which employs numerical integration over the PO currents. Our proposed method can be incorporated, for example, to simulation tools that produce radiocoverage patterns in urban outdoor environments, in which first or higher order scattering mechanisms are considered [1]. Assuming that the PEC plate simulates a wall-scatterer (for which an appropriate reflection coefficient has to be included, in

Manuscript received February 12, 2008; revised April 16, 2009. First published May 27, 2009; current version published January 04, 2010. C. G. Moschovitis, K. T. Karakatselos, E. G. Papkelis, I. C. Ouranos, and P. V. Frangos are with the School of Electrical and Computer Engineering, Division of Information Transmission Systems and Materials Technology, National Technical University of Athens, GR-15780 Zografou, Athens, Greece (e-mail: [email protected]; [email protected]). H. T Anastassiu is with the Hellenic Aerospace Industry, GR-32009 Schimatari-Tanagra/Viotia, Greece. A. Tzoulis was with the Department of Antennas and Scattering (AuS), Research Establishment for Applied Science (FGAN), Research Institute for High Frequency Physics and Radar Techniques (FHR), D-53343 Wachtberg, Germany. He is now with the New Technologies and Applications Group (NTAG) of TELETEL S.A., GR-11526 Athens, Greece. 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.2009.2024015 0018-926X/$26.00 © 2009 IEEE

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arbitrary observation point R(x; y; z ) in the region z > 0, according to the following:

A(x; y; z ) =

00 [^x(E0 cos  0 E0 cos  sin  ) 2 + y^(E0 sin  + E0pcos  cos  )] i



1

y

c

a

e (

i



jk x K

=d x =b

i





i

i

i

+y L) e0jk (x0x ) +(y0y ) +z 0 0 dx dy (x 0 x0 )2 + (y 0 y0 )2 + z2 (1)

Fig. 1. Scattering from a PEC rectangular plate of dimensions a-b and c-d.

order to account for its finite conductivity), our method can be used as a valuable tool in order to examine radiocoverage in those environments for frequencies of 1 GHz or higher, for possible applications to GSM, UMTS Mobile Communications, Wi-Fi or WiMax network simulations.

where o is the magnetic permeability of the vacuum (o = 41007 H=m), k is the propagation constant,  is the free space impedance ( = 0 ="0 = 377 ), Eo , Eo' are constants related to the incident EM field and its polarization components [2] and K; L are constants which depend on the angles of incidence (i ; 'i ) [2]. Integrals of the form of (1) are frequently encountered during the procedure of calculating the vector potential A in common electromagnetic problems and can be computed in various ways. In our case we apply the SPM approximation (a much faster method for computer implementations), we calculate the vector potential A, and subsequently, the electric field E at an arbitrary observation point R(x; y; z ) and finally in Section IV, we compare our results both with standard results in the far zone (sinc = sinx=x function results), with numerical results based on numerical quadrature [1] and with finite element boundary integral (FEBI) full-wave exact solution results ([6], [7]). D. Electric Field Calculation

II. THEORY AND FORMULATION BACKGROUND A. Electromagnetic Layout According to the layout in Fig. 1, an electromagnetic wave is considered to be incident on the direction of wave vector ki upon a PEC rectangular plate of negligible thickness. The plate vertices lie at the intersections of the lines x = a; b and y = c; d, as shown in Fig. 1. The direction of incidence is defined by the standard spherical angles, namely i and 'i . Initially, we are interested in calculating the vector potential A at an observation point R(x; y; z ), and eventually, in calculating the electric field E (with time dependence proportional to the factor e+j!t ) at the same observation point, which lies in the near, intermediate (Fresnel) or the far (Fraunhofer) field region as defined in [1]. We consider the specific problem as a prerequisite for modeling propagation in an urban outdoor environment, which consists of three dimensional scattering walls. Operating at the frequency of 1 GHz or higher, scatterers that appear in the above networks are considered to be electrically large, and current density may be calculated with good accuracy using the PO approximation. B. Physical Optics (PO) Current Density Assuming that the PEC rectangular plate is electrically large, the expression of the current density induced on its surface due to an incident plane wave with wavevector ki , according to the PO approximation, is provided in [1], [2]. C. Vector Potential A Calculation Applying the above calculated induced current density J s on the rectangular plate, the vector potential A(x; y; z ) can be calculated at an

Having calculated the vector potential A, we proceed to calculate the scattered electric field E as follows:

! E (x; y; z ) = 0j!A 0 j 2 grad (div (A)) : k

(2)

III. STATIONARY PHASE METHOD CALCULATIONS The SPM approximation, as applied to this work, is related to an asymptotic calculation of the definite double integral (DI) of the form c

I (k) = d

a

F (x; y )ejkf (x;y) dx dy:

(3)

b

At a frequency equal to f = 1 GHz, k = 2f=c  21 m01 , a value that can be considered relatively high for stationary phase calculations [3]. In other words, the electrical length of the rectangular scatterer p (diagonal of the plate) is kl = 40 2  1 where l is the maximum dimension of the scatterer (l equal to 20 in this calculation). Phase function f (x; y ) = x0 K + y 0 L 0 ((x 0 x0 )2 + (y 0 y 0 )2 + z 2 )1=2 and amplitude function F (x; y ) = ((x 0 x0 )2 + (y 0 y 0 )2 + z 2 )01=2 inside the integral of (1) fully comply with SPM conditions in [3]. Here it should be emphasized that the stationary point must be placed within the surface boundaries, i.e., b < xs < a, d < ys < c. The SPM calculation performed in this communication is taking into account the contribution from the finite limits of the plate (a; b; c; d). In [4], and for the case of a single integration (SI), the definite integral over the interval (a,b) is extended to the indefinite interval (01; +1) and correction terms related to the edge contributions a,b are provided. In this communication we generalize this method [4] for the case of

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double integrals (DI). We apply initially the equations of the single integral to appropriate integrand functions. Afterwards, by defining appropriate new integrand functions, we apply the SPM method regarding the second integration of the double integral, thus resulting finally to nine integration terms. The final result is the calculation of the double integral in a novel closed form (to our knowledge not documented to date in the literature). This form uses nine terms of stationary phase, which include the total information about the finite limits of the plate (a; b; c; d). Due to the complexity of the corresponding mathematical expressions (calculation of the partial derivatives, solution of complicated systems of equations and elaborate analytical formulas), this method was implemented in a standard MATLAB computational package, both for the case of single and double integral. Stationary phase method analytical calculations are very fast in terms of actual computation time, as it will be explained later in Section V, below. Explicit relative mathematical formulas and expressions, not published yet elsewhere, can be provided as a detailed mathematical report to interested readers.

235

the rectangular patch is being divided into four separate regions (regions 1–4, while region 0 corresponds to the entire x-y plane) and (3) is rewritten as

+ + 1

F (x0 ; y 0 )ejkf (x ;y ) dx0 dy 0

I (k) = 01 01

+ +

1

1

0

01

+

x

0

=a =b

x

1

y

0

F (x0 ; y 0 )ejkf (x ;y ) dx0 dy 0

=d

a

x

=b

01

A. SPM Double Integral Calculations

+

1

y

a

=c x =b

=d

y

I

0

=

=b

0

2



k

2 exp Ia = 0 0

Ib0 =

jkf (x ;y )

F (x ; y )e 0

1

jk

1

jk

@ f @x x

=x

=y

;y

F (a; y 0 )

=a;y =y

F (b; y 0 )

@f @x x

dx

0

= I0 0

0 f(02) y0

0 Ia 0 Ib 0

0

(5)

F (xo ; y 0 )

j kf (xo ; y 0 )+ @f @x x

0 0 F(02) (y ) =

(4)

a

x

Io0

I 0 dy 0

=b;y =y

 4

exp

exp

@2f @x02

sgn

jkf (a; y

jkf (b; y 0 )

0

)

;y

=y

2 + O (k )

(7)

k02

(8)

+O

0

x

=x

;y

=y

= 0:

(9)

B. Area Division Approach Equation (3) is extended properly, to include calculations involving (4)–(8). According to our proposed approach, the domain surrounding

(10)

0 jk1

=f

F (a; y 0 )

@f @x x

a; y 0

(11a)

=a;y =y

(11b)

0 F(02) (y02 )

@ f

(6)

where x0 = xo (y 0 ) is the curve determined by equating the first partial derivative of f with respect to x0 with zero, replacing the variables x0 = xo and y 0 = y 0 , and solving with respect to xo

@f (x0 ; y 0 ) @x0

2

k

=x

F (x0 ; y 0 )ejkf (x ;y ) dx0 dy 0 :

outer integration (for 01 < y 0 < +1) is performed again and the final result of this procedure is

I [1]  x

F (x0 ; y 0 )ejkf (x ;y ) dx0 dy 0

The first integral term of (10) (I[0]) is obtained in [3]. Note that for the EM scattering problem under consideration, one stationary point can exist at maximum, as expected from physical intuition, which corresponds to specular scattering. Regarding the calculation of the second integral term of (10), I[1], this is calculated in the following two steps: First the inner integral (from x0 = a to x0 = +1) is calculated. Then by defining new appro0 0 priate amplitude and phase functions (F02 and f02 ) by the following equations:

c

I (k) =

F (x0 ; y 0 )ejkf (x ;y ) dx0 dy 0

01 01

0

According to the proposed enhanced SPM method we rewrite the initial equation with slightly modified variables and referring to the amplitude function F and phase function f of (1), which are functions of two variables, we have to calculate a double integral, and the algebraic expressions are modified properly. The calculation is completed in two steps, in order to take into account the contribution of all four limits of the plate (a; b; c; d). The first step is related to the calculation of the inner (single) integral of (3), and it is described by (4)–(8) below

1

2 exp

@y

y

=y

0 j kf(02) (y02 ) +

 4

sgn

0 @ 2 f(02) @y 02

y

=y

(12)

where y02 is a modified stationary point defined by 0 0 @f(02) (y ) @y 0

y

=y

= 0:

(13)

In the same way integrals I[2], I[3], I[4] are being calculated. Regarding the above calculations of integral I(k) in (3), we note here that the modified stationary points y01 ; y02 ; y03 and x0 will be included in the final result, only if y01 ; y02 ; y03 2 (d; c) and x0 2 (b; a). In any other case, the contribution regarding these integral terms is zero. IV. NUMERICAL RESULTS The simulation parameters, according to the geometry of Fig. 1 are provided in Table I.

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TABLE I DOUBLE INTEGRAL SIMULATION PARAMETERS

As calculated from the corresponding Fresnel and far field area distances [1]

For a plate of dimensions 20 2 20, numerical results are displayed in Fig. 2(a)–(c) below for the far field, Fresnel zone and near field respectively. For all these cases, the maximum value for the main scattering lobe is observed for angle of observation s equal to angle of incidence i . Also, in the far field area we note that the number of the side lobes is proportional to the electrical size of the plate, as expected (not shown here).

A. Far Field, Fresnel and Near Field Area Comments From the above results [Fig. 2(a), (b)] it can be easily concluded that the proposed 3D analytical SPM method yields results of excellent accuracy for all angles of observation, except for a very narrow region of about 4 around the specular direction, where an error of about 2–4 dB is observed. Regarding this error the following remarks can be made. — This error is due to the vanishing of the denominators in the case that the stationary point, corresponding to the observation point of interest, approaches the boundaries of the plate. This error can be reduced by using higher order expansion terms in the development of our proposed SPM (Fresnel functions, see [4]—also see Section VI, below, regarding our related future research). — This error is reduced if the scatterer is electrically larger, e.g., 80 2 80 scatterer (not shown here), as expected since SPM is a high frequency asymptotic method. — This numerical error is much smaller than the error of about 30 dB, which appears at several observation angles between the standard numerical integration method and the analytical sinc function result, and which is due to the highly oscillatory behavior of the induced physical optics currents on the surface of the rectangular plate. In the case of near field observations, the results [Fig. 2(c) below] are still found to be satisfactory. Again, lower accuracy is found to occur in the region of the main lobe around the specular direction, with the maximum error being of about 15dB at a narrow window of about 3 . By physical intuition we expected such a larger numerical error at the near field area, since it appears that for this case the field contribution from only one stationary point and the diffracting edges of the scattering plate is not enough for very accurate calculations. Once again, these errors can be corrected either through the use of more accurate Fresnel functions in the development of SPM method (see Section VI below), or in the case when the scatterer is electrically larger.

Fig. 2. 1 GHz simulation results of magnitude of total electric field E as a function of observation angle  for plate of dimensions 20 20 and observation distances: (a) 600 m (far field area), (b) 100 m (Fresnel area), (c) 25 m (near field area).

2

V. RUN-TIME CONSIDERATIONS In Table II below, we compare the run-time needed for both methods (SPM and numerical integration—NI) for the cases of 20 and 80

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TABLE II TOTAL SIMULATION TIME FOR CALCULATIONS AT 90 DIFFERENT SCATTERING ANGLES ON A STANDARD DESKTOP PC

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(120,074 unknowns) is approximately 60 3 3600 sec, whereas the estimated memory required would approximately be equal to 130 GB of RAM. Regarding the accuracy provided by our proposed SPM method, as applied to the radio coverage problem described in [1], we may comment that the average path loss error in [1] being 3–8 dB at maximum, a small additional numerical error is expected through the use of our proposed SPM method. Note that this additional error can be fully corrected, when use of Fresnel functions is also employed (see Section VI, below). However, we consider that this is of minor importance, as compared to the large savings in computation time described above. VI. CONCLUSION—FUTURE RESEARCH

Personal computer with Pentium M 2GHz processor and 1 GB of RAM memory for the cases of 20 and 80 scatterers.

scatterers on a standard desktop personal computer with Pentium M 2 GHz processor and 1 GB of RAM memory. For the case of 20 scatterer we also provide the run-time for a full-wave exact method [boundary integral (BI)—multilevel fast multipole method (MLFMM)/ exact solution (ES)] [6], [7] simulated on an AMD Athlon XP 2800+. As shown in Table II, stationary phase method is much faster than any other method. Furthermore, as seen in Table II, we can observe that the run-time for the proposed 3D analytical SPM method is almost independent of the electrical length of the scatterer, unlike the numerical integration (NI) method where the run-time increases drastically with the electrical length of the scatterer. Moreover, for the proposed SPM the run-time is almost independent of the region of observation (far field or Fresnel area or near field), while for the numerical integration method the corresponding run-time increases drastically as the observation point moves from the far field to the near field area. Finally, it is clearly seen in Table II that full-wave exact methods are much slower in actual computer computation time than PO based methods, at least for the electrically large scatterers considered here. Using the above comparison results we can easily realize the usefulness of the proposed SPM method for the urban outdoor radio-coverage problem mentioned in Section I [1]. Assuming a typical 2D configuration [1] which consists, as a first approach, of two rectangular buildings, the following scattering mechanisms must be calculated: 8 scattering phenomena of first order (3 scattering mechanisms and 5 diffraction mechanisms), and a total of 93 phenomena of second order (31 scattering-diffraction mechanisms, 29 double scattering mechanisms and 33 diffraction-scattering mechanisms). Therefore, in this simple real world scenario of two buildings a total of 96 calculations of integrals related to scattering phenomena are required. For this simple scenario the acceleration in our radio coverage simulation tool will be 93 times the acceleration corresponding to that provided above for one single plate (for one scattering angle). Furthermore, realizing that the above acceleration rates in computation time refer only to one observation point, these figures have to be multiplied by the number of resolution cells in the urban scene in order to calculate the total acceleration in computation time for the actual radio coverage problem. Also, the comparison with full-wave exact solution results (BI-MLFMM FEBI [6], [7]), demonstrates faster rates up to 1000 times. In the case of an exact solution with Eventual BIM (pure MoM), a task not performed by our research group, the estimated run-time for the solution of the problem

In this communication we presented a 3D analytical method based on the stationary phase method (SPM) for the scattering of electromagnetic waves from electrically large conducting rectangular plates. The induced currents on the rectangular plate are calculated through the physical optics method and subsequently stationary phase method is used to calculate the contributions from the stationary point within the rectangular plate, as well as the contributions from the diffracting edges of the plate. This method was found to be accurate and very fast, compared to standard numerical integration methods. Furthermore, the method proposed here can be used, for example, for calculating the path loss in urban outdoor environments or other propagation scenarios [1], [5]. Our research group currently examines very carefully the problem of increasing the accuracy of our proposed method around the main scattering lobe using higher order approximations in the development of the proposed SPM method (use of Fresnel functions), as already discussed in Section IV [4]. We believe that this technique will further enhance the accuracy of the proposed method. APPENDIX FINITE CONDUCTIVITY OF THE PLATE For non perfect electric conductor (non PEC) rectangular plates, as is the case, for example, for the radio coverage problem in urban outdoor environment examined in [1], the scattered electric field as given by (2) should be multiplied, as a first approximation, with the appropriate Fresnel reflection coefficient. This coefficient depends on the electric characteristics of the surface and the operating frequency [5]. The values r ,  are chosen according to [5], namely "r = 15,  = 7 S=m for the ground and "r = 7,  = 0:005 S=m for the building walls (roofs are not considered at this point, once both transmitter and receiver in our prediction model [1] are considered to be placed well below rooftop level). Another approximate approach for the scattering from non-PEC rectangular plates might be implemented through the use of the impedance boundary condition method (IBCM, [2]).

REFERENCES [1] E. G. Papkelis, I. Psarros, I. C. Ouranos, C. G. Moschovitis, K. T. Karakatselos, E. Vagenas, H. T. Anastassiu, and P. V. Frangos, “A radio coverage prediction model in wireless communication systems based on physical optics and the physical theory of diffraction,” IEEE Antennas Propag. Mag., vol. 49, no. 2, pp. 156–165, Apr. 2007. [2] D. C. Jenn, Radar and Laser Cross Section Engineering. WA: American Institute of Aeronautics and Astronautics Inc., 1995, pp. 29–33 and pp. 47–57. [3] C. A. Balanis, Antenna Theory: Analysis and Design. New York: Wiley, 1996, pp. 922–926. [4] G. L. James, Geometrical Theory of Diffraction for Electromagnetic Waves. London, UK: Inst. Elect. Eng., 1976, pp. 30–42, 61, 90 and 117–123.

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[5] A. G. Kanatas, I. D. Kountouris, G. B. Kostaras, and P. Constantinou, “A UTD propagation model in urban microcellular environments,” IEEE Trans. Veh. Technol., vol. 46, no. 1, pp. 185–193, Feb. 1997. [6] J. Song, C.-C. Lu, and W. C. Chew, “Multilevel fast multipole algorithm for electromagnetic scattering by large complex objects,” IEEE Trans. Antennas Propag., vol. 45, no. 10, pp. 1488–1493, Oct. 1997. [7] A. Tzoulis, T. Vaupel, and T. F. Eibert, “Ray optical electromagnetic far-field scattering computations using planar near-field scanning techniques,” IEEE Trans. Antennas Propag., vol. 56, no. 2, pp. 461–468, Feb. 2008.

Experimental Characterization of UWB On-Body Radio Channel in Indoor Environment Considering Different Antennas Andrea Sani, Akram Alomainy, George Palikaras, Yuriy Nechayev, Yang Hao, Clive Parini, and Peter S. Hall

Abstract—An experimental investigation to characterize the transient and spectral behavior of the ultrawideband (UWB) on-body radio propagation channel for body-centric wireless communications is presented. The measurements were performed considering over thirty on-body links in the front of human body in the anechoic chamber, and in indoor environment. Two different pairs of planar antennas have been used, namely, CPW-fed planar inverted cone antennas (PICA), and miniaturized CPW-fed tapered slot antennas (TSA). A path loss model is extracted from measured data, and a statistical study is performed on the time delay parameters. The goodness of different statistical models in fitting the root mean square (RMS) delay has been evaluated. Results demonstrate that the TSA, due to its more directive radiation behavior is less affected from the reflections from body parts and surrounding environment. The antenna shows significant size reduction and improved time delay behavior, and hence is an ideal candidate for UWB body area networks (BAN). Index Terms—Body area networks, radio propagation, ultrawideband (UWB).

I. INTRODUCTION Ultrawideband (UWB) is a low-power, high data rate technology that provides immunity to multipath interference and has robustness to jamming because of its low-probability of detection [1]. Its low power requirement due to control over duty cycle allows smaller batteries and makes it suitable for wearable units. One of the most promising areas of UWB applications is the body-centric wireless communication where various units/sensors are scattered on/around the user. The human body is an uninviting and often hostile environment for the propagation of a wireless signal, so it is important to understand and characterize the on-body radio channel for the design of power as well as spectrum efficient wearable wireless systems [2]. In [3], on-body radio channel characterization was presented at the unlicensed Manuscript received November 03, 2008; revised March 23, 2009. First published June 10, 2009; current version published January 04, 2010. A. Sani, A. Alomainy, G. Palikaras, Y. Hao, and C. Parini are with the School of Electronic Engineering and Computer Science, Queen Mary, University of London, London E1 4NS, U.K. (e-mail: [email protected]). P. S. Hall and Y. Nechayev are with the Department of Electronic, Electrical and Computer Engineering, the University of Birmingham, Birmingham B15 2TT, U.K. 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.2009.2024969

frequency of 2.45 GHz, commonly used in the Bluetooth and Zigbee standards. Many efforts were made to characterize the UWB on-body radio channel [4]–[8]. In [4], a measurement campaign was performed using two different antennas exhibiting different radiation characteristics. In [6] the propagation around the trunk is fully analyzed and a statistical model is proposed. In [8] the effect of the indoor environment on the UWB body area channel was studied without considering the impact of antenna radiation characteristics. In this communication a measurement campaign is performed in the UWB frequency band 3–10 GHz using a pair of miniaturized coplanar waveguide (CPW) fed tapered slot antennas (TSA) [9], as well as a pair of modified CPW-fed planar inverted cone antennas (PICA)[10]. Free space and body mounted characterization of the aforementioned antennas has been presented in [11]. In this communication, the on-body radio propagation and its spectral and transient characterization are presented. The effect of the indoor environment on on-body UWB radio channel is fully investigated. For the proposed measurement setup, the propagation is quasi-line-of-sight: it consists of combination of line of sight, creeping wave, and reflections from body parts and surrounding scatterers. To enable modeling and prediction of the time delay characteristics of the radio channel, the measured data are fit to empirical statistical models. The goodness of different statistical distributions in fitting the root mean square delay spread has been evaluated, and results demonstrate that the Nakagami model provides the best fitting. When measurements were deployed in the indoor environment, more reflected components were collected, causing a degradation of the goodness of the statistical model. The miniaturized antenna (TSA), due to its more directive radiation [11], is less affected from the reflections from the human body and the indoor environment: it shows improved time delay behavior, and statistical model with higher goodness. Considering the significant size reduction, it was concluded that the TSA is an ideal candidate for UWB-BAN, where the connectivity between body-worn sensors is required. The rest of the communication is organized as follows: Section II shows the measurement setup, Section III discusses the characteristics of the measured UWB radio channels, and Section IV draws a conclusion. II. MEASUREMENT SETUP Fig. 1(c)-(d) shows the layout of the two different antennas used for the measurements. The two antennas are connected to a vector network analyzer (Hewlett Packard 8720ES-VNA) to measure the transmission response (S21) in the frequency range 3–10 GHz. Measurements are first performed in the anechoic chamber to eliminate multipath reflections from the surrounding environment, and then repeated in the Body-Centric Wireless Sensor Lab at Queen Mary, University of London [11] to consider the effect of the indoor environment on the on-body radio propagation channel. Fig. 1(b) shows the location of the two antennas during the measurements: the transmitting one is placed on the left side of the belt, while the receiver is moved along the front part of the body. Thirty-three different positions were measured (on the chest, legs and arms) to ensure sufficient data collection for channel characterization and modeling. III. ON-BODY RADIO PROPAGATION CHANNEL ANALYSIS A. Path Loss The path loss, which is given by the ratio between transmitted and received power is directly calculated from the measurements data,

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Fig. 1. (a) Setup of the proposed measurement; (b) location of the transmitting antenna and position of the 33 receiver; (c) layout of the PICA antenna; (d) layout of the TSA antenna.

averaging over the measured frequency transfers at each frequency points [12], and it can be represented as a function of the distance between transmitter and receiver as explained in [13]. The exponent

, known as path loss exponent, is useful to understand how fast the received power decays with the distance, in particular: P L / (d=d0 ) . It is well known from the Friis formula, that the exponent is equal to two for free space propagation; however for on-body propagation in the anechoic chamber, other factors including losses of human tissues, creeping waves, and reflections from human body parts, lead to higher exponent values compared to the free space one. A least square fit is performed on measured data to evaluate the average path loss, the exponent is the slope of the curve obtained. In this work, the exponent is 3.9 for the PICA and 3.0 for the TSA [see Fig. 2(a)]. When measurements are performed in the indoor environment, the reflection from the surrounding scatterers increases the received power, causing reduction of the path loss exponent. A reduction of 33% is experienced for the PICA case ( = 2:6), and 13% for the TSA ( = 2:6). The PICA having a more omnidirectional radiation than the TSA, is more affected from the multipath reflections, and the exponent reduction is more significant. The values of agree with the ones presented in [8], where it was found = 3:3 in free space, and = 2:7 in the office environment. To improve the accuracy of the path loss model, a zero mean, normal distributed statistical variable is introduced to consider the deviation of the measurements from the calculated average path loss (see Fig. 3). In the anechoic chamber the standard deviation of the normal distribution is  = 6:8 for the PICA, and  = 8:2 for the TSA. For the measurement setup proposed, the antenna radiation pattern is a function which weight the path loss: the more directive the antenna is, the less valid is the linear relation between path loss and logarithmic distance is valid, this is why for the TSA case the deviation of measurements from the linear fit is higher. In the indoor environment, the values of  are 8.0 and 6.7, respectively for PICA and TSA. In such scenarios, the reflections from surrounding scatterers are the main contributors to the deviation from the average path loss. This effect is more significant in the PICA case, where the radiation is more omni-directional. B. Time Delay Analysis The time domain dispersion of the received signal strongly affects the capacity of UWB systems [14]. This effect is characterized by the

Fig. 2. Measured and modelled path loss for on-body channel versus logarithmic Tx-Rx separation distance.

first central moment (mean excess delay m ), and the square root of the second order central moment (root mean square RMS ) of the power delay profile (PDP) [15]. The maximum excess delay is defined as the largest relative delay that a multipath component arrives with power

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TABLE I VALUES FOR THE ROOT MEAN SQUARE DELAY FOR MEASUREMENTS IN THE ANECHOIC CHAMBER; THE AKAIKE CRITERION HAS BEEN APPLIED TO EVALUATE THE GOODNESS OF 5 DIFFERENT DISTRIBUTIONS

1) Statistical Analysis: The Akaike information criteria is a method widely used to evaluate the goodness of a statistical fit [16]. The second order AIC (AICc ) is defined as

AICc = 02 ln(L) + 2K + 2nK0(KK +01)1 :

(1)

1i = AICc;i 0 min(AICc):

(2)

where L is the maximized likelihood, K is the number of parameters estimated for that distribution, n is the number of samples of the experiment. The criterion has been applied to evaluate the goodness of five different distributions commonly used in wireless communications that seem to provide the best fitting for our measurements, namely: Normal, Lognormal, Nakagami, Weibull, and Rayleigh. They are all two parameters distributions (K = 2), except the Rayleigh (K = 1). Smaller value of AICc means better statistical model, and the criterion is used to classify the models from the best to the worse; to facilitate this process the relative AICc is considered and results are normalized to the lowest value obtained:

Fig. 3. Deviation of measurements from the average path loss fitted to a normal distribution; (a) CPW-fed PICA and (b) CPW-fed tapered slot antenna.

greater than the threshold value. Channel impulse responses were calculated based on the measured frequency transfer functions which consist of 1601 frequency points using windowing and inverse discrete Fourier transform (IDFT). The applied time window can detect received multipath components up to 228 ns with a 50 ps resolution. Power delay profiles (PDP) were produced by averaging all impulse responses, considering only samples with the signal level higher than a selected threshold, and observing their delay respect to the peak sample (the direct pulse). Three different threshold levels have been considered: 20, 25, 30 dB below the peak power.

A zero value indicates the best fitness. In this analysis, two different antennas are compared. Furthermore the effect of the receiver sensitivity (the threshold applied to obtain the PDP) on the statistical model is considered. For both parameters the best case (the one used for normalization), is registered for the TSA in the anechoic chamber adopting a less sensitive receiver (threshold 020 dB). When measurements are performed in the indoor environment, more scattered components are considered, and the statistical model is less accurate (higher value of 1i ) than the anechoic chamber case. As shown in Table I, the receiver sensitivity (the threshold applied) affects the statistical model. Using a less sensitive system (020 dB), less reflected components are taken into account, and the model obtained is more deterministic. Comparing the two antennas, the TSA, which is more directive, collects less secondary components, producing a better statistical model. Nakagami distribution provides the better fitness for RMS delay spread data. The fact that the Nakagami is the best, is due to its adaptability to complex scenario with random arrival of pulses. 2) RMS Delay: The root mean square spread delay (RMS ) is a crucial parameter for multipath channels because it imposes a limit to the data rate achievable [15]. Fig. 4 shows the cumulative distribution of the delay spread (obtained applying a threshold of 020 dB) fitted to a Nakagami distribution. Table II summarize the average value and the standard deviation (respectively  and  ) of the Nakagami distributions for each case. The TSA, due to its directional radiation characteristics, is less affected by reflections from the human body parts and surrounding scatterers. When measurements are performed in indoor environment, the multipath effect produces higher delay spread. The PICA, picks up more multipath components due to its omnidirectional radiation patterns which leads to significant increase in the RMS delay

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demonstrate that for a BAN scenario, if a more omnidirectional antenna (PICA) is used, the goodness of the statistical model is lower, and the time delay behavior is degraded. Analyzing the indoor results, the PICA seems to be more sensitive to the reflections from the surrounding scatterers; the reduction of path loss exponent and the increment of time delay parameters are more significant. Furthermore the effect of the receiver sensitivity on the time delay profile of the received signal has been analysed; using a less sensitive receiver the statistical model is more deterministic. The TSA due to its smaller size, and improved time delay performance, is an ideal candidate for UWB-BAN. ACKNOWLEDGMENT Many thanks to Mr. J. Dupuy for his help with measurement campaign and antenna fabrications.

REFERENCES

Fig. 4. Cumulative distribution of the RMS spread delay fitted to a Nakagami distribution applying a threshold of 20 dB; (a) CPW-fed PICA and (b) CPW-fed tapered slot antenna.

0

TABLE II AVERAGE VALUE AND STANDARD DEVIATION OF THE NAKAGAMI DISTRIBUTIONS APPLIED TO FIT THE RMS DELAY MEASURED VALUES

spread. Furthermore using a more sensitive receiver, more secondary reflected components are considered, and the RMS delay is increasing.

IV. CONCLUSION A measurement campaign to evaluate the effect of human body and indoor environment on the UWB on-body radio propagation channel has been performed. The path loss and the time delay profile have been analyzed for propagation along the front of the body, and a statistical study on the RMS delay spread has been performed. Results

[1] J. Forester, E. Green, S. Somayazulu, and D. Leeper, “Ultra-wideband for short- or medium- range wireless communications,” Intel Technol. J., 2001. [2] Antennas and Propagation for Body-Centric Wireless Communications, P. S. Hall and Y. Hao, Eds. Boston, MA: Artech House, 2006. [3] Z. Hu, Y. Nechayev, P. Hall, C. Constantinou, and Y. Hao, “Measurements and statistical analysis of on-body channel fading at 2.45 GHz,” IEEE Antennas Wireless Propag. Lett., vol. 6, pp. 612–615, 2007. [4] A. Alomainy, Y. Hao, C. G. Parini, and P. S. Hall, “Comparison between two different antennas for UWB on-body propagation measurements,” IIEEE Antennas Wireless Propag. Lett., vol. 4, no. 1, pp. 31–34, Dec. 2005. [5] A. Alomainy and Y. Hao, “UWB on-body radio propagation and system modelling for wireless body-centric networks,” in Proc. IEEE Antennas and Propagation Society Int. Symp., Jul. 2006, pp. 2173–2176. [6] A. Fort, C. Desset, P. D. Doncker, and L. V. Biesen, “Ultrawideband body area propagation: From statistics to implementation,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 4, pp. 1820–1826, Jun. 2006. [7] A. Fort, P. D. Doncker, and L. V. Biesen, “An ultra wideband channel model for communication around the body,” IEEE J. Sel. Area Commun., vol. 24, no. 4, pp. 927–933, Apr. 2006. [8] T. Zasowski, F. Althaus, M. Stager, A. Wittneben, and G. Troster, “UWB for noninvasive wireless body area networks: Channel measurements and results,” presented at the IEEE Conf. on Ultra Wideband Systems and Technologies, Nov. 2003. [9] A. Rahman, A. Alomainy, and Y. Hao, “Compact body-worn coplanar waveguide fed antenna for UWB body-centric wireless-networks,” presented at the Eur. Conf. on Antennas and Propagation (EUCAP 07), Edimburgh, U.K., Nov. 2007. [10] A. Alomainy, Y. Hao, and C. G. Parini, “Transient and small-scale analysis of ultra-wide band on-body radio channel,” presented at the North America Radio Science Meeting URSI 2007, Ottawa, Canada, Jul. 22–26, 2007. [11] A. Alomainy, A. Sani, J. Santas, A. Rahman, and Y. Hao, “Transient characteristics of wearable antennas and radio propagation channels for ultrawideband body-centric wireless communications,” IEEE Trans. Antennas Propag, to be published. [12] J. A. Dabin, N. Ni, A. M. Haimovich, E. Niver, and H. Grebel, “The effects of antenna directivity on path loss and multipath propagation in UWB indoor wireless channels,” in Proc. IEEE Conf. Ultra Wideband Systems and Technologies, Newark, NJ, 2003, pp. 305–309. [13] S. S. Gassemzadeh, R. Jana, C. W. Rice, W. Turin, and V. Tarohk, “A statistical path loss model for in-home UWB channels,” in IEEE Conf. Ultrawide Band Systems and Technologies, Baltimore, 2002, p. 5964. [14] W. Ciccognani, A. Durantini, and D. Cassioli, “Time domain propagation measurements of the UWB indoor channel using PN sequence in the FCC-compliant band 3.66 GHz,” IEEE Trans. Antennas Propag., vol. 53, no. 4, pp. 1542–1549, Apr. 2005. [15] T. S. Rappaport, Wireless Communications: Principle and Practice. Englewood Cliffs, NJ: Prentice-Hall, 1999. [16] K. P. Burnham and D. R. Anderson, Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach. New York: Springer-Verlag, 2002.

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Corrections Corrections to “Modeling Antenna Noise Temperature Due to Rain Clouds at Microwave and Millimeter-Wave Frequencies” Frank Silvio Marzano

Abstract—Equation (B.4) and the associated text of the paper “Modeling antenna noise temperature due to rain clouds at microwave and millimeter-wave frequencies” by F. S. Marzano and published in the IEEE Transactions on Antennas and Propagation, Vol. 54, No. 4, pp. 1305–1317, April 2006, contain typographical errors which are now corrected.

In [1], (B.4) and the related text are incorrect as follows: Previous expressions can be sky-noise to a stratified atmosphere, as in Fig. 1, using the same recursive approach as in (18). The set of recursive equations to determine the upwelling TB (l01 ; 00 ) at the (l 0 1)th atmospheric level l01 is then the following:

TB (L ; 00 ) = esF (0 )Ts + [1 0 esF (0 )]TB (L ; 0 ) TB (l01 ; 00 ) 0( 0 )= = TB (l ; 00 )e wl gl 0 + t0l + t1l 0 + t1l (1 0 wl gl ) 2 1 0 e0( 0 )= 0 t1l l 0 l01e0( 0 )= 1 0 cl 0 gl 0 ( 0 ) + wl C1l e 1 + l 0 2 e ( 0 ) 0 e0( 0 )= 1 + cl 0 gl  ( 0 ) + wl C2l e 1 0 l 0 2 e0 ( 0 ) 0 e0( 0 )=

where l = L; L 0 1; . . . ; 1 and upwelling TB (s = L ; 00 ) at the surface is obtained from downwelling TB (s = L ; 0 ) by applying the boundary condition (B.2) at l = L . This should be corrected to read as follows: Previous expressions of satellite-based sky-noise can be extended to a stratified atmosphere, as in Fig. 1, using the same recursive approach as in (18). The set of recursive equations to determine the upwelling TB (l01 ; 00 ) at the (l 0 1)th atmospheric level l01 is then the following:

TB (L ; 00 ) = esF (0 )Ts + [1 0 esF (0 )]TB (L ; 0 ) TB (l01 ; 00 ) 0( 0 )= = TB (l ; 00 )e wl gl 0 + t0l + t1l 0 + t1l (1 0 wl gl ) 2 1 0 e0( 0 )= 0 t1l l 0 l01e0( 0 )= e0( 0 1 0 cl 0 gl + wl C1l 1 + l 0 2 e0  0 e0( 0 )= 0  1 + cl 0 gl + wl C2l 1 0 l 0   0 e0( 0 )= +  2 e

)=

(B.4)

where l = L; L 0 1; . . . ; 1 and upwelling TB (s = L ; 00 ) at the surface is obtained from downwelling TB (s = L ; 0 ) by applying the boundary condition (B.2) at l = L .

REFERENCES (B.4)

[1] F. S. Marzano, “Modeling antenna noise temperature due to rain clouds at microwave and millimeter-wave frequencies,” IEEE Trans. Antennas Propag., vol. 54, no. 4, pp. 1305–1317, Apr. 2006.

Manuscript received August 25, 2009. First published November 06, 2009; current version published January 04, 2010 The author is with the Department of Electronic Engineering, Sapienza University of Rome, Via Eudossiana 18-00184 Rome, Italy and also with the Center of Excellence CETEMPS, University of L’Aquila, L’Aquila, Italy (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2009.2036132

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List of Reviewers for 2009 Reviewing for the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION is an important service undertaken for the Antennas and Propagation Society. Those people who responded to the invitation to review, and who also submitted at least one review in the period between December 1, 2008 and December 1, 2009, are listed below. The names were compiled from information obtained from the ScholarOne Manuscript database. Grateful thanks are extended to all 1168 reviewers that are listed for their contribution to the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION over the past year. Digital Object Identifier 10.1109/TAP.2009.2038730

A Aas, Jon Anders Abd-Alhameed, Raed Abdallah, Esmat Aberle, James Abramovich, Yuri Abubakar, Aria Adams, Robert Adve, Raviraj Ahmad, Fauzia Ahmed, Shahid Ainsworth, Tomas Akkermans, Iwan Akyurtlu, Alkim Ala-Laurinaho, Juha Alayón-Glazunov, Andrés Albani, Matteo Ali, Mohammod Alitalo, Pekka Allard, Rene Alomainy, Akram Alu, Andrea Alvarez, Yuri Amadjikpe, Arnaud Amaya, Cesar Amendola, Giandomenico Ammann, Max Anastassiu, Hristos Andersen, Jorgen Ando, Makoto Andrenko, Andrey Andriulli, Francesco Anguera, Jaume Ansari, Davood Antar, Yahia Antipov, Yuri Antoniades, Marco Arai, Hiroyuki Arapoglou, Pantelis-Daniel Ares-Pena, Francisco Arima, Takuji Arndt, Fritz Athanasiadou, Georgia Austin, Andrew Aydin, Kultegin Azadegan, Reza

B Baccarelli, Paolo Bagci, Hakan Bagri, Durgadas Bailey, David Baker, Lynn Balanis, Constantine Balling, Peter Bandaru, Subbarao Bansal, Rajeev Bao, Xiulong Barba, Pedro Barbosa, Afonso Barka, Andre Barrios, Amalia Batchelor, John Baum, Carl Behdad, Nader Bekers, Dave Belov, Pavel Beneduci, Amerigo Bengtsson, Mats Berenger, Jean-Pierre Bernard, J. M. L. Bernhard, Jennifer Besnier, Philippe Besson, Olivier Bhattacharyya, Arun Bialkowski, Marek Bibby, Malcolm Biebl, Erwin Bikhazi, Nicholas Bilotti, Filiberto Bird, Trevor Blaunstein, Nathan Bleszynski, Marek Bluck, Mike Boag, Amir Boccia, Luigi Boeringer, Daniel Boix, Rafael Bolomey, Jean-Charles Boriskin, Artem Borja, Carmen Boryssenko, Anatoliy Bosisio, Ada Vittoria

Botha, Matthys Bourlier, Christophe Boutayeb, Halim Boyle, Kevin Bozza, Giovanni Bozzi, Maurizio Brachat, Patrice Branch, Karen Braunisch, Henning Brennan, Conor Brewitt-Taylor, Colin Brown, Gary Brown, Tim Bruno, Oscar Buch, Ujjval Buckwalter, James Budaev, Bair Buerkle, Amelia Bunger, Rainer Bunton, John Burghignoli, Paolo Burintramart, Santana Buris, Nicholas Burkholder, Robert C Cable, Vaughn Cahill, Robert Caloz, Christophe Camps, Adriano Canning, Francis Capalino, Filippo Capsoni, carlo Cardama, Angel Carlson, Blair Carlsson, Jan Carluccio, Giorgio Cassioli, Dajana Castaldi, Giuseppe Casula, Giovanni Catedra, Manuel Cavalcante, Charles Chaharmir, Reza Chai, Mei Chair, Ricky Chakraborty, Swagato

Chambers, Barry Chambers, David Chan, Chi Chang, Dau-Chyrh Chatterjee, Deb Chaudhuri, Sujeet Chee, Kin-Lien Chen, Chi-Chih Chen, Hongsheng Chen, Horng-Dean Chen, Hua-Ming Chen, Ji Chen, Kesong Chen, Qiang Chen, Ru shan Chen, Sun-ling Chen, Wen-Shan Chen, Wenhua Chen, Xiaodong Chen, Ye Chen, Zhi Ning Chen, Zhiming Chen, Zhizhang Chevalier, Timothy Chew, Weng Chiu, Tsenchieh Chiu, Frankie Cho, Yong Heui Choi, Charles Choi, Seungwon Christ, Andreas Christiansen, Snorre Christodoulou, Christos Chryssolallis, Michael Chu, Tah-Hsiung Chuang, Huey-Ru Chung, Boon Kuan Chung, David Chung, You Chung Ciattaglia, Matteo Cirstea, Silvia Civi, Aydin Ozlem Claudio, Elio Clavijo, Sergio Clemens, Markus Clenet, Michel

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Coetzee, Jacob Cohen, Gary Collado, Ana Collin, Robert Collins, Brian Colton, David Cools, Kristof Cooray, Francis Corona, Alonso Correia, Luis Cortes, German Costa, Emanoel Costa, Nelson Costa, Jorge Costantine, Joseph Costen, Fumie Craddock, Ian Craeye, Christophe Crane, Robert Cranganu-Cretu, Bogdan Creticos, Justin Crocco, Lorenzo Cui, Suomin Cui, Tie Jun Cummer, Steven Czink, Nicolai D Daniele, Vito Das, Nirod Dauvignac, Jean-Yves Davis, Lionel Davis, William De, Arijit De Grandi, Gianfranco D. de Oliveira, Rodrigo De Zutter, Daniel Debroux, Patrick Declercq, Frederick Deepu, V. Degauque, Pierre Degli-Esposti, Vittorio del Rio, Carlos Delgado, Carlos deng, fengshun Denidni, Tayeb Derneryd, Anders Di Giampaolo, Emidio Di Massa, Giuseppe Di Nallo, Carlo Dias, Ugo Dib, Nihad Dimitriou, Antonis Djordjevic, Antonije Dockery, George Donderici, Burkay Donelli, Massimo Dosopoulos, Stylianos

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 58, NO. 1, JANUARY 2010

Dou, Weiping Doyle, Rory Dragoman, Mircea Dreher, Achim Du, Jianyuan Du, Yang Duan, Baoyan Durgin, Gregory D’Urso, Michele Dussopt, Laurent Dyczij-Edlinger, Romanus E Eccleston, Kim Ederra, Inigo Eibert, Thomas Eleftheriades, George Ellgardt, Anders Ellis, Grant Ellis, Thomas Elnaggar, Michel El-Shenawee, Magda Eltawil, Ahmed Encinar, Jose Engheta, Nader English, Errol Eom, Hyo Ergin, Ahmet Ergul, Ozgur Erricolo, Danilo Erturk, Vakur Eshrah, Islam Essaaidi, Mohamed Estatico, Claudio F Faircloth, Daniel Fakharzadeh Jahromi, Mohammad Fang, Wen-Hsien Fang, Dagang Farr, Everett Farzaneh Koodiani, Sadegh Fasenfest, Benjamin Fathy, Aly Fear, Elise Feldner, Lucas Feng, Yijun Feresidis, Alexandros Fernandes, Carlos Fernandez, Mario Fernandez-Recio, Raul Ferrando, Prof. Miguel Ferreol, Anne Ferrer, Pere Ferrières, Xavier Fiebig, Uwe-Carsten Fikioris, George

Filipovic, Dejan Fink, Patrick Fisher, Rick Fleury, Bernard Flint, James Foltz, Heinrich Fordham, Jeffrey Forsyth, Anthony Fort, Andrew Fostier, Jan Fouad Hanna, Victor Franceschetti, Giorgio Franchois, Ann Francis, Michael Frangos, Panayiotis Freni, Angelo Fuchs, Benjamin Fujimoto, Mitoshi Fujio, Shohei Fuks, Iosif Fumeaux, Christophe Fung, Adrian Furse, Cynthia Fuschini, Franco Fusco, Vincent G Galdi, Vincenzo Galli, Alessandro Gan, Y. B. Gao, Steven Garcia, Enrique Garcia-Ariza, Alexis Garg, Ramesh Ge, Yuehe Gedney, Stephen Geissler, Matthias Gennarelli, Claudio Gentili, Guido Georgiadis, Apostolos Gerini, Giampiero Gershman, Alex Ghoraishi, Mir Ghorbani, Kamran Giannakopoulou, T Giannopoulos, Antonios Gilbert, Roland Ginn, James Giri, David Gjonaj, Erion Godara, Lal Golik, Wojciech Gong, Zhuqian Gong, Xun Gonzalez Garcia, Salvador Gonzalez-Arbesu, Jose Maria Gonzalo, Ramon Gorbachev, Anatoly

Goshi, Darren Goswami, Jaideva Goudos, Sotirios Gouesbet, Gerard Goussetis, George Graglia, Roberto Gragnani, Gian Luigi Granet, Christophe Gray, Derek Grbic, Anthony Griffin, Joshua Grzyb, Janusz Guglielmi, Marco Guha, Debatosh Guiffaut, Christophe Guinvarc’h, Régis Guo, Zhonghai Guo, Y. Jay Guo, Yongxin Gupta, Ramesh Gurel, Levent Gustafsson, Mats Guterman, Jerzy Gwarek, Wojciech H Habib, Mohamed Hadi, Mohammed Hallbjörner, Paul Han, Sok-Kyun Haneda, Katsuyuki Hansen, Thorkild Hansen, Robert Hanson, George Hao, Yang Haupt, Randy Hay, Stuart Hayman, Douglas He, Sailing He, Bo Heldring, Alex Hellicar, Andrew Hemmi, Chris Hendrantoro, Gamantyo Herscovici, Naftali Heyman, Ehud Hill, David Himdi, mohamed Hirata, Akimasa Hirokawa, Jiro Hirose, Masanobu Hislop, Gregory Holloway, Christopher Holter, Henrik Hong, Wei Honma, Naoki Hopkins, Glenn Hoppe, Reiner

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 58, NO. 1, JANUARY 2010

Hrabar, Silvio Hristov, Hristo Hsu, Powen Hu, Wenyi Hu, Zhirun Huang, Zhenyu Huang, Yi Huang, Yikun Huff, Gregory Huggard, Peter Hui, Hon Tat Hui, Ping Hum, Sean Hussain, Malek Hwang, Huan-Sheng Hwang, Ruey I Iiguni, Youji Iigusa, Kyoichi Imbriale, William Imperatore, Pasquale Ingvarson, Per Iodice, Antonio Isernia, Tommaso Ishimaru, Akira Islam, Rubaiyat Itoh, Tatsuo Ittipiboon, Apisak Ivashina, Marianna Iwai, Hiroshi Iyer, Ashwin J Jacob, Arne Jakobsen, Kaj Jakobus, Ulrich Jakoby, Rolf Jam, Shahrokh James, Geoff James, Graeme Jamnejad, Vahraz Jan, Jen-Yea Jensen, Frank Jensen, Michael Jiang, Peilin Jiao, Dan Jin, Jian-Ming Jin, Ya-Qiu Jofre, Lluis Jofre Roca, Luis Johnson, Joel Johnson, William Johnston, Ron Joler, Miroslav Jones, Stephen Jørgensen, Erik Josefsson, Lars

Joseph, Wout Joubert, Johan Judaschke, Rolf K Kabacik, Pawel Kabalan, Karim Kafesaki, Maria Kagoshima, Kenichi Kahng, Sungtek Kaifas, Theodoros Kaiser, Thomas Kaklamani, Dimitra Kalialakis, Chris Kalluri, Dikshitulu Kaltenberger, Florian Kan, Hing Kanellopoulos, John Kantartzis, Nikolaos Kara, Ali Karbeyaz, Ersel Karwowski, Andrzej Kastner, Raphael Kazim, Imran Kefauver, William Keizer, Will Kelley, David Kempel, Leo Kerby, Kiersten Kesteven, Michael Kharkovsky, Sergey Khodier, Majid Kiang, Jean-Fu Kildal, Per-Simon Kim, Bumman Kim, Kristopher Kim, Kyungjung Kim, Oleksiy Kingsley, Nickolas Kishk, Ahmed Kivekas, Outi Klemm, Maciej Knöchel, Reinhard Kobidze, Gregory Koh, Il-Suek Koh, Jinhwan Kolundzija, Branko Kondratiev, Alexander Kopilovich, Lazarus Kornbau, Thomas Kosmas, Panagiotis Koulouridis, Stavros Koutitas, George Koyanagi, Yoshio Kralovec, Jay Kramer, Brad Kress, Rainer Krishnasamy, Selvan

Krowne, Clifford Krzysztofik, Wojciech Kucharski, Andrzej Kumar, B. Preetham Kurner, Thomas Kuroda, Michiko Kuster, Niels Kuwahara, Yoshihiko Kuzuoglu, Mustafa Kwakkernaat, Maurice Kwon, Do-Hoon Kwon, Hyuck L Lacoste, Frederic Lager, Ioan Lail, Brian Laitinen, Tommi Lakhtakia, Akhlesh Lampérez, Alejandro García Landesa, Luis Lang, Roger Langley, Richard Lanne, Maria Las-Heras, Fernando Lau, Buon Kiong Lau, Ka Leung Laurin, Jean-Jacques Lazar, Steve Lazaropoulos, Athanasios Le Coq, Laurent Le Vine, David Leberer, Ralf Lee, Cheng-Jung Lee, J. Lee, Jin-Fa Lee, Kai-Fong Lee, Kun-Chou Lee, Robert Lee, Yongshik Lee, Yoonjae Leeper, David Leong, Hank Leong, Mook-Seng Lesselier, Dominique Letrou, Christine Leuchtmann, Pascal Leung, Kwok Leung, Yee Hong Leveque, Philippe Leviatan, Yehuda Levitas, Menachem Lheurette, Eric Li, Bin Li, Juan Li, Morui Li, RongLin Li, Ying

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Li, Yujia Liang, Chang-Hong Liang, Ming-Cheng Liang, Xu Liao, DaHan Liénard, Martine Lier, Erik Lim, Eng Hock Lim, Sungjoon Lim, Sungkyun Lin, Jau-Jr Lindmark, Bjorn Ling, Hao Linton, Christopher Liolis, Konstantinos Litman, Amelie Litschke, Oliver Liu, Dawei Liu, Qing Huo Liu, Wei Liu, Wen-Chung Liu, Yuan Liu, Zhijun Liu, Zhiwen Llombart Juan, Nuria Logani, Mahendra Loh, Tian Lomakin, Vitaliy Long, Stuart Losada, Vicente Loubaton, Philippe Lovat, Giampiero LoVetri, Joe Loyka, Sergey Lu, Caicheng Lui, Hoi-Shun Lu, Junwei Lu, Mingyu Lu, Yilong Luk, K. Luo, Chong Luxey, Cyril Lyalinov, Mikhail Lysko, Albert M Ma, Jinping Ma, Tzyh-Ghuang Maaskant, Rob Maci, Stefano MacKay, Tom Mahanfar, Alireza Maharaj, Sunil Mahmoud, Mohamed Mahmoud, Samir Mailloux, Robert Makarov, Sergey Makinen, Riku

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Malas, Tahir Mantel, Onno Manteuffel, Dirk Mao, Shi-Chun Maradudin, Alexei Marcano, Diogenes Marcos, Sylvie Marengo, Edwin Marhefka, Ronald Marklein, Rene Markley, Loïc Marques, Ricardo Marrocco, Gaetano Marti-Canales, Javier Martin, Ferran Martinez, Rene Martinez-Vazquez, Marta Martini, Enrica Marvin, Andy Massa, Andrea Matekovits, Ladislau Mateos, Rosa Materum, Lawrence Matricciani, Emilio Matsuzawa, Shin-Ichiro Maurer, Juergen Mayhew-Ridgers, Gordon Mazzarella, Giuseppe McGrath, Daniel McLean, James McNamara, Derek Meaney, Paul Medbo, Jonas Medina, Francisco Mei, Zicong Melamed, Timor Melapudi, Vikram Melde, Kathleen Mendes, Paulo Meng, Yu Song Menzel, Wolfgang Mesa, Francisco Mias, Christos Michalski, Krzysztof A. Michielssen, Eric Michishita, Naobumi Migliaccio, Claire Migliore, Marco Mikki, Said Miller, Philip Mirshekar, Dariush Mishchenko, Michael Mittra, Raj Miyashita, Hiroaki Moghaddam, Mahta Mohan, Ananda Mohanan, P. Molisch, Andreas

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 58, NO. 1, JANUARY 2010

Mongia, Rajesh Mongiardo, Mauro Monorchio, Agostino Montisci, Giorgio Montoya, Thomas Morabito, David Moraitis, Nektarios Moreira, Fernando Morioka, Takehiro Morishita, Hisashi Mosallaei, Hossein Mosca, Stefano Moschovitis, Charalampos Mouthaan, Koenraad Mumcu, Gokhan Munk, Ben N Nair, Naveen Nakano, Hisamatsu Nam, Sangwook Nanzer, Jeffrey Nefedov, Igor Nehra, C Nepa, Paolo Neto, Andrea Neve, Michael Nevels, Robert Newell, Allen Newman, Edward Ney, Michel Ng, Boon Ng Mou Kehn, Malcolm Nickel, Ulrich Nie, Zaiping Nikitin, Pavel Nikolaou, Symeon Nikolic, Nasiha Nikolova, Natalia Nikookar, Homayoun Nilavalan, Rajagopal Noghanian, Sima Noguchi, Keisuke Norgren, Martin Norrod, Roger Nosich, Alexander Notaros, Branislav O Odendaal, Johann Oestges, Claude Ogawa, Koichi Ohishi, Takafumi Öjefurs, Erik Okado, Hironori Okano, Yoshinobu Okhmatovski, Vladimir Oliveri, Giacomo

Olver, David Onishi, Teruo Orihashi, Naoyuki Orta, Renato Osipov, Andrey Otero, Pablo Oughstun, Kurt Ozturk, Alper P Paknys, Robert Palmer, Dev Palmer, Keith Pan, QingWei Pan, Bo Pan, Guang-Wen Pan, Helen Panagopoulos, Athanasios Papapolymerou, John Paraboni, Aldo Parfitt, Andrew Park, Seong-Ook Parker, Ted Parsa, Armin Pastorino, Matteo Pasveer, Frank Patnaik, Amalendu Paul, Dominique Paulotto, Simone Paulson, Kevin Pechac, Pavel Peeters, Joris Peng, Zhen Pereda, Jose Pereira-Filho, Odilon Peroulis, Dimitrios Perruisseau-Carrier, Julien Person, Christian Persson, Patrik Peterson, Andrew Petko, Joshua Petosa, Aldo Peyman, Azadeh Piazza, Daniele Piesiewicz, Radoslaw Pinchera, Daniele Pinel, Nicolas Pissoort, Davy Pivnenko, Sergey Poggio, Andrew Pogorzelski, Ronald Polemi, Alessia Polivka, Milan Polycarpou, Anastasis Ponchak, George Popov, Alex Popov, Alexandre Popovic, Milica

Potter, Mike Poulton, Geoff Pozar, David Prieto-Cerdeira, Roberto Priou, Alain Psychoudakis, Dimitris Q Qiang, Rui Qing, Xianming Qiu, Cheng-Wei R Raffetto, Mirco Rahman, Atiqur Rahmat-Samii, Yahya Railton, Chris Raisanen, Antti Raj, Rohith Rajo-Iglesias, Eva Ramahi, Omar Ramirez, Raul Randazzo, Andrea Rao, K. V. S. Rao, Patnam Rao, Qinjiang Rao, Sadasiva Rao, Sudhakar Rappaport, Carey Ravipati, Babu Rawat, Banmali Rawat, Vineet Rebeiz, Gabriel Reddy, C. J. Reineix, Alain Rengarajan, Sembiam Ribeiro, Marco Riccio, Daniele Richter, Andreas Rius, Juan Riva, Carlo Roblin, Christophe Rocca, Paolo Rocha, Armando Rodenbeck, Christopher Rodriguez, Jose Roederer, Antoine Rogers, David Rogier, Hendrik Rokhlin, Vladimir Romanofsky, Robert Romeu, Jordi Rossi, Marco Rothwell, Edward Row, Jeen-Sheen Rowe, Wayne Roy, Jasmin Rud, Leonid

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 58, NO. 1, JANUARY 2010

Rydberg, Anders Rylander, Thomas S Sagnard, Florence Sahalos, John Sahr, John Saily, Jussi Saito, Kazuyuki Saka, Birsen Sakaguchi, kei Sakakibara, Kunio Sakarellos, Vassilios Salem, Mohamed Salous, Sana Sanchez, Victor Sanchez-Hernandez, David Sangary, Nagula Sani, Andrea Santalla, Veronica Sarabandi, Kamal Sarkar, Tapan Sarris, Costas Sarvas, Jukka Sato, Kazuo Sato, Motoyuki Satoh, Akio Sauleau, Ronan Savov, Sava Scanlon, William Scharstein, Robert Schaubert, Daniel Scherbatko, Igor Schmidt, Carsten Schmidt, Stefan Schneider, John Schoenhuber, Michael Schuchinsky, Alex Schuhmann, Rolf Sebak, Abdel Seetharam, Venkatesh Segovia-Vargas, Daniel Selleri, Stefano Serebrennikov, Aleksey Sertel, Kubilay Sevgi, Levent Shafai, Lot Shahabadi, Mahmoud Shaker, George Shaker, Jafar Shamonina, Ekaterina Shanker, B. Shao, Wei Shaoqiu, Xiao Sharaiha, Ala Sharma, Rajesh Sharma, Satish Shastry, Prasad

Shavit, Reuven Shea, Jacob Shen, Jinjin Shen, Zhongxiang Sheng, Xin-Qing Shipley, Charles Shiroma, Wayne Shlager, Kurt Shlivinski, Amir Shu, Ting Shubitidze, Fridon Shuley, Nicholas Siakavarar, Katherine Sibille, Alain Sierra-Castañer, Manuel Sievenpiper, Daniel Sihvola, Ari Silveirinha, Mário Simons, Rainee Simovski, Constantin Simpson, Jamesina S¸ims¸ek, Ergün Sipus, Zvonimir Siwiak, Kai Sizun, H. Sjoberg, Daniel Skobelev, Sergei Skrivervik, Anja Smith, Glenn Smith, Jerry Smith, Paul Smith, Stephanie Smith, William Smolders, Bart Smulders, Peter Soldovieri, Francesco Song, Wei Song, Jiming Soras, Constantine Sorolla, Mario Spencer, Quentin Srikanth, S Stancil, Daniel Steinberg, Ben Sten, Johan Stephanson, Matthew Stevanovic, Ivica Steyskal, Hans Stockbroeckx, Benoit Stuart, Howard Stuchly, Maria Stupfel, Bruno Sturm, Christian Stutzman, Warren Subrahmanyan, Ravi Suh, Seong-Youp Sullivan, Dennis Sun, Guilin

Sun, Mei Sutinjo, Adrian Suzuki, Hajime Svantesson, Thomas Sydanheimo, Lauri T Tabatabaeenejad, Alireza Taflove, Allen Taillefer, Eddy Takahashi, Masaharu Tam, Wai-Yip Tan, Jilin Tan, Eng Tanaka, Toshiyuki Tang, Philip Tarng, J. Tavassolian, Negar Tayem, Nizar Taylor, Graham Teixeira, Fernando Tentzeris, Emmanouil Thevenot, Marc Thiel, David Thiel, Michael Thiele, Gary Thomae, Reiner Thomas, John Thornton, John Ting, Sioweng Tjelta, Terje Tjuatja, Saibun Toccafondi, Alberto Toh, Wee Kian Tomasic, Boris Tong, Kin Kai Tong, Mei Topa, Antonio Toporkov, Jakov Topsakal, Erdem Torlak, Murat Toupikov, Mikhail Tretyakov, Sergei Tricarico, Simone Trintinalia, Luiz Trott, Keith Trucco, Andrea Trueman, Christopher Tsalamengas, John Tsang, Leung Tsiboukis, Theodoros Tsoulos, George Tsuji, Mikio Tsukerman, Igor Tsunekawa, Koichi Tuovinen, Jussi Turkel, Eli Tyo, Scott

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U Ueda, Tetsuya Ukkonen, Leena Umul, Yusuf Uslenghi, Piergiorgio Usner, Brian V Valerio, Guido Valero-Nogueira, Alejandro Vallecchi, Andrea Vall-Llossera, Merce van Beurden, Martijn Van Caekenberghe, Koen van de Kamp, Max van den Berg, Peter M. Van der Vorst, Maarten van Genderen, Piet Van Lil, Emmanuel Vande Ginste, Dries Vandenbosch, Guy Vanjani, Kiran van’t Klooster, Kees Varadan, Vasundara Vasiliadis, Theodore Vaskelainen, Leo Vaughan, Rodney Vazquez Alejos, Ana Vecchi, Giuseppe Velamparambil, Sanjay Veliev, Eldar Ver Hoeye, Samuel Vescovo, Roberto Vigano, Maria Vinogradova, Elena Vinoy, Kalarickaparambil Vipiana, Francesca von Hagen, Jürgen Vorobyov, Sergiy Voronovich, Alexander Vouvakis, Marinos Vouyioukas, Demosthenes W Wagen, Jean-Frederic Wahid, Parveen Waldschmidt, Christian Wallace, Jon Wallén, Henrik Waller, Marsellas Wang, Bu-hong Wang, Chao-Fu Wang, Chien-Jen Wang, Gaofeng Wang, Hanyang Wang, Jianqing Wang, Junhong Wang, Shumin

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Wang, Xiande Wang, Yan Wang, Yide Ward, Jeffrey Warnick, Karl Waterhouse, Rod Watson, Robert Webb, J. Weile, Daniel Weily, Andrew Weller, Tom Wells, Mike Wen, Geyi Wen, Yinghong Werner, Douglas West, James Whitman, Gerald Wiart, Joe Wiesbeck, Werner Wildman, Raymond Wiltse, James Winton, Scott Wong, Hang Wong, Kainam Thomas Wong, Kin-Lu Wong, Man-Fai

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 58, NO. 1, JANUARY 2010

Wood, Aihua Worms, Josef Wu, Boping Wu, Dagang Wu, Ke Wu, Qun Wu, Xin Wu, Xuan Hui Wu, Zhen-Sen Wu, Zhipeng Wyglinski, Alexander X Xia, Mingyao Xie, Xin Xin, Hao XU, Lisheng Xu, Xiaojian Xu, Zheng-Wen Y Yaghjian, Arthur Yakovlev, Alexander Yamada, Hiroyoshi Yamada, Yoshihide Yamauchi, Junji

Yang, H. Y. David Yang, Fan Yang, Kehu Yang, Ning Yang, Shing Lung Steven Yang, Shiwen Yapar, Ali Yashchyshyn, Yevhen Yasumoto, Kiyotoshi Ye, Qiubo Ye, Zhongfu Yegin, Korkut Yeo, Junho Yeo, Tat Yi, Huiyue Yilmaz, Ali Yilmazer, Nuri Ying, Zhinong Yioultsis, Traianos Yla-Oijala, Pasi Young, Jeffrey Yu, Alfred Yu, Chun Yu, Kai Yu, Wenhua Yuan, Mengtao

Yun, Jane Yun, Zhengqing

Z Zaghloul, Amir Zaki, Kawthar Zapata, Juan Zedler, Michael Zekavat, Seyed Zeng, Qingsheng Zentner, Radovan Zhang, Hualiang Zhang, Y. Zhang, Yimin Zhao, Anping Zhao, Junsheng Zhou, Lei Zhao, Yan Zhao, Weixin Zhao, Zhiqin Zhu, Ning Yan Zieniutycz, Włodzimierz Ziolkowski, Richard Zurcher, J.-F. Zwick, Thomas

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Fritz Arndt Jean-Pierre Berenger Steven R. Best Christophe Craeye Peter de Maagt Tayeb A. Denidni George Eleftheriades Lal C. Godara

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Dallas R. Rolph, Editorial Assistant [email protected] +61 2 9372 4289 (voice) +61 2 9372 4446 (fax) Senior Associate Editor Graeme L. James Associate Editors Duixian Liu Stefano Maci Robert J. Paknys George W. Pan Athanasios Panagopoulos Matteo Pastorino Patrik Persson K.V. S. Rao

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Digital Object Identifier 10.1109/TAP.2009.2039675

INSTITUTIONAL LISTINGS The IEEE Antennas and Propagation Society is grateful for the support given by the organizations listed below and invites applications for Institutional Listings from other firms interested in the field of Antennas and Propagation.

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Digital Object Identifier 10.1109/TAP.2009.2039720