IEEE Transactions on Antennas and Propagation [volume 59 number 2]

Citation preview

FEBRUARY 2011

VOLUME 59

NUMBER 2

IETPAK

(ISSN 0018-926X)

PAPERS

Antennas The Parametric Optimization of Wire Dipole Antennas . ......... ......... ........ ......... .... J. Kataja and K. Nikoskinen Inductively-Loaded Yagi-Uda Antenna With Cylindrical Cover for Size Reduction at VHF-UHF Bands ....... ......... .. .. ........ ......... ... J. A. Tirado-Mendez, H. Jardon-Aguilar, R. Flores-Leal, M. Reyes-Ayala, and F. Iturbide-Sanchez Compact Coplanar Waveguide (CPW)-Fed Zeroth-Order Resonant Antennas With Extended Bandwidth and High Efficiency on Vialess Single Layer ..... ......... ........ ......... ......... ........ ......... ..... T. Jang, J. Choi, and S. Lim Increasing the Bandwidth of Microstrip Patch Antenna by Loading Compact Artificial Magneto-Dielectrics .. ......... .. .. ........ ......... ......... ........ ....... X. M. Yang, Q. H. Sun, Y. Jing, Q. Cheng, X. Y. Zhou, H. W. Kong, and T. J. Cui A New Design Method for Single-Feed Circular Polarization Microstrip Antenna With an Arbitrary Impedance Matching Condition ....... ......... ........ ......... ....... ... ........ ......... ......... ........ S. Maddio, A. Cidronali, and G. Manes and Resonators ....... ......... ......... .. Compact Band-Rejected Ultrawideband Slot Antennas Inserting With .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ... Z.-A. Zheng, Q.-X. Chu, and Z.-H. Tu Millimeter-Wave Microstrip Comb-Line Antenna Using Reflection-Canceling Slit Structure ... ........ ......... ......... .. .. ........ ......... ......... ........ ....... Y. Hayashi, K. Sakakibara, M. Nanjo, S. Sugawa, N. Kikuma, and H. Hirayama Frequency Steerable Two Dimensional Focusing Using Rectilinear Leaky-Wave Lenses ........ ........ ......... ......... .. .. ........ ....... J. L. Gómez-Tornero, F. Quesada-Pereira, A. Alvarez-Melcón, G. Goussetis, A. R. Weily, and Y. J. Guo Sub-Wavelength Profile 2-D Leaky-Wave Antennas With Two Periodic Layers ......... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... .... C. Mateo-Segura, G. Goussetis, and A. P. Feresidis Experimental Characterization of a Surfaguide Fed Plasma Antenna .... ........ ......... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ....... P. Russo, V. M. Primiani, G. Cerri, R. De Leo, and E. Vecchioni Experimental Study of the Effect of Modern Automotive Paints on Vehicular Antennas ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ .. B. D. Pell, E. Sulic, W. S. T. Rowe, K. Ghorbani, and S. John Array Antennas and Feeds A Toolset Independent Hybrid Method for Calculating Antenna Coupling ...... ......... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... . M. Kragalott, M. S. Kluskens, D. A. Zolnick, W. M. Dorsey, and J. A. Valenzi Array Noise Matching—Generalization, Proof and Analogy to Power Matching ........ ......... ........ ..... C. Findeklee Compact Two-Layer Rotman Lens-Fed Microstrip Antenna Array at 24 GHz .. ......... ... L. Lee, J. Kim, and Y. J. Yoon Bayesian Compressive Sampling for Pattern Synthesis With Maximally Sparse Non-Uniform Linear Arrays .. ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... ........ G. Oliveri and A. Massa Synthesis of Multi-Beam Sub-Arrayed Antennas Through an Excitation Matching Strategy .... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... L. Manica, P. Rocca, G. Oliveri, and A. Massa

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(Contents Continued on p. 349)

(Contents Continued from Front Cover) Investigations on the Efficiency of Array Fed Coherently Radiating Periodic Structure Beam Forming Networks ...... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ....... N. Ferrando and N. J. G. Fonseca Design and Implementation of Two-Layer Compact Wideband Butler Matrices in SIW Technology for Ku-Band Applications ... ......... ........ ......... ......... ........ ...... A. A. M. Ali, N. J. G. Fonseca, F. Coccetti, and H. Aubert Reflective Surfaces A Self-Tuning Electromagnetic Shutter .. ......... ........ ... R. O. Ouedraogo, E. J. Rothwell, S.-Y. Chen, and A. Temme Cross-Dipole Bandpass Frequency Selective Surface for Energy-Saving Glass Used in Buildings ...... ......... ......... .. .. ........ ......... ......... ........ ......... ......... .. G. I. Kiani, L. G. Olsson, A. Karlsson, K. P. Esselle, and M. Nilsson Numerical Techniques Multiscale Compressed Block Decomposition for Fast Direct Solution of Method of Moments Linear System ......... .. .. ........ ......... ......... ........ ......... ......... ....... A. Heldring, J. M. Rius, J. M. Tamayo, J. Parrón, and E. Ubeda Electromagnetic Scattering From Electrically Large Arbitrarily-Shaped Conductors Using the Method of Moments and a New Null-Field Generation Technique ....... ........ ......... ......... ... T. N. Killian, S. M. Rao, and M. E. Baginski Calderon Preconditioned Surface Integral Equations for Composite Objects With Junctions .... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... . P. Ylä-Oijala, S. P. Kiminki, and S. Järvenpää A Singularity-Free Boundary Equation Method for Wave Scattering .... ........ ......... ......... ........ .... I. Tsukerman A Complete Set of Linear-Phase Basis Functions for Scatterers With Flat Faces and for Planar Apertures ..... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ .... M. Casaletti, S. Maci, and G. Vecchi Stable Electric Field TDIE Solvers via Quasi-Exact Evaluation of MOT Matrix Elements ...... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ........ Y. Shi, M.-Y. Xia, R.-S. Chen, E. Michielssen, and M. Lu Boundary Diffracted Wave and Incremental Geometrical Optics: A Numerically Efficient and Physically Appealing Line-Integral Representation of Radiation Integrals. Aperture Scalar Case ... ......... ........ .. ........ ........ M. Albani On the Use of Series Expansions for Kirchhoff Diffractals ....... ......... ........ ......... ......... .. S. Perna and A. Iodice ˇ ´ , M. Casaletti, S. Maci, and S. B. Sørensen Complex Conical Beams for Aperture Field Representations .... .... S. Skokic Wireless Simulation and Measurement of Dynamic On-Body Communication Channels ......... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ...... M. Gallo, P. S. Hall, Q. Bai, Y. I. Nechayev, C. C. Constantinou, and M. Bozzetti Indoor Off-Body Wireless MIMO Communication With Dual Polarized Textile Antennas ...... ........ ......... ......... .. .. ........ ......... ......... ........ ... P. Van Torre, L. Vallozzi, C. Hertleer, H. Rogier, M. Moeneclaey, and J. Verhaevert Modal Network Model for MIMO Antenna in-System Optimization .... ........ ......... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ... J. Pontes, J. Córcoles, M. A. González, and T. Zwick Emulation of MIMO Rician-Fading Environments With Mode-Stirred Reverberation Chambers ....... ......... ......... .. .. ........ .... J. D. Sánchez-Heredia, J. F. Valenzuela-Valdés, A. M. Martínez-González, and D. A. Sánchez-Hernández Propagation of ELF Electromagnetic Waves in the Lower Ionosphere ... ......... ......... . K. Li, X. Y. Sun, and H. T. Zhai

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COMMUNICATIONS

A Printed Elliptical Monopole Antenna With Modified Feeding Structure for Bandwidth Enhancement ........ ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ....... J. Liu, S. Zhong, and K. P. Esselle Small Square Monopole Antenna With Enhanced Bandwidth by Using Inverted T-Shaped Slot and Conductor-Backed Plane ... ......... ......... ........ ......... ........ M. Ojaroudi, Sh. Yazdanifard, N. Ojaroudi, and M. Naser-Moghaddasi Coupled-Fed Shorted Monopole With a Radiating Feed Structure for Eight-Band LTE/WWAN Operation in the Laptop Computer ....... ......... ........ ......... ......... ........ ......... ... T.-W. Kang, K.-L. Wong, L.-C. Chou, and M.-R. Hsu Lower Bound for the Radiation of Electrically Small Magnetic Dipole Antennas With Solid Magnetodielectric Core . .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... .... O. S. Kim and O. Breinbjerg Leaky Coaxial Cable With Circular Polarization Property ....... ......... ........ ......... ......... ........ ......... . J. Wang Cavity-Backed Circularly Polarized Self-Phased Four-Loop Antenna for Gain Enhancement ... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... Q. Yang, X. Zhang, N. Wang, X. Bai, J. Li, and X. Zhao Using 3D Field Simulation for Evaluating UHF RFID Systems on Forklift Trucks ...... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ........ J.-F. Hoefinghoff, A. Jungk, W. Knop, and L. Overmeyer On the Singularity of the Closed-Form Expression of the Magnetic Field in Time Domain ... . H. A. Ülkü and A. A. Ergin Fast Computation of Sommerfeld Integral Tails via Direct Integration Based on Double Exponential-Type Quadrature Formulas ....... ......... ........ ......... ......... ........ .. R. Golubovic´ Nic´iforovic´, A. G. Polimeridis, and J. R. Mosig Diversity On-Glass Antennas for Maximized Channel Capacity for FM Radio Reception in Vehicles . ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... . S. Ahn, Y. S. Cho, and H. Choo

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CORRECTIONS

Correction to “3-D Thin-Wire FDTD Approach for Resistively Loaded Cylindrical Antennas Fed by Coaxial Lines” . .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... ........ S.-Y. Hyun and S.-Y. Kim

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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 59, NO. 2, FEBRUARY 2011

The Parametric Optimization of Wire Dipole Antennas Juhani Kataja and Keijo Nikoskinen, Senior Member, IEEE

Abstract—The shape representation of planar wire dipole antennas of fixed radius is discussed. Away from the feed point the shape can be always represented as a arc-length parametrizable Lipschitz-continuously differentiable curve. The representation is applied to classical directivity optimization design problems as well as impedance optimization problems. By such an optimization, it is shown that directive wire dipoles of length (3 2) can be tuned by shape optimization and that the real part of input impedance of 2 dipole is bounded from above if the imaginary part must be small. Index Terms—Antenna input impedance, directive antennas, electric field integral equation, optimal control, wire antennas and boundary integral equation.

I. INTRODUCTION

T

HE wire dipole is an important antenna type due to its simple design and the fact that its properties are well known. In this work we include the shape of the wire in the design parameter space in contrast to the classical situation where the parameter space consists solely of length and radius. In the general case, the shape of the antenna is represented by a plane curve. Such curves are, assuming certain regularity, completely defined with one scalar function [1] up to rotation and translation. We employ such an antenna representation to find optimal designs of directive and/or tuned antennas. More explicitly, we show that simultaneous optimization of the directivity and impedance of wire dipoles is possible within certain limits. As and dipoles examples we study design problems of and an antenna consisting of two elements. The shape optimization was, in part, inspired by [2], where Landstorfer presented a shaped wire dipole of length which has a much greater directivity than the straight antenna. Additionally, a similar problem has also been studied by Liang and Cheng [3] and others [4]. The field simulations are carried out using the electric field integral equation (EFIE) [5] for perfectly electrically conducting (PEC) objects discretized using Galerkin’s scheme with Rao-Wilton-Glisson (RWG) [6] basis functions. The simulations are verified by finite element analysis using COMSOL Multiphysics 3.5a software (COMSOL). Manuscript received August 21, 2009; revised June 15, 2010; accepted July 27, 2010. Date of manuscript publication December 03, 2010; date of current version February 02, 2011. The authors are with the Department of Radio Science and Engineering, Aalto University, School of Technology, FI-02150 Espoo, Finland (email: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2096181

The rest of the paper is divided into two parts. In the first part we describe the antenna modeling procedure as an arclength parametrizable curve and introduce the complete parameter space in an infinite dimensional setting. The introduced parametrization is the core of this paper. The first part concludes with a description how the shape space was discretized to be a . closed box in The second part is devoted to the presentation of the examples designs and discussion on the numerical aspects of the optimization procedure. In the appendices we review briefly the theory of the Lipschitz mappings since the shape function turns out to be Lipschitz continuous with a bounded Lipschitz constant and thus, using a standard Arzelà-Ascoli [7] argument, we obtain compactness results for the shape space yielding the existence of the extrema of continuous cost functions on the shape space. II. MODELING OF THE SHAPE We model the geometry of the curved wire dipole as a deformation of a reference cylinder of given radius and length. The and the length is left as an optimization radius is fixed to parameter. The feed port is modeled as a voltage-gap source. Given an arc-length parametrized curve , we define the deformation map that takes a straight reference cylinder to the curved one by (1) We define the reference cylinder

by (2)

where is half of the total length of the antenna and is the radius of the antenna wire. The assumption that is arc-length parametrizable does not restrict the set of possible geometries too much, since it is physically feasible to measure the length of a wire and attach a metric along it. is illustrated by The action of the deformation mapping Fig. 1. Immediately, it can be observed from (1) that the curve must be, at least, continuously differentiable and if the surface set of the antenna is required to be Lipschitz-continuous, then the derivative of must be Lipschitz-continuous. For a review of Lipschitz mappings, see Appendix A. Furthermore, requiring that is a one-to-one mapping yields a lower bound for the radius of curvature and a certain nearness measure of to be discussed later. If the radius of curvature of is less than the

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KATAJA AND NIKOSKINEN: THE PARAMETRIC OPTIMIZATION OF WIRE DIPOLE ANTENNAS

351

Fig. 3. A curve in

Since is

represented by a real function .

, it follows that for each such that

there (3)

Fig. 1. The deformation mapping F .

Moreover, since is continuous, we have that fixed yields a unique continuous by the theory of covering spaces [8]. Defining by the equation

and choosing , the following representation for

and

we have

(4)

Fig. 2. If the radius of curvature of p is too small, the resulting antenna shape intersects itself.

radius of the cross section, then the antenna shape intersects itself as illustrated in Fig. 2 and more importantly, the shape no longer represents a bent wire antenna. The lower bound for the curvature is respected if and only if the Lipschitz constant of is bounded from above. A. The Curve Let us now explicate the conditions that must satisfy in order for it to represent a wire dipole antenna: 1) Condition 1: The curve is continuously differentiable . and its derivative satisfies 2) Condition 2: The length of the curve is bounded from below and above. 3) Condition 3: The derivative is a function on taking . values on the unit circle and of 4) Condition 4: The component functions must satisfy and . , it holds that 5) Condition 5: For large . Without loss of generality we only need to consider positive arguments of the curve due to Condition 4 .

We fix in here, but in the case of multiple antenna elements would also be an optimization parameter. The choice is due to Conditions 1 and 4 , since can be that and the axis: if interpreted as the angle between then would not be continuous at the origin. The interpretation of is illustrated by Fig. 3. Thus far, we have shown that for each curve satisfying Consuch that (4) holds. ditions 1 – 4 , there is a unique pair On the other hand, let be such a pair that the following conditions hold: ; (i) and is continuous on ; (ii) (iii) . Then, the Formula (4) immediately yields a curve for which Conditions 1 to 3 hold. Condition 4 is enforced explicitly by and for . Likedefining wise, Condition is enforced explicitly on the corresponding curve as follows. to a curve Let us denote the mapping that takes data by (4) by

To a curve defined by

we associate a nearness function

(5)

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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 59, NO. 2, FEBRUARY 2011

where

is a positive constant making the argument of the function “large” thus making to quickly become the times . Note that distance function in

from which the approximate introduce the discrete space optimal shapes are sought by formula

(11) and where the upper formula on right hand side corresponds to large and the lower one corresponds to small . Finally, if (6) and

then for all , Condition 5 holds for the pair that corresponds to . It should be noted that this is just an arbitrary choice to enforce Condition 5 . This way, however, the functional is continuous when the space of those which satisfy conditions (i)-(iii) is equipped with the usual -metric. This makes it simple to show that the space is a compact metric space which in of admissible shapes turn yields existence of extrema for continuous cost functions. See Appendix B for details. Collecting results, we have the space of admissible shape data

(7)

(12) Note that (8) is replaced by (11), and for a suitable choice of , the latter implies the former. The condition (7) is replaced by a logarithmic barrier type [10] error function defined by (13) where is some positive constant dictating how close to itself is the positive part of the real number the curve can come, , i.e., if and if . The positive part is taken to ensure that the logarithm function does not interfere with the optimal shape if it is far from intersecting itself, and the second power in (13) is there to ensure differentiability when argument of the positive-part function is zero. is uniquely determined by the coefSince each element of we identify with the closed ficient vector box

(8) and (14) (9) and are any constants larger than . The lower where and upper bound for are due to fact that the antenna must have minimal and maximal length. No antenna can be of length and are arbitrary 0 or . However, the choices for as long as they are positive.

on which Let us now describe the actual cost function on be the directivity in the BFGS algorithm is applied to. Let direction, let be the input impedance and the positive and be the width and height of the bounding box let of the curve corresponding to the shape data in the plane. Now, the cost function on is defined by

B. Optimization and Approximation Our basic approach is to construct a single cost function on which the BFGS algorithm [9] is applied to in order to find a local minimum. The space is discretized by setting (10) where and is the usual hat-function basis, i.e., piecewise linear continuous functions with respect to uniconsisting of nodes. Thus, the apform partition of . We proximate shape is given by the vector denote the dimensional space of piecewise linear continuous . functions by The conditions (7) and (8) are impractical from the numerical point of view and are treated differently. To that end, let us first

(15) where , , and , , are the goal impedance, maximum width and height of the bounding box and weight factors, respectively. The weight factors are chosen . ad hoc to be in the interval The BFGS algorithm seeks local approximate solutions of the problem (16) under the assumption that is convex and twice differentiable. We denote the approximate local solution to this problem by . Thus we have converted the problem of finding optimal antenna shapes to a finite dimensional optimization problem on a . compact subset of

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TABLE I COMPARISON OF COMPUTED INPUT IMPEDANCES OF LINEAR =2 DIPOLE WITH DIFFERENT MESHES

III. EXAMPLE DESIGNS We apply the optimization methodology to design the proband directive long antennas of given lems of shaped input impedance. We also study the classical Landstorfer antenna [2] as an exemplary design to show that when the dimension of the shape space is increased, the shape seems to converge. antennas We show that it is possible to obtain having antenna length of about and curved dipoles, with directivity of about 7 dB just by bending the antenna wire. elements Finally we present a design consisting of two having an input impedance of 80 . The properties of the obtained antenna structures are verified by finite element analysis using COMSOL Multiphysics software. A. Numerics The field solutions for and were obtained by solving EFIE discretized with the RWG elements and the input port was modeled as a voltage-gap source. The actual deformation of the mesh was implemented by mapping the reference mesh by defined by (1). We used cylinders having digonal, pentagonal or octagonal cross sections as computational reference geometries. Their cross sections were scaled in such a manner their equivalent radius , defined by formula [11]

equals . Here is the side length of the regular -gon defining the cross-section, is the number of sides and is the gamma-function. The reason for using geometries having cross sections of regular -gon is that such geometries are easy to cover with the RWG basis functions and the elements of the impedance matrix can be accurately calculated efficiently [12]. However, the differences between the calculated input impedances of straight dipoles using meshes having octagonal, pentagonal and digonal cross-sections are about 2%. The comparison is presented in Table I. The local solutions of the optimization problem (16) were sought with the BFGS algorithm which was stopped when the or , where the metric is defined by formula (19) in the Appendix B. The derivatives required in the BFGS algorithm were computed using first order finite differences and the step size for the finite difference formula was chosen to be .

Fig. 4. The angle function  of optimal directive (3=2) antenna with 3 . . . 17 degrees of freedom.

Fig. 5. The shape of the optimal (3=2) directive antenna with 17 DOFs.

It should be noted that at the extrema of , its derivatives are zeros. Thus, when the solution of (16) with respect to is away from the boundary of the shape space and the penalty terms and in (15) are such that has an open neighborhood where they are zero, it follows that the sensitivities of with respect to the shape parameters are zeros. It turns out that in each of the following cases the approximate solutions enjoy these two aforementioned properties. However, the sensitivities of the obtained antennas are away from zero since the stopping criterion of the BFGS algorithm was not having the derivative of to be zero. The minimum of the nearness function was computed by disand taking the minimum over the nodes of the cretizing discretization. Furthermore, the coefficients in (13) and in (5) were chosen to be 100 and 10, respectively. B. The Landstorfer Antenna In this case we fix the antenna length to be exactly and study the convergence of the optimal shape in , , and . The convergence of the angle-function is illustrated by Fig. 4. As the result we obtain the antenna shown in Fig. 5 . having directivity of 7.0 dB and input impedance of and The weight factors as in (15) are chosen to be . The reference mesh for each of the antenna is chosen to be the one with digonal cross section having 255 degrees of freedom. The COMSOL simulation gives directivity of 7.1 dB and which yields 5% relative error in impedance of impedance and 2% in directivity. and in (16) and choosing By setting or , the algorithm finds tuned directive antennas

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Fig. 8. Optimal shapes due to impedance goals of 30 ; . . . ; 80 for dipoles. Impedance given in picture denotes the realized impedance.

=2

Fig. 6. (a) Shape and (b) impedance as a function of frequency of directional 95 . Center frequency is denoted by dashLandstorfer antenna tuned to Z dotted line.



Fig. 9. Input impedances for the optimization results with target impedances 30 ; . . . ; 80 . The dashed vertical line shows the position of the center frequency.

It should be noted that the wire width has a considerable effect on the directivity, but not that big on the optimal shape, e.g., yields 7.15 dB directivity for the equivalent radius of shape in Fig. 5 and 7.19 dB for a re-optimized shape. C. Input Impedance of

Fig. 7. (a) Shape and (b) impedance as a function of frequency of directional 50 . Center frequency is denoted by dashLandstorfer antenna tuned to Z dotted line.



with a gain near 7 dB. The shape in Fig. 6(a) corresponds to an impedance and antenna having 6.9 dB gain and the shape in Fig. 7(a) yields 6.8 dB gain and impedance. It seems that the optimization algorithm twists the ends of the wire to form a structure resembling top-hat loading. The simulated input impedances using COMSOL for 50 and 95 antennas are and , respectively and the directivities due to COMSOL simulation are 7.0 and 7.1 , respectively. The corresponding input impedance curves are presented in Figs. 6(b) and 7(b) together with the COMSOL simulation results. impedance target case, the self-intersection In the 50 penalty caused the antenna to be much more well behaved. Without it, the antenna would have almost self-intersected at the ends.

Dipoles

Next, we try to find dipoles with the desired input impedances without any requirements for the directivity. It seems that resistances higher than 73 are impossible to attain while still keeping the reactance near zero. In contrast, lower impedance values are easily obtainable. The computational reference mesh was chosen to be the one with octagonal cross section and 1468 degrees of freedom. The optimization target in this case was chosen so that the and the input antenna fits in a rectangle with side length of impedance target was set to , i.e., we set , , , and . The results of the optimization are presented in Fig. 8. It should be noted that the 80 impedance was never reached – the optimization algorithm stagnated at . The simulated impedances due to COMSOL had relative error of at most 4% and at least 0.4% at the center frequency and 4.2% across the frequency range at most and 1.5% at least. The input impedances of the resulting antennas are presented in Fig. 9. The directivities of these antennas range from 1.5 dB corresponding to the 30 antenna and increasing impedance antenna. Recall that the directivity is to 2.14 for the direction instead of maximal direction. measured to positive

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The representation as an arc-length parametrizable curve has several advantages, such as compactness properties of the shape space, and that the triangulation of the deformed mesh stays well behaved when the curvature radius is large enough. Furthermore, it requires minimal number of degrees of freedom to represent arbitrary physically feasible shapes. The antenna shapes presented here should be considered only as numerical examples and not final antenna designs since the input port modeling was rudimentary and only the metal parts of the antennas were simulated. Despite these shortcomings, the parametrization method showed great usability in the design problems of shaped dipole antennas. Not least for the fact that the field solutions and the differential of the objective function were obtained in a naive manner, since considerably better way would have been to us,e e.g., adjoint methods and automatic differentiation [13], [14] similarly as in [15]. APPENDIX A REVIEW ON LIPSCHITZ MAPPINGS A function from metric space to be -Lipschitz if

to metric space

is said (17)

Fig. 10. (a) Shape and (b) impedance as a function of frequency of =2 region 80 . Center frequency is denoted by dash2-element antenna tuned to Z dotted line.



However, the radiation pattern of each of the antennas in Fig. 8 dipole. is similar to that of the straight The algorithm seems to find antennas that have similar radiantenna near the ation characteristics to the usual straight center frequency but different input impedances. D.

Region 2-Element 80

Antenna

region dipole can be circumThe 73 upper limit of a vented by placing a second antenna wire as reflector element behind, with respect to main radiation direction, the main antenna element. We set the optimization goal to be maximal directivity and impedance of 80 . Additionally, the bounding box weight factor in (16) was set to be 1/10 and and . The reference mesh was the one having digonal cross section and 255 degrees of freedom. The resulting antennas has input impedance of 80 and directivity of 5.8 dB. The impedance curve and shape is presented in Fig. 10. The COMSOL simulation yields input impedance of and directivity of 5.8 dB which translates to a relative error of 2% for impedance and about 0.3% for directivity.

We say that is locally -Lipschitz at , if there is a neighborhood of such that is -Lipschitz in . Now we define and

In this paper, is and is or and the corresponding metrics are the ones defined by the euclidean norms. APPENDIX B COMPACTNESS OF THE SPACE OF ADMISSIBLE SHAPES The natural

-norm on

The subset of those

is defined by

for which (18)

holds is an equicontinuous set and its elements are pointwise by the uniformly bounded and thus it is precompact in Arzelà-Ascoli theorem. On the other hand, if each element of some converging sequence has the property (18), then the limit also has this property. Thus the subset

IV. CONCLUSION AND DISCUSSION We presented the complete parametrization for planar wire dipole antennas and a method for constructing approximations for them. Employing this parametrization we demonstrated approximate local solutions to the discrete optimization problems of finding optimal antennas of given input impedance, maximal directivity and given dimensions.

is closed and since it is precompact it is also compact. is a compact metric space in Finally, metric (19) where

and

.

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REFERENCES [1] W. Kühnel, Differential Geometry, 2nd ed. Providence, RI: American Mathematical Society, 2006. [2] F. Landstorfer, “Zur optimalen form von linearantennen,” Frequenz, vol. 30, no. 12, pp. 344–349, 1976. [3] C.-H. Liang and D. K. Cheng, “Directivity optimization for Yagi-Uda arrays of shaped dipoles,” IEEE Trans. Antennas Propag., vol. AP-31, pp. 522–525, May 1983. [4] W. Junhong and R. Lang, “Directivity optimization of curved surface dipole antennas,” J. Electron. (China), vol. 11, no. 4, pp. 322–331, Oct. 1994. [5] A. Poggio and E. Miller, “Integral equation solutions of three-dimensional scattering problems,” in Computer Techniques for Electromagnetics. Oxford, U.K.: Pergamon Press, 1973. [6] S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag., vol. 30, pp. 409–418, May 1982. [7] W. Rudin, Functional Analysis. New York: McGraw-Hill, 1973. [8] J. M. Lee, Introduction to Topological Manifolds. Berlin, Germany: Springer, 2000. [9] J. Nocedal and S. J. Wright, Numerical Optimization. Berlin, Germany: Springer, 1999. [10] D. G. Luenberger, Introduction to Linear & Nonlinear Programming. New York: Addison Wesley, 1973. [11] Y. T. Lo, “A note on the cylindrical antenna of noncircular cross section,” J. Appl. Phys., vol. 24, pp. 1338–1339, Oct. 1953. [12] S. Järvenpää, M. Taskinen, and P. Ylä-Oijala, “Singularity subtraction technique for high-order polynomial vector basis functions on planar triangles,” IEEE Trans. Antennas Propag., vol. 54, pp. 42–49, Jan. 2006. [13] J. Haslinger and R. A. E. Makinen, Introduction to Shape Optimization: Theory, Approximation, and Computation. Philadelphia, PA: Society for Industrial and Applied Mathematics, 2003.

[14] A. Griewank, “On automatic programming,” in Mathematical Programming: Recent Developments and Applications, M. Iri and K. Tanabe, Eds. Amsterdam, The Netherlands: Kluwer Academic Publishers, 1989, pp. 83–108. [15] J. I. Toivanen, R. A. E. Mäkinen, S. Järvenpää, P. Ylä-Oijala, and J. Rahola, “Electromagnetic sensitivity analysis and shape optimization using method of moments and automatic differentiation,” IEEE Trans. Antennas Propag., vol. 57, no. 1, pp. 168–175, 2009.

Juhani Kataja received the M.Sc.(Tech.) degree in electrical engineering from Helsinki University of Technology (TKK), Espoo, Finland, in 2008, where he is currently working toward the D.Sc.(Tech.) degree. Currently, he is with the Department of Radio Science and Engineering, Aalto University School of Technology, working as a Researcher.

Keijo Nikoskinen (M’85–SM’00) was born in Kajaani, Finland, in 1962. He received the Dipl.Eng., Lic.Tech., and D.Sc.(Tech.) degrees in electrical engineering from Helsinki University of Technology (TKK), Espoo, Finland, in 1986, 1989, and 1991, respectively. From 1991 to 1994, he was a Junior Scientist with the Academy of Finland. Since 1996, he has been a Professor of electromagnetics at TKK. His professional interest covers both the theory and applications of electromagnetics.

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Inductively-Loaded Yagi-Uda Antenna With Cylindrical Cover for Size Reduction at VHF-UHF Bands Jose Alfredo Tirado-Mendez, Hildeberto Jardon-Aguilar, Ruben Flores-Leal, Mario Reyes-Ayala, and Flavio Iturbide-Sanchez

Abstract—A new technique is employed to reduce the size of Yagi-Uda antennas. The technique consists of adding short circuited cylindrical covers to the structure, making the elements be inductively loaded, and as a result, increase the electric length. A prototype resonating at 660 MHz is developed, and compared to a conventional Yagi-Uda antenna. An effective area reduction of 35% is achieved without decreasing, considerably, the gain, preserving the bandwidth, and front to back lobe ratio. Index Terms—Reactive load, size reduction, wire antenna, YagiUda.

I. INTRODUCTION URRENTLY, communications systems have become more mobile and personal, that means gears are needed to be as smaller as possible. Low weight, reduced volume and low cost are some of the characteristics that must be accomplished in modern telecommunication equipment. It is well know that the antenna is one of the most important devices in the radiofrequency (RF) section. This element needs to perform effectively according to the necessities. To achieve low weight, small volume and low cost, the size needs to be reduced, but keeping the main features like directivity, gain, VSWR, among others. The size reduction of antennas can be made by several techniques according to the type of the element, for example, for wire antennas, there is the alternative of meander structures, where the elements are bent several times, which leads to the reduction of the length [1]. Similar to this technique is the particle swarm optimization [2] in which, the elements are also folded, in such a way that gain, front to back ratio are preserved.

C

Manuscript received March 12, 2010; revised June 29, 2010; accepted August 20, 2010. Date of publication December 03, 2010; date of current version February 02, 2011. J. A. Tirado-Mendez is with the Research and Advanced Studies Center, Inst. Politecnico Nacional (IPN), CINVESTAV, San Pedro Zacatenco D.F. 07360, Mexico and also with the Universidad Autonoma Metropolitana, Mexico City D.F. 02200, Mexico (e-mail: [email protected]). H. Jardon-Aguilar and R. Flores-Leal are with the Research and Advanced Studies Center, Inst. Politecnico Nacional (IPN), San Pedro Zacatenco D.F. 07360, Mexico. M. Reyes-Ayala is with the Universidad Autonoma Metropolitana, Mexico City D.F. 02200, Mexico (e-mail: [email protected]). F. Iturbide-Sanchez is with the I. M. Systems Group, Inc., NOAA/NESDIS/ Center for Satellite Applications and Research, Camp Springs, MD 20746 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2096397

On the other hand, some antennas can also be made over a substrate with permittivity . In this case, the size reduction is . This technique has obtained by a scaling factor close to been proposed some years ago [3]. Following this idea, many proposals have been made to reduce the dimensions of Yagi-Uda antennas over dielectric materials [4], [5]. Another form to obtain the size reduction of antennas is the employment of resonant structures over the device. This technique allows loading the structure inductively, increasing the electric length [6], [7]. Therefore, as far as authors’ knowledge [8], no simple and effective technique, other than bending the elements or loading with a reactance (disc hat, among others), is available to reduce the dimensions of wire antennas, preserving most of the element parameters. Disc hat load is a very effective technique, but the disc itself provides an extra area, which can be a drawback for multi element antennas, such as Yagi-Uda. In this work, following the idea introduced in [8], a new alternative to reduce the dimensions of wire Yagi-Uda antennas is proposed, without folding the device or increasing the permittivity in which the elements are built. The technique consists of adding a short-circuited cylindrical cover over the extremes of each element, including the driver, as explained in the next section. On the other hand, there are similar techniques like the one introduced in this work, but with some drawbacks, for example in [9] where a cylindrical hat is employed in a whip antenna. In that work, the reduced antenna has an inner stub which provides, as well as tuning, a reactive load to decrease the size. In spite of using a similar structure to the hat proposed in this paper, the reactive load is, mostly, generated by the stub instead of the hat. The use of this element increases the difficulty of the design, adding more parameters to be optimized, in comparison to the technique shown in this paper. Another similar technique is described in [10], where a coaxial line is employed with slots to perform as antenna. II. PROPOSED TECHNIQUE FOR YAGI-UDA SIZE REDUCTION The technique, employed in this work to reduce wire Yagi-Uda antenna dimensions, consists of adding a cylindrical cover at the extremes of the elements, and short-circuited at the end as show in Fig. 1 for one element. The cylindrical cover with length , called hat from this point, allows loading the element inductively, so the electric length of the wire is increased. This configuration can be modeled as a short-circuited lossless transmission line as depicted in Fig. 2

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Fig. 3. Current distribution over the driver and hat-cover.

TABLE I RESULTS VARYING THE DIELECTRIC PERMITTIVITY

Fig. 1. Transverse section of the proposed technique to reduce dimensions of wire elements. TABLE II RESULTS VARYING HAT’S RADIUS (Radiator radius = 0:5 mm)

Fig. 2. Short-circuited transmission line model. TABLE III RESULTS VARYING THE RADIATOR’S RADIUS

[8]. For a short-circuited transmission line, the input impedance behaves inductively for certain conditions. The input impedance is given as follows: (1) Where and are the characteristic impedance and the propagation constant of the line, respectively, is the hat length. Since the line is considered small enough, losses in this model are negligible, therefore they are not taken into account, and the attenuation constant, is set to zero. From (1), the associated inductance is obtained at a certain frequency as

(2) To satisfy the requirement of loading the antenna inductively, , where is the wavelength the hat length must be less than . Otherwise, the hat beat the operation frequency haves as a capacitor instead of an inductor. Making a current distribution analysis over the cylindrical hat, a slow-wave factor can be observed. As a result, the path in which the current goes through is, considerably, increased by the hat length, as depicted in Fig. 3. Fig. 3 shows a close up of the current distribution over the driver element cover by the cylindrical hat and short circuited

at the right-end. In this figure, the current over the driver goes from left to right hand, but at the right-end, the current suffers a change of direction, flowing over the hat, increasing the path length of the current. It is also found that the intensity of the current over the hat is less than that over the driver. This fact takes to the possibility of reducing the radiation field, and as a result, diminishing the antenna gain. This is a tradeoff which has to be taken into account for a given design. In [13], the relation of efficiency with antenna size was demonstrated. If the antenna is small, the drawback is a reduction in efficiency, when the bandwidth is kept. On the other hand, since the structure allows a variation of different factors, a parametric study was made by using HFSS, considering the permittivity of the dielectric, the radius of the hat and the radius of the inner conductor. To observe the behavior of these variations, a monopole structure like the one shown in Fig. 1 is employed. The radiator is 100 mm long, and the hat length is 60 mm. The results of this study are presented in Table I, Table II, and Table III.

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TABLE IV DIMENSIONS OF ANTENNA PROTOTYPES

Fig. 4. Proposed Yagi-Uda antenna with hat cover in all elements.

According to previous results, it is seen that increasing the permittivity, the resonance frequency and the bandwidth decreases, and the gain reduces. From Tables II and III, the resonance frequency varies, but the important result is that the bandwidth reduces considerably when the radii increase. This result can be explained from the point of view that the antenna, in this work, is modeled as a coaxial line and the associated inductance per length unit can be described by (3) Where and are the inner and outer radii of a coaxial line. From (3), if the inner radius increases, the associated inductance decreases and then, the resonance frequency is lowered. If the outer radius (in this case, the hat’s radius) increases, the associated inductance is higher and the resonance frequency is lower. All these results are demonstrated in Tables II and III. On the other hand, Table III shows reduction on the bandwidth, even when the radiator radius is increased, this is because, the hat’s radio was kept, making the opposite effect of increasing the bandwidth, due to the reactive load generated by the hat. Considering these results, taking into account a thin radiator, a small dielectric permittivity and, as possible, the smallest hat’s radius; as a case of study, a proposed 3-element Yagi-Uda antenna is depicted in Fig. 4. To compare the performance at VHF and UHF frequency bands, two conventional prototypes are compared to two antennas with the proposed technique. The VHF antenna is designed to operate at 180 MHz, while the UHF antenna operates at 660 MHz. The wire diameter for all elements of the 4 prototypes is 1 mm. For the conventional antennas [11], the dimensions are: The , is 0.5 , and the driver length, D, is 0.239 reflector length, . This length is the half of a dipole with no parasitic elements around. The director length is chosen to be 20% less than two times the driver (0.478 ). The reflector to driver distance, , is chosen to be 0.15 to 0.25 , according to the performance of the antenna. The driver to director distance, , can be chosen

The area was obtained considering the antenna as a trapezoid

from 0.2 to 0.35 . For the 660 MHz antenna, dimensions are as follows: , , , and . These dimensions were optimized by a simulation procedure considering the maximum achievable , , gain. For the 180 MHz antenna, , , . In the case of the proposed Yagi-Uda antennas with hats, the dimensions were obtained through a simulation process to optimize the dimensions of the elements, and these are as follows: , , For the 660 MHz prototype, , , and . To obtain a resonant frequency of 660 MHz, the hat length is 40 mm. The distances between elements were obtained by making a computer tuning process. An optimization of the distance, according the specifications mentioned above in function of the wavelength, was made. On the other hand, for the 180 MHz , , , prototype, and . The study of the performance was made by using HFSS Optimetrics, considering gain, port matching and front to back lobe ratio. Table IV shows a resume of the four antenna dimensions. With the proposed technique, the effective area of the antenna is reduced almost 35%, for the UHF antenna, and 22% for the VHF antenna, considering that the latter employed a hat length of 100 mm. For both reduced antennas, the hat length is about 10% of . Applying this technique, a reduction factor can be , but in improved by increasing the hat length to the limit this case, a tradeoff with gain and back to lobe ratio needs to be set. As observed, the structure uses hats over all elements, including the director and the reflector. However, it should be noticed that the director and reflector can be considered as “passive” elements, since no connection is made between these and

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TABLE V HFSS SIMULATION RESULTS OF NON-HAT ELEMENTS

Fig. 6. Simulation results for port coupling of reduced and conventional Yagi-Uda antennas at 180 MHz.

Fig. 5. Simulation results for port coupling of reduced and conventional Yagi-Uda antennas at 660 MHz.

the circuit of a transceiver. Then, it is valid to assume that no hats are needed in these parts of the antenna. However, these elements have a strong interaction in the radiation pattern and the impedance of the electromagnetic wave in the near field zone. A computer simulation study was made considering the use of hats in the reflector and the director by using HFSS. In all cases, the driver employed hats as proposed in Fig. 1. The results are shown in Table V. The previous results were obtained considering a three elements antenna, with diameter of 1 mm, and hat radius of 3.7 mm, working at 660 MHz, as the previous design. As observed in Table V, when no hat is employed in the reflector and radiator elements, the front to back lobe ratio is very poor, the bandwidth is reduced, and the coupling at the resonant frequency is diminished, as it will be compared in the next section, when hats are used in all antenna elements. III. SIMULATION RESULTS To corroborate the performance of the proposed reduced Yagi-Uda antennas, the prototypes were compared to the conventional ones. The antennas are made with thin wires with 1 mm diameter. The simulation results for port matching at the UHF band are displayed in Fig. 5, and Fig. 6 shows this parameter at the VHF band. From Figs. 5 and 6 it is concluded that the port coupling in reduced antenna is as deep as in the conventional one. The bandwidth is preserved, obtaining almost 50 MHz for the UHF element, and 10 MHz for the VHF one, which are good frequency ranges, considering that very thin wire antennas are being employed. This result is very important, since many techniques

Fig. 7. Normalized radiation pattern of reduced and conventional Yagi-Uda antennas obtained by simulation at 660 MHz.

which are employed to reduce dimensions, also have the disadvantage of decreasing the bandwidth [12]. Concerning the radiation pattern, Figs. 7 and 8 shows the comparison of the normalized gain of both prototypes in the X-Y plane, at 660 MHz and 180 MHz, respectively. In these pictures, a close convergence of results is seen. These figures show the disadvantage of the proposed Yagi-Uda antenna, where the gain is slightly reduced compared to the conventional one. However, the directivity and pattern shape are preserved. In this case, the antenna gain is decreased by 1.4 dB for the UHF element, and around 2 dB for the VHF element. Once again, the relation between efficiency-bandwidth and size is demonstrated [13]. In this case, the antenna is reduced by the use of the hats, but preserving the bandwidth. Therefore, the efficiency of the antenna is diminished, so is the gain. Then,

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Fig. 8. Normalized radiation pattern of reduced and conventional Yagi-Uda antennas obtained by simulation at 180 MHz.

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Fig. 10. Normalized measured radiation pattern.

Fig. 10 shows the measured radiation pattern of both prototypes. As observed, the reduced antenna gain is decreased by almost 1.3 dB, however, the reduction factor of the antenna area is 35%, which can be seen as a good tradeoff, specially for certain mobile communications equipment. However, the shape and directivity of the proposed prototypes is kept. V. CONCLUSION

Fig. 9. Measured port matching for reduced and conventional Yagi-Uda antennas.

as a result, there must be a tradeoff between size reduction and antenna efficiency. IV. MEASURED RESULTS After the computer design and optimization, to verify the performance of the proposed technique to decrease dimensions of wire antennas, the reduced prototype at 660 MHz was built and compared to a conventional Yagi-Uda structure operating at the same frequency. Fig. 9 shows the measured port matching of the structures. In this figure, almost the same bandwidth for these antennas is observed. In the case of the reduced device, the bandwidth is slightly increased by 5 MHz.

In this work, a novel technique is introduced to reduce dimensions of directive wire antennas. In this case, a Yagi-Uda antenna is designed and built, employing cylindrical-hat covers in all elements to load inductively each part of the structure. With this proposal, the effective area of the antenna is diminished almost 35%, preserving the bandwidth, which is a common drawback in other techniques employed to reduce antenna size. On the other hand, the main disadvantage obtained with this technique, is the small reduction of the gain, which is decreased almost 1.3 dB. All reduction size techniques applied in antennas has the disadvantage of modifying any antenna parameter, mainly the efficiency [13]. Therefore, there most be a tradeoff between size reduction and efficiency reduction. This gain reduction factor obtained in this work can be considered negligible, since typical gains in such antennas are around 6 to 8 dB. However, this compromise can be considered good enough for certain applications, where the total area is needed to be as small as possible. With this technique, there is an option to reduce wire antennas size without being a difficult tool to be employed, and no folding elements are needed, or even more, where materials with high dielectric permittivity are required. REFERENCES [1] D. K. C. Chew and S. R. Saunders, “Meander line technique for size reduction of quadrafilar helix antenna,” IEEE Antennas Wireless Propag. Lett., vol. 1, pp. 109–111, 2002.

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[2] Z. Bayraktar, P. L. Wegner, and D. H. Wegner, “The design of miniature three-element stochastic Yagi-Uda arrays using particle swarm optimization,” IEEE Antennas Wireless Propag. Lett., vol. 5, pp. 22–26, 2006. [3] J. R. James, A. J. Schuler, and R. F. Binham, “Reduction of antenna dimensions by dielectric loading,” Electron. Lett., vol. 10, no. 13, pp. 263–265, 1974. [4] D.-Z. Kim, S.-Y. Park, W.-S. Jeong, M.-Q. Lee, and W. Yu, “A small and slim printed Yagi antenna for mobile application,” in Proc. AP Microwave Conf., 2008, pp. 1–4. [5] E. Avila-Navarro, A. Segarra-Martinez, J. A. Carrasco, and C. Reig, “A low-cost uniplanar quasi-Yagi printed antenna,” Microw. Opt. Technol. Lett., vol. 50, no. 3, pp. 731–735, Mar. 2008. [6] E. G. Korkontzil, P. C. Liakou, and D. P. Chrissoulidis, “Dual frequency patch antenna, reduced in size by use of triangularly arranged peripheral slits,” in Proc. 1st Eur. Conf. on Antennas and Propagation (EuCAP 2006), 2006, pp. 1–6. [7] J. A. Tirado-Mendez, H. Jardon-Aguilar, F. Iturbide-Sanchez, I. Garcia-Ruiz, V. Molina, and R. Acevo-Herrera, “A proposed defected microstrip structure (DMS) behavior for reducing rectangular patch antenna size,” Microw. Opt. Technol. Lett., vol. 43, no. 6, pp. 481–484, Dec. 2004. [8] J. A. Tirado-Mendez, H. Jardon-Aguilar, R. Flores-Leal, and M. ReyesAyala, “A novel reduced lambda/4 resonant monopole with associated extra inductance,” Microw. Opt. Technol. Lett., vol. 52, no. 2, pp. 276–280, 2010. [9] F. Tefiku and K. Li, “Coupled retractable whip/stub antennas for mobile phone,” in Proc. IEEE/ACES Int. Conf. on Wireless Communications and Applied Computational Electromagnetics, 2005, pp. 952–955. [10] M. B. Perotoni and S. E. Barbin, “A slotted coaxial antenna as an alternative to wire dipole antenna,” in Proc. Progr. Electromagn. Res. Symp., Mar. 26–30, 2007, pp. 782–785. [11] W. L. Stutzman and G. A. Thiele, Antenna Theory and Design, 2nd ed. New York: Wiley, 1998. [12] A. K. Skrivervik, J.-F. Zürcher, O. Staub, and R. Mosig, “PCS antenna design: The challenge of miniaturization,” IEEE Antennas Propag. Mag., vol. 43, no. 4, pp. 12–27, Aug. 2001. [13] H. A. Wheeler, “Fundamental limitations of small antenna,” Proc. IRE, vol. 35, pp. 1479–1484, Dec. 1947.

Jose Alfredo Tirado-Mendez was born in Mexico City, Mexico. He received the B.S. degree in electronics and digital systems from Metropolitan Autonomous University (UAM)-Azcapotzalco, Mexico City, in 1999, and the M.Sc. and Ph.D. degrees in telecommunications from the Inst. Politecnico Nacional (IPN), CINVESTAV, San Pedro Zacatenco, Mexico, in 2001 and 2008, respectively. In 2002, he joined the Telecommunications Section, Inst. Politecnico Nacional (IPN), CINVESTAV, as a Researcher. In 2005, he joined the Electronics Department, UAM-Azcapotzalco. His research interests are electromagnetic compatibility and nonlinearities as well as RF and microwave circuit design, patch antennas and microstrip filters. Dr. Tirado-Mendez received the Arturo Rosenblueth Award in 2009 for the Best Ph.D. Engineering Thesis in CINVESTAV. He has level I in the SNI of Mexico.

Hildeberto Jardon-Aguilar was born in Tenancingo, Hidalgo, Mexico, in 1949. He received the B.S. degree in electrical engineering from ESIME-Inst. Politecnico Nacional (IPN), San Pedro Zacatenco, Mexico and the Ph.D. degree in radiosystems from the Moscow Technical University of Telecommunications and Informatics, Moscow, Russia. He is a Full Professor at the Center of Research and Advanced Studies, IPN, CINVESTAV, since 1985. His research interests include analysis of nonlinearities in RF and microwave circuits, electromagnetic compatibility and photonic systems. He is the author of five books and more than 100 technical papers in journals and symposiums. Prof. Jardon-Aguilar is a member of the Popov Society and has level III in the SNI of Mexico.

Ruben Flores-Leal was born in Puebla, Mexico, in 1979. He received the B.Sc. degree in electrical engineering from Puebla University, in 2004, and the M.Sc. degree in communications from the Research and Advanced Studies Center (CINVESTAV), Mexico City, in 2007. In 2007, he joined the Electrical Engineering Department, CINVESTAV. His research interests include development of radiocommunications systems, low noise amplifiers and active and retrodirective antennas.

Mario Reyes-Ayala received the B.S.E.E. degree from Metropolitan Autonomous University (UAM), Mexico City, Mexico, in 1994 and the M.S.E.E. degree from Inst. Politecnico Nacional (IPN), CINVESTAV, San Pedro Zacatenco, Mexico, in 2005. From 1994 to 1996, he was a Department Head at AEG Mexicana, Mexico City. In 1999, he joined the Electronic Department, UAM, where he has been engaged in studies on digital satellite communication and propagation models for radio communication systems. His research interests are also in theoretical analysis in satellite interference phenomena.

Flavio Iturbide-Sanchez was born in Mexico City, Mexico. He received the B.S. degree in electronics and digital systems from Metropolitan Autonomous University (UAM)-Azcapotzalco, Mexico City, in 1999, the M.Sc. degree in telecommunications from CINVESTAV, San Pedro Zacatenco, Mexico, in 2001, and the Ph.D. degree from the University of Massachusetts at Amherst, in 2008. In 2008, he joined the I. M. Systems Group, Inc., NOAA/ NESDIS/Center for Satellite Applications and Research, Camp Springs, MD. His research interests are geosciences and remote sensing, antennas and non-linear effects in electronics circuits for telecommunications.

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Compact Coplanar Waveguide (CPW)-Fed Zeroth-Order Resonant Antennas With Extended Bandwidth and High Efficiency on Vialess Single Layer Taehee Jang, Student Member, IEEE, Jaehyurk Choi, and Sungjoon Lim, Member, IEEE

Abstract—This paper presents the design and analysis of compact coplanar waveguide (CPW)-fed zeroth-order resonant (ZOR) antennas. They are designed on a CPW single layer where vias are not required. The ZOR phenomenon is employed to reduce the antenna size. The novel composite right/left-handed (CRLH) unit cell on a vialess single layer simplifies the fabrication process. In addition, the CPW geometry provides high design freedom, so that bandwidth-extended ZOR antennas can be designed. The antenna’s bandwidth is characterized by the circuit parameters. Based on the proposed bandwidth extension technique, symmetric, asymmetric, and chip-loaded antennas are designed. The ZOR characteristic and bandwidth extension are verified by a commercial EM simulator. Their performances are compared with those of previously reported metamaterial resonant antennas. They provide further size reduction, higher efficiency, easier manufacturing, and extended bandwidth. Index Terms—Co-planar waveguide (CPW), composite right/left-handed transmission line, metamaterials, small antenna.

I. INTRODUCTION

M

ETAMATERIALS have been widely studied for microwave circuit and antenna applications [1]–[7]. They have unique properties in comparison with conventional nature materials, such as anti-parallel phase and group velocities, and a zero propagation constant [1]–[5] and can be realized by means of split-ring resonators (SRRs) or composite right/left-handed (CRLH) transmission lines (TLs). Especially, the CRLH TL is able to easily achieve left-handed (LH) metamaterial properties by way of the circuit parameters [6]. By using the features of the anti-parallel phase and group velocities, CRLH TLs can be applied to dominant mode leaky-wave antennas radiating in the backward and forward directions [7]. In addition, due to the zero propagation constant inherent in the LH metamaterial Manuscript received March 11, 2010; revised June 10, 2010; accepted July 07, 2010. Date of publication December 03, 2010; date of current version February 02, 2011. This work was supported in part by the IT R&D program of MKE/KEIT [KI002084, A Study on Mobile Communication Systems for NextGeneration Vehicles with Internal Antenna Array] and in part by the National Research Foundation of Korea Grant funded by the Korean Government [20090071958]. The authors are with the School of Electrical and Electronics Engineering, College of Engineering, Chung-Ang University, Seoul, Korea (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2096191

properties, the resonator has an infinite wavelength and its resonant frequency is independent of the size of the resonator. Therefore, the zero propagation constant properties of resonant antennas enable them to be more compact than conventional half-wavelength antennas [8]–[10]. Although they offer the advantage of size reduction, it is difficult to apply them to modern wireless communication systems because of their narrow bandwidth. Recently, many researchers have attempted to solve the bandwidth problem of zeroth-order resonant (ZOR) antennas [11]–[15]. To accomplish this, a metamaterial ring antenna was reported in [11]. This antenna was implemented on a multi-layer structure in which a thick substrate with low permittivity is additionally employed. The substrate is supported by holding brackets and its bandwidth is increased up to 6.8% by means of a sleeve balun. Alternatively, the bandwidth of the ZOR antenna is increased by a strip matching ground [12]. In this case, the fractional bandwidth of the antenna was enhanced by up to 8%. It is also built on multiple substrates where a thin substrate with high permittivity is stacked on a thick substrate with low permittivity. The other method is to have two resonant frequencies close to each other [13]. Such an antenna consists of two resonators whose resonant frequencies are slightly different. The bandwidth is increased by up to 3.1%. This paper presents novel compact coplanar waveguide (CPW)-fed zeroth-order resonant antennas. Three types of antennas are proposed whose bandwidths are extended by up to 8.9% and they are built on a single layer without a via process. The size of the proposed antenna can be reduced, due to their infinite wavelength, while keeping a high radiation efficiency. According to [6], the Q-factor of the ZOR mode is determined by the shunt inductance and capacitance. In other words, the antenna’s bandwidth can be increased by manipulating its shunt reactance. Especially, since a CPW structure gives a lot of design freedom, it provides the benefits of easy design to implement the desired circuit parameters. Moreover, the use of a via free and single layer process results in a simpler fabrication process compared with that of the broad metamaterial resonant antennas proposed in [10]–[12]. This paper is organized as follows. Sections II and III demonstrate the principle of the CPW-based ZOR antenna and the bandwidth extension through the design parameters rather than the material properties. The input impedance, internal energy,

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Thus, the complex propagation constant are impedance

and characteristic

(5) (6) Because the CRLH TLs have periodic boundary conditions, the Bloch-Floquet theorem can be applied and its dispersion relation is determined by

(7)

Fig. 1. Illustration of (a) equivalent circuit model of the CRLH unit cell and (b) its dispersion curve.

where and are a sign function and the differential length, respectively. and can be unequal in the dispersion For instance, diagram of the unbalanced LC-based CRLH TL as shown in , an inFig. 1(b). At these resonant frequencies, where finite wavelength can be supported. According to the theory of the open-ended resonator with the CRLH TL [6], its resonance occurs when (8)

and quality (Q) factor are analyzed. In this paper, three antenna configurations are presented employing the proposed idea. The three antennas, viz. symmetric, asymmetric, and chip inductorloaded antennas, are proposed in Section IV. In Section V, their performances are compared with those of previously reported metamaterial resonant antennas. Finally, our conclusions are drawn in Section VI. II. ZOR ANTENNA THEORY As shown in Fig. 1, a general CRLH TL is composed of series capacitance and inductance as well as a shunt and inductance [1]. It is designed in a capacitance unit cells. The immitperiodic configuration by cascading tances of a lossy CRLH TL are given by

where , and are the physical length of the resonator, mode number, and number of unit cells, respectively. When is zero, the wavelength becomes infinite and the resonant frequency of the zeroth-order mode becomes independent of the size of the antenna, while the shortest length of the conventional openended resonator is one half of the wavelength. Thus, an antenna with a more compact size can be realized. As depicted in Fig. 1(b), for the unbalanced CRLH TL, two and , with are observed resonant frequencies, with a matched load. Considering the open-ended TL, where , the input impedance seen from one end of the resonator toward the other end is given by

(1) (2)

(9)

where and are the series resistance and shunt conductance of the lossy CRLH TL, respectively [17], [20]. The series and shunt resonant frequencies are given by

is the admittance of the CRLH unit cell. where Since, from (9), the input impedance of the open-ended times of the unit cell, resonator is equal to values are equal to , , the equivalent , , and , respectively [18]. Regardless of , the resonant frequency of the cascaded open-ended ZOR circuit is determined by the resonant frequency originating from the shunt LC . Thus, the open ended ZOR antenna’s resonant tank

(3) (4)

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frequency is given by (4), so that it depends only on the shunt parameters of the unit cell. III. PROPOSED BANDWIDTH EXTENSION TECHNIQUE A. Bandwidth Extension of ZOR Antenna Considering that the open ended resonator is only dependent of the unit cell, the average electric energy stored in on , is given by the shunt capacitor, (10) and the average magnetic energy stored in the shunt inductor, , is (11) where is the current through the inductor. Because resonance occurs when is equal to quality factor can be calculated as follows:

, the

(12) Therefore, in the open ended case, the fractional bandwidth of the resonator is given by

Fig. 2. Configuration of the proposed CRLH TL based on the CPW structure. (a) Unit cell design. (b) Dispersion diagram of symmetric and asymmetric unit cells.

(13) Although this equation does not consider the impedance matching at the input terminals, it provides an intuitive concept by means of which the bandwidth can be efficiently increased. Generally, ZOR antennas are known to have a narrow bandwidth problem compared to conventional resonant antennas. This is because the Q-factor of a ZOR antenna is only related to and . For example, in a microstrip structure, and are realized by the shorting pin (via) and parallel plate between in a microstrip the top patch and bottom ground. Since line (MSL) depends on the length of the via, the microstrip structure limits the value of . In addition, since the thickness and size of the substrate determine the capacitance of the parallel plate, the MSL has a large . According to (13), the narrow bandwidth is originated from the small and large . Therefore, the ZOR antenna in microstrip technology has a narrow bandwidth due to the structural problem. In order to extend the bandwidth of the microstrip structure, a thick substrate with low permittivity is generally utilized. However, this causes fabrication difficulties and reduces the design freedom.

In this paper, we focus on antennas with a large and small , which result in improved bandwidth without degrading the efficiency, due to the shunt conductance . Moreover, we suggest a structure which can be easily fabricated and offers more design freedom. Fig. 2(a) shows the proposed unit cell of the CPW-type CRLH TL. Through the parameter extraction of the proposed unit cell [20], the physical dimensions shown in Fig. 3 are determined. Based on the extracted parameters and (7), the dispersion diagrams for the symmetric and asymmetric unit cells are presented in Fig. 2(b). In addition, the values of the circuit parameters are extracted from the s-parameter results. The shunt parameters of the proposed CPW-type structure are obtained from the shunt capacitance between the top patch and CPW ground, and the shunt inductance of the shorted meander lines. When the unit cell is realized in the symmetric configuration, the total shunt inductance, , is equal to one half of the inductance of the meander line. This is because the two meander lines are connected in parallel . Since this CPW TL provides more design freedom for the reactive parameters compared with the MSL,

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both a wider bandwidth and smaller size can be achieved by using a proper design. Therefore, the meander lines facilitate . the realization of a short stub, as well as a large total Because the top patch is positioned far from the CPW ground, is smaller than that of the microstrip structure, which also enables the bandwidth to be extended. The inductance can be increased in proportion to the length of the shorted stub line. Therefore, the meander line enables to be realized in a limited space. In addition, the a large signal and ground planes are placed on the co-plane so that the shunt capacitance between the top patch and CPW ground patch can be easily adjusted. Thus, the CPW technology and meander lines provide a means of changing the shunt parameters very easily. In addition, in order to match the impedance, a stub on the top of the substrate and a partial bottom patch are also utilized. Both the width and length of the stub on the top play important roles in impedance matching. The bottom patch is placed under the top patch. Moreover, the bottom patch can be placed under the top patch to match the impedance. From Fig. 4(a), the and results in larger overlapping area of higher coupling capacitance in the feed network. In addition, when the size of the bottom ground patch is increased as shown in Fig. 4(b), the value of is increased. Therefore, good impedance matching can be obtained by adjusting the size and position of the bottom ground. IV. ANTENNA REALIZATION In this section, the three configurations of zeroth-order antennas are realized. Each antenna has two unit cells and their resonant frequencies are determined by the shunt parameters. A. Symmetric & Asymmetric Antenna Design The proposed CPW based ZOR metamaterial antennas are illustrated in Fig. 3. They are composed of top metallic patches, shorted meander lines, a CPW ground plane, and bottom patch. The proposed ZOR antennas are realized by cascading the , CPW-type unit cell periodically. In order to realize the the meander lines are connected between the top patch and the CPW grounds as the shorted stub. Fig. 3(a) shows the configuration of the symmetric antenna. The meander lines of the ZOR antenna are symmetrically aligned on both sides of the CPW GND. The bottom patch is placed under the top patch for the sake of impedance matching. In order to further reduce the resonant frequency of the antenna without changing the size, one side of the meander lines is removed, as shown in Fig. 3(b). Therefore, the meander lines in the unit cell are asymmetrically placed on only one side of the CPW GND. Since the shorted meander lines are connected in parallel, the shunt inductance of the antenna is approximately twice that of the symmetric case. Thus, the resonant frequencies of the symmetric and asymmetric antennas are given by

Fig. 3. Configuration of the proposed CPW-type ZOR antenna. (a) Top of symmetric unit cell. (b) Top of asymmetric unit cell. (c) Bottom of symmetric and asymmetric unit cells.

(15) where

(14)

is the inductance of the shorted meander line, and and are the resonant frequencies of the symmetric and asymmetric antenna, respectively.

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Fig. 4. Schematic explanation for the function of the bottom patch (a) the relationship between L and C (b) the relationship between W and C .

Fig. 6. Simulated and measured reflection coefficients of the proposed CPWtype ZOR antennas: (a) symmetric antenna, (b) asymmetric antenna.

Fig. 5. Prototype of (a) top and bottom view of the symmetric antenna, (b) top view of the asymmetric antenna.

B. Symmetric & Asymmetric Antenna Performance The proposed ZOR antennas are fabricated on Rogers RT/ Duroid 5880 substrates with a dielectric constant of 2.2 and thickness of 1.6 mm. A CPW feeding line of 50 and proximity coupling are used as the feed network so that the input impedance is matched to 50 . The resonant frequency of the proposed antenna is determined by and between the top patch and CPW ground. By changing the length of the meis changed so that the resonant frequency is ander line, the altered. These antennas are built in the CPW configuration and printed on the top and bottom of a substrate without vias. Thus,

the proposed antennas provide the features of easy fabrication and low profile configuration. The symmetric antenna’s dimensions are (unit: millimeter): , , , , , , , , as shown in Fig. 3(a). With these dimensions, the extracted values for the unit cell of 0.62 pF, of 9.26 nH, and of 0.0007. consist of a The fabricated prototype is shown in Fig. 5(a). The proposed antenna is designed to have its zeroth-order mode at 2.03 GHz. The electrical size of the unit cell of the antenna is at 2.03 GHz. The overall area of the radiating aperture is approximately . Fig. 6(a) shows the simulated and measured reflection coefficients where both resonant frequencies of the zeroth-order mode are 2.03 GHz. Moreover, the measured 10 dB bandwidth and radiation efficiency are 6.8% and 62%, respectively. Fig. 7 shows the simulated and measured radiation patterns on the Y-Z (E-plane) and X-Z (H-plane) planes at 2.03 GHz. Figs. 7(a) and (b) show the electric field distribution of and modes, the symmetric ZOR antenna in the mode respectively. The electric field distribution of the is 180 out of phase along the aperture. As shown in Fig. 8(b), the electric field distribution for the is in-phase.

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Fig. 7. Simulated and measured radiation patterns of the symmetric antenna at 2.03 GHz: (a) y-z plane (E-plane), (b) x-z plane (H-plane).

Fig. 8. Electric-field distribution of the symmetric antenna: (a) n = (b) n = 0 mode.

01 mode,

The asymmetric configuration can be implemented with a single meander line in each unit cell, while the symmetric one has two meander lines. Its dimensions are (unit: millimeter): , , , , , , , , . The fabricated prototype is shown in Fig. 5(b). It is designed to have the zeroth-order mode at 1.5 GHz. The electrical size of the unit cell of the anat 1.5 GHz. The overall area of the tenna is radiating patch is approximately . In Fig. 6(b), the simulated and measured reflection coefficients are shown, where the resonant

Fig. 9. Simulated and measured radiation patterns of the asymmetric antenna at 1.5 GHz (a) y-z plane (E-plane), (b) x-z plane (H-plane).

Fig. 10. Electric-field distribution of the asymmetric ZOR antenna (a) n = mode and (b) n = 0 mode.

01

frequency of the zeroth-order mode is observed to be 1.5 GHz and the 10 dB bandwidth is 4.8%. Figs. 9(a) and (b) show the simulated and measured radiation patterns on the Y-Z (E-plane) and X-Z (H-plane) planes at 1.5 GHz, respectively. The measured radiation efficiency is 42.5%. As shown in Fig. 10(a), mode is 180 out the electric field distribution in the of phase. Fig. 10(b) shows that the electric field distribution is . in-phase at The overall antenna performances of the proposed antennas are compared with those of the previously reported ZOR antennas [10]–[12] in Table I. Although the proposed antennas are

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TABLE I ANTENNA MEASUREMENT SUMMARY AND COMPARISON RESULTS OF PROPOSED AND REFERENCE ANTENNAS

Fig. 12. Simulated and measured reflection coefficients of the chip-loaded ZOR antenna.

Fig. 11. (a) Configuration of the proposed chip-loaded ZOR antenna, (b) Top view of the fabricated prototype.

realized on a single layer without vias, they provide competitively enhanced bandwidths and high efficiencies. Moreover, they are smaller size than the conventional ZOR antenna [10] and are easy to fabricate owing to the vialess single layer. C. Chip-Loaded Zeroth-Order Antenna The configuration and prototype of the proposed chip-loaded antenna are shown in Fig. 11. The chip-loaded antenna is designed with lumped elements (chip inductor) instead of shorted

stubs. Because this antenna is able to have a high shunt inductance, it is suitable for low frequency applications. It is realized with high frequency chip inductors having an inductance of 8.2 nH. This antenna has a zeroth-order resonant . The measured radiation efficiency frequency at was approximately 77.8%. The proposed at chip-loaded antenna is fabricated on a Rogers RT/Duroid 5880 substrate with a dielectric constant of 2.2 and thickness of 1.6 , mm as well. Its dimensions are (unit: millimeter): , , , , , , . It is also built with a CPW configuration and the shorted meander lines are replaced by the chip inductors. Thus, the resonant frequency can be tuned by changing the inductance values. The electrical size of the unit cell of the antenna is at 2.38 GHz. The overall area of the antenna is approximately . The simulated and measured reflection coefficients are shown in Fig. 12. The reflection coefficient is lower than 10 dB over the entire frequency range of 2.29–2.50 GHz so that a 10 dB bandwidth of 8.9% is achieved. Fig. 13 shows the measured and simulated radiation patterns on the Y-Z (E-plane) and X-Z (H-plane) planes at 2.38 GHz. The cross-polarization was lower than 11 dB.

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Fig. 13. Simulated and measured radiation patterns of the chip-loaded ZOR antenna at 2.38 GHz (a) y-z plane (E-plane), (b) x-z plane (H-plane).

TABLE II MEASUREMENT SUMMARY AND COMPARISON RESULTS FOR THREE PROPOSED ANTENNAS

V. SUMMARY AND COMPARISON The overall antenna performances of the three proposed antennas are compared in Table II. Their overall physical sizes are identical, except that the shorted meander lines of the unit cell

, , are differently designed. In other words, the values of in two of the models are the same and only the value of and is varied. The asymmetric antenna is designed by removing one side of the meander line of the unit cells from the symmetric antenna. of the asymmetric antenna is higher than that of Since the the symmetric antenna, its resonant frequency is decreased from 2.03 GHz to 1.5 GHz. Therefore, its electrical size can be further reduced in the asymmetric geometry. As a trade-off, the radiation efficiency is degraded due to the electrically smaller aperture size of the asymmetric antenna. Furthermore, the coupled slot mode originating from the discontinuity of the asymmetric antenna affects its radiation efficiency. The measured efficiencies of the symmetric and asymmetric antennas are 62% and 42.5%, respectively. In addition, their measured peak gains are 1.35 dBi and 2.15 dBi, respectively. The chip-loaded antenna is designed by replacing the meander lines by chip inductors. Since the chip inductance is easily adjustable, it has the advantage of easy realization at the desired frequency. However, the value of the chip inductance is not acceptable at high frequencies. The chip-loaded antenna shows the highest efficiency of 77.8% and its peak gain is 1.54 dBi at 2.38 GHz. The radiation efficiencies are obtained by measuring the total radiation power versus the input power. The symmetric, asymmetric, and chip-loaded antennas provide extended 10 dB bandwidths of 6.8%, 5%, and 8.9%, respec, its tively. Although the asymmetric antenna has a higher bandwidth is decreased. This is because the balancing of is not achieved and G is decreased. In terms of the radiation mechanisms of the proposed antenna, the magnitudes of the magnetic current densities in each patch edge are not equivalent. The dominant magnetic current source is from one slot which is located at the feeding line. The magnetic current sources from the other three slots are weaker because the signal plane is far from the ground plane. Therefore, the proposed antenna looks like an ideal magnetic dipole rather than the magnetic loop antenna described in [10]. At the zeroth-order resonant frequency, the resonant condition is independent of the aperture dimension. Fig. 14(a) clearly demonstrates that the resonant frequencies remain almost constant as the aperture dimension is increased. In a conventional resonant antenna, it is obvious that the resonant frequency is decreased as its size is increased. Fig. 14(b) shows the relationship between the gain and the number of unit cells. As shown in Fig. 14(a) and (b), the frequencies of these antennas do not vary much while their gains become higher as the number of the unit cells increases. Furthermore, the measured radiation patterns show that the cross-polarization levels of the proposed antennas are higher than the simulated ones. These differences are due to the fabrication limitation resulting from the fine meander lines and the measurement error resulting from the much smaller size of the aperture compared with that of the RF cable in the test environment.

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REFERENCES

Fig. 14. Comparison of the symmetric, asymmetric, and chip-loaded antennas (a) relationship between the frequency and the number of unit cells and (b) relationship between the gain and the number of unit cells.

VI. CONCLUSION In this study, we demonstrated the extended bandwidth of the proposed CPW-fed ZOR antennas. The size of the proposed ZOR antennas can be reduced, due to their zeroth-order resonance. When the ZOR antennas are realized using CPW technology, they allow for the design freedom of the shunt parameters in the equivalent circuit model. Thus, when a high shunt inductance and small shunt capacitance are realized, the symmetric and asymmetric ZOR antennas exhibit extended bandwidths of 6.8% and 5%, respectively. Alternatively, when chip inductors are applied to the proposed CPW ZOR antenna, its bandwidth can be improved by up to 8.9%. In order to analyze the principle of this bandwidth enhancement, the equivalent-circuit models were derived and analyzed. The proposed theory and experimental results show good agreement with each other. In addition, the proposed design has the advantages of easy fabrication, due to the via free structure, as well as a single layer process. Therefore, due to their features, the symmetric, asymmetric, and chip-loaded antennas are suitable for use in mobile wireless communication systems.

[1] V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of " and ,” Sov. Phys. Usp, vol. 10, no. 4, pp. 509–514, Jan.–Feb. 1968. [2] G. V. Eleftheriades, A. K. Iyer, and P. C. Kremer, “Planar negative refractive index media using periodically LC loaded transmission lines,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 12, pp. 2702–2712, Dec. 2002. [3] G. V. Eleftheriades, A. Grbic, and M. Antoniades, “Negative-refractive-index metamaterials and enabling electromagnetic applications,” in Proc. IEEE Int. Symp. Antennas and Propag., Monterey, CA, Jun. 2004, vol. 2, pp. 1399–1402. [4] G. V. Eleftheriades, “Enabling RF/microwave devices using negativerefractive-index transmission-line (NRI-TL) metamaterials,” IEEE Antennas Propag. Mag., vol. 49, no. 2, pp. 34–51, Apr. 2007. [5] G. V. Eleftheriades, M. A. Antoniades, and F. Qureshi, “Antenna applications of negative-refractive index transmission-line structures,” IET Microw., Antennas Propag., vol. 1, no. 1, pp. 12–22, Feb. 2007. [6] A. Lai, T. Itoh, and C. Caloz, “Composite right/left-handed transmission line metamaterials,” IEEE Microw. Mag., vol. 5, no. 3, pp. 34–50, Sep. 2004. [7] S. Lim, C. Caloz, and T. Itoh, “Metamaterial-based electronically controlled transmission-line structure as a novel leaky-wave antenna with tunable radiation angle and beamwidth,” IEEE Trans. Microw. Theory Tech., vol. 53, pp. 161–173, Jan. 2005. [8] A. Sanada, C. Caloz, and T. Itoh, “Novel zeroth-order resonance in composite right/left-handed transmission line resonators,” in Proc. Asia-Pacific Microwave Conf., Seoul, Korea, Nov. 2003, vol. 3, pp. 1588–1592. [9] C.-J. Lee, K. M. K. H. Leong, and T. Itoh, “Composite right/left-handed transmission line based compact resonant antennas for RF module integration,” IEEE Trans. Antennas Propag., vol. 54, pp. 2283–2291, Aug. 2006. [10] A. Lai, K. M. K. H. Leong, and T. Itoh, “Infinite wavelength resonant antennas with monopole radiation pattern based on periodic structures,” IEEE Trans. Antennas Propag., vol. 55, no. 3, pp. 868–875, Mar. 2007. [11] F. Qureshi, M. A. Antoniades, and G. V. Eleftheriades, “A compact and low-profile metamaterial ring antenna with vertical polarization,” IEEE Antenna Wireless Propag. Lett., vol. 4, pp. 333–336, 2005. [12] C.-J. Lee, K. M. K. H. Leong, and T. Itoh, Broadband Small Antenna for Potable Wireless Application iWat, pp. 10–13, Mar. 2008. [13] J. Zhu and G. V. Eleftheriades, “A compact transmission line metamaterial antenna with extended bandwidth,” IEEE Antenna Wireless Propag. Lett., vol. 8, pp. 295–298, 2009. [14] M. A. Antoniades and G. V. Eleftheriades, “A broadband dual-mode monopole antenna using NRI-TL metamaterial loading,” IEEE Antenna Wireless Propag. Lett., vol. 8, pp. 258–261, 2009. [15] T. Jang and S. Lim, “A novel broadband co-planar waveguide (CPW) zeroth order resonant antenna,” in Proc. Asia-Pacific Microwave Conf., Singapore, Dec. 2009, pp. 52–55. [16] A. Sanada, C. Caloz, and T. Itoh, “Planar distributed structures with negative refractive index,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 4, pp. 1252–1263, Apr. 2004. [17] D. M. Pozar, Microwave Engineering, 2nd ed. Toronto: Wiley, 1998, pp. 49–57. [18] R. L. Fante, “Quality factor of general ideal antennas,” IEEE Trans. Antennas Propag., vol. 17, no. 2, pp. 151–155, Mar. 1969. [19] A. D. Yaghjian and S. R. Best, “Impedance, bandwidth, and Q of antennas,” IEEE Trans. Antennas Propag., vol. 53, p. 1298, 2005. [20] C. Caloz and T. Itoh, Electromagnetic Metamaterials: Transmission Line Theory and Microwave Applications. New York: Wiley, Dec. 2005. [21] G. V. Eleftheriades and K. G. Balmain, Negative-Refraction Metamaterials: Fundamental Principles and Applications. New York: Wiley/ IEEE Press, Jun. 2005.

Taehee Jang (S’09) received the B.S. degree in the school of electrical and electronics engineering from the Chung-Ang University, Seoul, Korea, in 2009. Since 2008, he has been worked as a Student Researcher in the Microwave Wireless Communication Laboratory, Chung-Ang University, Seoul, Korea. His research interests include metamaterial applications and printed antennas.

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Jaehyurk Choi received the B.S. degree in the school of electrical and electronics engineering from the Chung-Ang University, Seoul, in 2009, where he is currently working toward the M.S. degree in electrical and electronics engineering. His research interests include metamaterial applications and reconfigurable antennas.

Sungjoon Lim (S’02–M’07) received the B.S. degree in electronic engineering from Yonsei University, Seoul, Korea, in 2002, and the M.S. and Ph.D. degrees in electrical engineering from the University of California at Los Angeles (UCLA), in 2004 and 2006, respectively. After a postdoctoral position at the Integrated Nanosystem Research Facility (INRF), the University of California at Irvine, he joined the School of Electrical and Electronics Engineering, Chung-Ang University, Seoul, Korea, in 2007, where he is currently an Assistant Professor. He has authored and coauthored more than 30 technical conference, letter and journal papers. His research interests include engineered metamaterial structures, printed antennas, arrays, and RF MEMS applications. He is also interested in the modeling and design of microwave/millimeter-wave circuits. Dr. Lim received the Institution of Engineering and Technology (IET) Premium Award in 2009.

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Increasing the Bandwidth of Microstrip Patch Antenna by Loading Compact Artificial Magneto-Dielectrics Xin Mi Yang, Quan Hui Sun, Ya Jing, Qiang Cheng, Xiao Yang Zhou, Hong Wei Kong, and Tie Jun Cui, Senior Member, IEEE

Abstract—We realize artificial magneto-dielectric loading for microstrip patch antenna by etching embedded meander-line (EML) array in the ground plane under the patch. The related artificial magneto-dielectric medium belongs to the waveguided metamaterial proposed previously. Both simulation and measurement results show that the proposed patch antenna with the EML array has wider impedance bandwidth than the conventional patch antenna with the same size. Though we have to increase the antenna profile by attaching an additional shield metal plate to suppress the back radiation, the proposed magneto-dielectric loading method requires lower fabrication complexity and lower cost than the existing techniques. Index Terms—Embedded meander line, magneto-dielectric, microstrip patch antenna, waveguided metamaterial.

A useful approach which was proposed a few years ago to construct the microstrip antenna with both small patch size and broad bandwidth was to fill in the volume between the patch and ground plane with magneto-dielectric material whose permeability and permittivity are both larger than one [2], [3]. The basic principle of this approach is that the enhanced magnetic response of magneto-dielectric material lowers the quality factor of patch antenna while its refractive index lowers the resonant frequency as pure dielectric material does. Consider the model of a microstrip patch antenna with a cubic magneto-dielectric material loaded in the substrate volume right under the antenna patch. The quality factor of the antenna at its resonance reads

I. INTRODUCTION

(1)

ICROSTRIP patch antenna has been widely used in wireless and mobile communications due to its simple structure and low cost. In recent years, the wide-band communication places requirements of fairly large bandwidth and small volume to antennas. For example, more than 200 MHz bandwidth is required for the WiMAX antenna. However, small size and broad bandwidth are often contradictory to each other for conventional microstrip patch antennas which are fabricated on the conventional dielectric substrate [1]. Normally, the patch size can be decreased by increasing the substrate permittivity but high permittivity will lead to low radiation efficiency (due to the large surface wave loss) and extremely narrow impedance bandwidth. On the other hand, many innovations to improve the impedance bandwidth performance give rise to the increase in size, height (or volume), and the fabrication complexity and cost, and hence are accompanied by degradation of other antenna characteristics.

M

Manuscript received November 30, 2009; revised April 29, 2010; accepted August 08, 2010. Date of publication December 06, 2010; date of current version February 02, 2011. This work was supported in part by a Major Project of the National Science Foundation of China (Fundamental Theories and Key Technologies of Metamaterials) under Grants 60990320 and 60990324, in part by Agilent Technologies, and in part by the 111 Project under Grant No. 111-2-05. X. M. Yang, Q. H. Sun, Q. Cheng, X. Y. Zhou, and T. J. Cui are with the State Key Laboratory of Millimeter Waves and the Institute of Target Characteristics and Identification, School of Information Science and Engineering, Southeast University, Nanjing 210096, China (e-mail: [email protected]; [email protected]; [email protected]; [email protected]; [email protected]). Y. Jing and H. W. Kong are with the Agilent Measurement Research Lab., Beijing 100102, China (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2096388

is the time averaged enwhere is the angular frequency, ergy residing in the near fields of the antenna, and is the power dissipated during one cycle. Both quantities can be quickly estimated by the approximately analytical method. Known from is mainly composed of the cavity model of patch antenna, the energy in the volume under the patch [3] (2) where and are relative permittivity and relative permeability of the magneto-dielectric material, respectively. By apmode resonance condition and making the inplying the tegration, (2) becomes (3)

where is substrate thickness, is the electric field amplitude at the radiation edge (with length ) of the patch and is the wave impedance in free space. As to , it is a good approximation to take only the radiation power into account: (4) where is the radiation conductance. Inserting (3) and (4) into (1), we get

0018-926X/$26.00 © 2010 IEEE

(5)

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Equation (5) clearly reveals that the quality factor decrease increases. [2] further gave the zero-order as the ratio VSWR=2 bandwidth of the square patch antenna (6) is the free space wavelength at antenna resonance. where Therefore one can enhance the antenna bandwidth by infor a given patch miniaturization factor creasing (constant ). Theoretically, the magneto-dielectric loading has little influence upon the radiation characteristics of the antenna. In the literatures, efforts have been made to investigate the magneto-dielectric-substrate-based patch antennas numerically or experimentally, where ferrites, ferromagnetic films, and macroscopically periodic structures were used as the magneto-dielectric substrates, respectively [4]–[10]. Ferrites or ferromagnetic films provide very high permeability but require complex fabrication technology and hence are relatively expensive. Besides, it is difficult to achieve low loss for these materials above a few GigaHertz. If we periodically arrange electrically small metal inclusions inside a dielectric matrix, the resulting electromagnetic composites may behave as effective medium with certain permittivity and permeability characteristics. Such artificial materials are usually called as metamaterials. The artificial magneto-dielectric materials comprise of metal inclusions which possess magnetic responses, such as spiral loops or split ring resonators. Though the available effective permeability of artificial magneto-dielectric materials is relative low, it is easy to design such materials with low loss at microwave frequencies. When it is loaded into patch antenna, the artificial magneto-dielectric material is usually stacked between the ground and patch metallization [5]–[7]. That configuration appears cumbersome and brings about inconvenience for the fabrication and system integration. In this paper, we present another configuration of artificial magneto-dielectric composites to improve the bandwidth of microstrip antenna. The antenna with such configuration requires simple single-layer PCB techniques for fabrication as the conventional microstrip patch antenna does. The proposed antenna shown in Fig. 1 consists of a dielectric substrate, a radiating patch, and a ground plane on the two sides of the substrate separately. It has additional etched patterns right under the patch in the ground metallization compared to the conventional patch antenna. The etched patterns include many identical electrically small planar unit cells, which are arranged periodically along the two orthogonal directions ( and axis in Fig. 1) in the ground plane. It is shown in the following sections that the etched patterns together with the dielectric substrate and top metallization (i.e., the metallic patch) also constitute the effective magneto-dielectric materials and are expected to make contribution to the antenna bandwidth. The rest of paper is organized as follows. In Section II, we introduce the ground etching configuration for the artificial magneto-dielectric material. The design of patch antenna with such artificial magneto-dielectric loading is presented in Section III, followed by Section IV where various performances including

Fig. 1. The illustration of microstrip antenna with the proposed magneto-dielectric loading. The yellow area indicates metallization and the green area indicates the supporting dielectric board (a) Front view. (b) Back view.

Fig. 2. The embedded meander line structure and the EML array in the planar waveguide environment.

the improvement of impedance bandwidth of the proposed patch antenna are demonstrated. We give further discussions and conclusions in Sections V and VI, respectively. II. EMBEDDED MEANDER-LINE STRUCTURE WITH MAGNETO-DIELECTRIC RESPONSE The planar unit cell of the etched patterns in the ground plane of the proposed patch antenna is called as the embedded meander-line structure (EML), which characterizes a meander line embedded in a square area defect in the ground, as shown in Fig. 2. When the EML array is loaded in the lower metal plate of the planar waveguide and TEM wave is applied (see Fig. 2), the cubic cell between each individual EML structure and the upper metal plane responds to the incident magnetic field and an enhanced magnetic moments align with the external magnetic field are induced near the resonance. The distributed capacitance and inductance provided by the EML structure contribute to the resonant loop inhabiting the cubic cell. Since the size of EML element is much smaller than the wavelength near resonance, the volume occupied by the EML array in the planar waveguide can be taken as an effective medium. We remark that the analogous effective medium configuration was also investigated in [11]–[13] and was named as waveguided metamaterial [12]. We show the relative effective medium parameters of the EML-based waveguided metamaterial in Fig. 2, in which the EML geometries are , , , , and . The EML array periodicity is 4.2 mm and directions, the height of the waveguide along the , and the permittivity and loss tangent of is supporting dielectric are 2.65 and 0.001. These effective medium parameters are obtained using full-wave simulations

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Fig. 4. The proposed antenna with an additional shield metal plate.

Fig. 3. The effective medium parameters of the EML-based waveguided metamaterial, where  and  are retrieved when the incident wave is from x direction and  and  are retrieved when the incident wave is from y direction.

(CST2006B) and the subsequent retrieval process. In the simulations, the TEM wave is applied in the planar waveguide as indicated in Fig. 2 and the simulated reflection and transmission coefficients are then used to retrieve the medium parameters. The detailed simulation and retrieval technique were introduced in [11] and [14], respectively. In view of the excitation manner, the retrieved parameters are the component of permittivity and the component of permeability which are represented by the solid lines in Fig. 3. We conclude the effective magneto-dielectric properties of the EML-based waveguided metamaterial as follows. • The resonance of EML structure is very weak and the effective parameters are almost non-dispersive before resonance; is enhanced as expected • The effective permeability before resonance); ( • The effective permittivity is lower than the dielectric con, which is favorstant of the supporting material able for increasing the factor in (6). The reason is that the defects in the lower metal plate decrease the capacitance between the upper and lower metal plates. and Fig. 3 also shows the effective medium parameters ( ) which are retrieved when the excitation TEM wave is from orthogonal direction (i.e., direction). The two sets of effective medium parameters with different excitation manner are very similar to each other, indicating that the effective medium behavior of the proposed EML structure is insensitive to rotation. III. DESIGN OF THE MICROSTRIP PATCH ANTENNA WITH EML LOADING When the patch antenna works at its dominant mode ( mode), the upper patch and lower ground metallization imitate the planar waveguide environment where the approximate TEM wave is guided. Hence the EML array in the ground, the patch and the supporting dielectric between them constitute the artificial magneto-dielectric material, acting as the substrate for the studied antenna (see Fig. 1).

The proposed antenna is designed to operate around 3.5 GHz. As the supporting dielectric parameters and the EML geometry are consistent with those in Section II, the effective medium parameters of the artificial substrate at the design frequency are and . The patch length can be estimated from the half wavelength resonance condition for the rectangular patch antenna as (7) where, is the light speed of free space, is the design freis the miniaturization factor for the rectanquency, and and , gular patch. When we neglect the imaginary part of and the estimation the miniaturization factor is of patch length is . Note that the miniaturization factor with the artificial magneto-dielectric substrate here is larger than that with the supporting dielectric only . The scale of EML array is chosen as 5 5 here, correarea in the ground plane. The sponding to a array is located completely under the patch, given that the and the patch length is more than patch width is 21 mm. This configuration is due to the fact that most of the resonant electromagnetic field is concentrated right under the patch as stated in Section I. and The imaginary parts of the effective parameters correspond to the leakage loss due to the defects in the ground plane. The energy leakage leads to increasing in back radiation and decreasing in antenna gain which are undesirable in many situations. To suppress the back radiation, an additional metal shield plate is added beneath and parallel to the antenna ground, as shown in Fig. 4. In our design, the sizes of shield plate are , , , and the distance between the shield and antenna ground is . We remark that the shield plate has a little influence on the magneto-dielectric medium behavior of the EML-based waveguided metamaterial. However, utilizing effective medium parameters without shield plate for the initial guess of patch size is acceptable. To get the final design, the patch antenna with above parameters are simulated in CST2006B, where the actual EML array is modeled. The patch length is optimized in the software to meet the resonance requirement around the specified frequency is 22 mm. Note that the patch 3.5 GHz. The final value of length is slightly longer than the EML array size in the direction so that the effect of EML array on the radiation conductance could be neglected. The dimensions of the feeding network are , , , determined as: , where and are the width and length and and of 50 Ohm microstrip feeding line, respectively, and

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Fig. 5. The simulated and measured reflection coefficient of the proposed antenna (solid line and dash dot line, respectively) and the simulated reflection coefficient of the control antenna (dashed line). Fig. 6. The simulated input admittance for the proposed antenna (the solid line) and the control antenna (the dash-dot line).

are the width and length of the quarter-wavelength matching line, respectively, as shown in Fig. 1. The overall sizes of the resulting antenna are 60 mm wide and 76.34 mm long. For comparison, we also design a conventional patch antenna with the purely dielectric substrate whose permittivity is tuned (with dielectric loss tangent 0.001) so that to be the proposed antenna and the control antenna have the same supporting material height, the same patch size, and the same overall size. IV. PERFORMANCE OF THE MICROSTRIP PATCH ANTENNA WITH EML LOADING Fig. 5 shows the simulated reflection coefficients of the antennas designed in Section III. From the figure, the proposed antenna and the control antenna have excellent matching at 3.486 impedance bandwidths are 77.7 MHz GHz and their and 42 MHz, respectively. Hence the improvement factor of impedance bandwidth of the proposed antenna over the conventional antenna is about 1.85. From (6), the bandwidth im, provement factor of 2.06 is predicted in considering and . The discrepancy between the theoretical and simulated results can be attributed to the inaccuracy of the effective medium parameters retrieved for the loaded EML array and the inaccuracy of the analytical model used to evaluate impedance bandwidth. The bandwidth improvement is better than those presented in [6]–[8] but is slightly worse than that in [4]. Fig. 6 shows the simulated input admittance with the reference plane located at the junction of the patch and feeding network. The susceptance curve for the proposed antenna varies more slowly around the resonant frequency than that of the control antenna, which indicates the improved impedance matching of the proposed antenna. To further verify that the impedance bandwidth improvement value of the artificial magnetois a result of enhanced dielectric substrate, we compare the simulated magnetic-field intensity at the resonance under the patch in Fig. 7 for the proposed antenna and the control antenna. When we take the patch as a wide transmission line (TL), the characteristic impedance of increases. Hence the antenna with the TL increases as

Fig. 7. The simulated magnetic-field intensity distribution under the patch at the resonance (a) The antenna with the EML array. (b) The control antenna. TABLE I COMPARISON OF RADIATION PERFORMANCES OF THE ANTENNA WITH THE EML LOADING AND THE CONTROL ANTENNA

the highsubstrate exhibits the weak magnetic-field intensity which leads to small energy storage under the patch. The quality factor of antenna is decreased as the energy stored is decreased, resulting in the increase of bandwidth [3]. It is obvious from Fig. 7 that the magnetic-field intensity of the proposed antenna is much smaller than that of the control antenna, validating the magneto-dielectric material effect of the EML loading. The periodic distribution of the magnetic intensity in Fig. 7(a) is due to the periodicity of the EML array. Simulations also show that the radiation pattern of the proposed antenna is close to that of the control antenna. They have almost the same radiation efficiency and forward co-polarization gain (see Table I) at the central working frequency. Compared with other magneto-dielectric loading techniques in the literatures [4], [6]–[8], the proposed EML loading technique provides more satisfactory radiation efficiency, as the

YANG et al.: INCREASING THE BANDWIDTH OF MICROSTRIP PATCH ANTENNA BY LOADING COMPACT ARTIFICIAL MAGNETO-DIELECTRICS

Fig. 8. The fabricated patch antenna with the EML array. The metal shelter is not shown in the figure (a) Front view. (b) Back view.

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half plane for the E plane, and the to the positive/negative value corresponds to the half plane for the H plane. The measured pattern shows that the radiation characteristics of the proposed antenna could compare favorably with the conventional antenna. The front-to-back ratios of the sample are 11.98 dB and 20.76 dB for E and H planes, respectively, and the cross-polarizations for E and and , respectively. H planes are about We remark that the substrate permittivity of the control antenna in simulation (3.51) is not available in the commercial high-frequency circuit boards, hence the measured results for the control antenna is not presented. However, we also fabricate a conventional patch antenna which has the same supporting substrate as the proposed antenna (dielectric constant 2.65) but a longer patch length to work at approximately the same frequency. The overall size and thickness of the conventional antenna are the same as those of the control antenna in the simulation. Hence the gain performance of the fabricated conventional path antenna changes little with respect to the control antenna. Experimental comparison shows that the forward co-polarization gain of the proposed antenna is 0.45 dB larger than that of the fabricated conventional patch antenna. It is clear that the EML loading together with metal shield plate has little influence on the gain of the patch antenna. V. DISCUSSIONS

Fig. 9. The measured radiation pattern of the patch antenna with the EML array at 3.494 GHz. In the E plane, the positive/negative  value corresponds to the  = half plane. In the H plane, the positive/negative  value = half plane. corresponds to the 

=0

= 180

= 90

= 270

studied antenna operates far away from the self-resonance frequency of the EML array. The actual loss (i.e., dielectric loss and conductor loss) of the antenna is very small and the visual loss (seen form the imaginary part of the retrieved medium parameters) originates from the energy leakage due to the defects in ground plane. In the presence of metal shield plate, the energy leakage is largely suppressed and changes to parallel plate mode between the ground plane and shield plate. The parallel plate mode contributes to the antenna’s back radiation and is unfavorable in terms of directivity. However, the proposed antenna still possesses a higher forward gain than the control antenna because the surface wave of the proposed antenna is smaller due to its lower supporting substrate permittivity. The above antenna with the EML array is fabricated and the sample is shown in Fig. 8. It is shown in Fig. 5 that the measured working frequency of the sample is 3.494 GHz and its bandwidth is 76 MHz, which are close to the simulation results. We also measured the radiation patterns of the sample at 3.494 GHz, which is shown in Fig. 9. In this figure, the spherical coordinate used agrees with the cartesian coordinate shown in Fig. 2 and the positive/negative value corresponds

In addition to the patch length, the patch width and substrate thickness are common design issues for patch antennas. The rule that decreasing patch width will result in the increased resonant resistance and the increased difficulty in the impedance matching is still valid for the patch antenna with the EML loading. Besides, decreasing the patch width will lower the design accuracy since the effective medium parameters are retrieved for infinitely wide EML array. Decreasing the substrate thickness will in the mean time alter the characteristic of the effective magneto-dielectric medium for the proposed of the EML array at the design freantenna. The ratio quency increases and the dispersion becomes more violent as the substrate thickness decreases. If the substrate thickness is too small, the bandwidth improvement factor becomes much smaller than that predicted by (6), indicating the contribution from the enhanced magnetic response to bandwidth improvement is mostly canceled out by the large dispersion because the substrate dispersion contributes to the stored energy of the patch antenna as stated in [7]. The decreased substrate thickness also leads to a decreased absolute bandwidth for the proposed patch antenna as the conventional patch antenna. It is interesting to address that as the effective medium property of the EML structure is insensitive to rotation (Fig. 3), rotating each EML unit in the above design around its central axis (pointing to the axis) by 90 degrees will not change the antenna performance significantly. Fig. 10 shows the simulated reflection coefficients of the original patch antenna with the EML array and the modified antenna. Both antennas share almost the same working frequency (3.486 GHz) and impedance bandwidth (77.7 MHz and 77.9 MHz, respectively). Other performances, like the radiation efficiency, directivity, cross-polarization and front-to-back ratio for such two antennas are also

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Fig. 10. The simulated reflection coefficient of the original and modified patch antenna with EML array (solid line and dash-dot line, respectively).

close to each other. Hence the proposed EML loading technique is suitable for dual polarization application. Compared with other magneto-dielectric loading techniques, the proposed EML loading technique requires more convenient fabrication process and lower cost. The fabrication of magnetodielectric fillings like ferrites, ferromagnetic films or periodically arranged metal inclusions is relatively expensive since those fillings are volumetric and require complicated fabrication art. Furthermore, the assembly of those filings with the patch antenna has additional costs. As to the proposed technique, there is no need to fabricate independent fillings and the realization of magneto-dielectric loading consists only of etching planar patterns in the antenna ground plane right under the patch. Once the etching process is finished, the integration of the magneto-dielectric fillings to antenna is also complete. Though the metal shield plate is required in the proposed technique, the total cost is still low if we utilize thin aluminum plate with low quality, which has negligible influence on the antenna performance. The main drawback of the proposed magneto-dielectric technique is that the EML-array length, and accordingly the patch length could not be continuously adjusted in the design process due to the periodicity of the EML array. Besides, the additional metal plate enlarges the profile of antenna. VI. CONCLUSIONS We have demonstrated the magneto-dielectric characteristics of the EML-based waveguided metamaterial. The design of patch antenna loaded with the EML-based waveguided metamaterial has been presented. It is verified by both simulations and measurements that such artificial magneto-dielectric loading improves the impedance bandwidth of the antenna but influences little on the radiation characteristic. The fabrication complexity and cost of the proposed antenna are low. It is anticipated that the proposed technique to improve the bandwidth will find more applications in the microwave region.

[2] R. C. Hansen and M. Burke, “Antenna with magneto-dielectrics,” Microw. Opt. Technol. Lett., vol. 26, no. 2, pp. 75–78, 2000. [3] P. Ikonen and S. A. Tretyakov, “On the advantages of magnetic materials in microstrip antenna miniaturization,” Microw. Opt. Technol. Lett., vol. 50, no. 12, pp. 3131–3134, 2008. [4] H. Mosallaei and K. Sarabandi, “Magneto-dielectrics in electromagnetics: Concept and applications,” IEEE Trans. Antennas Propag., vol. 52, pp. 1558–1567, 2004. [5] K. Buell, H. Mosallaei, and K. Sarabandi, “A substrate for small patch antennas providing tunable miniaturization factors,” IEEE Trans. Microw. Theory Tech., vol. 54, pp. 135–146, 2006. [6] H. Mosallaei and K. Sarabandi, “Design and modeling of patch antenna printed on magneto-dielectric embedded-circuit metasubstrate,” IEEE Trans. Antennas Propag., vol. 55, pp. 45–52, 2007. [7] P. M. T. Ikonen, S. I. Maslovski, C. R. Simovski, and S. A. Tretyakov, “On artificial magnetodielectric loading for improving the impedance bandwidth properties of microstrip antennas,” IEEE Trans. Antennas Propag., vol. 54, pp. 1654–1662, 2006. [8] P. M. T. Ikonen, K. N. Rozanov, A. V. Osipov, P. Alitalo, and S. A. Tretyakov, “Magnetodielectric substrates in antenna miniaturization: Potential and limitations,” IEEE Trans. Antennas Propag., vol. 54, pp. 3391–3399, 2006. [9] R. V. Petrov, A. S. Tatarenko, G. Srinivasan, and J. V. Mantese, “Antenna miniaturization with ferrite ferroelectric composites,” Microw. Opt. Technol. Lett., vol. 50, no. 12, pp. 3154–3157, 2008. [10] F. Grange, K. Garello, E. Benevent, S. Bories, B. Viala, C. Delaveaud, and K. Mahdjoubi, “Investigation of magneto-dielectric thin films assubstrates for patch antennas,” in Proc. 3rd Eur. Conf. on Antennas and Propagation EuCAP 2009, Mar. 23–27, 2009, pp. 1909–1913. [11] R. Liu, Q. Cheng, T. Hand, J. J. Mock, T. J. Cui, S. A. Cummer, and D. R. Smith, “Experimental demonstration of electromagnetic tunneling through an epsilon-near-zero metamaterial at microwave frequencies,” Phys. Rev. Lett., vol. 100, no. 2, p. 023903, 2008. [12] R. Liu, X. M. Yang, J. G. Gollub, J. J. Mock, T. J. Cui, and D. R. Smith, “Gradient index circuit by waveguided metamaterials,” Appl. Phys. Lett., vol. 94, no. 7, p. 073506, 2009. [13] Q. Cheng, R. Liu, J. J. Mock, T. J. Cui, and D. R. Smith, “Partial focusing by indefinite complementary metamaterials,” Phys. Rev. B, vol. 78, no. 12, p. 121102(R), 2008. [14] D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B, vol. 65, no. 19, p. 195104, 2002.

Xin Mi Yang, photograph and biography not available at the time of publication.

Quan Hui Sun, photograph and biography not available at the time of publication.

Ya Jing, photograph and biography not available at the time of publication.

Qiang Cheng, photograph and biography not available at the time of publication.

Xiao Yang Zhou, photograph and biography not available at the time of publication.

Hong Wei Kong, photograph and biography not available at the time of publication.

REFERENCES [1] R. Garg, P. Bhartia, I. Bahl, and A. Ittipiboon, Microstrip Antenna Design Handbook. Boston, London: ArtechHouse, 2001.

Tie Jun Cui, photograph and biography not available at the time of publication.

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379

A New Design Method for Single-Feed Circular Polarization Microstrip Antenna With an Arbitrary Impedance Matching Condition Stefano Maddio, Alessandro Cidronali, Member, IEEE, and Gianfranco Manes, Senior Member, IEEE

Abstract—This paper introduces a new analytical method suitable for the design of the single-feed circular polarization (CP) microstrip patch antenna. Specifically, the proposed method imposes simultaneously the circular polarization condition as well as an arbitrary input impedance matching condition. The two conditions are enforced by an analytical method derived from an equivalent circuit model of a quasi-symmetrical patch antenna, and manipulated to control the modal detuning. This method can be used as an aid to speed up the design procedure for CP antennas even working with a numeric CAD tool. The validation of this approach is proven designing and fabricating a prototype implementing an original design, which consists of a circular disc slotted by a concentric elliptical cut with coaxial feed and operating at the center frequency of 2.45 GHz. The fabricated prototype exhibited a return loss of about 19 dB, within an impedance bandwidth of 130 MHz, a measured gain of 3.85 dB and a 0 dB axial ratio at 2.45 GHz with a corresponding modal phase difference of 87 degrees. Index Terms—Antenna circuit model, circular polarization, microstrip patch antenna.

I. INTRODUCTION IRCULAR polarization (CP) antennas are largely involved in wireless systems embedded in complex environments as an effective way to mitigate multi-path effects [1]. The most common techniques to design compact and efficient CP microstrip patch antennas consist in exciting either two identical orthogonal modes of a symmetrical patch or in feeding two identical orthogonal patch antennas in quadrature. It is also possible to achieve the CP effect in a patch antenna, using a single probe that excites a quasi-symmetrical shape supporting two degenerated modes [2]–[4]. The two modes, determined by detuning the fundamental mode, are sources of two orthogonal polarized linear fields; CP radiation is achieved when these two modes have the same magnitude and are in phase quadrature. This can be accomplished by appropriately tuning the asymmetry of the patch’s shape and choosing the position of the probe with respect to the asymmetry. In literature an analysis of this kind of radiating structure was carried

C

Manuscript received November 28, 2008; revised November 19, 2009; accepted September 17, 2010. Date of publication December 03, 2010; date of current version February 02, 2011. The authors are with the Department of Electronics and Telecommunications Engineering, University of Florence, I-50139 Florence, Italy (e-mail: stefano. [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2096177

out either by directly manipulating the eigen-modes expression [2] or with a variational approach [3], [5]; always focusing on specific shapes, primarily squares and disks. In [6], a general multimodal analysis was proposed, although the results reached were not suitable for direct employment in either a problem of analysis or synthesis. Another promising but scarcely developed approach is based on the equivalent circuital model outlined in [7]. In literature very few works tried to develop this approach, in spite of its potential impact on design, and very few studies dealt with the simultaneous radiative and circuital aspects in the CP antenna design: to our knowledge the analytical question of CP and matching at the same frequency was never handled without concentrating on a specific patch shape. In this paper, the complex relations among the two main aspects of CP patch antenna are handled with an analytical approach based on the algebraic manipulation of circuital equivalent parameters. This approach can be considered a stand-alone instrument, which can be used for analysis purpose but it can be also employed as a tool for design task. Differently from the available design methods, the proposed approach leads to a procedure capable to control simultaneously the operative frequency, the axial ratio as well as input impedance in CP microstrip patch antenna, and this control is directly applicable in the early steps of the design flow. In literature the CP issue and the matching issue are almost always considered separated task, leading to lengthy cut-and-trial procedures to accomplish the goals in different steps or at most obtained through lengthy numerical optimization. Used in combination with a CAD tool like the electromagnetic solver HFSS [8], our approach can give physical insight of the operation principle and addresses the CP and matching design problems straightforwardly. The paper is organized as follows: Section II presents the proposed model, while Section III describes a design procedure based on the equivalent circuital model, which demonstrates how to impose both the CP and matching conditions consistently. Finally, Section IV presents the experimental results obtained with a prototype antenna working at 2.45 GHz designed by the described design procedure. II. ANALYTICAL TREATMENT This section deals with the analytical aspects of the proposed design method. We show that the chosen formulation is suitable to represent the polarization behavior of the antenna as a function of geometry and physical characteristics. First, an outline of the circuital equivalence of the antenna modes is presented,

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then the analytical function is defined based on the equivalent circuit adopted to enforce the CP condition. A. Modal Expansion and Circuital Equivalence Microstrip patch antennas are narrow-band radiators which can be modeled as lossy resonant cavities enclosed by perfect magnetic walls on the perimeter and bounded by perfect electric walls on the top and bottom [3]. The field distribution in the cavity can be described in terms of eigen-modes and with the , only additional hypothesis of a thin substrate layer , and are non-zero. Acthree components, namely cording to the Green function theory, [3], the electrical field , can be expressed by a set of homogeneous Helmholtz equation eigen-functions (1) the modal wave-number of the mode, a being support function related to the geometry of the probe and mode order and the position of the probe in reference to the patch, while represents the feeding current. Taking into account the various antenna losses, the wave-number is modified to a complex value defined as (2) thus, the impedance observed from the probe located in be expressed as

can

Fig. 1. Equivalent Parallel series circuit representing the modal expansion of the input impedance.

the static capacitance along with its relative losses. The modal resistance of each mode is theoretically frequency dependent but can be approximated to a constant value within the band of interest, due to the narrowness of the mode resonance bandrepresents the sum of the radiation, width. Furthermore, dielectric and metal losses, but for a well designed antenna the latter’s contribution is negligible. From a topological point of is proportional to the radiated field view the voltage across mode itself, while the resonant anthat originated from the -th mode results . gular frequency of the In the linear polarization analysis, only a single resonator cell is taken into account for the field propagation, while the effects of all the other eigen-modes and the feeder reactance are cumulated in a simple reactive LC series term. The analytical expresmight be very complicated for arbitrary geomesion of tries and consequently the circuital parameters as expressed in (5), but the circuital equivalence formulation is still valid even if the expressions are not in closed form, [9]. Hereinafter, when mode its impedance will be recast dealing with a single as

(3) (7) Letting (4) Thus we can introduce the equivalent circuital parameters for each mode as in [2]

(5)

is the normalized frequency response written in where and modal resonance frequency terms of the quality factor . We want to point out that this form is always appropriate to describe the fundamental modal response (within the limits of the cavity model) even if one does not exactly known the , , on the patch geometry and dependence of physical characteristics. In this phase we do not focus on a specific patch geometry, in fact: this approach is valid in a general way. B. Degenerated Modes Ratio

after some manipulations we derive

(6) Equation (6) suggests that the input impedance of the patch is equivalent to the response of a network consisting of an infinite series of parallel RLC resonators as shown in Fig. 1. Each resonant mode is equivalent to a single parallel RLC, exmode which consists only of cept for the fundamental

Let us consider quasi-symmetrical geometry where a single mode is detuned into two orthogonal and overlapping ones. Concerning the fundamental frequency the modal expansion and , from the can be limited to the first three terms: , which represents the whole fundamental detuning, and higher-order modes. The first two modal impedances will be and (with ); we finally consider indicated with their orientation respectively in the and direction. The total input impedance can thus be expressed as

(8)

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where and are modal dispersive functions that can be evaluated by either an analytical or numerical approach. To obtain a far-field CP the two detuned modes must combine with to the each other properly. Neglecting the contribution of radiated fields and assuming the quality factors for the two detuned modes are identical (in fact , is strongly dependent from the geometry of the patch and weakly from the detuned modes), and , we can concentrate on the relation between as defining the normalized impedance ratio function (9)

We can thus focus on the analytical behavior of this function following a graphical procedure similar to that in [10], although with a substantially different approach. From the definition it is the phase difference of becomes evident that the angle of the CP field components, while the magnitude is proportional to the axial ratio. Assuming that and significantly overlap, the Complex Cartesian diagram of is a quasi-circumference: in particular it coincides with a circumference in the range between and , (see Fig. 2). It is easy to realize that always lies on the straight line (varying the ratio )

Fig. 2. The locus in the complex uv plane.

Fig. 3. Comparison of three loci related to three different ratio ! =! .

(10) while

describes the circumference centered in with radius (11)

Moreover, once

is defined as the geometric mean of , holds that

in principle, but rather a function of the and through it a function of the angular frequency. Under the hypothesis of proxand , the pseudo-radius is practically constant in imity of the range of interest, as will be shown in the Appendix. The above-mentioned condition is commonly satisfied when the disand is less then 10% of . Hereinafter the locus tance of will be considered an exact circumference with the radius set as (see Appendix)

and (15) (12)

We can thus observe that two arbitrary frequencies and , whose geometric mean is , are conjugate and lie on the same . This is an important straight line satisfying is the unit geometrical relation, i.e., the locus of circle for a family of loci (13) the real and imaginary For a fixed term composed of parts of can be written as a function of two geometric parameters (14) The locus described in this way resembles a circumference centered in , but the pseudo-radius is not a constant value

In Section III we employ the locus properties to simultaneously impose the polarization and matching conditions. C. Imposing the Polarization Condition From the results of the previous section, it can be evidenced does not intersect the imaginary axis the two modes that if and cannot be in relative quadrature and the condismaller than tion for CP is not verified; this is the case of is tangent to the unity (sub-critical condition in Fig. 3). If and imaginary axis the modes are in quadrature in (critical condition, (13)). Lastly, for , it inter, where and at , sects the axis twice: at , with and the aswhere sociated angular frequencies, see Fig. 3. The two intersections and lay on the same straight line and hold condition, . We now want to find out the relationship be, and . There is a direct way to determine tween the two intersections from geometric inspection and elementary trigonometry. With reference to Fig. 4, the first interaction angle

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that corresponds Let us focus on the particular condition to the critical case. In this case CP is obtained at the geometrical ; the input impedance mean , assuming that can thus be written as (21) The imaginary part of the denominator of

computed in

is (22)

4

4 4 4

Fig. 4. Geometrical relationships of . Triangles B GC and B GC are equal right triangle sharing the GC cathetus. Also B F C and EF C are right triangle sharing the F C cathetus; they are both similar to AEF which is again similar to OEB . From this similitude follows that AEF F CE F CB ' = .

=

=

4

4

2

is twice the angle , while the second mentary to the first. With reference to the triangle

=

which is the pseudo-radius. In order to have the two modes excited with the same magnitude we also impose that . The input impedance can be thus written as (23)

is comple-

(16) (17) the intersection angular freManipulating the expression of quencies and the corresponding angles on the circumference are found to be linked as (see (35) in Appendix)

. Even by putting , being since the system impedance, we obtain the CP condition but the reactance of still remains and because of it, a mismatch of . At this point the parameter comes into play. If we consider the general case of two intersections the modal is impedance at the first angular frequency (24) Remembering (19) the input impedance is

(18) (25) Finally, from (16) and (18) and considering that it follows these coupled identities

,

(19) This tractation demonstrates that a precise relation holds for and pothe degenerated eigen-modes resonant frequencies . Satisfying either the first or the tential CP frequencies second row of (19) respect to (or respect to ) gives the value of the modal frequencies, which in turn lead to an exact quadrature condition at the design frequency. In this equation there is still the degree of freedom represented by the value (and the reciprocal ); additionally the modal resistances and are still unconstrained. Thanks to these degrees of freedom it is possible to impose the matching condition at the (or ) which satisfies the CP same operating frequency condition. This further imposition is discussed in the following.

In this case, unlike the previous one, the quadrature condition does not correspond to a unitary ratio condition; in fact . To obtain a perfect axial ratio, the different mode excitation levels, represented by the modal resistances, . With this conmust balance this value, i.e., depends only on straint, (26) is smaller than unity so it is At the first intersection, a resistive-inductive impedance, and if the higher-order modes (as in the case are dominated by a capacitive reactance of proximity coupling [11]), in order to obtain matching it is sufficient to impose that (27) which leads to

D. Imposing the Impedance Matching Condition From (8) the input impedance can be recast as

(28) (20)

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Fig. 5. Left: A double thin cut slit disc patch. Right: an elliptical slit disc patch. Because of the central symmetry, in each case there are two possible positions for the feed for each kind of RH/LH CP conditions.

Once is selected for a chosen operating CP frequency, the pair and , which guarantees the synthesis of this condition, are determined by the solution of (19)

(29)

If is an inductive reactance like in the case of coaxial is needed. In this case the input feeding, then a impedance is formally the same as in (26) but now is negative so it has a capacitive-resistive form; the matching conditions are formally identical to the previous and the two modal frequencies are solutions of a formally identical equation with the CP frequency at . This scenario is respect to the most common in practice, since higher-order modes and the coaxial probe bring an inductive term to the mode expansion. It can be remarked that geometry capable of two overlapping orthogonal modes can theoretically support two CP frequencies of the same type because of the two intersections, but it is very difficult to excite both of them due to different matching conditions (one inductive and one capacitive). In summary, we can impose the quadrature condition at the design frequency acting on the modal frequencies through (19); and we can impose thus, with the modal resistances the matching condition with (29), while keeping the quadrature condition. This leads to the control of both quadrature and impedance matching conditions in a consistent way. III. CIRCULARLY POLARIZED ANTENNA DESIGN The analysis technique discussed hereinabove can be applied to any patch shape supporting two orthogonal degenerated modes. In this section we want to demonstrate how to employ the proposed approach for the design process (synthesis). There are many shapes suited for the modal degeneration: in addition to the canonic rectangular, circular and triangular shapes [12] are good candidates. Focusing on the disc shape, a mode condirect way to split the radiator fundamental sists in cutting a thin slit through the center [7]. In this way the fields will not be affected significantly, since the fundamental mode exhibits a zero in this area. For the odd mode, oriented toward the cut, the perturbation is minimal if the cut width is surface current should not be afsufficiently narrow, i.e., the fected by this perturbation and the resonant angular frequency

Fig. 6. The conformal coordinate system for the proposed design and its equivalent Cartesian form.

remains almost the unperturbed of the mode. On the contrary, for the even mode, this perturbation is relevant, as surface current is forced to turn around the slot reducing the . Following this reasoning, a second orthogonal cut could be practiced in order to also increase the path of the odd mode. In this way the overall microstrip patch dimension decreases keeping the same central frequency for the two modes (Fig. 5, left). In [13] it was demonstrated that with annular shapes it is possible to enlarge the bandwidth because of the reduced amount of stored energy beneath the patch metallization, i.e., by lowering the quality factor. This effect is independent of the cuts and both modes observe an equivalent circuit affected in the same manner. A convenient way to combine the slit degeneration effect and the annular effect consists in using a central elliptical cut as a detuning element. While the disc radius is the main parameter for the determination of the central frequency, controlling the ellipse axes makes it possible to synthesize the two modal frequencies (Fig. 5, right). We have chosen to adopt this elliptical slitted disc for the design of our CP antenna in order to demonstrate the generality of our approach. For a gesolutions ometry like this there are no known analytical but it is reasonable to assume that the modal eigen-functions of and would still be a separation-variable type (like in the canonic annular elliptical disc), with a radial function deter, and an mining the maximum of the resonance curve angular function modulating the first, moving from one orthogonal axis (reference to one of the two modes) to the other. The radial function grows in absolute value moving toward the peripheral and vanishes at the center (like Bessel or radial Mathieu functions) while the angular one has a harmonic-like behavior (like trigonometric or angular Mathieu functions). Even if we do not know the exact form of the Green function, the equivalent circuital parameters , , , and are still functions of the patch characteristics and of the probe position and can be analyzed with a commercial electromagnetic engine solver [9]. With the aid of a full-wave EM simulator tool, it is possible to characterize a geometry like this, and thus obtain the functions which relate the patch characteristics to circuital parameters in a straightforward manner. Fig. 6 illustrates the conformal coordinate system for the proposed design. The radial coordinate is expressed through a normalized parameter called , while the angle is simply measured respect to the axis. To obtain a

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Fig. 7. Parametric simulation in function of feed position. Variation of the real and imaginary part of input impedance with (left) the radial parameter  and (right) with the angle  .

Fig. 8. Contour plot of R (; ) (left) and R (;  ) (right), defined over the patch domain. Dimensions expressed in mm. The dashed zone correspond to the minimum and maximum analyzed  . Outside them the plot is based on an extrapolation.

significant amount of data, a set of parametric simulations were executed over a grid of four points for (0.125, 0.375, 0.625, ; 0.875) and ten values for since the center-symmetry of the radiator this mesh of points characterizes the entire patch. To fix the idea, the disc radius is taken as 16 mm, (corresponding to a fundamental resonance around 2.45 GHz for a canonic disk), the internal ellipse axes are 5 mm 2 mm, while the substrate has a square shape with a side length of 45 mm. A full-wave analysis was executed for each mesh point. Since the central frequency of 2.45 GHz, the , for a total of 33 frequency range of analysis is points. In Fig. 7 (left) the behavior of the real part of the input impedance versus the angular variable is depicted. For only a peak is visible. This is the peak of the modal function insisting on the first modal axis. For an angular value about the middle of the range, both the peaks are visible. This is the zone where the combination of the modes leads to the best CP. When approaches the upper extreme we are near to the axis of the other modal function. The dual parametric simulation is depicted in Fig. 7 (right). The angular variable is now fixed at half of the sweep, while varies from the minimum of 0.125 to the maximum of 0.875. In this case the ratio of the peaks is almost the same over the entire range: both the peaks monotonically grow at almost the same rate. With the complete set of simulations the reconstruction of

, , , , , the three couples of circuital parameters as a function of the pin position (expressed by ) on the patch was finally possible. It is also possible to extract the , representing the probe reactance of the coaxial quantity probe feed (see (23)) Since the relative slowness of full-wave analysis it is extremely useful to infer the circuital parameters over the entire surface of the patch with the minimum amount of simulation grid-points, by means of an interpolation algorithm, for instance those implemented in commercial tools. The chosen parametric grid is in fact a compromise among accuracy and elaboration time. It is clear that simulations can be repeated for one other set of variables, for example for other axes dimension, to characterize the response of the patch versus all the possible geometric characteristic. and In Fig. 8 is reported the surface plot of the parameters as a function of the position on the patch. The double harmonic behavior in the direction and direc, and , in function is evidenced. The behavior of tion of the geometry of the patch can be derived in a similar way. With all the parameters extracted a synoptic view, like the one depicted in Fig. 9, it is very easy to handle. Apart from the obvious input impedance (a), and corresponding reflection coplot efficient plot (c), the most interesting plots is the and in complex form (b) and in dB/phase form (d). To demonstrate the effective advantages of using the proposed approach, Fig. 10 shows the comparison of a quantity obtained from our model versus the correspondent quantity given by a full-wave simulations, namely the commercial tool HFSS. plotted in dB form, while The first is the modulus of over . It the second is the numerically computed ratio of is opportune to remark that the beta modulus is obtained from the circuital parameters, which are in turn obtained by relaxed is obtained simulating and fast simulations. The ratio the same design with two increasing levels of mesh-refinement, both plotted in figure. The correspondence of our circuit-based model prediction with respect to the reliable field simulation of HFSS is satisfying. Once the modal functions are characterized in numerical form, the synthesis process is done satisfying the CP and matching conditions as explained in the previous section: the solution of (28) and (29) give the circuital parameter

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= 16 0 mm

385

= 5 3 mm

= 2 3 mm = 0 3875,  = 3:5). (a) Real and (!) versus frequency. The

Fig. 9. Synoptic view of a specific configuration for the proposed patch (A : ,L : ,L : , : imaginary part of Z . (b) Complex plot of ! and B ! . (c) Reflection coefficient in dB. (d) Modulus (in dB) and phase of B asterisks represent the limits of 10 dB return loss zone.

( )

Fig. 10. Analytical B grids.

( )

(!) in comparison with numerical Ey/Ex for two mesh-

values which in turn give the design constraint through the inversion of the (numerical) modal functions. In general the design procedure can be divided in two phases and described by the following steps. condi1) Proper geometry of the patch: prerequisites for tions. , • Starting from an estimation of the probe reactance value is obtained by using (28); the • Having estimated the quality factor , the two frequenare determined by (29); cies • Geometric parameters like the disc radius and the ellipse axes are chosen in order to synthesize the required frequencies for the modal functions, eventually with the aid of an interactive tuning procedure based on a direct

feedback (for example refreshing the synoptic view of Fig. 9). 2) Proper feed position: satisfaction of the axial ratio and impedance matching conditions. , the exact axial ratio at the • Imposing operating frequency is obtained. Changing the angular variable causes the variation of the modal resistance ratio, as seen in Fig. 7; • The radial distance from the center is proportional to and keeping the ratio almost constant (see both Fig. 7); adjusting this parameter, can be made equal as described in (28), thus achieving the matching to condition. At this point we can assess this final remark: normally the ground plane is chosen much larger than the patch dimension in order to improve directivity performance and to simplify the analysis. In addition, perfect CP can be achieved exactly only in the broadside direction since axial ratio deteriorates toward horizon. Short ground plane can slightly influence the design parameters [14], although a systematic investigation of this point is beyond the scope of this paper. Moreover, it was highlighted that the employment of short regular polygon shaped ground can help to maintain a good polarization condition for a wider angle range [15] so, also the possible asymmetry of the ground influences the project parameters. Anyway, the influence of actual ground dimensions and characteristics are included in the relations between geometry and circuital parameters, so they are implicitly considered. IV. EXPERIMENTAL VALIDATION The design procedure described in the previous section was adopted to design a prototype on an FR4 substrate operating at

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Fig. 11. The RH prototype of the elliptical slit disc antenna on square ground (left) and its quoted design (right), quotes are in millimeter; substrate: FR4, 1.6 mm thickness, 17  copper metallization thickness.

m

Fig. 14. Measured CP radiation pattern for three frequencies within the operative bandwidth.

Fig. 12. Comparison between measured and simulated reflection coefficient for an RH antenna.

Fig. 15. Measured axial ratio magnitude and modulus of Beta Locus versus frequency, both expressed in dB. A good consistency between measurement and theory is revealed.

Fig. 13. Measured input impedance on the Smith chart for the LH/RH cases. The circle delimits the 10 dB return loss zone.

2.45 GHz, with an input impedance of 50 , and suitable for WLAN applications. The resulting antenna layout and its dimensions, along with the prototype photograph, are shown in Fig. 11. Although the prototype was fabricated with the two vias pins, to drive the LH and RH polarizations respectively, the prototype in the picture adopts the RH polarization one. In Fig. 12 simulated and measured reflection coefficients are presented for the RH prototype, the achieved impedance matching at center frequency is better than 18 dB within a 10 dB RL bandwidth in excess of 130 MHz. The wide bandwidth exhibited by the prototype is due to the presence of the two resonant modes which,

being separated by several tenths of MHz, determines the double minimum in the matching frequency behavior. The same figure shows the results of a full-wave analysis, made with the reliable HFSS electromagnetic CAD, that shows good accuracy with the , for two experimental data. Fig. 13 illustrates the measured prototypes operating in LH and RH respectively. The two traces exhibit a cusp which represents the reflection coefficient and the corresponding frequency where the CP conditions are achieved, the difference between the traces is due to manufacturing inaccuracy. From the same figure a slight inductive residual is observed in the input impedance, although the above procedure of parameter compensation was adopted. The inductance is slightly different for the two prototypes since in practice it is impossible to exactly compensate the reactance of the coaxial probe which also depends on the coaxial probe mounting. Nevertheless the return loss at the center frequency is very good even with this residual, with a resistive part equal to nominal 50 ohm for both cases. In Fig. 14 the measured radiation patterns at are demonstrated in the azimuthal plane for 2.42–2.45–2.48 GHz ; in the measurement a circularly co-polarized test antenna was used. From the measurement a maximum and . The gain of 3.85 dB is achieved for gain value is affected by the sub-optimum low-cost substrate

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Fig. 16. Axial ratio phase versus frequency. Fig. 18. Realized gain of the prototype as function of the frequency in broadside direction.

range 2.1 GHz to 2.8 GHz. The experimental data in terms of are reported in Fig. 17 for the entire frequency range. As in the co-polarized link, it is evident from the figure that the S21 parameter reaches a broad maximum around 2.45 GHz as expected. Moving to the cross-polarized link, a deep glitch was registered with a minimum 31 dB lower than the value reached in the co-polarized link; the maximum rejection is reached at the frequency of 2.45 GHz. V. CONCLUSIONS Fig. 17. Link between prototype antennas in both co-polarized (diamond) and cross-polarized links (x).

adopted for the prototype manufacturing. In Fig. 15 the measured axial ratio is presented within the bandwidth of interest. It exhibits the perfect ratio between the even and odd modes at the frequency of 2.45 GHz, while the axial ratio is maintained below 5 dB from 2.42 GHz up to 2.48 GHz, demonstrating a bandwidth comparably higher than the previously reported design [4]. It is also remarkable that there is a substantial consistency with the modulus of beta locus expressed in dB form. Fig. 16 shows the relative phase between the two orthogonal modes, here the 87 degrees maximum is reached at 2.43 GHz while it is maintained within an error of 5 degrees between 2.41–2.45 GHz. Also in this case, measured phase and show a very good accord. The final angular behavior of evaluation of the design approach was conducted comparing the transmission links between two faced antennas both operating in broadside direction and placed 2.5 meters away from a laboratory anechoic chamber. Two set-ups were considered: the first, which acts as a baseline, is composed of two co-polarized (LH to LH) antennas, whereas the second involves two cross-polarized (LH to RH) antennas. As it is well known, due to the polarization diversity the expected transmission link should ideally vanish in the cross-polarized link situation; the isolation is an indirect measure of the polarization quality, whereas with the co-polarized link the ratio between the received and transmitted power levels is only proportional to the distance and the antenna gain versus frequency. The parameter selected to compare the two links was the transmission -parameter measured by a calibrated vector network analyzer employed in the frequency

An analytical method for studying circularly polarized microstrip antennas has been discussed. This method is applicable to both synthesis and analysis stage and it is based on a simple yet effective circuital equivalence of microstrip patch radiators. The proposed method enables the tuning of the circular polarization condition and the matching condition consistently. A prototype antenna based on the proposed method was designed, at the nominal center frequency of 2.45 GHz, fabricated and tested. The measurements showed a maximum gain of 3.87 dB and an unit axial ratio magnitude at the frequency of 2.45 GHz, where the phase error is lower than 3 degrees. The axial ratio was observed within 5 dB magnitude and 5 degrees phase in more than 30 MHz bandwidth around the design center frequency. The design method permitted to achieve an inherent 18 dB return loss at the central frequency and better than 10 dB within 130 MHz bandwidth. The CP-and-matching goal was accomplished without external circuitry: the pins for LH and RH excitation are directly available for connection with a coaxial cable. A single-poledouble-throw switch could be eventually integrated to control the polarization kind. APPENDIX In this Appendix we want to demonstrate that the Beta locus is practically coincident with a circumference in all the cases of interest. First of all we expand the Cartesian form of

(30)

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The relationship between and the primary circuital parameis rather simple. From (30) and (14) follows: ters (31) Isolating the term

and confronting with

results as (32)

The resulting function is fairly constant in the range from to , which is the range of interest for the application of CP method; thus we can write (33) It is remarkable that (and so ) is equal to multiplied and by the ratio between the arithmetic and the geometric. if are very close the two means are practically coincident; this suggests an interesting way to reformulate (32) (34) The pseudo-radius exhibits a minimum in and it is always , are slightly bigger than . To point out concave, so and is less a quantitative data, if the relative distance of then 10% then the geometric and arithmetic means are identical up to the third decimal. For the angular variable the reasoning is similar. With the aid of the trigonometric half angle identities and from the rela(31) tionships of the real and imaginary part of (35) Because of the periodicity of the tangent function, this equality is valid for , , but for the geometric definition of the variables the only valid choice for is

[4] B. Kim, B. Pan, S. Nikolaou, Y.-S. Kim, J. Papapolymerou, and M. M. Tentzeris, “A novel single-feed circular microstrip antenna with reconfigurable polarization capability,” IEEE Trans. Antennas Propag., vol. 56, no. 3, pp. 630–638, 2008. [5] H. Lee and W. Chen, Advances in Microstrip and Printed Antenna, New York: J. New York: Wiley, 1997. [6] J. James, C. Wood, and P. Hall, Microstrip Antenna Theory and Design. New York: IET, 1986. [7] J. Barbero, H. Lazo, F. Municio, and M. TeDeCe, “Model for the patch radiator with a perturbation to achieve circular polarization,” in Proc. Inst. Elect. Eng. Colloq. on Recent Developments in Microstrip Antennas, Feb. 1993, pp. 6/1–6/4. [8] Ansoft [Online]. Available: http://www.ansoft.com/products/hf/hfss/ [9] D. Neog, S. Pattnaik, D. Panda, S. Devi, B. Khuntia, and M. Dutta, “Design of a wideband microstrip antenna and the use of artificial neural networks in parameter calculation,” IEEE Antennas Propag. Mag., vol. 47, no. 3, pp. 60–65, 2005. [10] Y. Lo, B. Engst, and R. Lee, “Simple design formulas for circularly polarised microstrip antennas,” Proc. Inst. Elect. Eng. H, Microw., Antennas Propag., vol. 135, no. 3, pp. 213–215, 1988. [11] H. Iwasaki, “A circularly polarized small-size microstrip antenna with a crosss lot,” IEEE Trans. Antennas Propag., vol. 44, no. 10, pp. 1399–1401, 1996. [12] K. Wong, Compact and Broadband Microstrip Antennas. New York: Wiley, 2002. [13] A. Bhattacharyya and L. Shafai, “A wider band microstrip antenna for circular polarization,” IEEE Trans. Antennas Propag., vol. 36, no. 2, pp. 157–163, 1988. [14] A. Bhattacharyya, “Effects of finite ground plane on the radiation characteristics of a circular patch antenna,” IEEE Trans. Antennas Propag., vol. 38, no. 2, pp. 152–159, 1990. [15] B. Zheng and Z. Shen, “Effect of a finite ground plane on circularly polarized microstrip antennas,” in Proc. IEEE Antennas and Propagation Society Int. Symp., 2005, vol. 2, pp. 238–241. Stefano Maddio was born in Florence, Italy, on September 3, 1978. He received the Laurea degree in electronic engineering (cum laude) and the Ph.D. degree from the University of Florence, in 2005 and 2009, respectively. Afterwards, he joined the staff of the Microelectronic Laboratory, Department of Electronics and Telecommunications, University of Florence. His research activities cover the analysis and design of radiative system for microelectronics, especially in the field of smart antenna systems for wireless applications, with particular emphasis on the challenging issues of wireless positioning. His scientific interests cover also the area of numerical techniques for electromagnetic propagation.

(36) The pseudo-radius variation can be considered negligible in the range in the case of strong overlapping, and so it can be . assumed equal to the average This fact justifies our hypothesis. ACKNOWLEDGMENT The authors want to thank S. Maurri, technical assistant at the Department of Electronic and Telecommunications of the University of Florence, and G. Maddio for their assistance during the prototyping stage. REFERENCES [1] T. Rappaport and D. Hawbaker, “Wideband microwave propagation parameters using circular and linear polarized antennas for indoor wireless channels,” IEEE Trans. Commun., vol. 40, no. 2, pp. 240–245, 1992. [2] K. Carver and J. Mink, “Microstrip antenna technology,” IEEE Trans. Antennas Propag., vol. 29, no. 1, pp. 2–24, 1981. [3] R. Garg, Microstrip Antenna Design Handbook. Boston, MA: Artech House, 2001.

Alessandro Cidronali (M’89) received the Laurea and Ph.D. degrees in electronics engineering from the University of Florence, Florence, Italy, in 1992 and 1998, respectively. In 1993, he joined the Department of Electronics Engineering, University of Florence, where he became an Assistant Professor in 1999; in 2010 he qualified as Associate Professor. He teaches courses on electron devices and integrated microwave circuits. From 1999 to 2003, he was a Visiting Researcher with the Motorola Physics Science Research Laboratory. From 2002 to 2005, he was a Guest Researcher with the Non-Linear Device Characterization Group, Electromagnetic Division, National Institute of Standards and Technology (NIST). Under the frame of the ISTEUFP6 Network TARGET (IST-1-507893-NOE), he served as Workpackage Leader for the transmitters modeling/architectures for wireless broadband access work packages. His research activities cover the study of analysis and synthesis methods for nonlinear microwave circuits, the design of broadband monolithic microwave integrated circuits (MMICs) and the development of computer-aided design (CAD) and numerical modeling for microwave devices and circuits. Dr. Cidronali was recipient of the Best Paper Award presented at the 61st ARFTG Conference. From 2004 to 2006, he was an Associate Editor for the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES.

MADDIO et al.: A NEW DESIGN METHOD FOR SINGLE-FEED CP MICROSTRIP ANTENNA

Gianfranco Manes (M’01–SM’02) is a Full Professor with the Department of Electronics and Telecommunications, University of Florence, Florence, Italy. Since the early stages of his career, he has been involved in the field of surface acoustic wave (SAW) technology for RADAR signal processing and electronics countermeasure applications. His major contributions have been in the introduction of novel FIR synthesis techniques, fast analog spectrum analysis configurations, and frequency-hopping waveform synthesis. Since the early 1980s, he has been active in the field of microwave modeling and design. In the early 1990s, he founded and currently leads the Microelectronics Laboratory, University of Florence, where he is committed to research in the field of microwave devices. In 1982, he was committed to build up a facility for the design and production of SAW and microwave integrated circuit (MIC)/MMIC devices as a subsidiary

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of the radar company SMA SpA, Florence, Italy. In 1984, the facility became the stand-alone privately owned microwave company, Micrel SpA, operating in the field of defense electronics and space communications. From 1996 to 2000, he was involved in projects in the field of information technology applied to the cultural heritage, and was invited to orientation meetings and advisory panels for the Commission. He was founder and is currently President of MIDRA, a research consortium between the University of Florence and Motorola Inc. He is Director of the Italian Ph.D. School in Electronics. In November 2000, he was appointed Deputy Rector for the information system of the University of Florence. His current research interest is in the field of resonant inter-band tunneling diode (RITD) devices for microwave applications in a scientific collaboration with the group at the Physical Science Research Laboratories, Motorola Inc., Tempe, AZ. Dr. Manes is a member of the Board of the Italian Electronics Society.

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Compact Band-Rejected Ultrawideband Slot Antennas Inserting With =2 and =4 Resonators Zhi-An Zheng, Qing-Xin Chu, Member, IEEE, and Zhi-Hong Tu

Abstract—Two compact ultrawideband (UWB) slot antennas with band-rejected characteristics are proposed for UWB communication applications. The entire size of both antennas is as small as 15 mm 22 mm 0 8 mm, which is smaller than most of the proposed UWB antennas. The radiator is a stepped slot excited by a 50 feed line. By etching a half-wavelength and a quarter-wavelength slot from the ground plane of UWB slot antennas, respectively, two UWB antennas with band-notched characteristics can be obtained. The potential interference between UWB and WLAN/WiMAX can be reduced. Omnidirectional radiation patterns and constant gain can be observed in the pass bands. Time-domain characteristics of both proposed antennas are analyzed under different conditions. It demonstrates that the time-domain characteristics of the antennas are good enough for UWB applications. The results show that proposed antennas are very suitable for various portable UWB systems. Index Terms—Band-rejected UWB antenna, compact size, slot antenna, time-domain characteristics.

I. INTRODUCTION

S

INCE the Federal Communications Commission (FCC)’s allocation of the frequency band 3.1–10.6 GHz for commercial use [1], ultrawideband (UWB) system has attracted significant research power in the recent years. As one of main issue of UWB systems, UWB antenna has received increased attention [2]. Most of the proposed antennas have UWB performances of the impedance matching, radiation stability and constant gain. Usually, the size of such printed antenna with broad impedance bandwidth is about 40 mm 50 mm to 25 mm 25 mm [3]–[5]. As the requirement of reducing the size of the UWB communication system, especially those portable systems, the miniaturization of UWB antennas becomes a significant topic. Recently, some UWB antennas with more compact size have been proposed [6]–[9]. The size of the compact UWB antenna is about 13 mm 30 mm in [6], [7]. The antenna with smaller size was proposed, but the 10 dB impedance bandwidth is only from 3.1 to 5 GHz [8]. On the other hand, there are some other wireless communication systems in the UWB band, such as wireless local-area network (WLAN) IEEE 802.11a operating in the 5.15–5.825 GHz band. Besides WLAN, in some Europe and Asia countries, world interoperability for microwave access (WiMAX) service from 3.3 to 3.8 GHz also operates in Manuscript received March 19, 2010; revised June 24, 2010; accepted June 28, 2010. Date of publication December 03, 2010; date of current version February 02, 2011. This work was supported in part by the Science Fund of China (U0635004) and in part by the Science Fund of Guangdong (No. 60571056). The authors are with the School of Electronic and Information Engineering, South China University of Technology, Guangzhou 510641, China (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TAP.2010.2096399

the UWB band. As a result, the potential interference between UWB and the two wireless communication systems should be minimized. However, using filters will increase the complexity of the UWB systems. Therefore, various UWB antennas with one or dual even triple notched bands have been proposed in recent years [10]–[13]. The notched bands are mainly implemented by etching slots on the radiator or adding some stubs around the radiator or feed line. The lengths of those etched slots or additional stubs are about quarter-wavelength or halfwavelength corresponding to the designed notched frequencies [12]–[16]. Moreover, the notched band can be achieved by controlling the resonant frequency of the antenna [17]. It is well known that the first resonant frequency mainly depends on the size of the antenna [18]. With smaller dimensions, the first resonant frequency of the UWB antenna becomes higher. Therefore, to the same type antennas, if the start frequency of the antenna band is bigger than 3.1 GHz, the size of the antenna can be designed to be smaller than those whose band starts from 3.1 GHz. To miniaturize the antenna size and the potential interferences between UWB system and the two narrowband systems, the impedance bandwidth of UWB antenna is suggested to start from 3.8 GHz, but not 3.1 GHz as normal. The band of 3.1–3.3 GHz can be negligible compared to the whole UWB band. Therefore, the dimension of the antenna can be reduced. As the UWB antennas are employed to transmit/receive time domain transient pulse signal, the key is whether the time-domain characteristics will be affected or not. Some work has been done by authors in the study of time-domain characteristics of UWB antennas [19], [20]. Based on the previous work, the time-domain characteristics of proposed antennas are analyzed. In this paper, two band-notched UWB slot antennas with are proposed. They the size of are very small in dimension and simple in structure. With such a compact dimension, the 10 dB impedance bandwidth of the antenna covers the band of 3.8 to more than 11 GHz. A Z-shaped slot and a split rectangle ring slot were etched nearby the slot radiator in the ground plane, respectively. Both of them can achieve a notched band in frequency range from 5.15–5.825 GHz, and the notched band can be adjusted easily by varying the width and length of the slot. Good agreements are observed between the simulation and measurement. Moreover, the H-plane radiation patterns are nearly omnidirectional as the frequency increases. By analyzing the correlation factor and pulse wide stretch ratio, the time-domain characteristics of proposed antennas are demonstrated to be good enough. The results show that the proposed antennas are very suitable for the use of portable UWB systems.

0018-926X/$26.00 © 2010 IEEE

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Fig. 1. Geometry of the UWB slot antenna (antenna 1).

AND

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Fig. 3. Geometry of band-notched UWB slot antenna with half-wavelength slot (antenna 2).

The required notched band is 5.15–5.825 GHz, so the designed resonant frequency is set to be about 5.5 GHz. The wavelength can be calculated by

(1)

Fig. 2. Measured and simulated reflection coefficients of antenna 1.

II. ANTENNAS DESIGN Generally, the radiators with gradient, tapered or cambered shapes can implement wide-band function [12], [21]. Similarly, the slot antennas with those shapes can achieve wide-band characteristics as well [5], [22]. Fig. 1 is the geometry of a UWB and antenna printed on a substrate with permittivity thickness of 0.8 mm. A stepped slot in the ground plane is excited by a 50 microstrip line. The stepped radiator can result in a transition from one resonant mode to another. By using Ansoft’s High Frequency Structure Simulation (HFSS) software to optimize the parameters, good impedance match over a broad frequency range can be obtained. As shown in Fig. 2, the measured and simulated bandwidth of the antenna is from 3.8 to more than 11 GHz. To achieve band-notched characteristic, slots have been etched on UWB antenna patch as discussed in [12], [13]. Each slot acts as a resonator, and the resonant frequency mainly depends on the length of the slot. The length of the slot is about a half or a quarter of the wavelength corresponding to the resonant frequency [14].

where is the speed of the light, is the effective dielectric constant, and is the designed resonant frequency. Two slots were etched from the previous UWB slot antenna to implement the notched band, respectively. Both of the two slots can achieve the required notched band at 5.5 GHz. But the lengths and locations of the slots are different. If the slot is placed in the radiator, the length is about a half of the wavelength. While one end of the slot is open, the length is only about a quarter of the wavelength. The geometries of the antennas with band-rejected function are shown in Figs. 3, 4. The half-wavelength slot is a split rectangle ring with the two ends surround the radiating slot to enhance the coupling between them, and the radiating slot is in the middle of can the opening. The length of the split rectangle ring be deduced by

(2) The quarter-wavelength slot is a Z-shaped slot, and one end of can be the slot is open. The length of the Z-shaped slot deduced by

(3)

The optimized parameters obtained in Ansoft HFSS are as follows: , , , , , , , , , , , , , , ,

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Fig. 6. Measured and simulated reflection coefficients of antenna 2 and antenna 1. Fig. 4. Geometry of band-notched UWB slot antenna with quarter-wavelength slot (antenna 3).

Fig. 5. Photograph of fabricated antennas (front view and back views of antenna 1, antenna 2, antenna 3).

,

,

,

,

. III. RESULTS AND DISCUSSION For simplicity, the antennas in Fig. 1, Fig. 3 and Fig. 4 are named antenna 1, antenna 2, and antenna 3, respectively. The three antennas were fabricated and measured using Agilent N5230A vector network analyzer (VNA). The photographs of the three fabricated antennas are shown in Fig. 5. The designs of antenna 2 and antenna 3 are based on antenna 1. A. Frequency-Domain Characteristics of Antenna 2 The measured and simulated reflection coefficients of antenna 2 are plotted in Fig. 6. The simulated bandwidth is 3.8–5.15 and 5.9–11 GHz, and the measured result is 3.7–5.2 and 6.5–11 GHz. The measurement and simulations match well. Because of the machining accuracy, the centre frequency of the notched band is shifted a little, and the measured bandwidth is a little bigger than simulation. The simulated result of antenna 1 is added for comparison. Fig. 7 depicts the simulated reflection and . The centre coefficient of the antenna with different frequency of the notched band can be changed by adjusting the total length of the split rectangle ring slot. Increasing the length of the slot, the centre frequency of the notched band will accordingly decrease. The measured E-plane (XZ) and H-plane (YZ) radiation patterns at 4.2, 6.8 and 9.6 GHz are normalized and illustrated

Fig. 7. Simulated reflection coefficients of antenna 2 with different L and L .

in Fig. 8. Each radiation pattern is normalized to itself. The H-plane radiation patterns are observed to be nearly omnidirectional over the whole UWB frequency range but slightly deteriorated at high frequencies. The antenna was measured in the outdoor test system due to the limit of condition. As a result, the radiation patterns in some orientations are not good enough. The measured and simulated peak gains of the proposed antenna are plotted in Fig. 9. The antenna gain is nearly constant with sharp decrease occurs in the designed notched band. Moreover, the efficiency of the antenna has been measured in 4–6 GHz [23], [24], and the measured and simulated efficiencies are superimposed in the bottom right corner of Fig. 9. It can be observed that both the measurement and simulation decrease sharply in the frequency of about 5.5 GH, and the simulated results are bigger than the measurements. It is because of the dielectric loss of the substrate and the conductor loss, and the effect of the SMA connector and the soldering will influence the measurement as well. Furthermore, the variation trends of the measurement and simulation are the same. B. Frequency-Domain Characteristics of Antenna 3 The measured and simulated reflection coefficients of the antenna 3 are presented in Fig. 10. The simulated bandwidth is 3.76–5.12 and 5.84–11 GHz, and the measured result is

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Fig. 10. Measured and simulated reflection coefficients of antenna 3 and antenna 1.

Fig. 8. Normalized measured radiation patterns of antenna 2. (a) E-plane (X-Z plane); (b) H-plane (Y-Z plane).

Fig. 9. Measured and simulated peak gains of antenna 2. (measured and simulated efficiencies in the bottom right corner).

3.87–5.18 and 5.97–11 GHz. The measurement and simulations match well. Due to the machining accuracy, the first resonant frequency is shifted a little. The dielectric constant and dissipation factor are not stable with increasing frequency, so there are some differences between simulation and measurement in high frequency. The simulation of antenna 1 is added for comparison as well. The simulated reflection coefficient of

Fig. 11. Simulated reflection coefficients of antenna 3 with different L .

is shown in Fig. 11. The centre antenna 3 with different frequency of the notched band can be varied easily by adjusting the total length of the Z-shaped slot. With a shorter slot, the centre frequency of the notched band will correspondingly increase. The measured E-plane (XZ) and H-plane (YZ) radiation patterns at 4.1, 6.8 and 10 GHz are normalized and plotted in Fig. 12. It is noticed that the omnidirectivity of the H-plane radiation patterns is generally good but will be deteriorated as frequency increases. The measured and simulated peak gains are shown in Fig. 13. Sharp gain decrease occurs in the designed rejected-band as expected. Besides, the measured and simulated efficiencies of antenna 3 among 4–6 GHz are superimposed in the bottom right corner of Fig. 13. The results are similar to that of antenna 2. IV. TIME-DOMAIN CHARACTERISTICS OF PROPOSED ANTENNAS The UWB antennas are employed to transmit/receive timedomain transient pulse signals, so the time-domain characteristics of the UWB antennas are very important. The signal propagation characteristics of the UWB antennas have attached much attention [25], [26]. With excellent time-domain characteristics, little distortion will be introduced by the UWB antenna when

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Fig. 14. (a) Antenna input signal, (b) Fourier transform of the shrunken input signal. Fig. 12. Normalized measured radiation patterns of antenna 3. (a) E-plane (X-Z plane); (b) H-plane (Y-Z plane).

[27], [28]. This UWB signal is the 5th-derivative of the Gaussian pulse and is given by (4)

(4)

Fig. 13. Measured and simulated peak gains of antenna 3. (measured and simulated efficiencies in the bottom right corner).

transmitting or receiving pulse signals. Authors have done some work on it [19], [20]. In this paper, the time-domain characteristics analyses were carried out with CST microwave studio software. To satisfy the FCC spectral mask for indoor systems, the antenna is assumed to be excited by the UWB signal suggested in

Here, is a constant that can be chosen to comply with peak power spectral that FCC will permit and has to be 51 ps to ensure that the shape of the spectrum complies with the FCC spectral mask [26]. The input signal and the Fourier transform of the shrunken input signal are shown in Fig. 14. and pulse width stretch ratio In [20], correlation factor (SR) have been proposed to evaluate the time-domain characteristics of the UWB antennas. and is defined by Correlation factor of signals

(5) where is a delay which is varied to make numerator in (5) a maximum.

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For a signal , let the normalized cumulative energy function Es(t) be defined by

(6)

Then, the pulse width

for 90% energy capture is given by

(7) So the pulse width stretch ratio can be defined by the ratio of the width of the receiving antenna signal to the width of the transmitting antenna signal. That is

(8) The antenna transient performance parameters over the whole solid angles are investigated in the CST simulation. The analyses were done using far-field linear polarized electrical field probe over the antenna solid angle [29]. The probes are set to be 600 mm far away from the antenna. The analyzed antenna is excited with the signal shown in Fig. 14, then the probes around can receive signals. The correlation factors and SR values between the probe signals and the input signal are calculated and figured in Fig. 15. The correlation factors of the three antennas range from 0.8316 to 0.9899 in the E-plane and H-plane. The SR values of antenna 1 are stable and very close to 1, but the SR values of the other two antennas are not so good. It is because that the impedance mismatching in the notched band will introduce ringing distortion. It would be interesting to investigate the performance of pulse transmission between two UWB antenna. In the modeling, two identical antennas are oppositely positioned to each other. The distance between them is assumed to be 600 mm, which is approximately 8 wavelengths at the lower frequency of the considered band of operation. One is excited by a time-domain transient pulse to act as the transmitter, then the other can receive a time-domain signal with a delay of about 2 ns, which is the necessary time for the signal transmission. To analyze the time-domain characteristics of the antennas adequately, each antenna pair is studied in three cases, which are face to face, side by side and back to back. The received signals of the three antennas in different cases are presented in Fig. 16. The calculated correlation factors and pulse width stretch ratios are summarized in Table I and Table II, respectively. It can be observed that, the first period waveforms of the received signals are very similar to the input signal, and almost all the values of the correlation factors are bigger than 0.84. Only the correlation factor of antenna 2 pair is 0.7519 in the case of side by side. Therefore, during the UWB pulse transmission, little distortion will be introduced to the received signal compared with the input signal. The cases of face to face and back to back are and 0 , respectively, and the case corresponding to

Fig. 15. The correlation factors and SR values between the probe signals and the input signal. (a) Correlation factor; (b) pulse width stretch ratio.

of side by side is corresponding to . Taking the case of face to face for example. It can be observed from Table I that, the correlation factor of antenna 1 is bigger than that of the other two antennas. In Fig. 15(a), the correlation factor of antenna 1 is also bigger than that of the other two antennas when theta equals to 180 . Moreover, the situations of the other two cases are similar. They demonstrate that Fig. 15(a) and Table I are related to each other. Taking a good look at Fig. 16, it can be found that antenna 2 and antenna 3 will introduce ringing distortion, but antenna 1 will not. It is because of the impedance mismatching at the notched band. Not all the energy is concentrated in the vicinity of the peak and a little noise energy deviated from the peak introduces the ringing distortion on received signals. Consequently, the SR values of antenna 1 are all very close to 1, and the SR values of the other two antennas are not so good. It also can be observed from Fig. 15(b) that only the SR values of antenna 1 are closer to 1. It demonstrates that Fig. 15(b) and Table II are interrelated. It can be found that the attenuation of the ringing is fast. Since the time interval between two contiguous input pulses is big enough, the ringing distortion have nearly no harmful effect on the next received pulse. It demonstrates that the time-domain characteristics of both proposed antennas are good enough for the transmitting/receiving of the pulse signals.

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TABLE II PULSE WIDTH STRETCH RATIO (SR) BETWEEN INPUT AND OUTPUT SIGNALS OF THE ANTENNA PAIRS IN DIFFERENT CASES

very compact and simple. The radiation patterns in the H-plane are nearly omnidirectional over the entire UWB band. The gain is stable with a sharp decrease in the designed notched band. The interference of WLAN & WiMAX can be reduced. Moreover, the time-domain characteristics of both proposed antennas are very good. Consequently, the proposed antennas are expected to be a good candidate in various portable UWB systems. ACKNOWLEDGMENT Z.-A. Zheng would like to thank Dr. H.-Q. Ma for his help in this study. REFERENCES

Fig. 16. The received signals of antennas 1–3 in different cases, (a) face to face. (b) side by side. (c) back to back. Note that the waveform are shifted in Y-Axis for clearer distinction.

TABLE I CORRELATION FACTORS BETWEEN INPUT AND OUTPUT SIGNALS OF THE ANTENNA PAIRS IN DIFFERENT CASES

V. CONCLUSION In this paper, two compact UWB slot antennas with bandnotched characteristics have been proposed. The antennas are

[1] Revision of Part 15 of the Commission’s Rule Regarding Ultra-Wideband Transmission System FCC 0.2-48 FCC, 2002. [2] Q. Wu, R. Jin, J. Geng, and M. Ding, “Compact CPW-fed quasi-circular monopole with very wide bandwidth,” Electron. Lett., vol. 43, no. 2, pp. 69–70, Jan. 2007. [3] Z. N. Chen, T. S. P. See, and X. M. Qing, “Small printed ultrawideband antenna with reduced ground plane effect,” IEEE Trans. Antennas Propag., vol. 55, no. 2, pp. 381–388, Feb. 2007. [4] M. John and M. J. Ammann, “Antenna optimization with a computationally efficient multiobjective evolutionary algorithm,” IEEE Trans. Antennas Propag., vol. 57, no. 1, pp. 260–263, Jan. 2009. [5] A. Mehdipour, K. M. Aghdam, R. F. Dana, and M. R. K. Khatib, “A novel coplanar waveguide-fed slot antenna for ultrawideband applications,” IEEE Trans. Antennas Propag., vol. 56, no. 12, pp. 3857–3862, Dec. 2008. [6] M. Gopikrishna, D. D. Krishna, C. K. Aanandan, P. Mohanan, and K. Vasudevan, “Compact linear tapered slot antenna for UWB applications,” Electron. Lett., vol. 44, pp. 1174–1175, 2008. [7] Z. A. Zheng and Q. X. Chu, “CPW-fed ultra-wideband antenna with compact size,” Electron. Lett., vol. 45, no. 12, pp. 593–594, Jun. 2009. [8] T. S. P. See and Z. N. Chen, “A small UWB antenna for wireless USB,” in Proc. ICUWB 2007, pp. 198–203. [9] Z. N. Chen, “UWB antennas with enhanced performances,” in Proc. ICMMT, 2008, pp. 387–390. [10] H. G. Schantz and G. P. Wolenee, “Ultra Wideband Antenna Having Frequency Selectivity (Utility Patent),” U.S. Patent 6 774 859 B2, Aug. 10, 2004. [11] H. G. Schantz and G. P. Wolenee, “Frequency notched UWB antennas,” in Proc. UWBST 2003, pp. 214–218. [12] Q. X. Chu and Y. Y. Yang, “A compact ultrawideband antenna with 3.4/5.5 GHz dual band-notched characteristics,” IEEE Trans. Antennas Propag., vol. 56, no. 12, pp. 3637–3644, Dec. 2008. [13] Y. Zhang, W. Hong, C. Yu, Z. Q. Kuai, Y. D. Don, and J. Y. Zhou, “Planar ultrawideband antennas with multiple notched bands based on etched slots on the patch and/or split ring resonators on the feed line,” IEEE Trans. Antennas Propag., vol. 56, no. 9, pp. 3063–3068, Sep. 2008. [14] Z. A. Zheng and Q. X. Chu, “A CPW-fed ultrawideband antenna with dual notched bands,” in Proc. ICUWB 2009, pp. 645–648. [15] W. T. Li, X. W. Shi, and Y. Q. Hei, “Novel planar UWB monopole antenna with triple band-notched characteristics,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 1094–1098, 2009. [16] K. S. Ryu and A. A. Kishk, “UWB antenna with single or dual bandnotches for lower WLAN band and upper WLAN band,” IEEE Trans. Antennas Propag., vol. 57, no. 12, pp. 3942–3950, Dec. 2009. [17] H. G. Schantz, “Spectral Control Antenna Apparatus and Method (Utility Patent),” U.S. Patent 7 064 723 B2, Jun. 20, 2006.

ZHENG et al.: COMPACT BAND-REJECTED UWB SLOT ANTENNAS INSERTING WITH

[18] J. X. Liang, C. C. Chiau, X. D. Chen, and C. G. Parini, “Study of a printed circular disc monopole antenna for UWB system,” IEEE Trans. Antennas Propag., vol. 53, no. 11, pp. 3500–3504, Nov. 2005. [19] Y. Y. Yang, Q. X. Chu, and Z. A. Zheng, “Time domain characteristics of band-notched ultrawideband antenna,” IEEE Trans. Antennas Propag., vol. 57, no. 9, pp. 3426–3430, Oct. 2009. [20] Z. A. Zheng and Q. X. Chu, “A simplified modeling of ultrawideband antenna time-domain analysis,” in Proc. ICUWB 2009, pp. 748–752. [21] T. Dissanayake and K. P. Esselle, “Prediction of the notch frequency of slot loaded printed UWB antennas,” IEEE Trans. Antennas Propag., vol. 55, no. 11, pp. 3320–3325, Nov. 2007. [22] M. Gopikrishna, D. D. Krishna, C. K. Aanandan, P. Mohanan, and K. Vasudevan, “Compact linear tapered slot antenna for UWB applications,” Electron. Lett., vol. 44, no. 20, pp. 1174–1175, Sep. 2008. [23] R. H. Johnston and J. G. McRory, “An improved small antenna radiation-efficiency measurement method,” IEEE Antennas Propag. Mag., vol. 40, no. 5, pp. 40–48, Nov. 1998. [24] M. Geissler, O. Litschke, D. Heberling, P. Waldow, and I. Walff, “An improved method for measuring the radiation efficiency of mobile devices,” Proc. IEEE AP-S, vol. 4, pp. 743–746, 2003. [25] K. Siwiak, T. M. Babij, and Z. Yang, “FDTD simulations of ultra-wideband impulse transmissions,” presented at the RAWCON, Waltham, MA, Aug. 19–22, 2001. [26] K. Siwiak and Y. Bahreini, Radiowave Propagation and Antennas for Personal Communications. Boston, MA: Artech House, 2007, sec. 11.7. [27] H. Kim, D. Park, and Y. Joo, “All-digital low-power CMOS pulse generator for UWB system,” Electron. Lett., vol. 40, no. 24, pp. 1534–1535, Nov. 2004. [28] N. Telzhensky and Y. Leviatan, “Novel method of UWB antenna optimization for specified input signal forms by means of genetic algorithm,” IEEE Trans. Antennas Propag., vol. 54, no. 8, pp. 2216–2225, Aug. 2006. [29] J. R. Costa, C. R. Medeiros, and C. A. Fernandes, “Performance of a crossed exponentially tapered slot antenna for UWB systems,” IEEE Trans. Antennas Propag., vol. 57, no. 5, pp. 1345–1352, May 2009. Zhi-An Zheng was born in Xiajiang, Jiangxi, China, on June 3, 1985. He received the B.S. degree in electronic information engineering from Xidian University, Xi’an, China, in 2007, and is currently working toward the M.E. degree at South China University of Technology, Guangzhou. His research interests include the design and analysis of UWB antennas.

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RESONATORS

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Qing-Xin Chu (M’99) received the B.S., M.E., and Ph.D. degrees in electronic engineering from Xidian University, Xi’an, Shaanxi, China, in 1982, 1987, and 1994, respectively. He is currently a Full Professor of the School of Electronic and Information Engineering, South China University of Technology, China. He is also head of Research Institute of RF and Wireless Techniques of the school. He worked at the School of Electronic Engineering, Xidian University from 1982 to 2003, and was the Vice-Dean and a full Professor of the school from 1997 to 2003. He undertook his research in the Department of Electronic Engineering, Chinese University of Hong Kong as a Research Associate, from July 1995 to July 1997 and March to September 1998, and worked in the Department of Electronic Engineering, City University of Hong Kong, as a Research Fellow, from February to May 2001. He was a Visiting Professor of the Department of Electronic Engineering, Chinese University of Hong Kong, from July to October 2002, and the Department of Electronic Engineering, City University of Hong Kong, from December 2002 to March 2003. He visited the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore, from July to October 2004, with the Tan Chin Tuan Exchange Fellowship Award, and visited the Department of Electrical and Electronic Engineering, Okayama University, Japan, from January to March 2005, with a Fellowship awarded by the Japan Society for Promotion of Science (JSPS). His current research interests include UWB antennas and RF components, active integrated antennas, spatial power combining array, and computational electromagnetics. Q.-X. Chu received the first-class Educational Award of Shaanxi Province in 2003, the top-class Science Award of Education Ministry of China and secondclass Science and Technology Advance Award of Shanxi Province in 2002, the top-class Educational Award of Shanxi Province and the second-class Award of Science and Technology Advance of Electronic Industry Ministry in 1995. He is a senior member of the China Electronic Institute (CEI).

Zhi-Hong Tu received Ph.D. degree in 2007. She currently works at the School of Electronic and Information Engineering, South China University of Technology, Guangzhou, China. Her research interests include the synthesis theory and design of microwave filters and associated RF modules and antenna for microwave and millimeterwave applications research interests include the design and analysis of UWB antennas.

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Millimeter-Wave Microstrip Comb-Line Antenna Using Reflection-Canceling Slit Structure Yuki Hayashi, Kunio Sakakibara, Senior Member, IEEE, Morihiko Nanjo, Shingo Sugawa, Nobuyoshi Kikuma, Senior Member, IEEE, and Hiroshi Hirayama, Member, IEEE

Abstract—A microstrip comb-line antenna is developed in the millimeter-wave band. When the element spacing is one guide wavelength for the broadside beam in the traveling-wave excitation, reflections from all the radiating elements are synthesized in phase. Therefore, the return loss increases significantly. Furthermore, re-radiation from elements due to the reflection wave degrades the design accuracy for the required radiation pattern. We propose the way to improve the reflection characteristic of the antenna with arbitrary beam directions including strictly a broadside direction. To suppress the reflection, we propose a reflection-canceling slit structure installed on the feeding line around each radiating element. A 27-element linear array antenna with a broadside beam is developed at 76.5 GHz. To confirm the feasibility of the simple design procedure, the performance is evaluated through the measurement in the millimeter-wave band. Index Terms—Array antenna, comb-line antenna, microstrip antenna, millimeter-wave.

I. INTRODUCTION ILLIMETER-WAVE antennas have been developed for various applications such as broadband and high-speed wireless communication systems and automotive radar systems [1], [2]. Microstrip antennas are more advantageous than other millimeter-wave antennas in terms of its low profile and low cost. On the other hand, the feeding loss due to the transmission loss of the microstrip line is a significant problem in the array feeding. Therefore, microstrip array antennas are suitable for relatively low-gain applications such as a subarray of digital beam forming (DBF) systems [3]. A comb-line feeding system is effective for relatively low loss when compared with other microstrip patch array antennas fed by parallel or ordinary series feeding [4], [5]. When the element spacing is just one guide wavelength for the broadside beam in the traveling-wave excitation, reflections from all the radiating elements are synthesized in phase at the

M

Manuscript received August 21, 2009; revised March 26, 2010; accepted September 16, 2010. Date of publication December 03, 2010; date of current version February 02, 2011. Y. Hayashi was with the Department of Computer Science and Engineering, Nagoya Institute of Technology, Nagoya 466-8555, Japan. He is now with Toshiba Corporation, Tokyo 105 8001, Japan. K. Sakakibara, N. Kikuma, and H. Hirayama are with the Department of Computer Science and Engineering, Nagoya Institute of Technology, Nagoya 466-8555, Japan (e-mail: [email protected]). M. Nanjo was with the Department of Computer Science and Engineering, Nagoya Institute of Technology, Nagoya 466-8555, Japan. He is now with Yamaha Motor Co., Ltd., Shizuoka 438 8501, Japan. S. Sugawa with the Department of Computer Science and Engineering, Nagoya Institute of Technology, Nagoya 466-8555, Japan. He is now with Honda Motor Co., Ltd., Saitama 335 0031, Japan. Digital Object Identifier 10.1109/TAP.2010.2096180

feeding point. Therefore, the return loss increases significantly. Furthermore, re-radiation from the element due to the reflection wave affects the radiation pattern of the array antenna, which degrades the design accuracy. Matching characteristics have been improved in the conventional design by beam-tilting of several degrees, where reflections from all the elements are canceled out of phase at the input port [6], [7]. Beam directions to cancel reflections in the microstrip line are limited in some specific angles, where the broadside direction is identical to the nulls of the radiation pattern. The reflection-canceling pair element could be one of the solutions for array antennas to reduce reflection from each element. However, the element radiation pattern of the pair structure forms a cardioid directivity whose peak is the endfire direction [8]. The sidelobe levels increase in the endfire directions, resulting in an asymmetric radiation pattern. Furthermore, the mutual coupling between elements in the pair significantly reduces the total radiation from the elements, which must be taken into account in the design. Hence, a novel radiating element that can suppress the reflection without affecting the element radiation is required. We investigate how to improve the reflection characteristic of the antenna with arbitrary beam directions, including strictly a broadside direction. To suppress the reflection from each element, we propose a reflection-canceling slit structure installed on the feeding line around each radiating element [9]. The configuration of the antenna is described in Section II. The simple design procedure is produced in Section III. To confirm the feasibility of the proposed simple design procedure, a 27-element linear array antenna with a broadside beam is developed at 76.5 GHz. The measured antenna performance is reported in Section IV. II. CONFIGURATION A microstrip comb-line antenna is composed of several rectangular radiating elements that are directly attached to a straight feeding line printed on a dielectric substrate (Fluorocarbon resin , relative dielectric constant film, thickness and loss tangent ) with a backed ground plane as shown in Fig. 1. The width of the feeding microstrip line is 0.30 mm. The characteristic impedance of this line is 60 . The radiating elements are inclined 45 degrees from the feeding microstrip line for the polarization requirement of automotive radar systems [2]. The radiating elements with length and are arranged on the both sides of the feeding line, width which forms an interleaved arrangement in a one-dimensional is identical to a half guide wavearray. The resonant length is approximately a half guide length. The element spacing

0018-926X/$26.00 © 2010 IEEE

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Fig. 3. Structure of lower radiating element for electromagnetic analysis.

Fig. 1. Configuration of the proposed antenna.

Fig. 4. Reflection-canceling slit structure for electromagnetic analysis.

Fig. 2. Structure of radiating element with reflection-canceling slit for electromagnetic analysis.

wavelength so that all the elements on the both sides of the microstrip line are excited in phase. A matching element is designed to radiate in phase all the residual power at the termination of the feeding line. Coupling power of radiating elements is of the radiating element. Large power controlled by width radiates from a wide element [7]. A radiation pattern with a zero-degree broadside beam is often used in many applications. However, when all the radiating elements are designed to excite in phase, all the reflections are also in phase at the feeding point, thus significantly degrading the overall reflection characteristic of the array. In the conventional design with beam tilting by a few degrees, reflections are canceled at the feed point due to the distributed reflection phase of the radiating elements. This means that the design flexibility of beam direction is limited by the reflection characteristics. To solve this problem, we propose a reflection-canceling slit structure as shown in Figs. 1 and 2. A rectangular slit [10] is cut on the feeding line around the radiating element. A reflection from each radiating element is canceled with the reflection from the slit. As the reflection from a pair of a radiating element and a slit is suppressed in each element, a zero-degree broadside array can be designed without increasing the return loss of the array. Because the sizes of all the radiating elements are different for the required aperture distribution, the slit dimensions and the spacing of the slit from the radiating element are optimized for each radiating element. The array design requires a simple design procedure.

III. DESIGN In the array design for the comb-line antenna, required radiations are assigned to all the radiating elements for a given aperture distribution. Each radiating element is designed to realize the required radiation by electromagnetic analysis of finite element method. A reflection-canceling slit is introduced to suppress reflections from the radiating elements. Because the dimensions of all the radiating elements are different, the slit dimensions and the spacing of the slit from radiating element must be optimized for each radiating element. In this section, a simple design procedure for the array design of a comb-line antenna is proposed. A. Design Procedure of One Radiating Element The reflection amplitude and phase of the radiating elements and the slits are analyzed independently by an isolated simulation model shown in Figs. 3 and 4. To cancel the two reflections from the radiating element and the slit, the amplitude must be the same, and the phase must be opposite (180 degrees). The reflection amplitude is controlled by the dimensions of the slit, while the reflection phase is controlled by the spacing between the radiating element and the slit. The slit dimensions are designed for the same reflection amplitude as each radiating element. The spacing between the radiating element and the slit is obtained by calculation of a simple equation from the reflection phases and the path length required to cancel the reflections. First, the reflection amplitude and phase of a radiating element are simulated from the analysis model (Fig. 3). is obtained for each width of th The resonant length becomes minimum radiating element, where reflection at the design frequency. The length of each radiating element

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Fig. 5. Reflection characteristics 76.5 GHz.

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j

S

j

and

j

S

j

of radiating element at

Fig. 7. Radiation power from slit.

Fig. 8. Reflection characteristics of a radiating element with a reflectioncanceling slit.

Fig. 6. Reflection characteristics of microstrip slit at 76.5 GHz.

is optimized by electromagnetic simulation. Fig. 5 shows the reflection amplitude of the radiating element at 76.5 GHz. As the analysis model is not symmetrical for the input and output and are not identical; is larger than ports, for the same width . The radiating elements with a slit must in the both 45- and 135-debe designed for all widths and gree element angles. Next, the reflection amplitude of a slit are simulated by the analysis model shown phase in Fig. 4. The dependencies for the geometrical parameters of are shown in Fig. 6. The reflection becomes high when and are large. The maximum reflection level of a slit is is approximately equal to 0.70 mm, which is observed when a quarter guide wavelength of the microstrip line. The slit width is fixed as 0.15 mm for fabrication and reflection amplitude . Slit width is of a slit is initially controlled by slit length designed for fine adjustment afterwards. Radiation power from a slit is indicated in Fig. 7. Radiation powers from most slits including 0.15 mm width are less than 1% and are 2% maximum. Thus, radiation from slits can be neglected for radiation from the elements in the design. The reflection phase values of a radiating element and a slit are obtained by electromagnetic simulation. Then, slit position is obtained in quite simple manner by (1)

where means opposite phase to cancel reflections from the is the guide wavelength of the radiating element and the slit. microstrip line and is 2.88 mm at 76.5 GHz. and A design result of an element whose dimensions are 0.58 and 1.32 mm is shown here as an example. The reflection level of this radiating element is 17.2 dB at 76.5 GHz. To obtain the same reflection amplitude from a slit, we selected the and the slit slit dimensions as the slit length . Next, the slit position is obtained width and . by the (1) from their reflection phase values are obtained as 0.47 Consequently, the two slit positions and 0.97 mm, respectively. Fig. 8 shows the reflection characteristics of the designed radiating element with a slit. When is 0.47 mm, the resonant frequency is equal to the design is 0.97 mm, the resfrequency 76.5 GHz; however, when onant frequency shifts to a lower frequency. That is, the phase perturbation of a radiating element is large, which affects resonant frequency. On the other hand, the phase perturbation of the slit is small enough to be neglected in the element design. The variation of the reflection characteristics of the element and length are 0.50 and 1.34 mm dewhose width pends on the slit parameters investigated analytically by using the electromagnetic simulator. An analysis model is shown in Fig. 2 as well. The reflection characteristics from the variation between the radiating element and the slit of the spacing and the width of a slit are shown in Fig. 9. The length

HAYASHI et al.: MILLIMETER-WAVE MICROSTRIP COMB-LINE ANTENNA USING REFLECTION-CANCELING SLIT STRUCTURE

Fig. 9. Reflection characteristics in variation of spacing element and slit.

D

between radiating

Fig. 10. Variation of reflection characteristics depending on slit length

L

Fig. 11. Variation of reflection characteristics depending on slit width

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W

.

Fig. 12. Power relation for traveling-wave excitation. .

are fixed as 0.16 and 0.15 mm, respectively. The reflection coefficient of the radiating element with a slit is lower than 40 dB; the reflection is 17.6 dB at 76.5 GHz without a slit, as shown in Fig. 8. This confirms that the reflection from the radiating element is canceled sufficiently with the reflection from the slit. The resonant frequency shifts to a lower frequency when becomes large. The increase in the path length cancels the phase difference due to the increase in the guide wavelength. Fig. 10 shows the variation of reflection characteristics de. The spacing and the width pending on the slit length of a slit are fixed as 0.24 and 0.15 mm, respectively. The variation of reflection characteristics depending on the slit width is shown in Fig. 11. The spacing and the length of the slit are fixed as 0.24 and 0.16 mm, respectively. Reflection level at the resonant frequency changes because the resoor nant frequency of the radiating element changes when changes. This is due to amplitude variation of reflection from the slit. The reflection amplitude needs to be identical between the slit and the radiating element for them to cancel each other. A small shift in the resonant frequency is observed when changes. This is because phase shift of wave transmitting . Consethrough the slit changes depending on the length quently, in the optimization of the slit structure for each radiating element, resonant frequency is adjusted by the slit location and reflection amplitude is controlled by the slit dimensions to cancel the reflections from the slit and the radiating element. Coupling power to the air is calculated for each element width from the scattering parameters and obtained by

electromagnetic simulation of a single element with a slit, that is (2) where is simulated transmission of the microstrip line without the radiating element and the slit, which is identical to the loss of the line. Coupling power is 7.4% when the width and the length of the lower radiating element are 0.50 and 1.34 mm. B. Array Design for Traveling-Wave Excitation The microstrip comb-line antenna is designed to operate in traveling-wave excitation. The input power is gradually radiated from all the radiating elements during transmission from the input port toward the termination. An array design is implemented for Taylor distribution with sidelobe lower than 20 dB. The required coupling power is designed to be small near the input port and to increase toward the termination. The required variation of coupling is from 2.0 to 49.8% controlled by of the elements, where is the resonant length. the width A 27-element linear array antenna with broadside beam is designed for the experiments. In the traveling-wave excitation, the power gradually attenuates during transmitting along the feeding line due to the radiation from the elements and the transmission loss of the microstrip line. Power relation is indicated in Fig. 12. For the th element, (3)

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Fig. 14. Definition of element spacings d

and d

.

Fig. 13. Aperture amplitude distribution and coupling power assigned in the design.

where input, transmission and radiation powers are , and , respectively. Coupling power is given as (4) Furthermore, the transmission power of th element and the th element are related as input power of (5) where the transmission loss is , consisting of copper loss, dielectric loss and radiation loss. Radiation power from each element is assigned by taking both the radiation from the element and the transmission loss of the feeding line into account. The measured transmission loss of the microstrip line is 0.3 dB per cm. The required Taylor distribution and assigned coupling power for the design of 27-element array are shown in Fig. 13. Required range of coupling power is from 2.0 to 49.8%. The result of the analysis show that coupling power can be controlled from 1.6 to 36.0% for a 135-degree element and from 1.6 to 53.3% for of the radiating element a 45-degree element when width changes from a minimum of 0.1 mm for fabrication to 1.1 mm, where the width is the same with length. In the design of zero-degree broadside beam, the element spacing is designed by considering the phase perturbation. The element spacing is defined as a spacing between the corners of adjacent radiating elements attached to the center of the feeding is designed from the analyzed line. The element spacing as phase perturbation (6) is a wave number in the feeding microstrip line , is a wave number in free space and is a beam tilting angle from axis inclined to directions is obtained by defined in Fig. 1. The phase perturbation subtracting the transmission phase of the microstrip line with the same length from the transmission phase of the microstrip line with a radiating element and a slit. As the radiating elements attached on the both sides are not symmetrical to the feeding line, the effective element spacing is not the same with that of the defined element spacing. To design the element spacing for in-phase excitation between the elwhere

Fig. 15. Beam direction in the perpendicular plane to the feeding line depending on the element spacing d .

Fig. 16. Element spacing d element.

and d

depending on the width of radiating

ements on the both sides of the feeding line, the beam direction in the perpendicular plane to the feeding line of the two-element array is estimated by electromagnetic simulation. Analysis models are shown in Fig. 14. In this estimation, the dimensions of the two radiating elements with slits are the same. Two radiating elements with slits are arranged by element spacing . Element spacing and are defined as the distance between the corners of the radiating elements attached to the center line of the feeding line as shown in Fig. 14. This model is analyzed using an electromagnetic simulator. Fig. 15 shows the beam direction in the plane perpendicular to the feeding line . The element spacing for depending on the element spacing the broadside beam is not constant as the element width because the effective radiating point changes as well as the effective element spacing changes depending on the element width. for in-phase distribution is designed to Element spacing obtain the broadside beam. Fig. 16 shows the element spacing and for in-phase excitation depending on the width of

HAYASHI et al.: MILLIMETER-WAVE MICROSTRIP COMB-LINE ANTENNA USING REFLECTION-CANCELING SLIT STRUCTURE

Fig. 17. Element spacing d for in-phase excitation.

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Fig. 19. Simulated reflection characteristics of matching element (a = 1:31 mm, b = 0:26 mm, p = 0:40 mm, g = 0:24 mm, g = 0:25 mm).

Fig. 18. Configuration of matching element.

the radiating element. The change in the element spacing is significant, from 1.0 to 1.9 mm. The relationship of the element is applied to the design of the broadside beam anspacing tenna. Fig. 17 shows the element spacing of the array. The changes significantly around a half guide element spacing wavelength to excite all the radiating elements in phase. C. Matching Element The feeding line is terminated by a matching element to radiate all the residual power in the feeding line. Fig. 18 shows the structure of the matching element. The patch antenna forms a square on one side. Resonant frequency is controlled by longitudinal length in the feeding direction. Input impedance is controlled by insertion length of the microstrip line, and the width of the feeding microstrip line is 0.30 mm. The characteristic impedance of this line is 60 . As a result of the optimization by becomes minimum electromagnetic simulation, reflection , at the design frequency 76.5 GHz when , , , . The simulated reflection is shown in Fig. 19. is lower than 20 dB at the design frequency. The feeding line is bent 45 degrees to be the same polarization with all other radiating elements in the comb-line antenna. The bent effect on the impedance matching characteristic is inof the matching elements with vestigated. The simulated the 45-degree bend and without the bend are shown in Fig. 19. Since they are almost similar to each other, the effect due to the bend is very small.

Fig. 20. Photographs of the developed microstrip comb-line antenna.

IV. EXPERIMENTS A microstrip comb-line antenna with 27 elements and broadside beam is fabricated for experiments. Photographs of the developed antenna are shown in Fig. 20. As the coupling power from the radiating elements is controlled to realize the required aperture distribution, the widths of the radiating element are narrow near the input port and wide near the termination. The slit is cut on the feeding line at the opposite side from each radiating element. The fabricated antenna consists of two comblines fed from the waveguide through a microstrip-to-waveguide transition and a microstrip power divider. Fig. 21 shows the measured reflection characteristics of the fabricated antenna. Reflection level of the fabricated antenna is 12.9 dB at the design frequency 76.5 GHz. Although all the reflections are synthesized in phase, the reflection level is still low. Thus, the effect of the reflection-canceling slit is confirmed. The measured radiation patterns in transversal -plane and in longitudinal -plane are shown in Figs. 22 and 23, respectively. Measured beam direction, beam width and sidelobe level are almost the same with the designed array factor in the transversal plane. The beam direction in the longitudinal plane is 1.3 degree and the sidelobe level is 17.9 dB. The measured radiation pattern is close to the array factor. However, some errors are observed in the sidelobe level and the beam direction; the sidelobe

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Fig. 21. Measured reflection characteristics of fabricated antenna.

Fig. 22. Measured radiation pattern in transversal xz -plane.

Fig. 24. Measured amplitude distribution on the aperture.

Fig. 25. Measured phase distribution on the aperture.

Fig. 23. Measured radiation pattern in longitudinal yz -plane.

level grows by 2.1 dB, and beam direction shifts by 1.3 degree from the broadside direction. To clarify the cause, the aperture amplitude and phase distributions were measured. The fabricated antenna was set on the stage in the anechoic chamber. A waveguide probe scanned twice over the antenna 8 and 9 mm above the aperture. The spacing between the two planes 8 and 9 mm was determined to be 1 mm and approximately a quarter wavelength. Two complex electric field data were averaged to compensate the effect of the standing wave distribution in free space. The results are shown in Figs. 24 and 25, respectively. Fig. 24 shows that the growing sidelobe could be due to error of amplitude distribution at the input port and the termination. Fig. 25 shows the measured phase distribution and the calculated phase distribution for beam direction of 1.3 degrees. The slope of the measured phase distribution is close to the slope of 1.3 degree beam. Therefore, the error of beam direction is due to the phase distribution. We

Fig. 26. Simulated beam shift due to the mutual coupling and the rounded corners in the etching process.

confirmed this using an electromagnetic simulator that rounded the corners of the radiating elements; the etching process affects the phase perturbation during transmission through the radiating element. The change of phase perturbation due to the rounded corners could cause beam tilting. The size of the rounded corners of the radiating elements and the slits are measured by using an optical microscope. Fig. 26 shows the simulated radiation patterns of the comb-line antenna shown in Fig. 20 with and without the rounded corners including all mutual couplings in the antenna. The beam direction of the antenna without rounded corners is 0.3 degrees. It is still far from the measured beam direction 1.3 degrees. However, the beam direction of the antenna with the rounded corners is 1.6 degrees which is close to the measured beam direction. This

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[6] R. E. Collin, Antennas and Radiowave Propagation. New York: McGraw-Hill, ch. 4, p. 270. [7] Y. Owa, K. Sakakibara, Y. Tanaka, N. Kikuma, and H. Hirayama, “Low sidelobe millimeter-wave microstrip array antenna radiation-controlled by modification of feeding-line width,” in Proc. ISAP2005, Aug. 2005, pp. 1153–1156. [8] K. Sakakibara, J. Hirokawa, M. Ando, and N. Goto, “A linearly-polarized slotted waveguide array using reflection-canceling slot pairs,” IEICE Trans. Commun., vol. 77, no. 4, pp. 511–518, Apr. 1994. [9] Y. Hayashi, K. Sakakibara, N. Kikuma, and H. Hirayama, “Beamtilting design of microstrip comb-line antenna array in perpendicular plane of feeding line for three-beam switching,” presented at the IEEE Antennas and Propagation Society Int. Symp., Jul. 2008, 108.5. [10] W. J. R. Hoefer, “Equivalent series inductivity of a narrow transverse slit in microstrip,” IEEE Trans. Microwave Theory Tech., vol. MTT-25, pp. 822–824, Oct. 1977. Fig. 27. Measured frequency dependency of gain and efficiency.

means that the mutual coupling affects small to the beam shift. Major reason for the beam shift is the rounded corners occurred in the etching process during the antenna fabrication. Fig. 27 shows the frequency dependency of the gain and the antenna efficiency of the antenna composed of two feeding-lines with comb-line antennas (Fig. 20). The maximum antenna gain is 20.3 dBi, in which the antenna efficiency is 55.0% at the design frequency 76.5 GHz. Gain is decreased due to the beam tilting in other frequencies. Consequently, it is confirmed that the loss due to the slit structure could be small. The matching characteristic is improved independent of the beam direction using a reflection-canceling slit structure without increasing the loss of the transmission line. V. CONCLUSION To suppress reflection from each radiating element, we proposed the reflection-canceling slit structure. A 27-element linear array antenna with a broadside beam is designed and fabricated to confirm the feasibility of the simple design procedure. The reflection coefficient of the proposed antenna is comparable with the conventional antenna with beam-tilting technique, although the proposed antenna is designed for a broadside beam. The measured radiation pattern is close to the array factor of the design. However, the beam direction of the measured antenna is 1.3 degree. We also confirmed that the fabrication error of the etching process affects the beam direction. We estimated the gain and the antenna efficiency and confirmed that the loss due to the slit structure is small. As a result, the microstrip comb-line antenna with arbitrary beam direction can be designed without increasing the return loss through the use of a reflection-canceling slit structure.

Yuki Hayashi was born in Shiga, Japan, on February 3, 1984. He received the B.S. and M.S. degrees in computer science and engineering from Nagoya Institute of Technology, Nagoya, Japan, in 2009. He is currently working at Toshiba Corporation, Tokyo, Japan.

Kunio Sakakibara (M’94–SM’06) was born in Aichi, Japan, on November 8, 1968. He received the B.S. degree in electrical and computer engineering from Nagoya Institute of Technology, Nagoya, Japan, in 1991, and the M.S. and D.E. degrees in electrical and electronic engineering from Tokyo Institute of Technology, Tokyo, Japan, in 1993 and 1996, respectively. From 1996 to 2002, he worked at Toyota Central Research and Development Laboratories, Inc., Aichi, Japan, where he was engaged in development of antennas for millimeter-wave automotive radar systems. From 2000 to 2001, he was with the Department of Microwave Techniques in University of Ulm, Ulm, Germany, as a Guest Researcher. He was a Lecturer at Nagoya Institute of Technology, from 2002 to 2004, and is currently an Associate Professor. His research interest has been millimeter-wave antennas and circuits.

Morihiko Nanjo was born in Shizuoka, Japan, on October 24, 1985. He received the B.S. degree in electrical and electronic engineering and the M.S. degrees in computer science and engineering from Nagoya Institute of Technology, Nagoya, Japan, in 2008 and 2010, respectively. He is currently working at Yamaha Motor Co., Ltd., Shizuoka, Japan.

REFERENCES [1] S. Tokoro, “Automotive application systems using a millimeter-wave radar,” TOYOTA Tech. Rev., vol. 46, no. 1, pp. 50–55, May 1996. [2] K. Fujimura, “Current status and trend of millimeter-wave automotive radar,” in Microwave Workshops and Exhibition Digest, MWE 95, Dec. 1995, pp. 225–230. [3] Y. Asano, “Millimeter-wave holographic radar for automotive applications,” in Microwave Workshops and Exhibition Digest, MWE 2000, Dec. 2000, pp. 157–162. [4] J. R. James and P. S. Hall, Handbook of Microstrip Antennas, ser. IEE Electromagnetic Waves Series. London, U.K.: Peter Peregrinus, 1989, vol. 2. [5] H. Iizuka, T. Watanabe, K. Sato, and K. Nishikawa, “Millimeter-wave microstrip array antenna for automotive radar,” IEICE Trans. Commun., vol. E86-B, no. 9, pp. 2728–2738, Sep. 2003.

Shingo Sugawa was born in Toyama, Japan, on August 5, 1985. He received the B.S. degree in electrical and electronic engineering and the M.S. degree in computer science and engineering from Nagoya Institute of Technology, Nagoya, Japan, in 2008 and 2010, respectively. He is currently working at Honda Motor Co., Ltd., Saitama, Japan.

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Nobuyoshi Kikuma (M’83–SM’03) was born in Ishikawa, Japan, on January 7, 1960. He received the B.S. degree in electronic engineering from Nagoya Institute of Technology, Nagoya, Japan, in 1982, and the M.S. and Ph.D. degrees in electrical engineering from Kyoto University, Kyoto, Japan, in 1984 and 1987, respectively. From 1987 to 1988, he was a Research Associate at Kyoto University. In 1988, he joined Nagoya Institute of Technology, where he has been a Professor since 2001. His research interests include adaptive and signal processing array, multipath propagation analysis, mobile and indoor wireless communication, and electromagnetic field theory. Dr. Kikuma received the 4th Telecommunications Advancement Foundation Award in 1989.

Hiroshi Hirayama (S’01–M’03) received the B.E., M.E., and Ph.D. degrees in electrical engineering from the University of Electro-Communications, Chofu, Japan, in 1998, 2000, and 2003, respectively. Since 2003, he has been with the Nagoya Institute of Technology, Nagoya, Japan, where he is currently a Research Associate. His research interests include signal processing techniques and EMC/EMI.

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Frequency Steerable Two Dimensional Focusing Using Rectilinear Leaky-Wave Lenses José Luis Gómez-Tornero, Member, IEEE, Fernando Quesada-Pereira, Member, IEEE, Alejandro Alvarez-Melcón, Senior Member, IEEE, George Goussetis, Member, IEEE, Andrew R. Weily, Member, IEEE, and Y. Jay Guo, Senior Member, IEEE

Abstract—The concept of frequency steerable two-dimensional electromagnetic focusing by using a tapered leaky-wave line source embedded in a parallel-plate medium is presented. Accurate expressions for analyzing the focusing pattern of a rectilinear leakywave lens (LWL) from its constituent leaky-mode tapered propagation constant are described. The influence of the main LWL structural parameters on the synthesis of the focusing pattern is discussed. The ability to generate frequency steerable focusing patterns has been demonstrated by means of an example involving a LWL in hybrid waveguide printed-circuit technology and the results are validated by a commercial full-wave solver. Index Terms—Electromagnetic focusing, geometrical optics, leaky-wave antennas, lens waveguides, near fields.

I. INTRODUCTION OCUSING of microwave electromagnetic energy is desired for many applications, such as microwave hyperthermia therapy systems [1]–[4], non-contact (remote) sensing [5]–[7], and wireless power transmission [8], [9]. Normally, focusing is achieved by using phased-arrays antennas in waveguide [2], [3] or microstrip [6], [7], [10] technologies. These systems are less bulky and more flexible than others such as dielectric lenses [1] or reflectors systems [11], [12]. In general, large radiating apertures with quadratic phase and tapered amplitude distributions are needed to synthesize low sidelobes high-efficient focusing patterns [13]–[16]. Leaky-wave antennas (LWAs) have the ability of synthesizing large radiating apertures by simply exciting a leaky-mode [17]. Moreover, the amplitude and phase distribution of the radiating aperture can be adjusted by properly controlling the leaky-mode complex propagation constant, therefore avoiding complicated feeding networks associated with phased arrays [17]. LWAs have recently been proposed to synthesize focusing/diverging patterns

F

Manuscript received March 12, 2010; revised June 22, 2010; accepted July 09, 2010. Date of publication December 03, 2010; date of current version February 02, 2011. This work was supported in part by Spanish National project TEC2007-67630-C03-02/TCM, Regional Seneca project 08833/PI/08, and in part by Spanish scholarship “Salvador de Madariaga” (ref. PR2009-0336). J. L. Gómez-Tornero, F. Quesada-Pereira, and A. Alvarez-Melcón are with the Department of Communication and Information Technologies, Technical University of Cartagena, Cartagena 30202, Spain (e-mail: josel.gomez@upct. es). G. Goussetis is with the Queen’s University of Belfast, BT3 9DT Belfast, Northern Ireland. A. R. Weily and Y. Jay Guo are with the CSIRO ICT Centre, Epping, NSW 1710, Australia. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2096396

Fig. 1. Tapered rectilinear leaky-wave antenna for electromagnetic focusing.

[18]–[23]. The first approaches to leaky-wave lenses (LWL) were reported by Ohtera [18]–[20], who curved a leaky-waveguide structure in an equiangular spiral fashion so that the emitted rays converge and a focusing pattern is obtained. More recently, this bending technique has also been applied to microstrip LWAs [21]. Another approach to create a LWL is based on the tapering of the leaky-mode complex propagation constant along the length of a rectilinear LWA, as proposed by Burghignoli et al. [22] and illustrated in Fig. 1. This technique is preferred to curved-LWAs since the LWA does not need to be in any complicated and bulky curved shape. The focusing synthesis technique presented in [22] is based on a simplified geometrical-optics approach, but no focusing patterns were reported in the paper. A backfire-to-endfire LWA interface was proposed in [23] to synthesize a LWL, and a simplified nonlinear phased-array approach was used for a fast estimation of the focusing pattern. However, [22] and [23] did not present accurate analysis expressions or design results for practical tapered rectilinear LWLs. Finally, the frequency-scanning capability of LWAs [17] can provide steering of the focus by changing frequency. This can be of interest for certain applications, such as multi spot heating/sensing or spatial filtering [24], [25], but it has not been reported so far. In this paper, we investigate the ability of rectilinear tapered LWAs embedded in a parallel-plate medium to synthesize frequency steerable two-dimensional focusing patterns (see Fig. 2). The analysis theory is presented in Section II, and a parametric analysis is performed in Section III to understand the key aspects which determine the performance of tapered LWL of rectilinear geometry. Section IV presents validation results by designing a LWL in hybrid technology [26]–[28], illustrating for the first time the synthesis of a frequency steerable focusing pattern from a rectilinear LWA.

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the desired focal point [22]

can be easily derived from (7)

(8) On the other hand, the tapering function of the leakage rate will determine the amplitude illumination function and the radiation efficiency , as described by (3)–(5). The next well-known synthesis expression [17] allows one to obtain which provides a desired and the function Fig. 2. 3D Perspective of the rectilinear LWA embedded in parallel-plate medium.

(9)

II. THEORY The synthesis of LWL of rectilinear geometry is based on the use of a tapered leaky-wave mode [22], where the complex propagation wavenumber (phase constant and leakage rate ) is varied along the LWA longitudinal axis, (see Fig. 1) (1) The illumination created by such a tapered leaky-wave (LW) can be expressed as an x-directed equivalent surface electric curinvolving an amplitude and a phase rent density [29] distribution along the LWA radiating length, which can be computed from the tapered leaky-wavenumber (1) by the following expressions [17] (2) (3)

A more rigorous derivation of can be obtained by expressing the near fields in the vicinity of the LWL. The derivation of these fields is also needed for the accurate analysis and design of the focusing pattern in the following sections. For this purpose, we will assume that the LWA is embedded in a par, so that allel-plate waveguide (PPW) with separation only x-directed fields are radiated, as illustrated in Fig. 2. This assumption is valid for many of the stub-loaded LWAs with flexible control of and reported in literature [17], in which the leakage is induced by asymmetry. In this way, the evanescent fields associated with higher order modes of the PPW can be neglected, since they decay at a short distance from the leaky-PPW interface [17]. Under these conditions and by virtue of the symmetry created by the PPW, we can use the following two dimensional Green’s functions [29], [30] for an x-directed infinite electric current located at any position along the LWA aperture (see Fig. 2) (10)

(4)

(11)

(5)

(12) (13)

(6) where is the free-space wave impedance, is the power is the power travelling injected at the input of the LWA, along the LWA length, is the radiated power, and is the radiation efficiency. The complex illumination function , (2) induced by the leaky-wave along the aperture length ( see Fig. 1) will create radiated fields which can converge at the . The angle desired focal point, located at the position of radiation of the rays emitted from any longitudinal position of the LWA can be estimated by the next equation [17], [22]

The fields created by the whole LWA aperture at any observain the PPW can be computed by integrating the tion point contribution of a continuous distribution of x-directed infinite electric currents which are illuminated by the complex tapered leaky-wave illumination function (2), yielding (14)

(7) where is measured with respect to the y-axis (broadside direction), which is perpendicular to the LWA interface. Using a geometrical-optics approach in the scheme of Fig. 1, needed for the rays to converge at the tapering function

(15) In order to maximize the fields at the desired focal point, the contributions of all the radiating sections of the LWA must be

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in-phase at the focus . Introducing (2), (6), (10) in (14), and using the definition of the Hankel function of zero order and second kind (16) we obtain the following expression for the in-phase condition

(17) (18) where is an arbitrary phase. After derivation with respect to , (17) leads to (19) Applying some simple derivative rules and using the following Wronskian identity [30] for the derivative of Bessel functions of the first and second kind, we obtain (20) Fig. 3. (a) Tapered leaky-mode propagation functions, (z );  (z ) and (z ) obtained for a LWL with L = 20 ; z = 10 y = 10 and uniform amplitude distribution M (z ) with  = 90%. (b) Focusing near-field pattern.

(21) III. PARAMETRIC STUDY OF TAPERED LWL It should be noted that, applying the following identity [30] (22) , yielding the Equation (21) is reduced to (8) for same approximation as with simple geometrical optics. Once the and are obtained from leaky-mode tapering functions (21) and (9), the near fields distribution can be calculated using (14)–(15), thus yielding the focusing pattern of the designed rectilinear LWL. Another important parameter of the LWL is the near-field focal gain , defined as [4], [12]–[14]

(23) where is the power density at the focal point and is the by total radiated power, which is related to the input power (5). As can be seen from (23), a high efficiency LWL must make use of a LWA with high : typically, the maximum for tapered LWA is 90%, with the remaining 10% value of power being absorbed in a matched load to avoid the presence of any backlobe [17]. On the other hand, the value of the power needed to calculate (23) can be computed from density the leaky-radiated fields (14)–(15) (24)

The main parameters involved in the design of a rectilinear and the position of the LWL are the radiating length , which will determine the tapering of the focal point leaky-phase constant given by (21). The desired amplitude and radiation efficiency will illumination function (9). determine the tapering of the leaky-mode leakage rate All these parameters will influence the near-field focusing pattern (14)–(15) and the focusing efficiency (23)–(24) of the LWL, and they are studied in this section. As an example, Fig. 3(a) shows the normalized propagation and (together with the associated functions, pointing angle ) of the tapered leaky-mode, for a LWL , focal point at ( with length , and uniform amplitude distribution with %. The corresponding normalized focusing pattern with dB scale is plotted in Fig. 3(b). A detailed pattern with dB scale in a by region around the focal point is also shown. As can be seen in Fig. 3, by properly tapering the to leaky-mode pointing angle from , the fields are focused at the desired location. Moreover, the leakage rate must be exponentially tapered to produce the . Fig. 4 shows the desired uniform amplitude illumination axial and transverse cuts of the focusing pattern of Fig. 3. These and the halfcuts allow us to define the depth of focus [14], so that the power power beamwidth (or focus width) dB with respect to the focal point in density has decreased each plane. As is well known [13]–[16], focused apertures are

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Fig. 4. Axial (including focal gain) and transverse cuts for the LWL of Fig. 3.

Fig. 7. Effect of the focal distance y in the tapering of the leaky-mode propagation functions for constant L  and z L = .

= 20

Fig. 5. Effect of the LWL length L in the tapering of the leaky-mode propL = ;y . agation functions for a constant focal position z

=

2

= 10

=

2

Fig. 8. Axial and transverse cuts for the LWLs of Fig. 7.

When the focus is centered with respect to the LWL length , (25) is reduced to the following expression: (26)

Fig. 6. Axial and transverse cuts for the LWLs of Fig. 5.

more directive in the transverse plane than in the axial direction . The focal gain obtained is dB. A. Effect of the Leaky-Wave Lens Length Fig. 5 shows the leaky-mode tapering functions for several LWLs designed to provide a focal point at and uniform amplitude distribution % , for different values of ( and ). The axial and transverse cuts of the focusing patterns are shown in Fig. 6. As can be seen, more directive and efficient focusing patterns are obtained by in. creasing As will be shown in the next subsections, the important factor is the range of scanned angles emitted by the tapered leaky-mode. A wider range provides richer spatial diversity for the radiated fields, and thus a more focused interference pattern. This range can be computed from (8) as

(25)

As is increased for a fixed focal position , the range of scanned angles (26) is enlarged, tending to the maximum which provides the richest spatial range diversity and the highest focus directivity and gain, as shown in Figs. 5–6. Also, it is seen that the sidelobes increase and . As is the forelobes and aftlobes decrease for higher will determine shown by (26), the focal length ratio the range of scanned angles , and therefore the focusing patterns performance, as will be illustrated in the following subsection. B. Effect of the Focal Length Figs. 7–8 show the results obtained for several LWL with , and unifixed length with %. By varying the focus vertical form ( and ), the focal length ratio position is increased from to . The lens directivity, is ingain and sidelobe level decrease as the focal distance creased, which is due to the drop in the range of scanned angles (26) as is augmented. As tends to infinity, the range of scanned angles (26) reduce to , and a LWA radiating at broadside with very low focal gain is obtained.

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()

Fig. 9. Effect of the LWL length L in the tapering of the leaky-mode prop= z L = . agation functions for a constant focal ratio y =L

Fig. 11. Effect of the amplitude illumination M z in the tapering of the leakyy z  mode propagation functions (L %).

Fig. 10. Axial and transverse cuts for the LWLs of Fig. 9.

Fig. 12. Axial and transverse cuts for the LWLs of Fig. 11.

To provide a high gain focusing pattern as we increase the must be increased accordingly, so that focal distance and (26) are kept constant. This is illustrated , in Figs. 9–10 for several LWL with fixed (from to , as in and different focal distances is increased from to (as in Figs. 7–8), so that Figs. 5–6). As can be seen in Figs. 9, when the focal ratio is kept constant , the contribution of scanned angles is kept . From Fig. 10 it is observed that fixed to the focusing pattern performance (focus size and surrounding lobes level) is maintained, while the focal gain is increased for , as a result of the focal gain definition higher values of (23). These results are in accordance with general theory of focused apertures [14].

as can be seen in Figs. 11–12. Uniform illumination shows the best trade-off between axial and transverse lobe levels. These results are consistent with [15], [16]. The control of the level of lobes surrounding the focal point is of critical importance in certain applications where the fields must be minimized away from the prescribed focus, such as hyperthermia therapy [1]. As shown in Fig. 11(b), this can be (9), while done through proper tapering of the leakage rate unchanged. keeping the angular distribution

= 1 2( =

2)

C. Effect of the Amplitude Illumination All previous results were obtained for uniform amplitude il, in which constant power per unit of length is lumination radiated along all the LWA length. As is known [15], [16], tastrongly affects the pering of the amplitude illumination axial and transverse near-field patterns. Figs. 11–12 show the results obtained for a LWL with and % when is varied from the uniform case. When a cosine amplitude function is synthesized, most of the power is radiated from the central length of the LWA, and the forelobes and aftlobes in the axial pattern increase, while the sidelobe level in the transverse pattern decreases. On the contrary, inverse amplitude tapering in which the edges of the LWA contribute more to radiation than the center, reduces the forelobes and aftlobes, but increases the sidelobes,

= 20

90

=

= 10

=

D. Effect of the Focal Offset LWAs of uniform type are limited to radiation in a subset of the forward quadrant, so that the maximum and minimum to scanning angles are typically restricted from [17], [22]. As a consequence, the horizontal pois also limited according to (25), and sition of the focus the focus must be offset with respect to the centered position for practical designs of LWL based on forward scanning LWAs. Fig. 13(a) illustrates that the displacement of the focus posi(centered focus) to tion from allows a reduction in the range of scanned angles to practical values for LWAs of uniform type . As a result of this offset, the LWL focusing pattern (shown in Fig. 13(b)) is tilted and the depth of focus is enlarged, decreasing the lens directivity and the focal gain. IV. DESIGN OF A FREQUENCY-STEERABLE LWL The design of a LWL operating at 5.5 GHz is presented in this section using the leaky-mode theory developed above, and the results obtained with commercial full-wave 3D simulator

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= 20

Fig. 15. Tapered dimensions of hybrid LWL with L  to obtain the tapering of the leaky-mode wavenumber for the synthesis of a focusing pattern  y  , uniform M z ;  %. at 5.5 GHz with z

= 25

= 20

= 30

Fig. 13. (a) Effect of the focus offset z (L  ;y  , uni%) in the tapering of the leaky-mode propagation form M z ;  functions. (b) Focusing pattern.

()

= 90

Fig. 14. LWA in hybrid waveguide printed-slot-circuit technology (a) cross section dimensions, (b) non-tapered LWA, (c) tapered LWA to design a LWL.

CST MWS [31] will be compared with the proposed theory for validation. Also, the concept of frequency scanning of the focus will be illustrated and validated with this example. As shown in prior sections, the design of rectilinear LWLs depends on the use of a LWA which presents independent and simultaneous control of the leaky-mode pointing angle and leakage rate over a wide range of values. Several examples of rectilinear LWAs of uniform type in pure waveguide technology with flexible control of the leaky-mode complex propagation constant have been proposed in recent decades [17]. However, the tapering of these LWAs is based on the variation of the waveguide cross-section dimensions, therefore being difficult and expensive to realize. A much more flexible LWA is the one based on hybrid waveguide printed-circuit technology [26]–[28], shown in Fig. 14. In this case, there is no need to

= 25

()

= 90

modify the dimensions of the host waveguide in order to control the leaky-mode propagation constant; this can be achieved by simply tapering the dimensions of a planar printed circuit. mainly determines the As explained in [27], the slot width , while the slot popointing direction of the leaky-mode . Both the sition predominantly controls its leakage rate slot width and position can be consequently tapered to synthesize a focusing pattern. The variation of the slot dimensions along the LWL length to synthesize the requested tapered leaky-mode wavenumber can be obtained using the synthesis procedure described in [27] and from are shown in Fig. 15. Particularly, the variation of from 4.5 mm to 17 mm provides the requested tapering of to , which allows convergence of fields at the focal point according to the theory described in Section II. Uniform illumination and % is synthesized by tapering the slot position so that the requested function (given by (9)) is obtained. The near-fields obtained with the leaky-mode theory and those obtained with CST (using the time domain solver with waveport excitation and 3 passes of mesh adaption for dB scale, convergence) are shown in Fig. 16(a) in a and show a focusing region generated at the desired point . Axial and transverse cuts of the focusing patterns and represented with dotted ( lines in Fig. 16(a)) are plotted in Fig. 16(b) for precise comparison. Good agreement is observed in the depth and width of focus, and also in the sidelobe, forelobe and aftlobe levels, thus validating the proposed LWL analysis technique and the design concept. The focal region can be scanned by changing the frequency of operation. It is well known [17] that leaky-modes have a dispersive complex propagation constant, with increasing values of and decreasing values of as the frequency of operation is increased. The dispersive response of the leaky-mode depends on each particular type of LWA. In our case, the tapering of the leaky-mode complex propagation constant along the designed LWL in hybrid technology can be computed for different

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Fig. 16. Near-fields at 5.5 GHz (comparing theory and CST) for the LWL of Fig. 15 (a) 2D focusing pattern (ZY plane). (b) Axial and transverse cuts.

0

Fig. 18. Frequency scanning of the focusing pattern ( 6 dB scale).

Fig. 17. Variation of the tapered leaky-mode pointing direction and illumination with frequency.

frequencies using the method of moments full-wave technique described in [27] (27) From (27), the resulting tapered pointing direction (7) and amplitude illumination (3) can be derived for each frequency of interest, as plotted in Fig. 17. As can be seen in Fig. 17, changes from at 5.5 at 6.1 GHz, and to at 4.9 GHz. GHz to also changes strongly with frequency from the uniformly illuat 5.5 GHz. Particularly, as frequency is minated decreased from 5.5 GHz, the sections at the far end of the LWA

are below cut-off and do not contribute to radiation. As a result of these effects, the focusing region is scanned in the plane as frequency is increased from 4.7 GHz to 6.5 GHz, as shown in dB scale to highlight Fig. 18 (these plots are represented in the focal region in which 75% of energy is concentrated). The agreement observed between the leaky-mode theory and CST results for the entire scanning range validate the concept of frequency scanning of the focusing pattern. The focus width and depth and the surrounding lobes level are minimized in the 5.3–5.7 GHz band, obtaining the higher focal gain at these frequencies. For lower frequencies, the effective illuminated LWL length decreases as a result of the higher leakage rate (see in Fig. 17). However, the effective focal length becomes almost constant as a result of the decrease in the focus position , keeping nearly unchanged the focus size. The aftlobe level increases as frequency is decreased as a result of the change in the focus location and the perturbation of the illumination, which becomes stronger at the far end of the illumiin Fig. 17). These results are in accornated LWL (see dance with the parametric study performed in Section III. On the other hand, the focus offset becomes more evident for frequencies above 5.7 GHz, increasing the focus size and aftlobe level in accordance with Section III.D. The proposed theory is very efficient for the initial analysis and design of these electrically large LWL structures (the region by at shown in Figs. 16 and 18 covers an area of 5.5 GHz), taking only 1 minute per frequency point, whereas in CST the computational time is more than 2 hours and requires a large amount of memory. For the final accurate design and prediction of the LWL performance a much slower full-wave three-dimensional solver can be used.

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V. CONCLUSION The theory for the efficient analysis and design of two-dimensional frequency steerable leaky-wave lenses of rectilinear geometry has been presented in this paper. Using tapered leaky-mode complex propagation constant expressions, the illumination of these electrically large LWL structures and their focusing near-field patterns have been efficiently derived for the first time. Compared to phased-array lenses, LWL provide much simpler feeding, avoiding the use of expensive and lossy distribution networks. On the other hand, rectilinear LWL avoid the problematic curvature used in conformal LWL. The concept of frequency steering xof the focus, which is inherent to all LWLs, has been illustrated for the first time. It might have interesting applications for multispot heating/sensing, where several focusing regions are synthesized at the same time and independently illuminated by selecting the appropriate frequency. Also, LWLs can be used to improve the performance of spectral-spatial demultiplexers, such as those recently proposed to create millimeter-wave electrical prisms [24] and microwave analog real-time spectrum analyzers [25]. All these novel results have been validated by designing a rectilinear LWL in hybrid waveguide printed-circuit technology, and comparing the near-field patterns with those obtained with 3D full-wave electromagnetic simulations. Although these results are concerned with 2D focusing by a LWL embedded in a parallel-plate waveguide, the extension to 3D focusing can be obtained by arranging a linear phased array of LWLs. REFERENCES [1] C. H. Durney and M. F. Iskandar, , Y. T. Lo and S. W. Lee, Eds., “Antennas for medical applications,” in Antenna Hand Book: Theory, Applications, and Design. New York: Van Nostrand, 1988, ch. 24. [2] A. P. Anderson and M. Melek, “Feasibility of focused microwave array system for tumor irradiation,” Electron. Lett., vol. 15, pp. 564–565, Aug. 1979. [3] W. Gee, S. W. Lee, C. A. Cain, R. Mittra, and R. L. Magin, “Focused array hyperthermia applicator: Theory and experiment,” IEEE Trans. Biomed. Eng., vol. 31, pp. 38–46, Jan. 1984. [4] J. T. Loane, III and S. Lee, “Gain optimization of a near-field focusing array for hyperthermia applications,” IEEE Trans. Microw. Theory Tech., vol. 37, pp. 1629–1635, Oct. 1989. [5] E. Nyfors and P. Vainikainen, Industrial Microwave Sensors. Norwood, MA: Artech House, 1989. [6] M. Bogosanovic and A. G. Williamson, “Microstrip antenna array with a beam focused in the near-field zone for application in noncontact microwave industrial inspection,” IEEE Trans. Instrum. Meas., vol. 56, no. 6, pp. 2186–2195, Dec. 2007. [7] K. D. Stephan, J. B. Mead, D. M. Pozar, L. Wang, and J. A. Pearce, “A near field focused microstrip array for a radiometric temperature sensor,” IEEE Trans. Antennas Propag., vol. 55, no. 4, pp. 1199–1203, Apr. 2007. [8] W. C. Brown, “The history of power transmission by radio waves,” IEEE Trans. Microw. Theory Tech., vol. 32, pp. 1230–1242, Sep. 1984. [9] J. O. McSpaddan and J. C. Mankins, “Space solar power programs and microwave wireless power transmission technology,” IEEE Microw. Mag., vol. 3, no. 4, pp. 46–57, Dec. 2002. [10] S. Karimkashi and A. A. Kishk, “Focused microstrip array antenna using a Dolph-Chebyshev near-field design,” IEEE Trans. Antennas Propag., vol. 57, no. 12, pp. 3813–3820, Dec. 2009. [11] D. K. Cheng and S. T. Mosely, “On-axis defocus characteristics of the paraboloidal reflector,” IEEE Trans. Antennas Propag., vol. 3, pp. 214–216, Oct. 1955.

[12] L. Shafai, A. A. Kishk, and A. Sebak, “Near field focusing of apertures and reflector antennas,” in Proc. IEEE Conf. on Communications, Power and Computing WESCANEX97, Winnipeg, MB, May 22–23, 1997, pp. 246–251. [13] R. W. Bickmore and R. C. Hansen, “Antenna power densities in the Fresnel region,” Proc. IRE, vol. 47, pp. 2119–2120, Dec. 1959. [14] J. W. Sherman, “Properties of focused apertures in the Fresnel region,” IRE Trans. Antennas Propag., vol. 10, pp. 399–408, Jul. 1962. [15] W. J. Graham, “Analysis and synthesis of axial field patterns of focused apertures,” IEEE Trans. Antennas Propag., vol. 31, pp. 665–668, Jul. 1983. [16] R. C. Hansen, “Focal region characteristics of focused array antennas,” IEEE Trans. Antennas Propag., vol. 33, pp. 1328–1337, Dec. 1985. [17] A. A. Oliner, , R. C. Johnson, Ed., “Leaky-wave antennas,” in Antenna Engineering Handbook, 3rd ed. New York: McGraw-Hill, 1993, ch. 10. [18] I. Ohtera, “Focusing properties of a microwave radiator utilizing a slotted rectangular waveguide,” IEEE Trans. Antennas Propag., vol. 38, pp. 121–124, Jan. 1990. [19] I. Ohtera, “Diverging/focusing of electromagnetic waves by utilizing the curved leakywave structure: Application to broad-beam antenna for radiating within specified wide-angle,” IEEE Trans. Antennas Propag., vol. 47, pp. 1470–1475, Sep. 1999. [20] I. Ohtera, “Estimation of the radiation patterns of diverging/focusing type of leakywave antennas,” Microw. Opt. Technol. Lett., vol. 33, pp. 358–360, Jun. 2002. [21] O. Losito, “The diverging-focusing properties of a tapered leaky wave antenna,” in Proc. 3rd Eur. Conf. on Antennas and Propagation, EUCAP2009, Mar. 2009, pp. 1304–1307. [22] P. Burghignoli, F. Frezza, A. Galli, and G. Schettini, “Synthesis of broadbeam patterns through leaky-wave antennas with rectilinear geometry,” IEEE Antennas Wireless Propag. Lett., vol. 2, pp. 136–139, 2003. [23] I.-H. Lin, C. Caloz, and T. Itoh, “Near-field focusing by a nonuniform leaky-wave interface,” Microw. Opt. Technol. Lett., vol. 44, pp. 416–418, Mar. 2005. [24] O. Momeni and E. Afshari, “Electrical prism: A high quality factor filter for millimeter-wave and terahertz frequencies,” IEEE Trans. Microw. Theory Tech., vol. 57, no. 11, pp. 2790–2799, Nov. 2009. [25] S. Gupta, S. Abielmona, and C. Caloz, “Microwave analog real-time spectrum analyzer (RTSA) based on the spectral-spatial decomposition property of leaky-wave structures,” IEEE Trans. Microw. Theory Tech., vol. 57, pp. 2989–2999, Dec. 2009. [26] J. L. Gómez, D. Cañete, and A. Álvarez-Melcón, “Printed-circuit leaky-wave antenna with pointing and illumination flexibility,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 8, pp. 536–538, Aug. 2005. [27] J. L. Gómez, G. Goussetis, A. Feresidis, and A. A. Melcón, “Control of leaky-mode propagation and radiation properties in hybrid dielectric-waveguide printed-circuit technology: Experimental results,” IEEE Trans. Antennas Propag., vol. 54, no. 11, pp. 3383–3390, Nov. 2006. [28] J. L. Gómez, S. Martínez, and A. A. Melcón, “Simple analysis and design of a new leaky-wave directional coupler in hybrid dielectricwaveguide printed-circuit technology,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 9, pp. 3534–3542, Sep. 2006. [29] A. F. Peterson, S. L. Ray, and R. Mittra, Computational Methods for Electromagnetics. : IEEE Press, 1998, p. 451. [30] C. A. Balanis, Advanced Engineering Electromagnetics.. New York: Wiley, 1989. [31] CST Microwave Studio, CST 2006. Darmstadt, Germany.

José Luis Gómez Tornero (M’06) was born in Murcia, Spain, in 1977. He received the Telecommunications Engineer degree from the Polytechnic University of Valencia (UPV), Valencia, Spain, in 2001 and the “Laurea” Ph.D. degree (cum laude) in telecommunication engineering from the Technical University of Cartagena (UPCT), Cartagena, Spain, in 2005. In 1999, he joined the Radio Communications Department, UPV, as a Research Student, where he was involved in the development of analytical and numer-

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ical tools for the automated design of microwave filters in waveguide technology for space applications. In 2000, he joined the Radio Frequency Division, Industry Alcatel Espacio, Madrid, Spain, where he was involved with the development of microwave active circuits for telemetry, tracking and control (TTC) transponders for space applications. In 2001, he joined the Technical University of Cartagena (UPCT), Spain, as an Assistant Professor. From October 2005 to February 2009, he held the position of Vice Dean for Students and Lectures affairs in the Telecommunication Engineering Faculty, UPCT. Since 2008, he has been an Associate Professor at the Department of Communication and Information Technologies, UPCT. His current research interests include analysis and design of leaky-wave antennas and the development of numerical methods for the analysis of novel passive radiating structures in planar and waveguide technologies. Dr. Gómez Tornero received the national award from the foundation EPSONIbérica for the best Ph.D. project in the field of technology of information and communications (TIC) in July 2004 and the Vodafone Foundation-COIT/AEIT (Colegio Oficial de Ingenieros de Telecomunicación) award for the best Spanish Ph.D. thesis in the area of advanced mobile communications technologies in June 2006. This thesis was also awarded as the best thesis in the area of electrical engineering, by the Technical University of Cartagena in December 2006. He was appointed CSIRO Distinguished Visiting Scientist by the CSIRO ICT Centre, Sydney, in February 2010.

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

Fernando Daniel Quesada-Pereira (S’05–M’07) was born in Murcia, Spain, in 1974. He received the Telecommunications Engineer degree from the Technical University of Valencia (UPV), Valencia, Spain, in 2000 and the Ph.D. degree from the Technical University of Cartagena (UPCT), Cartagena, Spain, in 2007. In 1999, he joined the Radiocommunications Department, UPV, as a Research Assistant, where he was involved in the development of numerical methods for the analysis of anechoic chambers and tag antennas. In 2001, he joined the Communications and Information Technologies Department, UPCT, initially as a Research Assistant, and then as an Assistant Professor. In 2005, he spent six months as a Visiting Scientist with the University of Pavia, Pavia, Italy. In 2009, he was an invited researcher for five months at the Technival University of Valencia (iTeam), Spain. His current scientific interests include IE numerical methods for the analysis of antennas and microwave devices.

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

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

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

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Sub-Wavelength Profile 2-D Leaky-Wave Antennas With Two Periodic Layers Carolina Mateo-Segura, Student Member, IEEE, George Goussetis, Member, IEEE, and Alexandros P. Feresidis, Senior Member, IEEE

Abstract—A fast and accurate analysis and synthesis technique for high-gain sub-wavelength 2-D Fabry-Perot leaky-wave antennas (LWA) consisting of two periodic metallodielectric arrays over a ground plane is presented. Full-wave method of moments (MoM) together with reciprocity is employed for the estimation of the near fields upon plane wave illumination and the extraction of the radiation patterns of the LWA. This yields a fast and rigorous tool for the characterisation of this type of antennas. A thorough convergence study for different antenna designs is presented and the operation principles of these antennas as well as the radiation characteristics are discussed. Moreover, design guidelines to tailor the antenna profile, the dimensions of the arrays as well as the antenna directivity and bandwidth are provided. A study on the radiation efficiency for antennas with different profiles is also presented and the trade off between directivity and radiation bandwidth is discussed. Numerical examples are given throughout to demonstrate the technique. A finite size antenna model is simulated using commercial software (CST Microstripes 2009) which validates the technique. Index Terms—Artificial magnetic conductor, high-gain antennas, high impedance surfaces, leaky-wave antennas (LWAs), resonant cavities, sub-wavelength resonators.

I. INTRODUCTION ERIODIC planar 2-D leaky wave antennas (LWA) have attracted significant interest in recent years due to their high gain and efficiency performance in conjunction with the advantages in terms of the low fabrication complexity in the microwave and millimeter-wave region [1]–[3]. Typical implementations of high-gain 2-D LWAs consist of a single layer periodic metallo-dielectric array acting as a partially-reflecting surface (PRS) and forming a resonant cavity with a ground plane positioned at a distance of about half-wavelength [4]–[8]. The bandwidth and gain of such antennas depend on the reflection (amplitude and phase) of the PRS, which in turn is determined from the PRS geometrical characteristics

P

Manuscript received September 12, 2009; revised June 22, 2010; accepted September 22, 2010. Date of publication December 03, 2010; date of current version February 02, 2011. The work of G. Goussetis was supported by the Royal Academy of Engineering under a five year research fellowship. C. Mateo-Segura and A. P. Feresidis are with the Department of Electronic and Electrical Engineering, Loughborough University, Leicestershire LE11 3TU, U.K. (e-mail: [email protected]). G. Goussetis is with the Institute of Electronics Communications and Information Technology (ECIT), Queen’s University Belfast, Belfast, BT9 3DT, Northern Ireland (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2096384

[5]. Uniform 2-D LWAs have also been presented in the past [9]–[12] consisting of dielectric layers of alternating thickness and material constants, stacked over a ground plane. However, to achieve high gain a number of layers or particularly high dielectric constants are required. An efficient analysis of periodic 2-D LWAs has been recently presented based on the principle of reciprocity and full-wave spectral domain analysis [7], [8]. Highly directive planar antennas were achieved for PRSs consisting of metallic patches or apertures etched in a fully conducing sheet. In all cases however the antenna profile, which is mainly determined by the resonance condition of the cavity, has always been approximately half-wavelength. Over the last few years, planar periodic metallic arrays placed on a grounded dielectric substrate have been shown to behave as high impedance surfaces (HIS). Due to the effective high surface impedance, such surfaces reflect incident plane waves in-phase and are termed artificial magnetic conductors (AMC) [13], [14]. Recently, a planar HIS ground plane has been proposed as a means to reduce the profile of the high-gain resonant cavity 2-D LWA from approximately half wavelength to quarter wavelength [15] or to lower sub-wavelength values [16]. More recently, the LW analysis of thin subwavelength antennas based on a PRS and a HIS has been carried out using FDTD [17]. The aim of this paper is to present for the first time a fast and rigorous spectral-domain analysis of thin 2-D leaky-wave antennas consisting of a PRS array and a HIS ground plane (Fig. 1). A Method of Moments (MoM) analysis of 2-D leakywave antennas employing two periodic surfaces is initially used to determine the currents induced on the patches by an incident plane wave. Reciprocity is subsequently employed to calculate the far-field radiation patterns, reducing the calculations to the sampling of the near field within the antenna cavity. Based on this efficient method, design guidelines are produced for thin sub-wavelength antenna profiles. In addition, the effects of the profile reduction on the gain and radiation efficiency are studied. The paper is organized as follows. In Section II a brief overview of the reciprocity theory together with the spectral domain method of moment analysis of double-layer periodic arrays is presented. Following a thorough convergence analysis, the operation principles and radiation characteristics of these antennas are discussed. Section III provides design guidelines, addressing the design of the antenna profile, the dimensions of the HIS, the directivity and the bandwidth. A study on the antenna efficiency with decreasing profile is also presented. Lastly, a finite size antenna model is simulated using a 3-D electromagnetic software (CST Microstripes 2009) in order to

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MATEO-SEGURA et al.: SUB-WAVELENGTH PROFILE 2-D LEAKY-WAVE ANTENNAS WITH TWO PERIODIC LAYERS

Fig. 1. (a) Resonant cavity formed by metamaterial ground plane (HIS) and partially reflective surface (PRS) with excitation source inside the cavity. (b) Unit cell of a square patch PRS array and (c) HIS array.

check the agreement between the proposed technique and a realistic model. II. NEAR AND FAR FIELDS OF THE 2-D LWA The structure under investigation consists of a ground plane, that supports the first periodic a dielectric layer of thickness array, an air cavity of thickness and a second periodic array printed on another dielectric layer of thickness . A schematic drawing of the antenna is shown in Fig. 1. The top array acts as a partially reflective surface (PRS), while the combination of the lower array with the metal backing forms the HIS ground plane that produces a tailored reflection phase [16], [17]. In the following we employ a working example where both arrays are capacitive and consist of metallic square patches. The periodicities of the two arrays are assumed commensurate (Fig. 1). In order to provide high reflectivity, the PRS typically operates close to the array resonance. For this reason, square patches with dimensions of the order of half wavelength are employed. The HIS operation resembles that of an artificial magnetic conductor [15], which can be achieved with smaller unit cells. Here the periodicity of the HIS ground plane is half that of the PRS array in both directions. Reciprocity is invoked for the analysis of this LWA, which reduces the estimation of the far-field radiation characteristics to the calculation of the fields at an observation point inside the antenna when illuminated by a plane wave of fixed magnitude and . The technique is described a varying angle of incidence in [7] for antennas consisting of a single periodic layer and exploits the fact that the receiving and transmitting field patterns for an antenna are identical [18]. Without loss of generality, in the following we assume that the antenna is fed by a Hertzian dipole parallel to the -axis and located at the observation point , inside the cavity. By scanning the relative field strength, at the observation point, the (receiving) radiation pattern of the antenna can be obtained. In a practical implementation, a half

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wavelength dipole can be utilized as in [16] without significant perturbation in the antenna directivity. Using this technique in conjunction with a full-wave modeling tool, the radiation pattern of the antenna shown in Fig. 1 can be obtained in a fast and rigorous fashion [7]. A commonly employed technique for the modeling of cascaded layers of doubly periodic arrays, such as frequency selective surfaces (FSS), is based on applying the periodic spectral domain MoM for each array and subsequently cascading them using their generalized scattering matrix (GSM) [19], [20]. This method requires GSMs of increased order for more closely packed arrays, which eventually becomes inefficient for arrays at very close proximity. Other techniques such as the Finite-Difference Time-Domain (FDTD) or the Finite Element Method (FEM) can be time consuming particularly for a fine discretization of the geometry. Here we have applied a periodic spectral domain coupled electric field integral equation for the full-wave modeling of the entire structure shown in Fig. 1 [21]. Although less versatile, this technique is computationally more efficient since the coupling between the two arrays is inherently included in the algebraic formulation. The commensurate periodicities of the two arrays ensure that the set of Floquet Space Harmonics (FSH) associated with the PRS array is also suitable for expanding the fields at the HIS array. A finite summation of zero-ended orthogonal entire domain basis functions is employed to express the current on the patches, hence ensuring further computational efficiency. An in-house MoM software has been developed for the computation of the near fields excited at a point inside the antenna cavity under plane wave illumination. A. Convergence Study Although the application of the MoM for the analysis of structures such as the one depicted in Fig. 1 is well-established [19], a careful convergence analysis is necessitated due to some features specific to our study. In particular, this method has often been employed for the calculation of the far-field reflection or transmission properties of periodic arrays. At large distances from the array (typically a few wavelengths away), only the propagating FSH contributes to the fields [22]. Therefore obtaining the reflection and transmission of the fundamental FSH suffices if no grating lobes occur. A limited number of FSHs (in the order of 20 in each direction for 2-D periodic arrays) is typically sufficient to predict the interaction of the incoming plane wave with the periodic array and extract the far field reflection and transmission coefficient. In contrast to that case, here we are interested in the estimation of the fields in the proximity of the periodic array. Evanescent FSHs, that describe reactive fields bounded to the array, can contribute significantly to the overall field in the proximity of the array. In [23] it was demonstrated that increased number of FSHs are required for the estimation of the fields at distances increasingly close to the periodic arrays. In addition, the HIS array of Fig. 1(c) consists of electrically small and non-resonant elements, typically separated by narrow gaps. As will be shown in the following, accurate modeling of the currents in this case require increased number of entire domain basis functions.

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D= = 1 15 mm = 4 0 mm = 9 mm = 2 2 4 15mm = 4 25mm = 4 35mm = 4 45mm . = = 4 15

Fig. 2. Magnitude of the y-component of the electric field at the center of the , z h= ) for an antenna with cavity and the center of the unit cell (x y : ,h : , for the PRS: square patches L dimensions (in mm) D ,h : and " : and for the HIS: square patches L : , : and " : as calculated with MoM for increasing number of h entire domain basis functions in the HIS.

8

=15 = 1 15

= =0 = 2 = 9 0 = 5 45 = 2 55 = 22

The convergence requirements are here demonstrated by means of an example involving a PRS consisting of patches with edge 8.0 mm and periodicity 9.0 mm printed on a dielectric and relative permittivity 2.55. slab of thickness In the remaining, this will be referred to as PRS1. PRS1 is ( at 14 GHz) above located at a distance the HIS array, which consists of patches with edge 4.15 mm and and is printed on a dielectric slab of thickness relative permittivity 2.2. Fig. 2 shows the y-component of the (e.g., center of the cavity) and at electric near field at the center of the unit cell as estimated with increasing number of basis functions in the HIS patch and considering 120 FSHs in both -, and -directions when the antenna is excited by a normally incident plane wave. As shown, 30 basis functions are required for convergence at 1% level, which corresponds to about a tenfold increase compared to the number of entire domain basis functions employed in the modeling of typical dipole FSSs. Our study (not shown here for brevity) also showed that five basis functions suffice to model the current on the PRS patches. A further study such as the one conducted in [23] (not shown here for brevity) has been carried out in terms of the number of FSHs needed for convergence showing that 120 FSHs suffice to expand the fields in each direction, which is about thrice as many as commonly used in the far-field estimation of the reflection characteristics from this type of arrays. The total number increases with the square of the required FSH in each direction, which for this case is 14400 FSH for convergence level of 1%. Antennas with thinner profile can further increase the required number of FSHs for an accurate estimation of the near fields. Since HIS and PRS structures can be designed using dielectric substrates of various thicknesses, when referring to antenna profile ( in Fig. 1) the dielectric substrate of the HIS ground plane and PRS is not considered. In order to demonstrate the above, in the following we employ a series of broadside LWAs with different profiles, all designed to produce broadside radiation at 14 GHz. Detailed design guidelines for these LWAs will follow in the next section. An

Fig. 3. Reflection phase by HIS1, HIS2, HIS3, HIS4 and HIS5 with ," : ,h : : ,L and L : : L : : ,L and. L

antenna formed using PRS1 and no HIS array has a profile of and approximately equal to half wavelength in the following will be referred to as antenna #0. Antennas , , , with profile, , approximately equal to and have also been designed using the same PRS1 and . In the following these HISs with varying patch size, will be referred to as antennas #1–#5 respectively and the corresponding HISs are numbered accordingly. All HISs are printed on a dielectric substrate of thickness and . The patch dimensions for the HISs are (in mm) , , , and respectively. The thickness of the cavities, , for antennas #1–#5 are (in mm) , , , and respectively. The far-field reflection phase for normally incident plane wave in the vicinity of 14 GHz is shown in Fig. 3 for all five HISs. As shown, in the frequency range of interest the reflection phase reduces for HISs with larger patches. This gives rise to antennas with lower profile. The number of FSH in each direction required for a converged estimation of the near fields within 1% for increasingly thinner cavities is shown in Fig. 4. From this figure we see that for antenna #0 with half-wavelength profile 1600FSH (40FSH in x and y direction) suffice , a total of while for instance for antenna #4 with profile 28900 (170FSH in x and y direction) must be considered. As shown in Fig. 4 the number of FSH in each direction increases approximately linearly as the profile of the cavity antenna decreases. B. Operation Principle This type of LWA is known to operate at a mode which is a perturbation of the first order TE and TM modes of the parallel plate waveguide formed between the PRS and the ground plane [17], [24]. By substituting one of the two parallel plates by a PRS, the fast waveguide modes are allowed to radiate. For broadside radiation, the antenna operates close to the cutoff of the parallel plate waveguide TE/TM mode. At this frequency the distance between the parallel plates is equal to half wavelength, and a standing wave in the transverse direction is formed. This leads to a resonant Fabry-Perot type cavity model for this type

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Fig. 4. Minimum number of Floquet Space Harmonics required in each direction for 1% convergence in the calculation of the fields in the center of the antenna as a function of the profile, h.

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Fig. 6. 3-D radiation pattern of antenna #2 formed by PRS1 and HIS2 with profile h = 5:45 mm ( =4) at 14 GHz.



C. Radiation Characteristics of Thin 2-D LWA The radiation pattern for this type of LWAs at a fixed frequency can be obtained by scanning the near fields at the obserof the vation point inside the cavity for different angles incoming plane wave [7]. Once this procedure has been completed for all possible incoming directions, the integration of the obtained values can be used as reference in order to obtain the directivity, so that:

(1)

where

Fig. 5. Distribution of the y-component of the electric field at the center of the unit cell (x = y = 0) and varying z for six different antennas with different profile (shown in the legend as a function of the wavelength, ). All antennas are designed to produce a broadside beam at 14 GHz.

of antennas. For this reason, antennas formed with a single periodic array (only as PRS, [5]) typically have a profile of half wavelength. This is demonstrated in Fig. 5, where the -component of the electric field is plotted against the -axis for antenna #0 with at the center of the unit cell. The inprofile approximately troduction of a HIS ground plane modifies the reflection characteristics and hence the transverse resonance condition [15]. In this way, the HIS reduces the cutoff frequency of the parallel plate waveguide and eventually the antenna profile. Fig. 5 also shows similar field distributions for antennas #1–#5 with increasingly reduced profile. It is interesting to note that thinner antennas increasingly deviate from the typical half wavelength field distribution, exhibiting a field maximum at the plane of the HIS array. This is in agreement with the artificial magnetic boundary condition [25]. Moreover, significant near field enhancement is observed for thinner antennas which can be useful in, e.g., sensor applications.

is the directivity of the antenna in the direction and is the electric field at the observation point . The theoretical excited by a plane wave incident from background is well documented in [7] and therefore not further expanded here. As an example of this procedure, Fig. 6 shows the 3-D radiation pattern for antenna #2 with cavity profile approximately at 14 GHz. As shown, a pencil beam with directivity of 20.9 dB at broadside is produced. Since the analysis is based on an infinite array assumption, no sidelobes emerge below the grating lobe frequencies for a single fast propagating mode. To demonstrate the frequency dependence of this radiation pattern, Fig. 7 shows the H- and E-plane cuts of this radiation pattern for three frequency points. In agreement with the performance of half-wavelength antennas, as the frequency increases from that of broadside radiation, a conical pattern is obtained. III. DESIGN CONSIDERATIONS In this section we demonstrate the application of near-field calculation and reciprocity in the design of low-profile 2-D LWA with two periodic arrays. Several geometrical degrees of freedom are available, and the design procedure is shown below by means of examples. The five HISs of Section II (Fig. 3) are employed.

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Fig. 8. Magnitude of the y-component of the electric field in the center of the and at for an antenna with PRS1 and HIS4 for unit cell varying profile, .

(x = y = 0) h

z = h=2

Fig. 7. a) H-plane and b) E-plane cuts of antenna #2 radiation pattern at dif, , for PRS1: ferent frequencies. The dimensions (in mm) , and for HIS2: , . and

L = 8 h = 1 :5 1:15 " = 2:2

" = 2:55

D = 9:0 h = 5:45 = 4:15 h = L

A. Antenna Profile To a first approximation the profile of 2-D LWA with two periodic layers (Fig. 1) can be determined employing a ray optics model [15]. However for thinner cavities the single mode approximation becomes less accurate and therefore this model provides a worse estimation. Calculation of the near-fields inside the cavity in conjunction with reciprocity arguments can be employed towards a more accurate procedure for determining the antenna profile. Reciprocity suggests that at the central operating frequency of broadside antennas, the fields at the observation points are maximised for a given normally incident wave. Therefore, scanning the fields over the frequency yields the central operating frequency of the antenna at the peak of the observed curve. By modifying the antenna profile, it is then possible to tune the design to the specified frequency. This is shown here by means of an example involving an antenna formed by PRS1 and HIS4. The ray optics model suggests that for broadside operating frequency at 14 GHz, the antenna profile should be 3.0 mm. Commencing from this value, Fig. 8 shows the near fields at the center of the antenna cavity for different antenna profiles, . As expected from the ray optics model, lower cavity profiles yield operation at higher frequencies. The optimal profile for this antenna for broadside operation , which is approxat 14 GHz is determined to be imately . This value varies by 5% compared to the value obtained through the ray optics model where 30BF and 170FSH have been considered in order to calculate the reflection phase of HIS4 (Fig. 3).

Fig. 9. Magnitude of the y-component of the electric field in the center of the unit cell for an antenna with PRS1 for varying and at HIS element, .

(x = y = 0) L

z = h=2

different values of the lower array element size (corresponding to HIS1, HIS2, HIS3, HIS4 and HIS5) for normal incidence in Fig. 9. As shown, by increasing the lower array element size, the operating frequency of the antenna for broadside radiation reduces. This is in agreement with previous results [16] and to a ray optics approximation can be attributed to the reduced reflection phase provided by HIS arrays with larger element size (Fig. 3). This graph suggests that it is possible to design an antenna that operates at a specified frequency with a specific profile and a given PRS solely by modifying the lower array dimensions. Since the reflectivity of the PRS is typically directly associated with the directivity of the antenna, this process can be useful in the design of antennas that meet a required gain specification.

B. HIS Dimensions

C. Directivity

For a design procedure that commences from a given operating frequency, as well as antenna profile and a fixed PRS, reciprocity can be employed to determine the required HIS dimensions to meet the radiation requirements. Antenna #2 composed of PRS1 at a distance 5.45 mm above HIS2 is employed as an example. The fields at the observation point in the center of the cavity and the unit cell are shown versus frequency for five

Leaky wave theory suggests that among electrically large LWAs, those with lower leakage rates produce larger radiating apertures and therefore more directive patterns. Intuitively, it is plausible to expect that antennas formed by more reflective PRSs will produce lower leakage rates. Indeed, simple ray optics model predicts that a more reflective PRS results in a higher directivity [4]. A method to extract the directivity of the

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Fig. 10. Reflection coefficient (magnitude and phase) by two PRSs with di: , L : and L : . mensions (in mm)

D=90

=80

=52

LWA under consideration was discussed in the previous section. Here by means of an example employing two LWAs with the same profile that operate at the same frequency, we demonstrate the possibility to tailor the directivity by modifying the PRS. Following the technique outlined above, we design two antennas for broadband radiation at 14 GHz with equal profile, (profile ) employing two PRSs with different reflectivity at 14 GHz. In particular, we employ as one example antenna #2 of the previous section. The reflection magnitude and phase of PRS1 is shown in Fig. 10. Fig. 10 also shows the reflection magnitude and phase of PRS2, which shares the same geometrical features with PRS1, but the patch size is . As shown, a capacitive square patch PRS with fixed periodicity becomes increasingly reflective below resonance for larger patches. An antenna that employs PRS2 and produces a broadside beam at 14 GHz can be designed by tuning the HIS dimensions to 4.19 mm, as discussed above. The E and H plane cuts of the antennas directivity are shown in Fig. 11. As shown, the antenna formed using PRS1 produces directivity approximately 9.2 dB greater than that formed using PRS2. This is mainly attributed to the higher reflectivity of PRS1 at 14 GHz (Fig. 10). This result implies that it is possible to design a tailored directivity simply by modifying the reflectivity of the PRS. It is further instructive to study the directivity of antennas with a fixed PRS and varying profile. Fig. 12 shows the directivity of antennas #0–#5, all of which employ the same PRS but have increasingly reduced profile. As shown, antennas with increasingly lower profile produce reduced directivity for a given PRS. This is in agreement with the observations in [17], where full wave simulations of finite antennas were employed to demonstrate a reduced illuminating aperture for thinner arrays. D. Bandwidth In Section II it was shown that commonly to all LWA, the radiation patterns of the antennas under investigation change with frequency due to the dispersion of the radiating mode. Beyond matching considerations, the 3 dB pattern bandwidth of the antenna can be obtained as the bandwidth within which the fields at the observation point drop within 3 dB of their peak

Fig. 11. Directivity a) H-plane and b) E-plane of two LWA formed with a PRS: and L : ,h : and " square patches L : , and an HIS: square patches L : and L : : : , with profile 5.45 mm operating respectively, h and " both at 14 GHz.

2 55

=8 = 1 15 mm

= 5 2 mm = 4 15 =22

= 1 5 mm = = 4 19 mm

Fig. 12. Directivity at boresight of antenna#0–antenna#5 at the operating frequency, 14 GHz.

value. The directivity bandwidth is similarly defined as the frequency range within which the directivity of the antenna at a certain direction (typically broadside) varies within 3 dB of its maximum value. The pattern and directivity bandwidth for the two antennas of Fig. 10 have been calculated and are plotted in Fig. 13. As shown, the 3 dB directivity and pattern bandwidths for antenna # 2 formed with PRS1 are 90% and 88% narrower than those formed with PRS2 respectively. In agreement with

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Fig. 14. Gain of antenna #2 for different loss tangent values of the dielectric substrate.

Fig. 13. Plots showing the a) directivity and b) radiation bandwidth of the two antenna designs of Fig. 11.

the observation made in [26], the directivity bandwidth for each antenna is wider than the pattern bandwidth. Since PRS1 is more reflective than PRS2, the cavity formed by PRS1 is more weakly coupled to the incoming plane wave and therefore produces a higher external quality factor than the cavity formed by PRS2. As the external quality factor (in case of no ohmic losses) is inversely proportional to the 3 dB fractional bandwidth, a higher external quality factor implies that the 3 dB fractional bandwidth is narrower [27]. For the antenna shown in Fig. 1, increased directivity is therefore associated with reduced bandwidth. We note that in this case the reactive power stored in the resonator is also higher. The technique described in this and the previous subsections provide the designer with tools to address the trade off between bandwidth and directivity. E. Antenna Efficiency So far this study has been undertaken assuming zero thermal losses. In practice, losses reduce the antenna efficiency. The radiation efficiency, , (i.e., antenna efficiency neglecting the mismatch) can be obtained as the ratio of the total radiated power in the presence of thermal losses over the total radiated power in the absence of thermal losses

(2)

are the fields at the observation point for a where given plane wave incidence assuming/neglecting thermal losses. If the effect of losses is introduced in the MoM formulation, then the antenna radiation efficiency can be readily obtained for the LWA under consideration. In practice, the main source of thermal losses for this type of antennas in the microwave regime arises from the dielectrics [17]. To a good approximation, we have therefore assumed that the metallization is perfectly conducting. The accepted gain obtained for antenna #2 by increasing the loss tangent of the dielectric substrates involved is shown in Fig. 14. As shown, losses of about 3.6 dB are incurred when the loss tangent value increases to 0.02 (e.g. equal to that of the FR4 substrate). This is because part of the near field stored in the antenna is dissipated as heating in the dielectric substrates. As it was shown in Fig. 5, the near fields stored in the vicinity of the antenna cavity are generally increasing for cavities of reduced profile. Increased thermal losses are therefore predicted in the presence of non-perfect dielectrics for antennas of lower profile. This is presented in Fig. 15 showing the efficiency of the antennas #0–#5 when the loss tangent of the dielectrics is either 0.02 (equal to that of FR4) or 0.001 (equal to that of Teflon). As a result of the dielectric losses, antenna #4 with profile exhibits efficiency reduced by 54% and 4.8% for the two values antenna. It of the loss tangent respectively compared to the is worth noting that the efficiency appears to increase for very , which is in agreement with thin antennas in the order of the observations in [17]. F. Finite Antenna The antennas studied in the previous sections have been designed under the consideration of infinite lateral size. In order to check the agreement of the proposed technique with a finite size antenna realization, a finite size model of antenna #2 is simulated using a 3-D electromagnetic simulator (CST Microstripes 2009). The lateral dimensions of the antenna , i.e., just over . These dimensions are have been chosen in order to achieve a practical design with minimized edge effects that can be compared with the infinite model. Further studies have shown that smaller antenna sizes

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with reduced profile. An analysis of the field distribution inside the cavity has demonstrated the operation principles and the radiation characteristics of these antennas. Using these tools, design guidelines for tailoring the antenna profile, the HIS dimensions, the directivity and the bandwidth have been presented. The trade off between bandwidth and directivity has also been discussed. A study on the effect of the profile thickness on the antenna efficiency has been carried out. The proposed technique has been validated by comparison with 3D full-wave electromagnetic simulations. REFERENCES

Fig. 15. Antenna efficiency for the antennas in Fig. 5 (#0–#5) considering a dielectric substrate with tan  = 0:02 (equal to that of FR4) and a dielectric substrate with tan  = 0:001 (equal to that of TEFLON).

Fig. 16. Directivity of infinite and finite antenna #2 design using the proposed technique and CST microstripes.

shift the resonance to higher frequencies and also degrade the radiation performance in terms of higher sidelobes and lower directivity, with all these effects being more pronounced for sizes less than . For simulation purposes, a center fed 3 mm long wire model dipole is used to feed the cavity. The radius of the wire is 0.15 mm and the distance from the HIS substrate is . Fig. 16 shows 2.725 mm (i.e., center of the cavity, the boresight directivity as a function of frequency for antenna #2 calculated using the proposed technique as well as the corresponding finite size model. A very good agreement is observed, with a discrepancy of less than 0.2% in the frequency and 4% in the value of maximum directivity, which can be attributed to numerical tolerances of both techniques. This validates the proposed technique as a fast and accurate tool for the design of the proposed antennas. IV. CONCLUSION The analysis of sub-wavelength profile high-gain planar leaky-wave antennas employing two periodic surfaces (HIS and PRS) has been presented using MoM and invoking reciprocity. A careful convergence analysis has shown that more FSHs are necessary in order to achieve convergence for antennas

[1] A. Oliner, “Leaky-wave antennas,” in Antenna Engineering Handbook, R. C. Johnson, Ed., 3rd ed. New York: McGraw Hill, 1993. [2] C. Caloz and T. Itoh, Electromagnetic Metamaterials: Transmission Line Theory and Microwave Applications. Hoboken, NJ: Wiley, 2006. [3] J. R. James and P. S. Hall, Handbook of Microstrip Antennas. London, U.K.: Peter Peregrinus, Ltd., 1989. [4] G. V. Trentini, “Partially reflecting sheet array,” IRE Trans. Antennas Propag., vol. AP-4, pp. 666–671, 1956. [5] A. P. Feresidis and J. C. Vardaxoglou, “High-gain planar antenna using optimized partially reflective surfaces,” IEE Proc. Microw. Antennas Propag., vol. 148, no. 6, Feb. 2001. [6] Y. J. Lee, J. Yeo, R. Mittra, and W. S. Park, “Design of a high-directivity electromagnetic bandgap (EBG) resonator antenna using a frequency selective surface (FSS) superstrate,” Microw. Opt. Technol. Lett., vol. 43, no. 6, pp. 462–467, 2004. [7] T. Zhao, D. R. Jackson, J. T. Williams, H.-Y. D. Yang, and A. A. Oliner, “2-D periodic leaky-wave antennas-part I: Metal patch design,” IEEE Trans. Antennas Propag., vol. 53, no. 11, pp. 3505–3514, Nov. 2005. [8] T. Zhao, D. R. Jackson, J. T. Williams, H.-Y. D. Yang, and A. A. Oliner, “2-D periodic leaky-wave antennas-part II: Slot design,” IEEE Trans. Antennas Propag., vol. 53, no. 11, pp. 3515–3524, Nov. 2005. [9] D. R. Jackson and N. G. Alexopoulos, “Gain enhancement methods for printed circuits antennas,” IEEE Trans. Antennas Propag., vol. AP-33, no. 9, pp. 976–987, Sep. 1985. [10] H. Y. Yang and N. G. Alexopoulos, “Gain enhancement methods for printed circuits antennas through multiple superstrates,” IEEE Trans. Antennas Propag., vol. AP-35, no. 7, pp. 860–864, Jul. 1987. [11] D. R. Jackson, A. A. Oliner, and A. Ip, “Leaky-wave propagation and radiation for a narrow-beam multiple-layer dielectric structure,” IEEE Antennas Propag., vol. AP-41, no. 3, pp. 344–348, Mar. 1993. [12] T. Zhao, D. R. Jackson, J. T. Williams, and A. A. Oliner, “Simple CAD model for a dielectric leaky-wave antenna,” IEEE Antennas Wireless Propag., vol. 3, pp. 243–245, April 2004. [13] D. Sievenpiper, Z. Lijun, R. F. Broas, N. G. Alexopoulos, and E. Yablonovitch, “High-impedance electromagnetic surfaces with a forbidden frequency band,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 11, pp. 2059–2074, Nov. 1999. [14] Y. Zhang, J. von Younis, C. Fischer, and W. Wiesbeck, “Planar artificial magnetic conductors and patch antennas,” IEEE Trans. Antennas Propag., Special Issue on Metamaterials, vol. 51, no. 10, pp. 2704–2712, Oct. 2003. [15] A. P. Feresidis, G. Goussetis, S. Wang, and J. C. Vardaxoglou, “Artificial magnetic conductor surfaces and their application to low-profile high-gain planar antennas,” IEEE Trans. Antennas Propag., vol. 53, no. 1, pp. 209–215, Jan. 2005. [16] S. Wang, A. P. Feresidis, G. Goussetis, and J. C. Vardaxoglou, “Highgain subwavelength resonant cavity antennas based on metamaterial ground planes,” IEE Proc. Antennas Propag., vol. 153, no. 1, pp. 1–6, Feb. 2006. [17] J. R. Kelly, T. Kokkinos, and A. P. Feresidis, “Analysis and design of sub-wavelength resonant cavity type 2-D leaky-wave antennas,” IEEE Trans. Antennas Propag., vol. 56, no. 9, pp. 2817–2825, Sep. 2008. [18] C. A. Balanis, Antenna Theory: Analysis and Design, 2nd ed. New York: Wiley, 1997. [19] R. Mittra, C. H. Chan, and T. Cwik, “Techniques for analyzing frequency selective surfaces-a review,” Proc. IEEE, vol. 76, pp. 593–615, Dec. 1988. [20] W. Changhua and J. A. Encinar, “Efficient computation of generalized scattering matrix for analyzing multilayered periodic structures,” IEEE Trans. Antennas Propag., vol. 43, no. 11, pp. 1233–1242, Nov. 1995.

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[21] J. C. Vardaxoglou, Frequency Selective Surfaces Analysis and Design. New York: Wiley, 1997. [22] B. Munk, Frequency Selective Surfaces: Theory and Design. New York: Wiley, 2000. [23] C. Mateo-Segura, G. Goussetis, and A. P. Feresidis, “Resonant effects and near field enhancement in perturbed arrays of metal dipoles,” IEEE Trans. Antennas Propag., 10.1109/TAP.2010.2050416. [24] P. Kosmas, A. P. Feresidis, and G. Goussetis, “Periodic FDTD analysis of a 2-D leaky-wave planar antenna based on dipole frequency selective surfaces,” IEEE Trans. Antennas Propag., vol. 55, no. 7, pp. 2006–2012, Jul. 2007. [25] G. Goussetis, A. P. Feresidis, and R. Cheung, “Quality factor assessment of subwavelength cavities at FIR frequencies,” J. Opt. A, vol. 9, pp. s355–s360, Aug. 2007. [26] 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. [27] D. M. Pozar, Microwave Engineering, 2nd ed. New York: Wiley, 1998.

Carolina Mateo-Segura (S’08) was born in Valencia, Spain, in 1981. She received the M.Sc. degree in telecommunications engineering from the Polytechnic University of Valencia, Valencia, Spain, in 2006. She is currently working toward the Ph.D. degree jointly between the University of Edinburgh and Heriot-Watt University, Edinburgh, U.K. During the first half of 2006, she joined the Security and Defence Department of Indra Systems, Madrid, Spain, as a Junior Engineer. Currently, she is a Research Associate in the Wireless Communications Research Group, Department of Electronic and Electrical Engineering, Loughborough University, Leicestershire, U.K. Her research interests include the analysis and design of frequency selective surfaces, artificial periodic electromagnetic structures with applications on high-gain array antennas and medical imaging systems. Her research has been funded primarily by the Joint Research Institute for Integrated Systems, EPSRC, MRC and BBSRC. Ms. Mateo-Segura was awarded a prize studentship from the Edinburgh Research Partnership and the Joint Research Institute for Integrated Systems to join the RF and Microwave group at Heriot-Watt University, Edinburgh, Scotland, U.K.

George Goussetis (SM’99–M’02) graduated from the National Technical University of Athens, Greece, in 1998, and received the Ph.D. degree from the University of Westminster, London, U.K. and also received the B.Sc. degree in physics (first class honors) from University College London (UCL), U.K., in 2002. In 1998, he joined the Space Engineering, Rome, Italy, as an RF Engineer and, in 1999, joined the Wireless Communications Research Group, University of Westminster, U.K., as a Research Assistant. Between 2002 and 2006, he was a Senior Research Fellow at Loughborough University, U.K. Between 2006 and 2009, he was a Lecturer (Assistant Professor) with the School of Engineering and Physical Sciences, Heriot-Watt University, Edinburgh, U.K. He joined the Institute of Electronics Communications and Information Technology, Queen’s University Belfast, Northern Ireland, in September 2009 as a Reader (Associate Professor). In 2010, he was a Visiting Professor at UPCT, Spain. He has authored or coauthored over 100 peer-reviewed papers three book chapters and two patents. His research interests include the modelling and design of microwave filters, frequency-selective surfaces and periodic structures, leaky wave antennas, microwave heating as well numerical techniques for electromagnetics. Dr. Goussetis received the Onassis Foundation Scholarship in 2001. In October 2006, he was awarded a five-year research fellowship by the Royal Academy of Engineering, U.K.

Alexandros P. Feresidis (S’98–M’01–SM’08) was born in Thessaloniki, Greece, in 1975. He received the Physics degree from Aristotle University of Thessaloniki, Greece, in 1997, the M.Sc.(Eng.) degree in radio communications and high frequency engineering from the University of Leeds, U.K., in 1998, and the Ph.D. degree in electronic and electrical engineering from Loughborough University, U.K., in 2002. During the first half of 2002, he was a Research Associate and in the same year he was appointed Lecturer in Wireless Communications in the Department of Electronic and Electrical Engineering, Loughborough University, U.K., where, in 2006, he was promoted to Senior Lecturer. He has published more than 100 papers in peer reviewed international journals and conference proceedings and has coauthored three book chapters. His research interests include analysis and design of artificial periodic metamaterials, electromagnetic band gap (EBG) structures and frequency selective surfaces (FSS), array antennas, small/compact antennas, numerical techniques for electromagnetics and passive microwave/mm-wave circuits.

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Experimental Characterization of a Surfaguide Fed Plasma Antenna Paola Russo, Member, IEEE, Valter Mariani Primiani, Member, IEEE, Graziano Cerri, Member, IEEE, Roberto De Leo, and Eleonora Vecchioni

Abstract—The possibility of using a surfaguide device as plasma source for plasma antenna application has been experimentally investigated. The surfaguide was optimized, realized and used for the ignition of a plasma column to be used as a radiating structure: the coupling with the radiated signal network and plasma antenna efficiency were measured showing that a surfaguide can be effectively used to create and sustain the plasma conductive medium. A plasma diagnostic technique was also developed to evaluate the plasma column length and plasma conductivity with respect to the power supplied. These measurements highlighted that plasma antenna properties are strongly affected by the pump signal and therefore this signal has to be optimized in order to have the highest conductivity. Index Terms—Conductivity, surfaguide.

efficiency,

plasma antennas,

I. INTRODUCTION

T

HE idea of using a plasma element as the conductive medium in radio-frequency (RF) antennas and reflectors is not new [1]: several studies have demonstrated the feasibility of such devices [2]–[4]. In recent years, the scientific community has shown a growing interest in plasma antennas mainly because of their peculiar and completely innovative properties with respect to traditional metallic antennas. The electromagnetic characterization of a plasma antenna according to standard parameters requires the analysis of the physical aspects involved in the interaction mechanism between an electromagnetic field and a plasma; therefore new models and experimental techniques have to be developed [5]–[8]. A plasma antenna can be rapidly switched on or off by applying bursts of power to a tube filled with a low pressure gas: the power supplied ionizes the gas providing the conductive medium for the RF signal to be radiated; in plasma antenna application two signals are needed: the pump signal that creates and sustains the plasma column, and the signal to be radiated (in the following simply indicated as “radiated signal”) that has to be coupled to the plasma element. Thanks to this mode of operation, the plasma antenna offers several advantages over traditional metal antennas. Manuscript received November 17, 2009; revised June 30, 2010; accepted August 04, 2010. Date of publication December 03, 2010; date of current version February 02, 2011. P. Russo, V. M. Primiani, and G. Cerri are with the “Università Politecnica delle Marche,” Ancona I-60131, Italy (e-mail: [email protected]). R. De Leo is with the University of Ancona, Ancona I-60131, Italy. E. Vecchioni was with the “Università Politecnica delle Marche,” Ancona, Italy. She is now with the Software R&D Unit, Thermowatt, SpA, Arcevia 60011, Italy. Digital Object Identifier 10.1109/TAP.2010.2096387

The possibility to switch on and off the plasma makes plasma antennas suitable for the production of time varying radar cross section elements: when plasma is on, it behaves like a conductor, when it is off, it behaves like a dielectric material. This characteristic provides the possibility of creating reconfigurable array: an electrical control of the ignited elements allows the modification of the array geometry and so of its radiation characteristics. A simple linear plasma antenna can be created by using a tube filled with a gas. The effective antenna length of the column can easily be changed by controlling the power supplied to the pump signal. Finally, the electromagnetic characteristics of plasma can be used to realize frequency selective shields. The key point in the realization of plasma elements to be used as plasma antennas is the ignition of the plasma column: the pump signal network has to be optimized in order to obtain the highest plasma conductivity with the lowest power. At the same time, the realization of the feeding network must not degrade the antenna radiating properties. In the past, plasma was produced by DC or high frequency discharges from two electrodes at opposite ends of the column; [1] proposed a new way of producing microwave and RF discharges based on electromagnetic surface waves to sustain the discharge: in this way plasma could be driven from only one end of the column and electrodes should no longer be needed. Since the 1960s studies have shown that a surface wave can propagate along the interface between a plasma column and the tube containing it [9], [10] but the idea of using these waves to sustain a plasma column was only developed in the 1970s: in [11] a surface-wave-produced discharge was identified for the first time. On the basis of these studies, several surface-wave plasma sources have been developed: in [5], [8] a surface wave is launched by a capacitive coupling applying an intense field between a copper ring placed around the tube and a ground plane. The surfaguide is another device, first proposed in [12] and illustrated in several papers [13]–[17], which could be used as a plasma source, and it is by far the simplest of the surface-wave launchers: it is a wave-guide device able to excite a surface wave that propagates along the tube axis providing the power required to ignite the plasma. With respect to other plasma sources the surfaguide presents some advantages that can be favorably considered in designing plasma antennas: it is the most suitable device to propagate a power signal, confining its electromagnetic field in a closed structure; it is simple to realize, and matching stubs can easily be inserted in the design. Moreover, it can be used to feed several plasma elements realizing arrays of antennas. Finally, it offers the possibility of using the frequency of 2450 MHz for the

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pump signal because at this frequency high power is available with low cost. In this paper a complete characterization of the surfaguide as a feeding network for plasma antenna application is presented. This is a new application of an old structure that is usually used to produce plasma for different purposes. In this paper the geometric parameters of the surfaguide have been designed in order to optimize the radiation properties. In this study is also proposed a measurement set-up and described the experimental procedures followed to characterize both the surfaguide system and the radiated signal network. In particular a 2.45 GHz pump signal is used for antenna ignition, and the frequency of 430 MHz is chosen for the radiated signal, because both frequencies belong to the ISM frequency set and can be used without particular restrictions. The most critical aspect of the work is related to the strong coupling between the pump and the radiated signal networks. In fact, plasma antenna parameters have to be characterized, in terms of efficiency, effective length and conductivity, when the surfaguide is used to create and sustain the conductive medium. Moreover, plasma antenna properties are also strongly affected by the pump signal and therefore they have to be self-consistently determined: in particular, plasma conductivity, that determines the “metallic behavior” of plasma, relies on the ionization process ignited by the pump signal. The paper is organized as follows: Section II shows the design, the optimization and the realization of the surfaguide and the pump signal network used to supply the 2.45 GHz signal; Section III illustrates the signal network and describes the measurement of antenna efficiency; finally Section IV reports the measurement procedures to characterize the plasma column length and conductivity.

Fig. 1. Longitudinal section of the surfaguide. The vertical tube contains plasma to be ignited.

Fig. 2. Longitudinal section of the surfaguide in the coupling region with the glass tube: the simulated electric field is normalized to 1 W of incident power.

II. SURFAGUIDE: OPTIMIZATION AND REALIZATION This section describes the realization of the surfaguide system, designed to achieve an efficient ignition of the plasma column; this device represents the main difference with respect to traditional radiating systems, and is necessary to create the antenna. The whole feeding network set-up is also important, because it allows the power delivered to the antenna to be controlled and the effective power required for ignition to be measured. A plasma column is created by applying a pump signal to a tube containing a gas; the gas is ionized by a strong microwave electric field applied at one termination of the tube by a surfaguide device. The surfaguide launches an azimuthally symmetric electromagnetic surface wave that propagates along the tube creating and sustaining the plasma column. The surfaguide is basically a waveguide with a tapered section designed to increase the electric field strength in the reduced height region without affecting the impedance matching. Fig. 1 shows the longitudinal section of the surfaguide: it consists of two trunks L0 of a standard waveguide WR340, two transitions L1, and a waveguide L2 with a reduced height. The guide is terminated by a moving short, whose length Ls can be varied for matching when the plasma column is turned on. Two holes along the central axis of the reduced height guide allow the glass tube to be inserted: a commercially available

tube designed for lighting purposes was used to create the plasma column. The structure presented in Fig. 1 has to be optimized for the frequency 2.45 GHz: the surfaguide geometrical parameters need to be carefully chosen in order to have a very intense field coinciding with the holes where the tube is inserted. The optimization of the surfaguide dimensions was achieved using the commercial software CST Microwave Studio [18] to simulate electromagnetic field behavior before plasma ignition; the tube cylinder with thickness mm, diameter is a glass mm, filled with a dielectric having a relative dielectric constant . Fig. 2 shows the electric field in the longitufield of the waveguide dinal section of the surfaguide: the is well-coupled with the axial TM field of the surface wave along the glass tube that ignites the plasma: as GHz cm GHz cm (where is the plasma frequency) only the first TM mode of the surface wave is expected to be excited [14]. A parametric investigation of the field behavior was numerically carried out to determine the best values of the hole diameter D, guide height h, and transition length ; field values were normalized to 1 W of incident power and was set to get the maximum field which corresponds to the tube axis. To compare different situations the electric field was evaluated at three

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TABLE I MAXIMUM ELECTRIC FIELD INTENSITY

Fig. 3. Pump signal network.

TABLE II MAXIMUM ELECTRIC FIELD INTENSITY

Fig. 4. Signal connection network: the capacitive coupling between the copper ring and the metallic box is used to feed the antenna.

The set-up shown in Fig. 3 was used to investigate the nonlinear behavior of a plasma column as a function of the power: this aspect strongly affects the plasma antenna characteristics, in particular, efficiency, column length and conductivity. points: in the reduced height guide, on the internal face of the glass tube, at the tube centre. First of all a numerical investigation was conducted for the and and, field intensity with hole diameter D and fixed h, as expected, this showed that the narrower D is, the more intense the field is; therefore, D is chosen as small as possible to allow the tube to be inserted. The behavior of the field as a function of h for fixed D, and (Table I) is more interesting: a reduction in the guide height increases the field inside the tube, but beyond the optimal value a further reduction does not improve the field strength in the tube. Table II shows the behavior of the electric field as a function of the taper length and the final short termination distance: also in this case the optimal value was found, and finally the design mm, mm, mm. parameters were set: After realizing the surfaguide, the 2.45 GHz pump signal network was developed (Fig. 3). The power needed to ionize the gas was supplied by a magnetron generator; an isolator was inserted to prevent the high reflected power from arriving at the signal generator and a directional coupler was used to monitor the incident and reflected power. The minimum power necessary to ignite a small portion of plasma in the tube region crossing the waveguide is 2 W: on increasing the power it is possible to notice that the plasma column height also increases.

III. MEASUREMENT OF ANTENNA EFFICIENCY This measurement was carried out by comparing the power delivered by the radiated signal of a plasma antenna, and the same signal, radiated under the same conditions, by a copper antenna. The procedure is conceptually simple, but requires an accurate realization of the set-up in order to control: (i) the coupling between the pump and radiated signal networks to prevent instrument damage (Section III.A), (ii) the radiated signal power and matching conditions for measurement accuracy (Section III.B). A. Coupling Between Pump and Radiated Signal Networks A preliminary measurement of the coupling between the pump signal and the radiated signal network was performed: coupling occurs because the signal is connected to the plasma column. The signal to be radiated was coupled to the plasma antenna according to the set-up shown in Fig. 4, [5], [7]. The network was enclosed in a metallic cubic box with a side of 6 cm, placed below the surfaguide. The fluorescent tube comes out of the top wall of the box through a 19 mm diameter hole, and penetrates into the surfaguide for plasma ignition. A capacitive coupling is generated for the radiated signal between the box and a copper ring surrounding the tube; this provides the electric field that excites the signal current along the

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TABLE III CORRECTION FACTOR

Fig. 5. Measurement set-up used to characterize coupling between pump and signal networks.

the same length and fed with the same signal network, as shown in Fig. 6, but switching off the pump signal; relative efficiency is defined as

(1)

Fig. 6. Measurement set-up for antenna relative efficiency.

plasma antenna. The copper ring is placed at about 1 mm from the box upper wall. The pump and signal networks are coupled by the plasma generated in the tube. The pump generator generally provides a high power wave that could damage the signal network, and therefore it is necessary to quantify and reduce the coupling. The 2.45 GHz signal coupled to the signal network was measured straightforwardly with the set-up shown in Fig. 5: the isolator exhibits a band pass between 423 and 433 MHz for the 430 MHz direct signal and an attenuation of 13.8 dB for an inverse signal at 2.45 GHz; a further protection for the spectrum analyzer is given by the 20 dB attenuator and by the cable attenuation. The 2.45 GHz signal coupled to the signal network dBm (3,6 dBm at the insulator) for an incident pump is power of 44 dBm: this attenuation is high enough to ensure that the signal generator (later used in place of the spectrum analyzer) is not damaged for incident power up to 100 W (Fig. 6). B. Relative Efficiency Measurement A plasma column is characterized by a specific conductivity given by the free electrons of the ionized gas: if this conductivity is high enough, the plasma column can be efficiently used as the medium for an RF signal to be radiated. However, plasma conductivity is not as high as that of metal and therefore plasma antenna efficiency compared with the efficiency of a traditional metallic element has to be evaluated. The procedure consists in measuring the field radiated by a plasma column and the field radiated by a metallic element of

where and are the power received at the spectrum analyzer when plasma and copper respectively are used as radiating elements. The 430 MHz generator is set at the maximum available power (20 dBm) in order to have a good signal to noise ratio at the receiver. The radiated signal was measured with a loop placed in four different positions at the same distance from the radiating element in order to check the reliability of the results (Fig. 6). During the first stage, measurements were carried out after switching on the plasma element with 25 W of pump power, which allows the complete ignition of the column. The plasma column was then removed and substituted with a copper tube of the same length; the pump signal was switched off because not needed and the signal to be radiated was coupled to the copper element in the same way as the plasma column. Table III shows the matching conditions of the radiated signal coupling network (Fig. 4) at 430 MHz; this preliminary measurement is necessary in order to compare the two situations, since the efficiency has to be evaluated for the same effective signal power passing through the antenna input terminals. Measurements highlighted that the copper antenna is more mismatched than the plasma element, and therefore a correction factor of 0.9 dB was added to the power radiated by the copper element. Results of the power measured by the spectrum analyzer for the plasma and the copper elements in four different positions are reported in Table IV. It is important to point out that the measurement of the radiated signal is a narrowband measurement around 430 MHz, while the pump signal has a 2450 MHz frequency. This narrowband measurement reduces the 430 MHz noise floor (when the useful signal is switched off) about 30 dB below the useful signal peak. In Table IV, it is possible to notice that the average performance degradation of the plasma antenna with respect to the traditional metallic one is about 2.9 dB for all probe positions. This means that, for the analyzed structure, about half the power of the radiated signal is lost due to losses mechanism inside the plasma.

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TABLE IV RECEIVED POWER

Fig. 8. Set-up for the measurement of plasma column length and conductivity.

Fig. 7. Current probe for plasma diagnostics (a) to be put around the glass tube (b).

IV. PLASMA COLUMN HEIGHT AND CONDUCTIVITY MEASUREMENTS A direct measurement of the plasma column length H and conductivity is not possible. In fact a simple visual inspection of the light emitted by plasma [5] to evaluate H is susceptible to great measurement uncertainty. Moreover, the inner region of the tube is not accessible for a direct measurement of . As a consequence an indirect diagnostic technique to evaluate the plasma state along the column as a function of the pump power was developed. The idea was to design a loop probe to be inserted around the tube: the probe consists of a copper coil, Fig. 7(a), positioned as shown in Fig. 7(b) and connected to a network analyzer. depends on the maThe probe input impedance terial wrapped by the coil, which acts as a transformer: in particular R depends on the power dissipated in the plasma because of the currents induced by the probe itself. The idea is to relate R to the plasma state which coincides with the point where the probe is positioned; this method was developed considering the following assumptions: (i) conductivity depends only on the pump signal and not on the radiated signal or on the network analyzer signal: this assumption is well satisfied because the VNA signal is very low (10 mW) compared to the pump signal (several watts); (ii) at the measurement frequency the effects of the wire resistance and the loop radiation resistance are negligible with respect to the dissipation in the material filling the tube; finally, to make reasonable this assumption (iii) measurement has to be carried out at a frequency lower than the resonance of the coil in order to neglect the effect of the parasitic capacitances. Moreover, as the frequency approaches the resonance of the coil, the value of R depends not only on the conductivity , but also on more complicated factors (radiation, field penetration into

Fig. 9. Real part of the input impedance measured with the coil placed in different positions along the column: the point coinciding with the transition region is critical, and the corresponding curve (crossed line) is the average of a few repeated measurements.

the plasma) which affect its measurement. As a consequence of these assumptions, the frequency range for the network analyzer MHz. signal was Measurements were carried out as shown in Fig. 8: the probe was positioned around the glass tube and connected to the network analyzer by a low pass filter to prevent the 2.45 GHz signal from damaging the network analyzer. The plasma state of the column was achieved by retrieving the values of R moving the probe along the tube. A. Plasma Antenna Height The input impedance was first measured when no pump power was applied to the gas tube (gray line in Fig. 9). Subsequently the pump power was switched on and the gas inside the tube was ionized; as the power becomes greater, the plasma column length increases. Supplying a fixed pump power to the column, the plasma is ignited for only a certain length. The coil was then moved along the column to appreciate how the plasma state varies in the longitudinal direction: as the distance from the surfaguide increases, R becomes smaller, and, coinciding with the position where the conductivity is no longer significant (28 cm in Fig. 9), the real part is the same as measured when no pump signal is applied. This position at which the antenna is considered switched off (Fig. 9) determines the plasma column length. This result allows us to determine the profile of the plasma column length with respect to the pump power. Fig. 10 shows the experimental results for the plasma column height compared with those obtained with a different feeding network in [5]. In both situations the column height is proportional to the square

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Fig. 10. Plasma column length as a function of the absorbed power.

Fig. 12. Real part of the input impedance measured on test-tubes of known conductivity, numerical vs experimental.

Fig. 11. Copper coil simulated with CST.

root of the power; the difference in values is due to the different pump signal networks, gas pressure, and composition. Moreover, in this study the column height was determined by measuring a significant electrical parameter rather than by a simple visual inspection. Fig. 13. Relationship between R and  numerically recovered for three different plasma (layer) radial thicknesses.

B. Plasma Conductivity The same set-up shown in Fig. 8 was used to measure the plasma conductivity along the column: this is a key parameter because it affects all the radiation properties of plasma antennas. As a direct measurement is not possible, its value has to be inferred from R, determined for each probe position along the tube according to the procedure described in the previous section. A power balance between the power absorbed by the probe resistance R and the power lost in the plasma region surrounded by the coil allows us to recover a relationship between R and [19]

(2) In (2) the dependence of R on frequency and conductivity is explicitly written. In our case the value of the constant cannot be analytically calculated, therefore it has to be evaluated after a proper calibration of the probe. Calibration was performed by simulating the probe with the commercial software CST-Microwave Studio [18] as shown in Fig. 11. Numerical results were compared with some measurements in order to check the accuracy of the simulations and to provide the self-consistency of the procedure. was measured by putting the probe around some test-tubes filled with homogeneous solutions of known conductivity and permeability which were then simulated with the aforementioned numerical tool.

The frequency range chosen was lower than 150 MHz in order to be far enough from the resonance of the coil (380 MHz) and to be sure that any variation in the permittivity of the material would not affect the measurements of the real part of the input impedance. Fig. 12 reports the values measured and simulated for three different test-tubes showing a good agreement between the experimental and numerical results. Moreover, in the frequency and it is proportional to the range chosen, increases with conductivity predicted by (2). The numerical results obtained for different material filling the glass tube could be used to obtain the desired relationship with a good approximation, but in a plasma antenna it strongly depends on the charge distribution inside the tube. It is well-known from literature that plasma is mostly distributed along the inner surface of the tube [20], [21], but we are not able to appreciate experimentally the radial profile of conductivity. As an example, Fig. 13 shows the results obtained for a plasma uniformly distributed in an annular region of radial thickness mm, mm, mm respectively. The uncertainty of the conductivity radial profile also affects its distribution along the tube: Fig. 14 shows three different longitudinal profiles of plasma conductivity with respect to the

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TABLE V RECEIVED POWER

Fig. 14. Plasma conductivity with respect to tube length.

length of the tube, determined from the R measurement along the plasma tube. Results show that the knowledge of plasma thickness is essential for determining the conductivity profile. Actually the unknowns of the problem are two: plasma thickness and its conductivity. The knowledge of the loop input impedance is not sufficient to resolve the problem. It is necessary to add another independent parameter that depends on both the unknowns: the relative efficiency and R, measured along the tube, permit to recover both the unknowns.

C. Method of Moments (MoM) Simulation of Antenna Relative Efficiency The goal of the MoM simulation is the theoretical determination of plasma antenna efficiency at the frequency of the radiated signal (430 MHz) for different plasma conductivity profiles. A comparison between the efficiency obtained with the MoM simulation and the efficiency measured in Section III allows both an estimation of the value of and an estimation of the plasma thickness. In this section a dipole antenna with conductivity variation along its length, as in the case of a plasma antenna, is investigated. The classic approach based on the electric field integral equation (EFIE) was adopted, and thin wire approximation (TWA) was assumed; the antenna is considered as a monopole of length H over a ground plane. With reference to the coordinate system in Fig. 1, for the , the EFIE to be satisfied on the antenna scalar component surface can be written as

(3) The plasma conductivity is modeled as the straight interpolating line of Fig. 14

(4)

being the conductivity coinciding with the point where is the dipole current density of the pump signal is applied. the radiated signal

(5) with S(y) being the cross section where current flows with area

(6) is the plasma column radius and is the smallest value between the plasma layer thickness t and the skin MHz, . is the depth at antenna current flowing along the dipole axis according to the is TWA and is also the problem unknown. Finally, the magnetic current loop, wrapped around the dipole, placed in and representing the signal source. The method of moments (MoM) was applied using pulse functions as basis functions and the point matching condition; the use of the proper conductivity for the plasma column leads us to consider a varying cross section where the current density flows along the antenna. A numerical code was developed to solve the EFIE using MoM. Convergence tests led to the choice of 61 basis and weighting functions for MoM implementation. Both copper and plasma antennas were analyzed: in the first case an ideal conductor was assumed, whereas in the second case three different conductivity profiles were used. In particular the straight interpolating lines and the corresponding thickness of Fig. 14 were used. It is important to remember that these three profiles derive from the measured input impedance of the loop placed around the tube. Table V shows the power balance for a generator V, and the theoretical efficiency calculated for the three situations. These results highlight that the higher the conductivity is the greater the radiation is and therefore a better efficiency is achieved. mm is It is clear that the conductivity profile with the most acceptable because only this value implies a calculated dB similar to the measured one ( dB). efficiency

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The results of the entire recovering procedure show that curve mm of Fig. 14 best describes the antenna conductivity profile. We would like to underline that the plasma thickness : in fact t depends on the power and fredepends on the frequency of the pump signal, whereas quency of the radiated signal (430 MHz) and plasma conductivity. In this example the skin depth has a minimum value of 2.5 mm at the base of the antenna, therefore it is always greater than the estimated plasma thickness (0.5 mm). V. CONCLUSION Plasma antennas present some potential advantages compared with traditional metallic radiating systems, although a new approach for their characterization is required. This is due to the need for generating plasma, the physical support which allows the signal to be radiated. Even if the literature concerning plasma physics and application is extensive, very few papers deal with the specific subject of plasma antennas. In this context this paper is a step towards the definition of measurement techniques for the experimental characterization of this kind of antenna. Two main problems have been considered and solved: the first is the presence of two radio frequency signals at the same time on the same structure; the second is the design of a sensor for the characterization of the plasma state. In the former case a suitable measurement set-up was developed to evaluate and reduce the strong coupling between the pump and the radiated signals in order to prevent instrument damage and measurement errors; for the latter problem the measurement of the plasma state was complicated because the region where plasma is ignited (a glass tube) is not accessible. In this situation an indirect measurement procedure was carried out, involving a self-consistent technique that allows us to retrieve the value of plasma conductivity using experimental data and simulations. The complex solution provided for plasma antenna characterization is intrinsic to the physical phenomenon, because all the parameters strongly depend on each other according to nonlinear relations. Three important parameters were successfully measured: the efficiency of the plasma antenna, its length, and the column conductivity. Other parameters were also characterized in order to obtain the above mentioned quantities: matching conditions for both the pump signal and the radiated signal, coupling between the pump and signal networks. The experimental analysis showed that the surfaguide is an effective device to excite plasma antennas and is also suitable for array configurations. At the same time this study underlined the need not only to have a model to describe the interaction mechanism between a surface wave and plasma but also to carry out a parametric investigation of the problem: for this purpose a numerical tool was developed to help the optimization of all the parameters involved in plasma antenna design. REFERENCES [1] M. Moisan, A. Shivarova, and A. W. Trivelpiece, “Experimental investigations of the propagation of surface waves along a plasma column,” Phys. Plasmas, vol. 24, pp. 1331–1400, 1982.

[2] T. J. Dwyer, J. R. Greig, D. P. Murphy, J. M. Perin, R. E. Pechacek, and M. Raileigh, “On the feasibility of using an atmospheric discharge plasma as an RF antenna,” IEEE Trans. Antennas Propag., vol. 32, pp. 141–146, Feb. 1984. [3] G. G. Borg, D. G. Miljak, and N. M. Martin, “Application of plasma columns to radiofrequency antennas,” Appl. Phys. Lett., vol. 74, pp. 3272–3274, May 1999. [4] G. G. Borg, J. H. Harris, N. M. Martin, D. Thorncraft, R. Milliken, and D. G. Miljak, “Plasmas as antennas: Theory, experiment and applications,” Phys. Plasmas, vol. 7, pp. 2198–2201, May 2000. [5] J. P. Rayner, A. P. Whichello, and A. D. Cheetham, “Physical characteristics of plasma antennas,” IEEE Trans. Plasma Sci., vol. 32, pp. 269–281, Feb. 2004. [6] I. Alexeff, T. Anderson, S. Prameswaran, E. P. Pradeep, J. Hulloli, and P. Hulloli, “Experimental and theoretical results with plasma antennas,” IEEE Trans. Plasma Sci., vol. 33, pp. 166–171, Apr. 2006. [7] G. Cerri, R. De Leo, V. M. Primiani, C. Monteverde, and P. Russo, “Design and prototyping of a switching beam disc antenna for wideband communications,” IEEE Trans. Antennas Propag., vol. 54, pp. 3721–3726, Dec. 2006. [8] G. Cerri, R. De Leo, V. M. Primiani, and P. Russo, “Measurement of the properties of a plasma column used as a radiated element,” IEEE Trans. IMT, vol. 57, pp. 242–247, Feb. 2008. [9] A. W. Trivelpiece and R. W. Gould, “Space charge waves in cylindrical plasma columns,” J. Appl. Phys., vol. 30, pp. 1784–1793, Nov. 1959. [10] A. W. Trivelpiece, Slow-Wave Propagation in Plasma Waveguides. San Francisco: San Francisco Univ. Press, 1967. [11] D. R. Tuma, “A quiet uniform microwave gas discharge for lasers,” Rev. Sci. Instrum., vol. 41, pp. 1519–1520, Oct. 1970. [12] M. Moisan, Z. Zakrzewski, R. Panel, and P. Leprince, “A waveguidebased launcher to sustain long plasma columns through the propagation of an electromagnetic surface wave,” IEEE Trans. Plasma Sci., vol. 12, pp. 203–214, Sept. 1984. [13] M. Moisan and Z. J. Zakrzewski, “Plasma sources based on the propagation of electromagnetic surface waves,” J. Phys. D, Appl. Phys., vol. 24, pp. 1025–2048, July 1991. [14] M. Moisan and J. Pelletier, Microwave Excited Plasma. Amsterdam: Elsevier, 1992. [15] A. Böhle, O. Ivanov, A. Kolisko, U. Kortshagen, H. Schlüter, and A. Vikharev, “Pulsed discharges produced by high-power surface waves,” J. Phys. D.: Appl. Phys., vol. 29, pp. 369–377, Feb. 1996. [16] D. Czylkowski, M. Jasin´ski, J. Mizeraczyk, and Z. Zakrzewski, “Argon and neon plasma columns in continuous surface wave microwave discharge at atmospheric pressure,” Czech. J. Phys., vol. 56, pp. 684–689, Oct. 2006. [17] Y. Kabouzi, D. B. Graves, E. Castaños-Martínez, and M. Moisan, “Modeling of atmospheric-pressure plasma columns sustained by surface waves,” Phys. Rev. E, vol. 75, pp. 016402-1–016402-13, Jan. 2007. [18] CST-Computer Simulation Technology Microwave Studio, Bad Nuheimer Str. 19, 64289 Darmstadt, Germany. [19] S. Ramo, J. R. Whinnery, and T. Van Duzer, Fields and Waves in Communications Electronics. New York: Wiley, 1994. [20] G. Cerri, P. Russo, and E. Vecchioni, “Electromagnetic characterization of plasma antennas,” in Proc. EuCAP, Berlin, 2009, pp. 3143–3146. [21] O. A. Ivanov and V. A. Koldanov, “Self-consistent model of a pulsed air discharge excited by surface waves,” Plasma Phys. Rep., vol. 26, pp. 902–908, Oct. 2000. Paola Russo (S’98–M’00) was born in Turin, Italy, in 1969. She received the Ph.D. degree in electronic engineering from the Polytechnic of Bari, Italy, in April 1999. In 1999, she worked with a research contract at the Motorola Florida Research Lab. From 2000 to 2004, she worked with a research contract on the development of numerical tools applied to the coupling of electromagnetic field and biological tissue, and to different EMC problems, in the Department of Electronics, University of Ancona (now Università Politecnica delle Marche). Since January 2005, she is a Researcher at the Università Politecnica delle Marche, Italy, where she teaches EMC and antenna design. Her main research topics are on the application of numerical modeling to EMC problem, reverberation chamber, and new antenna design. Prof. Russo is a member of the IEEE EMC and AP societies and of the Italian Society of Electromagnetics SIEM.

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Valter Mariani Primiani (M’93) was born in Rome, Italy in 1961. He received the “Laurea” degree in electronic engineering in 1990. Currently, he is an Associate Professor of electromagnetic compatibility at the “Università Politecnica delle Marche,” Ancona, Italy, where he is a member of the DIBET Department, responsible for the EMC Laboratory. His area of interest in electromagnetic compatibility concerns the prediction of digital PCB radiation, the radiation from apertures, the ESD coupling effects modelling and the analysis of emission and immunity test methods. More recently he has extended his research activity in the field of the application of reverberation chambers for compliance testing and for metrology applications. Prof. Primiani is a member of the IEEE EMC and IM societies and of the Italian Society of Electromagnetics SIEM.

Graziano Cerri (M’93) was born in Ancona, Italy, in 1956. He received the Laurea degree in electronic engineering from the University of Ancona, in 1981. In 1983, after military service in the Engineer Corp. of the Italian Air Force, he became an Assistant Professor in the Department of Electronics and Control, University of Ancona where, from 1992, he was an Associate Professor of microwaves in the same Department, and is currently a Full Professor of electromagnetic fields in the DIBET Department, Università Politecnica delle Marche. His research is mainly devoted to EMC problems, to the analysis of the interaction between EM fields and biological bodies and to antennas. Prof. Cerri is a member of AEI (Italian Electrotechnical and Electronic Association). Since 2004, he is the Director of ICEmB (Interuniversity Italian Center for the study of the interactions between Electromagnetic Fields and Biosystems). He is also a Member of the Administrative and Scientific Board of CIRCE (Interuniversity Italian Research Centre for Electromagnetic Compatibility), the Scientific Board of CNIT (Interuniversity National Centre for Telecommunications), and the Scientific Board of SIEm (Italian Association of Electromagnetics).

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Roberto De Leo was born in Bari, Italy, in 1942. He received the Laurea degree in electronic engineering from the Politecnico di Torino, Turin, Italy, in 1965. From 1966 to 1975, he was an Assistant Professor of electronics on the Faculty of Engineering, University of Bari, Bari, Italy, where, in 1976, he was appointed Full Professor of Microwaves. In 1980, he was appointed Full Professor of electromagnetic field at the University of Ancona, Ancona, Italy, where, in 1992, he became a Full Professor of electromagnetic compatibility. His scientific interests are devoted to theoretical and experimental aspects of EMC. Prof. De Leo was an Associate Editor of the IEEE Transactions on Electromagnetic Compatibility. since 1976, he has been a Member of the Scientific Council of the Electromagnetic Group of the Italian National Research Council (CNR), and from 1989 to 1993, he was also the President of this Group. He is also a member of the Scientific Board of SIEm (Italian Association of Electromagnetics).

Eleonora Vecchioni was born in Macerata, Italy, in 1981. She received the Laurea degree in electronics engineering and the Ph.D. degree in electromagnetism from the Università Politecnica delle Marche, Ancona, Italy, in July 2006 and December 2009, respectively. Her research interests include computational electrodynamics and plasma physics, in particular the physical and numerical characterization of the electromagnetic properties plasma. In January 2010, she was collaborating with the Dipartimento Di Ingegneria Biomedica, Elettronica e Telecomunicazioni of Univpm, as an external collaborator and since June 2010, she is working in the Software R&D Unit, Thermowatt Company.

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Experimental Study of the Effect of Modern Automotive Paints on Vehicular Antennas Brendan D. Pell, Student Member, IEEE, Edin Sulic, Wayne S. T. Rowe, Member, IEEE, Kamran Ghorbani, Member, IEEE, and Sabu John

Abstract—Today’s automobiles are fitted with antennas for many wireless services, with modern vehicular antennas frequently configured in blade or “shark-fin” housings, or as planar roof mounted antennas. In these configurations vehicle manufacturers may wish to improve the appearance of the finished vehicle by painting these antennas or their coverings. This paper provides experimental results detailing the effect of two commonly used automotive paint chemistries both with and without the metallic particles used to create a “metallic paint” effect. Electrostatic primers are also considered. Narrowband and wideband antennas are investigated, and the effect of these coatings on impedance bandwidth and radiation is observed. Index Terms—Antennas, coatings, vehicles.

I. INTRODUCTION

M

ODERN vehicles are fitted with a growing number of electronic devices designed to aid the driver and enhance the driving experience. Many of these devices rely on wireless communication to connect the moving vehicle with the outside world. Antennas are a necessary part of any wireless communication system, enabling transmission and reception of signals in free-space. In past decades the use of antennas in vehicles was primarily limited to services such as AM and FM radio. Today’s vehicles are often fitted with antennas for additional purposes including cellular communications for voice calls, GPS antennas for satellite based positioning and navigation, along with antennas for remote keyless entry applications, and newer satellite radio services (i.e., SDARS). In the future it is likely that vehicles will also require antennas for such things as mobile internet and mobile video, collision avoidance radar, and vehicle-to-vehicle or vehicle-to-infrastructure communication. Each of these wireless services necessitates the incorporation of additional antennas into the vehicular platform. For many years the standard AM/FM radio antenna was a monopole antenna of about 75 cm in length which protruded Manuscript received October 09, 2009; revised June 09, 2010; accepted August 28, 2010. Date of publication December 03, 2010; date of current version February 02, 2011. This work was supported under the Australian Research Council’s Linkage Projects funding scheme (project LP0561868) in conjunction with Composite Materials Engineering Pty Ltd. B. D. Pell, W. S. T. Rowe, and K. Ghorbani are with the School of Electrical and Computer Engineering, RMIT University, Melbourne VIC 3001, Australia (e-mail: [email protected]; [email protected]; [email protected]). E. Sulic and S. John are with the School of Aerospace, Mechanical and Manufacturing Engineering, RMIT University, Bundoora VIC 3083, Australia (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2096182

vertically from the vehicle body. However, recent trends in vehicle design have seen a shift away from the traditional “mast” antenna towards more aesthetically pleasing antennas including smaller “bee-sting” type antennas [1], antennas incorporated in window or windshield glass [2], “shark-fin” designs [3] popularized primarily by the European marques, and conformal planar designs [4], [5]. In order to continue to hide the antennas as much as possible and provide aesthetic freedom to vehicle designers, it would be desirable to paint the antennas so they blend in with the vehicle exterior, yet without disturbing their radiation performance. Some existing “shark-fin” and conformal automotive antennas fitted to modern vehicles appear to have been painted, yet a detailed literature search reveals little about the effect of this paint on the antenna’s performance. An investigation of the effect of polyurethane-based metallic and non-metallic paints on a narrowband antenna at GPS frequencies was presented in [6]. The work focused on a very limited selection of paints for one specific application. This paper presents a detailed study on numerous additional paint types investigated over wider multi-service frequency ranges. References in the literature involving paint and antennas are sparse. The most complete are three investigations which involve paint applied to the surface of reflector antennas [7]–[9]. These papers pertain to testing at frequencies in excess of 10 GHz, and report observed effects in cross polar performance, noise temperature and gain loss. Other existing references to paint effects on antennas in the literature are side comments that excessive layers of paint have the potential to significantly affect automotive radar antennas in the mm-wave region [10], or that metallic paint has little effect for microwave frequency applications [11], [12]. To the authors’ knowledge, a thorough investigation of the effect of automotive paint on planar antennas has not been conducted. Section II of this paper details the paints used in this investigation and analyzes one of those paints in a Scanning Electron Microscope to determine the size of metallic flakes included in the paint which create a metallic appearance. Section III explains the testing method employed to determine the effect of each of the paints on various antennas, and Section IV presents the results of these tests. A discussion of the results is presented in Section V, while Section VI draws some conclusions. II. OVERVIEW OF PAINTS INVESTIGATED Paints with different chemical composition are used for different purposes around the world. This investigation focuses on paints that are used in automotive applications. Lacquers and enamels were traditionally used in the automotive industry, but

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TABLE I DETAILS OF PAINT SAMPLES INVESTIGATED

samples and antennas used in the investigation are shown in Fig. 1. The paint samples are presented in order from left to right, beginning in the upper left hand corner of the image with sample A (which is blank to represent “no superstrate”). A. Aluminium Flake Size in Metallic Paints

Fig. 1. Photo of the antennas and paint samples used in the investigation.

these chemistries had the disadvantages of being fragile, easily damaged, and creating excess amounts of pollution. In the past 50 years, developments in the realm of automotive paints have included waterborne paints, base coat/clear coat systems and polyurethane topcoats [13]. Water based paints in particular are considered to be the future of automotive coatings because of the fewer volatile organic compounds released by this paint type and the subsequent environmental benefits [14]. In this investigation we investigate these newer paint technologies which are used on vehicles today, and into the future. The paints used are from two families. The first is a two-pack polyurethane topcoat manufactured by BC Coatings in Australia, and marketed under the name VC800 Structure 105, while the second is a water based paint which uses the basecoat/clearcoat system, made by PPG Industries, a dominant supplier in the world of automotive coatings. We also investigate the effect of an electrostatic primer. Electrostatic paints are used on some production lines and in other painting facilities to obtain a higher yield during the painting process. An applied DC voltage creates an electrostatic attraction between the object to be painted and the paint particles leaving the painting gun. Once airborne, the paint particles are drawn towards the target object, reducing overspray and more efficiently utilizing the available paint. In order to enable electrostatic painting of inherently nonconductive polymeric panels, a partially conductive “electrostatic” primer must first be applied. The electrostatic primer investigated here is supplied by PPG Industries. Table I provides a list of the samples investigated, and assigns an identification letter to each paint for simplicity. The painted

Prior to examining the effect of these automotive paints on an antenna, a small section of metallic paint from sample D was closely studied to inspect the metallic inclusions which create the metallic effect. According to [15] the size of the aluminium flake in metallic paints typically ranges from 15 to 30 m. In order to determine the approximate size of the aluminium flakes in test sample D, a Scanning Electron Microscope (SEM) was used to obtain a highly magnified image of the paint. An SEM creates a high resolution image by scanning a beam of electrons across a sample inside a vacuum chamber, while a sensor measures the magnitude of electrons returning from the sample. A sample of paint was placed inside the microscope and it was found that the electrons would not travel through the paint binder, so observation of the flakes from the top surface was impossible. In order to overcome this observation difficulty, a small section of paint was lifted away from an off cut of the painted sample and placed in the microscope. This process of lifting the small section away from the main body of paint created a fractured edge along a small section of the paint. This fractured edge provides a useful surface for investigation in the SEM because it created flakes which protrude from the edge of the paint binder. By placing this small paint sample in the microscope, and rotating the stage, an image of the edge of the paint was obtained, which shows the protruding metallic flakes. A backscattered electron sensor was connected to the pole piece inside the microscope, aiding the determination of the materials of different chemical composition by creating visible variances in contrast. Fig. 2 shows the edge of the paint sliver with the protruding metallic flakes clearly visible. The lower portion of the image shows the conductive carbon tape used to adhere the paint to the sample holder in the microscope, while the upper portion of the image shows the top surface of the paint which is smooth in appearance. The horizontal band running through the middle of the image is the

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Fig. 2. SEM image of a section of paint removed from sample D.

Fig. 3. Paint test configuration. (a) Narrowband antenna and (b) wideband antenna.

area of interest and shows the fractured edge of the paint. Aluminium flakes can be seen to be protruding from the fractured edge and appear to be a lighter shade due to the backscattered electron sensor. The size of the flakes observed varies between 40 m and 55 m in maximum dimension, while the thickness of the paint layer is approximately 170 m for this sample. Close inspection of Fig. 2 reveals the high density packing of metallic inclusions in the paint under investigation. One may expect that given the density of aluminium flakes in the sample, significant alteration in antenna performance may be observed. Each metallic inclusion is a small flake of conductive aluminium, and it is intuitive to conclude that placing such a conductor in the near field of an antenna is likely to produce undesirable side effects. III. TESTING TECHNIQUE In order to examine the effect of the various paints on an antenna, the antenna architectures shown in Fig. 3 were employed. The antenna of Fig. 3(a) is a rectangular microstrip patch antenna, while Fig. 3(b) is a bowtie slot antenna. Patch antennas typically have a narrow impedance bandwidth. They provide unidirectional radiation across a relatively narrow range of frequencies. In contrast, the bowtie slot antenna has a wide impedance bandwidth and creates bi-directional radiation. These two kinds of antennas are each used in conjunction with removable coverings which serve a dual purpose. The coverings (or “superstrates”) primarily act as a carrier for the paint, but they also model the scenario of automotive applications, where a radiating element may be concealed under a painted dielectric material. Both antennas are planar, and are appropriately dimensioned to allow either the radiating patch or slot to be covered by the paint sample during testing. The L1-band GPS frequency of 1.575 GHz was chosen as the target frequency for the narrowband investigation because

of the ubiquitous satellite navigation facilities in modern vehicles. The required bandwidth of 20 MHz at 1.575 GHz leads to a narrow percentage bandwidth of 1.3%. In order to test the effect of paint on such a system a simple linearly polarized edge-fed square microstrip patch antenna was designed and fabricated using Ansoft HFSS. Rogers RT/duroid 5880 material with 0.062 thickness was chosen for the substrate. Identical material in the same thickness was chosen for the superstrates, with the antenna being designed so that its resonance would be close to the GPS L1-band frequency when covered by an unpainted reference sheet of RT/duroid 5880. The antenna was fabricated and tested on a Vector Network Analyzer and was found to have a centre frequency of 1.583 GHz in the presence of the reference RT/duroid 5880 superstrate (Fig. 3(a)). The slight variation from the centre frequency in the design is attributable to magnification inaccuracies in the photographic stage of fabrication. For the wideband investigation, an asymmetrical bow-tie slot antenna similar to [16] was designed and fabricated on FR-4 material (Fig. 3(b)). It operates from 1.4 GHz to 9.7 GHz, providing a bandwidth of 150% or approximately 7:1. A number of superstrates of equal size were cut from a sheet of Rogers RT/duroid 5880 material and painted in a specialised facility, each with one of the paints listed in Table I. Two additional samples were set aside for use as control samples. One was left unpainted, whilst one retained its original copper cladding to demonstrate the effect of a solid conductive sheet in the near field of an antenna, and enable a comparison of the aluminium flecks of the metallic paints with a continuous conductor. Double sided adhesive tape was used to hold the superstrates in place during testing. The tape is approximately 0.2 mm thick, and was located in areas of low surface current. The tape was present on the antennas during the measurement of sample A (when no superstrate was present) so any effects of the tape, however slight, were captured in the measurement data. Superstrates of 0.062 (1.575 mm) thickness result in paint samples which are reasonably rigid, and remain flat and straight when placed on the antenna. Extreme care was taken to ensure consistent x-y placement of the superstrates. The input impedance of both the narrowband and wideband antennas was recorded as each sample was placed on each antenna in turn. Changes in gain and radiation pattern caused by the addition of the painted superstrates were evaluated in an anechoic chamber. It is worth noting that the effect of superstrates on various kinds of planar antennas has been extensively studied in the past, but never superstrates bearing paint. Superstrate investigations have been conducted both theoretically [17]–[19] and practically [20], [21], and have considered patch antennas [22], cavity backed slot antennas [23], and linearly tapered slot antennas [24], among others. They report that careful selection of the permittivity and thickness of the superstrate can enhance the gain [18], [23], impact the radiation efficiency and input impedance [22], and increase the electrical length of a tapered slot antenna [24]. This study uses superstrates to enable comparison of the various paints under test, and to provide an approximation of a vehicular antenna mounted behind a painted non-metallic panel.

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However, the effect of the dielectric slab itself (which is the superstrate) can be assessed by comparison of sample A with sample B. The applied thickness of the paint is very thin relative to the wavelength of operation of the antennas in the frequency range considered here. This poses a challenge for the simulation of such scenarios. Computation time would be significantly increased due to the requirement to increase the mesh density in the plane of the paint to accurately model the thin material. This coupled with the difficulty of accurately determining the electrical properties of a thin layer of paint is the reason this research was conducted entirely by experimental means. IV. RESULTS A. Narrowband Antenna Results 1) Reflection Coefficient: The measured reflection coefficient of the narrowband antenna of Fig. 3(a) is shown in Fig. 4. The black curve shows the performance of the antenna without any superstrate present (representing sample A), and reveals that the antenna is resonant at approximately 1.613 GHz, yet has a narrow impedance bandwidth (defined as greater than 10 dB return loss) of 0.8%. A narrow impedance bandwidth is common for patch antennas. Placing sample B, the unpainted superstrate, on the antenna brings about a shift in resonant frequency due to the addition of the bulk dielectric material above the patch. The minimum of the reflection coefficient is shifted to 1.583 GHz, a shift of 2%, and the bandwidth remains unchanged at 0.8%. Painted samples C through to N are compositions of various paints, with and without primers and topcoats, yet all behave in a similar way, with centre frequencies ranging from 1.581 GHz through to 1.588 GHz, a fairly narrow spread of 0.44%. Interestingly, samples M and N (which feature an electrostatic primer) create a slightly wider bandwidth, of about 0.95%. The effect of samples M and N is examined further in the discussion section of this paper. The substitution of the copper clad sample of RT/duroid 5880, sample O, produces extremely degraded antenna performance as expected. The majority of the input signal is reflected back into the source by the presence of the conductive metallic sheet over the patch antenna. In addition to the effect of the various paints, some of the observed variation in impedance response between these samples may be due in part to permittivity tolerances in the RT/duroid superstrate itself (from which the samples are constructed), and any air gaps which may arise between the substrate and superstrate. 2) Radiation Patterns: Fig. 5 shows the co-polarized and cross-polarized radiation patterns of the patch antenna with various superstrates measured at the relevant centre frequency. The gain is measured by the standard gain substitution method and the plots are normalized to sample A, the maximum of the antenna without a superstrate. The antenna has hemispherical coverage, meaning that a roof mounted GPS antenna would receive signals from the sky. In general, a large degree of similarity is found between the radiation patterns of the narrowband antenna with the various painted superstrates. The radiation pattern is essentially unchanged. Some variation in shape is evident in the back lobes of the antenna and in the cross polarized patterns,

Fig. 4. Measured reflection coefficient of the narrowband antenna with various superstrates on (a) a wide scale, and (b) zoomed to a narrow scale.

Fig. 5. Measured radiation patterns of Antenna A with superstrates. The measurement is performed at the centre frequency for each sample. (a) x-z plane co-polarized, (b) x-z plane cross-polarized, (c) y-z plane co-polarized, (d) y-z plane cross-polarized.

yet these parts of the radiation contribute little to the received signal, as they are approximately 20 dB down on the main lobe at broadside, making any differences of low importance. Unsurprisingly, the antenna’s radiation performance is significantly degraded by the presence of the copper clad superstrate (sample O), which catastrophically alters the radiation pattern, and reduces the gain of the antenna by almost 20 dB.

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Fig. 6. Maximum gain of the narrowband antenna with painted superstrates, showing error bars to indicated measurement tolerance. Note that sample O is not presented here as it is out of range ( 17:5 dBi). Samples M and N provide a significant gain reduction.

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A valuable comparison of the performance of each of the superstrates is provided in examining the maximum gains produced by the narrowband antenna as displayed in Fig. 6. The gain is presented here in graphical form with error bars showing the repeatability of the gain measurements which was deterdB (as explained in Section V). mined to be approximately The gain of the narrowband antenna without any paint sample present was found to be 8 dBi. The maximum gain produced with the unpainted dielectric superstrate present (sample B) is measured to be 7.9 dBi. Application of various paint types was found to produce a reduction in gain of up to 0.5 dB (samples C to L). Interestingly samples M and N, bearing a partially conductive electrostatic primer, cause a reduction in antenna gain of approximately 1.5 dB. This is most notable in the main lobe of the co-polarized patterns (Fig. 5(a) and (c)). The shape of the radiation pattern is maintained, but at a reduced value of magnitude. The persistence of the pattern suggests minimal change in directivity has taken place. It is possible that the observed gain reduction is caused by a reduced efficiency due to the additional losses associated with the electrostatic primer. Once again, extreme disruption to the antenna’s performance is caused by the copper clad superstrate, as reflected by a large negative value for the gain. The results indicate that all painted samples produce a reduced value of gain to some degree as no measurements of painted samples exceed the gain of sample A or sample B, where no paint is present B. Wideband Antenna Results 1) Reflection Coefficient: To determine the input impedance variation created by each paint on a wideband antenna, the bow-tie antenna of Fig. 3(b) was connected to the vector network analyzer and the reflection coefficient (Fig. 7) was measured in the presence of each superstrate. As predicted in [6] the addition of dielectric material creates a band shifting effect. This shift is dependent upon the permittivity and thickness of the dielectric, and hence is slightly different for each paint sample. The shift appears to be larger at the upper end of the band, yet the percentage shift is approximately constant. Hence, although the wideband antenna’s frequency response is shifted by the paint, the antenna is likely to remain operational at frequencies of interest. The copper clad sample O is once again

Fig. 7. Measured reflection coefficient of the wideband antenna with each superstrate.

seen to have extreme consequences on the antenna’s impedance is matching. Narrow sections of bandwidth exist where dB, however the wideband response is lost. below 2) Radiation Patterns: The radiation pattern for the wideband antenna was investigated at numerous points across its bandwidth. Important communication bands in the automotive environment were chosen where possible. The first is the band reserved for the L1 Band of GPS signals broadcast by the orbiting satellites, at 1.575 GHz. The ISM band at 2.4 GHz is next and has found worldwide application for IEEE 802.15.1 (Bluetooth) and IEEE 802.11 b/g/n (Wi-Fi). The 5.9 GHz band has been reserved in many parts of the world for intelligent transportation systems applications. This includes Vehicle-to-Vehicle and Vehicle-to-Infrastructure communication, primarily for the purpose of transmitting and receiving safety messages. This technology is often referred to as wireless access in the vehicular environment (WAVE) or dedicated short range communication (DSRC), with the relevant IEEE standard being 802.11p. The wideband antenna is also characterized near the upper end of its bandwidth, at 9.5 GHz to examine the effect of these paints in the X-band. The radiation pattern for the wideband antenna at 1.575 GHz is shown in Fig. 8. The antenna radiates a broad beamwidth pattern in the x-z plane. Each of the superstrates can be seen to produce approximately equivalent performance as the unpainted superstrate (sample B) with the exception of sample O, which as previously noted, has a disruptive effect on antenna function. The wideband antenna’s radiation patterns at 2.4 GHz are shown in Fig. 9. Once again the copper clad superstrate of sample O is seen to interfere with proper antenna operation, while the other superstrates are closer in performance. Interestingly, sample A which denotes the antenna without any superstrate is seen to have slightly lower gain in the x-z plane, higher gain in the y-z plane, and lower cross-polarization levels. Radiation patterns for the wideband antenna at 5.9 GHz are shown in Fig. 10. At these frequencies the effect of painted superstrates on such an antenna are negligible, with each sample producing approximately equivalent gain and radiation characteristics. Fig. 7 revealed that sample O obtained an impedance match at this frequency, yet the radiation patterns are substantially different to the remainder of the samples. Of particular interest is Fig. 10(c), where the y-z plane co-polarized pattern level is enhanced in the endfire directions, even above the level of sample A (to which the figure was normalized). Simulated

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Fig. 8. Measured radiation patterns of the wideband antenna with each superstrate 1:575 GHz. (a) x-z plane co-polarized, (b) x-z plane cross-polarized, (c) y-z plane co-polarized, (d) y-z plane cross-polarized.

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Fig. 10. Measured radiation patterns of the wideband antenna with each superstrate 5:9 GHz. (a) x-z plane co-polarized, (b) x-z plane cross-polarized, (c) y-z plane co-polarized, (d) y-z plane cross-polarized.

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The radiation performance of the paint samples of the wideband antenna at 9.5 GHz is illustrated in Fig. 11. Inspection of the plots reveals ripples in the radiation associated with operating an electrically large structure at these higher frequencies. Once again, samples A and O are shown to provide varied performance from the painted superstrates. The departure of sample O from the norm is reduced compared to 1.575 GHz, 2.4 GHz and 5.9 GHz. V. DISCUSSION

Fig. 9. Measured radiation patterns of the wideband antenna with each superstrate 2:4 GHz. (a) x-z plane co-polarized, (b) x-z plane cross-polarized, (c) y-z plane co-polarized, (d) y-z plane cross-polarized.

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field results reveal that at this frequency a parallel plate waveguide mode is formed between the copper of the antenna and the copper of the superstrate. This creates a narrowband impedance match and leads to radiation at the edge of the plates.

The addition of paint creates an observable shift in the resonant frequency of both the narrowband and wideband antennas. Both polyurethane and water based paints are dielectric materials; hence their addition to the structure alters the effective relative permittivity, and therefore the electrical size of the antenna itself. This creates a shift in frequency, yet due to the thin layer of paint commonly applied the shift induced by the paint is only small, being less than 1%. A shift of this magnitude is negligible in many wideband systems, but may have adverse consequences for narrowband antennas. In such narrowband antennas where the overall bandwidth of the antenna itself may be only a few percent, a shift of up to 1% of the antenna’s impedance bandwidth may disrupt operation of the service in the desired frequency range. The impedance bandwidth of the wideband antenna of Fig. 3(b) was also altered by the addition of the various types of paint, yet its wideband nature means that possible loss of the desired wireless service by shifting out of band is somewhat less pronounced. This leads us to conclude that if an unknown paint is to be used to paint a radiating aperture, it is wise to use an antenna with greater bandwidth than what is required.

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Fig. 11. Measured radiation patterns of the wideband antenna with each superstrate 9:5 GHz. (a) x-z plane co-polarized, (b) x-z plane cross-polarized, (c) y-z plane co-polarized, (d) y-z plane cross-polarized.

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With respect to gain and radiation pattern, a great deal of care was taken to ensure the measurements obtained were as accurate as possible. Steps were taken to ensure the temperature of the anechoic chamber remained reasonably consistent throughout the measurement period. The paint samples were tested in succession, and a sweep of an ETS-Lindgren 3115 reference horn antenna was performed and recorded in between each measurement of a paint sample. This enabled an assessment of the fidelity of the measurements to be made at a later stage. Examination of the data produced by the horn antenna over the measurement period revealed that the measured value varied by not dB when the frequency was below 3 GHz, and more than dB at higher frequencies throughout the penot more than riod of testing. Hence, although the absolute value of the gain may be less certain, the data obtained reveals that relative measurements between different paint samples are repeatable within certain limits. Examination of the results reveals that the presence of the paints, whether metallic or non-metallic has only a very small impact on the narrowband antenna gain. It is difficult to be certain of the exact change in gain given the established tolerance dB, but it can be seen to be less than approximately of 0.5 dB where no electrostatic primer is used. The two-pack polyurethane paints and water based paints appear to have similar electromagnetic performance with little significant difference in the antennas’ impedance, radiation or gain performance. Likewise, the presence of a standard primer or clear topcoat has little measurable effect.

Those paints which are classified as metallic achieve this effect by the inclusion of small aluminium “flakes” as investigated in Section II of this paper. In our previous investigation [6], only a single variety of metallic paint was examined, hence it was difficult to conclude with certainty if the metallic inclusions were responsible for a reduction in gain. This extended investigation, with additional paints of both metallic and non-metallic varieties, demonstrates clearly that despite small variations in gain for each sample, there is no reason to believe that those paints with metallic inclusions alter the performance of the antenna in any significant way. It is therefore seen that the metallic paints behave as dielectrics despite the included metallic particles. The metallic particles are flakes of aluminium, and hence electrically conductive, however, they are precluded from creating a bulk conductive material by the material that surrounds them. The electrons are free to move within each individual flake, but these flakes are small (in the order of 50 m) and isolated from each other by the dielectric paint binder. The end result is that there are no truly free electrons in the metallic paints, and the material behaves like a dielectric in the microwave region. Given the small size of the particles, a very high frequency in the Terahertz region would be required to obtain wavelengths comparable to the particles dimension and elicit a different response from the material to an applied electromagnetic wave. Various metallic paint samples were investigated in the water based chemistry, as it is the dominant OEM automotive paint type. Only one metallic and non-metallic sample was tested from the polyurethane chemistry. The metallic polyurethane paint (sample D) was the thickest paint in the test, by a factor as high as six, yet it was found to have a smaller value of gain reduction than many of the other metallic paints which had been applied in thinner layers. It is possible that this result may indicate a difference between the two chemistries, but given the limited number of polyurethane samples investigated it is difficult to draw such a conclusion. Paint samples M and N which bear an electrostatic primer provide a clearly measurable change in the performance of the shows a narrowband antenna. Examination of the plot of slightly wider bandwidth when samples M and N are present. The electrostatic primer present on these samples is intended to provide a level of DC conductivity to enable electrostatic painting. At these microwave frequencies however, the electrostatic primer behaves like a lossy dielectric rather than a conductor. This leads to a reduction in the antenna’s quality factor, and a consequent widening of the antenna’s bandwidth. The radiation pattern of the narrowband antenna and corresponding gain plot reveal that samples M and N provide a similar radiation pattern in terms of shape, but at a reduced value of gain. Measured gains are approximately 1.5 dB lower than non-electrostatic samples. The persistence of the radiation pattern suggests that the directivity of the radiating structure may be unchanged. Recalling that antenna gain is the product of efficiency and directivity, it may be that the lower gain value is due to a reduced efficiency. This could be as a result of the additional lossy layer of electrostatic primer. Antenna efficiency measurements are very difficult to perform accurately, and no attempt to measure the efficiency directly has been made in this work. How-

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ever, it would seem that the effect of the addition of the partially conductive electrostatic primer is a widening of the impedance bandwidth and reduction in the gain of the narrowband antenna. The effects of the electrostatic primer are less prominent on the wideband antenna, with no significant gain reduction being visible in the presence of samples M and N. None of the paint samples provided truly disruptive shielding, as seen when the copper clad superstrate of sample O is applied to the antennas. The highly conductive layer of copper on the surface of this superstrate had extreme effects on the impedance bandwidth, gain and radiation of both antennas. Under the copper clad superstrate of sample O, the narrowband antenna’s impedance never matches to the 50 line, as shown remaining above the 10 dB line. This reby the curve of sults in very poor radiation performance, with the co-polarized patterns returning highly negative values of gain. Interestingly, on the wideband antenna the copper clad superstrate does achieve a matched impedance at several frequencies at the upper end of the band. Examination of the simulated fields at these frequencies reveal that this is likely a consequence of the energy guided down the CPW coupling into an alternative mode other than the usual radiation mode. This appears to be a parallel plate transmission line mode in between the metallic ground plane of the antenna and the copper cladding above, with the wave therefore propagating inside the Rogers RT/duroid 5880 material of the superstrate and radiating from the superstrate edges. For many applications it may be useful to simulate paints in electromagnetic software packages. The paint layer is very thin relative to the wavelength of interest which poses a challenge for the simulation of such scenarios due to the requirement to increase the mesh density in the plane of the paint to accurately model the thin material. A few investigative simulations were performed using the CST Microwave Studio Transient solver and the results indicated findings similar to those reported here, but required manual addition of local mesh which resulted in 30% more mesh cells than with no paint layer present. Note that for those paints which contain metallic flakes, it is not necessary to model the individual conductive particles since the material behaves like a bulk dielectric. VI. CONCLUSION The addition of superstrates painted with automotive-specific paints causes a shift in the frequency characteristics of both narrowband and wideband antennas. This is due to the change in the effective relative permittivity of the space surrounding the antenna. Hence the magnitude of the shift is a function of the dielectric constant of the paint and the thickness with which it is applied. The consequences of frequency shifting may be significant for narrowband antennas where a small shift could lead to loss of the desired communications service. If an unknown paint is to be applied to an antenna it would be advisable to employ an antenna with greater than required bandwidth. The automotive metallic paints examined appear to behave as dielectrics despite densely packed metallic inclusions. The size of each flake is small relative to the wavelength of interest, and the paint binder prevents electron flow between adjacent flakes under an applied electromagnetic field. The presence of metallic

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and non-metallic paint on the antenna superstrate leads to a very slight gain reduction on the narrowband antenna, less than or equal to 0.5 dB. Electrostatic primers are used globally in automotive production lines, yet to the authors’ knowledge no previous work in the literature has considered their effect on antennas. This work found that they may cause a reduction in the gain of an antenna due to the partially conductive nature of this primer. Our test showed a gain reduction of approximately 1.5 dB when applied to the narrowband patch antenna. ACKNOWLEDGMENT The authors wish to acknowledge the support of other members of the research group, K. Zhang and R. Gupta, and also D. Welch for his expert technical assistance. B. D Pell thanks Mr. P. Francis of the RMIT Microscopy and Microanalysis facility (RMMF). REFERENCES [1] M. Cerretelli and G. B. Gentili, “Progress in compact multifunction automotive antennas,” in Proc. Int. Conf. on Electromagnetics in Advanced Applications, 2007, pp. 93–96. [2] H. Lindenmeier, J. Hopf, and L. Reiter, “Active AM-FM windshield antenna with equivalent performance to the whip now as standard equipment in car production,” in Proc. Antennas Propag. Society Int. Symp., 1985, pp. 621–624. [3] J. Hopf, L. Reiter, and S. Lindenmeier, “Compact multi-antenna system for cars with electrically invisible phone antennas for SDARS frequencies,” in Proc. 2nd Int. ITG Conf. on Antennas, 2007, pp. 171–175. [4] L. Low, R. Langley, R. Breden, and P. Callaghan, “Hidden automotive antenna performance and simulation,” IEEE Trans. Antennas Propag., vol. 54, pp. 3707–3712, 2006. [5] E. Gschwendtner and W. Wiesbeck, “Ultra-broadband car antennas for communications and navigation applications,” IEEE Trans. Antennas Propag., vol. 51, pp. 2020–2027, 2003. [6] B. Pell, W. Rowe, E. Sulic, K. Ghorbani, S. John, R. Gupta, K. Zhang, and B. Hughes, “Experimental study of the effect of paint on embedded automotive antennas,” in Proc. IEEE Vehicular Technology Conf. VTC Spring, 2008, pp. 3057–3061. [7] T. Chu and R. Semplak, “A note on painted reflecting surfaces,” IEEE Trans. Antennas Propag., vol. 24, pp. 99–101, 1976. [8] V. Hombach and E. Kuhn, “Complete dual-offset reflector antenna analysis including near-field. paint-layer and CFRP-structure effects,” IEEE Trans. Antennas Propag., vol. 37, pp. 1093–1101, 1989. [9] T. Otoshi, Y. Rahmat-Samil, R. Cirillo, and J. Sosnowski, “Noise temperature and gain loss due to paints and primers: A case study of DSN antennas,” IEEE Antennas Propag. Mag., vol. 43, pp. 11–28, 2001. [10] E. Hoare and R. Hill, “System requirements for automotive radar antennas,” Inst. Elect. Eng. Colloq. on Antennas for Automotives, pp. 1/1–111, 2000. [11] C. Lyons and I. Taskin, “A low-cost MMIC based radar sensor for frontal, side or rear automotive anticipatory precrash sensing applications,” in Proc. 4th IEEE Intelligent Vehicles Symp., 2000, pp. 688–693. [12] C. Sabatier, “2 GHz compact antennas on handsets,” in Proc. Antennas Propag. Society Int. Symp., 1995, vol. 2, pp. 1136–1139. [13] J. Pfanstiehl, Automotive Paint Handbook: Paint Technology for Auto Enthusiasts and Body Shop Professionals. New York: HP Books, 1998, p. 13. [14] A. Robinson and A. Livesey, Repair of Vehicle Bodies. Burlington, MA: Butterworth-Heinemann, 2006, p. 614. [15] R. Lambourne and T. A. Strivens, Paint and Surface Coatings: Theory and Practice. Cambridge, U.K.: Woodhead Publishing, 1999, pp. 450–457. [16] C. Huang and D. Lin, “CPW-fed bow-tie slot antenna for ultra-wideband communications,” Electron. Lett., vol. 42, pp. 1073–1074, 2006. [17] N. Alexopoulos and D. Jackson, “Fundamental superstrate (cover) effects on printed circuit antennas,” IEEE Trans. Antennas Propag., vol. 32, pp. 807–816, 1984. [18] D. Jackson and N. Alexopoulos, “Gain enhancement methods for printed circuit antennas,” IEEE Trans. Antennas Propag., vol. 33, pp. 976–987, 1985.

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[19] H. Yang and N. Alexopoulos, “Gain enhancement methods for printed circuit antennas through multiple superstrates,” IEEE Trans. Antennas Propag., vol. 35, pp. 860–863, 1987. [20] R. Lee, A. Zaman, and K. Lee, “Effects of dielectric superstrates on a two-layer electromagnetically coupled patch antenna,” in Proc. Antennas Propag. Society Inte. Symp. Digest, 1989, vol. 2, pp. 620–623. [21] H. Chang-Hsiu and H. Powen, “Superstrate effects on slot-coupled microstrip antennas,” IEEE Trans. Magn., vol. 27, pp. 3868–3871, 1991. [22] R. Shavit, “Dielectric cover effect on rectangular microstrip antenna array,” IEEE Trans. Antennas Propag., vol. 42, pp. 1180–1184, 1994. [23] W. Tan, Z. Shen, and Z. Shao, “Radiation of high-gain cavity-backed slot antennas through a two-layer superstrate,” IEEE Antennas Propag. Mag., vol. 50, pp. 78–87, 2008. [24] R. Simons and R. Lee, “Linearly tapered slot antenna with dielectric superstrate,” in Proc. Antennas Propag. Society Int. Symp. Digest, 1993, vol. 3, pp. 1482–1485.

Brendan D. Pell (S’07) was born in Melbourne, Australia, in 1983. He received the B.E. (Hons.) degree in electronic engineering from RMIT University, Melbourne, Australia, in 2005, where he is currently working toward the Ph.D. degree. His research interests include wideband planar antennas, and polymeric composite materials used in automotive applications. Mr. Pell is a member of the IEEE AP and VT societies, and is also a member of SAE International (formerly the Society of Automotive Engineers). He was awarded the Highly commended student certificate at the Eleventh Australian Symposium on Antennas in 2009.

Edin Sulic was born in Bihac, Bosnia and Herzegovina, in 1977. He received the B.E. (Hons.) degree in manufacturing engineering and management and the B.E. degree in computer science from RMIT University, Melbourne, Australia, in 2003, where he is currently working toward the Ph.D. degree. He is currently employed as a Senior Product Engineer at Futuris Automotive in Melbourne, Australia. His research interests include use of multifunctional composite components in automotive applications and use of novel materials in NVH (noise, vibration and harshness) applications for electric/hydrogen vehicles. Mr. Sulic is a member of SAE International (formerly the Society of Automotive Engineers).

Wayne S. T. Rowe (M’04) received the Bachelor of Communication Engineering degree (Honors) and Ph.D. degree from RMIT University, Melbourne, Australia, in 1998 and 2002, respectively. He is currently a Senior Lecturer at RMIT University. He has performed research and consulting work for numerous industry partners, including Cochlear (Australia), DSTO Australia, Composite Materials Engineering Pty Ltd., SP Ausnet, and BAE Systems. His research interests include conformal, embedded and load bearing printed antennas and sensing structures, integrated antenna/microwave/photonic modules, tunable metamaterials, wireless power transmission, and RF sensors for high voltage power distribution lines. Dr. Rowe is a member of the IEEE AP and MTT societies, and is also a reviewer for the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION.

Kamran Ghorbani (S’96–M’01) received the B.E. degree in communication and electronic engineering (Honors) and Ph.D. degree (antennas and phase shifters) in communication engineering both from RMIT University, Melbourne, Australia, in 1995 and 2001, respectively. He worked as an RF Designer for AWA Defence Industries for two years, and then joined the RF and Photonic Research Group at RMIT University in 1996. He then worked for a telecommunication company (Tele_IP) as a Senior RF Engineer mainly working on RF circuit and system design from 1999 to 2001. He joined RMIT University in 2001 and for the last 9 years the focus of his research has been in the areas of broadband and printed antennas, tunable RF devices, microwave system and microwave photonic systems.

Sabu John was born in Kerala, India. He received the B.S. degree in mechanical engineering (Hons) from the University of Lagos, Nigeria, in 1982 and the M.S. degree in advanced applied mechanics and the Ph.D. degree from Imperial College, London, U.K., in 1983 and 1986, respectively. He is currently a Professor of smart materials and systems at RMIT University, Melbourne, Australia. His research interests include advanced composite materials, vibration control in smart structures, embedded communication devices in composite structures, and structural health monitoring of structures. He has also done work on signal processing and pattern recognition through artificial neural networks applied to feature detection in medicine and damage detection in large static and dynamic structures. He is currently working on resonator based wireless strain monitoring of static and dynamic structures and energy harvesting from piezoelectric devices.

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A Toolset Independent Hybrid Method for Calculating Antenna Coupling Mark Kragalott, Michael S. Kluskens, Member, IEEE, Dale A. Zolnick, Member, IEEE, W. Mark Dorsey, Member, IEEE, and John A. Valenzi

Abstract—Calculation of the electromagnetic interference (EMI) between electrically large antennas mounted on ships is important for a variety of Navy problems. This paper presents a toolset independent hybrid method for calculating the power at receive antenna terminals relative to the power incident on transmit antenna terminals. The hybrid method coupling results are validated against full-wave computational electromagnetic (CEM) simulations and measurements. An advantage of the proposed hybrid approach is that CEM calculations for antenna near-fields and propagation between antennas can be executed with user-preferred tools. In addition, transmit and receive antenna calculations are executed in transmit mode independent of ship structures. Thus, antenna calculations can be stored in a library for calculation reuse and optimization of antenna placement for EMI reduction. Index Terms—Electromagnetic coupling, electromagnetic interference (EMI), reciprocity theorem, surface equivalence.

I. INTRODUCTION

E

LECTROMAGNETIC interference (EMI) between transmit and receive antennas on Navy ships is an important consideration for relative antenna placement. Many ships in the Surface Fleet have more than one hundred on-board antennas operating collectively over several decades of bandwidth. Whether on-board or off-board, antennas that interfere reduce mission effectiveness. Owing to the electrically large geometry of ships over much of the operational bandwidth, an entire-domain full-wave computational electromagnetic (CEM) approach to directly determine coupling between the myriad of platform-mounted transmit and receive antenna pairs is presently prohibitive even if accelerated techniques are implemented on modern parallel computer platforms. An accurate alternative approach to the coupling problem is to solve for antenna near-fields using full-wave methods, and then to hybridize the antenna solutions with ship propagation solutions, which themselves are some combination of full-wave and asymptotic methods. For a given Navy ship, multiple government and corporate organizations typically make independent antenna calculations. A particular design group will utilize their preferred set of antenna tools, while other design groups employ different sets of Manuscript received April 19, 2010; revised June 24, 2010; accepted July 29, 2010. Date of publication December 03, 2010; date of current version February 02, 2011. The Authors are with the Radar Division, Naval Research Laboratory, Washington, DC 20375 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2096403

tools for other antennas on the same ship. Collating the diverse antenna data for the coupling calculation becomes problematical for the engineer. In some cases, the engineer may confront the costly decision of re-modeling the antennas and executing an antenna tool compatible with the available coupling method. An accurate and efficient antenna coupling method is needed to expedite the optimum relative positioning of antennas on a ship either in the initial design phase or for a sensor upgrade. The approach should allow antenna designers from independent groups to employ the data generation toolsets of preference including both CEM calculations and measurements. Further, the approach should not mandate that the antenna toolsets be re-executed as ship structures or antenna mount locations are altered. In the event of ship configuration modifications, the only tool that should normally require re-execution is the propagation tool. There are varied approaches in the literature to determine platform-mounted antenna-to-antenna coupling. CEM coupling approaches range from rudimentary methods to asymptotic, full-wave, or hybrid techniques. When computational resources are limited, entire-domain asymptotic coupling methods can be applied if antenna fields and platform propagation can be approximated with acceptable accuracy [1]–[5]. Entire-domain full-wave coupling methods are usually restricted either to rudimentary antenna coupling geometries [6] or to lower frequencies where the platform electrical size including the antennas is tractable given the available computational resources [7]–[9]. Hybrid coupling methods are a means of accurately modeling antenna-domain fields with full-wave methods while treating platform-domain propagation effects efficiently with either full-wave or asymptotic methods [10]–[18]. Hybrid methods often include innovative techniques for propagating fields between the antenna-domain and the exterior-domain [19]–[23]. While the literature on antenna coupling is substantial, the published approaches do not simultaneously accommodate the multiplicity of antennas, platform configurations, and toolsets required by the authors. This paper presents a toolset independent hybrid method to calculate antenna coupling. Assume the original problem geometry is too electrically large to be solved with entire-domain fullwave CEM methods employing the available computational resources. To make the coupling problem tractable for a variety of antennas mounted on an electrically large platform, the problem will be divided into independent calculation domains. Each domain problem will be solved with user-preferred CEM (or measurement) methods, and the solutions will be combined to find the coupling. In the first two independent calculation domains,

U.S. Government work not protected by U.S. copyright.

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transmit antenna and receive antenna near-fields are calculated on virtual surfaces enclosing each antenna with user-preferred full-wave CEM tools, where each antenna is in isolation but with electromagnetically significant platform-mounting structures inside the domain. In the near-field calculation, the receive antenna is excited with a test current source at its terminals. The surface equivalence theorem [24] is invoked to transform the transmit antenna virtual surface fields to equivalent currents. In the third calculation domain, the transmit antenna equivalent surface currents are employed as sources in a user-preferred CEM method to calculate the fields propagated to the receive antenna virtual surface in the presence of the platform. In the propagation calculation, the transmit antenna model is absent and the receive antenna model is usually absent. In the fourth and final calculation domain, the reciprocity theorem [24] is applied between the receive antenna virtual surface and receive antenna terminals. This operation leads to an equality between an inner product of the equivalent currents and fields integrated over the surface enclosing the receive antenna and the product of the test current and received voltage at the receive antenna terminals. The resulting equation requires the transmit antenna and receive antenna impedances along with antenna near-fields to determine coupling. However, if impedances are not available, an alternative is to estimate the mismatch losses of transmit and receive antennas. There are distinct advantages to the proposed coupling approach. The CEM tool employed to model the antennas is irrelevant provided that it can accurately calculate near-fields and impedances. Since the receive antenna virtual surface near-fields are solved in transmit-mode with a test source, the engineer avoids implementing complicated incident fields otherwise dictated by the platform field environment in a receive-mode calculation. The assumed platform independence of the antenna calculations implies that they need to be executed only one time. This enables the calculated set of antenna virtual surface fields and impedances to be stored in a library for antenna placement optimization on the platform. For purposes of library storage, the authors have found that a good choice for the near-field virtual surface is a rectangular box with regularly spaced fields because this configuration enables both ease of coding and compatibility with a wide range of commercially available CEM antenna codes. However, any near-field virtual surface that encloses the antenna and produces a field distribution with acceptable accuracy exterior to the surface is compatible with the hybrid method. Finally, the proposed hybrid method is compatible with measured near-fields at virtual surfaces in lieu of CEM calculated near-fields. Section II describes the hybrid coupling method. Section III presents the calculated results and compares them with both full-wave entire-domain CEM calculations and measurements. Section IV gives the conclusions. II. HYBRID COUPLING METHOD As shown in Fig. 1, the present problem is to compute the coupling between the antennas in the presence of nearby objects. The transmit antenna is excited by a generator that produces the . These fields couple to the receive antenna total fields

Fig. 1. Original coupling problem with antennas and nearby object.

and produce the voltage at the terminals which has looking toward the receiver. The nearby complex impedance objects could range from relatively simple geometries such as a reflector for an antenna feed to complicated platforms such as a ship. The coupling between two antennas can be described by [25] (1) is the power received at the receive where C is the coupling, is the power incident on the antenna terminals - , and is given by [26] transmit antenna terminals - .

(2) is the peak generator voltage, and is the real part where of the generator impedance. The power accepted by the transmit antenna at - is

(3) is transmit antenna impedance, is the generator impedance, and and are the complex voltage and current. The portion of incident power unavailable to the transmit antenna resulting from impedance mismatch reflection is found by subtracting (3) from (2). The power radiated by the transmit antenna is

where

(4) where and are the electric and magnetic fields, respecenclosing the transmit antenna. Subtively, on a surface tracting (4) from (3) gives the heating loss of the antenna. A. Transmit Antenna Calculation Fig. 2(a) shows the geometry for the transmit antenna doon main. The transmit antenna is enclosed by a virtual box which the fields produced by the generator are is also to be determined. The complex antenna impedance calculated at the terminals - . A box with faces parallel to . The field comCartesian coordinate planes is chosen for ponents tangential to are determined on each box face. Sur-

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a Thevenin equivalent voltage source. The fields are determined on the virtual box enclosing the receive is also found at antenna. The complex antenna impedance - . This calculation is executed in the absence of both nearby objects and the transmit antenna. Unlike the transmit antenna is required calculation, no equivalent current transform on in the receive antenna calculation. With this exception, all other comments in Section II.A that apply to the transmit antenna calculation also apply to the receive antenna calculation. C. Field Propagation Calculation

Fig. 2. The four calculation domains of the hybrid coupling method including (a) the transmit antenna domain, (b) the receive antenna (in transmit mode) domain, (c) the field propagation domain, and (d) the coupling domain.

face equivalence is then invoked to determine the equivalent currents , where , and is the outward-directed normal vector to . The fields are not determined in the presence of nearby objects or the receive antenna, so an approximation to the exact solution has been introduced. However, the transmit antenna model should include antenna platform-mounting structures that could significantly modify the near-fields of the antenna and thus impact coupling. Platform-mounting structures include, but are not limited to, the platform surface, radar absorbing structures (RAS), backplanes, radomes, antenna cavities, and edge tapers. could be a six-sided box for antennas The virtual surface mounted with no electrically large conducting backplane, a fivesided box for antennas mounted in front of a large conducting backplane, or a planar window for antennas mounted inside a cavity behind an aperture in a large conducting plane. The denis sufficient to ensure sity and distribution of field points on no aliasing of field data exterior to the . In general, the modeler should be able to calculate the near-fields exterior to the virtual surface by applying either a full-wave code to the original antenna problem or by radiating the virtual surface equivalent currents with a Green’s function appropriate to the structure exterior to the surface. If both the full-wave solution and equivalent currents produce near-fields converged to acceptable accuracy, there is adequate equivalent current distribution on the surface. B. Receive Antenna Calculation If the receive antenna domain has the same local geometry as the transmit antenna domain in every respect, including its platform mounting structures, then this calculation is not necessary, because the fields radiated by the receive antenna in transmit mode to the virtual box surface enclosing the antenna were determined in Section II.A. However, if the receive antenna or its mounting structure differ from the transmit antenna, then the receive antenna should be excited at antenna terminals - in transmit mode using the test current source as shown in Fig. 2(b) or by

Referring to Fig. 2(c), which shows the field propagation domain, a CEM method is now employed to determine the fields on produced by the equivalent current on the transmit antenna virtual box in sources the presence of nearby objects. The equivalent current sources were determined in Section II.A. To facilitate the should coupling calculation, the field point locations on in Section II.B. coincide with the field locations of using Equivalent currents are formed on and , where is the inward-directed normal produced by vector to . Since fields interior to are due only to scattering from objects exterior to , it is often a reasonable coupling approximation to ignore the in presence of the transmit antenna geometry interior to the propagation calculation. The propagation method can be , full-wave or asymptotic, but to obtain the total fields on the geometry should include both the nearby objects (e.g., ship structures) and a model of the receive antenna. However, in all the examples for which the authors have calculated coupling, incorporating a model of the receive antenna geometry in the propagation domain is not necessary to achieve accurate results. This is because the fields scattered by the receive antenna on are primarily outwardly propagating and thus do not usually contribute significantly to coupling at - . D. Coupling Calculation In the coupling calculation domain, the induced voltage across the receive antenna terminals due to the source from the transmit antenna is determined. Referring to the receive antenna is defined equivalent circuit in Fig. 3(a), the received power by

(5) where is the receiver impedance at terminals looking toward the receiver. interior to due Referring to Fig. 2(d), the fields to the equivalent current sources on could be diacross rectly calculated to determine the received voltage . However, in general most commercially available CEM codes do not efficiently calculate incident fields from multiple perimeter current sources. Moreover, the perimeter equivalent are platform-dependent so the incident field currents approach would require full-wave receive antenna recalculation every time the platform environment changed. To avoid these consequences, the reciprocity relation will be invoked so that the

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terminals - . The coupling, C, can be calculated by solving in (8) and in (5), substituting these results into for , and dividing by to arrive at (7) to find

Fig. 3. Equivalent circuit quantities of receive antenna described in Section II.D for (a) receive mode and (b) transmit mode.

(9) voltage can be found as a part of the relatively straightforward calculation of exciting the receive antenna with a current on . The source at - to find the radiated fields on was previously deprocedure to find the fields scribed in Section II.B. As an additional benefit, the reciprocity approach enables the receive antenna transmit mode virtual box near-fields, which are independent of the platform field environment, to be stored in a file library along with the transmit antenna near-fields for use in antenna placement optimization. In Fig. 2(d), the equivalent currents and fields that have been calculated on the receive virtual box using the procedures in Sections II.A–C are now employed in the reciprocity relation

is given by (2), and the square bracketed term is the where is transmission coefficient for the receive antenna. In (9), defined by

(10) is the power dissipated in heat by the receive where is the power radiated by antenna in transmit mode, and the receive antenna in transmit mode. is found from

(11)

(6)

in volume bounded by where the incident fields are produced by the currents on , and the fields on are produced by receive antenna test sources in . Carrying out the volume integration after in noting only electric sources are needed to determine and taking the magnitude squared, (6) becomes

(7)

is the test current source across the terminals where of the receive antenna as shown in Fig. 2(b). must be expressed by quantities readily available by the calculation made in Section II.B, which solved for the near fields of the receive antenna in transmit mode. Referring to the equivalent circuit in Fig. 3(b), the power accepted by the receive antenna in transmit mode is

(8) is the receive antenna impedance where looking into the antenna terminals - from the generator, is the current into the antenna, and is the voltage across the

does not appear in (9) The transmit antenna impedance which at first seems counter-intuitive. The explanation is that transmit antenna impedance mismatch losses are included in . the reaction integral in (9) through the amplitude of Polarization mismatch losses are also included in the reaction integral. The authors have found it convenient to normalize the transmit mode fields on each virtual box so that both transmit and receive antennas radiate unit power (i.e., one watt). The received power in watts is still readily found by multiplying the normalized received power by the power radiated in watts by the transmit antenna. The power incident on the transmit given in (2) can be re-written with the aid of antenna to become equivalent circuit theory and normalized by

(12) where is the normalized power incident on the transmit is the un-normalized power dissiantenna terminals, pated in heat by the transmit antenna, and the term in square brackets is the inverse of the transmit antenna transmission coefficient. In (12), the impact of the transmit antenna impedances is explicit. To normalize (9) for transmit antenna unit radiated with , and then power, (12) is used to substitute the equivalent currents in the numerator of (9) become normalized currents. Similarly, to normalize (9) for receive antenna (in transmit mode) unit radiated power, (10) is divided by

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and substituted for , while the numerator fields in (9) become normalized fields. Equation (9) for the coupling, C, is then

(13) and are found from where radiated powers un-normalized field quantities in (4) and (11) respectively. The superscript in the reaction integral indicates that the fields on produced by the receive antenna in transmit mode are normalized to radiate unit power, and the superscript also indicates are produced by transmit that the equivalent currents on antenna fields on normalized to radiate unit power. The and un-normalized antenna heating losses are determined by full-wave methods, but for most practical antennas the heating loss is small compared to the radiated power. Thus, the power ratio terms in (13) are in most cases close to unity. Each of the transmission coefficient terms in square brackets retain a factor of four so if an antenna is conjugate impedance matched to its generator or load impedance, the term goes to one. Polarization mismatch losses are included the reaction integral. III. RESULTS In this section, the hybrid coupling approach described in Section II is validated against entire-domain full-wave calculations for the cases of cavity to cavity coupling as well as dipole to dipole coupling. In addition, the hybrid coupling method is validated against measurements for the case of horn to horn coupling with and without the presence of a nearby object. For antenna domain modeling, the authors employed several different codes including the finite-difference time-domain (FDTD) code FDTD3D [27], the method of moments (MoM) code WIPL-D [28], the finite element method boundary integral equation (FEM-BIE) code CEMNAV [29], and the finite integral technique (FIT) code CST Microwave Studio [30]. These codes were chosen because of their familiarity and availability to the authors, although any full-wave CEM code capable of accurately calculating near-fields would have been appropriate. In addition, for calculating the fields propagated from the transmit antenna virtual surface to the receive antenna virtual surface, the half-space Green’s Function and the asymptotic physics code RTS [31] were employed owing to their availability to the authors. However, any full-wave or asymptotic code capable of accurately calculating propagating fields for the selected cases would have been appropriate. A. Cavity to Cavity Coupling Fig. 4(a) shows two identical cubical cavities, 0.4 m per side, that are separated edge-to-edge by 0.8 m and embedded in a perfect electric conducting (PEC) half-space that has a normal vector oriented in the -direction. A current source, located

Fig. 4. (a) The original cubical cavity coupling geometry and (b)–(d) the first three hybrid method calculation domains from the procedures described in Section II.A–C, respectively.

at the center of the cubical transmit cavity, produces fields that are to be determined at the center of the cubical receive antenna cavity from 0.25 GHz to 0.35 GHz. This example depicts rudimentary coupling between recessed antennas across a planar conducting ship deckhouse and tests the viability of the domain decomposition assumptions of the hybrid method described in Section II. With no antenna feeds present, the input impedances required by (13) are not available. As a consequence, (6) will be employed since that coupling expression involves only currents and fields. Following the procedure of Section II, the coupling is determined from calculations in the four domains. In the first calculation domain as shown in Fig. 4(b), the transmit cavity with the current source is modeled with the FDTD3D code [27] as a standalone structure embedded in a PEC half-space, and the (and resulting equivalent currents ) fields are determined on a transmit cavity virtual aperture surface at 0.05 m intervals on the dashed line. In the second calculation doproduced by test main shown in Fig. 4(c), the fields sources located at the center of the standalone receive cavity are calculated with FDTD3D at 0.05 m intervals on the virtual aper, and field components at the ture surface. To calculate center of the receive cavity using (6), the fields on the receive and cavity virtual aperture surface must be produced by test sources, respectively, at the center of the cavity. Because the receive cavity and transmit cavity geometries are identical, the fields due to a source were previously determined in the transmit cavity calculation. Thus, only the fields produced by and sources need to be determined on the receive aperture. For the field propagation domain shown in Fig. 4(d), the half-space Green’s function is employed to find the fields (and resulting equivalent currents) at the receive aperture virtual surface produced by equivalent currents at the transmit cavity aperture. Because the currents radiate with a half-space PEC surface, the geometry of each cavity is ignored. The PEC surface shorts out the electric equivalent currents at the transmit aperture, so only the tangential magnetic equivalent currents radiate in this problem. Only electric equivalent currents exist at the receive aperture because the tangential electric fields, which form

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Fig. 6. Strip dipole coupling geometry including virtual box dimensions for use with the hybrid coupling method.

Fig. 5. Comparison of calculated receive cavity coupled fields between an entire-domain FDTD3D simulation and the hybrid method employing FDTD3D for the aperture near-fields and the half-space Green’s function for the propagated fields.

currents, are zero on the PEC surface. In the final calculation domain, information from the other three calculation domains are employed in (6) to determine the field components at the center of the receive cavity. Fig. 5 shows the magnitude of the field components as a function of frequency determined both by the proposed hybrid approach and by a reference entire-domain FDTD3D simulation with a grid resolution of 20 cells per wavelength at 0.3 GHz. There is good agreement between the two calculation methods across the frequency range. Note that the hybrid results are accurate even though both the transmit cavity calculation and receive cavity transmit mode calculation ignored the presence of away over the frequency the other cavity, which is less than band. Moreover, accuracy is maintained even though the propagation calculation ignored the presence of both cavities. Thus, the domain decomposition assumptions of the hybrid method are valid for this class of problem. B. Dipole to Dipole Coupling Fig. 6 shows the geometry for coupling between identical transmit and receive strip dipoles in free space. The infinitesimally thin dipoles are 216 mm long and 6 mm wide, so they are one-half wavelength long at 0.694 GHz. The separation dism and m. The coutances considered are pling calculation will be carried out from 0.2 GHz to 0.8 GHz. Because the input impedance of this dipole varies considerably over the frequency range, the impedance terms as well as the reaction integral in (13) will be tested. The generator and load impedances of the antennas are assumed to be 50 ohms over the frequency range. The hybrid coupling method employs WIPL-D [28] to solve for dipole impedance and near-fields and RTS [31] to solve for the free-space propagated fields. As Sections II.A–B dictates, the dipole near-fields on a virtual box surrounding each antenna are calculated in isolation using WIPL-D. The near-fields are then normalized for unit power radiation. Equivalent currents

D=05

Fig. 7. Comparison of calculated strip dipole coupling at : m between entire-domain simulations using CEMNAV and WIPL-D and the hybrid method employing WIPL-D for the dipole near-fields and RTS for the free-space propagated fields.

are formed from the normalized near-fields on the transmit antenna box. Then, in accordance with Section II.C, RTS uses the equivalent currents from the transmit antenna box as sources to calculate the fields propagated to the receive antenna box. The RTS calculation environment is free-space, so the presence of both dipoles is ignored. Equivalent currents are then formed on the receive antenna surface from the propagated fields. Finally, the coupling is found from (13). Figs. 7 and 8 show the broadside strip dipole coupling for m and m respecthe separation distances of tively. For the 0.5 m case shown in Fig. 7, the Navy’s CEMNAV code [29] is used to obtain an entire-domain solution to validate the WIPL-D entire-domain reference solution. The two reference solutions are seen to differ by a slight frequency shift. The hybrid coupling solution employing WIPL-D and RTS exdB deviation) compared to the hibits reasonable accuracy ( entire-domain WIPL-D reference solution considering that the electrical separation between the virtual boxes is less than one wavelength and RTS is an asymptotic solution method. With the distance between antennas increased to 10.0 m as seen in dB Fig. 8, the hybrid coupling solution error decreases ( deviation) compared to the WIPL-D reference solution. The impedance terms in (13) have been validated in this example since the input impedance of this dipole antenna varies drastically over the frequency range.

KRAGALOTT et al.: A TOOLSET INDEPENDENT HYBRID METHOD FOR CALCULATING ANTENNA COUPLING

D = 10 0

Fig. 8. Comparison of calculated strip dipole coupling at : m between entire-domain simulations using CEMNAV and WIPL-D and the hybrid method employing WIPL-D for the dipole near-fields and RTS for the free-space propagated fields.

Fig. 9. Ku-band horn coupling geometries showing (a) the free-space measurement set-up and (b) the horns with aluminum plate and pole.

C. Horn to Horn Coupling Fig. 9 shows the geometries for coupling between identical circular aperture Ku-band horn antennas both in free-space and in the presence of an elevated rectangular aluminum plate connected to a square cross section aluminum pole. The two hybrid coupling calculations are performed within the impedance ) and the heating loss bandwidth of the horn (i.e., is small since the horn is constructed of metal. Thus, the reaction integral term of (13) will have the dominant quantitative impact on coupling. Measurement data is used for comparison with the hybrid method. The two horn broadside radiation directions are parallel to each other and the apertures are in the same plane. The separation distance between the centers of the apertures in both cases was 1 ft. measurements were performed in an NRL Coupling Radar Division anechoic chamber using an Agilent 8722ES network analyzer. The frequency-domain coupling signal was

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Fig. 10. Comparison of free-space E-plane to E-plane Ku-band horn coupling between measurements and the hybrid method using CST for the horn nearfields and RTS for the propagated fields.

transformed into the time domain through an inverse discrete Fourier transform (DFT) so that an appropriate pass band time gate could be applied. The time domain gate allows the primary coupling signal to pass without attenuation while minimizing the contribution of unwanted environmental reflections and multipath signals. The time-gated signal is transformed back into the frequency domain using a DFT, and the resulting frequency domain signal is compared to the analytical calculations. The signal-to-noise ratio (SNR) of the frequency domain measurement is improved through the use of this time gating technique by minimization of the effective noise floor of the measurement facility [32]. Subsequent improvements are obtained in the accuracy of the measured results. The measurement data was generated from several anechoic chamber experiments with the horns resting on a low relative permitfoam puck. The measurements were performed tivity over several days with the set-up dismantled and reassembled between days. The effective noise floor of the measurement dB across the frequency band, so the was approximately measurements cannot be employed as a benchmark below this level. Once again, the procedures outlined in Section II are followed to calculate coupling with the hybrid method. The isolated horn near-field calculation is performed using the FIT-based code CST [30]. Moreover, because transmit and receive antennas are identical, only a single calculation is necessary. The propagation calculation is executed by RTS using the transmit antenna box currents as sources with the horns not present. RTS propagates fields to the receive antenna virtual box both in free-space and in the presence of the plate and pole. Finally, the results from the calculations procedures are substituted into (13) to obtain the coupling results for each case. Fig. 10 shows the measured free-space E-plane to E-plane horn coupling plotted against the hybrid method. The coupling for the measured average, maximum, and minimum over several measurements are plotted against the hybrid method using CST and RTS. As the coupling approaches the noise floor of the measurements, the difference between the measured maximum and minimum increases. There is good agreement between the

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full-wave analysis after the antennas are mounted on a platform. This would eliminate the inherent efficiency advantage of using pre-calculated virtual box fields for antenna placement optimization on the platform. Moreover, the hybrid method has been shown to give accurate antenna coupling results at electrically short separation distances without feedback for typical coupling cases, including an example where a scattering object is electrically close to the antennas. A future problem to address is the file format of the virtual box fields. The Air Force Research Laboratory has developed a general CEM file format spec called CEMX [33] for consideration as an IEEE specification. A format specification would promote compatibility between different CEM tools. As new or existing antenna tools are developed, box fields could be written in the specified format as opposed to placing the format conversion burden on the engineer. ACKNOWLEDGMENT The authors thank M. Parent for the horn design, D. Taylor and R. Kindt for numerical data, and W. Pala, T. Ready, and R. Mittra for useful suggestions. REFERENCES

Fig. 11. Comparison of Ku-band horn coupling in the presence of metal pole and plate between measurements and the hybrid method using CST for the horn near-fields and RTS for the propagated fields.

free-space horn coupling measurements and the calculations, although the calculated coupling tends to track the upper bound of the measured coupling. The E-plane to E-plane horn orientation case is the only free-space coupling case that could be measured above the noise floor of the network analyzer, so that is the only case presented. Fig. 11 shows both E-plane to E-plane and H-plane to H-plane horn coupling in the presence of the metal plate and pole. The coupling in this case is dominated by specular scattering from the plate although the pole and plate edges also contribute. There is good agreement between the calculations and measurements over the frequency band. This example demonstrates that accurate calculations are achievable with the hybrid method even when an asymptotic propagation method is applied with objects in the near-zone of the transmit and receive antennas. The key point is to ensure that individual asymptotic sources on the virtual box are small enough in physical extent at the given wavelength to be in the far-zone of any nearby objects. IV. CONCLUSION The proposed toolset independent hybrid coupling method has been shown to be an accurate means of determining antenna coupling. Although the method has not been implemented with two-way feedback virtual field surfaces that would include re-excitation of the antennas from bounces off surrounding structures, it could be expanded to include these effects. However, implementing feedback would require additional

[1] B. J. Cown and J. P. Estrada, “SAF analysis for shipboard antenna performance, coupling and radhaz in complex near-field scattering environments,” in Proc. IEEE Antennas Propag. Soc. Int. Symp., 1995, vol. 1, pp. 6–9. [2] E. M. Koper, W. D. Wood, and S. W. Schneider, “Aircraft antenna coupling minimization using genetic algorithms and approximations,” IEEE Trans. Aerosp. Electron. Sys., vol. 40, pp. 742–751, Apr. 2004. [3] H. Miyashita, Y. Sunahara, R. Ishii, T. Katagi, and T. Hashimoto, “An analysis of antenna coupling between arrays on a polyhedron structure,” IEEE Trans. Antennas Propag., vol. 41, pp. 1242–1248, Sep. 1993. [4] R. M. Jha, P. R. Mahapatra, and W. Wiesbeck, “Surface-ray tracing on hybrid surfaces of revolution for UTD mutual coupling analysis,” IEEE Trans. Antennas Propag., vol. 42, pp. 1167–1175, Aug. 1994. [5] L. Matekovits, V. A. Laza, and G. Vecchi, “Synthetic-functions analysis of antennas and inter-antenna coupling in complex environments,” in Proc. IEEE Antennas Propag. Soc. Int. Symp., 2004, vol. 1, pp. 683–686. [6] J. Medbo, J. E. Berg, and M. Jovic, “Validation of antenna coupling and channel modeling in a real propagation environment,” in Proc. IEEE Indoor and Mobile Rad. Comm. Symp., 2009, pp. 1747–1751. [7] J. Peng and C. A. Balanis, “Coupling prediction of HF antennas mounted on helicopter structures using the NEC code,” in Proc. IEEE Antennas Propag. Soc. Int. Symp., 1995, vol. 1, pp. 380–383. [8] L. Juang-Lu, W. O. Price, and D. G. Chapman, “HF antenna coupling on the Boeing 767 AWACS,” in Proc. IEEE Electromagnetic Compat. Int. Symp., 1995, pp. 365–367. [9] B. Turetken, F. Ustuner, E. Demirel, and A. Dagdeviren, “EMI/EMC analysis of shipboard HF antenna by moment method,” in Proc. Int. Conf. on Mathematical Methods and Electromagnetic Theory, Kharkiv, Ukraine, 2006, pp. 350–352. [10] A. Taflove and K. Umashankar, “A hybrid moment method/finite-difference time-domain approach to electromagnetic coupling and aperture penetration into complex geometries,” IEEE Trans. Antennas Propag., vol. 30, pp. 617–627, Jul. 1982. [11] S. A. Davidson and G. A. Thiele, “A hybrid method of moments—GTD technique for computing electromagnetic coupling between two monopole antennas on a large cylindrical surface,” IEEE Trans. Electromagn. Compat., vol. EMC-26, pp. 90–97, May 1984. [12] P. Persson and L. Josefsson, “Calculating the mutual coupling between apertures on a convex circular cylinder using a hybrid UTD-MoM method,” IEEE Trans. Antennas Propag., vol. 49, pp. 672–677, Feb. 2001. [13] P. Bolli, G. G. Gentili, L. Lucci, R. Nesti, G. Pelosi, and G. Toso, “A hybrid perturbative technique to characterize coupling between a corrugated horn and a reflector dish,” IEEE Trans. Antennas Propag., vol. 54, pp. 595–603, Feb. 2006.

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[14] K. Tap, T. Lertwiriyaprapa, P. H. Pathak, and K. Sertel, “A hybrid MoM-UTD analysis of the coupling between large multiple arrays on a large platform,” in Proc. IEEE Antennas Propag. Soc. Int. Symp., 2006, vol. 4A, pp. 175–178. [15] X. Dahai and S. Donglin, “A hybrid method for analyzing isolation between antennas on an electrically large metallic surface,” in Proc. 7th Int. Symp. Antennas, Propag. EM Theory, 2006, pp. 1–4. [16] A. L. Drozd, I. P. Kasperovich, C. E. Carroll, and S. C. Hall, “Hybrid multi-resolution analysis of antenna-coupled electromagnetic interference for large resonant structures,” in Proc. IEEE Electromagnetic Compat. Soc. Int. Symp., 2003, vol. 1, pp. 300–303. [17] Z. Yu, C. Ming, D. Wei, and L. Changhong, “EMC analysis of antennas involving dielectric bodies with MoM-FDTD algorithm and network theory,” in Proc. EEE Microwave, Antennas, Propag. and EMC Tech. Wireless Comm. Int. Symp., 2005, vol. 1, pp. 768–771. [18] D. J. Riley, N. W. Riley, W. T. Clark, III, H. Del Aguila, and R. Kipp, “Electromagnetic coupling and interference predictions using the frequency-domain physical optics and the time-domain finite-element method [aircraft antenna applications],” in Proc. IEEE Antennas Propag. Soc. Int. Symp., 2004, vol. 1, pp. 886–889. [19] A. J. Parfitt and T. S. Bird, “Application of a near-field transform algorithm to antenna coupling using the FDTD method,” in Proc. IEEE Antennas Propag. Soc. Int. Symp., 1998, vol. 1, pp. 500–503. [20] Y. Álvarez, 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, no. 12, pp. 3460–3468, Dec. 2007. [21] F. Las-Heras, “Using equivalent currents to analyze antennas in complex environments,” Microw. Opt. Technol. Lett., vol. 31, no. 1, pp. 62–65, Oct. 2001. [22] J. Rubio, M. A. González, and J. Zapata, “Generalized-scattering-matrix analysis of a class of finite arrays of coupled antennas by using 3-D FEM and spherical mode expansion,” IEEE Trans. Antennas Propag., vol. 53, no. 3, Mar. 2005. [23] M. A. González de Aza, J. A. Encinar, and J. Zapata, “Radiation pattern computation of cavity-backed and probe-fed stacked microstrip patch arrays,” IEEE Trans. Antennas Propag., vol. 48, no. 4, Apr. 2000. [24] C. A. Balanis, Advanced Engineering Electromagnetics. New York: Wiley, 1989. [25] R. C. Johnson and H. Jasik, Antenna Engineering Handbook, 2nd ed. Long Island, NY: McGraw-Hill, 1984. [26] P. Penfield, Jr., “Noise in negative-resistance amplifiers,” IRE Trans. On Circuit Theory, vol. CT-7, pp. 166–170, Jun. 1960. [27] M. S. Kluskens, FDTD3D Code personal communication, NRL. [28] B. M. Kolundzija, J. S. Ognjanovic, and T. K. Sarkar, WIPL-D: Electromagnetic Modeling of Composite Metallic and Dielectric Structures: Software and User’s Manual. Boston, MA: Artech House, 2000. [29] R. W. Kindt, “Rigorous Analysis of Composite Finite Structures,” Ph.D. dissertation, EECS Dept., Univ. Michigan, Ann Arbor, 2004. [30] CST Microwave Studio 2010, v. 2020.00. [31] M. Busse, H. L. Toothman, and D. Zolnick, Radar Target Signature User Manual—Version 10 Naval Research Lab., Washington, DC, Rep. NRL/PU/5310–98-370, 1998. [32] M. Dadic and R. Zentner, “A technique for elimination of reflected rays from antenna measurements performed in echoic environment,” AEU—Int. J. Electron. Commun., vol. 61, pp. 90–94, Feb. 2007. [33] M. Gilbert, CEMX Format personal communication, AFRL.

Mark Kragalott received the B.A. degree in physics and economics from Kenyon College, Gambier, OH, in 1983 and the M.S. and Ph.D. degrees in electrical engineering from The Ohio State University, Columbus, in 1988 and 1993, respectively.

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As a Graduate Research Associate with the ElectroScience Laboratory, The Ohio State University, he developed method of moments solutions for material scatterers and extremely low frequency shields. Since 1994, he has been with the Electromagnetics Section, Analysis Branch, Radar Division, Naval Research Laboratory, Washington, DC, where he has researched electromagnetics topics ranging from numerical solution methods to ultrawideband radiation, scattering, and coupling.

Michael S. Kluskens (M’91) received the B.S. and M.S. degrees in electrical engineering from Michigan Technological University, Houghton, in 1984 and 1985, respectively, and the Ph.D. degree from The Ohio State University, Columbus, in 1991. From 1986 to 1991, he was a Graduate Research Associate at the ElectroScience Laboratory, Department of Electrical Engineering, The Ohio State University, where he conducted research on the method of moments and chiral media. Since 1991, he has been with the Radar Division, Naval Research Laboratory, Washington DC, and is currently with the Computational Electromagnetics Section, Analysis Branch, Radar Division. His primary research is in computational electromagnetics with emphasis on the method of moments, finite-difference time domain, and scattering from large complex structures.

Dale A. Zolnick (M’03) received the B.A. degree in physics and mathematics from Duke University, Durham, NC, in 1975 and the M.A. degree in physics from The Johns Hopkins University, Baltimore, MD, in 1979. He has worked for the Radar Division, Naval Research Laboratory, Washington DC, since 1983 and is currently Head of the Electromagnetics Section, Radar Analysis Branch. His primary areas of specialization are research on the radar scattering from complex military targets, especially low observable ships, and the development of computational methods, algorithms, and computer software to predict the radar signature of complex military targets. The Electromagnetics Section has developed the radar target signature (RTS) model to calculate and analyze the radar signatures of complex military targets and to calculate the electromagnetic interference (EMI) between antennas in a shipboard environment. He has worked on the development of the RTS model throughout his tenure at NRL.

W. Mark Dorsey (M’09) received the B.S. and M.S. degrees in electrical engineering with a focus in electromagnetics from the University of Maryland, Baltimore, in 2002 and 2005, respectively, and the Ph.D. degree in electrical engineering from Virginia Polytechnic Institute and State University, Blacksburg, in 2009. As a doctorate student, he researched dual-band, dual-polarized antenna elements and arrays. He has worked on antenna design, measurement, and integration for the Radar Division of the U.S. Naval Research Laboratory, Washington DC, since 1999. His primary research interests include reconfigurable and multifunction antenna design, ultrawideband (UWB) antenna design, antenna isolation studies, antenna measurement, and array calibration.

John A. Valenzi studied airborne radar systems while serving in the United States Air Force. He toured as an Airborne Radar Technician with the 965th AWACS and earned the Air Medal. He has worked in the Radar Division, Naval Research Laboratory, Washington, DC. He operates the Radar Division’s Antenna Measurement Facility, where he assembles/integrates antenna systems, performs antenna and radar cross-section measurements, and manages range resources.

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Array Noise Matching—Generalization, Proof and Analogy to Power Matching Christian Findeklee

Abstract—In array antennas with residual coupling, the signal-to-noise ratio (SNR) can suffer from noise coupling from the inputs of the low noise amplifiers (LNAs) to the other channels. As shown recently, the lost SNR can be retrieved by changing the individual matching circuits. This paper explains the theory behind array noise matching and generalizes it for non-equal LNAs and non-reciprocal sources. It also shows an analogy between noise matching of an array and power matching into the complex conjugate of the optimum noise impedances of the individual LNAs. This turns out to be useful for practical noise matching of mutually coupled arrays. In some cases it becomes impossible to reach the theoretical optimum matching with passive matching networks. Therefore an additional boundary condition will be introduced and investigated. Index Terms—Amplifier noise, antenna array mutual coupling, impedance matching, multiport circuits, nonreciprocal circuits, signal-to-noise ratio (SNR).

I. INTRODUCTION

N

OISE coupling is known to decrease the signal-to-noise ratio (SNR) in array applications like radar, mobile communication, radio astronomy [1], [2] or magnetic resonance imaging (MRI) [3]. If the signal distribution to the individual channels is unknown or if there are multiple source locations, the receiving elements have to be decoupled by lossless networks in order to obtain the theoretical optimal SNR [4]. However, an efficient decoupling typically adds significant losses and thus also decreases the SNR. For a given source i. e. a given signal distribution in the receiving elements, the SNR can be improved by matching to the so-called active antenna impedance [5], [6]. If all used low-noise-amplifiers (LNAs) share the same noise figure, the SNR-loss can be reduced to this noise figure, even with noise coupling present. This paper further investigates the array noise matching and generalizes it to non-reciprocal devices and non-equal LNAs. It also explains a practical solution to adjust the matching networks for optimum SNR [7]. This approach realizes the active antenna impedance matching1 without the knowledge of the antenna network matrix. It simplifies the adjustment especially in MRI-applications where the antenna does not provide a shared ground.

Manuscript received September 15, 2009; revised June 14, 2010; accepted July 01, 2010. Date of publication December 03, 2010; date of current version February 02, 2011. The author is with Philips Research Europe, Hamburg, Germany (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2010.2096183 1If

this can be achieved with passive matching circuits.

Fig. 1. Scalar noise matching: the gray symbols denote noise voltage and current sources. The antenna for receiving U is modeled by its impedance Z and the noise voltage U . The LNA is represented by its input impedance Z , a noise voltage U , and noise current source I at its input.

At first, the scalar noise matching is sketched. From that, equations are derived for obtaining an LNA model from four characteristic parameters (real and imaginary part of the optimum impedance, noise figure and one more parameter which will be chosen invariant to lossless matching circuits). The SNR will be written as a function of these parameters. It will be shown that SNR-optimized matching equals a weighted power matching to the complex conjugate of the individual optimal impedances of the LNAs. The weighting is reciprocal to the corresponding noise figures. In cases where the theoretical optimum can only be reached by active matching circuits the best solution using only passive circuits will be investigated. Each LNA is considered to have its own constant noise figure whereas its optimal impedance can be chosen arbitrarily, thus it includes a tunable lossless matching circuit.

II. SCALAR NOISE MATCHING Fig. 1 shows the model of a scalar receive chain. The antenna is modeled by a Thévenin equivalent circuit with an impedance and a noise voltage . The received signal voltage is . The LNA is modeled using an input referred noise model [8] , a noise current source , and with a noise voltage source and a noiseless input impedance . The two noise sources are in general assumed to be correlated. It can be shown that using this model the noiseless input impedance has no effect on the available SNR, defined by the SNR obtained behind the LNA using a noiseless following RF-chain. Hence we may [9], [10]. In most apwithout loss of generality assume plications, the noise generated behind an LNA can be neglected and the input impedance hardly effects the SNR.

0018-926X/$26.00 © 2010 IEEE

FINDEKLEE: ARRAY NOISE MATCHING—GENERALIZATION, PROOF AND ANALOGY TO POWER MATCHING

The SNR of the antenna is defined by the signal power divided by the average noise power

For other values of the antenna impedance can be derived [9] to be

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, the noise figure

(1) The -symbol means the complex conjugate (later in this paper also transpose) and overlined expressions denote average values for describing covariances. Underlined values are peak value phasors. Assuming a purely thermal noise, and an effective antenna temperature , the autocovariance of the noise voltage is proportional to the real part of the antenna impedance

(2) , the Boltzman’s constant and with2 width [11]. The LNA decreases the SNR to

the band-

(8) Using (8), the SNR can be expressed by

(9) As will be shown in Section V, (9) can be generalized to the case of an array with coupled antennas elements attached to preamplifiers and a linear combination method that maximizes the combined SNR from all channels.

III. EXTRACTION OF LNA PARAMETERS (3) with the noise figure

From (5), (6), (7), the covariances , expressed by the LNA parameters and the current autocovariance

and

can be

(10) (4) (11)

By differentiating (4) with respect to the real and imaginary part , we find the optimum noise match impedance of for an LNA to be

(12) By combining (11) and (12) to

(5) (13) and

we can also formulate

(6) which means, with this antenna impedance the LNA has the lowest possible noise figure

(7) 2In case of using root mean square (RMS) instead of peak values, c is given by 2kT B and some equations with power levels have to be changed by a factor of two in this paper.

(14)

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we find that the determinant of the noise covariance matrix is invariant

(21) From (7) we find with

(22) also

(23) Fig. 2. Transformation of a preamplifier via a linear (noiseless) matching network: the complete behavior (signal and noise) with respect to the (left) input port and the SNR at the (right) output is the same in both cases.

due to the obvious invariance of . With (11) we see that also the power absorbed in a noiseless antenna of optimal impedance stays invariant in case of infinite preamplifier input impedance

IV. LINEAR TRANSFORMATION NETWORK Fig. 2 shows the combination of an LNA and a linear 2-port and the resulting equivalent LNA. The transformation can be described [12] easily with the chain-matrix

(15)

containing the so-called -parameters corresponding to the transformation network, which is assumed noiseless here

(24) This is equivalent to the invariant parameter3 proposed in [13]. With (17), (19) also the general case for any LNA input can be derived to impedance

(25) since a lossless

showing (24) is valid for purely reactive linear transformation

(16) (26) (17)

(18) The tilde denotes the values of the transformed LNA corresponding to the second circuit in Fig. 2. A general linear lossless matching network can be expressed by

(19) with four free parameters, given by real and imaginary part of and the imaginary parts and of the diagonal elements in the corresponding impedance matrix (see also Appendix). Using (16), (19) LNA-parameters can be defined invariant to linear lossless matching networks. With

(20)

results always in an also purely reactive . To distinguish between variant and invariant parameters we can substitute (14) by

(27) variant and with to linear lossless matching.

,

invariant

V. ARRAY SNR Fig. 3 shows the model of a receive array, which is a general extension to the scalar case of Fig. 1. The -channel antenna is now represented by an impedance matrix and correlated . The received signals at the annoise voltages . Each LNA tenna ports are given by the voltages is again modeled with a noise voltage source and a noise current source . For any particular 3Lange only claimed the invariance in case of reciprocal lossless networks but as shown here, the network only needs to be lossless.

FINDEKLEE: ARRAY NOISE MATCHING—GENERALIZATION, PROOF AND ANALOGY TO POWER MATCHING

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The tilde denotes the voltages changed due to finite LNA input . Both, signaland noise voltages impedances are scaled with the same matrix . The transformed sum of both is combined in a linear way to form a scalar receive signal

(31) As can be found in [15], the maximum SNR of

(32) can be achieved for Fig. 3. Array noise matching: noise and signal sources add to the output voltages of the -port antenna. The antenna is connected to LNAs, modeled with noise voltage and noise current sources at the input. The LNA input impedances ... do not change the SNR in case of optimal channel combination.

N

Z

by using

N

Z

LNA, the noise sources and may be correlated. However, the noise sources of different LNAs are assumed to be uncorrelated. The resulting voltages at the LNA input impedances are denoted with . The thermal noise generated inside —or received with homogeneous isotropic thermal brightness by—a linear -port can be conmodeled with correlated voltage sources nected in series to the ports. These noise voltage sources are [14] related to the impedance matrix

(33) Note, that the transformation with due to the finite input impedances does not change the SNR achieved in case of opti, mized combination. Thus, we may again assume , which leads to an identity matrix for . By formulating (27) for a set of LNAs, assuming no correlation between the LNA-input referred noise sources of different channels, we obtain

(34) (28)

with

and

denoting empty and identity matrix. Inserting

If the antenna is reciprocal, this expression can be simplified using the real part of the corresponding impedance matrix

(29) The voltages

(35) and (28) in (32) together with the LNA parameters in (34) leads to the following array SNR expression:

at the input of the LNAs read

(36) (30) The first term inside the inverted brackets, Here and in the following equations, the squared brackets denote diagonal matrices derived from the parameters related to the individual channels describes the SNR-reduction by the noise figures as it would occur in case of perfect noise matching. The second term .. .

..

.

..

.

..

.

..

.

.. .

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describes the reduction in SNR due to mismatching. By changing the noise matching for the single LNAs, the diagonal , can be chosen such that the SNR matrices, in particular is maximized. This optimum will be discussed in Section VII.

We now assume that , and hence also its inverse, are positive definite

(41) VI. WEIGHTED RECEIVED POWER Before moving on towards maximizing the SNR (36), we like to show a relationship between generalized noise matching and generalized power matching. For this reason we calculate the target signal power received by each LNA. We ignore all the noise sources in Fig. 3 and calculate a weighted power sum recarceived by the now finite input impedances . Each individual received power is rying the currents divided by a real and positive weighting factor

In case of equal noise figures, this is obviously fulfilled, because is positive definite. The the antenna is lossy, and hence, is at least positive semidefiproduct (and its inverse) are always positive definite nite. The matrix and thus the big bracket in (40), and also its inverse are positive definite. This means, g can if the second term in (40) vanreach a maximum ishes. This occurs, if

(42) If no component of vanishes, we can formulate the optimized matching condition by a component-wise division (37) , an expression

Using and similar to (36) can be derived

(43) which means

(38) Maximizing (38) can be considered as a generalized power matching to receive as much weighted signal power as possible. VII. ARRAY MATCHING

(44) with

(45)

Equation (38) is very similar to (36); in case of and , the first term in the brackets is exactly the same, the second differs only in the central diagonal matrix. Both equations can be written as a function

(39) with a special diagonal matrix . As will be shown now, they typically share the same set of optimum impedances . Applying the Woodbury matrix identity [16], (39) becomes

Note that the diagonal matrix is not part of the solution for in (43), (44), (45). This means that noise matching is equivalent to the weighted power matching shown in Section VI. This relationship can be exploited for practical array noise matching (Section IX). In the special case of all amplifiers having the same noise is a scaled identity matrix and (43) is equivalent to figure, (45) in [6]. Interestingly, the antenna needs not to be reciprocal, does not need to be symmetric. However, a reciprocal i.e., antenna attached to a set of equal LNAs is a typical case, for which (43) reduces to

(46)

(40) with

.

with . The matching condition (43) also covers possible variances in the noise figures of the LNAs. If the array is well decoupled, has only diagonal elements, which have to be matched. If is not perfectly diagonal, elements with high signal or low

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noise figure are matched more precisely than others to the corresponding LNAs. VIII. IMPACT OF ELEMENT COUPLING ON SNR The available SNR offered by the antenna is with (32), (28) given as

(47) which is a generalized form of (1), (2). It can be shown that a lossless decoupling leads to a diagonal , but it does not affect the available SNR given in (47). As it has been shown in the last section, the optimal matching (for coupled or decoupled arrays) leads to the overall-SNR

(48) From this we see, that if the LNAs all have the same noise figure, the SNR is simply decreased by this noise figure, regardless of any element coupling. If the LNAs are different, the choice of . distributing the LNAs to the channels affects IX. PRACTICAL ARRAY NOISE MATCHING In a realistic environment, especially without a shared ground plane, it may be difficult to perform an adjustment according to of the antenna has to (43), since the exact impedance matrix be determined in advance. For large arrays this might become a complex and error-prone measurement. , a weighted power In cases where it is difficult to obtain matching can be used as an alternative. An implementation of this method is sketched in Fig. 4. The LNAs are modified or exchanged such that the complex conjugate of their is offered individual optimum impedances to adjustable lossless matching circuits (switches open). The are evaluated for the power amplification factors individual modified LNAs (taking into account any losses of this modification). The input ports of the matching circuits are connected to the antenna. Behind the LNAs, each signal power is measured and divided by the corresponding power amplification factor (of the modified amplifier) and optimal noise figure (of the unmodified amplifier). The sum of the scaled powers is given at the output. Since the matching circuits are also fed into the input ports of the are lossless, ), and maximizing matching circuits (offering the weighted receive signal leads to the maximum of (38) . After adjustment, the matching without the knowledge of circuits are tuned such, that they transform the complex con, jugate of the optimal LNA impedances, to . This implies, that at the input ports of if we would connect the matching networks, we would measure into the output ports of the matching circuits and thus the unmodified LNAs would be noise matched, if connected with

Fig. 4. Practical array noise matching: for adjusting the matching circuits, the switches are opened, and thus the LNA inputs are modified such, that they offer the complex conjugate of their optimum impedance. After maximizing the weighted power sum, the switches are closed again, and the array is optimized for SNR.

the switches closed. This also means, that each combination of matching circuit and LNA, considered as a new LNA, would , and thus, with closed be noise matched to connectors, i. e. without the modifications, the array is noise matched according to (43). This way of array matching was verified for a two channel receive array in [7]. A special case is given, if the complex conjugate of the LNAs of the meaoptimum impedances equal the cable impedance surement hardware (e.g. 50 ). In that case the switches in Fig. 4 are not needed and the power can be measured with standard probes directly behind the matching networks. The measured power levels have to be divided only by the noise figures in that case. Sometimes it happens, that the theoretical optimum can not be reached. In case of strong element coupling or distinct asymmetry of the antenna, the weighted sum of the off-diagonal elements in (44) can lead to a negative real part in one or more . In this case, the optimum matching might not values of (Section VII) is no be found with (44) because the matrix longer positive definite. However, this cannot be reached with passive matching anyway. It now makes sense not to use all of the elements for receiving a signal. A general matching circuit therefore also has to enable terminating single channels with a tunable (noiseless) reactance.

X. GENERALIZED MATCHING TO PURELY REACTIVE IMPEDANCES As mentioned in the last section, the mathematical solution (43), (44), (45) can result in negative real components for the . This could only be realized with matching impedances active circuits that add additional noise and therefore does not offer a practical solution to the matching problem. A lossless matching transforms an input impedance as described in (26) and the noise-optimized impedance via

(49)

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as can be seen with (18). Thus (assuming pureley reactive corresponds to we find with (26)

) a . In that case

(50) independent of the attached LNA . As can be found in [17], is just the reciprocal of the impedance matrix of vanishes and the LNA does the matching network and thus not receive any signal. Therefore in can simply be skipped and the matching network may be replaced by an impedance without changing the SNR. The network and the signals may be described with respect to the residual ports then. Assuming the block matrix structure

(51) of the antenna impedance matrix and the signal voltages

(52) and without loss of generality choosing reactive impedances for the last port numbers this transforms the antenna to

(53) (54) which can be used to evaluate SNR and power (36), (38) by reducing the diagonal matrices to the first ports accordingly. Thus the global optimum passive matching, allowing also the just described reactive termination, is also the global optimum of the reduced (53), (54) versions of (36), (38), which can be found by (43), (44), (45). Therefore the global optimum of noise matching can also be found by the power matching method proposed in Section IX. XI. EXAMPLE CALCULATION As an example we now want to investigate the matching of a two-channel array for different signal distributions. The LNAs , . The are equal and described by antenna is given by

Fig. 5 shows the resulting noise figure of the array for given signal voltage ratios . The resulting noise figure is the factor of SNR decrease due to overall LNA-noise

(55)

Fig. 5. Resulting noise figure for a lossless matched two-element array for different signal voltage ratios U =U : in the white area, a perfect matching is possible. In the gray areas, this cannot be realized with passive matching netand works. In that case one LNA is replaced by an optimized reactance X the other is matched perfectly. The effective noise figure increases above the single LNA noise figure there.

In the white area of Fig. 5, a perfect matching is possible with passive components, because (43), (44), (45) leads to positive real parts in the optimum impedances. In the gray areas, this is not possible any more and one LNA is replaced by a reactance . This was varied in a nonlinear optimization to achieve maximum SNR for the array in case of perfectly matching the other channel. As expected from theory the contour lines of are continued by those of (shown only for and . In case of equal signal levels in both channels (same absolute voltage) perfect matching is always possible in this example. the gray area (circle in Fig. 5) For a ratio of is touched at4 . Due to the symmetry in this example, the lower gray area is touched at the reciprocal voltage ratio. XII. CONCLUSION The matching of array antennas can often be done in such way, that noise coupling does not reduce the SNR, if the signal distribution to the individual channels is known and the LNAs share the same noise figure. With (41) fulfilled, array matching can also be generalized for nonreciprocal networks and unequal LNAs. In some cases the theoretical optimum can not be reached with passive matching networks but SNR-reduction due to noise 4For reciprocal networks (Z = Z ) attached to equal LNAs it can be shown, that for each voltage vector U proportional to a column of Z the LNA corresponding to the column number has the only nonzero real part in the optimal impedance (43).

FINDEKLEE: ARRAY NOISE MATCHING—GENERALIZATION, PROOF AND ANALOGY TO POWER MATCHING

coupling can be decreased by skipping single LNAs and terminating the corresponding channels with optimized reactances then. A weighted power matching to the complex conjugate of the optimal impedances of the LNAs is equivalent to noise matching. This enables array noise matching in applications where the antenna impedance matrix is unknown or difficult to access. It is also cheaper since it only needs a power measurement per channel instead of a complete vector network analysis of the antenna. APPENDIX GENERAL LOSSLESS LINEAR TWOPORT An impedance matrix

is lossless, if (56)

hence, a general lossless two-port can be described by four real degrees of freedom

(57) From this we can derive with lossless ABCD-matrix

[17] also the general

(58) The four degrees of freedom are now given by the real and the complex .

,

ACKNOWLEDGMENT The author would like to thank J. Koskela, J. Lampe, O. Lips, K. M. Lüdeke, P. Mazurkewitz, C. Possanzini, and A. Reykowski for inspiring discussions. REFERENCES [1] R. Maaskant, E. Woestenburg, and M. Arts, “A generalized method of modeling the sensitivity of array antennas at system level,” in Proc. 34th Eur. Microwave Conf., Amsterdam, The Netherlands, Oct. 2004, pp. 1541–1544.

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[2] C. Craeye, B. Parvais, and X. Dardenne, “MoM simulation of signal-tonoise patterns in infinite and finite receiving antenna arrays,” IEEE Trans. Antennas Propag., vol. 52, no. 12, pp. 3245–3256, Dec. 2004. [3] A. Reykowski and J. Wang, “Rigid signal-to-noise analysis of coupled MRI coils connected to noisy preamplifiers and the effect of coil decoupling on combined SNR,” in Proc. 8th Annu. Meeting of ISMRM, Denver, CO, Apr. 2000, p. 1402. [4] K. F. Warnick and M. A. Jensen, “Optimal noise matching for mutually coupled arrays,” IEEE Trans. Antennas Propag., vol. 55, no. 6.2, pp. 1726–1731, Jun. 2007. [5] R. Maaskant and E. Woestenburg, “Applying the active antenna impedance to achieve noise match in receiving array antennas,” in Proc. IEEE Antennas and Propagation Society Int. Sym., Honolulu, HI, Jul. 2007, pp. 5889–5892. [6] K. F. Warnick, B. Woestenburg, L. Belostotski, and P. Russer, “Minimizing the noise penalty due to mutual coupling for a receiving array,” IEEE Trans. Antennas Propag., vol. 57, no. 6, pp. 1634–1644, Jun. 2009. [7] C. Findeklee, “Improving SNR by generalizing noise matching for array coils,” in Proc. 17th Annu. Meeting of ISMRM, Honululu, HI, Apr. 2009, p. 507. [8] H. Rothe and W. Dahlke, “Theory of noisy fourpoles,” Proc. IRE, vol. 44, no. 6, pp. 811–818, Jun. 1956. [9] J. Engberg and T. Larsen, Noise Theory of Linear and Nonlinear Circuits. New York: Wiley, 1995, 0-471-94825-X. [10] B. Schiek, I. Rolfes, and H.-J. Siweris, Noise in High-Frequency Circuits and Oscillators. New York: Wiley, 2006, 0-471-70607-8. [11] H. Nyquist, “Thermal agitation of electric charge in conductors,” Phys. Rev., vol. 32, pp. 110–113, Jul. 1928. [12] H. Hillbrand and P. H. Russer, “An efficient method for computer aided noise analysis of linear amplifier networks,” IEEE Trans. Circuits Syst., vol. CAS-23, no. 4, Apr. 1976, (Correction in November 1976). [13] J. Lange, “Noise characterization of linear two ports in terms of invariant parameters,” IEEE J. Solid-State Circuits, vol. 2, no. 2, pp. 37–40, Jun. 1967. [14] R. Q. Twiss, “Nyquist’s and Thevenin’s theorems generalized for nonreciprocal linear networks,” J. Appl. Phys., vol. 26, no. 5, pp. 599–602, May 1955. [15] S. P. Appelbaum, “Adaptive arrays,” IEEE Trans. Antennas Propag., vol. 24, no. 5, pp. 585–598, Sep. 1976. [16] M. A. Woodbury, “Inverting Modified Matrices,” Statistical Research Group, Princeton Univ., Princeton, NJ, Memo. Rep. 42, Jun. 1950. [17] D. A. Frickey, “Conversions between S, Z, Y, h, ABCD, and T parameters which are valid for complex source and load impedances,” IEEE Trans. Microw. Theory Tech., vol. 42, no. 2, pp. 205–211, Feb. 1994.

Christian Findeklee received the Dipl.-Ing. degree (summa cum laude) in electrical engineering from the Hamburg University of Technology (TUHH), Hamburg, Germany, in 2000. Currently, he is with Philips Research Europe-Hamburg, working mainly on MRI technology.

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Compact Two-Layer Rotman Lens-Fed Microstrip Antenna Array at 24 GHz Woosung Lee, Student Member, IEEE, Jaeheung Kim, Senior Member, IEEE, and Young Joong Yoon, Senior Member, IEEE

Abstract—This paper presents a new design to realize a compact Rotman lens-fed antenna array. The lens-fed antenna has the form of two layers, which is a new approach to reducing the size of the Rotman lens. The approach is demonstrated at 24 GHz aiming for an automotive sensing radar. The lens consists of a top metal layer, a dielectric, a common ground, a dielectric, and a bottom metal layer, in sequential order. The layout of the lens body is placed on the bottom layer and the antennas are placed on the top layer. Both of them are electrically connected through slot transitions. This two-layer structure reduces not only the total size of the lens, but also the loss of the delay lines because the lines can be designed to be as short and straight as possible. The two-layer Rotman lens-fed antenna array is measured in terms of scattering parameters and beam patterns. From the scattering parameters, the power efficiencies of the beam port 1, 2, and 3 at 24 GHz are obtained as 32.3%, 48.5%, and 50.8%, respectively. The measured beam patterns show that the beam directions are 28 1 14 9 , 0 , 15.5 , and 28.6 and the beamwidths are 13.4 , 13.2 , 12.8 , 13.5 , and 13.0 . The measurements confirm that the compact two-layer Rotman lens has been successfully demonstrated. Index Terms—Multilayer, Rotman lens, system-in-package, 24 GHz radar.

I. INTRODUCTION

S

ENSING radars are being proposed in the range of millimeter-wave due to the advance of microelectronics at higher frequencies. They have beneficial aspects such as compact size and wide bandwidth, which are favorable for both communications and radars. A beamformer is required to implement a sensing radar, and a compact design is desirable for deployment on the surface of a vehicle. One of the best candidates for beamformers is a Rotman lens [1]. The Rotman lens is a constrained lens in which a wave is guided along constrained paths upon design equations. It can generate multiple beams with phase relationships that are determined from the path length of the wave passing through the lens. Recently, application of the lens has expanded to commercial radars, such Manuscript received July 29, 2009; revised June 24, 2010; accepted September 16, 2010. Date of publication December 03, 2010; date of current version February 02, 2011. This work was supported in part by the Korea Science and Engineering Foundation (KOSEF) grant funded by the Korea government (R01-2007-000-11294-0) and in part by National Research Foundation of Korea Grant funded by the Korean Government (KRF-2008-D00792). The authors are with the Department of Electrical and Electronic Engineering, Yonsei University, Seoul 120-749, Korea (e-mail: [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2096380

as automotive short range radar at 24 GHz, radar at 77 GHz, and millimeter wave imaging radar at 94 GHz [2]. At the system level, electronic packaging technology is driving towards packing more components in limited space to facilitate the design of compact systems. The packaging technology is focused on system-in-package (SiP) in which a system is typically implemented in the form of a multilayer structure such as low temperature co-fired ceramics (LTCC) [3]. Of particular interest in this research is how to package the lens in the appropriate form of SiP. So far, most Rotman lenses and antennas have been designed and fabricated on a single layer structure. In this structure, the constrained lines between the Rotman lens and the antennas are considerably long, resulting in more loss and more space requirement [4]. A previous attempt was made [5] to design a Rotman lens and a series patch antenna array in different layers to avoid beam squint in the E-plane. However, there was no change to the Rotman lens design, hence size reduction and performance improvement were not significant. In this paper, a two-layer Rotman lens-fed microstrip antenna array has been proposed to design a compact beam forming lens-fed antenna array at 24 GHz. The multilayer implementation of the lens-fed antenna array would be favorable in the era of SiP, which is an easy way to realize a beamforming module. The design of the constrained lines and the connections between the Rotman lens and radiating elements is a new contribution, and will be demonstrated. It will also be useful for applications in which a surface mount is required because the lens system has a low profile and occupies a relatively small area. For example, at least in an automobile radar system, four radars have to be mounted on each side of a vehicle to cover 360 degrees. Therefore, the design might be very favorable for implementing an automobile radar system. II. DESIGN OF A TWO-LAYER ROTMAN LENS A. Two-Layer Rotman Lens Since the Rotman lens was invented in 1963, the lens equation has been well explained and modified in many publications [1], [6], [7]. The typical geometry and design parameters are shown in Fig. 1. It consists of three parts: a lens body, delay lines, and an antenna array. The lens body is shaped by the focal arc and the array curve. The lengths of the delay lines from the array curve to the antenna array are constrained to have three on the focal arc. Based perfect focal points at , G, and on initially chosen design parameters, that are the focal angle, the scan angle, the ratio of on-axis to off-axis focal length, and the electric property of the substrate, a designer can calculate

0018-926X/$26.00 © 2010 IEEE

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Fig. 1. Geometry and design parameters of a Rotman lens.

the length of the delay line and the array port coordinate from the design equations of the lens. The overall size and shape of the Rotman lens can be finalized by an interactive process considering the focal length (or the scaling factor) and the pathlength error [6]. So far, the Rotman lens is typically designed on a single layer with antennas. A simple approach to reduce the size is to use a high dielectric substrate, but the degree of reduction is limited [8]. Thus, in this paper, a two-layer structure has been proposed to implement a compact lens-fed antenna array system. The lens-fed antenna array was designed in the form of a two-layer structure as shown in Fig. 2. The lens body is placed on a bottom metal layer, and the antennas are placed on a top metal layer. Therefore, the lens body and the antenna array are separated by a common ground plane, and the slot-coupled transitions are used to feed the lens and antenna array through the common ground. The concept of a two-layer structure is an elegant approach for size reduction. It has been widely used in many components [9], [10]. This structure has advantages in both size and performance. As shown in Fig. 2, the twofold design can considerably reduce the total size, which is desirable for SiP. Moreover, the proposed two-layer Rotman lens separates the antenna array from the lens body and the constrained lines by a ground plane. The separation is helpful to prevent spurious radiation which may affect the radiation pattern. In addition to the two-layer structure, this work proposed a new approach to shorten the delay lines with less bending. Fig. 3 shows the alignment of the delay lines of the two-layer Rotman lens. denotes the length of the th curved line from the corresponding array port to the start of the straight line, and denotes the length of the th straight line to the center of the slot. Because the total length from the array curve to the array plane has to satisfy the condition of delay line, the value of can be obtained from the relationship

(1) where is the length of the th delay line. This method can minimize the length of the delay lines and eliminate the meandering part in the delay lines. This is a very effective way to

Fig. 2. Geometry of the proposed two-layer Rotman lens-fed antenna array that consists of a top metal layer, a dielectric substrate, a common ground plane, a dielectric substrate, and a bottom metal layer.

reduce the size of the Rotman lens, since the delay lines generally occupy considerable area in the lens system [11], [12]. The second advantage is the performance improvement due to short, less bent lines. In the case of a bent line, there are spurious radiation and unexpected phase shift along a rapidly bent section. Although a microstrip line is a guided wave structure, spurious radiation and surface wave can be generated around discontinuities and bent sections. This would not be avoidable in a typical design of the Rotman lens because most Rotman lenses have bent lines to satisfy the constraints on the delay lines. This problem would be worse at millimeter-wave frequency because the spurious radiation becomes more serious at higher frequency. In the two-layer lens, the spurious radiation and the surface wave can be suppressed because the lines are designed as short and straight as possible. The last advantage is that the lens’ size can be reduced without violating the true-time-delay condition. This characteristic is important in a wideband pulse-modulated system since a wideband signal passing through the lens antenna will suffer frequency dispersion, limiting wideband operation even though the lens is well designed. The prototype of the lens consists of seven array ports, five beam ports, and six dummy ports. It was fabricated on an RO3003 substrate whose electric permittivity is 3.0, thickness is 0.508 mm, and loss tangent is 0.0013. The design parameters of the proposed lens are summarized in Table I. Both the focal angle and the corresponding scanning angle are 30 and the

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Fig. 3. Geometrical path from the array curve to the array plane of the proposed two-layer Rotman lens. The delay from the slot-coupled transition is not considered because all paths include the same transitions.

TABLE I DESIGN PARAMETERS OF THE PROPOSED ROTMAN LENS

Fig. 4. The slot transition. (a) The geometry and (b) the picture.

spacing between antennas is 0.6 at 24 GHz. The diameter of the lens medium is approximately 27 mm and the overall size of the lens including the lens body, ports, and transitions is 75 mm 80 mm. The lens is fed by 50 microstrip lines whose corresponding width is 1.28 mm. All ports except the array ports are connected to high performance SMA connectors that are guaranteed to operate up to 26 GHz. In the calculation of beam patterns, the lens-fed antenna array generates five beams , 14.8 , 0 , 14.8 , and 29.2 , respectively. Each beam at has a half-power beamwidth of 13.8 , 12.5 , 12.0 , 12.5 , and 13.8 , respectively.

when the width of the slot is 0.9 mm. Next, as shown in Fig. 4(a), the slot stub was modified to a radial stub to a have wider bandwidth [14], [15]. The angle of the radial stub is 90 . To verify the performance of the transition, a back-to-back transition was designed and fabricated as shown in Fig. 4(b). The measured insertion loss and return loss are compared to the simulated results in Fig. 5(a). The simulation was done using a method of moment based tool. The measured insertion losses are less than 1.85 dB and the return losses are over 10 dB from 22 GHz to 26 GHz, whereas the simulation results are 0.59 dB and 22 dB, respectively. These results show that the designed transition has good performance in the frequency band of interest. In addition, the phase change through the transition was checked as shown in Fig. 5(b) to verify that the transition is a true-time-delay line. The simulated and measured results show that the phase varied linearly over a wide frequency range, which means that the transition does not break the condition of a true-time-delay line. Therefore, the slot transition is suitable for the two-layer Rotman lens, both in terms of insertion loss and phase condition.

B. Transition Between the Top Layer and the Bottom Layer

C. Series-Fed Microstrip Patch Array

The performance of the transition across layers is critical to the overall performance of a multilayer system. Typically, the transition can be implemented in forms of vias or slot couplings. The via can transit through more than two layers and the slot coupling can transit only between adjacent layers. Both can be implemented with insertion loss of less than 1 dB at millimeterwave frequency [13], [14]. In this paper, the slot transition has been chosen to demonstrate the two-layer lens since it can be easily fabricated in an etching process. To design the transition, a quarter wave length slot transition was initially considered. The insertion loss is small and stable

In packaging, a microstrip patch antenna is the most favorable candidate because of its low profile and planar structure. To utilize more area of the common ground, a series-fed microstrip patch array has been employed instead of a single microstrip patch antenna. The array was designed based on previous reports [16]–[18]. Fig. 6 shows the designed series-fed microstrip patch array on an RO3003 substrate. The width of the patch is 4.4 mm and the length is 3.4 mm, designed to resonate at 24 GHz. The series-fed patch array has four patches. The patches are connected in series by a narrow microstrip line because each patch has an

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Fig. 6. Fabricated series-fed microstrip patch array.

Fig. 7. Simulated radiation pattern of the series-fed patch array.

Fig. 5. Simulated and measured two-port S-parameters of the designed back-to-back slot transition. (a) Magnitude and (b) phase.

input impedance of 360 ; hence the width and the length of the line are sensitive to impedance matching and phase distribution. The distance between patches is 3.0 mm and the patches are positioned periodically with a spacing of 6.4 mm (0.512 at 24 GHz). The antenna was designed to have a tilting angle from the Z-axis as shown in Fig. 7 to meet the requirement of the 24 GHz automotive radars [19]. A method of moment based tool was used to compute the reflection coefficient and radiation pattern. As shown in Fig. 8, the measured reflection coefficient of the series-fed patch array is in good agreement with the simulation result. The center frequency of the fabricated array met the designed frequency of 24 GHz, with bandwidth from 23 GHz to 25 GHz. Finally, the series-fed microstrip patch arrays were connected to each array port, resulting in a 7 by 4 antenna array as shown in Fig. 9(c). III. FABRICATION AND MEASUREMENT The proposed lens-fed antenna array was realized in the form of a two-layer structure using two Rogers RO3003 substrates. Before assembling the two substrates, the geometry of the lens body, the slots, and the antenna array were deliberately patterned on the substrates in a high precision etching process. The common ground was implemented on the substrate of the lens. The two substrates were tightly attached in alignment with the slots. Fig. 9 shows the etched patterns on the bottom layer, the

Fig. 8. Simulated and measured reflection coefficients of the series-fed patch array.

ground plane, and the top layer of the fabricated lens-fed antenna array, respectively. All ports are connected with high performance SMA connectors. Before measuring the radiation pattern, the lens efficiency was checked from the measured scattering parameters of a lensonly-structure. Since the prototype shown in Fig. 9 does not have array ports, the lens-only-structure including slot-coupled transitions was redesigned to check on the efficiency. The twoport scattering parameters were measured by a network analyzer (HP8722D) from 22 GHz to 26 GHz. All ports except the port under test were terminated with commercial 50 loads (Mini-circuits ANNE-50X ). The measurement reveals that the return losses at the beam and array ports are better than 10 dB

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Fig. 9. The fabricated two-layer Rotman lens-fed antenna: (a) The bottom layer, (b) the ground plane, and (c) the top layer.

Fig. 10. Measured power efficiency of the proposed two-layer Rotman lens at beam port 1, 2, and 3.

32.3%, 48.5%, and 50.8% respectively. The power efficiency at the outer beam port (beam port 1) is lower than those at the other beam ports, which can be expected from the fact that the center beam port tends to transmit more power to all array ports than to the dummy ports. The radiation patterns of the fabricated lens antenna were measured in a near field measurement system as shown in Fig. 11. The lens antenna was mounted on thick styrofoam. An open-ended waveguide (WR-42) probe was used and the from the lens to the probe is about 50 mm (4 ) distance which is beyond the reactive near field region [20]. The probe mm to 106 mm in the Z-axis (vertically) scanned from mm to 121 mm in the Y-axis (horizontally), and from corresponding to by . The sample spacing is 5 mm, satisfying the criterion given by Yaghjian [20] (2)

Fig. 11. Photograph of the measurement setup.

over the band of interest, 23 GHz–25 GHz, indicating that all ports are well matched. The efficiency can be defined as the ratio of the sum of the output power at all array ports to the power fed at one of the beam ports. The obtained efficiencies are plotted in Fig. 10. The power efficiencies of beam ports 1, 2, and 3 at 24 GHz are

All the other ports except the port under test were terminated with 50 loads as shown in Fig. 11. The 2-D patterns are measured at 24 GHz and are shown in Fig. 12(a), (b), and (c) at B1, B2, and B3, respectively. Even though the 2-D patterns at B4 and B5 are not shown due to the symmetry of the lens, the patterns were measured and shown in Fig. 12(d). The measured beam patterns are compared to the calculated patterns in Fig. 12(d). The calculated patterns are the product of the pattern of the array and the element pattern of the patch antenna [21]. , the H-plane patterns Because the main beam is tilted at were plotted at this angle . In Fig. 12, means the normal direction to the antenna array which corresponds to the direction of the X-axis in Fig. 11. Measurements showed , 0 , 15.5 , and that the beam directions are 28.6 , respectively and the beamwidths are 13.4 , 13.2 , 12.8 , 13.5 , and 13.0 . These values are summarized in Table II and compared with the calculated ones. The measured results are in good agreement with the calculated results from ray-optics. The beamwidths are slightly wider than that of the calculated results due to error in the phase and amplitude distribution of the lens. The normalized gains were obtained from calculation and measurement. Although the gain of the outer beams (B1 and

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Fig. 12. The measured beam patterns of the proposed lens antenna at 24 GHz. 2-D radiation patterns at (a) Beam port 1, (b) Beam port 2, and (c) Beam port 3. . (d) The measured beam patterns are compared to the calculated beam As the main beams were inclined to the Z-axis, maximum power is obtained at  patterns at this angle  .

= 78

( = 78 )

TABLE II CALCULATED AND MEASURED BEAM DIRECTIONS, BEAMWIDTHS, AND NORMALIZED GAINS (CALC. CALCULATED, MEAS. MEASURED)

=

=

B5) are lower due to relatively low efficiency and aberration at the outer beam ports, the inner beams (B2, B3, and B4) show good agreement between calculated and measured results. IV. CONCLUSION We have successfully demonstrated a compact design of the Rotman lens-fed antenna array. From a new design of the constrained lines and the connections, the size of the lens-fed antenna array can be dramatically reduced with short and

straight delay lines. In addition, this approach is useful for suppressing spurious radiation and undesired phase deviations along a sharply bent section. The measured results show that the overall performance of the proposed lens is in good agreement with the calculated results. In conclusion, with the advancement in packaging technologies such as SiP, the proposed compact design should be a useful approach to the implementation of printed lens-fed multi-beam arrays. This approach can also be applied to a waveguide Rotman lens-fed horn antenna array, via a precise milling process. REFERENCES [1] W. Rotman and R. F. Turner, “Wide-angle microwave lens for line source applications,” IEEE Trans. Antennas Propag., vol. AP-11, pp. 623–632, Nov. 1963. [2] J. Schoebel, T. Buck, M. Reimann, M. Ulm, M. Schneider, A. Jourdain, G. J. Carchon, and H. A. C. Tilmans, “Design considerations and technology assessment of phased-array antenna systems with RF MEMS for automotive radar applications,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 6, pp. 1968–1975, Jun. 2005. [3] R. K. Ulirch and W. D. Brown, Advanced Electronic Packaging, 2nd ed. Hoboken, NJ: Wiley, 2006, ch. 14.

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[4] I. S. Song, J. Kim, D. Y. Jung, K. C. Eun, J. J. Lee, S. J. Cho, H. Y. Kim, J. Bang, I. Oh, and C. S. Park, “60 GHz Rotman lens and new compact low loss delay line using LTCC technology,” in Proc. IEEE Radio and Wireless Symp., 2009, pp. 663–666. [5] C. Metz, J. Grubert, J. Heyen, A. F. Jacob, S. Janot, E. Lissel, G. Oberschmidt, and L. C. Stange, “Fully integrated automotive radar sensor with versatile resolution,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 12, pp. 2560–2566, Dec. 2001. [6] J. Kim and F. S. Barnes, “Scaling and focusing of the Rotman lens,” in Proc. IEEE AP-S Int. Symp. Dig., Jul. 2001, pp. 773–776. [7] T. Katagi, “An improved design method of Rotman lens antennas,” IEEE Trans. Antennas Propag., vol. AP-32, no. 5, pp. 524–527, May 1984. [8] D. H. Archer, “Lens-fed multiple beam arrays,” Microw. J., pp. 171–195, Sep. 1984. [9] T. Tanaka, K. Tsunoda, and M. Aikawa, “Slot-coupled directional couplers between double-sided substrate microstrip lines and their applications,” IEEE Trans. Microw. Theory Tech., vol. 36, no. 12, pp. 1752–1757, Dec. 1988. [10] W. Schwab and W. Menzel, “On the design of planar microwave components using multilayer structures,” Trans. Microw. Theory Tech., vol. 40, no. 1, pp. 67–72, Jan. 1992. [11] S. Weiss and R. Dahlstrom, “Rotman lens development at the Army Research Lab.,” in Proc. IEEE Aerosp. Conf., Mar. 2006, pp. 1–7. [12] D. Nußler, H. H. Fuchs, and R. Brauns, “Rotman lens for the millimeter wave frequency range,” in Proc. Eur. Microw. Conf., Oct. 2007, pp. 696–699. [13] F. P. Casares-Miranda, C. Viereck, C. Camacho-Penalosa, and C. Caloz, “Vertical microstrip transition for multilayer microwave circuits with decoupled passive and active layers,” IEEE Microw. Wireless Compon. Lett., vol. 16, no. 7, pp. 401–403, July 2006. [14] O. Lafond, M. Himdi, J. P. Daniel, and N. Haese-Rolland, “Microstrip/ thick-slot/microstrip transitions in millimeter waves,” Microw. Optical Tech. Lett., vol. 34, no. 2, pp. 100–103, July 2002. [15] K. C. Gupta, R. Garg, I. Bahl, and P. Bhartia, Microstrip Lines and Slotlines, 2nd ed. Norwood, MA: Artech House, 1996, sec. Sec. 5.6. [16] T. Metzler, “Microstrip series arrays,” IEEE Trans. Antennas Propag., vol. AP-29, no. 1, pp. 174–178, Jan. 1981. [17] K.-L. Wu, M. Spenuk, J. Litva, and D.-G. Fang, “Theoritical and experimental study of feed network effects on the radiation pattern of series-fed microstrip antenna arrays,” IEE Proc. H Microw., Antennas and Propag., vol. 138, no. 3, pp. 238–242, 1991. [18] C. Karnfelt, P. Hallbjorner, H. Zirath, and A. Alping, “High gain active microstrip antenna for 60-GHz WLAN/WPAN applications,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 6, pp. 2593–2603, Jun. 2006. [19] T. H. Ho and S. J. Chung, “A compact 24 GHz radar sensor for vehicle sideway-looking applications,” in Proc. Eur. Radar Conf., Oct. 2005, pp. 351–354. [20] A. D. Yaghjian, “An overview of near-field antenna measurements,” IEEE Trans. Antennas Propag., vol. AP-34, no. 1, pp. 30–45, Jan. 1986.

[21] C. A. Balanis, Antenna Theory: Analysis and Design, 3rd ed. York: Wiley, 2005, ch. 14.

New

Woosung Lee (S’08) received the B.S. and M.S. degrees in electrical and electronic engineering from Yonsei University, Seoul, Korea, in 2005 and 2007, respectively, where he is currently working toward the Ph.D. degree. Since 2005, he has been working as a Research Assistant at Yonsei University involved in the projects of millimeter-wave lens antenna and packages. His research interests include beamforming arrays, small antennas, and millimeter-wave antennas.

Jaeheung Kim (S’98–M’02–SM’07) received the B.S. degree in electronic engineering from Yonsei University, Seoul, Korea, in 1989, and the M.S. and Ph.D. degrees in electrical and computer engineering from the University of Colorado at Boulder, in 1998 and 2002, respectively. From 1992 to 1995, he was with the DACOM Corporation, where he was involved with wireless communication systems. From 2002 to 2006, he was with the Department of Electrical and Electronic Engineering, Kangwon National University, Chuncheon, Korea. From 2006 to 2008, he was with the Information and Communications University, Daejeon, Korea. In 2008, he joined the Department of Electrical and Electronic Engineering, Yonsei University. His research interests include beamforming arrays, millimeterwave sensing and imaging, and high-efficiency active circuits.

Young Joong Yoon (M’93–SM’10) received the B.S. and M.S. degrees in electronic engineering from Yonsei University, Seoul, Korea, in 1981 and 1986, respectively, and the Ph.D. degree in electrical engineering from the Georgia Institute of Technology, Atlanta, in 1991. From 1992 to 1993, he was a Senior Researcher with the Electronics and Telecommunications Research Institute (ETRI), Daejeon, Korea. In 1993, he joined the faculty of Yonsei University, where he is currently a Professor with the Department of Electrical and Electronics Engineering. And, currently, he is a vice president at the Korean Institute of Electromagnetic Engineering & Science (KIEES). His research interests are antennas, RF devices, and radio propagations.

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Bayesian Compressive Sampling for Pattern Synthesis With Maximally Sparse Non-Uniform Linear Arrays Giacomo Oliveri, Member, IEEE, and Andrea Massa, Member, IEEE

Abstract—A numerically-efficient technique based on the ) for the design of maxiBayesian compressive sampling ( mally-sparse linear arrays is introduced. The method is based on a probabilistic formulation of the array synthesis and it exploits a fast relevance vector machine ( ) for the problem solution. The proposed approach allows the design of linear arrangements fitting desired power patterns with a reduced number of non-uniformly spaced active elements. The numerical validation assesses the effectiveness and computational efficiency of the proposed approach as a suitable complement to existing state-of-the-art techniques for the design of sparse arrays. Index Terms—Array synthesis, Bayesian compressive sampling (BCS), linear arrays, relevance vector machine, sparse arrays.

I. INTRODUCTION YNTHESIZING antenna arrays with a minimum number of elements is a problem of high importance in those applications (e.g., satellite communications, radars, biomedical imaging, acoustics, and remote sensing) where the weight, the consumption, and the hardware/software complexity of the radiating device have a strong impact on the whole cost of the overall system [1], [2]. Non-uniform arrangements have potential advantages with respect to uniform layouts [3] such as (a) significantly increased resolution (i.e., decreased mainlobe width) [4], (b) sidelobe level control/reduction [5], and (c) enhanced efficiency in dealing with physically-constrained geometries (e.g., conformal architectures) [6]. However, sparsening array elements has the main drawback of reducing the control of the beam shape [1]–[7] and several approaches for the design and optimization of sparse arrangements have been proposed in the last 50 years [1]–[31] to properly address such an issue. Dealing with beam shape control, two different problems are usually considered in the state-of-the-art literature [20]: by (I) the minimization of the peak sidelobe level determining a fixed set of element positions over an aperture and sometimes the corresponding weights; (II) the synthesis of a maximally-sparse array1 radiating a desired pattern. A

S

Manuscript received March 19, 2010; revised June 26, 2010; accepted August 14, 2010. Date of publication December 03, 2010; date of current version February 02, 2011. The authors are with the Department of Information Engineering and Computer Science, University of Trento, Povo 38050 Trento, Italy (e-mail: giacomo. [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2096400 1An array with the minimum number of active elements, (regular or irregular) of N positions.

P , over a lattice

wide set of methods concerned with Problem I [2] has been investigated including random approaches [11], [15], dynamic -filter design [16], stochastic optiprogramming [12], mization methods [17], [18], [20], [24], [27]–[29], analytical techniques [22], [31], and hybrid algorithms [25], [30], as well. On the contrary, Problem II has received less attention and few methods have been developed [2], [3], [13], [14], [19]–[21], [23], [26]. Because of the limitations of available computers, first attempts relied on techniques requiring as few computational resources as possible such as the steepest descent method [13] and the iterative least-square technique [14]. However, those approaches have strong limitations as, for example, the need to a-priori know the number of active elements of the array and the aperture size [13], [14]. In order to overcome these drawbacks, a technique exploiting the simplex search was developed in [3] to find the sparsest array matching a given reference pattern. Moreover, a mixed linear programming approach was introduced in [19] with the same aim. Further developments ranging from a recursive inversion algorithm based on the Legendre transform [21], [26] up to the use of a stochastic optimizer based on the simulated annealing technique [20] or a generalized Gaussian quadrature approach [23] have been successively analyzed. More recently, Problem II has been solved by means of an innovative technique based on the matrix pencil method [7]. Thanks to its efficiency, generally outperforms other synthesis techniques the in terms of convergence speed and array performances [7]. Despite its effectiveness, such an approach presents some limitations as follows. , of the active elements of 1) The locations , the array are proportional to the complex values of the nonzero roots of the generalized eigenvalue problem described in [7]. Consequently, unphysical complex solutions (i.e., ) can be generated [7] and an approximation [i.e., ] is required (p. 2957—[7]) whose impact on the array performances cannot be a-priori estimated nor neglected; 2) No requirements on the element positions [7] can be stated. Thus, no geometrical regularity or user-desired geometric features on the synthesized array can be a-priori enforced; 3) The method may fail in synthesizing/matching shaped , beam patterns because of the imaginary parts of are not usually negligible ([7, p. 2958]). This paper is aimed at proposing an innovative, flexible, and computationally-efficient complement to the existing synthesis methods that solve Problem II. The method, based on [32] (a robust the Bayesian Compressive Sampling and theoretically solid technique to produce sparse models in

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regression and classification problems [33]–[35]),2 is devoted to find the maximally-sparse array with the highest a-posteriori probability to match a user-defined reference pattern. Towards solver exploiting a fast relevance this end, an efficient algorithm [32], [36] is adopted. vector machine The outline of the paper is as follows. Section II is aimed at mathematically formulating the synthesis problem and describing an algorithm for minimizing a suitable cost function that depends on the degree of sparseness of the array and the mismatch between the desired power pattern and the actual one. Section III provides a selected set of numerical results to validate the proposed approach as well as to compare its performances with state-of-the-art techniques. Finally, some conclusions are drawn (Section IV).

To recast the problem at hand as a problem, the following three steps are necessary. Let us first rewrite the -norm as3 [35] constraint (2) where is a zero mean Gaussian error vector [32], [34], [35] with an user-defined variance proportional to the mismatching with the reference pattern (i.e., ). Then, through a Gaussian likelihood let us model

(3) to recast the original problem as the following linear regression one with sparseness constraints

II. MATHEMATICAL FORMULATION

LRSC Problem—Given find and which maximize the a-posteriori probability subject to the constraint that is maximally-sparse.

Formulation

A.

Let us consider a symmetric linear arrangement of ( if an even number of elements is at hand, otherwise) isotropic elements, being the real . The synexcitation of the -th element pair thesis problem is that of finding the set of array weights such that (a) the radiated pattern is sufficiently close to a given ref, and (b) the number of active (i.e., erence one, , , being the Kronecker function) array elements is as small as possible [3]. Towards this formulation is considered and similarly to [3] the end, the following assumptions are taken into account: (a) the reference angular posipattern is approximated in an arbitrary set of , within the visible range ; tions , (b) the set of active positions are constrained to a large, but (i.e., ) candidate locations finite, user-chosen set of not necessarily belonging to a regular lattice. Mathematically, the problem can be formulated as follows: Synthesis Problem—Given a set of samples of the , and a fidelity factor reference pattern, find the set of array weights, , which is maximally sparse subject to where

is

the

-norm, ,

, whose -th entry is given , being the by the distance of the -th location from the array wavelength, center ( if ), and is the Neumann’s number [9] if , and otherwise. defined as The synthesized pattern samples can be then expressed as (1)

Finally, the sparseness of [34], [35] is enforced. As regards the Bayesian formulation, such a task is accomplished by introducing a sparseness prior4 over [32]. Hereinafter, the Gaussian hierarchical prior [33]–[35] is invoked (4) where and is the -th independent hyperparameter controlling the strength of the prior [33]. To fully specify (4), the hyperpriors over [i.e., over ] and [i.e., ] have to be defined. The Gamma distributions are here considered [33] (5) and (6) is the -th scale prior, , and is the gamma function [33]. Thanks to (4), (5), and (6), the original synthesis problem can be finally formulated as where

BCS Problem—Given , and which maximize B.

Solver—The

, find

, .

Procedure

-th element is given by

In order to determine the desired sparse solution (i.e., the , , and ), the unknown parameters method [32], [33], which theoretically guarantees to solve the

full treatment of BCS in terms of convergence theory, performances in benchmark and illustrative problems and relations with other classification and regression techniques can be found in [33]–[36].

3It is worth pointing out that (2) and the ` -norm constraint are mathematically equivalent [35]. 4In Bayesian inference, a prior represents the a-priori knowledge about an unknown quantity in probabilistic terms.

where 2A

and its .

OLIVERI AND MASSA: BCS FOR PATTERN SYNTHESIS WITH MAXIMALLY SPARSE NON-UNIFORM LINEAR ARRAYS

BCS Problem [34], is applied. Towards this end, let us consider that the posterior over all unknowns can be expressed as

turns

out

to

be

equal

469

to

the

posterior given by

mean

of

(14) (7) Moreover, because of (3) and (4), the posterior distribution over III. (8) turns out to be equal to the following multivariate Gaussian distribution [35]

(9) where the posterior mean and the covariance are given by and , respectively, being . As for the second term on the right-hand side of (7), the delta-function approximation is used [33] to model the hyperparameter posterior (10) are the most probable values, , also called hyperparameter posterior modes. In order to determine their values, let us consider that where

and

(11) and and let us assume uniform scale priors. Then, become constant values [33] and the maximization of (11) is , whose logaequivalent to maximize the term rithm is given by [33]

SYNTHESIS METHOD—ALGORITHMIC IMPLEMENTATION

-based pattern The algorithmic implementation of the synthesis consists of the following steps: , the grid 1) Input Phase—Set the reference pattern ), the set of of admissible locations ( ; ), the target pattern sampling points ( ; of the error term , and its initial estimate variance for the sequential solver of the algorithm (see the Appendix); , 2) Matrix Definition—Fill the entries of the matrices , , and ; 3) Hyperparameter Posterior Modes Estimation—Find by maximizing (12) as described in the Appendix; by (14); 4) Array Weights Estimation—Find 5) Output Phase—Return the estimated array weights, , the number of active array elements, ,5 and the corresponding hyperparameter . modes Starting from an user-required pattern (i.e., its sam), the control parameters of the synpled representation ; thesis process are the following variables: (a) , ; (c) , and (d) . Consequently, it is (b) , possible to synthesize arbitrary reference patterns specifying the pattern matching accuracy (c) and the sequential solver initialmethod allows one to enforce ization (d). Moreover, the pattern constraints within the whole or in a subset of the visible range (b) as well as to set suitable geometrical features of the array arrangement (a). IV. NUMERICAL ANALYSIS AND ASSESSMENT

(12) . It is worthwhile to point out where that it is not possible to perform the maximization of the “marginal likelihood” (12) in an exact fashion, but a type-II maximum likelihood procedure [35] can be profitably exploited for . Such determining an iterative re-estimation of a technique, whose Matlab implementation is available in [37], is summarized in the Appendix. Finally, by substituting (9) and (10) in (7), one obtains that

(13) The posterior over all unknowns results a multivariate Gaussian function (9) only depending on the unknown set once have been determined. Therefore, the value of

This section is devoted to numerically assess potentialities approach for the design and limitations of the proposed of sparse linear arrays. The numerical analysis is carried out by considering a set of representative/benchmark reference patin terns to evaluate the effectiveness and reliability of the approximating a user-desired pattern. In order to evaluate the “degree of optimality” of the array designs, the following metrics and pattern descriptors are used: the matching error defined as6 (15) the aperture length , the mean inter-element spacing , and the minimum spacing . 5In this paper kxk is the ` -norm of x (i.e., the number of non-zero elements of x). 6Only u

2 [0; 1] is considered in the definition of  for symmetry reasons.

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Sensitivity Analysis

As a first numerical experiment, the synthesis of a non-uniform array matching a Dolph-Chebyshev pattern [2] is considand ered. A broadside Dolph-Chebyshev pattern with is assumed as reference. Let us notice that such a pattern can be synthesized through a uniform array with -spaced elements. The synthesis has at points ( , been carried out by sampling , ) and assuming the following grid of admissible locations (16) results by reporting the matching Fig. 1(a) describes the error versus the number of active elements for dif, ferent values of the control parameters: , , and . The Pareto is indicated, as front of the solution set in the plane well. As it can be observed, different trade-off solutions are obtained with accuracy and element number in the range and , respectively. By comparing the patterns related to three representative points of the Pareto front ) with the reference one [Fig. 1(b)], it (i.e., elements provides a turns out that the solution with , while a reliable reconvery poor matching is yielded choosing the solution struction having [Fig. 1(b)] with a non-negligible saving -spaced uniform array of array elements with respect to the ). As a general by-product, it results (i.e., that a value of the accuracy index around the threshold identifies an optimal trade-off solution, whereas lower values usually require more radiating elements [ , —Fig. 1(b)] without significant/relevant improvements in the matching of the reference pattern. As regards the resulting layouts, it is worth pointing out that the optimal array has an aperture and an excitation displacement [Fig. 1(c)] close to those of the uniform array. This proves the effective non-uniform sampling of the ideal . Otherwise, different current distribution affording vs. ] apertures [e.g., and weights [Fig. 1(c)] are synthesized in correspondence with greater values of . As for the element arrangement, a positive feature of the arrays is the enlarged inter-element spacing with respect to the corresponding uniform array [Fig. 1(c)] despite the closely-spaced admissible locations [(16)]. In order to provide a deeper understanding about the senperformances on the control parameters, sitivity of the Figs. 2 and 3 summarize the results of a comprehensive numerical analysis. More specifically, the matching error has , or , or by been evaluated as a function of , or setting the other parameters to the values used to obtain the (i.e., , , optimal trade-off with , ). For completeness, the behavior of has been reported, as well. As expected [Fig. 2(a)], the pattern matching improves as the number of samples of increases. However, does not further decreases slightly above the Nyquist beyond a threshold value

BCS Sensitivity Analysis (Dolph-Chebyshev: L = 9:5, PSL = 020 dB)—Plot of the representative points of a set of BCS solutions in the

Fig. 1.

) plane (a). Power patterns (b) and corresponding layouts (c) of the ( , P reference and of a set of representative BCS arrays.

threshold ( ) even though the corresponding still grows. A sampling value number of array elements between and turns out to be a reliable choice as confirmed by the behaviour of the plots of for [Fig. 3(a)], as gives the poorest fitting well. Indeed, the lowest value of —Fig. 3(a)], while satisfactory reconstructions [ .A are obtained when

OLIVERI AND MASSA: BCS FOR PATTERN SYNTHESIS WITH MAXIMALLY SPARSE NON-UNIFORM LINEAR ARRAYS

Fig. 2.

BCS Sensitivity Analysis (Dolph-Chebyshev: L = 9:5, PSL = 020 dB)—Behaviours of  and P

further increment of only marginally enhances the accuracy [ —Fig. 3(a)]. , the integral error has Concerning the sensitivity to , while it sharply insmall variations for creases afterwards [Fig. 2(b)] as pointed out by the plots in correspondence with a set of of (i.e., ) representative values of of [Fig. 3(b)]. More sparse arrays are synthesized in corresponat the expense of higher values dence with larger values of [Fig. 2(b)]. Good tradeoffs between accuracy and element . Such an reduction then arise by setting performances are significantly outcome indicates that the less sensitive to than to . As a matter of fact, a reduction of of about one order in magnitude requires a variation of of about 10–20% [Fig. 2(a)], while the same effect holds true for a variation of of more than two orders in magnitude [Fig. 2(b)]. Similar deductions can be drawn from the behaviour of the integral error versus . Moreover, the matching error increases values are almost monotonically with , whereas low obtained within the range [Fig. 2(c)]. Such a range can be also assumed as reference guideline since smaller values only marginally improve the , —Fig. 3(c)], matching accuracy [ , while higher values do not allow reliable syntheses [ —Fig. 3(c)].

471

versus (a) K , (b)  , (c)  , and (d) N .

Finally, the plots in Fig. 2(d) are concerned with the sension . By analyzing the behaviour of , tivity of the it comes out that great care must be exercised on the choice of to obtain a sparse array matching with a good accuracy the reference one. A good receipt coming also from other heuristic . analyses suggests to choose B.

Assessment—Synthesis of Broadside Patterns

The second set of experiments is aimed at assessing in a more when dealing exhaustive fashion the performances of the with broadside patterns. More specifically, Dolph-Chebyand shev reference patterns with have been used and the Pareto fronts of the solutions are shown in Fig. 4(a). As expected, wider apertures require more elements to reach the (e.g., , accuracy threshold , and ). On the contrary, does not generally change when varying the peak sidelobe level (e.g., ). The method allows a saving of about 30–35% of the array elements with respect to the corresponding uniformly -spaced array still keeping a very accurate pattern matching (i.e., ) [Table I]. This implies an increasing of the average inter-element distance

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= 9 5 PSL = 020 dB)—Plots of E

Fig. 3. BCS Sensitivity Analysis (Dolph-Chebyshev: L : , computed at different values of (a) K , (b)  , (c)  , and (d) N .

and, usually, of the minimum spacing between adjacent eleexcept for the case with ments ( and ). However, it is worth observing that the array element saving does not yield a significant —Table I). directivity reduction ( Despite the lower number of elements, the directivity of the resulting sparse array is very close to that of the corresponding ability to fully-populated arrangement thanks to the match a reference pattern with a high accuracy. Therefore and unlike previous array thinning techniques, no specific constraints (e.g., on the maximum percentage of antenna elements that can be thinned from an array) have to be enforced to guarantee a good directivity. On the other hand, the array aperture only slightly reduces when and (e.g., ) since it controls the mainlobe pattern matching. As far as the “shape” of the Pareto front is concerned [Fig. 4(a)], the plot of the matching error shows a step-like beconditions. Morehaviour whatever the array aperture and below which the over, it exists a threshold value of cannot provide an accurate matching for a given . For shows that example, the case decreases of more than two orders in magnitude passing from to . This is visually pointed out in

j

(u) E 0

(u)

j

of representative

BCS solutions

for Fig. 4(c) where the plots of are compared to the reference pattern. Such a behaviour is further confirmed by the results in Fig. 4(b) where Taylor patterns [1] with transition index and different sizes (i.e., ) and s ) are taken into account. (i.e., Also in this case, a small variation of leads to a significant improvement of the reconstruction accuracy . provide also for Taylor The reliable solutions with syntheses an accurate matching of the reference pattern with negligible errors confined to very low sidelobes, far from the mainlobe [see the inset of Fig. 4(d)], which do not contain relevant portions of the radiated power. -spaced As for the element saving with respect to the arrangement, the values in Table I confirm that as well as the conclusion drawn for the Dolph-Chebyshev patterns on the distribution of the array ) and on the arising elements (i.e., —Table I). Condirectivity ( cerning the computational issues, the turns out to be very —Table I) whatever the broadside efficient ( reference pattern, despite the non-optimized implementation of the Matlab code.

OLIVERI AND MASSA: BCS FOR PATTERN SYNTHESIS WITH MAXIMALLY SPARSE NON-UNIFORM LINEAR ARRAYS

Fig. 4. BCS Assessment (Broadside Pattern Synthesis)—Pareto fronts in the (; P when matching (a), (c) Dolph-Chebyshev and (b), (d) Taylor reference patterns.

) plane (a), (b) and power patterns (c), (d) of representative

473

BCS solutions

TABLE I

BCS Assessment (Broadside Pattern Synthesis)—ARRAY PERFORMANCE INDEXES

In order to complete the analysis of the performance of the approach when dealing with broadside patterns, comparisons with state-of-the-art techniques have been carried approach [7]7 out, as well. Towards this purpose, the has been considered because of its efficiency and the enhanced matching accuracy compared to similar methods such as the

Prony technique [7]. The results from the analysis of different Dolph-Chebyshev references are summarized in Fig. 5 8 for both and where the plots of versus arrays are shown. Let us consider the test case characterized defined over by a reference pattern with a linear aperture of length [Fig. 5(a)]. In such

7A MATLAB implementation of the MPM has been used for the numerical tests (mpencil function—http://www.mathworks.se/matlabcentral/index. html) by setting the default parameters as suggested in [7].

8Please notice that only the MPM arrays with SV D -truncation parameter below 10 have been reported in order to guarantee an accurate pattern matching [7].

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Fig. 5. BCS Assessment (Broadside Pattern Synthesis)—Representative points in the (; P ) plane of BCS and MPM solutions synthesized when matching the reference Dolph-Chebyshev patterns characterized by: (a) L = 9:5 PSL = 30 [dB], (b) L = 14:5 PSL = 30 [dB], (c) L = 19:5 PSL = 30 [dB], (d) L = 19:5 PSL = 20 [dB], and (e) L = 19:5 PSL = 40 [dB].

0

0

0

0 0

a case, the provides a more accurate fitting than whatever the number of array elements (e.g., the vs. [7]) and the generally requires a larger to satisfy the ( condition vs. ). The performances come closer to those of the as increases —Fig. 5(b) and —Fig. 5(c)] and some[ outperforms the in terms of fitting times the

0 0

0

0

0

index for both small and large values of [Fig. 5(b) and (c)]. Moreover and with reference to Fig. 5(c)–(e), it results enhances when rethat the efficiency of the overcomes the duces. As a matter of fact, the when and [Fig. 5(d)], for the aperture with while [Fig. 5(e)] as also pictorially pointed out and synthesized with by the plots of -element arrangement [inset of the corresponding

OLIVERI AND MASSA: BCS FOR PATTERN SYNTHESIS WITH MAXIMALLY SPARSE NON-UNIFORM LINEAR ARRAYS

475

Fig. 6. BCS Assessment (Broadside Pattern Synthesis)—Representative points in the (; P ) plane of BCS and MPM solutions synthesized when matching the reference Taylor patterns characterized by: (a) L = 9:5 PSL = 30 [dB], (b) L = 14:5 PSL = 30 [dB], (c) L = 19:5 PSL = 30 [dB], (d) L = 19:5 PSL = 20 [dB], and (e) L = 19:5, PSL = 40 [dB].

0

0

0

0

0

Fig. 5(e)]. As it can be observed, the properly matches the reference pattern within the entire visible range, while accuracy worsen near the mainlobe and in the far the sidelobes. Similar conclusions hold true when dealing with Taylor ref(Fig. 6) still inerence patterns. The behavior of versus outperforms the concerning the dicates that the when minimum to reach the matching threshold

0

0

0

0

dealing with small arrays and high s[ vs. —Fig. 6(a)], while the betters the performance for larger with low peak sidelobe levels [ vs. —Fig. 6(e)]. This is further confirmed by the patterns of the optimal trade-off solutions displayed in the insets of the pictures of Fig. 6.

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=54

(; P ) plane of BCS and MPM solutions synthesized when = 030 [dB], and (c) PSL = 040 [dB]. Array excitations (d).

Fig. 7. BCS Assessment (Shaped Pattern Synthesis: L :  [38])—Representative points in the , (b) PSL matching the reference Shaped patterns [38] characterized by: (a) PSL

= 020 dB

TABLE II

BCS Assessment (Shaped Pattern Synthesis: L = 5:4 [38])—ARRAY PERFORMANCE INDEXES

C.

Assessment—Synthesis of Shaped Patterns

In order to evaluate the flexibility of the proposed approach, numerical tests concerned with shaped patterns have been also performed. The first experiment deals with the reconstruction of flat top patterns defined over an aperture of with s as in [38]. The plots of as a function of different show that neither the nor the is able to reduce the number of array elements of the uniform array (being its inter-element distance) synthesized in [38] still keeping a

good accuracy, although the [ —Fig. 7(a)] reduces the array aperture with respect —Table II). On the contrary, the to [38] ( defines wider arrangements , as shown in Fig. 7(d), without yielding a good matching with the reference patterns ( —Table II). The is also pointed out by the enhanced accuracy of the , , and in the insets of plots of Figs. 7(a)–7(c) related to the arrays with . For completeness, the distributions of the array excitations

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477

TABLE III

BCS Assessment (Shaped Pattern Synthesis: L = 8:5 [39])—ARRAY PERFORMANCE INDEXES

=85

Fig. 8. BCS Assessment (Flat-Top Pattern Synthesis: L :  [39])—Representative points in the ; P plane of BCS and MPM solutions (a), optimal trade-off beampatterns (b), and associated array excitations (c).

(

)

versus show that the faithfully reconstructs the elereference pattern synthesizing an array of —Fig. 8(a)] with a reduction of ments [ about 1/3 of the array elements with respect to the uniform . As a side effect of the approximation, layout trade-off slightly improves the of the the optimal vs. reference pattern ( —Table III), as well. On the contrary, synthesis in [39] and the pattern genboth the elements do not provide an accurate erated with —Fig. 8(a)], fitting [ ), and unless using more antenna elements (e.g., significantly worsen the as highlighted by the plots of the associated patterns [Fig. 8(b)]. For completeness, the behaviour of the array excitations and the corresponding figures of merit are reported in Fig. 8(c) and Table III, respectively. As for the computational costs, the still retains the numerical efficiency proved in synthesizing broadside patterns (Table III). Similar conclusions can be also drawn when considering wider reference apertures. For example, with reference to a Woodward reference pattern with [Fig. 9(a)], the yields an accurate approximation with less elements than the ( vs. ). Moreover, significantly worsens when using the accuracy of the the same number of active elements of the solution , [ vs. , —Table IV and provides Fig. 9(b)]. As for the array arrangement, the a more widely-spaced design characterized by the following parameters: and (Table IV). D.

along the array extension are given in Fig. 7(d). As it can be observed and also predicted in [7], the worsening of the is mainly due to the errors in performances of the estimating the element positions caused by the non-negligible values of the imaginary parts of the non-zero roots of the associated eigenvalue problem. The second experiment considers as reference the Woodward pattern with analyzed in [39]. The plots of

Assessment—Constrained Synthesis

This section is devoted to assess the reliability of the approach in solving constrained synthesis problems (i.e., matching a reference pattern under some explicit geometric and/or radiation constraints). Towards this aim, the synthesis of a Dolph-Chebyshev pattern with and under different synthesis constraints has been addressed. The first test case has been formulated by enforcing the pattern matching constraints in the angular

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TABLE IV

BCS Assessment (Shaped Pattern Synthesis: L = 19:5)—ARRAY PERFORMANCE INDEXES

= 19 5

Fig. 9. BCS Assessment (Flat Top Pattern Synthesis: L : )— Representative points in the ; P plane of BCS and MPM solutions (a), optimal trade-off beampatterns (b), and associated array excitations (c).

(

)

region , being and . As trade-off solution desired, the pattern of the optimal —Table V) fits in a faithful way the reference ( one within the constrained region as well as in the transition regions close to the unconstrained angular range [Fig. 10(b)]. It is also of interest to observe that the distribution of the array excitations of the synthesis and those of the uniform array quite significantly differ [Fig. 10(a)].

Fig. 10. BCS Assessment [Constrained Synthesis—Dolph-Chebyshev: L : , u 62 : ; : ]—Array excitations (a) and power patterns (b).

19 5

(0 45 0 55)

=

To further verify the efficiency of the to include pattern constraints in the synthesis process without affecting the reliability of the matching in the remaining portion of the pattern, the constraint has been moved in another region of the visible range by setting and . As expected, the trade-off pattern carefully matches the reference in the constrained region —Table V), while uncontrolled lobes appear ( [Fig. 11(b)]. The use of a directive element [e.g., a for

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479

TABLE V

BCS Assessment (Constrained Synthesis—Dolph-Chebyshev: L = 19:5)—ARRAY PERFORMANCE INDEXES

=

Fig. 11. BCS Assessment (Constrained Synthesis—Dolph-Chebyshev: L : , u 62 : ; : )—Array excitations (a) and power patterns when using isotropic or directive elements (b).

19 5

(0 8 1 0]

radiating element] might then enable the control of the sidelobes in the whole visible region [Fig. 11(b)] with a significant saving of active elements in comparison with the uniform array synthesizing the entire Dolph pattern ( vs. ). The last part of the numerical assessment is aimed at analyzing the capability of the approach to also take into account geometrical constraints. Towards this end and considering the same reference pattern of the previous experiments, two different aperture-blockage problems have been defined: (i) and (ii) . The plots of the synthesized trade-off arrangements assess the effectivetechnique in constraining the ness and reliability of the element positions to desired locations [Figs. 12(a) and 13(a)],

Fig. 12.

BCS Assessment [Constrained Synthesis—Dolph-Chebyshev: L =

19:5, d 62 (5:3; 6:5)]—Array excitations (a) and power patterns (b).

while designing sparse arrangements ( —Table V) with reduced apertures , as well. It is also worthwhile to point out that, notwithstanding the non-negligible reduction of the admissible spatial region for the array elpattern ements (more than 10% in both cases), the matches the reference with a great care [Fig. 12(b) and Fig. 13(b)] as confirmed by the values of the matching index and (ii) —Table V]. [(i) V. CONCLUSIONS In this paper, the has been proposed as an innovative, flexible, and computationally-efficient complement to the existing state-of-the-art methods for the synthesis of sparse arrays with desired radiation properties. The pattern matching problem

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• despite no specific constraints (e.g., on the maximum percentage of antenna elements that can be thinned) have been enforced and unlike previous array thinning techniques, the sparse arrays is very close to that of directivity of their fully-populated counterparts (Table I–IV); • usually outperforms when dealing with shaped beampatterns (Section III-C); • application-specific constraints on either the radiation pattern or the geometrical characteristics of the array can be easily and efficiently taken into account (Section III-D). Subjects of future researches will be the analysis of the mutual coupling effects in the presence of realistic array elements as well as an enhanced exploitation of directive elements. Further extensions, out-of-the-scope of the present paper, will concern with complex excitations and non-symmetric layouts. Moreover, further works will be done on the sensitivity on control parameters, performances and on the reduction of the computational complexity of the method. APPENDIX

Fig. 13. : , d

19 5

BCS Assessment [Constrained Synthesis—Dolph-Chebyshev: L = 62 (0:0; 1:0)]—Array excitations (a) and power patterns (b).

has been properly reformulated in a suitable Bayesian framework and successively solved with a fast solver. An extensive numerical validation has been carried out dealing with different reference patterns, array sizes, and constraints to assess the feasibility and reliability of the approach as well as its efficiency, flexibility, and accuracy. Selected comparisons with state-of-the-art techniques have highlighted the advantages and (in some special cases) limitations of the synthesis in terms of sensitivity on control parameters, performances, and computational complexity. The proposed technique has shown the following main features: • several tradeoffs solutions can be easily obtained by means of simple modifications of the control parameters ( , , , and ) (Section III-A); • favorably compares with state-of-the-art techniques such as the [7] in terms of accuracy, array sparseness, and computational burden when matching reference broadside patterns (Section III-B); • on average the number of active elements in a array turns out to be smaller than the corresponding uniform arrangement still providing a high accuracy in matching the reference pattern (i.e., );

Sequential Solver for the Maximization of : The marginal likelihood maximization algorithm proposed in [35] is hereinafter customized to deal with user-defined pattern and matching problems. Starting from the knowledge of , the following sequence is iteratively ( being the iteration index) applied: —Set 1) Initialization and the -th entry of the diagonal matrix as follows (17) if and otherwise, and being ranand the -th column domly picked integers within of , respectively; and 2) Update—Evaluate to compute the sparsity factors , and the quality , where factors ; 3) Candidate Basis Vector Evaluation—Select the -th can, , and compute didate basis vector9 . If , then update the value of by means of (17), otherwise set ; 4) Convergence Check—Compute the value of . If ( being the tolerance factor [37]), then terminate. Otherwise, upusually set to date the iteration index and go to step 2. ACKNOWLEDGMENT The authors wish to thank Dr. S. Ji, Dr. Y. Xue, and Prof. L. Carin for sharing the code online. 9Please

refer to [35] for a review of the strategies for candidate selection.

OLIVERI AND MASSA: BCS FOR PATTERN SYNTHESIS WITH MAXIMALLY SPARSE NON-UNIFORM LINEAR ARRAYS

REFERENCES [1] C. A. Balanis, Antenna Theory: Analysis and Design, 2nd ed. New York: Wiley, 1997. [2] R. J. Mailloux, Phased Array Antenna Handbook, 2nd ed. Norwood, MA: Artech House, 2005. [3] R. M. Leahy and B. D. Jeffs, “On the design of maximally sparse beamforming arrays,” IEEE Trans. Antennas Propag., vol. 39, no. 8, pp. 1178–1187, Aug. 1991. [4] D. King, R. Packard, and R. Thomas, “Unequally spaced, broadband antenna arrays,” IRE Trans. Antennas Propag., vol. AP-8, pp. 380–384, Jul. 1960. [5] A. Maffett, “Array factors with nonuniform spacing arrays,” IRE Trans. Antennas Propag., vol. AP-10, pp. 131–136, Mar. 1962. [6] N. Balakrishan, P. Murthy, and S. Ramakrishna, “Synthesis of antenna arrays with spatial and excitation constraints,” IEEE Trans. Antennas Propag., vol. AP-29, pp. 690–696, Sep. 1962. [7] Y. Liu, Z. Nie, and Q. H. Liu, “Reducing the number of antenna elements in a linear antenna array by the matrix pencil method,” IEEE Trans. Antennas Propag., vol. 56, no. 9, pp. 2955–2962, Sep. 2008. [8] R. F. Harrington, “Sidelobe reduction by nonuniform element spacing,” IEEE Trans. Antennas Propag., vol. AP-9, p. 187, Mar. 1961. [9] M. G. Andreasan, “Linear arrays with variable interelement spacings,” IEEE Trans. Antennas Propag., vol. AP-10, pp. 137–143, Mar. 1962. [10] A. Ishimaru, “Theory of unequally-spaced arrays,” IEEE Trans. Antennas Propag., vol. AP-11, pp. 691–702, Nov. 1962. [11] Y. T. Lo, “A mathematical theory of antenna arrays with randomly spaced elements,” IEEE Trans. Antennas Propag., vol. 12, no. 3, pp. 257–268, May 1964. [12] M. I. Skolnik, G. Nemhauser, and J. W. Sherman, “Dynamic programming applied to unequally-space arrays,” IRE Trans. Antennas Propagat., vol. AP-12, pp. 35–43, Jan. 1964. [13] J. Perini and M. Idselis, “Note on antenna pattern synthesis using numerical iterative methods,” IEEE Trans. Antennas Propag., vol. 19, no. 2, pp. 284–286, Mar. 1971. [14] R. W. Redlich, “Iterative least-squares of nonuniformly spaced linear arrays,” IEEE Trans. Antennas Propag., vol. AP-21, no. 1, pp. 106–108, Jan. 1973. [15] B. Steinberg, “The peak sidelobe of the phased array having randomly located elements,” IEEE Trans. Antennas Propag., vol. 20, no. 2, pp. 129–136, Mar. 1972. [16] P. Jarske, T. Sramaki, S. K. Mitra, and Y. Neuvo, “On the properties and design of nonuniformly spaced linear arrays,” IEEE Trans. Acoust., Speech, Signal Processing, vol. 36, pp. 372–380, Mar. 1988. [17] R. L. Haupt, “Thinned arrays using genetic algorithms,” IEEE Trans. Antennas Propag., vol. 42, no. 7, pp. 993–999, Jul. 1994. [18] V. Murino, A. Trucco, and C. S. Regazzoni, “Synthesis of unequally spaced arrays by simulated annealing,” IEEE Trans. Signal Processing, vol. 44, no. 1, pp. 119–123, Jan. 1996. [19] S. Holm, B. Elgetun, and G. Dahl, “Properties of the beampattern of weight- and layout-optimized sparse arrays,” IEEE Trans. Ultrason., Ferroelectr., Freq. Control, vol. 44, no. 5, pp. 983–991, Sep. 1997. [20] A. Trucco and V. Murino, “Stochastic optimization of linear sparse arrays,” IEEE J. Ocean. Engrg., vol. 24, no. 3, pp. 291–299, Jul. 1999. [21] B. P. Kumar and G. R. Branner, “Design of unequally spaced arrays for performance improvement,” IEEE Trans. Antennas Propag., vol. 47, pp. 511–523, Mar. 1999. [22] D. G. Leeper, “Isophoric arrays—Massively thinned phased arrays with well-controlled sidelobes,” IEEE Trans. Antennas Propag., vol. 47, no. 12, pp. 1825–1835, Dec. 1999. [23] F. B. T. Marchaud, G. D. de Villiers, and E. R. Pike, “Element positioning for linear arrays using generalized Gaussian quadrature,” IEEE Trans. Antennas Propag., vol. 51, no. 6, pp. 1357–1363, Jun. 2003. [24] D. G. Kurup, M. Himdi, and A. Rydberg, “Synthesis of uniform amplitude unequally spaced antenna array using the differential evolution algorithm,” IEEE Trans. Antennas Propag., vol. 51, pp. 2210–2217, Sep. 2003. [25] S. Caorsi, A. Lommi, A. Massa, and M. Pastorino, “Peak sidelobe reduction with a hybrid approach based on GAs and difference sets,” IEEE Trans. Antennas Propag., vol. 52, no. 4, pp. 1116–1121, Apr. 2004. [26] B. P. Kumar and G. R. Branner, “Generalized analytical technique for the synthesis of unequally spaced arrays with linear, planar, cylindrical or spherical geometry,” IEEE Trans. Antennas Propag., vol. 53, pp. 621–633, Feb. 2005. [27] T. G. Spence and D. H. Werner, “Thinning of aperiodic antenna arrays for low side-lobe levels and broadband operation using genetic algorithms,” in Proc. IEEE Antennas Propag. Society Int. Symp., Jul. 9–14, 2006, pp. 2059–2062.

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[28] T. G. Spence and D. H. Werner, “Design of broadband planar arrays based on the optimization of aperiodic tilings,” IEEE Trans. Antennas Propag., vol. 56, no. 1, pp. 76–86, Jan. 2008. [29] R. L. Haupt and D. H. Werner, Genetic Algorithms in Electromagnetics. Hoboken, NJ: Wiley, 2007. [30] P. J. Bevelacqua and C. A. Balanis, “Minimum sidelobe levels for linear arrays,” IEEE Trans. Antennas Propag., vol. 55, pp. 2210–2217, Dec. 2007. [31] G. Oliveri, M. Donelli, and A. Massa, “Linear array thinning exploiting almost difference sets,” IEEE Trans. Antennas Propag., vol. 57, no. 12, pp. 3800–3812, Dec. 2009. [32] S. Ji, Y. Xue, and L. Carin, “Bayesian compressive sensing,” IEEE Trans. Signal Process., vol. 56, no. 6, pp. 2346–2356, Jun. 2008. [33] M. E. Tipping, “Sparse bayesian learning and the relevance vector machine,” J. Machine Learning Res., vol. 1, pp. 211–244, 2001. [34] A. C. Faul and M. E. Tipping, “Analysis of sparse Bayesian learning,” in Advances in Neural Information Processing Systems (NIPS 14), T. G. Dietterich, S. Becker, and Z. Ghahramani, Eds., 2002, pp. 383–389 [Online]. Available: http://citeseer.ist.psu.edu/faul01analysis.html [35] M. E. Tipping and A. C. Faul, “Fast marginal likelihood maximization for sparse Bayesian models,” in Proc. 9th Int. Workshop Artificial Intelligence and Statistics, C. M. Bishop and B. J. Frey, Eds., 2003 [Online]. Available: http://citeseer.ist.psu.edu/611465.html [36] M. E. Tipping, “The relevance vector machine,” in Advances in Neural Information Processing Systems, S. A. Solla, T. K. Leen, and K.-R. Muller, Eds. Cambridge, MA: MIT Press, 2000, vol. 12, pp. 652–658. [37] S. Ji, Y. Xue, and L. Carin, Bayesian Compressive Sensing Code 2009 [Online]. Available: http://people.ee.duke.edu/~lihan/cs/ [38] F. Ares and E. Moreno, “The convolution applied on the synthesis shaped beam,” in Proc. 20th Eur. Microwave Conf., Oct. 1990, vol. 2, pp. 1491–1494. [39] S. Yang, Y. Liu, and Q. H. Liu, “Combined strategies based on matrix pencil method and tabu search algorithm to minimize elements of non-uniform antenna array,” Progr. Electromagn. Res. B, vol. 18, pp. 259–277, 2009.

Giacomo Oliveri (M’10) received the B.S. and M.S. degrees in telecommunications engineering and the Ph.D. degree in space sciences and engineering from the University of Genoa, Italy, in 2003, 2005, and 2009 respectively. Since 2008, he has been a member of the Electromagnetic Diagnostic Laboratory, University of Trento, Italy. His research work is mainly focused on cognitive radio systems, electromagnetic direct and inverse problems, and antenna array design and synthesis.

Andrea Massa (M’96) received the “Laurea” degree in electronic engineering and Ph.D. degree in electronics and computer science from the University of Genoa, Genoa, Italy, in 1992 and 1996, respectively. From 1997 to 1999, he was an Assistant Professor of electromagnetic fields in the Department of Biophysical and Electronic Engineering, University of Genoa, teaching the university course, Electromagnetic Fields 1. From 2001 to 2004, he was an Associate Professor at the University of Trento, Trento, Italy, where, since 2005, he has been a Full Professor teaching electromagnetic fields, inverse scattering techniques, antennas and wireless communications, and optimization techniques. He is also the Director of the ELEDIALab at the University of Trento and Deputy Dean of the Faculty of Engineering. Since 1992, his research has focused on electromagnetic direct and inverse scattering, microwave imaging, optimization techniques, wave propagation in presence of nonlinear media, wireless communications and applications of electromagnetic fields to telecommunications, medicine and biology. Prof. Massa is a member of the IEEE Society, the PIERS Technical Committee, and the Inter-University Research Center for Interactions Between Electromagnetic Fields and Biological Systems (ICEmB). He is the Italian representative in the general assembly of the European Microwave Association (EuMA).

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Synthesis of Multi-Beam Sub-Arrayed Antennas Through an Excitation Matching Strategy Luca Manica, Paolo Rocca, Member, IEEE, Giacomo Oliveri, Member, IEEE, and Andrea Massa, Member, IEEE

Abstract—An innovative synthesis procedure to design sub-arrayed antennas affording multiple patterns is presented in this paper. The approach is based on an excitation matching procedure aimed at generating one optimal pattern and multiple compromises close as much as possible to user-defined reference beams. -means clustering algorithm A suitable modification of the integrated into a customized version of the contiguous partition method is used to efficiently sample the solution space looking for the best compromise excitations. A set of representative numerical results is reported to give some indications on the reliability, potentialities, and limitations of the proposed approach. Index Terms—Linear arrays, multi-beam antennas, sub-arraying.

I. INTRODUCTION

T

HE synthesis of switchable multi-beam antennas has always received a great attention from the scientific community because of the wide range of applications. Multi-beam antennas constitute the radiating part of monopulse radar trackers [1] to determine the positions of moving targets from the information collected by two different patterns (i.e., a sum pattern and a difference one). Furthermore, cellular base stations and communication satellites are also equipped with antennas generating multiple radiation patterns [2], [3]. Typical configurations are based on reflector antennas equipped with a cluster of feeds generating multiple independent beams at different directions. The main limitations are the difficulty of generating low sidelobe patterns and the limited scanning capability. Other solutions based on lenses of Butler matrices have been considered, as well [4]. Nowadays, the use of arrays of (direct) radiating elements is preferred since they enable the direct control of the illumination on the aperture for beam forming, the electronic steering of the patterns, and a more compact antenna realization. Several analytical methods have been developed to determine element excitations able to generate optimal sum patterns [5]–[7], difference patterns [8], [9], and patterns with Manuscript received September 15, 2009; revised July 04, 2010; accepted August 19, 2010. Date of publication December 03, 2010; date of current version February 02, 2011. L. Manica, P. Rocca, and G. Oliveri are with the Department of Information Engineering and Computer Science, University of Trento, 38123 Trento, Italy (e-mail: [email protected]; [email protected]; [email protected]). A. Massa is with the Department of Information Engineering and Computer Science, ELEDIA Research Group – DISI, University of Trento, I-38123 Trento, Italy (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2096383

arbitrary shapes [10], [11]. Unfortunately, the synthesis of a switchable antenna affording multiple optimal patterns implies the use of different and independent feeding networks. The total beamforming network (BFN) is usually characterized by a complex layout with a large number of active elements and high implementation costs. It is often more convenient to define compromise solutions with suitable trade-offs between costs and radiation performances. In this framework, a-priori fixed excitation amplitudes and optimized phase distributions [12]–[14] as well as partially-shared apertures [15] have been considered. Another alternative is the use of sub-arrayed antennas [4]. The elements of the array are grouped into clusters which are properly weighted to generate “best” compromise patterns. The price to pay for the simplification of BFN is an unavoidable reduction of the pattern performances [16] to be limited thanks to a careful design of the sub-arrayed network and an optimization of the sub-array weights. Different synthesis approaches have been proposed to generate a single compromise beam pattern [17]–[19] and the design of sum and difference patterns has been dealt with [20]–[26], as well. In this latter, one pattern (typically the sum pattern) is generated by means of optimal excitations analytically-computed, while the difference beam is obtained throughout the sub-arrayed BFN. As regards sum-difference compromises, excitation matching strategies [20], [25], approaches based on evolutionary algorithms [21]–[23], [26], and hybrid techniques [24], [27] have been used. Of course, the sub-arraying strategy can be also extended to the synthesis of multi-beam antennas [29], but such a potential has not been yet deeply investigated. By supposing the generation of patterns and exploiting the guidelines of [25], once the excitations of the main pattern have been set through the primary feeding network, sub-arrayed transmission lines can be designed in a serial way [serial approach, – Fig. 1(a)] to generate the sub-optimal beam patterns. Even if the number of active elements is reduced with respect to the complete BFN independent transmission lines, the antenna manhaving ufacturing could still be impracticable or very complex due to the number of circuit crossing. The use of a common sub-array feed network can further simplify the complexity of the antenna design [parallel approach, – Fig. 1(b)]. In such a case, the feed networks can be more easily implemented using microstrip technology by considering either single-layer or multi-layer design [28] as well as coaxial cables. This paper deals with a synthesis method based on the parallel approach for the design of multi-beam antennas. More specifically, patterns are generated throughout a compromise BFN

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MANICA et al.: SYNTHESIS OF MULTI-BEAM SUB-ARRAYED ANTENNAS THROUGH AN EXCITATION MATCHING STRATEGY

of them [Fig. 1(b)]. The

483

sets of compromise real excitations , , are given by

(1) where is an integer index that identifies the subarray membership of the th array element to the th sub-array. The whole sub-array configuration is mathematically described [23]. Morethrough the integer vector over, is the weight coefficient of the th sub-array related to is the Kronecker delta function ( the th beam and if , otherwise) [25]. Following the guidelines of the optimal matching approach presented in [25] and here properly customized to the generation of multiple patterns, the problem is recast as the definition of the sub-array aggregation, , and of sets of real sub, , array weights, such that the compromise excitations, , , are close (in the least square sense) as much as possible to sets of optimal and independent real excitations , , affording reference patterns [5]–[11]. The cost function that quantifies such a mismatch is given by (2) Fig. 1. Sketch of the multi-beam sub-arrayed antenna: (a) serial architecture and (b) parallel architecture.

where composed by a common sub-array architecture, whereas the sub-array weights are independently computed for each beam. Likewise [25], the solution of the problem at hand is formulated as the definition of compromise patterns close as much as possible to reference beams by means of an excitation matching strategy. The paper is organized as follows. In Section II, the problem is mathematically formulated and the adopted metric as well as the solution searching procedure are presented. The results of a set of representative experiments are reported in Section III to describe the synthesis process and to assess the effectiveness of the proposed method. Finally, some conclusions are drawn (Section IV).

and

(3) By substituting (1) into (3) and after simple mathematical manipulations, it turns out that

(4)

As shown in [25], once the sub-array configuration the weights , defined as follows:

is set, , are

II. MATHEMATICAL FORMULATION Let us consider a uniform linear array of elements with inter-element distance . In order to generate different beams on the same antenna aperture, the sub-arraying technique [20] is considered. One pattern, called main pattern, is generated by means of a set of optimal real excitations . The other compromise patterns are obtained by aggregating the array sub-arrays and assigning weights to each elements into

(5)

, , are the reference weights where optimal patterns [25], namely those coefficients generating when using independent BFNs.

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In order to optimize (2), let us first define , , as

reference vectors,

and their values are sorted on a line [Fig. 2(c)] to determine the list

(6) Unlike [25], where the Contiguous Partition Method (CPM) has ), been proposed to synthesize one compromise pattern ( we are now aimed at extending the CPM to deal with sub-optimal patterns ( – CPM). Unfortunately, the guidelines of [30] suitably exploited in [25] cannot be applied here since a sorting , does not property for the reference vectors exist. However, it is still expected that is obtained aggregating in the same sub-array those elements whose refer, are closer in such that ence vectors the values of the sub-array weights (5) are “good” approximations of the corresponding reference weights. Towards this is defined as the weighted (with weights , aim, since ) arithmetic mean of those , belonging to the th sub-array, the problem at hand is reformuaggregating the lated as “the search of the best grouping reference vectors into disjoint sub-sets , ( ) such that their internal variances, computed as (4), are minima”. State-of-the-art literature refers this problem as the unsupervised clustering problem [31]. Several techniques have been proposed to deal with it and the guidelines of the -means (here referred as -means) Clustering Algorithm [32], [33] are considered hereinafter because of its convergence rate and the implementation simplicity. Starting from an initial grouping , the memberships of the vectors , to the different clusters are changed until their maximum internal variance (2) is minimized. More specifically, the proposed algorithm works as follows: • Step 0 – Initial Step Reference Excitations Selection – The excitations of the main pattern, , as well as the reference excitations of the , are chosen; compromise beams, , , Initialization – The reference vectors, [Fig. 2(a)] are computed and the iteration counter is initialized ( ). If the elements are not positive, they are translated of the quantity

(9) The initial sub-array configuration is obtained by cut points among the randomly choosing inter-element spaces of the list [Fig. 2(d)], then defining

, being . Moreover, the Euclidean distance between each couple of reference vectors is computed

(10) The sequence index is set to ; • Step 1 – Cost Function Evaluation – The cost function is evaluated by means of of the current aggregation (3),

to obtain the set of translated reference vectors , , where [Fig. 2(b)]. Successively, the norms of the vectors , are computed

, and compared with the

best cost function value obtained up-till now, . If

•

•

• •

(7)

,

the initial subsets

) by also optimal cost function is updated ( setting , elsewhere ; ( Step 2 – Convergence Check – If being the maximum number of iterations) or the solution is stationary for iterations (i.e., , ), then the optimization process is stopped; Step 3 – Sequence Updating – The sequence index is up) and if then the process jumps to dated ( Step 5; Step 4 – Iteration Updating – The iteration index is updated ( ) and the sequence index is reset ( ); Step 5 – Border Element Identification – The vector related to the list element is selected. It is a border vector, , if

,

(11) where

(8)

then the

is the reference vector given by

(12)

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485

K N = 12, K = 2, Q = 3). (a) Reference vectors V = v ; k = 1; . . . ; K , n = 1; . . . ; N , (b) translated V = v^ ; k = 1; . . . ; K n = 1; . . . ; N , (c) generation of the list L of the norm values of the references vectors, and (d) element aggreC.

Fig. 2. Parallel – CPM – Synthesis Process ( , reference vectors gation and definition of the sub-array configuration,

and belonging to the subset , . If (11) holds true then the algorithm goes to Step 6. Otherwise, the Step 3 is repeated; • Step 6 – Aggregation Updating – The border element is aggregated to the subset (and to the corresponding sub-array) to obtain a new trial configuration . If , then (i.e., , ) and the Step 1 is iterated. Otherwise, the algorithm goes to Step 3. III. NUMERICAL RESULTS In this section, the results of representative simulations are reported to show the behavior of the – CPM synthesis process as well as the performances of the proposed approach. In order to provide quantitative information, the mainlobe beamwidth, , the position of the first pattern null, , and the peak sidelobe level, SLL, have been evaluated for the compromise patterns and compared to those of the reference ones. Furthermore, the matching indexes [25]

(13)

have been used to quantify the degree of matching with referand are the normalized ences. In (13), th reference array pattern and that synthesized with the proposed approach, respectively. For comparative purposes, the so– lution synthesized with the serial implementation of the CPM is given, as well. A. Test 1 – Double-Sum and Difference Patterns Synthesis Let us consider a linear array of with and the generation of three beams ( ). The main pattern excitations have been set to those of a Dolph-Chebyshev pattern , while the reference co[5] with efficients for the first compromise pattern, ,

and

the

second

one,

, have been chosen to afford a Zolotarev difference pattern [9] with and a Taylor sum pattern [7] with and , . respectively. The number of sub-arrays has been set to By virtue of the symmetries among the excitation coefficients, only half array has been involved in the synthesis process ). ( – CPM, the reference At the first step of the parallel vectors (6) are computed. Since all the terms are posi. The values of the reference tive, it follows that

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Fig. 3. Parallel .

i = i = 10

K – CPM Analysis (Test 1: N = 20, K = 2, Q = 3) – Sub-array configuration synthesized at (a) i = 0, (b) i = 1, (c) i = 3 and (d)

K – CPM Analysis (Test 1: N = 20, K = 2, Q = 3) – Relative power patterns synthesized at iteration (a)(b) i = 0, (c)(d) i = 1, (e)(f) i = 3 i = i = 10. Difference compromise pattern, k = 1 (left column) and sum compromise pattern, k = 2 (right column).

Fig. 4. Parallel and (g)(h)

vectors and their norms (8) are reported in Table I. Starting from the initial randomly-chosen configuration equal to , the clustering is iteratively updated. The evolution of the sub-array aggregations is shown – Fig. 3(a); – Fig. 3(b); – Fig. 3(c) in Fig. 3 [ – Fig. 3(d)]. The corresponding patterns are reported and – first row; – second row). It in Fig. 4 ( is worth noting that the initial aggregation leads to a compromise difference beam far from the target [Fig. 4(a)], whereas the second beam is close to the corresponding reference one [Fig. 4(b)]. Such a situation is confirmed by the values of the and cost function in Fig. 5. At the convergence ( ), the trade-off solution shown in Fig. 4(g)–(h) is obtained. The synthesized patterns ) identified by the label “ ” are shown in Fig. 6(a) ( ) along with the solution from the serial and Fig. 6(c) ( implementation of the CPM (line “ ”). The corresponding HW layouts are also given in Fig. 6(b) and (d), as well. For completeness, the sub-array configurations and weights are listed in Table II, whereas the values of the pattern indexes are reported in Table III. As it can be observed, both implementations do not exactly match the reference difference pattern [Fig. 6(a) – Table III (Pattern 1)], while a good fitting is achieved in . Moreover, the same correspondence with the pattern compromise difference beam ( ) is generated by the – CPM architectures, while the pattern matching for two the sum beam [Fig. 6(c)] slightly worsens with the parallel solution against a significant reduction of the circuit complexity

K

N = 20, K = 2, Q = 3) –

Fig. 5. Parallel – CPM Analysis (Test 1: and of the terms Behavior of the cost function iterative synthesis process ( : iteration index).

i

9

9

and

9

during the

( vs. , being the crossing count in the BFN). In order to assess the reliability of the proposed strategy, Fig. 7 gives some indications on the asymptotic behavior of the and method performances. More specifically, the values of [Fig. 7(a)] and of the indexes and [Fig. 7(b)] versus are reported for both implementations. As expected, the plots present a monotonic decreasing behavior and when . As regards the multiple beams in Fig. 6 ( ), it is worth noting that the compromise pattern synthesized through the matching of the Taylor excitations is quite similar to the main pattern [Fig. 6(c)]. To better test the potentialities of proposed

MANICA et al.: SYNTHESIS OF MULTI-BEAM SUB-ARRAYED ANTENNAS THROUGH AN EXCITATION MATCHING STRATEGY

K

N = 20 K = 2 Q = 3

K

487

k = 1 (a) and k = 2 (c).

Fig. 6. – CPM Multi-Beam Synthesis (Test 1: , ) – Patterns synthesized with the , – CPM techniques at [5]. Array layouts: (b) serial architecture and (d) parallel architecture. Main beam set to a Dolph-Chebyshev pattern with

TABLE I

PARALLEL

K – CPM MULTI-BEAM SYNTHESIS (TEST 1: N = 20, K = 2, Q = 3) – REFERENCE VECTORS AND THEIR NORMS

SLL = 025 dB

the proposed method. The features of the compromise beams are close to those of the previous case (Table III) confirming the effectiveness of the proposed excitation matching strategy. B. Test 2 – Flat Top, Sum and Difference Patterns Synthesis

TABLE II

K – CPM MULTI-BEAM SYNTHESIS (TEST 1: N = 20, K = 2, Q = 3) – SUB-ARRAY CONFIGURATIONS AND SUB-ARRAY WEIGHTS

approach, a Dolph-Chebyshev pattern [5] with has been then selected as main beam, while the reference excicompromise patterns are equal to those of tations for the the previous case. Fig. 8 shows the solutions synthesized with

The second example deals with the synthesis of a linear array elements ( ) generating with a flat-topped main beam and two compromise patterns. The flat-topped pattern is characterized by ripples within the main and . It is aflobe region of amplitude forded by a set of symmetrical real excitations available in [10]. The reference excitations for the first and the second sub-arrayed beams have been chosen to generate a Zolotarev pattern and a Dolph-Chebyshev pattern [5] [9] with . The reference excitations are given in with Table IV (rows 2–4). The number of sub-arrays has been set to . The final aggregations and the corresponding weights synthesized with the proposed parallel – CPM approach are , , and , respectively. In this case, the same result is obtained by the serial approach as confirmed by the value of the cost function as well as from the matching indexes (Table V). The convergence patterns layouts of both arare shown in Fig. 9 along with the vs. ). As far as the pattern chitectures ( performance is concerned (Table V), the sum pattern presents a of almost 5 dB above the value of the ref[deg] vs. erence beam. Moreover, [deg]. A better matching has been yielded for the difference

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TABLE III

K – CPM MULTI-BEAM SYNTHESIS (TEST 1: N = 20, K = 2, Q = 3) – PERFORMANCES INDEXES

K

N = 20 K = 2

Fig. 7. – CPM Asymptotic Analysis ( , ) – Behavior of (a) and and of (b) the matching indexes the cost function terms and versus ( ).

1

9 9 Q Q = 2; . . . ; 10

1

pattern since vs. and [deg] vs. [deg]. As for arrays of direct radiating elements able to generate both narrow-beam and shaped-beam patterns, solutions based on Butler matrices have been also considered [28]. Likewise the proposed method, Butler matrices are suitable for narrow bandwidth systems, but they generate symmetrical orthogonal beams of the same shape as many as the number of array elements to be properly combined to synthesize shaped beams [2]. When using Butler matrices, the number of active elements (i.e., phase shifters, attenuators/amplifiers) of the BFN grows proportionally with the number of radiating elements. On the contrary, the proposed method allows one to generate an arbitrary number

K

N = 20, K = 2, Q = 3) – k = 1 (a) and k = 2 (b). SLL = 040 dB [5].

Fig. 8. – CPM Multi-Beam Synthesis (Test 1: Patterns synthesized with the – CPM techniques at Main beam set to a Dolph-Chebyshev pattern with

K

of patterns of different shapes with a BFN complexity growing as the number of sub-arrays, usually much lower than the total number of elements especially in large arrays. To better and further point out the effectiveness of the proposed method in reducing the BFN complexity, the synthesis of large sub-arrayed arrays providing multiple beams is then addressed. C. Test 3 – Large Array Synthesis The last test case is concerned with the synthesis of a large elements ( ) with linear array having

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489

TABLE IV

PARALLEL

K

K – CPM MULTI-BEAM SYNTHESIS (TEST 2: N = 12, K = 2, Q = 4) – REFERENCE EXCITATIONS AND REFERENCE VECTORS

N = 12 K = 2 Q = 4

Fig. 9. – CPM Multi-Beam Synthesis (Test 2: , , )– Optimal and compromise patterns (a). Layouts derived from the serial approach (b) and the parallel approach (c).

a compromise feed network of sub-arrays. The main patand tern has been set to a Taylor pattern with [7]. Two reference difference patterns have been chosen, namely a modified Zolotarev pattern with

and [34] and a difference pattern providing maximum directivity whose excitations have been computed as proposed in [35]. The sub-array configuration and the corresponding weights – CPM approaches are reported in synthesized with the Table VI. A pictorial representation of the element aggregation in the reference vector space is shown in Fig. 10 where the sub-arrays is indicated by means of membership to the different point styles . As expected, the searching procedure is able to aggregate in the same sub-array the array elements whose reference vectors are closer by minimizing the internal clusters. In Fig. 11, the patterns radiated variance of the – CPM solution are shown. For the sake of by the parallel clarity, only the envelopes are plotted.1 Fig. 12 compares the – CPM patterns with the reference [Fig. 12(a)] and ones in correspondence with [Fig. 12(b)], respectively. As it can be observed, the parallel solution gets worse than the serial implementation dealing with – Table VII) when the matching the difference pattern ( with the reference one is also not very accurate [Fig. 12(a)] especially outside the angular region close to the mainlobe (i.e., ). On the other hand, it is worthwhile to notice the strong reduction of the layout complexity obtained with the parvs. . As regards allel architecture since [Fig. 12(b)], both – CPM patterns have the same pattern features of the reference beam (i.e., and [deg]) even though the maximum directivity vs. ). slightly reduces ( As far as the computational burden is concerned, Table VIII (dimension of the solution summarizes the main issues: (number of iterations), (number of cost space), (CPU time). Despite the wide function evaluations), and dimension of the solution space with adlayout missible alternatives, the process for defining the and it requires cost takes just [sec]. In such function evaluations performed in only a case, the extra computation time with respect to the serial ) is mainly related to the sorting implementation ( process of the reference vectors. IV. CONCLUSIONS In this paper, an innovative approach for the synthesis of multiple-beam sub-arrayed antennas has been presented. The solution procedure is based on an excitation matching algorithm 1Each envelope is obtained from the corresponding beam pattern drawing the shape of a lobe until the height of the (spatially) successive secondary lobe is reached and connecting the two points at the same height with a straight line.

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TABLE V

K – CPM MULTI-BEAM SYNTHESIS (TEST 2: N = 12, K = 2, Q = 4) – PERFORMANCES INDEXES

TABLE VI

K – CPM MULTI-BEAM SYNTHESIS (TEST 3: N = 100, K = 2, Q = 8) – SYNTHESIZED SUB-ARRAY CONFIGURATIONS AND WEIGHTS

K

N = 100 K = 2 Q = 8

Fig. 10. – CPM Multi-Beam Synthesis (Test 3: , , ) – Representation in the reference vector space of the sub-array configurations synthesized with the – CPM techniques.

K

K

K

N = 100, K = 2, k = 1, 2).

Fig. 11. Parallel – CPM Multi-Beam Synthesis (Test 3: ) – Envelope of the main and compromise patterns (

Q=8

aimed at defining an optimal pattern through a set of independent excitations and synthesizing multiple compromise beams

N = 100 K = 2 Q = 8) K–

Fig. 12. – CPM Multi-Beam Synthesis (Test 3: , , – Envelope of the reference and compromise patterns synthesized with the and (b) . CPM techniques: (a)

k=1

k=2

by using a common sub-array feed network and independent sub-array weights for each pattern. A fast searching procedure exploiting a suitable integration of the CPM with a customized

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TABLE VII

K – CPM MULTI-BEAM SYNTHESIS (TEST 3: N = 100, K = 2, Q = 8) – PERFORMANCES INDEXES

TABLE VIII

K – CPM MULTI-BEAM SYNTHESIS (TEST 3: N = 100, K = 2, Q = 8) – COMPUTATIONAL INDEXES

version of the -means clustering algorithm has been used to effectively sample the space of admissible solutions. The obtained results have proved the feasibility of the proposed method as well as its reliability in fitting multiple reference patterns with satisfactory performances and a limited circuit complexity. The computational efficiency of the approach has been pointed out dealing with large linear arrays, as well. REFERENCES [1] I. M. Skolnik, Radar Handbook, 3rd ed. New York: McGraw-Hill, 2008. [2] C. A. Balanis, Antenna Theory: Analysis and Design, 3rd ed. Hoboken, NJ: Wiley, 2005. [3] M. Barba, J. E. Page, J. A. Encinar, and J. R. Montejo-Garai, “A switchable multiple beam antenna for GSM-UMTS base stations in planar technology,” IEEE Trans. Antennas Propag., vol. 54, no. 11, pp. 3087–3094, Nov. 2006. [4] R. J. Mailloux, Phased Array Antenna Handbook. Boston, MA: Artech Hause, 2005. [5] C. L. Dolph, “A current distribution for broadside arrays which optimises the relationship between beam width and sidelobe level,” Proc. IRE, vol. 34, pp. 335–348, 1946. [6] T. T. Taylor, “Design of line-source antennas for narrow beam-width and low side lobes,” Trans. IRE, vol. AP-3, pp. 16–28, 1955. [7] A. T. Villenueve, “Taylor pattern for discrete arrays,” IEEE Trans. Antennas Propag., vol. AP-32, no. 10, pp. 1089–1093, Oct. 1984. [8] E. T. Bayliss, “Design of monopulse antenna difference patterns with low sidelobes,” Bell Syst. Tech. J., vol. 47, pp. 623–640, 1968. [9] D. A. McNamara, “Direct synthesis of optimum difference patterns for discrete linear arrays using Zolotarev distribution,” IEE Proc. H Microw. Antennas Propag., vol. 140, no. 6, pp. 445–450, Dec. 1993. [10] Y. U. Kim and R. S. Elliott, “Shaped-pattern synthesis using pure real distributions,” IEEE Trans. Antennas Propag., vol. 36, no. 11, pp. 1645–1649, Nov. 1988. [11] F. Ares, R. S. Elliott, and E. Moreno, “Synthesis of shaped line-source antenna beams using pure real distributions,” Electron. Lett., vol. 30, no. 4, pp. 280–281, Feb. 1994. [12] M. Durr, A. Trastoy, and F. Ares, “Multiple-pattern linear antenna arrays with single prefixed amplitude distributions: Modified WoodwardLawson synthesis,” Electron. Lett., vol. 36, no. 16, pp. 1345–1346, Aug. 2000. [13] A. Trastoy, Y. Rahmat-Samii, F. Ares, and E. Moreno, “Two-pattern linear array antenna: Synthesis and analysis of tolerance,” IEE Proc. Microw. Antennas Propag., vol. 151, no. 2, pp. 127–130, Apr. 2004. [14] M. Comisso and R. Vescovo, “Multi-beam synthesis with null constraints by phase control for antenna arrays of arbitrary geometry,” Electron. Lett., vol. 43, no. 7, pp. 374–375, Mar. 2007. [15] M. Alvarez, J. A. Rodriguez, and F. Ares, “Synthesising Taylor and Bayliss linear distributions with common aperture tail,” Electron. Lett., vol. 45, no. 1, pp. 18–19, Jan. 2009. [16] R. J. Mailloux, “Phased array theory and technology,” IEEE Proc., vol. 70, no. 3, pp. 246–302, Mar. 1982.

[17] K. S. Rao and I. Karlsson, “Low sidelobe design considerations of large linear array antennas with contiguous subarrays,” IEEE Trans. Antennas Propag., vol. 35, no. 4, pp. 361–366, Apr. 1987. [18] N. Toyama, “Aperiodic array consisting of subarrays for use in small mobile earth stations,” IEEE Trans. Antennas Propag., vol. 53, no. 6, pp. 2004–2010, Jun. 2005. [19] R. L. Haupt, “Optimized weighting of uniform subarrays of unequal size,” IEEE Trans. Antennas Propag., vol. 55, no. 4, pp. 1207–1210, Apr. 2007. [20] D. A. McNamara, “Synthesis of sub-arrayed monopulse linear arrays through matching of independently optimum sum and difference excitations,” IEE Proc. H Microw. Antennas Propag., vol. 135, no. 5, pp. 371–374, 1988. [21] F. Ares, S. R. Rengarajan, J. A. Rodriguez, and E. Moreno, “Optimal compromise among sum and difference patterns,” J. Electromag. Waves Appl., vol. 10, pp. 1143–1555, 1996. [22] P. Lopez, J. A. Rodriguez, F. Ares, and E. Moreno, “Subarray weighting for difference patterns of monopulse antennas: Joint optimization of subarray configurations and weights,” IEEE Trans. Antennas Propag., vol. 49, no. 11, pp. 1606–1608, Nov. 2001. [23] S. Caorsi, A. Massa, M. Pastorino, and A. Randazzo, “Optimization of the difference patterns for monopulse antennas by a hybrid real/integercoded differential evolution method,” IEEE Trans. Antennas Propag., vol. 53, no. 1, pp. 372–376, Jan. 2005. [24] M. D’Urso, T. Isernia, and E. F. Meliadò, “An effective hybrid approach for the optimal synthesis of monopulse antennas,” IEEE Trans. Antennas Propag., vol. 55, no. 4, pp. 1059–1066, Apr. 2007. [25] L. Manica, P. Rocca, A. Martini, and A. Massa, “An innovative approach based on a tree-searching algorithm for the optimal matching of independently optimum sum and difference excitations,” IEEE Trans. Antennas Propag., vol. 56, no. 1, pp. 58–66, Jan. 2008. [26] Y. Chen, S. Yang, and Z. Nie, “The application of a modified differential evolution strategy to some array pattern synthesis problems,” IEEE Trans. Antennas Propag., vol. 56, no. 7, pp. 1919–1927, Jul. 2008. [27] P. Rocca, L. Manica, R. Azaro, and A. Massa, “A hybrid approach to the synthesis of subarrayed monopulse linear arrays,” IEEE Trans. Antennas Propag., vol. 57, no. 1, pp. 280–283, Jan. 2009. [28] C. A. Balanis, Modern Antenna Handbook. New York: Wiley, 2008. [29] W. Wen-Chang, L. Chun-Jing, L. Feng, and L. Lei, “A new multi-beam forming method for large array,” in Proc. IEEE Radar Conf., Pasadena, CA, May 4–8, 2009, pp. 1–4. [30] W. D. Fisher, “On grouping for maximum homogeneity,” Amer. Stat. J., pp. 789–798, 1958. [31] R. O. Duda, P. E. Hart, and D. G. Stork, Pattern Classification, 2nd ed. New York: Wiley, 2000. [32] J. B. MacQueen, “Some methods for classification and analysis of multivariate observation,” in Proc. 5th Berkley Symp. on Mathematical Statistics and Probability, Berkeley, 1967, pp. 281–297. [33] S. P. Lloyd, “Least square quantization in PCM,” IEEE Trans. Inf. Theory, vol. 28, no. 2, pp. 129–137, Mar. 1982. [34] D. A. McNamara, “Discrete -distribution for difference patterns,” Electron. Lett., vol. 22, no. 6, pp. 303–304, Mar. 1986. [35] D. A. McNamara, “Excitations providing maximal directivity for difference arrays of discrete elements,” Electron. Lett., vol. 23, no. 15, pp. 780–781, Jul. 1987.

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Luca Manica was born in Rovereto, Italy. He received the B.S., M.S., and Ph.D. degrees in telecommunication engineering from University of Trento, Italy, in 2004, 2006 and 2010, respectively. He is a member of the Electromagnetic Diagnostic (ELEDIA) Research Group, University of Trento. His main interests are the synthesis of the antenna array patterns and fractal antennas.

Paolo Rocca (M’08) received the B.S. and M.S. degrees in telecommunications engineering and the Ph.D. degree in information and communication technologies from the University of Trento, Italy, in 2004, 2005, and 2008, respectively. Since 2003, he has been a member of the Electromagnetic Diagnostic (ELEDIA) Research Group, University of Trento. He is currently an Assistant Professor at the Department of Information Engineering and Computer Science of the University of Trento. His main interests are in the framework of antenna array synthesis and design, electromagnetic inverse scattering, and optimization techniques for electromagnetics.

Giacomo Oliveri (M’09) received the B.S. and M.S. degrees in telecommunications engineering and the Ph.D. degree in space sciences and engineering from the University of Genoa, Italy, in 2003, 2005, and 2009 respectively. Since 2008, he has been a member of the Electromagnetic Diagnostic Laboratory, University of Trento, Italy. His research work is mainly focused on cognitive radio systems, electromagnetic direct and inverse problems, and antenna array design and synthesis.

Andrea Massa (M’96) received the “Laurea” degree in electronic engineering and Ph.D. degree in electronics and computer science from the University of Genoa, Genoa, Italy, in 1992 and 1996, respectively. From 1997 to 1999, he was an Assistant Professor of electromagnetic fields at the Department of Biophysical and Electronic Engineering, University of Genoa, teaching the university course of Electromagnetic Fields 1. From 2001 to 2004, he was an Associate Professor at the University of Trento. Since 2005, he has been a Full Professor of electromagnetic fields at the University of Trento, where he currently teaches electromagnetic fields, inverse scattering techniques, antennas and wireless communications, and optimization techniques. At present, he is the Director of the ELEDIALab, University of Trento and Deputy Dean of the Faculty of Engineering. His research work since 1992 has been principally on electromagnetic direct and inverse scattering, microwave imaging, optimization techniques, wave propagation in presence of nonlinear media, wireless communications and applications of electromagnetic fields to telecommunications, medicine and biology. Prof. Massa is a member of the IEEE Society, of the PIERS Technical Committee, of the Inter-University Research Center for Interactions Between Electromagnetic Fields and Biological Systems (ICEmB) and Italian representative in the general assembly of the European Microwave Association (EuMA).

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Investigations on the Efficiency of Array Fed Coherently Radiating Periodic Structure Beam Forming Networks Nicolas Ferrando and Nelson J. G. Fonseca, Senior Member, IEEE

Abstract—The concept of coherently radiating periodic structure beam forming network (C-BFN) has been introduced recently in the literature as a possible mean to feed linear or planar arrays. Losses are expected in such a structure as it produces non-orthogonal excitation laws, motivating the investigations on C-BFN efficiency presented in this paper to guide antenna designers. Monoand multi-beam applications are considered, including also beam steering capability. These investigations help define some useful design rules depending on the application. Some evolutions are also proposed that may overcome some limitations of the original concept. All the presented results are supported by both simulations and measurements of various prototypes designed on printed technology at 6 GHz. Index Terms—Beam forming network, beam steering, coherently radiating periodic structure beam forming network (CORPS-BFN or C-BFN), linear and planar arrays.

I. INTRODUCTION

M

ULTIPLE beam forming networks (M-BFN) are of great interest to improve antennas flexibility and/or system capacity. They are characterized by multiple inputs feeding multiple outputs, each output being generally connected to one elementary radiator of an array antenna. Each input produces one beam. Well-known applications are electronically scanned beams and space diversity multiple access (SDMA) antennas. These M-BFNs are usually based on specific associations of directional couplers, power dividers/combiners and eventually phase-shifters. Well-known examples are the Butler [1], Blass [2] and Nolen [3] matrices. It was demonstrated that a lossless matrix is necessarily orthogonal, meaning that the output vectors produced by feeding each input port are orthogonal according to the inner product [4], [5]. To increase the flexibility, it is often preferred to work with coherent (in-phase) output excitations associated with amplitude and/or phase controls. Coherent networks associated with an Manuscript received January 25, 2010; revised June 28, 2010; accepted September 17, 2010. Date of publication December 03, 2010; date of current version February 02, 2011. N. Ferrando was with the Antenna Department, French Space Agency (CNES), 31401 Toulouse cedex 9, France. He is now with the Antenna Studies Department, Thalès Alénia Space, Toulouse, France. N. J. G. Fonseca was with the Antenna Department, French Space Agency (CNES), 31401 Toulouse, cedex 9, France. He is now with the Antenna and Sub-millimetre Wave section, European Space Agency/ESTEC, Noordwijk, The Netherlands (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2096392

array eventually placed in the focal plane of a reflector enable a refined beam steering and may also offer a beam shaping capability, these features being particularly attractive for next generation telecommunication satellite systems. A typical topology for a coherent beam forming network is based on the association of parallel power dividing sections followed by parallel power combining sections [6]. Recently, a topology named coherently radiating periodic structure beam forming network (CORPS-BFN or C-BFN) was proposed with the particularity of mixing power dividers and combiners [7]–[11]. This structure produces a coherent Gaussian amplitude tapered excitation law. However an M-BFN producing only coherent output excitations must have losses. Very limited information is found on that matter in the papers describing C-BFNs. A very similar structure based on hybrid junctions was previously introduced in [12] for cosine and cosine-squared illuminations with clearly stated efficiency reduction (half of the energy is dissipated for a cosine illumination design and three fourth for a cosine-squared illumination design), thus emphasizing the need for deeper investigations on C-BFN efficiency as provided in this paper to help antenna designers select the proper BFN topology and required amplification level for a given application. Some design rules come out of these investigations to keep optimal or reasonable performances. Some evolutions are also proposed to reduce the losses in some specific applications. The paper is organized as follows. Section II gives a general description of C-BFN and proposes a calibration of our models based on the measurement of an elementary component. Monoand multi-beam applications are then detailed in Section III with resulting losses depending on the C-BFN topology. When applicable, beam-steering capability and resulting impact on the loss budget are investigated. This paper is concluded by recommendations that can be taken as design rules and some improvements that may be considered for specific applications. II. C-BFN AND BASIC CONSTITUTING COMPONENT A. C-BFN Description The basic principle of C-BFN lies on the binomial linear array excitation law, the corresponding coefficients being determined by Pascal’s triangle [13]. Consequently, the C-BFN is implemented using an alternate arrangement of power combiners (C) and power dividers (D) as illustrated in Fig. 1. Typical input ports and output ports, C-BFN implementation has being greater than . The number of layers required is equal

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component has the following ideal -parameters matrix, often referred to as the Wilkinson’s matrix:

(1)

Fig. 1. C-BFN concept.

When used as a power divider, the sum of the output signals at ports 2 and 3 is equal in power to the input signal at port 1. When used as a power combiner, losses appear depending on relative amplitudes and phases of the two incident signals at ports 2 and 3 due to the resistive load(s) required to achieve matching condition at all ports. The resulting combination efficiency is a significant contributor to the losses in C-BFN. Typical coherent applications consider only unbalanced amplitudes at the input ports and . Then according to (1) and omitting 2 and 3, namely phase to simplify notations, the signal at the output of port 1 is (2) The power efficiency of the component is defined as (3) This can be simplified as follows:

Fig. 2. C-BFN for multi-beam applications.

(4) to . Each node of the structure is either a power divider or a power combiner. Each layer is an alternate arrangement of power combiners and power dividers, power combiners of one layer being connected to the power dividers of upper layer. The resulting output ports excitation law is close to a binomial amplitude law in some specific cases, but more generally we better describe the amplitude distribution using a Gaussian function due to the inherent losses of the structure that increase the dynamic variation in amplitude when compared to a standard binomial distribution. The general topology can be simplified depending on the number of beams to be produced. As illustrated in Fig. 2, one can see that the energy of one input port is confined in a triangular zone (dashed area) if we assume power combiners with perfect isolation between input ports. As a result, all the components beyond this limit are not used for this specific beam. Once the number of beams is defined, one can determine the components needed depending mainly on the number of layers, and suppress all the others. At the edges of the structure, loads are needed to keep balanced behavior for all the beams.

B. Basic Constituting Component The same 3-port component is used for both power division and power combination. A schematic representation of this component and associated notations are given in Fig. 2. This

where is the power ration between the inputs. One of the two input signals equal to zero ( is zero or infinity) leads to a worst case efficiency of 0.5 corresponding to a 3 dB insertion loss. For specific applications detailed in Section III, one may also find a phase difference, , between input ports. To take phase difference into account, (4) is updated as follows: (5) Ohmic and substrate losses are also considered in the simulation model for a more accurate loss figure. The model is calibrated based on the measurement of a basic constituting component. We selected a Circular In-phase Hybrid Ring Power Divider (CIHRPD) [8], [9] because this 5-port component has wideband performances. The three main ports are matched and the two remaining are isolated ports (see Fig. 3). The advantages of this 5-port component are more extensively described in [14]. Simulations are performed using the method of moment based commercial Tool Momentum in ADS (Agilent) software. The layout and characteristic dimensions are given in Fig. 3. The corresponding prototype (see Fig. 4) was manufactured on a NY9208 Neltec substrate, characterized by a dielectric constant of 2.08 and a thickness of 0.762 mm. Centre design frequency is set at 6 GHz. The 50 line width is 2.53 mm. Isolated ports were matched with 50 Vishay FC series resistors. Final values

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The maximum directivity is achieved when all the signals , corresponding to sum in phase in the same direction a phase delay excitation described by

(7) When this condition is satisfied, the resulting maximum array factor value is simply (8) Fig. 3. Designed ring power divider.

Taking as reference the equi-amplitude case, we introduce an “amplitude distribution efficiency” defined as follows: (9) The impact of this term on the maximum directivity will be further detailed through the examples reported in this paper. All the theoretical computations provided in this paper come from a simplified matrix formulation, detailed in Appendix. Each layer of a C-BFN can be expressed as a rectangular matrix with very simple formulation that only depends on the configuration (mono- or multi-beam) and on the index of the layer, the final output law being evaluated as the matrix product of all the elementary layers matrices.

Fig. 4. Ring power divider manufactured at 6 GHz.

A. Mono-Beam Linear Array

Fig. 5. Mono-beam C-BFN feeding a 6-element linear array.

considered in the simulation model are 0.0008 for substrate tangential losses, while standard copper characteristics were used for the metal. III. LINEAR ARRAY APPLICATIONS In this section, we investigate the losses generated by C-BFN in four different linear array configurations, including monoand multi-beam designs as well as beam steering capability. But first, we may give some reminders on linear arrays. The array factor of an N-element linear array in the angular direcis given by the following expression: tion (6) where is the th source amplitude and the wavenumber and the array spacing.

its phase delay,

First application to be investigated in this paper is the C-BFN fed mono-beam linear array. The general C-BFN structure as reported in Fig. 1 is simplified to the one illustrated in Fig. 5. All electrical paths need to be equivalent for coherent operation. Accordingly, the power combiners at the edge of the structure are replaced by phase shifters with similar insertion phase. Still, unbalanced power combinations appear inside of the structure as the number of layers increase. This is detailed in Fig. 5, in which components with unbalanced combination are highlighted. When these combination losses are limited, the resulting amplitude excitation is really close to a binomial power distribution as defined by Pascal’s triangle. But as the number of layers and the resulting losses increase, tapering increases and the amplitude distribution is better described as a Gaussian distribution. The combination losses actually appear in all the combiners out of the C-BFN axis of symmetry. It can also be noted that a C-BFN with 2-layers is lossless, but it reduces its use to 3-element linear arrays which is quite limited. Phase shifters on the edge of the first layer can be removed as there is no power combiner in this layer. More generally, the losses of a mono-beam C-BFN versus the number of layers are reported in Fig. 6. These losses are quite reasonable even for large linear arrays. For instance, a 20-layer C-BFN (21 outputs) has intrinsic losses around 2 dB. Obviously, insertion losses are to be added to this value. In micro-strip, technology, these losses increase quite fast and represent almost half of the simulated losses. To illustrate the impact on the radiation pattern, we considfed by ered a theoretical linear array with a spacing of 0.785

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Fig. 6. C-BFN intrinsic losses in mono-beam configuration.

a mono-beam C-BFN as described in this section. The elementary radiating pattern is described by a Gaussian function with a half power beam-width of 74 resulting in a radiating element directivity of 7.4 dB, which is a typical value for a resonant antenna such as a square patch. Computed array gain is reported in Fig. 7. It details intrinsic losses of C-BFN and losses due to amplitude distribution efficiency when compared to uniform amplitude distribution as defined in (9). The resulting gain appears to converge quickly with the number of layers, thus having no benefit in increasing significantly the size of the array. Including the insertion losses due to the materials, the gain even slowly decreases above 8 radiating elements. Computed half power beam-width (HPBW) and side lobe level (SLL) are reported in Fig. 8. As expected, the Gaussian-like amplitude distribution of a C-BFN improves significantly SLL compared to . HPBW uniform linear arrays, with typical SLL around is wider than that of an equivalent uniform linear array. All these results are easily understood as the amplitude taper drastically increases with the number of layers. Consequently, adding more elements at the edges contribute little to the gain and HPBW above 8 to 10 layers. This determines the typical range (up to 10 layers) of application for mono-beam C-BFN fed linear arrays. To illustrate the large difference in amplitude between elements, a 10-output C-BFN has a maximum amplitude difference of 20.4 dB. B. Steered Mono-Beam Linear Array An electronic beam steering capability can be implemented on a linear array using variable phase shifters. Typical technical solution requires one phase shifter per radiating element to produce the desired linear phase progression. In C-BFN, it can be drastically simplified as only one variable phase shifter is needed to produce a regular phase progression [11]. As illustrated in Fig. 9, a single variable phase shifter is added after the first power divider to control the beam pointing. Doing so, the maximum relative phase between the first element and the last element of the linear array is equal to the insertion phase of this variable phase shifter, which means that a large phase shift may be required for significant beam deviation. Furthermore,

Fig. 7. Gain of a C-BFN fed mono-beam linear array.

Fig. 8. Half power beam-width and side lobe level of a C-BFN fed monoand multi-beam linear arrays versus uniform amplitude distribution linear array (0.785  ).

this phase shift breaks the C-BFN symmetry. So, all power combiners have losses due to unbalanced signals, either in amplitude, phase or both. Losses versus the number of layers are given in Fig. 10 for different variable phase shift values. As expected, minimum losses are achieved for the 0 phase shift. Losses increase significantly with the phase shift within the range 0 to 90 , even for small number of layers. According to (7), optimal beam steering requires a linear phase progression at the output ports. We investigated the effective phase progression of C-BFN with only one phase control. Simulation results are reported in Fig. 11 for different linear array sizes with a same variable phase shift value. The phase progression is linear for a 3-element linear array, and remains linear in the central portion (bold lines in Fig. 11) of the array for larger arrays. This non-linear phase progression

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Fig. 9. Steered mono-beam C-BFN fed 6-element linear array.

Fig. 11. Normalized phases in mono-beam C-BFN with beam steering capability (each curved line refers to a linear array of corresponding dimension, while bold section highlights the radiating elements within a dynamic amplitude range of 15 dB compared to the maximum amplitude).

C. Multi-Beam Linear Array

Fig. 10. C-BFN losses in steered mono-beam configuration for different variable phase shifts (from 0 to 90 per step of 10 ).

introduce further maximum directivity losses as discussed in the beginning of this section, but these losses will be limited as the non-linear sections concern mainly the radiating elements at the edges of the array which have less power compared to central ones (thinner lines in Fig. 11 relate to radiating elements 15 dB under the central radiating element(s)). This constraint of linear phase progression is also the reason why we limited our investigations to a maximum phase shift of 90 in Fig. 10. We actually noticed that beyond this value, phase distribution is no longer linear or quasi-linear. Then, the beam steering capability is lost. This obviously limits the maximum phase dynamic between first and last elements of the linear array and as a consequence the maximum beam deviation. To conclude on mono-beam C-BFN, one may suggest using it for small linear arrays: 8 to 10 radiating elements seem a reasonable upper limit for passive configuration and 5 radiating elements with beam steering capability. Of course, this is to be balanced, as some applications may require larger arrays but then the consequences in the overall performances are clearly identified here.

As already discussed in Section II, C-BFN is particularly adapted for multi-beam configuration. The topology produces a natural overlap between adjacent beams which increases with the number of layers, thus providing an efficient radiating aperture reuse. Since multi-beam C-BFN is the general form, we decided to investigate them experimentally assuming that if the model proved accurate in the multi-beam configuration it would most probably be also accurate in mono-beam configuration. Two prototypes, illustrated in Fig. 12, were manufactured. Good agreement was found between simulation and measurements apart from a slight frequency shift due mainly to the high component integration. But since the elementary component is wideband, this did not affect performances at central design frequency. The two prototypes having each 3 beams, it was also possible to confirm the good reproducibility of the performances. Compared to mono-beam configuration, edge losses are now added due to the combiners at the edge of the triangular zone in which the energy is distributed. This is properly illustrated in Fig. 13, to be compared with Fig. 5. As in mono-beam configuration, the components along the axis of symmetry of the power distribution per beam are lossless for this beam. The edge losses introduced in the multi-beam topology are already present at first layer level, while intrinsic losses would only appear at third layer for mono-beam topology. These edge losses are detailed in Fig. 14. It easily appears that they may be described by a geometric series. As the number of layers increases, the edge losses decrease and converge to zero. Consequently, the first layers are the main contributors to these edge losses. From (4), the first layer efficiency is only 50%, resulting in a 3 dB loss. Summation of all the edge losses for an N-layer C-BFN is given by the following formula: (10)

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Fig. 15. Losses in multi-beam C-BFN.

Fig. 12. Manufactured 3-beam C-BFN with (a) 2 layers and (b) 4 layers.

Fig. 13. Losses in a 3-beam C-BFN.

layer alone introduces 3 dB losses because the corresponding combiners are totally unbalanced: power only arrives from one side. Measured losses for the 2- and 4-layer C-BFNs were respectively 4.63 and 6.26 dB. Corresponding values in simulation are 4.52 and 6.16 dB respectively. The consequence on the array gain is reported in Fig. 16. The increased losses in multi-beam configuration actually cancel the array gain meaning that the gain of the array will be roughly the gain of one radiating element. Interestingly, inner losses are not significantly affected by the edge losses, resulting in values quite similar between inner losses in multi-beam configuration and total losses in mono-beam configuration. Losses due to non-uniform amplitude law are slightly higher in multi-beam configuration when compared to mono-beam configuration because edge losses tend to increase the amplitude taper. Including insertion losses due to micro-strip technology, the array gain becomes negative, meaning that having two radiating elements would actually result in lower gain than that of a single element. Accordingly, three to four layers appear as a maximum for potential applications. D. Multi-Beam Design With Independent Beam Steering

Fig. 14. Edge losses progression in a multi-beam C-BFN.

The total losses of a C-BFN including inner and edge losses versus the number of layers are plotted in Fig. 15. Measured results show very good agreement with simulated ones. Edge losses appear to be the main contributor, although they conaccording to (10). First verge quite rapidly toward

Beam steering capability described in Section III-B can be extended to multi-beam configuration adding variable phase shifters between the first and second layers as illustrated in Fig. 17. Consequently, the beam pointing is no longer defined by only one phase shifter as in the case of mono-beam configuration but by the phase difference between the two adjacent phase shifters related to one beam (within the dashed triangle area in Fig. 17), except for one beam at the edge (the beam on the left in Fig. 17). This evolution is possible because only relative phase is meaningful in steered beam phased array design. Additional losses due to the phase unbalance appear and are similar to those appearing in steered mono-beam configuration. Edge losses are not affected by the phase and remain similar to those of multi-beam configuration without steering. Conclusions on the potential use of this concept are then similar to the previous ones. Due to the high edge losses

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Fig. 18. Components suppressed to improve efficiency of a multi-beam C-BFN.

Fig. 16. Array gain of a multi-beam C-BFN fed linear array. Fig. 19. Manufactured 2-beam C-BFN with suppressed components for improved efficiency.

Fig. 17. C-BFN for steered multi-beam applications.

imposed by the multi-beam configuration, a reduced number of layers is recommended resulting in limited applicability to phased arrays, although this application is the one that most likely would require independent beam steering. IV. POSSIBLE EVOLUTIONS Previous section highlighted some specific limitations of C-BFN structures: intrinsic losses in multi-beam configuration and limited beam steering capability for large mono-beam configurations. Evolutions proposed in this section try to overcome in some way these two limitations. A. Reduced Losses in Multi-Beam Configuration Based on previous section results, the main contributors to the losses in multi-beam configuration are the two combiners of the first layer. They introduce a systematic 3 dB loss. A simple but quite efficient idea is just to remove them. This results in the configuration illustrated in Fig. 18. Consequence is that one beam over two is suppressed and overlap between adjacent beams is reduced. The simplification can be extended to the upper layers,

for instance suppressing two beams over three, three beams over four and so one. The impact on the design will mainly depend on the number of layers. For larger number of layers this may be acceptable, but for most practical applications, one should limit the suppression of beams to maintain the advantages of C-BFN. A practical example is illustrated in Fig. 19 with a 2-beam C-BFN having four layers with the first one reduced according to the evolution proposed in this section. Losses were equal to 3.44 dB in measurement. Compared with the 6.26 dB of the standard 4-layer 3-beam C-BFN previously described, this confirms the loss reduction with a 2.82 dB reduction, very close to the theoretical 3 dB loss reduction expected. B. Improved Beam-Steering Capability in Mono-Beam Configuration The solution proposed in the literature for steered mono-beam C-BFN fed linear array is obviously attractive because only one phase shifter is required but limited beam steering is achieved [11]. One natural evolution could be an intermediate solution between this one and the conventional one which is one phase shifter per output port. This is illustrated in Fig. 20 where a layer of phase shifters is inserted between two layers of the C-BFN. One phase shifter inserted between first and second layers corresponds to configuration in Fig. 9. We investigated the impact of the phase shifter layer location on the output phase dynamic range and total losses for a 5-layer C-BFN associated to a 6-element linear array. As already discussed, we are limited to a maximum of 90 relative phase shift between two adjacent phase shifters. Fig. 21 illustrates the output phase variation versus the phase shifter row location (an phase shifter row is necessarily

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Fig. 20. Mono-beam C-BFN with beam steering produced by phase shifters inserted between two layers. Fig. 22. Output phase linearity with an intermediate row of phase shifters in mono-beam C-BFN with beam steering capability.

Fig. 21. Output phase dynamic range with an intermediate row of phase shifter in mono-beam C-BFN with beam steering capability. Fig. 23. Losses in mono-beam C-BFN with intermediate row of phase shifters.

after the th layer) assuming a constant 60 phase shift between two adjacent phase shifters. As expected, the total dynamic range varies linearly with the number of phase shifters. Phase progression linearity also improves with the number of phase shifters. To further investigate the impact of this configuration on the output phase linearity, Fig. 22 illustrates results assuming a constant dynamic range set to 90 . Phase difference between two adjacent phase shifters is then adjusted to produce this dynamic range at the output. Reported results confirm the fact that increasing the number of phase shifters increases also the output phase progression linearity although the improvement is quite limited: linearity is quite acceptable for small C-BFN structures even with only one phase shifter. Consequently, the proposed intermediate configuration is mainly to improve the total dynamic range of the output phase and thus the maximum scan angle achievable. Losses versus the location of the row of phase shifters are reported in Fig. 23 for different relative phase differences between adjacent phase shifters. This highlights the capability of the proposed evolution to reduce losses while maintaining

reduced complexity compared to a more conventional phased array configuration. For example, to achieve a 90 dynamic range at the output with one phase shifter introduces in theory 2.32 dB losses. Using a configuration with 3 phase shifters having a phase difference of 30 , which also results in a 90 dynamic range at the output, reduces losses down to 0.82 dB. This 3-phase shifter configuration even permits to double the output dynamic range with losses slightly lower (2.05 dB) than the 1-phase shifter configuration. Accordingly, the proposed intermediate solution might be preferred for some specific designs. V. CONCLUSION This paper has introduced a deep investigation on C-BFN capability and efficiency. Theoretical losses due to the non-orthogonal nature of this specific BFN were detailed for different configurations including mono- and multi- beam design as well as beam steering capability. A simple matrix formulation was also introduced. Presented results proved C-BFN to have their most interesting configurations when limited in size: 3 to 4 layers

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are a reasonable upper limit in multi-beam configuration, but it might go up to 10 layers in mono-beam configuration. Interestingly, side lobe level reduction is still achieved even with small C-BFNs. These conclusions are compliant with previous developments found in the literature of C-BFNs having limited number of layers. We also proposed design evolutions that overcome some of the identified limitations and which may be of interest for specific applications. They can even be combined to some extent if necessary. C-BFN will most likely be used in association with reflectors to compensate for the naturally low gain of this structure in association with a linear array. Potential extension of the concept to planar arrays has already been given some attention in the literature and might be a promising solution. Using a set of coherent signals as inputs in a multi-beam configuration may also introduce some amplitude control capability, enabling complex beam shaping with a limited number of controls. The periodic arrangement of components characterizing C-BFN structures can also find applications in circular or cylindrical array design.

The second layer is composed of two phase shifters, assumed to be theoretically lossless, two power dividers and one power combiner

APPENDIX

To introduce beam steering capability in the calculation, one simply add between two consecutive layers, let say layer and , a square matrix with size which diagonal layer contains the proper phase shifts and zeros otherwise. The main difference in multi-beam configuration is introduced by the edge losses that only affect limited values in the matrix, actually the coefficients located at (1, 1) and for the transfer matrix of index . Resulting general matrix form for the multi-beam application is given by

In order to evaluate the output law of a C-BFN, used as the excitation law for the corresponding linear array, we propose a generic matrix formulation. Notations assume that the matrix is defined in transmit, but similar formulation may be derived in receive due to reciprocity. , with an index We introduce a transfer matrix per layer equal to the number of input ports, such that

(14)

In this matrix, one can say to simplify that the rows enable the power combining function while the columns enable the power dividing function. A general form quickly appears and can be written as:

.. .. .

.. .

.

..

.. . .

(15)

(11) and are respectively the output and input vecwhere tors of considered layer. Subscripts in brackets define the size of the matrices. Notation for the output vector highlights the fact that it is at the same time the input vector of the following layer. input ports and output ports Considering a C-BFN with can and using (11) sequentially, the output excitation law be related to the input vector by

(12) where , , are the transfer matrices defining each layer of the structure. In multi-beam designs, the layer number corresponding to a given transfer matrix with index for a C-BFN having inputs . Accordingly, the first layer has is simply given by an index equal to , while the last layer has an index . equal to Let now define the transfer matrices for both mono- and multi-beam applications. The matrices are defined based on the transfer coefficients of the elementary component as given in (1). We start with mono-beam configuration, meaning that the input vector is reduced to only one value. The first layer has only one power divider resulting in the transfer matrix (13)

.. .. .

.. .

.

..

.. . .

(16)

Independent beam pointing capability can also easily be added using a square matrix containing the phase shifts in the diagonal. The only constraint in multi-beam configuration is that this matrix is necessarily added between the first and second layer, , othresulting in a square matrix with size erwise adjacent beam phase shifts affect each other. REFERENCES [1] J. Butler and R. Lowe, “Beam-Forming matrix simplifies design of electronically scanned antennas,” Electron. Design, pp. 170–173, Apr. 1961. [2] J. Blass, “Multidirectional antenna, a new approach to stacked beams,” in IRE Int. Conf. Record, 1960, vol. 8, pp. 48–50. [3] J. Nolen, “Synthesis of Multiple Beam Networks for Arbitrary Illuminations,” Ph.D. dissertation, Bendix Corporation, Radio Division, Baltimore, 1965. [4] J. L. Allen, “A theoretical limitation on the formation of lossless multiple beams in linear arrays,” IRE Trans. Antennas Propag., vol. 9, no. 4, pp. 350–352, Jul. 1961. [5] N. J. G. Fonseca, An Investigation of Blass and Nolen Matrices 2007 [Online]. Available: http://cct/cct13/infos/notestech.htm [6] E. H. Kadac, “Conformal Array Beam Forming Network,” U.S. Patent 3868695, Jul. 18, 1973.

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[7] D. Betancourt and C. del Rio, “A novel methodology to feed phased array antennas,” IEEE Trans. Antennas Propag., vol. 55, no. 9, pp. 2489–2494, Sep. 2007. [8] D. Betancourt and C. del Rio, “A beamforming network for multibeam antenna arrays based on coherent radiating periodic structures,” in Proc. 2nd Eur. Conf. on Antennas and Propagation, Nov. 11–16, 2007, pp. 1–4. [9] D. Betancourt and C. del Rio, “Novel circular in-phase hybrid,” Microw. Opt. Tech. Lett., vol. 49, no. 9, pp. 1317–2314, 2007. [10] D. Betancourt and C. del Rio, “Using CORPS-BFN to feed multibeam antenna systems,” presented at the 29th ESA Antenna Workshop, 2007. [11] D. Betancourt and C. del Rio, “Designing feeding networks with CORPS: Coherently radiating periodic structures,” Microw. Opt. Tech. Lett., vol. 48, no. 8, pp. 1599–1602, 2006. [12] W. D. White, “Pattern limitation in multiple-beam antennas,” IRE Trans. Antennas Propag., vol. 10, no. 4, pp. 430–436, Jul. 1962. [13] C. A. Balanis, Antenna Theory Analysis and Design, 3rd ed. New York: Wiley, 2005. [14] G. F. Mikucki and A. K. Agrawal, “A broad-band printed circuit hybrid ring power divider,” IEEE Trans. Microw. Theory Tech., vol. 37, no. 1, pp. 112–117, Jan. 1989. Nicolas Ferrando was born in Sète, France, in 1986. He received the Engineering degree in electronics and signal processing from Ecole Nationale Supérieure d’Electrotechnique, d’Electronique, d’Informatique, d’Hydraulique et des Télécommunications (ENSEEIHT) and the Master degree in microwave, electromagnetism and opto-electronics from the Institut National Polytechnique de Toulouse (INPT), Toulouse, France, both in 2009. He joined the Antenna Studies Department, Thalès Alénia Space, Toulouse, France, in 2009. His current

technical interests are related to C/Ku telecommunication antennas for space applications.

Nelson J. G. Fonseca (M’06–SM’09) was born in Ovar, Portugal, in 1979. He received the Electrical Engineering degree from Ecole Nationale Supérieure d’Electrotechnique, d’Electronique, d’Informatique, d’Hydraulique et des Telecommunications (ENSEEIHT), Toulouse, France and the Master degree from the Ecole Polytechnique de Montreal, Quebec, Canada, both in 2003, and the Ph.D. degree from the Institut National Polytechnique de Toulouse Université de Toulouse, France, in 2010. He worked as an Antenna Engineer successively in the Antenna Studies Department, Alcatel Alénia Space (now Thalès Alénia Space—France) and in the Antenna Department, French Space Agency (CNES), Toulouse. In 2009, he joined the Antenna and Sub-Millimetre Wave Section, European Space Agency (ESA), Noordwijk, The Netherlands. His technical interests cover antenna design for space applications, beam forming network theory and design as well as new enabling technologies such as fractals, metamaterials and membranes applied to antenna design. He has authored or coauthored more than 70 papers in journals and conferences, including two CNES Technical Notes. He holds six patents and has four patents pending. Dr. Fonseca received several prizes including the best young engineer paper award at the 29th ESA Workshop on Antennas in 2007. He is currently serving as a Technical Reviewer for the Journal of Electromagnetic Waves and Applications—Progress in Electromagnetic Research (PIER), MIT and the IEEE Microwave and Wireless Components Letters (MWCL). He is listed in Who’s Who in the World.

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Design and Implementation of Two-Layer Compact Wideband Butler Matrices in SIW Technology for Ku-Band Applications Ahmed Ali Mohamed Ali, Nelson J. G. Fonseca, Senior Member, IEEE, Fabio Coccetti, Member, IEEE, and Hervé Aubert, Senior Member, IEEE

Abstract—The design and realization of a novel wideband twolayer 4 4 Butler matrix in substrate integrated waveguide (SIW) technology are addressed. The two-layer SIW design is exploited through a two-fold enhancement approach. The two-layer topology is first explored in a simple matrix layout with minimum number of components. A space saving design is then proposed making optimum use of the two-layer topology and the SIW technology leading to a significant size reduction. A two-level, low-loss, wideband SIW transition is designed and optimized using its equivalent circuit model. The two corresponding Butler matrix prototypes are optimized, fabricated and measured. Measured and simulated results are in good agreement. Isolation characteristics better than 15 dB with input reflection levels lower than 12 dB are experimentally validated over 24% frequency bandwidth centered at 12.5 GHz. Measured transmission magnitudes and phases exhibit good dispersive characteristics of 1 dB, around an average value of 6.8 dB, and 10 with respect to the theoretical phase values, respectively, over the entire frequency band. The impact of the measured transmission phases and magnitudes on the radiation pattern of a 4-element antenna array is also investigated. Index Terms—Beam forming networks, Butler matrix, multilayer, substrate integrated waveguide (SIW) technology.

I. INTRODUCTION ULTIBEAM antennas (MBA) have become a key element in nowadays wireless communication systems where increased channel capacity, improved transmission quality with minimum interference and multipath phenomena are severe design constraints. One way to implement a MBA is to use an antenna array fed by a multiple beam forming network (beamformer) (M-BFN) [1]. Such antennas are extensively used in space division multiple access applications [2]. M-BFNs are also used in multi-port amplifiers for distributed amplification used in satellite communication systems [3]. Since the early 1960s, different solutions have been proposed, such as Blass matrix [4], Nolen matrix [5], Rotman lens [6] and

M

Manuscript received January 07, 2010; revised June 09, 2010; accepted July 29, 2010. Date of publication November 18, 2010; date of current version February 02, 2011. This work was supported in part by the Centre National des Etudes Spatiales (CNES). A. A. M. Ali, F. Coccetti and H. Aubert are with the Laboratory of Analysis and Architecture of Systems (LAAS-CNRS) and the University of Toulouse, Toulouse 31400, France (e-mail: [email protected]). N. J. G. Fonseca is with the European Space Agency (ESA), Noordwijk 2200AG, The Netherlands. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2093499

Butler matrix [7]. Among these matrices, the Butler matrix has received particular attention in literature as it is theoretically lossless and employs the minimum number of components [8]. However, the Butler matrix has a main design problem which is the presence of path crossings. To overcome this problem, some specific designs have been suggested but they are not easily scalable for larger matrices [9], [10]. Another method for single-layer realizations is to employ extra 0 dB couplers by means of back-to-back pairs of 3 dB couplers to produce the crossover transfer function [3], [11]. This leads to increased number of components with increased losses especially for larger matrices. In [12] a planar design with wideband performances is reported. However, the complete structure is rather adapted to microstrip technology as it requires wire bonding. Two-layer designs are also found in literature. In [13], a waveguide-based structure has been reported for a narrow-band Butler matrix. Although a compact area layout was proposed, the use of classical waveguides leads to a bulky structure. Good performances have also been obtained using a multilayered design with suspended stripline technology [14]. However, the circuit suffers from a linear phase variation with frequency beside its fabrication complexity. A coplanar waveguide-based two-layer structure is suggested in [15]. However, this solution suffers from being narrow band, consequently it is very sensitive to technological inaccuracies. In this paper, a two-layer compact 4 4 Butler matrix is proposed in SIW technology offering wideband performances for both transmission magnitudes and phases with good isolation and input reflection characteristics. The use of SIW technology combines the advantages of classical rectangular waveguides while being compatible with standard low-cost printed circuit board technology [16]. Owing to its attractive features, a number of implementations for BFNs have been recently developed in SIW technology [17]–[20]. In this paper, the use of the two-layer SIW implementation is explored through a two-fold enhancement approach. On the one hand, the two-layer topology offers an inherent solution for the crossing problem allowing more flexibility for wideband phase compensation. On the other hand, the two-layer SIW technology is exploited through an optimized space saving design by implementing common SIW lateral walls for the matrix adjacent components for maximum size reduction. A two-layer, transverse slot-coupled SIW transition is used as a low-loss guiding element between the two layers of the matrix. The transition is analyzed and optimized using the equiv-

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Fig. 1. General block diagram of a 4 4 Butler matrix with 3 dB/90 couplers schematically mapped to a two-layer topology.

Fig. 2. The developed two-layer SIW coupler structure with microstrip to SIW : ,L : , : , transitions. a , slot : : : ,L ,w and w . ( is the propagation phase delay between an input port and its corresponding direct port).

15

= 10 2 mm = 10 7 mm d width = 1 mm = 2 6 mm = 4 9 mm

= 2 1 mm = = 1 29mm

alent circuit model of [21]. To demonstrate the validity of our design, two different configurations of the 4 4 Butler matrix are designed, optimized and implemented showing good agreement between simulated and measured results. This paper is organized as follows. Section II presents a brief overview on the design considerations of the Butler matrix. Section III addresses the design and realization of the two-layer SIW coupler. Section IV is devoted to the design and analysis of the employed two-layer SIW transition. Experimental results 4 Butler matrices are for two different configurations of 4 detailed in Section V. In Section VI, the measured transmission magnitudes and phases are used to calculate the radiation pattern for a hypothetical antenna array followed by conclusions in Section VII.

thors in [22]. The matrix components are a wideband two-layer SIW coupler [23], variable width compensated length wideband SIW phase shifters and a low-loss two-layer SIW transition which will be presented in the following sections. In this paper, the FEM-based HFSS simulator is used for simulations and optimizations. The substrate used is the , thickness Rogers 6002 with dielectric constant and loss tangent tan . The via hole diameter is 0.78 mm while the pitch size is 1.3 mm. Conductor and dielectric losses have been considered in the electromagnetic simulations. Microstrip to SIW access transitions have been de-embedded in the measurements [16]. All measurements have been performed on an Anritsu 37397D network analyzer with standard TRL calibration.

II. BUTLER MATRIX: ARCHITECTURE AND DESIGN CONSIDERATIONS

III. WIDEBAND TWO-LAYER SIW HYBRID COUPLER

Fig. 1 shows the layout of a 4 4 Butler matrix using 3 dB/90 hybrids. The figure is a direct schematic mapping for Butler matrix the matrix in a two-layer topology. For an employing 90 hybrid couplers, each input port is associated to a specific linear phase at the output ports orienting therefore the main beam towards a corresponding direction such that (1) (2) is assigned to each input port. is the inOne value of terelement spacing of the antenna array and is the free space wavelength. The theoretical scattering parameters relating the outputs to the inputs of the matrix of Fig. 1 is given by (3). The 8 8 scattering matrix of the entire Butler matrix consists of (3) as the lower left quadrant and its transpose as the upper right one. The two diagonal quadrants are zeros representing the theoretical reflection and isolation coefficients

The structure of the two-layer SIW coupler consists of two parallel waveguides coupled through two narrow offset inclined slots in the common broad wall, Fig. 2. Based on a detailed study for the coupler, recently presented by the authors in [23], the coupler has been optimized for a 3 dB coupling value over the 11–14 GHz frequency band with 90 phase difference between the coupled and direct ports. Simulated and measured S-parameters of the designed coupler are shown in Fig. 3. The simulated results (access transitions are not included in the simulations) show a peak-to-peak error for the coupling coefficient less than 0.5 dB over the 11–14 GHz frequency band while isolation and reflection coefficients are below 20 dB with a relative phase difference which is al. Measured results are in good agreement most constant at with the simulated ones. The measured reflection and isolation coefficients are lower than 20 dB over the entire band with a measured peak-to-peak error of 0.6 dB for the coupling coefficient. On the other hand, the measured and simulated phase differences are also in good agreement with less than 5 peak-to peak error over the entire frequency band. IV. TWO-LAYER SIW TRANSITION

(3)

Preliminary results based on ADS simulation of the assembled HFSS simulated components have been reported by the au-

In this section, the two-layer SIW transition is presented. The transition is used as a low-loss interconnect between the two layers of the matrix. A preliminary design with only simulation results has been proposed by the authors in [22]. The transition is hereby analyzed based on the wide-band equivalent circuit

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The power voltage definition is hereby adopted for the waveand is given by (5), [25] guide impedance (4) (5) is the SIW equivalent width, is the waveguide Where mode cutoff frequency. height and is the For the equivalent circuit of Fig. 4(b), the reflection and transmission scattering parameters are given by (6)

(7) Where and are the input impedances of short and ended waveguide sections with waveguide impedance lengths and respectively, is the slot admittance and is given by Fig. 3. Simulated and measured S-parameters for the two-layer SIW hybrid coupler: (a) Reflection and isolation magnitudes with the phase difference between the direct and coupled ports. (b) Direct and coupled magnitudes.

Fig. 4. Two layer transverse slot-coupled waveguide transition. (a) Schematic cross section. (b) Zero-thickness wall equivalent circuit model.

model of [21], and experimentally verified. The two-level transition consists of two parallel waveguides sharing a common broad wall with a narrow transverse coupling slot, Fig. 4(a). The and , slot is bounded by two shorting walls at distances from the slot center. Fig. 4(b) illustrates the equivalent circuit model. The wall thickness of the slot is neglected, as for the case of SIW implementation, it is represented by the thickness of the in our case). The slot (presented substrate copper coating (9 by the parallel LC) is coupled to the upper and lower domains via an impedance transformer, which is unity in our case as both upper and lower SIWs have the same impedance. By considering the parallel waveguide broadwall transverse slot coupling problem (encircled part of Fig. 4(b)) and following [21], the slot inductance and capacitance can be accurately calculated by evaluating (4) at two different frequencies. The real and imaginary parts employed in (4) are obtained through HFSS simulation of the broadwall coupler problem using the same slot dimensions as well as the same waveguide structural parameters employed in the transition. The reference plane in the simulation of all four ports of the coupler must be at the slot center level.

(8) Fig. 5 shows the magnitude and phase transmission characteristics versus frequency of the transition in Fig. 4 for different values of available for the Rogers 6002 substrate. To avoid filtering effects and/or increased insertion losses that may be introduced by the shorted stubs at both sides of the tranand have been kept constant and sition slot, distances equal to the minimum distance allowed by the available tech, while the slot dimennology, such that sions are . The SIW as well as the substrate parameters are those mentioned in Section II with SIW width , used in Fig. 5 is therefore equal to 9.9 mm. The calculated values for and corresponding to the above mentioned slot dimensions with the different values of of Fig. 5 are summarized in Table I. The calculated results of Fig. 5 are obtained upon determining the values of and using (4)through (8). The calculated values based on the circuit model are in good agreement with those obtained from the HFSS simulations. The simulated phase results of Fig. 5(b) are obtained by setting the reference planes of both ports at the center of the slot, thus accounting only for the phase introduced by the slot. By considering the magnitude and phase characteristics of Fig. 5, it can and are good candidates for the transition to be seen that be used in the matrix owing to their low insertion loss and their relatively small, low frequency-dependent associated transmishas the minimum insersion phase. Although the thickness tion loss, its main disadvantage for the foreseen matrix implementation would be its larger associated phase with larger slope of the phase curve. In fact, the lower the slope of the transmission phase-frequency curve with smaller absolute phase values is, the easier would be the integration of the corresponding transition in the phase delay arms of the matrix for wideband fixed phase shift.

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modifying the effective length of the outer SIW. Upon determining the proper substrate thickness, the two-level transition is integrated to the SIW arms and then optimized together for wideband relative phase shift with respect to the matrix inner reference SIW arms to fit the overall geometrical design. This can be given by phase shift

(9)

Fig. 5. Transmission coefficient versus frequency for the two-layer transition : ,h : , of Fig. 4 for different substrate heights, h h : ,h : . (a) Magnitude of S . (b) Phase of S .

= 1 524 mm

= 3 05 mm

= 0 508 mm

= 0 787 mm

TABLE I TRANSVERSE SLOT L AND C VALUES DETERMINED UPON SIMULATION OF THE PARALLEL WAVEGUIDE TRANSVERSE SLOT BROAD WALL STRUCTURE

V. DEVELOPED 4

and are the effective widths of the rectanwhere gular waveguides equivalent to the SIW outer and inner arms is the phase delay of the transition respectively. transverse slot which is the argument of (7). and are the effective path lengths of the outer and inner SIW arms, respectively. As mentioned before, two different 4 4 wideband Butler matrix configurations have been designed and optimized to operate in the 11–14 GHz frequency band. The substrate used and the SIW parameters are those mentioned in Section II. The coupler parameters are those of Section III. The SIW widths of the outer arms of the phase shifters of configuration 1 and configuration 2 will be denoted by and , respectively. will be fixed for both conThe width of the inner arms figurations to the same value as that of the employed coupler, . Both configurations are described, therefore illustrating their corresponding simulation and measurement results in the following subsections. Simulation results account only for the central part of the matrix, whose length and width and respectively, the microstrip tranwill be denoted by sitions are not considered either.

4 BUTLER MATRIX

In this section, the previously designed components are combined to form the foreseen two-layer 4 4 Butler matrix. Seeking wideband performance for both magnitude and phase implies wideband characteristics for all matrix components. The phase shifter becomes therefore a crucial component in the matrix in order to provide the required wideband phase shift with low insertion loss. Referring to Fig. 1, the required phase delay can be achieved by properly increasing the SIW width of the outer arms with respect to that of the inner ones, denoted by . The phase curves of both arms will maintain almost the same slope over a wide frequency range [17], [24]. However, as the twolevel transition is integrated within the former SIW arms, additional frequency-dependent phase delay with a different slope due to the transverse slot will be added to their transmission phases degrading therefore the bandwidth of the relative phase shift. Owing to the phase characteristics of the transition studied in the previous section, the frequency dependent phase delay of the employed slot for a specific substrate thickness (0.508 mm in our case) and slot dimensions can be compensated by properly

A. Configuration 1 Owing to the two-layer design, this configuration is a direct two-layer mapping of the block diagram of Fig. 1. Fig. 6 illustrates the developed layout of the overall matrix while Fig. 7(a) shows that of the employed SIW phase shifters. Exploiting the two-layer topology, the proposed design of the Butler matrix ensures a geometrical flexibility by exploring the lateral dimension on each layer while keeping the same physical longitudinal length for the phase shifting arms. Fig. 7(b) shows the simulated and measured results for the insertion loss of the employed phase shifting arms with the asso. Simulated results show a wideband ciated phase difference performance for the relative phase shift over the 11–14 GHz frequency band with a phase peak-to-peak error of 10 with respect to the required 45 phase difference. The optimized simulated insertion loss is kept below 0.6 dB for both arms over the entire frequency band. The additional insertion loss observed for the outer arm with respect to that of the inner one is due to the presence of the two-level transition in the former one. Measured results are in good agreement with the simulated ones for both

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2

Fig. 6. Complete layout of the developed configuration 1 for the 4 4 two-layer SIW Butler matrix, including phase-compensated (at the out:  , puts) microstrip to SIW transitions. L W : mm  ,L : ,L : , : ,W : , : . is the L waveguide wavelength at 12.5 GHz and is equal to 25.8 mm.

= 77 2 ( 3 j ) = 9 11 mm = 0 5 mm

= 76 67 mm( 3 j ) = 10 2 mm = 30 6 mm = 0 5 mm j

Fig. 8. Scattering parameters versus frequency for the matrix configuration 1 for feeding from port 1. (a) Simulated (left axis) and measured (right axis) coupling magnitudes. (b) Simulated and measured phase characteristics at the output ports with respect to that of port 5.

Fig. 9 Simulated and measured results for the isolation and reflection magnitudes versus frequency when feeding from port 1 for configuration 1. Fig. 7. (a) Layout of the SIW phase shifting arms employed in configuration 1. L : : : : ,L ,L ,L , : : : : L ,a ,a , . (b) Simulated and measured insertion loss for the inner and outer arms together with the corresponding simulated and measured phase difference 1 versus frequency.

= 51 2 mm = 20 4 mm

= 10 4 mm = 10 4 mm

= 19 32 mm = 5 84 mm = 10 2 mm = 1 06 mm

magnitude and phase. The slight increase observed in the measured insertion loss is due to the dispersive dielectric loss whose value was considered constant in the simulations. Fig. 8 shows the simulated and measured results for transmission phase and magnitude parameters versus frequency when port 1 is fed. The wideband performance is clearly verified over the 11–14 GHz frequency with a good agreement between sim-

ulations and measurements. The maximum dispersion in the simulated transmission magnitudes is less than 1 dB, with an average value of 6.5 dB. Measured transmission magnitudes have almost the same dispersion characteristic as the simulated ones with an average value around 7 dB. Simulated and measured phase characteristics have peak-to peak dispersions less than 7 and 10 , respectively, with respect to the required theoretical scattering parameters given by (1). The simulated reflection and isolation losses are below 15 dB over the entire frequency band, Fig. 9. The measured reflection coefficient is lower than 12 dB while the isolation coefficients are below 15 dB over the entire band.

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2 = 83 18 mm( 3 2 j ) = 9 15 mm

Fig. 11. Complete layout of the developed 4 4 two-layer SIW Butler matrix, configuration 2, including microstrip to SIW transitions. Layout of the : : , employed SIW phase shifting arms, L W : : , L : , W , : . :

= 36 25 mm( 1 4 j = 0 25 mm 0 5 mm

) =

Fig. 10 Scattering parameters versus frequency for the matrix configuration 1 for feeding from port 2. (a) Simulated (left axis) and measured (right axis) coupling magnitudes. (b) Simulated and measured phase characteristics at the output ports with respect to that of port 6.

Due to the symmetrical structure of the used coupler and the geometry of the matrix, see Fig. 6, the transmission dispersion characteristics as well as the reflection and isolation losses discussed above are expected to be almost the same when feeding from the other ports. This is verified from the results of Fig. 10 which also show good agreement between the simulated and measured results. A slight increase in the measured insertion loss is observed for both cases (feed at port 1 and feed at port 2). This is attributed to the dispersive substrate losses as well as possible minor misalignment between the two layers. B. Configuration 2 Fig. 11 illustrates the proposed layout for configuration 2. In this configuration, a further improvement in the matrix layout is achieved by the implementation of common SIW walls between the adjacent hybrid couplers and parts of the SIW phase shifters. As they have the same SIW width of that of the hybrids, the central part of the inner SIW phase shifting arms are in fact a replica in the lateral diof the coupler waveguides shifted by rection. This allows a two-layer overlapped implementation of these arms, each in a separate layer. The outer arms are slightly shifted by the distance for low insertion loss, relative wideband phase shift, and share one of the SIW walls of the inner arms. The overall width is thus reduced to half its value leading to a size reduction of almost 50% with respect to configuration 1. The two-level transition parameters are shown in the insets in Fig. 11. Fig. 12(a) shows the layout of the employed phase shifting arms, while Fig.12 (b) presents their corresponding simulated

Fig. 12. (a) Layout of the SIW phase shifting arms employed in configuration 2. L : ,a : , : . (b) Simulated and measured insertion loss for the inner and outer arms together with the corresponding simulated and measured phase difference 1 versus frequency.

= 51 2 mm

= 10 44 mm

= 1 53 mm

and measured results for the insertion loss and the associated phase difference . Fig. 13 and Fig. 15 show the simulated and measured results for transmission phase and magnitude parameters versus frequency when feeding from ports 1 and 2, respectively. The simulated and measured reflection and isolation parameters are shown in Fig. 14. By considering the results of Figs. 12(b)to Fig. 15, the wideband behavior is clearly verified with a good agreement between simulation and measurement results.

ALI et al.: DESIGN AND IMPLEMENTATION OF TWO-LAYER COMPACT WIDEBAND BUTLER MATRICES IN SIW TECHNOLOGY

Fig. 13. Scattering parameters versus frequency for the matrix configuration 2 for feeding from port 1. (a) Simulated (left axis) and measured (right axis) coupling magnitudes. (b) Simulated and measured phase at the output ports with respect to that of port 5.

Fig. 14. Simulated and measured results for the isolation and reflection magnitudes versus frequency when feeding from port 1, configuration 2.

As it can be expected, the matrix performances are quite similar to those of configuration 1. A slight enhancement ( 0.2 dB) in the measured insertion losses for this configuration is generally observed compared to those of the former one validating the losses introduced due to slight misalignment between the two different layers of the matrix. For both matrix configurations, the two layers have been fabricated separately. The coupling slots for the 3-dB coupler and the two-layer transitions have been etched in one layer only while corresponding larger windows have been etched in the other substrate layer to reduce losses due to alignment inaccuracies which is illustrated in Fig. 16 for the case of configuration 2.

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Fig. 15. Scattering parameters versus frequency for the matrix configuration 2 for feeding from port 2. (a) Simulated (left axis) and measured (right axis) coupling magnitudes. (b) Simulated and measured phase characteristics at the output ports with respect to that of port 6.

Fig. 16. Photograph of the common layer between the upper and lower substrates for the matrix Configuration 2.

To avoid the use of extra adhesive layers, copper eyelets (hollow pins) are used to attach the two layers. VI. USE OF THE DEVELOPED BUTLER MATRIX TO FEED A LINEAR ANTENNA ARRAY In this section, the developed 4 4 Butler matrix is used as a feeding network for an antenna array to investigate the impact of the phase and magnitude dispersions on the radiation pattern over the entire frequency band. The measured results of configuration 2 have been adopted to feed a theoretical 4-element linear at array. The array inter-element spacing has been set to

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Fig. 17. Fabricated matrices. (a) Configuration 1 and (b) Configuration 2.

the design center frequency (12.5 GHz), while a small, low directivity antenna was selected as the elementary radiator. The radiation pattern of the adopted array element is defined analytis the observation angle with respect to the ically by normal direction of the array plane, with set to 1.2. This antenna type has been chosen with the aforementioned value of in order to reduce the grating lobes especially for those beams that have the highest values of , while maintaining realistic values. Fig. 18 shows the calculated array radiation patterns versus frequency for the different input ports. According to (1) and (2) and the value of , the theoretical pointing angles of the main , , beam produced at 12.5 GHz are and when feeding from ports 1 to 4, respectively. The calculated diagrams are therefore in good agreement with the theoretical ones at the design center frequency with values of , , and when feeding at of ports 1 to 4 respectively. On the other hand, as it could be expected owing to the wideband phase performance of the developed matrix, a relatively small main lobe beam squint less than 5 is observed over the 11.5–13.5 GHz frequency band. Interestingly, the natural beam squint of a linear array, due to the fact that the inter-element spacing normalized to the wavelength varies with the frequency, is of the same order. This means that the phase dispersion of the matrix over the operating

Fig. 18. Calculated radiation patterns versus frequency of a 4-element linear array fed by the developed Butler matrix, for different input ports. (a) Port 1. (b) Port 2. (c) Port 3. (d) Port 4. Dashed line: 11.5 GHz, solid line: 12.5 GHz and dotted line: 13.5 GHz.

bandwidth has almost no impact on the beam squint. The side lobe levels (SLL) remain around 10 dB. Typical value for a uniform magnitude array distribution is around 13 dB on the array factor. The element factor of the aforementioned array element has naturally an impact on the overall radiation pattern as it tends to degrade the side lobe levels of the beams with larger . This explains the SLL values between 10 and 9 dB of Fig. 18(b) and (c). One can also note the existence of grating lobes in Fig. 18(b) and (c). The levels of grating lobes can be lowered by reducing the inter-element spacing down to but this is limited by the elementary radiator size and the coupling between elements. A preferred solution to further reduce

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the grating lobes would be to increase the directivity of the elementary radiator, using stacked patch to keep similar in-plane dimensions but this would affect the SLL. VII. CONCLUSION A novel compact two-layer Butler matrix has been presented in SIW technology. Two 4 4 prototypes have been designed, fabricated and measured. A compact design has been also validated, exploiting the flexibility of the two-layer SIW technology. The developed matrices have shown good dispersion characteristics in terms of insertion, reflection, isolation losses and phase characteristics over 24% bandwidth centered at 12.5 GHz. The two proposed designs have similar performances emphasizing that high component density is acceptable for small size subsystems, with limited impact on isolation between electrical paths. Furthermore, the proposed implementation offers a robust, compact design, suitable for larger matrices. Future work will focus on further size reduction exploiting the multi-layer SIW as well as the design of larger matrices and integrated antenna arrays.

[14] M. Bona, L. Manholm, J. P. Starski, and B. Svensson, “Low-loss compact Butler matrix for a microstrip antenna,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 9, pp. 2069–2075, Sep. 2002. [15] M. Nedil, T. A. Denidi, and L. Talbi, “Novel Butler matrix using CPW multilayer technology,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 1, pp. 499–507, Jan. 2006. [16] D. Deslandes and K. Wu, “Integrated microstrip and rectangular waveguide in planar form,” IEEE Microw. Wireless Comp. Lett., vol. 11, pp. 68–70, Feb. 2001. [17] E. Sbarra, L. Marcaccioli, R. V. Gatti, and R. Sorrentino, “A novel Rotman lense in SIW technology,” in Proc. Eur. Microw. Conf., Munich, Oct. 2007, pp. 1515–1518. [18] Y. Cheng, W. Hong, K. Wu, Z. Q. Kuai, C. Yu, J. X. Chen, J. Y. Zhou, and H. J. Tang, “Substrate integrated waveguide (SIW) Rotman lens and its Ka-band multibeam array antenna applications,” IEEE Trans. Antennas Propag., vol. 56, pp. 2504–2513, 2008. [19] T. Djerafi, N. Fonseca, and K. Wu, “Architecture and implementation of planar 4 4 Ku-band Nolen matrix using SIW technology,” presented at the Asia Pacific Microw. Conf., Hong Kong, Dec. 2008. [20] S. Yamto, J. Hirokawa, and M. Ando, “A beam switching slot array with a 4-way Butler matrix installed in a single layer post-wall waveguide,” in Proc. IEEE AP-S Int. Symp., San Antonio, TX, Jun. 2002, pp. 138–141. [21] I. A. Eshrah, A. A. Kishk, A. B. Yakovlev, A. W. Glisson, and C. E. Smith, “Analysis of waveguide slot-based structures using-wideband equivalent circuit model,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 12, pp. 2691–2696, Dec. 2004. [22] A. Ali, N. Fonseca, F. Coccetti, and H. Aubert, “Novel two-layer broadband 4 4 Butler matrix in SIW technology for Ku-band applications,” in Proc. Asia Pacific Microw. Conf., Hong Kong, Dec. 2008, pp. 1–4. [23] A. Ali, H. Aubert, N. Fonseca, and F. Cocetti, “Wideband two-layer SIW coupler: Design and experiment,” Electron. Lett., vol. 45, no. 13, pp. 687–689, Jun. 2009. [24] Y. Cheng, W. Hong, and K. Wu, “Novel substrate integrated waveguide fixed phase shifter for 180-degree directional coupler,” in Proc. IEEE MTT Symp., Honolulu, HI, Jun. 2007, pp. 189–192. [25] P. A. Rizzi, Microwave Engineering: Passive Circuits. Englewood Cliffs, NJ: Prentice-Hall, 1988.

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ACKNOWLEDGMENT The authors would like to thank Prof. K. Wu for providing the fabrication facilities and T. Djerafi for his help in the fabrication process. REFERENCES [1] P. S. Hall and S. J. Vetterlein, “Review of radio frequency beamforming techniques for scanned and multiple beam antennas,” IEE Proc., vol. 137, no. 5, pp. 293–303, 1990. [2] A. El-Zooghby, “Potentials of smart antennas in CDMA systems and uplink improvements,” IEEE Antennas Propag. Mag., vol. 43, no. 5, pp. 172–177, Oct. 2001. [3] B. Piovano, L. Accatino, A. Angelucci, T. Jones, P. Capece, and M. N Butler maVotta, “Design and breadboarding of wideband N trices for multiport amplifiers,” in Proc. Microw. Conf., Brazil, 1993, pp. 175–180, SBMO. [4] J. Blass, “Multi-directional antenna: New approach top stacked beams,” in IRE Int. Convention Record, 1960, pp. 48–50, Pt 1. [5] J. Nolen, “Synthesis of multiple beam networks for arbitrary illuminations,” Ph.D. dissertation, Bendix Corporation, Radio Division, Baltimore, MD, Apr. 1965. [6] W. Rotman and R. Tuner, “Wide-angle microwave lens for line source applications,” IEEE Trans. Antennas Propag., vol. 11, pp. 623–632, 1963. [7] J. Butler and R. Lowe, “Beam-forming matrix simplifies design of electrically scanned antennas,” Electron Design, pp. 170–173, 1961. [8] W. White, “Pattern limitations in multiple-beam antennas,” IRE Trans. Antennas Propag., vol. 10, pp. 430–436, Jul. 1962. [9] C. Dall’Omo, T. Monediere, B. Jecko, F. Lamour, I. Wolk, and M. Elkael, “Design and realization of a 4 4 microstrip Butler matrix without any crossing in millimeter waves,” Microw. Opt. Tech. Lett., vol. 38, no. 6, pp. 462–465, Sep. 2003. [10] H. Hayashi, D. A. Hitko, and C. G. Sodini, “Four-element planar Butler matrix using half-wavelength open stubs,” IEEE Microw. Wireless Comp. Lett., vol. 12, no. 3, Mar. 2002. [11] F. Alessandri, M. Dionigi, and L. Tarricone, “Rigorous and efficient fabrication-oriented CAD and optimization of complex waveguide networks,” IEEE Trans. Microw. Theory Tech., vol. 45, no. 12, pp. 2366–2374, Dec. 1997. [12] T. N. Kaifas and J. Sahalos, “On the design of a single-layer wideband Butler matrix for switched-beam UMTS system applications,” IEEE Trans. Antennas Propag., vol. 48, pp. 193–204, Dec. 2006. [13] J. Hirokawa, F. Murukawa, K. Tsunekawa, and N. Goto, “Double-layer structure of rectangular-waveguides for Butler matrix,” in Proc. Eur. Microw. Conf., Milan, Oct. 2002, pp. 1–4.

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Ahmed Ali Mohamed Ali was born in Cairo, Egypt, in January 1981. He received the B.Sc. and the M.Sc. degrees in electrical engineering from Cairo University, in 2003 and 2005, respectively, and the Ph.D. degree from the University of Toulouse, Toulouse, France and the LAAS-CNRS, in 2010. From 2003 to 2007, he worked as a Teaching Assistant and then as a Lecturer Assistant at the French University in Egypt, El-Shorouk. In 2007, he joined the Laboratory of Analysis and Architecture of Systems, CNRS (LAAS-CNRS), Toulouse, France, where he conducted research towards his Ph.D. Since 2010, he has been working at the LAAS-CNRS as a Postdoctoral Research Engineer. His current research interests include substrate integrated waveguide technology, microwave circuits and subsystems, beamforming matrices, antenna design, antenna miniaturization techniques and composite right/left handed propagation.

Nelson J. G. Fonseca (M’06–SM’09) received the Electrical Engineering degree from Ecole Nationale Supérieure d’Electrotechnique, Electronique, Informatique, Hydraulique et Telecommunications (ENSEEIHT), Toulouse, France and the Master degree from the Ecole Polytechnique de Montreal, Quebec, Canada, both in 2003. He is currently working toward the Ph.D. degree at the Université de Toulouse—Institut National Polytechnique de Toulouse, France. He works at the Antenna and Sub-Millimetre Wave Section, European Space Agency (ESA), Noordwijk, The Netherlands, since 2009. His interests cover the telecommunication antennas, beam forming network designs, and new enabling technologies such as fractals, metamaterials and membranes applied to antenna design. He has authored or coauthored more than 70 papers in journals and conferences, including two CNES Technical Notes. He holds two patents and has eight patents pending.

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Mr. Fonseca received the Best Young Engineer Award at the 29th ESA Workshop on Antennas in 2007. He was also co-recipient of the Best Application Paper Award at the 30th ESA Workshop on Antennas in 2008.

Fabio Coccetti (M’10) received the Laurea (M.S.) degree in electrical engineering from the University of Perugia, Perugia, Italy and the Ph.D. degree from the Technische Universität München, Germany, in 1999 and 2004, respectively. From February to July 2000, he was a Visiting Scientist at the University of Michigan, Ann Arbor. Since September 2004, he has been working as a Research Scientist at LAAS-CNRS, Toulouse, France. During 2005 to 2007, he was the Coordinator of the EU NoE-AMICOM. His research interests include computational electromagnetic modeling and optimization and RF-micro and nanosystems technology and characterization for reconfigurable microwave and millimeterwave circuits.

Hervé Aubert (SM’99) was born in Toulouse, France, in July 1966. He received the Eng. Dipl. and the Ph.D. degree (with high honors) both in electrical engineering from the Institut National Polytechnique (INPT), Toulouse, France, in July 1989 and January 1993, respectively. Since February 2001, he has been a Professor at INPT. In February 2006, he joined the Laboratoire d’Analyse et d’Architecture des Systèmes, National Center for Scientific Research, Toulouse, France. From April 1997 to March 1998, he was a Visiting Associate Professor at the School of Engineering and Applied Science, University of Pennsylvania, Philadelphia. He is a contributor to the books Fractals: Theory and Applications in Engineering (Springer, 1999), Micromachined Microwave Devices and Circuits (Romanian Academy Edition, 2002), and New Trends and Concepts in Microwave Theory and Techniques (Research Signpost, 2003). He has authored or coauthored one book, 44 papers in refereed journals and over 120 communications in international conferences. He holds four international patents in the area of antennas. Dr. Aubert is the Secretary of the IEEE Antennas and Propagation French Chapter and was the Vice Chairman of this Chapter from 2004 to 2009 and Secretary from 2001 to 2004. He is a member of URSI Commission B.

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A Self-Tuning Electromagnetic Shutter Raoul O. Ouedraogo, Student Member, IEEE, Edward J. Rothwell, Fellow, IEEE, Shih-Yuan Chen, Senior Member, IEEE, and Andrew Temme, Student Member, IEEE

Abstract—A self-tuning electromagnetic shutter is introduced, consisting of a slotted metallic surface with computer controlled switches placed across the slots. By opening and closing the switches, the transmissivity of the surface may be adjusted over a broad range of frequencies. In particular, the surface may be placed into open and closed states, creating an electronically-controllable shutter. The ability of the shutter to act as an open or a closed surface over a broad range of frequencies, incidence angles, and polarization states is investigated using simulations and measurements, and the feasibility of the system is thereby demonstrated. Index Terms—Electromagnetic scattering, frequency selective surfaces (FSSs), genetic algorithms (GAs), slot antennas, tunable filters.

I. INTRODUCTION

T

RADITIONAL frequency selective surfaces (FSSs) used as electromagnetic filters are periodic planar structures composed of metallic unit cells usually backed by one or several dielectric layers [1]. Several variables, such as the shape, spacing and orientation of the unit cells, and the properties of the dielectric, must be simultaneously adjusted during the design process to achieve acceptable stopband or passband characteristics. Although traditional FSSs exhibit a single, narrow stopband or passband characteristic, multiband FSSs have been demonstrated [2]–[10] that use multiply resonant structures such as fractals, or multiple layers of elements, with two or more FSS screens backed by dielectric layers. The design costs for these structures is significant, requiring a time consuming trial and error process in simultaneously tuning the elements of the periodic surface and the characteristics of the dielectric. To reduce design costs, nature based search algorithms such as genetic algorithms (GAs) have been used to synthesize FSSs [11]–[15], but the designs are fixed and thus lack the ability to adapt to changing conditions such as orientation or coupling with nearby structures. Several researchers [16]–[23] have proposed reconfigurable FSSs that use lumped elements and MEMS switches that can be placed into a number of preconfigured states, and thus adjust for changing conditions such as operational frequency. Unfortunately, these FSSs are based on a periodic array of identical unit cells much larger than a wavelength in extent, with a large Manuscript received March 26, 2010; revised June 22, 2010; accepted June 24, 2010. Date of publication December 03, 2010; date of current version February 02, 2011. E. J. Rothwell, R. O. Ouedraogo, and A. Temme are with the Department of Electrical and Computer Engineering, Michigan State University, East Lansing, MI 48824 USA (e-mail: [email protected]). S.-Y. Chen is with the Department of Electrical Engineering, National Taiwan University, Taipei, 10617 Taiwan (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2010.2096401

number of switches interspersed between elements, and suffer from a performance that is highly dependent on the polarization and arrival angle of the incident field. Importantly, they cannot respond to an environment that changes in an unpredictable way. In this paper the authors propose an adjustable surface based on the underlying concepts of both reconfigurable FSSs and self-structuring antennas [24]–[31], but with a much simpler operating mission. By using a feedback system, the surface is adjusted to be either transparent or opaque to incident electromagnetic waves, and thus can be viewed as an “electromagnetic shutter.” When placed into an open state, electromagnetic waves pass through the surface with relatively little attenuation; in the closed state the surface effectively blocks the penetration of the waves. The shutter comprises a slotted metallic surface with computer-controllable switches placed across the slots at strategic points. The size of the surface can be smaller than a half wavelength on a side, and the number of switches is much less than the number of nodes in a traditional FSS. A sensor is used to determine the field passed through the surface and an efficient binary search algorithm is used to shift through the large number of possible configurations to either minimize field penetration (producing a closed state) or maximize penetration (producing an open state). The shutter geometry is chosen so as to provide a large number of possible configurations not dependent on any particular operational requirements, and thus the shutter can respond to a wide range of environmental conditions, such as the polarization and arrival angle of the incident field, or coupling to nearby structures, over a large tunable bandwidth. A proof of concept is performed in simulations using the numerical electromagnetic code (NEC4), and practical performance is investigated using a prototype shutter. Results demonstrate that the controllable surface can be used effectively as a tunable EM shutter over a wide range of frequencies, arrival angles, and polarization states. The shutter should prove useful in conditions where sensitive sensors must be shielded from intermittent incident fields of extreme strength, or when it is desired to hide a sensor from probing fields. II. SHUTTER DESIGN The proposed shutter is a slotted metallic surface with computer controlled switches placed across the slots. Due to the binary nature of the problem (each switch is either on or off), the , where number of possible switch configurations is equal to represents the number of switches. The shutter described in or more than 4 billion this paper uses 32 switches leading to different switch configurations. Since the electrical characteristics of the shutter depend greatly on the switch configuration, a modest number of switches can create a huge diversity of shutter characteristics. However, the large number of possible switch

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Fig. 2. Shutter simulation setup. Fig. 1. Top view of the simulated shutter template with numbered switches.

configurations requires using a robust and efficient search algorithm capable of finding suitable characteristics in a reasonable number of looks. A GA is used in this work since it has been demonstrated that nature based search algorithms such as GAs are appropriate for similar binary problems encountered in self-structuring antenna systems. To determine the performance of the shutter for a given switch configuration, the device is used to seal the opening of a conducting box containing one or more monopole probes. The probes are used to measure the strength of the field that passes through the shutter into the box, and the performance of the shutter is determined by comparing this field to the value measured when the shutter is removed so that the top is completely open. Simulations of the conducting box, including the monopole probe and shutter, were used to establish proof-of-principle before constructing a prototype, and to explore the ability of a GA to find appropriate switch configurations. All simulations were performed using the Numerical Electromagnetics Code (NEC4) [32]. Although more sophisticated commercial simulation packages such as HFSS, FEKO or CST could have been used, NEC4 was chosen because of its numerical Green’s function option that enables the user to model a specific structure and save the interaction matrix to a file. When a change is made to the structure, only the interactions with the altered portions need to be recomputed. Thus the interactions between the shutter, conducting box, and probe need not be recomputed as the configurations change, and only the interactions with the wires containing the switches are required at each step in the optimization process. This leads to a significant reduction of the total simulation time. For the model used in this work, the simulation time for each configuration is reduced by a factor of five when using the numerical Green’s function (all simulations were performed on a computer with a 2 GHz processor and 2 Gb of RAM). The shutter template, shown in Fig. 1, is a square surface at the lowest freof side length 27.3 cm (approximately quency of operation) made of 12 conducting patches interconnected with 32 switches, with the gaps between the patches creating the desired slots. The conducting patches were modeled in

NEC4 using a wire grid model determined by the equal area rule [33]–[38]. Each cell of the wire grid is a square with a segment length of 9.75 mm and a wire radius of 1.9 mm. The width of the slots was chosen to be 9.75 mm since this dimension corresponds to the width of the switches used for the prototype discussed in Section IV. The number and specific geometry of the conducting patches, which determine the shape of the slots, are not crucial provided the final design produces a wide diversity of configurations and the switches are spread out over the entire surface of the template. To avoid the occurrence of identical switch configurations, the switches are placed in an asymmetric manner on the surface. Such an approach has proven successful with self-structuring antennas and tunable resonator designs. See [24]–[31]. To determine the performance of the shutter, a cubical conducting open box of side length 27.3 cm with a wire segment length of 9.75 mm and a wire radius of 1.9 mm is used (producing a grid cell size identical to that used to model the shutter). A single probe is used to measure the field inside the box. The probe was chosen to be 9 cm long with a radius of 0.56 mm, and is loaded with a 50 resistor representing the receiver impedance. The probe is connected perpendicularly to the middle of a side wall of the box as shown in Fig. 2. To determine the ability of the shutter to provide an open or closed surface, the box is first excited by an incident plane wave of unit amplitude with the shutter removed and the top of the box open, and the current induced in the loaded segment of the probe, , is recorded. The shutter is then used to seal the opening of the box and the structure is again excited using the same plane wave. With the current induced on the probe defined as , the shutter effectiveness is computed using (1) and a closed Then, an open surface is sought by maximizing surface by minimizing . The genetic algorithm tool GA-NEC [39] was used to perform all optimizations. III. SIMULATION RESULTS The ability of the shutter to effectively create an open or closed surface was evaluated at five different frequencies from

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TABLE I GA PARAMETERS USED IN THE SIMULATIONS

TABLE II OPTIMIZED SHUTTER EFFECTIVENESS FOR NORMAL PLANE WAVE INCIDENCE DETERMINED FROM SIMULATIONS

Fig. 3. Frequency sweep of the shutter optimized at 625 MHz. Normal incidence. TABLE III OPTIMIZED SHUTTER EFFECTIVENESS FOR OBLIQUE PLANE WAVE INCIDENCE DETERMINED FROM SIMULATIONS

625 MHz to 725 MHz with an increment of 25 MHz. Excitation is provided by a plane wave under both normal and oblique incidence. The characteristics of the plane wave are described following the drawing in Fig. 2. The direction of the incoming wave vector is given by and and the polarization is given by the angle between the vectors and . ( and are in a plane normal to ). Under normal incidence the wave vector , ), while under oblique incidence has angles ( , ). In each case the the angles are chosen to be ( incident field is polarization matched to the probe. The shutter is optimized to create an open or closed surface using GA-NEC with the GA parameters given in Table I. To seek a switch configuration capable of creating a closed surface, the fitness function of the GA is set to minimize the current on the probe while an open surface is sought by setting the fitness function to maximize the current on the probe. Once an acceptable configuration is found, the shutter effectiveness as a function of frequency is computed with the switches fixed in position. Results for the optimized shutter at the five chosen frequencies are shown in Tables II and III for normal and oblique plane wave incidence, respectively. It can be seen from Table II that a is obtained at each shutter effectiveness no greater than frequency when the shutter is optimized to create a closed surface under normal illumination. A value no greater than is obtained at 625, 650 and 700 MHz. This represents a reduction of at least three orders of magnitude compared to the open box, attesting to the ability of the shutter to effectively seal the box from incoming waves. Optimization of the shutter to create an open surface returned values of 0 dB or higher at all frequencies except 675 MHz, where a shutter effectiveness of was obtained. A shutter effectiveness greater than 0 dB implies

Fig. 4. Frequency sweep of the shutter optimized at 625 MHz. Oblique incidence.

that the presence of the shutter serves to focus the fields in the region where the probe is located. Similar results are obtained for oblique incidence of the plane wave. Under oblique incidence, the optimizer was able to find switch configurations that produce open states with significantly higher probe currents than present when the box is uncovered. A frequency sweep of the shutter optimized at 625 MHz is shown in Figs. 3 and 4 for normal and oblique incidences, respectively. It can be seen from these figures that an open shutter operates effectively across a wide instantaneous band, but that a closed shutter has a fairly narrow instantaneous bandwidth. Of course, both the open and closed shutters have a wide tunable bandwidth as shown in Tables II and III. Extensive simulations where not undertaken for incidence angles beyond 60 . However, preliminary results suggest that acceptable performance should be possible for strongly oblique incidence, although some performance degradation is expected near grazing for the closed shutter configuration. For instance, an optimization performed at 625 MHz for an incidence direc, produced a closed tion characterized by

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TABLE IV LOCATIONS OF THE FOUR PROBES USED IN THE EXPERIMENT TO MEASURE THE FIELD INTERNAL TO THE BOX. POSITIONS ARE RELATIVE TO THE COORDINATE SYSTEM SHOWN IN FIG. 5. ALL POSITIONS ARE IN MM

shutter effectiveness of and an open shutter effectiveness of 16.42 dB. Comparing these results with those ob, ), it can be seen tained for normal incidence ( that the closed shutter effectiveness is somewhat degraded (from to ) while the open shutter effectiveness is enhanced (from 1.23 dB to 16.42 dB). Similar observations can be made by comparing the results from Tables II and III. The closed shutter effectiveness, though degraded, is maintained . below the target value of IV. PROTOTYPE AND EXPERIMENT A prototype was fabricated to investigate the practical implementation of the EM shutter concept. The template of the prototype shutter was fabricated using a single sided FR4-epoxy copper clad circuit board of sides 27.1 cm by 27.1 cm. Slots of width 9.75 mm were milled into the conducting surface of the board to produce a pattern of 12 conducting patches identical to the simulation model shown in Fig. 1. The switches used in the experiment are COTO single-inline-package reed relay switches (series 9011-05-10). The two outer pins of each switch are soldered across a slot to provide the electrical connection between the patches, while the two inner pins are connected to a control board through a cable assembly. A six-inch (152 mm) 64-line ribbon cable connects the control board to a desktop computer. The control board serves as interface between the shutter and the computer and is needed to provide the necessary current to drive the switches. As a means to measure the shutter effectiveness, a cubical open box of side length 27.1 cm was built from aluminum plates of thickness 1.57 mm. In the simulations the field within the box was sampled by a single probe placed in the center of one side of the box, polarization-aligned to the incident field. A closed shutter was then established by minimizing the current on the probe. A problem arises since this approach does not guarantee that the field everywhere inside the box is small. Thus in the experiments four orthogonal probes were used and the probe responses combined using a four-port power combiner. An open or a closed shutter was then established by optimizing the combined voltage received by the four probes. The probes have a length of 9 cm and a radius of 0.5 mm, and their positions are given in Table IV. A diagram of the measurement system is shown in Fig. 5. The box is placed in an anechoic chamber and illuminated using a wideband TEM horn with a bandwidth of 0.5–6 GHz. The operating band of the power combiner (500–800 MHz) presents a practical limitation on the measurement band explored during the experiments. Optimizations to establish open and closed shutter configurations were performed at an incidence of ( , ), with vertical ( ), horizontal ( ) and

Fig. 5. Shutter measurement setup diagram.

skewed ( ) polarizations of the incident field at frequencies of 500, 600, 700 and 800 MHz. The shutter effectiveness was computed in a manner similar to that in (1), but using the ratio of the measured voltage with the shutter present to the measured voltage with the shutter absent (at identical polarizations of the incident wave). The genetic algorithm used to optimize the prototype shutter is fairly standard. The shutter effectiveness is used as the fitness function in the optimization, and is evaluated for each of the 200 randomly selected switch configurations in the initial population. Once the fitness evaluation of the initial population is completed, the best 20% are selected and a two point crossover and single bit mutation are performed on the selected 20% until a new population of 200 switch configurations is created. This process is repeated until the stopping criteria is reached. If the goal of the optimization is to create an open surface, then the optimization is stopped when a fitness value of 0 dB or higher is obtained. If the goal of the optimization is to create a closed surface, then the optimization is stopped when a fitness value or lower is obtained. In both cases the maximum of number of generations was set to 100. Results obtained for both open and closed surface optimizations are shown in Fig. 6. Analysis of this plot shows that for a shutter optimized to provide a closed surface, at each frequency (except at 500 MHz for the skew polarization case) values of or lower are observed for all three polarizations. This suggests that the prototype shutter is effective at reducing the fields within the box over a wide range of frequencies and polarizations of the incident wave. With the shutter optimized to create an open surface, the shutter effectivenesses obtained are of the 0 dB line except for a value of within at 600 MHz (with vertical polarization) and 3.36 dB at 700 MHz (with skew polarization). To test the sensitivity of the shutter effectiveness to probe position, optimizations were done with the probes moved to other locations within the box and results very similar to those shown in Fig. 6 were observed. The closed shutter effectiveness was while the open shutter effectiveness was mostly below of the 0 dB line. found to usually be within

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Fig. 6. Measured shutter effectiveness for optimization at various frequencies and polarizations.

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Fig. 8. Measured instantaneous shutter effectiveness for the shutter optimized at 600 MHz. Horizontal polarization. TABLE V BEST SWITCH CONFIGURATIONS OBTAINED AT 600 MHZ FOR HORIZONTAL AND SKEW POLARIZATION

Fig. 7. Measured instantaneous shutter effectiveness for the shutter optimized at 600 MHz. Skew polarization.

The frequency response of the shutter is different for each switch configuration. For example, sweeps of the shutter effectiveness for the best switch configurations obtained at 600 MHz are shown in Figs. 7 and 8 for skew and horizontal polarizations of the incident field, respectively. It is seen that the closed shutter effectiveness in the skew case is not as low as that in the horizontal case, but that the instantaneous bandwidth is wider in the skew case. This suggests that broadband optimizations of the shutter effectiveness should be possible, although they are not explored in this paper. The best switch configurations obtained at 600 MHz for skew and horizontal polarizations of the incident field are shown in Table V. In this table a zero represents an OFF switch state while a one represents an ON switch state. The locations of the switches on the template are shown in Fig. 1 with switch number 1 referring to the first switch in the Table V entries and switch number 32 referring to the last switch in the Table V entries. Since no obvious pattern is present in these configurations, it is doubtful that the elimination of any particular switch will allow for simplification of the template. To investigate the performance of the shutter as a function of the arrival angle of the incident field, the transmit antenna

Fig. 9. Measured shutter effectiveness for optimization at various frequencies and incidence angles.

was fixed for vertical polarization ( ) at and the and (rotation in the box was rotated through - plane). For each incidence angle the shutter was optimized at each of the frequencies 500, 600, 700, and 800 MHz. Results of the optimization for open and closed shutters are shown or in Fig. 9. For each angle a shutter effectiveness of lower was found for the closed shutter, and 0 dB or higher was obtained for the open shutter. In both the simulation and the experiment, the GA examined only a small fraction of the total number of switch configurations. With the prototype, the GA examined at most 16000 switch configurations, which represent 0.0003% of the total

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number of switch states. Moreover, in most optimizations the desired fitness values were found by the GA before all 100 generations were completed. Thus, the fitness values reported here may not be the best that can be achieved; examining more switch configurations or using a more sophisticated optimizer might lead to better results. V. CONCLUSION A self tuning electromagnetic shutter capable of producing an electromagnetically transparent or opaque surface is introduced. The shutter consists of a slotted conducting surface with computer-controlled switches extending across the slots. It is shown through simulations and through measurements of a prototype system that the switch configurations can be optimized to successfully create a closed or open shutter over a broad range of frequencies, incidence angles, and polarization states. Experimental results indicate that it may be possible to control the transmissivity of the shutter between the extremes of open and closed surfaces. This suggests that the shutter could also be used as a controllable iris, with the desired level of attenuation of the incident field determined by choosing an appropriate switch configuration. Exploration of this concept is left for future studies. ACKNOWLEDGMENT The authors are grateful to Dr. J. Ross for providing the use of the program GA-NEC. REFERENCES [1] B. A. Munk, Frequency Selective Surfaces: Theory and Design. New York: Wiley, 2000. [2] T. K. Wu, “Double-square-loop FSS for multiplexing four (S/X/Ku/Ka) bands,” in IEEE Int. Symp. on Antennas and Propagation Digest, Jun. 1991, vol. 3, pp. 1885–1888. [3] T. K. Wu and S. W. Lee, “Multiband frequency selective surface with multiring patch elements,” IEEE Trans. Antennas Propag., vol. 42, pp. 1484–1490, Dec. 1994. [4] T. K. Wu, “Four-band frequency selective surface with double-squareloop patch elements,” IEEE Trans. Antennas Propag., vol. 42, pp. 1659–1663, Dec. 1994. [5] J. Romeu and Y. Rahmat-Samii, “Dual band FSS with fractal elements,” IEEE Electron. Lett., vol. 35, no. 9, pp. 702–703, Apr. 1999. [6] J. Romeu and Y. Rahmat-Samii, “Fractal FSS: A novel dual-band frequency selective surface,” IEEE Trans. Antennas Propag., vol. 48, pp. 1097–1105, Jul. 2000. [7] J. P. Gianvittorio, J. Romeu, S. Blanch, and Y. Rahmat-Samii, “Selfsimilar prefractal frequency selective surfaces for multiband and dualpolarized applications,” IEEE Trans. Antennas Propag., vol. 51, pp. 3088–3096, Jul. 2003. [8] D. H. Kim and J. I. Choi, “Frequency selective surface for the blocking of multiple frequency bands,” in IEEE Int. Symp. on Antennas and Propagation Digest, Jul. 2006, vol. 1, pp. 4195–4198. [9] M. Lambea and J. A. Encinar, “Analysis of multilayer frequency selective surfaces with rectangular geometries,” in IEEE Int. Symp. on Antennas and Propagation Digest, Apr. 1995, vol. 1, pp. 528–531, 407. [10] C. Wan and J. A. Encinar, “Analysis of multi layered FSS by efficient computation of generalized scattering matrix,” in IEEE Int. Symp. on Antennas and Propagation Digest, Jun. 1994, vol. 3, pp. 2270–2273. [11] J. M. Johnson and Y. Rahmat-Samii, “Genetic algorithms in engineering electromagnetics,” IEEE Antennas Propag. Mag., vol. 39, no. 4, pp. 7–21, Aug. 1997. [12] G. Manara, A. Monorchio, and R. Mittra, “Frequency selective surface design based on genetic algorithm,” IEEE Electron. Lett., vol. 35, no. 17, pp. 1400–1401, Aug. 1999. [13] E. A. Parker, A. D. Chuprin, J. C. Batchelor, and S. B. Savia, “GA optimization of crossed dipole FSS array geometry,” IEEE Electron. Lett., vol. 37, no. 16, pp. 996–997, Aug. 2001. [14] Y. Yuan, C. H. Chan, K. F. Man, and R. Mittra, “A genetic algorithm approach to FSS filter design,” in IEEE Int. Symp. on Antennas and Propagation Digest, Jul. 2001, vol. 4, pp. 688–691.

[15] S. Chakravarty and R. Mittra, “Application of the micro-genetic algorithm to the design of spatial filters with frequency-selective surfaces embedded in dielectric media,” IEEE Trans.n Electromagn. Compat., vol. 44, pp. 338–346, May 2002. [16] A. C. Lima, E. A. Parker, and R. J. Langley, “Tunable frequency selective surface using liquid substrates,” IEEE Electron. Lett., vol. 30, no. 4, pp. 281–282, Feb. 1994. [17] G. Y. Li, Y. C. Chan, T. S. Mok, and J. C. Vardaxoglou, “Analysis of frequency selective surfaces on biased ferrite substrate,” in IEEE Int. Symp. on Antennas and Propagation Digest, Jun. 1995, vol. 3, pp. 1636–1639. [18] J. P. Gianvittorio, J. Zendejas, Y. Rahmat-Samii, and J. Judy, “Reconfigurable MEMS-enabled frequency selective surfaces,” IEEE Electron. Lett., vol. 38, no. 25, pp. 1627–1628, Dec. 2002. [19] C. Mias, “Frequency selective surfaces loaded with surface-mount reactive components,” IEEE Electron. Lett., vol. 39, no. 9, pp. 724–726, May 2003. [20] J. A. Bossard, D. H. Werner, T. S. Mayer, and R. P. Drupp, “A novel design methodology for reconfigurable frequency selective surfaces using genetic algorithms,” IEEE Trans. Antennas Propag., vol. 53, pp. 1390–1400, Apr. 2005. [21] J. A. Bossard, D. H. Werner, T. S. Mayer, and R. P. Drupp, “Reconfigurable infrared frequency selective surfaces,” in IEEE Int. Symp. on Antennas and Propagation Digest, Jun. 2004, vol. 2, pp. 1911–1914. [22] X. Liang, L. Li, J. A. Bossard, and D. H. Werner, “Reconfigurable frequency selective surfaces with silicon switches,” in IEEE Int. Symp. on Antennas and Propagation Digest, Jul. 2006, pp. 189–192. [23] F. Bayatpur and K. Sarabandi, “A tunable, band-pass, miniaturized-element frequency selective surface: Design and measurement,” in IEEE Int. Symp. on Antennas and Propagation Digest, Jun. 2007, pp. 3964–3967. [24] C. M. Coleman, E. J. Rothwell, and J. E. Ross, “Self-structuring antenna,” in IEEE Int. Symp. on Antennas and Propagation Digest, Jul. 2000, vol. 3, pp. 1256–1259. [25] B. T. Perry, C. M. Coleman, B. F. Basch, E. J. Rothwell, and J. E. Ross, “Self-structuring antenna for television reception,” in IEEE Int. Symp. on Antennas and Propagation Digest, Jul. 2001, vol. 1, pp. 162–165. [26] C. M. Coleman, E. J. Rothwell, J. E. Ross, and L. L. Nagy, “Selfstructuring antenna,” IEEE Antennas Propag. Mag., vol. 44, no. 3, pp. 11–23, Jun. 2002. [27] C. M. Coleman, E. J. Rothwell, and J. E. Ross, “Investigation of simulated annealing, ant-colony optimization, and genetic algorithm for self-structuring antenna,” IEEE Trans. Antennas Propag., vol. 52, pp. 1007–1014, Apr. 2004. [28] J. E. Ross, E. J. Rothwell, and S. Preschutti, “A complimentary selfstructuring antenna for use in a vehicle environment,” in IEEE Int. Symp. on Antennas and Propagation Digest, Jun. 2004, vol. 3, pp. 2321–2324. [29] B. T. Perry, E. J. Rothwell, L. L. Nagy, and J. E. Ross, “Self-structuring antenna concept for FM-band automotive backlight antenna design,” in IEEE Int. Symp. on Antennas and Propagation Digest, Jul. 2005, vol. 1B, pp. 92–95. [30] B. T. Perry, E. J. Rothwell, and L. L. Nagy, “Analysis of switch failures in a self-structuring antenna system,” IEEE Antennas Wireless Propag. Lett., vol. 4, pp. 68–70, Jun. 2005. [31] R. O. Ouedraogo, E. J. Rothwell, S.-Y. Chen, and B. J. Greetis, “An automatically tunable cavity resonator system,” IEEE Trans. Microw. Theory Tech., vol. 58, pp. 894–902, Apr. 2010. [32] G. Burke and A. Poggio, Numerical Electromagnetic Code-Method of Moment Lawrence Livermore National Laboratory, Livermore, CA, 1981, Rep. no. UCID-18834. [33] A. Sarolic, B. Modlic, and D. Poljak, “Measurement validation of ship wiregrid models of different complexity,” in Proc. IEEE Int. Symp. on Electromagnetic Compatibility, Aug. 2001, vol. 1, pp. 147–150. [34] M. McKaughan, “Coast guard applications of NEC,” in IEEE Int. Symp. on Antenna and Propagation Digest, Jun. 2004, vol. 3, pp. 2879–2882. [35] C. W. Trueman and S. J. Kubina, “Fields of complex surfaces using wire grid modeling,” IEEE Trans. Magn., vol. 27, pp. 4262–4267, Sep. 1991. [36] A. Rubinstein, M. Rubinstein, and F. Rachidi, “A physical interpretation of the equal area rule,” IEEE Trans. Electromagn. Compat. Digest, vol. 48, no. 2, pp. 258–263, Jun. 2006. [37] A. C. Ludwig, “Wire grid modeling of surfaces,” IEEE Trans. Antennas Propag., vol. 35, pp. 1045–1048, Sep. 1987. [38] A. Rubinstein, C. Rostamzadeh, M. Rubinstein, and F. Rachidi1, “On the use of the equal area rule for the wire-grid representation of metallic surfaces,” in IEEE Int. Symp. on Electromagnetic Compatibility Digest, Feb. 2006, pp. 212–215. [39] GA-Suite With NEC Analysis7.0 ed. John Ross & Associate, 2010.

OUEDRAOGO et al.: A SELF-TUNING ELECTROMAGNETIC SHUTTER

Raoul O. Ouedraogo (S’08) was born in Ouagadougou, Burkina Faso, in 1982. He received the B.Sc. degree in electrical engineering from Southern Illinois University, in 2006, and the M.Sc. in electrical engineering from Michigan State University, East Lansing, in 2008, where he is currently working toward the Ph.D. degree. His current research interests include metamaterials, small antennas, self-structuring devices, electromagnetic radiation, and scattering.

Edward J. Rothwell (S’84–M’85–SM’92–F’05) was born in Grand Rapids, MI, on September 8, 1957. He received the B.S. degree in electrical engineering from Michigan Technological University, Houghton, in 1979, the M.S. degree in electrical engineering and the degree of electrical engineer from Stanford University, Stanford, CA, in 1980 and 1982, respectively, and the Ph.D. degree in electrical engineering from Michigan State University, East Lansing, in 1985, where he held the Dean’s Distinguished Fellowship. From 1979 to 1982, he was with the Microwave and Power Tube Division, Raytheon Co., Waltham, MA, working on low power traveling wave tubes, and, in 1985, he was with MIT Lincoln Laboratory, Lexington, MA. He was at Michigan State University, East Lansing, from 1985 to 1990, as an Assistant Professor of electrical engineering, from 1990 to 1998, as an Associate Professor, and, since 1998, as a Professor. Prof. Rothwell received the John D. Withrow award for teaching excellence from the College of Engineering at Michigan State University, in 1991, 1996 and 2006, the Withrow Distinguished Scholar Award in 2007, the MSU Alumni Club of Mid Michigan Quality in Undergraduate Teaching Award, in 2003, and in 2005 he received the Southeast Michigan IEEE Section Award for Most Outstanding Professional.

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Shih-Yuan Chen (M’05–SM’11) was born in Changhua, Taiwan, in May 1978. He received the B.S. degree in electrical engineering in 2000, and the M.S. and Ph.D. degrees in communication engineering in 2002 and 2005, respectively, all from the National Taiwan University, Taipei, Taiwan. From 2005 to 2006, he was a Post Doctorate Research Fellow with the Graduate Institute of Communication Engineering, National Taiwan University, working on the 60-GHz switched-beam circularly-polarized antenna module. In July 2006, he joined the faculty of the Department of Electrical Engineering and Graduate Institute of Communication Engineering, National Taiwan University, and served as an Assistant Professor. From August 2008 to July 2009, he visited the Department of Electrical and Computer Engineering, Michigan State University, East Lansing. His current research interests include the design and analysis of slot antennas/arrays, dielectric lens antennas, reconfigurable antennas, near-field communication systems, and metamaterial-inspired antennas.

Andrew Temme (S’07) is originally from Casper, WY. He received the B.S. degree in electrical engineering from Michigan State University, East Lansing, in 2010, where he is working toward the M.S. degree. As an undergraduate he worked in the Smart Microsystems Laboratory, Michigan State University, before joining the Electromagnetics Research Group in 2010. His interests include through-wall radar, antenna design, and microwave measurement techniques. Mr. Temme is the current president of the MSU chapter of Tau Beta Pi.

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Cross-Dipole Bandpass Frequency Selective Surface for Energy-Saving Glass Used in Buildings Ghaffer I. Kiani, Member, IEEE, Lars G. Olsson, Anders Karlsson, Member, IEEE, Karu P. Esselle, Senior Member, IEEE, and Martin Nilsson

Abstract—Energy-saving glass is becoming very popular in building design due to their effective shielding of building interior against heat entering the building with infrared (IR) waves. This is obtained by depositing a thin layer of metallic-oxide on the glass surface using special sputtering processes. This layer attenuates IR waves and hence keeps buildings cooler in summer and warmer in winter. However, this resistive coating also attenuates useful microwave/RF signals required for mobile phone, GPS and personal communication systems etc. by as much as 30 dB. To overcome this drawback, a bandpass aperture type cross-dipole frequency selective surface (FSS) is designed and etched in the coatings of energy-saving glass to improve the transmission of useful signals while preserving IR attenuation as much as possible. With this FSS, 15–18 dB peak transmission improvement can be achieved, for waves incident with 45 from normal for both TE and TM polarizations. Theoretical and measured results are presented. Index Terms—Coating, cross-dipole, energy-saving glass, etching, frequency selective surface (FSS), GSM, heat, infrared, isolation, personal communication systems (PCS), visible spectrum, WiFi, wireless broadband, wireless network, WLAN, 3G.

I. INTRODUCTION

E

NERGY-SAVING glass is becoming very popular in the modern day building design due to their low-emissivity properties [1]–[7]. This energy-saving property is achieved by applying a thin coating of metallic-oxide on one side of ordinary float glass. This coating provides good thermal isolation to the buildings by blocking infrared rays while being almost transparent to the visible part of the spectrum. Therefore, the visibility through the glass remains good while the buildings can be kept cool for a longer period of time in the summer. The reverse is true for the winter. However, there is one drawback associated with these energy-saving glass panels. Due to the presence of the metallicoxide coating, these glass panels also attenuate many useful

Manuscript received August 24, 2009; revised June 23, 2010; accepted August 08, 2010. Date of publication December 03, 2010; date of current version February 02, 2011. This work was supported in part by Macquarie University Sydney, the Australian Research Council, Lund University Sweden and in part by Pilkington Glass Sweden. G. I. Kiani was with the Department of Electronic Engineering, Macquarie University, Sydney NSW 2109, Australia. He is now with the CSIRO Information and Communication Technologies Centre (ICT), Epping NSW 1710, Australia (e-mail: [email protected]). L. G. Olsson, A. Karlsson, and M. Nilsson are with the Department of Electrical and Information Technology, Lund University, SE-221 00, Sweden (e-mails: [email protected]; [email protected]; [email protected]). K. P. Esselle is with the Department of Electronic Engineering, Macquarie University, Sydney NSW 2109, Australia (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2010.2096382

RF/microwave signals through them [2]–[7]. Most of these signals fall within the frequency band of 800 MHz to 2200 MHz. Signals of GSM mobile phones, personal communication systems (PCS), GPS and 3G/wireless broadband systems are examples. In order to improve the transmission of these useful signals through coated glass, an aperture type frequency selective surface (FSS) can be used [8]–[10]. Such an FSS can provide good transmission improvement in the desired band while maintaining IR attenuation to an acceptable level. Research has been carried out recently to provide an FSS solution for the above mentioned problem [2]–[4]. In this research, the authors have presented interesting results and ideas for FSS design and measurement but some important parts of this research were either based on assumptions or no direct comparison between theoretical and measured results was presented. For example, the relative permittivity of glass was chosen arbitrarily from 3–7 [4] and no measured values of glass conductivity or coating properties was used in the simulations. Therefore, accurate measurements of relative permittivity and conductivity of glass and the electrical properties of coating surface were essential for a reliable design of band pass FSS for energy-saving glass. We have presented the experimental determination of these important parameters in our previous research in which the dielectric constant of the ordinary (float) glass was found to be about 7. Based on measurements, the coating was represented by a resistive sheet having a surface resistance of 6 per square [5]. A small finite conductivity (0.00005 S/m) was also found in float glass [6]. To confirm these measured values, a quantitative comparison between measured and modeled results has been presented recently [6]. In this paper we propose a bandpass FSS solution to the above mentioned problem. Since FSSs are spatial filters, the incoming wave may strike the FSS surface from any arbitrary angle and may have different polarizations. Therefore, the FSS should have a reasonably consistent frequency response for waves having parallel (TM) or perpendicular (TE) polarizations incident either at normal or at oblique incidence, at least up . We have chosen a cross-dipole as the FSS aperture to due to its design simplicity and sufficiently stable frequency response against angle and polarization variations. HP Suncool [11], an energy-saving glass from Pilkington [12], has been analyzed in this research. Both theoretical and measured results show a reasonably stable frequency response and hence justify the usefulness of the proposed FSS design. II. FREQUENCY SELECTIVE SURFACE MODELLING Fig. 1 shows the unit cell of the cross-dipole bandpass FSS designed for HP Suncool. CST Microwave Studio commercial

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coating been a good conductor, the use of loop type FSS elements may have been required to meet the bandwidth and angular stability requirements. However, in our research, the use of cross-dipole FSS yielded the required results and proved easier to be etched than other complex FSS elements. The reason being that the milling machines use standard tools to etch ordinary printed circuit boards (PCBs) with copper coatings. However, glass is more fragile and it is very difficult to etch complex FSS elements on it because the tools are not able to handle the bends and tend to break. Even if the fabrication was possible, there was no need to design a complex FSS as a simple cross-dipole provided the desired results. IV. 2 Fig. 1. The dimensions of the cross-dipole frequency selective surface unit cell.

software (frequency domain solver) has been used for its simulation. The periodicity of unit cell is 100 mm in both horizontal and vertical directions. The length and the width of each segment of the cross-dipole aperture are 64 mm and 8 mm, respectively. The FSS periodic structure is tuned to 1.3 GHz. The thickness of the glass is 6 mm while the relative permittivity is 6.9. The conductivity in the glass and the surface resistivity of the coating are same as described in [5]. The value of surface resistivity for metallic-oxide coating for different glass samples is provided by Pilkington. The details of the coating can not be disclosed due to commercial confidentiality. First, the unit cell shown is Fig. 1 was simulated with full coating (without FSS). The coating is modelled as an impedance boundary condition in CST Microwave Studio. The simulation is carried out from 800 MHz to 6 GHz, to obtain the effect of coating in the desired band (800–2200MHz) as well for WLAN bands (2.45 GHz and 5.25 GHz). This is due to the fact that in some applications it is desirable to attenuate the WLAN signals (for network confinement or security) [13] while allowing mobile telephone signals, GPS and personal communication signals to pass through the FSS. Next, the cross-dipole FSS aperture is removed from the coated side of the unit cell as shown in Fig. 1. The results are presented in the next subsections for dif. It can ferent polarizations and angles of incidence up to be noticed that the frequency response of this FSS is reasonably stable for the range of incident angles and polarizations considered.

III. SELECTION OF FSS ELEMENT The choice of an FSS element is mostly based on the required bandwidth, the frequency response and the frequency response stability with respect to incident angles and polarizations. Our prior research indicated that the cross-dipole FSS provides the required bandwidth and an acceptable angular stability [14], [15]. Therefore, it was chosen for FSS due to its simplicity and relatively better performance. Also, it was noted that due to the resistive nature of the coating, the angular stability was inherently better as compared to a highly conducting FSS. Had the

2 CROSS-DIPOLE FSS

One of the main requirements of the bandpass frequency selective solution for energy-saving glass is to allow minimal amount of IR transmission, while providing improved transmission in useful RF/microwave band. Therefore, it is desirable to remove a little of the metallic oxide coating when etching the FSS since IR transmission will increase in direct proportion to the removed area. This process is simpler and may cost less as opposed to etching on a full glass pane during both pre and post manufacturing processes as described in Section IX. To analyze this concept, we etched a finite array of 2 2 cross-dipole FSS on a part of glass sample having a length and a width of 400 mm, by removing only 2.4% of coating. A LPKF Protomat 95S/II milling machine from LPKF (Laser and Electronics) [16] was used for etching. In such cases, the improvement in microwave transmission is uniform near the FSS area but may not be uniform over the whole panel area. For measurement purposes, the angle of incidence was defined at the center of the array. However, to find the practical benefit of this concept, an FSS should be etched on a part of a larger glass sample and transmission needs to be measured in various locations in a more practical building environment. V. MEASUREMENT SETUP Fig. 2 shows the measurement setup for glass, with the 2 2 FSS etched on the coated side of the glass sample. HyperLog 7060 log periodic antennas from Aariona were used for measurements [17]. These antennas operate between 700 MHz to 6 GHz and were well suited for the experiment. The antennas were connected to the network analyzer (Rohde & Schwarz ZVC, 20 KHz to 8 GHz). The 2 2 periodic array was illuminated with a far field distance of 173 mm (at 800 MHz) and 25 mm (at 6 GHz). It was observed that there was no major difference in the transmission results whether the measurements were made inside or outside an anechoic chamber. Therefore, for ease of handling of heavy glass panels, the measurements were carried outside the chamber and the results are presented in Sections VI & VII. VI. RESULTS A. TE Theoretical Results Fig. 3 shows the theoretical transmission through energysaving glass for perpendicular (TE) polarization. Let us first consider results with full coating. At 0 , 30 and 45 incidence

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Fig. 2. The measurement setup showing 2 2 cross-dipole FSS etched on a part of glass sample fixed in a metallic window, positioned between two logperiodic antennas (TM Polarization).

Fig. 3. Theoretical TE transmission results of HP Suncool with full coating and with a cross-dipole FSS etched in the coating.

angles, an average transmission loss of 30 dB, 31 dB and 33 dB can be observed in the useful frequency band (800–2200 GHz), respectively. The transmission loss increases with the increase in incidence angle. For 2.45 GHz and 5.25 GHz WLAN bands, the attenuation for normal incidence is 28 dB and 24 dB, respectively. With the increase in incident angle, the attenuation at 5.25 GHz remains almost unchanged. At 2.45 GHz, it rises to 29 dB and 31 dB for 30 and 45 incident angles, respectively. It is an added advantage of energy-saving glass that they provide good WLAN confinement and security due to this attenuation; therefore it is recommended that when designing an FSS, this property should not be allowed to degrade significantly. Fig. 3 also shows the transmission response of the crossdipole FSS on HP Suncool. These results are obtained by unit cell simulation with the periodic boundary condition (i.e. for an infinite periodic structure). The dimensions of the unit cell and cross-dipole FSS have been selected in such a way that it resonates at 1.3 GHz. This resonance frequency is slightly less than the center frequency of the desired transmission band of

Fig. 4. Measured TE transmission results of HP Suncool with full coating and with 2 2 cross-dipoles etched in the coating.

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800–2200 MHz. As the coating is not a good conductor, it is not possible to obtain a sharp roll-off. Therefore, it is intended that the transmission curve starts to roll-off before the limiting frequency (2.2 GHz) of the desired transmission band to maintain a high transmission loss at 2.45 GHz WLAN band. At the resonance frequency of 1.3 GHz, the transmission loss at 0 , 30 and 45 incidence angles is 9 dB, 10 dB and 13 dB, respectively. Therefore the transmission improvement at resonance frequency is 21 dB, 21 dB and 20 dB, respectively. On average, about 15 dB transmission improvement is predicted in the desired transmission band for normal incidence. Due to the FSS, there is also an increase in transmission at WLAN bands. On average, about 6 dB and 5 dB increase in transmission is noticed at 2.45 GHz and 5.25 GHz, respectively. For an FSS on a resistive coating (not a PEC), these results are very encouraging as they indicate the possibility of maintaining WLAN confinement at a reasonable level when the transmission in the desired transmission band is improved. B. TE Measured Results Fig. 4 shows the measured results for perpendicular (TE) polarization. For glass covered with full coating, an average transmission loss of 26.7 dB, 27.9 dB and 30 dB can be observed in the useful frequency band for 0 , 30 and 45 incident angles, respectively. The transmission loss in this case is slightly less than the theoretical results due to diffraction from the edges of the measurement setup. For 2.45 GHz and 5.25 GHz WLAN bands, the attenuation at 0 incidence angle is 24.9 dB and 21 dB, respectively. With the increase in incident angle, there is a slight change in transmission at 5.25 GHz while it is 26.5 dB and 27.8 dB at 2.45 GHz for 30 and 45 , respectively. In this case as well, the theoretical and measured results are in good agreement. Fig. 4 also shows the transmission response of the finite 2 2 cross-dipole FSS etched on the coated side of HP Suncool glass sample (shown in Fig. 2). The resonance is observed at about 1.4 GHz which is close to the theoretical prediction. The

KIANI et al.: CROSS-DIPOLE BANDPASS FREQUENCY SELECTIVE SURFACE FOR ENERGY-SAVING GLASS USED IN BUILDINGS

Fig. 5. Theoretical TM transmission results of HP Suncool with full coating and with cross-dipole FSS in the coating.

comparison between the measured transmission for normal incidence and corresponding theoretical result is very good in the desired band especially close to resonance frequency. Also, the measured frequency response of this finite FSS is reasonably stable with the angle of incidence but not as stable as the theoretical results of the infinite FSS presented earlier. At the resonance frequency, the transmission loss for 0 , 30 and 45 incident angles is 9 dB, 12 dB and 15.1 dB, respectively. Therefore the transmission improvement at resonance frequency is 18.5 dB, 16.5 dB and 15.4 dB, respectively. On average, about 11.3 dB transmission improvement is achieved in the desired band for the normal incidence, compared with the 15 dB improvement in the theoretical results. Moreover, about 6 dB average increase in transmission (for all angles considered) is noticed at 2.45 GHz. There is about 2 dB decrease in transmission loss at 5.25 GHz. Therefore, the measured results are very encouraging for WLAN confinement and security. C. TM Theoretical Results Fig. 5 shows the theoretical results for parallel (TM) polarization. With full coating, an average transmission loss of 30 dB, 28 dB, and 26 dB can be observed in the useful frequency band for 0 , 30 and 45 incident angles, respectively. The transmission loss decreases with the increase in incident angle due to Brewster effect. For 2.45 GHz and 5.25 GHz WLAN bands, the attenuation for normal incidence is 28 dB and 24 dB, respectively. With the increase in incident angle, the attenuation at 5.25 GHz remains almost unchanged while it’s 27.5 dB and 26 dB at 2.45 GHz for 30 and 45 , respectively. The transmission response of the cross-dipole FSS (with periodic boundary condition) for TM polarization is also depicted in Fig. 5. At the resonance frequency of 1.3 GHz, the transmission loss for 0 , 30 and 45 incident angles is 9 dB, 8 dB and 7 dB, respectively. In this case the transmission improvement at the resonance frequency is 21 dB, 21 dB and 20 dB, respectively (i.e., same as TE case). On average, about 15 dB transmission improvement is achieved in the desired transmission band for

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Fig. 6. Measured TM transmission results of HP Suncool with full coating and with 2 2 cross-dipoles etched in the coating.

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normal incidence. In this case as well, there is an increase in transmission in WLAN bands. On average, about 4 dB and 5 dB increase in transmission is noticed at 2.45 GHz and 5.25 GHz, respectively. This is quite similar to the TE case. Therefore, this particular FSS design provides good transmission improvement in the desired transmission band while keeping a WLAN confined and secure. D. TM Measured Fig. 6 shows the measured results for parallel (TM) polarization. For fully-coated glass, an average transmission loss of 26.7 dB, 25.5 dB and 24.6 dB is found in the desired frequency band for 0 , 30 and 45 incident angles, respectively. Brewster effect can be noticed in this case as well. For 2.45 GHz and 5.25 GHz WLAN bands, the attenuation at 0 incidence angle is 24.5 dB and 20.5 dB, respectively. The measured transmission response of the cross-dipole FSS prototype for TM polarization is also depicted in Fig. 6. Again, the measured results for normal incidence compare well with the theoretical results for frequencies close to the resonance frequency. The resonance frequency for TM case is 1.4 GHz and the transmission loss at resonance is about 9 dB at normal incidence. Unlike in theoretical results, an increase in transmission with incident angle is observed only for certain frequencies in the desired band. The transmission improvement at resonance frequency of 1.4 GHz is 18.2 dB, 17.5 dB and 15.7 dB, respectively, for incident angles of 0 , 30 and 45 . On average, about 11.3 dB transmission improvement is achieved in the desired transmission band for normal incidence. WLAN confinement and security are quite good in this case as well for both 2.45 GHz and 5 GHz WLAN bands. VII. EFFECT ON INFRARED ATTENUATION Another important aspect of FSS design for such energy-saving glass is the increase in IR transmission as a result of etching the FSS in the coating. Our research indicates that the relationship between the increase in IR transmission and the percentage of coating area removed by FSS elements is

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approximately linear [18]. For approximately 20 dB improvement of RF/microwave transmission at 1.3 GHz through a glass panel, 9.6% of coating area should be removed. This increases IR transmission by 10% (from 23.8% to 33.8%). A narrower (4 mm) FSS improves microwave transmission only by 16 dB at the resonance frequency but the increase in IR transmission is only 6.2% in that case. The area removed due to etching of this narrower FSS is 4.96%. Therefore, in cases where a 10% increase in IR transmission is not acceptable, a narrower FSS design may be considered [18].

while achieving a reasonable transmission improvement, a 2 cross-dipole FSS was employed in our small, finite, 2 experiments by removing only 2.4% of the coating. However, to confirm the benefit of this concept, the FSS should be etched on a part of a larger glass sample and transmission need to be measured in a more practical building environment. The FSS design presented here can be customized to any frequency band (with a comparable percentage bandwidth) in which a transmission improvement is required. It also provides good confinement and security at 2.45 GHz and 5.25 GHz WLAN bands.

VIII. FABRICATION ISSUES It can be noticed in Fig. 2 that due to the etching of FSS pattern in the coating of energy-saving glass, the aesthetic of glass is degraded. Therefore, a 64 mm long and 8 mm wide cross-dipole FSS element may not seem to be practical if fabricated using a standard milling process. This is due to the fact that the milling tool also cuts into the glass (not just the coated surface) due to the very small thickness of metallic oxide coating. However, the length and the width of cross-dipole could not be avoided because the transmission of RF/microwave signals can only be improved to an acceptable level using these particular dimensions. Hence, for practical FSS design with greater FSS elements widths, two different methods can be considered for mass production of energy-saving glass with an FSS solution: pre-production or post-production. In the pre-production process, the FSS can be made on the glass during sputtering process, by applying an FSS mask pattern on the side of the glass that needs to be coated. The glass can then be passed through vacuum chambers where the oxide layers are deposited. Once the glass product is ready, the mask can be removed, leaving the FSS pattern as apertures in the coating of the energy-saving glass. If the color of the glass and the coating are not same, the apertures may be coated with a lossless material that is colormatched to the coating in order to maintain aesthetics. In the post-production process, the FSS can be made in the coating after the glass is manufactured with coating. In this case, the techniques like laser beams or sand blasts (by applying an aperture type mask on coated side) can be considered for the production process. Again, the apertures may be coated with a lossless, color-matched material. In both cases, the aesthetic of the glass will not be severely degraded. IX. CONCLUSION For RF and microwave signals from 0.8–6 GHz, the transmission loss due to the resistive coating in energy-saving is 25–30 dB. A solution to improve the transmission of RF/microwave signals through energy-saving glass is presented here. It is achieved by designing and etching a cross-dipole bandpass FSS in the coatings of energy-saving glass. In addition the large increase in RF/microwave transmission, etching of FSS in energy-saving glass also leads to an unwanted small increase in IR transmission and associated heat transfer. It is required to keep the latter to a minimum level by etching as less coating as possible. In the infinite FSS presented here, 9.6% of coating (by area) is removed in each unit cell. This will cause about 10% increase in the IR transmission. To minimize the heat transfer

ACKNOWLEDGMENT The authors would like to thank A. Rolandsson and R. Borjesson from Pilkington, Halmstad, Sweden, for supporting the research and providing different types of commercial glass panels for the project. Many thanks to Prof. N. Shuley, Director International Development Research Group, University of Queensland, Australia, for his valuable guidance and support during this project. REFERENCES [1] C. Mias, C. Tsakonas, and C. Oswald, An investigation into the feasibility of designing frequency selective windows employing periodic structures The Nottingham Trent Univ., Nottingham, U.K., 2001, Final Rep., Radio Communications Agency, Ref. AY3922. [2] M. Philippakis, C. Martel, D. Kemp, M. C. S. M. R. Allan, S. Appleton, W. Damerell, C. Burton, and E. A. Parker, “Application of FSS structures to selectively control the propagation of signals into and out of buildings,” ERA Technology, Leatherhead, Surrey, U.K., 2004, Tech. Rep. [3] B. Widenberg and J. V. R. Rodriguez, “Design of energy-saving windows with high transmission at 900 MHz and 1800 MHz,” Lund Inst. Technol., Dept. Electrosci., Lund, Sweden, Tech. Rep. LUTEDX/(TEAT-7110)/114/(2002), 2002 [Online]. Available: www.es.lth.se [4] M. Gustafsson, A. Karlsson, A. P. P. Rebelo, and B. Widenberg, “Design of frequency selective windows for improved indoor outdoor communication,” IEEE Trans. Antennas Propag., vol. 54, no. 6, pp. 1897–1900, Jun. 2006. [5] G. I. Kiani, A. Karlsson, L. Osslon, and K. P. Esselle, “Glass characterization for designing frequency selective surfaces to improve transmission through energy-saving glass windows,” presented at the Asia Pacific Microwave Conf., Bangkok, Dec. 2007. [6] G. I. Kiani, A. Karlsson, L. Osslon, and K. P. Esselle, “Transmission analysis of energy saving glass windows for the purpose of providing FSS solutions at microwave frequencies,” presented at the IEEE Antennas Propag. Society Int. Symp., San Diego, 2008. [7] G. I. Kiani, A. Karlsson, L. Osslon, and K. P. Esselle, “Transmission improvement of useful signals through energy saving glass windows using frequency selective surfaces,” presented at the Workshop on Applications of Radio Science (WARS) Conf., Gold Coast, Queensland, Australia, Feb. 10–12, 2008. [8] B. A. Munk, Frequency Selective Surfaces: Theory and Design. New York: Wiley, 2000. [9] G. I. Kiani, K. L. Ford, K. P. Esselle, A. R. Weily, C. Panagamuwa, and J. C. Batchelor, “Single-layer bandpass active frequency selective surface,” Microw. Opt. Technol. Lett., vol. 50, no. 8, pp. 2149–2151, May 2008. [10] H. Liu, K. L. Ford, and R. J. Langley, “Miniaturised bandpass frequency selective surface with lumped components,” Electron. Lett., vol. 44, no. 18, pp. 1054–1055, Aug. 2008. [11] [Online]. Available: www.pilkington.com/applications/products2006/ english/bybenefit/solarcontrol/suncool/default.htm [12] [Online]. Available: www.pilkington.com [13] G. I. Kiani, A. R. Weily, and K. P. Esselle, “A novel absorb/transmit FSS for secure indoor wireless networks with reduced multipath fading,” IEEE Microw. Wireless Compon. Lett., vol. 16, no. 6, pp. 378–380, Jun. 2006.

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[14] G. I. Kiani, K. L. Ford, K. P. Esselle, A. R. Weily, and C. Panagamuwa, “Oblique incidence performance of a novel frequency selective surface absorber,” IEEE Trans. Antennas Propag., vol. 55, no. 10, pp. 2931–2934, Oct. 2007. [15] G. I. Kiani, K. L. Ford, K. P. Esselle, A. R. Weily, and C. Panagamuwa, “Angle and polarization independent bandstop frequency selective surface for indoor wireless systems,” Microw. Opt. Technol. Lett., vol. 50, no. 9, pp. 2315–2317, Sep. 2008. [16] [Online]. Available: http://www.lpkfusa.com/RapidPCB/Circuitboard Plotters/95sii.htm [17] [Online]. Available: http://www.elektrosmog.de/Gutachten/Hyper LOG7000-E.pdf [18] G. I. Kiani, A. Karlsson, L. Osslon, K. P. Esselle, and M. Nilson, “Transmission of infrared and visible wavelengths through energy-saving glass due to etching of frequency selective surfaces,” IET Microw., Antennas Propag., vol. 4, no. 7, pp. 955–961, Jul. 2010. Ghaffer I. Kiani (M’09) received the B.Sc. degree in electrical and electronic engineering (with first-class distinction) from the Islamic University of Technology, Dhaka, Bangladesh, in September 1997, the M.Sc. degree (with first-class high distinction) in electronic engineering from Ghulam Ishaq Khan (GIK) Institute of Engineering Sciences and Technology, Topi, Pakistan, in May 2003, and the Ph.D. degree in electronic engineering from Macquarie University, Sydney, Australia, in May 2009. He is currently pursuing a Postdoctoral Fellowship in very-high throughput wireless communication systems at CSIRO ICT Centre, Sydney, Australia. He is also an Honorary Associate in the Department of Electronic Engineering, Macquarie University. His current research interests include frequency selective surfaces, metamaterials, electromagnetics, antenna design, micorwave polarizers, MEMS, NEMS and THz modulators. Dr. Kiani was awarded the Best Student Paper Award at the 2008 Workshop on Applications of Radio Science (WARS) Conference, Gold Coast, Australia, for the paper, “Transmission Improvement of Useful Signals through Energy Saving Glass Windows using Frequency Selective Surfaces.”

Lars G. Olsson was born on January 15, 1945. He received the M.Sc.EE and Ph.D. degrees from Lund University, Lund, Sweden, in 1973 and 1980, respectively. In 1980, he joined the Department of Applied Electronics, Lund University, as a Lecturer, becoming Professor pro tempore from 1982 to 1986, and Associate Professor thereafter, and until this day. Additionally, he was Deputy Head of the Department of Applied Electronics and Director of Studies from February 1995 to June 1996, and Director of Studies of the EE program from July 1996 to December 2005. His research has been on different aspects of radio: “Precise time and frequency” and “propagation of radio waves in personal communications applications.” His Ph.D. dissertation was on the development of a new receiver to extract precise time from the Loran-C signal—time and frequency being the research area of the department at that time.

Anders Karlsson (M’85) was born in 1955 in Gothenburg, Sweden. He received the M.Sc. and Ph.D. degrees from Chalmers University of Technology, Gothenburg, Sweden, in 1979 and 1984, respectively. Since 2000, he has been a Professor in the Department of Electrical and Information Technology, Lund University, Lund, Sweden. His research activities include comprehend scattering and propagation of waves, inverse problems, and time-domain methods. Currently he is involved in projects concerning frequency selective structures, physical limitations of antennas, and scattering of light from soot particles.

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Karu P. Esselle (M’92–SM’96) received the B.Sc. degree in electronic and telecommunication engineering (First Class Honors), from the University of Moratuwa, Sri Lanka, and the M.A.Sc. and Ph.D. degrees in electrical engineering from the University of Ottawa, Canada, with a nearly perfect GPA. He is a Professor of electronic engineering at Macquarie University, Sydney, Australia. He was the Immediate Past Associate Dean—Higher Degree Research and the Founding Director of the Postgraduate Research Committee in the Division of Information and Communication Sciences. He held these positions from 2003 to 2008 and was also a member of the Division Executive. He served in all Macquarie University HDR-related committees at the highest level. He is the Director of Electromagnetic and Antenna Engineering, and the Deputy Director of the Research Centre on Microwave and Wireless Applications, which was recently expanded after being recognized as a Concentration of Research Excellence. He has been invited to serve as an international expert/research grant assessor by several overseas nationwide research funding bodies from The Netherlands, Finland, Hong Kong, and Chile. In Australia, he has been invited to assess grant applications submitted to the nation’s most prestigious schemes such as Australian Federation Fellowships and Australian Laureate Fellowships. His industry experience includes full-time employment as Design Expert by the Hewlett Packard Laboratory, USA, and several consultancies for local and international companies, including Cisco Systems (USA), Cochlear, Optus Networks, Locata (USA)/QX Corporation, ResMed, FundEd and Katherine-Werke (Germany) through Peter-Maxwell Solicitors. He was an Assistant Lecturer at the University of Moratuwa, a Canadian Government Laboratory Visiting Postdoctoral Fellow at Health Canada, a Visiting Professor of the University of Victoria and a Visiting Scientist of the CSIRO ICT Centre. He is an Editor of the International Journal of Antennas and Propagation. He has authored over 250 scientific publications, including six invited book chapters and over 15 invited conference presentations. His current research interests include metamaterials and their microwave applications, photonic crystals and photonic band gap (PBG)/electromagnetic band gap (EBG) structures, millimeter-wave EBG MMIC devices, antennas based on EBG, periodic structures including frequency selective surfaces, antennas for mobile and wireless communication systems including multi-band WiFi, WiMAX, HyperLAN, and ultra-wideband systems, antennas for multi-signal location and navigation systems, dielectric-resonator (DR) antennas, broadband and multi-band printed antennas, smart antenna systems, hybrid antennas, theoretical methods, lens and focal-plane-array antennas for radio astronomy, moment methods, FDTD methods for periodic structures and closed-form Green’s functions for layered structures. Prof. Esselle’s recent awards include the 2009 Vice Chancellor’s Award for Excellence in Higher Degree Research Supervision (the first such award ever offered in Macquarie University) and the 2004 (Inaugural) Innovation Award for best invention disclose. Since 2002, he was involved with research grants and contracts worth about five million dollars, and his research team members attracted further grants worth about a million dollars. The CELANE, which he founded, has provided a stimulating research environment for a strong team of researchers including six postdoctoral fellows. His mentees have been awarded six extremely competitive postdoctoral fellowships. Nine international experts who examined the theses of his recent five Ph.D. graduates ranked them in the top 5% or 10% in the world. He has served on technical program committees or international committees for many international conferences. He will be chairing the Technical Program Committee of APMC 2011; he was the Publicity Chair of the APMC 2000. He is the Chair of the IEEE New South Wales (NSW) MTT/AP Joint Chapter, an Editor of the Macquarie University Engineering Colloquia (MQEC), the past Chair of the Educational Committee of the IEEE NSW, and a member of the IEEE NSW Committee.

Martin Nilsson, photograph and biography not available at the time of publication.

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Multiscale Compressed Block Decomposition for Fast Direct Solution of Method of Moments Linear System Alex Heldring, Juan M. Rius, Senior Member, IEEE, José M. Tamayo, Josep Parrón, and Eduard Ubeda

Abstract—The multiscale compressed block decomposition algorithm (MS-CBD) is presented for highly accelerated direct (non iterative) solution of electromagnetic scattering and radiation problems with the method of moments (MoM). The algorithm 2 computational complexity and is demonstrated to exhibit 1 5 , for electrically large storage requirements scaling with objects. Several numerical examples illustrate the efficiency of the method, in particular for problems with multiple excitation vectors. The largest problem presented in this paper is the monostatic RCS of the NASA almond at 50 GHz, for one thousand incidence angles, discretized using 442,089 RWG basis functions. Being entirely algebraic, MS-CBD is independent of the Greens function of the problem. Index Terms—Computational electromagnetics, fast solvers, impedance matrix compression, method of moments (MoM), numerical simulation.

I. INTRODUCTION widely used method for the numerical solution of scattering and radiation problems in electromagnetics is the method of moments (MoM) [1], applied to a surface integral equation (SIE) in the frequency domain and using local basis functions. The MoM gives rise to a linear system with a full coefficient matrix (the impedance matrix). For electrically large objects, when the discretization of the problem geometry is fixed with respect to the wavelength (for example 100 basis functions per ), the size of the impedance matrix grows as the squared electrical size of the problem geometry. Straightforward storage and reand solution time of the linear system scale with spectively, which places a rather steep upper bound on the size of problems that can be solved on a given computer. To solve electrically large problems, several accelerating methods have been developed, such as AIM [2], MLFMA [3],

A

Manuscript received September 22, 2009; revised June 29, 2010; accepted July 01, 2010. Date of publication December 06, 2010; date of current version February 02, 2011. This work was supported by the Spanish Interministerial Commission on Science and Technology (CICYT) under projects TEC2009-13897-C03-01, TEC2009-13897-C03-02 and TEC2010-20841-C04-02 and CONSOLIDER CSD2008-00068 and by the “Ministerio de Educación y Ciencia” through the FPU fellowship program. A. Heldring, J. M. Rius, J. M. Tamayo, and E. Úbeda are with the AntennaLab, Department of Signal Processing and Telecommunications, Universitat Politecnica de Catalunya, 08034 Barcelona, Spain (e-mail: [email protected]. edu). J. Parrón is with the Department of Telecommunications and Systems Engineering, Universitat Autónoma de Barcelona, 08193 Bellaterra, Spain. Digital Object Identifier 10.1109/TAP.2010.2096385

MLMDA [4]–[6] and SVD-MDA [7]. These methods have in common an efficient approximative representation of the impedance matrix, with highly reduced storage requirements. This representation can be used inside an iterative method such as GMRES, often with an appropriate preconditioner [8]. All these methods achieve an important reduction of the scaling factor for storage and solution time as a function of , the most efficient one being the MLFMA with both storage and solution . Here, is the number of iterations time scaling as needed, which generally increases with , in an unpredictable way, but it is typically much smaller than , in particular with a good preconditioner. As a result, these methods allow the solution of problems that are orders of magnitude larger than before. While there is no doubt that for very large problems, iterative methods like the ones above are the only option, there are a number of reasons why, if the problem size allows it, a direct (non-iterative) method may be preferable. Firstly, direct methods always yield a solution in a fixed time. In contrast, as mentioned above, iterative methods often show unpredictable convergence and may even fail to converge altogether, when the impedance matrix is particularly badly conditioned. A good preconditioner can help, but the construction of such a preconditioner may become the bottleneck of the computation [9], [10]. Secondly, most iterative methods and preconditioners depend on several parameters that may need to be adjusted for every specific problem. Thirdly, if a solution is required for many different excitation vectors (for example, monostatic RCS computations), the iteration process has to be restarted for every excitation (although seeded Krylov solvers may accelerate the solution for subsequent excitations [11]). In direct methods, the bulk of the work is independent of the excitation, and once this is done, the solution per excitation is obtained very fast. In view of the above, depending on the specifics of the problem and the available computational resources, a turning point exists (in terms of ), below which direct solution may be preferable to the fast iterative methods. For straightforward LU decomposition, this turning point is quite low, typically in the order of a few thousand unknowns. Beyond this point, either the matrix simply complexity becomes prohibitive. cannot be stored or the In [12], the authors of this paper presented a new direct solution method, the compressed block decomposition algorithm (CBD), which greatly reduces the cost of the direct solution, both in terms of storage and computational complexity. In [13], for storage and with CBD was shown to scale with for complexity, such that for very large problems the

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HELDRING et al.: MCBD FOR FAST DIRECT SOLUTION OF MOMs LINEAR SYSTEM

fast iterative methods remain the only option. However, in our experience, CBD is competitive with fast iterative methods up to several tens of thousands of unknowns, even for problems with a single excitation vector, and considerably higher when the solution is needed for many excitation vectors. In [14], we presented a multiscale version of the CBD for electrostatic proband the lems and showed that the storage scales with complexity with . This very favorable behavior is due to the fact that in electrostatics, the compression rate is independent of the size of the problem. This is not true for the electrodymamic case, but, as this paper will show, a multiscale (MS) implementation of the CBD is worthwhile nonetheless. In Section II the concept of compressed impedance matrix sub blocks, underlying both the CBD and the MS-CBD, is addressed. Section III introduces the multiscale, or hierarchical, CBD (MS-CBD) for electromagnetic problems. In Section IV, for storage and with the MS-CBD is shown to scale with for complexity. It is therefore asymptotically faster than the CBD. A further important advantage of the MS-CBD with respect to CBD is that the only parameter that needs to be adjusted is a threshold value that governs the tradeoff between the compression rate and the accuracy of the result. Section V-A illustrates the computational complexity for the case of PEC spheres. The stability of EFIE-MoM solvers at low frequencies is addressed in Section V-B. Finally Section V-C illustrates the efficiency of the proposed method when dealing with multiple independent vectors, presenting the computation of the monostatic RCS of the NASA almond at 50 GHz, discretized with 442,089 RWG basis functions.

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Fig. 1. Recursive binary space subdivision: at level-1 there are two subdomains, at level-2 there are four and at level-3 there are 8. At any level, the two children subspaces are not necessarily of the same size.

In most fast solver implementations, like [7] or [17], the resulting block-compressed matrix is used to compute the matrix vector products in an iterative solver. In the CBD [12] and in the MS-CBD proposed here, the block-compressed matrix is factorized, retaining the compression, to find a direct solution. In the subdoCBD, the problem geometry was sub-divided into mains, leading to sub-blocks, all of which except for the diagonal blocks were compressed. One of the problems was the determination of the optimum value of . In contrast, in the MS-CBD the geometry is sub-divided in two subdomains at level one, each of which is again sub-divided in two subdomains at level two, etc., (see Fig. 1). III. MULTISCALE CBD A. The Algorithm In this section we present the MS-CBD algorithm, which is essentially a nested implementation of the well known partitioned inverse formulas [24], [25]. Let us consider the linear system resulting from the MoMs discretization of the electric field integral equation (EFIE) [1]

II. IMPEDANCE MATRIX SUB-BLOCK COMPRESSION Both the CBD and the MS-CBD make use of the concept of impedance matrix sub-block compression. This concept has been exploited in many methods [7], [15]–[18]. It is based on the fact that for an impedance matrix sub-block representing the interaction between two geometrically separated groups of elementary scatterers (basis functions) the number of significant degrees of freedom drops with the square of the distance between the groups (see e.g. [19]). Consequently, one can subdivide the entire problem geometry into several non-overlapping subdomains, and all impedance matrix blocks representing the interaction between two different subdomains can be approximated with a low rank representation (a compressed block), for example a truncated SVD decomposition. While the SVD is a computationally expensive operation, fortunately there exist fast block compression methods yielding the same result. In [7] and [12] we used the MDA method from [4]. In this paper we use the Adaptive Cross Approximation technique (ACA, [21]) which, unlike the MDA, can be used even for compressing blocks representing adjacent subdomains. In both cases, an SVD post-compression can be applied ([12], [22]) to obtain the optimum SVD representation with unitary bases and a diagonal matrix of singular values. In fact, since the ACA yields a less accurate approximation than full SVD, it is recommendable to use ACA with a safety margin (a threshold that is lower than required, typically ten times lower), and then post-compress with SVD with the actual required threshold.

(1) We can partition the unknown vector X, the excitation vector Y and the impedance matrix Z as (2) , , In order to have compressible off-diagonal blocks the matrix and vectors partitioning must correspond to a domain subdivision of the volume containing the object. In our implementation, the subdivision is performed as follows: The Cartesian axis along which the subdomain is largest is determined, then the subdivision is made at the coordinate which divides the subdomain in two parts with equal (or as equal as possible) numbers of basis functions (level-1 in Fig. 1). Consequently, the subdomains are not necessarily of equal size, but at a given level all subdomains contain (almost) the same number of elements. In 3-D surface meshes space subdomains do not correspond to contiguous row and column indices, and therefore the rows and columns of sub-vectors and sub-matrices in (2) are interleaved. The partitioned inverse matrix of Z is [24], [25] (3) with (4)

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Equations (4)-(7) can be rewritten for symmetric matrices as (11) (12) (13) We wish to compute (14) which in partitioned form (2), (3) is (15) (16)

Fig. 2. Impedance matrix of PEC sphere, R = 2:86, 49152 RWG basis functions, with 7-level subdivision. The colors denote the rank of the depicted subblocks after SVD compression with threshold 0.001. Only the upperdiagonal part of the symmetric matrix is needed. The thick black lines show the multiscale partitioning for the MS-CBD decomposition (for the first three levels).

Using (11), (12) and (13) we can compute the partitioned inverse operator B

in terms of (17) (18) (19) (20)

(5) (6) (7) . The result is inTo compute (3) one starts by inverting . serted into (7) and a second inversion follows to determine Consequently, the inversion of the total matrix is replaced by two inversions of the size of the two diagonal sub-blocks, plus a number of matrix multiplications and summations. Of course, the partitioned inverse algorithm can be applied recursively to compute the two matrix inversions in (4)–(7). This leads to a recursive subdomain partition in which at each level the off-diagonal sub-blocks are compressed, while the diagonal sub-blocks are partitioned again. Accordingly, level-1 subdomains are split in two level-2 subdomains, which are again split in two level-3 subdomains, etc., as depicted in Fig. 1, until we reach a preestablished block size (see Section III-C below). Fig. 2 shows the Z matrix partition along the thick black lines: for example, the upper-right quarter of the matrix in the figure at the first level. The resulting hierarchically subdidenotes vided compressed matrix is similar to the compressed matrices of the IES3 software [15] and to the -matrices of [23]. We will present the algorithm for symmetric matrices ( , , ) since, in this paper, we only apply it to the electric field integral equation (EFIE). The extension to unsymmetrical matrices is trivial. Let us define (8) (9) (10) where ator B.

,

and

represent the partitioned inverse oper-

If we compute

before (21) (22)

The algorithm , given below, returns the operators from (8), (9) and (10) that together three permit fast reconstruction of the partitioned inverse

When is called, it calls itself recursively to compute the partitioned inverse operators for the two diagonal sub-blocks of Z. When the diagonal sub-blocks are not further partitioned, at the finest level, an LU decomposition is returned instead of algoa partitioned inverse. In the second line of the is called, which returns the rithm, a function product of , generated in the first line, and . This is a special function rather than a simple matrix multiplication because is itself recursively partitioned and what actually does is to use the components of to reconstitute (through (11)–(13)) and immediately multiply this with . The function is given below. Once the entire recursively partitioned impedance matrix Z has been converted into the recursively partitioned inverse algorithm, the solution of the operator B by the system (1) is obtained with (21), (22). In fact, the same function that appears inside the algorithm, but now with the entire inverse operator B as the first argument and the excitation matrix Y as the second argument, does exactly this. This is not surprising, because in both cases, a partitioned inverse operator is left-multiplied to another matrix.

HELDRING et al.: MCBD FOR FAST DIRECT SOLUTION OF MOMs LINEAR SYSTEM

In order to compute X given an excitation matrix Y and the partitioned inverse operator B of Z, the algorithm is

Since and are operators that return the inverse of a and times a matrix in (21) and matrix, the products of (22) are computed by applying recursively the function. The recursion is stopped when, at the finest level, the matrix blocks are small enough (order about 600 in our implementation). At this level, the inverse matrices are stored in LU form and applied by forward and backsubstitution [25]. B. Compressed Storage One observes that inside the and routines to build the partitioned inverse of Z, only the submatrices at different levels are needed, together with the LU decomposition of diagonal blocks at the finest level. Since the submatrices correspond to interactions between distinct source and field subdomains, they are stored in compressed form, ideally an SVD (see Section II): (23) where S is a diagonal matrix and U, V are orthonormal matrices. The columns of S, U and V matrices corresponding to singular values smaller than a threshold are discarded, thus obtaining the truncated SVD compression. is first Since the SVD is computationally very expensive, pre-compressed with ACA [17], [21] (24) where the number of columns in F and G is much smaller than the number of rows. and As far as the block-multiplications occurring in are concerned, these can be done efficiently for form yielding a well compressed result in the matrices in same form. However, the block additions are more difficult: the result needs to be re-compressed in order to maintain optimum compression. This is essential in multi-scale CBD because small losses in compression rate will accumulate across the levels. For optimum compression, a true SVD representation is required. To obtain this, first a QR decomposition [25] is applied to the F and G matrices (25) (26) are orthonormal matrices and upper-triwhere angular matrices. The inner matrices are re-compressed with truncated SVD and, finally, the outer orthonormal matrices are multiplied: (27)

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(28) to obtain very efficiently a compressed representation of with outer orthonormal matrices and and an inner diagonal matrix S, equivalent to the direct SVD (23). As mentioned earlier (Section II), the ACA error threshold is ten times lower than the SVD threshold. All the matrix blocks, both of Z and of B, appearing in the algorithms above are in one of four representations or ‘forms’. At all levels in the recursion, except the finest level, all blocks are either partitioned blocks, denoted from here on as a P-form, or SVD decompositions, denoted from here on as an S-form. For example, a block can be a P-form, consisting of four S-forms, but also a P-form, consisting of two S-forms and two P-forms. Diagonal blocks, at all levels except the finest, are always P-forms, since they can not be compressed in their entirety. At the finest level, all non-diagonal blocks are S-forms, and all diagonal blocks of Z are full matrices, or M-forms, while all diagonal blocks of B are LU decompositions (LU-forms). As the algorithms show, the necessary operations on the matrix blocks are transpose, multiplication and addition (and LU, but only at the finest level). The transpose operation is trivial. For the multiplications and additions, a set of operators has to be defined for all possible cases: Multiplying two S-forms efficiently (and retaining optimum compression) is explained in [12], as is adding two S-forms and recompressing the result. If one argument is a P-form and the other is an S-form, the latter has to be partitioned to ‘fit’ to the first, possibly recursively, and the multiplications/additions are performed block-by-block. If both arguments are P-forms, all the operations are performed block-by-block. Consequently, once the original matrix Z is recursively partitioned according to some rules explained in Section III-C, the operations taking place inside the and routines automatically define the partitioning of the partitioned inverse B, which closely follows, but is not necessarily identical to that of Z.

C. Implementation Issues Below we discuss a number of implementation issues concerning the algorithm. • Number of levels: We recursively subdivide the problem geometry into rectangular subdomains, based on the number of basis functions inside each domain. This means that at every level, the number of basis functions that belong to each subdomain is practically the same. We found that below approximately 400 elements, further partitioning is not more efficient than LU decomposition, unknowns so the number of levels for a problem with is given by (29) where ceil(x) denotes the first integer above x. This procedure generates a ‘binary tree’ recursive subdivision for the problem.

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• Maximum rank of the sub-blocks: For the initial matrix compression we use a strategy similar to [15]. At every level, the largest possible pairs of subdomains are determined, whose minimum distance is larger than the sum of the radii of the smallest spheres enclosing all their elements. Such sets are called ‘non-overlapping’. ACA is used to construct the S-form of their interaction. If the sets are overlapping, the same procedure is applied to their 2 times 2 ‘child-sets’, and so on recursively. This procedure does not guarantee a fixed rank for all submatrices at all levels. Therefore, every time one S-form is generated, and it is , it found that its rank exceeds a chosen threshold is chopped into four sub-blocks, according to the binary tree mentioned above. At the same time, larger blocks are more efficient in the algorithm, as long as their rank remains bounded. Therefore, every time four S-forms of 2 times 2 ‘child-sets’ are constructed, and it turns out that all , the four blocks four sub-blocks have a rank below are merged into one S-form. We found that, independently of the problem specifics, or of its size, choosing (30) leads to maximum computation speed in our implementation. It should be noted that, in reality, following the above procedure most blocks will have rank . This is illustrated in Fig. 2, which was constructed according to (30). • MS-CBD: In order to preserve the optimum compression during the MS-CBD decomposition, the output of the multiplication and addition operators must be adequately defined. Specifically, for multiplication, the rank of the result is at most the minimum rank of the input arguments, and therefore the result of the multiplication of any two types of matrix (M-, S- or P-form) can always safely be cast in S-form, except in the case of multiplying two P-forms. If this is the case, the result is subject to the same test as above (and multiplication of an M-form with with threshold an M- or P-form does not occur in the algorithm). For the addition operation, the rank of the result will generally be at least that of the higher-rank argument. Again, to determine the format of the output we apply the test with threshold . Using the above rules, we have observed that the compression rate of the MS-CBD is of the same order of magnitude as that of the original matrix, for a large variety of problems and problem sizes. As an example, compare Fig. 2 and Fig. 3. IV. COMPUTATIONAL COMPLEXITY The computational complexity of an algorithm is the asymptotic scaling of the scalar operation count with the number of unknowns . In electromagnetics, there are two separate cases to consider. The first case concerns refining the discretization for a fixed frequency. In theory, the presented algorithm scales for this case since the rank of the blocks does not dewith pend on the discretization size. In practise this is not always observed (see Section V-B). This section deals with the second case, scaling the frequency while maintaining a fixed discretization with respect to the wave length.

Fig. 3. MS-CBD decomposition of the matrix of Fig. 2. The colors denote the rank of the depicted subblocks after SVD compression with threshold 0.001.

and the reWe start by addressing the relation between quired storage of the compressed matrix. The number of degrees of freedom or rank of the interaction between two non-overlapping subdomains of approximately equal size is asymptotically proportional to [19], [20] (31) where is the wave number, the typical dimension (“diameter”) of the subdomains and is the distance between the subdomain-centers. The first factor in (31) is related to the number of spherical harmonics needed to resolve the “sources,” the second term represents the solid angle subtended by the “observation domain” as seen from the “source domain” center (by reciprocity, the “source domain” and “observation domain” are interchangeable). From (31) we see that the minimum distance at which has reduced below some fixed value scales with (32) For a MoM surface discretization with fixed discretization size, the number of elements in a subdomain scales with the electrical surface, or (33) Consider a large problem, of unknowns (elements), subdi, not to vided at the finest block-size-level (with index be mixed up with the recursion levels in the MS-CBD algosubdomains each with approximately elerithm) into is independent of , we have ments. Since the choice of and . We wish to represent the whole interaction matrix in blocks with . This is obviously . But beyond a given mutual disfulfilled for all blocks at it will also be fulfilled if we combine two sourcetance and two observation subdomains into subdomains with approximately elements. At even larger distance, we

HELDRING et al.: MCBD FOR FAST DIRECT SOLUTION OF MOMs LINEAR SYSTEM

can construct general

subdomains with

elements, etc. In

(36)

. At level , the work per block-block multiplilevel . If both P-forms were entirely composed of level cation is blocks, the total number of block-block multiplications and thus the total work would scale with . But we have seen blocks scales with . Therefore that the number of level the probability that at least one of the two blocks involved in block every multiplication is actually an existing level scales with and the total work scales with . The subroutine Apply_Inverse contains two multiplications ( and two calls to itself. The work for the multiplications is is a constant) and the work for the recursive calls is . This series converges, so the work for Apply_In. Exactly the same argument shows that the work verse is , and therefore, when applied to the for MS-CBD is also . entire problem, it is Using ACA to construct the compressed matrices, the conoperations for a block of struction process only requires dimension , because all the blocks are rank order one S-forms. Consequently, the computational complexity of the matrix con. The function Apply_Instruction equals the storage, or verse, applied to one independent vector, only uses all of its ele. The ments once, so the backsubstitution also scales with above scale factors are illustrated numerically for the case of a PEC sphere in Section V-A.

units

V. NUMERICAL EXAMPLES

(34) The observation subdomains that interact with a given source subdomain at a given level are those that lay at a distance beand from the source group center. tween For surface discretization, the total number of elements within that range scales with . With (34) this means that moving up one level, approximately four times more elements and twice as many observation subdomains are covered per source domain. Because there are also two times less source domains, the number of blocks per level is approximately constant. Considthis number is easily deduced to ering the case of level . Now since every block accounts for elements, we be can estimate the maximum block-size requiring that (35) which leads to

Since all the blocks are , every block requires of storage. Consequently the total storage

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(37) In order to determine the complexity of the MS-CBD algorithm, first we address the complexity of the different algebraic operations that are used, in particular when applied to recursive P-forms of asymptotically large dimension and storage . They are: LU decomposition (but this is only applied ), transpose, addition and multiplication. As explained at above, the entire P-form is built up of rank order one S-forms. Addition and multiplication of two such S-forms are operations. Addition and multiplication of P-forms depend on re-compression and re-orthogonalization using QR and SVD, and operations, respectively. The but these are additions (and transpose) of P-forms only address every stored . element once, therefore they scale as the storage, or as This leaves us with the determination of the complexity of the multiplications, which is addressed in the next paragraph. We have seen above that the number of blocks is approximately the same for all levels. The only way to accommodate for this is to require that on average, one in every four blocks at every level is subdivided. Therefore, the probability that, given two blocks at level to be multiplied, neither is subdivided, . If at least one of the blocks is subequals . Asdivided, 8 multiplications must be executed at level suming we have block multiplications at level , the work at . The work at level level itself scales with scales with . Consequently, the work decreases exponentially with , which means that the work for the multiplication is asymptotically dominated by the work at

In all the presented numerical experiments, when reference is made to a relative error in a computed (vector) parameter , this error has been calculated according to

where is the 2-norm and is an independently created reference solution. All the numerical experiments reported in this section have been performed on a PC with 20 GB of RAM and an AMD Opteron processor at 2.2 GHz, except for the ones reported in Section V-C, which were done on a PC with 64 GB of RAM and a Dual Intel Xeon X5482 processor at 3.20 GHz The used code is written in MATLAB 7.3.0. The computations were done in single precision except those reported in Section V-B which addresses the well known problem of bad conditioning of the EFIE at low frequencies. A. Complexity We illustrate the complexity analysis from Section IV by measuring the respective calculation times and matrix sizes for a series of three PEC spheres with growing by a factor four and the frequency correspondingly by a factor two (such that , does not change). The the average discretization size, of results are presented in Table I. All SVD thresholds were set to . The accuracy of the results is verified by comparing the computed bistatic RCS of the largest sphere with the exact Mie-series solution, see Fig. 4 (there is a very small discrepancy, regarding the depth of the nulls, probably attributable to the MoM discretization error). In Figs. 5 to 7 the various performance parameters from Table I are plotted as a function of . All the parameters scale approximately according to the theory in Section IV. In fact, all of them approach the asymptotic slope

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TABLE I PERFORMANCE PARAMETERS FOR MS-CBD APPLIED TO PEC SPHERES WITH RADIUS 1 m.  IS THE WORKING WAVELENGTH. N IS THE NUMBER OF RWG BASIS FUNCTIONS. L IS THE NUMBER OF MS-CBD LEVELS. THE BACK-SUBSTITUTION TIME IS FOR ONE INDEPENDENT VECTOR

Fig. 6. Compressed matrix size and MS-CBD size versus number of unknowns , see Section IV. for PEC spheres. Theoretical slope: N

Fig. 4. Bistatic RCS of PEC sphere, radius 1 m, wavelength 0.175 m, exact RWG basis functions. solution (Mie-series) and MS-CBD with N

= 196608

Fig. 7. MS-CBD time versus number of unknowns for PEC spheres. Theoretical slope: N , see Section V-A

B. Low Frequency EFIE

Fig. 5. Compressed matrix build time versus number of unknowns for PEC , see Section IV. spheres. Theoretical slope: N

from above. This has a simple explanation: the problems are not asymptotically large. Therefore the higher levels (with smaller blocks) contribute significantly to the overall time and storage. And the smaller blocks do not necessarily satisfy (31).

It is well known that the EFIE-MoM impedance matrix becomes badly conditioned at low frequencies (see, e.g., [27]), which causes iterative solution methods to converge very slowly or not at all. For this reason, the development of efficient lowfrequency preconditioners for the EFIE is a very active field of research (e.g., [28]). Direct LU solution does not suffer from this problem: as long as the condition number of the matrix (the ratio between the largest and the smallest singular values) does not exceed the inverse of the machine precision (typically for double precision, for single prein the order of cision), LU decomposition with pivoting is stable [29]. In this section we investigate the behavior of MS-CBD with respect to this problem (the computations in this paragraph have been done in double precision). On the one hand, the MS-CBD algorithm may be less robust than LU decomposition with pivoting. On

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TABLE II CONDITION NUMBER OF IMPEDANCE MATRIX FOR PEC SPHERE (3072 RWG BASIS FUNCTIONS) VERSUS AVERAGE DISCRETIZATION LENGTH

Fig. 10. Z matrix build time versus SVD threshold for MS-CBD analysis of PEC spheres.

Fig. 8. Relative error in the surface current versus SVD threshold for MSCBD  denotes the ratio of RWG discretization length analysis of PEC spheres. and wavelength.

1=

Fig. 11. Decomposition time versus SVD threshold for MS-CBD analysis of PEC spheres.

Fig. 9. Compressed matrix size versus SVD threshold for MS-CBD analysis of PEC spheres.

the other hand, and probably more importantly, a badly conditioned matrix is generally more sensitive to perturbations of the individual elements, such as the approximations introduced in the compression. , to a PEC sphere of unit We have applied MS-CBD radius discretized into 3072 RWG basis functions, with an av(variance ), excited by erage edge length (a common discretizaa plane wave with wavelength tion size for MoM), and subsequently increasing in steps of a factor ten. For every case, we computed the condition number

of the full impedance matrix and the direct LU solution as a reference. Table II shows the condition number as a function of . The table clearly indicates that for large , the condition number grows proportionally to . In Fig. 8, the relative error in the surface current is plotted against the SVD threshold used in the MS-CBD algorithm. The figure confirms that for badly conditioned matrices, such as those of low frequency EFIE, a smaller threshold (less perturbation) is necessary. In order to see whether the above results depend on , we repeated the experiment for a larger sphere, RWG basis functions and . The results, with also shown in Fig. 8, indicate a minimal dependence on the size of the problem. Figs. 9, 10 and 11 show the corresponding compressed matrix sizes, matrix build times and decomposition times, respectively.

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TABLE IV MULTISCALE CBD PERFORMANCE ON MONOSTATIC RCS COMPUTATION OF NASA ALMOND AT 50 GHz. N = 442; 089, L = 10, 1000 INCIDENCE ANGLES

Fig. 12. Monostatic RCS of NASA almond at 7 GHz; ( = 90 ), computed with MS-CBD, 8862 unknowns.

TABLE III MULTISCALE CBD PERFORMANCE ON MONOSTATIC RCS COMPUTATION OF NASA ALMOND AT 7 GHz. N = 8862, L = 4, 1000 INCIDENCE ANGLES

Fig. 13. Monostatic RCS of NASA almond at 50 GHz; ( = 90 ), computed with MS-CBD, 442.089 unknowns.

C. Monostatic RCS Among the most important advantages of the MS-CBD is the fast solution for multiple incident fields. As was noted in [18], which presents a method similar to the single level CBD, not only the impedance matrix can be compressed, but also the matrix that represents multiple incident fields, and the resulting matrix of surface current coefficients will be in compressed form too. The algorithms from Section III need no adaptation to implement this feature, all that is necessary is to use the ACA method to convert the matrix of incident field vectors into S-form. The simultaneous solution for a large number of excitation vectors is therefore much faster than times the solution for one independent vector. A common benchmark for monostatic RCS computations is the NASA almond (see, e.g., [30]). This is a PEC geometry with a length of 9.936 inch, shown in the inset of Fig. 12. The polarization of the incident field is along the -axis in the figure. We used the MS-CBS to compute the monostatic RCS at 7 GHz ). The perfor(discretized with an average edgelength of mance of the MS-CBD for these computations as a function of the SVD threshold is given in Table III (As mentioned in Section II, the ACA threshold is always 10 times smaller). The results are shown in Fig. 12. They compare well with [30]. The figure also shows that the result has converged for (meaning that we would obtain the same result for , which is identical to uncompressed LU decomposition). We have repeated this simulation at 50 GHz, using a correspondingly larger number of unknowns. The performance is

given in Table IV and the results are shown in Fig. 13. In this is needed to reach convergence everywhere, case, including in the low RCS region around (the tip of the almond). This is not because of a loss of precision for larger problems, but simply because for these incidence angles the surface currents must be computed to a very high accuracy (and even so these very low RCS values are affected by the RWG discretization error). VI. CONCLUSION The multiscale compressed block decomposition (MS-CBD) has been presented. This method allows for the direct solution of scattering and radiation problems with the MoMs, with highly reduced computational and storage requirements compared to conventional LU decomposition. For scattering and radiation problems in free space, the computational complexity of the algorithm is shown, both theoretically and through a numerand the storage requirements ical example, to be of order of order . Numerical examples show that for badly conditioned problems such as low frequency EFIE, the error can be adequately controlled by adapting the compression threshold. The efficiency of the method when a solution is needed for multiple excitation vectors is demonstrated with monostatic RCS computations: the total computation time for the NASA almond , and one thousand incidence anat 50 GHz, with gles, was about 12 hours and 20 minutes (with SVD threshold and ACA threshold ).

HELDRING et al.: MCBD FOR FAST DIRECT SOLUTION OF MOMs LINEAR SYSTEM

In this paper only perfectly conducting structures in free space have been addressed. However, the algorithm is entirely algebraic, and therefore applicable to almost any electromagnetic problem provided it is discretized with local basis functions, to allow for blocked compression of the linear system.

REFERENCES [1] R. F. Harrington, Field Computation by Moment Methods. New York: MacMillan, 1968. [2] E. Bleszynski, M. Bleszynski, and T. Jaroszewicz, “Adaptive integral method for solving large-scale electromagnetic scattering and radiation problems,” Radio Sci., vol. 31, no. 5, pp. 1225–1251, Sep.–Oct. 1996. [3] 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. [4] E. Michielsen and A. Boag, “A multilevel matrix decomposition algorithm for analyzing scattering from large structures,” IEEE Trans. Antennas Propag., vol. 44, no. 8, pp. 1086–1093, Aug. 1996. [5] J. M. Rius, J. Parrón, E. Úbeda, and J. R. Mosig, “Multilevel matrix decomposition algorithm for analysis of electrically large electromagnetic problems in 3-D,” Microw. Opt. Technol. Lett., vol. 22, no. 3, pp. 177–182, Aug. 5, 1999. [6] J. Parrón, J. M. Rius, and J. R. Mosig, “Application of the multilevel decomposition algorithm to the frequency analysis of large microstrip antenna arrays,” IEEE Trans. Magn., vol. 38, no. 2, pp. 721–724, Mar. 2002. [7] J. M. Rius, J. Parrón, A. Heldring, J. M. Tamayo, and E. Ubeda, “Fast iterative solution of integral equations with method of moments and matrix decomposition algorithm—Singular value decomposition,” IEEE Trans. Antennas Propag., Special Issue on Large and Multiscale Computat. Electromagn., vol. 56, no. 8, pp. 2314–2324, Aug. 2008. [8] Y. Saad, Iterative Methods for Sparse Linear Systems. Boston, MA: PWS Publishing, 1996. [9] A. Heldring, J. M. Rius, L. P. Ligthart, and A. Cardama, “Accurate numerical modeling of the TARA reflector system,” IEEE Trans. Antennas Propag., vol. 52, no. 7, pp. 1758–1766, Jul. 2004. [10] A. Heldring, J. M. Rius, and L. Ligthart, “New block ILU preconditioner scheme for numerical analysis of very large electromagnetic problems,” IEEE Trans. Magn., vol. 38, no. 2, pp. 337–340, Mar. 2002. [11] I. S. Duff, L. Giraud, J. Langou, and E. Martin, “Using spectral low rank preconditioners for large electromagnetic calculations,” Int. J. Numer. Meth. Engng., vol. 62, pp. 416–434, 2005. [12] A. Heldring, J. M. Rius, J. M. Tamayo, J. Parrón, and E. Ubeda, “Fast direct solution of method of moments linear system,” IEEE Trans. Antennas Propag., vol. 55, no. 11, pp. 3220–3228, Nov. 2007. [13] A. Heldring, J. M. Rius, and J. M. Tamayo, “Comments on fast direct solution of method of moments linear system,” IEEE Trans. Antennas Propag., to be published. [14] A. Heldring, J. M. Rius, J. M. Tamayo, and J. Parron, “Compressed block decomposition algorithm for fast capacitance extraction,” IEEE Trans. Comput. Aided Design, vol. 27, no. 2, pp. 265–271, Feb. 2008. [15] S. Kapur and D. E. Long, “IES3: Efficient electrostatic and electromagnetic simulation,” IEEE Comput. Sci. Engng., vol. 5, no. 4, pp. 60–67, Oct. 1998. [16] S. M. Seo and J. F. Lee, “A single-level low rank IE-QR algorithm for PEC scattering problems using EFIE formulation,” IEEE Trans. Antennas and Propag., vol. 52, no. 8, pp. 2141–2146, Aug. 2004. [17] K. Zhao, M. N. Vouvakis, and J.-F. Lee, “The adaptive cross approximation algorithm for accelerated method of moments computations of EMC problems,” IEEE Trans. Antennas Propag., vol. 47, no. 4, pp. 763–773, Nov. 2005. [18] J. F. Shaeffer, “LU factorization and solve of low rank electrically large MOM problems for monostatic scattering using the adaptive cross approximation for problem sizes to 1,025,101 unknowns on a PC workstation,” presented at the Proc. IEEE AP-S Int. Symp., Honolulu, HI, Jun. 10–15, 2007. [19] A. S. Y. Poon, R. W. Brodersen, and D. N. C. Tse, “Degrees of freedom in multiple-antenna channels: A signal space approach,” IEEE Trans. Inf. Theory, vol. 51, no. 2, pp. 523–536, Feb. 2005.

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[20] O. M. Bucci, C. Gennarelli, and C. Savarese, “Representation of electromagnetic fields over arbitrary surfaces by a finite and nonredundant number of samples,” IEEE Trans. Antennas Propag., vol. 46, no. 3, pp. 351–359, Mar. 1998. [21] M. Bebendorf, “Approximation of boundary element matrices,” Numer. Math., vol. 86, pp. 565–589, 2000. [22] M. Bebendorf, “Hierarchical LU decomposition-based preconditioners for BEM,” Computing, vol. 74, pp. 225–247, 2005. [23] W. Hackbusch, “A sparse matrix arithmetic based on H-matrices. Part I: Introduction to H-matrices,” Comp., no. 62, pp. 89–108, 1999. [24] T. Banachiewicz, “Zur Berechnung der Determinanten, wie auch der Inversen und zur darauf basierten Auflosung der Systeme linearer Gleichungen,” Acta Astronom. Ser. C, vol. 3, pp. 41–67, 1937. [25] W. H. Press et al., Numerical Recipes in C, 2nd ed. Cambridge, U.K.: Cambridge Univ. Press, 1994. [26] R. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shapes,” IEEE Trans. Antennas ropag., vol. 30, no. 3, pp. 409–418, May 1982. [27] W. Wu, A. W. Glisson, and D. Kajfez, “A study of two numerical solution procedures for the electric field integral equation at low frequency,” Appl. Computat. Electromagn. Soc. J., vol. 10, no. 3, pp. 69–80, Nov. 1995. [28] J. Zhao and W. C. Chew, “Integral equation solution of Maxwell’s equations from zero frequency to microwave frequencies,” IEEE Trans. Antennas Propag., vol. 48, no. 10, Oct. 2000. [29] L. N. Trefethen and D. Bau, Numerical Linear Algebra. Philadelphia, PA: SIAM, 1997. [30] A. C. Wool, H. T. G. Wan, M. J. Schuh, and M. L. Sanders, “Benchmark radar targets for the validation of computational electromagnetics programs,” IEEE Antennas Propag. Mag., vol. 35, no. 1, pp. 84–89, Feb. 1993.

Alex Heldring was born in Amsterdam, The Netherlands, on December 12, 1966. He received the M.S. degree in applied physics and the Ph.D. degree in electrical engineering from the Delft University of Technology, Delft, The Netherlands, in 1993 and 2002, respectively. He is presently working as an Associate Professor at the Telecommunications Department, Universitat Politecnica de Catalunya, Barcelona, Spain. His special research interests include integral equation methods for electromagnetic problems and wire antenna analysis.

Juan M. Rius (SM’00) received the “Ingeniero de Telecomunicación” degree in 1987 and the “Doctor Ingeniero” degree in 1991, both from the Universitat Politècnica de Catalunya (UPC), Barcelona, Spain. In 1985, he joined the Electromagnetic and Photonic Engineering Group, Department of Signal Theory and Communications (TSC), UPC, where he currently holds a position of “Catedrático” (equivalent to Full Professor). From 1985 to 1988, he developed a new inverse scattering algorithm for microwave tomography in cylindrical geometry systems. Since 1989, he has been engaged in the research for new and efficient methods for numerical computation of electromagnetic scattering and radiation. He is the developer of the graphical electromagnetic computation (GRECO) approach for high-frequency RCS computation, the Integral Equation formulation of the Measured Equation of Invariance (IE-MEI) and the multilevel matrix decomposition algorithm (MLMDA) in 3D. Current interests are the numerical simulation of electrically large antennas and scatterers. He has held positions of “Visiting Professor” at EPFL (Lausanne) from May 1, 1996 to October 31, 1996; “Visiting Fellow” at City University of Hong Kong from January 3, 1997 to February 4, 1997; “CLUSTER chair” at EPFL from December 1, 1997 to January 31, 1998; and “Visiting Professor” at EPFL from April 1, 2001 to June 30, 2001. He has more than 48 papers published or accepted in refereed international journals (27 in IEEE TRANSACTIONS) and more than 142 in international conference proceedings.

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José M. Tamayo was born in Barcelona, Spain, on October 23, 1982. He received the Mathematics degree and the Telecommunications Engineering degree from the Universitat Politècnica de Catalunya (UPC), Barcelona, Spain, both in 2006, where he is currently working toward the Ph.D. degree. In 2004, he joined the Telecommunications Department, UPC. His current research interests include accelerated numerical methods for solving electromagnetic problems and parallelization.

Josep Parrón was born in Sabadell, Spain, in 1970. He received the Telecommunication Engineer degree and the Doctor Engineer degree from the Universitat Politécnica de Catalunya (UPC), Barcelona, Spain, in 1994 and 2001, respectively. From 2000 to 2002, he was with the Electromagnetic and Photonic Engineering Group, Signal Theory and Communication Department, UPC, as an Assistant Professor. Since 2002, he has been an Associate Professor in the Telecommunication and Systems Engineering Department, Universitat Autónoma de Barcelona (UAB), Spain. His research interests include efficient methods for numerical computation of electromagnetic scattering and antenna radiation. He is the author or coauthor of more than 50 technical journal articles and conference papers.

Eduard Ubeda was born in Barcelona, Spain, in 1971. He received the Telecommunication Engineer degree and the Doctor Ingeniero degree from the Polytechnic University of Catalonia (UPC), Barcelona, Spain, in 1995 and 2001, respectively. In 1996, he was with the Joint Research Center, European Commission, Ispra, Italy. From 1997 to 2000, he was a Research Assistant in the Electromagnetic and Photonic Engineering Group, UPC. From 2001 to 2002, he was a Visiting Scholar in the Electromagnetic Communication Laboratory, Electrical Engineering Department, Pennsylvania State University (PSU). Since 2003, he is at UPC. He is author of 15 papers in international journals and 35 in international conference proceedings. His main research interests are numerical computation of scattering and radiation using integral equations.

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Electromagnetic Scattering From Electrically Large Arbitrarily-Shaped Conductors Using the Method of Moments and a New Null-Field Generation Technique Tyler N. Killian, Sadasiva M. Rao, Fellow, IEEE, and Michael E. Baginski

Abstract—In this work, a new numerical procedure is developed to apply the well-known method of moments (MoM) formulation to electrically large conducting bodies of arbitrary shape. The numerical procedure involves developing a combination of subdomain-entire domain basis functions which result in a sparse moment matrix as opposed to a full matrix in the traditional method. Moreover, the zeros in the MoM matrix are precisely at the same locations where one would have encountered the most significant values. The solution of the new matrix may be obtained using the simple Gauss-Seidel iterative procedure with only two or three iterations. All the traditional advantages of the MoM procedure are retained including the solution for multiple incident fields. Several numerical results are presented to illustrate the validity of the new approach. Index Terms—Fast method, large-body problems, method of moments (MoM), RWG basis.

I. INTRODUCTION HE solution of large problems with the method of moments (MoM) [1] is typically limited by excessive computational and memory requirements. Direct solvers, such as LU decomposition, require that we store the entire matrix. Since where is the number of this requirement grows as unknowns, the storage requirement will exceed even available hard disk space. Furthermore, directly solving a linear system operations making this approach highly inefrequires ficient for large problems. Also, iterative methods, such as the bi-conjugate gradient solver do not converge for large numbers of unknowns and typically require preconditioning. Fast solvers, such as the fast multipole method [2] make use of matrix approximations in order to accelerate the solution procedure. These approximations may introduce noise into the solution and make it difficult to calculate the exact current solution and therefore find sensitive values such as input impedance. In this work, we present a new procedure to solve the MoM problems for electrically large bodies. The MoM matrix generated in this new procedure is a block-wise sparse matrix, im-

T

Manuscript received January 12, 2010; revised June 14, 2010; accepted August 03, 2010. Date of publication December 03, 2010; date of current version February 02, 2011. T. N. Killian is with the Georgia Tech Research Institute, Atlanta, GA 30332 USA (e-mail: [email protected]). S. M. Rao is with the Naval Research Laboratory, Washington DC 20375 USA (e-mail: [email protected]). M. E. Baginski is with the Electrical and Computer Engineering Department, Auburn University, Auburn, AL 36849 USA. Digital Object Identifier 10.1109/TAP.2010.2096186

plying that several blocks of elements are identically zero, as opposed to a full matrix in the traditional MoM solution. A set of new basis functions are developed using the traditional sub-domain basis functions to achieve the required sparsity. The null elements in the new matrix are not random but are designed such that these elements would have been the most significant elements, representing the near-field interaction between the source and the observer, in the traditional MoM matrix. The validity, accuracy, and efficiency of this new procedure for two dimensional bodies is presented in [3] and, in this work, we apply it to three-dimensional complex bodies. In the next section, a detailed description of the numerical procedure is presented. In Section III, several important guidelines to efficiently apply the numerical procedure and a few important observations are presented. In Section IV, several numerical examples are described and quantitatively tested for accuracy. Finally, some conclusions are presented in Section V. II. DESCRIPTION OF THE NUMERICAL PROCEDURE To begin the numerical procedure, we divide the given large structure into several smaller substructures (groups). For a disjoint group of objects, such as phased-array antennas, each antenna element may be treated as one group. It is also possible to gather fewer elements into each group and treat each element as multiple groups. Furthermore, there is no requirement that each group must be the same size. Next, each substructure is approximated using planar triangular patches and the standard RWG [4] basis functions are used as primary basis functions for further numerical processing. Here, we note the substructures induce a partitioning on the system matrix. Now, let us consider a substructure and call it a source block. The groups adjacent to and surrounding the source block may be termed as near-field blocks. The goal here is to set the MoM matrix elements corresponding to the interaction between the source block and the near-field blocks to zero. Note that these neighboring blocks represent near-field terms which are strongly coupled to the source block and therefore represent the dominant terms in the classical MoM procedure. In the new procedure, these terms are reduced to zero. We can achieve this goal by using a linear combination of classical RWG functions on the near-field blocks with unknown coefficients. The unknown coefficients are obtained by enforcing the condition that the interaction terms must be zero. Thus, we effectively eliminate the dominant near field interactions between each group and the neighboring groups. Note that the far-field interaction terms are not zero in this new method. However, the far-field

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Fig. 1. Disjoint bodies illuminated by EM Plane wave.

interactions are weak and have less influence on the solution of the system matrix. The overall result of this new procedure is the generation of a block-diagonally dominant matrix, which will converge very quickly with an iterative procedure, typically in one or two iterations if the groups are chosen properly. Furthermore, we can construct an initial guess for the solution that will greatly speed up the iterative process. The method may be explained further as follows: Referring to Fig. 1, we have four disjoint bodies each mod, and sub-domain RWG functions and let eled by . Let represent the classical RWG basis functions which are also used as testing functions, for the sake of simplicity of description. Obviously, a conventional MoM matrix procedure with basis and testing functions generates a Z-matrix, given given by by (1)

represents the coupling between Note that the submatrices bodies and , respectively. In the new procedure, let us consider Body #1 as the source block, Bodies #2 and #3 as near-field blocks and Body #4 as a far-field block, respectively. Similarly, when we consider Body #2 as the source block, we will treat Bodies #1, #3, and #4 as near-field blocks. For this case, we do not have a far-field block. As stated before, our goal is to generate a new MoM matrix and must be where the matrix elements of submatrices identically equal to zero. We achieve such a goal as follows. 1) Define a set of new basis functions (2) where and are unknown coefficients. 2) Write the following equations using the new basis functions and testing with the classical testing functions on Bodies #2 and #3. Thus, we have:

(3) for where the integral operator on the expansion functions.

is

3) Note that the (3) is a system of equations of dimension . Further, the RHS of (3) represents multiple right-hand side vectors for each value of . 4) Obviously, (3) can be solved exactly for the coefficients and . , the new Once we determine the unknown coefficients are defined. basis functions 5) Now, the new basis functions, when used in the MoM procedure, results in a new -matrix where in the submais a full matrix and submatrices and are trix remains non-zero but it represents null matrices. Also, far-field interaction terms. 6) Using a similar procedure, the submatrices , and can also be set to zero. 7) The new -matrix will be of the form

(4)

which can be easily solved to obtain the solution. 8) The new MoM matrix can be most efficiently solved by iterative methods. For the initial guess, one can set all the non-zero off-diagonal blocks to zero and obtain the unknown currents. The current values can be improved by using the matrix presented in (4) and using a simple block Gauss-Seidel method. 9) Note that the solution converges very quickly, typically in one or two iterations, since the strongest interaction terms are set to zero. III. GUIDELINES AND OBSERVATIONS In this section, we discuss a few important considerations to develop an efficient algorithm. The algorithm development involves five basic steps enumerated as follows: 1) Group basis functions—The geometry of the problem is divided up into disjoint groups as shown in Fig. 2. Each group consists of a set of RWG sub-domain basis functions. The sub-domain functions are necessary so that functions near to one another in the geometry will be strongly coupled and those that are distant will be weakly coupled. For the arbitrary three-dimensional conducting case, a good rule of thumb is that the group sizes should be around 1 . There are a few good reasons for this. First, all the functions in the group should be strongly coupled with one another to ensure the final matrix is diagonally dominant. Next, surrounding null-field points are picked according to a nearest-neighbor group criteria. In other words, null field points are picked up one group at a time rather than on an individual basis function level. When solving for the new basis functions, a linear system with unknowns must be is the number of test points for a given solved, where group. Thus, if the number of test points is too large, then this matrix system will be too expensive to solve. These properties indicate that there is an upper limit to how large the groups should be. However, the groups should still be made as large as possible since only one linear system per group must be solved. In this way, functions within a group

KILLIAN et al.: ELECTROMAGNETIC SCATTERING FROM ELECTRICALLY LARGE ARBITRARILY-SHAPED CONDUCTORS USING THE MOMs

can share information and therefore reduce the computaappears to meet the tional requirements. A choice of above requirements and serves as a good rule when there is no other obvious way to form the groups. 2) Basis transformation—We replace each function in a group with a linear combination of the original sub-domain functions such that they create null-fields everywhere around the group within a certain radius. The radius divides the geometry into a “near-field” and a “far-field” region as seen in Fig. 2. A good choice for the conducting case is around . In theory, the larger the radius, the more effective the “decoupling” process becomes. However, if the radius includes too many points, solving the aforementioned matrix system may become too expensive. 3) Create new system matrix—Since we have performed a basis transformation, we must create a new system matrix representing the coupling between the new functions. Furthermore, since the new functions are linear combinations of the original sub-domain functions, the new matrix elements are linear combinations of the original system matrix elements. We can represent this step with the following equation: (5) where is the standard MoM matrix, is the newly created matrix, and is a sparse matrix representing the basis transformation. The coefficients for the new basis functions to perform the basis are placed along the columns of transformation. 4) Make initial guess—To begin the solution procedure, we invert the diagonal blocks of to create an initial guess. This step is computationally inexpensive and accounts for all the interaction within each group as well as all reflections between each group and the surrounding geometry. Since it represents the dominant portion of the interaction exactly, this solution will have decent accuracy. Further accuracy can be obtained through a simple iterative procedure. 5) Use iterative solver for more accuracy—Here, we use the block version of the Gauss-Seidel method. This solver requires a diagonally-dominant matrix which we obtain through the basis transformation. Furthermore, each iteration may be interpreted as a kind of time step. Interpreting it this way, we see that each step accounts for reflections between the groups on the structure. Thus, since the local interaction is solved exactly and only a few reflections occur between elements that are separated by a long distance, only a few iterations are necessary here.

A. Operations Scaling There are three main procedures to observe in order to determine how the operations count scales. In step 2, we must solve a linear system for each group in order to compute the coefficients for the new basis functions. These systems will be small with respect to the overall system matrix. Next, we can assume that each group size is roughly the same and that each group

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Fig. 2. Groups for an arbitrary three-dimensional conducting surface.

includes roughly the same amount of null-field points. Furthermore, the number of groups will scale linearly with the problem . The size. Thus the operations count for this step scales as next procedure, given in step 3, is to create a new system matrix due to the basis change. We can view the new system matrix as being the product given by the right hand side of (5). So, in this step and then procedure, one must fill , which is an perform the matrix multiplication. Typically, matrix multiplicaprocedure, but in this case our basis functions tion is an do not span the entire structure. Thus, is a highly sparse matrix and the operations count for this stage is . Finally, in step 5, we must solve the system to obtain the currents. One iteration of the Gauss-Seidel solver is roughly the same as a matrix-vector multiplication. Furthermore, only a few iterations are step. So the overall solution proceapplied so this is an . Finally, each one of the above steps dure scales with may be performed in a highly parallel fashion to fully utilize computer resources. B. Memory Requirements Memory usage is largely dependent upon how the algorithm is implemented and of course it may be implemented differently for varying requirements. As implemented in this work, there are three major parts requiring significant memory usage. First, we must store the coefficients for the new basis functions. The memory requirement here is a total of (6) is the total number of groups, is complex numbers where is the number of test points for the size of the th group, and group . Note this is the storage requirement for the matrix mentioned earlier. Next, we must allocate enough scratch space to solve for the new basis coefficients. If is the maximum number of test points for any given group, then we must allomatrix of complex values. Finally, we imcate a single plemented the iterative solver by storing one block row of at a time as we progress through each iteration step. This requires matrix where is the largest group in the storing an is the total number of unknowns. Also, the masystem and trix for the test points and the matrix slice appear in two different parts of the algorithm and need not be allocated simultaneously. Thus, the memory requirement is approximately (7)

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2 5.5 m square plate.

C. Block Gauss-Seidel Method Here, our iterative solver is the block version of the GaussSeidel method [5]. The Gauss-Seidel solver is a standard algorithm for solving a linear system:

Fig. 4. Real part of current for 5.5 m x 5.5 m square.

(8) If we have a partitioned system and a solution guess , the iterative step for obtaining a more accurate set of currents is as follows: (9)

and are th subvectors of and , rewhere each spectively, each is a submatrix of , and there are column blocks in . Now, the part contained in parentheses in (9) indicates that the summation represents the scattered fields from the previous set of currents. Therefore, the new current solution is due to energy that is being reflected between the groups on the structure. Next, due to the nature of the null-field producing basis functions, the interaction within each group and with its surrounding groups is accounted for exactly. Further iteration takes into account reflections between functions that are very weakly coupled due to their geometrical separation and therefore very few significant reflections will occur. Thus, the number of necessary iterations will be very low, typically around 2 or 3 iterations for scattering cases.

Fig. 5. Imaginary part of current for 5.5 m x 5.5 m square.

D. Convergence of Current Solution To demonstrate the convergence of current values on the structure, we solve a system that allows one to apply the null-field technique as well as perform a full LU decomposition in memory. Here, we choose a 5.5 m 5.5 m square, as shown at an angle in Fig. 3 with an incident wave of and a frequency of 300 MHz. The total of unknown count is 10225. The square may be easily divided into groups using a grid as shown in Fig. 3. We use a 6 6 , close to our rule grid since it gives us a group size of . The plots in Figs. 4 and 5 show the real and of thumb of imaginary parts of the current for a part of the square. The first three iterations and the full LU decomposition results are given. Only a small subset of the basis function coefficients have been shown for readability. Next, in Fig. 6, we show the maximum error per term over several iterations for various choices of near-field distances. The

Fig. 6. Maximum term error for 5.5 m x 5.5 m square for 10 iterations and multiple near-field radii.

maximum error per term is defined as the maximum as seen in the left hand side of (10). Fig. 7 shows the total error (sum of all term errors) for the same settings. As would be expected, choosing a larger near-field radius produces faster convergence. Furthermore, the solution is seen to converge to the exact answer after only a few iterations. E. Choice of Group Size and Near Field Radius We can make further observations based on the 5.5 m 5.5 m square given in the previous section. In Table I, we show run-

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TABLE I

RUNTIMES FOR 5.5 M

2 5.5 M SQUARE PLATE FOR VARIOUS GROUP SIZES AND NEAR-FIELD REGIONS

near-field distance does not affect the runtime because the actual solution procedure is not changing. Also, since the group size is so large, including an additional neighboring group may result in a dramatic increase in the runtime. However, we may use the information given to obtain a decent rule of thumb that group size and a will work fine in most cases. In fact, a near-field radius appears to work well for most cases. This combination permits a nice tradeoff between runtime, rate of convergence, and sensitivity to choice of near-field radius. F. Parallel Processing Fig. 7. Total error for 5.5 m x 5.5 m square for 10 iterations and multiple nearfield radii.

times for various choices of group size as well as near-field radii. Runtimes are given for 4 threads running on a 2.67 GHz Intel Core i7 processor. Those labeled as “n/a” are not valid combinations since the near-field radius too small to include any surrounding groups in those cases. Places marked with a “D” are valid combinations, but the iterative solution was divergent in those cases. Note that the divergent solutions form a diagonal line across the table. This table gives insight as to a proper choice of group size and near-field radius. For a given near-field radius, decreasing the group size (moving down a column) results in higher runtimes since information is reused less and more linear systems must be solved in order to determine the new set of basis functions. Increasing the group size (moving up a column) decreases the runtimes, but moves the choice closer to the divergent solutions along the diagonal. Thus, it is necessary to increase the near-field radius in order to regain a properly convergent solution. As can be seen in the table, the proper choice of group size and near-field radius may not be unique, but there are other considerations to take into account. For example, the run times may be the same for two choices, but their convergence rates may not be. In general, increasing the group size leads to faster convergence since, for a given group, its nearest neighboring groups are also enlarged. Including those groups then has a larger impact on the matrix conditioning since more terms will be eliminated. However, the group size may not be increased arbitrarily. This is due to the fact that as the group size increases, the centroids of adjacent groups are moved further apart. Thus, a significant increase in near-field choice may not include any extra groups. This can be seen in the first few rows of the table where, moving across a row, an increase in

Since each part of the solution procedure can be further divided into computationally separate pieces, the method is highly amenable to parallel processing. In this section, we will discuss how each part of the algorithm may be implemented in a parallel scheme. In order to construct the new system matrix , we must first generate the elements of the original matrix . Each one of the matrix elements may be computed independently from the other elements. Furthermore, the elements may be distributed evenly across each of the processors allowing for equal work load. Therefore, the speed of this part of the algorithm tends to scale linearly with the number of processors used. In the present work, only one block-row of was stored in memory at a time. The row was broken up according to the group choice and then the column blocks corresponding to the groups were divided amongst the processors. Once has been computed, the mamust be carried out. For this step, columns of trix product may be distributed over the processors so that this becomes a highly parallel matrix multiply. Thus, creating is a highly parallel process. When solving for the coefficients used to construct the new basis functions, each source group can be considered separately. Each processor may be assigned a set of groups, each of which requires a single matrix inversion. If the groups are similar in size, they may simply be distributed evenly across the processors. If they are not similarly sized, the processors can each get a different number of groups in order to balance the load. Here, dynamic scheduling may be used so that when a CPU becomes available, it simply advances to the next group and begins solving for weights. In this type of situation, if one CPU becomes occupied with a large set of coefficients, the remaining CPUs can continue to work on smaller groups. Good load balancing can be achieved in this way. For example, one CPU can solve for the coefficients on one group with N null-field points, while another can solve eight groups, each of which has N/2

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null-field points. Here, each group solution carries a complexity where N is the number of null-field points that must of be produced for that source group. Typically, one iteration of the solver takes roughly the same amount of time as does one matrix-vector multiply with the same dimensions as the system size. Therefore, the iteration procedure generally does not require parallel processing since it is a very fast procedure and requires only a few iterations. In this work, this procedure was not made parallel. However, it is possible to create a parallel version if necessary. If the Gauss-Seidel version is used, then as we move from one group to the next solving for currents, we may take each group and assign subsets of the basis functions to each processor. Each processor can then form the summations given in (9). The inverse in (9) may then be done by a single CPU. Alternatively, one could use a block Jacobi iterative solver, which is essentially the same as the Gauss-Seidel solver with the exception that once a set of currents are found for a group, they are not reused until the next iteration. The Gauss-Seidel version uses those new current values immediately, leading to faster convergence. In a parallel scheme, the block Jacobi solver would have the benefit that each group of currents could be found independently and thus the algorithm is slightly more efficient than the Gauss-Seidel version. The graphs in Figs. 8 and 9 demonstrate the parallel efficiency of the arbitrary three-dimensional code. The test problem was a linear array of cubes, each one has dimensions of and they are each spaced apart. Each cube has 72 unknowns and there are 40 cubes for a total of 2880 unknowns. Nulls were placed on adjacent boxes so that the two cubes on the ends of the array generated 72 null points and those within the array generated 144. Using different numbers of CPUs, the entire program was timed to display the parallel efficiency of the overall algorithm. This involves generating the new basis coefficients, creating the new system matrix, a single matrix fill, and one iteration of the block Gauss-Seidel solver. Two test machines were used. The first has 6 CPUs, each of which is a 64-bit 1.4 GHz Intel Itanium 2 processor. Here, the times were taken for 1, 2, 4, and 6 CPUs. For each case, we plot the ratio of the time for a single processor to that of the time for the given case. The results are shown in Fig. 8. The second machine has 8 2.3 GHz AMD Opterons. For this machine, 1, 2, 4, 6, and 8 processors were used. The results for this test are given in Fig. 9. The ideal ratio for each case is also given. This test demonstrates that the overall algorithm is almost ideally parallel which is difficult to achieve in standard methods due to typical solvers such as LU decomposition. IV. NUMERICAL RESULTS In this section, we present a few numerical results to verify the accuracy of the algorithm. Here, we consider two cases viz. a) single large body and b) a finite periodic array. A. Electrically Large Conducting Body First, using the combined field integral formulation (CFIE) formulation [6], we show a bistatic RCS calculation for a 5 radius sphere, where is the wavelength for the given frequency. In this case, we have 92 550 unknowns. The structure is diin area. For each source vided into 314 groups, each roughly

Fig. 8. Ratio of single CPU time to 1, 2, 4, and 6 processors for machine one.

Fig. 9. Ratio of single CPU time to 1, 2, 4, 6, and 8 processors for machine two.

group, all groups within a 2 distance were included as nearfield groups. Fig. 10 shows the bistatic RCS for 1 and 2 iterations as well as the RCS when computed with a MoM BOR (Body of Revolution) code. The sphere is centered at the origin and has an at an angle of with a incident field of frequency of 300 MHz. The RCS is calculated for the x-z plane, and is shown in Fig. 10. Since there is very little difference between the plots, a single iteration suffices in this case. Also, note that, for this code, the initial guess mentioned in Section III is implemented as the first iteration of the solver. Thus, the first iteration only includes near-field interactions. A good way to determine the quality of our solution is to calculate the following: (10) where is the excitation vector, is the standard moment matrix using RWG functions, and is the current vector obtained using the present solution. The total error is then given by: (11) where is the number of unknowns for the system. The average error per term is then given by: (12)

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Fig. 11. Incident wave and coordinate system for 12

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2 12 square plate.

Fig. 10. Bistatic RCS for 5 radius sphere. First and second iterations and MoM BOR code.

For the case of the sphere, the average error per term is .096 for 1 iteration, while it is only .0142 for two iterations. So, the spherical case compares very well with a standard Method of Moments solution. The RCS was also compared to a MoM BOR (Body of Revolution) code and shows excellent agreement. Also, when computed on 8 3.0 GHz AMD Opteron CPUs, the solution took approximately 25 wall clock hours and less than 4.5 GB of storage. This includes the time taken for a serialized RCS calculation on a single CPU (around 2 hours). Storing the full matrix in memory would require approximately 64 GB of storage. Furthermore, an LU decomposition would require an estimated 19 days and 11 hours to solve on 8 CPUs assuming a perfectly parallel solver. Also, the weight generation step shows proper scaling when using multiple CPUs. For a single CPU, the weight generation procedure takes 49 hours. For 4 CPUs, this step took 9 hours and 14 mins. It required only 4 hours, 28 mins for 8 CPUs. For the next example, we compute the bistatic RCS of a 12m 12 m square plate with 42883 unknowns. The plate -plane, with a corner of the plate at the is placed in the origin of the coordinate system as seen in Fig. 11. Here, we have an open structure and therefore utilize only the electric field integral equation (EFIE) [4]. The plate is excited with an at an angle of with a incident wave frequency of 300 MHz. The RCS is computed for the x-z plane. The groups were formed by dividing the plate into a square in size and having on grid of 144 groups, each roughly average 298 unknowns. Using a near-field criteria, the solver went through two iterations and the RCS for each is shown in Fig. 12. The average error for the first iteration is 0.612 while the average error dropped to 0.232 in the second iteration. Also, when computed on 8 3.0 GHz AMD Opteron CPUs, the solution took approximately 2 hours 45 minutes wall clock time and less than 2 GB of storage. Storing the full matrix in memory would require 13.7 GB of memory. Furthermore, computing a LU decomposition on 8 CPUs would take approximately 1 day and 22 hours assuming the solver is fully parallel. As a final example for continuous body problems, we also calculate the bistatic RCS of a French Mirage fighter jet. The geometry and coordinate system may be seen in Fig. 13. The jet points direction while points straight above the aircraft. For in the

Fig. 12. Bistatic RCS for 12 tions are shown.

2 12 square plate. The first and second itera-

this case, we have an open structure (the rear of the jet is not closed off) and therefore use the EFIE only. From end to end, for an incident frequency of 750 the jet is approximately in width. Also, MHz. Across the wings, it is approximately from the bottom to the highest point on the plane (vertical fin), high. The total surface area is . There are it is about 159293 unknowns and so a full LU decomposition in this case would be computationally expensive both in terms of memory and CPU time. In fact, the required storage for a full LU decomposition would be 189 GB. Furthermore, the LU decomposition itself would take approximately 99 days 5 hours assuming the solver was fully parallel. For this case, we formed the groups by placing the aircraft inside a cubical grid with each cell having di. All functions within a given cell were mensions of grouped together. There are 702 groups with an average group size of 226 functions and all groups within 2 of a given source group were decoupled with null field points. Using an incident at an incident angle of with a wave frequency of 750 MHz, we calculate the bistatic RCS for the x-z plane as shown in Fig. 14. After 2 iterations, the average error per term for the first iteration is 0.224 while the error for the second iteration is 0.218. Also, note that the grouping scheme was generated automatically and thus it is possible to automate the entire process for an arbitrary three-dimensional conductor. B. Periodic Arrays In this section, we use our method to solve a special case of arbitrary 3D conductors. Here, we solve for the case of finite con-

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Fig. 16. Elimination of groups within near-field for finite periodic array.

Fig. 13. Triangular mesh for French Mirage aircraft geometry.

Fig. 17. Geometry for 50

Fig. 14. Bistatic RCS for aircraft. The first and second iterations are shown.

Fig. 15. Finite periodic conducting array periodic in two dimensions.

ducting periodic arrays. The motivation behind this special case is to demonstrate that the geometry behind certain problems may be utilized to further enhance the solution efficiency. Although each of the array elements may be any arbitrary three-dimensional surface, the arrays themselves are periodic only in two dimensions as shown in Fig. 15. However, the concepts applied here may easily be extended to cases where there is periodicity in three dimensions.

2 50 0.5

square plate finite periodic array.

For the case of periodic arrays, the element grouping may done in a number of ways. Although nothing prevents each array element from being subdivided into groups, we are assuming here that each element forms one and only one group. Null fields are produced on those groups that are horizontally and vertically adjacent to a given source group as shown in Fig. 16. For groups on the edges and corners of the array, there will only be three and two adjacent groups, respectively. The finite periodic array is a special case in the general method because the weights used to produce the null fields are largely redundant. In fact, for two-dimensional arrays, one must solve for at most 9 sets of weights. These are each of the four corners, one element along each of the four sides, and one element internal to the structure. Once these weights have been computed, they can easily be copied to the other groups in the system. It greatly simplifies the programming and is computationally inexpensive to compute all the corners separately. Note that while the physical structure may show symmetry for edge groups and corner groups, this symmetry may not exist numerically due to the meshing of each element and the basis functions defined there. So, in general, one cannot solve for the coefficients at one corner of the structure and then directly copy those weights to the other three corners. The same reasoning applies to elements along the edges of the array. Here we present a bistatic RCS calculation for a finite perisquare plate, each odic array. Each element is a expanded with 64 RWG basis functions. There are 2500 plates arranged on a 50 50 grid for a total of 160000 unknowns. Each plate is spaced 0.75 from each of the adjacent plates. The array is excited with a 300 MHz incident wave with at an angle of . Finally, the bistatic RCS is calculated for the x-z plane. Fig. 17 displays the geometry for the problem. Fig. 18 shows a comparison between the first and second iterations of the Gauss-Seidel scheme. Note that there

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[6] J. R. Mautz and R. F. Harrington, “H-field, E-field, and combined-field solutions for conducting bodies of revolution,” Archiv fuer Elektronik und Uebertragungstechnik, vol. 32, pp. 157–164, Apr. 1978. [7] S. M. Rao, “Electromagnetic Scattering and Radiation of ArbitrarilyShaped Surfaces by Triangular Patch Modeling,” Ph.D. dissertation, Univ. Mississippi, University, 1980. [8] T. K. Sarkar and S. M. Rao, “An iterative method for solving electrostatic problems,” IEEE Trans. Antennas Propag., vol. 30, pp. 611–616, Jul. 1982. [9] T. K. Sarkar and S. M. Rao, “The application of the conjugate gradient method for the solution of electromagnetic scattering from arbitrarily oriented wire antennas,” IEEE Trans. Antennas Propag., vol. 32, pp. 398–403, Apr. 1984.

Fig. 18. Bistatic RCS result for finite periodic array after 1 and 2 iterations.

is a negligible difference between the RCS values. When compared to the standard MoMs solution, the average error per current term for 1 iteration was 0.288 while it was 0.122 for 2 iterations. Here the average error per term is calculated in the same manner as done for the arbitrary case above. When computed on 8 3.0 GHz AMD Opteron CPUs, the solution took approximately 19.5 wall clock hours and less than 500 MB of storage. This includes the time taken for a serialized RCS calculation on a single CPU (around 3 hours). A full LU decomposition of this problem would require 190 GB of storage and would have a solution time of approximately 100 days and 13 hours assuming the solver is fully parallel. Again, note that, for this code, the initial guess mentioned in Section III is implemented as the first iteration of the solver. Thus, for this case, the simulation shows that the currents are almost entirely determined by the near-field interactions on the structure. V. CONCLUSION In this work, a new algorithm, based on the method of moments solution procedure, is presented and is applicable to electrically large bodies and large but finite phased array structures. The algorithm can be fully automated and, if implemented properly, results in substantial savings in terms of computer memory and execution times. It may be possible to increase the accuracy by using other iterative solvers, such as steepest decent method [8] and conjugate gradient method [9], and reduce the number of iterations. Although not tested here, it is envisaged that the algorithm becomes more and more efficient for larger and larger bodies. Finally, due to the algebraic nature of the algorithm, the method should apply to other subdomain basis functions and formulations as well. REFERENCES [1] R. F. Harrington, Field Computation by Moment Methods Classic Reissue. New York: IEEE Press, 1993. [2] R. Coifman, V. Rokhlin, and S. Wandzura, “The fast multipole method: A pedestrian prescription,” IEEE AP-S Mag., vol. 35, pp. 7–12, 1993. [3] T. N. Killian, S. M. Rao, and M. E. Baginski, “A new method of moments solution procedure to solve electrically large electromagnetic scattering problems,” Comput. Model. Engrg. Sci., vol. 46, no. 3, pp. 255–269, 2009. [4] S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces or arbitrary shape,” IEEE Trans. Antennas Propag., vol. 30, pp. 409–418, May 1982. [5] J. R. Westlake, A Handbook of Numerical Inversion and Solution of Linear Equations. New York: Wiley, 1968.

Tyler N. Killian received the B.E.E., M.S., and Ph.D. degrees in electrical engineering from Auburn University, Auburn, AL. He currently works at the Signature Technology Laboratory, Georgia Tech Research Institute, Atlanta. His research interests include numerical methods in antenna and scattering problems.

Sadasiva M. Rao (F’00) received the Bachelors degree in electrical communication engineering from Osmania University, Andhra Pradesh, India, in 1974, the Masters degree in microwave engineering from Indian Institute of Sciences, India, in 1976, and the Ph.D. degree with specialization in electromagnetic theory from University of Mississippi, University, in 1980. He served as an Assistant Professor in the Department of Electrical Engineering, Rochester Institute of Technology, from 1980 to 1985, Senior Scientist at Osmania University, from 1985 to 1987, and currently, as a Professor in the Electrical Engineering Department, Auburn University, Auburn, AL. He also held visiting Professorships at University of Houston, Osmania University, and Indian Institute of Science. He worked extensively in the area of numerical modeling techniques as applied to electromagnetic/acoustic scattering. He published/presented over 150 papers in international journals/conferences. His research interests are in the area of numerical methods applied to antennas and scattering. Dr. Rao and his team at the University of Mississippi, were the original researchers to develop the planar triangular patch model and to solve the problem of EM scattering by arbitrary shaped conducting bodies. For this work, he received the best paper award for the period 1979–1981 from SUMMA Foundation. For his contributions in numerical electromagnetic problems, he was awarded the status of Fellow of IEEE. Further, he was recognized as a highly cited researcher by Thomson ISI in 2001.

Michael E. Baginski received the B.S., M.S., and Ph.D. degrees, all in electrical engineering, from Pennsylvania State University, University Park. He is currently an Associate Professor of electrical engineering at Auburn University, Auburn, AL, where he has resided since the completion of his doctorate. His research interests include analytic and numerical solutions to transient electromagnetic problems, transient heat flow and solid state structural analysis using finite element routines, EMI and EMC characterization of MCMs and PCBs, simulation of rapid thermal expansion of metals under the action of large electric currents, S-parameter permittivity extraction routines, synthetic aperture radar (SAR) design and data processing routines, and the use of genetic algorithms for antenna optimization. Dr. Baginski is a member of Eta Kappa Nu, Sigma Xi, the New York Academy of Sciences, and the IEEE Education and Electromagnetic Compatibility Societies. He is also a member of Who’s Who in Science and Engineering and Who’s Who Among Americas Teachers.

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Calderon Preconditioned Surface Integral Equations for Composite Objects With Junctions Pasi Ylä-Oijala, Sami P. Kiminki, and Seppo Järvenpää

Abstract—A Calderon preconditioned (CP) surface integral equation method is developed for the analysis of scattering by piecewise homogeneous dielectric and composite metallic and dielectric objects. The method is based on the electric current formulation (ECF), which only uses the electric surface currents as unknowns. In the ECF the ill-conditioned electric field integral operator and the well-conditioned magnetic field integral operator appear on separate equations, making the application of the CP much easier than e.g., in the PMCHWT formulation where these operators are mixed. In particular, using ECF, Calderon multiplicative preconditioner (CMP) can be straightforwardly extended for composite objects with junctions. Numerical examples demonstrate that the developed formulation, CMP-ECF, is well-conditioned on a very broad frequency range. Index Terms—Composite objects, electromagnetic scattering, method of moments, preconditioning, surface integral equations.

I. INTRODUCTION

EVELOPING efficient numerical methods for analyzing electromagnetic scattering and radiation by arbitrarily shaped composite metallic and dielectric structures have attained a lot interested lately. If the dielectric region is homogeneous, or piecewise homogeneous, the surface integral equation (SIE) method is an attractive choice, since the analysis and the unknowns can be focused to the boundary surfaces of the homogeneous regions. The SIEs can be formulated in many different ways [1]–[4]. However, all formulations that contain the electric field integral operator (EFIO) become ill-conditioned for fine mesh densities and at low frequencies [5] and lead to problems with the iterative solution of the matrix equation. Well-known examples are the electric field integral equation (EFIE) for the PEC objects and the Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT) formulation for dielectric objects, see, e.g., [6] and [7]. There are a few formulations that do not contain the EFIO, e.g., the magnetic field integral equation (MFIE) for (closed)

D

Manuscript received March 19, 2010; revised June 15, 2010; accepted August 11, 2010. Date of publication December 03, 2010; date of current version February 02, 2011. This work was supported by the Academy of Finland project numbers 125979 and 124979. The authors are with the Department of Radio Science and Engineering, School of Electrical Engineering, Aalto University, FI-00076 AALTO, Finland (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2096192

PEC objects [8] and the Müller formulation for homogeneous dielectric objects [9]. For composite objects with junctions, current and charge integral equations (CCIE) [10] use both the equivalent surface current and charge densities as unknowns and give well-conditioned matrices on a broad frequency range. Common feature on all of these formulations is that they are integral equations of the second kind. As known, in the case of non-smooth surfaces with sharp edges and singular currents the solution accuracy of the integral equations of the second kind can be unsatisfactory [11]. Recently, so called Calderon preconditioning (CP), has been studied intensively to remove the problems associated with the EFIO. CP is based on the self-regularizing property of the EFIO and Calderon integral identities [12]. Multiplying the EFIO by itself gives a well-conditioned integral equation of the second kind. The first numerical applications of the CP were presented in [13] and in [14]. In [13] the CP was applied to the combined field integral equation (CFIE) for PEC objects and Nyström method was used to discretize the equations. In [14] Adams and Champagne applied the CP for the EFIE in the case of PEC objects. They discretized the equations with the Galerkin method and developed special intermediate spaces to properly map the range of the first EFIO onto the domain of the second one. The resulting formulation, called modified EFIE, however, requires calculation of new matrix elements and usually has poorer solution accuracy than the original EFIE (assuming that the EFIE is solvable). In [15] a matrix multiplicative form of the CP preconditioner, CMP, was developed, making the method easily and straightforwardly applicable for many existing MoM codes based on the Rao-Wilton-Glisson (RWG) functions. Another benefit of the CMP compared to the earlier CP implementations is that it can maintain the solution accuracy. After the fundamental paper by Andriulli et al. [15], several authors have considered CMP. In [16] Ba˘gci et al. used the same approach for the combined field integral equation (CFIE) formulation of PEC objects. In [17], [18] CMP was generalized for the PMCHWT formulation and in [19], [20] for the single-integral formulation. Both consider scattering by homogeneous penetrable objects. Similar formalism has also been applied for the coupled surface-volume integral equation formulation for scattering by composite metallic and inhomogeneous dielectric objects [21]. In addition, CMP has been extended for high-order [22] and for curvilinear basis functions [23]. Stephanson and Lee have investigated physical properties of the CMP-EFIE [24] and developed a dual loop-star decomposition [25]. The loop-star decomposition in the conjunction with the CMP-EFIE has also been investigated by Yuan et al.

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YLÄ-OIJALA et al.: CALDERON PRECONDITIONED SURFACE INTEGRAL EQUATIONS FOR COMPOSITE OBJECTS WITH JUNCTIONS

[26], [27]. Calderon preconditioning has also been investigated in the time-domain [28], [29]. However, all previous applications of the CMP for SIEs consider either PEC objects or single homogeneous penetrable objects. In this paper we develop a Calderon preconditioned SIE method that can be used to analyze electromagnetic scattering by piecewise homogeneous penetrable objects as well as by composite metallic and dielectric objects. The method is based on the electric current formulation (ECF) [2]. In the ECF the EFIO and the magnetic field integral operator (MFIO) appear in separate equations, making the use of CMP much easier than e.g., in the PMCHWT formulation. In particular, this property, together with the fact that in ECF the currents on the opposite sides of an interface are not combined, allows us to straightforwardly generalize the CMP for composite objects with junctions. The major drawback of the ECF is that it suffers from the problem of internal resonances. II. FORMULATION ) electromagnetic Consider time-harmonic (time factor scattering by a piecewise homogeneous object in free space. denote the exterior and , the homoLet geneous regions of the object. The exterior and the regions are and characterized by constant electromagnetic parameters . The regions can also be perfect electric conductors (PEC). Let denote the surface of with the unit we denote the normal vector pointing into . By incident fields with the sources in the exterior. In this paper the electric current formulation (ECF) [2] is applied to solve the scattering problem. In the ECF the scattered secondary electromagnetic fields within each homogeneous reare expressed in terms of unknown electric current dengion sities on the boundary surface of the region. By adding the incident fields the total fields in can be expressed as (1) (2) with non-zero incident fields only in the region define the EFIO and the MFIO

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Fig. 1. A homogeneous penetrable object.

(8) where and is the identity operator. The surface integral equations of the ECF are obtained by expressing the fields separately in each non-metallic region with (1)–(8) and by requiring that the total fields satisfy the boundary conditions on the boundary surfaces of the regions. It is important to note that in the ECF the currents in each region are considered as independent unknowns and composite objects are considered as combinations of individual objects separated by zero distance [30]. Hence, complex composite structures with junctions can be treated much easily than, e.g., in the PMCHWT formulation where both the currents and the fields are combined. In the following three subsections, the integral equations of the ECF are presented and Calderon preconditioners (CP) are developed in three cases of a homogeneous penetrable object, a piecewise homogeneous penetrable object and a composite PEC-penetrable object. The formulation can be generalized for more complex structures, too. A. Homogeneous Object Consider first scattering by a single homogeneous penetrable object in free space. Let denote the exterior and the interior of the object. The setting is illustrated in Fig. 1. By requiring that the tangential components of the total electromagnetic fields are continuous on the surface gives

. Let us next

(9) (10)

(3)

Using expressions (7) and (8), in (9) and (10), gives the equations of the ECF in the case of a single homogeneous penetrable object

(4) where

(11) (5)

of region . With these operators the rotated tangential components of the scattered secondary fields can be expressed on as the surface

and . Similarly with as in all other surface integral equation formulations containing also in the ECF (11) the matrix becomes ill-condithe EFIO tioned at low frequencies and at fine mesh densities, if the equations are discretized with standard basis and testing functions, like the RWG functions. In order to avoid these problems we next develop a CP for (11). CP is based on the Calderon identity [31]

(7)

(12)

is the single-layer operator with the Green’s function (6)

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Fig. 2. Piecewise homogeneous penetrable object.

Fig. 3. Composite metallic (PEC) and homogeneous penetrable object.

In other words, EFIO (3) squared is an integral equation of the second kind. As shown in [15] this self-regularizing property of the EFIO can be used to regularize the EFIE in the case of PEC objects. Next we utilize (12) to regularize ECF (11). Since the problematic EFIO appears only on the first equation of (11), i.e., on the EFIE part of the ECF, the CP for ECF (11) is obtained by multiplying (11) with (13) The resulting CP-ECF formulation reads (14) , is well-condiThe first operator on the first equation, tioned due to the Calderon identity (12). The second operator on , can also be shown to be well-conthe first equation, ditioned [13], [17]. Obviously the operators on the second equation of (14) are well-conditioned. Hence, we may conclude that the proposed CP-ECF (14) for penetrable objects is well-conditioned. In the following sections the formulation is generalized for composite objects with junctions.

(15) Here the first and third equations are enforced on and the second and the fourth ones on . By substituting the expressions of the secondary fields (7) and (8) into (15) gives the ECF for a piecewise homogeneous object with two regions shown in (16), at the bottom of the page. Here the first subindex on the operators indicates the target (testing) surface, the second one the and source surface and on . The CP for (16) is obtained by

(17)

By the same arguments as in the previous section, we may deduce that by multiplying (16) with (17) results as a well-conditioned formulation. C. Composite Metallic and Penetrable Object

B. Piecewise Homogeneous Penetrable Object Consider next scattering by a piecewise homogeneous peneand with surtrable object that consists of two regions and . faces and and with constant parameters The currents on the opposite sides of the surface are denoted by , see Fig. 2. Similarly as in [30] the definition region of the is the whole and the currents in the intersection currents of two regions are not combined, rather considered as independent unknowns. By using a similar approach as in the previous section, the and are obtained following equations on the surfaces

As the third case we consider scattering by a composite metallic and homogeneous penetrable object. The object consists of two regions, the first one is homogeneous and is PEC, see Fig. 3. penetrable, and the second one vanish, the current can be reSince the fields inside moved. Using a similar approach as in the previous case and by requiring that the tangential component of the total electric field vanishes on , gives equations

(18)

(16)

YLÄ-OIJALA et al.: CALDERON PRECONDITIONED SURFACE INTEGRAL EQUATIONS FOR COMPOSITE OBJECTS WITH JUNCTIONS

Fig. 4. Bistatic RCS for a dielectric sphere, "

= 4 and a =  =2.

Fig. 5. Bistatic RCS for a dielectric sphere at 20 kHz, "

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= 2:6 and a = 1 m.

Here we applied the electric field integral equation (EFIE) on the (the second equation). The other two equations are surface enforced on . By substituting the expressions of the secondary fields into the equations gives the ECF for a composite metaldielectric object

(19) The CP for (19) reads (20) Fig. 6. Bistatic RCS for a homogeneous cube at 20 kHz, "

In the next section we consider discretization of the above developed CMP-ECF formulations. III. DISCRETIZATION The equations derived in the previous sections are discretized and the Buffa-Christiansen, BC, by using the RWG functions using a similar approach as proposed in [15] and [18]. The unknown electric current densities are expanded with the RWG functions, the EFIE parts of the formulations are tested functions, the MFIE parts of the formulawith the functions and the outer EFIO of tions are tested with the the CMP is discretized with BC functions. For example, in the case of a single homogeneous object the matrix equation (21) (shown at the bottom of the page) is obtained, where is the identity matrix, denotes the vector of the coefficients of the

RWG functions of the current trices are

=2:6 and L =1 m.

and the elements of the ma(22) (23) (24)

Above and are either the RWG functions and functions

or the BC

(25)

(21)

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=2 6

Fig. 7. Bistatic RCS for a homogeneous cube at 20 kHz, " : and L modeled as a piecewise homogeneous object with two equal regions.

Fig. 8. Bistatic RCS for a PEC-dielectric cube at 20 kHz, " m.

1

=1 m,

Fig. 9. Condition numbers of the matrices as functions of frequency. A dielecand a m. tric sphere with "

=4

=1

= 2:6 and L = Fig. 10. The numbers of GMRES iterations required to obtain relative residual as functions of frequency. A sphere, " ;a of m.

10

=4 =1

The excitation vectors contain the following elements (26) (27) The elements of the matrices and the vectors of (21) are calculated using the translation matrices presented in [15]. Hence, all matrix elements can be expressed in terms of the matrix elements containing the RWG functions on the barycentrically refined mesh only. This gives the Calderon multiplicative preconditioner for the ECF, CMP-ECF. The matrix equations of the CMP-ECF formulation for the other cases are obtained using a similar approach. IV. NUMERICAL EXAMPLES Next the developed CMP formulations are verified and their performances are studied with numerical examples. Consider first scattering by a homogeneous dielectric sphere with

and radius , where is the wavelength in the exterior (vacuum). The incident wave is a linearly polarized planewave. The same example is considered in [34] (Fig. 1 a). Fig. 4 shows the bistatic RCS in the E plane scaled with and computed with the PMCHWT, the modified N-Müller (mNMüller) [4], and with the ECF and the CMP-ECF formulations. . The PMCHWT, the mN-Müller The backscattering is at and the ECF formulations are discretized using the Galerkin method and the RWG functions. The solutions of all other formulations expect mN-Müller seem to agree well with the Mie series solution. Next we consider scattering at a low frequency. Fig. 5 shows the bistatic RCS in the E plane for a dielectric sphere with and m at 20 kHz. The same example is considered in [33] (Fig. 3(a)). Clearly, the PMCHWT formulation fails to give correct result due to the low frequency breakdown. The results also show that the ECF, even without CMP, performs significantly better at low frequencies than the PMCHWT.

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Fig. 11. A composite box with two regions. The dimensions are in meters.

Fig. 14. Condition numbers of the matrices as functions of frequency. A metaldielectric box of Fig. 11.

Fig. 12. Condition numbers of the matrices as functions of frequency. A piecewise homogeneous dielectric box of Fig. 11.

Fig. 15. The numbers of GMRES iterations as functions of frequency. A metaldielectric box of Fig. 11.

Fig. 13. The numbers of GMRES iterations as functions of frequency. A piecewise homogeneous dielectric box of Fig. 11.

To further validate our approach we considered scattering by non-smooth and inhomogeneous objects. Consider first scattering by a homogeneous cube with the edge length of 1 m and

at 20 kHz. Figs. 6 and 7 show relative permittivity the bistatic RCS in the E plane when the cube is consider as a homogeneous object (Fig. 6) and as a piecewise homogeneous object made of two equal homogeneous regions having the same (Fig. 7). The same cases are considered permittivity in [32] (Fig. 4). The ECF and the CMP-ECF seem to give the same solution in both cases, whereas the PMCHWT formulation fails in both cases due to the low frequency breakdown and the mN-Müller gives the correct solution only in the first case. As an inhomogeneous object, a PEC-dielectric cube with edge length of 1m is considered. The same geometry is considered in [32] (Fig. 5) and in [33] (Fig. 7). Fig. 8 shows the bistatic RCS in the E plane at 20 kHz. Now only the solution of the CMP-ECF resembles the solutions of [32] and [33] obtained with the loop-tree decomposition. Next we investigate the conditioning of the matrix equations and the number of iterations required to solve the matrix equations iteratively with the GMRES method. GMRES method is

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TABLE I THE NUMBER OF ITERATIONS (TOL ) AS A FUNCTION OF PERMITTIVITY OF A DIELECTRIC SPHERE. THE SECOND AND THIRD COLUMN AT 20 KHZ AND MHZ THE LAST TWO COLUMNS AT

= 10

150

Fig. 16. Bistatic RCS for a piecewise homogeneous dielectric box of Fig. 11 . with k L

=1

Fig. 17. Bistatic RCS for a metal-dielectric box of Fig. 11 with k L

= 1.

applied without restarting and the iteration is stopped when the or the number of iterations exrelative residual error is ceeds 1000. Figs. 9 and 10 show the spectral condition numbers of the matrices and the numbers of the GMRES iterations as functions of frequency in the case of a homogeneous sphere with . The number of unknowns is 1500. Both the mN-Müller and the CMP-ECF formulations are well-conditioned at low frequencies. However, at very low frequencies the CMP-ECF is found to break. The reasons and remedies for this are discussed elsewhere [25], [27]. The same analysis is then repeated in the case of an inhomoalong the geneous object, a box of dimensions and coordinate axes, where m. The incident wave is a linearly polarized planewave propagating along the axis and the electric field is parallel to the axis. The box is divided into two equal parts in the direction, so that the first one on the negative axis has the permittivity of 2 and the second one on the positive axis has the permittivity of 4. The geometry is illustrated in Fig. 11.

Fig. 18. Condition number as a function of the diameter of a sphere, "

= 4.

Figs. 12 and 13 show the spectral condition numbers and the number of GMRES iterations as the function of frequency in the case of a piecewise homogeneous box of Fig. 11. The number of unknowns in the CMP-ECF is 1344. In this case also the mM-Müller formulation becomes ill-conditioned at low frequencies and only the CMP-ECF is well-conditioned. In the next example the geometry is the same as in the previous example, expect the second region is PEC. Figs. 14 and 15 show the spectral condition number and the number of GMRES iterations as functions of frequency. The number of unknowns in the CMP-ECF is 1008. Again only the CMP-ECF remains well-conditioned on a broad frequency range. To validate the develop CMP-ECF formulation for inhomogeneous objects also at higher frequencies Figs. 16 and 17 show the bistatic RCS in the E plane for an inhomogeneous dielec, tric box and for a composite PEC-dielectric box when where is the wavenumber in vacuum. The geometry is the same as in Fig. 11. All formulations do not give identical solutions, because the solution accuracy of the integral equations of

YLÄ-OIJALA et al.: CALDERON PRECONDITIONED SURFACE INTEGRAL EQUATIONS FOR COMPOSITE OBJECTS WITH JUNCTIONS

the first kind (PMCHWT) and of the second kind (Müller) can be very different in the case of objects with sharp wedges and junctions. The ECF is a mixture of integral equations of the first and second kinds (the EFIE part is of the first kind and the MFIE part is of the second kind), and hence, its solution accuracy is between the accuracy of the PMCHWT and the Müller formulations. Next we investigate the performance of the ECF and CMP-ECF as functions of the permittivity. Table I shows the number of GMRES iterations at 20 kHz and at ( MHz) in the case of a homogeneous dielectric sphere. At low frequencies the performance of the CMP-ECF seems not to be sensitive to the permittivity contrast, whereas at higher frequencies the number of GMRES iterations seems to increase as the permittivity contrast is increased. Finally, we show that both the ECF and the CMP-ECF suffer from the problem of internal resonances. Fig. 18 shows the condition numbers of the ECF and the CMP-ECF formulations in . The the case of a homogeneous dielectric sphere with first resonance is at ([34] Fig. 3), where is the is the wavelength in vacuum. radius of the sphere and V. CONCLUSION We have developed a Calderon preconditioned (CP) surface integral equation method for analyzing electromagnetic scattering by piecewise homogeneous penetrable and composite metallic and penetrable objects. The method is based on the electric current formulation (ECF). In the ECF the electric field integral operator and the magnetic field integral operator appear on separate equations, making the use of CP easier and more straightforward than e.g., in the PMCHWT formulation. In particular, this property together with the fact that in ECF the currents on the opposite sides of an interface are not combined, makes it possible to develop a multiplicative CP for piecewise homogeneous and composite metallic and dielectric objects with junctions. Numerical examples show that the developed CMP-ECF formulations give well-conditioned matrix on a broad frequency range. The major drawback of the ECF is that it has the problem of internal resonances. REFERENCES [1] A. J. Poggio and E. K. Miller, “Integral equation solutions of three-dimensional scattering problems,” in Computer Techniques for Electromagnetics, R. Mittra, Ed. Oxford, U.K.: Pergamon Press, 1973. [2] R. F. Harrington, “Boundary integral formulations for homogeneous material bodies,” J. Electromag. Waves Applicat., vol. 3, no. 1, pp. 1–15, 1989. [3] B. M. Kolundzija and A. R. Djordjevic, Electromagnetic Modeling of Composite Metallic and Dielectric Structures. Boston, MA: Artech House, 2002. [4] P. Ylä-Oijala, M. Taskinen, and S. Järvenpää, “Surface integral equation formulations for solving electromagnetic scattering problems with iterative methods,” Radio Sci., vol. 40, no. 6, RS6002, Nov. 2005. [5] D. R. Wilton and A. W. Glisson, “On improving the stability of the electric field integral equation at low frequencies,” in Proc. URSI Radio Science Meeting, Los Angeles, CA, Jun. 1981, p. 24. [6] J.-S. Zhao and W. C. Chew, “Integral equation solution of Maxwell’s equations from zero frequency to microwave frequencies,” IEEE Trans. Antennas Propag., vol. 48, no. 10, pp. 1635–1645, Oct. 2000. [7] S. Y. Chen, W. C. Chew, J. M. Song, and J.-S. Zhao, “Analysis of lowfrequency scattering from penetrable scatterers,” IEEE Trans. Geosci. Remote Sensing, vol. 39, no. 4, pp. 726–735, Apr. 2001.

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[8] Y. Zhang, T. C. Cui, W. C. Chew, and J.-S. Zhao, “Magnetic field integral equation at very low frequencies,” IEEE Trans. Antennas Propag., vol. 51, no. 8, pp. 1864–1871, Aug. 2003. [9] P. Ylä-Oijala and M. Taskinen, “Well-conditioned Müller formulation for electromagnetic scattering by dielectric objects,” IEEE Trans. Antennas Propag., vol. 53, no. 10, pp. 3316–3323, Oct. 2005. [10] M. Taskinen and P. Ylä-Oijala, “Current and charge integral equation formulation,” IEEE Trans. Antennas Propag., vol. 54, no. 1, pp. 58–67, Jan. 2006. [11] P. Ylä-Oijala, M. Taskinen, and S. Järvenpää, “Analysis of surface integral equations in electromagnetic scattering and radiation problems,” Eng. Anal. Boundary Elements, vol. 32, pp. 196–209, 2008. [12] R. J. Adams, “Physical and analytical properties of a stabilized electric field integral equation,” IEEE Trans. Antennas Propag., vol. 52, no. 2, pp. 362–372, Feb. 2004. [13] H. Contopanagos, B. Dembart, M. Epton, J. J. Ottusch, V. Rokhlin, J. L. Fisher, and S. M. Wandzura, “Well-conditioned boundary integral equations for three-dimensional electromagnetic scattering,” IEEE Trans. Antennas Propag., vol. 50, no. 12, pp. 1824–1830, Dec. 2002. [14] R. J. Adams and N. J. Champagne II, “A numerical implementation of a modified form of the electric field integral equation,” IEEE Trans. Antennas Propag., vol. 52, no. 9, pp. 2262–2266, Sep. 2004. [15] F. P. Andriulli, K. Cools, H. Ba˘gci, F. Olyslager, A. Buffa, S. Christiansen, and E. Michielssen, “A multiplicative Calderon preconditioner for the electric field integral equation,” IEEE Trans. Antennas Propag., vol. 56, no. 8, pp. 2398–2412, Aug. 2008. [16] H. Ba˘gci, F. P. Andriulli, K. Cools, F. Olyslager, and E. Michielssen, “A Calderon multiplicative preconditioner for the combined field integral equation,” IEEE Trans. Antennas Propag., vol. 57, no. 10, pp. 3387–3392, Oct. 2009. [17] F. Olyslager, K. Cools, and J. Peeters, “A Calderon multiplicative preconditioner for the PMCHWT integral equation,” presented at the IEEE Int. Symp. on Antennas and Propagation, Charleston, SC. [18] K. Cools, F. P. Andriulli, and F. Olyslager, “A Calderon preconditioned PMCHWT equation,” presented at the Int. Conf. on Electromagnetics in Advanced Applications, ICEAA’09, Torino, Italy. [19] F. Valdes, F. P. Andriulli, H. Ba˘gci, and E. Michielssen, “On the discretization of single source integral equations for analyzing scattering from homogeneous penetrable objects,” presented at the IEEE Int. Symp. on Antennas and Propagation, 2008. [20] F. Valdes, F. P. Andriulli, H. Ba˘gci, and E. Michielssen, “On the regularization of single source combined integral equations for analyzing scattering from homogeneous penetrable objects,” presented at the IEEE Int. Symp. on Antennas and Propagation, Charleston, SC, 2009. [21] H. Ba˘gci, F. P. Andriulli, K. Cools, F. Olyslager, and E. Michielssen, “CMP-based discretization of the coupled surface volume electric field integral equations,” presented at the 2009 IEEE Int. Symp. on Antennas and Propagation, Charleston, SC. [22] F. Valdes, F. P. Andriulli, K. Cools, and E. Michielssen, “High-order quasi-curl conforming functions for multiplicative Calderon preconditioning of the EFIE,” presented at the IEEE Int. Symp. on Antennas and Propagation, Charleston, SC, 2009. [23] S. Yuan, J.-M. Jin, and Z. Nie, “Implementation of the Calderon multiplicative preconditioner for the EFIE solution with curvilinear triangular patches,” presented at the IEEE Int. Symp. on Antennas and Propagation, Charleston, SC, 2009. [24] J.-F. Lee and M. B. Stephanson, “Some practical and theoretical aspects of the Calderon preconditioned EFIE,” presented at the Int. Conf. on Electromagnetics in Advanced Applications, ICEAA’09, Torino, Italy, 2009. [25] M. B. Stephanson and J.-F. Lee, “Preconditioner electric field integral equation using Calderon identities and dual loop/star basis functions,” IEEE Trans. Antennas Propag., vol. 57, no. 4, pp. 1274–1279, Apr. 2009. [26] S. Yuan and J.-M. Jin, “Analysis of low-frequency electromagnetic problems using the EFIE with a Calderon multiplicative preconditioner and loop-star decomposition,” presented at the IEEE Int. Symp. on Antennas and Propagation, Charleston, SC, 2009. [27] S. Yuan, J.-M. Jin, and Z. Nie, “EFIE analysis of low-frequency problems with loop-star decomposition and Calderon multiplicative preconditioner,” IEEE Trans. Antennas Propag., vol. 58, no. 3, pp. 857–867, Mar. 2010. [28] K. Cools, F. P. Andriulli, F. Olyslager, and E. Michielssen, “Time domain Calderon identities and their application to the integral equation analysis of scattering by PEC objects Part I: Preconditioning,” IEEE Trans. Antennas Propag., vol. 57, no. 8, pp. 2352–2364, Aug. 2009.

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[29] K. Cools, F. P. Andriulli, F. Olyslager, and E. Michielssen, “Time domain Calderon identities and their application to the integral equation analysis of scattering by PEC objects Part II: Stability,” IEEE Trans. Antennas Propag., vol. 57, no. 8, pp. 2365–2375, Aug. 2009. [30] R. Aghajafari and H. Singer, “Transient source current formulation for the analysis of arbitrarily shaped dielectric/conducting composite objects,” Engrg. Analy. Boundary Elements, vol. 34, pp. 440–446, 2010. [31] C. C. Hsiao and R. E. Kleinman, “Mathematical foundations for error estimation in numerical solutions of integral equations in electromagnetics,” IEEE Trans. Antennas Propag., vol. 45, no. 3, pp. 316–328, 1997. [32] S. Chen, J.-S. Zhao, and W. C. Chew, “Analyzing low-frequency electromagnetic scattering from a composite object,” IEEE Trans. Geosci. Remote Sensing, vol. 40, no. 2, pp. 426–433, Feb. 2002. [33] Y. Chu, W. C. Chew, J. Zhao, and S. Chen, “A surface integral equation formulation for low-frequency scattering from a composite object,” IEEE Trans. Antennas Propag., vol. 51, no. 10, pp. 2837–2844, Oct. 2003. [34] X.-Q. Sheng, J.-M. Jin, J. Song, W. C. Chew, and C.-C. Lu, “Solution of combined-field integral equation using multilevel fast multipole algorithm for scattering by homogeneous bodies,” IEEE Trans. Antennas Propag., vol. 46, no. 11, pp. 1718–1726, Nov. 1998.

Pasi Ylä-Oijala received the M.Sc. and Ph.D. degrees in applied mathematics from the University of Helsinki, Helsinki, Finland, in 1992 and 1999, respectively. Currently he is working as a Senior Researcher with the Department of Radio Science and Engineering, School of Electrical Engineering, Aalto University, Finland. His field of interest includes integral equation and fast methods in computational electromagnetics.

Sami P. Kiminki received the M.Sc. degree in electrical engineering from Helsinki University of Technology, Espoo, Finland, in 2009. Currently he is working toward the D.Sc. degree at Aalto University. His research interests are in the field of computational electromagnetics, focusing on development of well-conditioned integral equation methods to analyze antenna performance and circuit-level parameters of RF structures.

Seppo Järvenpää received the M.Sc. and Ph.D. degrees in applied mathematics from the University of Helsinki, Helsinki, Finland, in 1992 and 2001, respectively. Currently, he is working as a Senior Researcher in the Department of Radio Science and Engineering, School of Electrical Engineering, Aalto University, Finland. His field of interest includes numerical techniques in computational electromagnetics based on the integral equation and finite element methods.

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A Singularity-Free Boundary Equation Method for Wave Scattering Igor Tsukerman

Abstract—Traditional boundary integral methods suffer from the singularity of Green’s kernels. The paper develops, for a model problem of 2D scattering as an illustrative example, singularity-free boundary difference equations. Instead of converting Maxwell’s system into an integral boundary form first and discretizing second, here the differential equations are first discretized on a regular grid and then converted to boundary difference equations. The procedure involves nonsingular Green’s functions on a lattice rather than their singular continuous counterparts. Numerical examples demonstrate the effectiveness, accuracy and convergence of the method. It can be generalized to 3D problems and to other classes of linear problems, including acoustics and elasticity. Index Terms—Boundary difference equations, boundary element methods, boundary integral equations, difference equations, diffraction, discrete transforms, flexible local approximation, Green’s functions, scattering.

I. INTRODUCTION OUNDARY equation methods have a long history, with practical applications dating back to the 1960s. An interesting historical account given by Cheng and Cheng [1] includes the work on wave scattering and radiation in 1962–1967 by Friedman and Shaw, Chen and Schweikert, Banaugh and Goldsmith, Mitzner, and others [2]–[8]. In eletromagnetics, boundary integral techniques became very popular due primarily to Harrington’s work published in 1967–68 [9], [10] (see also [11]–[15]). In traditional boundary integral methods, linear boundary value problems of field analysis are transformed into integral equations with respect to equivalent sources residing on the boundaries. In the simplest example of capacitance calculation [9], [10], the distributed charge density on conducting plates becomes the principal unknown variable. By equating the Coulomb potential of that charge to the given potential of the conductors, one obtains an integral equation. It can then be discretized using variational techniques (moment methods), collocation and Galerkin methods being particular cases of those. As all numerical methods, boundary integral techniques do carry some trade-offs. Their key advantage is the lower dimensionality of the problem: 3D analysis is reduced to equivalent

B

Manuscript received February 21, 2010; revised June 16, 2010; accepted July 19, 2010. Date of publication December 03, 2010; date of current version February 02, 2011. The author is with the Department of Electrical and Computer Engineering, The University of Akron, OH 44325-3904 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2096189

sources on 2D boundaries and 2D analysis—to 1D contours. Another advantage is a natural treatment of unbounded problems (e.g., wave scattering and radiation), without the artificial domain truncation unavoidable in differential methods such as finite difference (FD) schemes and the finite element method (FEM). Integral equation methods have, in general, two major disadvantages. First, the matrices of the discrete system are almost always full. This is due to the fact that a source at any point on the boundary contributes to the field at all other points. In contrast, FD and FE matrices are sparse, with very efficient system solvers available (iterative: multilevel methods, incomplete factorization and other effective preconditioners; direct: minimum degree, nested dissection and others; see e.g., [16] and references there). Cases where Green’s functions decay rapidly in space, giving rise to quasi-sparse integral equations, are exceptional (e.g., periodic structures in the electromagnetic band gap regime [17]). Another disadvantage is that the integral kernels in field analysis are singular. At the surface points, the kernel singularity can usually be handled analytically, and the fields remain bounded as long as the surfaces are smooth. However, for points in the vicinity of the surface, the evaluation of the integral is problematic, as analytical expressions are usually unavailable and numerical quadratures require extreme care. The same is true for two adjacent surfaces with a narrow gap in between. Significant progress in fast multipole methods (FMM) [18]–[22] has helped to alleviate the first disadvantage of boundary methods. FMM accelerates the computation of fields due to distributed sources—or equivalently, matrix-vector multiplications for the dense system matrices. The second disadvantage is more difficult to overcome. Singular kernels are inherent in boundary integral methods because the fields of point sources are unbounded. However, a drastic change in the computational procedure leads to a singularityfree method; this is accomplished by reversing the sequence of stages in the boundary techniques. The standard sequence is

The alternative sequence is

0018-926X/$26.00 © 2010 IEEE

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Discretization of the differential problem is performed on an infinite regular grid and yields an FD scheme. This scheme is converted—as explained in the remainder of the paper—to a boundary problem that involves discrete fundamental solutions (Green’s functions) on the grid. Discrete Green’s functions, unlike their continuous counterparts, are always nonsingular. This general idea is not new. In fact, there are two related but independently developed methodologies for boundary difference equations. The first one, put forward and thoroughly studied by Ryaben’kii, Reznik, Tsynkov and others [34]–[37], is known as the method of difference potentials and can be viewed as a discrete analog of the Calderon projection operators in functional analysis [34]. The second methodology, called boundary algebraic equations by Martinsson and Rodin [23], is at least 50 years old (Saltzer [25]) and is a discrete analog of first- or second-order Fredholm boundary integral equations for potential problems [23]. In comparison with [23], the method of this paper has several novel features. First, the paper deals—to my knowledge, for the first time—with boundary difference equations for electromagnetic wave scattering. In [23], a simple model problem is considered: the Laplace equation (e.g., electrostatics or heat transfer) in a homogeneous domain with Dirichlet boundary conditions; the focus of [23] is on the mathematical analysis of the respective boundary difference operators, their spectral properties and the appropriate iterative solvers. One key distinction between the methodology of Ryaben’kii [34] and this paper’s is in the choice of the main unknown: the boundary field/potential (Ryaben’kii) vs discrete sources on the boundary (the present paper). The treatment via sources parallels that of the continuous boundary integral method [10]–[12], [14], [15] and should therefore be intuitive to applied scientists and practitioners. Further analysis and comparison of these methodologies will be presented elsewhere. An additional novelty of this paper is the use of high-order Flexible Local Approximation MEthods (FLAME, see Section IV-A) in the context of boundary difference equations. Also, this is the first application of FLAME to a 2D boundary of a generic shape; this is done by approximating this boundary locally by its osculating circle at any given point. II. BOUNDARY DIFFERENCE EQUATIONS FOR A MODEL PROBLEM

Fig. 1. Setup of the scattering problem for the E -mode.

For definiteness, let us focus on the -mode (TM- or -mode) governed by the familiar equation for the electric field with a single -component: (1) where the standard notation for the angular frequency , the magnetic permeability , the dielectric permittivity and the inside the scatterer wavenumber is used ( is equal to outside; assumed). Equation (1) should be and to supplemented by the standard radiation boundary conditions for at infinity. The incident field the scattered field is a plane wave (2) where the convention for complex phasors is implied. In a departure from the boundary integral methodology, we now proceed, prior to formulating a problem on the boundary of the scatterer, to FD discretization. To this end, let us introduce an infinite lattice with a grid size , for simplicity the same in both and directions. Although infinite lattices are not a very common computational tool, they were already featured prominently in Martinsson and Rodin’s work [23], [39] as well as in the much earlier report by Saltzer [25]. The actual computation, clearly, never involves an infinite amount of data on the lattice; in fact, the unknowns are ultimately confined only to the boundaries. As an auxiliary device, we need to consider the wave (1) in the homogeneous space with a constant generic parameter . Various FD discretizations of this equation are available; see e.g., Harari’s review [24] for further information and references. Here we settle for the simplest five-point scheme

A. Formulation and Setup To fix ideas and explore the potential of the proposed approach, let us consider the classical 2D case of electromagnetic wave scattering as a model problem. It should be emphasized from the outset that the method has a much broader range of applicability; possible generalizations are discussed in Section V-C. Consider a plane wave impinging from the air on a homogeneous dielectric cylinder (Fig. 1) with a given isotropic dielec. The cross-section of the cylinder could be tric permittivity arbitrary, but for the sake of simplicity we shall assume that its surface is smooth (no edges or corners).

(3) where is the field value at a grid point characterized . As reflected by an integer double index in the notation, the coefficients of the difference operator depend on the mesh size and on the wavenumber; this may not be explicitly indicated if there is no possibility of confusion. defined Associated with is its Green’s function as the solution of (4)

TSUKERMAN: A SINGULARITY-FREE BOUNDARY EQUATION METHOD FOR WAVE SCATTERING

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For the outer boundary, (7) The discrete convolution in the equations above is defined in the usual way as (8)

Fig. 2. Discrete boundary with 196 nodes. Squares: ; circles:

.

with the boundary condition (5) Here is the discrete delta-function (equal to one at the origin is the continuous and zero elsewhere) and Green function, being the Hankel function. Without getting into the mathematical theory of lattice Green functions (see [20], [39] and Section III), let us note some features critical for our purposes. • In contrast with its continuous counterpart , the discrete Green function (4) is bounded everywhere, including the origin. • The discrete Green function differs significantly from the continuous one only within a spatial window of several grid layers around the origin. Therefore only a relatively small amount of information needs to be stored—namely, the values of the Green function within that window. This and for data can be precomputed for any given value of each linear medium in a given problem. The discrete boundary of the scatterer can be defined in a natural way following Ryaben’kii [34]. Each grid node with discrete coordinates has four immediate neighbors from the respective five-point stencil in the difference scheme (3). If node lies inside the scatterer (or exactly on its boundary) but some of its neighbors are outside, this grid node is said to belong to the discrete inner boundary . Likewise, if the central node of the stencil lies strictly outside the scatterer but at least one of its neighbors is inside (or exactly on the boundary), this node . The comis said to belong to the discrete outer boundary plete discrete boundary has (loosely speaking) two layers, Fig. 2. (For larger multipoint stencils, the discrete boundary can be composed of several layers.) The number of . These nodes can nodes on the boundary is or, alternatively, by be referred to by pairs of indexes some global numbers from 1 to . The order of this numbering makes no principal difference but may slightly affect the practical implementation of the method. B. Boundary Sources The critical step is to express the lattice-based field in terms of fictitious discrete sources that are nonzero only on the discrete boundary . For the inner boundary, (6)

Note that for the nodes on each side of the boundary the field is described via the respective discrete Green function. Note also that, by construction, the field (6), (7) satisfies the homogeneous difference equation with operator (3) at all lattice nodes lying strictly inside or outside the domain (i.e., not on the discrete boundary ). Formally, this is easily deduced by applying the difference operator , with respect to index , to convolution (8). Informally, (6) and (7) are analogous to standard convolution integrals in potential theory or, in a different area, to the forced response of linear time-invariant systems, where the impulse response and the input signal play the roles of Green’s function and the source, respectively. The auxiliary sources need not have a direct physical interpretation, although ultimately they are indirectly related to the equivalent electric and magnetic surface currents of traditional boundary integral methods [13], [14]. However, the fields derived from these sources are physical. The convolutions in (6), (7) can be interpreted as (discrete) scattered fields. It can be shown that any discrete field satisfying the FD wave equation on the lattice can indeed be expressed via convolution with some fictitious boundary sources as stipulated above, except possibly for special resonance cases (see Appendix). C. Boundary Difference Equations By construction, the electric fields defined by (6), (7) satisfy the respective discretized wave equation on each side of the boundary. What remains to be done then is to impose the boundary conditions; this will lead to a system of equations from which the sources can be found. To this end, one may use another difference scheme, , that approximates the boundary conditions; we shall call it a “boundary test scheme.” The simplest example is the five-point scheme

(9) In this second-order scheme, the value of is taken at the midpoint of the stencil. A more accurate alternative is the ninepoint FLAME (flexible local approximation method) proposed in [16], [26], [27]). Both types of schemes are used in the numerical examples of Section IV. The FLAME coefficients are computed as the nullspace of a matrix comprising the nodal values of a set of basis functions on the stencil (Section IV-A and [16], [26], [27]). Applying a given boundary test scheme on to fields (6), (7), one obtains a system of boundary-difference equations of the form

(10)

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where is the discrete version of the incident field, i.e., its values at the lattice nodes. It is implied here that, in accordance strictly outside the with (6) and (7), only the values of scatterer are included in (10); the values inside the scatterer are indicates that different schemes ignored. The superscript with different coefficients can in principle be used over different stencils. More explicitly, denoting the coefficients of the boundary test with (where index runs over all nodes of scheme the grid stencil centered at node ), one can write the boundary (10) as (11) where

is an

Re( )

=1 =17

= 50

Fig. 3. g for k ,h = ,M . Note that the discrete Green function is nonsingular everywhere; in fact, its magnitude in this example is quite moderate.

matrix with the entries

and

The meaning of the terms above are as follows: are the global numbers1 of • , nodes and on the discrete boundary . • are the coefficients of the boundary test scheme corresponding to node . (In principle, different schemes could be used at different nodes. One may even envision an adaptive procedure where the order of the scheme will vary in accordance with local accuracy estimates.) if node is on the inner boundary and • if it is on the outer boundary . • is the value of the incident field at node of stencil , for nodes strictly outside the scatterer; zero otherwise. We shall call the numerical procedure leading to (11) the boundary difference method (BDM). III. THE LATTICE GREEN FUNCTION As evident from the description of the BDM, lattice Green’s functions play a central role in it and must be computed accurately. There are at least two general ways to do so: Fourier analysis and finite difference solutions. A detailed exposition for the Laplace equation has been given by Martinsson and Rodin [20], [23], [39]. Similar ideas can be immediately applied to the wave equation as well, although a more elaborate analysis would be desirable in the future. (discrete physical space Applying Fourier transform continuous reciprocal space) to the difference (4) with the fivepoint operator (3), one obtains

where 1Not

,

are the Fourier parameters in the square

=(

.

)

to be confused with the Euclidean lengths of vectors n n ; n and ; these lengths are irrelevant and never appear in our analysis.

m = (m ; m

)

Fig. 4.

Re(g) for k = 2, h = 1=7, M = 50.

The inverse Fourier transform may then serve as a staring point for an asymptotic analysis similar to Martinsson’s [20], [39] and for practical computation of Green’s function . However, this Fourier analysis is quite involved and must be performed with great care, especially in 2D where Green’s functions decay slowly and regularization of Fourier integrals is necessary [20], [39]. For the purposes of this paper, a more straightforward route is sufficient. The finite difference problem (4) for the Green function can be solved directly, with condition (5) imposed on the boundary of a large enough square . This can be done efficiently with fast Fourier transforms over the square, but the computational cost in 2D is so moderate that any other reasonable solver can be applied. Obviously, one can also take advantage of the symmetries to reduce the size of the computational problem. The following plots illustrate the behavior of the lattice Green function and its computation. All of the plots were generated for as an example. Surface plots of the real the grid size part of for wavenumbers and are shown in Figs. 3 and 4, respectively; Green’s function was computed in with . the spatial window of the window need not Fig. 5 demonstrates that the size be too large. Indeed, lattice Green’s functions for and are quite close. The numerical experiments reported . Even assuming an in Section IV were performed with and 10 different materials in a given overkill value practical problem, one ends up with less than 1 MB of data to be stored. In 3D, if one takes advantage of the symmetries of , the memory requirements are still reasonable, even for vector fields and dyadic Green’s functions, except for problems where the number of different materials is unusually large.

TSUKERMAN: A SINGULARITY-FREE BOUNDARY EQUATION METHOD FOR WAVE SCATTERING

Fig. 5. Lattice Green’s function g (m ; m ) vs m for kh = 1=7, m = 0. The results for two different values of M , M = 50 and M = 100, are close.

Fig. 6. Color plot of the E field for a circular cylinder with  with n = 460.

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= 4. BDM

IV. NUMERICAL SIMULATION A. FLAME The theory, implementation and various applications of FLAME have been discussed in a number of previous publications [16], [26]–[33], and therefore only a brief summary is given here. FLAME replaces the usual Taylor expansions, the key tool of standard finite-difference analysis, with much more accurate local (quasi-)analytical approximations of the solution. Such approximations can be obtained, for example, via cylindrical or spherical harmonics, plane waves, etc. Since the local behavior of the field is “built into” the difference scheme, the accuracy often improves dramatically. FLAME has already been applied to the simulation of colloidal and plasmonic particles [16], [26], [27], negative-index materials [30], the computation of Bloch bands of photonic crystals [28], including complex bands for plasmonic systems and other dispersive media [28], [29]. FLAME can work even on irregular grid stencils [31], although in BDM only regular lattices are used. For the model problems in this paper, local analytical bases for FLAME are available via Bessel/Hankel functions. More specifically, in the vicinity of a dielectric cylinder with a circular cross-section centered for convenience at the origin of a polar , these approximating functions—the coordinate system FLAME basis —are [16], [26], [27]

where is the Bessel function of order , is the Hankel function of the second kind, and , are coefficients to be determined. These coefficients are found via the standard conditions on the boundary of the cylinder [16], [26], [28]. Index runs over all grid stencils where the FLAME scheme is generated, while index runs over all basis functions in a given stencil . In this paper, the 9-point (3 3) stencil with a grid size is . The eight basis functions are obtained used and , two harmonics of by retaining the monopole harmonic orders , 2, 3 (i.e., dipole, quadrupole and octupole), and

. This set of basis functions one of the harmonics of order produces a nine-point scheme as the null vector of the respective matrix of nodal values [26], [27]. For the test problem with an elliptical cylinder (Section IV-C), it is still possible to use the same Bessel-Hankel basis functions in FLAME. Toward this end, a piece of the ellipse straddled by a given grid stencil is approximated by its osculating circle. While this approach for constructing FLAME bases is fairly straightforward, it has never been used previously. (In the past, the primary motivation was to apply FLAME on very coarse grids that carry almost no information about the shape of the boundary [16], [26], [27].) For the ellipse, the osculating circle can easily be found analytically; for more complicated boundaries, the curvature could approximately be evaluated numerically—for example, as the best local fit to a piece of the discrete boundary . In yet more complex cases—especially in 3D where there are two radii of curvature—one could use piecewise-planar approximations and Fresnel-formula FLAME bases [30]. B. Circular Cylinder For verification, let us first consider a circular scattering cylinder, as in this case a well-known analytical solution via cylindrical harmonics exists. In all numerical experiments mode was considered. The wavenumber for the below, the ; the wavenumber for incident wave was normalized at (i.e., ). The incident the scatterer was taken as plane wave propagates in the positive direction. The color plot of the electric field in BDM with is shown in Fig. 6. The numerical error as a function of the BDM grid size is plotted in Fig. 7 for two boundary test schemes : the standard five-point scheme and the nine-point FLAME (these schemes were briefly described in the previous sections). The dashed line in the figure serves only as a visual aid indicating the second order convergence of the method for both schemes. Not surprisingly, the numerical error for FLAME is about an order of magnitude lower than that of the five-point scheme. However, the order of convergence is still limited by the secondorder five-point difference scheme used to compute the discrete

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Fig. 7. The relative error in BDM as a function of grid size. Discrete boundary

with 196 nodes. Quadratic convergence, commensurate with the order of the scheme for the lattice Green function, is observed. The FLAME results are about an order of magnitude more accurate than for the standard five-point scheme, even around h 0:05 where the FLAME data points exhibit some scatter.



Fig. 9. The numerical results for different cases are seen to be in a very good agreement. The real part of the electric field is plotted; the agreement for the imaginary part is similar. Discrete boundary with 196 nodes.

well. Nine-point FLAME and standard five-point schemes were ; results applied on several grids with sizes for and are shown. V. DISCUSSION AND CONCLUSION A. Summary

Fig. 8. Re(E ) obtained with BDM, 9-point FLAME scheme. Discrete boundary with 196 nodes.

Green function (Section III). The relative error was calculated , where and are as the numerical and the quasi-exact solutions on the grid, respectively; the norms are Euclidean. The quasi-exact solution was computed via the standard expansion into cylindrical harmonics up to order 50. C. Elliptical Cylinder The simulations have been repeated for an elliptical cylinder, with the same physical parameters as above, and with the ratio of the axes 1.5:1. Fig. 8 is a color plot of the real part of the electric field obtained with BDM, 9-point FLAME scheme, discrete boundary with 196 nodes. FLAME was generated as described at the end of Section IV-A: by locally approximating a piece of the elliptic boundary with a circle and using the respective Bessel/Hankel bases. Fig. 9 demonstrates that the field distributions obtained with different methods are in a very good agreement. Plotted in the ) and vs figure is the real part of the electric field vs. (at (at ). The imaginary parts are not plotted but agree equally

The boundary difference method described and implemented in this paper for wave scattering avoids the singularities inherent in traditional boundary integral methods. This is accomplished by reversing the sequence of stages in the procedure. Traditionally, the differential equations are first reduced to boundary integrals with respect to equivalent sources on the boundary and then discretized; the kernels of the underlying integral equations are singular due to the infinite self-fields of concentrated sources. In BDM, the differential problem is first discretized on a regular grid to obtain a finite-difference approximation that is then reduced to a boundary difference equation with respect to auxiliary sources on the discrete boundary. The field of these sources can be expressed by convolution with the discrete Green function that, unlike its continuous counterpart, is finite at all points. Thus no singularities ever arise. Technically, the underlying grid is infinite. The computational procedure, however, involves only the boundary nodes of the grid and a finite spatial window where the discrete Green function is precomputed, which can be done once and for all for a given set of parameters. The validity of BDM has been demonstrated using 2D scattering from dielectric cylinders as a model problem. Convergence of the method as a function of the grid size has been established and is commensurate with the order of finite-difference schemes used. B. Trade-Offs Since the proposed approach has common features with the traditional integral equation methods, some of the usual tradeoffs between differential and integral techniques [38] apply. The differential methods lead to sparse matrices, whereas the boundary methods produce dense ones. This drawback can be partly alleviated via fast multipole acceleration [18], [20]–[23].

TSUKERMAN: A SINGULARITY-FREE BOUNDARY EQUATION METHOD FOR WAVE SCATTERING

Its use in conjunction with BDM is relatively straightforward. Indeed, FMM relies on a recursive splitting of the solution into near- and far-field components. The far field in BDM is essentially the same as in the continuous problem, by construction of the discrete Green function; see (5). It is only in the near field that discrete and continuous Green functions may differ significantly, but this makes little difference in FMM algorithms because the near-field contribution is computed directly. As already emphasized, BDM completely dispenses with singular integral kernels, an inherent drawback of integral methods. The price to pay for that is the need to precompute discrete Green functions. In practice, this price can be expected to be modest, because the number of different materials in any given problem is limited and the computation involves a relatively small number of grid layers around Green’s point source. In any event, this computational overhead is independent of the size of the problem being solved and therefore does not affect the overall asymptotic complexity of the procedure. For unbounded problems, differential methods such as FEM and FD require artificial domain truncation with absorbing boundary conditions or matched layers. No such truncation is needed in boundary methods. At the same time, differential methods are generally better suited for nonlinear problems that call for volume discretization, in which case the boundary methods usually lose their effectiveness. The key source of numerical errors in traditional BEM is approximation of singular integrals (typically, by piecewise-polynomial functions of low order, including piecewise-constant approximations in the simplest case). In BDM, the error is due to the finite-difference approximation of the boundary conditions and of the lattice Green function. If the order of these approximations is increased, the overall numerical error of the method can be reduced accordingly. In all numerical tests the condition number of the system maor less, so no comtrix was very moderate, on the order of plications arose due to ill-conditioning. This is consistent with the theoretical and numerical results presented by Martinsson and Rodin [23], albeit for the Laplace equation in their case. C. Generalizations and Future Directions Boundary difference schemes developed in this paper lend themselves to generalization in quite a natural way. Unlike traditional boundary integral methods, BDM is automatic, in the sense that it does not require the suitable sets of equivalent boundary sources (electric or magnetic surface currents, surface charges, etc.) and the respective equations to be worked out in advance. Instead, one introduces discrete boundary sources that need not even have a specific physical meaning; but once computed, they can be used to find physical fields by convolution with Green’s functions on the lattice. In particular, the -mode (TE-or -mode) of electromagnetic wave scattering is treated in BDM exactly the same way as the -mode. (As a side note, for the -mode the classical five-point control volume scheme would only be of order one at the boundary, but this is a feature of that scheme, not of BDM as a whole.) Further, extension to 3D vector problems is also conceptually straightforward, although clearly algorithmic challenges do arise. This line of research is currently being pursued.

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The boundary difference method does in general require a spatially uniform grid. Although this grid is “virtual,” in the sense that the actual computation involves only the nodes on the discrete boundary and not the volume nodes, the uniformity of the grid may still be a limiting factor in some problems. However, if several scatterers are present and well-separated (in practice, by at least a few grid layers), then each of them may be meshed separately. Indeed, in that case the interactions between different scatterers are numerically in the far field, where the continuous Green function can be used as a good proxy for the discrete one. Finally, the method is not limited to electromagnetics and can be extended to other classes of linear problems, including acoustics and elasticity. It may even be applied to micro-, nanoand molecular-scale models on a discrete lattice (e.g., Haq et al. [40]), when continuous equations may not even be available. APPENDIX REPRESENTATION OF THE FIELD VIA DISCRETE SOURCES Let us show that any discrete field on the boundary can be represented via convolution of the Green functions with some auxiliary sources on the same boundary, except possibly for some special cases of interior resonance. More precisely, let be the scattered component of a lattice field that satisfies the discretized wave equation both inside and outside the scatterer. is defined as outside the scatterer and as ( inside.) Further, let represent the values of on the discrete boundary . We intend to show that (12) for some source on . The discrete convolution in (12) can be viewed as a linear to fields , also in operator that maps functions in . It is then sufficient to demonstrate that this operator is nonsingular or, equivalently, that an identically zero field on the discrete boundary can be produced only by zero sources on that boundary. is identically zero. Then Let us thus assume that must be zero, too, everywhere in the outside region. This is true because, by its construction as convolution with sources only on , this field satisfies the homogeneous difference equation in the outside region and also, by assumption, the Dirichlet conare zero. Similar considerations hold for ditions for it on in the inside region away from the interior resonance, i.e., is not an eigenvalue of the wave problem inside as long as the scatterer, with zero Dirichlet conditions. Thus the convolution in (12) must be identically zero on the whole lattice, from which it immediately follows (e.g., via Fourier transforms) that . A similar proof can be found in [36]. ACKNOWLEDGMENT The author is very grateful to S. V. Tsynkov for informative and illuminating discussions that helped to catalyze the research reported in the paper. He also thanks P.-G. Martinsson and G. Rodin for helpful comments. The time and effort of the anonymous reviewers are greatly appreciated.

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REFERENCES [1] H.-D. Alexander and D. T. Cheng, “Heritage and early history of the boundary element method,” Engrg. Analy. With Boundary Elements, vol. 29, pp. 268–302, 2005. [2] M. B. Friedman and R. Shaw, “Diffraction of pulse by cylindrical obstacles of arbitrary cross section,” Trans. ASME J. Appl. Mech., vol. 29, pp. 40–46, 1962. [3] R. P. Banaugh and W. Goldsmith, “Diffraction of steady acoustic waves by surfaces of arbitrary shape,” J. Acoust. Soc. Am., vol. 35, pp. 1590–1601, 1963. [4] L. H. Chen and D. G. Schweikert, “Sound radiation from an arbitrary body,” J. Acoust. Soc. Am., vol. 35, pp. 1626–1632, 1963. [5] G. Chertock, “Sound radiation from vibrating surfaces,” J. Acoust. Soc. Am., vol. 36, pp. 1305–1313, 1964. [6] L. G. Copley, “Integral equation method for radiation from vibrating bodies,” J. Acoust. Soc. Am., vol. 41, pp. 807–816, 1967. [7] R. P. Shaw, “Diffraction of acoustic pulses by obstacles of arbitrary shape with a Robin boundary condition,” J. Acoust. Soc. Am., vol. 41, pp. 855–859, 1967. [8] K. M. Mitzner, “Numerical solution for transient scattering from a hard surface of arbitrary shape-retarded potential technique,” J. Acoust. Soc. Am., vol. 42, pp. 391–397, 1967. [9] R. F. Harrington, “Matrix methods for field problems,” Proc. IEEE, vol. 55, pp. 136–149, 1967. [10] R. F. Harrington, Field Computation by Moment Methods. New York: Wiley-IEEE Press, 1993. [11] O. V. Tozoni and I. D. Mayergoyz, Raschet Trehmernyh Electromagnitnyh Polej (Computation of Three-Dimensional Electromagnetic Fields). Kiev, Ukraine: Tehnika, 1974. [12] W. C. Chew, M. S. Tong, and B. Hu, Integral Equation Methods for Electromagnetic and Elastic Waves. New York: Morgan and Claypool, 2007. [13] W. C. Gibson, The Method of Moments in Electromagnetics. New York/Boca Raton, FL: Chapman & Hall/CRC, 2008. [14] A. F. Peterson, S. L. Ray, and R. Mittra, Computational Methods for Electromagnetics. New York: IEEE Computer Society Press, 1997. [15] M. Bonnet, Boundary Integral Equation Methods for Solids and Fluids. New York: Wiley, 1999. [16] I. Tsukerman, Computational Methods for Nanoscale Applications: Particles, Plasmons and Waves. Berlin: Springer, 2007. [17] D. Pissoort, E. Michielssen, and A. Grbic, “An electromagnetic crystal Green function multiple scattering technique for arbitrary polarizations, lattices, and defects,” J. Lightw. Technol., vol. 25, no. 2, pp. 571–583, 2007. [18] L. Greengard and V. Rokhlin, “A fast algorithm for particle simulations,” J. Comput. Phys., vol. 73, no. 2, pp. 325–348, 1987. [19] H. Cheng, L. Greengard, and V. Rokhlin, “A fast adaptive multipole algorithm in three dimensions,” J. Comp. Phys., vol. 155, no. 2, pp. 468–498, 1999. [20] P.-G. Martinsson, “Fast Multiscale Methods for Lattice Equations,” Ph.D. dissertation, Texas Inst. Comput. Appl. Math., Univ. Texas at Austin, Austin, 2002. [21] L. X. Ying, G. Biros, and D. Zorin, “A kernel-independent adaptive fast multipole algorithm in two and three dimensions,” J. Comp. Phys., vol. 196, no. 2, pp. 591–626, 2004. [22] W. Fong and E. Darve, “The black-box fast multipole method,” J. Comp. Phys., vol. 228, no. 23, pp. 8712–8725, 2009. [23] P.-G. Martinsson and G. J. Rodin, “Boundary algebraic equations for lattice problems,” Proc. Royal Soc. A—Math. Phys. Eng. Sci., vol. 465, no. 2108, pp. 2489–2503, 2009. [24] I. Harari, “A survey of finite element methods for time-harmonic acoustics,” Comput. Methods Appl. Mech. Engrg., vol. 195, pp. 1594–1607, 2006.

[25] C. Saltzer, “Discrete Potential Theory for Two-Dimensional Laplace and Poisson Difference Equations,” National Advisory Committee on Aeronautics, 1958 [Online]. Available: http://ntrs.nasa.gov/archive/ nasa/casi.ntrs.nasa.gov/19930085135_1993085135.pdf [26] I. Tsukerman, “Electromagnetic applications of a new finite-difference calculus,” IEEE Trans. Magn., vol. 41, pp. 2206–2225, 2005. [27] I. Tsukerman, “A class of difference schemes with flexible local approximation,” J. Comp. Phys., vol. 211, no. 2, pp. 659–699, 2006. ˇ [28] I. Tsukerman and F. Cajko, “Photonic band structure computation using FLAME,” IEEE Trans. Magn., vol. 44, pp. 1382–1385, 2008. [29] I. Tsukerman, “Quasi-homogeneous backward-wave plasmonic structures: Theory and accurate simulation,” J. Opt. A: Pure Appl. Opt., vol. 11, no. 114025, 2009. ˇ [30] F. Cajko and I. Tsukerman, “Flexible approximation schemes for wave refraction in negative index materials,” IEEE Trans. Magn., vol. 44, pp. 1378–1381, 2008. [31] I. Tsukerman, “Trefftz difference schemes on irregular stencils,” J. Comput. Phys., vol. 229, pp. 2948–2963, 2010. [32] H. Pinheiro, J. P. Webb, and I. Tsukerman, “Flexible local approximation models for wave scattering in photonic crystal devices,” IEEE Trans. Magn., vol. 43, pp. 1321–1324, 2007. [33] H. Pinheiro and J. P. Webb, “A FLAME molecule for 3-D electromagnetic scattering,” IEEE Trans. Magn., vol. 45, pp. 1120–1123, 2009. [34] V. S. Ryaben’kii, The Method of Difference Potentials and Its Applications (in Russian), 2nd ed. Moscow, Russia: Fizmatlit, 2002. [35] A. A. Reznik, “Approximation of surface potentials of elliptic operators by difference potentials,” Dokl. Acad. Nauk, SSSR, vol. 263, pp. 1318–1321, 1982. [36] S. V. Tsynkov, “On the definition of surface potentials for finite-difference operators,” J. Sci. Comput., vol. 18, pp. 155–189, 2003. [37] S. V. Tsynkov, Private Communication. [38] A. Konrad and I. A. Tsukerman, “Comparison of high- and low-frequency electromagnetic field analysis,” J. Phys. III France, vol. 3, pp. 363–371, 1993. [39] P. G. Martinsson and G. Rodin, “Asymptotic expansions of lattice Green’s functions,” Proc. Royal Soc. A, vol. 458, pp. 2609–2622, 2002. [40] S. Haq, A. B. Movchan, and G. J. Rodin, “Analysis of interphases in lattices,” Acta Mech. Sin., vol. 22, pp. 323–330, 2006.

Igor Tsukerman received the combined B.Sc./M.Sc. degree (with honors) in control systems and the Ph.D. degree in electrical engineering from Polytechnical University of Leningrad, Russia, in 1982 and 1988, respectively. From 1990 to 1995, he worked in the Department of Electrical and Computer Engineering, University of Toronto, Toronto, ON, Canada. He is currently a Professor of electrical and computer engineering at the University of Akron, Akron, OH, where he has been a faculty member since 1995. His research is focused on the simulation of nanoscale systems, applied electromagnetics and photonics. He teaches a variety of undergraduate and graduate courses (signals & systems, circuits, electromagnetic fields, digital signal processing, random signal analysis, simulation of nanoscale systems, and others), has over 140 refereed publications and is the author of the book Computational Methods for Nanoscale Applications: Particles, Plasmons and Waves (Springer). He is currently co-editing the book Plasmonics and Plasmonic Metamaterials: Analysis and Applications (World Scientific, to be published in 2011).

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A Complete Set of Linear-Phase Basis Functions for Scatterers With Flat Faces and for Planar Apertures Massimiliano Casaletti, Stefano Maci, Fellow, IEEE, and Giuseppe Vecchi, Fellow, IEEE

Abstract—Entire-domain basis functions are introduced for the analysis of scattering from bodies with flat portions, and of radiation from planar apertures; they are defined via the application of the sampling theorem of Shannon. These basis functions are particularly convenient for polygonal contours; however, arbitrary contours can be treated in the same framework with a very minor pre-processing time. A non-redundant set of basis functions is determined a priori by explicit approximate formulas either for currents and for radiated field. Numerical results are presented to shown the accuracy of the method. Index Terms—Aperture antennas, integral equation methods, method of moments (MoM), physical optics, radiation, scattering.

I. INTRODUCTION HE integral equation (IE) approach combined with the method of moments (MoM) discretization scheme is widely used in the prediction of the electromagnetic scattering from a large complex objects. The conventional MoM formulations are severely limited by the problem size; iterative approaches exploiting fast factorization methods—and notably the multilevel fast multiple approach (MLFMA)—are the common choice to overcome these limitation. Iterative methods still have to overcome the conditioning issue, especially relevant when the structure presents geometrical details together with the usual large size, and the cost associated to multiple excitations. Because of this, alternative, iteration free approaches [2]–[5] have been presented in recent years. All these approaches are of the domain-decomposition (DD) type, and rely on segmenting the structure in smaller parts, called blocks; each block are first separately analyzed in the presence of auxiliary sources that represent the effect of the other blocks. The responses to these auxiliary sources on the isolated blocks are then used to construct basis functions with support over the entire extension of each block [6]; these are subsequently used in the full-wave MoM analysis of the entire structure. In the following, we will adopt the terminology “synthetic functions” (SF) to denote the entire-domain basis functions generated

T

Manuscript received May 13, 2009; revised June 18, 2010; accepted July 06, 2010. Date of publication December 03, 2010; date of current version February 02, 2011. M. Casaletti and S. Maci are with the Department of Information Engineering, University of Siena, 50124 Siena, Italy (e-mail: [email protected]; [email protected]). G. Vecchi is with the Department of Electronic Engineering, Politecnico di Torino, Turin, Italy (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2096178

from the solution of the electromagnetic problem for the block in isolation, under excitation by suitably defined “generating sources.” The latter can be either plane waves, or point sources. A key point in the construction of the synthetic basis functions is the application of a singular value decomposition (SVD) [2], [4], [5] to select the relevant and linearly independent responses. Within the previous scheme, the introduction of special types of basis functions for treating large flat portions of PEC objects appears a quite useful complement to the general framework of the domain-decomposition methods. The specialized treatment starts, as in the general scheme, from the application of the equivalence theorem. This isolates the flat portion from the rest by a virtual equivalence surface, and an appropriate set of generating sources will account for the presence of the external world. When solving for the internal part problem, the arbitrary medium that fills the external part can be chosen in order to simplify as much as possible the internal Green’s function. For flat plates, one possible choice is the continuation of the flat surface into an infinite plate. If on one hand, this choice implies difficulties in achieving the right edge singularities for true edges, it naturally avoids the problem of spurious edge singularities in the basis functions for fictitious edges. The responses to plane waves are in such a case simply the PO currents due to the generating sources, i.e., functions with linear-phase spatial dependences. In [7], a comparisons has been presented between plane wave and point-wise (dipole) generating sources. In general [8], the use of an SVD orthogonalization and ranking is still a necessary ingredient of the scheme. The aim of the present work is two-fold; on the one hand, it hinges on the generation of synthetic functions on large, flat portions of the overall structure under analysis; in this sense it aims at adding generality and efficiency to the existing schemes, and in particular those in [4], [7]–[10] (the commonly employed PO basis functions are a special, physically meaningful, case). On the other hand, the formulation is not limited to the case of scatterers nor of PEC material, being applicable to non-PEC objects and to field radiated by apertures as well. Indeed, here we will show that the plane-wave-type, linear-phase spatial dependences constitute a complete, discrete basis for any kind of equivalent currents on a flat finite surface. In this sense the basis behaves as an alternative of other wave-object bases to expand aperture fields, like the Gabor expansion [11]. The present results appear as the application of the well-known sampling theorem by C. Shannon [12], extended to deal with non-separable domains. We will introduce a systematic sampling approach that will allow to select the necessary phasings to form a basis without having recourse to an SVD.

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As in the usual sampling theorem, there remains to find a criterion to truncate a (convergent) series of contributions on the basis of some “accuracy” specification. In the presented setting, the issue will become an alternative (and somehow extended) approach to the determination of the degrees of freedom of the source [13]; also, this work shares some of the objectives of [14]. Finally, we observe that here we consider the planar domain in isolation, as it is enough for generating basis functions for domain-decomposition methods for faceted structures as reported in the literature. If desirable, the extension to generate basis functions for the association of two or more plates may be similarly treated by a technique of spectral rotation [15], [16]; it is however outside the scopes of the present work, and the subject of future communications. The presentation of the paper is organized as follows. In Section II, the formulation is presented, based on the application of sampling theorem for non-separable 2D domains. The process leads to a complete set of linear-phase functions (LPF). Section III discusses the concept of degrees of freedom of the field, namely the criteria to determine the sufficient and non redundant number of basis functions to have a desired accuracy at decreasing distance from the planar surface. In Section IV, the independent LPF are orthogonalized in closed form by a Gram-Smith process. Numerical results are presented in Section V, and conclusions are drawn in Section VI.

II. BASIS GENERATION VIA AN EXTENDED SAMPLING THEOREM In this section we employ the 2D version of the classical sampling theorem to the currents (and fields) on a flat arbitrarily contoured surface. We are interested in finding a representation for the fields on the surface, but we do not assume any material property, and in this general setting is independent on the specific (integral) equation that these fields/current will satisfy. The term “extended” refers to the fact that the usual Shannon sampling theorem [12] is one-dimensional, and its extension is obvious for 2D separable, i.e., rectangular, domains; in this endeavor, instead, we consider an arbitrary contour. We use a sampling in the spectral domain because the concerned functions are compact-supported in the spatial domain, thus reversing the meaning of time and frequency of the original setting of the sampling theorem.

Fig. 1. Geometry of the problem.

Fig. 2. Spatial representation of the sampled current function for polygonal contour.

In the following, we will also use the term “compactly supported” to refer to this property. Fourier (2D) transformed funcare denoted by tions (spectra) of the spectral variables a tilde, e.g., The spatial limitation of implies the applicability of the usual sampling (Shannon) theorem to express the Fourier of as a function of an infinite set of spectrum spectral samples. To this end, let us first construct the infinite of the current periodic repetition

(2) and is equal to the maximum Note that since the period dimension of the plate, the representation in (2) is not affected by current aliasing. It is known (see the Appendix) that the periodic repetition of the spatial currents possesses a spectrum equal to the sampling of the spectrum of the current at spectral sampling points (3) (the spectral lattice is shown in Fig. 3), i.e.,

A. Formulation Let us consider the electromagnetic scattering problem in Fig. 1, constituted of a generic flat surface , lying in the plane, and denote by and the maximal widths along the and axis, respectively. For the sake of simplicity, we will , understood that analogous refer here to electric current steps can be followed for possibly additional magnetic currents in case of aperture or impedance surfaces. The induced current is a spatially limited function, that is

(4) The exact non-periodic current can be therefore reconstructed by filtering the periodic space representation by the of the surface , namely characteristic function (5)

(1)

(6)

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IV.C); general shapes can be treated at the expenses of defining via a line integration or via FFT (Section IV.D). B. Spatial and Spectral Completeness Relationship We now investigate on the completeness properties of the representation, by analyzing the representation of the spatial Dirac delta. By inserting a constant (unit) spectrum in (7), one finds the representation of the delta

(10) Fig. 3. Lattice of the sampling spectral points for sampling the current spectra. The dashed circle is the boundary of the visible region. The mesh is referred to and .

D = 3

D = 4

Using (4) in (5), leads to

(7)

(One can ascertain that this is indeed the delta by using the usual distribution theory; by projecting (10) over any test function compactly supported in , one finds the unit property of the delta.) This means that the representation in (7) is complete, and any Fourier-transformable function that vanish outside can be represented in terms of the space-windowed exponential functions in (7). Likewise, by inverse transform of (10), we see that the summation of the translated spectra of the characteristic function leads to unity, i.e.,

Transforming both sides of (7) in the spectral domain, one has (11)

(8)

that is the spectral version of the completeness relation. This latter relation will be used next to determine the numbers of sufficient and non redundant spectral samples needed for the complete description of the scattered field.

where C. Spectrum of the Characteristic Unit Function for Polygonal Contours (9)

is the spectrum of the characteristic function . For a rectangular domain, the pair of (7) and (8) reduces to the usual Shannon sampling theorem for 2D separable domains; in it, lead to products of “sinc” functions. the functions The latter appear to be the ideal interpolating functions, in that they achieve the Nyquist limit. The pair (7)–(8) constitutes the extension of the Shannon sampling theorem to non-separable planar domains. In it, the compact-supported currents are represented by spectral samples at sampling steps inversely proportional to the maximum dimension of the source. In this representation for non-separable domains, the representing functions appear the spectra of the characteristic . We emphasize here that function of the domain, general shapes of the contour can be framed in the same formulation. For polygonal contours the spectral characteristic funcpossesses an analytical closed form (Section tion

To derive a closed form expression of the characteristic function spectrum, let us denote by the positions of the vertexes of the polygonal contour (see Fig. 4(a)). By simple algebraic manipulations one can find

(12)

where

and

(13) Note that the above spectra are regular for . An example of characteristic function spectrum for a triangular domain is shown in Fig. 4(b). This spectrum is relevant to a triangular domain with dimension maximum dimension equal

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Fig. 5. Pictorial representation of the “generating plane waves” (linear phasing terms) of the LPF.

W k ;k

Fig. 4. (a) Geometry of a polygonal flat surface; (b) amplitude of the spectrum ~( ) of the characteristic function for a triangular plate.

to . The circle depicted in the figure represents the boundaries of the visible region. D. Spectrum of the Characteristic Unit Function for an Arbitrary Contour We begin by noting that an arbitrary contour can be approximated by a polygonal contour, with a number of vertices essentially depending on its electrical length; this gives an explicit of the unit characteristic basis expression for the spectrum function in terms of (12) with a linear complexity. can be calculated in terms of a line integraAlternatively, tion through the Stokes theorem directly applied to the definition of the Fourier Transform in (9). The integrand can be set up in the exact form

spatially windowed (compactly supported) plane-wave basis functions; we will call them “linear phase functions” (LPF). The necessary phasing are set by the spectral sampling points in (3), thus, the angular density of the necessary plane waves is imposed, and this allows one to establish a priory the set of independent functions without redundancy. We observe that the LPF correspond to PO currents induced on the polygonal surface by plane waves incident with appropriate wavevectors (Fig. 5). However, we are not using their physical meaning—indeed we have not requested or enforced that the considered quantities be Maxwellian. For the same reason, we need not be concerned with the vector (“polarization”) properties of these “generating plane waves,” since the vector amplitudes in (8) are those appropriate for the representation of a given function, or will be the unknowns of an integral equation when using the representation to provide basis functions for the method of moments. Thus, we will refrain from the most common TE-TM decomposition, and simply consider—without loss of generality—to two independent components along and , normalized in such a way to have unit amplitude, i.e.,

(14) , from which one can exwhere press the r.h.s. of (14) in terms of a contour integration through the Stokes theorem

(16) The coefficients of the LPF current expansion are given by (7) via the spectrum of the exact current, namely

(15) (17) where is the tangent vector to the contour and is the ele. Note that for circular surface one mental portion of contour obtains again closed form. An alternative to (15) one can use a . For a circular con2D FFT to evaluate the spectrum tour of radius centered in the origin, one has , where is the first order Bessel function. E. Linear-Phase Functions (LPF) The exponential terms in (7) have the clear meaning of phase of a plane wave observed on the planar domain of interest. Therefore, the representation in (7) implies that any planar current can be represented exactly in terms of a discrete set of

where is the th scalar component of the spectral current (in typical applications, this spectrum will be unknown). The vector LPF in (17) for polygonal contours possess closed-form spectra given by (18) III. DEGREES OF FREEDOM AND NON REDUNDANT BASIS The results in Section II state that we can represent a general function on a compact support via an infinite, discrete, set of

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basis functions. The relevant problem at this point is to find the number of basis functions required to have a representation of the field radiated by an arbitrary current within a specified accuracy. By using the well established terminology introduced in [13] this number is equal to the degrees of freedom (DoF) of the field. We observe that we must be able to represent the current produced by a general (unspecified) source and the scattered field produced in whatever point of space. In fact, the number of basis functions can be limited if the source of the scattered field is known a priori [17]. We address the question coherently with the formulation introduced here, by analyzing the spectrum of the radiated field. We consider here the case when the observation point is out from the reactive region and gradually approach the radiating surface. A. Field Outside the Radiating Surface Let us consider the field radiated by the currents at a certain distance from the plane in its spectral radiation integral representation [18]; for simplicity, we retain only the contribution directly proportional to the vector potential, i.e.,

(19)

where

(20) is the z-dependent Green’s function spectrum, is the 2D spectral variable in (13), and are the free-space wavenumber and impedance, respectively. We are interested in the minimal spectral content that is necessary to represent the scattered field in (19) within a given accuracy, for an arbitrary (spatially compact-supported) current . When the observation distance is non-zero, the Green’s function acts as a low-pass filter for its exponential decay factor outside the visible region ; the decay rate is set by the minimum observation distance . Therefore, the field produced by an arbitrary current can be represented within a given accuracy with a finite spectral content. This clearly indicates that the most critical case for us is when the scattered field is observed on the surface, i.e., . This case will be addressed in the next subsection. By setting a threshold value to the Green’s function spectral attenuation factor in (19), the significant spectrum lies inside a region of area where the parameter is found from the exponential term of (20) as

(21)

Fig. 6. Gradual filling of the spectral circle of radius area k ( = 2:1) by increasing the number of spectral LPF functions with (a) wavenumbers within a circle of 1:5k ; (b) wavenumbers within a circle of radius 2k , (c) final setting of  = 2:5 with error " = 0:03 with respect to unity.

A good empirical threshold has been seen to be , that implies . Since the LPF constitute a set of independent functions, the ability to reconstruct an arbitrary function is measured by the ability to reconstruct the spectrum of the delta function inside the region . The spectral completeness relationship in (11) leads to a useful tool to establish a non redundant basis set. To this end, it is convenient to define the quantity

(22) that is the truncated version of the rhs of (11) with summation extended to the spectral samples inside . From (11), one has . We therefore define the truncation region so that

(23)

Namely, as a criterion of truncation to define the degrees of freedom, we impose that in the average quadratic error in reconstructing unity is less than a fixed threshold by summing spectral LPF with wavenumbers inside . Fig. 6 shows, for the triangular plate in the inset, the amplitude of when gradually increasing till the value . Gradual filling of the spectral circle of radius is obtained by increasing the number of spectral LPF till obtaining the required flatness.

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Substituting (22) in (23), and for numerical convenience (albeit not strictly necessary), approximating the integral in (23) by staircase summation on the spectral generating lattice, leading to

(24) where and is the number . Equation (24) deof lattice samples in the circular region as a function of and of the fines the truncation limit . The non redundant LPF threshold; namely are therefore those for which

(25) The number of degrees of freedom can be defined as the multiplied by a factor 2 to account number of LPF inside for the two polarizations, thus leading to the approximation

(26)

surface, we employ the results in [16], that fit well in our spectral sampling framework. In that work, it was observed that a spectrally truncated Green’s function in was enough to obtain the correct MoM solution. (This led to a spatial Green’s function free of the near-field singularity.) In terms of our present analysis, this means that we can rein (19) with where is a window place function that is zero, or negligible, after a spectral threshold . The relevant issue is that the spectral truncation is determined by the (spatial) mesh density [16]. A secondary finding of [16] was that a spectral roll-off in the window is necessary (as opposed to sharp truncation). The degree of sophistication of this spectral windowing depends on the accuracy requirements on the unknown current; for scattering problems (where only the scattered field away from the surface is of interest) an exponential roll-off was enough, while a Gaussian or Blackman window functions where necessary when the near-field of the unknown current was of importance (antenna problems). The exponential windowing has the appeal of a cogent physical interpretation, since it corresponds to remove the source from the equivalent surface [16], or equivalently, to observe the scattered from the equivalent currents; this field at a certain distance “stand-off” distance is directly proportional to the mesh size. On the basis of the last consideration, we suggest here the following “truncation criterion for currents.” To establish the number of LPF to be used for currents (or aperture fields) within the approximation of a numer, we use (25) with ical meshing of minimum mesh size in (27) and obtained by setting and in (21).

wherein is the free-space wavelength. The approximation in (26) is more accurate for increasing dimension of the plate in wavelength. When the observer is out of the reactive region (let us say , it is found that loses the dependence on and and can be approximated (for convex contours) as

By using the above semi-empirical approximation, the following non redundant number of LPF is obtained

(27)

(28)

where is the area of the plate. For instance, the case of a and , rectangular and triangular plate respectively. In summary, we propose the following “truncation criterion for field”

Some remarks are useful to put this, and other similar endeavors in proper perspective. One could observe that the criterion above for on-surface observation leads to essentially the same number of DoF as one would have using conventional space-sampling basis functions (RWG) for a given desired mesh density. Indeed, the results in [16] and the present extension of the Shannon sampling theorem can be phrased as follows. The (electric) overall size of the structure will determine the spectral sampling rate (in the extended sense) and the shape of the spectral basis functions; in a conventional MoM, the necessary spatial mesh size (spatial sampling) will determine the overall spectral extent necessary to give the same accuracy in the representation of the solution and then the number of spectral function to be used to cover the same extent. It can be observed that the above framework parallels and extends the usual setting of the representation of a band-limited (1D) function. Therefore, no reduction of the DoF is expected per se from this approach, inasmuch as that the quantity of information of a general signal does change passing from time to frequency representation.

Equation (21) is used to establish which LPF have to be included in the expansion of the field at distance from the aperture. Inside the reactive region is established by (24) with use of (21). Outside this region is approximated with (27). The number of LPF (equal to the degrees of freedom of the field) is established by (26). B. Representation of On-Surface Radiated Field The spectral process presented in the previous section held for representing the field at a finite (albeit small) distance from the surface. In handling the present situation of observer at the

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On the other hand, the use of domain-decomposition approaches [7]–[10] does appear to reduce the DoF; this is not in contrast with the above. As discussed in [17], the reduction of the DoF in actual aggregate basis function approaches derives from the fact that a significant degree of information on the solution is injected as one generates the basis functions, that are obtained by solving the EM problem for a portion of the overall structure. This added physical information allows one to reduce the sufficient number of the synthetic functions also when describing the currents—as opposed to radiated fields only. For instance, introducing the edge information in a synthetic function by aggregation of RWG functions keeps the right spectral information of the evanescent spectrum of a portion of structure. In some special cases, on the other hand, the number of degrees of freedom might be intrinsically reduced [20], [21]. It is obvious that the same amount of physical information can be also injected if one uses the LPF to construct synthetic basis functions by direct MoM solution or by interpolation of solution obtained with div-conforming basis functions, like RWG. In the latter case the number of LPF needed for the interpolation would be comparable with those of the used RWG, since proportional to the inverse of the mesh size If needed in the specific MoM implementation, one can add basis functions with the right edge behavior, or localized standard functions along the edges [10] to improve the descriptive ability of the LPF. To this end, the LPF can be completed by additional fringe basis functions that contain the right edge singularities for the edge-normal and edge-parallel components. To this end, one should refer to [22] for both edge and vertex based fringe current contribution. The additional basis functions satisfy a separate “fringe” integral equation as shown in [23]. In a general perspective, the issue of the actual minimum number of degrees of freedom in MoM appears to be not completely closed, and problem-dependent. In the perspective of the present work, on the other hand, is important to remark that our setting allows to catch all possible scenarios, adding a systematic framework to select the non-redundant number of basis functions. IV. ORTHOGONAL MODES In this section, we show how to make the orthogonalization of the LPF. While not strictly necessary for use with the MoM (the LPF are already a basis) the orthogonalization is useful for improving the MoM matrix condition number, and in the present case it will be shown to be achievable in closed form; second, in case the aperture distributions is known a priori, the orthogonalization facilitate the construction of a generalized Fourier expansion by a simple projection process. A complete orthogonal set of basis functions can be obtained by a Gram-Schmidt procedure applied to the sufficient and nonredundant number of LPF for a given observation distance . We will term these orthogonal functions “modes”, and denote them as . Since the adopted orthogonalization is hierarchic, the process starts with the LPF whose associated wavenumber is closest to the origin and proceeds on a spiral squared path by

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the adding one more LPF at each step. Denoting by starting function, the orthogonal modes are constructed as (29)

(30) (31) where, as usual, we have denoted

(32)

. and used the L2 norm The coefficients in (30) can all be expressed in closed form, since one easily verifies that

(33) where is the Kronecker delta. According to what prescribed in the previous section, we can include in the Gram-Schmidt process a number of LPF equal to the degrees of freedom of the field we want to represent, namely those LPF inside the circle of radius , with defined by (24) or by the simplified formula (33) when applicable. The use of the orthogonal functions set in (31) allows the expansion of the space currents through generalized Fourier coefficients, namely (34) , we use a spectral In unwrapping the index triplet ordering; the spectral index pair is mapped into a sequential single index by starting from the origin of the sampling lattice and moving in counterclockwise sense along a rectangular spirals; discrimination between the two polarizations indicated by the last index will be achieved by even or odd values of the global index . The colored maps of some orthogonal modes in spatial and spectral domains associated to a pentagonal plate included in a square of sides are shown in Fig. 7. The pictures does not include the dominant mode, which appears trivially uniform in amplitude. The spectra of the orthogonal modes are concentrated close to the boundary of the visible region; their spatial domain counterpart are localized at the boundary edges. V. NUMERICAL RESULTS In the following we analyze the convergence of the LPF expansion of currents and corresponding radiated field. This will

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Fig. 8. Current amplitude (x-components) on the H-plane scan for the pentagonal plate illuminated by a horizontal electric dipole placed at (x; y; z ) = (0; ; =6). The MoM results with RWG functions (dotted line) is compared with a LPF expansion with 578 term (continuous line). The PO current is also reported for comparison (dashed line).

0

Fig. 7. Some spatial (a) and spectral domain (b) orthogonal basis functions for a pentagon bounded by a square of side 1:8

be done by projecting the full-wave solution onto the modal functions introduced in the previous section on modes. The full-wave solutions have been obtained by using a commercial software (FEKO) implementing a standard method of moments (MoM) with RWG basis functions. We consider a first problems of a conducting pentagonal flat plate with maximal widths , illuminated by a short electric dipole very close to one edge (this critical situation is chosen to enhance the current amplitude at one edge). Figs. 8 and 9 compares the currents on a E-plane cut (Fig. 8) and the relevant radiated E-field at from the plane (Fig. 9). The number of LPF for currents and fields are chosen according to the truncation criteria established in Section III.B and III.A, respectively. To have a comparable agreement all over the plate, almost the same number of LPF (578) and RWG functions (563) are needed. For completeness, curves for currents and fields are also reported for the standard PO approximation (Figs. 8 and 9), that exhibits an expected significant deviation from the correct behaviors. The next example is relevant to a circular perfectly conducting circular disk with diameter , illuminated by a TM -polarized plane wave coming from a direction that forms 80 with the normal z-axis. Figs. 10(a) (E-plane cut) and (b) (H-plane cut) compare the MoM currents (with 3446 unknowns) with the relevant truncated LPF expansion and with a circular waveguide mode (CWM) expansion. The latter representation is added in order to check the relative efficiency

Fig. 9. Scattered field on the E-plane and H-plane for the problem in Fig. 8 at z = 2 (rectilinear scan). Curve for LPF (450 terms) and RWG (560 terms) are undistinguishable within the used scale. PO field is also reported.

of the LPF with another type of entire-domain basis functions, that is directly conformed to the shape of the body, and that also conveys significant physical information. It is seen that a good accuracy is obtained by using 1260 LPF. With the same number of terms, the CWM expansion exhibits a comparable accuracy, except at the edges, where the LPF expansion works better. Fig. 11 shows the field scattered by the same circular disc at two different heights ( and . For the (very near field) case the curve relevant to MoM (with 3446 unknowns), LPF (1260 terms) and CWM (1260 terms) are undistinguishable within the vertical range of the chosen scale. For the case the same agreement is obtained by choosing 1470 MoM unknowns, 450 LPF terms and 450 CWM terms. At a distance the number of necessary and non redundant LPF decreases to 200. As expected, PO yields a very poor approximation (due to the near-grazing incidence).

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Fig. 11. Scattered field on a rectilinear scan parallel to the x axis for the ;  and ;  For the problem in Fig. 10. Two cases are shown: case z :  the curve relevant to MoM (with 3446 unknowns), LPF (1260 terms) and CWM (1260 terms) are undistinguishable within the vertical range of the chosen scale. For the case z :  the same agreement is obtained by choosing 1470 MoM unknowns, 450 LPF terms and 450 CWM terms. PO field is also reported.

z=01

= 01

z=05

=05

Fig. 10. Induced current amplitude on the E-plane (a) and H plane (b) for a , geometry shown circular plate illuminated close to grazing incidence ( in the inset of (a)). Comparison among RWG (3446), LPF expansion (1260), and Circular waveguide mode (CWM) expansion (1260). The inset of Fig (b) shows the behavior of  and  as a function of the observation level.

= 80

As a final example we address a 3D body, a perfectly conducting corner reflector composed by two orthogonal square plates of side illuminated by a plane wave; we use LPF on each of the two composing plates, without adding any (“attachment”) term to adjoin the functions on the two parts of the body. According with the previous simplified formula, we have seen that the agreement between the MoM solution with 2157 RWG is reached by using 2116 LPF. This example shows the potential for applicability of the LPF expansion to 3D cases. VI. CONCLUSION A new complete set of linear-phase basis functions has been introduced, based on a suitable application of the Shannon the-

orem generalized to non separable domains; it is able to represent any current on a arbitrarily contoured planar surface. For simplicity, the notations and the examples have been referred to electric currents, while the process is not limited to PEC surfaces, and valid for any planar surface like apertures and/or dielectric facets. The generalization of the Shannon theorem refers to the fact that its 2D version, obvious for a rectangular domain, is here applied to arbitrary contours. Furthermore, sampling in spectral domain rather than in space domain has been used because the concerned functions are compact-supported in the spatial domain; thus, reversing the meaning of time and frequency of the original setting of the sampling theorem. A systematic criterion to select a priori the sufficient and non redundant number of LPF (degrees of freedom) has been introduced to represent the field. This criterion is indeed based on a uniform filling of a circular spectral region by the set of spectral basis functions, and it corresponds to the spectral version of the delta function completeness relationship. When the observer is out of the reactive field and the surface is large in terms of the wavelength, the necessary number of LPF tends to the degrees of freedom defined by Bucci et al. [19]. A truncation criterion has been also defined when looking at the currents more than at the field. In this case, the truncation criterion is unavoidably linked to the degree of approximation of the current, and therefore to the minimum size of the meshing. After having defined the sufficient and non redundant number of functions, an orthogonal set of modes has been constructed by a closed-form Gram-Smith process applied to the LPF. The numerical examples have shown that the LPF afford the predicted economy in representing non-reactive fields, and that they are also able to correctly represent the currents (i.e., the fields on the surface itself), including their edge-related strong

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Before concluding, we mention the following two important aspects. 1. The formulation presented here becomes appealing in view of the fact that the field radiated at any point (not only in the far field) by a LPF can be evaluated in exact form by a simple contour line integration; to this end, the formulation in [24] can be adopted. 2. The same approach used here can be extended to the study of curved surface [25]. The points above are under progress and will be the subject of future communications. APPENDIX in (2) may be rewritten in The periodic function terms of convolution product with of a collection of spatial delta functions

(A1) where the asterisk denotes space convolution product. The spectral domain version of the above expression is given by

(A2) The series at the r.h.s. can be seen as the Fourier series representation of the periodic repetition of spectral Dirac functions, thus leading to

(A3) is the spectrum of where spectral sampling point defined in (3). Fig. 12. Currents on a perfectly conducting corner reflector composed by two orthogonal square plates of side 2, illuminated by a plane wave. (a) Map of the MoM currents (2157 unknowns). (b) Map of the LPF expansion (2116 terms). (c) Comparison of current amplitude on the bottom face.

variations. In this case, however, their number is not much less than the number of RWG functions used in MoM. This is primarily due to the fact that the LPF do not incorporate edge singular behavior (that is well known to be non-radiative); if one insists in improving their ability to represent this stronger spatial variations, they should be completed with additional fringe current contribution.

evaluated at the

REFERENCES [1] W. C. Chew, J. M. Jin, E. Michielssen, and J. Song, Fast and Efficient Algorithms in Computational Electromagnetics. Boston, MA: Artech House, 2001. [2] L. Matekovits, G. Vecchi, G. Dassano, and M. Orefice, “Synthetic function analysis of large printed structures: The solution space sampling approach,” in Proc. IEEE AP-S Soc. Int. Symp., Boston, MA, Jul. 2001, pp. 568–571. [3] S. E. Suter and J. Mosig, “A subdomain multilevel approach for the MoM analysis of large planar antennas,” Microw. Opt. Technol. Lett., vol. 26, no. 4, pp. 270–277, Aug. 2000. [4] V. S. Prakash and R. Mittra, “Characteristic basis function method: A new technique for fast solution of integral equations,” Microw. Opt. Technol. Lett., pp. 95–100, Jan. 2003.

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[5] L. Matekovits, V. A. Laza, and G. Vecchi, “Analysis of large complex structures with the synthetic-functions approach,” IEEE Trans. Antennas Propag., vol. 55, no. 9, pp. 2509–2521, Sep. 2007. [6] S. Rao, D. Wilton, and A. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag., vol. 30, no. 5, pp. 409–418, May 1982. [7] M. Casaletti, S. Maci, and G. Vecchi, “Diffraction-like synthetic functions to treat the scattering from large polyhedral metallic object,” Appl. Comput. Electromagn. Society J., vol. 24, no. 2, 2009. [8] E. Lucente, G. Tiberi, A. Monorchio, G. Manara, and R. Mittra, “The characteristic basis function method (CBFM): A numerically efficient strategy for solving large electromagnetic scattering problems,” Turkish J. Elect. Eng., vol. 16, no. 1, 2008. [9] C. Delgado, R. Mittra, and F. Cátedra, “Efficient multilevel approach for the generation of characteristic basis functions for large scatters,” IEEE Trans. Antennas Propag., vol. 56, no. 7, pp. 2134–2137, Jul. 2008. [10] C. Delgado, R. Mittra, and F. Cátedra, “Accurate representation of the edge behavior of current when using PO-derived characteristic basis functions,” IEEE Antennas Wireless Propag. Lett., vol. 7, pp. 43–45, 2008. [11] J. J. Maciel and L. K. B. Felsen, “Discretized Gabor-based beam algorithm for time-harmonic radiation from two-dimensional truncated planar aperture distribution: I Formulation and solution,” IEEE Trans. Antennas Propag., vol. 50, no. 12, Dec. 2002. [12] J. G. Proakis and M. Salehi, Communication Systems Engineering, 2nd ed. Englewood Cliffs, NJ: Prentice Hall, 2002. [13] O. M. Bucci and G. Franceschetti, “On the degree of freedom of scattered fields,” IEEE Trans. Antennas Propag., vol. 37, no. 7, pp. 918–926, Jul. 1989. [14] O. Bucci and G. Franceschetti, “On the spatial bandwidth of the scattered fields,” IEEE Trans. Antennas Propag., vol. 35, no. 12, Dec. 1987. [15] G. Tiberi, S. Rosace, A. Monorchio, G. Manara, and R. Mittra, “A matrix-free spectral rotation approach to the computation of electromagnetic fields generated by a surface current distribution,” IEEE Antennas Wireless Propag. Lett., vol. 4, pp. 918–926, 2005. [16] F. Vipiana, A. Polemi, S. Maci, and G. Vecchi, “A mesh-adapted closed-form regular kernel for 3D singular integral equations,” IEEE Trans. Antennas Propag., vol. 56, no. 6, pp. 1687–1698, June 2008. [17] L. Matekovits, G. Vecchi, and F. Vico, “Physics-based aggregate-functions approaches to large MoM problems,” Appl. Comput. Electromagn. Society J., vol. 24, no. 2, 2009. [18] L. B. Felsen and N. Marcuwitz, Radiation and Scattering of Waves. New York: Wiley-IEEE Press, 1994. [19] O. M. Bucci, C. Gennarelli, and C. Savarese, “Representation of electromagnetic fields over arbitrary surfaces by a finite and nonredundant number of samples,” IEEE Trans. Antennas Propag., vol. 46, no. 3, pp. 351–359, Mar. 2008. [20] S. N. Chandler-Wilder and S. Langdon, “A Galerkin boundary element method for high frequency scattering by convex polygons,” SIAM J. Numer. Analy., vol. 45, no. 2, pp. 610–640, Feb. 2007. [21] C. P. Davis and W. C. Chew, “Frequency-independent scattering from a flat strip with TEz-polarized fields,” IEEE Trans. Antennas Propag., vol. 56, no. 4, pp. 1008–1016, Apr. 2008. [22] S. Maci, M. Albani, and F. Capolino, “ITD formulation for the currents on a plane angular sector,” IEEE Trans. Antennas Propag., vol. 46, no. 9, pp. 1318–1327, Sept. 1998. [23] A. Neto, S. Maci, G. Vecchi, and M. Sabbadini, “A truncated Floquet wave diffraction method for the full-wave analysis of large phased arrays. Part II: Generalization to the 3D case,” IEEE Trans. Antennas Propag., vol. 48, no. 3, pp. 601–611, Apr. 2000. [24] M. Albani and S. Maci, “An exact line integral representation of the PO radiation integral from a flat perfectly conducting surface illuminated by elementary electric or magnetic dipoles,” Elektrik, Turkish Int. J. Electron. Comput. Sci., vol. 10, pp. 291–305, 2002. [25] M. Casaletti, S. Maci, and G. Vecchi, “Generalized Shannon basis functions for curved surfaces,” presented at the EuCAP 2010, Barcelona, Apr. 12–16, 2010.

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Massimiliano Casaletti was born in Siena, Italy, in 1975. He received the Laurea degree in telecommunications engineering and the Ph.D. degree in information engineering from the University of Siena, Italy, in 2003 and 2007, respectively. From September 2003 to October 2005, he was with the research center MOTHESIM, Les Plessis Robinson (Paris, FR), under an EU grant RTN-AMPER (RTN: Research Training Network, AMPER: Application of Multiparameter Polarimetry). Since 2006, he has been a Research Associate at the University of Siena. His research interests include electromagnetic band-gap structures, polarimetric radar, rough surfaces and numerical methods for electromagnetic scattering and beam waveguides. Dr. Casaletti was awarded the Best Antenna Poster Paper Prize at the 3rd European Conference on Antennas and Propagation EuCap-2009, Berlin, Germany, in March 2009. In April 2010, he was awarded an Honorable Mention for Antenna Theory at the 4th European Conference on Antenna and Propagation EuCap-2010, Barcelona, Spain.

Stefano Maci (S’98–F’04) received the Laurea degree (cum laude) in electronic engineering from the University of Florence, Italy. Since 1998, he has been with the University of Siena, Italy, where he presently is a Full Professor. His research interests include EM theory, antennas, high-frequency methods, computational electromagnetics, and metamaterials. He was a coauthor of an incremental theory of diffraction for the description of a wide class of electromagnetic scattering phenomena at high frequency, and of a diffraction theory for the analysis of large truncated periodic structures. He was responsible and international coordinator of several research projects funded by the European Union (EU), by the European Space Agency (ESA-ESTEC), by the European Defence Agency, and by various European industries. He was the founder and presently is the Director of the European School of Antennas (ESoA), a post-graduate school that comprises 30 courses on antennas, propagation, and EM modeling though by 150 teachers coming from 30 European research centers. He is principal author or coauthor of more than 100 papers published in international journals, (among which 60 are IEEE journals), 10 book chapters, and about 350 papers in proceedings of international conferences. Prof. Maci was an Associate Editor of the IEEE TRANSACTIONS ON EMC and the IEEE TRANSACTION ON ANTENNAS AND PROPAGATION, the latter of which he was twice a Guest Editor. He is presently a member of the IEEE AP-Society AdCom, a member of the Board of Directors of the European Association on Antennas and Propagation (EuRAAP), a member of the Technical Advisory Board of the URSI Commission B, a member of the Italian Society of Electromagnetism and of the Advisory Board of the Italian Ph.D. school of Electromagnetism. He was recipient of several national and international prizes and best paper awards.

Giuseppe Vecchi (M’90–SM’07–F’10) received the Laurea and Ph.D. (Dottorato di Ricerca) degrees in electronic engineering from the Politecnico di Torino, Torino, Italy, in 1985 and 1989, respectively, with doctoral research partly carried out at the Polytechnic University, Farmingdale, NY. He was a Visiting Scientist at Polytechnic University from 1989 to 1990. In 1990, he joined the Department of Electronics, Politecnico di Torino, as an Assistant Professor (Ricercatore) where, from 1992 to 2000, he was an Associate Professor and, since 2000, he has been a Professor. He was a Visiting Scientist at the University of Helsinki, Finland, in 1992, and has been an Adjunct Faculty in the Department of Electrical and Computer Engineering, University of Illinois at Chicago, since 1997. His current research activities concern analytical and numerical techniques for analysis, design and diagnostics of antennas and devices, RF plasma heating, electromagnetic compatibility, and imaging.

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Stable Electric Field TDIE Solvers via Quasi-Exact Evaluation of MOT Matrix Elements Yifei Shi, Student Member, IEEE, Ming-Yao Xia, Senior Member, IEEE, Ru-Shan Chen, Member, IEEE, Eric Michielssen, Fellow, IEEE, and Mingyu Lu, Senior Member, IEEE

Abstract—Prior theoretical studies and experience confirm that the stability of marching-on-in-time (MOT) solvers pertinent to the analysis of scattering from free-standing three-dimensional perfect electrically conducting surfaces hinges on the accurate evaluation of MOT matrix elements resulting from a Galerkin discretization of the underlying time domain integral equation (TDIE). Unfortunately, the accurate evaluation of the four-dimensional spatial integrals involved in the expressions for these matrix elements is prohibitively expensive when performed by computational means. Here, a method that permits the quasi-exact evaluation of MOT matrix elements is presented. Specifically, the proposed method permits the analytical evaluation of three out of the four spatial integrations, leaving only one integral to be evaluated numerically. Since the latter has finite range and a piecewise smooth integrand, it can be evaluated to very high accuracy using standard quadrature rules. As a result, the proposed method permits the fast evaluation of MOT matrix elements with arbitrary (user-specified) accuracy. Extensive numerical experiments show that an MOT solver for the electric field TDIE that uses the proposed quasi-exact method is stable for a very wide range of time step sizes and yields solutions that decay exponentially after the excitation vanishes. Index Terms—Exact integration, late time instability, marching-on-in-time (MOT), time domain integral equations.

I. INTRODUCTION

VER since their inception in the 1970s, marching-on-intime (MOT) schemes for solving time domain integral equations (TDIEs) pertinent to the analysis of electromagnetic surface scattering phenomena [1] have been plagued by late time instabilities. Methods proposed to prevent these instabilities include averaging/filtering [2], [3], implicit time stepping

E

Manuscript received March 31, 2010; revised June 21, 2010; accepted June 24, 2010. Date of publication December 03, 2010; date of current version February 02, 2011. This work was supported in part by the U.S. AFRL CONTACT program (FA 8650-07-2-5061), the Major State Basic Research Development Program of China (973 Program 2009CB320201), NSFC Projects 60825102 and 60771001, the Air Force Office of Scientific Research under Grant MURI F014432-051936, and National Science Foundation Grant DMS 0713771. Y. Shi and R.-S. Chen are with the Department of Communication Engineering, Nanjing University of Science and Technology, Jiangsu 210094, China (e-mail: [email protected]). M.-Y. Xia is with the School of Electronic Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China. E. Michielssen is with the Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI 48109 USA (e-mail: [email protected]). M. Lu is with the Department of Electrical Engineering, University of Texas at Arlington, Arlington, TX 76019 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2096402

[4], using smooth temporal basis functions [5], [6] and exact integration schemes [7], and Calderon preconditioning [8]. Significant progress notwithstanding, to date no unconditionally stable and accurate MOT-TDIE solvers have been developed. Given the availability of fast MOT-TDIE convolution methods [9], [10], the development of truly robust MOT-TDIE schemes that are immune to late time instabilities appears to be the principal hurdle preventing their widespread acceptance among CEM (computational electromagnetics) practitioners. Among the aforementioned techniques to tackle late time MOT-TDIE instabilities, exact integration methods appear to hold the most promise. “Exact integration” here refers to the analytical evaluation of MOT-TDIE matrix elements [7], [11], [12]. When a space-time Galerkin MOT-TDIE solver is used to analyze scattering from free-standing three-dimensional (3D) objects residing in a lossless background medium, the temporal integrals in the expressions of the MOT-TDIE matrix elements are easily evaluated in closed form as the Green function involved is impulsive in time. Unfortunately, the analytical evaluation of the remaining 4D spatial integrals is very difficult [13]. In recent years, several hybrid analytical-numerical methods for evaluating the 4D spatial integrals have been developed [12], [14], [15]. Specifically, in [12] all 2D source integrals are evaluated analytically, while 2D test integrals are computed numerically (assuming Rao-Wilton-Glisson (RWG) spatial [16] and Lagrange polynomial temporal [17] basis functions). Schemes similar to that in [12] were developed for acoustic TDIEs in [11], [18]. A recent paper reports a fully analytical method for evaluating matrix elements in MOT-TDIE solvers for analyzing scattering from thin wires, where the source and test integrals are both 1D [19]. The above studies unequivocally demonstrate that, as the evaluation of the MOT-TDIE matrix elements is performed ever more accurately, the solver becomes increasingly stable (that is, instabilities are pushed further and further into late time). In this paper, a novel quasi-exact method for evaluating MOTTDIE matrix elements is proposed. This method is similar to that in [12] in that it assumes RWG spatial and Lagrange polynomial temporal basis functions. The proposed scheme improves on that in [12] by inching closer to a complete analytical evaluation of the MOT-TDIE matrix elements. Specifically, the proposed method permits the analytical evaluation of three out of the four spatial integrals, leaving only one integral to be evaluated numerically. Since the latter has a finite range and a piecewise smooth integrand, it can be evaluated to very high precision using standard quadrature rules. As a result, the proposed scheme permits the fast evaluation of MOT matrix elements

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The total electric field, viz. the sum of the incident and the scattered fields, satisfies

(2) Fig. 1. Illustration of electromagnetic scattering from a PEC object.

with arbitrary (user-specified) accuracy. The quasi-exact integration scheme proposed in this paper is incorporated into an MOT-based electric field TDIE solver, well-known to be highly vulnerable to late time instabilities [20], [21]. Extensive numerical experiments show that the resulting MOT-TDIE solver is stable for a very wide range of time step sizes and yields solutions that always decay exponentially after the excitation vanishes. In the numerical examples included in this paper, the time step size is made more than 10 times smaller than is typical for MOT-TDIE solvers without compromising stability even after more than half a million time steps. Moreover, results obtained using the MOT-TDIE solver are in excellent agreement with analytical results or data obtained using a frequency domain method of moments (MOM) solver. In addition, the solver’s stability is unaffected by the order of the temporal basis function used. To the best of the authors’ knowledge, no MOT solvers for electric field TDIEs with comparable performance have been reported before. This paper is organized as follows. Section II presents an electric field TDIE pertinent to the analysis of scattering from perfect electrically conducting (PEC) objects and its MOT-based discretization. Section III describes the proposed quasi-exact method for evaluating the MOT-TDIE matrix elements. Section IV presents numerical results that demonstrate the accuracy and stability of the proposed scheme. Section V presents our conclusions and avenues for future research.

is the unit vector normal to at (Fig. 1), and Here, ” extracts the total electric field’s components operator “ tangential to . Substitution of (1) into (2) results in an electric field TDIE for , which can be solved by MOT [21], [22]. To that end, is approximated by triangular patches and is expanded in terms of RWG spatial basis functions [16] and Lagrange temporal basis functions (of order ) as (3) spans two triangular patches

RWG spatial basis function and that share edge

(4) Here, and spans

is the length of edge is the area of patch , is patch ’s free vertex. The temporal basis function , where and is the time step size, consecutive time steps:

.. .

(5)

(6)

II. MOT SOLVER OF TIME DOMAIN SURFACE ELECTRIC FIELD INTEGRAL EQUATIONS Consider a PEC scatterer with surface that resides in free induces the space (Fig. 1). The incident electric field surface electric current density on . This current in turn produces the scattered electric field

(7) When function

, (5) reduces to the well-known linear interpolation

(8) An explicit expression for Enforcing (2) at time the MOT equation

is available in [17]. via spatial Galerkin testing yields

(1) (9) and act on the observation and source position where vectors and and are the permeability is the speed and permittivity of free space, and and current of light in free space. The incident field are assumed vanishingly small when for density all and on .

where

and

are

-vectors with

(10)

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, are

matrices with

Throughout this paper, all triple integrals are over the entire and are generalized indicator 3D space. In (14), functions defined such that (15)

(11)

MOT Equation (9) can be solved for once the current coare known. The next section details a efficient vectors quasi-exact method for evaluating the four-fold integrals in (11). III. QUASI-EXACT INTEGRATION METHOD FOR THE FOUR-FOLD INTEGRALS IN (11) The two-dimensional observation (source) integral in (11) is . Because RWG function over and its surface divergence are piecewise linear and constant , the right hand side of (11) can across be expressed as a linear combination of integrals with the following two forms:

(12)

for any continuous . The first transition in (14) uses and to extend the surface integrals over and to the entire 3D space. In the second step of (14), the new variable is introduced to cast the 3D integral in terms of the correlation function (16) As the correlation between and has finite spatial support (that is, it is non-zero within a finite region only). In and are assumed not parallel Subsections III.A to III.D, occuto each other; as a result, the spatial support of pies a volume. The special case when and are parallel to each other is touched upon in Subsection III.E. To facilitate the visualization of , a new Cartesian coordinate system is introduced. Let and denote unit direction vectors normal to and , respectively. The Cartesian cooris anchored to with axes along dinate system , and (Fig. 2). In coordinate system, , where the for and 0 otherwise, and is Dirac (a slice of delta function. For fixed ) is the correlation between triangle and a line segment

(17)

(13) In (12) and (13), can be either of the observation or ( or ), and is the (source) patches . In this section, a quasi-exact scheme for free vertex of evaluating in (13) is presented in Subsections III.A to III.D; in (12) is straightthe extension of this scheme to evaluate forward. In the interest of readability, some implementation details are skipped in Subsections III.A to III.D, and they are summarized in Subsection III.E. A. Transformation from “Double Surface Integral” to “a Volume Integral” The double surface integrals in (13) can be cast in terms of volume integrals as

(14)

where

is the angle between

and (18)

and and are the coordinates (the intersection of of the two endpoints of line segment and the plane ) (Fig. 2). The spatial support of in the plane is shown in Fig. 3(a). For can be obtained by correlating two fixed in Fig. 3(b) results line segments. Specifically, line segment and .” Obviously, the from “correlating line segments resulting correlation value is not constant across line segment . To arrive at a mathematical description of amenable to analytical integration, its support is subdivided into is a polygonal blocks, across each of which sum of continuous multinomials . The derivation of explicit multinomial expressions for is fairly straightforward (though lengthy), and not detailed here. In Fig. 4, the spatial support of is sketched by stacking representations of for various along the -direction. Note that, in the new Cartesian coordinate system in (14). Hence, the last line of (14) implies that

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Fig. 2. Two triangle patches in (u; v; w ) coordinate system.

Fig. 4. Depiction of the correlation function .

B. Decomposition of the “Volume Integral” into “Line and Surface Integral” The volume integral in the last line of (14) can be decomposed into “line and surface integral” as

(19) with (20)

Fig. 3. Illustration of (u; v; w = w ). (a) Correlation between a line segment and a triangle, (b) Line segment EF is resulted from correlating AB and CD.

can be interpreted as the field due to volume source distribuobserved at the spatial origin. tion

The temporal basis function , its temporal derivative , are piecewise polynomials in , having disand its integral continuous temporal derivatives only when It immediately follows that the functions with are piecewise polynomials in , having discontinuous spatial derivatives across the surfaces of shells

(21)

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Upon decomposing

as

(22) in (20) can be expressed as (23) where (24) Fig. 5. Planar source spatial origin.

It immediately follows that

(u; v; w = w ) with respect to an observer at the

(25) To derive expressions for grals are needed

with

, and

, two new inte-

(26) In Subsection III.C, it is shown that in (24) can be evaluated analytically. Further in Subsection III.D, it is argued that is a piecewise smooth function of , which implies that the -integral in (26) can be evaluated rapidly and accurately using standard one-dimensional integration rules. As a result, the potentially singular four-dimensional integral in (13) is reduced to a one-dimensional integral in (26), which can be evaluated to arbitrary precision using readily available techniques. C. Analytical Evaluation of Surface Integral

(29) and

(30) Obviously

As discussed in Subsection III.A, for a fixed value ’s spatial support consists of a few polygonal blocks, can be expressed as a sum of over each of which . Consequently, in multinomials (24) can be evaluated as linear combination of the following integrals (27)

(31) (32) (33) The above procedure can be carried out recursively to derive with higher orders. To be specific, with the aid of

where denotes a generic polygonal block of at a fixed value. The intersection of and is in (27) can be evaluated depicted in Fig. 5. The integrals in closed form following the procedure in [12]. Indeed, Fig. 5 is almost the same as Fig. 2 of [12], and complete formulations for the following three groups of integrals are already available in [12]: (34)

(28) it is obvious that In the remainder of this subsection, methods for evaluating , and are derived first, and then the derivation is for arbitrary and . generalized to that of

(35) (36)

SHI et al.: STABLE ELECTRIC FIELD TDIE SOLVERS VIA QUASI-EXACT EVALUATION OF MOT MATRIX ELEMENTS

The surface integrals in (34) can be converted to contour integrals [12], [23] over straight line segments and arcs (Fig. 5), which can be calculated analytically via a straightforward extension of the procedures in [12]. D. Numerical Evaluation of the Line Integration The line integral in (26) has a finite integration range. De, and , note the coordinates of ’s three nodes as . From Fig. 4, it is obvious that with for , where and . is a piecewise smooth function of , At the same time, only having discontinuous derivatives when the support of changes abruptly; this happens at the “critical points” described below. • The spatial support of has 14 edges, 7 of which are not perpendicular to the axis (Fig. 4). The possible intersecare critical tions of these 7 edges with the surfaces of points. • The surface of the spatial support of has 8 faces, 6 of which are not perpendicular to the axis (Fig. 4). If these , arcs 6 faces have intersections with the surfaces of result. The points along these arcs with maximum and minimum are critical points. reaches extrema at • On the spherical surfaces of , and . Any of these points residing within the spatial support of is a critical point. After all critical points have been identified, their coordinates are sorted (in ascending order). Also, , and are incorporated into this list as . Then, the integral in (26) is evaluated as

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segments are correlated with the observation patch, and the resulting polygons are (virtually) stacked onto one anand are other. This procedure remains valid when close to or coincide with each other; that is, the presence ” singularity poses no problems for the proof the “ posed method. As a matter of fact, the quasi-exact integration method in this paper draws on methods detailed in [23] that were specifically designed to analytically integrate the “ ” singularity. In other words, the proposed scheme naturally accounts for all singularities: their possible presence does not require any special treatment and has little impact on accuracy. • To not complicate the exposition in Subsections III.A to III.D, we omitted bounds for all the summation indices as they depend on the orientations of the source and observation patches, the time step size, and so on, in a somewhat complicated way. It is important to note however that all summations only nonvanishing terms. involve • Compared to integral , integral in (12) involves vectors and . Their presence can be easily accounted for in the proposed quasi-exact integration scheme. Some vector operations aside, these two vectors only impact the scheme by increasing the orders and required for the multinomials appearing in (27). IV. NUMERICAL RESULTS This section presents numerical results that demonstrate the accuracy of the proposed quasi-exact integration method (Subsection IV.A) and its application to various scattering problems (Subsections IV.B through IV.E). In all scattering problems considered below, a PEC surface is excited by a plane wave pulse with propagation vector and polarization parameterized as vector

(37)

(38)

is a smooth function. Within each interval, the integrand As a result, the above integrations can be performed with high accuracy by efficient and readily available quadrature rules. In -order Gauss-Legendre rules our implementation, regular [24] are used for each interval. As will be shown in Section IV, often yields 5 digits of accufor typical MOT meshes, racy.

Here, is the pulse’s delay. The frequency spectrum is centered about frequency and its width is inof versely proportional to . Specifically, this incident wave is apwith proximately bandlimited to frequency range and ; for frequen’s spectrum is 40 dB cies outside this band the power in below that of its peak at . In our MOT solver, is chosen is termed the temporal oversampling as ratio. To interpret this time step in terms of the spatial discretiza, where tion, we also define implicit ratio is the maximum edge length in the mesh. As shrinks, increases and the MOT scheme becomes increasingly implicit. It improves the stability of typical is known that increasing MOT solvers while decreasing their accuracy [4]. In a typical MOT solver, is usually chosen between 0.5 and 2. In the scattering examples in this section, values for as small as 0.04 are used to demonstrate the stability of an MOT solver leveraging the proposed quasi-exact integration method.

E. Several Implementation Details Several implementation details skipped in Subsections III.A to III.D are summarized in this subsection. and are parallel to each other, the spatial sup• When port of is planar instead of volumetric. The procedure in Subsections III.A and III.D still goes through, however. To be specific, when the spatial support of is planar, it is considered a “volume with zero thickness.” The source triangle patch is decomposed into line segments, these line

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TABLE I ACCURACY AND TIMING OF THE QUASI-EXACT INTEGRATION METHOD

TABLE II ACCURACY AND TIMING OF A CONVENTIONAL NUMERICAL METHOD

Fig. 6. Relative error in MOT elements for a pair of triangles.

A. Accuracy in the Evaluation of MOT Elements with vertices with CarteConsider an observation patch sian coordinates (0, 0.169 m, m), (0.126 m, 0.188 m, m), and (0.108 m, 0.098 m, m). Likewise, conwith vertices with Cartesian coordinates sider a source patch (0, 0.169 m, m), (0.126 m, 0.188 m, m), and m, 0.239 m, m). The following two integrals ( are evaluated

(39)

In

(39),

the

two

free

vertexes

are

and , respecns. “Exact” values for these two tively; and integrals are obtained by brute-force numerical integration with a very large number of quadrature points; the integrals obtained using the algorithm of Section III are compared to their exact values in Fig. 6. The relative errors drop rapidly as increases, and accuracy is reached for as low as 7. Based on extensive numerical tests, it is observed that the error associated with the quasi-exact integration algorithm in is 4. In the Section III is rarely greater than 0.01 when is always MOT examples in Subsections IV.B to IV.E, chosen to be 4, which generally guarantees stability of our MOT solver. To demonstrate the efficiency of the proposed quasi-exact integration scheme, some accuracy and timing data associated are tabulated in Tables I (quasi-exact with the evaluation of method) and II (a conventional numerical integration scheme,

Fig. 7. Current density results for scattering from a sphere.

which is unaware of the presence of discontinuities across surfaces of ). Table I clearly shows that the relative error of the without quasi-exact method drops quickly with increasing significant effect on CPU time. Table II, in contrast, shows that substantial CPU resources are required for a conventional numerical method to achieve reasonable accuracies. B. MOT Analysis of Scattering from a Sphere Consider a PEC sphere with radius 1 meter that is centered at the spatial origin. The sphere is excited by the incident field MHz, and in (38) with s, and current on the sphere surface is modeled spatial basis functions. Fig. 7 compares the using exact transient current density on the sphere surface at to that obtained from the MOT solver using first-order temporal basis function with

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Fig. 10. Current density results for scattering from two-parallel-plate. Fig. 8. Error in MOT solutions for scattering from a sphere (with p

= 1).

Fig. 11. Stability performance of MOT solutions for scattering from two-parallel-plate. Fig. 9. Stability performance of MOT solutions for scattering from a sphere ). (with p

=1

ns ( and ) and ns ( and ). Fig. 8 shows the difference between the exact ns, ns ( and MOT results for and ), and ns. Obviously, the error in the MOT results drops as the time step size decreases. The stability of our MOT solver is demonstrated in Fig. 9. For all ns of the above considered time step sizes and ( and ), the MOT code is stable; note that includes 437 000 time steps. After the the curve for main waveform vanishes, the MOT result decays exponentially in time, albeit slower for smaller time step sizes. C. MOT Analysis of Scattering from Two Parallel Plates In this subsection, scattering from two parallel PEC plates is analyzed. Both plates measure 1 m by 1 m and reside parallel to the - plane. The plates are centered about (0.5 m, 0,

0.5 m) and (0.5 m, 0.1 m, 0.5 m), and illuminated by the incident field in (38) with MHz, and s. Current spatial basis funcon the plates is modeled using m) tions. Fig. 10 shows the current density at (0.081 m, 0, along direction obtained by our MOT solver with ns ( and ) and quadratic tem. As expected, multiple reflections poral basis functions between the two plates produce a long tail in the temporal waveform, which requires the MOT solver to be executed for many time steps. Fig. 11 shows that our MOT solver remains stable for many more time steps than necessary not only for the above ns considered temporal discretization but also for ( and ) and . It is worth noting that the ns in Fig. 11 involves more curve associated with than 800 000 time steps. To verify the MOT solver’s accuracy, the MOT results are Fourier transformed and the plates’ (frequency dependent) radar cross sections (RCS) is compared to

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= 0 for scattering from two-parallel-plate. (a) = 174:7 MHz, (c) f = 354:6 MHz, (d) f = 751:5

Fig. 13. Current density results for scattering from an open-ended rectangular waveguide.

Fig. 12. RCS results when  f : MHz, (b) f MHz.

= 5 29

that obtained using a frequency domain MOM code in Fig. 12. It is observed that the MOT and MOM data match very well not MHz] only within the frequency band [ but also outside this band, implying that the MOT solver is capable of tracking temporal waveforms with very high accuracy.

D. MOT Analysis of Scattering from an Open-Ended Rectangular Waveguide Consider the open-ended PEC rectangular waveguide shown in the inset in Fig. 13 with dimensions 4 m by 0.1 m by 1 , and directions); the two faces parallel to m (along the the - plane are open. The waveguide is excited by the incident field in (38) with MHz, and s. The field that enters the waveguide apertures bounces back and forth inside the waveguide, making this problem even more challenging spathan the previous two-plate one. A total of tial basis functions are used to model the current density on the waveguide. Figs. 13 and 14 show the current density at location . MOT re(0.13 m, 0.075 m, 0) along direction sults from three simulations are displayed: ns ( and ) and ; i) ns ( and ) and ; ii) ns ( and ) and . iii) In Fig. 13, MOT results from simulations (i) and (iii) are compared to inverse Fourier transformed MOM (IFT-MOM) data. Overall the three curves shown match each other well; as expected, the MOT results from simulation (iii) (with smaller time step size and higher order temporal basis function) are closer to the MOM results. Fig. 14 shows that the MOT results in all three simulations exhibit exponential decay at late time.

Fig. 14. Stability performance of MOT solutions for scattering from an openended rectangular waveguide.

E. MOT Analysis of Scattering from an Almond-Shaped Scatterer The last example involves the almond-shaped scatterer depicted in the inset of Fig. 15; for a detailed description of this geometry, see [25]. Altogether 1 965 RWG basis functions are used to discretize the current across the almond surface. In contrast to the meshes used in the previous few examples, that for the almond is highly non-uniform; specifically, many tiny triangles are required to precisely model the almond’s tip, which tends to make the MOT system ill-conditioned, and thereby the MOT analysis more prone to instability. The almond is excited by the incident field in (38) with GHz, and s. Figs. 15 and 16 show current m, 0.003 m) along on the almond at location (0.11 m, direction . The two curves in Figs. 15 and 16 result from using first order temporal basis functions with ps ( and ) and ps (

SHI et al.: STABLE ELECTRIC FIELD TDIE SOLVERS VIA QUASI-EXACT EVALUATION OF MOT MATRIX ELEMENTS

Fig. 15. Current density results for scattering from an almond-shaped scatterer.

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= 0 for scattering from an almond-shaped scat= 49:9 MHz, (b) f = 1:65 GHz, (c) f = 3:34 GHz,

Fig. 17. RCS results when  ). (a) f terer (with p (d) f : GHz.

=1 = 5 19

the remaining numerical integral covers a finite range and involves a piecewise smooth integrand, it can be evaluated to very high accuracy using readily available quadrature rules. The accuracy and stability of an MOT solver that incorporates the proposed quasi-exact integration method were verified through extensive numerical experiments. No instabilities were ever observed; moreover, the MOT solutions exhibit exponential decay after the main waveforms vanish, for a wide range of time step sizes and temporal basis functions’ orders. Our current research efforts are aimed at extending this exact integration scheme to magnetic and combined field TDIEs. Accelerating the proposed MOT solver using plane-wave-time-domain algorithm is also under investigation. Fig. 16. Stability performance of MOT solutions for scattering from an almond-shaped scatterer.

and ), respectively. Fig. 16 demonstrates the stability of the MOT solver for a simulation covering a time period of ps and 2.5 s, which requires 250 000 time steps for ps. To verify the accuracy of 625 000 time steps for our MOT results, they are Fourier transformed to the frequency domain and compared with RCS data obtained using an MOM code; Fig. 17 shows that, once again, the RCS results obtained using the MOT and MOM solvers match very well over the enGHz]. tire band [ V. CONCLUSION A novel quasi-exact integration scheme for evaluating matrix elements arising in the MOT-based solution of electric field TDIEs pertinent to the analysis of scattering from PEC surfaces was presented. A one-dimensional integral aside, all integrals (and notably, also differentiations) involved in setting up the MOT system of equations are evaluated analytically. Since

ACKNOWLEDGMENT The authors would like to acknowledge Texas Advanced Computing Center (TACC) for granting access to its computational facilities. REFERENCES [1] C. L. Bennett and W. L. Weeks, “Transient scattering from conducting cylinders,” IEEE Trans. Antennas Propag., vol. 18, no. 5, pp. 627–633, Sep. 1970. [2] P. J. Davies, “On the stability of time marching schemes for the general surface electric field integral equation,” IEEE Trans. Antennas Propag., vol. 44, no. 11, pp. 1467–1473, Nov. 1996. [3] D. A. Vechinski and S. M. Rao, “A stable procedure to calculate the transient scattering by conducting surfaces of arbitrary shape,” IEEE Trans. Antennas Propag., vol. 40, no. 6, pp. 661–665, Jun. 1992. [4] S. J. Dodson, S. P. Walker, and M. J. Bluck, “Implicit and stability of time domain integral equation scattering analysis,” Appl. Comput. Electromagn. Society J., vol. 13, pp. 291–301, 1997. [5] J.-L. Hu, C. H. Chan, and Y. Xu, “A new temporal basis function for the time-domain integral equation method,” IEEE Microw. Wireless Compon. Lett., vol. 11, pp. 465–466, 2001. [6] D. S. Weile, G. Pisharody, N.-W. Chen, B. Shanker, and E. Michielssen, “A novel scheme for the solution of the time-domain integral equations of electromagnetics,” IEEE Trans. Antennas Propag., vol. 52, no. 1, pp. 283–295, Jan. 2004.

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[7] T. Abboud, J.-C. Nedelec, and J. Volakis, “Stable solution of the retarded potential equations,” presented at the 17th Annual Review of Progress in Applied Computational Electromagnetics, Monterey, CA, Mar. 2001. [8] F. P. Andriulli, K. Cools, F. Olyslager, and E. Michielssen, “Time domain Calderon identities and their application to the integral equation analysis of scattering by PEC objects Part II: Stability,” IEEE Trans. Antennas Propag., vol. 57, no. 8, pp. 2365–2375, Aug. 2009. [9] G. Kobidze, G. Jun, B. Shanker, and E. Michielssen, “A fast time domain integral equation based scheme for analyzing scattering from dispersive objects,” IEEE Trans. Antennas Propag., vol. 53, no. 3, pp. 1215–1226, Mar. 2005. [10] A. E. Yilmaz, J. M. Jin, and E. Michielssen, “Time domain adaptive integral method for surface integral equations,” IEEE Trans. Antennas Propag., vol. 52, no. 10, pp. 2692–2708, Oct. 2004. [11] T. Ha-Duong, “On retarded potential boundary integral equations and their discretisation,” in Topics in Computational Wave Propagation: Direct and Inverse Problems, M. Ainsworth, P. Davies, D. Duncan, P. Martin, and B. Rynne, Eds. Berlin: Springer-Verlag, 2003, pp. 301–336. [12] B. Shanker, M. Lu, J. Yuan, and E. Michielssen, “Time domain integral equation analysis of scattering from composite bodies via exact evaluation of radiation fields,” IEEE Trans. Antennas Propag., vol. 57, no. 5, pp. 1506–1520, May 2009. [13] B. Shanker, A. A. Ergin, M. Lu, and E. Michielssen, “Fast analysis of transient scattering phenomena using the multilevel plane wave time domain algorithm,” IEEE Trans. Antennas Propag., vol. 51, no. 3, pp. 628–641, Mar. 2003. [14] A. C. Yucel and A. A. Ergin, “Exact evaluation of retarded-time potential integrals for the RWG bases,” IEEE Trans. Antennas Propag., vol. 54, no. 5, pp. 1496–1502, May 2006. [15] J. Pingenot, S. Chakraborty, and V. Jandhyala, “Polar integration for exact space-time quadrature in time-domain integral equations,” IEEE Trans. Antennas Propag., vol. 54, no. 10, pp. 3037–3042, Oct. 2006. [16] S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag., vol. 30, no. 3, pp. 409–418, May 1982. [17] G. Manara, A. Monorchio, and R. Reggiannini, “A space-time discretization criterion for a stable time-marching solution of the electric field integral equation,” IEEE Trans. Antennas Propag., vol. 45, no. 3, pp. 527–532, Mar. 1997. [18] P. J. Davies and D. B. Duncan, “Stability and convergence of collocation schemes for retarded potential integral equations,” SIAM J. Numer. Anal., vol. 42, no. 3, pp. 1167–1188, 2004. [19] G. H. Zhang, M. Y. Xia, and X. M. Jiang, “Transient analysis of wire structures using time domain integral equation method with exact matrix elements,” Progr. Electromagn. Res., vol. 92, pp. 281–298, 2009. [20] M. J. Bluck and S. P. Walker, “Time-domain BIE analysis of large three-dimensional electromagnetic scattering problems,” IEEE Trans. Antennas Propag., vol. 45, no. 5, pp. 894–901, May 1997. [21] B. Shanker, A. A. Ergin, K. Aygun, and E. Michielssen, “Analysis of transient electromagnetic scattering from closed surfaces using a combined field integral equation,” IEEE Trans. Antennas Propag., vol. 48, no. 7, pp. 1064–1074, July 2000. [22] M. Lu and E. Michielssen, “Closed form evaluation of time domain fields due to Rao-Wilton-Glisson sources for use in marching-on-intime based EFIE solvers,” presented at the IEEE Antennas Propag. Society Int. Symp., San Antonio, TX, Jun. 2002. [23] D. R. Wilton, S. M. Rao, A. W. Glisson, D. H. Schaubert, O. M. Al-Bundak, and C. M. Butler, “Potential integrals for uniform and linear source distributions on polygonal and polyhedral domains,” IEEE Trans. Antennas Propag., vol. 32, no. 3, pp. 276–281, Mar. 1984. [24] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran 90, 2nd ed. Cambridge, U.K.: Cambridge Univ. Press, 1996. [25] A. C. Woo, H. T. G. Wang, M. J. Schuh, and M. L. Sanders, “EM programmer’s notebook-benchmark radar targets for the validation of computational electromagnetics programs,” IEEE Antennas Propag. Mag., vol. 35, pp. 84–89, 1993.

Yifei Shi (S’09) was born in Jiangsu, China, in December 1981. He received the B.S. degree in electrical engineering from Nanjing University of Technology, Jiangsu, China, in 2004. He is currently working toward the Ph.D. degree at Nanjing University of Science and Technology, Jiangsu, China. His research interests include time domain integral equation and its fast methods.

Ying-Mao Xia (M’00–SM’03) received the Master and Ph.D. (first-class honors) degrees in electrical engineering and from the Institute of Electronics, Chinese Academy of Sciences (IECAS), in 1988 and 1999, respectively. From 1988 to 2002, he was with IECAS as an Engineer and a Senior Engineer. He was a Visiting Scholar at the University of Oxford, U.K., from October 1995 to October 1996. From June 1999 to August 2000 and from January 2002 to June 2002, he was a Senior Research Assistant and a Research Fellow, respectively, with the City University of Hong Kong. He joined Peking University as an Associate Professor in 2002 and was promoted to Full Professor in 2004. He moved to the University of Electronic Science and Technology of China as a Chang-Jiang Professor nominated by the Ministry of Education of China in 2010. His research interests include many aspects of electromagnetic theory and applications, such as wave propagation and scattering, microwave remote sensing, antennas and microwave components. Dr. Xia was a recipient of the Young Scientist Award of the URSI in 1993. He was awarded the first-class prize on Natural Science by the Chinese Academy of Sciences in 2001. He was the recipient of the Foundation for Outstanding Young Investigators presented by the National Science Foundation of China (NSFC) in 2008.

Ru-Shan Chen (M’01) was born in Jiangsu, China. He received the B.Sc. and M.Sc. degrees from Southeast University, China, in 1987 and 1990, respectively, and the Ph.D. degree from City University of Hong Kong, in 2001. He joined the Department of Electrical Engineering, Nanjing University of Science and Technology (NUST), China, where he became a Teaching Assistant in 1990 and a Lecturer in 1992. Since September 1996, he has been a Visiting Scholar with the Department of Electronic Engineering, City University of Hong Kong, first as Research Associate, then as a Senior Research Associate in July 1997, a Research Fellow in April 1998, and a Senior Research Fellow in 1999. From June to September 1999, he was also a Visiting Scholar at Montreal University, Canada. In September 1999, he was promoted to Full Professor and Associate Director of the Microwave and Communication Research Center in NUST, and in 2007, he was appointed Head of the Department of Communication Engineering, NUST. His research interests mainly include microwave/millimeter-wave systems, measurements, antenna, RF-integrated circuits, and computational electromagnetics. He has authored or coauthored more than 200 papers, including over 140 papers in international journals. Dr. Chen is a Senior Member of the Chinese Institute of Electronics (CIE). He received the 1992 third-class science and technology advance prize given by the National Military Industry Department of China, the 1993 third-class science and technology advance prize given by the National Education Committee of China, the 1996 second-class science and technology advance prize given by the National Education Committee of China, and the 1999 first-class science and technology advance prize given by Jiangsu Province, as well as the 2001 second-class science and technology advance prize. At NUST, he was awarded the Excellent Honor Prize for academic achievement in 1994, 1996, 1997, 1999, 2000, 2001, 2002, and 2003. He is the recipient of the Foundation for China Distinguished Young Investigators presented by the National Science Foundation (NSF) of China in 2003. In 2008, he became a Chang-Jiang Professor under the Cheung Kong Scholar Program awarded by the Ministry of Education, China.

SHI et al.: STABLE ELECTRIC FIELD TDIE SOLVERS VIA QUASI-EXACT EVALUATION OF MOT MATRIX ELEMENTS

Eric Michielssen (M’95–SM’99–F’02) received the M.S. degree in electrical engineering (summa cum laude) from the Katholieke Universiteit Leuven (KUL), Belgium, in 1987 and the Ph.D. in electrical engineering from the University of Illinois at Urbana-Champaign (UIUC), in 1992. He joined the faculty of the Department of Electrical and Computer Engineering, UIUC, in 1993, reaching the rank of Full Professor in 2002. In 2005, he joined the University of Michigan as a Professor of electrical engineering and computer science. Since 2009, he directs the University of Michigan Computational Science Certificate Program. He authored or coauthored over one 160 journal papers and book chapters and over 250 papers in conference proceedings. His research interests include all aspects of theoretical and applied computational electromagnetics. His research focuses on the development of fast frequency and time domain integral-equation-based techniques for analyzing electromagnetic phenomena, and the development of robust optimizers for the synthesis of electromagnetic/optical devices. Dr. Michielssen received a Belgian American Educational Foundation Fellowship in 1988 and a Schlumberger Fellowship in 1990. Furthermore, he was the recipient of a 1994 International Union of Radio Scientists (URSI) Young Scientist Fellowship, a 1995 National Science Foundation CAREER Award, and the 1998 Applied Computational Electromagnetics Society (ACES) Valued Service Award. In addition, he was named 1999 URSI United States National Committee Henry G. Booker Fellow and selected as the recipient of the 1999 URSI Koga Gold Medal. He also was awarded the UIUC’s 2001 Xerox Award for Faculty Research, appointed 2002 Beckman Fellow in the UIUC Center for Advanced Studies, named 2003 Scholar in the Tel Aviv University Sackler Center for Advanced Studies, and selected as UIUC 2003 University and Sony

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Scholar. He is a Fellow of the IEEE (elected 2002) and a member of URSI Commission B. He served as the Technical Chairman of the 1997 Applied Computational Electromagnetics Society (ACES) Symposium (Review of Progress in Applied Computational Electromagnetics, March 1997, Monterey, CA), and served on the ACES Board of Directors (1998–2001 and 2002–2003) and as ACES Vice-President (1998–2001). From 1997 to 1999, he was as an Associate Editor for Radio Science, and from 1998 to 2008, he served as an Associate Editor for the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION.

Mingyu Lu (M’03–SM’08) received the B.S. and M.S. degrees in electrical engineering from Tsinghua University, Beijing, China, in 1995 and 1997, respectively, and the Ph.D. degree in electrical engineering from the University of Illinois at Urbana-Champaign, in 2002. From 1997 to 2002, he was a Research Assistant in the Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, where, from 2002 to 2005, he was a Postdoctoral Research Associate in the Electromagnetics Laboratory. In 2005, he joined the faculty of the Department of Electrical Engineering, University of Texas at Arlington, as an Assistant Professor. His current research interests include radar systems, antenna design, computational electromagnetics, and microwave remote sensing. Dr. Lu was the recipient of the first prize award in the student paper competition of IEEE AP-S International Symposium, Boston, MA in 2001. He received Outstanding Service Award from IEEE Fort Worth Chapter in 2008.

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Boundary Diffracted Wave and Incremental Geometrical Optics: A Numerically Efficient and Physically Appealing Line-Integral Representation of Radiation Integrals. Aperture Scalar Case Matteo Albani, Senior Member, IEEE

Abstract—This paper presents a novel formulation to reduce radiation integrals to line integrals. Such a reduction is exact for Kirchhoff aperture radiation integrals and physical optics (PO) scattering from flat soft/hard (perfectly conducting) plates, illuminated by a spherical source, but can be effectively extended in an approximate version to more general configurations. The advantage of our approach is that the integrand of the line integral along the rim of the radiating surface is free from singularities and can be easily integrated at all the observation aspects, including geometrical optics shadow boundaries. Conversely, at those aspects, existing formulations exhibit, in the integrand, a pole singularity that renders the numerical integration inaccurate or time consuming, since it requires adaptive integration routines. This was a main concern in the use of this kind of approach for the time reduction in the numerical calculation of aperture/scattering radiation integrals, which is overcome by our approach. Also, the novel result presents a neat ray interpretation which is physically appealing and allows for the heuristic extension of the approach to non-exact cases (e.g., arbitrary impedance boundary conditions or curved surfaces) using standard ray approximations. Beside the already known boundary diffraction wave (BDW), which is an incremental wave excited by the incident field and arising from the rim of the surface, a further term called incremental geometrical optics (IGO) is introduced. This novel term is an elementary portion of the direct field arising from the source and impinging at the observation point; it is able to cancel the BDW singularity thus rendering the whole integrand smooth. For the sake of simplicity, theory is here presented with reference to the the simplest scalar case of aperture radiation.

BDW + IGO

Index Terms—Acoustic radiation, acoustooptic diffraction, apertures, physical optics, shadow boundary.

I. INTRODUCTION HE possibility of reducing radiation integrals, which are essentially surface integrals, to line integrals is known in the literature and can be accomplished in a number of ways, some of them exact and others approximated. It was demonstrated first by Maggi [1], and later independently by Rubinowicz [2] (also reported in [3], section 8.9), for the case of the scattering by a hole in a perfectly absorbing screen. All

T

Manuscript received June 23, 2010; accepted August 11, 2010. Date of publication December 03, 2010; date of current version February 02, 2011. This work was supported in part by IDS SpA, Pisa, Italy. The author is with the Dip. Ingegneria dell’Informazione, Università di Siena, 53100 Siena, Italy (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2010.2096404

the pioneering “ancient” literature, well summarized in [4] either for the opto-acoustic scalar [5]–[10] or the electromagnetic case [11]–[13], had the objective to demonstrate the existence of boundary waves (beugungswelle) responsible for the formation of diffracted field. Indeed, in this description, the field scattered by an aperture is split into the sum of a direct geometrical optics (GO) contribution, that abruptly vanishes inside the shadow region, and a diffracted field contribution expressed by the line integration of a vector potential [4]–[6] along the rim of the radiating aperture. The diffracted field exhibits the proper discontinuity that compensates the discontinuous behavior of GO, thus restoring the total field continuity across the shadow boundaries. The reduction of radiation integrals to line integrals has been considered more recently in [14]–[29], where either numerical acceleration techniques are applied to evaluate radiation integrals over apertures or PO is applied to evaluate scattering from electrically large objects. The reduction of PO to a line integral along the rim of the structure is particularly attractive also because, to augment its accuracy, PO is usually used in synergy with incremental diffraction techniques [30]–[36] which intrinsically presents the same line integral form. Therefore the two line integrals can be combined resulting in a very efficient algorithm. Also, the reduction of PO to a line integral can be applied within hybrid asymptotic-moment method codes for electrically large objects, to accelerate the computation of reaction integrals between large domain basis functions, whose shape is chosen accordingly to asymptotic theory [37]–[39]. However, when the scattered field is constructed as the sum of GO and diffracted field, the discontinuous behavior of the diffracted field is obtained integrating a singular vector potential (or dyadic potential for the EM case). Despite being analytically integrable, such a singular function seriously affects the numerical effort for its numerical evaluation, thus partially or totally reducing the time saving. In [14], Gordon theoretically proves the existence of singularity-free representations and gives some general directions concerning how one might use the exterior calculus of differential forms to calculate it; however he prefers to cast his final results in a singular form conformable to the original Maggi-Rubinowicz results. Only Asvestas practically addressed this problem in [18]–[20], focusing on the derivation of a suitable representation for an efficient numerical calculation and providing an alternative mathematically based formulation. The final obtained formula can be a posteriori interpreted in terms

0018-926X/$26.00 © 2010 IEEE

ALBANI: BOUNDARY DIFFRACTED WAVE AND IGOs: A NUMERICALLY EFFICIENT AND PHYSICALLY APPEALING

of GO and boundary rays, for a specific choice of the unit vector ; namely, in (1.13) of [19]. In this paper, we present a new formulation for the reduction of a radiation integral into a line integral, based on a geometrical construction and the use of Huygens’ principle. The new formulation provides an exact line integral representation for the radiation integral that describes the scattering from an aperture in a black screen or by an arbitrary shaped flat plate, when illuminated by a point source, and coincides with that in [19] for . The major advantage of the new the special choice formulation comes from joining Asvestas’ numerical efficiency with the neat ray interpretation and ease of implementation of Rubinowicz’ approach. As a matter of fact, the present formulation provides a result that overcomes numerical impairments of traditional techniques and, at the same time, a clear ray interpretation of the result. A ray picture interpretability allows the heuristic extension of the formulation to the scattering by impenetrable or transparent impedance objects, by using typical high frequency ray approximations, i.e., by weighting incremental rays with the relevant GO reflection/transmission coefficient. For the sake of simplicity, in the present paper we derive the new formulation with reference to the scalar opto-acoustic radiation by an aperture in a black screen. Future works will be devoted to the exact and heuristic extensions to the scalar opto-acoustic scattering by flat arbitrarily shaped plates with generic boundary conditions and to the electromagnetic (either aperture or PO scattering) case, whose formulation will be reduced to the scalar case using Kottler’s version of the equivalence principle [40]. The present paper is organized as follows: the formulation is derived in Section II for a generic illumination field. Then, in Section III, the result is specialized for spherical wave illumination, for which the vector potential is calculated in a closed form and its behavior is analyzed. The plane wave illumination case is explicitly considered in Section IV. In Section V, some numerical examples are used to check properties and behavior of the new formulation, highlighting the correctness of the line-integral reduction and its numerical efficiency. Section VI ends the paper with some concluding remarks. II. FORMULATION We consider the case of the radiation through an aperture in a black (i.e., perfectly absorbing) screen [40]. For such a configuration, the Kirchhoff approximation for the aperture field (i.e., aperture field equals incident field) is assumed to be exact. Equivalently, we may assume the screen not to be perfectly absorbing, and consider the aperture radiated field under Kirchhoff approximation. We derive an exact line integral representation for the scalar field radiated by an aperture , in terms of boundary diffracted waves (BDW) and incremental geometrical optics (IGO) field. The field radiated by the aperture at is expressed by the Kirchhoff radiation integral

(1) denotes the integration point on , is an where illuminating field satisfying Helmholtz wave equation

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Fig. 1. Two different cases are considered in the formulation depending on observation point position.

Fig. 2. Frustum cone arrangement for case 1.

outside the source region, and is the free space Green’s function. Here and in what follows, an arrow over two points denotes a vector from to . Furthermore, a hat will . denote unit vectors, e.g., The screen divides the space into two half-spaces, the halfspace before the screen, which contains the source or the direction of incidence of the illuminating field, and the half-space behind the screen. Depending on the position of the observation point with respect to the source of the field, we distinguish two cases (Fig. 1). A. Case 1, Observation Before the Screen First, we consider the case in which lies before the screen; despite being usually not of interest in aperture problems, this case is useful for the extension to the scattering from a plate. In this case we follow the procedure outlined in [6]. We construct a frustum cone projecting the observation point through each will extend point of the aperture rim (Fig. 2). The volume in the half-space behind the screen, i.e., to the opposite side with respect to the illuminating source, therefore is free from sources. Applying the Green’s second identity to the source free volume we obtain

(2) Hence, in light of Huygens’ equivalence principle, the field radiated by the aperture can be alternatively calculated as the negative of the field radiated by the lateral surface of the cone. The latter integral can be remapped as a line integral along the aperture rim of an inner integral along spherically spread semi-inare mapped by . For finite strips, whose points

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in which the negative sign accounts for the fact that the normal . Note (internal to ) to the strip is now in the integration end point in (6) is not that the limit critical since is not a critical point of the integrand. The approach outlined for (6) was previously obtained in (2.13) of [19], and in [16] but in the framework of Fresnel diffraction. can be calculated, as an alternative to [18], The solid angle using again an appropriate geometrical construction. We extend the cone across the vertex and we consider the other infi(Fig. 3). As before, the tip of the cone nite fold denoted by is removed by the vanishing radius sphere. Applying Green’s we obtain second identity to the source free volume

Fig. 3. Double cone arrangement for case 2.

such a mapping we have cause of ad hoc construction of , , whereas (internal to ) to the strip is obtain the known result [5], [6]

; furthermore, bevanishes on , where the normal . Hence, we

(3)

B. Case 2, Observation Behind the Screen Second, we consider the case 2 in which lies behind the screen. Applying a modified formulation, we provide a geometrical interpretation for a particular case of the procedure outlined in [18]–[20]. We construct the cone connecting the observation point to each point of the aperture rim (Fig. 3). The volume will extend in the half-space behind the screen and therefore it will be again free from sources. To avoid calculation problems due to the Green’s function singularity at , we remove the tip of the cone with a sphere with vanishing radius . Applying the Green’s second identity to the source free volume we obtain

(7) where is the lateral surface of and the spherical endcan be remapped as a line intecap. Again, the integral over gral over the aperture rim of an inner integral along semi-into finite spherically spread strips, extending from the rim of is calculated anal(minus) infinity; whereas the integral over ogously to (5). Hence, (7) reduces to (8) is not critical in which, as before, the end point limit . Equation (8) is introbecause the integrand is regular at duced in (6), and the two integrals are merged together into a to , which is then combined together single integral from with (3), thus obtaining the final expression (9) in which the Miyamoto-Wolf vector potential [5] is given by

(4) Hence, in light of Huygens’ equivalence principle, the field radiated by the aperture can be alternatively calculated as the negative of the sum of the field radiated by the lateral surface of the cone and by the spherical end-cap of the cone. As in the previous case, the integral over can be remapped as a line integral along the aperture rim of an inner integral along spherically spread strips, that now extend from the rim of to the rim of the aperture . The surface integral over the spherical surface can be evaluated as an integral over the solid angle . In the limit , collapses in and the integral can be simply calculated as

(10) where the or end point applies to case 1 or 2, respectively. On the basis of the proposed formulation, the vector potential (10) is physically interpreted as the elementary field radiated by each infinitely thin strip, defined in the geometrical construction. The field radiated by each strip is associated to its endpoint on the rim in the incremental distribution along the rim (9). For case 1, the vector potential and the associated strip are the same as in [6]; conversely, a new strip and a novel physical interpretation is defined in case 2. III. SPHERICAL WAVE ILLUMINATION

(5)

When the incident field is a spherical wave produced by an , by using isotropic scalar source at , the integral in (10) becomes

Hence, in this second case, we obtain

(6)

(11)

ALBANI: BOUNDARY DIFFRACTED WAVE AND IGOs: A NUMERICALLY EFFICIENT AND PHYSICALLY APPEALING

where

.

Equation (11) admits closed form antiderivatives [2], [4]

(12) . Note that (12) vanishes in which is used in the range . Hence, by using (12), (11) is calculated as a conas , i.e., at a point on the aperture tribution at the endpoint rim , and, only in case 2, contributions at the two end points and , i.e., at the observation point ; i.e.,

(13)

in which , and is a unit step function that equals one or vanishes for case 2 or 1, respectively. Finally, the terms between brackets are collected together by using and , and the Miyamoto-Wolf vector potential in (10) is split into the sum of the two closed form terms (14) where (15) and (16) A physical interpretation of this result is given in the following. Thanks to the geometrical construction, all the radiating points on a strip are aligned in the observation point direction and contribute to the field at with a progressive phase delay ; in addition the spherical spread of the strip compensates for the Green’s function spherical spreading factor . Indeed the incoherence between adjacent points along the strip results in end-point radiation effect, therefore the field radiated by the strip (i.e., the vector potential) only comprises and, only spherical waves arising from the strip endpoints in case 2, . This reduction is exact for the spherical source illumination, but it is possible asymptotically for any generic illuminating field. Note that only when the field impinges from the illuminating field phase and the the strip direction radiation phase delay to compensate each other and all the point of the strip coherently contribute to . Since the strip is semiinfinite, this result in a caustic of the strip field, i.e., a singularity of the vector potential. However, thanks to the swap between case 1 and 2 this aspect is avoided in our formulation.

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A correspondence of our formulation with the Fermat Principle can also be established. It is well known that the surface aperture integral (1) is asymptotically dominated by critical points dictated by the phase behavior. Indeed, points for which the phase delay time is minimum (or more generally stationary) define a ray path for light propagation, in accordance to the Fermat Principle (or its Generalized version). Namely, a stationary phase point on the aperture results in a direct GO ray; partial stationary phase points on the rim of the aperture correspond to edge diffracted rays; and those points where the aperture rim presents some discontinuity of the derivative (i.e., of the tangent vector) lead to vertex diffracted rays. In [41] it is shown how, by using Stoke’s theorem, the stationary phase point of the surface integral, which corresponds to the asymptotic leading ray GO contribution, is converted into a singularity of the equivalent line integral. Therefore the IGO contribution exactly corresponds to the stationary phase point in the original surface integral which is the minimum delay ray path. In our approach the line integration is performed numerically, however its analytical asymptotic evaluation would result in edge and vertex contributions at partial stationary phase points and corners on the rim, as detailed in [41]. A. Wave Analysis In this paragraph the behavior of expression (9) is analyzed for the point source illumination case also providing mathematical details for the above summarized physical behavior. The integrand (14) of (9) consists of two terms, i.e., (15) and (16). 1) Boundary Diffracted Wave: The first term (15) is the same obtained in [2]–[6] and, once integrated alone along the rim of the aperture, provides the diffracted field. It is recognized to be a spherical wave arising from the point on the rim, excited by the incident wave at the same point. Therefore, it is regarded as a boundary diffracted wave (BDW). An inclination factor also appears, involving the local impinging and observation directions. The integration of these BDWs along the rim becomes numerically critical at the shadow boundary line (SBL) where the integrand function presents a singular behavior. Despite this numerical impairment, the BDW singularity is analytically integrable, and, once integrated, provides the discontinuous behavior of the diffracted field at the shadow boundary. In the standard formulation [2]–[6], the total field is split into the sum of this BDW contribution, i.e., the rim diffracted field, and the GO contribution. The jump of the diffracted field across the shadow boundary compensates for the opposite jump discontinuity there exhibited by the GO term, thus providing a continuous total field. 2) Incremental Geometrical Optics: Since in our formulation the entire total aperture field is expressed by (9), it is easy to realize that the integration along the rim of the second term (16) alone provides the GO contribution. Indeed, the second term contains a spherical wave corresponding to the incident field at . This term can be interpreted as an elementary (incremental) portion of the incident field. Indeed, the incident field is multiplied by the elementary angle (Fig. 4) (17)

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Fig. 4. Geometry for the interpretation of IGO contributions.

Fig. 5. Projection of the geometrical scenario on a plane orthogonal to the direct ray, for the analysis of the integration of the IGO term alone. Various cases are considered. Observation point in the GO lit region (a); in the shadow region (b); exactly at the GO shadow boundary relevant to a regular point (c); and, at a corner of the rim (d).

measured on a plane perpendicular to the direct ray , corresponding to the elementary edge portion . Hence, with a change of variable, the integration of (16) can be rearranged as

singularity, i.e., exhibits opposite residue at the same shadow boundary aspect. Hence, the novel IGO contribution compensates for the singularity of the BDW contribution, resulting in a regular whole integrand whose integration provides a continuous total field. To summarize, in our description (9), the incremental field (14) is the superposition of two rays, namely the BDW and the IGO contribution. The whole integrand is always regular, even when observing at a SBL. Despite the fact that expressions (14)–(16) allow for the above physical interpretation for the vector potential, they are not suitable for an efficient numerical computation. This happens because the singularity removal requires the exact cancellation of two poles and this at cancellation is exact only for analytical evaluations, but it is not robust for numerical computations. Hence, the computation of the integrand function near the SBL may still cause problems. Therefore, in the following section, an alternative expression is given to overcome such an impairment. B. Numerical Considerations In order to achieve a numerically stable expression in the neighborhood of the SBL at , by using and , the Miyamoto-Wolf vector potential (14) is rearranged as

(18) It is easy to verify that if the direct ray passes through the aperture , i.e., the observation point is in the GO lit region, the angular integral extends over a complete circle and (Fig. 5(a)); whereas when is in the GO shadow region the angular integral extends from a minimum angle to a maximum angle and then from back to , so that (Fig. 5(b)). In conclusion, the integral that multiplies the incident field at the right-hand side of (18) builds a unit step function that vanishes in the GO shadow region, and bounds the incident field inside the GO lit region, thus providing the GO contribution. Thereby (16) is called Incremental GO (IGO) contribution. Note that, if the direct ray exactly intercepts the aperture at a point of the rim, i.e., the observation point exactly lies at the SBL, the integration of the IGO term provides either of the incident field, when is regular at (Fig. 5(c)), or of the incident field, when is a corner of with angle (Fig. 5(d)). The above results for the cases of observation point in the lit region (a), in the shadow region (b), and exactly at the shadow boundary for either a regular (c) or a corner point (d) of the rim, are straightforwardly calculated with our geometric of the construction and exactly reproduce the leading term asymptotic evaluation of the surface aperture radiation integral at the double stationary phase point (if present). Likewise the BDW term (15), the IGO term (16) is also sin, and the integration of the singular at the SBL where gular integrand provides a discontinuous GO field. In the same way in which the diffracted field jump perfectly matches the GO discontinuity (resulting in a smooth continuous total field), at incremental level the BDW singularity perfectly matches the IGO

(19) This clearly shows the symmetry between source and observation (reciprocity) also for the IGO contribution, which is not explicit in (16). Next, following the outline of [19] Appendix B, by using (15) in (19) and close to and at the SBL where (case 2 applies), one obtains

(20) in which for for

(21)

expresses the removable singularity of at the SBL in terms of a closed form function which can be easily calculated to any desired accuracy through a Maclaurin expansion. Therefore, (20) can be used alternatively to (14)–(16) for a numerically stable evaluation of the vector potential close to the SBL. Equation (20) also shows the smooth continuous behavior at , where it vanishes like of

(22)

ALBANI: BOUNDARY DIFFRACTED WAVE AND IGOs: A NUMERICALLY EFFICIENT AND PHYSICALLY APPEALING

We adopted a simple standard rule for the numerical integration without experiencing any numerical difficulties; however, it is worth noting that in [42] a specific numerical integration strategy was developed for the numerical calculation of integrals whose integrand function exhibits a removable singularity. The use of such approach can further improve the time efficiency of the integral calculation in the neighborhood of these points. Finally we compare our formulation to that in [19] from the numerical point of view. Concerning the robustness, the two methods are equivalent as both express the removable singularity in terms of a sinc function (21). Concerning the efficiency, our expressions (14)–(16) and (20) require fewer calculations than their counterparts (3.8) and (4.3) in [19]; in addition, the method in [19] also requires the calculation of the solid angle contribution. Therefore our method is more efficient. It is also to note that in the present aperture case, the backscatis not critical. Indeed, this aspect falls tering direction , and is smooth as . into case 1 so that

591

Fig. 6. Aperture scattered field for the geometry of Fig. 7. Surface integral (circles) and line integral (BDW + IGO) (continuous line) are overlapped, as expected. The line integration of BDW alone (dashed line) is critical around the shadow boundaries  73 and it provides only the diffracted field that must be added to the GO (dotted line) to give the total field.

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IV. PLANE WAVE ILLUMINATION The result for the plane wave illumination can be directly obtained as the limit of the spherical wave case when the point source location is moved to infinity while the field amplitude in the aperture region is kept constant. For a unit amplitude plane propagating in the wave incident field direction, with phase reference at , the aperture field is still expressed by (9) and (14) with (23)

Fig. 7. Geometry for the scattering by a circular aperture in a black screen.

and (24) It is evident how the interpretation of the two terms as BDW and IGO contribution still applies in the plane wave case. The method to treat the numerical evaluation of the vector potential close to the SBL is analogous to the spherical wave case, and it reduces to

(25) For this plane wave case, the computational effort of our expressions (23)–(25) is about the same of that of their counterparts (3.11) and (4.1) in [19]. V. NUMERICAL EXAMPLES In this section we provide some numerical examples in order to check the correctness of the IGO formulation and its numerical efficiency. We simply adopt a trapezoid quadrature rule with a uniform step, both for the line and the surface integral. A. Circular Aperture First we consider a circular aperture in a black screen with ra, with denoting the free space wavelength (Fig. 6). dius

The aperture is illuminated by a point source placed at a disfrom the center of the aperture; the observation tance , tracked by the angle point scans a circle with radius . In Fig. 7 the aperture scattered field is shown. The surface integral (1) (circles) and its line integral exact reduction (9) are indistinguishable (continuous line), as expected. Note that the , 180 observation aspects are caustics of diffracted rays where a discrete, i.e., not incremental, ray techniques would fail. The line integration along the rim of the BDW term (15) alone, as prescribed in standard Maggi-Rubinowicz approach [2]–[6], provides the diffracted field (dashed line) whose calculation is . Indeed, at critical around the shadow boundaries those aspects the observation point lies close to the SBL of an incremental contribution (Fig. 6) and the line integrand (15) becomes singular leading to a numerical instability. The spikes in the diffracted field may be reduced or even eliminated by refining the numerical integration around the singularities but this requires an increased computational effort. Such effect is better clarified in Fig. 8, which shows the integrand of the line integral representation vs. the integration variable , which maps the circular rim (Fig. 6), for different observation aspects . Data (a) and standard Maggi-Rubinowicz for both our (b) formulations are presented. Note that when observing at , the observation point approaches the SBL of the in; there, the cremental contribution lying on the rim at Maggi-Rubinowicz integrand (15) exhibits a pole singularity. at . An analogous singularity occurs for

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Fig. 8. Integrand of the line integral representation (9) for the geometry of Fig. 7 vs. the integration variable  along the rim, at various observation aspects  . (a) Novel BDW + IGO formulation (14), and (b) standard Maggi-Rubinowicz approach (15). The latter presents a singularity at  = 0, 180 when observing at the SBLs  73 , 73 .



0

Fig. 9. Geometry for the scattering by a square aperture in a black screen.

Conversely, adding the IGO term (16) allows the compensation of these singularities, so that the integrand (14) is smooth everywhere. This eliminates the numerical instability in the integral calculation when observing at the shadow boundaries, thus providing a robust algorithm and an accurate solution without increasing the number of integration points or both the calculation time. Using an integration step in the rim-line and in the aperture-surface integrations leads to surface samples versus samples on the rim. Despite a reduction in the amounts of integration , this results in a CPU time points of a factor , because the line integrand inreduction of volves some additional calculation for each point. The saving of CPU time is obviously linearly dependent on the aperture dimension, and becomes larger for increasing apertures. B. Square Aperture Next we consider the diffraction through a square aperture with side (Fig. 9). Source position and observation scan intervals are the same as in the previous example; i.e., and ; the aperture rim is mapped by a length coordinate ranging from 0 to along the square perimeter. Now, SBLs , are encountered when the observation point approach . At these aspects, the incremental point which becomes singular lies at a corner of the aperture rim. Hence we check here

Fig. 10. Aperture scattered field for the geometry of Fig. 9. Surface integral (circles) and line integral (BDW + IGO) (continuous line) are overlapped, as expected. Line integration of BDW alone (dashed line) is critical around the shadow boundaries  59 and provides only the diffracted field that has to be added to the GO (dotted line) to give the total field.

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the behavior of the solution when the singularity exactly falls at a non-regular point of the integration contour. The relevant aperture scattered field is plotted in Fig. 10. Again, this numerical check confirms the correctness of the line integral reduction, in fact the surface (circles) and the line integral (continuous line) calculations provide overlapping results. Also in this case, the integration of the BDW term alone (dashed line) is found to be critical around the SBL, because the fine but not adaptive integration scheme is not sufficiently accurate and leaves spurious spike in the total field prediction around the SBL, when added to GO (dotted line). Similarly to the previous case, the analysis of the integrand vs. the integration variable , for various observation aspects , is presented in Fig. 11 for both our novel (a) and standard Maggi-Rubinowicz (b) formulations. The discontinuous behavior of the integrand at aperture , , , , are due to the abrupt change of rim corners direction along the integration contour; this does not represent

ALBANI: BOUNDARY DIFFRACTED WAVE AND IGOs: A NUMERICALLY EFFICIENT AND PHYSICALLY APPEALING

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Fig. 11. Integrand of the line integral representation (9) for the geometry of Fig. 9 vs. the integration variable ` along the rim, at various observation aspects  . (a) Novel BDW + IGO formulation (14), and (b) standard Maggi-Rubinowicz approach (15). The latter presents a singularity at ` = 2, , 6 when observing at the SBLs  59 , 59 .



0

a numerical impairment when the integration is a priori subdivided into the sum of the integrals along each side, which are regular curves. Note, again, that Maggi-Rubinowicz standard approach requires the numerical integration of a singular integrand when observation approach , , conversely approach always provide a finite, smooth inour tegrand. In the present test the line integral reduction allows to samples throughout the aperture to reduce from only samples along the rim, corresponding to an . As a conintegration point reduction factor . sequence, the CPU time reduction factor is With reference to a square aperture like that of the last example, if is the aperture side and the spatial sampling points, rate, the surface integration requires , with a sample whereas the line integration only . The CPU number reduction factor of time reduction factor is roughly proportional to this value , with the constant depending on the ratio between the number of floating point operations involved in the two different integrand functions. Since (15) and (16) involve some geometrical construction and vector operations, the calculation of the line integrand function appears to be more time consuming than the surface counterpart. The value of also depends on the specific numerical implementation of the algorithm. For our simple Matlab code, with which the previous tests were obtained, it is found , with a weak dependence on the specific run and on the state of the PC (presence of other tasks, memory availability). Nevertheless, because of the linear dependence with respect to the aperture size , for sufficiently large apertures the line integral becomes very soon more efficient than the surface integral. In Fig. 12, the CPU time required for the calculation of the field scattered from a square aperture, along a 360 observation point scan, is plotted for various aperture sides . Both the surface and line numerical integration are performed using spatial step. For increasing a trapezoidal rule with a aperture size, the CPU time grows almost quadratically for the surface integral approach, but linearly for the line integral

Fig. 12. CPU time for the calculation of the scattered field form a square aperture with side a on a 360 points scan, using a 1 = =20 numerical integration step.

formulation; in agreement with the previous discussion. The line integral reduction starts to be convenient for apertures as small as ; it has to be noticed that for smaller apertures the Kirchhoff approximation itself is arguable. VI. CONCLUSIONS In this paper a novel formulation for the reduction of surface radiation integrals to line integrals was presented, with reference to the scalar case of Kirchhoff aperture radiation. The present approach is numerically efficient because it eliminates the singular behavior at SBLs. It is also physical appealing because it admits a neat ray interpretation. In the future, upcoming papers will extend the same approach to the Physical Optics scattering from flat plates and to the electromagnetic case. ACKNOWLEDGMENT G. De Mauro and A. Italiano are acknowledged for their contribution to the development of the IGO approach during their

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Laurea thesis at the University of Siena and at the University of Messina, respectively. The author is very thankful to one anonymous reviewer whose careful and constructive comments helped improve this manuscript. REFERENCES [1] G. A. Maggi, “Sulla propagazione libera e perturbata delle onde luminose in un mezzo isotropo,” Annali di Matem. II , vol. 16, pp. 21–48, 1888. [2] A. Rubinowicz, “Die beugungswelle in der Kirchhoffschen theorie der beugungsercheinungen,” Ann. Physik, vol. 53, pp. 257–278, 1917. [3] M. Born and E. Wolf, Principle of Optics. Oxford: Pergamon Press, 1964, pp. 449–453. [4] A. Rubinowicz, “The Miyamoto-Wolf diffraction wave,” Progr. Opt., vol. 4, pp. 331–377, 1965. [5] K. Miyamoto and E. Wolf, “Generalization of the Maggi-Rubinowicz theory of the boundary diffraction wave—Part I and Part II,” J. Opt. Soc. Am., vol. 52, no. 6, pp. 615–625, Jun. 1962. [6] A. Rubinowicz, “Simple derivation of the Miyamoto-Wolf formula for the vector potential associated with a solution of the Helmholtz equation, and geometric derivation of the Miyamoto-Wolf formula for the vector potential associated with a solution of the Helmholtz equation,” J. Opt. Soc. Am., vol. 52, no. 6, pp. 717–718, Jun. 1962. [7] E. W. Marchand and E. Wolf, “Boundary diffraction wave in the domain of the Rayleigh-Kirchhoff diffraction theory,” J. Opt. Soc. Am., vol. 52, no. 7, pp. 761–767, Jun. 1962. [8] A. Rubinowicz, “Beugungswelle im Falle einer Beliebigen Einfallenden Lichtwelle,” Acta Phys. Polon., vol. 21, pp. 61–87, 1962. [9] A. Rubinowicz, “Über Miyamoto-Wolfsche vectorpotentiale, diemit der Lösung eines Randwertproblems im Gebiete der schwingungsgleichung Verknüpft sind,” Acta Phys. Polon., vol. 28, pp. 361–387, 1965. [10] A. Rubinowicz, “Darstellung der Sommerfeldeschen Beugungswelle in einer Gestalt, die Beitrage der einzelnen elemente der Beugende Kante zur Gesamten Beugungswelle Erkennen last,” Acta Phys.Polon., vol. 28, pp. 361–387, 1965. [11] O. Laporte and J. Meixner, “Kirchhoff-Youngsche theorie der Beugung elektromagnetischer Wellen,” Zeitschrift für Physik, vol. 153, pp. 129–148, 1958. [12] B. Karczewski, “Boundary wave in electromagnetic theory of diffraction,” J. Opt. Soc. Am., vol. 53, no. 7, pp. 878–879, Jul. 1963. [13] A. Kujawski, “On the Kirchhoff-Young diffraction theory of electromagnetic waves,” Bull. Acad. Polon., Sci. Math., Astr. Phys., vol. XI, no. 2, pp. 67–72, 1963. [14] W. B. Gordon, “Vector potentials and physical optics,” J. Math. Phys., vol. 16, pp. 448–454, 1975. [15] W. Gordon, “Far-field approximations to the Kirchoff-Helmholtz representations of scattered fields,” IEEE Trans. Antennas Propag., vol. 23, pp. 590–592, Jul. 1975. [16] J. H. Hannay, “Fresnel diffraction as an aperture edge integral,” J. Modern Optics, vol. 47, no. 1, pp. 121–124, 2000. [17] R. Meneghini, P. Shu, and J. Bay, “Several Maggi-Rubinowicz representations of the electric field,” IEEE Trans. Antennas Propag., vol. 30, pp. 516–520, May 1982. [18] J. S. Asvestas, “Line integrals and physical optics. Part I. The transformation of the solid-angle surface integral to a line integral,” J. Opt. Soc. Am. A, vol. 2, no. 6, pp. 891–895, Jun. 1985. [19] J. S. Asvestas, “Line integrals and physical optics. Part II. The conversion of the Kirchhoff surface integral to a line integral,” J. Opt. Soc. Am. A, vol. 2, no. 6, pp. 896–902, Jun. 1985. [20] J. S. Asvestas, “The physical optics fields of an aperture ona perfectly conducting screen in terms of line integrals,” IEEE Trans. Antennas Propag., vol. 34, pp. 1155–1159, Sep. 1986. [21] P. M. Johansen and O. Breinbjerg, “An exact line integral representation of the physical optics scattered field: The case of a perfectly conducting polyhedral structure illuminated by electric Hertzian dipoles,” IEEE Trans. Antennas Propag., vol. 43, pp. 689–696, Jul. 1995. [22] A. C. Brown Jr. and W. K. Khan, “Comparison of various image induction (II) methods with physical optics (PO) for the far-field computation of flat-sectioned segmented reflectors,” IEEE Trans. Antennas Propag., vol. 44, pp. 1133–1141, Aug. 1996. [23] F. Mioc, M. Albani, P. Focardi, and S. Maci, “Line-Integral representation of the field radiated by a rectangular waveguide modal current distribution,” IEEE Trans. Antennas Propag., vol. 47, pp. 408–410, Feb. 1999.

[24] K. Sakina and M. Ando, “Line integral representation for diffracted fields in physical optics approximation based on field equivalence principle and Maggi-Rubinowicz transformation,” IEICE Trans. Comm., Sep. 2001. [25] G. Pelosi, G. Toso, and E. Martini, “PO near-field expression of a penetrable planar structure in terms of a line integral,” IEEE Trans. Antennas Propag., vol. 48, pp. 1274–1276, Aug. 2000. [26] W. B. Gordon and H. J. Bilow, “Reduction of surface integrals to contour integrals,” IEEE Trans. Antennas Propag., vol. 50, pp. 308–311, Mar. 2002. [27] P. Meincke, O. Breinbjerg, and E. Jorgensen, “An exact line integral representation of the magnetic physical optics scattered field,” IEEE Trans. Antennas Propag., vol. 51, pp. 1395–1398, Jun. 2003. [28] L. Infante and S. Maci, “Near-field line-integral representation of the Kirchhoff-type aperture radiation for a parabolic reflector,” Antennas Wireless Propag. Lett., vol. 2, no. 1, pp. 273–276, 2003. [29] E. Martini, G. Pelosi, and S. Selleri, “Line integral representation of physical optics scattering from a perfectly conducting plate illuminated by a Gaussian beam modeled as a complex point source,” IEEE Trans. Antennas Propag., vol. 51, pt. 2, pp. 2793–2800, Oct. 2003. [30] C. E. Ryan and L. Peters Jr., “Evaluation of edge-diffracted fields including equivalent currents for the caustic regions,” IEEE Trans. Antennas Propag., vol. AP-17, pp. 292–299, May 1969. [31] K. M. Mitzner, Incremental Length Diffraction Coefficients Aircraft Division Northrop COT, 1974, Tech. Rep. No. MAL-TR-73-296. [32] A. Michaeli, “Equivalent edge currents for arbitrary aspects of observation,” IEEE Trans. Antennas Propag., vol. AP-23, pp. 252–258, Mar. 1984. [33] R. A. Shore and A. D. Yaghjian, “Incremental diffraction coefficients for planar surfaces,” IEEE Trans. Antennas Propag., vol. 36, pp. 55–70, Jan. 1988. [34] P. Y. Ufimtsev, Method of Edge Waves in the Physical Theory of Diffraction U.S. Air Force Syst. Command Foreign Technol., Office Document ID no. FTD-HC-23-259-71 (translation Russian orig.). [35] P. Y. Ufimtsev, “Elementary edge waves and the physical theory of diffraction,” Electromagn., vol. 11, no. 2, pp. 125–159, Apr.–Jun. 1991. [36] R. Tiberio, A. Toccafondi, A. Polemi, and S. Maci, “Incremental theory of diffraction: A new-improved formulation,” IEEE Trans. Antennas Propag., vol. AP-52, pp. 2234–2243, Sep. 2004. [37] G. Tiberi, A. Monorchio, G. Manara, and R. Mittra, “A spectral domain integral equation method utilizing analytically derived characteristic basis functions for the scattering from large faceted objects,” IEEE Trans. Antennas Propag., vol. 54, no. 9, Sep. 2006. [38] D. H. Kwon, R. J. Burkholder, and P. H. Pathak, “Efficient MoM formulation for large PEC electromagnetic scattering problems using asymptotic phasefront extraction,” IEEE Trans. Antennas Propag., vol. 49, pp. 583–591, Apr. 2001. [39] K. Tap, R. J. Burkholder, P. H. Pathak, and M. Albani, “Methods for efficiently computing the MoM impedance matrix for APEx type basis functions,” in Proc. IEEE Antennas and Propag. Society Int. Symp., Jul. 9–14, 2006, pp. 4119–4122. [40] F. Kottler, “Diffraction at a black screen. Part II: Electromagnetic theory,” Progr. Opt., vol. 6, pp. 331–377, 1967. [41] G. Carluccio, M. Albani, and P. H. Pathak, “Uniform asymptotic evaluation of surface integrals with polygonal integration domains in terms of UTD transition functions,” IEEE Trans. Antennas Propag., vol. 58, pp. 1155–1163, Apr. 2010. [42] J. S. Asvestas, “A class of functions with removable singularities and their application to the physical theory of diffraction,” Electromagnetics, vol. 15, pp. 143–155, 1995.

Matteo Albani (M’76–SM’10) was born in Florence, Italy, in 1970. He received the Laurea degree in electronic engineering (cum laude) and the Ph.D. degree in telecommunications engineering from the University of Florence, Italy, in 1994 and 1999, respectively. He was a Research Associate at the University of Siena, Italy, in 2001, he joined the College of Engineering, University of Messina, Italy, as an Assistant Professor. In 2006, he moved to the University of Siena where he currently is an Adjunct Professor teaching microwaves and optics at the College of Engineering. His research interests are concerned with asymptotic methods for electromagnetics, antenna modeling and design, and metamaterials. He has authored more than 30 journal papers and 110 conference contributions. Dr. Albani was awarded the “Giorgio Barzilai” prize for the Best Young Scientist paper at the Italian National Conference on Electromagnetics (XIV RiNEm) in 2002.

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On the Use of Series Expansions for Kirchhoff Diffractals Stefano Perna, Member, IEEE, and Antonio Iodice, Senior Member, IEEE

Abstract—In recent years, some methods have been devised to evaluate the field scattered by natural surfaces modeled by using fractals. In particular, use of the Kirchhoff approximation allows expressing the field scattered by a fractal surface in terms of two different series expansions. In this paper, the practical applicability of these expansions is addressed. To this aim, we first of all reformulate the derivation of the two series in order to clearly identify the key parameters on which the series behavior depends. Then, we perform a rigorous analysis of the properties of the two series. Based on such an analysis, we present suitable truncation criteria which allow understanding how to practically employ the two series expansions to compute the scattered field with a controlled error. A deep analysis of the range of applicability of the presented truncation criteria is also included. This allows providing a criterion which, given the surface and illumination parameters, and given the required accuracy and the computer floating-point format, allows us to choose which of the two series, if any, can be used, and how it can be properly truncated. Based on the presented analysis, we verify that for values of surface parameters of practical interest and for which the Kirchhoff approach can be used, for reasonable values of the required accuracy, and if the IEEE standard floating-point double-precision numbering format is used, then there is always at least one of the two series that provides an approximation of the scattering integral with the required accuracy. Index Terms—Fractals, scattering, series expansions.

I. INTRODUCTION T is well-known that natural surfaces can be very accurately described in terms of fractal geometry [1]. In fact, natural surfaces exhibit statistical scale-invariance properties that are very well modeled by using the 2D fractional Brownian motion (fBm) stochastic process [1], [2]. Therefore, for all of the applications in which it is of interest to compute the electromagnetic scattering from natural surfaces (i.e., remote sensing, wireless telecommunications, etc.), it is certainly useful to devise methods to evaluate the field scattered by a surface whose roughness is an fBm process. In the last two decades, several attempts have been made in this direction [3]–[7]. In particular, the Kirchhoff approach for scattering from dielectric fBm surfaces was introduced in [6] and better detailed in [7]. It was verified

I

Manuscript received September 15, 2009; revised March 14, 2010; accepted July 28, 2010. Date of publication December 03, 2010; date of current version February 02, 2011. S. Perna is with the Dipartimento per le Tecnologie (DIT), Università degli Studi di Napoli “Parthenope,” Napoli 80143, Italy and also with the Istituto per il Rilevamento Elettromagnetico dell’Ambiente (IREA), Italian National Research Council (CNR), Napoli 80128, Italy (e-mail: [email protected]). A. Iodice is with the Dipartimento di Ingegneria Biomedica Elettronica e delle Telecomunicazioni (DIBET), Università degli Studi di Napoli “Federico II,” Napoli 80125, Italy (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2096381

that use of the fBm surface model leads to scattering results that are in better agreement with measurements than those obtained by using classical, non-fractal, surface models. Scattering from fractal surfaces has been sometimes termed as “diffractal” [3], [5], as a contraction of “diffraction from a fractal.” By using the Kirchhoff approach in conjunction with an fBm description of the scattering surface, it turns out [7] that the scattered power density is analytically expressed by a scattering integral that can be expanded in terms of two different series, which are, to some extent, complementary: in fact, one is con, the other for vergent for surface fractal dimension ; in addition, one is an asymptotic expansion of the scattering integral for near specular directions/high frequency, the other for far from specular direction/low frequency. In practice, this approach is useful if, for at least one of the two series, a good approximation of the scattering integral is obtained with a sufficiently small number of terms. However, experiments show that numerical evaluation of these series is sometimes problematic. In this paper, a rigorous analysis of the behavior of the two series is performed, in order to derive their “region of practical applicability”. In particular, to this aim we properly reformulate the series expressions, so that the key parameters on which the series behavior depends are clearly identified: these parameters have also the desirable feature of showing an interesting physical meaning. Then, we theoretically analyze properties of the two series. Based on such an analysis, by making use of the Leibniz [8], [9], Braden [9], and Stieltjes [10] criteria, we present suitable truncation criteria which allow understanding how to practically employ the two series expansions to compute the scattered field with a controlled error. An original modified version of the Stieltjes criterion is also derived to properly deal with one of the series, and a deep analysis of the range of applicability of the presented truncation criteria is included. This allows us to obtain a criterion which, given the surface and illumination parameters, and given the required accuracy and the computer floating-point format, allows us to choose which of the two series (if any) can be used, and how it can be properly truncated. Based on the presented analysis, we verify that for values of surface parameters of practical interest (i.e., values actually encountered in real natural surfaces and for which the Kirchhoff approach can be used), for reasonable values of the required accuracy, and if the IEEE standard floating-point double-precision numbering format is used, then there is always at least one of the two series that provides an approximation of the scattering integral with the required accuracy. In other words, the union of the “regions of practical applicability” of the two series includes all the natural surfaces of practical interest. We finally underline that the obtained criterion for the choice of the series can be implemented in a software code for the efficient evaluation of scattering from natural surfaces.

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II. SCATTERING FROM FRACTAL SURFACES describes an fBm surface if, for A stochastic process every , , , , it satisfies the following relation:

(1) Fig. 1. Scattering geometry.

where, , is the Hurst coefficient, and is a characteristic length of the fBm surface, called topothesy. It can be demonstrated [2] that a process satisfying and that an fBm sample surface has a (1) exists if . Furthermore, it can be verified by fractal dimension using (1) that chords joining points on the surface at fixed dis, tance have a root mean square (rms) slope equal to so that topothesy can be defined as the distance over which chords joining points on the surface have a rms slope equal to unity. The fractal dimension (and hence ) is a measure of the observed surface roughness: at any given observation scale, the higher (or, equivalently, the smaller ), the rougher the surface appears. A surface satisfying (1) for any value of is sometimes referred to as mathematical fBm. However, in surface scattering problems, it is sufficient that the scattering surface satisfies (1) , with of the order of on a range of scales the linear size of the illuminated area and of the order of a . Such a surface fraction of the incident wavelength , say is sometimes called a physical fBm. A discussion on the applicability of the Kirchhoff approach to physical fBm surfaces and on its validity limits is reported in [6], [7]. We here recall that use of the Kirchhoff approach in the small slope regime leads to the following expression of the mean square value of the scattered field (i.e., the scattered power density):

In order to better understand the behavior of the integral in (2), , so that (2) can it is useful to make the substitution be recast as

(4) where (5) The integral appearing in (2) or (4) can be evaluated in a simple closed form for , as shown in [6], [7], whereas, for different from 0.5, two series expansions can be used. We discuss such two expansions in Sections III and IV, respectively. Here we only add that the parameter has an interesting physical meaning. In fact, it is proportional (through a constant depending on incident and scattering directions) to , i.e., to the surface rms slope at the wavelength scale. III. FIRST SERIES EXPANSION A. Region of Convergence and Asymptotic Behavior As shown in detail in [5]–[7], use of the Taylor expansion for the zero order Bessel function [11], and subsequent integration by series allow us to rearrange the integral (4) as follows:

(2)

where is the zero order Bessel function [11], and where, is the wavenumber of the incident see also Fig. 1, wave, and may each stand for or , i.e., horizontal or is the incident electric field, is the vertical polarization, distance from the surface to the receiver, is the area of the depends on incident and scatilluminated surface, angles, on polarization and on the surface dielectering tric constant, its complete expression being reported in [6], [7], and

(6) where is the Gamma function [11]. The region of convergence of the series in (6) can be obtained by applying the ratio test [8]. From (6) and Appendix I–(46), (47) we have

(3) (7)

PERNA AND IODICE: ON THE USE OF SERIES EXPANSIONS FOR KIRCHHOFF DIFFRACTALS

which easily leads to if if

(8)

According to the ratio test, from (8) we get that the series (6) converges when (for every , but for ), whereas (for every , but for ). it diverges when Let us now consider the asymptotic behavior of this series. As reported in [5]–[7] and demonstrated in [12], the series of functions (6) is an asymptotic expansion [13] for the integral in , i.e., (6) as

(9) , i.e., This limiting case is obtained, see (3), (5), when in near-specular scattering directions, and/or for , i.e., at high frequencies (or, equivalently, high values of the surface rms slope at the wavelength scale). We explicitly note that this asymptotic behavior is obtained for any value of , i.e., it is independent of the series convergence. The importance of this asymptotic behavior in the practical use of the series (6) will be clarified in the Section III-C. B. Truncation: Leibniz Criterion Let us consider the region of convergence of the series (6), that is and . In such a region, the sequence is infinitesimal, and it is definitely decreasing, i.e., an integer exists such that for the sequence is decreasing (see (8)). Accordingly, in all the points of this region it is possible to apply the Leibniz criterion [8], [9], which allows approximating, with a controlled error, the sum of an infinite alternating series by the sum of a proper truncation of the series itself. More precisely, according to such a Leibniz criterion, we have that

597

employed computer numbering format [14]. For instance, use of an IEEE standard floating-point format [14] allows representing numbers with a finite number of decimal places depending on the number of bits devoted to the memory field referred to as mantissa. Accordingly, computation of the finite sum in (10) , provided can be achieved with a required accuracy, say that, in addition to (11), also the following condition

(12) is satisfied 1. Condition (12) imposes an upper threshold, depending on both the required sum accuracy and the employed computer numbering format, to the maximum acceptable amplitude of the terms involved in the finite sum in (10). Accordingly, although analytical application of the Leibniz criterion in (10) requires only condition (11) to be satisfied, its practical application requires condition (12) to be satisfied as well. Thus, in the following, the region in which conditions (11) and (12) are simultaneously satisfied will be referred to as the Region of Practical Applicability (RPA) of the Leibniz criterion for the series in (6). Its computation is addressed in the following. From (8) and (11) it turns out that such a RPA is included in the region of convergence of the series itself, that is and . Moreover, according to (12), computation of such a RPA requires fixing the order of magnitude of the absolute accuracy required for the computation of the integral in (6): an absolute will be considered in the accuracy in the range of following. Furthermore, according to (11), computation of such a RPA requires the knowledge of the monotonic properties of for all the indexes (not necessarily large, as in (8)). To this aim, it is useful to manipulate the ratio in (11) as follows: (13) where, see (6)

(10) provided that the index

(14)

is chosen in such a way that (11)

According to (10) and (11), in all the points of the region of convergence of the series (6) it is possible to compute the sum of the series in (10) with arbitrary accuracy by properly setting of the finite sum in (10): the higher the value of the index , the smaller the value of the right hand term in (10) (see (11)), and the more accurate is the approximation in (10). , the numerical comHowever, for large values of putation of the finite sum in (10) may strongly be impaired by the finite computer precision, thus leading to completely wrong results. More precisely, as it is well known, to reach a given accuracy in the numerical computation of sums, it is necessary that the magnitude of the involved terms is smaller than a threshold which depends on both the required sum accuracy itself and the

and to introduce the following inequalities: (15) which easily come from (13) (note that ). The expressions in (15), strictly related to the condition (11), show the attractive feature to allow separation between the and parameters. 1IEEE Floating-Point representation has three basic components: sign, exponent and mantissa. The mantissa, also known as significand, represents the precision bits of the number. In particular, in the single-precision Floating-Point standard, the number of bits devoted to the field mantissa is equal to 23 (which leads to 2 8 10 ), whereas in double-precision format, the number of bits of the mantissa is equal to 52 (which leads to 2 4 10 )

 2

 2

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Fig. 2. The bound B (H ) versus H , for different absolute sum accuracies (circles: sum accuracy of 10 ; diamonds: sum accuracy of 10 ) and for different IEEE floating-point numbering formats (dashed line: single-precision; dotted line: double-precision). The solid line represents the function  (H ) versus H .

Fig. 3. Pictorial behavior of the sequence H.

 (H )

for different ranges of

We verified (via numerical investigation) that for the sequence is decreasing. Thus,

(i.e.,

the value of for ), which is plotted in Fig. 2 (solid line), divides the region of convergence into two regions: and , characterized by two different behaviors of the sequence . In fact, if , as it is shown in the pictorial graph of Fig. 3(c), the inequality (15) (b) is satisfied for all the indexes : accordingly the alteris absolutely decreasing for all the nating sequence indexes , and the condition (11) is thus always satisfied. Moreover, considering that in such a region, the following inequality:

(16) is always satisfied, we can conclude that, in practice, also the condition (12) is always satisfied for any reasonable absolute accuracy, and for the most common numbering formats

Fig. 4. Behavior of the sequence fj8 (H; S )jg for n = 0; 1; . . . 90, H = 0:6 and for four values of S in the range 0.8–1.

currently employed by computers (see the values of in Fig. 2). Thus, the couples such that and certainly belong to the RPA of the Leibniz criterion. Conversely, if , then, as shown (again) in the pictorial graph of Fig. 3(c), the inequality (15) (b) is satisfied only for greater than a proper index : thus, for , the alternating sequence is , whereas it absolutely increasing for gets absolutely decreasing for . To better show such a behavior, we plot in Fig. 4 the sequence for and for four values of picked up from this region. An additional smaller value of is considered 2 in Fig. 5. Similar behaviors are obtained for other values of . It turns out that, for a fixed value of , as decreases, the number of terms needed to reach the maximum of increases, and the maximum value of also increases. Accordingly, if we fix the required sum accuracy and the employed numbering format, 2In order to limit the overflow problems occurring in the computation of the n-th term of the sequence fj8 (H; S )jg, it is convenient to rearrange its expression as follows:

j8

H; S )j = S

2

(

exp

ln 0

+

n

n+1 H H

102

H

H 0 2 ln [0(n + 1)]

ln[2] 0

n H ln[S ]

2

As a matter of fact, we note that in the plot of Fig. 5, carried out by employing a double-precision floating-point numbering format, use of such an expression (dashed line) makes it possible to compute values of fj8 (H; S )jg much greater than those computable by using the expression (6) (thick line). We also underline that with an IEEE standard floating-points numbering format [14], the overflow limit must not be confused with the threshold of (12). Indeed, the overflow limit depends on the number of the bits devoted to the field exponent, whereas the threshold of (12) depends on the achievable precision and thus on the number of the bits of the mantissa. For instance, with a double-precision format, the overflow limit is of the order of 10 , whereas the threshold in (12), for an accuracy of the order of 10 0 10 , is of the order of 10 0 10 , and thus it is much less than 10 .

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are monotonically provided that the late terms of is chosen in such a way that increasing and the index (18)

Fig. 5. Behavior of the sequence fj8 0 6, = 0 58.

: S

:

(H;S )jg for n = 0; 1; . . . ; 3000. H =

then for each a bound exists, such that the condition (12) is satisfied for all , whereas it is violated for all : the RPA of the Leibniz criterion such for the series in (6) is thus given by the couples that and . can be computed, for a given sum accuThe bound racy and for an employed numbering format, by fixing (in the region ) and searching for the smallest such that the condition (12) is satisfied. In Fig. 2 we plot the value of computed by setting different absolute sum accuracies (circles correspond to sum accuracy of , diamonds to sum accuracy of ) and considering the number of bits of the mantissa of the single-precision (dashed line) and double-precision (dotted line) formats. Of course, for a fixed sum accuracy, the double-precision format makes it possible to achieve a RPA wider than that in the single-precision case. Moreover, for a fixed numbering format, the less accurate is the approximation required for the Leibniz truncation, the wider is the RPA of the method.

Some comments on such a Stieltjes criterion are now needed. First of all, we observe that, similarly to the Leibniz criterion, the Stieltjes one makes it possible to approximate the integral in (17) (that is, the integral in (6)) via a proper truncation of the series in (6). However, differently from the Leibniz case, for a such that the conditions (18) can be satisfixed couple fied, the finite sum in (17) does not allow to approximate the integral in (17) with arbitrary accuracy: indeed, from (17) and (18) it follows that such an accuracy is limited by the minimum , which cannot be forced to be as small as we of want (once that and have been fixed). Accordingly, in order to use the finite sum in (17) to compute the integral in (17) with , it is necessary that, for the consida given accuracy, say ered values of and , in addition to (18), also the condition (19) is satisfied. An additional comment is now needed. Let us suppose that the terms of the alternating sequence of functions progressively decrease in magnitude, , reach a minimum and thereafter then, for a certain index increase, thus satisfying condition (18). In this case, it can be shown that

(20) for every index we easily obtain that

Accordingly, from (17) and (20),

C. Truncation: Stieltjes Criterion Contrarily to what could be believed at first sight, practical use of the series (6) outside its region of convergence, which means for and , is possible. In fact, as shown in Section III-A, the series of functions (6) is an asymptotic . According to expansion for the integral in (6) as the theory of the asymptotic expansions, and considering that the series in (6) is alternating, it is thus possible to apply the Stieltjes criterion [10], [13] according to which

(21) where use of the triangular inequality has been made. From the inequality in (21) we obtain that when conditions (18) and (19) are simultaneously satisfied, it may be unnecessary to use the finite sum in (17) to compute the integral in (17) with the given , whereas the finite sum in (21), involving a accuracy smaller number of elements, may be appropriate as well, provided that the following condition: (22)

(17) is satisfied.

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Fig. 6. The bound B (H ) versus H , for different absolute sum accuracies (circles: sum accuracy of 10 ; diamonds: sum accuracy of 10 ). For 0 < H < 0:45 the solid line represents the function  (H ) versus H , whereas  (H ) versus H . for 0:45 < H < 0:5 it represents the function 

By summarizing, correct application of the Stieltjes criterion requires, differently from the Leibniz truncation, that the terms progressively deof the sequence of functions crease in magnitude, then reach a minimum which satisfies the condition (19), and thereafter increase. In the following, the region in which conditions (18) and (19) are simultaneously satisfied will be referred to as the RPA of the Stieltjes criterion for the series in (6). Computation of such a RPA is now addressed. From (8) it turns out that the Stieltjes truncation cannot be and , i.e., in the region of conapplied when vergence of the series in (6), because in such a region the late are not monotonically increasing. Acterms of cordingly, such a RPA is included in the region of non-conver. gence of the series in (6), that is Similarly to the Leibniz case, also for computation of the RPA of the Stieltjes criterion use of the expressions in (15) is helpful. and . Two cases are of interest: is now in order. The case We verified (via numerical investigation) that for the sequences is increasing. Thus, similarly to what observed in the previous Subsection, the threshold , which is plotted in Fig. 6 (solid line), divides also the region into two regions, and . , as shown in the pictorial graph of Fig. 3(a), If the inequality (15) (a) is satisfied for all the indexes : accordis absolutely iningly, the alternating sequence creasing for all the indexes , and the conditions (18) can never such that be satisfied. Accordingly, the couples and certainly do not belong to the RPA of the Stieltjes criterion. Conversely, if , as shown in the pictorial graph of Fig. 3(a), an index exists, such that for

the inequality (15) (b) holds, whereas for the inequality (15) (a) holds. For , the alternating sequence is thus absolutely decreasing , whereas it gets absolutely increasing for , so that reaches its minimum for , and it is always possible to satisfy the value at . However, condiconditions (18) by letting tion (19) may be satisfied or not, depending on the values of and . In fact, by varying , for a fixed value of , as decreases, the value of increases. Accordingly, for each a bound exists, such that the condition (19) is satisfied for all , whereas it is violated for all : the RPA of the Stieltjes crite, by the rion for the series (6) in is thus given, when couples such that . The bound depends (similarly to ) on the required sum accuracy, whereas (differently from ) it does not depend on the employed computer numbering format: (in the it can be computed, for a given accuracy, by fixing region ) and searching for the greatest such that the condition (22) is not satisfied. In Fig. 6 we plot the value of computed by setting different absolute accuracies for the computation of the integral in (17) (as in Fig. 2, , diamonds to an accircles correspond to an accuracy of ). Again, the less accurate is the approximation curacy of required for the Stieltjes truncation, the wider is the RPA of the method. Of course, differently from the plot of Fig. 2, the numbering format has no impact on the extension of the RPA of the Stieltjes criterion for the series in (6). is now in order. The case We verified (via numerical investigation) that for the terms of the sequence progressively decrease, then, for a certain index, reach a minimum, and thereafter increase (as depicted in the pictorial graph of Fig. 3(b). The extension of the considerations carried out for to this case is straightforward: we only need to note that the role of is now played by (23) A last consideration is now needed. It is important to note that the number of terms needed to approximate the series by using the Stjeltjes criterion , see (21)–(22). may be much smaller than To better show this point, we plot in Fig. 7 the sequence . In this case, we have and : condition (19) is thus satisfied for any reasonable accuracy, which means that belongs to the RPA of the Stieltjes criterion for the series in (6). In addition, in this case, if a reasonable ) for the absolute accuracy is required (up to the order of computation of the integral in (17), it is sufficient that condition ) is satisfied: this implies that computation (22) (with and subsequent summation of only 10 terms (see again Fig. 7) is actually necessary in this case. of the sequence

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exponential function [11], and subsequent integration by series allow us to rearrange the integral (4) as follows:

(24)

Fig. 7. Behavior of the sequence fj8 0 4, = 3 3.

: S

:

(H;S )jg for n = 0; 1; . . . 3000. H =

Evaluation of the region of convergence of the series in (24) can be obtained by application of the root test [8]. For large values of we can write (see Appendix I– (49))

D. RPA of the Leibniz and Stieltjes Criterion: A Joint Analysis In this Subsection, the results shown in the two previous Subsections are jointly analyzed, in order to get the RPA of the First Series Expansion, that is, the region in which the series in (6) can be practically employed for computation of the scattered power density in (4). The RPA of the First Series Expansion is given by the union of the RPA of the Leibniz criterion in (10)–(12) for the series in (6) and the RPA of the Stieltjes criterion in (17)–(19) for the same series. To this regard, we just recall here that, as shown in Section III-B, the RPA of the Leibniz criterion for the series ; in (6) is given by the couples differently, as shown in Section III-C, the RPA of the Stieltjes criterion for the same series is given by the couples . Accordingly, we can easily , such that the RPA conclude that a bound exists, say . of the series in (6) is given by the couples Such a bound, which is equal to when , and to when , depends first of all on the accuracy needed to compute the scattered power density in (4) (see Figs. 2 and (6)); furthermore, when it also depends on the employed computer numbering format (see again Fig. 2). It is worth noting that the RPA of the First Series Expansion in general neither corresponds to, nor includes or is included in the region of convergence (that in practice depends only by ) of the series itself; differently, it strongly depends on the parameter introduced in Section II.

IV. SECOND SERIES EXPANSION A. Region of Convergence and Asymptotic Behavior As shown in detail in [5] and [7], deformation of the integral path in the complex plane, use of the Taylor expansion for the

(25)

From (25) we have that (26) moreover, if

and

, infinite

exist such that (27)

According to the root test, we can thus conclude that, for every , the series (24) converges for value of such that whereas it diverges for and . It is interesting to underline that the region of convergence of the , is almost complementary to that series (24), which is and . of the series (6), which is As for the asymptotic behavior of the series (24), see [5], [7] and [12], it is an asymptotic expansion [13] for the integral in , i.e., (24) as

(28) , i.e., at This limiting case is obtained, see (3), (5), when low frequencies (or, equivalently, for low values of the surface is not vanishing, i.e., rms slope at the wavelength scale), and sufficiently far from the specular scattering direction. Similar to what noted for the series (6), this asymptotic behavior of the series (24) is obtained for any value of , i.e., it is independent of the series convergence. The importance of this asymptotic behavior in the practical use of the series (24) will be clarified in the Section IV-C.

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For the analysis of truncation criteria of this series, reported in Sections IV-B–IV-D, it is useful to rearrange the series (24) as follows:

(29) where (30) is From (29)–(30) it follows that the sequence given by the multiplication of a bounded sequence by the that shows some attractive features that sequence will be used in Sections IV-B–IV-D to analyze the way to employ the series in (24). Such features are now listed. , also First of all, similarly to the sequence is an alternating sequence. Second, by considering the following ratio:

B. Truncation: Braden Criterion Let us consider the region of convergence, that is and . In general, in this region it is not possible to apply the Leibniz criterion in (10) and (11) (differently from what has been done in the Section III-B for the series in (6)), because, as noted above, in such a region, the sequence is infinitesimal, but it does not get monotonically decreasing and, more important, the sequence is not alternating. Notwithstanding, we can observe that in is positive, decreasing this region the sequence (see and such that (33)); moreover, in this region, the series is convergent. Accordingly, in all the points of this region it is convenient to verify if it is possible to use the Braden criterion [9], which states that the sum of an infinite positive convergent series can be approximated with a controlled error by the sum of a proper truncation of the series itself. More precisely, according to such a criterion, we have

(34)

(31) (see (30)), we can get the monotonic properties of . Indeed, for large values of , the expression (31) can be rearranged as follows (see Appendix I, (46)–(48)):

(32)

provided that the index

is chosen in such a way that

(35) Use of the inequality (34) along with the triangular inequality allows us to write the following:

which leads to if if (33) From (33) it follows that in the region of convergence of the , the alternating sequence series (24), i.e., for gets monotone decreasing (and infinitesimal). This implies that in such a region is infinitesimal as well, whereas nothing can be said about its monotonic properties. and From (33) we also obtain that for the series is convergent (according to the ratio test). In the region of non-convergence of the series (24), i.e., and , the sequence gets for monotonically increasing (and divergent), which means that , although not regular, is given by the product of a bounded sequence by an absolutely divergent sequence. It is interesting to underline that the behavior of the alteris somehow dual with respect to nating sequence . that of the alternating sequence In the following, we will exploit the above listed properties to employ the series expansion in (24) for comof putation of the scattered power density in (4).

(36)

which, of course, holds if conditions (35) are satisfied. Some considerations on such a Braden criterion are now in order. First of all, from (33) it turns out that in the region of convergence of the series in (24) it is always (which means for every such that both finite value of ) possible to find an index the inequalities in (35) are satisfied. Second, from (33) it also follows that in the region of convergence of the series in (24) the right hand term in (36) can be forced to be as small as we want by properly setting the index . In other words, similarly to what shown for the Leibniz criterion in and (10), the (11) finite sum in (36) allows, provided that conditions (35) are satisfied, computing the sum of the se, ries in (24) with arbitrary accuracy: the higher the value of the smaller the value of the right hand term, and the more accurate is the approximation.

PERNA AND IODICE: ON THE USE OF SERIES EXPANSIONS FOR KIRCHHOFF DIFFRACTALS

Fig. 8. Pictorial behavior of the sequence of H .



H)

(

603

for different ranges

Third, the Braden criterion in (34)–(36) could be applied also to the series in (6) in its region of convergence: however, by doing so, for a fixed truncation index, the estimation of the error involved in the truncation would be less accurate than that in (10), as it can be seen by comparing the right hand terms in (10) and (36) (provided that the obvious substitutions and are carried out in (36)). Of course, similarly to what shown in Section III, for large , and thus of , the numerical values of computation of the finite sum in (36) may strongly be impaired by the finite computer precision. Accordingly, in order to reach , also in this case it is neca required sum accuracy, say essary that condition (12) is satisfied as well (where the obvious and must be done). Thus, substitutions although analytical application of the Braden criterion in (36) requires only condition (35) to be satisfied, its practical application requires condition (12) (with the substitutions mentioned above, which hereafter in this Section will be always applied) to be satisfied as well. In the following, the region in which conditions (12) and (35) are simultaneously satisfied will be referred to as the RPA of the Braden criterion for the series in (24). Its computation is now addressed. From (33) and (35) it turns out that such a RPA is included in the region of convergence of the series in (24). Moreover, according to condition (12), computation of such a RPA requires again fixing the order of magnitude of the absolute accuracy required for the computation of the integral in (24): similarly to what done in Section III, an absolute accuracy in will be considered in the following. the range of Furthermore, according to (35), computation of such a RPA requires the knowledge of the monotonic properties of for all the indexes (not necessarily large, as in (33)). To this aim, it is useful to manipulate (31) as follows: (37) where

Fig. 9. The bound B (H ) versus H , for different absolute sum accuracies (circles: sum accuracy of 10 ; diamonds: sum accuracy of 10 ) and for different IEEE floating-point numbering formats (dashed line: single-precision; dotted line: double-precision). The solid line represents the function  (H ) versus H .

and to introduce the following inequalities:

(39) which represent the Second Series counterpart of the inequalities (15). the We verified (via numerical investigation) that for sequence is increasing, as depicted in the pictorial graph of Fig. 8(a). Use of inequalities in (39) along with Fig. 8(a) allow computing the RPA of the Braden criterion for along the same lines shown in Section III-B, provided that some differences, which are listed in the following, are properly taken into account. and in First of all, the roles of Section III-B are now played by and (that is, the value of for , which is plotted in Fig. 9, solid line), respectively. Moreover, the considerations carried out in Section III-B for are now valid for , whereas those carried out for are 3 now valid for . Finally, as shown in Fig. 10 , for , as increases, the number of terms a fixed value of needed to reach the maximum of increases, also increases (difand the maximum value of ferently from what observed in Section III-B, where a similar is observed as decreases). behavior for As a consequence, if we fix the required sum accuracy and the employed numbering format, then for each we can compute a bound , such that the RPA of the Braden 3In order to limit the overflow problems occurring in the computation of the n-th term of the sequence fj2 (H; S )jg, similarly to what has been done in Section III (note 2), it is convenient to rearrange (30) as follows:

j2 =

(38)

H; S )j =

(

2



nH + 1)) 0 ln(0(n + 1)) 0 2n[(1 0 H ) ln(2) 0 ln(S ) + n ln(2)]g:

expf2 ln(0(

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2 (H; S )

Fig. 10. Behavior of the sequence fj jg for and for four values of in the range 2.6–3.3.

H = 0:4

S

n = 0; 1; . . . ; 3000,

criterion for the series in (24) is given by the couples such that and . computed by setting In Fig. 9 we plot the value of different absolute sum accuracies (circles correspond to sum ac, diamonds to sum accuracy of ) and concuracy of sidering the number of bits of the mantissa of the single-precision (dashed line) and double-precision (dotted line) formats. Of course, the same considerations carried out in Section III are valid for as well. In fact, on the bound also in this case, for a fixed sum accuracy, the double-precision format makes it possible to achieve a RPA wider than that in the single-precision case. Moreover, for a fixed numbering format, the less accurate is the approximation required for the Braden truncation, the wider is the RPA of the method. C. Truncation: Pseudo-Stieltjes Criterion Practical use of the series (24) outside its region of converand , is now addressed. gence, which means for As shown in Section IV-A, the series of functions (24) is an asymptotic expansion for the integral in (24) as . Unfortunately, in this case the series in (24) is not alternating: accordingly, we are not allowed to use the Stieltjes criterion shown in Section III-A. Notwithstanding, in Appendix II it is shown that in this case the following inequality:

(40) can be reasonably expected4, provided that the index chosen in such a way that the following condition:

is

(41) 4A rigorous upper bound for the truncation error in (40) can be provided as well, as described in Appendix II.

is satisfied. Hereafter, the truncation criterion in (40)–(41), which extends to non-alternating series the Stieltjes criterion in [10], will be referred to as pseudo-Stieltjes criterion. Indeed, conditions (40)–(41) are formally equivalent, but for a constant amplitude factor of the error, to conditions (17)–(18) introduced in Section III-A and, therefore, the same considerations carried out for the Stieltjes truncation can be done also in this case More precisely, we can say that it is possible to approximate the integral in (40) via a proper truncation of the series in (24); also in this case, for a fixed couple such that the conditions (41) can be satisfied, the finite sum in (40) does not allow to approximate the integral in (40) with arbitrary accuracy, because this latter is limited by the minimum of , which and cannot be forced to be as small as we want (once that have been fixed). Accordingly, in order to use the finite sum in (40) to compute the integral in with a given accuracy, say , it is necessary that, for the considered values of and , in addition to (41), also the following condition is satisfied: (42) By summarizing, correct application of the truncation in (40) requires, similarly to the Stieltjes criterion, that the terms of the sequence of functions progressively decrease in magnitude, then reach a minimum which satisfies condition (41) and thereafter increase. In the following, the region in which conditions (41) and (42) are simultaneously satisfied will be referred to as the RPA of the pseudo-Stieltjes criterion for the series in (24). Its computation is addressed now in order. Similarly to the Braden case, also for computation of such a RPA use of the expressions in (39) is helpful. From the inequalities in (39) and Fig. 8(a) it turns out that for and the term decreases in magnitude for all the indexes , whereas for and it progressively increases in magnitude, then reaches a maximum, and thereafter decreases. Accordingly, in all the points of the region of convergence of the series in (24) the pseudo-Stieltjes criterion in (40), (41) cannot be applied, because conditions (41) can never be satisfied. Thus, the searched RPA is included in the region of non-convergence of the series in (24). In such a region, two cases are of interest: and . is now considered. The case We verified (via numerical investigation) that for the sequence is decreasing, as depicted in the pictorial graph of Fig. 8(c). Use of inequalities in (39) along with Fig. 8(c) allow computing the RPA of the pseudo-Stieltjes criterion for along the same lines shown in Section III-C, provided that differences already listed in Section IV-B are properly taken into account. In particular, the roles of and in Section III-C are now played by and (which is plotted in Fig. 11, solid line), respectively. Moreover, similarly to what observed in Section IV-B, the considerations carried out in Section III-C for are now valid for , whereas those carried out for are now valid for . Finally, by varying and , for a fixed value of , as increases, the value

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We verified (via numerical investigation) that for the terms of the sequence progressively increase, then, for a certain index, reach a maximum, and thereafter decrease (as depicted in the pictorial graph of Fig. 8(b). to The extension of the considerations carried out for this case is straightforward: we only need to note that the role of is now played by (43)

D. RPA of the Braden and Pseudo-Stieltjes Criterion: A Joint Analysis

Fig. 11. The bound B (H ) versus H , for different absolute sum accuracies (circles: sum accuracy of 10 ; diamonds: sum accuracy of 10 ). For 0:58 < H < 1 the solid line represents the function  (H ) versus H , whereas for 0:5 < H < 0:58 it represents the function  (H ) versus H .

Fig. 12. Behavior of the sequence fj2 (H; S )jg for n = 0; 1; . . . 5000, H = 0:6 and for S = 0:58.

of increases (differently from what observed is in Section III-C, where a similar behavior for observed as decreases). As a consequence, if we fix the required sum accuracy, then for each we can compute a bound , such that the RPA of the pseudo-Stieltjes criterion for the series in (24) is given by the couples such that and . The bound , , see Section III) only on the required sum depends (as accuracy, whereas it does not depend on the employed computer numbering format, and it is plotted in Fig. 11 for different absolute accuracies (as usual, circles correspond to an accuracy of , diamonds to an accuracy of ). Again, the less accurate is the approximation required for the pseudo-Stieltjes truncation, the wider is the RPA of the method. in the region The behavior of the sequence and is shown in Fig. 12 where the has been considered. couple Let us now move to the case .

In this Subsection, the results shown in the two previous Subsections are jointly analyzed, in order to get the RPA of the Second Series Expansion, that is, the region in which the series in (24) can be practically employed for computation of the scattered power density in (4). The RPA of the Second Series Expansion is given by the union of the RPA of the Braden criterion in (35)–(36) and the RPA of the pseudo-Stieltjes criterion in (40)–(42). To this regard, we just recall here that, as shown in Section IV-B, the RPA of the Braden criterion for the series (24) in is given by ; differently, as shown in the couples Section IV-C, the RPA of the pseudo-Stieltjes criterion for the . same series is given by the couples Accordingly, in this case we can easily conclude that a bound , such that the RPA of the series in (24) is exists, say . Such a bound, which given by the couples is equal to when , and to when , depends, similarly to , first of all on the accuracy needed to compute the scattered power density in (4) (see Figs. 9 and 11); furthermore, when it also depends on the employed computer numbering format (see again Fig. 9). It is worth noting that the RPA of the Second Series Expansion in general neither corresponds to, nor includes or is included in the region of convergence (that in practice depends only by ) of the series itself; differently, it strongly depends on the parameter introduced in Section II. V. JOINT ANALYSIS OF THE REGIONS OF PRACTICAL APPLICABILITY OF THE TWO SERIES EXPANSIONS From the results shown in the previous Sections it easily follows that the region in which the integral in (4) can be evaluated at least by means of one of the two series in (6) and (24) is given by the union of the RPA of the First Series Expansion, computed in Section III, and the RPA of the Second Series Expansion, computed in Section IV. To this regard, we recall here that in Section III it has been got , such that the RPA of the First a bound, referred to as . Differently, Series is given by the couples in Section IV it has been got a bound, referred to as , such that the RPA of the Second Series is given by the couples . Thus, it turns out that for a fixed , if the following condition is satisfied: (44)

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then, for each it is always possible to use at least one of the two series in (6) and (24) to compute the scattered power density in (4). Further considerations on the inequality in (44) are now in order. First of all, as shown in Sections III and IV, the bounds and (and thus the RPA of the First and of the Second Series) depend on one side on the accuracy required for computation of the integral in (4), and on the other side on the employed floating-point numbering format; accordingly, satisfaction of the inequality in (44) depends on both the required accuracy and the employed computer numbering format, too. Second, it must be noted that when condition (44) is satiscorresponding to scattering surfaces of practical fied for all , [1], [6] (and interest, that is for all such that references therein), then, the union of the RPA of the First Series Expansion and that of the Second Series Expansion allows relevant to scatcovering all the possible combinations tering surfaces of practical interest. In order to better investigate this latter issue, we set an abso, consider the number of bits of the lute sum accuracy of mantissa of the double-precision floating-point format, that is, 52 (see footnote 1), and plot in Fig. 13 the corresponding bounds (solid line) and (dashed line), which have been achieved by picking up from Figs. 2, 6, 9, 11 such curves that are relevant to the considered accuracy and computer numbering format. From Fig. 13 it turns out that condition (44) is , which means that if a satisfied for all such that double-precision floating-point format is employed, it is always (which means for all the natural surfaces of practical interest) possible to use at least one of the two series in (6) and (24) to , compute the integral in (4) with an absolute accuracy of and in particular both series can be used in the region of overlapping between the two RPAs (bright-gray region in Fig. 13). To , and then we have combetter show this, we have fixed puted for different values of the integral in (4) by means of the two series in (6) and (24). The results, achieved by employing a floating-point double-precision numbering format and setting a are collected in Fig. 14. Two cases are of sum accuracy of interest: and . , that is, for (see Fig. 13), For we have employed the First Series Expansion with the Stieltjes , truncation (solid line in Fig. 14), whereas for that is, for (see again Fig. 13), we have employed the Second Series Expansion with the Braden truncation (dotted of line in Fig. 14). In the former case, the truncation index the employed finite sum in (21) has been chosen in such a way to satisfy the following condition: (45) which implies condition (22) (with ), thus guaranteeing by employing only to achieve an absolute sum accuracy of (instead of ) terms of the finite sum in (21), see (22). In the latter case, the truncation index of the employed finite sum has been chosen in such a way to simultaneously satisfy the second inequality in (35) and render the absolute sum ac. It is interesting to note that for curacy in (36) less than

Fig. 13. The bounds B (H ) (solid line) and B (H ) (dashed line) versus H , for an absolute sum accuracy of 10 , and for a double-precision floatingpoint format. The RPA of the First Series is given by the union of the bright-gray and black regions; the RPA of the Second Series is given by the union of the bright-gray and dark-gray regions.

Fig. 14. Computation of the integral in (1), for H = 0:3 and H = 0:7, and for different values of S , via the First Series Expansion (solid line) and the Second Series Expansion (dotted line). Absolute sum accuracy: 10 ; floating-point format: double-precision.

, i.e., in the region in which both the two series in (6) and (24) can be employed to compute the , a comparison integral in (4) with an absolute accuracy of between the results achieved by means of the two different Series (and truncation criteria) can be carried out. To this aim, the difference between the dotted and solid curves of Fig. 14 has : it turned out been computed for , in total agreement with the expected to be of the order of accuracy. Results similar to those collected in Fig. 14 have been . From this achieved also for the other values of set of results, we plot again in Fig. 14 also those achieved for , in order to show what happens for . Again,

PERNA AND IODICE: ON THE USE OF SERIES EXPANSIONS FOR KIRCHHOFF DIFFRACTALS

Fig. 15. The bounds B (H ) (bottom) and B (H ) (top) versus H , for an absolute sum accuracy of 10 , double-precision floating-point format and a relative sum accuracy of ten per cent. In the gray areas the fixed absolute sum accuracy is not appropriate to achieve the required relative sum accuracy. A logarithmic scale has been used for the S -axis.

two cases are of interest: and . , that is, for (see again Fig. 13), For we have employed the First Series Expansion with the Leibniz , that is, for truncation (solid line), whereas for (see again Fig. 13), we have employed the Second Series Expansion with the pseudo-Stieltjes truncation (dotted line). In the former case, the corresponding truncation index has been chosen in such a way to simultaneously satisfy condition (11) and render the absolute sum accuracy in (10) less . In the latter case, the truncation index of the than employed finite sum has been chosen in such a way to satisfy ). Again, the difference conditions (41) and (42) (with between the dotted and solid curves has been computed for : it turns out to be of the order of , in total agreement with the expected accuracy. It is worth underlining that the absolute sum accuracy of which has been set in the plots of Figs. 13–14 is appropriate provided that the computed sum is at least one order of magnitude greater than the absolute accuracy itself. For instance, in the case plotted in Fig. 14, the absolute sum accuracy of turns out to be not appropriate for and if a relative sum accuracy of ten per cent is required; in the case plotted in Fig. 14, it turns out that the absolute accuracy of is not appropriate for and . More generally, once the absolute and relative sum accuracies have been both and an upper bound exist, fixed, a lower bound such that only for the fixed absolute accuracy is appropriate to achieve the required relative accuracy. Computation of these two bounds can be easily performed by means of plots similar to those presented in Fig. 14 and relevant to different values of . In Fig. 15, the bounds and have been computed for an absolute sum accuracy and a relative sum accuracy of ten per cent. For couples of belonging to the gray areas of Fig. 15, use of (at least) one of the two aforementioned series expansions must be thus verified by computing a map similar to that in Fig. 13, but characterized by higher absolute sum accuracy. It is now worth investigating what is the maximum absolute accuracy for which RPAs of the two series cover all the

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Fig. 16. The bounds B (H ) (solid line) and B (H )(dashed line) versus H , for an absolute sum accuracy of 10 , and for a single-precision floatingpoint format. The RPA of the First Series is given by black region; the RPA of the Second Series is given by the dark-gray region.

pairs of practical interest, when a fixed floating-point numbering format is used. By retaining the IEEE double-precision floating-point format, and for absolute and computing the bounds accuracies higher than that considered in Fig. 13, it has been veris ified that an absolute sum accuracy of the order of 0.5 the highest possible that allows the bright-gray region of Fig. 13 to persist for all the values of practical interest. Conversely, if we move to consider the single-precision (solid line) and floating-point format, the bounds (dashed line) of Fig. 16 are achieved. They correspond to the curves of Figs. 2, 6, 9, 11 that are relevant to these considered accuracy and computer numbering format. We observe that in this single-precision case there is a region (white area in Fig. 16) of no overlapping between the RPA of the First Series and that of the Second series, for which use of neither the series in (6) nor that in (24) allows computing the . integral in (4) with an absolute accuracy of of pracOf course, satisfaction of condition (44) for all tical interest requires the absolute sum accuracy to be relaxed when IEEE single-precision floating-point format is employed. In particular, it has been verified that an absolute sum accuracy is the highest possible that allows the white area of of Fig. 16 to vanish. Finally, a few words on computational time are needed. Computation of the curves of Fig. 14 (where 1000 samples of the variable have been considered) took at worst about half second by using a 2.4 Ghz Intel Core 2 Duo CPU with 3 GB RAM: computational time is thus negligible when the scattering integral (4) in is computed by means of two series expansions in (6) and (24) via the most appropriate truncation criterion. However, we stress again that the main advantage of using the proper truncation criterion is not the reduction of computational time, but rather the fact that it ensures achieving the requested accuracy, and hence it avoids obtaining completely wrong evaluations of the integral in (4).

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VI. CONCLUSIONS

APPENDIX I

In previous works [5]–[7], it was shown that use of the Kirchhoff approximation allows expressing the mean square modulus of the field scattered by a fractal surface in terms of two series expansions. In this paper, the practical applicability of these expansions has been addressed. We have expressed these series in terms of two parameters with a clear physical meaning: the Hurst coefficient , which is a measure of the surface roughness, and the parameter , which is proportional (through a constant depending on incident and scattering directions) to the surface rms slope at the wavelength scale. We have theoretically analyzed the series behavior, in order to devise appropriate truncation criteria which allow understanding how to practically employ the two series expansions to compute the scattered field with a controlled error. A deep analysis of the range of applicability of the presented truncation criteria has also been included. As a result, we have obtained a criterion which, given the surface and illumination parameters, and given the required accuracy and the computer floating-point format, allows us to choose which of the two series (if any) can be used, so that the RPA of each series can be identified. In particular, it has turned out that the first series, (6), can be (i.e., in truncated by using the Leibniz criterion if its convergence region), or the Stjelties criterion if (i.e., outside its convergence region). In the former case, conditions (11) and (12) must be satisfied, whereas in the latter case conditions (18) and (19) must be fulfilled. We have veris ified that these conditions are met by the first series if sufficiently large. In a complementary way, the second series, (24), can be truncated by using the Braden criterion if (i.e., in its convergence region) or a pseudo-Stjelties criterion, if (i.e., outside its convergence region). In the former case, conditions (12) and (35) must be satisfied, whereas in the latter case conditions (41) and (42) must be fulfilled. It is worth underlining that the aforementioned pseudo-Stjelties criterion extends the Stieltjes criterion to the case of non-alternating asymptotic series, and represents one of the original contributions of the work. Based on the presented analysis, the regions of practical applicability (RPA) of the two series have been computed. In particular, we have plotted, as a function of , and for different values of the required accuracy and different numbering formats, the thresholds , that defines the RPA of the first series, and , that defines the RPA of the second series (see Figs. 13 and 16). We have hence verified that, for values of surface parameters of practical interest and for which the Kirchhoff approach can be used, for reasonable values of the required accuracy, and if the IEEE standard floating-point double-precision numbering format is used, then there is always at least one of the two series that provides an approximation of the scattering integral with the required accuracy. We finally underline that the obtained criterion for the choice of the series can be implemented in a software code for the efficient evaluation of scattering from natural surfaces. This last issue will be matter of future work.

function, used in In this appendix some properties of the different Sections of the work, are listed. Use of the following asymptotic formula [15]:

(46) has been done in Sections III and IV. In particular, from (46) we have (47) which leads to the approximation in (7) (Section III-A). From (46) we also have (48) which, substituted in (31), leads to (32) (Section IV-C). Use of the following asymptotic formula [16], which holds for large values of , (49) has been done to write (25) (Section IV-A). For (49) reduces to the Stirling formula, i.e., , used in (25) as well.

,

APPENDIX II In this appendix a theorem (the proof of which is shown in [13]) regarding asymptotic expansions is reported, and subsequently applied to show that the inequality in (40) (introduced in Section IV-C), subject to condition (41), can be reasonably expected. be a function, and be Theorem 1: Let a sequence of positive functions, such that (50) It can be shown that, similarly to the Stieltjes truncation (which holds for alternating series, the late terms of which are monocan be approximated, with a contonic increasing), trolled error, by a proper truncation of the series in (50). More precisely, the following inequality holds: (51) provided that the index

is chosen in such a way that

(52)

PERNA AND IODICE: ON THE USE OF SERIES EXPANSIONS FOR KIRCHHOFF DIFFRACTALS

Let us now consider the following series:

(53)

where

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is chosen in such a way provided that the index that conditions (52) (where the obvious substitution must be done) are satisfied. From (58), (59) we thus obtain

is defined in (30) and (54) (60)

It can be shown [18] that it is always possible to find a proper such that function Use of the inequality (56) in (60) leads to (55)

Considering that the sequence is positive, we can apply to the asymptotic expansion in (55) the Theorem1, which leads to the following: (61)

(56) provided that the index is chosen in such a way that conditions (52) (where the obvious substitution must be done) are satisfied. In addition, from (28) and (55) it follows [19] that

(57)

where the use of the following: (58) has been done (see (24) and (30)). In (57), the sequence is clearly positive: accordingly, we can apply also to the asymptotic expansion in (57) the Theorem 1, thus obtaining the following:

(59)

and . Considwhere ering the characteristics of the bounded sequence , it is reasonable to assume that the in (56) is approximately equal in (60): in this case the inequality in (61) reduces to to the that in (40) (subject to the condition (41)). A last comment is now needed. Numerical assessment of the first condition in (52) requires, in practice, a proper threshold to be fixed, and the condition to be verified. Of course, special care must be taken in setting the threshold , since too small values as well as too big values of could lead to wrong results. Accordingly, from the numerical point of view, the first condition in (52) could be not straightforward to be verified. Differently, numerical assessment of the second condition in (52) is very easy to be performed. These considerations suggest the inequality in (40) to be preferred to that in (61), although this latter has been rigorously derived. To better clarify this, we observe that due to (see Section IV-C), the behavior of the sequence computation of in (59) requires, in practice, only the second condition in (52) to be verified, since this latter implies also the first condition in (52) to be satisfied as well. Differently, due to , the oscillating behavior of the sequence computation of in (56) would require also the first condition in (52) to be verified. Then, use of (61), although theoretically rigorous, requires the computation of both (which is simple in practice) and , which in practice is subject to a significant uncertainty. Conversely, use of (40), although based on the reasonable but undemonstrated relation , only requires the computation of , which can be accurately computed in practice. Accordingly, in practice (40) and (61) have similar degrees of reliability, and in some cases use of (40) may even lead to more accurate results then use of (61).

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REFERENCES [1] B. B. Mandelbrot, The Fractal Geometry of Nature. New York: W.H. Freeman, 1983. [2] K. Falconer., Fractal Geometry. Chichester, U.K: Wiley, 1990. [3] M. V. Berry and T. M. Blackwell, “Diffractal echoes,” J. Phys. A Math. Gen., vol. 14, pp. 3101–3110, 1981. [4] D. L. Jaggard, “On fractal electrodynamics,” in Recent Advances in Electromagnetic Theory, H. N. Kritikos and D. L. Jaggard, Eds. Berlin, Germany: Springer-Verlag, 1990, pp. 183–223. [5] O. Y. Yordanov and K. Ivanova, “Kirchhoff Diffractals,” J. Phys. A Math. Gen., vol. 27, pp. 5979–5993, 1994. [6] G. Franceschetti, A. Iodice, M. Migliaccio, and D. Riccio, “Scattering from natural rough surfaces modelled by fractional Brownian motion two-dimensional processes,” IEEE Trans. Antennas Propag., vol. 47, pp. 1405–1415, 1999. [7] G. Franceschetti, A. Iodice, and D. Riccio, “Fractal models for scattering from natural surfaces,” in Scattering, R. Pike and P. Sabatier, Eds. London: Academic Press, 2001, pp. 467–485. [8] Knopp and Konrad, Infinite Sequences and Series. New York: Dover, 1956. [9] B. Braden, “Calculating sums of infinite series,” Amer. Math. Monthly, vol. 99, no. 7, pp. 649–655, 1992. [10] T. J. Stieltjes, “Recherches sur quelques series semi-convergentes,” Ann. Sci. Ecole norm. Sup. 3, pp. 201–258, 1886. [11] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products. New York: Academic Press, 1980. [12] S. Perna and A. Iodice, “Asymptotic behavior of two series used for the evaluation of Kirchhoff Diffractals,” IEEE Trans. Antennas Propag., to be published. [13] R. B. Dingle, Asymptotic Expansions. Their Derivation and Interpretation. London: Academic Press, 1973. [14] C. Hamacher, Z. Vranesic, and S. Zaky, Computer Organization. New York: McGraw-Hill, 2002. [15] Tricomi and Erdely, “The asymptotic expansion of a ratio of Gamma functions,” Pacific J. Math., pp. 132–142, 1951. [16] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions. New York: Dover, 1964. [17] N. Bleistein and R. A. Handelsman, Asymptotic Expansions of Integrals. New York: Dover, 1986. [18] H. A. Lauwerier, Asymptotic Analysis. Amsterdam: Mathematish Centrum, 1974. [19] A. Erdélyi, Asymptotic Expansions. New York: Dover, 1956.

Stefano Perna (S’03–M’07) received the Laurea degree (summa cum laude) in telecommunication engineering and the Ph.D. degree in electronic and telecommunication engineering, both from the Università degli Studi di Napoli “Federico II,” Naples, Italy, in 2001 and 2006, respectively. He was with Wise S.p.A., Naples, from 2001 to 2002. In 2003, 2005, and 2006, he received grants from CNR (Italian National Research Council) to be spent at the Istituto per il Rilevamento Elettromagnetico dell’Ambiente (IREA), Naples, for research in the field of remote sensing. In 2003 and 2006, he visited Orbisat Remote Sensing, Brazil, for repeat pass interferometric processing of airborne synthetic aperture radar (SAR) data. Since 2006, he has been with the Dipartimento per le Tecnologie (DIT), Università degli Studi di Napoli “Parthenope,” Naples, where he is currently a Researcher in electromagnetics. He currently also holds the position of Adjunct Researcher at IREA-CNR, Naples. His main research interests are in the field of microwave remote sensing and electromagnetics: airborne SAR data modelling and processing, airborne differential SAR interferometry, modelling of electromagnetic scattering from natural surfaces and, more recently, synthesis of antenna arrays.

Antonio Iodice (S’97–M’00–SM’04) was born in Naples, Italy, on July 4, 1968. He received the Laurea Degree (cum laude) in electronic engineering and the Ph.D. degree in electronic engineering and computer science, both from the University of Naples “Federico II,” Naples, Italy, in 1993 and 1999, respectively. In 1995, he received a grant from CNR (Italian National Council of Research) to be spent at IRECE (Istituto di Ricerca per l’Elettromagnetismo e i Componenti Elettronici), Naples, Italy, for research in the field of remote sensing. He was with Telespazio S.p.A., Rome, Italy, from 1999 to 2000. Since 2000, he has been with the Department of Electronic and Telecommunication Engineering of the University of Naples “Federico II,” where he is currently a Professor of electromagnetics. His main research interests are in the field of microwave remote sensing and electromagnetics: modelling of electromagnetic scattering from natural surfaces and urban areas, simulation and processing of synthetic aperture radar (SAR) signals, SAR interferometry, and electromagnetic propagation in urban areas. He is the author or coauthor of about 170 papers published on refereed journals or on proceedings of international and national conferences. Prof. Iodice received the “2009 Sergei A. Schelkunoff Prize Paper Award” from the IEEE Antennas and Propagation Society.

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Complex Conical Beams for Aperture Field Representations ˇ ´ , Massimiliano Casaletti, Stefano Maci, Fellow, IEEE, and Stig Busk Sørensen Sinisa Skokic

Abstract—A method is presented for computing aperture radiated fields by means of new conical beams with azimuth phase variation. These beams are generated in a natural way starting from the spectral-domain radiation integral, by expanding the electric field spectrum in the aperture plane in a Fourier series, and by approximating the obtained Fourier series coefficients by a sum of complex exponentials using the generalized pencil-of-function method. This transforms the radiation integral to a simpler form which can be evaluated analytically. Two types of wave objects are derived, both of them arising from the same spectral GPOF process, that possess different properties. Aperture fields obtained via the new approach are successfully compared to those calculated via direct near field integration or asymptotic evaluation. Index Terms—Aperture antennas, beams, complex point sources, propagation.

Section III introduces the formulation of a first type of conical beams, which arises naturally from the angular FFT and radial GPOF expansion of the aperture spectrum. Section IV presents the formulation of a second type of beams, directly related to the analytical continuation of spherical wave modes. Section V illustrates the vector form of the conical beams, obtained by enforcing Maxwell’s equations. Section VI shows how the two different typologies of these new wave objects can be used to reconstruct the near and far field radiated by both circular and rectangular apertures. The results are compared to those obtained via direct integration of the radiation integral and to those obtained via Gauss-Hermite beams. II. FIELD REPRESENTATION IN TERMS OF BEAMS

I. INTRODUCTION

I

N this paper we introduce new types of Maxwellian conical beams and discuss their properties and potential advantages in expanding aperture fields over other commonly used solutions. These conical beams rigorously respect the wave equation and, in their vector form, Maxwell’s equations. The beam expansion of an aperture radiated field is simply obtained by first expanding the field spectrum in the aperture plane in terms of an angular Fourier series and next representing the radial spectral coefficients in terms of complex exponentials obtained by the generalized pencil-of-function method (GPOF) [1] method. The starting radiation integral is this way reduced to a double sum of wave objects which can be computed analytically in both space and spectral domain. The vector form is then obtained through the use of vector potentials. This paper will only deal with the development, the basic properties analysis, and the propagation of the new beams. Their integration into the framework of the analysis of reflector antenna systems will be presented in a subsequent communication. After a review of the literature and the motivation for our analysis presented in Section II,

Manuscript received July 26, 2009; revised June 23, 2010; accepted June 25, 2010. Date of publication December 03, 2010; date of current version February 02, 2011. This work was supported by the European Space Agency (ESAESTEC, Noordwijk, The Netherlands) European Antenna Modelling Library programme (EAML2, ESA contract n. 18802). ˇ ´ is with the Department of Wireless Communications, Faculty of S. Skokic Electrical Engineering and Computing, University of Zagreb, Croatia (e-mail: sinisa.skokic@ fer.hr). M. Casaletti and S. Maci are with the Department of Information Engineering, University of Siena, Siena, Italy (e-mail: [email protected]; [email protected]. it). S. B. Sørensen is with Ticra, Copenhagen, Denmark (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2010.2096379

Generally speaking, a wave object is a spatial function which exhibits wave-like behavior in that its amplitude and phase fronts can be identified. When the amplitude of a wave object is confined to a region of space around a certain direction of propagation, one normally refers to it as a beam. Desirable properties for beams are: 1) availability of spatial domain (and may be spectral domain) expression in a closed form; 2) existence of a systematic methodology for generating and truncating the beam expansion; 3) respect of the wave equation in a largest possible region of space. Different beam expansions approaches may require a different number of beams to represent a given field. This property should be viewed in light of the degrees of freedom [2], which establishes a natural criterion for the sufficient and non-redundant number of wave functions able to fully reconstruct the field in a given region within a predetermined error. The optimal (minimal) number of wave functions is approximately equal to introduced by Bucci et al. [2] the degrees of freedom (1) where is the surface of the minimum convex volume enclosing the source(s), and the factor 2 arises from the need of representing the fields by two orthogonal polarizations. In case degenerates into a planar aperture, then the total surface is twice the area of the aperture (if is interpreted as the area of the identifies the degrees of freedom of the upper aperture, half-space field only). The definition of the degrees of freedom in (1) is consistent with the Landau-Pollak bound [3] where, for a quasi spatially limited aperture of approximate area , a band-limited aperture field of spectral bandwidth K can be obbeams (the factor tained by expansion of 2 is relevant to the 2 orthogonal polarizations). The relationship between the Landau-Pollak bound and the degrees of freedom of

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the radiated fied in (1) can be obtained by approximating with , namely by the area of the spectral visible region, as reasonable for large apertures in terms of wavelengths when observing out of the reactive field region. This leads to (the redundancy factor obtained in (1) is due to the men). tioned approximation Any representation requiring more beams than the number prescribed by (1) has some redundancy. We emphasize that (1) gives a sampling criterion for representing the fields sufficiently well without any a priori knowledge of the actual sources. If, instead, the sources are known, field-matched wave objects may be used and their number can be much lower than (1), and this is what happens in the formulation presented here. A number of different beams have been introduced in the past [4]–[18]: Gaussian Beams (GB), Gaussian-ray basis functions (GRBF), higher-order Gauss-Laguerre (GL) or Gauss-Hermite (GH) beams, complex source points (CSP), Bessel beams (BB). GB [4], [5], have the limitation of satisfying Maxwell’s equations only in the paraxial region. GRBF [6], [7] do not satisfy wave equations, and are obtained by adding an empirical extra parameter in the formulation of a Gaussian beam, in order to manually control beam width at a given distance from the source. GL or GH beams [8], [9], [21] deal with expansions around a preferred axis of propagation with the higher order terms representing the off-axis variations. They have the advantage of constituting an orthogonal set, and are often used as basis for a mode matching technique so they can be considered as global expansions of the aperture. However, their descriptive capability are still restricted to the paraxial region. To expand the field in the whole space, Complex Source Points (CSP) [10]–[12] may be used. CSP have been used for expanding space domain Green’s functions of the mixed potentials for layered dielectric media [13], [14]. To this end, the spectral Green’s function is represented in terms of exponentials by using Prony’s method (or GPOF method) thus leading to the CSP expansion by using the Sommerfeld integral. A similar technique is used to regularize the Kernel of 3D integral equation [15]. We use here a similar method, with the difference that the azimuth field variFourier-seation requires here an additional step of azimuth ries and a generalization of the Sommerfeld integrals to higher order Bessel spectral functions. In some optical laser framework, the field is represented by a discrete spectrum of Bessel beams [16]. Each beam introduced in this paper is composed by a continuous spectrum of Bessel beams over a spectral wave-number, that removes the non-physical behavior of the discrete Bessel beam spectra (Bessel beams do not spread out as they propagate). Contrary to the above mentioned global expansions, there are also Gabor-type (or phase-space) expansions [17]–[20], whereby the field is expanded using a lattice of beams that emerge from a set of points in the aperture plane and propagate from each point in a lattice of directions. These beam basis functions describes the local radiation properties of the aperture distribution, and hence the beam amplitudes are determined by the local radiation properties (the local spectrum) of the aperture near the lattice points. This beam representation is therefore localized near the phase-space Lagrange manifold of the field, and can be viewed as a “local” expansion of the aper-

ture field. It is known that for off-axis observation the localized nature of the spectral elements implies more rapid convergence of the field representation than with global expansions [3]. In the Gabor-based Gaussian beam expansion, the basis set is complete; This poses a restriction on the choice of the spatial and spectral resolutions. The Gabor-frame scheme in [18], [19] (termed there “windowed Fourier transform frame”) relaxes this restriction by using overcomplete sets of GB’s, thus enabling the user to choose the spatial and spectral resolutions so that they best fit the local properties of the source distribution; the guidelines for choosing these parameters are discussed in these references. Furthermore, the flexibility gained by using the overcomplete frame expansion allows an efficient representation of ultra wide band (UWB) fields [18], [19], as well as direct analysis in the time domain using a discrete spectrum of pulsed beams [20]. In this paper, we introduce two new types of conical beams, herein after referred to as “Formulation A” and “Formulation B”. We will be using the term “beam” in a more general sense, i.e., even though the new wave objects are not collimated around a single axis, but rather around a conical surface. This property makes them non-local, at least for high-order, when they form a wide cone. In such cases, they cannot a priori be expected to maintain their structure once they hit a reflector or propagate through an inhomogeneous medium, and are therefore strictly speaking not true beams. It is worth mentioning that conical Gaussian beams naturally arise in connection with radiation from a line source distribution [22]. The new conical beams introduced here arise from an analytical continuation of a closed form solution of the wave equation, with complex displacements automatically generated by the GPOF technique for both formulations. For this reason, they will be denoted as complex conical beams (CCBs). Both formulations have potential advantages over other approaches for the following properties: 1) the CCB’s respect the wave equation in all space; 2) the generation of CCB’s is obtained in efficient and natural way; 3) the CCB’s possess analytical expressions in both spatial and spectral domain. In light of the above properties, we note that unlike GL or GH beams, the CCB’s can be used to expand both collimated and non-collimated aperture field, due to the fact they respect the wave equation exactly (and not only in the paraxial region). Both CCB-A and CCB-B reduce to a Complex Source Point beam at zero-order, and a set of higher-order beams with azimuth phase variation describe the off-axis field. Therefore, they can be regarded as a general and physically correct alternative to the GL (or GH) expansion. III. COMPLEX CONICAL BEAMS-FORMULATION A Let us consider the Fourier-type spectral radiation integral [23]

(2) denotes an aperture spectrum of either where electric or magnetic field or of a scalar potential, while

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, is the spectral-domain representation of the free-space Green’s function. In cylindrical coordinates, the same integral is rewritten as

where

, . Due to the inherent periodicity in , can be expanded in a Fourier series

(3) , and

(4) and the integral in yielding

in (3) can be evaluated in a closed form,

(5)

It should be noted that for the integral in (9) to converge, the real part of complex coordinate should be positive. With , this translates to the condition , meaning that in a general realistic situation, whereby in (7) are generated by the GPOF fitting procedure, there may exist a minimal distance for the validity of this expansion. The reason of , this apparent incongruence is now explained. If where then (6) diverges exponentially . This implies that (6) can be applied only in the range , where is set a priori within the GPOF procedure and represents the evanescent spectrum portion of the aperture-field that one decides to include in the expansion [24]. However, (6) is inserted in (5) and integrated over the semi-infinite integration range to give the final representation (7)–(9) and the corresponding space-domain closed form given next. The need to maintain the infinite integration in (5) implies that the accuracy of (7), (9) is preserved for ; in fact, under this condition the exponential attenuation term of the spectral Green’s function actually filters-out the spectral components of (5) larger than , where (6) is inaccurate. This, however, does not present a practical limitation in most cases. It is evident that (9) obey the wave equation in cylindrical coordinates, (10)

where

is the Bessel function of th order. The coefficients are represented by using the generalized pencil of function (GPOF) method [1] as (6) where and are the output residues and poles of the is represented as a function GPOF algorithm. Note that of . Using (6) in (5) leads to (7) where

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since the integrand in (9) does it for any because of the definition of the Bessel function. Before proceeding further, we observe that the procedure outlined here permits the calculation of the beam expansion prois available. vided that the aperture Fourier spectrum The latter should not necessarily be known in analytical form, but can also be obtained through a FFT of space field samples. A. Space Domain Closed Form Expression It is possible to find an analytical solution to the integral in (9). This is achieved by relating it to a similar integral whose closed-form solution is known [25] (Eq. 6.637). In Appendix A it is demonstrated that for greater than 1, the following recurrence formula holds

(8) and

for complex values of

is defined by

(9) (11) From here on, coordinates labeled with are intended to be the complex analytical continuation of the corresponding real is evaluated at a point whose -coordinate variable. In (7), has been displaced in the complex plane by . Equation (9) is the defining equation of the wave object that we shall refer to as the conical beam, for reasons which will become obvious in the following sections.

where and are the Bessel function and the Hankel function of the second kind of order , respectively, and (12) (13)

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W n = 0; . . . 4 at a distance r = 5 from  coordinate for b = 0.

Fig. 1. Normalized amplitude of the origin as a function of the spherical

where and For

with the condition . We note that for , , , where is the conventional spherical radial coordinate. less than 1 the recurrence relation is completed by (14)

The starting terms

and

of the recurrence are given by (15)

that is found through (9) by the well known Sommerfeld identity, and

(16) and noticing that which comes from (11) by setting . in the Fig. 1 shows normalized values of as a function of (real) at the distance case from the origin. From (15), we see that is the scalar Green’s function of free space, producing the same beam amplitude in , all higher order beams possess a zero all directions. Unlike on the -axis and a maximum at an angle which increases for increasing . Furthermore, for increasing orders , one can observe the increasing flatness of the beam around the origin. If a purely imaginary parameter is added to the real coordinate , we gain a degree of freedom in describing the shapes of , with the beams. Figs. 2(a)–(d) shows the variation of (Fig. 2(a)), (Fig. 2(b)), (Fig. 2(c)) and (Fig. 2(d)). Note that a complex displacement in leads to the conventional quasi-Gaussian behavior with increasing directivity for increasing complex displacement. For beams of order different from zero, the beams become narrower for increasing and the lateral lobes get more attenuated, thus leading to a more directive conical beam shape. The imaginary part of has a minor influence on the direction of maximum field, since it in (6) comes from only shifts the phase center (note that the GPOF process, so it can have both real and imaginary part). The phase distribution on the beam is determined by the term in front of the integral in (9). It is interesting to notice that

W on the imaginary part of complex parameter b. (a) W , at a distance r = 5 from the origin as a function

Fig. 2. Dependence of , (b) , (c) , (d) of for .

W



W W b=0

in the double sum given in (7), each -term ( th Fourier harmonic) contains complex exponents, which all generate wave

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objects of the same order , thus rendering the process particularly suited for circular apertures with a low number of azimuth harmonics. If, for example, the angular variation of the aperture field is such that only one harmonic is present, then all beams will be of that very order, regardless of the radial variation of the field. In fact, any radial field variation can be achieved by summing up wave objects of only one order. In an equivalent one-dimensional case, one could interpret as though each of these beams of the same order accounts for one of the local minima or maxima in the field distribution. The use of GPOF expansion procedure ensures that the minimal number of these terms is required, by . This is the basis of the generating best fitted coefficients efficiency of the exposed method. B. CCB’s Close to the Aperture We emphasize that, since the zero order beam is coincident with a conventional scalar spherical wave, its analytical continuation for complex value of coordinate leads to the same space-domain interpretation as that of the complex point source. solution exhibits a branch surFor imaginary ( real), face at with a singularity on the circle . The perturbation due to the singularity is highly concentrated around , hence moving by a small fraction of wavelength from a well behaved function. A similar situthe aperture makes ation occurs for the higher order terms , namely the same branch disc of even sharper singularity occurs. For increasing , the aperture field is less concentrated outside of the . aperture disc To illustrate this behavior, Fig. 3 shows the normalized am, as a function of , at displitude of tance from the aperture, and for different values of and . Fig. 3(a) and (b) show how the aperture field associated to the beam becomes wider for increasing and for increasing . This means that the far field distance becomes larger for larger . The far-field limit of the conical beams can be derived with some elaborate manipulations; however, since the far-field distance depends on the beam index, its consequence is that the beam’s asymptotic approximation becomes impractical, and is therefore omitted here. We do make a note, however, that despite apparent similarities between CCB’s and GL beams, the CCB’s paraxial asymptotic form is different from GL beams. As with the conventional complex source beam, the field is in the aperture plane . singular at the branch circle However, as discussed later, a displacement of the order of a fraction of wavelength is sufficient to gain full accuracy. When the objective is to represent the aperture field in the reactive relarger gion, the portion of the aperture field spectrum with than (invisible region) becomes significant; sometimes, the spectrum in the invisible region is not known and should be estimated by analytical continuation (for instance, in the case where the information is obtained from far field). To this end, one can use the same GPOF expansion in (6); in fact, although this approach does not allow the error control of the estimated analytical continuation it has been empirically found sufficiently accurate for observing the space field out of the reactive region of the aperture. This is consistent with the fact (observed in Section III)

=0 3 =5

Fig. 3. Aperture distribution as a function of = at distance z ;  from  (b). the aperture, for different value of n and for b  (a) and b

=

that we cannot observe the field too close to the aperture because of the condition . IV. COMPLEX CONICAL BEAMS-FORMULATION B The essential property of the conical beams introduced and elaborated in Formulation A, is that they are constructed naturally starting from the radiation integral (2), (3). However, a different formulation is possible, giving rise to another type of CCB’s. The modified formulation is less intuitive than the one in Section III, but results at the end in wave objects that are somewhat easier to handle and strictly related to the conventional spherical modes. It should however be pointed out that both types of wave objects satisfy the wave equation without restrictions, and that they can represent the radiated field equally well from very close distances to very far distances from the aperture, as will be shown in Section VI. With the starting aperture field spectrum (3) in mind, we mul. One of these two tiply and divide the integrand by factors is included in the Fourier expansion yielding the coeffidefined by (4). The subsequent GPOF expansion cients is -dependent (17) because (18)

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Fig. 4. Normalized amplitude of the origin as a function of  for b

,n

= 0.

= 0; . . . 4 at a distance r = 5 from

The initial radiation integral, as in the first case, reduces to the double sum (19) where , and the new wave objects assume a different form compared to (9), due to the presence in the integrand of the term

(20) A closed form solution to this integral can be constructed by using formulas in [25, Eqs. 6.737.5–6.737.6], yielding

(21) where kind and obtained as

is the spherical Hankel function of the second . The solution for negative orders is again

9

(22) One can also notice that the new wave objects denoted by constitute a special subset of the classical spherical harmonics when ; namely [26] (23) It is worth noting that for source point

, we again obtain a complex

(24) Fig. 4 shows normalized values of as a function of at the distance case

in the from the

9

on the imaginary part of complex parameter b. (a) Fig. 5. Dependence of , (b) , (c) , at a distance r  from the origin as a function of  for different values of the parameter b.

9

9

=5

origin. Contrary to what happens for the CCB’s in Formulation A (CCB’s-A) (see Fig. 1) the CCB’s-B do not exhibit any oscillations in for . However, they retain for increasing order . the increasing flatness around Similarly to what happens to the CCB’s-A, the complex displacement along the -coordinate actually causes the occurrence of the conical zone of maximum directivity in the radiation pattern of CCB’s-B. , The present CCB’s still possess singular behavior at . In contrast to the previously defined CCB’s, however, , they are significantly attenuated on the aperture plane for for real.

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As concluding remarks to this section, we note that the CCB’s in Formulation B have a closed-form representation in the spatial domain that does not imply any iterative computation. The resulting formulation is thus simpler than the formulation A in Section III and presents computational advantage once the GPOF coefficients have been calculated. On the other hand, unlike in Formulation A, the GPOF expansion requires attention . close to the origin V. VECTOR FIELD EXPANSION In order to expand a vector field into beams, it is necessary to determine how (3) relates to fields or vector potentials, de. pending on the choice of the spectrum The simplest approach is to define (26)

=0 2 =5

Fig. 6. Aperture distribution as a function of = at distance z ;  from  (b). the aperture, for different value of n and for b  (a) and b

=

Fig. 6 shows the aperture field associated to (the and . small shift serves to avoid the singularity), for The plots show that the CCB’s of type B for a given value of parameter remain confined to approximately the same surface area regardless the order . As a consequence, the far field region is well defined for each , and the asymptotic evaluation of beams can be used in practice. The far-field approximation of the CCB-B beams can be calculated as

where are the Cartesian tangential components of the electric fields; the remaining component of the EM field can be derived from spectral domain Maxwell’s equations. This turns (3) into the classical inverse Fourier-transform, whereby each is calculated from Cartesian field component in the plane . The above procedure, howits spectrum in the plane ever, implies a GPOF expansion of a function proportional to , necessitating both a fine GPOF sampling for and a higher number of exponential terms. An alternative way, that is revealed more convenient, is based on eliminating the square root in (26) via a -derivative. This is equivalent to assuming in (3) (27) and interpreting in (3) as the Cartesian spectral components of the vector potential associated to twice the equivalent magnetic current at the aperture. This leads to [29]: (28) where coefficients

and

and the are obtained by CCB expansion of

(25) A special remark is concerned with Formulation B. In (18), the , possesses function to be expanded by GPOF, denoted by apparently a th order pole for . Actually, it turns out is analytical at , its Fourier coefficients that, if have a th order zero at . A rigorous demonstration of this general property can be found in [27]. In our case, is regular at . This th order zero this implies that cancels the introduced th order pole, and the resulting function is well-behaved in the whole visible spectrum region. However, for numerical errors this cancellation can be critical, therefore in the practical implementation it is necessary to in the GPOF expansion. exclude the origin

Finally, we obtain (29)

(30) where the positive sign in (29) is related to -component and the negative sign to the -component, respectively.

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Concerning Formulation B, the expressions are identical with . Explicit expressions for the derivathe substitutions tives are given in Appendix B. VI. EXAMPLES OF APERTURE FIELD EXPANSION The described approach is applied here to calculate the radiation from both circular and rectangular apertures. A. Circular Aperture Let us consider a [29]

circular waveguide mode defined by

(31) and are the th order Bessel function and its where derivative, respectively, and , where is the is the radius of the circular waveguide (i.e. aperture), and th zero of the th order Bessel function. This particular aperture field distribution has been selected because it possesses closed form expressions in both spectral and spatial domain, therefore allowing a simple construction of the reference solution. The Cartesian components of the aperture E-field spectrum can be found in closed form as

Fig. 7. (a) E-plane ( = 0) and H-plane ( = 90 ) cuts of the radiated total electric field magnitude by a TM circular waveguide mode, sampled on a sphere of radius 300; (b) corresponding field representation error.

(32) illuminated by a An aperture with radius mode has been considered. For this aperture size, this particular mode is the closest one to cut-off, i.e., it possesses the eigenclosest to the visible region boundary; this results in a value fast aperture field variation. The spectrum is sampled only in the visible spectrum region. The aperture field in (32) is described by 4 harmonics of the Fourier expansion (4). A total of 64 and 52 CCB’s have been automatically generated by the GPOF procedure to represent the field in terms of W and , respectively. Figs. 7(a), 8(a) present successful comparative results obtained using both formulations A and B; the reference solution is provided by the direct integration of (2). Figs. 7(b) and 8(b) present the absolute errors between normalized fields. In particular, Fig. 7(a) refers to the total electric field radiated in both and cut-planes sampled on a sphere of radius ; Fig. 8(a) shows the vertical component of the electric by plane parallel to the aperture field calculated on a distance. at It can be noticed that the number of CCB’s obtained in the field expansion for both formulations is significantly lower than . the number of degrees of freedom defined by (1)

This latter aspect can be explained by the natural aptness of CCB’s for describing cylindrical apertures and by the fact that GPOF procedure adapts the waist and direction of each beam to the specific aperture field being expanded. B. Rectangular Aperture The cylindrical nature of CCB’s provides them with a natural ability to describe a circular aperture field. However, we demonstrate here that they can also represent aperture fields that do not posses any cylindrical symmetry. To this end, a rectangular aperture of dimensions is considered, with uniform aperture field amplitude and linear , . phase, such as to radiate the main beam toward After the Fourier series expansion (4), using an energy-based selection criteria, 39 azimuthal-harmonics have been considered and used to generate in total 181 CCB-A beams and 99 CCB-B beams, respectively. Figs. 9(a), 10(a) present comparative results between the electric field obtained using both formulations A and B and the reference solution provided by the direct integration of (2). The comparisons are presented in the most significant cutand refer respectively to the total electric field at plane and to the vertical component of the field sampled on

ˇ ´ et al.: COMPLEX CONICAL BEAMS FOR APERTURE FIELD REPRESENTATIONS SKOKIC

Fig. 8. (a) Vertical component of the E-field radiated by a TM circular waveguide mode calculated in a plane at a distance of 5 from the aperture plane; (b) corresponding field representation error.

a rectilinear scan plane parallel to the aperture plane at . The small absolute errors shown in Figs. 9(b) and 10(b) confirm the excellent capability of CCB’s to treat even rectangular phased aperture radiation. In order to make a comparison between the CCBs and a standard beam expansion, we have considered a well collimated aperture case constituted by the rectangular aperture presented above with uniform aperture field (both phase and amplitude). , In such a case, the radiated main beam will be toward . Since we are representing a rectangular domain aperture field, a Gauss-Hermite (GH) expansion has been performed. All GH beams are characterized by the same complex exponential parameter that controls the beamwidth and the wavefront curvature, which was chosen according to criteria [21] based on matching the properties of the aperture field. In the CCB expansion, on the other hand, each beam is identified by its own “opwhich is extracted timal” complex exponential parameter from the data using GPoF. A total of 91 CCB-A beams and 58 CCB-B beams have been obtained, respectively. Figs. 11(a) and (b) present a comparison between the electric field obtained using the Gauss-Hermite expansion, CCB type A and B and the reference solution provided by the direct integration, in two principal planes. The results provided by the CCB expansion agree perfectly with the reference solution for every observation angle, whereas

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Fig. 9. (a) E-plane ( = 0) cuts of the total electric field magnitude radiated by a 5 2 phased rectangular aperture, sampled on a sphere of radius 300; (b) corresponding field representation error.

2

the results obtained with GH are correct only in a paraxial region. Even by increasing the number of GH beams used in the expansion (from 225 to 3600) it is still not possible to improve the accuracy of that result. This is a direct consequence of the fact that the GH beams are only a paraxial solution of the wave equation. VII. CONCLUSIONS A novel method of calculating aperture radiated fields via complex conical beams has been developed. The beams are obtained starting from a standard spectral domain radiation integral, upon applying GPOF expansion to the angular spectral Fourier coefficients of the electric field spectrum in the aperture plane. These manipulations allow reducing the initial double integral to a double sum of conical wave objects, whose analytical solution has been presented in this paper. If the aperture distribution is collimated, the GPoF will automatically give rise to collimated CCB’s, which have beamwidth and slope similar to the Gauss-Laguerre beams. The numvary from bers of beams M and the complex displacement case to case without detectable rules: they are “automatically” chosen by the GPOF spectral matching. If on one hand this implies that our set does not constitute an a priori set of orthogonal beams, on the other hand it actually constitute a point of force of the method, because no a priori parameterization is required.

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2

Fig. 10. (a) Vertical component of the E-field radiated by a 5 2 rectangular aperture field in a plane at a distance of 5 from the aperture plane; (b) corresponding field representation error.

Fig. 11. (a) E-plane ( = 0) and (b) H-plane ( = 90) cuts of the total electric field magnitude radiated by a 5 2 phased rectangular aperture, sampled on a sphere of radius 300.

2

APPENDIX A We have introduced two new kinds of conical beams denoted by A and B. Both kinds of beams exactly satisfy the wave equation. Moreover, constructing the vector solution through the use of auxiliary vector potentials, leads to vector forms that satisfy Maxwell’s equations. Both types of beams have closed form expressions in both space and spectral domains, and their formulations are found to be very accurate for expanding any aperture field into beams (not only circular aperture but also rectangular apertures). Formulation B offers the additional advantage of simplicity. On the other hand, Formulation A offers more stable numerical fitting with the GPOF expansion. For representation outside the aperture field reactive region, Formulation B requires a lower number of beams than Formulation A, and normally well below the number of degrees of freedom of the aperture. This is due to the GPOF process, which automatically optimizes the combination of the CCB’s spectra to any a-priori known field spectrum. However, an a priori choice of beams (namely of a set of complex displacements) would not imply the use of the same number of beams for an unknown aperture field. Presently, we are investigating about the possibility to select, on the basis of the sole information of the aperture geometry, an a displacements for obtaining a non-repriory set of complex dundant (namely equal to the degrees of freedom) number of beams.

From [25, Eq. 6.637], we have

(A1) where is a Bessel function of the first kind and th order, while and are modified Bessel functions of the first . Substituting for and second kind, respectively, and order , , , , and , we get

(A2)

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where and . The only difference between (A2) and (9) is the additional factor in the integrand, which is easily introduced by differentiating (A2) with respect to . By using the properties of the Bessel functions, this leads to

(12) and the where the Bessel functions are evaluated at (13). In order to define the recurrence, Hankel functions at the explicit derivatives for the first two terms of the recurrence, and are namely needed. These are given as follows:

(A3) , the following recurrence relation is

(B11)

Multiplying (A3) by obtained

(B12) (A4)

Equation (11) is obtained by calculating analytically the -derivative of the right hand side of (A2).

(B13) where

APPENDIX B

(B14)

The recurrence relation can be rewritten as in (A4) and differentiated with respect to (B1) where (we introduce to simplify the notation and suppress the subscript). Differentiating with respect to gives

(B15) As for Formulation B, one has

(B16)

(B2) The derivative with respect to is formally identical upon inter. After tedious, but straightforward algebraic changing manipulation, one obtains (B3) (B4) where

(B5)

(B6)

(B7) (B8)

(B9)

(B10)

(B17)

ACKNOWLEDGMENT The authors wish to acknowledge M. Sabbadini of ESA ESTEC for useful suggestions during the project development. The authors also thank an anonymous reviewer for the useful comments that have improved the quality of the paper. REFERENCES [1] T. K. Sarkar and O. Pereira, “Using the matrix pencil method to estimate the parameters of a sum of complex exponentials,” IEEE Antennas Propag.Mag., vol. 37, no. 1, pp. 48–55, Feb. 1995. [2] O. M. Bucci and G. Franceschetti, “On the degrees of freedom of scattered fields,” IEEE Trans. Antennas Propag., vol. 37, no. 7, pp. 918–926, Jul. 1989. [3] J. M. Arnold, “Phase-space localization and discrete representation of wave fields,” J. Opt. Soc. Am. A, vol. 12, no. 1, pp. 111–123, Jan. 1995. [4] G. A. Deschamps, “The Gaussian beam as a bundle of complex rays,” Electron. Lett., vol. 7, no. 23, pp. 684–685, 1971. [5] N. J. McEwan and P. F. Goldsmith, “Gaussian beam techniques for illuminating reflector antennas,” IEEE Trans. Antennas Propag., vol. 37, no. 3, pp. 297–304, Mar. 1989. [6] H.-T. Chou, P. H. Pathak, and R. J. Burkholder, “Application of Gaussian-ray basis functions for the rapid analysis of electromagnetic radiation from reflector antennas,” IEE Proc. Microw. Antennas Propag., vol. 150, pp. 177–183, 2003. [7] H.-T. Chou and P. H. Pathak, “Uniform asymptotic solution for electromagnetic reflection and diffraction of an arbitrary Gaussian beam by a smooth surface with an edge,” Radio Sci., vol. 32, no. 4, pp. 1319–1336, Jul./Aug. 1997.

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[8] W. A. Imbriale and D. J. Hoppe, “Recent trends in the analysis of quasioptical systems,” presented at the Millenium Conf. on Antennas Propag., Davos, Switzerland, 2000. [9] S. Withington, J. A. Murphy, and K. G. Isaak, “Representation of mirrors in beam waveguides as inclined phase-transforming surfaces,” Infrared Phys. Technol., vol. 36, no. 3, pp. 723–734, Apr. 1995. [10] L. B. Felsen, “Complex-source-point solutions of the field equations and their relation to the propagation and scattering of Gaussian beams,” in Proc. Symp. Math, 1975, vol. 18, pp. 39–56. [11] Y. Dezhong, “Complex source representation of time harmonic radiation from a plane aperture,” IEEE Trans. Antennas Propag., vol. 43, no. 7, pp. 720–723, Jul. 1995. [12] A. Polemi, G. Carluccio, M. Albani, A. Toccafondi, and S. Maci, “Incremental theory of diffraction for complex point source illumination,” Radio Sci., vol. 42, 2007. [13] Y. L. Chow, J. J. Yang, D. G. Fang, and G. E. Howard, “A closedform spatial Green function for the microstrip substrate,” IEEE Trans. Microw. Theory Tech., vol. 39, no. 3, pp. 588–592, Mar. 1991. [14] J. He, T. Yu, N. Geng, and L. Carin, “Method of moments analysis of electromagnetic scattering from a general three-dimensional dielectric target embedded in a multilayered medium,” Radio Sci, vol. 35, no. 2, pp. 305–313, 2000. [15] F. Vipiana, A. Polemi, S. Maci, and G. Vecchi, “A mesh-adapted closed-form regular kernel for 3D singular integral equations,” IEEE Trans. Antennas Propag., vol. 56, no. 6, pp. 1687–1698, Jun. 2008. [16] , H. E. Hernàndez-Figueroa, M. Zamboni-Rached, and E. Recami, Eds., Localized Waves. Hoboken, NJ: Wiley-Interscience, 2008, ch. 6. [17] B. Z. Steinberg, E. Heyman, and L. B. Felsen, “Phase space methods for radiation from large apertures,” Radio Sci., vol. 26, pp. 219–227, 1991. [18] A. Shlivinski, E. Heyman, A. Boag, and C. Letrou, “A phase-space beam summation formulation for ultrawideband radiation: A multiband scheme,” IEEE Trans. Antennas Propag., vol. 52, no. 8, pp. 2042–2056, Aug. 2005. [19] A. Shlivinski, E. Heyman, and A. Boag, “A phase-space beam summation formulation for ultrawideband radiation—Part II: A multiband scheme,” IEEE Trans. Antennas Propag., vol. 53, no. 3, pp. 948–957, Mar. 2005. [20] A. Shlivinski, E. Heyman, and A. Boag, “A pulsed beam summation formulation for short pulse radiation based on windowed radon transform (WRT) frames,” IEEE Trans. Antennas Propag., vol. 53, no. 9, pp. 3030–3048, Sep. 2005. [21] E. Heyman and I. Beracha, “Complex multipole pulsed beams and Hermite pulsed beams: A novel expansion scheme for transient radiation from well-collimated apertures,” J. Opt. Soc. Am., vol. A 9, pp. 1779–1793, 1992. [22] M. Katsav and E. Heyman, “A beam summation representation for 3-D radiation from a line source distribution,” IEEE Trans. Antennas Propag., vol. 56, no. 2, pp. 602–605, Feb. 2008. [23] L. B. Felsen and N. Marcuwitz, Radiation and Scattering of Waves. New York: Wiley-IEEE Press, 1994. ˇ ´ , “Analysis of Reflector Antenna Systems by Means of New [24] S. Skokic Conical Wave Objects,” Ph.D. dissertation, University of Zagreb, Croatia, 2010. [25] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products. London: Academic Press, 1994. [26] J. A. Stratton, Electromagnetic Theory. Piscataway, NJ: Wiley-IEEE Press, 2007. [27] J. P. Boyd, Chebyshev and Fourier Spectrum Methods, 2nd ed. New York: Dover, 2001. [28] A. Sommerfeld, Partial Differentia Equations in Physics. New York: Academic Press, 1964. [29] C. A. Balanis, Advanced Electromagnetic Engineering. New York: Wiley, 1989, pp. 470–482.

of Antennas. Currently he is a Research Assistant at the Department of Wireless Communications, Faculty of Electrical Engineering and Computing, University of Zagreb. His research focuses on numerical methods in electromagnetic with applications to curved antennas and reflector systems. ˇ ´ was awarded with the Best Antenna Poster Paper Prize at the 3rd Dr. Skokic European Conference on Antennas and Propagation EuCap-2009, Berlin, Germany, in March 2009 and an Honorable Mention for Antenna Theory at the 4th European Conference on Antenna and Propagation EuCap-2010, Barcelona, Spain, in March 2009.

ˇ ´ was born in Zagreb, Croatia, in 1978. Sinisa Skokic He received the M.Sc. and Ph.D. degrees in electrical engineering from the University of Zagreb, in 2005 and 2010, respectively. In 2004 and 2007, he was a Visiting Researcher at the University of Siena, Italy, working on the development of new numerical methods for analyzing curved reflector systems and frequency selective radomes. He has also been actively involved in the organization of several Ph.D. courses within the ACE/MCA Doctoral Programme European School

Stig Busk Sørensen was born in Denmark, in 1958. He received the M.Sc. degree and the Industrial Research Education degree from the Technical University of Denmark, Lyngby, in 1982 and 1985, respectively. Since 1983, he has been with TICRA, Copenhagen, Denmark, where he has worked with analysis and synthesis of reflector antennas. He has special experience with numerical techniques for optimization and analysis of shaped reflector antennas.

Massimiliano Casaletti was born in Siena, Italy, in 1975. He received the Laurea degree in telecommunications engineering and the Ph.D. degree in information engineering from the University of Siena, Italy, in 2003 and 2007, respectively. From September 2003 to October 2005, he was with the research center MOTHESIM, Les Plessis Robinson (Paris, FR), under EU grant RTN-AMPER (RTN: Research Training Network, AMPER: Application of Multiparameter Polarimetry). Since 2006, he has been Research Associate at the University of Siena, Italy. His research interests include electromagnetic band-gap structures, polarimetric radar, rough surfaces and numerical methods for electromagnetic scattering and beam waveguides. Dr. Casaletti was awarded the Best Antenna Poster Paper Prize at the 3rd European Conference on Antennas and Propagation EuCap-2009, Berlin, Germany, in March 2009 and an Honorable Mention for Antenna Theory at the 4th European Conference on Antenna and Propagation EuCap-2010, Barcelona, Spain, in April 2010.

Stefano Maci (S’98–F’04) received the Laurea degree (cum laude) in Electronic engineering from the University of Florence, Italy. Since 1998, he is with the University of Siena, Italy, where he presently is a Full Professor. His research interests include EM theory, antennas, high-frequency methods, computational electromagnetics, and metamaterials. He was a coauthor of an Incremental Theory of Diffraction for the description of a wide class of electromagnetic scattering phenomena at high frequency, and of a diffraction theory for the analysis of large truncated periodic structures. He was responsible and international coordinator of several research projects funded by the European Union (EU), by the European Space Agency (ESA-ESTEC), by the European Defence Agency, and by various European industries. He was the founder and presently is the Director of the European School of Antennas (ESoA), a postgraduate school that comprises 30 courses on antennas, propagation, and EM modeling though by 150 teachers coming from 30 European research centers. He is principal author or coauthor of more than 100 papers published in international journals, (among which 60 on IEEE journals), 10 book chapters, and about 350 papers in proceedings of international conferences. Prof. Maci was Associate Editor of the IEEE TRANSACTIONS ON EMC, two times Guest Editor of IEEE TRANSACTION ON ANTENNAS AND PROPAGATION (IEEE-TAP), Associate Editor of IEEE-TAP. He is presently a member of the IEEE AP-Society AdCom, a member of the Board of Directors of the European Association on Antennas and Propagation (EuRAAP), a member of the Technical Advisory Board of the URSI Commission B, a member of the Italian Society of Electromagnetism and of the Advisory Board of the Italian Ph.D. school of Electromagnetism. He was the recipient of several national and international prizes and best paper awards.

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Simulation and Measurement of Dynamic On-Body Communication Channels Michele Gallo, Member, IEEE, Peter S. Hall, Fellow, IEEE, Qiang Bai, Yuriy I. Nechayev, Member, IEEE, Costas C. Constantinou, Member, IEEE, and Michele Bozzetti

Abstract—A study is presented of wireless on-body communication links, with the body in motion. This paper compares simulations and measurements of the path gain (PG) on moving male and female bodies. Simulations using avatars derived from an animation software have been performed at 2.45 GHz. Male and female avatars walking and a male rising from a chair have been simulated and the results are compared with measurements carried out in an anechoic chamber. The results show that good agreement between simulation and measurement of slow fading features can be achieved for a reasonable computational effort. Index Terms—Body area networks, computer simulations, on-body propagation channel, wireless communication systems.

I. INTRODUCTION HE growing miniaturization of electronic devices combined with the recent developments in wearable computer technology have been leading to the creation of a wide range of devices which can be carried in a pocket or attached to a user’s body [1]. These applications have allowed the removal of the need for wired interconnections and have led to the rise of the concept of the wireless body area network (WBAN) [2], [3]. Propagation channels may extend from the body to a local base station or between two antennas on the body. In this work the on-body channels are considered, although the conclusions drawn may also be applicable to off-body communication links with a local base station. In either case, a full understanding of the channel is vital for the design of very low power transceivers and associated baseband processing, which maximize the battery life of body worn equipment. WBAN channel characterization has been performed using measurements on human subjects

T

Manuscript received July 15, 2009; revised June 30, 2010; accepted August 03, 2010. Date of publication November 18, 2010; date of current version February 02, 2011. This work was supported by EPSRC Grant EP/E029922/1. M. Gallo was with the Communications Engineering Group, University of Birmingham, Birmingham B15 2TT, U.K., on leave from the Department of Electrotechnical and Electronic, Polytechnic of Bari, Bari, Italy. He is now with the Department of Electrotechnical and Electronic, Polytechnic of Bari, Bari, Italy (e-mail: [email protected]). P. S. Hall, Y. I. Nechayev, and C. C. Constantinou are with the School of Electronic, Electrical and Computer Engineering, University of Birmingham, Birmingham B15 2TT, U.K. (e-mail: [email protected]; [email protected]; [email protected]). Q. Bai is with the Department of Electronic and Electrical Engineering, University of Sheffield, Sheffield S1 3JD, U.K. (e-mail: [email protected]). M. Bozzetti is with the Department of Electrotechnical and Electronic, Polytechnic of Bari, Bari, Italy (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2093498

performing a number of common activities [4], [5]. Movement of the body in a scattering environment causes channel fading. This has been characterized into long and short term fading. Long term fading is due to relative movement of the various body parts which leads to shadowing of the antennas by the body parts as well as changes of the path length and of the relative orientation of the antennas (thus changing the antenna gain). Short term fading is due to multipath interference effects, where multipath may be present due to the different signal paths around the body and from scattering (or multiple reflections) of signal power off the local environment. Simulation has the potential to allow channel characterization and hence reduce the time and cost of system design. However, its application to WBANs is complicated by the need to model body movement, which plays an important part in the channel behavior. There are two issues. Firstly, there is a need to get a body phantom which can assume the relevant posture of the human body during its motions. Secondly, the simulation of microwave antennas on the body is a multi-scale problem requiring long durations of computational time. Much work on the latter problem has been performed. Finite difference time domain (FDTD) and finite element method (FEM) have been used for simulation of whole bodies with wearable antennas [6], for a single or a few stationary postures. Advanced computational methods have been used to improve performance, including conformal grid methods [7], and equivalent source models [8]. Various numerical phantoms are available, including the Taro and Hanako Japanese human voxel models [9], or Norman human model [10]. Several software packages used to manipulate posture are also available [11], [12]. However, extraction of channel fading characteristics requires modeling of small changes in body posture as it moves. The sampling rate of the motion is determined based on whether it is desired to extract long term or short term fading information. To the authors' knowledge, the modeling of small changes in body posture to characterize on-body communication channels has not been demonstrated. It is the objective of this paper to demonstrate the feasibility of an approach that combines the use of animation software with conventional electromagnetic solvers to allow long term channel fading to be characterized [13]. The method is, in principle, extendable to short term fading characterization. The simulation methodology has been verified by comparison with measurements. The achieved results are found to depend on the quality of the phantom that has been utilized. A comparison of the transmission path gains (PG) for static postures of different types of phantom are presented. The results show that using a crude geometry for the phantom

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It is clear that a monopole-like antenna is suitable for many on-body links, having an omni-directional pattern, which obviates the need for directional orientation. In addition, polarization perpendicular to the body surface provides optimum PG. However, the monopole is less compact and rugged than other antenna types, such as printed inverted-F antennas, planar inverted-F antennas, patches, etc. [4]. Results for static postures using printed inverted F antennas are provided in Section III. B. Simulations

Fig. 1. Avatars used in XFDTD simulations. This posture is one frame out of 30 available which was obtained from POSER 6; the monopole on the belt is the transmitting antenna and the others are receiving antennas.

is sufficient to improve computational efficiency although care should be exercised to correctly model large features of the body. II. STATISTICS OF CHANNELS ON A MOVING HUMAN BODY A. Simulated Scenarios A generic body-centric wireless network consists of several nodes placed on a body. The design of wearable antennas that can be integrated into these nodes is a critical consideration in a WBAN because they can suffer from reduced efficiency, radiation pattern fragmentation and variations in impedance at the feeds. Investigated antenna channels have been chosen based on the locations of commonly used on-body communication devices, such as head-mounted displays, headsets, wristwatches, was placed on the belt in the etc. The transmitting antenna middle of the WBAN which is representative of where a mobile phone might be worn. In movement simulations, two receiving antennas were mounted on the wrist, representative of a device inside a wristwatch and on the ankle, giving a channel similar to the Nike iPoD device [14], as is shown in Fig. 1. Three different cases have been investigated: a walking movement performed by a man, a walking movement performed by a woman, and a standing up movement from a chair. The behavior of the on-body channels depends on the parts of the body at which the transmitting and receiving antennas are attached and on the conducted activity. For example, a trunk-to-limb link is expected to be subject to significant fading due to the movement of the limb while a trunk-to-trunk link is more stable. Simulations and measurements of the dynamic body channels have been carried out at a frequency of 2.45 GHz, which is within the unlicensed band used by Bluetooth systems. The quarter-wavelength monopole antennas used in this study had a length of 28 mm and a wire diameter of 1.1 mm, located above a 78 mm by 78 mm square ground plane. The antennas were spaced 25 mm away from the surface of the body, which although greater than what might be used in practice, reduces the sensitivity of the characteristics of the antennas to proximity of the body.

In order to define the shape of the body phantom, we have used the animation software POSER 6 from E-Frontier [15]. POSER can create male or female avatars, of various sizes and shapes, and is designed specifically to enable realization of a realistic motion of the avatar, including walking which was used in this work. Various other objects, such as floors, walls and furniture can also be created. The objects, including the avatar and the environment items, can be exported using widely supported formats. The animation of the movement can be stored in the form of a movie, with the geometry of each frame of the movie specified in a separate file. Movies of male and female avatars, called James and Jessica, respectively, were prepared in this way and exported into the electromagnetic simulation program XFDTD from REMCOM [16]. Within XFDTD phantom material properties were assigned and antennas added. Simulation of the channel PG was then performed. Details of the process are as follows. Avatar geometries corresponding to each of the simulated postures during their movement were created in POSER and exported into XFDTD as “dxf” files. In XFDTD, the avatars were then filled, forming a homogeneous phantom, using the parameters of skin tissue defined in the human voxel model supplied with XFDTD, i.e., the relative permittivity of 37.05 and the conductivity of 1.31 S/m. The monopole antennas were added and positioned manually, as the location of the belt, wrist or ankle in each frame were difficult to extract. Simulations of the scattering parameter between the ports of the transmitting and receiving antennas were then run using a Gaussian pulse source with a convergence dB (the level relative to the peak value of the threshold of total energy in the simulation space to be reached before simulation completes). Fig. 1 shows the male and female avatars with the antennas, during their walking movement. The avatar sizes were selected to match those of the subjects used in the measurements described in the following section. In each movement, 30 frames (postures) were simulated. In the walking activity, shown in Fig. 2(a), these frames corresponded to the period of one walking pace (2 steps) starting with the right foot and the left hand forward. In the standing up movement, shown in Fig. 2(b), the period was the elapsed time it took to move up from a seated posture to a standing position. The floor was not imported from POSER, but it was added in XFDTD as a perfectly conducting sheet. However, the chair and coffee table shown in Fig. 2(b) were exported from POSER and were assigned the properties of a perfect electric conductor (PEC) in XFDTD. This provides for a good example of a worst case scenario when there is a strong scattering object in the

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Fig. 2. Simulated body movements: (a) avatar Jessica walking, (b) avatar James standing up.

vicinity of the human body, and can be simulated much faster than any real materials with finite conductivity. C. Measurements Measurements were carried out to allow evaluation of the suitability of the simulation process in predicting the effects of the movement of the body on the PG. The test subject (male or female) was wearing the transmitting antenna (Tx) and the receiving antenna (Rx) as shown in Fig. 1. The measurements were performed in a semi-anechoic chamber, in which, there are no absorbers on the metallic floor. The antennas were connected paramto an HP 8720C vector network analyzer and the eter was measured as the test subjects performed a movement in the chamber. The power fed to the transmit antenna was approximately 0 dBm. The whole-body specific absorption ratio (SAR) was calculated using XFDTD and was found to be 1.2 mW/kg, which is well below the safety limit of 0.08 W/kg specified in [18]. In each of the three scenarios studied, each movement (walk or standing up) was repeated 60 times and lasted parameter was sampled 2 s each, during which time the every 10 ms. The dimensions of the test subjects were as folcm and cm for the man, and lows: cm and cm for the woman. The primary problem encountered was that of ensuring that the movements of the test subject and the avatar are similar and synchronous. In order to maintain synchronism between the avatar's and subject's movements, the test subject had to start the movement with the same posture as the first posture of the avatar and had to maintain the same walking rhythm. Due to the difficulty in satisfying these requirements in practice, the postures of the moving test subject did not always match the corresponding postures of the avatar at the same time instant. This problem was overcome by recording the video of the subject's movement and comparing each POSER frame with the video snapshots. The measured data points concurrent with the snapshots that best matched each simulated frame were then used for comparison with the simulations of that frame. This allowed for a correction to be made to the systematic error due to the lack of synchronism. A complete resemblance between the avatar in the POSER frames and the body of the subject in the measurements was not possible to achieve because the limbs were not perfectly coordinated. Therefore, for each frame, the snapshot with the positions of the arms of the avatar that best matched that of the subject was chosen when the belt-to-wrist channel was considered. Likewise, the snapshot with the positions of the legs of the avatar that best matched that of the subject was

Fig. 3. Simulated and measured S a walking man.

for the left-belt-to-right-wrist channel for

Fig. 4. Simulated and measured S a walking man.

for the left-belt-to-right-ankle channel for

chosen for the case of the belt to-ankle channel. The procedure for matching the simulation frames to the measurement points has been described in detail in [13]. The matching and averaging process was highly time intensive, taking approximately 2 days per frame. Therefore, it was not carried out for every frame but for a number of frames sufficiently large to assess the quality of the simulations. It is estimated that, in this matching process, the differences between the positions of the human body and the avatar were of the order of 5 cm, primarily due to the difficulty of controlling the posture of the subject. Due to this slight possible mismatch, five adjacent data samples centered on the instance that better matched the simulated and measured frames were taken. For each simulated frame, on average, 40 such matched instances were found out of the 60 available data sets. Finally, these (approximately) 40 5 values were averaged to give the triangle points displayed in Figs. 3–8, that show the simulated and meavalues in dB for the two studied channels at 2.45 GHz sured in the three considered scenarios. The error bars around the measured points represent the spread of the (approx. 40 5) measured values assigned to

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Fig. 5. Simulated and measured S a walking woman.

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for the left-belt-to-right-wrist channel for

Fig. 8. Simulated and measured S the standing up movement.

for the left-belt-to-right-ankle channel for

exactly match the simulations. The average difference between measured and simulated data is equal to 5 dB. D. Analysis of the Results

Fig. 6. Simulated and measured S a walking woman.

for the left-belt-to-right-ankle channel for

Fig. 7. Simulated and measured S the standing up movement.

for the left-belt-to-right-wrist channel for

each simulated frame by the video snapshot matching. They demonstrate the difficulty of performing measurements that

For a stationary body, the PG has been shown to be determined primarily by the path length in addition to antenna orientation and shadowing [17], [19], [20]. Figs. 3–8 show variation of the PG with avatar's movement. Fading greater than 15 dB is observed between the 12th and 19th frames of the belt-to-wrist channel of Fig. 3, due to the limb shadowing encountered during the movement. Indeed, the walk starts with the right foot slightly forward and the arms close to the body so that, between the 12th and 19th frame, the right wrist is shadowed by the body. Fig. 4 shows that fading of over 25 dB occurs in the belt-to-ankle channel, caused by the high mobility of the legs and by the scattering from the right arm and from the knees. Furthermore, the strongest fading of this channel is probably due not only to the leg movement but also to the multipath propagation caused by signal reflections from the floor. Figs. 5 and 6 show that the PG in the woman's walk has a variation of up to 20 dB for both the simulated and measured data. This change is again caused by the movement of the arms that can significantly affect the mean path. For example, in Fig. 5, at instances 6 to 10 corresponding to the arm moving into the dB to dB shadow, a drop of the received power from is recorded. The error bars show power fluctuations of about dB from the median value. Table I shows the mean values of the PG (expressed in dB) and its standard deviations (STD) for the male and female walking. The mean and the STD in the table were calculated for: 1) the 30 simulated frames; 2) all of the measured data for the 60 walks; and 3) the selected measured data for which the synchronism between the subjects and the avatars were maintained. The results show that the difference between the mean values of the simulated and measured PGs is within 3–4 dB. The standard deviation, being within 1.5 dB, indicates that the spreads of the simulated and measured data are similar. One of the primary difficulties of this approach has been making the body used in the measurements take the posture of

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TABLE I MEAN PG AND STANDARD DEVIATION FOR FOUR ON-BODY CHANNELS DURING WALKING

Fig. 9. Three phantoms: (a) James, (b) JamesS, (c) Mike.

that WBAN designers can use these values as a benchmark to set the transmitted power (within accepted safety limits) in order to avoid drops of signal, to optimize the channel and to prolong the battery life of the on-body worn devices. III. EFFECT OF PHANTOM SHAPE AND ANTENNA TYPE ON SIMULATION ACCURACY the simulation. Nevertheless, it is important to point out that the results (mean and standard deviation) derived from all the measured data (2nd rows in Table I) are very close to those derived from the data chosen by comparing every frame to the video (3rd rows). This suggest that complete similarity of the walking styles is not necessary for accurate predictions of the PG statistics using simulations. Table I also shows that, even if they execute the same movement under the same environmental conditions, the PG statistics for the male and female bodies are different. A difference of the (wrist) mean PG within 9 dB is observed for the receivers while a difference within 4 dB is shown for the receivers (ankle). Moreover, the STD is higher in the belt-to-wrist channel than in the belt-to-ankle channel for James and vice versa for Jessica. These differences are believed to be due to the different shapes of the body, walking styles and distances between the antennas. In Figs. 7 and 8, the PG of the standing up movement is shown when the antennas are mounted on the right wrist and the right ankle respectively. In this movement, the belt-to-wrist channel is not affected by deep fading, because during the movement a transition from shadowing to line of sight scenario occurs at about frame 18, as the body stands up. It was found that accurate repetition of this movement was much more difficult for the subject. Thus, repeatability of the measurements became worse, and, as a consequence, the error bars are bigger and the agreement of the simulated PG with the measured averages also deteriorates. Figs. 3–8 show that significant errors can occur if simulations are used to predict PG for single postures, primarily, it is assumed, due to the differences between the simulated and actual posture. However, the mean PG and its STD show good agreement, over what are relatively limited posture changes. Thus, simulation is seen to be useful for extraction of long term statistics. This conclusion is extremely important because this means

A. Phantom Shape In the previous section, the simulations were performed on a phantom with a realistic body shape. This phantom features a great deal of small detail. Creating and manipulating such phantoms is complicated and requires use of specialized software. Subsequent simulations of them can be extremely time consuming, particularly if the geometry mesh is sufficiently small to reproduce the small features of the body shape accurately. Therefore a less detailed body shape of a phantom is desirable provided it does not lead to significant errors. Therefore, in addition to James, Fig. 9(a), two other phantoms were used to simulate three on-body channels, and the results were then compared with the measurements, for single standing postures. The first of these phantoms, JamesS, Fig. 9(b), was derived by smoothing out fine details of James. This is provided by POSER, exporting it as a POSER 4 avatar. The other phantom, Mike, Fig. 9(c), was constructed from simple geometrical shapes joined together and rounded to avoid sharp edges. Mike was constructed in Microwave Studio CST software and involves 17 articulation variables at the joints. It is possible to change the variables between successive simulations to create movement. The three channels investigated included the transmitting antenna on the belt, and the receiving antennas on the head, wrist and ankle, as shown in Fig. 9. The PG for each on-body channel was measured in an anechoic chamber by using a vector network analyzer. The subject remained standing still on an absorbing platform during the measurement. A polystyrene tube with 30 mm thickness was used to separate the antenna from the body surface. Two identical monopoles, used as the transmitter and receiver, were connected to the network analyzer with 5 meter long coaxial cables. reFor each channel measurement spanning 10 seconds, the sponse was measured every 0.05 second. The final result is the average of all 201 sampled values.

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TABLE II MEASURED AND SIMULATED PATH GAINS (DB) FOR THREE ON-BODY CHANNELS USING MONOPOLE ANTENNAS

Table II compares the simulated PGs, when monopole antennas are used, with the measurements. With the exception of the head channel simulation on Mike, there is good agreement between all the simulations and measurements. This demonstrates that even a relatively simple phantom can provide acceptable accuracy of the simulated results. The error in the head channel simulated on Mike is believed to be due to the unnatural shape of shoulders and neck in Mike, which can be improved by performing additional modeling of the phantom.

Fig. 10. Inverted-F antenna: (a) front view, (b) back view.

TABLE III MEASURED AND SIMULATED PATH GAINS (DB) FOR THREE ON-BODY CHANNELS WITH INVERTED-F ANTENNAS

B. Antenna Type Section III.A shows that good accuracy of simulated PGs is achieved when monopole antennas are used on the body. These antennas have relatively large ground planes and were placed far from the body surface, thus they were not significantly affected by the body presence. In order to study a more realistic scenario, the three on-body channels shown in Fig. 9 were also simulated with inverted-F antennas (IFA) shown in Fig. 10 [21]. Each antenna is printed on a 30 30 mm piece of FR4 substrate of 0.8 mm thickness. Simulations with CST Microwave Studio showed that these IFAs are tuned at 2.45 GHz with an 8% impedance bandwidth. The simulated total efficiency and directivity are 76% and 2.2 dBi, respectively. These antennas are not shielded from the body by a ground plane and therefore are subject to detuning and extra losses when placed near a human body. However, the separation from the body was kept to about 27–28 mm, in which case, as simulations showed, the return loss exceeds 9.5 dB. Comparison of the simulated results on the Mike phantom with measured PGs is provided in Table III. The table shows that, for the wrist and ankle channels, the agreement is better than 3 dB. The large discrepancy of 11 dB between measurements and simulations for the head channel is due to the shape of Mike's shoulders and neck, as was the case for the monopole antennas. In addition to the PG results for the homogeneous Mike, Table III also shows the PG obtained with the multi-layered Mike, which has the same shape as Mike but whose internal structure consists of several layers of different tissues. This does not seem to make much difference in the case of head and wrist channels. However, for the ankle channel, there is a big difference of 8 dB. This indicates that in some cases the internal multilayer structure of the human body cannot be neglected. IV. LINK BUDGET IMPLICATIONS In Table IV, we present typical system characteristics of two popular WBAN technologies, namely Bluetooth [22] and

TABLE IV EXAMPLE WBAN SYSTEM CHARACTERISTICS

Zigbee [23], together with a commercial Zigbee transceiver implementation [24] found in SUN SPOT devices [25]. In the case of, say, the Chipcon SmartRF CC2420 transceiver [24] being used for the belt-to-wrist WBAN channel in some hypothetical implementation, we can see from Table I that the dB. mean PG for the wrist channel on James phantom is The receiver sensitivity (defined as the minimum received power required for the CC2420 to maintain a packet error ratio (PER) dBm, which would better than 1%) is, from Table IV, allow successful operation of a channel with a maximum PG dB and a transmitted maximum power of 0 dBm. Thereof fore, such a WBAN implementation will have a fade margin of dB. If we were to use power control to avoid transmitting 0 dBm and instead transmit the dBm (see specification in [24]), the fade minimum possible margin would reduce to 21.7 dB. In this case the maximun PG for successful operation becomes dB. In order to demonstrate a calculation of the probabilities of signal fading, we assume a log-normal distribution for the PG

GALLO et al.: SIMULATION AND MEASUREMENT OF DYNAMIC ON-BODY COMMUNICATION CHANNELS

Fig. 11. Path gain probability distribution from simulations of James phantom walking.

(or normal distribution for the PG expressed in dB). This is justified because the walking phantoms were simulated on their own, without any surrounding environment. Therefore multipath fading is likely to be weak and the PG is expected to be log-normally distributed. Fig. 11 shows cumulative distribution functions (CDF) with the means and the standard deviations derived from the simulations of James walking (see Table I). Fig. 11 shows that, for the wrist channel, we should expect that dB level coronly 0.2% of the time the PG will lie below responding to the receiver sensitivity. Therefore, for only 0.2% of the time we shall not satisfy the quality of service criterion %. For the ankle channel this value is considerably of PER smaller. V. CONCLUSION A methodology for electromagnetic simulation of channels on a moving human body has been presented. Results for two channels at 2.45 GHz made on male and female avatars for two activities, walking and standing up from a chair, have been shown and compared to measurements made on human bodies in an anechoic chamber. Results for channel PG were in the dB to dB. Agreement for individual movement range frames was in general poor, due to the sensitivity of the PG to small differences in the body shape and position between the avatar and the human subject. However good agreement was found for the PG mean and the standard deviation, which suggests that the simulation methodology will be useful in channel prediction for body area networks. The simulation process involved the use of animation software, in which it is relatively easy to create avatars of various shapes and sizes, and to create motion sequences in the form of multiple files containing snapshots of the body shape during the movement. In addition, objects forming an environment, such as floor, walls and furniture can also be exported from the animation software. This provides great scope for future simulation possibilities, which could include complete rooms and multiple figures. With some further work, this approach could be also extended to simulate inhomogeneous avatars with a layered internal structure. This may be especially necessary for simulation of implanted antennas [26] or at lower frequencies. We have shown that avatar modeling gives good results for signal mean and standard deviation in the absence of a local scattering environment, and the results are not sensitive to the

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presence of small features of the body such as nose, ears etc. Extraction of fading statistics appears to be possible also, but would require sampling of the avatar motion at much higher rates than 15 frames/s as done here, in order to capture shortterm signal variation due to multipath fading. Likewise, creation of a local environment (which can significantly affect the short-term fading) in the animation software is also possible, but it is likely that the computational demands will grow significantly if this is done. However, system designers may consider using statistical models, derived from measurements for short-term fading, and the simulation approach presented in this paper, for deriving the long-term fading statistics of on-body channels. The possibility of automating the exporting of the frame data to the electromagnetic simulator and the insertion of the antennas, suggests that the methodology might be suitable for adoption onto large clusters of computers, in which the motion frame simulations could be distributed. ACKNOWLEDGMENT The authors are grateful to I. Buil for the measurements in the anechoic chamber. REFERENCES [1] S. I. Woolley, J. W. Cross, S. Ro, R. Foster, G. Reynolds, C. Baber, H. Bristow, and A. Schwirtz, “Forms of wearable computer,” presented at the Inst. Elect; Eng. Eurowearable’03, Birmingham, U.K., 2003. [2] , P. S. Hall and Y. Hao, Eds., “Antennas and Propagation for On-Body Communication at Microwave Frequencies,” in Antennas and Propagation for Body Centric Communications Systems. Norwood, MA: Artech House, 2006, ch. 3, pp. 39–63. [3] P. S. Hall, Y. Hao, Y. I. Nechayev, A. Alomainy, C. C. Constantinou, C. G. Parini, M. R. Kamarudin, T. Z. Salim, D. T. M. Hee, R. Dubrovka, A. S. Owadally, and W. Song, “Antennas and propagation for on-body communication systems,” IEEE Antennas Propag. Mag., vol. 49, no. 3, pp. 41–58, Aug. 2007. [4] G. A. Conway and W. G. Scanlon, “Antennas for over-body-surface communication at 2.45 GHz,” IEEE Trans. Antennas Propag., vol. 57, no. 4, pt. 1, pp. 844–855, Apr. 2009. [5] 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, no. 4, pp. 956–963, Apr. 2009. [6] C.-Q. Zhang, J.-H. Wang, and Y.-N. Han, “Coupled planar dipole UWB antenna design for wearable computer,” presented at the Microwave and Millimeter Wave Technology Symp. (ICMMT’07), Apr. 18–21, 2007. [7] J. Joseph and R. Mittra, “On the use of conformal grids for propagation and scattering problems in finite-difference time-domain computations,” in Antennas and Propagation Society Int. Symp. AP-S Digest, Jun. 26–30, 1989, vol. 1, pp. 38–41. [8] F. Las-Heras, B. Galocha, and J. L. Besada, “Equivalent source modelling and reconstruction for antenna measurement and synthesis,” in IEEE Antennas and Propagation Society Int. Symp. Digest, Jul. 13–18, 1997, vol. 1, pp. 156–159. [9] K. Ito, “Human body phantoms for evaluation of wearable and implantable antennas,” in Proc. IET Seminar on Antennas and Propagation for Body-Centric Wireless Communications, Apr. 24, 2007, pp. 6–12. [10] P. J. Dimbylow, “Current densities in a 2 mm resolution anatomically realistic model of the body induced by low frequency electric fields,” Phys. Med. Biol., vol. 45, pp. 1013–1022, 1999. [11] [Online]. Available: http://www.remcom.com/varipose [12] [Online]. Available: http://www.semcad.com/simulation/applications [13] M. Gallo, P. S. Hall, Y. I. Nechayev, and M. Bozzetti, “Use of animation software in simulation of on-body communications channels at 2.45 GHz,” IEEE Antennas Wireless Propag. Lett., vol. 7, pp. 321–324, 2008. [14] [Online]. Available: http://www.apple.com/ipod/nike [15] [Online]. Available: http://my.smithmicro.com/win/graphics.html [16] [Online]. Available: http://www.remcom.com

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[17] Z. H. Hu, Y. I. Nechayev, P. S. Hall, C. C. Constantinou, and H. Yang, “Measurements and statistical analysis of on-body channel fading at 2.45 GHz,” IEEE Antennas Wireless Propag. Lett., vol. 6, pp. 612–615, 2007. [18] IEEE Standard for Information Technology-Telecommunications and Information Exchange Between Systems—Local and Metropolitan Area Networks-Specific Requirements—Part 11: Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) Specifications, IEEE Std 802.11-2007, Status: Active, Publication Date: Jun. 12 2007. [19] P. S. Hall, M. Ricci, and T. M. Hee, “Measurements of on-body propagation characteristics,” in Proc. IEEE Antennas and Propagation Society Int. Symp., 2002, vol. 2, pp. 310–313. [20] Y. I. Nechayev and P. S. Hall, “Multipath fading of on-body propagation channels,” presented at the Proc. IEEE Antennas and Propagation Society Int. Symp. (AP-S 2008), Jul. 5–11, 2008. [21] Z. H. S. Hu, “Antenna Design and Channels Characterization for Body Centric Communication,” M.Phil. dissertation, Univ. Birmingham, Birmingham, U.K., Oct. 2007. [22] Wireless Medium Access Control (MAC) and Physical Layer (PHY) Specifications For Wireless Personal Area Networks (WPANs), IEEE Std 802.15.1-2005 (Revision of IEEE Std 802.15.1-2002), 2005. [23] Wireless Medium Access Control (MAC) and Physical Layer (PHY) Specifications for Low-Rate Wireless Personal Area Networks (WPANs), IEEE Std 802.15.4-2006 (Revision of IEEE Std 802.15.4-2003), 2006. [24] [Online]. Available: URL http://inst.eecs.berkeley.edu/~cs150/Documents/CC2420.pdf (last accessed on 15 Oct. 2009) [25] [Online]. Available: URL http://www.sunspotworld.com/docs/index. html (last accessed on 15 Oct. 2009) [26] B. Zhen, K. Takizawa, T. Aoyagi, and R. Kohno, “A body surface coordinator for implanted biosensor networks,” presented at the IEEE Int. Conf. on Communications, Dresden, Germany, Jun. 14, 2009. Michele Gallo (M’10) was born in Bari, Italy, in 1979. He received the M.S. and Ph.D. degrees in electronic engineering from the Politecnico of Bari, Italy, in 2004 and 2008, respectively. In 2005, he joined the Electromagnetic Field Research Group, Politecnico di Bari, where he is currently working as a Postdoctoral Researcher. From February 2006 to February 2007, he was a Guest Researcher in the Communications Engineering Group, University of Birmingham, Birmingham, U.K. His research interests include antenna for wireless on-body communications and antenna diversity.

Peter S. Hall (F’01) received the Ph.D. degree in antenna measurements from Sheffield University, Sheffield, U.K. in 1973. He spent three years with Marconi Space and Defense Systems, Stanmore, U.K., working on a European Communications satellite project. He then joined The Royal Military College of Science, Swindom, U.K., as a Senior Research Scientist, progressing to Reader in Electromagnetics. In 1994, he joined The University of Birmingham, Birmingham, U.K., 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 antennas, propagation and antenna measurements. He has published five books, over 300 learned papers and taken various patents. His current research interests include body centric wireless communications, reconfigurable antennas, and antennas with left handed loading. Professor Hall is a Fellow of the IET (formerly Inst. Elect Eng. – IEE) and the IEEE and a past IEEE Distinguished Lecturer. His publications have earned six Inst. Elect. Eng. Premium Awards, including the 1990 IEE Rayleigh Book Award for the book, Handbook of Microstrip Antennas. He is a member of the IEEE AP-S Fellow Evaluation Committee, and has served on the organizing committee of numerous conferences.

Qiang Bai received the M.S. degree in electronics engineering from the University of Kent, Canterbury, U.K., in 2006 and the M.Phil. degree in body centric communications from the University of Birmingham, Birmingham, U.K., in 2009. He is currently working toward the Ph.D. degree at the University of Sheffield, Sheffield, U.K. His research interests include on-body communications and flexible antenna for wearable applications, electromagnetic bandgap structures and compact UWB antennas.

Yuriy I. Nechayev (M’03) was born in Kurakhovo, Ukraine, in 1974. He received the degree of Specialist in physics (Hons.) from Kharkiv State University, Kharkiv, Ukraine, in 1996 and the Ph.D. degree in electronic and electrical engineering from the University of Birmingham, Birmingham, U.K., in 2004. Since 2003, he has been a Research Associate and then a Research Fellow with the University of Birmingham, working on the problems of on-body channel characterization. He is an author of a book chapter, an IEEE magazine article, and a number of technical papers on radio propagation in urban environments and around the human body. His research interests include radio wave propagation modelling and measurements, propagation in random media, wave scattering, antennas and electromagnetics.

Costas C. Constantinou (M’92) was born in Famagusta, Cyprus, in 1964. He received B.S. (electronic and communications engineering) and Ph.D. (electronic and electrical engineering) degrees from the University of Birmingham, U.K., in 1987 and 1991, respectively. In 1989, he joined the Faculty of the Department of Electronic, Electrical and Computer Engineering, University of Birmingham, as a full-time Lecturer and, subsequently as a Senior Lecturer. He currently heads the radio-wave propagation research activity in the Communications Engineering Research Group, Birmingham. His research interests include optics, electromagnetic theory, electromagnetic scattering and diffraction, electromagnetic measurement, radiowave propagation modelling, mobile radio, and communications networks.

Michele Bozzetti was born in Bari, Italy, in 1950. He graduated in electric engineering from the University of Study of Bari, in April 1975. Between 1975 and 1977, he was Gardiamarina in the Italian Navy and participated in experiments on wire-guided torpedoes. Between 1977 and 1983, he was an Assistant Professor on the Faculty of Engineering, University of Bari, where he delivered the course of electronic components and microwave. Between 1984 and 1987, he was an Associate Professor on the Faculty of Engineering, University of Ancona, teaching microwaves. Since 1987, he is an Associate Professor of electromagnetic fields in the Electrotechnical and Electronic Department, Politecnico di Bari. Since 2000, he is responsible for the Laboratories of EMC at the Politecnico di Bari. His research activity pertains to dosimetry, EMC, microwave antennas, use of metamaterials for antennas on UWB communications. He has coauthored over 80 publications, some of which published in high impact factor journals and international conferences.

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Indoor Off-Body Wireless MIMO Communication With Dual Polarized Textile Antennas Patrick Van Torre, Luigi Vallozzi, Carla Hertleer, Hendrik Rogier, Senior Member, IEEE, Marc Moeneclaey, Fellow, IEEE, and Jo Verhaevert

Abstract—Off-body data communication for firefighters and other rescue workers is an important area of development. The communication with a moving person in an indoor environment can be very unreliable due to channel fading. In addition, when considering off-body communication by means of textile antennas, propagation is affected by shadowing caused by the human body. By transmitting and receiving signals using multiple-input, multiple-output antennas (MIMO communication) a large improvement in reliability of the wireless link is obtained. In this contribution, the performance of wireless data communication using quadrature phase shift keyed (QPSK) modulated data in the 2.45 GHz ISM-band is evaluated in the case of firefighters walking indoor and communicating by means of a compact dual-pattern dual-polarization diversity textile patch antenna system integrated into their clothing. Simultaneous transmit diversity (at the firefighter) and receive diversity (at the base station) up to fourth order are achieved by means of orthogonal space-time codes, providing a maximum total diversity order of 16. The measurements confirm that MIMO techniques drastically improve the reliability of the wireless link. Measurements are compared for three test persons of significantly different sizes. For equal transmitted power levels, the bit error rates for the 2 2 and 4 4 links are much lower than for a system without diversity, with the 4 4 system clearly providing the best performance. Index Terms—Body-centric, diversity, ISM band, multiple-input multiple-output (MIMO) systems, space-time codes, textile antennas.

I. INTRODUCTION HE performance of a multiple-input multiple-output (MIMO) wireless off-body data communication link is studied for the case of a firefighter working in a building and

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Manuscript received December 14, 2009; revised June 29, 2010; accepted August 04, 2010. Date of publication December 03, 2010; date of current version February 02, 2011. This work was supported by the Fund for Scientific Research – Flanders (FWO-V) by Project “Advanced space-time processing techniques for communication through multi-antenna systems in realistic mobile channels.” The work of H. Rogier was supported by a Postdoctoral Fellowship. P. Van Torre is with the Information Technology Department (INTEC), Ghent University, 9000 Ghent, Belgium and also with INWE Department, Hogeschool Gent, 9000 Gent, Belgium (e-mail: [email protected]). L. Vallozzi and H. Rogier are with the Information Technology Department (INTEC), Ghent University, 9000 Ghent, Belgium (e-mail: [email protected]; [email protected]). C. Hertleer is with the Department of Textiles, Ghent University, 9052 Zwijnaarde, Belgium (e-mail: [email protected]). M. Moeneclaey is with the Department of Telecommunications and Information Processing (TELIN), Ghent University, 9000 Ghent, Belgium (e-mail: [email protected]). J. Verhaevert is with the INWE Department, Hogeschool Gent, 9000 Gent, Belgium. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2096389

communicating by means of multiple textile patch antennas integrated in his/her clothing. Earlier measurement campaigns, involving a single transmit antenna at the base station and multiple receive antennas in the garment of the moving firefighter, were performed by our team in the same environment. These scenarios achieved second-order receiver diversity [1] and fourth-order receiver diversity at the firefighter using two dual-polarized antennas [2]. However, this paper discusses our first measurement campaign using true MIMO communication, combining transmit diversity at the firefighter (using space-time codes sent over textile antennas) and receive diversity at the base station. Bit error rates documenting the real-time behavior of an actual data transmission are presented. Recent measurement campaigns related to body-centric wireless communication with multiple antennas have been documented in [3]–[17] but to our knowledge, no measurements transmitting real blocks of data off-body, via a textile antenna system, over a 4 4 MIMO communications link have been published before. The 2.45 GHz ISM frequency band was chosen for the transmissions, providing a sufficiently large bandwidth and allowing to design antennas of a convenient size, given the wavelength of 12 cm. The integration of textile antennas in the garment is not straightforward, as the equipment carried by the firefighter severely limits the number of suitable locations for an antenna. The indoor environment consists of a fading channel where multipath effects and shadowing make the signal levels fluctuate significantly as a function of position. Channel measurements in the same environment revealed Rayleigh type small-scale fading. Using multiple antennas and combining the received signals by means of diversity techniques significantly improves the performance of the wireless link. The use of multiple antennas is also very effective against degradation of the communication due to the shadowing effect of the human body. Using two dual polarized textile antennas, on the front and back of the firefighter, fourth order diversity is realized using only two patch antennas. The base station uses the same type of antennas, placed one meter apart, and also achieves fourth-order diversity. The resulting 4 4 MIMO channel provides a maximum total diversity order of 16. The measurements in this paper pertain to the situation where the firefighter is transmitting data to the base station. Sections II–III document the measurement setup and results. At the transmitter, a sequence of uncoded QPSK symbols enters antenna a space-time encoder, whose outputs are applied to antenna ports, are ports. The received signals, captured by properly combined according to the particular space-time code, making use of the estimated channel gains. In our experimental

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setup we consider (being the most realistic sit, 2, 4. The uation for a bidirectional wireless link), with resulting signal to noise ratios (SNRs) at the detector and the associated bit error rates (BERs) clearly demonstrate the advantages of MIMO communication. In a real-life scenario, the effective diversity for the practical range of signal-to-noise ratios is degraded by the correlation between the signals and by unequal channel gain. The impact of antenna coupling on MIMO communication has been studied in [3] and specifically for dual-polarized antennas in [4]. Channel measurements with dual polarized transmissions are presented in [5]; in our measurements the polarization of the off-body antennas varies, due to changes in body posture of the rescue worker in action. The correlation of signals received by dual polarized antennas in an indoor environment was studied in [17]. In [14] measurements of a body-worn antenna system are performed in open space and the effects of a multipath environment are indirectly assessed. By means of the cumulative distribution function for the instantaneous SNR at the detector input we compare the 10% outage probability levels for the 1 1, 2 2 and 4 4 systems. The corresponding bit error characteristics resulting from the measurement are compared to the theoretical BER characteristics for independent identically distributed Rayleigh fading channels. The measured correlation between the signal levels for different antennas and polarizations indicates to which degree the channels vary independently. In Section IV, the channel is modeled using the Kronecker and Eigenbeam channel models. Based on these models, the BER characteristics are accurately regenerated using a matrix of i.i.d. pseudo random values having the same statistical distribution as the measured signals. The differences in distribution have a significant impact on BER characteristics, as shown theoretically in [18] for Nakagami distributed signals. Additional measurements with three test persons of significantly different sizes are presented in Section V, resulting in similar performance gains by using MIMO communication. Due to the nature of the measurements, with a real test person performing a random walk in an active office environment during working hours; each measurement session will always be different. Despite this variation, the use of diversity techniques always results in a significant performance gain. II. MEASUREMENT DETAILS A. Measurement Setup We consider the uplink scenario, where the mobile firefighter transmits and the base station receives. The measurement setup is composed of two fixed dual-polarized patch antennas connected to the base station, resulting in a total of four received signals. The wearable antenna system under test consists of two similar dual-polarized textile antennas, resulting in four simultaneously transmitted signals. The proposed wearable antenna system is realized by integrating two textile antennas, as documented in Section II-B, into the front and back side of a firefighter jacket, worn by a test person, as shown in Fig. 1. All antennas are then connected to a Signalion-HaLo 430 measurement testbed, operated by our Matlab software. The transmitted

Fig. 1. Positions of front (ports 1,2) and back (ports 3,4) transmitting antennas integrated into the firefighter jacket (on the inside, the antennas are actually not visible).

space-time encoded data blocks consist of QPSK symbols, modulated on an RF carrier frequency of 2.45 GHz at a baud rate of 1 Msymbols/s. The corresponding complex baseband signals are generated in Matlab and then up converted to RF by the testbed transmitter. The testbed receiver down converts to baseband the signals received by the textile antenna system and samples the resulting baseband signals. These samples are post-processed in Matlab, in order to perform carrier frequency offset estimation and correction, timing estimation and correction, channel estimation, space-time decoding, demapping and calculation of BERs and SNRs. B. Wearable Textile Antennas In this measurement, dual-polarized textile patch antennas are used [19], enabling the implementation of 4th-order diversity in a compact dual-pattern dual-polarized system. The dual-polarized wearable antenna, shown in Fig. 2, is a patch antenna consisting entirely of textile materials, suitable for integration into protective clothing such as firefighter suits. The substrate material is a protective, water-repellent, fire-retardant foam, commonly used in firefighter garments, whereas the ground plane and patch are made out of FlecTron and ShieldIt, respectively, two breathable and highly conductive textile materials. The layout of the dual-polarized patch antenna consists of a rectangular patch with a slot. The antenna possesses two feed points, each one corresponding to an antenna terminal or port, located on the patch diagonals. The topology and feeding structure ensure the excitation of two signals with different polariza, tions. The wearable antenna, at center frequency was designed to transmit/receive two quasi-linearly polarized waves, which are almost orthogonal in space, with the two polar. The radiation izations oriented at tilt angles of about pattern of the antenna has been verified by measurement in the anechoic chamber [2]. The antenna radiates most of its power away from the body and approximately covers a half-space. The transmitting patch antenna is located in the firefighter’s and when the jacket and aligned for polarizations of

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Fig. 2. Layout of the textile antenna with two feed points, to excite signals with orthogonal quasi-linear polarizations.

user is in the vertical position. Applying two such dual-polarized antennas, one at the front and one at the back of the test person, adds front-to-back diversity, allowing a total of four signals to be transmitted. Front-to-back diversity is very important in body-centric communications since the human body shadows the RF signals significantly, causing the front and back antennas to virtually cover two complementary half-spaces [1], [2].

C. The Indoor Environment A floor plan of the indoor environment where the measurements were performed is displayed in Fig. 3. The path followed by the test person during the measurements is marked, as well as the position of the receiver. The considered cases are listed here as a function of the labels shown on Fig. 4 in Section III-A. : the test person walking towards the re1) Path ceiver from a distance of 15 m and ending at 3 m from the receiver. : walking away from the receiver, in the 2) Path opposite direction of the first path. : walking sideways, along a path 3) Path perpendicular to the receiver, at up to 18 m of distance. In the first two cases, a line-of-sight path is present to a varying degree. In case 3, the receiving conditions are most unfavorable, since the transmitted signals experience many obstacles and the transmitter-receiver distance is large. The measurements in this article focus mainly on the sideways path in case 3, as previous measurements [1], [2] have clearly confirmed the weak signals in this area and the Rayleigh-like fading due to multipath propagation.

Fig. 3. Floor plan of the multipath environment. The sideways path is non line-of-sight.

D. Operation of Transmitter and Receiver In the MIMO link considered, the firefighter simultaneously transmits four signals in the same frequency range. The receiver synchronously captures these signals on its four antenna ports. receive and In a space-time coded MIMO system with transmit antenna ports, the received signal corresponding to a codeword can be represented as

(1) , , , and are matrices of dimensions , , and , respectively; with equal to the number of time slots in the codeword. The quantity is the complex channel gain between the -th receive and -th transmit antenna port; is a space-time matrix with orthogonal rows, whose elements are linear functions of information symbols and their complex conjugates. The information symbols are QPSK symbols, with variance . The elements of are i.i.d. complex-valued Gaussian random the noise matrix variables; their real and imaginary parts are independent and . The quantity denotes the have variance where

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2

2

2

2

Fig. 4. Signal-to-noise ratios for the 16 channels present in the 4 4 MIMO link. Bit error rates for 1 1, 2 2 front-to-back diversity, 2 2 polarization diversity and 4 4 transmissions (log scale). The transmitted power is chosen very low to create a sufficient number of bit errors for illustration (see text).

2

signal captured by the -th receive antenna port during the -th time slot of duration . In our measurement setup, we restrict our attention to , with , 2, 4. : All quantities in (1) are scalars, with reducing • to a single QPSK information symbol. : We use the Alamouti code [20], defined as •

which depends on two QPSK information symbols and . : we use a rate 3/4 complex orthogonal space-time • code [21, p. 194, Eq. (5.143)], defined as

which depends on three QPSK information symbols , and . The corresponding total (sum over all transmit antennas) , total transmitted energy per infortransmit power mation bit and information bitrate are given in Table I. configurations, we obtain Note that, for the considered . For a QPSK constellation, the total instantaneous received with energy per information bit equals denoting the Frobenius norm of . Similarly, the per antenna . port received energy per information bit is The corresponding average energies per information bit are

TABLE I TRANSMITTED POWER, BITRATE AND ENERGY PER BIT

(total) and (per receive antenna port). , the receiver conBy linearly combining the quantities structs decision variables on which symbol-by-symbol decision is performed to obtain the detected information symbols. The related to the information symbol can decision variable where the noise is a combe decomposed as plex-valued Gaussian random variable with i.i.d. real and imag, 2, 4 the corresponding inary parts. For the above cases instantaneous SNR at the detector is given by

(2) The resulting instantaneous BER for QPSK information symbols [22] is given by (3) For a fair comparison, the values of for the 1 1, 2 2 and 4 4 configurations are adjusted such that the total transmitted is the same for all configurations power

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TABLE II STRUCTURE OF THE TRANSMITTED SIGNALS, INDICATING TIMING OF (P)ILOT AND (D)ATA TRANSMISSION

To compare the performance of different MIMO schemes in similar conditions, bursts are transmitted that contain a sequence of data blocks using the following structure. • A transmission without diversity, using only one of the polarizations on the front antenna (1 TX signal); • The Alamouti space-time code for pattern diversity using one of the polarizations on both front and back antenna (2 TX signals, F/B: front-to-back diversity); • The Alamouti space-time code for polarization diversity on the front antenna (2 TX signals); • The rate 3/4 space-time code (4 TX signals). The signals, transmitted simultaneously on multiple antennas, are received on up to four antenna ports. An estimation is performed for of the complex-valued channel gains all 16 combinations of . These channel gain estimates are needed to compute the decision variables by properly combining the demodulated signals. The received bit stream is obtained by symbol-by-symbol detection on the decision variables, followed by demapping. In order to perform the initial estimation of the 16 channels, pilot symbols are transmitted by the four antennas, without overlap in time. These pilots are also used to determine the different carrier frequency offsets for the signals received from different transmit antennas. Table II illustrates the structure of the transmitted signals. The signals consist of 300 BPSK pilot symbols and 396 QPSK information symbols per transmit antenna for each data block. In our experiment a large overhead is created by transmitting the pilot symbols, because for measurement purposes an accurate channel estimation is preferred. Further tracking of the time-varying channel (during the course of the data burst) is performed using decision oriented feedback. III. MEASUREMENT RESULTS The signal levels and bit error rates in Section III-A result . The power was from a total transmit power chosen very low, in order to generate an illustrative amount of bit errors during the measurement. The results from Sections Sections III-B, III-D and III-E are derived from accurate channel measurements. To minimize the influence of the background noise, the total transmitted power was raised to 100 mW, during an additional set of measurements along the sideways path (case 3). A. Signal Levels and Bit Error Rates The recorded instantaneous SNRs for each of the 16 channels of the 4 4 transmission are displayed in Fig. 4. The SNR related to the channel from the -th transmitting port to the -th

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, where receiving port is defined as denotes the variance of the pilot symbols. Along the line-of-sight path of cases 1 and 2, the difference in signal levels for transmissions from the front and back antennas is clearly visible. When the firefighter is walking towards the receiver, the signals from the front antenna are significantly stronger than those from the back antenna. The opposite is true when the firefighter is walking away from the receiver. This illustrates the complementarity of the antennas, with their radiation patterns pointing in opposite directions and their isolation by the shadowing of the body. In the same graph the BER per burst is displayed for each type of diversity listed in the previous section (the BER plots are listed in the same order). This graph is included as an illustration of the actual wireless MIMO link in operation. The total transmitted power is constant for all diversity types; 100 for the 1 1 link, 50 per antenna for the 2 2 links and 25 per antenna for the 4 4 link. The following considerations are important for a correct interpretation of the graph. • The signal levels plotted for each burst are calculated based on the average received power during the transmission of the pilot symbols. This is only an estimate of the received SNR, as the channels are not invariant during the transmission of the burst. • The signal levels vary drastically during the measurement, due to the path walked by the firefighter. Although this is useful to demonstrate the shadowing effect of the body in the to/from cases, there is also a downside. As the bit errors result from a single implementation based on a limited number of bursts, the bit error rates in this particular graph only indicate the order of magnitude of the statistical BER to be expected with the type of diversity used. • The performance of the detection at very low signal levels is compromised by inaccurate channel estimation, due to the small transmit power and the limited number of pilot symbols per burst. • For all of the above reasons the graph fails to point out the better performance of 2nd order (at the transmitter) front-to-back over polarization diversity, as the BER values for both cases are in the same order of magnitude. A more accurate comparison with calculated BER characteristics is deferred to Section III-E. For the 1 1 configuration (no diversity), errors occur even along the line-of-sight path (cases 1 and 2), in spite of strong average signal levels. Along this path no errors occur when using 2 2 or 4 4 MIMO systems transmitting the same power. Further, even in this short measurement series, the superiority of the 4 4 MIMO link over both 2 2 systems is illustrated by the lower BER values recorded along the sideways path (non line-of-sight; case 3). B. Cumulative Distribution Functions Based on the pilot symbols received along the sideways path by each receiving antenna, all 16 channels are estimated using a total transmit power of 100 mW. The different channel gain magnitudes are approximately Rayleigh distributed but with very different average powers (average taken over the sideways

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TABLE IV MINIMUM, MEDIAN AND MAXIMUM FOR 10% OUTAGE PROBABILITY IN THE 1 1,2 2 AND 4 4 CASES MEASURED ALONG THE SIDEWAYS PATH

2 2

2

E =N

TABLE V ESTIMATED NAKAGAMI PARAMETERS FOR THE 16 SISO CHANNELS

Fig. 5. Cumulative distribution functions for signals measured along the sideways path, from left to right for 1 1, 2 2 and 4 4 links. The CDFs for all combinations are displayed, with the bold lines representing possible the median cases, based on 10% outage probability levels.

N 2N N 2N

2

2

2

TABLE III AVERAGE POWER LEVELS [DB] RECEIVED ALONG THE SIDEWAYS PATH USING THE 4 4 SYSTEM. THE VALUES ARE NORMALIZED WITH RESPECT TO THE STRONGEST SIGNAL

2

path). The normalized average power levels for the 16 combinations of TX and RX antennas are presented in Table III. Normalization has been performed such that the largest channel gain corresponds to a level of 0 dB. The cumulative distribution at the input of of the instantaneous , 2, 4. the detector is displayed in Fig. 5, for The individual CDFs for the 16 possible 1 1 cases are displayed, with the median case shown in bold. Similarly, for 2 2 diversity, 36 combinations of 2 transmitting antenna ports with 2 receiving antenna ports are possible; all of them are displayed in the graph, with the median CDF as a bold line. The median case is selected based on the median 10% outage probability levels for each individual case. The right-most thick line displays the CDF for 4 4 diversity. We observe that in our measurement the 4 4 diversity performs significantly better than any possible combination using 2 2 diversity and certainly better than the median realization for 2 2 diversity. Based on Fig. 5, the performance gain w.r.t. the 1 1 system is quantified by comparing the 10% outage probability power levels,[8]. These power levels define the 10th percentile in the CDF; the power will be higher than these values 90% of the time. Comparing these values, the 4 4 and median 2 2 systems perform better than the median 1 1 system by 15.0 dB and 9.9 dB, respectively. Note that the results apply to transmissions using the same total transmit power level. Table IV lists the 10% outage probability levels expressed in dB for the best (MAX), worst (MIN) and median as diversity cases. The 4 4 system still performs 1.5 dB better than the best 2 2 case and 6.2 dB better than the best 1 1 case. Note that the “best” scenario refers to the specific path

walked by the firefighter, for the specific orientations of receive and transmit antennas. In practice, it is impossible to rely on the best scenario, as the movements of the mobile user are not known a priori. C. Estimated Nakagami Parameters for the CDF The CDF’s for all 16 SISO channels were fitted to the Nakagami distribution and the parameters producing the best fit are listed in Table V. The shape factor varies between 0.7 and 1.8, with an average of 1.06. The actual distribution of a set of measured signals at a specific antenna is the result of a large number of factors, including fading, shadowing (by the body as well as by the environment) and changes in orientation of the antennas. This is consistent with the results obtained in [6], where it was found that a difference in antenna height above the floor level results in different shadowing conditions. Inaccuracies can also result from , the the limited set of measurements. With on average signals are considered to be approximately Rayleigh distributed. Note that the highest -values occur for the signals related to receiving antenna 2 and/or transmit antenna 3. However, as seen in Table III, the average power levels associated with these antennas are also lower. Therefore the impact of the higher -values on the bit error characteristics presented in Section III-E will be limited. The values are comparable to those listed in [23] for NLoS off-body communication in office environments at 868 MHz. D. Spatial Correlation Some correlation exists between the signals received from different channels. This correlation is partially caused by the propagation environment and partially by mutual coupling between both feeds of the dual polarized antennas. However, the

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TABLE VI CORRELATION COEFFICIENTS FOR THE TX CHANNEL GAINS, AS RECEIVED BY EACH INDIVIDUAL RX ANTENNA. CORRELATION FOR SIGNALS DIFFERING ONLY IN POLARIZATION IS MARKED IN BOLD

TABLE VII CORRELATION COEFFICIENTS FOR THE RX CHANNEL GAINS, FOR SIGNALS TRANSMITTED BY EACH INDIVIDUAL TX ANTENNA. CORRELATION FOR SIGNALS DIFFERING ONLY IN POLARIZATION IS MARKED IN BOLD

correlation coefficient is low enough to still achieve a substantial diversity gain by combining the multiple signals [2]. The correlation between the four transmit channel gains is determined for each of the different receiving ). antenna ports ( For a given port the correlation for transmit antenna ports is defined as

Table VII displays the correlation matrices of the channel gains for each of the four transmitting antenna ports ). As the receiving patch antennas are mounted ( next to each other at one meter distance, isolation by the body (as for the transmit antennas) is not present here. Moreover the antennas are now oriented in the same direction, providing spatial diversity but no pattern diversity (both antennas now receive signals with similar angles of arrival). Therefore, the correlation between signals from different patch antennas at the receiving side is significantly higher than at the transmitting side. These correlations correspond to the matrix elements (1, 3), (1, 4), (2, 3) and (2, 4). Due to the unequal power of the different received signals and the correlation between them, the obtained performance gain will be lower than the theoretical optimum, which is achieved for independent identically distributed fading channels. However it is clear that the channel gains fluctuate in a partially independent way while the firefighter is walking in the indoor environment. Combining the different received signals will result in a significant improvement of the reliability of the communication, as compared to a 1 1 configuration.

(4)

Table VI displays the channel gain correlation matrices, as seen by each of the four receiver ports. The correlation is higher between signals originating from the same patch antenna and differing only in polarization (values marked in bold). The correlation between front and back signals is lower because the shadowing of the human body isolates the antennas from each other and moreover, their radiation patterns are oriented in opposite directions. For this reason, front to back diversity results in more diversity gain than does polarization diversity. This is confirmed by the bit error characteristics from Section III-E. The correlation between the four receiving channel gains is determined for each of the different trans). For a given port the mitting antenna ports ( is defined as correlation for receiving antenna ports

(5)

E. Bit Error Rate Characteristics The BER characteristics can be calculated based on the set of received Signal-to-Noise Ratios . For these calculations only measurement data recorded along the sideways track were used. In this measurement series the path loss is nearly constant. Inevitably some shadowing will be present, making the signal worse than Rayleigh distributed. , Since the measurements were performed with for the 1 1 and 2 2 links multiple combinations of transmit and receive antenna ports are possible. For a fair comparison, instead of just selecting one possible combination of ports, all

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Fig. 6 shows the resulting BER curves and displays diversity gain as well as array gain. The curve “no diversity” refers to the average BER for a 1 1 configuration. This characteristic is calculated for the average channel, involving all 16 possible transmit/receive port combinations. The curve for the average channel approximates the theoretical curve for 1 1 communication with a Rayleigh distributed signal. The theoretical characteristics with diversity are calculated [22, p. 825], by taking into account that there is array gain at the receiver but not at the transmitter, assuming independent identically distributed Rayleigh fading channels, as Fig. 6. Bit error graphs showing diversity and array gain. The theoretical characteristics are calculated including receive array gain but no transmit array gain (the total transmitted power is constant).

(9) possible combinations that yield a given type of diversity are used in the calculation of the BER characteristics. Assuming the channel amplitude to be approximately invariant during the time of one received burst, the instantaneous , for the -th burst and -th combit error rate bination of antenna ports, is calculated from (2) and (3) with where , 2, 4 for the diversity is the channel matrix corresponding to cases considered. the -th burst and the particular antenna port selection indexed by . For example, the four possible antenna port combinations that , provide 2 2 polarization diversity are , and . The four corresponding 2 2 complex channel matrices include only the complex channel gains which are relevant to the considered combination. The average BER, averaged over all bursts and all antenna port combinations that yield a given diversity order is calculated as

(6) with the number of bursts and the number of signal combinations considered for that particular diversity case. This BER is expressed as a function of the average per receive antenna port, given by

(7) with

(8) The BER characteristics, as a function of tained by computing (6) to (8) for a range of

, are obvalues.

with

the bit error probability,

the diversity order and

(10) The use of the Alamouti code to transmit and receive two signals, results in two characteristics for 2 2 MIMO communicombinations of 2 2 out of cation. A total of 4 4 channels is possible, of which 4 cases correspond to polarization diversity and 16 cases to pattern diversity (the remaining cases correspond to pattern diversity on one side and polarization diversity on the other side). The BER characteristics are substantially better with 2 2 diversity, but not as good as the theoretical curve for Rayleigh fading with fourth order diversity. Pattern diversity (front-to-back diversity at the transmitter and receiving two signals from different patch antennas) performs better than polarization diversity (transmitting two signals from one patch antenna and receiving two signals in the same way). Using the orthogonal space-time code for 4 4 MIMO communication results in the largest performance gain. The improvement is, compared to the 2 2 system, very significant. However, the performance is not as good as for the theoretical curve based on Rayleigh fading with 16th-order diversity. An important conclusion is that the measured characteristics are clearly better for each increase in the order of diversity, illustrating the practical benefits of using transmit as well as receive diversity. In the 2 2 diversity system, front-to-back diversity performs better than polarization diversity, because of the lower correlation between the transmitted signals. ratios in dB needed to obTable VIII displays the tain a given bit error rate for the recorded signals with varying degrees of diversity. Values marked ( ) are based on the theoretical characteristics for Rayleigh fading, the required for the recorded signals will not be smaller. Due to the absence of a sufficiently large number of channel measurements an accurate calculation for the recorded signals is not possible in these cases. The table clearly illustrates that, using MIMO techniques, a given bit error rate can be achieved using a significantly lower total transmitted power.

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TABLE VIII REQUIRED FOR A GIVEN BER FOR VARYING DEGREES OF DIVERSITY USING THE RECORDED SIGNALS. 2 2 DIVERSITY VALUES FOR POLARIZATION DIVERSITY (POL.) AND FRONT-TO-BACK DIVERSITY (F/B)

E

=N

2

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for 16 channels and 325 bursts, is used as a reference for the generation of the matrix. B. The Eigenbeam Model The eigenbeam model [25], treats the influence of the antennas and environment by means of eigenbases and a coupling matrix. and the eigenbases of the unparameterized With one-sided correlation matrices of sides A and B of the link (correlation as perceived from the other side of the link), a MIMO channel realization is generated as

IV. MIMO CHANNEL MODEL The bit error characteristics in the previous section illustrated that using MIMO techniques, the performance of the system improves significantly compared to a SISO system. However the performance is not as good as predicted by the theory for Rayleigh fading channels with diversity. The performance of the MIMO system is limited due to the correlation between the channels and the unequal gain of the receiver’s four input amplifiers. Several MIMO channel models that include this correlation are available. In this section two models will be applied to the measurement data recorded along the sideways path. A. The Kronecker Model The Kronecker model [24], uses separate transmit and receive correlation matrices. The model assumes that the full channel correlation matrix is given by the Kronecker product of the transmitter correlation matrix

(11) and the receiver correlation matrix

(12) as

(13) A MIMO channel realization is generated by the model as

(15) with the Hadamard (entry-wise) product of , which is a matrix of i.i.d. random zero-mean complex-normal distributed values, and a coupling matrix . The coefficients of specify the mean amount of energy that is coupled from the th eigenvector of side A to the th eigenvector of side B. C. BER Characteristics of the Measured and Modeled Channels Bit error characteristics are generated for the measured and the modeled channels in a similar way as the one described in Section III-E, however, the BER is now displayed as a function to show only the diversity gain and not the array of total gain. For the measured channels 325 recorded MIMO channel realizations are used. For the channels reconstructed by the models realizations are generated, to minimize differences due to the random values in the matrix and in this way producing a curve for the average model-based channel realization. The bit error characteristics are compared to curves for uncorrelated Rayleigh fading to verify the effective diversity order. The associated bit error graphs, displaying the diversity gain, are displayed in Fig. 7 The characteristics display a good match of the models to the measurements, indicating that the correlation properties of the actual MIMO channel were indeed correctly reproduced by both models. The eigenbeam model, having more parameters, matches the measurements slightly better than the Kronecker model. V. PERFORMANCE FOR DIFFERENT BODY SIZES

(14) with the trace of a matrix, the transpose and the Kronecker product. Theoretically the matrix contains i.i.d. random zero-mean complex-normal distributed values. As the distribution of our measured signals is nearly but not exactly Rayleigh distributed this produces a deviation in the characteristics (measured versus modeled channel) of up to 2 dB. Therefore a matrix containing i.i.d. pseudo-random values having the same distribution as the measured signals is generated. The overall distribution of all 5200 signal levels, measured

To assess the influence of a person’s body size, an extra measurement series was performed with three persons of substantially different stature and weight. The results illustrate that the gains obtained by using MIMO techniques are similar for all three persons and also for both measurement campaigns, despite some differences in propagation conditions. The signal distribution is different from the previous campaign as the office environment has changed considerably in the mean time (more equipment and people present). All measurements for this section were performed along the sideways (NLoS) path, in similar operating conditions as the previous measurements. Fitting the signal values to the Nakagami distribution resulted in the average -values listed in Table IX, accompanied by

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Fig. 7. Diversity gain for the measured and modeled channels, using measurement-like distributed pseudo random values for instantiating the modeled channels.

TABLE IX LENGTH AND WEIGHT OF THE TEST PERSONS. BEST FITTING NAKAGAMIVALUES FOR THE SIGNAL DISTRIBUTION. MINIMUM, MEDIAN AND MAXIMUM FOR 10% OUTAGE PROBABILITY IN THE 1 1,2 2 AND 4 4 CASES MEASURED ALONG THE SIDEWAYS PATH

E =N

m

2 2

2

Fig. 8. Bit error graphs for firemen of different sizes, showing diversity and array gain.

When the antennas are worn at a different height above the floor (due to body size) they consistently experience a slightly different fading pattern, even for a random walk. Body mass is not expected to have a significant impact as the torso of even a small adult is still much wider than the size of the antenna patch. The main conclusion is that for all three persons of different size, MIMO techniques significantly enhance the performance of the wireless link. While the persons experience a different fading distribution, the gains obtained when comparing the 1 1, 2 2 and 4 4 systems are similar. This is the case for both the outage probability levels and the BER characteristics.

VI. CONCLUSIONS

the minimum, median and maximum 10% outage probability values. While the distribution of the signal levels is now significantly ) for all three persons, MIMO “worse than Rayleigh” ( communication offers again a substantial performance gain for each increase in the degree of diversity. Based on the median values and for all three persons, the 2 2 system offers 10.8 – 11.9 dB of diversity gain, compared to the 1 1 system. Similarly, the 4 4 system offers 15.3 – 18.9 dB diversity gain, compared to the 1 1 system. These results are similar to those of the previous measurement campaign, with 9.9 dB and 15.0 dB gain for the 2 2 and 4 4 systems, respectively (Section III-B, Table IV). The BER characteristics in Fig. 8 also display a significant improvement in performance for each increase in the link’s diversity order. These results are consistent for all three test persons. A spread of up to 3 dB is present when comparing the BER characteristics for the same diversity order between different test persons. The order of the curves with respect to each other is consistent with the average Nakagami- values. Some difference is unavoidable as the persons can never walk exactly the same trajectory. For the performance of an off-body communication system in real life, variation in performance is to be expected due to many factors, including the environment, body posture and the specific path followed.

The measurements confirm that in a multipath environment, an off-body wireless data link is more reliable when implementing MIMO receive and transmit diversity. The most practical evidence is provided by the bit error rates obtained by demodulation and detection of the data transmitted and received with various orders of diversity. 1 1, 2 2 and 4 4 transmissions were tested, resulting in fewer bit errors for each increase of diversity order, while keeping the transmitted power equal. These BER results include the combination of fading, shadowing, path loss, Doppler spread and channel estimation errors. Cumulative distribution functions allow a comparison of the 10% outage probability levels, providing a quantitative indication of the gain realized by the MIMO system. Measured median gains (relative to a 1 1 system) are 9.9 dB and 15.0 dB for the 2 2 and 4 4 systems, respectively. Signals with a higher cross correlation provide less diversity gain when combined. However, the correlation coefficients are low enough to achieve a significant performance gain in practice. Bit error characteristics are derived for the different orders of diversity and are compared to theoretical characteristics for Rayleigh fading channels with diversity. In a practical system the diversity gain is compromised due to signal correlation and unequal receiver channel gain. The off-body MIMO channel was represented by the Kronecker and Eigenbeam models and the bit error graphs were accurately reconstructed from these models.

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Additional measurements with persons of different sizes illustrate the MIMO systems consistently increase the performance of the off-body communication link, even for persons of significantly varying sizes.

REFERENCES [1] P. Van Torre, L. Vallozzi, C. Hertleer, H. Rogier, M. Moeneclaey, and J. Verhaevert, “Dynamic link performance analysis of a rescue worker’s off-body communication system using integrated textile antennas,” IET Sci., Meas. Technol., vol. 4, no. 2, pp. 41–52, Mar. 2010. [2] L. Vallozzi, P. Van Torre, C. Hertleer, H. Rogier, M. Moeneclaey, and J. Verhaevert, “Wireless communication for firefighters using dual-polarized textile antennas integrated in their garment,” IEEE Trans. Antennas Propag., vol. 58, no. 4, pp. 1357–1368, Apr. 2010. [3] B. Clerckx, C. Craeye, D. Vanhoenacker-Janvier, and C. Oestges, “Impact of antenna coupling on 2 2 MIMO communications,” IEEE Trans. Veh. Technol., vol. 56, no. 3, pp. 1009–1018, May 2007. [4] C. Oestges, B. Clerckx, M. Guillaud, and M. Debbah, “Dual-polarized wireless communications: From propagation models to system performance evaluation,” IEEE Trans. Wireless Commun., vol. 7, no. 10, pp. 4019–4031, Oct. 2008. [5] R. Parviainen, J. Ylitalo, R. Ekman, P. H. K. Talmola, and E. Huhka, “Measurement based investigations of cross-polarization characteristics originating from radio channel in 500 MHz frequency,” in Proc. IEEE Antennas and Propagation Society Int. Symp., Jul. 2008, pp. 1–4. [6] S. L. Cotton and W. G. Scanlon, “Channel characterization for singleand multiple-antenna wearable systems used for indoor body-to-body communications,” IEEE Trans. Antennas Propag., vol. 57, no. 4, pp. 980–990, Apr. 2009. [7] Y. Ouyang, D. J. Love, and W. J. Chappell, “Body-worn distributed MIMO system,” IEEE Trans. Veh. Technol., vol. 58, no. 4, pp. 1752–1765, May 2009. [8] Y. Ouyang and W. J. Chappell, “Diversity characterization of bodyworn textile antenna system at 2.4 GHz,” in IEEE Proc. Antennas and Propagation Society Int. Symp., Jul. 2006, pp. 2117–2120. [9] 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, no. 4, pp. 956–963, Apr. 2009. [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, pp. 361–363, 2007. [11] 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, no. 1, pp. 248–255, Jan. 2009. [12] T. F. Kennedy, P. W. Fink, A. W. Chu, N. J. Champagne, G. Y. Lin, and M. A. Khayat, “Body-worn e-textile antennas: The good, the low-mass, and the conformal,” IEEE Trans. Antennas Propag., vol. 57, no. 4, pp. 910–918, Apr. 2009. [13] S. He, X. Dong, Z. Tian, T. C.-K. Liu, M. Ghoreishi, M. L. McGuire, S. W. Neville, and N. Tin, “On the empirical evaluation of spatial and temporal characteristics of ultra-wideband channel,” in Proc. IEEE 69th Vehicular Technology Conf. VTC Spring, Apr. 2009, pp. 1–5. [14] D. Psychoudakis, G. Y. Lee, C.-C. Chen, and J. L. Volakis, “Estimating diversity for body-worn antennas,” in Proc. 3rd Eur. Conf. on Antennas and Propagation, Mar. 2009, pp. 704–708. [15] 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. [16] J. Karedal, A. J. Johansson, F. Tufvesson, and A. F. Molisch, “A measurement-based fading model for wireless personal area networks,” IEEE Trans. Wireless Commun., vol. 7, no. 11, pp. 4575–4585, Nov. 2008. [17] M. Hajian, H. Nikookar, F. v. der Zwan, and L. P. Ligthart, “Branch correlation measurements and analysis in an indoor Rayleigh fading channel for polarization diversity using a dual polarized patch antenna,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 9, pp. 555–557, Sep. 2005. [18] G. Femenias, “BER performance of linear STBC from orthogonal designs over MIMO correlated Nakagami- fading channels,” IEEE Trans. Veh. Technol., vol. 53, no. 2, pp. 307–317, Mar. 2004.

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[19] L. Vallozzi, H. Rogier, and C. Hertleer, “Dual polarized textile patch antenna for integration into protective garments,” IEEE Antennas Wireless Prop. Lett., vol. 7, pp. 440–443, 2008. [20] S. M. Alamouti, “A simple transmit diversity technique for wireless communications,” IEEE J. Sel. Areas Commun., vol. 16, no. 8, pp. 1451–1458, 1998. [21] C. Oestges and B. Clerckx, MIMO Wireless Communications: From Real-World Propagation to Space-Time Code Design. New York: Academic Press, 2007. [22] J. G. Proakis et al., Digital Communication. New York: Osborne-McGraw-Hill, 2001. [23] S. L. Cotton and W. Scanlon, “Characterization and modeling of the indoor radio channel at 868 MHz for a mobile bodyworn wireless personal area network,” IEEE Antennas Wireless Propag. Lett., vol. 6, 2007. [24] 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. 1211–1226, Aug. 2002. [25] 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., vol. 5, no. 1, pp. 90–100, Jan. 2006.

Patrick Van Torre received the M.Sc. degree in electrical engineering from Hogeschool Gent, Ghent, Belgium, in 1995. For three years he worked as a hardware development Engineer in the private sector. Since November 1998, he has been active as an educator in electronics and researcher in the field of ultrasound technology. He is currently employed by Hogeschool Gent where he teaches theory courses in analog electronics, organizes project oriented lab sessions and is involved in public relations activities and hardware development projects for third parties. He is a part-time researcher, affiliated with the Department of Information Technology, Ghent University. His current research focuses on body-worn multiple-input multiple-output wireless communication systems.

Luigi Vallozzi was born in Ortona, Italy, in 1980. He received the Laurea degree in electronic engineering from the Universita Politecnica delle Marche, Ancona, Italy, in 2005 and the Ph.D. degree in electrical engineering from Ghent University, Ghent, Belgium, in 2010. His research focuses on design and prototyping of antennas for wearable textile systems, and the modeling and characterization of multiple-input multipleoutput wireless communication systems.

Carla Hertleer received the M.Sc. degree in textile engineering and the Ph.D. degree in engineering (on the research topic of textile-based antennas) from Ghent University, Ghent, Belgium, in 1990 and 2009, respectively. For three years, she worked in a vertically integrated textile company that produced terry cloth. She worked six years in a bank office but in June 2000, decided to return to her roots: textiles. Since then, she works as a Researcher at the Textile Department, Ghent University. She has given classes in weaving and Jacquard technology, but her recent activities are mainly concentrated on smart textiles, more specifically the research and development of textile sensors for integration in biomedical clothing and textile antennas. The latter research is carried out in collaboration with the Department of Information Technology, Ghent University. Her research is carried out in the framework of national and European projects.

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Hendrik Rogier (SM’10) was born in 1971. He received the Electrical Engineering and the Ph.D. degrees from Ghent University, Gent, Belgium, in 1994 and in 1999, respectively. He is currently a Postdoctoral Research Fellow of the Fund for Scientific Research Flanders (FWO-V), Department of Information Technology, Ghent University where he is also Associate Professor with the Department of Information Technology. From October 2003 to April 2004, he was a Visiting Scientist at the Mobile Communications Group of Vienna University of Technology. He authored and coauthored about 50 papers in international journals and about 70 contributions in conference proceedings. His current research interests are the analysis of electromagnetic waveguides, electromagnetic simulation techniques applied to electromagnetic compatibility (EMC) and signal integrity (SI) problems, as well as to indoor propagation and antenna design, and in smart antenna systems for wireless networks. Dr. Rogier is serving as a member of the Editorial Boarding of IET Science, Measurement Technology and acts as the URSI Commission B representative for Belgium. He was twice awarded the URSI Young Scientist Award, at the 2001 URSI Symposium on Electromagnetic Theory and at the 2002 URSI General Assembly.

Marc Moeneclaey (F’02) received the Diploma of electrical engineering and the Ph.D. degree in electrical engineering from Ghent University, Gent, Belgium, in 1978 and 1983, respectively. He is a Professor at the Department of Telecommunications and Information Processing (TELIN), Gent University. His main research interests are in statistical communication theory, (iterative) estimation an detection, carrier and symbol synchronization, bandwidth-efficient modulation and coding, spreadspectrum, satellite and mobile communication. He is the author of more than 400 scientific papers in international journals and confer-

ence proceedings. He coauthored the book Digital Communication Receivers – Synchronization, Channel Estimation, and Signal Processing (Wiley, 1998). Dr. Moeneclaey is the co-recipient of the Mannesmann Innovations Prize 2000. Since 2002, he has been a Fellow of IEEE. During 1992 to 1994, he was Editor for Synchronization, for the IEEE TRANSACTIONS ON COMMUNICATIONS. He served as Co-Guest Editor for special issues of the Wireless Personal Communications Journal (on Equalization and Synchronization in Wireless Communications) and the IEEE Journal on Selected Areas in Communications (on Signal Synchronization in Digital Transmission Systems), in 1998 and 2001, respectively.

Jo Verhaevert received the Engineering degree and the Ph.D. degree in electronic engineering from the Katholieke Universiteit Leuven, Belgium, in 1999 and 2005, respectively. He teaches courses on telecommunication at the Department of Applied Engineering Sciences, University College Ghent, Ghent, Belgium, where he also performs research. His research interests include indoor wireless applications (e.g., wireless sensor networks), indoor propagation mechanisms and smart antenna systems for wireless systems.

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Modal Network Model for MIMO Antenna in-System Optimization Juan Pontes, Member, IEEE, Juan Córcoles, Miguel A. González, and Thomas Zwick, Senior Member, IEEE

Abstract—The analysis of MIMO systems is described with the aid of a novel modal network model. For this purpose the capacity performance of typical base station and mobile station antennas in a simulated macro-cellular scenario with varying antenna inter element spacings and antenna rotation will be studied. The model is based on the modal description of typical receiving and transmitting antennas. In this manner a significant simulation time reduction is achieved which allows for faster analysis and optimization. To prove this the effects of both the mobile and base station antennas are investigated. Moreover, for the more restrictive case of base station antennas, a fully modal descriptive model is proven to yield very similar results as those from measured commercial antennas. It is found that the modal approach improves simulation speed without loss of accuracy or generality. Simulations are done for the city of Karlsruhe with a three-dimensional Ray-tracing tool. Index Terms—Antenna arrays, antenna optimization, multipleinput multiple-output (MIMO), MIMO network model, network theory, path-based channel models, spherical mode expansion.

I. INTRODUCTION

M

ULTIPLE antenna systems, mostly multiple-input multiple-output (MIMO) systems, have in general been subject of investigation for several years now. In this time, several advances in the field have been done, and a more general knowledge of the antenna and channel effects has been gained. However, in spite of the great interest this subject has been given, there seems to be no general agreement for the optimal antenna spacing, configuration and polarization for a specific scenario. This is mainly because of the fact that all works are done under different frameworks which in most cases are not comparable or flexible enough, such as those resulting from analytical channel models [1]. Moreover, attempts to provide results obtained with measured data, even though significant in number, have also proven to be inflexible. These studies are limited to certain antenna configurations or scenarios and in most cases focus primarily on simulation validation rather than the study of different MIMO systems.

Manuscript received July 03, 2009; revised June 16, 2010; accepted July 30, 2010. Date of publication December 03, 2010; date of current version February 02, 2011. This work was supported by Kathrein-Werke KG. J. Córcoles is with the Departamento de Matemáticas, Facultad Ciencias, Universidad Autónoma de Madrid, 28049 Madrid, Spain. M. A. González is with the Departamento de Electromagnetismo y Teoría de Circuitos, Universidad Politécnica de Madrid. E.T.S.I. Telecomunicación, 28040 Madrid, Spain. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2096179

In response to this, a research trend towards physical channel models [1] has been observed as a more realistic and pliant simulation and evaluation alternative for MIMO systems. Such approaches have proven valuable from a simulative point of view and more recently also for the channel estimation of measured scenarios [2] since they make it possible to analyze different antenna types and configurations. However, so far, the complexity of such systems has been an impediment leading to consideration of real multi-path environments in an oversimplified or not sufficiently detailed manner [1]. In the case of channel estimation from measurements the significant amount of data to be processed as well as the considerable difficulties it poses have also limited more generalized studies. Therefore, a need for easy integration of double-directional multipath propagation models with flexible and realistic system evaluation seems at hand. In this work, we set out to provide a faster, general and flexible simulation framework for multiple antenna systems, studied in conjunction with path-based propagation channels, for which comparisons and extensive studies become feasible and computationally less expensive. For this purpose we make use of network theory, which is a convenient way to describe the interaction of fields and waves through power waves and scattering parameters, and we integrate it to a modal description of the field of the transmitting and receiving antennas [3]. Network models for MIMO systems have already been proposed in the literature for path-based channels [4], [5]. In both cases a general framework for the use of scattering parameters was presented for computation of the system capacity focusing on mutual coupling effects. When considering a small number of antennas or for propagation channels with few propagation paths, these path-based approaches pose no problem. However, in the case of multiple channel realizations (i.e., multiple links per channel), of several thousands paths each, the computational cost of extensive antenna evaluations becomes steep. To address this need, in this paper, these previous efforts are extended, without loss of generality, through the inclusion of a modal description of the antenna fields. A modal description of antennas for investigation of communication systems has been already proposed in the literature [6]–[10]. In [6] it is proven that a modal description of the base station antenna results in very similar radiation patterns to those of measured antennas. Yet, no integration into the modeling of the MIMO system is discussed. In [7], on the other hand, a complete MIMO system based on the modal description of dipoles is shown, whereas in [8] spiral antennas are used. In these works it is proven that the use of a spherical mode expansion can reproduce accurately the channel behavior with respect to capacity and correlation. However, this is done on the basis of a current

0018-926X/$26.00 © 2010 IEEE

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Fig. 1. Signal flow graph of generalized network model.

based spherical expansion, similar to that of [11], which poses a considerable limitation when dealing with typical antennas for which no current distribution is known. Moreover, in [7], [8] no typical base station antennas were handled or discussed. In a similar manner in [9] also a very simplified channel with non realistic antenna configurations is used. Nonetheless, a simpler expression for the antenna electric field in terms of spherical waves (without currents) is found for dipole antennas. Finally, in [10] a MIMO system based on the spherical description of dipoles under consideration of mutual coupling is introduced for short range MIMO applications. Even though [10] provides an accurate representation of the communication problem it does not consider complex propagation scenarios such as the urban one and does not provide a system simulation framework. In fact, this is a common issue among all the previous discussed works [6]–[10]. Therefore, in this work the use of a modal description of typical base station and mobile station antennas is done in conjunction with a complex propagation scenario, without knowledge of the antennas currents. In this manner multipath processing becomes significantly reduced and study of multiple antennas and/or orientations becomes feasible. The proposed modal network model is validated for the very complex case of a 3D path-based urban channel simulated with a non-commercial Ray-tracing tool. In the simulations 3D measured complex patterns are used. Our goal is to show how this new simulation framework allows optimization of MIMO systems. With this in mind an extensive comparison of 2 2 MIMO systems with different antenna types and at different antenna inter element spacings is done. This paper is outlined as follows. First, in Section II, a generalized Network Model [4] of the whole communication chain, comprising the antennas, matching network and propagation channel, is presented. Afterwards, in Section III, a modal extension of this model is introduced along with the simulation assumptions and criteria used in this work. In Section IV a modal analysis of the receiver unit is shown. In addition, a thorough study of antenna inter element spacings and polarization effects at the receiver for different antenna configurations at both the base station and mobile station is presented. In Section V the modal analysis is extended for the base station and a modal-based description of typical base station antennas is given and compared with the measured commercials one used in Section IV. Finally in the conclusions, an outlook on this method potential is given along with the most relevant results.

II. GENERALIZED NETWORK MODEL As already mentioned, at least two rigorous network models for path-based channels have been independently published in the past [4], [5]. In this section we will present the generalized network model of [4], further explained in [12]. It consists on a network theory model of the whole transmission chain, i.e., transmitter unit (signal source), transmit antennas, physical channel, receive antennas, and receiver unit (signal drain) modeled as networks described by scattering matrices as seen , where the S-Parameter matrix of in Fig. 1 for antenna pair is of the form the propagation channel (1) with and being the number of receiving and transmitting antennas for the system under test. With this model it is possible to take into account effects like signal correlation or antenna coupling for capacity computations and, in addition, consider additional components such as matching networks [5]. However, in order to use the previous model an accurate description of all S-Parameters needs to be given. In the case of the antennas and the physical channel, this can only be done through measurements or use of numerical methods that deterministically model the interaction between the antennas and the scenario. This is only possible, however, with a path-based description of the propagation, which results in a great computational cost and significant complexity. Therefore, in order to simplify the understanding of the whole transmission chain, it makes sense to use a joint network for the antennas and channel, as done in [4], resulting in the extended channel of Fig. 2. This compact form of the generalized network model results , from two assumptions: (1) there is no back transmission i.e., unilateral propagation channel, and (2) there are no scatterers in the proximity of both the transmit and receive antennas, and respectively. It thus follows that, after some algebraic transformations of the original model (Fig. 1), the merged inner network can be expressed as (2) Here, the term given by results from the transmission coefficients between all transmitting and receiving

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III. MODAL NETWORK MODEL

Fig. 2. Signal flow graph of generalized network model in its compact form, i.e., transmit antennas, physical channel and receive antennas merged into one network.

Fig. 3. Equivalent circuit of transmit and receive antennas.

The main drawback of the previous modeling approach, used throughout the literature, is that computation of the transmission for a certain configuration requires the processing of matrix every single path for every communication link . In addition, if it is considered that for statistic purposes in each studied scenario there are channel realizations, i.e., possible antenna positions with several thousand paths each, the computational cost becomes restrictive. A way in which this computational effort can be reduced is of the extended channel is if the transmission coefficient expressed as a sum of basis functions (basis channels). In this way, once the basis functions have been determined, computing the channel response for another antenna reduces to finding an adequate set of coefficients for the basis functions. However, this does not look practical in general since both the antennas and the physical channel change depending on the configuration and the channel realization. An alternative approach is found by expressing the antennas radiation pattern as a summation of the normalized outgoing spherical wave functions [3] (5)

ports and constitutes the transmission submatrix of the extended . channel scattering matrix For the case of a signal transmitted from port to port under matched conditions yields the channel coefficient , i.e., the ratio between the voltage induced by the and the excitation voltage incoming field

where are the wave coefficients of the radiated pattern approximated by the first spherical wave functions, thus, not an identity. And where the normalized spherical wave function can be expressed as (6)

(3) with the link

given by

and are the and polar components of the where normalized spherical wave function. In this way the radiation patterns of both transmit and receive antennas in (3) can be rewritten according to (5) thus resulting in the following expresand the link transmission coefficient of the sion for extended channel

(4) (7) where the sum is done over the total number of paths of the link. Then, once all channel coefficients have been computed the channel matrix is found. Equations (3) and (4) show together with Fig. 3 that once a , expressed here through the full certain physical channel polarimetric transmission matrix , is known for a certain communication link, the gains and and the normalized and of both receive antenna and radiation patterns transmit antenna at the direction of arrival and the direction of departure have to be included for all paths. Therefore, for each combination of antenna types, orientations or pohas to be calculated. The latter larizations of interest a new greatly limits the amount of possible configurations that can be studied and thus represents a great bottleneck common to both simulations and measurements when dealing with multi-path propagation channels. To overcome this difficulty, the inclusion of a modal antenna representation is suggested.

(8) where defined as

constitutes the

subfunction of the link

(9) and

(10) is a multiplicative factor resulting from (3).

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Fig. 4. Signal flow graph of modal network model.

It follows from (7) that through the use of antenna propagation modes, in this case spherical wave functions, a pseudo modal expression for the transmission coefficient of the exfunctions tended channel has been found, where the build the scenario-dependent basis set. As result the extended channel transmission coefficient of any desired antenna pair can now be expressed as a weighted sum of basis functions for the specific scenario considered (assuming the number of modes and accurately describe the antennas to be studied, see Sections V and VI). The resulting modal network model antennas pair and the modes is shown in Fig. 4 for the pair. In Fig. 4 the notation of the antennas S-Parameter matrix used in Fig. 1 and [4] has been replaced, for conformity, with the notation used by Hansen [3] for spherical wave propagation, where the antenna scattering matrix is of the form (11) In addition, since the assumption of no channel backscattering and no back transmission matrices is taken over from the generalized network model, the submatrices , , and vanish. Therefore, it can be seen in Fig. 4 that the inclusion of the signal source and signal drain remains unchanged in spite of the newly defined model. In consequence, (7) allows the computation of the extended channel matrix in a modal manner without adding complexity to the evaluation of port mismatches. Because of this, the study of feeding network effects will be omitted in the following. It is seen from Fig. 4 that the complexity of the system is increased since the interactions of every received (incoming) and transmitted (outgoing) mode and related to all scattered (outgoing) and reflected (incoming) mode and need to be known at both the channel and the antennas. This suggests that if too many modes are used for the antennas description, the modal approach presented may become unfavorable. To explore this issue, we consider the computational cost of the modal implementation versus the traditional one. In the generalized netand , evaluation work model of Section II, regardless of and different receiving and transmitting antennas reof quires computations of . This yields operaat each channel realtions for each antenna (position) pair ization ((4)). For the modal case, computation of all necessary operations ((9)) and computation basis functions requires

antennas of interest additional of the leads to the inequality

((7)). This (12)

Considering now that the number of paths is significantly greater that the number of modes to be considered, (12) reduces to (13) This means that for the modal network model to be efficient more antennas than modes per antenna have to be considered. can be defined as the It follows that the simulation gain quotient between total number of antennas to be studied and total number of modes (14) It follows that the simulation time of each antenna link is reduced times. For the case of dipoles, it will be shown at least three different antenna that in order to have orientations need to be studied, since dipoles can be exactly described at all desired orientations with only three spherical modes. When considering different antenna types, the number of antennas to be studied have to exceed the number of modes required for adequate description, which depends on the antenna size as discussed in [13]. With this modal network model it is now possible to save valuable time in otherwise very lengthy simulations and, more importantly, significantly increase the number of configurations that can be studied. For example, if a modal description of the base station is available for outdoor channels, in addition to being able to consider different antenna types, different cell sectorizations can be considered with only a one time . In the case of the mobile station, computation of the link on the other hand, different orientations can be analyzed. Based on this in the following section a thorough study of the mobile station antenna effects on an urban MIMO scenario will be presented. IV. SIMULATION SETUP A. Physical Channel As shown in (3) the transmission submatrix of the extended channel depends on both the antennas and the propagation channel. Even though the latter is accounted for through the full

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B. Evaluation Criteria For this work three different power schemes: waterfilling, uniform power distribution and beamforming, will be used. In all cases the ergodic capacity will be investigated for a fixed signal to noise ratio (SNR) of 10 dB. Furthermore, channel matrices will be normalized with respect to a reference MIMO system (15) Fig. 5. Digital model of the city of Karlsruhe. Simulation area is shaded and base station position is denoted with an . In addition, the shaded region is divided into three sectors.

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polarimetric matrix of the path and the communication , which describes the path attenuation and time delay, link and direction of arrival the path direction of departure are also needed to determine the antennas influence on the system, or in the case of the proposed model, the modes’ contribution to the received signal. In consequence, for the modal model suggested a path-based description of the channel for every mode pair is needed. In order to validate the modal approach proposed in this paper, we have chosen to work with a double-directional three dimensional ray tracing model developed at “Universität Karlsruhe” based on a ray-optical approach [14]. The model has been validated with wide-band channel measurements at 2 and 5.2 GHz showing high accuracy for fixed to mobile and mobile to mobile communications in macrocell scenarios. For the simulation, a digital description of the city of Karlsruhe by means of a vector database will be used containing the exact position and size of buildings, trees and other objects. It is generated based on two-dimensional plan information of the buildings along with a digital height profile of the environment. For brevity only one base station location is considered, which corresponds to a current location of the German Vodafone provider. The simulated area, shaded region in Fig. 5, spans about 1 km in each direction. The base station antenna is positioned 30 m above street level. Mobile station locations, on the other hand, are given by a 17.7 17.7 m grid rotated 45 . This means that the system is evaluated at 3000 channel realizations for each studied antenna setup. Furthermore at each channel realization up to 8000 paths can be found. In order to obtain a more realistic description of the actual site, influenced to a lesser extent by the base station look direction, a three-sectorized cell is used. This means that effectively three different propagation environments for the base station are considered, namely the ones resulting from each sector landscape. If the cell orientation was rotated or the antenna position shifted then many more scenarios would be considered. However, this is very lengthy and time consuming. Because of this, the decision was made to keep the antennas orientation fixed. A depiction of the Karlsruhe scenario together with the sectorization used is shown in Fig. 5. In the following the results obtained with the described setup for the mobile station and base station will be shown.

in order to fairly compare the different antenna setups regardless in (15) denotes the Frobeof received power. Where nius Norm of the reference system, chosen here to be the same 2 2 system under study but based on ideal omnidirectional radiating sources instead. C. Antennas During this work two exemplary commercial base station antennas of the antenna manufacturer Kathrein will be considered: the 742215 vertically polarized antenna and the 742445 polarized antenna. In both cases complex measured patterns will be used. For the mobile station, on the other hand, dipoles rotated at different angles will be used. V. MODAL DESCRIPTION OF MOBILE STATION In the previous section it was shown how the whole transmission chain of a certain communication link could be more easily computed through the modal description of the transmit and receive antennas. Furthermore, it was shown that by doing so it is possible to obtain the channel matrix of any desired configuration as a weighted sum of the transmission coefficients of the basis modes. In the general case, these modes correspond to both the transmit and receive antennas, but this is not always the case. For example, it might be of interest to evaluate several mobile station antennas referenced to only one base station. In this case (7) becomes (16) with (17)

and are the and components of the radiation where pattern of the transmitting base station. In consequence, only a modal approach at the mobile station will be implemented. In the following, (16) will be used to investigate the effect of mobile station placement and polarization in an urban scenario for modal described ideal dipoles at the mobile station. In the next subsection the modal description of an arbitrarily oriented dipole will be introduced and afterward results using the proposed modal network model and the dipole modal description will be given.

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A. Arbitrarily Oriented Dipoles In [3] the modal description of dipoles oriented along the , and axis is given. Out of it, the coefficients needed to express an arbitrary oriented dipole can be deduced to be: (18a) (18b) (18c) (18d) where is the rotation angle with respect to the axis, i.e., rotation angle in elevation, and is the rotation angle with respect to the axis, i.e., rotation angle in azimuth. Thus, for the ideal link in (7) can be written as a function of and dipole, the as well, i.e., , for which only three modes are needed in order to consider all possible orientations, i.e., polarizations. Moreover, for the specific case of a vertically oriented dipole, , only one mode is needed, as seen from (18). The potential of this will be now used to evaluate the effects of antenna placement and rotation. B. Antenna Placement 1) 2 2 Systems: For the study of antenna placement effects on 2 2 MIMO systems the antennas inter element spacings at the base station and mobile station are simultaneously , polarized varied between 0.5, 1, 2 and . In the cases of of is also included. The resulting ergodic antennas a for all and all power schemes capacity versus varying for two 742445 vertically polarized Kathrein antennas at the base station is shown in Fig. 6. As there is a significant number of cases considered for each configuration (up to 25 cases per plot), these curves are intended to give the reader an overview of the general trends for each configuration rather than insight of each considered case. Fig. 6(a) shows first the case with two vertically polarized dipoles at the mobile unit, i.e., only one spherical mode is used. Out of it, considering that no mutual coupling effects are taken into account, three main conclusions can be drawn: 1) ergodic , yet larger capacity is in general insensitive to changes in spacings at both the mobile and base station result in higher capacities, 2) increased base station spacing results in greater signal decorrelation at the mobile station and is relevant only , 3) a beamforming scheme where only the best at available subchannel is used yields no improvement regardless and , whereas a uniform power distribution departs of spacings from a beamforming-like performance at small towards an optimum power allocation for increased base station spacings. If now the performance of the vertically polarized base sta, polarized dipoles (cross dipoles) tion in the presence of needs to be studied three modes have to be simulated, i.e., two additional modes. This is shown in Fig. 6(b). At first glance simhas ilar trends as those observed in Fig. 6(a) are noticed: 1) on capacity and 2) signal decorrelation a larger impact than , thus yielding higher capacities. Howincreases at larger ever, two significant differences are observed: 1) a reduction of

Fig. 6. Ergodic capacity versus horizontal antenna inter element spacing at the base station of two 742445 vertically polarized Kathrein antennas with (a) two parallel dipoles and (b) two perpendicular dipoles.

the overall capacity due to polarization mismatch and 2) the deis now inverted. The latter means pendence of capacity on that, contrary as was the case with vertically polarized dipoles, reduces the capacity. Moreover, evaluation of the increased beamforming capacity, in which no multiplexing effects are conimproves diversity but that this sidered, shows that smaller improvement is reduced at larger (seen as bundling of the capacity curves). 2) 4 4 Systems: Up to this point only 2 2 MIMO systems have been considered. Now, in order to show the applicability of this modal approach in the simulation of larger sized systems, 4 4 MIMO systems will be studied as well. In this case the same two types of base station antennas and mobile station antennas will be investigated in order to show the impact of polarization and orientation. Similar to 2 2 systems, antennas inter element and mobile station will be spacings at the base station and simultaneously varied. In this case, however, both will range between 0.5, 1 and . Spacings of have been purportedly omitted in order to avoid discussion of unrealistic dimensioned arrays. Here, when using vertically polarized antennas at either the base station or mobile station 4 antennas are used spanning a maximum array length of . On the other hand, when using polarized base dual polarized antennas (cross dipoles or station antennas) two antennas at each position (one antenna per polarization) are considered. This means that for dual polarized antenna arrays the maximum array length is . The simulated capacities are shown in Figs. 7 and 8 for the case of two 742215 polarized vertically polarized antennas and four 742445 antennas, respectively.

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Fig. 9. Mobile station antenna setups. (a) Two parallel dipoles rotated with respect to the z-axis and (b) two perpendicular dipoles rotated with respect to their starting position as vertically and horizontally polarized dipoles.

Fig. 7. Ergodic capacity versus horizontal antenna inter element spacing at the base station of four 742445 vertically polarized Kathrein antennas with (a) two parallel dipoles and (b) two perpendicular dipoles.

at all antenna spacings and regardless of power scheme. Furthermore, capacity curves resulting from waterfilling and uniform power distribution approaches exhibit greater dependence than on . Nonetheless, this capacity improvement on polarized base station antennas. is less pronounced with Capacity curves under use of beamforming, on the other hand, attain almost identical values regardless of antenna type or antenna spacing. Furthermore, capacity dependence on both and is the same in all cases, i.e., larger antenna spacings lead to better capacities. It should be noted though, that this was not the case for 2 2 systems where slightly better capacities at smaller antenna spacings were found when perpendicular dipoles at the mobile station were used. Finally, it can be stated that for these larger sized systems the use of cross dipoles at the . mobile station only reduces the dependence on C. Antenna Rotation Effects

Fig. 8. Ergodic capacity versus horizontal antenna inter element spacing at the base station of two 742215 45 polarized Kathrein antennas with (a) two parallel dipoles and (b) two perpendicular dipoles.

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From these curves it is seen that for 4 4 systems polarized base station antennas yield the best overall capacities

The previous results have been novel in the sense that they extend previous network modeling efforts to urban scenario with 3D measured patterns. Nonetheless, up to this point, in spite of the results obtained, no additional benefit seems to have been gained from the modal approach with respect to the known network model. This follows from the fact that in the previous section three different modes and three different antenna orientawhere used per base station antenna, tions, vertical and which results in the same number of computations for both network modeling approaches. The key difference is, however, that with the modal method it is now possible to describe all possible antenna orientations, since all modes needed to describe an arbitrarily oriented dipole have already been computed, i.e., all in (16) are known. In this way, the otherwise prohibitive study (because of the big computational load it implies) of antenna rotation becomes straightforward as will be now shown. 2 Systems: Here, the two different mobile station 1) 2 antenna setups seen in Fig. 9 will be used: a) two parallel dipoles rotated with respect to the z-axis and b) two perpendicular dipoles rotated with respect to their starting position as vertically and horizontally polarized dipoles. In consequence a total of 12 orientations will be considered from 0 to 90 . In all cases the three previously simulated spherical modes will be used to model the rotated dipoles. As result and according to (14) our , modal network model attains a simulation gain of i.e., simulations take only one quarter of the time otherwise needed with the traditional network model. In the following , polarized base station antennas will be the vertical and

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Fig. 10. Ergodic capacity versus varying rotation angle  of two 742445 vertically polarized Kathrein antennas with (a) two parallel dipoles and (b) two perpendicular dipoles.

used. Here, the ergodic capacity will be shown plotted against rotation angle . In Fig. 10 the 742445 vertically polarized Kathrein antenna at the base station is studied. More specifically in Fig. 10(a) the case with two parallel dipoles rotated around the z-axis is considered. Here it is seen that there is a cosine-like dependence on the angle resulting in decreased capacity for all angles greater than 0 . Nevertheless, no significant influence of rotation angle and is seen. In Fig. 10(b) the same on the behavior of base station antenna with two perpendicular dipoles is studied. and In this case, the capacity dependence on varying also remains constant, but in addition almost no dependence with respect to rotation angle is seen. Fig. 11(a) and (b) show , polarized Kathrein the equivalent results for the 742215 antenna. Contrary to the vertically polarized base station, when two parallel dipoles are used at the mobile station a slight maxand imum appears at a rotation angle of 30 for almost all configurations. On the other hand, a setup of two perpendicular dipoles yields a similar performance as in Fig. 10(b) with almost no dependence of capacity on rotation angle. It thus follows that regardless of base station antenna type for 2 2 systems crossed dipoles are much more robust to changes in orien. tation and deliver the best results when 2) 4 4 Systems: Analogous to 2 2 systems, for 4 4 systems the mobile station antenna setups seen in Fig. 9 will be used. In this case, however, two such antenna pairs will be used in tandem, leading to four parallel dipoles and one pair of two crossed dipoles. As result, the performance improvement becomes even more dramatic given that the simulation time is proportional to the system size.

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Fig. 11. Ergodic capacity versus varying rotation angle  of 742215 45 polarized Kathrein antenna with (a) two parallel dipoles and (b) two perpendicular dipoles.

Fig. 12. Ergodic capacity versus varying rotation angle  of four 742445 vertically polarized Kathrein antennas with (a) two parallel dipoles and (b) two perpendicular dipoles.

Figs. 12 and 10 show the resulting curves for both base station antenna types. In both cases it is seen that when parallel dipoles

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are used at the mobile station increasing angles reduce capacity significantly. For the pair of crossed dipoles, on the other hand, no dependence on is noticed contrary to Fig. 10(b). Overall, has a larger impact than . However, for 4 4 systems changes in antenna spacing have a lesser impact on the achievable capacities than 2 2 systems. Paradoxically though dual polarized configurations in this larger sized systems are and (cf. Fig. 11). Finally, also in more influenced by polarized this case, larger capacities are achieved when antennas are used at the base station. From the previous results follows that regardless of system size perpendicular polarized dipoles are robust toward changes in rotation angle . Furthermore, has little to no impact on the influence of antenna spacing with respect to capacity at either the mobile or base station. This exemplifies how through the use of the proposed modal network approach, greater insight into the workings of communication systems can be gained. More importantly, its potential is not limited to urban settings but extends to all communication systems. VI. BASE STATION So far, a measured pattern has been considered for the base station, so that (7) turned into (16) to perform the modeling of different mobile station configurations. However, it is also desirable to have a modal base station model in order to investigate different antenna types. This poses one major problem: base stations are usually big antennas, so the number of spherical modes needed to have their radiated field rigorously characterized may be prohibitive [3], [13] to be inserted into the proposed network model. This is confirmed in [6] where a typical base station is successfully synthesized with 96 spherical modes. In order to overcome this problem, we can take advantage of the knowledge of some base station configurations, such as the one of Kathrein, to expand the base station field into a number of pseudomodes. Here, the main objective is to obtain a main beam width as close as possible to the measured pattern. In this case, the major variation of the field from the base station will be in the elevation plane, as it is mainly given by the array factor. In this section we will focus on the modal representation of the 742445 vertically polarized base station antenna only. Given the fact that this antenna consists of 10 dipole pairs arranged vertically and , in front of a a metallic plane, separated approximately by we can carry out the expansion of the radiated field by focusing on the radiation pattern of one dipole pair. Therefore, our task consists in finding a modal representation for the single antenna element (dipole pair) for later multiplication with the antenna array factor. For this, the elevation radiation pattern of one dipole in front of an infinite ground plane is used (19) whereas in the azimuthal plane, the direct measured plane from the base station (for example, in the broadside direction) is used to synthesize the necessary coefficients. By applying a projection synthesis method it follows that the coefficients that reproduce the radiated pattern of one antenna element of the base

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Fig. 13. Ergodic capacity versus varying rotation angle  of two 742215 45 polarized Kathrein antennas with (a) two parallel dipoles and (b) two perpendicular dipoles.

station can be synthesized from their projection on the set of normalized outgoing spherical wave functions: (20) In our case, 35 spherical modes are needed to satisfactorily represent the single antenna configuration. At this point, to get the base station radiation pattern from these coefficients, the spherical modes must be multiplied by the array factor in order to obtain the “pseudomodal” representation of the base station. It is seen that when compared to [6], where singular value decomposition is used for synthesis instead, a significant reduction in the number of modes is achieved. Fig. 14 shows the elevation and azimuth plane patterns achieved with this procedure, compared to the measured patterns from the vertically polarized Kathrein base station antenna. As can be seen, a good degree of compliance is reached. In order to see the real benefits of this approach, in Fig. 15 the ergodic capacity for a 2 2 MIMO configuration based on the newly synthesized antennas is shown. Here, the gain is adjusted to match that of the 742445 Kathrein antenna. Remarkably, little difference with respect to Fig. 6(a) is seen. This shows the great potential for MIMO system analysis behind the modal description of the antennas in general and not only those that can be perfectly decomposed in propagation modes. In addition, it opens a significant window of opportunity since different antenna sectorizations can also be considered for improved statistics. Furthermore, it lays ground to an approximation scheme, which as

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station has been validated against system simulations with measured patterns and its performance has been proven to be satisfactory. In this manner exhaustive simulations, otherwise computationally restrictive, for 2 2 and 4 4 MIMO systems in urban scenarios with varying antenna type, spacing and orientation have been done. Thereby, previous results found in the literature have been confirmed and more insight has been gained. In particular the robustness of perpendicular polarized dipoles against mobile station rotation has been proven and it has been shown that capacity dependence on antenna spacing is different for parallel and perpendicular polarized configurations. ACKNOWLEDGMENT The authors acknowledge helpful discussions with R. Gabriel. REFERENCES

Fig. 14. Comparison of synthesized radiation pattern and 742445 vertically polarized Kathrein antennas in (a) azimuth and (b) elevation.

Fig. 15. Ergodic capacity versus horizontal antenna inter element spacing at the base station for two vertically polarized synthesized antennas and two vertically oriented dipoles.

seen in the previous figures is validated not only with pattern comparison but with system simulations as well. VII. CONCLUSION The goal of this work was to provide a simplified and more efficient simulation framework for multiple antenna systems in general. To achieve this goal a modal network model was implemented, capable of reducing simulation time of different antenna types and/or antenna orientations. In this context not only has a modal description of the mobile station proven to be of great benefit, but also a pseudomodal description of the base

[1] P. Almers, E. Bonek, A. Burr, N. Czink, M. Debbah, V. Degli-Esposti, H. Hofstetter, P. Kyösti, D. Laurenson, G. Matz, A. Molisch, C. Oestges, and H. Özcelik, “Survey of channel and radio propagation models for wireless MIMO systems,” EURASIP J. Wireless Commun. Network., vol. 2007, 2007, Article ID 19070. [2] R. Thomä, D. Hampicke, M. Landmann, A. Richter, and G. Sommerkorn, “Measurement-based parametric channel modelling (mbpcm),” presented at the Int. Confe on Electromagnetics in Advanced Applications, ICEAA 2003, Torino, Italy, Sep. 2003. [3] J. E. Hansen, Spherical Near-field Antenna Measurements: Theory and Practice. London, U.K.: IET, 1988. [4] C. Waldschmidt, S. Schulteis, and W. Wiesbeck, “Complete RF system model for analysis of compact MIMO arrays,” IEEE Trans. Veh. Technol., vol. 53, no. 3, pp. 579–586, May 2004. [5] J. W. Wallace and M. A. Jensen, “Mutual coupling in MIMO wireless systems: A rigorous network theory analysis,” IEEE Trans. Wireless Commun., vol. 3, no. 4, pp. 1317–1325, Jul. 2004. [6] Y. Adane, A. Gati, M.-F. Wong, C. Dale, J. Wiart, and V. Hanna, “Optimal modeling of real radio base station antennas for human exposure assessment using spherical-mode decomposition,” IEEE Antennas Wireless Propag. Lett., vol. 1, pp. 215–218, 2002. [7] O. Klemp, S. K. Hampel, and H. Eul, “Study of MIMO capacity for linear dipole arrangements using spherical mode expansions,” presented at the 14th IST Mobile&Wireless Comm. Summit, 2005. [8] O. Klemp, G. Armbrecht, and H. Eul, “Computation of antenna pattern correlation and MIMO performance by means of surface current distribution and spherical wave theory,” Adva. Radio Sci. vol. 4, pp. 33–39, 2006 [Online]. Available: http://www.adv-radio-sci.net/4/33/2006/ [9] M. Gustafsson and S. Nordebo, “Characterization of MIMO antennas using spherical vector waves,” IEEE Trans. Antennas Propag., vol. 54, no. 9, pp. 2679–2682, Sep. 2006. [10] J. Corcoles, J. Pontes, M. A. Gonzalez, and T. Zwick, “Modelling line-of-sight coupled MIMO systems with generalised scattering matrices and spherical wave translations,” Electron. Lett., vol. 45, no. 12, pp. 598–599, Jun. 4, 2009. [11] Y. Chen and T. Simpson, “Radiation pattern analysis of arbitrary wire antennas using spherical mode expansions with vector coefficients,” IEEE Trans. Antennas Propag., vol. 39, no. 12, pp. 1716–1721, Dec. 1991. [12] C. Waldschmidt, “Systemtheoretische und experimentelle charakterisierung integrierbarer antennenarrays,” Ph.D. dissertation, Universität Karlsruhe, Germany, 2004, Fak. f. Elektrotechnik und Informationstechnik. [13] J. Rubio, M. A. Gonzalez, and J. Zapata, “Generalized-scattering-matrix analysis of a class of finite arrays of coupled antennas by using 3D FEM and spherical mode expansion,” IEEE Trans. Antennas Propag., vol. 53, no. 3, pp. 1133–1144, Mar. 2005. [14] T. Fügen, J. Maurer, T. Kayser, and W. Wiesbeck, “Capability of 3-D ray tracing for defining parameter sets for the specification of future mobile communications systems,” IEEE Trans. Antennas Propag., vol. 54, pp. 3125–3137, Nov. 2006.

PONTES et al.: MODAL NETWORK MODEL FOR MIMO ANTENNA IN-SYSTEM OPTIMIZATION

Juan Pontes (M’10) was born in Caracas, Venezuela, in 1981. received the Dipl.-Ing. (M.S.E.E.) from the Universität Karlsruhe (TH), Karlsruhe, Germany, in 2005 and the Dr.-Ing. (Ph.D.E.E.) degree from the Karlsruhe Institute of Technology, Karlsruhe, in 2010. Since 2005, he has worked as a Research Assistant at the Institut für Hochfrequenztechnik und Elektronik (IHE), Karlsruhe Institute of Technology. His research topics include MIMO systems, antenna synthesis and smart antennas for urban communications.

Juan Córcoles was born in Albacete, Spain, in 1981. He received the Ingeniero de Telecomunicación degree in 2004 and the Doctor Ingeniero de Telecomunicación (Ph.D.) degree in 2009, both from the Universidad Politécnica de Madrid, Spain. Since September 2005, he has been collaborating with the Departamento de Electromagnetismo y Teoría de Circuitos, Universidad Politécnica de Madrid. At present, he is with the Departamento de Matemáticas, Universidad Autónoma de Madrid. His current research interests include the application of numerical and analytical methods, as well as optimization techniques, to the analysis and design of antennas, especially antenna arrays.

Miguel A. González was born in Madrid, Spain. He received the Ingeniero de Telecomunicación and Ph.D. degrees, both from the Universidad Politécnica de Madrid, Spain, in 1989 and 1997, respectively. Since 1990, he has been with the Departamento de Electromagnetismo y Teoría de Circuitos, Universidad Politécnica de Madrid, first as a Research Assistant, as an Assistant Professor from 1992 to 1997, and Associate Professor from 1997. His main research interests include analytical and numerical techniques for the analysis and design of antennas and microwave passive circuits.

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Thomas Zwick (S’95–M’00–SM’06) received the Dipl.-Ing. (M.S.E.E.) and the Dr.-Ing. (Ph.D.E.E.) degrees from the Universität Karlsruhe (TH), Karlsruhe, Germany, in 1994 and 1999, respectively. From 1994 to 2001, he was a Research Assistant at the Institut für Höchstfrequenztechnik und Elektronik (IHE), Universität Karlsruhe (TH), Germany. In 2001, he joined IBM as a Research Staff Member at the IBM T. J. Watson Research Center in Yorktown Heights, NY. From October 2004 to September 2007, he was with Siemens AG, Lindau, Germany. During this period he managed the RF development team for automotive radars. In October 2007, he was appointed Full Professor at Karlsruhe Institute of Technology (KIT), Germany, where he is Director of the Institut für Hochfrequenztechnik und Elektronik (IHE). His research topics include wave propagation, stochastic channel modeling, channel measurement techniques, material measurements, microwave techniques, millimeter wave antenna design, wireless communication and radar system design. He participated as an expert in the European COST231 Evolution of Land Mobile Radio (Including Personal) Communications and COST259 Wireless Flexible Personalized Communications. For the Carl Cranz Series for Scientific Education he served as a Lecturer for Wave Propagation. He is the author or coauthor of over 100 technical papers and over 10 patents. Prof. Zwick received the Best Paper Award at the International Symposium on Spread Spectrum Techn. and Appl. ISSSTA 1998. In 2005, he received the Lewis Award for the outstanding paper at the IEEE International Solid State Circuits Conference. Since 2008, he has been the President of the Institute for Microwaves and Antennas (IMA).

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Emulation of MIMO Rician-Fading Environments With Mode-Stirred Reverberation Chambers Juan D. Sánchez-Heredia, Juan F. Valenzuela-Valdés, Antonio M. Martínez-González, and David A. Sánchez-Hernández, Senior Member, IEEE

Abstract—Some recent publications have extended the emulating capabilities of mode-stirred reverberation chambers, which are now able to emulate Ricean-fading and non-isotropic environments. Either the need to physically modify existing chambers or multiple sets of measurements is required for these enhancements. In this paper a novel post-processing hybrid tool is presented for the transformation of a Rayleigh-fading emulated environment into a Rician one. The model is tested and compared to outdoor measurements and simulations through the K-factor, correlation, diversity gain and MIMO capacity. Results show an excellent matching performance with no hardware modifications of mode-stirred chambers with just one set of measurements. The method is patent protected by EMITE Ing. Index Terms—Channel capacity, diversity gain, multiple-inputmultiple-output (MIMO) systems, radiation efficiency.

I. INTRODUCTION

A

single-cavity mode-stirred reverberation chamber, also known as a reverberation chamber (RC), is a metal cavity sufficiently large to support many resonant modes. Although RCs have been extensively used in electromagnetic compatibility problems [1], their use for antennas and propagation-related problems have only acquired attention in the last few years. This is due to their ability to artificially generate a repeatable multipath environment. RCs were initially thought to provide an isotropic and randomly polarized measuring scenario. This was done by perturbing the modes with stirrers and rotating platforms. In this way a fading environment similar to the ones found in indoor and urban environments (Rayleigh) but with a uniform elevation distribution of the incoming waves is obtained [2], [3]. The real and imaginary parts of the received signal become then normally (Gaussian) distributed, with the associated magnitude following a Rayleigh distribution and the phase following a uniform distribution over

Manuscript received October 26, 2009; revised June 16, 2010; accepted October 25, 2010. Date of publication December 03, 2010; date of current version February 02, 2011. This work was supported in part by Fundación Séneca, the R&D unit of the Autonomous Region of Murcia (Spain) under project reference 11783/PI/09 and in part by the Spanish National R&D Programme through TEC2008-05811. J. Valenzuela-Valdés is with the EMITE Ingeniería SLNE, Edificio CEEIM, E-30100 Espinardo, Murcia, Spain (e-mail: juan.valenzuela@emite-ingenieria. es). J. D. Sánchez-Heredia, A. M. Martínez-González, and D. A. SánchezHernández are with the Departamento de Tecnologías de la Información y Comunicaciones, Universidad Politécnica de Cartagena, Cartagena E-30202, Spain (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2096185

. While avoiding cumbersome outdoor measurements, compact-size RCs provide accurate, repeatable and reliable ways of validating measurements for antenna systems and diversity schemes [4]. Yet, real propagating scenarios rarely follow an ideal Rayleigh-fading environment. The K-factor of more general Rician-fading environments changes as the distance of receiver to transmitter changes [5]. Macrocells usually offer a greater K-factor than microcells [6] while rich multipath environments provide K-factors typically close to 0 [7]. Consequently, recent research efforts have concentrated on extending the original capabilities of first-generation RCs. Good examples are the recent extension to non-isotropic environments [8] and the emulation of the effects of metallic windows and other artefacts, trees and walls in buildings [9], although this last one only for Rayleigh-fading scenarios. Unlike reverberation chambers (RCs), where only one cavity is used, MSCs may contain more than one metal cavity. Cavities are coupled by diverse means and a rich fading environment consisting on diverse clusters with different fading characteristics can be obtained. The MSC also contains a set of mode stirrers that change the boundary conditions of the main cavity within the chamber. This causes a multi-reflective environment which is repeatable and can be statistically studied. MSCs can be used as a very fast, easy, and accurate tool to measure a wide variety of MIMO parameters for different extension of the performance of conventional RCs [10]–[12]. With the use of appropriate equipment and through adequate processing of the measured scattering (S) parameters, radiation efficiency, self-impedance, total radiated power, total isotropic sensitivity and several MIMO channel performance variables, including bit error rate (BER), can now be evaluated using MSCs. The extension to emulate Rician-fading [13] has also been available recently. In this Rician-fading emulation technique, however, either complex hardware alterations or large sets of measurements are required. In [13], the Rician-fading environment is obtained at the cost of varying the chamber characteristics and/or the antenna configuration. There is an inherently higher hardware complexity to this option. In this paper, a novel mode-processing technique which allows for an emulation of Rician-fading environments from the data taken in a mode-stirred reverberation chamber with Rayleigh-fading emulated environment is presented. The method is not intended to describe in detail the response of a device to a change in the angle of arrival (AoA), but rather uses a deterministic component of a Rician signal via post processing to observe the associated changes in the conventional post-processing techniques used to determine general MIMO parameters of the device, such as diversity gain and MIMO

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is the measured parameter in the mode-stirred rewhere is the radius of verberation chamber for each antenna pair, is the distance of centroid of cluster from cluster data and the origin. A scatter plot would result in the data clustered in a circle and centered about the origin for pure Rayleigh-fading . As the direct line of sight (LoS) energy environments became comparable to the non-LoS energy, the cluster of data would move away from the origin, and the fading environment becomes a Rician one. Scatter plots are useful not only for identifying the LoS behavior of the fading environments, but also to quantify the data dispersion through the K-factor. The data set measured with the chamber has to be large enough to ensure a Rayleigh-fading scenario. While this is a typical use of RCs, it also means that both the direct and non-direct coupling paths are included in the data set in a more complex mode-coupling muticavity MSC. It is clear from (1) that by increasing the direct component the K-factor will increase. Consequently, we can move the cluster data away from the origin by adding an offset continuous component. The offset has to be a complex number by

Fig. 1. Sketch of measurements with the E200 MIMO analyzer.

capacity. The method allows the MIMO designer to evaluate several Rician K-factors environments with a single measurement set in the MSC. With this technique a considerable reduction of chamber cost and complexity as well as measuring time is provided, alleviating the hardware complexity of other techniques. Section II describes the proposed measurement and emulation technique for Rician-fading environments. The method is validated through a comparison to other simulated and measured results. Channel eigenvalues, correlation, diversity gain and system capacity results for several scenarios are described in Section III. Finally, some conclusions are outlined in Section IV. II. EMULATING AND MEASURING TECHNIQUE

(2) For a target K-factor defined by

, the required offset can be

(3) The phase of the offset in (3) is obtained from the averaged phase of all data samples in the initial set. Equation (3) can be re-written in terms of the S-parameters by (4)

Measurements illustrated in this paper have been performed with the EMITE Ing 8 8 MIMO Analyzer Series E200a in connection to the Rohde & Schwarz ZVRE Vector Network Analyzer (9 kHz to 4 GHz). The MIMO Analyzer is a second generation two-cavity mode-stirred reverberation chamber with external dimensions of 0.82 m 1.275 m 1.95 m, 8 exciting antennas, polarization stirring due to aperture-coupling and to the different orientation of the antenna exciting elements, 3 mechanical and mode-coupling stirrers, 1 holder-stirrer and variable iris-coupling. The MIMO Analyzer was set-up for 3 holder positions with 15 different mechanical stirrer positions for each holder position, 12 iris-coupling aperture stirring and 20 MHz frequency stirring. Measurements were performed at 1800 MHz and half-wave dipoles were used as MIMO antennas. A sketch of the measurement setup can be observed from Fig. 1. The K-factor, defined as the ratio of the direct path component to the scattered component can be calculated from measured S-parameter in a mode-stirred reverberation chamber by [13],

(1)

In order to obtain the desired results, the added offset would have to be phase-coherent to the selected radius in the way (5) The calculated offset has to be added to all samples that are used for the Rician emulation. In this way the new Rician-fading samples keep the standard deviation of the original Rayleighfading ones, and therefore the distance can be altered to be adapted to the target K-factor. Fig. 2 depicts several scatter plots for the original Rayleigh-fading samples and different Rician, 15 and 100 using the proposed techfading ones with nique. It is clearly observed from this figure that the modified samples keep the standard deviation while their radius varies in a proportional way to the target K-factor. Fig. 3 shows a comparison between measured histograms for diverse sample sets with their associated probability distribution functions and several emulated sample sets. Originally-emulated samples can clearly be associated to a Rayleigh-fading scenario, while modified samples conform very well to measured Rician-fading

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Fig. 4. Measured and emulated capacity versus SNR for 3

Fig. 2. Scatter plots for mode-stirred and mode-processed data.

2 3 MIMO systems.

in Fig. 4. Good agreement is observed between outdoor measurements and measurements emulated with the proposed technique. Thus, this method does not intend to describe the specific response of the device to a change in the specific angle of arrival (AoA) characteristic of a fading scenario, but rather to describe the changes in the general performance of the device when the scenario contains an increased LoS component. In Fig. 4, the method closely follows the real evaluation of a device for changing K factors in outdoor measurements made in [14]. This is clear evidence that the technique represents a step forward to the goal of obtaining real-world performance from a mode-stirred chamber test. III. MEASURED RESULTS A. Correlation Results

Fig. 3. Histograms for measured and modified data.

scenarios with , 15 and 100. It seems clear, however, that the method relies on the previous estimation of the initial factor of the sample set. Consequently, the uncertainties of this method are linked to the uncertainties of the method employed to estimate the initial factor of the sample set. In order to validate the proposed technique, 3 different inidipoles tial 3 3 MIMO systems were measured. Three in a vertical position were employed as antennas in reception. The difference between the tested arrays as the spatial separation between the receiving dipoles, as shown in Table I. The initial measured K-factor was always below 0.0001 for the 3 systems, that is, clearly a Rayleigh-fading environment. The initial K-factors were then incremented slightly to between and using the proposed technique. In this way, with only 3 system measurements in a Rayleigh-fading environment, it was possible to emulate and evaluate their performance for any Rician-fading environment. This drastically reduces both R&D costs and time. The emulated results were compared with the outdoor measurements results in [14]. This comparison is shown

A traditional parameter to be analyzed is the correlation between antennas, which gives us an idea of signal similarities. This parameter has been postulated as sufficient to characterize the MIMO systems in some popular models such as the Kronecker model [15]–[18]. Recent analyses, however, have raised the need of more information to accurately predict system capacity [19], [20]. With the aid of the proposed technique, the correlation properties in terms of the fading environment could be studied. In this way we can analyze how does the correlation coefficient change from Rayleigh-fading environments as K-factors are increased. Fig. 5 shows the correlation coefficient between adjacent antennas for the different systems in Table I. From this figure one can observe that different correlation characteristics exist for the different systems. As it was expected, the system with highest correlation coefficient is system A. This is because of its smallest separation distance between adjacent vertically-polarized dipole antennas, which leads to the highest correlation coefficient. In consequence, system C depicts the lowest correlation coefficients. It is also important to note from Fig. 5 that the correlation coefficients change with changes in the K-factor. While the increment of correlation in system A with increasing K factor is almost imperceptible (0.97 to 0.98), system C nearly doubles this parameter (0.18 to 0.39), while system B provides for a 22.4% increment (0.58 to 0.71). Therefore, one can conclude that when increasing the K-factor

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Fig. 5. Emulated correlation coefficients between adjacent antennas for the MIMO systems in Table I.

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Fig. 6. Cumulative probability density function versus relative power for system B with different K-factors.

TABLE I MEASURED MIMO ARRAYS

the correlation coefficient also increases, and this increment is more noticeable when the correlation factor is low. In other words, the influence of the K-factor is more important for systems with low correlation between adjacent antennas. It is also important to stress that the increase in the change in the correlation coefficient that occurs when varying the K-factor has a limitation. For K-factors below 0.01 the correlation remains constant. For K-factors between 0.01 and 0.1 the correlation increases very quickly with increasing K-factor. This K-factor range accounts for 10% of the total correlation increment. For K-factors between 0.1 and 10, the correlation increment slope is more pronounced, achieving in this K-factor range a 90% of the total increase in correlation. Finally, for K-factors above 10 the correlation coefficient again remains constant. This means that the environment influences the correlation factor in a specific K-factor range. In other words, the K-factor plays a role . In this on correlation coefficients for K-factor range a small change in the fading environment has a direct impact on correlation. Once the NLOS fading environ, a further degradament degraded to LoS with tion of the NLOS environment has no effect on the correlation coefficient. B. Diversity Gain Results With the proposed technique the influence of the K-factor on diversity gain can also be analyzed. Fig. 6 shows the cumulative probability density function versus relative power for system B with different K-factors. From this figure it becomes clear that cannot be differentithe different curves with ated, representing a typical Rayleigh-fading distribution. From

Fig. 7. Diversity gain versus K-factor with probability level as parameter, for all tested systems.

K-factors between 1 and 10000, the curves approach a perpendicular line to the -axis. Consequently, diversity gain decreases with increasing K-factors, which was expected but tested in a MSC for the first time. Fig. 7 depicts the diversity gain versus K-factor with probability level as parameter for all tested systems. In a similar way to what happened for correlation, Fig. 7 also provides for a three-stage effect of the K-factor, this time for diversity gain. Roughly constant diversity gain values are observed for K-factors below 0.1. From until a moderate decline in the diversity gain happens. This decline until . Finally, becomes more pronounced for there is no diversity gain as this value approaches for 0 for any given probability level and array type. Fig. 7 also offers an interesting comparison between tested systems. For a 5.8 dB of diversity gain and a cumulative probability of 1%, the highest possible K-factor is 0.001, 2 and 3.5 for system A, B and C respectively. This means that system C, with the lowest correlation but the largest volume, represents a stronger design against a degradation of the NLOS characteristics of the fading scenario. Yet, it is also observed from these figures that the inherent advantages of system types are mitigated with increasing K-factor.

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Fig. 8. Mean eigenvalues for tested system B. Fig. 9. Mean eigenvalues versus K-factor for all tested systems.

For example the difference between selection combined of and a cumulative probability of system A and C for 0.2% is 9 dB, while the difference between these same systems and the same probability is only 1 dB. This means for that with the influence of the array geometry is very important, while the influence of the array geometry for onwards is very small. This could be taken into account by MIMO engineers to decide when it may be necessary to use only a selection of separated antennas or the whole receiving array as a functions of fading scenarios where the system is under operation. As a general recommendation, it can be said that for K-factors below 10, an increase in the spacing between antennas may be useful. Also as a general rule, it can be established that for lower cumulative probability and lower K-factors, the effect of MIMO array geometry on diversity gain is more pronounced. Fig. 10. Measured capacity of all tested systems versus SNR for different K-factors.

C. Channel Eigenvalues The eigenvalues limit the capacity of the channel. Therefore, this limitation can be observed by studying their evolution with changing K-factor. Fig. 8 illustrates the mean eigenvalues for tested system B. It can be seen that the first eigenvalue is always constant regardless of the K-factor value. Yet, the second and third eigenvalues change with changing K-factor. It is important to note that for system B the eigenvalues do not change . Fig. 9 shows the eigenvalues response versus when K-factor for all tested systems. From Fig. 9 it can be seen that while the first eigenvalue is practically equal in all three systems, eigenvalues 2 and 3 decrease with the same slope for all . That is, with the eigensystems when values of the channel matrix remain constant. The eigenvalues for all tested of the channel matrix starts changing when systems. This indicates that the evolution of the eigenvalues depends only on the K-factor and not so strongly on the specific characteristic of the MIMO array. D. MIMO Capacity Results The capacity of MIMO channels can be written as [21], [22]

(6)

with the channel matrix, the number antennas at the transmitter, the number antennas at the receiver and as the idendenote tity matrix with dimension . Let matrix, ’ the nonzero eigenvalues of the with , then the capacity in (6) can be written as [23]

(7) Fig. 10 shows the measured capacity for all tested systems versus SNR for different K-factors. Fig. 11 depicts the same measured capacity versus K-factor for different SNR values. Fig. 10 clearly shows that system capacity decreases with increasing K-factor, as expected. The decrement is more noticeable for very high SNR values, as it was also expected. As an , the capacity loss example, for system B and a , 1 and 10000 is 1.6 with regard to the case with bit/s/Hz (7%), 6.1 bit/s/Hz (26.9%) and 11 bit/s/Hz (48.7%), respectively. For very high K-factors all systems exhibit the same capacity, as it was also expected. Similarly, the minimum SNR

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-

with . The same change but with , however, provides for similar capacity incre. This means that larger ments but this time for spatial separations in MIMO arrays provide a better benefit for high SNR scenarios. IV. CONCLUSIONS

Fig. 11. Measured capacity of all tested systems versus K-factor for different SNR values.

TABLE II REQUIRED SNR FOR DIFFERENT CAPACITIES

TABLE III CAPACITY INCREASE (%) WHEN CHANGING MIMO SYSTEMS

Accurate emulation of any Rician-fading environment has been performed with a mode-stirred reverberation chamber for the first time. The new technique allows for greater versatility and a more detailed study of the influence of the K-factor on MIMO performance. Correlation, diversity gain and MIMO capacity have been analyzed for three different MIMO systems under a variety of Rician-fading environments with different K-factors. Different effects have been evaluated. Results confirm previously-published influence of K-factors for outdoor measurements, yet this time with the use of a mode-stirred chamber for the first time. The proposed technique allows for a better study of K-factor influence on MIMO performance for wireless communications systems, avoiding cumbersome outdoor measurements. The results presented in this paper have been employed to extend the measuring capabilities of EMITE Ing second generation mode-stirred reverberation chambers, in addition to the emulation of non-isotropic environments. The technique represents a step forward to the goal of obtaining real-world performance from a mode-stirred reverberation chamber test. The technique is patent protected by EMITE Ing. Future research works include the hybridization of the new technique with some sample selection algorithm so as to further increase accuracy while reducing representative test samples to those important for each specific fading scenario. REFERENCES

required for achieving a specific system capacity can be extracted from Fig. 11. For example, in order to reach a system , 16.3, 20, 23.2 and B capacity of 10 bits/s/Hz, a 25 would be required for K-factors of 0.001, 0.1, 1, 10, 100 and 1000, respectively. Some examples of SNR values required to reach specific capacities at different K-factors are listed in Table II. For example, if a capacity greater than 15 bit/s/Hz is required, and or a this can be done with a and . This requirement would not . Finally, it is also clear from be fulfilled for Fig. 11 that for high K-factors and given a specific SNR value, all systems achieve the same capacity. The capacity increase when changing a MIMO system to a different one out of Table I is illustrated in Table III with respect to the K-factor. From this Table it is also extracted that the change from system A to B produce a considerable capacity increase (over 15%) for

[1] C. Bruns and R. Vahldieck, “A closer look at reverberation chambers-3D simulation and experimental verification,” IEEE Trans. Electromagn. Compat., vol. 47, pp. 612–626, Aug. 2005. [2] K. Rosengren and P. S. Kildal, “Theoretical study of distributions of modes and plane waves in reverberation chamber for characterization of antennas in multipath environment,” Microw. Opt. Technol. Lett., vol. 30, pp. 386–391, 2001. [3] P. S. Kildal, K. Rosengren, J. Byun, and J. Lee, “Definition of effective diversity gain and how to measure it in a reverberation chamber,” Microw. Opt. Technol. Lett., vol. 34, no. 1, pp. 56–59, Jul. 2002. [4] P. S. Kildal and K. Rosengren, “Correlation and capacity of MIMO systems and mutual coupling, radiation efficiency, and diversity gain of their antennas: Simulations and measurement in a reverberation chamber,” IEEE Commun. Mag., pp. 104–112, Dec. 2004. [5] L. Greenstein, S. Ghassemzadeh, V. Erceg, and D. G. Michelson, “Theory, experiments, and statistical models,” presented at the WPMC’99 Conf., Amsterdam, Sep. 1999. [6] E. Green, “Radio link design for microcellular systems,” BT Tech. J., vol. 8, no. 1, pp. 85–96, 1990. [7] V. R. Anreddy and M. A. Ingram, “Capacity of measured Ricean and Rayleigh indoor MIMO channels at 2.4 GHz with polarization and spatial diversity,” in Proc. IEEE Wireless Communications and Networking Conf. (WCNC’06), Apr. 2006, vol. 2, pp. 946–951. [8] J. F. Valenzuela-Valdés, A. M. Martínez-González, and D. A. SánchezHernández, “Emulation of MIMO non-isotropic fading environments with reverberation chambers,” IEEE Antennas Wireless Propag. Lett., vol. 7, pp. 325–328, 2008. [9] Z. Yun and M. F. Iskander, “MIMO capacity for realistic wireless communications environments,” in Proc. IEEE Antennas and Propagation Society Int. Symp., Jun. 2004, pp. 1231–1234. [10] P. Hallbjorner and K. Madsen, “Terminal antenna diversity characterization using mode stirred chamber,” Electron. Lett., vol. 37, no. 5, pp. 273–274, Mar. 2001.

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[11] J. F. Valenzuela-Valdés et al., “Evaluation of true polarization diversity for MIMO systems,” IEEE Trans. Antennas Propag., vol. 57, pp. 2746–2755, Sep. 2009. [12] J. D. Sánchez-Heredia, M. Grudén, J. F. Valenzuela-Valdés, and D. A. Sánchez-Hernández, “Sample selection method for arbitrary fading emulation using mode-stirred chambers,” IEEE Antennas and Wireless Propag. Lett., 2010, to be published. [13] C. L. Holloway, D. A. Hill, J. M. Ladbury, P. F. Wilson, G. Koepke, and J. Coder, “On the use of reverberation chambers to simulate a Rician radio environment for the testing of wireless devices,” IEEE Trans. Antennas Propag., vol. 54, pp. 3167–3177, Nov. 2006. [14] M. Kang and M. S. Alouini, “Capacity of MIMO Rician channels,” IEEE Trans. Wireless Commun., vol. 5, pp. 112–122, Jan. 2006. [15] C. Chen-Nee, J. M. Kahn, and D. Tse, “Capacity of multi-antenna array systems in indoor wireless environment,” in Proc. IEEE Global Telecommunications Conf. GLOBECOM 98, 1998, vol. 4, pp. 1894–1899. [16] D. Chizhik, G. J. Foschini, and R. A. Valenzuela, “Capacities of multi-element transmit and receive antennas: Correlations and keyholes,” Electron. Lett., vol. 36, no. 13, pp. 1099–1100, Jun. 2000. [17] K. I. Pedersen, J. B. Andersen, J. P. Kermoal, and P. Mogensen, “A stochastic multiple-input-multiple-output radio channel model for evaluation of space-time coding algorithms,” in Pro. IEEE Vehicular Technology Conf., Sep. 2000, vol. 2, pp. 893–897. [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, pp. 1211–1226, Aug. 2002. [19] D. Chizhik, J. Ling, P. W. Wolniansky, R. A. Valenzuela, N. Costa, and K. Huber, “Multiple-input-Multiple-output measurements and modelling in Manhattan,” IEEE J. Sel. Areas Commun., vol. 21, pp. 321–331, 2003. [20] 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., vol. 5, pp. 90–100, Jan. 2006. [21] G. Foschini and M. Gans, “On limits of wireless communication in a fading environment when using multiple antennas,” Wireless Personal Commun., pp. 311–335, Mar. 1998. [22] E. Telatar, “Capacity of multi-antenna Gaussian channels,” Eur. Trans. Telecommun., vol. 10, no. 6, pp. 585–595, Nov./Dec. 1999. [23] J. Winters, “On the capacity of radio communication systems with diversity in a Rayleigh fading environment,” IEEE J. Sel. Areas Commun., vol. 5, pp. 871–878, Jun. 1987. Juan D. Sánchez-Heredia was born in Lorca, Spain. He received the Telecommunication Engineering Degree from the Universidad Politécnica de Cartagena, Spain, in 2009, and is working toward the Masters degree in information technologies at the Universidad de Murcia, Spain. In 2007, he worked at General Electric (Cartagena), and was involved in several projects in relation with the network infrastructure. In 2009, he joined the Department of Information Technologies and Communications, Universidad Politécnica de Cartagena, Spain. His current research areas cover MIMO communications, multimode-stirred chambers and electromagnetic dosimetry.

Juan F. Valenzuela-Valdés was born in Marbella, Spain. He received the Degree in Telecommunications Engineering from the Universidad de Malaga, Spain, in 2003 and the Ph.D. from Universidad Politécnica de Cartagena, in May 2008. In 2004, he worked at CETECOM (Malaga). In 2004, he joined the Department of Information Technologies and Communications, Universidad Politécnica de Cartagena, Spain. In 2007, he joined EMITE Ing as Head of Research. His current research areas cover MIMO communications, multimode-stirred chambers and SAR measurements.

Antonio M. Martínez-González received the Dipl.-Ing. degree in telecommunications engineering from the Universidad Politécnica de Valencia, Spain, in 1998 and the Ph.D. degree from the Universidad Politécnica de Cartagena, Spain, in 2004. From 1998 until September 1999, he was employed as a Technical Engineer at the Electromagnetic Compatibility Laboratory, Universidad Politécnica de Valencia, where he developed assessment activities and compliance certifications with European directives related with immunity and emissions to electromagnetic radiation from diverse electrical, electronic and telecommunication equipment. Since September 1999, he has been an Associate Professor at the Universidad Politécnica de Cartagena. At present, his research interest is focused on electromagnetic dosimetry, radioelectric emissions and mode stirred chambers applied to MIMO systems. In December 2006, he was one of the founders of EMITE Ing, a technological spin-off company founded by telecommunication engineers and doctors of the Microwave, Radiocommunications and Electromagnetism Research Group (GIMRE), Technical University of Cartagena (Spain). Prof. Martínez-González was awarded with the Spanish National Prize from Foundation Airtel and Colegio Oficial de Ingenieros de Telecomunicación de España for the best final project on mobile communications in 1999. In 2006 and 2008, the second i-patentes prize for innovation and technology transfer in the Region of Murcia (Spain) was awarded to the founders of EMITE Ing.

David A. Sánchez-Hernández (M’00–SM’06) received the Dipl.-Ing. degree in telecommunications engineering from the Universidad Politécnica de Valencia, Spain, in 1992 and the Ph.D. degree from King’s College, University of London, London, U.K., in early 1996. From 1992 to 1994, he was employed as a Research Associate for The British Council-CAM, King’s College London, where he worked on active and dual-band microstrip patch antennas. In 1994, he was appointed EU Research Fellow at King’s College London, working on several joint projects at 18, 38 and 60 GHz related to printed and integrated antennas on GaAs, microstrip antenna arrays, sectorization and diversity. In 1997, he returned to Universidad Politécnica de Valencia, Spain, where was co-leader of the Antennas, Microwaves and Radar Research Group and the Microwave Heating Group. In early 1999, he received the Readership from Universidad Politécnica de Cartagena, and was appointed Vice Dean of the School for Telecommunications Engineering and Leader of the Microwave, Radiocommunications and Electromagnetism Engineering Research Group. In late 1999, he was appointed Vice Chancellor for Innovation & Technology Transfer at the Universidad Politécnica de Cartagena and a member of several Foundations and Societies for promotion of R&D in the Autonomous Region of Murcia, in Spain. In May 2001, he was appointed official advisor in technology transfer and a member of The Industrial Advisory Council of the Autonomous Government of the Region of Murcia, Spain. In May 2003 he was appointed Head of Department and, in June 2009, he received the Chair at the Universidad Politécnica de Cartagena, Spain. He has published three international books, over 45 scientific papers and over 90 conference contributions, and is a reviewer of several international journals. He holds six patents. His current research interests encompass all aspects of the design and application of printed multi-band antennas for mobile communications, electromagnetic dosimetry issues and MIMO techniques for wireless communications. Dr. Sánchez-Hernández is a Chartered Engineer (CEng), IET Fellow, IEEE Senior Member, CENELEC TC106X member, and is the recipient of the R&D J. Langham Thompson Premium, awarded by the Institution of Engineering and Technology, the i-Patentes award to innovation and technology transfer, the Emprendedor XXI award to innovative entrepreneurship, granted by the Spanish National Innovation Entity (ENISA), as well as other national and international awards.

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Propagation of ELF Electromagnetic Waves in the Lower Ionosphere Kai Li, Xue Yan Sun, and Hou Tao Zhai

Abstract—In this paper, propagation of ELF electromagnetic waves in the lower ionosphere is treated analytically. Because of the fact that the lower ionosphere varies greatly with the height, the lower ionosphere is regarded as a horizontally stratified anisotropic plasma for ELF electromagnetic waves. Especially, in the analysis and computation the effects by heavy ions are considered. Finally, some new numerical results and discussions are also given. Index Terms—ELF electromagnetic waves, heavy ions, lower ionosphere.

I. INTRODUCTION

B

ECAUSE of the existence of the earth’s magnetic field, the ionosphere behaves as remarkably anisotropic properties in the LF/VLF/ELF frequency ranges. In the past 60 years, the propagation of LF/VLF/ELF waves in a homogeneous ionosphere has been investigated widely, while the investigation on LF/VLF/ELF wave propagation in an inhomogeneous ionosphere is seldom reported. In the past two decades, few attempts of making the experiments on the direct excitation of VLF/ELF waves in ionosphere were undertaken [1]–[3]. It has been known that the new programmes of the active VLF/ELF wave experiments with the uses of large loop antenna or line antenna onboard a spacecraft are carried out [1], [2]. Also, the programmes, which the space borne VLF/ELF experiments are also used to study earthquake, are also carried out [4]. In the past two decades, the relations between the ionospheric propagation of LF/VLF/ELF waves and earthquake have been treated widely by many investigators [4]–[9]. It is seen that LF/VLF/ELF wave propagation will penetrate the lower ionosphere from the ionospheric -layer to Earth’s surface or from the Earth’s surface to the ionospheric -layer. Therefore, it is necessary to carry out the LF/VLF/ELF wave propagation in the lower ionosphere analytically. Manuscript received January 19, 2010; revised June 30, 2010; accepted August 28, 2010. Date of publication December 03, 2010; date of current version February 02, 2011. This work was supported in part by the National Science Foundation of China under Grants 60971057 and 60831002, and in part by the Innovation fund of State Key Lab of Millimeter Waves under Grant K201010. K. Li and X. Y. Sun are with the Department of Information Science and Electronic Engineering, Zhejiang University, Hangzhou 310027, China and also with the State Key Lab of Millimeter Waves, Southeast University, Nanjing 210096, China (e-mail: [email protected]). H. T. Zhai is with the Department of Information Science and Electronic Engineering, Zhejiang University, Hangzhou 310027, China. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2096391

In the past century, the theory and experiment of ELF wave propagation have been treated by many investigators and the findings are well summarized by Wait [10] and Galejs [11]. In the paper by Xia and Chen [12], by using the matrix-exponent formulations, the wave radiation and propagation in a stratified anisotropic plasma are analyzed and computed numerically. In the previous work by the first author [13], the solution of the propagation of VLF electromagnetic waves penetrating the lower ionosphere is carried out, and the analytical formulas are obtained for the electromagnetic field components on the sea surface generated by a space borne loop antenna. In this paper the lower ionosphere is regarded as a horizontally stratified anisotropic plasma, considering the effects by both the electrons and heavy ions, the propagation of ELF electromagnetic waves in the lower ionosphere is treated analytically. Some new numerical results and discussions are also carried out. is assumed In the whole text, the dependence of and suppressed. II. FORMULATIONS OF THE PROBLEM The geometry and notation of consideration are shown in Fig. 1. The earth’s magnetic field , which has the angle with -direction, is in the - plane. The ionosphere above the height of 120 km is regarded as a homogeneous anisotropic plasma, which is characterized by a tensor permittivity [14]. The lower ionosphere below the height of 120 km, considering the changes of the ionospheric parameters, is divided into -layers. Each layer of the lower ionosphere can be characterized by a tensor permittivity . It is (1) is 3 3 unit matrix, where is the free-space permittivity, is the susceptibility of the -layer of the ionosphere, which is given by

(2) is the effective electron collision frewhere quency of the ionosphere,

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If all layers are very thin, each layer can be treated as a homogenous anisotropic medium. Then, the expression of the electromagnetic field in the -layer is written in the following form.

Fig. 1. The geometry and notation of physical model.

and are the gyrofrequency of the electrons and the angular plasma frequency of the ionosphere, respectively; are the corresponding parameters of heavy ions; and are the directional cosine of the earth’s magnetic field in the and direction, respectively. In the -layer of the lower ionosphere, the electromagnetic fields should be satisfied the following Maxwell’s equations:

(3) (4)

(7) where and are the eigenvalues and the corresponding of the matrix , respectively. The eigenvectors are the roots of the characteristic equation of the eigenvalues and stand for matrix . The coefficients the down-going and up-going waves, respectively. Rewriting (7), the following expression is obtained readily.

Then, a matrix equation is obtained readily

(8) where (5) (9) (10)

where and are the wave number and wave impedance is the column matrix of free-space, respectively. , and is the 4 4 matrix given by

(6)

where

(11)

From (8), we write

(12) is the propagation matrix where from to . In free space, the electromagnetic field components can be written as follows: (13) It is noted that the subscript 0 stands for free-space, and . We define the 2 2 reflection

LI et al.: PROPAGATION OF ELF ELECTROMAGNETIC WAVES IN THE LOWER IONOSPHERE

matrix to indicate the relationship between the down-going and up-going waves. It is

(14) Then, we have

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may be easily determined by using the numerical method. For the ELF waves traveling in the ionosphere, the ELF wave is mainly confined within a very narrow range of the incident angle near the boundary between the ionosphere and the air. So that we may only consider the case which is near the normal of and . Thus, the matrix is boundary, namely, simplified as follows:

(15) (22) With (14), (13) is rewritten in the form where (16) In the ionospheric region above the lower ionosphere, only the up-going wave exists. The 2 2 transmission matrix is defined by

(23) (24) (25)

(17) . where The characteristic equation of the matrix form

Then, we have

is written in the

(18) where the subscript stands for in the homogeneous ionospheric region above the lower ionosphere. , we have With the boundary condition

(26) where

(27) (28)

(19)

By solving the above characteristic equation, the eigenvalues of the matrix can be obtained readily. They are

Obviously, the relation between reflection matrix and the transmission matrix is obtained readily.

(20) (29) where the propagation matrix

is defined by

(21)

III. THE QUASI-LONGITUDINAL APPROXIMATION

(30)

In what follows, we will attempt to treat the case of the quasilongitudinal approximation. When the horizontal wave numbers and are given, the eigenvalues and eigenvectors of matrix

(31) (32)

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The corresponding eigenvectors are given by (33) (34) (35) (36) where

(37) (38) In the case that , and , the effects by the heavy ions can be neglected, then the eigenvalues are simplified as follows:

Fig. 2. The reflection coefficients Hz. operating frequency f

= 80

R

and

R

versus the angle

 with the

R

and

R

versus the angles

 with the

(39)

(40) (41) (42) The corresponding eigenvectors are (43) (44) (45) (46) where is the sign function. By now, the eigenvalues and eigenvectors are derived analytically for the case of the quasi-longitudinal approximation. Evidently, following the derivations in Section II, both the reflection coefficients and the transmission coefficients can be obtained readily. IV. NUMERICAL RESULTS AND DISCUSSIONS In this section, some new numerical results are carried out. and can be computed readily. It The propagation factors is known that the extraordinary wave, which is corresponded with a small imaginary part, can to the propagation factor propagate a long distance in the ionosphere. The ordinary wave, with a large which is corresponded to the propagation factor imaginary part, is the evanescent wave.

Fig. 3. The reflection coefficients operating frequency f Hz.

= 80

When the ionosphere is treated as a homogeneous plasma and the effects by the ions are not considered, the corresponding re, and are computed and fection coefficients shown in [15]. It is found that, for the case of the homogeneous ionosphere, the numerical results by using the proposed method are agreement to those by Pan [15]. In this paper, the lower ionosphere with the heights of 70–120 km is an inhomogeneous plasma. The electron density is almost same as the ion density in the lower ionosphere, as shown in [16, Fig. 2]. The electron density in the lower ionosphere can be described by a two-parameter exponential profile [17], [18]

(47) where the two parameters in kilometers and in km control the altitude of the profile and the sharpness of the ionospheric transition, respectively. Both the electron collision fre-

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above 120 km is treated as a homogeneous anisotropic plasma . Aswith , suming that the Earth’s magnetic field is and the electron gyro-frequency is Hz, the refec, and are computed readily. tion coefficients From Figs. 2 and 4, it is seen that both the magnitude of and that of are close to 1, which means that the ELF waves are reflected intensively between the boundary the lower ionosphere and the air. From Figs. 3 and 5, it is seen that both the and that are not too small, which means magnitude of that the coupling between the ordinary wave and the extraordinary wave should be considered in the ELF range. REFERENCES Fig. 4. The reflection coefficients R cies with  .

and R

versus the operating frequen-

Fig. 5. The reflection coefficients R . cies with 

and R

versus the operating frequen-

= 45

= 45

quency and the ion collision frequency are described in [19] and [20], respectively. We write (48) (49) Considering that the lower ionosphere is regarded as a stratified anisotropic plasma, from (47)–(49), the ionospheric parameters are determined readily. In the following computations, the lower ionosphere is divided into four layers and each layer is regarded as a homogeneous plasma. For the first layer of the lower ionosphere, the electron density and the ion density are while the effective collision frechosen as quency of electron and that of ion are chosen as and , respectively. For the rest three layers, the parameters are chosen as . The ionosphere

[1] N. A. Armand et al., “Experimental research in the ionosphere of the Earth of the radiation of loop antenna in a range VLF waves, installed onboard the orbital complex ‘Progress-28’-‘Souz TM-2’,” Radiotechnika i electronica vol. 33, pp. 2225–2233, 1988. [2] P. B. Bannister et al., “Orbiting transmitter and antenna for space borne communications at ELF/VLF to submerged submarine,” in Proc. AGARD Conf., Jul. 1992, vol. 529, pp. 33.1–33.14. [3] D. S. Lukin, V. B. Presniakov, and P. P. Savtchenko, “The calculation of wave field in the near-zone of the loop VLF radiator in the uniform magnetoplasma,” Geomagn. Aeron., vol. 27, no. 2, p. 262, 1988. [4] O. N. Serebryakova, S. V. Bilichenko, V. M. Chmyrev, M. Parrot, J. L. Rauch, F. Lefeuvre, and O. A. Pokhotelov, “Electromagnetic ELF radiation from earthquake region as observed by low-altitude satellite,” Geophys. Res. Lett., vol. 19, pp. 91–94, 1992. [5] O. A. Molchanov, O. A. Mazhaeva, A. N. Goliavin, and M. Hayakava, “Observations by intercosmos-24 satellite of ELF/VLF electromagnetic emissions associated with earthquakes,” Ann. Geophy., vol. 11, pp. 431–440, 1993. [6] T. R. Henderson, W. S. Sonwalker, R. A. Helliwell, U. S. Inan, and A. G. Fraser-Smith, “A search for ELF/VLF emissions induced by earthquakes as observed in the ionosphere by the DE-2 satellite,” J. Geophys. Res., vol. 98, pp. 9503–9514, 1993. [7] M. Parrot, “Statistical study of ELF/VLF emissions recorded by a lowaltitude satellite during seismic events,” J. Geophys. Res., vol. 99, pp. 23 339–23 347, 1994. [8] V. M. Chmyrev, N. V. Isaev, O. N. Serebryakova, V. M. Sorokin, and Ya. P. Sobolev, “Small-scale plasma inhomogeneities and correlated ELF emissions in the ionosphere over an earthquake region,” J. Atmosph. Solar-Terres. Phys., vol. 59, pp. 967–974, 1997. [9] M. Ozaki, I. Nagano, S. Yagitani, and K. Miyamura, “Ionospheric propagation of ELF/VLF waves radiated from earthquake,” in Proc. AsiaPacific Radio Science Conf., 2004, pp. 539–542. [10] J. R. Wait, Electromagnetic Waves in Stratified Media. New York: Pergamon Press, 1962. [11] J. Galejs, Terrestrial Propagation of Long Electromagnetic Waves. New York: Pergamon Press, 1972. [12] M. Y. Xia and Z. Y. Chen, “Matrix-exponent formulations for wave radiation and propagation in anisotropic stratified ionosphere,” IEEE Trans. Antennas Propag., vol. 48, no. 2, pp. 339–341, 2000. [13] K. Li and W. Y. Pan, “Propagation of VLF electromagnetic waves penetrating the low ionosphere,” Indian J. Radio Space Phys., vol. 33, pp. 87–94, 1999. [14] K. G. Budden, Radio Waves in the Ionosphere. Cambridge: Cambridge Univ. Press, 1961. [15] W. Y. Pan, LF/VLF/ELF Wave Propagation (in In Chinese). Chengdu, China: UESTC Press. [16] S. A. Cummer, “Modeling electromagnetic propagation in the earthionosphere waveguide,” IEEE Trans. Antennas Propag., vol. 48, no. 9, pp. 1420–1429, 2000. [17] K. Rawer, D. Bilitza, and S. Ramakrishnan, “Goals and status of the international reference ionosphere,” Rev. Geophys., vol. 16, pp. 177–181, 1978. [18] S. A. Cummer, U. S. Inan, and T. F. Bell, “Ionospheric D region remote sensing using VLF radio atmospherics,” Radio Sci., vol. 33, pp. 1781–1792, Nov.–Dec. 1998. [19] J. R. Wait and K. P. Spies, Characteristics of the earth-ionosphere waveguide for VLF radio waves Nat. Bur. of Stand., Boulder, CO, Tech. Note 300, 1964.

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[20] D. G. Morfitt and C. H. Shellman, “MODESRCH, an Improved Computer Program for Obtaining ELF/VLF/LF Mode Constants in an Earth-Ionosphere Waveguide,” National Technical Information Service, Springfield, VA, Naval Electron. Lab. Center Interim Rep. 77T, NTIS Accession No. ADA032573, 1976, 22161.

Kai Li was born in Xiao County, Anhui, China, on February 10, 1968. He received the B.Sc. degree in physics from Fuyang Normal University, Anhui, China, in 1990, the M.Sc. degree in radio physics from Xidian University, Xi’an, Shanxi, China, in 1994, and the Ph.D. degree in astrophysics from Shanxi Astronomical Observatory, Chinese Academy of Sciences, Shaanxi, China, in 1998, respectively. From August 1990 to December 2000, he was on the faculty of the China Research Institute of Radiowave Propagation (CRIRP). From January 2001 to December 2002, he was a Postdoctoral Fellow at the Information and Communications University (ICU), Taejon, South Korea. From January 2003 to January 2005, he was a Research Fellow in the School of Electrical and Electric Engineering, Nanyang Technological University (NTU), Singapore. Since January 2005, he has been a Professor with the Department of Information Science and Electronic Engineering, Zhejiang University, Hangzhou, China. His current research interests include classic electromagnetic theory and radio wave propagation.

Dr. Li is a senior member of Chinese Institute of Electronics (CIE) and a member of Chinese Institute of Space Science (CISS).

Xue Yan Sun was born in Yantai, Shandong, China in July, 1985. She received the B.Eng. degree in electronic science and technology from Shandong Jianzhu University, Jinan, China, in 2008. Currently, she is working towards the M.Eng. degree at Zhejiang University, Hangzhou, China. Her research interests include radio wave propagation theory and applications.

Hou Tao Zhai was born in Jurong, Jiangsu, China, in July, 1983. He received the B.Sc. degree in electronic and information engineering from Nanjing University of Science and Technology, Nanjing, Jiangsu, China, in 1990, and the M.Sc. degree in electromagnetic field and microwave technology from Zhejiang University, Hangzhou, China, in 2007, respectively. Since July 2007, he has been an Engineer. His research interests include radio wave propagation in plasma.

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Communications A Printed Elliptical Monopole Antenna With Modified Feeding Structure for Bandwidth Enhancement Jianjun Liu, Shunshi Zhong, and Karu P. Esselle

Abstract—A printed extremely wideband antenna is studied and results presented. The proposed antenna is composed of a trapezoid ground plane and an elliptical monopole patch which is fed by a modified tapered CPW line. The simulated and measured results agree well and it indicates the antenna has a measured impedance bandwidth from 1.02 GHz to 24.1 GHz . It is shown that the antenna radiation pattern is with a nearly omni-directional over the entire frequency band and the antenna is suitable for wireless communication applications.

VSWR

2

Index Terms—Multiple-feed, printed monopole, triple-feed, ultrawideband (UWB), wideband.

I. INTRODUCTION With the development of wireless technology, many systems now can operate in more than one frequency band, satellite navigation systems, wireless LANs, ultrawideband (UWB) systems and some combinations of them are examples. Their capability of operating in multiple, diverse frequency bands eventually depend on their antennas performance. To fulfill this requirement, multiple antennas are implemented in many devices, and each one covers a specific operating band or several bands. However, these antennas together occupy much space, which is at a premium in most devices, and also increase the system complexity. Such installations of multiple antennas prevent future system upgrades that require the use of currently unsupported bands. Therefore, a single antenna that has an impedance bandwidth that is wide enough to cover the operating frequency bands of multiple wireless communication systems is more desirable. Such an antenna should have stable radiation-pattern characteristics over the entire frequency range. To expand the impedance bandwidth of printed antennas, many designs have been extensively investigated. The first wideband antenna, traced back to 1898, is the biconical antenna. After that, various types of wideband antennas have been discussed, like the conical monopole antenna, spheroidal antenna, coaxial horn antenna, volcano smoke antenna and metal-plate antenna, etc., [2]–[8]. The wideband antenna design standard was proposed from investigation of the volcano smoke antenna. The dimension of the main radiation body for a wideband antenna should be about a quarter wavelength at the lowest operating Manuscript received March 16, 2010; revised July 02, 2010; accepted July 31, 2010. Date of publication December 03, 2010; date of current version February 02, 2011. J. Liu is with the School of Communication and Information Engineering, Shanghai University, Shanghai 200072, China and also with the Centre for Microwave and Wireless Applications, Electronic Engineering, Macquarie University, NSW 2019, Australia (e-mail: [email protected]). S. Zhong is with the School of Communication and Information Engineering, Shanghai University, Shanghai 200072, China (e-mail: [email protected]). K. P. Esselle is with the Centre for Microwave and Wireless Applications, Electronic Engineering, Macquarie University, NSW 2019, Australia (e-mail: [email protected]). Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2096398

Fig. 1. (a) Configuration of the proposed antenna. (b) Photograph of the manufactured elliptical antenna.

frequency [9]. The above-mentioned antennas are based on three-dimensional structures or need perpendicular ground planes. Their bulky shapes are undesirable for many modern sleek electronic devices. Recently, many printed wideband antennas with parallel ground planes have been proposed. They have lower profiles and can be easily integrated with wireless systems. In [10], a microstrip-fed printed triangular-ring antenna with a 2:1 bandwidth is presented. A printed circular disc monopole antenna, proposed in [11] for ultrawideband (UWB) systems, is fed by a microstrip line and has a bandwidth of about 3.5:1. A circular monopole antenna with a trapezoid ground plane, presented in [12], has a 2:1 VSWR bandwidth of more than 10:1. In this communication, a printed elliptical monopole antenna with a trapezoid ground plane, fed by a modified, tapered CPW line, is presented. The measurements indicate it has an extremely wide impedance bandwidth of 1.02–24.1 GHz. The antenna can support many existing wireless services, including GPS (1.57–1.58 GHz), GSM1800 (1.71–1.88 GHz), PCS1900 (1.93–1.99 GHz), WLAN (2.5 or 5–6 GHz), multi-band GNNS and UWB (3.1–10.6 GHz). The proposed antenna design parameters are analyzed and results are discussed. II. ANTENNA DESIGN It has been found that the incorporation of multiple feeding lines to a planar metal-plate monopole antenna can efficiently promote vertical current distribution in the monopole patch while restraining the horizontal current distribution and the result is an improvement in impedance bandwidth [13]. The proposed antenna design, featured in Fig. 1(a), has been developed on this basis. The wideband performance

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TABLE I THE DIMENSIONS OF THE PROPOSED ANTENNA [MM]

Fig. 3. Measured and predicted VSWR of the proposed antenna.

Fig. 2. Smith plot of antenna impedance.

is reached by adding two feeding branches, optimizing the elliptical radiation patch, and modifying the ground plane shape. Both the monopole and the ground plane are etched on the same side of the substrate. In this example, substrate thickness h = 1:524 mm and relative permittivity "r = 3:48. The elliptical monopole is fed by a modified, tapered CPW line in the middle of the ground plane. The gap width w of the uniform section of the CPW line is fixed at 3.0 mm. The connection between the CPW feed line and the elliptic radiator is comprised of three metal branches: one central branch connected to point A and two side branches of semicircular shape connected to points C and D. The semicircular branches have an outer radius of d. The width of the CPW central strip at the top end is wtop = 1.0 mm, corresponding to a characteristic impedance of 100 and the width at the bottom end is wbot = 2:6 mm, corresponding to a characteristic impedance of 50 . The widths of the semicircular-shaped feeding branches are set to 1 mm in this study. The tapered CPW line from point B to Point A gradually transforms the impedance from 50

to 100 . The other parameter values are listed in Table I. A more uniform current distribution is expected in the monopole patch because the modified feed has three feeding points symmetrically connected to the bottom of the elliptical radiation patch. The tapered CPW line, the semicircular feeding branch and the elliptical patch are integrated together. By adjusting the outer radius of the feeding branch and other relative parameters, a greatly enhanced impedance bandwidth can be achieved. The ground plane is a part of impedance matching network which can lead to wideband multi-resonance characteristics of the input impedance at the top point A and it contributes to radiation as well. The photograph of the fabricated antenna is shown in Fig. 1(b). III. RESULTS AND DISCUSSION By simulating the antenna using CST Microwave Studio software, which is based on the finite integration method, the characteristics of the proposed antenna have been investigated. Antenna impedance matching is illustrated with the help of the Smith chart in Fig. 2. It is observed that multiple resonant loops are excited and the resonances are close to each other. Since these loops associated with various modes are within the VSWR = 2 circle, a large bandwidth is achieved.

Fig. 4. VSWR comparison between the proposed antenna and the common tapered CPW-fed elliptical monopole antenna.

The measured results are obtained using a Wiltron 37369A vector network analyzer (40 MHz–40 GHz). The calculated 2:1 VSWR bandwidth covers a frequency range from 0.5 to 25 GHz with a ratio bandwidth of 50:1, while the measured bandwidth covers a frequency range from 1.02 to 24.1 GHz with a ratio bandwidth of more than 23:1 (see Fig. 3). According to [8], the ratio theoretical bandwidth for the proposed antenna should be 40:1 when the widths of the radiation patch and the gap between the two ground planes equal to 120 mm and 3 mm, respectively (120=3 = 40). The lower frequency limit should be 25=40 = 0:625 GHz which coincides with the simulation result. However, it is difficult to fully reach the theoretical lower frequency limit with current practical approach. There is some discrepancy between the simulation and the experiment at the lower and higher frequency limits due to fabrication tolerances and SMA connector effects. But, as a whole, both the simulated and measured results agree well, almost over the whole frequency band. And because the gap between the radiation patch and the ground plane provides a smooth path for guided wave, low reflection can be achieve from the open end and the antenna is well-matched to space within a broad frequency range. The VSWR of the proposed antenna is compared with that of a standard elliptical monopole antenna with an ordinary tapered CPW line in Fig. 4. The results indicate that the feeding arrangement proposed in

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Fig. 5. VSWR of the antenna with different radii d.

Fig. 6. VSWR of the antenna for different values of the major axis a of the elliptical patch.

this communication dramatically extends the upper limit of the bandwidth, whereas there is no significant change to the lower limit of the bandwidth. Fig. 5 shows the VSWR of the antenna for different outer radii d of the semicircular-shaped feeding branch. It is found that the impedance bandwidth first increases with an increase of d. However, the VSWR exhibits an intense fluctuation, sometimes even exceeding the value 2, when d is more than 4.1 mm. When the radius is equal to 4.1 mm, the impedance bandwidth reaches the maximum. Fig. 6 shows the VSWR variation with different major axis lengths of the elliptical patch. It is shown that the lower limit decreases from 980 MHz to 500 MHz when a is increased from 30 mm to 120 mm, improving the impedance ratio bandwidth effectively. Considering the impedance bandwidth and the size of the monopole antenna, the value of a is fixed at 120 mm in the example design. The measured radiation patterns of the antenna, at 1, 5, 10, 15 and 20 GHz, are shown in Fig. 7. These patterns indicate the antenna has nearly omni-directional radiation pattern at lower frequencies in the H-Plane (from 1 GHz to 5 GHz). The E-plane radiation pattern shows a typical figure-of-eight at lower frequency of 1 GHz, which is similar to a conventional dipole or a biconical antenna. The cross-polarization is low at lower frequencies and the value of cross-polarization goes up with an increase in frequency, which is a result of the increasing horizontal component of the surface currents. The measured gain and computed radiation efficiency of the proposed antenna at 1, 5, 10, 15 and 20 GHz

Fig. 7. Measured radiation patterns (a) f GHz, (d) f = 15 GHz, (d) f = 20 GHz.

= 1

GHz, (b) f

= 5

GHz, (c) f

= 10

are displayed in Fig. 8. Some gain fluctuations are witnessed within the operating bandwidth of the antenna. The gain gradually increases from 1 GHz to 15 GHz, while there is a monotonic decline after 15 GHz. The radiation efficiency is fairly constant in low frequency range, but then it drops off with a minimum value of 76% at 20 GHz due to the high

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[10] T. Dissanayake, K. Esselle, and Y. Ge, “A printed triangular-ring antenna with a 2:1 bandwidth,” Microwave Opt. Tech. Lett, vol. 44, pp. 51–53, 2005. [11] L. Jianxin, C. C. Chiau, C. Xiaodong, and C. G. Parini, “Study of a printed circular disc monopole antenna for UWB systems,” IEEE Trans. Antennas Propag, vol. 53, pp. 3550–3554, 2005. [12] X. L. Liang, S. S. Zhong, and W. Wang, “UWB printed circular monopole antenna,” Microw. Opt. Tech. Lett., vol. 48, pp. 1532–1534, 2006. [13] K. L. Wong, C. H. Wu, and S. W. Su, “Ultrawideband square planar metal-plate monopole antenna with a trident-shaped feeding strip,” IEEE Trans. Antennas Propag., vol. 53, pp. 1262–1269, 2005.

Fig. 8. Measured gain and computed efficiency of proposed antenna.

frequency loss of the RF substrate, which is one of the major reasons for the decrease in gain after 15 GHz. Moreover, at very high frequencies, the antenna operates in several higher order modes instead of the base mode, and the inverse current will excite anti-resonances at some points, which offset the vertical current on the monopole patch. This results in degrading the performance of antenna in the far field and the gain decreases significantly at higher frequencies. This also contributes to the stronger ripples in the radiation patterns at higher frequencies. IV. CONCLUSION A bandwidth enhancement technique for printed monopole antenna is introduced. It is found that the proposed antenna design exhibits an extremely wideband impedance matching and modifying the feeding arrangement is an effective method to promote antenna performance for multi-system support. Its 2:1 VSWR ratio bandwidth is over 23:1, covering frequencies from 1.02 to 24.1 GHz. In addition, the proposed triple-segment feeding arrangement can be easily integrated with any patch-type planar monopoles, thus the proposed antenna can be fabricated at a low cost.

REFERENCES [1] H. Schantz, “A brief history of UWB antennas,” Aerosp. Electron. Syst. Mag., vol. 19, no. 4, pp. 22–26, 2004. [2] G. Dubost and S. Zisler, Antennas a Large Band. New York: Masson, 1976, pp. 128–129. [3] J. A. Evans and M. J. Ammann, “Planar trapezoidal and pentagonal monopoles with impedance bandwidths in excess of 10:1,” in Proc. IEEE Antennas Propag. Symp., 1999, vol. 3, pp. 1558–1561. [4] P. V. Anob, K. P. Ray, and G. Kumar, “Wideband orthogonal square monopole antennas with semi-circular base,” in Proc. IEEE Antennas Propag. Symp., 2001, vol. 3, pp. 294–297. [5] E. Antonino-Daviu and M. Cabedo-Fabres, “Wideband double-fed planar monopole antennas,” Electron. Lett., vol. 39, pp. 1635–1636, 2003. [6] M. J. Ammann and Z. N. Chen, “A wide-band shorted planar monopole with bevel,” IEEE Trans. Antennas Propag., vol. 51, pp. 901–903, 2004. [7] S. Y. Suh, W. L. Stutaman, and W. A. Davis, “A new ultrawideband printed monopole antenna: The planar inverted cone antenna (PICA),” IEEE Trans. Antennas Propag., vol. 52, pp. 1361–1365, 2004. [8] J. D. Kraus and R. J. Marhefka, Antennas, for All Applications, 3rd ed. New York: McGraw-Hill, 2002, p. 43. [9] L. Paulsen, J. B. West, W. T. Perger, and J. Kraus, “Recent investigation of the volcano smoke antenna,” in Proc. IEEE Antennas Propag. Symp., 2003, vol. 3, pp. 845–848.

Small Square Monopole Antenna With Enhanced Bandwidth by Using Inverted T-Shaped Slot and Conductor-Backed Plane M. Ojaroudi, Sh. Yazdanifard, N. Ojaroudi, and M. Naser-Moghaddasi

Abstract—We present a novel printed monopole antenna for ultra wideband applications. The proposed antenna consists of a square radiating patch with an inverted T-shaped slot and a ground plane with an inverted T-shaped conductor-backed plane, which provides a wide usable fractional bandwidth of more than 130% (2.91–14.1 GHz). By cutting a modified inverted T-shaped slot with variable dimensions on the radiating patch and also by inserting an inverted T-shaped conductor-backed plane, additional resonances are excited and hence much wider impedance bandwidth can be produced, especially at the higher band. The designed antenna has a small size of 12 18 mm . Simulated and experimental results obtained for this antenna show that it exhibits good radiation behavior within the UWB frequency range. Index Terms—Inverted T-Shaped conductor-backed plane, inverted T-shaped slot, square monopole antenna.

I. INTRODUCTION Commercial UWB systems require small low-cost antennas with omnidirectional radiation patterns and large bandwidth [1]. It is a wellknown fact that planar monopole antennas present really appealing physical features, such as simple structure, small size and low cost. Due to all these interesting characteristics, planar monopoles are extremely attractive to be used in emerging UWB applications, and growing research activity is being focused on them. In UWB communication systems, one of key issues is the design of a compact antenna while providing wideband characteristic over the whole operating band. Consequently, number of planar monopoles with different geometries have been experimentally characterized [2], [3] Manuscript received October 11, 2009; revised June 29, 2010; accepted October 07, 2010. Date of publication December 03, 2010; date of current version February 02, 2011. M. Ojaroudi and N. Ojaroudi are with the Department of Electrical Engineering, Islamic Azad University, Ardabil Branch, Ardabil, Iran (e-mail: [email protected]; [email protected]). Sh. Yzadanifard and M. Naser-Moghaddasi are with the Faculty of Engineering, Islamic Azad University, Science and Research branch, Tehran, Iran (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2096386

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Fig. 2. (a) The ordinary square antenna, (b) the square antenna with inverted T-shaped slot, (c) the square antenna with inverted T-shaped slot and parasitic structure. Fig. 1. Geometry of proposed antenna with inverted T-shaped slot and conductor-backed plane.

and automatic design methods have been developed to achieve the optimum planar shape [4], [5]. Moreover, other strategies to improve the impedance bandwidth have been investigated [6]–[8]. This letter focuses on a square monopole antenna for UWB applications, which combines the Square-patch approach with an inverted T-shaped slot, and the ground plane with an inverted T-shaped conductor backed plane that achieves a fractional bandwidth of more than 130%. In [6]–[8], three new small wideband printed monopole antennas with notched ground plane using an inverted T-shaped, rectangular, and trapezoid notch in the upper edge of ground plane to achieve the maximum impedance bandwidth were proposed, but in this letter, for the first time, by inserting an inverted T-shaped conductor backed plane in the feed gap distance, without notch in the ground plane structure (this structure has an ordinary rectangular ground plane configuration), and also by cutting a modified inverted T-shaped slot with variable dimensions on the radiating patch, additional resonances (third and fourth resonances) are excited. By obtaining the third and fourth resonances, the usable upper frequency of the monopole is extended from 10.3 GHz to 14.4 GHz. The designed antenna has a small size of 12 2 18 mm2 , and the impedance bandwidth of the designed antenna is higher than the UWB antennas reported recently [6]–[8]. In the proposed structure, by cutting the inverted T-shaped slot of suitable dimensions at the monopole’s patch a double fed structure can be constructed. This structure has a novel feeding configuration that consists of a splitting network connected to two symmetrical ports on its base. Using the Theory of Characteristic Modes it has been demonstrated that the insertion of two symmetric feed ports prevents the excitation of horizontal currents and assures that only the dominant vertical current mode is present in the structure [9]. As a result, unlike other antennas reported in the literature to date [10], [11], the proposed antenna displays a good omni-directional with low cross-polarization level radiation pattern even at higher frequencies. II. ANTENNA DESIGN The square monopole antenna fed by a microstrip line is shown in Fig. 1, which is printed on a FR4 substrate of thickness 1.6 mm, permittivity 4.4, and loss tangent 0.0018. The width f of the microstrip feedline is fixed at 2 mm. The basic antenna structure consists of a square patch, a feedline, and a ground plane. The square patch has a width . The patch is connected to a feed line of width f and length f , as shown in Fig. 1. On the other side of the substrate, a conducting ground plane of width sub and length gnd is placed. The proposed antenna is connected to a 50 SMA connector for signal transmission

W

L

W

W

L

W

To design a novel antenna, the antenna with inverted T-shaped slot and inverted T-shaped conductor-backed plane is proposed. Based on the analysis of current distribution in UWB frequency band, it is observed that the currents at low frequency are distributed on the vertical plane at the monopole’s bottom edge but the currents at high frequency are distributed on the horizontal plane [12]. By cutting the inverted T-shaped notch of suitable dimensions ( S S S1 and S1 ) at the square radiating patch, it is found that much enhanced impedance bandwidth can be achieved for the proposed antenna. In addition, the conductor-backed plane is playing an important role in the broadband characteristics of this antenna, because it can adjust the electromagnetic coupling effects between the patch and the ground plane, and improves its impedance bandwidth without any cost of size or expense. This phenomenon occurs because, with the use of a conductor-backed plane structure in air gap distance, additional coupling is introduced between the bottom edge of the square patch and the ground plane [13], [14]. The optimal dimensions of the designed antenna are as follows: = 10 mm, f = 2 mm, f = 7 sub = 12 mm, sub = 18 mm, mm, S = 4 mm, S = 2 5 mm, S1 = 8 mm, S1 = 0 5 mm, =2 P = 4 mm, P = 0 5 mm, P 1 = 10 mm, P 1 = 1 mm, mm, PS = 2 mm and gnd = 3 5 mm.

W ;L ;W

W W W L

L

L

L

:

:

L

W

W :

L

W L L

W

L : L

III. RESULTS AND DISCUSSIONS In this Section, the planar monopole antenna with various design parameters were constructed, and the numerical and experimental results of the input impedance and radiation characteristics are presented and discussed. The parameters of this proposed antenna are studied by changing one parameter at a time and fixing the others. The simulated results are obtained using the Ansoft simulation software highfrequency structure simulator (HFSS) [15]. Fig. 2 shows the structure of the various antennas used for simulation studies. Return loss characteristics for ordinary square patch antennas (Fig. 2(a)), with an inverted T-shaped slot ( S = 4 mm, S = 2 5 mm, S1 = 8 mm and S1 = 0 5 mm, Fig. 2(b)), and with inverted T-shaped slot and conductor-backed plane ( S = 4 mm, S = 2 5 mm, S1 = 8 mm, S1 = 0 5 mm, P = 4 mm, P = 0 5 mm, P 1 = 10 mm and P 1 = 1 mm, Fig. 2(c)) are compared in Fig. 3. As shown in Fig. 3, it is observed that by using this modified element including an inverted T-shaped slot cut in the radiating patch and inverted T-shaped conductor-backed plane inserted on the other side of substrate, additional third and fourth resonances are excited respectively, and hence the bandwidth is increased. As shown in Fig. 3, in the proposed antenna configuration, the ordinary square monopole can provide the fundamental and next higher resonant radiation band at 4 and 7.9 GHz, respectively, in the

W W W

L L

L

: :

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: : :

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Fig. 5. Simulated surface current distributions on the radiating patch and ground plane for the proposed antenna at, (a) 4 GHz, (b) 8 GHz, and (c) 12 GHz.

Fig. 3. Simulated return loss characteristics for antennas shown in Fig. 2.

Fig. 6. Simulated return loss characteristics for various values of d

Fig. 4. Simulated surface current distributions on the radiating patch and ground plane for (a) the square antenna with inverted T-shaped slot at third resonance frequency (12.2 GHz), (b) the square antenna with inverted T-shaped slot and parasitic structure at fourth resonance frequency (14.4 GHz).

absence of the inverted T-shaped slot and conductor-backed plane. The upper frequency bandwidth is significantly affected by using the inverted T-shaped slot in the radiating patch. This behavior is mainly due to the change of surface current path by the dimensions of inverted T-shaped slot as shown in Fig. 4(a). In addition, by inserting inverted T-shaped conductor-backed plane on the other side of substrate the impedance bandwidth is effectively improved at the upper frequency [13]. The inverted T-shaped conductor backed plane can be regarded as a parasitic resonator electrically coupled to the square monopole. As shown in Fig. 4(b), the current concentrated on the edges of the interior and exterior of the inverted T-shaped conductor-backed plane at fourth resonance frequency (14.4 GHz). This figure shows that the electrical current for the fourth resonance frequency [Fig. 4(b)] does change direction along the bottom edge of the square radiating patch. Therefore, the antenna impedance changes at this frequency, the radiating power and bandwidth will increases. Furthermore, the radiation efficiency will increase. However, the resonant resistance is decreased [7]. The proposed antenna has a slightly higher efficiency rather than ordinary square antenna throughout the entire radiating band, which is mainly owing to the electromagnetic coupling and the new resonant

.

properties. Results of the calculations using the software HFSS indicated that the proposed antenna features a good efficiency, being greater than 82% across the entire radiating band. As shown in Fig. 4(b), the direction of the simulated surface current distributions on the patch for the proposed antenna with inverted T-shaped slot and parasitic structure is downward. The change of this current direction is due to the selected frequency (14.4 GHz), and it has not any relationship with inverted T-shaped slot and parasitic structure. To clarify this point Fig. 5 shows the current distributions for the same proposed antenna (with inverted T-slot and parasitic structure) at three typical frequencies in the band. It can be seen that the current directions in Fig. 5(a) and (c) are upward, and the current directions in Fig. 5(b) are downward. By properly tuning the dimensions and spacing (dPS ) to semi-ground plane for the T-shaped conductor backed plane, the antenna can create the fourth resonant frequency in individual resonant radiation band based on an over-coupling condition. Fig. 6 shows the effects of the feed gap distance dPS (as shown Fig. 2, dPS = Lf 0 Lgnd 0 Lps 0 LP 1 ) of the square patch and dimension of the inverted T-shaped conductor-backed plane on the impedance. As illustrated in Fig. 6, the feed gap distance dPS is an important parameter in determining the sensitivity of impedance matching. By adjusting dPS , the electromagnetic coupling between the bottom edge of the square patch and the ground plane can be properly controlled [6]. Fig. 7 shows the measured and simulated return loss characteristics of the proposed antenna. The fabricated antenna satisfies the 10-dB return loss requirement from 2.91 to 14.1 GHz. As shown in Fig. 7, there exists a discrepancy between measured data and the simulated results

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Fig. 7. Measured and simulated return loss for the proposed antenna.

Fig. 9. Measured and simulated radiation patterns of the proposed antenna. (a) Third resonance frequency (12.2 GHz), (b) fourth resonance frequency (14 GHz).

(Fig. 8) and at 12.2 and 14.4 GHz (Fig. 9). Similar agreement is obtained at higher frequencies. It can be seen that the radiation patterns in - plane are nearly omni-directional even at higher frequencies, and also the cross-polarization level is low for the four frequencies. The measured radiation patterns agree very well with the simulated results in the four frequencies.

xz

IV. CONCLUSION

Fig. 8. Measured and simulated radiation patterns of the proposed antenna. (a) First resonance frequency (4 GHz), (b) second resonance frequency (10 GHz).

this could be due to the effect of the SMA port. In order to confirm the accurate return loss characteristics for the designed antenna, it is recommended that the manufacturing and measurement process need to be performed carefully. Figs. 8 and 9 show the simulated and measured radiation patterns at resonance frequencies including the co-polarization and cross-polarization in the -plane ( - plane) and -plane ( - plane). The main purpose of the radiation patterns is to demonstrate that the antenna actually radiates over a wide frequency band. Reasonable agreement between simulations and measurements is demonstrated at 4 and 10 GHz

H

xz

E

yz

In this letter, a novel compact printed monopole antenna (PMA) has been proposed for UWB applications. The fabricated antenna satisfies the 10-dB return loss requirement from 2.9 to 14.1 GHz. By cutting a modified inverted T-shaped slot with variable dimensions on the radiating patch and also by inserting an inverted T-shaped conductor-backed plane, additional resonances are excited and hence much wider impedance bandwidth can be produced, especially at the higher band. The proposed antenna has a simple configuration and is easy to fabricate. Experimental results show that the proposed antenna could be a good candidate for UWB application. ACKNOWLEDGMENT The authors thank Microwave Technology Company staff for their beneficial and professional help. (www.microwave-technology.com)

REFERENCES [1] H. Schantz, The Art and Science of Ultra Wideband Antennas. Boston, MA: Artech House, 2005. [2] M. J. Ammann, “Impedance bandwidth of the square planar monopole,” Microw. Opt. Technol. Lett., vol. 24, no. 3, Feb. 2000.

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[3] J. A. Evans and M. J. Ammann, “Planar trapezoidal and pentagonal monopoles with impedance bandwidths in excess of 10:1,” in Proc. IEEE Antennas and Propagation Society Int. Symp., Jul. 1999, vol. 3, pp. 1558–1561. [4] S. Y. Suh, W. L. Stutzman, and W. A. Davis, “A new ultrawideband printed monopole antenna: The planar inverted cone antenna (PICA),” IEEE Trans. Antennas Propag., vol. 52, no. 5, pp. 1361–1364, May 2004. [5] A. J. Kerkhoff, R. L. Rogers, and H. Ling, “Design and analysis of planar monopole antennas using a genetic algorithm approach,” IEEE Trans. Antennas Propag., vol. 2, pp. 1768–1771, Jun. 2004. [6] M. Ojaroudi, G. Kohneshahri, and Ja. Noory, “Small modified monopole antenna for UWB application,” IET Microw, Antennas Propag., vol. 3, no. 5, pp. 863–869, Aug. 2009. [7] M. Ojaroudi, Ch. Ghobadi, and J. Nourinia, “Small square monopole antenna with inverted t-shaped notch in the ground plane for UWB application,” IEEE Antennas Wireless Propag. Lett., vol. 8, no. 1, pp. 728–731, 2009. [8] M. Ojaroudi, Gh. Ghanbari, N. Ojaroudi, and Ch. Ghobadi, “Small square monopole antenna for UWB applications with variable frequency band-notch function,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 1061–1064, 2009. [9] E. Antonino-Daviu, M. Cabedo-Fabres, M. Ferrando-Bataller, and A. Valero-Nogueira, “Wideband double fed planar monopole antennas,” Electron. Lett., vol. 39, no. 23, pp. 1635–1636, Nov. 2003. [10] C.-W. Ling and S.-J. Chung, “A simple printed ultrawideband antenna with a quasi-transmission line section,” IEEE Trans. Antennas Propag., vol. 57, no. 10, pp. 3333–3336, 2009. [11] S. T. See and Z. N. Chen, “A planar UWB antenna with a broadband feeding stru,” IEEE Trans. Antennas Propag., vol. 57, no. 6, pp. 1597–1605, 2009. [12] J. Jung, W. Choi, and J. Choi, “A compact broadband antenna with an L-shaped notch,” IEICE Trans. Commun., vol. E89-B, no. 6, pp. 1968–1971, Jun. 2006. [13] C.-Y. Pan, T.-S. Horng, W.-S. Chen, and C.-H. Huang, “Dual wideband printed monopole antenna for WLAN/WiMAX applications,” IEEE Antennas Wireless Propag. Lett., vol. 6, pp. 149–151, 2007. [14] H. D. Chen, H. M. Chen, and W. S. Chen, “Planar CPW-fed sleeve monopole antenna for ultra-wideband operation,” IET Microw. Antennas Propag., vol. 152, no. 6, pp. 491–494, Dec. 2005. [15] Ansoft High Frequency Structure Simulation (HFSS). Pittsburgh, PA: Ansoft Corporation, 2005, Ver. 10.

Coupled-Fed Shorted Monopole With a Radiating Feed Structure for Eight-Band LTE/WWAN Operation in the Laptop Computer Ting-Wei Kang, Kin-Lu Wong, Liang-Che Chou, and Ming-Ren Hsu

Abstract—A coupled-fed shorted monopole with its feed structure as an effective radiator for eight-band LTE/WWAN (LTE700/GSM850/900/1800/ 1900/UMTS/LTE2300/2500) operation in the laptop computer is presented. The radiating feed structure capacitively excites the shorted monopole. The feed structure mainly comprises a long feeding strip and a loop feed therein. The loop feed is formed at the front section of the feeding strip and connected to a 50- mini-cable to feed the antenna. Both the feeding strip and loop feed contribute wideband resonant modes to combine with those generated by the shorted monopole for the desired eight-band operation. The antenna size above the top shielding metal wall of the laptop display is and is suitable to be embedded inside the casing of the laptop computer. The proposed antenna is fabricated and tested, and good radiation performances of the fabricated antenna are obtained.



4 10 80 mm

Index Terms—Coupled-fed antennas, internal laptop computer antennas, LTE antennas, shorted monopoles, WWAN antennas.

I. INTRODUCTION The capability of penta-band WWAN (wireless wide area network) operation in the GSM850/900/1800/1900/UMTS (824  896=880  960=1710  1880=1850  1990=1920  2170 MHz) bands [1] has been provided in many laptop computers to enhance their functionality in wireless internet access. For this application, promising penta-band WWAN internal laptop computer antennas have been devised and reported in published papers [2]–[9]. Very recently, owing to the introduction of the LTE (long term evolution) operation [10] which can provide much higher data rate than the WWAN operation, it is now required in some laptop computers that their internal antennas should cover the LTE operation which includes the LTE700 (698  787 MHz), LTE2300 (2300  2400 MHz) and LTE2500 (2500  2690 MHz) bands. The internal antennas that can cover the eight-band LTE/WWAN (including three LTE and five WWAN bands) operation are therefore preferred for the modern laptop computers. However, for the present, the available internal laptop computer antennas that have been reported [2]–[9] cannot provide wide operating bands to cover the eight-band operation. In this communication, we present a coupled-fed shorted monopole with a radiating feed structure to provide wide operating bands for the desired eight-band LTE/WWAN operation in the laptop computer. The radiating feed structure not only serves as a capacitive feed to couple the energy to the shorted monopole, but also functions as an effective radiator to contribute wideband resonant modes for the antenna. The resonant modes contributed by the radiating feed structure combine Manuscript received January 05, 2010; revised June 24, 2010; accepted August 07, 2010. Date of publication December 03, 2010; date of current version February 02, 2011. T.-W. Kang and K.-L. Wong are with the Department of Electrical Engineering, National Sun Yat-sen University, Kaohsiung 804, Taiwan (e-mail: [email protected]). L.-C. Chou and M.-R. Hsu are with the High Frequency Business Unit, Yageo Corporation Nantze Branch, Kaohsiung 811, Taiwan. Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2096390

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Fig. 1. Geometry of the proposed coupled-fed shorted monopole with a radiating feed structure for eight-band internal LTE/WWAN laptop computer antenna.

with those generated by the shorted monopole to lead to two wide operating bands for the antenna to respectively cover the desired 698  960 MHz band for the LTE700/GSM850/900 operation and the 1710  2690 MHz band for the GSM1800/1900/UMTS/LTE2300/2500 operation. With the eight-band operation obtained, the antenna size above the top shielding metal wall of the laptop display is 4 2 10 2 80 mm3 and is suitable to be embedded inside the casing of the laptop computer. The operating principle of the proposed antenna is discussed in detail in the communication. The antenna is also fabricated and tested. Obtained results are presented and analyzed. II. PROPOSED ANTENNA FOR THE LTE/WWAN OPERATION Fig. 1 shows the geometry of the proposed eight-band internal LTE/ WWAN laptop computer antenna. The antenna is to be mounted along the top shielding metal wall (5 mm in width) of the supporting metal frame (a 0.2-mm thick copper plate measuring 260 2 200 mm2 in this study) of the laptop display. The top shielding metal wall is usually added to provide some isolation between the internal antenna and the circuitry on the back side of the laptop display. However, the presence of the shielding metal wall usually results in some degrading effects on the impedance matching of the internal antenna, which decreases the bandwidths of the antenna. By including the shielding metal wall in the antenna design, the proposed antenna with a small volume of 4 2 10 2 80 mm3 can still generate two wide operating bands to respectively cover the desired eight-band operation. Also, since the central region is usually reserved for the lens of the embedded digital camera, the proposed antenna is placed with a spacing of 30 mm to the center line of the laptop display. The proposed antenna mainly comprises a shorted monopole and a radiating feed structure. The shorted monopole is fabricated from a 0.2-mm thick copper plate and consists of two radiating arms of different lengths (one longer arm and one shorter arm). The longer and shorter arms are short-circuited to the shielding metal wall through a shorting strip of length 25 mm and capacitively excited by the radiating feed structure which comprises a long feeding strip and a loop feed (loop ACD shown in the figure) formed at the front section of the feeding strip. The radiating feed structure is printed on a 0.4-mm thick FR4 substrate of relative permittivity 4.4 and size 10 2 80 mm2 in the study. It not only serves as a capacitive feed to couple the energy

Fig. 2. Photo of the fabricated antenna.

to the shorted monopole, but also functions as an effective radiator to contribute wideband resonant modes for the antenna. This characteristic in functioning as a feed and a radiator is an advantage over that of the feeding strip of the traditional coupled-fed antennas [11]–[23], in which the feeding strip does not contribute additional resonant modes. In the proposed antenna, the longer arm and shorter arm of the shorted monopole are excited by the radiating feed structure to respectively contribute a wideband resonant mode at about 900 and 1800 MHz. The long feeding strip and the loop feed in the radiating feed structure will also contribute additional wideband resonant modes at about 2100 and 2500 MHz, respectively. These excited resonant modes form two wide operating bands to cover the desired 698  960 MHz and 1710  2690 MHz bands for the eight-band LTE/WWAN operation. Note that for practical applications, the proposed antenna is fed by a 50- mini-cable (diameter 1.37 mm), where the center conductor and outer grounding sheath are respectively connected to point A in the loop feed and point B at the shielding metal wall. In this study, the 50- mini-cable is aligned on the shielding metal wall as it usually done in practical applications. III. RESULTS AND DISCUSSION The photo of the fabricated antenna is shown in Fig. 2. The simulated return loss obtained using HFSS version 12 [24] and the measured data are presented in Fig. 3. From the results, the measured data are similar to the simulated results. The discrepancies between the measured and simulated results are related to the coupling between the mini-cable and the antenna in the experiment. (It has been noted that when the mini-cable is not aligned along the shielding metal wall and placed

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Fig. 3. Measured and simulated return loss for the proposed antenna.

Fig. 5. Measured radiation efficiency and antenna gain of the fabricated antenna. (a) The lower band. (b) The upper band.

Fig. 4. Measured radiation patterns at (a) 830 MHz, (b) 1940 MHz and (c) 2500 MHz for the fabricated antenna.

close to the FR4 substrate as shown in Fig. 1 and is moved away to be parallel to the center line of the display ground, the measured return loss becomes much closer to the simulated result which is obtained using a

gap source in this study.) From the measurement, the 3:1 VSWR (6-dB return loss) bandwidth, which is widely used as the practical design specification of the internal WWAN antennas for mobile communications [1]–[9], is about 320 MHz (640  960 MHz) in the lower band and about 1000 MHz (1695  2695 MHz) in the upper band. That is, two wide operating bands are obtained, and the desired eight-band LTE/WWAN operation can be covered by the proposed antenna. Also note that the plotted marks in the figure throughout the communication are used to distinguish each line and are not the data points. Fig. 4 shows the measured radiation patterns at 830, 1940, and 2500 MHz for the fabricated antenna. In the figure, the E and E components are shown. From the results, it can be obtained that the total power patterns (E and E together) will show no nulls in the azimuthal plane (x 0 y plane). This is an advantage for laptop computer applications. No large variations in the radiation patterns versus frequency in each operating band of the eight-band LTE/WWAN operation have also been observed. Stable radiation patterns are therefore obtained for the proposed antenna for LTE/WWAN operation. The measured radiation efficiency and antenna gain of the fabricated antenna are presented in Fig. 5. Results for the lower and upper bands are respectively shown in Fig. 5(a) and (b). The measured results show that the radiation efficiency is all larger than 50% over the eight operating bands. The measured antenna gain varies from about 2.0  3.5 dBi over the lower band and about 2.0  6.1 dBi over the upper band. The operating principle of the proposed antenna is also analyzed. Fig. 6 shows the simulated return loss for the proposed antenna and the case without the loop feed (Ref1). It is clearly seen that the loop feed can contribute an additional wideband resonant mode at about 2500 MHz to help achieve a very wide upper band to cover the desired frequency range of 1710  2690 MHz. Without the loop feed, the bandwidth of the antenna’s upper band is far from that is required. Fig. 7 shows the simulated return loss for the proposed antenna, the case with the feeding strip only (Ref2), the case with the feeding strip

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Fig. 8. Simulated return loss for the proposed antenna positioned at different locations along the top edge of the supporting metal frame of the laptop display.

Fig. 6. Simulated return loss for the proposed antenna and the case without the loop feed (Ref1).

Fig. 9. Simulated return loss for the proposed antenna as a function of the width w of the shielding metal wall.

Fig. 7. Simulated return loss for the proposed antenna, the case with the feeding strip only (Ref2), the case with the feeding strip and the loop feed only (Ref3), and the case with the feeding strip, the loop feed and the longer arm of the shorted monopole (Ref4).

and the loop feed only (Ref3), and the case with the feeding strip, the loop feed and the longer arm of the shorted monopole (Ref4). Ref2 is with the feeding strip only, without the loop feed and the shorted monopole. Ref2 generates its fundamental resonant mode with poor impedance matching at about 850 MHz and its higher-order resonant mode at about 2000 MHz. By further adding the loop feed (Ref3), it generates an additional resonant mode at about 2500 MHz. This resonant mode combines with the one contributed by the feeding strip to obtain an enhanced bandwidth for the upper band, which however is still not wide enough to cover the desired 1710  2690 MHz band. Next, by further adding the longer arm of the shorted monopole (Ref4), which is capacitively excited by the proposed radiating feed structure (the feeding strip and the loop feed), a dual-resonance excitation for the antenna’s lower band is obtained. However, some degradation in the impedance matching for frequencies at around 2000 MHz is also caused by the adding of the longer arm of the shorted monopole. Finally, by adding the shorter arm of the shorted monopole, an additional resonant mode at about 1800 MHz is excited, and the impedance matching for frequencies at around 2000 MHz is also re-claimed to

be with an acceptable level. In this case, wide lower and upper bands are obtained for the proposed antenna to cover the desired eight-band LTE/WWAN operation. Effects of the locations at which the antenna is positioned are analyzed in Fig. 8. The simulated results of the return loss for the proposed antenna positioned near the right edge (Fig. 1), at the center, and near the left edge along the top edge of the supporting metal frame of the laptop display are shown in the figure. Large effects of different locations on the impedance matching in the lower band are seen. Degraded impedance matching in the lower band for the proposed antenna positioned at the center and near the left edge is seen. This is reasonable since the antenna dimensions in this study are adjusted for the proposed antenna to be positioned near the right edge as seen in Fig. 1, which is the usual case for the internal WWAN or LTE/WWAN antenna embedded in the laptop computer. As for the center region, it is usually reserved for the lens of the embedded digital camera, and the left region is usually for the embedded WLAN antennas. Effects of the width w of the shielding metal wall are studied in Fig. 9. The simulated results of the return loss for the width w varied from 0 to 7 mm are shown in the figure. When there is no shielding metal wall, better impedance matching or wider bandwidth is in general obtained for the proposed antenna, which is similar to the observations in [25]. Fig. 10 shows the simulated return loss for the proposed antenna and the case with a metal pad replacing the loop feed (Ref5). Note that the metal pad in Ref5 has the same size and shape as the loop feed. In this case, the resonant mode at about 2500 MHz seen in the proposed antenna cannot be generated for Ref5. Fig. 11 shows the simulated return loss for the proposed antenna and the case with a simple rectangular loop feed (Ref6). Some degradation in the impedance matching for frequencies in the lower band and around 2.1 GHz in the upper band is seen. This can be improved by slightly adjusting the shape of the loop feed as seen in Fig. 1 to reduce

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Fig. 10. Simulated return loss for the proposed antenna and the case with a metal pad replacing the loop feed (Ref5).

Fig. 11. Simulated return loss for the proposed antenna and the case with a simple rectangular loop feed (Ref6).

Fig. 13. Simulated return loss for the proposed antenna, the case with the radiating feed structure only (Ref3 in Fig. 7), and the case with the shorted monopole not coupled-fed by the radiating feed structure (Ref7).

for the proposed antenna with the keyboard ground is still possible by slightly adjusting the antenna dimensions. Effects of the coupling excitation of the radiating feed structure on the shorted monopole are analyzed in Fig. 13. Results of the simulated return loss for the proposed antenna, the case with the radiating feed structure only (Ref3 in Fig. 7), and the case with the shorted monopole not coupled-fed by the radiating feed structure (Ref7) are presented. Results clearly indicate that without the coupling excitation of the radiating feed structure, the shorted monopole cannot be excited as it is in the proposed antenna.

IV. CONCLUSION

Fig. 12. Simulated return loss for the proposed antenna with and without the keyboard ground.

the mutual coupling between the loop feed, the feeding strip, and the longer arm of the shorted monopole. Effects of the keyboard ground [7], [26], [27] on the proposed antenna are studied in Fig. 12. Results of the simulated return loss for the antenna with and without the keyboard ground are shown. The keyboard ground has the same dimensions as those of the supporting metal frame (260 2 200 mm2 ) in Fig. 1, and the antenna dimensions for the two cases are the same. Small variations in the simulated return loss are seen, when the keyboard ground is present. This is reasonable, since the supporting metal frame has very large dimensions compared to those of the proposed antenna. Hence, with the presence of the keyboard ground, small effects can be expected. Improved impedance matching

A wideband coupled-fed shorted monopole for eight-band internal LTE/WWAN laptop computer antenna has been proposed, fabricated, and tested. The wideband operation for the proposed antenna is achieved by using a radiating feed structure to capacitively excite a shorted monopole. The radiating feed structure further contributes additional wideband resonant modes to combine with those generated by the shorted monopole such that two wide operating bands are obtained for the antenna to respectively cover the LTE700/GSM850/900 operation (698  960 MHz) and the GSM1800/1900/UMTS/LTE2300/2500 operation (1710  2690 MHz). Good radiation characteristics for frequencies over the eight operating bands have also been obtained. A detailed description of the operating principle of the proposed antenna in exciting the resonant modes for the desired eight-band operation has also been provided. The proposed antenna occupies a small volume of 3 4 2 10 2 80 mm at the top shielding metal wall of the laptop display and is suitable to operate as an internal LTE/WWAN antenna in the laptop computers.

REFERENCES [1] K. L. Wong, Planar Antennas for Wireless Communications. New York: Wiley, 2003. [2] X. Wang, W. Chen, and Z. Feng, “Multiband antenna with parasitic branches for laptop applications,” Electron. Lett., vol. 43, pp. 1012–1013, Sep. 13, 2007. [3] C. H. Chang and K. L. Wong, “Internal coupled-fed shorted monopole antenna for GSM850/900/1800/1900/UMTS operation in the laptop computer,” IEEE Trans. Antennas Propag., vol. 56, pp. 3600–3604, Nov. 2008.

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[4] K. L. Wong and S. J. Liao, “Uniplanar coupled-fed printed PIFA for WWAN operation in the laptop computer,” Microwave Opt. Technol Lett., vol. 51, pp. 549–554, Feb. 2009. [5] K. L. Wong and L. C. Lee, “Multiband printed monopole slot antenna for WWAN operation in the laptop computer,” IEEE Trans. Antennas Propag., vol. 57, pp. 324–330, Feb. 2009. [6] C. Zhang, S. Yang, S. El-Ghazaly, A. E. Fathy, and V. K. Nair, “A lowprofile branched monopole laptop reconfigurable multiband antenna for wireless applications,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 216–219, 2009. [7] K. L. Wong and F. H. Chu, “Internal planar WWAN laptop computer antenna using monopole slot elements,” Microwave Opt. Technol Lett., vol. 51, pp. 1274–1279, May 2009. [8] C. W. Chiu, Y. J. Chi, and S. M. Deng, “An internal multiband antenna for WLAN and WWAN applications,” Microwave Opt. Technol Lett., vol. 51, pp. 1803–1807, Aug. 2009. [9] C. T. Lee and K. L. Wong, “Study of a uniplanar printed internal WWAN laptop computer antenna including user’s hand effects,” Microwave Opt. Technol Lett., vol. 51, pp. 2341–2346, Oct. 2009. [10] , S. Sesia, I. Toufik, and M. Baker, Eds., LTE, The UMTS Long Term Evolution: From Theory to Practice. New York: Wiley, 2009. [11] C. T. Lee and K. L. Wong, “Uniplanar printed coupled-fed PIFA with a band-notching slit for WLAN/WiMAX operation in the laptop computer,” IEEE Trans. Antennas Propag., vol. 57, pp. 1252–1258, Apr. 2009. [12] C. H. Wu and K. L. Wong, “Ultra-wideband PIFA with a capacitive feed for penta-band folder-type mobile phone antenna,” IEEE Trans. Antennas Propag., vol. 57, pp. 2461–2464, Aug. 2009. [13] C. T. Lee and K. L. Wong, “Internal WWAN clamshell mobile phone antenna using a current trap for reduced groundplane effects,” IEEE Trans. Antennas Propag., vol. 57, pp. 3303–3308, Oct. 2009. [14] K. L. Wong and C. H. Huang, “Bandwidth-enhanced internal PIFA with a coupling feed for quad-band operation in the mobile phone,” Microwave Opt. Technol. Lett., vol. 50, pp. 683–687, Mar. 2008. [15] C. H. Chang, K. L. Wong, and J. S. Row, “Coupled-fed small-size PIFA for penta-band folder-type mobile phone application,” Microwave Opt. Technol. Lett., vol. 51, pp. 18–23, Jan. 2009. [16] C. T. Lee and K. L. Wong, “Uniplanar coupled-fed printed PIFA for WWAN/WLAN operation in the mobile phone,” Microwave Opt. Technol. Lett., vol. 51, pp. 1250–1257, May 2009. [17] T. W. Kang and K. L. Wong, “Simple small-size coupled-fed uniplanar PIFA for multiband clamshell mobile phone application,” Microwave Opt. Technol. Lett., vol. 51, pp. 2805–2810, Dec. 2009. [18] C. H. Chang and K. L. Wong, “Printed /8-PIFA for penta-band WWAN operation in the mobile phone,” IEEE Trans. Antennas Propag., vol. 57, pp. 1373–1381, May 2009. [19] K. L. Wong and C. H. Huang, “Compact multiband PIFA with a coupling feed for internal mobile phone antenna,” Microwave Opt. Technol. Lett., vol. 50, pp. 2487–2491, Oct. 2008. [20] W. J. Tseng, “Coupled-Fed Antenna Device,” U.S. Patent 2007/ 0257842 A1, Nov. 8, 2007. [21] L. Lu and J. C. Coetzee, “A modified dual-band microstrip monopole antenna,” Microwave Opt. Technol. Lett., vol. 48, pp. 1401–1403, Jul. 2006. [22] R. Borowiec and P. M. Slobodzian, “A miniaturized antenna for 2G/3G frequency-based applications,” Microwave Opt. Technol. Lett., vol. 48, pp. 399–402, Feb. 2006. [23] C. R. Rowell and R. D. Murch, “A compact PIFA suitable for dualfrequency 900/1800-MHz operation,” IEEE Trans. Antennas Propag., vol. 46, pp. 596–598, Apr. 1998. [24] [Online]. Available: http://www.ansoft.com/products/hf/hfss/Ansoft Corporation HFSS [25] S. Y. Lin, “Multi-band folded planar monopole antennas for mobile handset,” IEEE Trans. Antennas Propag., vol. 52, pp. 1790–1794, Jul. 2004. [26] C. Zhang, S. Yang, S. El-Ghazaly, A. E. Fathy, and V. K. Nair, “A lowprofile branched monopole laptop reconfigurable multiband antennas for wireless applications,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 216–219, 2009. [27] J. Guterman, A. A. Moreira, C. Peixeiro, and Y. Rahmat-Samii, “Wrapped microstrip antennas for laptop computers,” IEEE Antennas Propag. Mag., vol. 51, pp. 12–39, Aug. 2009.

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Lower Bound for the Radiation of Electrically Small Magnetic Dipole Antennas With Solid Magnetodielectric Core Oleksiy S. Kim and Olav Breinbjerg Abstract—A new lower bound for the radiation Q of electrically small spherical magnetic dipole antennas with solid magnetodielectric core is derived in closed form using the exact theory. The new bound approaches the Chu lower bound from above as the antenna electrical size decreases. For ka < 0:863, the new bound is lower than the bounds for spherical magnetic as well as electric dipole antennas composed of impressed electric currents in free space. Index Terms—Chu lower bound, electrically small antennas, fundamental limitations, magnetic dipole, quality factor.

I. INTRODUCTION An ideal electrically small magnetic dipole antenna composed of an impressed electric current on the surface of an imaginary sphere exhibits a radiation three times the Chu lower bound [1], [2]. When the surface current encloses a magnetic core the internal stored magnetic energy reduces, and the Chu lower bound [3] can be reached for a vanishingly small antenna. However, the behavior of a finite-size magnetic dipole antenna is more complicated due to the internal resonances in the magnetic core. Exact analytical expressions derived in [4] show that the minimum achievable is a function of the antenna electrical size. This function represents the lower bound for the radiation of a magnetic dipole antenna with a magnetodielectric core. In this communication, taking account of the rigorous theory presented in [4], we derive an analytical expression for this lower bound and define its range of validity.

Q

Q

Q

II. LOWER BOUND FOR THE QUALITY FACTOR We consider an impressed electric current on the surface of a material spherical core with relative permeability r and permittivity r . In a ) the surface current density radiating spherical coordinate system ( the TE10 spherical mode (magnetic dipole mode) is



r

"

J = a^ J0 sin 

(1)

J

^ is the azimuthal unit vector and 0 is the amplitude (A/m). where a Closed-form expressions for the stored magnetic H and electric E energies as well as for the radiated power rad are derived in [4]. From those, the radiation quality factor can be expressed in compact form, subject to H  E , as

W W Q(ka;  ; " ) =

W

P

Q

W

1 (2)  3(ks a) Q (ka) where k is free-space wave number, a is the radius of the antenna, ks = p"  k is the wave number in the core material, the function 3(ks a) 1 + (ks a)j (ks a)y (ks a) (3) 3(ks a) = 01 2j (ks a) r

r

r

1+

Chu

r

r

0 2 1

0

Manuscript received February 17, 2010; revised June 24, 2010; accepted August 31, 2010. Date of publication December 03, 2010; date of current version February 02, 2011. This work was supported by the Danish Research Council for Technology and Production Sciences within the TopAnt project. The authors are with the Department of Electrical Engineering, Electromagnetic Systems, Technical University of Denmark, DK-2800 Kgs. Lyngby, Denmark (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TAP.2010.2096394

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TABLE I ZEROS u OF THE FUNCTION F u IN (7) AND THE CORRESPONDING CONSTANTS  IN (9)

()

Fig. 1. Optimum permeability (8) as a function of the antenna electrical size ka for n and " , 4, 12.

=1

=1

jn (x) and yn (x) are the n-order spherical Bessel functions of the first and second kind, respectively, and QChu is the Chu lower bound [3], [5] QChu (ka) =

1

(ka)3

+

1

ka

:

(4)

In the limiting case of a vanishingly small antenna, expression (2) reduces to the well-known formula by Wheeler [1] lim Q(ka; r ; "r ) = Q(ka; r ) =

ka!0

1+

2

r

1

(ka)3

:

(5)

For a finite-size antenna, the function 3(ks a) exhibits singularities at zeroes of the Bessel function j1 (ks a), and, consequently, so does the quality factor Q in (2). The singularities originate from the internal TE10 -mode resonances in the spherical core, when the external fields, and thus the radiated power, are zero. To find the minimum Q between the resonances and the corresponding optimal permeability of the core, we have to solve the following equation @Q(ka; r ; "r ) @r

= 0:

(6)

Substituting (2) into (6) and letting u = ks a, we obtain u @ 3(u) 2

@u

0 3(u)  FF12 ((uu)) = 0:

2

un

(7)

(8)

"r (ka)2

and illustrated in Fig. 1. Obviously, for each solution un there is a corresponding quality factor Qn (ka), which can be obtained from (2) taking into account (8) as Qn (ka; "r ) =



2

1 + (ka) "r

2

3(un )

2 un

QChu (ka)

1 + (ka) "r n QChu (ka):

=

=

The first four values of un and the corresponding constants n are given in Table I. From Table I as well as from Fig. 2, which illustrates the expression (9), it is clear, that the first value u1 yields the lowest n , and, consequently, the lowest Qn (ka; "r ). Since the Qn (ka; "r ) also reduces as the permittivity "r decreases, the optimal core must be pure magnetic with relative permeability r (ka) =

(9)

2

u1

(ka)2

:

(10)

Finally, the lower bound for magnetic dipole antennas can be written from (9) as QLB (ka) = Q1 (ka; "r = 1) = =

This equation has an infinite number of solutions—zeroes un (n = 1; 2; 3; . . .) of the oscillating function F1 (u). The corresponding optimal permeability is then found as r (ka; "r ) =

Fig. 2. Normalized quality factor Q =Q (9) corresponding to the optimum permeability (8) for n 1, 2, 3 and " 1, 4, 12.

2

1 + (ka) 1 QChu (ka) 1

(ka)3

+

1 + 1 ka

+ (ka)1 :

(11)

The conclusion is that for a given antenna electrical size ka there is an optimum magnetic core permeability (10) that ensures the lowest possible Q as given in (11). This Q is thus the lower bound for magnetic dipole antennas with solid magnetodielectric core. As illustrated in Fig. 3, the new bound (11) is higher than the Chu lower bound, but approaches it as ka ! 0. At the same time, it shows that with a properly selected magnetic core an electrically small magnetic dipole antenna exhibits a Q that is lower not only as compared to the same antenna with air core, but also as compared to an air-core electric dipole antenna1 of the same size. We note, however, that in a region 0:863 < ka < 1:325—although, in this region an antenna can hardly be considered electrically small—an air-core electric dipole antenna performs slightly better. 1Here, and throughout the communication, we assume that air-core magnetic and electric dipole antennas are composed of electric current distributions on an imaginary spherical surface in free space, that is, no other materials or objects, neither inside, nor on the surface, nor outside of the antenna, are present.

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Fig. 3. The lower bound for magnetic dipole antennas with magnetodielectric core is compared to the bounds for air-core electric and magnetic dipole an. tennas [6], [7]. The bounds are normalized to the Chu lower bound

Q

ka;  W