REFERENCES

1. D.M. Pozar, Input impedance and mutual coupling of rectangular mi-crostrip antennas, IEEE Trans Antennas Propagat AP-30 (1982), 1191–1196.

2. S. Sagiroglu and K. Guney, Calculation of resonant frequency for anequilateral triangular microstrip antenna with the use of artificial neuralnetworks, Microwave Opt Technol Lett 14 (1997), 89–93.

3. S.S. Pattnaik, D.C. Panda, and S. Devi, Input impedance of microstrippatch antenna using artificial neural networks, Microwave Opt TechnolLett, 32 (2002), 381–383.

4. V. Rao and H. Rao, C�� neural networks and fuzzy logic, BPB, 1996,p. 336.

© 2002 Wiley Periodicals, Inc.

EXTENSION OF IMPEDANCE MATRIXCOMPRESSION METHOD WITHWAVELET TRANSFORM FOR 2-DCONDUCTING AND DIELECTRICSCATTERING OBJECTS DUE TOOBLIQUE PLANE-WAVE INCIDENCE

Jin Yu and Ahmed A. KishkDepartment of Electrical EngineeringUniversity of MississippiUniversity, Mississippi 38677

Received 7 January 2002

ABSTRACT: The impedance matrix compression (IMC) technique isapplied to analyze the square method–of-moments (MoM) matrix arisingfrom the surface integral equation for 2-D conducting and dielectricobjects with oblique plane-wave incidence. The induced current compo-nents are processed separately by using wavelet basis functions. Thecomparison between cylinders with circular and square cross section ispresented to show the effectiveness of IMC, which depends on the geom-etry of the object and the wavelet transform matrix. © 2002 Wiley Pe-riodicals, Inc. Microwave Opt Technol Lett 34: 53–56, 2002;Published online in Wiley InterScience (www.interscience.wiley.com).DOI 10.1002/mop.10371

Key words: wavelet transforms; impedance matrix compression; methodof moments

1. INTRODUCTION

One of the most important applications of wavelet transform in themethod of moments is the use of few wavelet basis functions toexpress the unknown induced current components on the bound-ary. Recently, applications of wavelet transform have been re-garded to be synonymous with data compression [1]. Only signif-icant and necessary wavelet basis expansion functions for theunknown currents are iteratively selected in the impedance matrixcompression (IMC), which is used to analyze the scattering prob-lems of conducting objects with normal TMz incident wave andwhere the original MoM matrix of size M�N is not square; M�2N[2]. IMC leads to a significant decrease in the size of the wavelet-transformed MoM matrix. IMC with two-stage wavelet packettransformations is used to analyze a truncated periodic array [3].The correlation between wavelet expansion of the error satisfyingthe boundary conditions and wavelet expansion of the unknowninduced current plays a key role in choosing the associated waveletbasis functions for the next iteration in the algorithm [2]. Onlyconducting objects are considered, and only one current compo-nent exists on the boundary of the objects in References 1–3.

In this Letter, the analysis of conducting and dielectric infinitecylinders with circular and square cross section with oblique planewave incidence with the use of IMC are presented, and the MoMmatrices [4] for these objects are square: N�N, and with numberof wavelet basis functions, Q (Q �1, 2, 3, 4) added into the nextiteration. Because of the oblique wave incidence on the scatteringobjects, electric current components (Jz, Jt) and magnetic currentcomponents (Mz, Mt) exist on the boundary of the dielectric object,and (Jz, Jt) exist on the boundary of the conducting objects, wherez and t stand for vertical and transverse direction, respectively. Thetransform matrix [W] is constructed based on the number of theequivalent current components, which are referred to as waveletsubtransform, and is used to initialize the iteration.

In Section 2, the formulation and a method of improving theefficiency of IMC are presented. Numerical results for circularcylinder and square cross cylinder are given in Section 3.

2. FORMULATION

The method of moments is applied to the electromagnetic surfaceintegral equation with pulse basis expansion functions and pointmatching, and a conventional square matrix equation is obtained[4]:

�Z��I� � �V�, (1)

Figure 1 Flow chart for the algorithm of the modified impedance matrixcompression (IMC)

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 34, No. 1, July 5 2002 53

where [Z] is MoM matrix, [I] is unknown induced current vector,and [V] is excitation vector. If wavelet transform is carried out withwavelet subtransform matrix [W], one gets

�Z���I�� � �V��, (2)

where [Z]� � [W][Z][W]T, [I]� � [W][I], [V]� �[W][V], and [W]T

stands for the transpose of [W]. Because [W] is orthogonal,[W]�1� [W]T. The pulse basis functions are transformed intowavelet basis functions; thus the current [I]� is expressed in termsof wavelet basis functions.

The algorithm shown in Figure 1 is initialized with scalingfunctions, which means that the iterative compression starts withthe coarsest approximation. In Step 2 the wavelet transform matrix[W�] is constructed from these selected wavelet functions in iter-ation step �, and the matrix equation becomes

�Z�]�[J�]�[E]. (3)

Equation (3) is solved in the least-squares sense for the solution of[J�] in iteration step �, because the dimensions of [Z�]�, [J�], and[E] are N�N�, N� �1, and N�1, respectively. [Z�]�, [J�], and [E]

represent wavelet-transformed MoM matrix, wavelet-transformedcurrent vector, and wavelet-transformed excitation source vector initeration step �. The error vector satisfying the boundary condi-tions is calculated as

�e�]�[E]�[Z�]�[J�]. (4)

The equivalent current with pulse basis functions are obtainedfrom

I��[W�][J�]. (5)

The relative error err is calculated as

err ���e�]��[B]� ; (6)

Then err is used in Step 5 to check the exit criteria with a verysmall positive number �.

The most important step in the algorithm is Step 4. In Step 4,the first largest Q (Q �1, 2, 3, 4) elements in the error vector aresearched. If the associated wavelet basis functions are alreadycontained in this set of wavelet basis, there will be a search for thenext largest ones to reduce the relative error err. The original MoMmatrices are square, so the corresponding wavelet basis functions

Figure 2 The scattering objects with oblique plane-wave incidence, (a)circular cylinder, (b) square cross-sectional cylinder

Figure 3 Selected wavelet basis functions for (a) Q � 1 and Q � 2, (b)Q �3 and Q �4. Inw: normalized wavelet-transformed current vector

54 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 34, No. 1, July 5 2002

are chosen according to the positions of those largest elements inthe error vector. For example, if the positions of those largestelements are {a, b, c, d}, which must be put in an ascending order,four rows of [W], numbered {a, b, c, d}, are added into [W�] toconstruct an updated transform matrix [W��1].

In Step 6, the exit criteria are checked. If the relative err is lessthan a predefined value � or the iteration counter � is larger thanthe number �stop, the iteration will be stopped and I� is defined asthe solution. Otherwise the iteration has to be done again.

3. NUMERICAL RESULTS

In this Letter, conducting or dielectric infinite cylinders withcircular and square cross sections are considered. �i is the polar-ization angle (the angle of the electric field with the plane ofincidence), as shown in Figure 2. The excitation is an obliqueplane wave incidence with �i � 45°, �i � 180o, and �i � 45° forthe circular cylinder and �i � 0° (TMz), 45°, and 90° (TEz) for thesquare cross-sectional cylinder.

Daubechies wavelet filters with p � 8 vanishing moments areused throughout this Letter. The perimeters of the circular cylindervary from 3 to 24, and the support of the pulse basis function(PBF) is selected to be about /10 and /21. The matrix sizes of theproblems vary from 128 to 1024. It is found that the IMC does notdepend on the size of the geometry, but it depends on the basisfunction rate (BFR: number of segments per wavelength). Theperimeter of the square-cross cylinder is 8, which means that thelength of each side is 2. The square cross cylinder is used toanalyze the influence of the discontinuous current singularityaround the square corners on the IMC method.

When Q is chosen to be from 1 to 4, the selected wavelet basisfunctions are found to almost exactly correspond to the significantcoefficients in the wavelet-transformed current vector (Inw). Oneexample for a dielectric cylinder with a contour length � 3 andBFR � 21.33 is shown in Figure 3. It is clear that the significantcoefficients of the normalized transformed current vector (Inw) aretotally covered by the selected wavelet basis functions, which areindependent of Q. If the wavelet basis function is selected, themagnitude of its corresponding coefficient is 1; otherwise themagnitude is 0. The curves of Q � 1 and Q � 2 are perfectlyoverlapped, and the curves of Q � 3 and Q � 4 are almost inagreement with each other (89 wavelet basis functions are selectedwith Q � 3 and 92 wavelet basis functions are selected withQ � 4).

The equivalent current distribution is computed with the use ofthe full dense MoM matrix and compared with the current solution

Figure 4 The magnitude of the electric current density on the boundary of the conducting cylinder with err � 1% for (a) Jt, (b) Jz

Figure 5 The error vector satisfying the boundary condition

TABLE 1 Number of Basis Functions in IterativeCompression with Q � 1 and err � 0.2% for ConductingCircular Cylinder

BFR 10.67 21.33

Ka () 6 12 24 6 12 24Nsize 128 256 512 256 512 1024Nbasis 60 127 226 65 119 273�r 0.86% 0.97% 0.51% 0.11% 0.23% 0.1%

Note: Ka: length of the perimeter in wavelength, Nsize: matrix size; Nbasis:number of wavelet basis functions; �r: relative error of the current.

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 34, No. 1, July 5 2002 55

obtained from IMC, as shown in Figure 4. The corresponding errorvector [e�], calculated in the same iteration, is shown in Figure 5,which shows a maximum percentage error of less than 1%. TablesI and II show the number of the selected wavelet basis functionsfor the conducting and dielectric circular cylinder with fixed rel-ative error err, respectively. Tables III and IV present the numberof selected wavelet basis functions for the conducting and dielec-tric square-cross section cylinder with fixed err, respectively. InTables I and II, the numbers of selected wavelet basis functions forthe same scattering object with different BFR are close to eachother. When the matrix size N � 1024 and BFR �21.33 forconducting cylinder, only 240 wavelet basis functions are required.If Q � 4, the total iteration steps are

��(240�2�8)/4�1�57.

In Tables III and IV, the number of the selected wavelet basisfunctions is greater than half of the size of the current vector forsquare cross-section cylinder. That may be due to the presence ofedges on the square cross-section cylinder; the current is peakingat the square edges. Therefore, more wavelet basis functions areneeded to express such variations of the current density around theedges.

If more effective wavelet transform matrix [W] renderingsparser [Z]�could be constructed, fewer wavelet basis functionswould be required, which means the number of iteration stepswould be reduced and more memory and CPU time would besaved. For example, 83 wavelet basis functions using waveletsub-transform are required for the dielectric cylinder with contourlength of 3 and BFR � 10.67, but 91 wavelet basis functions areneeded for wavelet transform.

4. CONCLUSION

The IMC is an efficient technique for the solution of the MoMmatrix equation. The IMC is extended to treat MoM matrices dueto oblique plane-wave incidence, which induced equivalent surfacecurrents with more than one current component. This indicates thatit is easy to apply this method for MoM matrix arises from 3Dproblems. This method reduced the matrix size of the circularcylinder by half or less, but it is a little larger for the squarecross-sectional cylinder. Therefore, it can be concluded that thereduced matrix size depends on the geometry of the scattering

object, or, in other words on the current variations around theobject.

REFERENCES

1. Z. Baharav and Y. Leviatan, Impedance matrix compression usingadaptively constructed wavelet basis, IEEE Trans Antennas PropagatAP-44 (1996), 1231–1238.

2. Z. Baharav and Y. Leviatan, Impedance matrix compression usingiteratively selected wavelet basis, IEEE Trans Antennas PropagatAP-46 (1998), 226–233.

3. Y. Shiftman, Z. Baharav, Y. Leviatan, Analysis of truncated periodicarray using two-stage wavelet-packet transformations for impedancematrix compression, IEEE Trans Antennas Propagat AP-47 (1999),630–636.

4. A. A. Kishk, P. Slattman, and P.S. Kildal, Radiation from 3D sources inthe presence of 2D objects of arbitrary cross-sectional shape, ACES J 14(1999), 17–24.

© 2002 Wiley Periodicals, Inc.

DEFINITION OF EFFECTIVE DIVERSITYGAIN AND HOW TO MEASURE IT IN AREVERBERATION CHAMBER

Per-Simon Kildal,1 Kent Rosengren,2 Joonho Byun,3 andJuhyung Lee3

1 Chalmers University of Technology, Sweden2 Intenna Technology AB, Sweden3 Wireless Terminal Division, Samsung Electronics Co. Ltd., SouthKorea

Received 4 January 2002

ABSTRACT: The performance of cellular phones and other mobile orwireless terminals operating in multipath propagation environment canbe greatly improved by introducing different diversity schemes. The im-provement is characterized in terms of a diversity gain. An effective di-versity gain is defined here. This is an absolute measure of diversitygain and can therefore be used to compare different diversity antennas.The Letter also shows how the effective diversity gain can be measuredin a reverberation chamber. Measured effective diversity gains agreemuch better with theoretical diversity gains than measured values pub-lished previously. © 2002 Wiley Periodicals, Inc. Microwave OptTechnol Lett 34: 56–59, 2002; Published online in Wiley InterScience(www.interscience.wiley.com). DOI 10.1002/mop.10372

Key words: antenna effective diversity gain; reverberation chamber;antenna measurements

1. INTRODUCTION

Reverberation chambers are used to generate a statistically uni-form field distribution, needed for certain EMC tests. The statis-tical properties are obtained by mechanically stirring the modes inthe chamber. It has previously been shown that the multiple-modefields inside reverberation chambers have characteristics similar to

TABLE 2 Number of Basis Functions in IterativeCompression with Q � 1 and err � 1% for DielectricCircular Cylinder

BFR 10.67 21.33

Ka () 3 6 12 3 6 12Nsize 128 256 512 256 512 1024Nbasis 84 128 240 88 128 240�r 0.66% 0.2% 1.51% 0.69% 0.43% 1.81%

TABLE 3 Number of Basis Functions in IterativeCompression With Q � 4 and err � 0.02% for ConductingSquare Cross Section Cylinder With BFR � 16, �i � 45°, and�i � 180°

�i 0° (TM) 90° (TE) 45°

Nsize 256 256 256Nbasis 172 184 180�r 1.18% 0.007% 0.04%

TABLE 4 Number of Basis Functions in IterativeCompression With Q � 4 and err � 0.1% for DielectricSquare-Cross Cylinder With BFR � 16, �i � 45°, and �i � 180°

�i 0° (TM) 90° (TE) 45°

Nsize 512 512 512Nbasis 396 416 416�r 0.2% 0.23% 0.19%

56 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 34, No. 1, July 5 2002