A note on the choice weighting functions in the method of moments

The objective is to show from a mathematical standpoint that there are certain rules that must be followed in the choice of weighting functions used in the method of moments (MM). It is shown that for a particular problem it is the operator that dictates the method (Galerkin's method or another method such as the method of least squares) to be applied, and it is not computational considerations only. For example, it is shown that in solving Hallen's and Pocklington's equation by the method of moments, it is unnatural to choose the weighting functions which are zero at the ends of the domain of the solution. The deficiency of certain weighting functions is presented based on mathematical reasoning, and a numerical example is given to illustrate the effect of the choice of the weighting functions on the rate of the convergence of the solution.     

Numerical solution of the continuous waveguide transition problem

A simple transition between two sizes of rectangular waveguide is analyzed using the generalized telegraphist's equation. Solutions are obtained using a new moment method technique, a Runge-Kutta algorithm, and an iterative numerical integration technique. The results are compared to previously published experimental and numerical data. It is found that the numerical stability, accuracy, and consistency of the results are critically dependent on the choice of weighting and expansion functions. The best results for a simple rectangular-to-rectangular transition were obtained when Galerkin's method and triangle expansion functions were applied to several short sections which were then cascaded. Unlike the Runge-Kutta technique or the integration technique, the Galerkin's method procedure did not result in instabilities with the inclusion of evanescent modes. The programs can, in fact, be extended to any number of modes, the only apparent limitations being the obvious ones of computer time and memory     

SOLUTION OF ARBITRARY CROSS-SECTION WAVEGUIDE USING METHOD OF EIGEN WEIGHTED BOUNDARY INTEGRAL EQUATION

Based on the second kind of Green's identity,a boundary integral equation forarbitrary cross-section waveguide is transformed to a system of linear homogeneous algebraicequations by means of expansion of boundary bases and by using the eigenfunctions of a fictitiousregular boundary as weighting functions,which corresponds to less algebraic equations than BEMand simpler coefficients than the modified BEM.The numerical results for some typical metallicwaveguides are given by using the method of eigen-weighted boundary integral equation,and theyare accurate enough with fast convergence.     

本征权边界积分法解任意截面波导传输问题

本文选取本征函数作为权函数,由格林第二恒等式建立边界积分方程,并采用边界基形式得到线性齐次方程组。这种方法不仅降低了系数矩阵的维数,而且使其各项仍保持为简单一项,减少了计算量.本文提供了几种常见金属波导的例子,计算结果既收敛快、又足够准确。     

On the reflectivity of complex mesh surfaces [spacecraft reflectorantennas]

Poorer than expected surface reflectivity was observed in an early Tracking and Data Relay Satellite System antenna utilizing a tricot mesh weave. This poor reflectivity was determined to be caused by inadequate electrical contact at wire crossover points. A proper mathematical and numerical approach to assess the impact of wire junctions on reflectivity performance is developed. A mathematical method is presented for computing the surface reflectivity of complex mesh configurations like those on unfurlable-type spacecraft antennas. The method is based on the Floquet mode expansion to establish an integral equation for mesh wire currents. The equation is solved using the method of moments with triangular basis functions. It is observed that it is necessary to give special attention to the junction treatment among different branches of the mesh configurations. A vector junction current approach that resulted in satisfactory solutions for the current is described. The results of numerical simulations are compared against measured data and excellent agreement is observed     

Analytic method of arbitrary cross section open groove guide

Groove guide is one of the waveguides that have been used at millimeter and submillimeter waveband. This paper analyzes arbitrary groove guides by means of the eigen-weighted boundary integral equation method that uses the eigenfunctions of a fictitious regular boundary as weighting functions. In comparison with the theoretical and experimental results published, the numerical results for rectangular, circular and V-groove guides using this method are exact enough with fast convergence and less calculation.     

A Moment Method with Mixed Basis Functions for Scatterings by Waveguide Junctions

A moment method with mixed basis functions is introduced. In this formulation, modal basis functions are used for the expansion of the currents corresponding to the scattered propagating modes, while pulse basis functions are used for the expansion of the current corresponding to the scattered envancescent waves. This, together with the Dirac /spl delta/ weighting functions, reduces the number of total basis functions needed while retaining the simplicity and versatility of the method to cover junctions of an arbitrary shape. This method is applied to study examples of homogeneous and inhomogeneous waveguide junctions of parallel-plate waveguide propagating TE waves. It is found that for junctions that are not electrically large the convergence of the solutions is good. An appendix is included to transform and quicken the numerical integration of the modal basis functions.     

Effect of detector weighting functions on the point spread function of high-resolution PET tomographs: a simulation study

The point spread function (PSF) of a ring PET tomograph is known to be spatially variant and difficult to obtain because it must be reconstructed from projections. A mathematical model was developed to simulate the data acquisition from a point source and to reconstruct the PSF, taking into account weighting functions to describe the detector response functions. In order to investigate the effect of the detector weighting function on the PSF, the reconstruction, based on the filtered backprojection algorithm, was implemented with three classes of weighting functions of decreasing complexity: exact, locally invariant, and constant. Significant differences are observed to result from the three hypotheses, and this is shown to lead to distorted PSFs and to erroneous estimates of the intrinsic resolution off the center of the tomograph.     

A diagonalized multilevel fast multipole method with spherical harmonics expansion of the k-space Integrals

Diagonalization of the fast multipole method (FMM) for the Helmholtz equation is usually achieved by expanding the multipole representation in propagating plane waves. The resulting k-space integral over the Ewald sphere is numerically evaluated. Storing the k-space quadrature samples of the method of moments (MoM) basis functions constitutes a large portion of the overall memory requirements of the resulting algorithm for solving the integral equations of scattering and radiation problems. In this paper, it is proposed to expand the k-space representation of the basis functions by spherical harmonics in order to reduce the sampling redundancy introduced by numerical quadrature rules. Aggregations, plane wave translations, and disaggregations in the realized multilevel fast multipole method (MLFMM) are carried out using the k-space samples of a numerical quadrature rule. However, the incoming plane waves on the finest MLFMM level are expanded in spherical harmonics again. Thus, due to the orthonormality of spherical harmonics, the testing integrals for the individual testing functions are simplified into series over products of spherical harmonics expansion coefficients. Overall, the resulting MLFMM can save a considerable amount of memory without compromising accuracy and numerical speed.     

矩量法中阻抗矩阵的稀疏化研究

将高阶叠层矢量基函数及最大正交高阶矢量基函数应用于电磁场积分方程方法,提出将阻抗矩阵按稀疏阵处理的方法.通过文中的处理,使得存储阻抗矩阵的内存需求量和求解矩阵方程的迭代求解时间大为降低.本文还结合适当算例,分析了判断门限的选取对阻抗矩阵的存储量与迭代法求解的计算量的影响.     

Receive and transmit mode aperture field of the parallel plate-fedslot antenna

Two numerical solutions are given for an integrodifferential equation governing the electric field in the aperture of a parallel-plate waveguide-loaded slot antenna excited by an obliquely incident plane wave. Resultant aperture field distributions obtained via Galerkin's method with a basis of edge-condition weighted Chebyshev-polynomials are compared with the results from pulse expansion with point matching. It is shown that the relative numerical convergence of the Chebyshev-Galerkin method and of the pulse expansion-point matching applied to the receive mode of the parallel plate-fed slot antenna are qualitatively similar to a previously reported transmit case, that is, the full domain expansion with the built-in edge condition behavior converges faster than the sub-domain method, and agreement between the two independent results serves as a necessary, albeit not strictly sufficient, check on the accuracy of the implementation     

卫星赋形波束天线优化方法的改进

本文采用了最小ｐ乘法和Ｍｉｎｉｍａｘ法对卫星多波束天线的赋形问题进行了研究，提出了用归一化的功率偏差值作为目标函数，使各点的偏差得以平衡，权值的选取变得十分简单。同时根据上述两种方法的目标函数特点，采用了相应有效的算法。最后用一个偏置抛物面多波束天线对本文改进后的方法进行了验证，并和以前的方法进行了比较，说明用归一化的功率偏差值作为目标函数来研究天线的赋形问题是非常合适的     

Full wave analysis of microstrip floating line structures bywavelet expansion method

A full wave analysis of microstrip floating line structures by wavelet expansion method is presented. The surface integral equation developed from a dyadic Green's function is solved by Galerkin's method, with the integral kernel and the unknown current expanded in terms of orthogonal wavelets. Using the orthonormal wavelets (and scaling functions) with compact support as basis functions and weighting functions, the integral equation is converted into a set of linear algebraic equations, with the matrices nearly diagonal or block-diagonal due to the localization, orthogonality, and cancellation properties of the orthogonal wavelets. Limitations inherited in the traditional orthogonal basis systems are released: The problem-dependent normal modes have been replaced by the problem-independent wavelets, preserving the orthogonality; the trade-off between orthogonality and continuity (e.g. subsectional basis functions including pulse functions, roof-top functions, piecewise sinusoidal functions, etc.) is well balanced by the orthogonal wavelets. Numerical results are compared with measurements and previous published data with good agreement     

导体线面连接问题中奇异函数积分的计算

在线面连接问题中，电流展开函数包含体展开函数、线展开函数和连接点展开函数三类。求解电场积分方程的积分项是电流基函数及其散度分别与自由空间格林函数的乘积，由于连接点展开函数含有一个奇异点，所以被积函数中含有两个奇异点。本文通过积分变换消除了奇异点，并将二重积分化为一重积分，使计算精度得到提高。计算实例验证了本文方法的正确性。     

The use of the frill generator in thin-wire integral equations

For a frill-generator feed, certain fundamental mathematical properties of Hallen's and Pocklington's equations with the approximate kernel are stated and derived. For a particular moment-method procedure (Galerkin's method with pulse functions applied to Hallen's equation), the consequences to the numerical solutions are carefully examined. Generalizations to other numerical methods with subsectional basis functions are discussed. Many of our results come from studying the simpler problem of the infinite dipole analytically and applying the understanding thus obtained to the case of the finite dipole.     

Weighted least mean square design of 2-D FIR digital filters: thegeneral case

This paper solves the weighted least mean square (WLMS) design of two-dimensional (2-D) finite impulse response (FIR) filters with general half plane symmetric frequency responses and nonnegative weighting functions. The optimal solution is characterized by a pair of coupled integral equations, and the existence and uniqueness of the WLMS solution for 2-D FIR filter design are established. Two efficient numerical algorithms using a 2-D fast Fourier transform (FFT) are proposed to solve the WLMS solution. One is based on the contraction mapping and fix point theorem characterizing the coupled integral equation; the other uses conjugate gradient techniques, which guarantees finite convergence. The associated computational complexity is analyzed and compared with existing algorithms. Examples are used to illustrate the effectiveness of the proposed design algorithms. The selection of weighting functions to improve the minimax performance of the filter is also discussed     

Direct extrapolation of a causal signal using low-frequency and early-time data

In this paper, we provide three direct procedures to extrapolate the early-time and the low-frequency response of a causal signal simultaneously in the time-and frequency domain. Compared with the extrapolation by orthonormal basis functions, direct extrapolation is straightforward and we do not need to evaluate the basis functions and search for the optimal scaling factor and the optimal number of basis functions. We show that the extrapolation introduced by Adve and Sarkar is equivalent to a Neumann-series solution of an integral equation of the second kind. It is further shown that this iterative Neumann expansion is an error-reducing method. We propose to solve this integral equation efficiently by employing a conjugate gradient iterative scheme. The convergence of this scheme is also demonstrated. We provide the matrix equations and show the equivalence to the integral equations, and demonstrate that the method of singular value decomposition (SVD) of solving the matrix equation provides accurate and stable results. Finally, a number of illustrative numerical examples are presented and the performances of the three direct methods are compared.     

Efficient calculation of the electromagnetic dyadic Green's function in spherical layered media

A numerical methodology is proposed for the evaluation of the electromagnetic fields of an electric or magnetic dipole of arbitrary orientation in spherical stratified media. The proposed methodology is based on the numerical solution of the differential equation in the radial coordinate that is satisfied by each coefficient in the spherical harmonics expansion of the governing Helmholtz equation. More specifically, the finite-difference solution of this equation may be cast in a pole-residue form that allows the analytic evaluation of the spherical harmonics series by means of the Watson transformation. Thus, a closed-form expression is obtained for the electromagnetic fields in terms of a short series of associated Legendre functions P/sup m//sub /spl nu//(cos(/spl theta/)) of integer order m and complex degree /spl nu/. The number of terms in the series is strongly dependent on the angle /spl theta/, and decreases very fast when the point of observation moves away from the source. The method allows for arbitrary variation in the permittivity and permeability profiles in the radial directions.     

An efficient solver for the three-dimensional capacitance of theinterconnects in high speed digital circuit by the multiresolutionmethod of moments

The computation of the equivalent capacitances for three-dimensional (3-D) interconnects features large memory usage and long computing time. In this paper, a matrix sparsification approach based on multiresolution representation is applied with the method of moments (MoM) to calculate 3-D capacitances of interconnects in a layered media. Instead of direct expansion of the charge distribution by the orthogonal wavelet basis functions, the large full matrix resulting from discretization of the integral equations is taken as a discrete image and sparsified by two-dimensional (2-D) multiresolution representations. The inverse of the obtained sparse matrix is efficiently implemented by Schultz's iterative approach. Several numerical examples are given and the results obtained show that the proposed method significantly sparsifies the matrix equation and the capacitance parameters computed by the matrix equation with high sparsity agree well with the results of other reports and those computed by an established capacitance extractor FASTCAP     

A New Technique for the Analysis of the Dispersion Characteristics of Microstrip Lines

Dispersion characteristics of shielded microstrip lines are investigated using a new technique. The method utilizes the well-known singular integral equation approach for deriving an alternate form of eigenvalue equation with superior convergence properties. It is shown that accurate numerical results may be obtained from this eigenvalue equation using only a 2x2 matrix equation. In comparison, the conventional formulation of the problem requires the use of matrices that are much larger in size. Aside from the numerical efficiency, the simplicity of the method makes it possible to conveniently extract higher order modal solutions for the propagation constants that affect the high-frequency application of microstrip lines. Even though the derivation of the determinantal equation requires some intricate mathematical manipulations, the user may bypass these completely and use the final eigenvalue equation which is programmable on the computer.