Impedance Matrix Compression Using Adaptively-Constructed Basis Functions

Wavelet expansions have been employed recently in numerical solutions of commonly used frequency-domain integral equations. In this paper we propose a novel method for integrating wavelet-based transforms into existing numerical--solvers. The newly proposed method differs from the presently used ones in two ways. First, the transformation is effected by means of a digital filtering approach. This approach enables a much faster implementation of the transform. It also renders the transform algorithm adaptive and facilitates the derivation of a basis which best suits the problem at hand. Second, the conventional thresholding procedure applied to the impedance-matrix is substituted for by a compression process in which only the significant terms in the expansion of the (yet-unknown) current are retained and subsequently derived. Numerical results for a few TM scattering problems are included to demonstrate the advantages of the proposed method over the presently used ones. 1 Introduction ..

Scattering Analysis Using Fictitious Wavelet Array Sources

In this paper we study the incorporation of wavelet--transforms into the source-model technique (SMT) for efficient analysis of electromagnetic scattering problems. The idea is to divide the discrete sources into groups of arrays with wavelet amplitude distributions. We refer to these array sources as fictitious wavelet array sources. They can be readily formed by applying appropriate wavelet transformations to the original matrix equation obtained based on a conventional SMT solution. The transformed impedance-matrix obtained in this manner is then compressed and thus a substantially smaller matrix equation has to be solved. The conventional as well as the windowed Fourier transform variant of the wavelet transform are considered. The ease with which one can adjust the expansion for resolution of small features and for handling small perturbations in the scatterer geometry is demonstrated. A comparison with a conventional method of moments solution is presented to show the advantages ..

Impedance Matrix Compression (IMC) Using Iteratively Selected Wavelet Basis for MFIE Formulations

In this paper, we present a novel approach to incorporating wavelet expansions in method of moments (MoM) solutions for scattering problems described by a magnetic field integral equation (MFIE) formulation. In this approach, we utilize the fact that when the basis--functions used are wavelet-type functions, only a few terms in a series expansion would be needed to represent the unknown quantity. An iterative procedure is suggested to determine these dominant expansion functions. The new approach combined with the iterative procedure yields a new algorithm which has many advantages over the presently used methods for incorporating wavelets. Numerical results which illustrate the approach are presented. 1 Introduction Wavelet expansions have been employed recently in method of moments solutions of frequencydomain integral equations [1, 2, 3, 4, 5]. In these solutions the unknown quantity of interest (usually the current on the scatterer) is first represented in terms of a set of wavel..

Impedance Matrix Compression (IMC) Using Iteratively Selected Wavelet-Basis

In this paper we present a novel approach for the incorporation of wavelets into the solution of frequencydomain integral equations arising in scattering problems. In this approach, we utilize the fact that when the basis functions used are wavelet-type functions, only a few terms in a series expansion would be needed to represent the unknown quantity. Moreover, an iterative procedure is suggested to determine these dominant expansion functions. The new approach combined with the iterative procedure yields a new algorithm which has many advantages over the presently used methods for incorporating wavelets. Numerical results which illustrate the approach are presented for three scattering problems. I. Introduction W AVELET expansions have been employed recently in numerical solutions of commonly used frequencydomain integral equations [1], [2], [3], [4], [5]. In the conventional approach to the solution of these integral equations [1], the unknown quantity of interest (usually the cur..

Improving Impedance Matrix Localization By A Digital Filtering Approach

Wavelet bases have been employed recently in numerical solutions of integral equations encountered in electromagnetic scattering problems. The advantage of these bases lies in their ability to render the problem impedance matrix more localized. In this paper, we propose a digital filtering approach for integrating wavelet transforms into existing electromagnetic--scattering numerical--solvers. The suggested approach facilitates a much faster implementation of the transform. It also allows the incorporation of other ideas such as wave--packets and best-basis from the discipline of digital signal processing. A physical interpretation of the basis functions obtained by a few selected structures of digital filters is given, and their usefulness is explained. A numerical example is given for the case of TM scattering by a square cylinder. 1. INTRODUCTION Integral equations encountered in electromagnetic scattering problems have been recently solved using wavelet-based methods [1, 2]. In t..

Resolution Enhancement and Small Perturbation Analysis using Wavelet Transforms in Scattering Problems

this paper we suggest a novel method which combines the use of models of fictitious sources [1] and wavelet transforms [2]. The idea is to integrate wavelet transforms into the simple and efficient source-model technique and thereby obtain accurate numerical results while using a highly sparse approximation to the typically full impedance matrix. The wavelet transform is applied to both the unknown current vector and the excitation vector. It can be effected either by matrix multiplication or by a hierarchical structure of digital filters. The use of digital filtering is preferable because it makes the computation more efficient and allows one to adapt the transform to the problem at hand. The proposed method facilitates a convenient way to go into higher resolution levels only where necessary. Moreover, if the original scatterer is slightly deformed, there will be no need to recompute the whole matrix; minor add-ons to the matrix will suffice. This is in contrast to the conventional on-surface wavelet expansion approach in which the whole matrix has to be recomputed with every change in the length-parameter on the circumference. Another important feature of this method is that different variants of the wavelet-transform can be used in different regions of the scatterer. Thus, for example, near the smooth parts of the scatterer the sources will be transformed to generate wave-packet basis functions, whereas near the parts of high-curvature the sources will be transformed to generate regular wavelet basis functions. The idea presented in this paper is illustrated by a study of the problem of TM plane-wave scattering by an infinite cylinder of elliptic cross section. This cylinder is analyzed at a certain level of resolution. Then, the cylinder surface is assumed to have ..

Impedance Matrix Compression Using Wavelet Expansions

Wavelet expansions have been used recently in numerical solutions of integral equations encountered in various electromagnetic scattering problems. In these solutions one utilizes the power of the wavelet basis functions to make the problem impedance matrix localized. Thus, after computing the impedance matrix, it can be rendered sparse via a thresholding procedure, and the resultant matrix equation can be solved in a much faster way without any significant loss in accuracy. In this paper we propose a novel approach, where instead of thresholding the impedance matrix in a conventional manner, it is compressed to a reduced-size form. This is effected by first singling out a small number of basis functions which are expected to accurately represent the unknown and keeping only the matrix elements needed for finding out the coefficients of these basis functions. A method to carry out this matrix compression automatically is described. Numerical examples are given for the case of TM scatte..

Scattering Analysis Using Fictitious Wavelets

In this paper we study the incorporation of wavelet--transforms into the source-model technique (SMT) for efficient analysis of electromagnetic scattering problems. The discrete nature of the fictitious sources in the SMT allows the use of wavelet-transform methods in which the transformation is effected in a rapid way by means of digital filters. The source functions obtained upon integrating the wavelet transform methods into the SMT are referred to as fictitious wavelets. The transformed impedance-matrix obtained in this manner is then compressed and thus a substantially smaller matrix equation needs to be solved. The conventional as well as variants of the wavelet transform (e.g., windowedFourier -transform and adaptive-basis selection) are considered. The ease with which one can enhance the resolution in a given solution and the simple way by which small perturbations in the scatterer geometry can be treated are both demonstrated. A comparison with a conventional method of moments..