Z. Baharav

Impedance matrix compression (IMC) using iteratively selected wavelet basis

We present a novel approach for the incorporation of wavelets into the solution of frequency-domain 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 are needed to represent the unknown quantity. To determine these dominant expansion functions, an iterative procedure is devised. The new approach combined with the iterative procedure yields a new algorithm that has many advantages over the presently used methods for incorporating wavelets. Numerical results which illustrate the approach are presented for three scattering problems.

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 affected by means of a digital filtering approach. This approach 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.

Analysis of electromagnetic scattering using arrays of fictitious sources

The use of models of fictitious elemental current sources, located inside the scatterer to simulate the scattered field, has proved to be an efficient computational technique for analyzing scattering by metallic bodies. This paper presents a novel modification of the technique in which the omnidirectional elemental sources are arranged in groups of array sources with directional radiation patterns, and the boundary testing points are arranged in groups of testing arrays with directional receiving patterns. This modification which is motivated by physical understanding is equivalent to mathematical basis transformations. It renders the system matrix more localized and thereby enables the analysis of larger bodies. The new approach is applied to the case of TM scattering by a perfectly conducting square cylinder with side-length of 20/spl lambda/. Reduction of 50% in the number of the nonzero elements of the system matrix is achieved with virtually no degradation in the accuracy of the radar cross section (RCS) calculations. >

Wavelets in electromagnetics: the impedance matrix compression (IMC) method

The use of wavelet expansions in numerical solutions of electromagnetic frequency-domain integral equation formulations is steadily growing. In this paper we review the recently suggested impedance matrix compression (IMC) method for a more effective integration of wavelet-based transforms into existing numerical solvers. The difference between the IMC method and the previous approaches to applying wavelets in computational electromagnetics is twofold. Firstly, the transformation is effected by means of a digital filtering approach. This approach renders the transform algorithm adaptive and facilitates the derivation of a basis which best suits the problem at hand. Secondly, 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 hence a substantially smaller number of coefficients has to be determined. A few numerical results are included to demonstrate the advantages of the presented method over the currently used ones. The feasibility of ensuring a slow growth in the number of unknowns even when there is a rapid increase in the problem complexity is shown by an illustrative example.

Impedance matrix compression (IMC) using iteratively selected wavelet basis for MFIE formulations

Wavelet expansions have been employed in method of moments (MoM) solutions of frequency-domain integral equations. In these solutions the unknown quantity of interest (usually the current on the scatterer) is first represented in terms of a set of wavelet basis functions, and then the difference between the two sides of the equation is forced to be orthogonal to a set of wavelet testing functions. While this approach generally yields satisfactory solutions, it has two disadvantages. We propose a new approach which overcomes these two drawbacks. The new approach concerns the incorporation of wavelets into MoM solutions to magnetic field integral equation (MFIE) formulations of scattering problems. In this approach, rather than merely resorting to the sparseness of the operator in the wavelet expansion, we also utilize the sparse representation of the (yet unknown) current in the wavelet expansion. It is well known that wavelets can represent non-stationary signals with just a few terms. This fact has also been used in the context of computational electromagnetics, where the idea of impedance matrix compression (IMC) has been introduced. The new approach, combined with an iterative procedure, yields a new algorithm, that has many advantages over the present methods for incorporating wavelets.

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 and disadvantages of ...

Analysis of truncated periodic array using two-stage wavelet-packet transformations for impedance matrix compression

A novel method of moments procedure is applied to the problem of scattering by metallic truncated periodic arrays. In such problems, the induced current shows localized behavior within the unit cell and at the same time exhibits cell-to-cell periodicity. In order to select a set of expansion functions that may account for such behavior, a two-stage basis transformation, of which the first stage is an ordinary wavelet transformation performed independently on each unit-cell, has been applied to a pulse basis. The resultant basis functions at the first stage are regrouped and retransformed to reveal the periodicity of their coefficients. Expansion functions are then iteratively selected from this newly constructed basis to form a compressed impedance matrix. The compression ratios obtained in this manner are higher than the compression ratio achieved using a basis constructed via an ordinary single-stage wavelet transformation, where compression is the ratio between the number of nonzero elements in the matrix used to solve the problem and the number of elements in the original matrix. An even higher compression is attained by considering, in addition, functions that reveal array-end related features and iteratively selecting the expansion from an overcomplete dictionary.

Resolution enhancement and small perturbation analysis using wavelet transforms in scattering problems

Summary form only. We suggest a novel method which combines the use of models of fictitious sources and wavelet transforms. 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. 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 a small protruding part and the analysis is repeated for a higher resolution level using additional sources, transformed to generate wavelet basis functions of finer scale, but only in the vicinity of the protrusion.

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.