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

In this paper we suggest a novel method which combines the use of models of ctitious sources 1] and wavelet transforms 2]. The idea is to integrate wavelet transforms into the simple and eecient 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 eeected either by matrix multiplication or by a hierarchical structure of digital lters. The use of digital ltering is preferable because it makes the computation more eecient 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 suuce. 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 diierent variants of the wavelet-transform can be used in diierent 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 innnite 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 ner scale, but only in the vicinity of the protrusion.