Jaideva C. Goswami

Semi-orthogonal versus orthogonal wavelet basis sets for solving integral equations

The two categories of wavelets, orthogonal and semi-orthogonal, are compared and it is shown that the semi-orthogonal wavelet is best suited for integral equation applications. The Battle-Lemarie orthogonal wavelet and the spline generated semi-orthogonal wavelet are each used to solve for the current distribution on an infinite strip illuminated by a transverse magnetic (TM) plane wave and a straight thin wire illuminated by a normally incident plane wave. The grounds for comparison are accuracy in characterizing the current, matrix sparsity, computation time, and singularity extraction capability. The limitations and advantages of solving integral equations with each of the two wavelet categories are discussed.

An application of fast integral wavelet transform to waveguide mode identification

We introduce a very efficient method for computing the integral wavelet transform, with any compactly supported spline-wavelet as the analyzing wavelet, on a dense set of the time-scale domain. While the mathematical analysis of this algorithm will be presented elsewhere, the objective of this paper is to describe the computational scheme along with its computer implementation, and to demonstrate its effectiveness in the identification of mode propagation in a rectangular waveguide. >

Cylindrical cavity-backed suspended stripline antenna-theory and experiment

The problem of a cylindrical cavity-backed suspended stripline (SSL) antenna is viewed as a transition of the SSL to a circular cylindrical waveguide opening into an infinite ground plane. The fields in the waveguide are expanded in terms of TE and TM modes. The effect of the radiating aperture on the modal expansion of the fields is taken into account by introducing reflection coefficients for each mode. The current on the SSL probe is assumed to have sinusoidal distribution. These simplifications reduce the original problem to that of a known radially oriented current residing on a dielectric sheet inside a circular-cylindrical cavity whose top wall has known impedances corresponding to different modes. The Greens function for this modified structure is found and is used to obtain a general expression for the input impedance. This expression is specialized to the case where the SSL probe and the radiating aperture are coupled through the dominant TE/sub 11/ mode only. This input impedance is translated to the measurement plane of the antenna. The computed and measured results are found to be in good agreement. >

On a Spline-Based Fast Integral Wavelet Transform Algorithm

Multiresolution properties of the cardinal B-splines and the high-degree of vanishing moments of their corresponding B-wavelets, along with the flexibility and near-optimality of their time-frequency windows, make them suitable for the time-frequency analysis of signals consisting of a wide range of frequency components. In most of the applications, one needs to compute the integral wavelet transform (IWT) of the signal only at certain scales. The standard wavelet decomposition algorithm (sometimes called the fast wavelet transform (FWT) algorithm), based on certain digital samples, say s(k/2 N ), k ∈ ℤ, of a signal s(t), can be applied with real-time capability to give the IWT values of s(t) on the scale levels a = 2−j , and j ≤ N - 1. However, this information on the IWT of s(t), on such a sparse set of dyadic points (k/2 j , 1/2 j ) in the time-scale domain, is sometimes insufficient to give the desirable time-frequency analysis of the function s(t).

An analysis of two-dimensional scattering by metallic cylinders using wavelets on a bounded interval

Summary form only given. The method of moments (MOM) when applied to integral equations results into a fully-populated matrix which is often ill-conditioned, causing numerical instability and poor convergence in the case of iterative techniques. The condition number increases with the decrease in the step-size of the discretisation. It is now well known that multigrid methods offer a solution to such difficulties. Wavelets, because of their multiresolution properties, are naturally suited for multigrid methods. Their applications to integral equations lead to a well-conditioned matrix. Furthermore, the resultant matrix is sparse because of the local supports and the vanishing moment property of wavelets. The purpose of the paper is to demonstrate the application of semi-orthogonal compactly supported spline-wavelets on a bounded interval to resolve integral equations encountered in the two-dimensional electromagnetic scattering by metallic cylinders. In order to achieve a higher degree of sparsity, one must use higher order spline wavelets. However, higher order wavelets have higher spatial supports. The authors restrict themselves to the use of linear and cubic spline wavelets. The results obtained using the wavelet approach and the conventional MOM approach are presented.<<ETX>>

Wavelet Methods for Solving Integral and Differential Equations

The sections in this article are 1 Wavelet Preliminaries 2 Integral Equations 3 Matrix Equation Generation 4 Intervallic Wavelets 5 Numerical Results 6 Semiorthogonal Versus Orthogonal Wavelets 7 Differential Equations