## 1. Introduction

[2] Radiated emission of electromagnetic energy is one of the major aspects of the electromagnetic compatibility (EMC) problem. To study this problem, it is necessary to understand the electromagnetic field distribution in the surrounding environment, which may consist of various dielectric media and conducting objects. Generally, EMC problems are relatively complicated and usually cannot be solved analytically, so that numerical analysis techniques must be used. Numerical analysis techniques include method of moment (MoM) [*Harrington*, 1968; *Miller and Landt*, 1980; *Bernardi et al*., 1996], the finite difference time domain method [*Tirkas et al.*, 1993; *Taflove and Umashankar*, 1989], the transmission line matrix method [*Christopoulos and Herring*, 1993; *D'amore and Sarto*, 1996], the finite element method [*Dixon et al.*, 1993; *Sacks and Lee*, 1995], and the finite volume time domain method [*Holland et al.*, 1991]. These methods have been successfully applied in solving electromagnetic scattering problems in EMC-related fields such as antennae, microwave, and millimeter wave circuits and radar cross sections.

[3] MoM is a well-established numerical method and is probably the most widely used technique for solving integral equations in electromagnetics. In conventional MoM the boundary of integration is approximated by discretizing it into many segments, and then the unknown function is expanded in terms of known basis functions with unknown coefficients. However, classical subsectional bases, when applied directly to the integral equations, generally produce a dense impedance matrix. The dense matrix often becomes computationally unmanageable owing to the large memory requirement and CPU time required to invert the dense matrix. Recently, the use of wavelets and wavelet-like basis functions for the efficient solution of electromagnetic integral equations has received considerable attention. The wavelet bases have been used to overcome the major drawback mentioned above and to sparsify the matrix, primarily because of local supports and vanishing moment properties of the wavelet bases.

[4] *Beylkin et al.* [1991] first applied wavelets to the solution of integral equations having essentially smooth, nonoscillatory kernels, such as those encountered in electrostatics. For electrodynamic problems, however, the Green's function (kernel) is oscillatory in the spectral frequency domain which leads to a dense impedance matrix. Hence many other researchers applied different wavelet transform methods to effectively sparsify the impedance matrix in MoM. *Kim et al.* [1996] used spectral domain wavelet transform to increase matrix sparsity. *Golik* [1998] applied discrete wavelet packet transform and presented an adaptive algorithm for the selection of the near-best basis transform to sparsify the matrix. *Deng and Ling* [1999a, 1999b] employed adaptive wavelet packet transform and predefined wavelet packet bases to further sparsify the matrix.

[5] In addition, many papers have applied Daubechies wavelets to obtain the matrix sparsification in MoM. Recently, many studies have used different kinds of wavelets to efficiently solve electromagnetic (EM) integral equations. *Pan et al.* [1998, 1999] use the Coifman intervallic wavelets as the basis and testing functions in MoM for fast construction of wavelet-sparisfied matrices. *Huang et al.* [2000] applied Vaidyanathan wavelets based on the quadrature mirror filter to construct the wavelet transform matrix. However, it seems that most of the previous studies did not specify how to choose a proper wavelet for targeting electromagnetic integral equations with oscillatory kernels.

[6] In this paper, we exploit a concept of “visible energy” as a criterion for choosing suitable wavelets in solving electrodynamic scattering problems. Visible energy is defined as the energy of all dilations of a single mother wavelet for an arbitrary translation in the spectral domain over the entire visible region in which the spatial frequency is smaller than the free space spatial frequency (wave number). In the visible region the spectral content of wavelet current becomes the wavelet source that will produce far-field radiation. Since the elements of the wavelet-domain impedance matrix represent the interaction between wavelet sources and receivers, the interaction becomes stronger when more spectral content leaks into the visible region, i.e., greater visible energy. Thus we expect that the quantity of visible energy may reflect the sparsified extent of the impedance matrix. To investigate our concept, numerical simulations are used to solve electromagnetic scattering problems. Different shapes of two-dimensional conducting scatterers such as a circular cylinder, an L-shaped scatterer, and a duct are considered. The combined field integral equation (CFIE) is employed and is solved by the wavelet MoM (WMoM). Note that all the different wavelet basis vectors used here are orthogonal.