Investigation of the discrete wavelet transform as a change of basis operation for a moment method solution to electromagnetic integral equations

The moment method as a solution method for electromagnetic integral equations has been modified extensively this decade to promote faster performance with controlled error. One modification has been to convert the dense impedance matrix to a sparse form by thresholding the matrix when wavelets are used as basis functions. There are currently two approaches to introducing wavelets as basis functions. The integral equation has been directly expanded and tested with orthogonal wavelets by Steinberg and Leviatan (1993) and with semi-orthogonal wavelets by Goswami et. al. (see IEEE Trans. Antennas Propagat., vol.AP-43, p.614-22, 1995). This requires considerable numerical effort to efficiently evaluate the integrals. In fact, for wavelets that do not have closed form expressions, an efficient calculation may still be numerically prohibitive. The other approach is to use a conventional basis and testing functions and then perform a discrete wavelet transform (DWT) on the impedance matrix, source and current vectors. This has also been termed a change of basis operation. We review the mathematics of the DWT and show how the resultant matrix elements compare to the direct expansion of the integral equation into the same wavelet basis. This analysis is carried out with orthogonal wavelet families.