This paper introduces the construction of a family of complex-valued scaling functions and wavelets with symmetry/antisymmetry, compact support and orthogonality from the Daubechies polynomial, and applies them to solve electromagnetic scattering problems. For simplicity, only two extreme cases in the family, maximum-localized complex-valued wavelets and minimum-localized complex-valued wavelets are investigated. Regularity of root location of the Daubechies polynomial in spectral factorization are also presented to construct these two extreme genus of complex-valued wavelets. When wavelets are used as basis functions to solve electromagnetic scattering problems by the method of moment (MoM), they often lead to sparse matrix equations. We will compare the sparsity of MoM matrices by the real-valued Daubechies wavelets, minimum-localized complex-valued Daubechies and maximum-localized complex-valued Daubechies wavelets. Our research summarized in this paper shows that the wavelets with smaller signal width will result in a more sparse MoM matrix, especially when the scatterer is with many corners.

In this paper, we apply the discrete wavelet transform (DWT) and the discrete wavelet packet transform (DWPT) with the Daubechies wavelet of order 16 to effectively solve for the electromagnetic scattering from a one-dimensional inhomogeneous slab. Methods based on the excitation vector and the [Z] matrix are utilized to sparsify an MoM matrix. As we observed, there are no much high frequency components of the field in the dielectric region, hence the wavelet coefficients of the small scales components (high frequency components) are very small and negligible. This is different from the case of two-dimensional scattering from perfect conducting objects. In the excitation-vector-based method, a modified excitation vector is introduced to extract dominant terms and achieve a better compression ratio of the matrix. However, a smaller compression ratio and a tiny relative error are not obtained simultaneously owing to their deletion of interaction between different scales. Hence, it is inferior to the [Z]-matrix-based methods. For the [Z]-marix-based methods, our numerical results show the column-tree-based DWPT method is a better choice to sparsify the MoM matrix than DWT-based and other DWPT-based methods. The cost of a matrix-vector multiplication for the wavelet-domain sparse matrix is reduced by a factor of 10, compared with that of the original dense matrix.

Effective wavelets to solve electromagnetic integral equations are proposed. It is based on the same construction procedure as Daubechies wavelets but with mix-phase to obtain maximum sparsity of moment matrix. These new wavelets are proved to have excellent performance in non-zero elements reduction in comparison with minimum-phase wavelet transform (WT). If further sparsity is concerned, wavelet packet (WP) transform can be applied but increases the computational complexity. In some cases, the capability of non-zero elements reduction by this new wavelets even better than WP with minimum-phase wavelets and with less computational efforts. Numerical experiments demonstrate the validity and effectiveness of the new wavelets.