This paper presents a new class of complex-valued compact-supported orthonormal symmlets. Firstly, some properties of complex-valued compact-supported orthonormal symmlets are investigated, and then it is shown that complex-valued symmlets can be generated by real-valued half-band filters. Therefore, the construction of complex-valued symmlets can be reduced to the design of real-valued half-band filters. Next, a design method of real-valued half-band FIR filters with some flatness requirements is proposed. For the maximally flat half-band filters, a closed-form solution is given. For the filter design with a given degree of flatness, the design problem is formulated in the form of linear system by using the Remez exchange algorithm and considering the given flatness condition. Therefore, a set of filter coefficients can be easily computed by solving a set of linear equations, and the optimal solution is obtained through a few iterations. Finally, some design examples are presented to demonstrate the effectiveness of the proposed method.

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.