A zero-assignment approach to two-channel filter banks and wavelets

Abstract

It is well-known that subband decomposition and perfect reconstruction of an arbitrary input signal is possible by a proper design of four lters. Besides having a wide range of applications in signal processing, perfect reconstruction lter banks have a strong connection with wavelets as pointed out by Mallat. Daubechies managed to design minimal order, maximally at lters and she proposed a cascade algorithm to construct compactly supported orthogonal wavelets from the orthogonal perfect reconstruction lter banks. The convergence of the cascade algorithm requires at least one zero at z = 1 and z = 1 for the lowpass and the highpass lters, respectively. This thesis focuses on the design of two-channel lter banks with assigned zeros. The fact that causal, stable and rational transfer functions form a Euclidean domain is used to pose the problem in an abstract setup. A polynomial algorithm is proposed to design lter banks with lters having assigned zeros and a characterization of all solutions having the same zeros in terms of a free, even, causal, stable and rational transfer function is obtained. A generalization of Daubechies design of orthogonal lter banks is given. The free parameter can be used to improve the lter bank design and the design of corresponding orthogonal or biorthogonal wavelets. The results also nd an application in examining the robustness of regularity of minimal length compactly supported wavelets with respect to perturbation of lter zeros at 1 and -1.