Factorization Approach to Unitary Time-Varying Filter Bank Trees and Wavelets

Abstract

A complete factorization of all optimal (in terms of quick transition) time-varying FIR unitary filter bank tree topologies is obtained [8]. This has applications in adaptive subband coding, tiling of the time-frequency plane and the construction of orthonormal wavelet and wavelet packet bases for the half-line and interval [14], [21], [6]. For an M-channel filter bank the factorization allows one to construct entry/exit filters that allow the filter bank to be used on finite signals without distortion at the boundaries. One of the advantages of our approach is that an efficient implementation algorithm comes with the factorization. The factorization can be used to generate filter bank tree-structures where the tree topology changes over time. Explicit formulas for the transition filters are obtained for arbitrary tree transitions. The results hold for tree structures where filter banks with any number of channels or filters of any length are used. Time-varying wavelet and wavelet packet bases are also constructed using these filter bank structures. Our construction of wavelets is unique in several ways: 1) the number of entry/exit functions is equal to the number of entry/exit filters of the corresponding filter bank; 2) these functions are defined as linear combinations of the scaling functions—other methods involve infinite product constructions [6], [14]; 3) the functions are trivially as regular as the wavelet bases from which they are constructed. © 1995 IEEE