FACTORIZATION APPROACH TO UNITARY TIME-VARYING FILTER BANK TREES AND WAVELETS

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 wavele t 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 w ithout distortion at the boundaries. One of the advantages of our appr oach is that an efficient implementation algorithm comes with the fact orization. The factorization can be used to generate filter bank tree- structures where the tree topology changes over time. Explicit formula s for the transition filters are obtained for arbitrary tree transitio ns. The results hold for tree structures where filter banks with any n umber of channels or filters of any length are used. Time-varying wave let and wavelet packet bases are also constructed using these filter b ank structures. Our construction of wavelets is unique in several ways : 1) the number of entry/exit functions is equal to the number of entr y/exit filters of the corresponding filter bank; 2) these functions ar e defined as linear combinations of the scaling functions-other method s involve infinite product constructions [6], [14]; 3) the functions a re trivially as regular as the wavelet bases from which they are const ructed.