A Hilbert-Space Approach to the Complete Investigation of Vaidyanathan’s Procedure Applied to the Design of Unitary Filterbanks for the Generation of Orthogonal Wavelet Bases

The initial parameters we use in constructing a unitary filter bank are the number of channels N and the flatness M of the squared magnitude of the frequency response. With N = 2 and M ≥ 1 we get the classical Daubechies wavelet family, including the class of Haar wavelets for M = 1. Any choice of these two parameters leads to a polynomial of degree n = N · M − 1 in the variable z−1 containing the vector of impulse responses of the filterbank at time k. Introducing polyphase components it can be transferred into a polynomial of degree M − 1 in the variable z−N . This polynomial can be factorized into M − 1 factors, called elementary building blocks, as described in P. P. Vaidyanathan’s book. From these building blocks the entire filter bank can be derived by constructing a special unitary constant matrix. Under certain conditions additional elementary building blocks can be inserted into this product, increasing the overall complexity of the system without changing the polynomial degree of the considered matrix polynomial and leaving the low pass unchanged.