Easy-to-compute parameterizations of all wavelet filters: input-output and state-space

TL;DR

This paper presents a straightforward method to characterize all rational wavelet filters using linear systems theory, enabling easy construction and state-space realization of these filters with a parameter called the index.

Contribution

It introduces a new input-output parameterization of rational wavelet filters based on elementary factorizations and provides a step-by-step approach for their state-space realization.

Findings

The parameterization simplifies the characterization of wavelet filters.

The method allows systematic construction of wavelet filters with increasing index.

State-space models can be derived directly from the parameterization.

Abstract

We here use notions from the theory linear shift-invariant dynamical systems to provide an easy-to-compute characterization of all rational wavelet filters. For a given N bigger or equql to 2, the number of inputs, the construction is based on a factorization to an elementary wavelet filter along with of m elementary unitary matrices. We shall call this m the index of the filter. It turns out that the resulting wavelet filter is of McMillan degree $N((N-1)/2+m). Rational wavelet filters bounded at infinity, admit state space realization. The above input-output parameterization is exploited for a step-by-step construction (where in each the index m is increased by one) of state space model of wavelet filters.