Linear Phase Perfect Reconstruction Filters and Wavelets with Even Symmetry

TL;DR

This paper characterizes all IIR linear phase filters that generate symmetric wavelets with any number of vanishing moments, providing explicit formulas and a new design parameterization for advanced wavelet filter construction.

Contribution

It introduces a comprehensive classification and explicit construction of IIR linear phase filters for symmetric wavelets with customizable properties, extending the FIR wavelet design framework.

Findings

Explicit formulas for IIR filter coefficients, zeros, and poles.

A new parameterization enabling tailored filter properties.

Method to implement IIR filters as FIR filters for practical use.

Abstract

Perfect reconstruction filter banks can be used to generate a variety of wavelet bases. Using IIR linear phase filters one can obtain symmetry properties for the wavelet and scaling functions. In this paper we describe all possible IIR linear phase filters generating symmetric wavelets with any prescribed number of vanishing moments. In analogy with the well known FIR case, we construct and study a new family of wavelets obtained by considering maximal number of vanishing moments for each fixed order of the IIR filter. Explicit expressions for the coefficients of numerator, denominator, zeroes, and poles are presented. This new parameterization allows one to design linear phase quadrature mirror filters with many other properties of interest such as filters that have any preassigned set of zeroes in the stopband or that satisfy an almost interpolating property. Using Beylkin's approach, it is indicated how to implement these IIR filters not as recursive filters but as FIR filters.