New results on approximate Hilbert pairs of wavelet filters with common factors

TL;DR

This paper analyzes the design of approximate Hilbert pairs of wavelet filters using the Thiran common-factor approach, providing explicit bounds on their analyticity and demonstrating the existence of filters with perfect reconstruction and many vanishing moments.

Contribution

It offers explicit expressions and bounds for the analyticity of wavelet filters constructed via the common-factor approach and proves the existence of such filters with perfect reconstruction and multiple vanishing moments.

Findings

Explicit bounds for the analyticity approximation of wavelet filters.

Proof of existence of filters with perfect reconstruction and many vanishing moments.

Quantitative measures for the analyticity of approximate Hilbert wavelet pairs.

Abstract

In this paper, we consider the design of wavelet filters based on the Thiran common-factor approach proposed in Selesnick [2001]. This approach aims at building finite impulseresponse filters of a Hilbert-pair of wavelets serving as real and imaginary part of a complexwavelet. Unfortunately it is not possible to construct wavelets which are both finitelysupported and analytic. The wavelet filters constructed using the common-factor approachare then approximately analytic. Thus, it is of interest to control their analyticity. Thepurpose of this paper is to first provide precise and explicit expressions as well as easilyexploitable bounds for quantifying the analytic approximation of this complex wavelet.Then, we prove the existence of such filters enjoying the classical perfect reconstructionconditions, with arbitrarily many vanishing moments.