On Filter Banks and Wavelets Based on Chebyshev Polynomials

TL;DR

This paper introduces Chebyshev wavelets derived from Chebyshev polynomials, exploring their properties, implementation, and application in signal denoising, highlighting their good selectivity despite non-orthogonality.

Contribution

It presents a new family of wavelets based on Chebyshev polynomials, analyzing their properties and demonstrating their effectiveness in signal denoising tasks.

Findings

Chebyshev wavelets have compact support and good selectivity.

They are not orthogonal but satisfy perfect reconstruction conditions.

Wavelet cascade convergence is proven using Markov chain properties.

Abstract

In this paper we introduce a new family of wavelets, named Chebyshev wavelets, which are derived from conventional first and second kind Chebyshev polynomials. Properties of Chebyshev filter banks are investigated, including orthogonality and perfect reconstruction conditions. Chebyshev wavelets have compact support, their filters possess good selectivity, but they are not orthogonal. The convergence of the cascade algorithm of Chebyshev wavelets is proved by using properties of Markov chains. Computational implementation of these wavelets and some clear-cut applications are presented. Proposed wavelets are suitable for signal denoising.