Convolution Based Special Affine Wavelet Transform and Associated Multi-resolution Analysis

TL;DR

This paper introduces a novel special affine wavelet transform based on convolution in the special affine Fourier domain, exploring its properties, orthogonality, and multi-resolution analysis for advanced time-frequency analysis.

Contribution

It develops the first comprehensive framework for special affine wavelet transform and its multi-resolution analysis, including basis construction and fundamental properties.

Findings

Derived fundamental properties and orthogonality relations.

Established conditions for basis formation in the transform domain.

Extended the framework to multi-resolution analysis with orthogonal wavelets.

Abstract

In this paper, we study the convolution structure in the special affine Fourier domain to combine the advantages of the well known special affine Fourier and wavelet transforms into a novel integral transform coined as special affine wavelet transform and investigate the associated constant Q-property in the joint time-frequency domain. The preliminary analysis encompasses the derivation of the fundamental properties, orthogonality relation, inversion formula and range theorem. Finally, we extend the scope of the present study by introducing the notion of multi-resolution analysis associated with special affine wavelet transform and the construction of orthogonal special affine wavelets. We call it special affine multi-resolution analysis. The necessary and sufficient conditions pertaining to special affine Fourier domain under which the integer shifts of a chirp modulated functions form a Riesz basis or an orthonormal basis for a multi-resolution subspace is established.