Novel Special Affine Wavelet Transform and Associated Uncertainity Inequalities

TL;DR

This paper introduces a new wavelet transform based on the special affine Fourier transform to capture local signal information, extending harmonic analysis results and establishing uncertainty inequalities.

Contribution

It proposes the novel special affine wavelet transform (NSAWT) and extends key harmonic analysis results and uncertainty principles to this new framework.

Findings

Established fundamental properties of NSAWT including Moyal's principle and inversion formula.

Derived Heisenberg and Pitt's inequalities for SAFT and NSAWT.

Extended uncertainty principles to the proposed NSAWT framework.

Abstract

{.2in} {\small {\bf Abstract.} Due to the extra degrees of freedom, special affine Fourier transform (SAFT) has achieved a respectable status within a short span and got versatile applicability in the areas of signal processing, image processing,sampling theory, quantum mechanics. However, due to its global kernel, SAFT fails to obtain local information of non-transient signals. To overcome this, we in this paper introduce the concept of novel special affine wavelet transform (NSAWT) and extend key harmonic analysis results to NSAWT analogous to those for the wavelet transform. We first establish some fundamental properties including Moyal's principle, Inversion formula and the range theorem. Some Heisenberg type inequalities and Pitt's inequality are established for SAFT and consequently Heisenberg uncertainity principle is derived for NSAWT.