Special Affine Wavelet Transforms and the Corresponding Poisson Summation Formula

TL;DR

This paper introduces the special affine wavelet transform (SAWT), a new time-frequency analysis tool that overcomes SAFT's limitations in local feature extraction, and establishes its mathematical properties and relationships.

Contribution

The paper proposes the SAWT, explores its properties, discrete form, relation to Wigner distribution, and derives a Poisson summation formula for it.

Findings

SAWT possesses the constant Q-property in joint time-frequency domain.

The paper provides inversion, range characterization, and discrete reconstruction formulas for SAWT.

A Poisson summation formula analogue for SAWT is established.

Abstract

The special affine Fourier transform (SAFT) is a promising tool for analyzing non-stationary signals with more degrees of freedom. However, the SAFT fails in obtaining the local features of non-transient signals due to its global kernel and thereby make SAFT incompetent in situations demanding joint information of time and frequency. To circumvent this limitation, we propose a highly flexible time-frequency transform namely, the special affine wavelet transform (SAWT) and investigate the associated constant $Q$-property in the joint time-frequency domain. The basic properties of the proposed transform such as Rayleigh's theorem, inversion formula and characterization of the range are discussed using the machinery of special affine Fourier transforms and operator theory. Besides this, the discrete counterpart of SAWT is also discussed and the corresponding reconstruction formula is obtained. Moreover, we also drive a direct relationship between the well known special affine Wigner distribution and the proposed transform. This is followed by introducing a new kind of wave packet transform associated with the special affine Fourier transform. Towards the end, an analogue of the Poisson summation formula for the proposed special affine wavelet transform is derived.