The Affine Wigner Distribution

TL;DR

This paper explores the affine Wigner distribution's mathematical structure, connecting it to group theory, the Mellin transform, and ambiguity functions, and discusses its properties, applications, and conjectures in signal analysis.

Contribution

It provides a rigorous connection between affine Wigner distributions, the Mellin transform, and introduces an affine ambiguity function, advancing the theoretical understanding of affine time-frequency analysis.

Findings

The affine Wigner distribution is never an analytic function.

The scalogram can be expressed as an affine convolution of affine Wigner distributions.

Deviation from being an affine Wigner distribution relates to intrinsic operator properties.

Abstract

We examine the affine Wigner distribution from a quantization perspective with an emphasis on the underlying group structure. One of our main results expresses the scalogram as (affine) convolution of affine Wigner distributions. We strive to unite the literature on affine Wigner distributions and we provide the connection to the Mellin transform in a rigorous manner. Moreover, we present an affine ambiguity function and show how this can be used to illuminate properties of the affine Wigner distribution. In contrast with the usual Wigner distribution, we demonstrate that the affine Wigner distribution is never an analytic function. Our approach naturally leads to several applications, one of which is an approximation problem for the affine Wigner distribution. We show that the deviation for a symbol to be an affine Wigner distribution can be expressed purely in terms of intrinsic operator-related properties of the symbol. Finally, we present an affine positivity conjecture regarding the non-negativity of the affine Wigner distribution.