On the symplectic covariance and interferences of time-frequency distributions

TL;DR

This paper investigates the covariance properties of quadratic time-frequency distributions, especially the Wigner distribution, and their relation to interference damping, providing characterizations and numerical illustrations.

Contribution

It characterizes the unique covariance property of the Wigner distribution among quadratic distributions and explores the trade-off with interference damping.

Findings

Wigner distribution's covariance property uniquely characterizes it.

Covariance and interference damping are in competition.

Numerical experiments illustrate theoretical results.

Abstract

We study the covariance property of quadratic time-frequency distributions with respect to the action of the extended symplectic group. We show how covariance is related, and in fact in competition, with the possibility of damping the interferences which arise due to the quadratic nature of the distributions. We also show that the well known fully covariance property of the Wigner distribution in fact characterizes it (up to a constant factor) among the quadratic distributions $L^{2}(\mathbb{R}^{n})\rightarrow C_{0}({ \mathbb{R}^{2n}})$. A similar characterization for the closely related Weyl transform is given as well. The results are illustrated by several numerical experiments for the Wigner and Born-Jordan distributions of the sum of four Gaussian functions in the so-called "diamond configuration".