Mutual space-frequency distribution of Gaussian signal

TL;DR

This paper introduces a new mutual space-frequency distribution that generalizes Wigner and Weyl distributions, providing analytical expressions for Gaussian signals and revealing their geometric relationship in the conjugate coordinate space.

Contribution

It proposes a novel mutual distribution framework that encompasses Wigner and Weyl distributions as special cases, with analytical results for Gaussian signals.

Findings

Wigner and Weyl distributions are special cases of the mutual distribution.

The mutual distribution for Gaussian signals is derived analytically.

Wigner distribution is shown to be a rotational displacement of Weyl distribution.

Abstract

Mutual space-frequency distribution is proposed and it is shown that Wigner and Weyl distribution functions are only particular cases of these distribution. Mutual distribution for Gaussian signal is analytically obtained. The simple connection between Wigner and Weyl distributions is established. It is shown that Wigner distribution forms as the rotational displacement of Weyl distribution on informational diagram of conjugate coordinates $(x;p)$ on an angle proportional to the mutual parameter $t$. The results of direct calculations of mutual distribution for Gaussian signal in the mutual domain are presented.