# Linear perturbations of the Wigner transform and the Weyl quantization

## TL;DR

This paper explores quadratic time-frequency representations derived from linear perturbations of the Wigner transform, analyzing their properties, associated pseudodifferential calculus, and relation to Weyl quantization.

## Contribution

It introduces a new class of time-frequency representations, characterizes their relation to Cohen's class, and develops a corresponding pseudodifferential calculus.

## Key findings

- These representations satisfy Moyal's formula.
- They generally do not belong to Cohen's class.
- The paper establishes a link with Weyl calculus.

## Abstract

We study a class of quadratic time-frequency representations that, roughly speaking, are obtained by linear perturbations of the Wigner transform. They satisfy Moyal's formula by default and share many other properties with the Wigner transform, but in general they do not belong to Cohen's class. We provide a characterization of the intersection of the two classes. To any such time-frequency representation, we associate a pseudodifferential calculus. We investigate the related quantization procedure, study the properties of the pseudodifferential operators, and compare the formalism with that of the Weyl calculus.