The wave front set of the Wigner distribution and instantaneous frequency

TL;DR

This paper establishes a formula linking the phase gradient of a function to a frequency moment of its Wigner distribution, under specific regularity and wave front set conditions.

Contribution

It introduces a new formula connecting phase gradients to Wigner distribution moments and analyzes the wave front set of the Wigner distribution for tempered distributions.

Findings

The phase gradient can be expressed as a normalized frequency moment of the Wigner distribution.

The formula applies to functions with compactly supported Fourier transforms or in certain Sobolev spaces.

Conditions on the wave front set ensure the well-definedness of the Wigner distribution at fixed time.

Abstract

We prove a formula expressing the gradient of the phase function of a function $f: \mathbb R^d \mapsto \mathbb C$ as a normalized first frequency moment of the Wigner distribution for fixed time. The formula holds when $f$ is the Fourier transform of a distribution of compact support, or when $f$ belongs to a Sobolev space $H^{d/2+1+\epsilon}(\mathbb R^d)$ where $\epsilon>0$. The restriction of the Wigner distribution to fixed time is well defined provided a certain condition on its wave front set is satisfied. Therefore we first study the wave front set of the Wigner distribution of a tempered distribution.