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arXiv · 1007.0874

The wave front set of the Wigner distribution and instantaneous frequency

Paolo Boggiatto·Alessandro Oliaro·Patrik Wahlberg
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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 $H^{d/2+1+ε}(\mathbb R^d)$ where $ε>0$. 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.

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Paolo Boggiatto, Alessandro Oliaro, Patrik Wahlberg. 2010-07-06. The wave front set of the Wigner distribution and instantaneous frequency. https://arxiv.org/abs/1007.0874

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