We prove a formula expressing the gradient of the phase function of a function $f :\mathbb{R}^d \rightarrow \mathbb{C}$ 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 theWigner distribution to fixed time is well defined provided a certain condition on its wave front set is satisfied. Therefore we first need to study the wave front set of the Wigner distribution of a tempered distribution.

### The wave front set of the Wigner Distribution and Instantaneous Frequency

#### Abstract

We prove a formula expressing the gradient of the phase function of a function $f :\mathbb{R}^d \rightarrow \mathbb{C}$ 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 theWigner distribution to fixed time is well defined provided a certain condition on its wave front set is satisfied. Therefore we first need to study the wave front set of the Wigner distribution of a tempered distribution.
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http://arxiv.org/pdf/1007.0874v1.pdf
Wigner distribution; Microregularity; Wave front set; Restriction of distributions; Instantaneous frequency
P. Boggiatto; A. Oliaro; P. Wahlberg
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/2318/89608