The wave front set of the Wigner Distribution and Instantaneous Frequency

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.