Time-Frequency Reassignment Formula

If $X(t,\omega)$ denotes the STFT at frame-center-time $t$ and bin-center-frequency $\omega$ , then the new (``reassigned'') location of the point $X(t,\omega)$ is computed as

$$\begin{eqnarray*} t' &=& t - \frac{\partial \angle X(t,\omega)}{\partial \omega}\\ [5pt] % \isdefs t - \grad_\omega X(t,\omega) \omega' &=& \frac{\partial \angle X(t,\omega)}{\partial t} % \isdefs \grad_t X(t,\omega) \end{eqnarray*}$$

We see that

$$\begin{eqnarray*} t' &\!=\!& t\,+ \mbox{ \emph{group delay} of spectral neighborhood $X(t,\omega\pm\epsilon)$}\\ \omega' &\!=\!& \mbox{\emph{instantaneous frequency} of temporal neighborhood} \\ && X(t\pm\epsilon,\omega)\\ && \mbox{(a complex narrowband signal centered at frequency $\omega$)} \end{eqnarray*}$$

(Instantaneous frequency = Fourier dual of group delay)