# Interpretation of instaneous time and instaneous frequency in time-frequency plane?

In this paper: Sparse time-frequency representations, Given signal $x(t)$, the time-frequency representation was got by Gabor transform and was denoted by $\chi(t,\omega)=|\chi(t,\omega)|\exp(\mathrm{i}\,\phi(t,\omega))$. Then it give two quantities:

instantaneous frequency:$$\omega_{ins}(\omega,t)=\frac{\partial\,\phi}{\partial\,t}$$ instantaneous time: $$t_{ins}(\omega,t)=t-\frac{\partial\,\phi}{\partial\,\omega}$$

My questions is: I have learned the interpretation of instaneous frequency of a signal $x(t)$ and the group delay of a filter $H(\omega)$, but I do not know how to interpret these two quantities in time-frequency representation. Any kind of thought is welcomed.

• dunno about "instantaneous time". dunno why it would be different (conceptually) than $t$. i understand that $t_\text{ins}$ is different than $t$, but i am not sure what the significance is of that. – robert bristow-johnson Jan 11 '18 at 22:07