# Reconstruct signal from its Spectrogram only

The spectrogram of a signal throws away the phase information, but it is said to be possible to reconstruct the signal only from the spectrogram [1] via $$s(t) = \frac{1}{2\pi s^*(0)}\int_{-\infty}^{\infty}\frac{M_{SP}(\theta,t)}{A_h(-\theta,t)}e^{-j\theta t/2}d\theta$$ where $M_{SP}$ is the chariteristic function of spectrogram and ${A_h(\theta,t)}$ is the ambiguity function of the window.

The formula seems to work in theory. Could anybody explain why it works? Some phase information is thrown away after all. Also, is it a good reconstruction tool used in practice? It seems to me that nobody uses this as the reconstruction tool.

[1] L. Cohen, Time-frequency analysis. Englewood Cliffs, N.J: Prentice Hall PTR, 1995.

• I've got that book. Which page? – Marcus Müller Mar 14 '18 at 11:28
• In my edition (1995), p. 108, section 7.9, formula 7.94 – Laurent Duval Mar 14 '18 at 21:31
• Does anybody know why? – ZHUANG Mar 21 '18 at 3:14
• What are the integration limits? What variable are we supposed to integrate ($\mathrm{d}\theta$, I suppose)? – Tendero Mar 22 '18 at 13:24
• Sorry for missing this information. It has been edited. – ZHUANG Mar 23 '18 at 1:08