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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.

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

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