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I have a speech signal and have calculated the magnitude spectrum for the signal. Is it possible to get a good estimate of the original signal from the magnitude spectrum, seeing that phase information has been discarded after the FFT?

I have looked at the Wiener-Khintchine theorem, but I am unable to find any code examples of it in use, and I have not managed to get it to work.

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    $\begingroup$ A useful answer here: Reconstruction of audio signal from Spectrogram. $\endgroup$ – Gilles Nov 30 '17 at 17:28
  • $\begingroup$ Your question is very fundamental. What limit do you set to what you consider, and would use, as a spectrum? Which side information are you ok to work on $\endgroup$ – Laurent Duval Nov 30 '17 at 20:46
  • $\begingroup$ For historical context: I believe that Bode (or Bode plot fame) proved some theorems on this in the 1930's. It involved assumptions like the all the poles in the left-hand plane and such, but was lucid and gave a constructive formula. $\endgroup$ – rrogers Dec 6 '17 at 14:04
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Apart from the stochastic literature, there are a set of deterministic algorithms that claim exact(?) recovery of a signal form its partial Fourier description, collectively referred to as:

signal reconstruction from DTFT phase or magnitude alone

They are generally iterative in nature and exact convergence is hard to achieve in practice. Monson Hayes [et al] published relevant IEEE papers to propose several algorithms that reconstruct a real signal from its DTFT phase function alone, which is based on the phase sufficiency claim. The magnitude is a little more diffcult as I know. Have a look at those papers for your benefit.

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