Iterative reconstruction of speech from short-time Fourier transform phase and magnitude spectra

In this paper, we consider the topic of iterative, one dimensional, signal reconstruction (specifically speech signals) from the magnitude spectrum and the phase spectrum. While this topic has been extensively researched and documented, we wish to recast some well-established results for the benefit of new researchers and those who desire a short, yet comprehensive, review of the subject. The three main points of the review are: (i) a signal can be reconstructed to within a scale factor from its phase spectrum, (ii) a signal cannot be reconstructed to within a scale factor from its magnitude spectrum, and (iii) a signal can be reconstructed to within a scale factor from its magnitude spectrum when the phase-sign (i.e., one bit of phase spectrum information) is known. Through a number of illustrative examples, we first demonstrate how the algorithms work when the spectral information is determined over the entire duration of the signal. We then demonstrate that the algorithms are equally valid for reconstruction of a signal from the spectra obtained from short-time segments. In addition, we present the results of some further experimentation in which we have attempted to reconstruct a speech signal from only partial phase spectrum information (in the absence of all magnitude spectrum information). We make the following observations: (i) intelligible signal reconstruction (albeit noisy) is possible from knowledge of only the phase spectrum sign information, (ii) an intelligible signal cannot be reconstructed from knowledge of only the phase spectrum frequency-derivative or only the phase spectrum time-derivative, and (iii) an intelligible signal can be reconstructed from the combined knowledge of both the phase spectrum frequency-derivative and time-derivative.