Magnitude-phase relationships for short-time Fourier transforms based on Gaussian analysis windows

The short-time Fourier transform (STFT) formally represents the output of a filter-bank spectrum analyzer as a two-dimensional function of time and frequency. For the case of a Gaussian analysis window, the log magnitude and phase of the STFT are related by a coupled pair of first order linear partial differential equations. Moreover, the log magnitude and phase independently satisfy second order linear partial differential equations. Because not all functions of time and frequency are STFT's, the second order equations provide a test for determining whether a particular magnitude or phase function corresponds to the magnitude or phase of a STFT obtained with a Gaussian analysis window. Furthermore, given a valid magnitude (or phase) function, the pair of first order equations can be integrated to determine the corresponding phase (or magnitude) function.