Optimum oversampling in the rectangular Gabor scheme

The windowed Fourier transform of a time signal is considered, as well as a way to reconstruct the signal from a sufficiently densely sampled version of its windowed Fourier transform using a Gabor representation; following Gabor, sampling occurs on a two-dimensional time-frequency lattice with equidistant time intervals and equidistant frequency intervals. For sufficiently dense sampling, the synthesis window (which appears in Gabor´s reconstruction formula) may be constructed such that it resembles a rather arbitrarily given function; this function may or may not be proportional to the analysis window (which is used in the windowed Fourier transform). It is shown that the resemblance can already be reached for a rather small degree of over-sampling, if the sampling distances in the time and frequency directions are properly chosen. A procedure is presented with which the optimum ratio of the sampling intervals can be determined.