On optimum oversampling in the Gabor scheme

The windowed Fourier transform of a time signal is considered, as well as a way to reconstruct the signal from a su ciently densely sampled version of its windowed Fourier transform using a Gabor representation; followingGabor, sampling occurs on a two-dimensional time-frequency lattice with equidistant time intervals and equidistant frequency intervals. In the limit of innitely dense sampling, the optimum synthesis window (which appears in Gabor's reconstruction formula) becomes similar to the analysis window (which is used in the windowed Fourier transform). It is shown that this similarity can already be reached for a rather small degree of oversampling, 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. The theory is elucidated by nding the optimum ratio in the cases of a Gaussian and an exponential analysis window.