Oversampling in Gabor´s signal expansion by an integer factor

Gabor´s (1946) expansion of a signal into a discrete set of shifted and modulated versions of an elementary signal is reviewed and its relation to sampling of the sliding-window spectrum is shown. It is indicated how Gabor´s expansion coefficients can be found as samples of the sliding-window spectrum, where the window function, which still has to be determined, is related to the elementary signal. Gabor´s critical sampling as well as the case of oversampling by an integer factor are considered. The Zak (1967) transform is introduced and its intimate relationship to Gabor´s signal expansion is demonstrated. It is shown how the Zak transform can be helpful in determining Gabor´s expansion coefficients and how it can be used in finding window functions that correspond to a given elementary signal. An arrangement is described which is able to generate Gabor´s expansion coefficients of a rastered, one-dimensional signal by coherent-optical means