Gabor's signal expansion and the Zak transform, with oversampling by an integer factor

Gabor's 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 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.