Gabor's expansion and the Zak transform for continuous-time and discrete-time signals

Gabor's expansion of a signal into a discrete set of shifted and modulated versions of an elementary signal is introduced and its relation to sampling of the sliding-window spectrum is shown. It is shown how Gabor's expansion coefficients can be found as samples of the sliding-window spectrum, where - at least in the case of critical sampling - the window function is related to the elementary signal in such a way that the set of shifted and modulated elementary signals is bi-orthonormal to the corresponding set of window functions. 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 the window function that corresponds to a given elementary signal and how it can be used to find Gabor's expansion coefficients. The continuous-time as well as the discrete-time case are considered, and, by sampling the continuous frequency variable that still occurs in the discrete-time case, the discrete Zak transform and the discrete Gabor transform are introduced. It is shown how the discrete transforms enable us to determine Gabor's expansion coefficients via a fast computer algorithm, analogous to the well-known fast Fourier transform algorithm. Not only Gabor's critical sampling is considered, but also - for continuous-time signals - the case of oversampling by an integer factor. It is shown again how in this case the Zak transform can be helpful in determining a (no longer unique) window function corresponding to a given elementary signal. An arrangement is described which is able to generate Gabor's expansion coefficients of a rasterd, one-dimensional signal by coherent-optical means