Gabor's signal expansion and the Zak transform

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, in which the window function is related to the elementary signal in such a way that the set of shifted and modulated elementary signals is biorthonormal 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 and the discrete-time cases 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 by a fast computer algorithm, which is analogous to the well-known fast Fourier-transform= algorithm.