Gabor transform and Zak transform with rational oversampling

Gabor´s expansion of a signal into a set of shifted and modulated versions of an elementary signal is introduced, along with the inverse operation, i.e. the Gabor transform, which uses a window function that is related to the elementary signal and with the help of which Gabor´s expansion coefficients can be determined. The Zak transform — with its intimate relationship to Gabor´s signal expansion — is introduced. 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. In particular, a simple proof is presented of the fact that the window function with minimum L2 norm is identical to the window function whose difference from the elementary signal has minimum L2 norm, and thus resembles best this elementary signal, and that this window function yields the Gabor coefficients with minimum L2 norm.