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 L
2
norm is identical to the window function whose difference from the elementary signal has minimum L
2
norm, and thus resembles best this elementary signal, and that this window function yields the Gabor coefficients with minimum L
2
norm.