Gabor's signal expansion and the Zak transform with non-critical sampling

Gabor's expansion of a signal into a set of shifted and modulated versions of a synthesis window is introduced, along with the inverse operation, i.e. the Gabor transform, which uses an analysis window with the help of which Gabor's expansion coefficients can be determined. The Zak transform is introduced and it is shown how this transform can be helpful in finding analysis windows that correspond to a given synthesis window, both in the case of critical sampling and in the case of rational oversampling. In particular, it is shown how an analysis window can be found with minimum L₂ norm, and that this window resembles best the synthesis window and yields the Gabor coefficients with minimum L₂ norm. Moreover, it is shown how an analysis window can be found that resembles best a function that is different from the synthesis window. The effects of different amounts of oversampling in the time and the frequency direction are considered.