On the discrete version of Gabor's signal expansion, the Gabor transform, and the Zak transform

Gabors expansion of a discrete-time signal into a set of shifted and modulated versions of an elementary signal or synthesis window is introduced, along with the inverse operation, i.e., the Gabor transform, which uses an analysis window that is related to the synthesis window and with the help of which Gabors expansion coefficients can be determined. The restriction to a signal and an analysis window that both have finite-support, leads to the concept of a discrete Gabor expansion and a discrete Gabor transform. After introduction of the discrete Fourier transform and the discrete Zak transform, it is possible to express the discrete Gabor expansion and the discrete Gabor transform as matrix-vector products. Using these matrix-vector products, a relationship between the analysis window and the synthesis window is derived. It is shown how this relationship enables us to determine the optimum synthesis window in the sense that it has minimum L_2 norm, and it is shown that this optimum synthesis window resembles best the analysis window.