On the discrete Gabor transform and the discrete Zak transform

Gabor's expansion of a discrete-time signal into a set of shifted and modulated versions of an elementary signal (or synthesis window) and the inverse operation - the Gabor transform - with which Gabor's expansion coefficients can be determined, are introduced. It is shown how, in the case of a finite-support analysis window and with the help of an overlap-add technique, the discrete Gabor transform can be used to determine Gabor's expansion coefficients for a signal whose support is not finite. The discrete Zak transform is introduced and it is shown how this transform, together with the discrete Fourier transform, can be used to represent the discrete Gabor transform and the discrete Gabor expansion in sum-of-products forms. It is shown how the sum-of-products form of the Gabor transform enables us to determine Gabor's expansion coefficients in a different way, in which fast algorithms can be applied. Using the sum-of-products forms, 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.