The Zak Transform: a Signal Transform for Sampled Time-continuous Signals

In this paper we study the Zak transform which is a signal transform relevant to time-continuous signals sampled at a uniform rate and an arbitrary clock phase. Besides being a time-frequency representation for timecontinuous signals, the Zak transform is a particularly useful tool in signal theory. It can be used to provide a framework within which celebrated themes like the Fourier inversion theorem, Pars eval's theorem, the Nyquist criterion, the Shannon sampling theorem, and the Gabor representation problem fit. Moreover, the Zak transform is of particular use for solving linear integral equations, the kernel of which is the autocorrelation function of a cyclostationary random process. This paper lists and interprets the numerous properties of the Zak transform; to one of these properties, the occurrence of zeros of Zak transforms, special attention is paid. Furthermore, the relation with other time-frequency representations such as the Wigner distribution and the radar ambiguity function is given, and the Gabor representation problem is tackled. Finally, an example of how to use the Zak transform is given by solving data transmission problems where a robust filter, with or without decision feedback, has to be designed with optimal performance properties with respect to intersymbol interference and noise surpression.