THE ZAK TRANSFORM - A SIGNAL TRANSFORM FOR SAMPLED TIME-CONTINUOUS SIGNALS

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 time-continuous signals, the Zak transform is a useful tool in signal theory. It can be used to provide a framework within which celebrated themes like the Fourier inversion theorem, Parseval's theorem, the Nyquist criterion, the Shannon sampling theorem, and the Gabor representation problem fit. The Zak transform is of 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. 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. 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.