A Discrete Zak Transform

A discrete version of the Zak transform is defined and used to analyze discrete Weyl–Heisenberg frames, which are nonorthogonal systems in the space of square-summable sequences that, although not necessarily bases, provide representations of square-summable sequences as sums of the frame elements. While the general theory is essentially similar to the continuous case, major differences occur when specific Weyl–Heisenberg frames are evaluated. In particular, it is shown that while Weyl–Heisenberg frames in the continuous case are bases only if they are generated by functions that are not smooth or have poor decay, it is possible in the discrete case to construct Weyl–Heisenberg frames that are bases and are generated by sequences with good decay. The sampled Gaussian provides an example of such a signal.