DISCRETE ZAK TRANSFORMS, POLYPHASE TRANSFORMS, AND APPLICATIONS

We consider three different versions of the Zak transform (ZT) for dis crete-time signals, namely, the discrete-time ZT, the polyphase transf orm, and a cyclic discrete ZT. In particular, we show that the extensi on of the discrete-time ZT to the complex z-plane results in the polyp hase transform, an important and well-known concept in multirate signa l processing and filter bank theory. We discuss fundamental properties , relations, and transform pairs of the three discrete ZT versions, an d we summarize applications of these transforms. In particular, the di screte-time ZT and the cyclic discrete ZT are important for discrete-t ime Gabor expansion (Weyl-Heisenberg frame) theory since they diagonal ize the Weyl-Heisenberg frame operator for critical sampling and integ er oversampling, The polyphase representation plays a fundamental role in the theory of filter banks, especially DFT filter banks. Simulatio n results are presented to demonstrate the application of the discrete ZT to the efficient calculation of dual Gabor windows, tight Gabor wi ndows, and frame bounds.