Discrete Zak transforms, polyphase transforms, and applications

Abstract

Abstract-We consider three different versions of the Zak transform (ZT) for discrete-time signals, namely, the discretetime ZT, the polyphase transform, and a cyclic discrete ZT. In particular, we show that the extension of the discrete-time ZT to the complex z-plane results in the polyphase transform, an important and well-known concept in multirate signal processing and filter bank theory.We discuss fundamental properties, relations, and transform pairs of the three discrete ZT versions, and we summarize applications of these transforms. In particular, the discrete-time ZT and the cyclic discrete ZT are important for discrete-time Gabor expansion (Weyl-Heisenberg frame) theory since they diagonalize the Weyl-Heisenberg frame operator for critical sampling and integer oversampling. The polyphase representation plays a fundamental role in the theory of filter banks, especially DFT filter banks. Simulation results are presented to demonstrate the application of the discrete ZT to the efficient calculation of dual Gabor windows, tight Gabor windows, and frame bounds.

Frame-theoretic analysis of oversampled filter banks

“…We consider a DFT FB (see Section VII-A) with channels and a 192-tap lowpass analysis prototype filter . The simulation results were obtained by performing all calculations within the framework of cyclic DFT FB's (cyclic Weyl-Heisenberg frames) [39] with period 192. The dual windows and the frame bounds we obtained are, hence, approximations to the true (i.e., noncyclic) dual windows and frame bounds.…”

“…Proof: Using (6), it can easily be shown that , where : denotes the polyphase transform (Zak transform) operator, i.e., the operator mapping a signal to the polyphase domain with . Since is a unitary transformation [39], it follows that and are unitarily equivalent. Therefore, and have the same eigenvalues [44].…”

“…Integrating both sides of this inequality with respect to the frequency parameter and using [39], we obtain (10).…”

“…Consequently, a FB corresponding to a UFBF is "diagonal in the polyphase domain" if the UFBF matrix is a diagonal matrix diag with It follows from (8) that the polyphase matrix of the minimum norm synthesis FB is given by diag (19) We can see that the calculation of the minimum norm synthesis FB, which in general requires the inversion of the UFBF Integer oversampled or critically sampled DFT FB's [4], [6], [7], [11], [16], [24], [39], [54], [58] are an important example of FB's that are diagonal in the polyphase domain. The corresponding UFBF type is the important class of Weyl-Heisenberg frames [16], [25], [28], [30], [33], [34], [39], [59], [60]. In a DFT FB, the analysis filters are modulated versions of a single analysis prototype filter , i.e., with .…”

Equivalence of two methods for constructing tight Gabor frames

“…) and critical sampling, the Gabor frame operator can conveniently be expressed using the Zak transform (ZT) [6], [2], [14], [15]. For , the ZT of a signal is defined as (7) In this letter, we will be only concerned with the case and .…”