The Zak transform and decimated time-frequency distributions

In this paper, the interrelation between the Zak transform and the generalized discrete time-frequency distribution (GDTFD) is examined. Starting with the discrete Zak transform, its definition is broadened to include arbitrary windows creating the windowed Zak transform (WZT). The WZT is then combined with the spectrogram. It is demonstrated that the spectrogram based upon the WZT, called the Zak spectrogram (ZS), is a generalization of the standard spectrogram. Next, building upon the idea of the weighted spectrogram, the weighted ZS is used to produce a new class of GDTFD, called the decimated GDTFD (D-GDTFD). The decimated GDTFD is similar to the GDTFD, except it trades bandwidth for computational speed and requires significantly less storage in order to be implemented. The reduction in discrete bandwidth is from 2π for the GDTFD to 2π/m for the D-GDTFD. An important attribute of the DGDTFD is that it requires significantly less storage than the GDTFD. The D-GDTFD requires only 1/m² of the storage of the GDTFD. An example using the binomial distribution is given to illustrate the connection between the D-GDTFD and the GDTFD. Throughout the paper, examples of multirate systems are given, which could be used to implement the building blocks of the DGDTFD. This makes the D-GDTFD practical to implement using slower and less expensive elements