| Signal processing is concerned with the representation, manipulation, and transformation of signals and the information that they carry in such a way that enhance our ability to extract the aspect of interest from the information that a signal carries, since the same signal has different aspects of interest for different users and/or different applications, this makes the task of choosing a signal representation more difficult and more crucial; an efficient representation in the sense of easy implementation, provide meaningful and easy interpretation, and fast processing is the heart of many signal processing problems and applications. Time-domain and frequency-domain representations are the most prominent representations in signal processing for describing a signal in time or frequency domains, not in both. Because time and frequency representations are related via Fourier transform, the signal time and frequency behaviors are not independent. Based on the frequency behavior signals can be grouped into two classes. First class, signals whose frequency contents do not change with time and called stationary signals. For this class Fourier analysis techniques (frequency-domain representations) provide efficient meaningful and easy interpretable representations that precisely characterize the signal frequency behavior. The second class, signals whose frequency contents evolve with time and called non-stationary signals. For this class classical Fourier analysis leads to physically meaningless descriptions, and thus the need for alternative representations become a necessity. Such representations which provide simultaneous time-frequency information are called joint time-frequency representations (JTFRs).; Throughout the course of searching for a JTFR of high time-frequency resolution and capability of fully characterizing a multicomponent non-stationary transient signals many joint time-frequency representations were implemented and tested to discover their advantages and their inherent limitations, including bilinear and linear TFRs. Finite Zak transform (FZT) and Weyl-Heisnberg (W-H) expansions as a powerful linear time-frequency representations outperform all the bilinear TFRs for representing and detecting the multicomponent signal. Furthermore, efficient algorithms to compute FZT of R-D signals were formulated using tensor product notations. The intimate relationship of FZT to W-H expansion is utilized extensively to characterize W-H systems and their linear span, to establish the necessary and sufficient conditions that guarantee the existence and convergence of the W-H coefficients, and to design efficient stable algorithms to compute the W-H expansion coefficients in Zak space. Moreover, an orthogonal projection algorithm to project W-H expansion coefficients onto subspaces as a tool for data compression and potential tool for multiresolution analysis is presented. |
