Fractional Zak transform: Theory and applications

The Zak transform (ZT) emerges as a powerful tool for time–frequency analysis. However, the ZT is inherently limited in its ability to capture local time–frequency features of non-stationary or time-varying signals, due to its nature as a global spectral analysis tool. In this article, we present a comprehensive exploration of the properties, discretization and potential applications of the fractional ZT (FRZT). We begin by elucidating the mathematical foundation of the FRZT, highlighting its ability to capture the time–frequency characteristics of signals with enhanced flexibility. Furthermore, we explore the features of the FRZT, including its basic properties, convolution theorem, and the Weyl–Heisenberg frames associated with FRZT. Additionally, for practical applications where the signals are discrete, we develop two discretization algorithms for the FRZT. Finally, the experiments with simulated chirp signals and potential applications demonstrate the utility of the FRZT in various signal processing tasks through theoretical exposition and practical examples.