Performance of Discrete Fractional Fourier Transform Classes in Signal Processing Applications

Abstract

Given the widespread use of ordinary Fourier transform in science and engineering, it is important to recognize this integral transform as the fractional power of FT. Indeed, it has been this recognition, which has inspired most of the many recent applications replacing the ordinary FT with FrFT (which is more general and includes FT as special case) adding an additional degree of freedom to problem, represented by the fraction or order parameter a. This in turn may allow either a more general formulation of the problem or improvement based on possibility of optimizing over a (as in optimal wiener filter resulting in smaller mean square error at practically no additional cost). The FrFT has been found to have several applications in the areas of optics and signal processing and it also lead to generalization of notion of time (or space) and frequency domains which are central concepts of signal processing. In every area where FT and frequency domain concepts are used, there exists the potential for generalization and implementation by using FrFT. With the advent of computers and enhanced computational capabilities the Discrete Fourier Transform (DFT) came into existence in evaluation of FT for real time processing. Further these capabilities are enhanced by the introduction of DSP processors and Fast Fourier Transform (FFT) algorithms. On similar lines, so there arises a need for discretization of FrFT. Furthermore, DFT is having only one basic definition and nearly 200 algorithms are available for fast computation of DFT. But when FrFT is analysed in discrete domain there are many definitions of Discrete Fractional Fourier Transform (DFrFT). These definitions are broadly classified according to the methodology of computation adopted. In the current study the various class of DFrFT algorithms are studied and compared on the basis of computational complexity, deviation factor, properties of FrFT retained by particular class and constraints on the order or fraction parameter a etc. As discussed earlier