Applications of the fractional Fourier transform in optics and signal processing: a review

Abstract

The fractional Fourier transform The fractional Fourier transform is a generalization of the common Fourier transform with an order parameter a. Mathematically, the ath order fractional Fourier transform is the ath power of the fractional Fourier transform operator. The a = 1st order fractional transform is the common Fourier transform. The a = 0th transform is the function itself. With the development of the fractional Fourier transform and related concepts, we see that the common frequency domain is merely a special case of a continuum of fractional domains, and arrive at a richer and more general theory of alternate signal representations, all of which are elegantly related to the notion of space-frequency distributions. Every property and application of the common Fourier transform becomes a special case of that for the fractional transform. In every area in which Fourier transforms and frequency domain concepts are used, there exists the potential for generalization and improvement by using the fractional transform.

A Novel way of image encrption with piplining pixel scrambling and fractional fourier transform

“…FRFT is a generalization of FT [1,15]. It is not only richer in theory and more flexible in application, but is also not expensive in implementation.…”

Optimal representation and processing of optical signals in quadratic-phase systems

“…See [1] for subtleties in the definition of fractional operator powers as well as alternative ways of defining the FRT. Here…”

“…We will first determine the spatial extent of the output signal^( ) g x observed at z¼d, given an input signal^( ) f x at z ¼0. It is known that the Wigner distributions of^( ) f x and^( ) g x have the following relationship [1,[33][34][35][36]:…”

“…Quadratic-phase systems (QPSs), also known as first-order optical systems or ABCD systems, are a very general family of optical systems encompassing arbitrary concatenations of various components such as thin lenses and sections of free space in the Fresnel approximation, as well as quadratic graded-index media [1][2][3][4]. Mathematically, QPSs are referred to as linear canonical transforms [5,7,6,[8][9][10][11].…”

The discrete fractional random cosine and sine transforms

“…The DFRNT is a kind of discrete transform with fractional order originated from the fractional fourier transform (FrFT) [2] and especially the discrete fractional Fourier transform (DFrFT) [3], and thus has the most excellent mathematical properties as FrFT and DFrFT have. Meanwhile, however, the result of transform itself can be inherently random.…”