Introduction to the fractional fourier transform and its applications

Abstract

This chapter is an introduction to the fractional Fourier transform and its applications. The fractional Fourier transform is a generalization of the ordinary Fourier transform with an order parameter a. Mathematically, the ath order fractional Fourier transform is the ath power of the Fourier transform operator. The a = 1st order fractional transform is the ordinary Fourier transform. In essence, the ath order fractional Fourier transform interpolates between a function f(u) and its Fourier transform F(μ). The 0th order transform is simply the function itself, whereas the 1st order transform is its Fourier transform. The 0.5th transform is something in between, such that the same operation that takes us from the original function to its 0.5th transform will take us from its 0.5th transform to its ordinary Fourier transform. More generally, index additivity is satisfied: The a2th transform of the a1th transform is equal to the (a2 + a1)th transform. The –1th transform is the inverse Fourier transform, and the –ath transform is the inverse of the ath transform.