Sampling theorem for two dimensional fractional Fourier transform

Abstract

The fractional Fourier transform, which is a generalization of the Fourier transform, has become the focus of many research papers in recent years because of its applications in electrical engineering and optics. The fractional Fourier transform has been extended to n dimensions using tensor product of n copies of the one-dimensional transform.Recently, a new two dimensional fractional Fourier transform that is not a tensor product of two one-dimensional transforms was introduced. The definition of the new transform, which depends on two independent angles α and β, is based on the fact that the Hermite functions of two complex variables are eigenfunctions of the Fourier transform.The aim of this paper is to derive sampling theorem for this new transform. Unlike the sampling theorem in the tensor product case, where the sampling function is a product of two Sinc functions, one in each of the transform variables, in the new sampling theorem the sampling function is a product of two Sinc functions whose arguments are not the variables of the transform but a weighted sum and a weighted difference of the transform variables. Furthermore, the sample points depend on the sum and the difference of the transform angles, α,β, which leads to a more interesting configuration and distribution of the sample points.