A New Perspective on the Two-Dimensional Fractional Fourier Transform and Its Relationship with the Wigner Distribution

Abstract

The fractional Fourier transform \(F_{\theta }(w)\) with an angle \(\theta \) of a function f(t) is a generalization of the standard Fourier transform and reduces to it when \(\theta =\pi /2. \) It has many applications in signal processing and optics because of its close relations with a number of time-frequency representations. It is known that the Wigner distribution of the fractional Fourier transform \(F_{\theta }(w)\) may be obtained from the Wigner distribution of f by a two-dimensional rotation with the angle \(\theta \) in the \(t-w\) plane The fractional Fourier transform has been extended to higher dimensions by taking the tensor product of one-dimensional transforms; hence, resulting in a transform in several but separable variables. It has been shown that the Wigner distribution of the two-dimensional fractional Fourier transform \(F_{\theta ,\phi }(v,w)\) may be obtained from the Wigner distribution of f(x, y) by a simple four-dimensional rotation with the angle \(\theta \) in the \(x-y\) plane and the angle \(\phi \) in the \(v-w\) plane. The aim of this paper is two-fold: (1) To introduce a new definition of the two-dimensional fractional Fourier transform that is not a tensor product of two copies of one-dimensional transforms. The new transform, which is more general than the one that exists in the literature, uses a relatively new family of Hermite functions, known as Hermite functions of two complex variables. (2) To give an explicit matrix representation of a four-dimensional rotation that verifies that the Wigner distribution of the new two-dimensional fractional Fourier transform \(F_{\theta ,\phi }(v,w)\) may be obtained from the Wigner distribution of f(x, y) by a four-dimensional rotation. The matrix representation is more general than the one for the tensor product case and it corresponds to a four-dimensional rotation with two planes of rotations, one with the angle \((\theta +\phi )/2\) and the other with the angle \((\theta -\phi )/2\).