XFT: An Extension of the Discrete Fractional Fourier Transform

TL;DR

This paper introduces XFT, an extension of the fractional Fourier transform, providing a fast discretization method using a Gaussian-like quadrature based on Hermite polynomials, and extends its range to complex values.

Contribution

It derives a new Gaussian-like quadrature for the fractional Fourier transform, enabling a fast, closed-form discretization and extending the transform's range to complex values inside the unit circle.

Findings

Provides a fast discretization of the fractional Fourier transform.

Extends the transform to complex values inside the unit circle.

Offers a more accurate version of FFT for non-periodic functions.

Abstract

In recent years there has been a growing interest in the fractional Fourier transform driven by its large number of applications. The literature in this field follows two main routes. On the one hand, the areas where the ordinary Fourier transform has been applied are being revisited to use this intermediate time-frequency representation of signals, and on the other hand, fast algorithms for numerical computation of the fractional Fourier transform are devised. In this paper we derive a Gaussian-like quadrature of the continuous fractional Fourier transform. This quadrature is given in terms of the Hermite polynomials and their zeros. By using some asymptotic formulas, we rewrite the quadrature as a chirp-fft-chirp transformation, yielding a fast discretization of the fractional Fourier transform and its inverse in closed form. We extend the range of the fractional Fourier transform by considering arbitrary complex values inside the unitary circle and not only at the boundary. We find that, the chirp-fft-chirp transformation evaluated at z=i, becomes a more accurate version of the fft which can be used for non-periodic functions.