XFT: An Extension of the Discrete Fractional Fourier Transform
This work offers a new numerical method for computing the fractional Fourier transform, which is relevant for signal processing applications requiring time-frequency representations.
The paper derives a Gaussian-like quadrature for the continuous fractional Fourier transform using Hermite polynomials, leading to a fast chirp-FFT-chirp discretization. This method extends the transform to complex values inside the unit circle and provides a more accurate FFT for non-periodic functions.
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.