XFT：离散分数阶傅里叶变换的扩展
XFT: An Extension of the Discrete Fractional Fourier Transform
AI总结：
本文提出一种基于Hermite多项式的高斯型求积公式来离散化连续分数阶傅里叶变换，并将其转化为chirp-fft-chirp变换，从而得到快速且封闭形式的离散分数阶傅里叶变换及其逆变换，同时将变换参数扩展到单位圆内的任意复数。
摘要：
近年来，由于分数阶傅里叶变换的大量应用，人们对其兴趣日益增长。该领域的文献主要遵循两条路线。一方面，重新审视普通傅里叶变换已应用的领域，以使用这种信号的中间时频表示；另一方面，设计用于数值计算分数阶傅里叶变换的快速算法。在本文中，我们推导了连续分数阶傅里叶变换的高斯型求积公式。该求积公式以Hermite多项式及其零点表示。通过使用一些渐近公式，我们将该求积公式重写为chirp-fft-chirp变换，从而得到分数阶傅里叶变换及其逆变换的快速离散化封闭形式。我们通过考虑单位圆内（而不仅仅是边界上）的任意复数值，扩展了分数阶傅里叶变换的范围。我们发现，在z=i处计算的chirp-fft-chirp变换成为FFT的一个更精确版本，可用于非周期函数。
英文摘要：
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.
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