线性正则变换的快速算法
A fast algorithm for the linear canonical transform
AI总结：
提出一种基于chirp-FFT-chirp变换的O(N log N)算法，通过分数傅里叶变换的收敛求积公式计算线性正则变换，并推广到复值情况。
摘要：
近年来，由于线性正则变换（LCT）在光学和信号处理中的大量应用，人们重新关注寻找快速算法以准确计算给定函数的LCT。众所周知的积分变换：傅里叶变换、分数傅里叶变换、双边拉普拉斯变换和菲涅耳变换都是LCT的特例。本文通过分数傅里叶变换的收敛求积公式产生的chirp-FFT-chirp变换，得到了一种O(N log N)算法来计算LCT。该公式给出了闭式形式的酉离散LCT。在分数傅里叶变换的情况下，该算法可以计算单位圆内任意复数值的变换，而不仅仅是在边界上。在普通傅里叶变换的情况下，该算法改进了FFT的输出。
英文摘要：
In recent years there has been a renewed interest in finding fast algorithms to compute accurately the linear canonical transform (LCT) of a given function. This is driven by the large number of applications of the LCT in optics and signal processing. The well-known integral transforms: Fourier, fractional Fourier, bilateral Laplace and Fresnel transforms are special cases of the LCT. In this paper we obtain an O(N*Log N) algorithm to compute the LCT by using a chirp-FFT-chirp transformation yielded by a convergent quadrature formula for the fractional Fourier transform. This formula gives a unitary discrete LCT in closed form. In the case of the fractional Fourier transform the algorithm computes this transform for arbitrary complex values inside the unitary circle and not only at the boundary. In the case of the ordinary Fourier transform the algorithm improves the output of the FFT.
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