A fast algorithm for the linear canonical transform

TL;DR

This paper introduces a fast O(N log N) algorithm for computing the linear canonical transform (LCT), which is applicable in optics and signal processing, and improves upon existing methods by using a chirp-FFT-chirp approach.

Contribution

It presents a novel, efficient algorithm for the LCT that works for arbitrary complex values and enhances Fourier transform computations.

Findings

Computes the LCT in O(N log N) time using a chirp-FFT-chirp method.

Provides a unitary discrete LCT in closed form based on a convergent quadrature formula.

Improves Fourier transform outputs compared to standard FFT.

Abstract

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.