Fast algorithm for chirp transforms with zooming-in ability and its applications

Abstract

A general fast numerical algorithm for chirp transforms is developed by using two fast Fourier transforms and employing an analytical kernel. This new algorithm unifies the calculations of arbitrary real-order fractional Fourier transforms and Fresnel diffraction. Its computational complexity is better than a fast convolution method using Fourier transforms. Furthermore, one can freely choose the sampling resolutions in both x and u space and zoom in on any portion of the data of interest. Computational results are compared with analytical ones. The errors are essentially limited by the accuracy of the fast Fourier transforms and are higher than the order 10 Ϫ12 for most cases. As an example of its application to scalar diffraction, this algorithm can be used to calculate near-field patterns directly behind the aperture, 0

Generalizing, optimizing, and inventing numerical algorithms for the fractional Fourier, Fresnel, and linear canonical transforms

“…1(a). This assumption is used in all of the algorithms [1][2][3][4][5][6][7][8][9][10][11] in the literature. In this case the input CCM is…”

“…(17) and should use whatever a priori knowledge is available to define the correct initial elements of the CCM. In Section 5 we will optimize the existing algorithms [11][12][13][14][15][16][17][18][19][20][21] to allow for a nonrectangular initial WDF.…”

“…As part of this paper we explicitly discuss more than ten numerical algorithms [1][2][3][4][5][6][7][8][9][10][11] known in the literature that describe different methods of calculating the output of optical systems (including free-space propagation) given some input wave field. We begin our introduction by highlighting this fact because it illustrates (a) the practical importance of these algorithms and (b) what appears to be the complexity of a problem that requires the derivation of so many different numerical techniques.…”

EURASIP Book Series on Signal Processing and Communications

“…Modification of this transform for zero-shift parameters is also known under the names of "chirp z-transform" (see [11,12]) and "fractional discrete Fourier transform." The first name is associated with the way to compute it efficiently (see (3.30)).…”

Reevaluation of the direct method of calculating Fresnel and other linear canonical transforms

“…As such, issues relating to their numerical approximation have been explored in depth by the optics community, in terms of sampling the signals [3][4][5][6][7][8][9], defining discrete approximations to the transforms [10][11][12][13][14][15], and developing fast algorithms to evaluate these discrete LCTs [4,8,11,12,16]. Most of the proposed fast algorithms decompose the optical system into an equivalent concatenation of subsystems for which established algorithms are known.…”