Discrete Fractional Fourier Transform Based on New Nearly Tridiagonal Commuting Matrices

Abstract

Based on discrete Hermite-Gaussian-like functions, a discrete fractional Fourier transform (DFRFT), which provides sample approximations of the continuous fractional Fourier transform, was defined and investigated recently. In this paper, we propose a new nearly tridiagonal matrix, which commutes with the discrete Fourier transform (DFT) matrix. The eigenvectors of the new nearly tridiagonal matrix are shown to be DFT eigenvectors, which are more similar to the continuous Hermite-Gaussian functions than those developed before. Rigorous discussions on the relations between the eigendecomposition of the newly proposed nearly tridiagonal matrix and the DFT matrix are described. Furthermore, by appropriately combining two linearly independent matrices that both commute with the DFT matrix, we develop a method to obtain DFT eigenvectors even more similar to the continuous Hermite-Gaussian functions (HGFs). Then, new versions of DFRFT produce their transform outputs closer to the samples of the continuous fractional Fourier transform, and their applications are described. Related computer experiments are performed to illustrate the validity of the works in this paper.

Fractional Fourier transform as a signal processing tool: An overview of recent developments

“…However, authors in [67] believe that the discrete time counterparts of the continuous time Hermite-Gaussians maintained the same properties, since these discrete time counterparts exhibited better approximations than the other proposed approaches [68]. In order to resolve this issue about the approximation of Hermite-Gaussian functions, a nearly tri-diagonal commuting matrix of the DFT and a corresponding version of the DFRFT was proposed in [69]. Most of the eigenvectors of this proposed nearly tridiagonal matrix result in a good approximation of the continuous Hermite-Gaussian functions by providing a smaller approximation error in comparison to previous approaches.…”

DFRFT: A Classified Review of Recent Methods with Its Application

“…For the DFRFT of type 2, the parameter " " is used to control the variation of the chirp in frequency domain [13]. The DFRFT of type 2 also needed "2 + ( /2)log 2 " multiplication operations.…”

“…The authors had derived another type of discrete fractional Fourier transform as reported in [13,[18][19][20][21][22][23][24][25] by searching the eigenvectors and eigenvalues of the DFT matrix followed by computing the fractional power of the DFT matrix based on Hermite function. This type of DFRFT worked very similarly to the continuous FRFT, and it fulfills the properties of orthogonality, additivity, and reversibility.…”

“…However, authors in [19] believe that the discrete time counterparts of the continuous time Hermite-Gaussians maintained the same properties because these discrete time counterparts exhibited better approximations than the other proposed approaches [18]. In order to resolve this issue about the approximation of Hermite-Gaussian functions, a nearly tridiagonal commuting matrix of the DFT and a corresponding version of the DFRFT were proposed in [13]. Most of the eigenvectors of this proposed nearly tridiagonal matrix result in a good approximation of the continuous HermiteGaussian functions by providing a smaller approximation error in comparison to the previous approaches.…”

“…Pei et al [13] uses the method together with the Grunbaum's matrix [30] as a linear combination, in which it has been observed that Hermite-Gauss-like eigenvectors of ( + 15 ) are more accurate than both and alone. Serbes and Durak-Ata [28] proposed that eigenvectors of linear combinations of , , and matrices as 1 + 1 + 1 produce more accurate eigenvectors.…”