A Comprehensive Survey on Fractional Fourier Transform

Issue title: Theory and Applications of Fractional Fourier Transform and its Variants

Guest editors: Yudong Zhang, Xiao-Jun Yang, Carlo Cattani, Zhengchao Dong, Ti-Fei Yuan and Liang-Xiu Han

Article type: Research Article

Authors: Zhang, Yudong^{a; *} | Wang, Shuihua^{b; †; ‡} | Yang, Jian-Fei^c | Zhang, Zheng^d | Phillips, Preetha^e | Sun, Ping^f | Yan, Jie^g

Affiliations: [a] School of Computer Science and Technology, Nanjing Normal University, Nanjing, Jiangsu 210023, China | [b] School of Computer Science and Technology, Nanjing Normal University, Nanjing, Jiangsu 210023, China. wangshuihua@njnu.edu.cn | [c] Jiangsu Key Laboratory of 3D, Printing Equipment and Manufacturing, Nanjing, Jiangsu 210042, China | [d] Viterbi School of Engineering, University of Southern California, Los Angeles, CA 90089, USA | [e] School of Natural Sciences and Mathematics, Shepherd University, Shepherdstown, WV 25443, USA | [f] Department of Electrical Engineering, The City College of New York, CUNY, New York, NY 10031, USA | [g] Department of Applied Physics, Stanford University, Stanford, CA 94305, USA

Correspondence: [†] Address for correspondence: School of Computer Science and Technology, Nanjing Normal University, Nanjing, Jiangsu 210023, China

Note: [*] Also affiliated at: Jiangsu Key Laboratory of 3D Printing Equipment and Manufacturing, Nanjing, Jiangsu 210042, China

Note: [‡] Also affiliated at: School of Electronic Science and Engineering, Nanjing University, Nanjing, Jiangsu 210046, China

Abstract: The Fractional Fourier transform (FRFT) is a relatively novel linear transforms that is a generalization of conventional Fourier transform (FT). FRFT can transform a particular signal to a unified time-frequency domain. In this survey, we try to present a comprehensive investigation of FRFT. Firstly, we provided definition of FRFT and its three discrete versions (weighted-type, sampling-type, and eigendecomposition-type). Secondly, we offered a comprehensive theoretical research and technological studies that consisted of hardware implementation, software implementation, and optimal order selection. Thirdly, we presented a survey on applications of FRFT to following fields: communication, encryption, optimal engineering, radiology, remote sensing, fractional calculus, fractional wavelet transform, pseudo-differential operator, pattern recognition, and image processing. It is hoped that this survey would be beneficial for the researchers studying on FRFT.

Keywords: fractional Fourier transform, discrete fractional Fourier transform, optimal order, time-frequency analysis, fractional wavelet transform, signal processing, fractional calculus

DOI: 10.3233/FI-2017-1477

Journal: Fundamenta Informaticae, vol. 151, no. 1-4, pp. 1-48, 2017