Eigenfunctions of Fourier and Fractional Fourier Transforms With Complex Offsets and Parameters

• Published in: IEEE transactions on circuits and systems. 1, Fundamental theory and applications Fundamental theory and applications, 2007-07, Vol.54 (7), p.1599-1611
• Main Authors: Pei, Soo-Chang, Ding, Jian-Jiun
• Format: Article
• Language: English
• Subjects: Approximation; Circuits; Councils; Cryptography; Discrete Fourier transforms; Discrete transforms; Eigenfunctions; Eigenvalue; Eigenvalues and eigenfunctions; eigenvector; Fourier analysis; Fourier transforms; fractional Fourier transform (FRFT); fractional Laplace transform; fractional Z -transform; Karhunen-Loeve transforms; Laplace equations; Laplace transforms; linear canonical transform (LCT); Modulation; offset discrete FT (DFT); Offsets; Optical devices; Optical modulation
• ISSN: 1549-8328; 1057-7122; 1558-0806
• DOI: 10.1109/TCSI.2007.900182

In this paper, we derive the eigenfunctions of the Fourier transform (FT), the fractional FT (FRFT), and the linear canonical transform (LCT) with (1) complex parameters and (2) complex offsets. The eigenfunctions in the cases where the parameters and offsets are real were derived in literature. We extend the previous works to the cases of complex parameters and complex offsets. We first derive the eigenvectors of the offset discrete FT. They approximate the samples of the eigenfunctions of the continuous offset FT. We find that the eigenfunctions of the offset FT with complex offsets are the smoothed Hermite-Gaussian functions with shifting and modulation. Then we extend the results for the case of the offset FRFT and the offset LCT. We can use the derived eigenfunctions to simulate the self-imaging phenomenon for the optical system with energy-absorbing component, mode selection, encryption, and define the fractional Z-transform and the fractional Laplace transform.