Eigenfunctions of Fourier and Fractional Fourier Transforms With Complex Offsets and Parameters
Journal
IEEE Transactions on Circuits and Systems I: Regular Papers
Journal Volume
54
Journal Issue
7
Pages
1599-1611
Date Issued
2007
Author(s)
Abstract
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. © 2007 IEEE.
Subjects
Eigenvalue; Eigenvector; Fractional Fourier transform (FRFT); Fractional Laplace transform; Fractional Z-transform; Linear canonical transform (LCT); Offset discrete FT (DFT)
SDGs
Other Subjects
Discrete Fourier transforms; Image processing; Laplace transforms; Modulation; Optical systems; Z transforms; Fractional fourier transform; Hermite-Gaussian functions; Linear conical transform; Eigenvalues and eigenfunctions
Type
journal article
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