Eigenfunctions of Fourier and Fractional Fourier Transforms With Complex Offsets and Parameters
Eigenfunctions of Fourier and Fractional Fourier Transforms With Complex Offsets and Parameters
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DOI：
10.1109/tcsi.2007.900182
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发表时间：
2007-07
期刊：
影响因子：
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通讯作者：
S. Pei;Jian-Jiun Ding
中科院分区：
文献类型：
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作者：
S. Pei;Jian-Jiun Ding
在本文中，我们推导了具有（1）复参数和（2）复偏移的傅里叶变换（FT）、分数阶傅里叶变换（FRFT）和线性正则变换（LCT）的本征函数。在参数和偏移为实数的情况下的本征函数已在文献中推导出来。我们将先前的工作扩展到复参数和复偏移的情况。我们首先推导了偏移离散傅里叶变换的本征向量。它们近似于连续偏移傅里叶变换本征函数的样本。我们发现具有复偏移的偏移傅里叶变换的本征函数是具有平移和调制的平滑厄米 - 高斯函数。然后我们将结果扩展到偏移FRFT和偏移LCT的情况。我们可以使用推导出的本征函数来模拟具有能量吸收组件的光学系统的自成像现象、模式选择、加密，并定义分数Z变换和分数拉普拉斯变换。
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.