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On the effects of windowing on the discretization of the fractional Fourier transform

机译：关于加窗对分数阶傅里叶变换离散化的影响

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The eigenvalue degeneracy problem inherent in the discrete Fourier transform (DFT) matrix operator and the development of a full basis of orthogonal eigenvectors have been addressed via a commuting matrix, devoid of the aforementioned eigenvalue degeneracy problem, that also serves as a discrete version of the Gauss-Hermite (G-H) differential operator. This G-H operator is however, is not bandlimited, and existing discretization efforts run into distortion problems that manifest as deviation from the ideal linear eigenvalue spectrum, aliasing in the eigenvectors, and as a non-invertible peak to parameter mapping associated with the discretization restricting its ability to uniquely represent multicomponent chirp signals. Existing approaches do not account for the effects of windowing on discretization. In this paper, we focus on distortion issues associated with the discretization of the G-H operator and their sources. We specifically analyze the discrete version of the G-H operator based on quantum mechanics in finite dimensions (QMFD), by computing its underlying peak to parameter mapping and its invertibility to subsequently present a representation of the operator with improved mapping invertibility via use of suitable windowing of the eigenvalue spectrum.
机译：离散傅里叶变换（DFT）矩阵算子固有的特征值简并性问题和正交特征向量的完整基础的开发已通过换向矩阵解决，它没有上述特征值简并性问题，该问题也可以用作离散特征。高斯-赫尔米特（GH）微分算子。但是，此GH算子不受带宽限制，并且现有的离散化工作遇到了失真问题，表现为与理想线性特征值谱的偏离，特征向量的混叠以及与离散化相关的不可逆峰到参数映射，从而限制了其离散化。独特地代表多分量线性调频信号的能力。现有方法没有考虑窗口化对离散化的影响。在本文中，我们关注与G-H算子及其源离散化相关的失真问题。我们通过计算有限元（QMFD）的潜在峰到参数映射及其可逆性来具体分析GH算子的离散版本，方法是通过使用其适当的窗口化，随后通过改进的映射可逆性来表示GH算子的表示形式。特征值谱。

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