This paper focuses on analyzing the fractional spectrum of nonuniform sampling which is affected by clock jitter and timing offset. Both cases of single-channel and recurrent samplings with limited bandwidth in the fractional Fourier domain are considered, for which we study two types of discrete-time fractional Fourier transforms (DTFrFTs). For the first-type DTFrFTs, we perform their statistical analyses, and meanwhile establish relationships between the fractional spectra of the input and output signals of non-ideal single/multi-channel analog-to-digital converter. Such relationships indicate that the statistical mean of each fractional spectrum contains periodic replicas of the spectrum for the input signal, and these replicas are particularly modulated by the characteristic function of the perturbations introduced by the clock jitter. Moreover, we interpret the first-type DTFrFTs from the perspective of linear frequency modulation signal decomposition for nonuniform sampling. The second-type DTFrFTs serve as approximate DTFrFTs, and they especially deal with the general situation when the nonuniform sampling instants are unknown. To this end, we study modified forms of the first-type DTFrFTs with known inputs in kernels, based on which the statistical mean of the approximate fractional spectra for both cases are derived. Then we compensate the fractional spectrum bias with high accuracy by developing optimal filters that minimize the mean square error between the original and the compensated fractional spectra for both cases. Different from single-channel sampling, the compensation for recurrent sampling involves an additional step before optimal filtering, which gives better accuracy. Simulation results show that the proposed methods outperform existing methods.

中文翻译：

---

存在时钟抖动和时序偏移的非均匀采样的分数频谱分析

本文重点分析受时钟抖动和时序偏移影响的非均匀采样的分数谱。考虑了分数傅里叶域中带宽有限的单通道和循环采样两种情况，为此我们研究了两种类型的离散时间分数傅里叶变换 (DTFrFT)。对于第一类DTFrFT，我们对其进行了统计分析，同时建立了非理想单/多通道模数转换器的输入和输出信号的分数谱之间的关系。这种关系表明，每个分数频谱的统计平均值包含输入信号频谱的周期性副本，并且这些副本特别受时钟抖动引入的扰动的特征函数调制。此外，我们从非均匀采样的线性调频信号分解的角度解释了第一类 DTFrFT。第二类 DTFrFT 用作近似 DTFrFT，它们特别处理非均匀采样时刻未知时的一般情况。为此，我们研究了在内核中具有已知输入的第一类 DTFrFT 的修改形式，在此基础上推导出两种情况的近似分数谱的统计平均值。然后我们通过开发优化滤波器来高精度地补偿分数光谱偏差，该滤波器在两种情况下最小化原始和补偿分数光谱之间的均方误差。与单通道采样不同，对循环采样的补偿包括在最优滤波之前的一个额外步骤，这提供了更好的准确性。仿真结果表明，所提出的方法优于现有方法。