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Period and signal reconstruction from the curve of trains of samples
IET Signal Processing ( IF 1.7 ) Pub Date : 2021-11-16 , DOI: 10.1049/sil2.12086
Marek W. Rupniewski 1
Affiliation  

A finite sequence of equidistant samples (a sample train) of a periodic signal can be identified with a point in a multidimensional space. Such a point depends on the sampled signal, the sampling period, and the starting time of the sequence. If the starting time varies, then the corresponding point moves along a closed curve. We prove that such a curve, i.e., the set of all sample trains of a given length, determines the period of the sampled signal, provided that the sampling period is known. This is true even if the trains are short, and if the samples comprising trains are taken at a sub-Nyquist rate. The presented result is proved with a help of the theory of rotation numbers developed by Poincaré. We also prove that the curve of sample trains determines the sampled signal up to a time shift, provided that the ratio of the sampling period to the period of the signal is irrational. Eventually, we give an example which shows that the assumption on incommensurability of the periods cannot be dropped.

中文翻译:

从样本序列曲线重建周期和信号

可以用多维空间中的一个点来识别周期信号的有限等距样本(样本序列)序列。这样的点取决于采样的信号、采样周期和序列的开始时间。如果开始时间发生变化,则对应点沿闭合曲线移动。我们证明了这样一条曲线,即给定长度的所有样本序列的集合,决定了采样信号的周期,前提是采样周期是已知的。即使火车很短,并且如果包含火车的样本是以亚奈奎斯特速率采集的,也是如此。所呈现的结果在 Poincaré 开发的旋转数理论的帮助下得到证明。我们还证明了样本序列的曲线决定了采样信号的时间偏移,前提是采样周期与信号周期之比是不合理的。最后,我们给出一个例子,表明不能放弃关于周期不可通约性的假设。
更新日期:2021-11-16
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