Period and signal reconstruction from the curve of trains of samples

Abstract

Abstract 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 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 this curve, that is, the set of all sample‐trains of a given length, determines the signal period, provided that the sampling period is known and is smaller than half of the signal period. The presented result is proved with the help of the theory of rotation numbers developed by Poincaré. We also prove that the curve of sample‐trains determines the signal up to a time shift, provided that the ratio of the sampling period to the period of the signal is irrational. A counterexample shows that the assumption on the incommensurability of the periods cannot be dropped. Eventually, we show how to estimate the period of a signal from a finite number of its sample‐trains.

Keywords

• dynamical system
• period estimation
• rotation number
• signal sampling
• time delay embedding