Unified framework for detecting phase synchronization in coupled time series.

Phase synchronization (PS) has drawn increasing attention in recent years for its extensive applications in analyzing time series observed from coupled systems. In this paper, we examine the detection of PS in bivariate time series from the viewpoints of signal processing and circular statistics. Several definitions of instantaneous phase (IP) are first revisited and further unified into a framework, which defines IP as the argument of the signal with a specific bandpass filter applied. With this framework, the constraints for IP definition are discussed and the effect of noise in IP estimation is studied. The estimate error of IP, which is due to noise, is shown to obey a scale mixture of normal (SMN) distributions. Further, under the assumption that the SMN of IP error can be approximated by a particular normal distribution, the estimate of mean phase coherence of bivariate time series is shown to be degraded by a factor, which is determined by only the level of in-band noise. Finally, simulations are provided to support the theoretical results.