Analysis and processing of nonlinear time series - from speech to neurophysiological signals.

摘要

This thesis presents new methods of nonlinear signal analysis and processing and their applications. In particular, these methods are inspired by multiple disciplines (nonlinear time series analysis, signal processing, chaos theory, and circular statistics), and applied to analyze, characterize, and process complicated observed signals such as speech signals, laser data, EEG data, and those measured from coupled chaotic systems. Three topics, which are different but related to each other, have been studied.The first topic is noise reduction for chaotic time series and its application in speech enhancement. The local projection (LP) method is powerful in reducing white noise for chaotic time series. But for the case with coloured noise, LP is no longer effective. By investigating the energy distributions of coloured noise and chaotic time series in the local phase space reconstructed by time delay embedding, a two-step extension of the LP method is proposed. Experimental results show that this extension can reduce coloured noise for chaotic time series effectively. Further, this extension is adapted to enhance speech signals which are contaminated by environmental noise. Comparison shows that this scheme is comparable to the state-of-art algorithm of speech enhancement.The second topic is time-frequency analysis. First, the reference phase point and its neighbours in the phase space reconstructed by time delay embedding are shown to cover data segments with similar waveform. To exploit the redundant information possessed by the neighbors, a neighbourhood-based spectral estimator is proposed for chaotic flow. With this estimator, the theory of time delay embedding is bridged to the frequency domain. Then time-frequency analysis with the spectral estimator is performed for chaotic time series. It is shown that the hidden frequency of chaotic systems can be detected by this method reliably and noisy chaotic time series can be distinguished from colored noise which has similar spectra by their different ridge patterns in the time-frequency plane.The last topic is synchronization analysis. Synchronization is a cooperative behaviour by which coupled systems evolve with the same rhythm. It can help to understand the underlying mechanism and gain new applications such as providing clinical evidence. Our contributions include four aspects. First, a neighbourhood-based method is proposed to estimate instantaneous phase (IP) in the phase space reconstructed by time delay embedding. Simulations show that this method is robust to noise and can avoid overestimation of the degree of phase synchronization (PS). Second, several definitions of IP are revisited and further unified into a framework which defines IP by combing the Hilbert transform with specific filter. Third, an analytical study of the effect of noise in IP estimation and PS detection is performed. The distribution of IP error induced by noise is shown to be a scale mixture of normal distribution. Fourth, a band-weighted synchronization index is proposed based on the PS index in each frequency band specified by a bank of filter. It is tested by toy models and further applied to EEG signals, yielding positive results.