Study On The Theory And Application Of Parameter Estimation And Spectral Analysis Of α-stable Distribution

With the development of science and technology, the theory and application of non-Gaussian stochastic signal processing gain widen regards and developments. In practice, various non-Gaussian signals and noises have distinct spiky and impulsive characteristics, leading them deviate from Gaussian distribution. If such processes are still modeled with Gaussian distribution, the designed signal processor will degenerate for the miss-match between the models and signals, while α-stable distribution is the useful tool for these processes.α-stable distribution is a kind of generalized Gaussian distribution. It can exactly describe the Gaussian and non-Gaussian α-stable distribution signals and noises in reality. The algorithms based on assumption of α-stable distribution is robust under uncertain characteristics of signals and noises. The study on the theory of a -stable distribution signal processing helps to the development of the theory of signal processing from second order statistics to both higher order and fractional lower order statistics, consequently form a integrated theory system. This dissertation focuses on the study of the parameter estimation and spectral analysis of α-stable distribution, and designs robust time delay estimation methods and evoked potential (EP) latency change estimation methods according to the characteristics of α -stable distribution. The main researches and conclusions are listed as follows:(1) In order to ensure the accuracy of models and the reliability of algorithms, it is important to obtain precise estimation of parameters from one sample realization of α-stable distribution. Considering of the condition of time variant parameters of the models, two dynamic parameter estimation methods based on the negative moment and the logarithm are proposed in order to realize dynamic estimation. The dynamic parameter estimation methods can track the changes of parameters effectively. On the latency change estimation of EP, this dissertation improves both Direct Least Mean p-norm (DLMP) algorithm and Adaptive Fractional Lower-order Covariance (AFLC) algorithms by combining the dynamic parameter estimation methods with the existing EP latency change estimation algorithms. The new algorithms overcome the drawbacks arosed by the parameter selection, and ensure the reliability of the algorithms. Further more, a new EP latency change estimation method based on neural network preprocessing is proposed. Theoretical analysis and simulation resultsshow that this method suppresses the impulsive noises in EP signals and realize reliable estimation of EP latency changes.(2) According to the theory of a -stable distribution signal processing and fractional lower order statistics, this dissertation analyze the frequency domain characteristics of a-stable distribution. A definition of fractional lower order covariance spectrum is proposed, and the properties and the corresponding proofs of fractional lower order covariance spectrum are given in the dissertation. A thorough study is conducted on the estimation of fractional lower order covariance spectrum. The direct method, indirect method, weighted overlap average method and character decompose method for the estimation of fractional lower order covariance spectrum is proposed. Theoretical analysis and simulation results show that the fractional lower order covariance spectrum is an effective tool of frequency domain analysis for a -stable distribution processes. Moreover, the output characteristic in time and frequency domains of linear time invariant system excited by a -stable distribution processes is discussed.(3) According to the degeneration of traditional second statistics based time delay estimation methods under a -stable distribution environments, this dissertation proposes weighted time delay estimation methods and adaptive weighted time delay estimation methods suitable for a -stable distribution environments. Moreover the new methods of multi-source time delay estimation are designed. The fractional lower order covariance spectrum based methods and the nonlinear transform based weighted time delay estimation methods perform robust under both Gaussian and non-Gaussian a -stable distribution environments because of the nonlinear preprocessing for noisy signals and the suppression of the spiky and impulsive noises. The adaptive weighted time delay estimation method based on minimum dispersion (MD) criterion and nonlinear transform is needless of priori-knowledge of signals and noises, and possesses the ability of dynamic tracking. The new methods perform robust under both Gaussian and non-Gaussian a -stable distribution environments because of the MD criterion in stead of the minimum mean square error (MMSE) criterion is adopted as well as the nonlinear transform method is adopted to suppress the spiky and impulsive noises.