Research On The Parameters Estimation Of α-Stable Distributions And Adaptive Filtering Algorithms

The signal processsing literature has traditionally been dominated by the Gaussiandistribution models. However, a broad and increasingly important class of signals or noisesoften encountered in practice can be characterized by its significant impulsive nature, and itsstatistical properties subject to non-Gaussian α-stable distributions. α-stable distributions aregeneralized Gaussian distributions, it can exactly decribe many Gaussian and non-Gaussiansignals and noises in reality. The signal processing algorithms based on the assumption ofα-stable distributions are very robust to the uncertainty of characterisitics of signal and noise.The study on the theory of α-stable distributions signal processing will contribute to thedevelopment of the theory of signal processing from both second and higher order statistics tothe fractional lower order statistics, consequently form a integrated theory system. Thisdissertation focuses on the study of parameters estimation of α-stable distributions modelsand adaptive filtering algorithms in the circumstances. The main works are as follows:Three definitions of α-stable distributions are discussed, different effects of the fourcharacteristic parameters for α-stable distributions are explained. The properties,applications, the theory of fractional lower order statistics and linear theory of α-stabledistributions are studied. Pointed out that all the fractional lower order moments of α-stabledistributions are limited, minimizing the dispersion and α-norm and p-norm of the errors areequivalent, and these are the basis of this thesis. Four different parameterizations and theirrealationships are discussed, the problem of generation of α-stable random variables subjectto arbitrary four parameters in the standarded parameterizaiton is explained. Finally, theimpulsive nature in time-domain and heavy tails in probability density functions are shown bysimulations.Parameters estimation is the key point of modeling impulsive signals or noises anddesigning signal processing algorithms based on the assumption of α-stable distributions.Aimed at the difficulties to estimate the parameters of α-stable distributions using classicalmethods such as maximum likelihood and so on, and to estimate location parameter usingfractional lower order moments method, a method was proposed based on Metropolis-Hastings(M-H) sampling. Firstly, Bayesian inference model is constructed using Bayes’ theorem, allunknown parameters were viewd as variables under Bayesian framework in order to changethe parameters estimation into probability calculation, and then dynamically generate Markovchain by properly choosing propocal distribution and by using M-H sampling, which achievedaccurate estimation for all the four parameters of α-stable distributions. The simulationresults verify the validity and accuracy of this method. Aimed at an improper choice of proposal distribution for M-H algorithm may lead tounexpected results, three improved methods for parameters estimaiton of α-stable distributionsare proposed by introduing adaptive sampling strategy. One is condidered as a way ofcombining different kernels for M-H by delayed rejection (DR), which can improve theefficiency of parameters estimation by local adaptation; the other is adaptive Metropolis (AM)algorithm, which can globally adapt the size and spatial orientation of the proposal distributionsusing all the cumulated information of Markov chain. According to the complementarycharacteristics of the above algorithms, a method combining DR and AM (DRAM) algorithmsis proposed. The simulation results indicate that the proposed methods improved efficiencyand performed accurately and robustly.The principle of classical time-domain adaptive filter are analyzed, a new variablestep-size normalized least mean p-norm (NLMP) algorithm is proposed based on the Euclideannorm of smoothed gradient vector, which can track the variation of the mean square deviation.According to the idea of data block filtering, adaptive data block NLMP algorithm and datareusing NLMP algorithm are proposed by using more input signals and errors, thusconvergence is improved. For the problem of tap-length is unknown or variable in practicalapplications, the cost function of variable tap-length algorithm is redefined, and a fractionaltap-length least mean p-norm (FTLMP) algorithm is proposed for α-stable distributionsenvironments. An improved variable iteration parameter FTLMP algorithm is proposed afterperformance analysis, in which the iteration parameter is controlled by smoothed errors, so theconvergence speed of tap-length is improved. Simulation results show the effectiveness of thealgorithms.The correlated input signals can often result in significant convergence degradation fortime-domain adaptive filtering. According to the idea of transform domain adaptive filteringand fractional lower order statistics, a kind of transform domain least mean p-norm (TDLMP)algorithm is proposed. The algorithm have a preprocessing stage to decorrelate the inputsignals by unitary orthogonal transformations followed by a power normalization operation,and then perform adaptive filtering with transformed signals, a faster convergencecharacteristics can be achieved. The decorrelation ability of various orthogonal transforms isstudied by simulations. From the points of filtering and geometrical approach intuitivelyexplained how transform domain LMP algorithms can improve convergence property of thestandard LMP algorithms. Finally, the effectiveness of the algorithms is improved bysimulations.