Some aspects of signal processing in heavy tailed noise
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This thesis addresses some problems that arise in signal processing when the noise is impulsive and follows a heavy tailed distribution. After reviewing several of the more well known heavy- tailed distributions the common problem of which of these hest models the observations is considered. To this end, a test is proposed for the symmetric alpha stable distribution. The test threshold is found using both asymptotic theory and parametric bootstrap resampling. In doing so, some modifications are proposed for Koutrouvelis' estimator of the symmetric alpha stable distributions parameters that improve performance. In electrical systems impulsive noise is generated externally to the receiver while thermal Gaussian noise is generated internally by the receiver electronics, the resultant noise is an additive combination of these two independent sources. A characteristic function domain estimator for the parameters of the resultant distribution is developed for the case when the impulsive noise is modeled by a symmetric alpha stable distribution. Having concentrated on validation and parameter estimation for the noise model, some problems in signal detection and estimation are considered. Detection of the number of sources impinging on an array is an important first. step in many array processing problems for which the development of optimal methods can be complicated even in the Gaussian case. Here, a multiple hypothesis test for the equality of the eigenvalues of the sample array covariance is proposed.The nonparametric bootstrap is used to estimate the distributions of the test statistics removing the assumption of Gaussianity and offering improved performance for heavy tailed observations. Finally, some robust estimators are proposed for estimating parametric signals in additive noise. These are based on M-estimators but implicitly incorporate an estimate of the noise distribution. enabling the estimator to adapt to the unknown noise distribution. Two estimators are developed, one uses a nonparametric kernel density estimator while the other models the score function of the noise distribution with a linear combination of basis functions.
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