Maximum likelihood methods and symmetric alpha-stable environments

Many signal processing techniques have been developed by modeling with Gaussian noise. These methods have proved useful for myriad tasks, such as bearing estimation, adaptive filtering, and others. Unfortunately, the use of the Gaussian assumption generally causes them to be sensitive to impulsive noise. The presence of a few such outlying values decreases accuracy and can even render the results completely meaningless. However, there are other distribution functions that can be used.; The Symmetric Alpha-Stable {dollar}rm (Salpha S){dollar} family of distributions arises from a generalization of the Gaussian characteristic function and has been shown to accurately model a number of impulsive noise sources. Except for the Gaussian case, the members have an infinite variance and cause erratic behavior in many traditional algorithms. The family provides a difficult and important environment in which to test new algorithms.; This dissertation uses maximum-likelihood to modify traditional signal processing techniques so that they only require unimodal symmetric densities with continuous nonzero second derivatives near the origin. We present a locally-optimal maximum-likelihood parametric bearing estimation method, an {dollar}rm Salpha S{dollar} filter theory with a generalized Wiener-Hopf filter equation, and the Recursive Local Orthogonality (RLO) adaptive filter, a maximum-likelihood generalization of recursive least squares. These algorithms are only slightly more computationally intensive than their Gaussian counterparts. The solution of the {dollar}rm Salpha S{dollar} Wiener-Hopf equation gives the theoretical final value of RLO taps. In the course of our research, we have also found an efficient maximum-likelihood parameter {dollar}rm Salpha S{dollar} estimation algorithm, the effect of outliers in hypothesis testing, and the closed form solution to the minimum dispersion {dollar}rm Salpha S{dollar} adaptive filter with independent inputs.