Nonlinear signal processing in the complex domain and higher dimensions

This dissertation is an attempt to bridge this gap and provide a framework for nonlinear signal processing to thrive in the complex domain.; The weighted median is first extended onto the complex domain by applying the "phase coupling" technique, according to which the complex weights have two roles: their phases are used to rotate original samples in the complex plane, whilst their magnitudes are used to emphasize or de-emphasize these modified samples just as in real signal space. As a result, phase-coupled complex weighted median and its suboptimal implementation marginal phase-coupled complex weighted median are defined based on this powerful concept. Constructed on complex threshold decomposition and the so-called complex differentiation operators, without which the regular rigorous complex differentiation methods will lead us to no solutions at all, the optimal filter design for marginal phase-coupled complex WM is formulated.; A complete package of complex myriad tools are developed following a similar process, where complex weighted myriad filters are defined once again by applying phase coupling. Unlike medians, myriads need extra effort in finding their fast computation and their optimization, due to their complicated structures. This is solved by exploiting the local minima properties of the myriad objective function. Later, combining the constant modulus criterion the myriad structure is fused into the blind equalization problem, creating a robust blind equalizer against impulsive noise encountered in communication channels. Two variations of the myriad equalizer are introduced to reduce the computation complexity based on a novel impulsiveness index---the running kurtosis.; In multichannel signal processing, linear approaches also suffer from impulsive degradation. A close examination on the multivariate Gaussian model indicates that, the popular weighted vector median is in fact cross-channel blind, in that it can not fully exploit the cross-channel information. Moreover, we learned that the structure of the correlation matrix in the model determines the best filtering architecture. Hence two novel multichannel filtering structures are proposed under this notion, making them best matches in structure for different signal correlation scenarios. We then go one step further by making each channel complex. The optimization for all these filters are included, and they are tested through a series of simulations in an array processing setup.; Robust correlation estimates are crucial to frequency estimation in array processing, delay estimation in speech processing etc., since these applications are known for their impulsive environments. We show that the Maximum Likelihood estimates of location under the Laplacian model and the Cauchy model, which form the basis for weighted median filters and weighted myriad filters respectively, can be generalized to correlation estimates. (Abstract shortened by UMI.)...