| From all the available models for non-Gaussian impulsive interference, algebraic-tailed distributions present perhaps the most desirable characteristics in terms of both fidelity and mathematical appeal. We introduce the foundations of a new theory of statistics which is well defined over all processes with algebraic or lighter tails. These Zero-Order Statistics or ZOS, as we call them, provide a common ground for the analysis of basically any distribution of practical use known today. Three new parameters, namely the geometric power, the zero-order location and the zero-order dispersion, constitute the underpinnings of ZOS theory. They play roles similar to those played by the power, the expected value and the standard deviation, in the theory of second-order processes. The location estimation problem in the ZOS framework gives rise to the discovery of a novel mode-type estimator with optimality properties under very impulsive noise.; Non-Gaussian {dollar}alpha{dollar}-stable distributions are with no doubt one of the most important models with heavy algebraic tails. Since they are the only distributions that can be the limit of normalized sums of i.i.d impulsive variables, they can arise in practice as a result of physical principles, much as the Gaussian distribution does. In the same way as the Gaussian distribution has largely motivated the development of linear filtering theory, we introduce the class of Weighted Myriad Filters motivated by the statistical properties of {dollar}alpha{dollar}-stable distributions. The foundation of the proposed filtering algorithms lies in the definition of the sample myriad as a generalized maximum likelihood estimator derived from the Cauchy distribution. We show that the sample myriad presents important properties that make it a very desirable estimator in the presence of impulsive noise. In particular, by tuning a simple parameter, its behavior can easily range from highly robust mode-type operation to the familiar average-type operation of the sample mean. We extend the formulation of the sample myriad to allow the use of weights, and provide different methodologies for optimal filter tuning and design. Finally, we develop a myriad-based receiver structure with strong potential for the robust estimation of QAM and DS-CDMA signals. This matched myriad filter, as we call it, can adaptively adjust its behavior according to both the strength and the impulsiveness of the channel interference, resulting in a powerful receiver with optimality properties over a wide spectrum of channel statistics. |
