Micro Statistics in Signal Decomposition and the Optimal Filtering Problem

最优滤波是一个具有重大工程意义的统计信号处理问题。从历史上看，最常用的信号处理工具本质上是线性的，但尽管有丰富的线性系统理论，许多信号处理问题并没有通过使用线性方案得到令人满意的解决。这项研究旨在发展一大类基于观测向量、分类观测向量或一般观测向量的非线性变换的组合的非线性滤波器。因此，我们实现了非线性滤波器的响应特性，但可以利用机械的线性系统理论对其进行优化和设计。最佳解决方案需要对分解后的信号集(微观统计)进行统计表征。滤波问题归结为对分解后的信号进行运算的一组滤波器(微统计滤波器)，其中输出是分解后的滤波信号的加权和。在这项工作中，将发展一种稳健的微观统计滤波器的理论，其中线性运算是在排序的观测矢量上执行的。对于未知或非平稳特征的环境，我们开发了自适应微统计滤波器，并解决了收敛、格子结构、快速算法和复杂性等问题。

Optimal Filtering is a statistical signal processing problem of major engineering importance. Historically, the most frequently used signal processing tools have been linear in nature, but despite rich linear systems theories, many signal processing problems have not been satisfactorily addressed through the use of linear schemes. This research aims at the theoretical development of a large class of non-linear filters which are based in combination of either the observation vector, the sorted observation vector, or in general a non-linear transformation of the observation vector. Thus, we achieve non-linear filter response characteristics but with the machinery of linear systems theory available for their optimization and design. The optimal solution requires the statistical characterization of the set of decomposed signals (micro-statistics). The filtering problem reduces to a set of filters operating on the decomposed signals (micro-statistic filters) where the output is a weighted sum of the decomposed filtered signals. In this work, a theory will be developed for robust micro-statistic filters in which linear operations are executed in a sorted observation vector. For environments with unknown or non-stationary characteristics, we develop adaptive micro-statistic filters and address issues such as convergence, lattice structures, fast algorithms, and complexity.