Weighted Myriad Filters and Their Applications in Communications

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Linear Filtering theory has been largely motivated by the characteristics of Gaussian random processes. In the same manner, research in myriad filtering theory is motivated by the statistical properties of alpha-stable processes which describe an important class of processes, impulsive in nature, that obey a Generalized Central Limit Theorem; thus these processes can arise in practice as a result of physical principles. The foundation of myriad filtering algorithms lies in the definition of the sample myriad as the location estimate of a class of alpha-stable distributions. The main goals of this research are: (1) To study the theoretical properties of the sample myriad and to exploit its geometrical structure for the development of efficient computational algorithms. (2) To solve the optimal "weighted myriad" filtering problem - the problem analogous to the design of the optimal FIR (Wiener) filter and the optimal weighted median filter. (3) To develop adaptive filtering algorithms for the simple design of weighted myriad filters for applications where the statistical characteristics of the underlying signals may be unknown or varying. (4) To extend the filter formulation to the case where the signals are complex or multivariate, providing us with the tools needed to develop robust CDMA multiple access detectors. This work can have a significant impact on applications requiring robust estimation and filtering. This is particularly the case in mobile and personal communication systems, to be deployed in the near future, where the underlying statistics of the noise and interferences closely follow the models used in this research rather that the traditional models used in practice today. The results of this research can also be applied to a wide range of problems including remote sensing imaging of the environment and non-destructive evaluation of materials.