#### Technical Report USC-SIPI-204

“Fuzzy Adaptive Filters, with Application to Nonlinear Channel Equalization”

by Li-Xin Wang and Jerry M. Mendel

October 1992

A fuzzy adaptive filter is constructed from a set of fuzzy IF-THEN rules which change adaptively to minimize some criterion function as new information becomes available. In this paper, two fuzzy adaptive filters are developed: one uses a recursive least squares (RLS) adaptation algorithm, and the other uses a least mean squares (LMS) adaptation algorithm. The RLS fuzzy adaptive filter is constructed through the following four steps: 1) define fuzzy sets in the filter input space U _ Rn whose membership functions cover U; 2) construct a set of fuzzy IF-THEN rules which either come from human experts or are determined during the adaptation procedure by matching input-output data pairs; 3) construct a filter based on the set of rules; and, 4) update the free parameters of the filter using the RLS algorithm. The design procedure of the LMS fuzzy adaptive filter is similar. The most important advantage of the fuzzy adaptive filters is that linguistic information (in the form of fuzzy IF-THEN rules) and numerical information (in the form of input-output pairs) can be combined into the filters in a uniform fashion. Finally, these two fuzzy adaptive filters are applied to nonlinear communication channel equalization problems; the simulation results show that: 1) without using any linguistic information, the RLS and LMS fuzzy adaptive filters are well-performing nonlinear adaptive filters (similar to polynomial and neural-net adaptive filters); 2) by incorporating some linguistic description (in fuzzy terms) about the channel into the fuzzy adaptive filters, the adaptation speed is greatly improved; and, 3) the bit error rates of the fuzzy equalizers are close to that of the optimal equalizer.

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