Complex Membership Grades with an Application to the Design of Adaptive Filters

Abstract

In this paper, complex membership grades are introduced for the extension of fuzzy set theory to the complex domain. This model is based on the idea of viewing the complex domain in a linguistic manner, where two linguistic terms are required to define an object. Thus, as opposed to Buckley's model, after fuzzification the two- dimensionality of the universe of discourse is still apparent. One form for representing a complex fuzzy set is using the Cartesian Complex Fuzzy Set representation, which produces complex sets of the form [Z\tilde]c = [X\tilde] + j[Y\tilde]. The motivation for this aberrant representation is oriented from the limitations in using a direct extension to Zadeh, that Buckley introduced. These limitations pose the guidelines for Complex Membership Grades and, therefore, are initially discussed in this paper. Complex Fuzzy Sets are defined and a technique for converting between Complex Fuzzy Sets and Fuzzy Relations is developed based on Cylindrical Extensions and Projections defined by Zadeh. Next, linguistic coordinate transformations are discussed and exemplified by a rule-base coordinate transformation between Polar and Cartesian Complex Fuzzy Sets. Arithmetic operations and defuzzification are demonstrated. The simplicity of these latter operations is crucial when considering implementation prospects. Finally, Complex Membership Grades are applied to the design of adaptive filters. It is shown that a logically derived rule-base can be described, using the linguistic complex domain, for the adaptation process. Emphasis, in this part, is put on the unique characteristics of the complex membership grades model.