Complex fuzzy sets with applications in signals

Abstract

A complex fuzzy set is characterized by a membership function, whose range is not limited to [0, 1], but extended to the unit circle in the complex plane. In this paper, we introduce some new operations and laws of a complex fuzzy set such as disjunctive sum, simple difference, bounded difference, distributive law of union over intersection and intersection over union, equivalence formula, symmetrical difference formula, involution law, absorption law, and idempotent law. We introduce some basic results on complex fuzzy sets with respect to standard complex fuzzy intersection, union, and complement functions corresponding to the same functions for determining the phase term, and we give particular examples of these operations. We use complex fuzzy sets in signals and systems, because its behavior is similar to a Fourier transform in certain cases. Moreover, we develop a new algorithm using complex fuzzy sets for applications in signals and systems by which we identify a reference signal out of large number of signals detected by a digital receiver. We use the inverse discrete Fourier transform of a complex fuzzy set for incoming signals and a reference signal. Thus, a method for measuring the exact values of two signals is provided by which we can identity the reference signal.