FAST IMPLEMENTATIONS OF FUZZY ARITHMETIC OPERATIONS USING FAST FOURIER-TRANSFORM (FFT)

In engineering applications of fuzzy logic, the main goal is not to si mulate the way the experts really think, but to come up with a good en gineering solution that would (ideally) be better than the expert's co ntrol. In such applications, it makes perfect sense to restrict oursel ves to simplified approximate expressions for membership functions. If we need to perform arithmetic operations with the resulting fuzzy num bers, then we can use simple and fast algorithms that are known for op erations with simple membership functions. In other applications, espe cially the ones that are related to humanities, simulating experts is one of the main goals. In such applications, we must use membership fu nctions that capture every nuance of the expert's opinion; these funct ions are therefore complicated, and fuzzy arithmetic operations with t he corresponding fuzzy numbers become a computational problem. In this paper, we design a new algorithm for performing such operations. This algorithm uses Fast Fourier Transform (FFT) to reduce computation tim e from O(n(2)) to O(n log(n)) (where n is the number of points x at wh ich we know the membership functions mu(x)). To compute FFT even faste r, we propose to use special hardware. The results of this gaper were announced in the work of Kosheleva et al. [Proc. 1996 IEEE Int. Conf. on Fuzzy Systems, Vol. 3, pp. 1958-1964]. (C) 1997 Elsevier Science B. V.