Prime Length Symmetric FFTS and Their Computer Implementations

Abstract

Since its rediscovery in 1965 by Cooley and Tukey¹, the fast Fourier transform (FFT) has become one of the most widely used computational tools in science and engineering. The term FFT, initially associated to the Cooley-Tukey FFT for sequences of period N = 2^k, has become after the efforts of many researchers over the years, the generic name of a whole family of efficient DFT numerical methods. Each member in the FFT family is specialized in computing the DFT of a particular class of periodic sequences. This period is also referred as the transform’s length and the DFT (FFT) of length N is usually called N-point DFT (FFT). The first member in this family is actually an extension of Cooley and Tukey’s idea to N-point DFTs where N is factorizable. These N-point FFTs compute the N-point DFT through nested sequences of DFTs whose lengths are the factors of N. The Good-Thomas algorithm² improves over the extended Cooley-Tukey FFT for highly composite transform’s length. Rader’s algorithm³, on its turn, is designed for computing prime length DFTs. These algorithms, all members of the family of traditional FFTs, reduce the N-point DFT arithmetic complexity from O(N ²) to O(N log N).