Prime Length Symmetric FFTS and Their Computer Implementations
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).
KeywordsFast Fourier Transform Discrete Fourier Transform Fast Fourier Transform Algorithm Prime Length Core Procedure
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- 1.Cooley J. and Tukey J. “An algorithm for the Machine Calculation of the Complex Fourier Series”, IEEE Trans. Comt. AC-28 (1965), pp 819–830.Google Scholar