Computer Science 2 pp 57-65 | Cite as

# Prime Length Symmetric FFTS and Their Computer Implementations

## Abstract

Since its rediscovery in 1965 by Cooley and Tukey^{1}, 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 algorithm^{2} improves over the extended Cooley-Tukey FFT for highly composite transform’s length. Rader’s algorithm^{3}, 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*).

## Keywords

Fast Fourier Transform Discrete Fourier Transform Fast Fourier Transform Algorithm Prime Length Core Procedure## Preview

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