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

Unable to display preview. Download preview PDF.

## References

- 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
- 2.Good I. “The Interaction Algorithm and Practical Fourier Analysis”, J. Royal Stat. Soc. Ser. B 20 (1958), pp 361–375.MathSciNetzbMATHGoogle Scholar
- 3.Rader C. “Discrete Fourier Transforms when the Number of Data Points is Prime”, Proc. IEEE 56 (1968), pp 1107–1108.CrossRefGoogle Scholar
- 4.Cooley J., Lewis P. and Welch P. “The Fast Fourier Transform Algorithm: Programming considerations in the Calculation of sine, Cosine and Laplace transforms”, J. Sound Vib. 12 (1970), pp 315–337.zbMATHCrossRefGoogle Scholar
- 5.Dollimore J. “Some Algorithms for use with the Fast Fourier Transform”, J. Inst. Math. Appl. v. 12 (1973), pp 115–117.MathSciNetzbMATHCrossRefGoogle Scholar
- 6.Swarztrauber P. “Symmetric FFTs”, Math. Comp. 47 (1986), pp 323–346.MathSciNetzbMATHCrossRefGoogle Scholar
- 7.Briggs W. “Further Symmetries of In-place FFTs”, SIAM J. Sci. Stat. Comp. 8 (1987), pp 644–654.MathSciNetzbMATHCrossRefGoogle Scholar
- 8.Van Loan, C. “Computational Frameworks for the Fast Fourier Transform”, SIAM, Philadelphia, 1992.zbMATHCrossRefGoogle Scholar
- 9.Otto J. “Symmetric Prime Factor Fast Fourier Transform Algorithm”, SIAM J. Sci. Stat. Comput. 10(3) (1989), 419–431.MathSciNetzbMATHCrossRefGoogle Scholar