MFTA: The Prime Case

  • Richard Tolimieri
  • Chao Lu
  • Myoung An
Part of the Signal Processing and Digital Filtering book series (SIGNAL PROCESS)


For transform size p, p a prime, Rader [1] developed an FT algorithm based on the multiplicative structure of the indexing set. The main idea is as follows. For a prime p, Z/p is a field and the unit group U(p) is cyclic. Reordering input and output data relative to a generator of U(p), the p-point FT becomes essentially a (p-1) x (p-1) skew-circulant matrix action. We require 2(p-1) additions to make this change. Rader computes this skew-circulant action by the convolution theorem that returns the computation to an FT computation. Since the size (p-1) is a composite number, the (p-1)-point FT can be implemented by Cooley-Tukey FFT algorithms. The Winograd algorithm for small convolutions also can be applied to the skew-circulant action. (See problems 3, 4 and 5 for basic properties of skew-circulant matrices.)


Discrete Fourier Transform Fundamental Factorization Permutation Matrix Real Multiplication Additive Stage 
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  1. [1]
    Rader, C. M. “Discrete Fourier Transforms When the Number of Data Samples Is Prime”, Proc. IEEE, 56, 1968, pp. 1107–1108.CrossRefGoogle Scholar
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    Winograd, S. “On Computing the Discrete Fourier Transform”, Proc. Nat. Acad. Sci. USA, 73(4), April 1976, pp. 1005–1006.MathSciNetzbMATHCrossRefGoogle Scholar
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    Winograd, S. “On Computing the Discrete Fourier Transform”, Math. Comput., 32, Jan. 1978, pp. 175–199.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    Blahut, R. Fast Algorithms for Digital Signal Processing, Addison-Wesley Pub. Co., 1985, Chapter 4.Google Scholar
  5. [5]
    Heideman, M. T. Multiplicative Complexity, Convolution, and the DFT, Springer-Verlag, 1988, Chapter 5.Google Scholar

Copyright information

© Springer Science+Business Media New York 1997

Authors and Affiliations

  • Richard Tolimieri
    • 1
  • Chao Lu
    • 2
  • Myoung An
    • 3
  1. 1.Department of Electrical EngineeringCity College of CUNYNew YorkUSA
  2. 2.Department of Computer and Information SciencesTowson State UniversityTowsonUSA
  3. 3.A.J. Devaney AssociatesAllstonUSA

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