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MFTA: The Prime Case

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

Abstract

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.)

Keywords

Discrete Fourier Transform Fundamental Factorization Permutation Matrix Real Multiplication Additive Stage 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  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
  2. [2]
    Winograd, S. “On Computing the Discrete Fourier Transform”, Proc. Nat. Acad. Sci. USA, 73(4), April 1976, pp. 1005–1006.MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    Winograd, S. “On Computing the Discrete Fourier Transform”, Math. Comput., 32, Jan. 1978, pp. 175–199.MathSciNetMATHCrossRefGoogle 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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