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MFTA: The Prime Case

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

Abstract

For transform size p, p a prime, Rader introduced an approach to construct algorithms which depends on the multiplicative structure of indexing set. In fact, for a prime p, Z/p is a field and the unit group U(p) is cyclic. Reordering input and output data corresponding to a generator of U(p), the p-point FFT becomes essentially a (p−1) × p−1) skew-circulant matrix. We require 2(p−1) additions to make this change. Rader computes this skew-circulant action by the convolution theorem which returns the computation to an FFT computation. Since the size (p−1) is a composite number, the (p−1)-point FT can be handled by Cooley-Tukey FFT algorithms. The Winograd algorithm for small convolutions can also be applied to the skew-circulant action.

Keywords

Discrete Fourier Transform Unit Group Fundamental Factorization Permutation Matrix Real Multiplication 
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):1107–1108.CrossRefGoogle Scholar
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    Winograd, S. “On Computing the Discrete Fourier Transform”, Proc. Nat. Acad. Sct. USA., vol. 73. no. 4, (April 1976):1005–1006.MathSciNetzbMATHCrossRefGoogle Scholar
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    Winograd, S. “On Computing the Discrete Fourier Transform”, Math. of Computation, Vol.32, No. 141, (Jan. 1978).Google Scholar
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    Blahut, R.Fast Algorithms for Digital Signal Processing Addison-Wesley Pub. Co. 1885, Chapter 4.Google Scholar
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    Heideman, M. T. Multilicative Complexity, Convolution, and the DFT, Chapter 5. Springer-Verlag, 1988.Google Scholar
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    Johnson, R.W., Lu, Chao and Tolimieri, R.:“Fast Fourier Algorithms for the Size of Product of Distinct Primes and implementar tions on VAX”. Submitted to IEEE Trans. Acout., Speech, Signal Proc.Google Scholar

Copyright information

© Springer Science+Business Media New York 1989

Authors and Affiliations

  • R. Tolimieri
    • 1
  • Myoung An
    • 1
  • Chao Lu
    • 1
  1. 1.Center for Large Scale ComputingCity University of New YorkNew YorkUSA

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