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Rationality

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

Abstract

Multiplicative character theory provides a natural setting for developing the complexity results of Auslander, Feig and Winograd [1]. The first reason for this is the simplicity of the formulas describing the action of FT on multiplicative characters. We will now discuss a second important property of multiplicative characters. In a sense defined below, the spaces spanned by certain subsets of multiplicative characters are rational subspaces. As a consequence, we will be able to rationally manipulate the FT matrix F(p m ) into block diagonal matrices where each block action corresponds to some polynomial multiplication modulo a rational polynomial of a special kind. This is the main result in the work of Auslander, Feig and Winograd. Details from the point of view of multiplicative character theory can be found in [2].

Keywords

Rational Basis Discrete Fourier Transform Basis Matrix Vandermonde Matrix Multiplicative Character 
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]
    Auslander, L., Feig, E. and Winograd, S. “The Multiplicative Complexity of the Discrete Fourier Transform”, Adv. in Appl. Math., 5, 1984, pp. 31–55.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    Tolimieri, R. “Multiplicative Characters and the Discrete Fourier Transform”, Adv. in Appl. Math., 7, 1986, pp. 344–380.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    Rader, C. “Discrete Fourier Transforms When the Number of Data Samples is Prime”, Proc. IEEE, 56, 1968, pp. 1107–1108.CrossRefGoogle Scholar
  4. [4]
    Winograd, S. “Arithmetic Complexity of Computations”, CBMS Regional Conf. Ser. in Math. SIAM, 33, Philadelphia, 1980.Google Scholar
  5. [5]
    Tolimieri, R. “The Construction of Orthogonal Basis Diagonalizing the Discrete Fourier Transform”, Adv. in Appl. Math., 5, 1984, pp. 56–86.MathSciNetzbMATHCrossRefGoogle 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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