Generalized Fourier Transforms

  • Thomas Beth
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 296)


In the process of specifying and modelling data structures for wide classes of real-time applications (signal-processing, pattern recognition, communications engineering) it has become a helpful tool to choose these structures as algebras which allow a group of automorphisms acting on the set of data.

With this data algebras being suchwise enhanced many of the above mentioned applications find a natural description.

In this lecture we will show that this description can successfully be used for a unifying approach to fast algorithms for many applications e.g. by reproducing the known classes of FFT's. The concept of the generalized Fourier-Transform can also be used in this context to generate such algorithms almost mechanically, as we will show in some examples.


Normal Subgroup Irreducible Representation Cyclic Group Solvable Group Finite Abelian Group 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

9. Literature

  1. [1]
    A. Aho, J.E. Hopcroft, J.D. Ullman: "The design and Analysis of Computer Algorithms", Addison-Wesley 1974Google Scholar
  2. [2]
    M.D. Atkinson: "The complexity of Group Algebra Computations", Theoretical Computer Science, I, p.205–209, 1977Google Scholar
  3. [3]
    T. Beth: "Verfahren der Schnellen Fourier Transformation", Teubner Verlag Stuttgart, 1984Google Scholar
  4. [4]
    J.W. Cooley, J.W. Tukey: "An Algorithm for the machine calculation of complex Fourier series", Math. Comp., nr. 19, p.297–301, 1965Google Scholar
  5. [5]
    C.W. Curtis, I. Reiner: "Methods of Representation Theory", Wiley-Interscience, 1981Google Scholar
  6. [6]
    W. Feit, J.G. Thompson: "Solvability of Groups of Odd Order", Pacific J.Math., nr.13, p.755–1029, 1963Google Scholar
  7. [7]
    I.J. Good: "The interaction algorithm and practical Fourier series", J.Royal Stat. Soc., ser.B, nr.20, p.361–372, 1958Google Scholar
  8. [8]
    B. Huppert: "Endliche Gruppen I", Springer Verlag, 1967Google Scholar
  9. [9]
    N. Jacobson: "Basic Algebra I", 1974, "Basic Algebra II", 1980, FreemanGoogle Scholar
  10. [10]
    M.G. Karpovski: "Fast Fourier Transforms of Finite Non-Abelian Groups", IEEE Trans. Computers, vol.26, nr.10, p.1028–1030, 1977Google Scholar
  11. [11]
    S. Lang: "Algebra", Addison-Wesley, 1965Google Scholar
  12. [12]
    F.J.McWilliams, N.J.A. Sloane: "The Theory of Error-Correcting Codes", North-Holland, 1977Google Scholar
  13. [13]
    J. Morgenstern: "Note on a lower bond of the linear complexity of the fast Fourier Transform", J.ACM, vol.20, nr.2, p.305–306, 1973CrossRefGoogle Scholar
  14. [14]
    H.J. Nussbaumer: "Fast Fourier Transform and Convolution Algorithms", Springer Verlag, 1981Google Scholar
  15. [15]
    F. Pichler: "Analog Scrambling by the General Fast Fourier Transform", Comp. Science Lect. Notes, vol.149, p.173–178, 1983Google Scholar
  16. [16]
    J.P. Serre: "Linear Representations of Finite Groups", Springer Verlag, 1971Google Scholar
  17. [17]
    S. Winograd: "Arithmetic Complexity of Computations", S.I.A.M. CBMS-NSF Regional Conference Series in Applied Mathematics, 1980Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1988

Authors and Affiliations

  • Thomas Beth
    • 1
  1. 1.Institut für Algorithmen und Kognitive SystemeUniversität KarisruheKarisruhe 1

Personalised recommendations