Advertisement

Linear Filtering Computation of Discrete Fourier Transforms

Chapter
  • 421 Downloads
Part of the Springer Series in Information Sciences book series (SSINF, volume 2)

Abstract

The FFT algorithm reduces drastically the number of arithmetic operations required to compute discrete Fourier transforms and is easily implemented on most existing computers. Thus, it is usually advantageous to compute linear filtering processes via the circular convolution property of the DFT with the FFT algorithm. Under these conditions, it would seem paradoxical to develop linear filtering algorithms for the computation of the DFT. This may explain why some algorithms which have been introduced in 1968 by Bluestein [5.1, 2] and Rader [5.3] have long been regarded as a curiosity.

Keywords

Discrete Fourier Transform Primitive Root Reduction Modulo Cyclotomic Polynomial Circular Convolution 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 5.1
    L. I. Bluestein: “A Linear Filtering Approach to the Computation of the Discrete Fourier Transform”, in 1968 Northeast Electronics Research and Engineering Meeting Rec., pp. 218–219Google Scholar
  2. 5.2
    L. I. Bluestein: A linear filtering approach to the computation of the discrete Fourier transform. IEEE Trans. AU-18, 451–455 (1970)Google Scholar
  3. 5.3
    C. M. Rader: Discrete Fourier transforms when the number of data samples is prime. Proc. IEEE 56, 1107–1108 (1968)CrossRefGoogle Scholar
  4. 5.4
    S. Winograd: On computing the discrete Fourier transform. Proc. Nat. Acad. Sci. USA 73, 1005–1006 (1976)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.5
    L. R. Rabiner, R. W. Schafer, C. M. Rader: The Chirp z-transform algorithm and its application. Bell Syst. Tech. J. 48, 1249–1292 (1969)MathSciNetGoogle Scholar
  6. 5.6
    G. R. Nudd, O. W. Otto: Real-time Fourier analysis of spread spectrum signals using surface-wave-implemented Chrip-z transformation. IEEE Trans. MTT-24, 54–56 (1975)Google Scholar
  7. 5.7
    M. J. Narasimha, K. Shenoi, A. M. Peterson: “Quadratic Residues: Application to Chirp Filters and Discrete Fourier Transforms”, in IEEE 1976 Acoust., Speech, Signal Processing Proc., pp. 376–378Google Scholar
  8. 5.8
    M. J. Narasimha: “Techniques in Digital Signal Processing”, Tech. Rpt. 3208–3, Stanford Electronics Laboratory, Stanford University (1975)Google Scholar
  9. 5.9
    J. H. McClellan, C. M. Rader: Number Theory in Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N. J. 1979 )Google Scholar
  10. 5.10
    H. J. Nussbaumer, P. Quandalle: Fast computation of discrete Fourier transforms using polynomial transforms. IEEE Trans. ASSP-27, 169–181 (1979)MathSciNetGoogle Scholar
  11. 5.11
    I. J. Good: The interaction algorithm and practical Fourier analysis. J. Roy. Stat. Soc. B-20, 361–372 (1958); 22, 372–375 (1960)zbMATHMathSciNetGoogle Scholar
  12. 5.12
    I. J. Good: The relationship between two fast Fourier transforms. IEEE Trans. C-20, 310–317 (1971)Google Scholar
  13. 5.13
    D. P. Kolba, T. W. Parks: A prime factor FFT algorithm using high-speed convolution. IEEE Trans. ASSP-25, 90–103 (1977)Google Scholar
  14. 5.14
    C. S. Burrus: “Index Mappings for Multidimensional Formulation of the DFT and Convolution”, in 1977 IEEE Intern. Symp. on Circuits and Systems Proc., pp. 662–664Google Scholar
  15. 5.15
    S. Winograd: “A New Method for Computing DFT”, in 1977 IEEE Intern. Conf. Acoust., Speech and Signal Processing Proc., pp. 366–368Google Scholar
  16. 5.16
    S. Winograd: On computing the discrete Fourier transform. Math. Comput. 32, 175–199 (1978)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 5.17
    H. F. Silverman: An introduction to programming the Winograd Fourier transform algorithm (WFTA). IEEE Trans. ASSP-25, 152–165 (1977)Google Scholar
  18. 5.18
    R. W. Patterson, J. H. McClellan: Fixed-point error analysis of Winograd Fourier transform algorithms. IEEE Trans. ASSP-26, 447–455 (1978)Google Scholar
  19. 5.19
    L. R. Morris: A comparative study of time efficient FFT and WFTA programs for general purpose computers. IEEE Trans. ASSP-26, 141–150 (1978)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1982

Authors and Affiliations

  1. 1.Department d’ElectricitéEcole Polytechnique Fédérale de LausanneLausanneSwitzerland

Personalised recommendations