Advertisement

The Fast Fourier Transform

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

Abstract

The object of this chapter is to briefly summarize the main properties of the discrete Fourier transform (DFT) and to present various fast DFT computation techniques known collectively as the fast Fourier transform (FFT) algorithm. The DFT plays a key role in physics because it can be used as a mathematical tool to describe the relationship between the time domain and frequency domain representation of discrete signals. The use of DFT analysis methods has increased dramatically since the introduction of the FFT in 1965 because the FFT algorithm decreases by several orders of magnitude the number of arithmetic operations required for DFT computations. It has thereby provided a practical solution to many problems that otherwise would have been intractable.

Keywords

Fast Fourier Transform Discrete Fourier Transform Real Multiplication Fast Fourier Transform Algorithm Inverse Discrete Fourier Transform 
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. 4.1
    B. Gold, C. M. Rader: Digital Processing of Signals ( McGraw-Hill, New York 1969 )zbMATHGoogle Scholar
  2. 4.2
    E. O. Brigham: The Fast Fourier Transform (Prentice-Hall, Englewood Cliffs, N. J. 1974 )Google Scholar
  3. 4.3
    L. R. Rabiner, B. Gold: Theory and Application of Digital Signal Processing (PrenticeHall, Englewood Cliffs, N. J. 1975 )Google Scholar
  4. 4.4
    A. V. Oppenheim, R. W. Schafer: Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N. J. 1975 )Google Scholar
  5. 4.5
    A. E. Siegman: How to compute two complex even Fourier transforms with one transform step. Proc. IEEE 63, 544 (1975)CrossRefGoogle Scholar
  6. 4.6
    J. W. Cooley, J. W. Tukey: An algorithm for machine computation of complex Fourier series. Math. Comput. 19, 297–301 (1965)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 4.7
    G. D. Bergland: A fast Fourier transform algorithm using base 8 iterations. Math. Comput. 22, 275–279 (1968)zbMATHMathSciNetGoogle Scholar
  8. 4.8
    R. C. Singleton: An algorithm for computing the mixed radix fast Fourier transform. IEEE Trans. AU-17, 93–103 (1969)Google Scholar
  9. 4.9
    R. P. Polivka, S. Pakin: APL: the Language and Its Usage (Prentice-Hall, Englewood Cliffs, N. J. 1975 )Google Scholar
  10. 4.10
    P. D. Welch: A fixed-point fast Fourier transform error analysis. IEEE Trans. AU-17, 151–157 (1969)Google Scholar
  11. 4.11
    T. K. Kaneko, B. Liu: Accumulation of round-off errors in fast Fourier transforms. J. Assoc. Comput. Mach. 17, 637–654 (1970)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 4.12
    C. J. Weinstein: Roundoff noise in floating point fast Fourier transform computation. IEEE Trans. AU-17, 209–215 (1969)Google Scholar
  13. 4.13
    C. M. Rader, N. M. Brenner: A new principle for fast Fourier transformation. IEEE Trans. ASSP-24, 264–265 (1976)Google Scholar
  14. 4.14
    S. Winograd: On computing the discrete Fourier transform. Math. comput. 32, 175–199 (1978)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 4.15
    K. M. Cho, G. C. Temes: “Real-factor FFT algorithms”, in IEEE 1978 Intern. Conf. Acoust., Speech, Signal Processing, pp. 634–637Google Scholar
  16. 4.16
    H. J. Nussbaumer, P. Quandalle: Fast computation of discrete Fourier transforms using polynomial transforms. IEEE Trans. ASSP-27, 169–181 (1979)MathSciNetGoogle Scholar
  17. 4.17
    G. Bonnerot, M. Bellanger: Odd-time odd-frequency discrete Fourier transform for symmetric real-valued series. Proc. IEEE 64, 392–393 (1976)CrossRefGoogle Scholar
  18. 4.18
    G. Bruun: z-transform DFT filters and FFTs. IEEE Trans. ASSP-26, 56–63 (1978)Google Scholar
  19. 4.19
    G. K. McAuliffe: “Fourier Digital Filter or Equalizer and Method of Operation Therefore”, US Patent No. 3 679 882, July 25, 1972Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1982

Authors and Affiliations

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

Personalised recommendations