Fast Fourier Transforms

  • Gerlind Plonka
  • Daniel Potts
  • Gabriele Steidl
  • Manfred Tasche
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)


As shown in Chap.  3, any application of Fourier methods leads to the evaluation of a discrete Fourier transform of length N (DFT(N)). Thus the efficient computation of DFT(N) is very important. Therefore this chapter treats fast Fourier transforms. A fast Fourier transform (FFT) is an algorithm for computing the DFT(N) which needs only a relatively low number of arithmetic operations.


  1. 3.
    A. Akavia, Deterministic sparse Fourier approximation via approximating arithmetic progressions. IEEE Trans. Inf. Theory 60(3), 1733–1741 (2014)MathSciNetzbMATHGoogle Scholar
  2. 8.
    M. Arioli, H. Munthe-Kaas, L. Valdettaro, Componentwise error analysis for FFTs with applications to fast Helmholtz solvers. Numer. Algorithms 12, 65–88 (1996)MathSciNetzbMATHGoogle Scholar
  3. 30.
    L. Berg, Lineare Gleichungssysteme mit Bandstruktur und ihr asymptotisches Verhalten (Deutscher Verlag der Wissenschaften, Berlin, 1986)zbMATHGoogle Scholar
  4. 34.
    S. Bittens, Sparse FFT for functions with short frequency support. Dolomites Res. Notes Approx. 10, 43–55 (2017)MathSciNetzbMATHGoogle Scholar
  5. 36.
    R.E. Blahut, Fast Algorithms for Digital Signal Processing (Cambridge University Press, New York, 2010)zbMATHGoogle Scholar
  6. 47.
    E.O. Brigham, The Fast Fourier Transform (Prentice Hall, Englewood Cliffs, 1974)Google Scholar
  7. 50.
    G. Bruun, z-Transform DFT filters and FFT’s. IEEE Trans. Acoust. Speech Signal Process. 26(1), 56–63 (1978)Google Scholar
  8. 60.
    D. Calvetti, A stochastic roundoff error analysis for FFT. Math. Comput. 56(194), 755–774 (1991)zbMATHGoogle Scholar
  9. 67.
    C.Y. Chu, The fast Fourier transform on the hypercube parallel computers. Ph.D. thesis, Cornell University, Ithaca, 1988Google Scholar
  10. 73.
    J.W. Cooley, J.W. Tukey, An algorithm for machine calculation of complex Fourier series. Math. Comput. 19, 297–301 (1965)MathSciNetzbMATHGoogle Scholar
  11. 84.
    C.J. Demeure, Fast QR factorization of Vandermonde matrices. Linear Algebra Appl. 122–124, 165–194 (1989)MathSciNetzbMATHGoogle Scholar
  12. 91.
    P. Duhamel, H. Hollmann, Split-radix FFT algorithm. Electron. Lett. 20(1), 14–16 (1984)Google Scholar
  13. 92.
    P. Duhamel, M. Vetterli, Fast Fourier transforms: a tutorial review and a state of the art. Signal Process. 19(4), 259–299 (1990)MathSciNetzbMATHGoogle Scholar
  14. 122.
    M. Frigo, S.G. Johnson, The design and implementation of FFTW3. Proc. IEEE 93, 216–231 (2005)Google Scholar
  15. 123.
    M. Frigo, S.G. Johnson, FFTW, C subroutine library (2009).
  16. 128.
    W. Gentleman, G. Sande, Fast Fourier transform for fun and profit, in Fall Joint Computer Conference AFIPS, vol. 29 (1966), pp. 563–578Google Scholar
  17. 130.
    A.C. Gilbert, M.J. Strauss, J.A. Tropp, A tutorial on fast Fourier sampling. IEEE Signal Process. Mag. 25(2), 57–66 (2008)Google Scholar
  18. 131.
    A. Gilbert, P. Indyk, M. Iwen, L. Schmidt, Recent developments in the sparse Fourier transform: a compressed Fourier transform for big data. IEEE Signal Process. Mag. 31(5), 91–100 (2014)Google Scholar
  19. 132.
    G. Goerzel, An algorithm for the evaluation of finite trigonometric series. Am. Math. Mon. 65(1), 34–35 (1958)MathSciNetGoogle Scholar
  20. 134.
    G.H. Golub, C.F. Van Loan, Matrix Computations, 3rd edn. (Johns Hopkins University Press, Baltimore, 1996)zbMATHGoogle Scholar
  21. 138.
    I.J. Good, The interaction algorithm and practical Fourier analysis. J. R. Stat. Soc. Ser. B 20, 361–372 (1958)MathSciNetzbMATHGoogle Scholar
  22. 156.
    H. Hassanieh, The Sparse Fourier Transform: Theory and Practice (ACM Books, New York, 2018)zbMATHGoogle Scholar
  23. 157.
    H. Hassanieh, P. Indyk, D. Katabi, E. Price, Simple and practical algorithm for sparse Fourier transform, in Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms (ACM, New York, 2012), pp. 1183–1194Google Scholar
  24. 163.
    M.T. Heideman, D.H. Johnson, C.S. Burrus, Gauss and the history of the fast Fourier transform. Arch. Hist. Exact Sci. 34(3), 265–277 (1985)MathSciNetzbMATHGoogle Scholar
  25. 164.
    S. Heider, S. Kunis, D. Potts, M. Veit, A sparse Prony FFT, in Proceedings of 10th International Conference on Sampling Theory and Applications, vol. 9 (2013), pp. 1183–1194Google Scholar
  26. 168.
    N.J. Higham, Accuracy and Stability of Numerical Algorithms, 2nd edn. (SIAM, Philadelphia, 2002)zbMATHGoogle Scholar
  27. 174.
    M.A. Iwen, Combinatorial sublinear-time Fourier algorithms. Found. Comput. Math. 10(3), 303–338 (2010)MathSciNetzbMATHGoogle Scholar
  28. 175.
    M.A. Iwen, Improved approximation guarantees for sublinear-time Fourier algorithms. Appl. Comput. Harmon. Anal. 34(1), 57–82 (2013)MathSciNetzbMATHGoogle Scholar
  29. 179.
    S.G. Johnson, M. Frigo, A modified split radix FFT with fewer arithmetic operations. IEEE Trans. Signal Process. 55(1), 111–119 (2007)MathSciNetzbMATHGoogle Scholar
  30. 222.
    D. Lawlor, Y. Wang, A. Christlieb, Adaptive sub-linear time Fourier algorithms. Adv. Adapt. Data Anal. 5(1), 1350003 (2013)MathSciNetGoogle Scholar
  31. 246.
    J. Morgenstern, Note on a lower bound of the linear complexity of the fast Fourier transform. J. Assoc. Comput. Mach. 20, 305–306 (1973)MathSciNetzbMATHGoogle Scholar
  32. 257.
    H.J. Nussbaumer, Fast Fourier Transform and Convolution Algorithms, revised edn. (Springer, Berlin, 1982)Google Scholar
  33. 263.
    S. Pawar, K. Ramchandran, Computing a k-sparse n-length discrete Fourier transform using at most 4k samples and \(o(k \log k)\) complexity, in Proceedings of the IEEE International Symposium on Information Theory (ISIT) (2013), pp. 464–468Google Scholar
  34. 265.
    T. Peter, G. Plonka, A generalized Prony method for reconstruction of sparse sums of eigenfunctions of linear operators. Inverse Prob. 29, 025001 (2013)MathSciNetzbMATHGoogle Scholar
  35. 269.
    M. Pippig, PFFT, Parallel FFT subroutine library (2011).
  36. 271.
    M. Pippig, PFFT: an extension of FFTW to massively parallel architectures. SIAM J. Sci. Comput. 35(3), C213–C236 (2013)MathSciNetzbMATHGoogle Scholar
  37. 274.
    G. Plonka, K. Wannenwetsch, A deterministic sparse FFT algorithm for vectors with small support. Numer. Algorithms 71(4), 889–905 (2016)MathSciNetzbMATHGoogle Scholar
  38. 275.
    G. Plonka, K. Wannenwetsch, A sparse fast Fourier algorithm for real non-negative vectors. J. Comput. Appl. Math. 321, 532–539 (2017)MathSciNetzbMATHGoogle Scholar
  39. 278.
    G. Plonka, K. Wannenwetsch, A. Cuyt, W.-S. Lee, Deterministic sparse FFT for m-sparse vectors. Numer. Algorithms 78(1), 133–159 (2018)MathSciNetzbMATHGoogle Scholar
  40. 295.
    D. Potts, G. Steidl, M. Tasche, Numerical stability of fast trigonometric transforms - a worst case study. J. Concr. Appl. Math. 1(1), 1–36 (2003)MathSciNetzbMATHGoogle Scholar
  41. 298.
    D. Potts, M. Tasche, T. Volkmer, Efficient spectral estimation by MUSIC and ESPRIT with application to sparse FFT. Front. Appl. Math. Stat. 2, Article 1 (2016)Google Scholar
  42. 299.
    E. Prestini, The Evolution of Applied Harmonic Analysis. Models of the Real World, 2nd edn. (Birkhäuser/Springer, New York, 2016)zbMATHGoogle Scholar
  43. 302.
    C. Rader, Discrete Fourier transforms when the number of data samples is prime. Proc. IEEE 56(6), 1107–1108 (1968)Google Scholar
  44. 304.
    G.U. Ramos, Roundoff error analysis of the fast Fourier transform. Math. Comput. 25, 757–768 (1971)MathSciNetGoogle Scholar
  45. 314.
    C. Runge, Über die Zerlegung einer empirisch gegebenen periodischen Funktion in Sinuswellen. Z. Math. Phys. 48, 443–456 (1903)zbMATHGoogle Scholar
  46. 321.
    J.C. Schatzman, Accuracy of the discrete Fourier transform and the fast Fourier transform. SIAM J. Sci. Comput. 17(5), 1150–1166 (1996)MathSciNetzbMATHGoogle Scholar
  47. 350.
    M. Tasche, H. Zeuner, Roundoff error analysis for fast trigonometric transforms, in Handbook of Analytic-Computational Methods in Applied Mathematics (Chapman & Hall/CRC Press, Boca Raton, 2000), pp. 357–406zbMATHGoogle Scholar
  48. 351.
    M. Tasche, H. Zeuner, Worst and average case roundoff error analysis for FFT. BIT Numer. Math. 41(3), 563–581 (2001)MathSciNetzbMATHGoogle Scholar
  49. 362.
    C.F. Van Loan, Computational Frameworks for the Fast Fourier Transform (SIAM, Philadelphia, 1992)zbMATHGoogle Scholar
  50. 363.
    M. Vetterli, P. Duhamel, Split- radix algorithms for length-p m DFTs. IEEE Trans. Acoust. Speech Signal Process. 37(1), 57–64 (1989)MathSciNetzbMATHGoogle Scholar
  51. 377.
    S. Winograd, Some bilinear forms whose multiplicative complexity depends on the field of constants. Math. Syst. Theory 10, 169–180 (1977)MathSciNetzbMATHGoogle Scholar
  52. 378.
    S. Winograd, On computing the discrete Fourier transform. Math. Comput. 32(141), 175–199 (1978)MathSciNetzbMATHGoogle Scholar
  53. 379.
    S. Winograd, Arithmetic Complexity of Computations (SIAM, Philadelphia, 1980)zbMATHGoogle Scholar
  54. 381.
    X. Wu, Y. Wang, Z. Yan, On algorithms and complexities of cyclotomic fast Fourier transforms over arbitrary finite fields. IEEE Trans. Signal Process. 60(3), 1149–1158 (2012)MathSciNetGoogle Scholar
  55. 384.
    P.Y. Yalamov, Improvements of some bounds on the stability of fast Helmholtz solvers. Numer. Algorithms 26(1), 11–20 (2001)MathSciNetzbMATHGoogle Scholar
  56. 387.
    R. Yavne, An economical method for calculating the discrete Fourier transform, in Proceedings of AFIPS Fall Joint Computer Conference, vol. 33 (1968), pp. 115–125Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Gerlind Plonka
    • 1
  • Daniel Potts
    • 2
  • Gabriele Steidl
    • 3
  • Manfred Tasche
    • 4
  1. 1.University of GöttingenGöttingenGermany
  2. 2.Chemnitz University of TechnologyChemnitzGermany
  3. 3.TU KaiserslauternKaiserslauternGermany
  4. 4.University of RostockRostockGermany

Personalised recommendations