Discrete Fourier Transforms

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


This chapter deals with the discrete Fourier transform (DFT). In Sect. 3.1, we show that numerical realizations of Fourier methods, such as the computation of Fourier coefficients, Fourier transforms or trigonometric interpolation, lead to the DFT. We also present barycentric formulas for interpolating trigonometric polynomials. In Sect. 3.2, we study the basic properties of the Fourier matrix and of the DFT. In particular, we consider the eigenvalues of the Fourier matrix with their multiplicities. Further, we present the intimate relations between cyclic convolutions and the DFT. In Sect. 3.3, we show that cyclic convolutions and circulant matrices are closely related and that circulant matrices can be diagonalized by the Fourier matrix. Section 3.4 presents the properties of Kronecker products and stride permutations, which we will need later in Chap.  5 for the factorization of a Fourier matrix. We show that block circulant matrices can be diagonalized by Kronecker products of Fourier matrices. Finally, Sect. 3.5 addresses real versions of the DFT, such as the discrete cosine transform (DCT) and the discrete sine transform (DST). These linear transforms are generated by orthogonal matrices.


  1. 2.
    N. Ahmed, T. Natarajan, K.R. Rao, Discrete cosine transform. IEEE Trans. Comput. 23, 90–93 (1974)MathSciNetCrossRefGoogle Scholar
  2. 6.
    T.M. Apostol, Introduction to Analytic Number Theory (Springer, New York, 1976)zbMATHGoogle Scholar
  3. 7.
    A. Arico, S. Serra-Capizzano, M. Tasche, Fast and numerically stable algorithms for discrete Hartley transforms and applications to preconditioning. Commun. Inf. Syst. 5(1), 21–68 (2005)MathSciNetzbMATHGoogle Scholar
  4. 31.
    J.-P. Berrut, L.N. Trefethen, Barycentric Lagrange interpolation. SIAM Rev. 46(3), 501–517 (2004)MathSciNetCrossRefGoogle Scholar
  5. 44.
    R.N. Bracewell, The Hartley Transform (Clarendon Press/Oxford University Press, New York, 1986)zbMATHGoogle Scholar
  6. 46.
    W.L. Briggs, V.E. Henson, The DFT. An Owner’s Manual for the Discrete Fourier Transform (SIAM, Philadelphia, 1995)Google Scholar
  7. 78.
    P.J. Davis, Circulant Matrices (Wiley, New York, 1979)zbMATHGoogle Scholar
  8. 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)MathSciNetCrossRefGoogle Scholar
  9. 166.
    P. Henrici, Barycentric formulas for interpolating trigonometric polynomials and their conjugates. Numer. Math. 33(2), 225–234 (1979)MathSciNetCrossRefGoogle Scholar
  10. 169.
    R.A. Horn, C.R. Johnson, Matrix Analysis, 2nd edn. (Cambridge University Press, Cambridge, 2013)zbMATHGoogle Scholar
  11. 239.
    J.H. McClellan, T.W. Parks, Eigenvalue and eigenvector decomposition of the discrete Fourier transform. IEEE Trans. Audio Electroacoust. 20(1), 66–74 (1972)MathSciNetCrossRefGoogle Scholar
  12. 247.
    P. Morton, On the eigenvectors of Schur’s matrix. J. Number Theory 12(1), 122–127 (1980)MathSciNetCrossRefGoogle Scholar
  13. 306.
    K.R. Rao, P. Yip, Discrete Cosine Transform: Algorithms, Advantages, Applications (Academic, Boston, 1990)CrossRefGoogle Scholar
  14. 315.
    C. Runge, H. König, Vorlesungen über Numerisches Rechnen (Springer, Berlin, 1924)CrossRefGoogle Scholar
  15. 343.
    G. Strang, The discrete cosine transform. SIAM Rev. 41(1), 135–147 (1999)MathSciNetCrossRefGoogle Scholar
  16. 356.
    L.N. Trefethen, Approximation Theory and Approximation Practice (SIAM, Philadelphia, 2013)zbMATHGoogle Scholar
  17. 367.
    Z.D. Wang, Fast algorithms for the discrete W transform and the discrete Fourier transform. IEEE Trans. Acoust. Speech Signal Process. 32(4), 803–816 (1984)MathSciNetCrossRefGoogle Scholar
  18. 376.
    M.V. Wickerhauser, Adapted Wavelet Analysis from Theory to Software (A K Peters, Wellesley, 1994)Google Scholar

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© 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

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