Transformations of Functions and Signals

  • Simon Širca
  • Martin Horvat
Part of the Graduate Texts in Physics book series (GTP)


The discrete Fourier transformation is one of the most important tools in the analysis of functions and signals, but its detailed numerical aspects are often disregarded or neglected. Fourier and sampling theorems, Parseval’s equality, and power spectral densities are introduced, and the concepts of signal uncertainty, aliasing, and leakage are discussed, along with the caveats due to different notations and nomenclatures. Transformations with orthogonal polynomials (Legendre and Chebyshev) are described, and their relation to quadrature formulas is explained. The numerical computation of the Laplace transformation and its inverse is introduced in the context of differential equations for transfer functions of physical systems. The continuous and discrete Hilbert transformations are discussed next, introducing the concept of the analytic signal, and their relation to the Kramers–Krönig dispersion relations is given. The last Section deals with the continuous wavelet transform. The Examples and Problems include the multiplication of polynomials by FFT, the Fourier analysis of realistic acoustic signals and the Doppler effect, and the analysis of brain potentials by the Hilbert transform.


Orthogonal Polynomial Discrete Wavelet Transform Discrete Fourier Transform Wavelet Transformation Chebyshev Polynomial 
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.


  1. 1.
    B.D. MacCluer, Elementary Functional Analysis (Springer, New York, 2010) Google Scholar
  2. 2.
    C. Shannon, Communication in the presence of noise. Proc. IEEE 86, 447 (1998) (reprint) CrossRefGoogle Scholar
  3. 3.
    M. Unser, Sampling-50 years after Shannon. Proc. IEEE 88, 569 (2000) CrossRefGoogle Scholar
  4. 4.
    H. Nyquist, Certain topics in telegraph transmission theory. Proc. IEEE 90, 280 (2002) (reprint) CrossRefGoogle Scholar
  5. 5.
    C. Canuto, M.Y. Hussaini, A. Quarteroni, T.A. Zang, Spectral Methods. Fundamentals in Single Domains (Springer, Berlin, 2006) zbMATHGoogle Scholar
  6. 6.
    D. Donnelly, B. Rust, The fast Fourier transform for experimentalists, part I: concepts. Comput. Sci. Eng. Mar/Apr, 80 (2005) CrossRefGoogle Scholar
  7. 7.
    F.J. Harris, On the use of windows for harmonic analysis with the discrete Fourier transform. Proc. IEEE 66, 51 (1978) ADSCrossRefGoogle Scholar
  8. 8.
    M. Frigo, S.G. Johnson, The design and implementation of FFTW3. Proc. IEEE 93, 216 (2005); documentation can be found at CrossRefGoogle Scholar
  9. 9.
    W.H. Press, B.P. Flannery, S.A. Teukolsky, W.T. Vetterling, Numerical Recipes: The Art of Scientific Computing, 3rd edn. (Cambridge University Press, Cambridge, 2007). See also the equivalent handbooks in Fortran, Pascal and C, as well as zbMATHGoogle Scholar
  10. 10.
    P. Duhamel, M. Vetterli, Fast Fourier transforms: a tutorial review and a state of the art. Signal Process. 19, 259 (1990) MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    J.C. Schatzman, Accuracy of the discrete Fourier transform and the fast Fourier transform. SIAM J. Sci. Comput. 17, 1150 (1996) MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    G. Szegö, Orthogonal Polynomials (Am. Math. Soc., Providence, 1939) CrossRefGoogle Scholar
  13. 13.
    T.J. Rivlin, The Chebyshev Polynomials (Wiley, New York, 1974) zbMATHGoogle Scholar
  14. 14.
    V. Rokhlin, A fast algorithm for discrete Laplace transformation. J. Complex. 4, 12 (1988) MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    J. Strain, A fast Laplace transform based on Laguerre functions. Math. Comput. 58, 275 (1992) MathSciNetADSzbMATHCrossRefGoogle Scholar
  16. 16.
    M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions, 10th edn. (Dover, Mineola, 1972) zbMATHGoogle Scholar
  17. 17.
    S. Zhang, J. Jin, Computation of Special Functions (Wiley-Interscience, New York, 1996) zbMATHGoogle Scholar
  18. 18.
    W.E. Boyce, R.C. DiPrima, Elementary Differential Equations and Boundary Value Problems, 5th edn. (Wiley, New York, 1992) zbMATHGoogle Scholar
  19. 19.
    J. Abate, P.P. Valkó, Multi-precision Laplace transform inversion. Int. J. Numer. Methods Eng. 60, 979 (2004) zbMATHCrossRefGoogle Scholar
  20. 20.
    A.M. Cohen, Numerical Methods for Laplace Transform Inversion (Springer, New York, 2007) zbMATHCrossRefGoogle Scholar
  21. 21.
    B. Davies, B. Martin, Numerical inversion of the Laplace transform: a survey and comparison of methods. J. Comput. Phys. 33, 1 (1979) MathSciNetADSzbMATHCrossRefGoogle Scholar
  22. 22.
    D.G. Duffy, On the numerical inversion of Laplace transforms: comparison of three new methods on characteristic problems from applications. ACM Trans. Math. Softw. 19, 333 (1993) zbMATHCrossRefGoogle Scholar
  23. 23.
    C.L. Epstein, J. Schotland, The bad truth about Laplace’s transform. SIAM Rev. 50, 504 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    R. Piessens, Gaussian quadrature formulas for the numerical integration of Bromwich’s integral and the inversion of the Laplace transform. J. Eng. Math. 5, 1 (1971) MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    L.N. Trefethen, J.A.C. Weideman, T. Schmelzer, Talbot quadratures and rational approximations. BIT Numer. Math. 46, 653 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    J. Abate, W. Whitt, The Fourier-series method for inverting transforms of probability distributions. Queueing Syst. 10, 5 (1992) MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    P. den Iseger, Numerical transform inversion using Gaussian quadrature. Probab. Eng. Inf. Sci. 20, 1 (2006) zbMATHGoogle Scholar
  28. 28.
    A. Yonemoto, T. Hisikado, K. Okumura, Accuracy improvement of the FFT-based numerical inversion of Laplace transforms. IEEE Proc., Circuits Devices Syst. 150, 399 (2003) CrossRefGoogle Scholar
  29. 29.
    J. Abate, G.L. Choudhury, On the Laguerre method for numerically inverting Laplace transforms. INFORMS J. Comput. 8, 413 (1996). This paper presents efficient algorithms based on the expansion of the function f in terms of Laguerre polynomials and the corresponding computation of \(\mathcal{L}[f]\) and \(\mathcal{L}^{-1}[F]\) zbMATHCrossRefGoogle Scholar
  30. 30.
    E. Titchmarsh, Introduction to the Theory of Fourier Integrals, 2nd edn. (Clarendon, Oxford, 1948) Google Scholar
  31. 31.
    L. Grafakos, Classical and Modern Fourier Analysis (Prentice Hall, New Jersey, 2003) Google Scholar
  32. 32.
    M.L. Glasser, Some useful properties of the Hilbert transform. SIAM J. Math. Anal. 15, 1228 (1984) MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    R. Wong, Asymptotic expansion of the Hilbert transform. SIAM J. Math. Anal. 11, 92 (1980) MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    S.L. Hahn, Hilbert Transform in Signal Processing (Artech House, Boston, 1996) Google Scholar
  35. 35.
    W.J. Freeman, Origin, structure, and role of background EEG activity, part 1: analytic amplitude. Clin. Neurophysiol. 115, 2077 (2004) CrossRefGoogle Scholar
  36. 36.
    W.J. Freeman, Origin, structure, and role of background EEG activity, part 2: analytic phase. Clin. Neurophysiol. 115, 2089 (2004) CrossRefGoogle Scholar
  37. 37.
    W.J. Freeman, Origin, structure, and role of background EEG activity, part 3: neural frame classification. Clin. Neurophysiol. 116, 1118 (2005) CrossRefGoogle Scholar
  38. 38.
    W.J. Freeman, Origin, structure, and role of background EEG activity, part 4: neural frame simulation. Clin. Neurophysiol. 117, 572 (2006) CrossRefGoogle Scholar
  39. 39.
    M. West, A. Krystal, EEG, Duke University Google Scholar
  40. 40.
    C. Kittel, Introduction to Solid State Physics, 8th edn. (Wiley, New York, 2005) Google Scholar
  41. 41.
    M. Nandagopal, N. Arunajadai, On finite evaluation of finite Hilbert transform. Comput. Sci. Eng. 9, 90 (2007) CrossRefGoogle Scholar
  42. 42.
    T. Hasegawa, T. Torii, Hilbert and Hadamard transforms by generalized Chebyshev expansion. J. Comput. Appl. Math. 51, 71 (1994) MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    W. Gautschi, J. Waldvogel, Computing the Hilbert transform of the generalized Laguerre and Hermite weight functions. BIT Numer. Math. 41, 490 (2001) MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    V.R. Kress, E. Martensen, Anwendung der Rechteckregel auf die reelle Hilberttransformation mit unendlichem Intervall. Z. Angew. Math. Mech. 50, 61 (1970) MathSciNetCrossRefGoogle Scholar
  45. 45.
    F.W. King, G.J. Smethells, G.T. Helleloid, P.J. Pelzl, Numerical evaluation of Hilbert transforms for oscillatory functions: a convergence accelerator approach. Comput. Phys. Commun. 145, 256 (2002) MathSciNetADSzbMATHCrossRefGoogle Scholar
  46. 46.
    J.A.C. Weideman, Computing the Hilbert transform on the real line. Math. Comput. 64, 745 (1995). Attention: the authors use a non-standard definition sign(0)=1; in Eq. (22) correct 1/N→1/(2N) MathSciNetADSzbMATHCrossRefGoogle Scholar
  47. 47.
    X. Wang, Numerical implementation of the Hilbert transform. PhD thesis, University of Saskatchewan, Saskatoon, 2006 Google Scholar
  48. 48.
    H. Boche, M. Protzmann, A new algorithm for the reconstruction of the band-limited functions and their Hilbert transform. IEEE Trans. Instrum. Meas. 46, 442 (1997) CrossRefGoogle Scholar
  49. 49.
    A.V. Oppenheim, R.W. Schafer, Discrete-time Signal Processing, 2nd edn. (Prentice Hall, New Jersey, 1989) zbMATHGoogle Scholar
  50. 50.
    R. Bracewell, The Fourier Transform and Its Applications, 2nd edn. (McGraw-Hill, Reading, 1986) Google Scholar
  51. 51.
    F.W. King, Hilbert Transforms, Vols. 1 & 2. Encyclopedia of Mathematics and Its Applications, vols. 124 & 125 (Cambridge University Press, Cambridge, 2009) CrossRefGoogle Scholar
  52. 52.
    C. Torrence, G.P. Compo, A practical guide to wavelet analysis. Bull. Am. Meteorol. Soc. 79, 61 (1998) ADSCrossRefGoogle Scholar
  53. 53.
    D. Jordan, R.W. Miksad, E.J. Powers, Implementation of the continuous wavelet transform for digital time series analysis. Rev. Sci. Instrum. 68, 1484 (1997) ADSCrossRefGoogle Scholar
  54. 54.
    P.S. Addison, The Illustrated Wavelet Transform Handbook (Institute of Physics, Bristol, 2002) zbMATHCrossRefGoogle Scholar
  55. 55.
    H.-G. Stark, Wavelets and Signal Processing. An Application-Based Introduction (Springer, Berlin, 2005) Google Scholar
  56. 56.
    G. Kaiser, A Friendly Guide to Wavelets (Birkhäuser, Boston, 1994) zbMATHGoogle Scholar
  57. 57.
    G. Kaiser, Physical wavelets and their sources: real physics in complex space-time. J. Phys. A, Math. Gen. 36, R291 (2003). A physicist might find particular pleasure in physical (acoustic and electro-magnetic) wavelets discussed in this paper MathSciNetADSzbMATHCrossRefGoogle Scholar
  58. 58.
    J.F. Kirby, Which wavelet best reproduces the Fourier power spectrum? Comput. Geosci. 31, 846 (2005) ADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Simon Širca
    • 1
  • Martin Horvat
    • 1
  1. 1.Faculty of Mathematics and PhysicsUniversity of LjubljanaLjubljanaSlovenia

Personalised recommendations