Skip to main content
SpringerLink
Log in

An algorithm for fast Hilbert transform of real functions

  • Published:
Advances in Computational Mathematics Aims and scope Submit manuscript

Abstract

A simple and accurate algorithm to evaluate the Hilbert transform of a real function is proposed using interpolations with piecewise–linear functions. An appropriate matrix representation reduces the complexity of this algorithm to the complexity of matrix-vector multiplication. Since the core matrix is an antisymmetric Toeplitz matrix, the discrete trigonometric transform can be exploited to calculate the matrix–vector multiplication with a reduction of the complexity to O(N log N), with N being the dimension of the core matrix. This algorithm has been originally envisaged for self-consistent simulations of radio-frequency wave propagation and absorption in fusion plasmas.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Brambilla, M.: Kinetic Theory of Plasma Waves. Oxford University Press (1998)

  2. Bilato, R., Brambilla, M., Maj, O., Horton, L., Maggi, C., Stober, J.: Simulations of combined neutral beam injection and ion cyclotron heating with the toric-ssfpql package. Nucl. Fusion. 51(10), 103034 (2011) http://stacks.iop.org/0029-5515/51/i=10/a=103034

    Article  Google Scholar 

  3. Brambilla, M., Bilato, R.: Advances in numerical simulations of ion cyclotron heating of non-maxwellian plasmas. Nucl. Fusion. 49(8), 085004 (2009) http://stacks.iop.org/0029-5515/49/085004

    Article  Google Scholar 

  4. Huang, N.E., Shen, Z., Long, S.R., Wu, M.C., Shih, H.H., Zheng, Q., Yen, N.-C., Tung, C.C., Liu, H.H.: The empirical mode decomposition and the hilbert spectrum for nonlinear and non-stationary time series analysis. Proc. Math. Phys. Eng. Sci. 454(1971), 903–995 (1998) http://www.jstor.org/stable/53161

    Article  MATH  MathSciNet  Google Scholar 

  5. Huang, N.E., Wu, Z.: A review on hilbert-huang transform: Method and its applications to geophysical studies. Rev. Geophys. 46, 8755–1209 (2008)

    Article  Google Scholar 

  6. Goswami, J.C., Hoefel, A.E.: Algorithms for estimating instantaneous frequency. Signal Process. 84(8) (1423). doi:10.1016/j.sigpro.2004.05.016, http://www.sciencedirect.com/science/article/pii/S0165168404001033

  7. Knockaert, L., Dhaene, T.: Causality determination and time delay extraction by means of the eigenfunctions of the hilbert transform. In: 12th IEEE Workshop on Signal Propagation Interconnects, 2008. SPI 2008, pp. 14 (2008). doi:10.1109/SPI.2008.4558337

  8. Weideman, J.A.C.: Computing the hilbert transform on the real line. Math. Comput. 64, 745 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  9. Zhou, C., Yang, L., Liu, Y., Yang, Z.: A novel method for computing the hilbert transform with haar multiresolution approximation. J. Comput. Appl. Math. 223(2), 585–597 (2009). doi:10.1016/j.cam.2008.02.006, http://www.sciencedirect.com/science/article/pii/S0377042708000526

    Article  MATH  MathSciNet  Google Scholar 

  10. Micchelli, C., Xu, Y., Yu, B.: On computing with the hilbert spline transform. Adv. Comput. Math. 1–24 (2012). doi:10.1007/s10444-011-9252-x

  11. Wright, J.C., Valeo, E.J., Phillips, C.K., Bonoli, P.T., Brambilla, M.: Full wave simulations of lower hybrid waves in toroidal geometry with non-maxwellian electrons. Communincations Comput. Phys. 4, 545 (2008)

    Google Scholar 

  12. Frigo, M., Johnson, S.: The design and implementation of fftw3. Proc. IEEE 93(2), 216 –231 (2005). doi:10.1109/JPROC.2004.840301

    Article  Google Scholar 

  13. Press, W.H., Flannery, B.P., Teukolsky, S.A., Vetterling, W.T.: Numerical Recipes in Fortran. Cambridge University Press (1992)

  14. Potts, D., Steidl, G.: Optimal trigonometric preconditioners for nonsymmetric toeplitz systems. Linear Algebra Appl. 281(13), 265–292 (265). doi:10.1016/S0024-3795(98)10042-3, http://www.sciencedirect.com/science/article/pii/S0024379598100423

    MathSciNet  Google Scholar 

  15. Taylor, M.E.: Partial Differential Equations I: Basic Theory. Springer, New York (1996)

    Book  Google Scholar 

  16. Poppe, G.P.M., Wijers, C.M.J.: More efficient computation of the complex error function. ACM Trans. Math. Softw. 16(1), 38–46 (1990)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Max Planck Institute for Plasma Physics, EURATOM Association, Boltzmannstr. 2, 85748, Garching, Germany

    Roberto Bilato, Omar Maj & Marco Brambilla

Authors
  1. Roberto Bilato
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Omar Maj
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Marco Brambilla
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Roberto Bilato.

Additional information

Communicated by: Zydrunas Gimbutas

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bilato, R., Maj, O. & Brambilla, M. An algorithm for fast Hilbert transform of real functions. Adv Comput Math 40, 1159–1168 (2014). https://doi.org/10.1007/s10444-014-9345-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10444-014-9345-4

Keywords

Mathematics Subject Classifications (2010)

Search

Navigation