Advances in Computational Mathematics

, Volume 24, Issue 1–4, pp 171–195 | Cite as

A B-spline approach for empirical mode decompositions

  • Qiuhui Chen
  • Norden Huang
  • Sherman Riemenschneider
  • Yuesheng Xu


We propose an alternative B-spline approach for empirical mode decompositions for nonlinear and nonstationary signals. Motivated by this new approach, we derive recursive formulas of the Hilbert transform of B-splines and discuss Euler splines as spline intrinsic mode functions in the decomposition. We also develop the Bedrosian identity for signals having vanishing moments. We present numerical implementations of the B-spline algorithm for an earthquake signal and compare the numerical performance of this approach with that given by the standard empirical mode decomposition. Finally, we discuss several open mathematical problems related to the empirical mode decomposition.


B-splines nonlinear and nonstationary signals empirical mode decompositions Hilbert transforms 


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  1. [1]
    E. Bedrosian, A product theorem for Hilbert transform, Proc. IEEE 51 (1963) 868–869. Google Scholar
  2. [2]
    C. Bennett and R. Sharpley, Interpolation of Operators (Academic Press, Boston, 1988). MATHGoogle Scholar
  3. [3]
    C. de Boor, A Practical Guide to Splines (Springer-Verlag, New York, 1978). MATHGoogle Scholar
  4. [4]
    L. Cohen, Time-Frequency Analysis (Prentice-Hall, Englewood Cliffs, NJ, 1995). Google Scholar
  5. [5]
    I. Daubechies, Ten Lectures on Wavelets, CBMS 61 (SIAM, Philadelphia, 1992). MATHGoogle Scholar
  6. [6]
    C. Diks, Nonlinear Time Series Analysis (World Scientific, Singapore, 1997). Google Scholar
  7. [7]
    P.A.M. Dirac, The Principles of Quantum Mechanics (Oxford University Press, New York, 1935). Google Scholar
  8. [8]
    M. v. Golitschek, On the convergence of interpolating periodic spline functions of high degree, Numer. Math. 19 (1972) 146–154. MATHMathSciNetCrossRefGoogle Scholar
  9. [9]
    P. Flandrin, G. Rilling and P. Goncalves, Empirical mode decomposition as a filterbank, IEEE Signal Process. (2003) in press. Google Scholar
  10. [10]
    N.E. Huang, Empirical mode decomposition for analyzing acoustic signal, US Patent 10-073857 (August, 2003) Pending. Google Scholar
  11. [11]
    N.E. Huang et al., The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis, Proc. Roy. Soc. London A 454 (1998) 903–995. MATHGoogle Scholar
  12. [12]
    N.E. Huang, S.R. Long and Z. Shen, The mechanism for frequency downshift in nonlinear wave evolution, Adv. Appl. Mech. 32 (1996) 59–111. CrossRefGoogle Scholar
  13. [13]
    N.E. Huang, C.C. Chern, K. Huang, L. Salvino, S.R. Long and K.L. Fan, Spectral analysis of the Chi-Chi earthquake data: Station TUC129, Taiwan, September 21, 1999, Bull. Seismol. Soc. Amer. 91 1,310-1,338. Google Scholar
  14. [14]
    N.E. Huang, C.C. Tung and S.R. Long, The probability structure of the ocean surface, The Sea 9 (1990) 335–366. Google Scholar
  15. [15]
    N.E. Huang, Z. Shen and S.R. Long, A new view of nonlinear water waves: The Hilbert spectrum, Ann. Rev. Fluid Mech. 31 (1999) 417–457. MathSciNetCrossRefGoogle Scholar
  16. [16]
    N.E. Huang, M.-L.C. Wu, S.R. Long, S.S.P. Shen, W. Qu, P. Gloersen and K.L. Fan, A confidence limit for the empirical mode decomposition and Hilbert spectral analysis, Proc. Roy. Soc. London A 459 (2003) 2317–2346. MathSciNetMATHCrossRefGoogle Scholar
  17. [17]
    N.E. Huang, Z. Wu, S.R. Long, K.C. Arnold, K. Blank and T.W. Liu, On instantaneous frequency, Preprint (2003). Google Scholar
  18. [18]
    H. Kantz and T. Schreiber, Nonlinear Time Series Analysis (Cambridge University Press, Cambridge, 1997). MATHGoogle Scholar
  19. [19]
    A.H. Nuttall, On the quadrature approximation to the Hilbert transform of modulated signals, Proc. IEEE 54 (1966) 1458–1459. CrossRefGoogle Scholar
  20. [20]
    S. Olhede and A.T. Walden, The Hilbert spectrum via wavelet projections, Proc. Roy. Soc. London (2003) in press. Google Scholar
  21. [21]
    I.J. Schoenberg, On interpolation by spline functions and its minimal properties, ISNM Approx. Theory 5 (1964). Google Scholar
  22. [22]
    I.J. Schoenberg, Notes on spline functions I: The limits of the interpolating periodic spline functions as their degree tends to infinity, Indag. Math. 34 (1972) 412–422. MathSciNetGoogle Scholar
  23. [23]
    I.J. Schoenberg, On remainders in and the convergence of cardinal spline interpolation for almost periodic functions, in: Studies in Spline Functions and Approximation, eds. S. Karlin et al. (Academic Press, New York, 1976) pp. 277–303. Google Scholar
  24. [24]
    I.J. Schoenberg, A new approach to Euler splines, J. Approx. Theory 39 (1983) 324–337. MATHMathSciNetCrossRefGoogle Scholar
  25. [25]
    H. Tong, Nonlinear Time Series Analysis (Oxford University Press, Oxford, 1990). Google Scholar
  26. [26]
    B. Windrows and S.D. Stearns, Adaptive Signal Processing (Prentice-Hall, Upper Saddle River, NJ, 1985). Google Scholar
  27. [27]
    Z. Wu and N.E. Huang, A study of the characteristics of white noise using the empirical mode decomposition method, Proc. Roy. Soc. London (2003) in press. Google Scholar

Copyright information

© Springer 2006

Authors and Affiliations

  • Qiuhui Chen
    • 1
  • Norden Huang
    • 2
  • Sherman Riemenschneider
    • 3
  • Yuesheng Xu
    • 4
  1. 1.Faculty of Mathematics and Computer ScienceHubei UniversityWuhanP.R. China
  2. 2.Laboratory for Hydrospheric Process/Oceans and Ice BranchNASA Goddard Space Flight CenterGreenbeltU.S.A.
  3. 3.Department of MathematicsWest Virginia UniversityMorgantownU.S.A.
  4. 4.Deparment of Mathematics, Syracuse University, Syracuse, NY 13244, U.S.A. and Institute of Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of SciencesBeijingP.R. China

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