Skip to main content
SpringerLink
Log in

On the Convergence of Iterative Filtering Empirical Mode Decomposition

  • Chapter
  • First Online:
Excursions in Harmonic Analysis, Volume 2

Part of the book series: Applied and Numerical Harmonic Analysis ((ANHA))

Abstract

Empirical mode decomposition (EMD), an adaptive technique for data and signal decomposition, is a valuable tool for many applications in data and signal processing. One approach to EMD is the iterative filtering EMD, which iterates certain banded Toeplitz operators in l ∞(ℤ). The convergence of iterative filtering is a challenging mathematical problem. In this chapter we study this problem, namely for a banded Toeplitz operator T and x∈l ∞(ℤ) we study the convergence of T n(x). We also study some related spectral properties of these operators. Even though these operators don’t have any eigenvalue in Hilbert space l 2(ℤ), all eigenvalues and their associated eigenvectors are identified in l ∞(ℤ) by using the Fourier transform on tempered distributions. The convergence of T n(x) relies on a careful localization of the generating function for T around their maximal points and detailed estimates on the contribution from the tails of x.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Bartle, R.: The Elements of Real Analysis, 2nd edn. Wiley, New York (1976)

    MATH  Google Scholar 

  2. Böttcher, A., Grudsky, S.: Toeplitz Matrices, Asymptotic Linear Algebra, and Functional Analysis. Birkhäuser (2000)

    Google Scholar 

  3. Chen, Q., Huang, N., Riemenschneider, S., Xu, Y.: B-spline approach for empirical mode decomposition, Adv. Comput. Math. 24, 171–195 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cordoba, A.: Dirac combs. Lett. Math. Phys. 17, 191–196 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  5. Echeverria, J.C., Crowe, J.A., Woolfson, M.S., Hayes-Gill, B.R.: Application of empirical mode decomposition to heart rate variability analysis. Med. Biol. Eng. Comput. 39, 471–479 (2001)

    Article  Google Scholar 

  6. Folland, G.: Real Analysis. Modern techniques and their application. 2nd ed. John Wiley and Sons Inc., New York, (1999)

    Google Scholar 

  7. Hörmander, L.: The Analysis of Linear Partial Differential Operators I. Springer (1983)

    Google Scholar 

  8. Huang, N., et al.: The empirical mode decomposition and the Hilbert spectrum for nonlinear nonstationary time series analysis. Proceedings of Royal Society of London A 454, 903–995 (1998)

    Article  MATH  Google Scholar 

  9. Huang, N., Shen, Z., Long, S.: A new view of nonlinear water waves: the Hilbert spectrum. Annu. Rev. Fluid Mech. 31, 417–457 (1999)

    Article  MathSciNet  Google Scholar 

  10. Hughes, J., Mao, D., Rockmore, D., Wang, Y., Wu, Q.: Empirical mode decomposition analysis of visual stylometry, preprint

    Google Scholar 

  11. Lagarias, J.: Mathematical quasicrystals and the problem of diffraction. In: Baake, M., Moody, R.V. (eds.) Directions in Mathematical Quasicrystals, CRM Monograph Series, Amer. Math. Soc., vol. 13, pp. 61–93. Providence, RI (2000)

    Google Scholar 

  12. Lin, L., Wang, Y., Zhou, H.: Iterative filtering as an alternative algorithm for empirical mode decomposition. Adv. Adapt. Data Anal. 1(4), 543–560 (2009)

    Article  MathSciNet  Google Scholar 

  13. Liu, B., Riemenschneider, S., Xu, Y.: Gearbox fault diagnosis using empirical mode decomposition and hilbert spectrum, preprint

    Google Scholar 

  14. Mao, D., Wang, Y., Wu, Q.: A new approach for analyzing physiological time series, preprint

    Google Scholar 

  15. Pines, D., Salvino, L.: Health monitoring of one dimensional structures using empirical mode decomposition and the Hilbert-Huang Transform. In: Proceedings of SPIE 4701, pp. 127–143 (2002)

    Article  Google Scholar 

  16. Yu, Z.-G., Anh, V., Wang, Y., Mao, D.: Modeling and simulation of the horizontal component of the magnetic field by fractional stochastic differential equation in conjunction with epirical mode decomposition. J. Geophys. Res. Space Phys. to appear

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Michigan State University, East Lansing, MI, 48824, USA

    Yang Wang & Zhengfang Zhou

Authors
  1. Yang Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhengfang Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yang Wang .

Editor information

Editors and Affiliations

  1. , Department of Mathematics, University of Maryland, Norbert Wiener Center, College Park, 20742-0001, Maryland, USA

    Travis D. Andrews

  2. Norbert Wiener Center, Department of Mathematics, University of Maryland, College Park, 20742, Maryland, USA

    Radu Balan

  3. Department of Mathematics, University of Maryland, College Park, 20742-4015, Maryland, USA

    John J. Benedetto

  4. Department of Mathematics, University of Maryland, College Park, 20742-4015, Maryland, USA

    Wojciech Czaja

  5. , Department of Mathematics, University of Maryland, College Park, 20742, Maryland, USA

    Kasso A. Okoudjou

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Birkhäuser Boston

About this chapter

Cite this chapter

Wang, Y., Zhou, Z. (2013). On the Convergence of Iterative Filtering Empirical Mode Decomposition. In: Andrews, T., Balan, R., Benedetto, J., Czaja, W., Okoudjou, K. (eds) Excursions in Harmonic Analysis, Volume 2. Applied and Numerical Harmonic Analysis. Birkhäuser, Boston. https://doi.org/10.1007/978-0-8176-8379-5_8

Download citation

Publish with us

Policies and ethics

Search

Navigation