ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)

For citation:

Pavlov A. N., Filatova A. E., Hramov A. E. Time­frequency analysis of nonstationary processes: concepts of wavelets and empirical modes. Izvestiya VUZ. Applied Nonlinear Dynamics, 2011, vol. 19, iss. 2, pp. 141-157. DOI: 10.18500/0869-6632-2011-19-2-141-157

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Full text:
(downloads: 144)
Article type: 

Time­frequency analysis of nonstationary processes: concepts of wavelets and empirical modes

Pavlov Aleksej Nikolaevich, Saratov State University
Filatova Anastasija Evgenevna, Saratov State University
Hramov Aleksandr Evgenevich, Innopolis University

A comparation of wavelets and empirical modes concepts is performed that represent the most perspective tools to study the structure of nonstationary multimode processes. Their advantages over the classical methods for time series analysis and restrictions of both approaches are discussed that needs to be known for correct interpretation of the obtained results. New possibilities in the study of signals structure at the presence of noise are illuctrated for digital single-channel experimental data of prospecting seismology.

  1. Bendat J, Piersol A. Random Data: Analysis and Measurement Procedures. Wiley; 1971. 640 p.
  2. Press WH, Teukokolsky SA, Vetterling WT, Flanney BP. Numerical Recipes in C: The Art of Scientific Computing. Cambridge: Cambridge University Press; 1992. 994 p.
  3. Dremin IM, Ivanov OV, Nechitailo VA. Wavelets and their uses. Phys. Usp. 2001;44(5):447–478. DOI: 10.1070/PU2001v044n05ABEH000918.
  4. Addison PS. The Illustrated Wavelet Transform Handbook: Applications in Science, Engineering, Medicine and Finance. Philadelphia: IOP Publishing; 2002. 472 p.
  5. Mallat SG. A Wavelet Tour of Signal Processing. New York: Academic Press; 1998. 805 p.
  6. Grossman A, Morlet J. Decomposition of Hardy functions into square integrable wavelets of constant shape. SIAM J. Math. Anal. 1984;15(4):723–736. DOI: 10.1137/0515056.
  7. Meyer Y. Wavelets: Algorithms and Applications. Philadelphia: S.I.A.M.; 1993. 133 p.
  8. Meyer Y. Wavelets and Operators. Cambridge: Cambridge University Press; 1993. 223 p. DOI: 10.1017/CBO9780511623820.
  9. Daubechies I. Ten Lectures on Wavelets. Philadelphia: S.I.A.M.; 1992. 350 p. DOI: 10.1137/1.9781611970104.
  10. Torrence C, Compo GP. A practical guide to wavelet analysis. Bull. Amer. Meteor. Soc. 1998;79(1):61–78. DOI: 10.1175/1520-0477(1998)079<0061:APGTWA>2.0.CO;2.
  11. Astaf’eva NM. Wavelet analysis: basic theory and some applications. Phys. Usp. 1996;39(11):1085–1108. DOI: 10.1070/PU1996v039n11ABEH000177.
  12. Van den Berg JC, editor. Wavelets in Physics. Cambridge: Cambridge University Press; 1993. 453 p. DOI: 10.1017/CBO9780511613265.
  13. Vetterli M, Kovacevic J. Wavelets and Subband Coding. NJ: Prentice Hall; 1995. 488 p.
  14. Foufoula-Georgiou E, Kumar P, editors. Wavelets in Geophysics. Academic Press; 1994. 373 p.
  15. Stolnitz E, DeRose T, Salesin D. Wavelets for Computer Graphics. Theory and Applications. Morgan Kaufmann; 1996. 245 p.
  16. Koronovskii AA, Khramov A.E. Continuous wavelet analysis and its applications. Moscow: Fizmatlit; 2003. 176 p. (in Russian).
  17. Sitnikova E, Hramov AE, Koronovskii AA, Luijtelaar EL. Sleep spindles and spike-wave discharges in EEG: Their generic features, similarities and distinctions disclosed with Fourier transform and continuous wavelet analysis. Journal of Neuroscience Methods. 2009;180(2):304–316. DOI: 10.1016/j.jneumeth.2009.04.006.
  18. Hramov AE, Koronovskii AA, Ponomarenko VI, Prokhorov MD. Detection of synchronization from univariate data using wavelet transform. Phys. Rev. E. 2007;75(5):056207. DOI: 10.1103/physreve.75.056207.
  19. Pavlov AN, Makarov VA, Mosekilde E, Sosnovtseva OV. Application of wavelet-based tools to study the dynamics of biological processes. Briefings in Bioinformatics. 2006;7(4):375–389. DOI: 10.1093/bib/bbl041.
  20. Sosnovtseva OV, Pavlov AN, Mosekilde E, Yip KP, Holstein–Rathlou NH, Marsh DJ. Synchronization among mechanisms of renal autoregulation is reduced in hypertensive rats. Am. J. Physiol. Renal Physiol. 2007;293(5):F1545–F1555. DOI: 10.1152/ajprenal.00054.2007.
  21. Pavlov AN, Anishchenko VS. Multifractal analysis of complex signals. Phys. Usp. 2007;50(8):819–834. DOI: 10.1070/PU2007v050n08ABEH006116.
  22. Kumar P, Foufoula-Georgiou E. Wavelet analysis for geophysical applications. Reviews in Geophysics. 1997;35(4):385–412. DOI: 10.1029/97RG00427.
  23. Huang NE, Shen Z, Long SR, Wu MC, Shi HH, Zheng Q, Yen NC, Tung CC, Liu HH. The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis. Proc. R. Soc. Lond. A. 1998;454(1971):903–995. DOI: 10.1098/rspa.1998.0193.
  24. Huang NE, Shen Z, Long SR. A new view of nonlinear water waves: the Hilbert spectrum. Annu. Rev. Fluid Mech. 1999;31:417–457. DOI: 10.1146/annurev.fluid.31.1.417.
  25. Huang NE, Shen SP, editors. Hilbert–Huang Transform and Its Applications. Singapore: World Scientific; 2005. 324 p. DOI: 10.1142/5862.
  26. Flandrin P, Goncalves P. Empirical mode decompositions as data-driven wavelet-like expansion. Int. J. Wavelets Multiresolut. Inform. Process. 2004;2(4):477–496. DOI: 10.1142/S0219691304000561.
  27. Neto EP, Custaud MA, Cejka CJ, Abry P, Frutoso J, Gharib C, Flandrin P. Assessment of cardiovascular autonomic control by the empirical mode decomposition. Method. Inform. Med. 2004;43(1):60–65.
  28. Huang NE, Wu Z. A study of the characteristics of white noise using the empirical mode decomposition method. Proc. R. Soc. London Ser. A. 2004;460(2046):1597–1611. DOI: 10.1098/rspa.2003.1221.
  29. Gabor D. Theory of communication. J. Inst. Electr. Eng. London. 1946;93(3):429–457.
  30. Baskakov SI. Radio Circuits and Signals. Moscow: Vysshaya Shkola; 2005. 448 p.
  31. Ville J. Theorie et applications de la notion de signal analytique. Cables et Transm. 1948;2A(1):61–74.
  32. Wigner EP. On the quantum correction for thermodynamic equilibrium. Phys. Rev. 1932;40(5):749–759. DOI: 10.1103/PhysRev.40.749.
  33. Hramov AE, Koronovskii AA. Time scale synchronization of chaotic oscillators. Physica D. 2005;206(3–4):252–264. DOI: 10.1016/j.physd.2005.05.008.
  34. Hramov AE, Koronovskii AA. An approach to chaotic synchronization. Chaos. 2004;14(3):603–610. DOI: 10.1063/1.1775991.
  35. Ponomarenko VI, Prokhorov MD, et al. Diagnostics of the synchronization of self-oscillatory systems by an external force with varying frequency with the use of wavelet analysis. J. Commun. Technol. Electron. 2007;52(5):544–554 (2007). DOI: 10.1134/S1064226907050087.
  36. Hramov AE, Koronovskii AA, Popov PV, Rempen IS. Chaotic synchronization of coupled electron-wave systems with backward waves. Chaos. 2005;15(1):013705. DOI: 10.1063/1.1857615.
  37. Koronovskiy AA, Khramov AE. An introduction to continuous wavelet analysis for specialists in the field of nonlinear dynamics. Part 1. Fundamentals, numerical implementation and model signals. Izvestiya VUZ. Applied Nonlinear Dynamics. 2001;9(4–5):3 (in Russian).
  38. Koronovskii AA, Khramov AE. An introduction to continuous wavelet analysis for specialists in the field of nonlinear dynamics. Part 2. Paths to chaos from the point of view of wavelet analysis. Izvestiya VUZ. Applied Nonlinear Dynamics. 2002;10(1–2):3 (in Russian).
  39. Pavlov AN. Wavelet-­analysis and examples of it's applications. Izvestiya VUZ. Applied Nonlinear Dynamics. 2009;17(5):99–111 (in Russian). DOI: 10.18500/0869-6632-2009-17-5-99-111.
  40. Pavlov AN, Filatova AE. Method of empirical modes and wavelet­filtering: application in geophysical problems. Izvestiya VUZ. Applied Nonlinear Dynamics. 2011;19(1):3–13 (in Russian). DOI: 10.18500/0869-6632-2011-19-1-3-13.
  41. Hramov AE, Koronovskii AA, Midzyanovskaya IS, Sitnikova E, Rijn CM. On-off intermittency in time series of spontaneous paroxysmal activity in rats with genetic absence epilepsy. Chaos. 2006;16(4):043111. DOI: 10.1063/1.2360505.
  42. van Luijtelaar G, Hramov AE, Sitnikova E, Koronovskii AA. Spike-wave discharges in WAG/Rij rats are preceded by delta and theta precursor activity in cortex and thalamus. Clin. Neurophysiol. 2011;122(4):687–695. DOI: 10.1016/j.clinph.2010.10.038.
  43. Ovchinnikov A, Luttjohanna A, Hramov A, van Luijtelaar G. An algorithm for real-time detection of spike-wave discharges in rodents. J. Neurosci. Methods. 2010;194(1):172–178. DOI: 10.1016/j.jneumeth.2010.09.017.
  44. Sitnikova EY, Hramov AE, Koronovskii AA, van Luijtelaar G. Sleep spindles and spike-wave discharges in EEG: Their generic features, similarities and distinctions disclosed with Fourier transform and continuous wavelet analysis. J. Neurosci. Methods. 2009;180(2):304–316. DOI: 10.1016/j.jneumeth.2009.04.006.
  45. Lee TW. Independent Component Analysis: Theory and Applications. Boston: Kluwer Academic Publishers; 1998. 210 p. DOI: 10.1007/978-1-4757-2851-4.
  46. Yilmas O. Seismic Data Analysis. Vol. I, II. Tulsa: Society of Exploration Geo-physicists; 2001.
  47. Gurvich II, Boganik GN. Seismic Exploration. Moscow: Nedra; 1980. 551 p. (in Russian).
  48. Filatova AE, Artemev AE, Koronovskii AA, Pavlov AN, Hramov AE. Progress and prospect of wavelet transform application to the analysis of nonstationary nonlinear dates in contemporary geophysics. Izvestiya VUZ. Applied Nonlinear Dynamics. 2010;18(3):3–23 (in Russian). DOI: 10.18500/0869-6632-2010-18-3-3-23.
  49. Filatova AE, Ovchinnikov AA, Koronovskiy AA, Khramov AE. Application of wavelet transform for diagnostics of noise-waves of sound and surface types according to digital data of ground seismic prospecting. Vestnik TSU. 2010;15(2):561–565 (in Russian).  
Short text (in English):
(downloads: 59)