Empirical Mode Decomposition
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  • 1. Empirical Mode Decompositionand Hilbert-Huang Transform
    Emine Can 2010
  • 2. nonlinear process
    Data Analysis
    Fourier Spectral Analysis
    Data Processing Methods
    • Spectrogram
    • 3. Wavelet Analysis
    • 4. Wigner-Ville Distribution (Heisenberg wavelet)
    • 5. Evolutionary spectrum
    • 6. EmpiricalOrthogonalFunction Expansion (EOF)
    • 7. Other methods
    Energy-frequency distributions =Spectrum≈Fourier Transform of the data
    Restrictions: * the system must be linear
    * the data must be strictly periodic or stationary
    10.2010
    2
    Empirical Mode Decomposition and Hilbert-Huang Transform
    Modifications of Fourier SA
  • 8. Hilbert
    Transform
    Instantaneous
    Frequency
    10.2010
    3
    Empirical Mode Decomposition and Hilbert-Huang Transform
    Empirical Mode Decomposition
    Complicated
    Data Set
    Intrinsic Mode Functions
    (Energy-Frequency-Time)
  • 9. A method that any complicated data set can be decomposedinto
    a finiteand oftensmallnumber of `intrinsicmode functions' that
    admitwell-behaved HilbertTransforms.
    10.2010
    4
    Empirical Mode Decomposition and Hilbert-Huang Transform
    Emperical Mode Decomposition (EMD)
    Intrinsic Mode Functions(IMF)
    IMF is a function that satisfies two conditions:
    1- In the whole data set, the number of extrema and the number of zero crossings musteither equal or differ at most by one
    2-At any point, the mean value of theenvelope defined by the local maxima and the envelope defined by the local
    minima is zero
  • 10. The empirical mode decomposition method: the sifting process
    10.2010
    Empirical Mode Decomposition and Hilbert-Huang Transform
    5
  • 11. 10.2010
    Empirical Mode Decomposition and Hilbert-Huang Transform
    6
  • 12. The sifting process
    Complicated Data Set x(t)
  • 13. The sifting process
    1. identify all upperextrema of x(t).
  • 14. The sifting process
    2. Interpolate the local maxima to form an upper envelope u(x).
  • 15. The sifting process
    3. identify all lowerextrema of x(t).
  • 16. The sifting process
    4. Interpolate the local minima to form an lower envelope l(x).
  • 17. The sifting process
    5. Calculate the mean envelope: m(t)=[u(x)+l(x)]/2.
  • 18. The sifting process
    6. Extract the mean from the signal: h(t)=x(t)-m(t)
  • 19. The sifting process
    7. Check whether h(t) satisfies the IMF condition.
    YES: h(t) is an IMF, stop sifting. NO: let x(t)=h(t), keep sifting.
  • 20. The sifting process
  • 21. The sifting process
  • 22. The sifting process
  • 23. The sifting process
  • 24. The sifting process
  • 25. The sifting process
  • 26. 10.2010
    Empirical Mode Decomposition and Hilbert-Huang Transform
    21
  • 27. 10.2010
    Empirical Mode Decomposition and Hilbert-Huang Transform
    22
    The signal is composed of
    a “high frequency” triangular waveform whose amplitude is slowly (linearly) growing.
    a “middle frequency”sine wave whose amplitude is quickly (linearly) decaying
    a “low frequency” triangular waveform
  • 28. 10.2010
    Empirical Mode Decomposition and Hilbert-Huang Transform
    23
    The sifting process
    Stop criterion
    A criterionfor the sifting process to stop: Standard deviation, SD, computed from the two consecutive sifting results is in limited size.
    :residue after the kth iteration of the 1st IMF
    A typical value for SD can be set between 0.2 and 0.3.
  • 29. 10.2010
    Empirical Mode Decomposition and Hilbert-Huang Transform
    24
    Hilbert Transform
    *
    Analytic Signal:
    Instantaneous Frequency:
  • 30. Advantages
    *Adaptive,highly efficient,applicable to
    nonlinear and non-stationary processes.
    10.2010
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    Empirical Mode Decomposition and Hilbert-Huang Transform
  • 31. Applications of EMD
    10.2010
    Empirical Mode Decomposition and Hilbert-Huang Transform
    26
    nonlinear wave evolution,
    climate cycles,
    earthquake engineering,
    submarine design,
    structural damage detection,
    satellite data analysis,
    turbulence flow,
    blood pressure variations and heart arrhythmia,
    non-destructive testing,
    structural health monitoring,
    signal enhancement,
    economic data analysis,
    investigation of brain rythms
    Denoising

  • 32. References
    “The empirical mode decomposition and theHilbert spectrum for nonlinear and non-stationary time series analysis”Huanget al., The Royal Society, 4 November 1996.
    Rilling Gabriel, FlandrinPatrick ,Gon¸calv`es Paulo, “On Empirical Mode Decomposition and Its Algorithms”
    Stephen McLaughlin and YannisKopsinis.ppt “Empirical Mode Decomposition:A novel algorithm for analyzingmulticomponent signals” Institute of Digital Communications (IDCOM)
    “Hilbert-Huang Transform(HHT).ppt” Yu-HaoChen, ID:R98943021, 2010/05/07
    10.2010
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    Empirical Mode Decomposition and Hilbert-Huang Transform