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Empirical Mode Decomposition: a Filter Bank Viewpoint

Explore the theory and algorithms of Empirical Mode Decomposition in signal processing, offering a data-driven wavelet-like expansion for signal analysis. Obtain AM-FM type representation from signal observations using EMD.

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Empirical Mode Decomposition: a Filter Bank Viewpoint

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  1. Empirical Mode Decomposition:a Filter Bank Viewpoint Paulo Gonçalves INRIA Rhône-Alpes, France On leave @ IST – ISR (2003-2005) Joint work with: Patrick Flandrin (CNRS, ENS-Lyon, France) Gabriel Rilling (PhD, ENS-Lyon, France) ― Jean-Claude Nunes (Post-doc, Univ. Tech. Compiegnes) IST-ISR February 2005

  2. Empirical Mode DecompositionRelated papers Related papers • Empirical Mode Decompositions as data-driven wavelet-like expansions, P. F. and P. G. • Int. J. of Wavelets, Multiresolution and Information Processing, Vol. 2(4), pp. 477--496, 2004. • Empirical Mode Decomposition as a Filter Bank, P. F., G. R. and P. G. • IEEE Sig. Proc. Letters, Vol. 11(2), pp. 112--114, 2004. • EMD Equivalent Filter Banks, from Interpretation to Applications, P. F., P. G. and G. R. • Hilbert-Huang Transform: Introduction and Applications (N.E. Huang and S.S.P. Shen, eds.), World Scientific, 2004. To appear. • Empirical Mode Decomposition, Fractional Gaussian Noise and Hurst Exponent Estimation, G. R., P. F. and P. G. • IEEE-ICASSP, March 19-23, Philadelphia, USA 2005. • Detrending and denoising with Empirical Mode Decompositions, P. F., P. G. and G. R. • Eusipco, 12th European Signal Processing Conference, Vienna, Austria 2004. • Sur la décomposition modale empirique, P. F. and P. G. • GRETSI, Paris, France, 2003. • On empirical mode decomposition and its algorithms, G. R., P. F. and P. G. • IEEE-EURASIP Workshop on Nonlinear Signal and Image Processing NSIP-03, Grado (I), 2003. • www.ens-lyon.fr/~flandrin/emd.html

  3. EMD : principle & algorithmic definition Equivalent filter bank : a stochastic approach Equivalent filter bank : a deterministic approach Scaling exponent estimation Denoising-Detrending signal + noise mixture Open issues Outline

  4. Objective— From one observation of x(t), get a AM-FM type representation : K x(t) = Σ ak(t) Ψk(t)k=1 with ak(.) amplitude modulating functions and Ψk(.) oscillating functions. Idea— “signal = fast oscillations superimposed to slow oscillations”. Operating mode —(“EMD”, Huang et al., ’98) (1) identify locally in time, the fastest oscillation ; (2) subtract it from the original signal ; (3) iterate upon the residual. Empirical Mode DecompositionPrinciple

  5. Empirical Mode DecompositionAlgorithmic definition A LF sawtooth + A linear FM =

  6. Empirical Mode DecompositionAlgorithmic definition S I F T I N G P R O C E S S

  7. Empirical Mode DecompositionAlgorithmic definition S I F T I N G P R O C E S S

  8. Empirical Mode DecompositionAlgorithmic definition S I F T I N G P R O C E S S

  9. Empirical Mode DecompositionAlgorithmic definition S I F T I N G P R O C E S S

  10. Empirical Mode DecompositionAlgorithmic definition S I F T I N G P R O C E S S

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  20. Empirical Mode DecompositionAlgorithmic definition S I F T I N G P R O C E S S

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  22. Empirical Mode DecompositionAlgorithmic definition S I F T I N G P R O C E S S

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  27. Empirical Mode DecompositionAlgorithmic definition S I F T I N G P R O C E S S

  28. Empirical Mode DecompositionAlgorithmic definition S I F T I N G P R O C E S S

  29. Empirical Mode DecompositionAlgorithmic definition S I F T I N G P R O C E S S

  30. Empirical Mode DecompositionAlgorithmic definition S I F T I N G P R O C E S S

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  34. Empirical Mode DecompositionAlgorithmic definition S I F T I N G P R O C E S S

  35. Empirical Mode DecompositionAlgorithmic definition S I F T I N G P R O C E S S

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  38. Empirical Mode DecompositionAlgorithmic definition

  39. Empirical Mode DecompositionIntrinsic Mode Functions • Quasi monochromatic harmonic oscillations • #{zero crossing} = #{extrema} ± 1 • symmetric envelopes around the y=0 axis • IMF ≠ Fourier mode and, in nonlinear situations, IMF = several Fourier modes • Output of a self-adaptive time-varying filter (≠ standard linear filter)ex: 2 sinus FM + gaussian wave packet

  40. Empirical Mode DecompositionIntrinsic Mode Functions Signal time Spectrum frequency Time-Frequency representation

  41. Empirical Mode DecompositionIntrinsic Mode Functions

  42. Empirical Mode DecompositionIntrinsic Mode Functions frequency time Signal

  43. Empirical Mode DecompositionIntrinsic Mode Functions frequency time Signal 1st IMF 2nd IMF 3rd IMF

  44. Empirical Mode DecompositionIntrinsic Mode Functions t 1st IMF 1 2 3 3rd IMF 2nd IMF

  45. Empirical Mode DecompositionA mathematical approach (V. Vatchev, 2002) Definition― A twice differentiable function f is an IMF if it is a solution of a self-adjoint ODE of the type: (Pf’)’ + Qf = 0for some P(t) > 0 and Q(t) > 0 for t є [a,b] Ensures that #{zero crossing) = #{extrema} ± 1 AND Q(t) = 1 / P(t) Ensures that upper and Lower envelopes U(t) and L(t) are symmetric

  46. Empirical Mode DecompositionA mathematical approach (V. Vatchev, 2002) • Theorem―If f is solution of a self-adjoint ODE, with Q(t) = 1 / P(t) • then, f is an oscillating function with constant amplitude ! • ________ • In practice, to overcome the envelope restriction: • require |U(t) − L(t)| < ε, for some prescribed ε > 0 • modify the construction of the envelopes (???)

  47. Empirical Mode DecompositionEquivalent Filter bank: a stochastic approach • Goal― Consider EMD as a filter bank and identify for each IMF an “equivalent” frequency response • Model―fractional Gaussian noise (fGn – derivative of fractional Brownian motion fBm) with Hurst exponent0 < H < 1 : • H = 1/2 : white gaussian noise • H < 1/2 : short range correlation • H > 1/2 : long range correlation • Power spectral density (PSD) ~ f 1-2H

  48. Empirical Mode DecompositionEquivalent Filter bank: a stochastic approach • Goal― Consider EMD as a filter bank and identify for each IMF an “equivalent” frequency response • Model―fractional Gaussian noise (fGn – derivative of fractional Brownian motion fBm) with Hurst exponent0 < H < 1 : • H = 1/2 : white gaussian noise • H < 1/2 : short range correlation • H > 1/2 : long range correlation • Power spectral density (PSD) ~ f 1-2H

  49. Empirical Mode DecompositionEquivalent Filter bank: a stochastic approach • Goal― Consider EMD as a filter bank and identify for each IMF an “equivalent” frequency response • Model―fractional Gaussian noise (fGn – derivative of fractional Brownian motion fBm) with Hurst exponent0 < H < 1 : • H = 1/2 : white gaussian noise • H < 1/2 : short range correlation • H > 1/2 : long range correlation • Power spectral density (PSD) ~ f 1-2H

  50. Empirical Mode DecompositionEquivalent Filter bank: a stochastic approach • Goal― Consider EMD as a filter bank and identify for each IMF an “equivalent” frequency response • Model―fractional Gaussian noise (fGn – derivative of fractional Brownian motion fBm) with Hurst exponent0 < H < 1 : • H = 1/2 : white gaussian noise • H < 1/2 : short range correlation • H > 1/2 : long range correlation • Power spectral density (PSD) ~ f 1-2H

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