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On Empirical Mode Decomposition and its Algorithms. G. Rilling, P. Flandrin (Cnrs - Éns Lyon, France) P. Gonçalvès (Inria Rhône-Alpes, France). outline. Empirical Mode Decomposition (EMD) basics examples algorithmic issues elements of performance evaluation perspectives. basic idea.

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on empirical mode decomposition and its algorithms

On Empirical Mode Decomposition and its Algorithms

G. Rilling, P. Flandrin (Cnrs - Éns Lyon, France)

P. Gonçalvès (Inria Rhône-Alpes, France)

IEEE-EURASIP Workshop on Nonlinear Signal and Image Processing, Grado (I), June 9-11, 2003

outline
outline
  • Empirical Mode Decomposition (EMD) basics
  • examples
  • algorithmic issues
  • elements of performance evaluation
  • perspectives
basic idea
basic idea
  • « multimodal signal = fast oscillations on the top of slower oscillations »
  • Empirical Mode Decomposition (Huang)
    • identify locally the fastest oscillation
    • substract to the signal and iterate on the residual
    • data-driven method, locally adaptive and multiscale
huang s algorithm
Huang’s algorithm
  • compute lower and upper envelopes from interpolations between extrema
    • substract mean envelope from signal
    • iterate until mean envelope = 0 and #{extrema} = #{zero-crossings} ± 1
  • substract the obtained Intrinsic Mode Function (IMF) from signal and iterate on residual
emd and am fm signals
EMD and AM-FM signals
  • quasi-monochromatic harmonic oscillations
  • self-adaptive time-variant filtering
  • example : 2 sinus FM + 1 Gaussian wave packet
nonlinear oscillations
nonlinear oscillations
  • IMF ≠ Fourier mode and, in nonlinear situations, 1 IMF = many Fourier modes
  • example : 1 HF triangle + 1 MF tone + 1 LF triangle
issues
issues
  • algorithm ?
    • intuitive but ad-hoc procedure, not unique
    • several user-controlled tunings
  • performance ?
    • difficult evaluation since no analytical definition
    • numerical simulations
algorithmic issues
algorithmic issues
  • interpolation
    • type ? cubic splines
    • border effects ? mirror symmetry
  • stopping criteria
    • mean zero ? 2 thresholds
    • variation 1 : « local EMD »
  • computational burden
    • about log2 N IMF ’s for N data points
    • variation 2 : « on-line EMD »
performance evaluation
performance evaluation
  • extensive numerical simulations
  • deterministic framework
    • importance of sampling
    • ability to resolve multicomponent signals
  • a complement to stochastic studies
    • noisy signals + fractional Gaussian noise
    • PF et al., IEEE Sig. Proc. Lett., to appear
experiment 1
experiment 1
  • 256 points tone, with 0 ≤ f ≤1/2
  • error = normalized L2 distance comparing tone vs. IMF #1
  • minimum when 1/f even multiple of the sampling period
experiment 2
experiment 2
  • 256 points tones, with 0 ≤ f1 ≤1/2 and f2 ≤ f1
  • error = weighted normalized L2 distance comparing tone #1 vs. IMF #1, and tone #2 vs. max{IMF #k, k ≥ 2}
experiment 3
experiment 3
  • intertwining of amplitude ratio, sampling rate and frequency spacing
  • dominant effect when f1 ≤ 1/4 : constant-Q (wavelet-like) « confusion band »
concluding remarks
concluding remarks
  • EMD is an appealing data-driven and multiscale technique
  • spontaneous dyadic filterbank structure in « stationary » situations, stochastic (fGn) or not (tones)
  • EMD defined as output of an algorithm: theoretical framework beyond numerical simulations?
p reprints matlab codes and demos www ens lyon fr flandrin emd html

(p)reprints, Matlab codes and demoswww.ens-lyon.fr/~flandrin/emd.html

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