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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









Zero mean tone imf
zero mean : tone = IMF







Non zero mean tone imf
non zero mean : tone ≠ IMF


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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