On Empirical Mode Decomposition and its Algorithms

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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)

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outline

• Empirical Mode Decomposition (EMD) basics
• examples
• algorithmic issues
• elements of performance evaluation
• perspectives

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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

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Huangs 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

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how EMD works
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EMD and AM-FM signals

• quasi-monochromatic harmonic oscillations
• self-adaptive time-variant filtering
• example 2 sinus FM 1 Gaussian wave packet

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EMD
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time-frequency signature
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time-frequency signature
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time-frequency signature
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time-frequency signature
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nonlinear oscillations

• IMF ? Fourier mode and, in nonlinear situations,
  1 IMF many Fourier modes
• example 1 HF triangle 1 MF tone 1 LF
  triangle

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EMD
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issues

• algorithm ?
• intuitive but ad-hoc procedure, not unique
• several user-controlled tunings
• performance ?
• difficult evaluation since no analytical
  definition
• numerical simulations

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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 

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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

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EMD of fractional Gaussian noise
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1. EMD and (tone) sampling
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case 1 oversampling
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equal height maxima
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constant upper envelope
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equal height minima
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constant lower envelope
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zero mean tone IMF
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case 2 moderate sampling
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fluctuating maxima
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modulated upper envelope
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fluctuating minima
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modulated lower envelope
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non zero mean tone ? IMF
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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

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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. maxIMF k, k
  2

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experiment 3

• intertwining of amplitude ratio, sampling rate
  and frequency spacing
• dominant effect when f1 1/4 constant-Q
  (wavelet-like)  confusion band 

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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?

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