1. 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

2. outline Empirical Mode Decomposition (EMD) basics examples
algorithmic issues
elements of performance evaluation
perspectives

3. 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

4. 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

5. how EMD works

82. EMD and AM-FM signals quasi-monochromatic harmonic oscillations
self-adaptive time-variant filtering
example : 2 sinus FM + 1 Gaussian wave packet

83. EMD

84. time-frequency signature

85. time-frequency signature

86. time-frequency signature

87. time-frequency signature

88. nonlinear oscillations
IMF ≠ Fourier mode and, in nonlinear situations, 1 IMF = many Fourier modes
example : 1 HF triangle + 1 MF tone + 1 LF triangle

89. EMD

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

91. algorithmic issues interpolation stopping criteria
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 »

92. 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

93. EMD of fractional Gaussian noise

94. 1. EMD and (tone) sampling

95. case 1 — oversampling

96. equal height maxima

97. constant upper envelope

98. equal height minima

99. constant lower envelope

100. zero mean : tone = IMF

101. case 2 — moderate sampling

102. fluctuating maxima

103. modulated upper envelope

104. fluctuating minima

105. modulated lower envelope

106. non zero mean : tone ≠ IMF

107. 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

108. 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}

109. experiment 3
intertwining of amplitude ratio, sampling rate and frequency spacing
dominant effect when f1 ≤ 1/4 : constant-Q (wavelet-like) « confusion band »

110. 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?

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