EMPIRICAL MODE DECOMPOSITION AN INTRODUCTION
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Transcript of EMPIRICAL MODE DECOMPOSITION AN INTRODUCTION
1
AN INTRODUCTION TO EMPIRICAL MODE DECOMPOSITION
By Harikrishna satish.T
M.E. Electrical engg,Jadavpur university.
2
Empirical mode decomposition
Method for processing non stationary signals and signals produced by nonlinear processes and Decomposition of the signal into a set of Intrinsic Mode Functions (IMF) which are defined as
1. In the whole set of data, the numbers of local extrema
and the numbers of zero crossings must be equal or differ by 1 at most.
2. At any time point, the mean value of the “upper envelope” (defined by the local Maxima) and the “lower envelope” (defined by the local minima) must be zero.
3
The simplest model for a signal is given by circular functions of the type
Such “Fourier modes” are of particular interest in the case of stationary signals and linear systems.
(Type I)
Why Empirical Mode Decomposition?
4
we can think of representing these signals in terms of amplitude and frequency modulated (AM–FM) components
However, many physical situations are known to undergo non stationary and/or nonlinear behaviors.
The rationale for such a modeling is to compactly encode possible non stationarities in a time variation of the amplitudes and frequencies of Fourier-like modes.
(Type II)
5
More generally, signals may also be generated by nonlinear systems for which oscillations are not necessarily associated with circular functions, thus suggesting decompositions of the following form
(Type III)
Each of the components has to have physical and mathematical meaning.
6
Empirical Mode Decomposition (EMD) is designed primarily for obtaining representations of Type II or Type III in the case of signals which are oscillatory, possibly non stationary or generated by a nonlinear system, in some automatic, fully data-driven way.
7
Main differences between EMD and traditional data analysis methods
Stationary Non-stationary Linear Non-linear Theory
Fourier
Wavelets
Time-series
EMD
Traditional methods EMD
- Not appropriate for nonlinear & non stationary signals.
- Predefined basis and/or system model.
- Distorted information extracted.
- Full theoretical basis.
- Adequate for both nonlinear & non stationary.
- Adaptive – data driven basis.
- Preserves physical meaning.
- Sharper spectrum
- Lack of theoretical analysis.
8
Basic Parts of the Empirical Mode Decomposition
• Interpolation technique (cubic spline).
• Sifting process to extract and identify intrinsic modes.
• Numerical convergence criteria (mainly to stop the iterative process of identifying every IMF as well as the whole set of IMFs)
9
The starting point of EMD is to consider oscillatory signals at the level of their local oscillations and to formalize the idea that:
“signal = fast oscillations superimposed to slow oscillations”
10
“signal = fast oscillations superimposed to slow oscillations”
Iterate on the slow oscillations component considered as a new signal.
11
Decomposition of the signal:-
Thus , we can say that the original signal is the combination of all the EMF’s decomposed and the residue.
12
• Decomposition:
• Mode: Intrinsic Mode Functions (IMF’s) -Represents the oscillation modes embedded in the data.
• Empirical: The Sifting process is essentially defined by an algorithm.EMD lacks theoretical foundations.
Empirical Mode Decomposition
13
1. identify all extrema of x(t).
2. Interpolate the local maxima to form an upper envelope u(x).
3. Interpolate the local minima to form an lower envelope l(x).
4. Calculate the mean envelope:m(t)=[u(x)+l(x)]/2.
5. Extract the mean from the signal:h(t)=x(t)-m(t)
6. Check whether h(t) satisfies the IMF condition. YES: h(t) is an IMF, stop sifting. NO: let x(t)=h(t), keep sifting.
How to find one Intrinsic Mode Functions of a signal?Sifting procedure
14
10 20 30 40 50 60 70 80 90 100 110 120
-2
-1
0
1
2
IMF 1; iteration 0Residue = s(t)
I1(t) = Residue
i = 1
k = 1
while Residue not equal zero or not
monotone
while Ii has non-negligible local
mean
U(t) = spline through local
maxima of IiL(t) = spline through local
minima of IiAv(t) = 1/2 (U(t) + L(t))
Ii(t) = Ii(t) - Av(t)
i = i + 1
end
IMFk(t) = Ii(t)
Residue = Residue - IMFk
k = k+1
end
Huang’s “Sifting Process”.
15
10 20 30 40 50 60 70 80 90 100 110 120
-2
-1
0
1
2
IMF 1; iteration 0Residue = s(t)
I1(t) = Residue
i = 1
k = 1
while Residue not equal zero or not
monotone
while Ii has non-negligible local
mean
U(t) = spline through local
maxima of Ii
L(t) = spline through local
minima of IiAv(t) = 1/2 (U(t) + L(t))
Ii(t) = Ii(t) - Av(t)
i = i + 1
end
IMFk(t) = Ii(t)
Residue = Residue - IMFk
k = k+1
end
Huang’s “Sifting Process”.
16
10 20 30 40 50 60 70 80 90 100 110 120
-2
-1
0
1
2
IMF 1; iteration 0Residue = s(t)
I1(t) = Residue
i = 1
k = 1
while Residue not equal zero or not
monotone
while Ii has non-negligible local
mean
U(t) = spline through local
maxima of Ii
L(t) = spline through local
minima of IiAv(t) = 1/2 (U(t) + L(t))
Ii(t) = Ii(t) - Av(t)
i = i + 1
end
IMFk(t) = Ii(t)
Residue = Residue - IMFk
k = k+1
end
Huang’s “Sifting Process”.
17
10 20 30 40 50 60 70 80 90 100 110 120
-2
-1
0
1
2
IMF 1; iteration 0Residue = s(t)
I1(t) = Residue
i = 1
k = 1
while Residue not equal zero or not
monotone
while Ii has non-negligible local
mean
U(t) = spline through local
maxima of IiL(t) = spline through local
minima of Ii
Av(t) = 1/2 (U(t) + L(t))
Ii(t) = Ii(t) - Av(t)
i = i + 1
end
IMFk(t) = Ii(t)
Residue = Residue - IMFk
k = k+1
end
Huang’s “Sifting Process”.
18
10 20 30 40 50 60 70 80 90 100 110 120
-2
-1
0
1
2
IMF 1; iteration 0Residue = s(t)
I1(t) = Residue
i = 1
k = 1
while Residue not equal zero or not
monotone
while Ii has non-negligible local
mean
U(t) = spline through local
maxima of IiL(t) = spline through local
minima of Ii
Av(t) = 1/2 (U(t) + L(t))
Ii(t) = Ii(t) - Av(t)
i = i + 1
end
IMFk(t) = Ii(t)
Residue = Residue - IMFk
k = k+1
end
Huang’s “Sifting Process”.
19
10 20 30 40 50 60 70 80 90 100 110 120
-2
-1
0
1
2
IMF 1; iteration 0Residue = s(t)
I1(t) = Residue
i = 1
k = 1
while Residue not equal zero or
not monotone
while Ii has non-negligible
local mean
U(t) = spline through local
maxima of IiL(t) = spline through local
minima of IiAv(t) = 1/2 (U(t) + L(t))
Ii(t) = Ii(t) - Av(t)
i = i + 1
end
IMFk(t) = Ii(t)
Residue = Residue - IMFk
k = k+1
end
Huang’s “Sifting Process”.
20
10 20 30 40 50 60 70 80 90 100 110 120
-2
-1
0
1
2
IMF 1; iteration 0
10 20 30 40 50 60 70 80 90 100 110 120
-1.5
-1
-0.5
0
0.5
1
1.5
residue
Residue = s(t)
I1(t) = Residue
i = 1
k = 1
while Residue not equal zero or
not monotone
while Ii has non-negligible local
mean
U(t) = spline through local
maxima of IiL(t) = spline through local
minima of IiAv(t) = 1/2 (U(t) + L(t))
Ii(t) = Ii(t) - Av(t)
(“residue”-->)
i = i + 1
end
IMFk(t) = Ii(t)
Residue = Residue - IMFk
k = k+1
end
Huang’s “Sifting Process”.
21
10 20 30 40 50 60 70 80 90 100 110 120
-1.5
-1
-0.5
0
0.5
1
1.5
IMF 1; iteration 1
10 20 30 40 50 60 70 80 90 100 110 120
-1.5
-1
-0.5
0
0.5
1
1.5
residue
Residue = s(t)
I1(t) = Residue
i = 1
k = 1
while Residue not equal zero or
not monotone
while Ii has non-negligible
local mean
U(t) = spline through local
maxima of IiL(t) = spline through local
minima of IiAv(t) = 1/2 (U(t) + L(t))
Ii(t) = Ii(t) - Av(t)
i = i + 1
end
IMFk(t) = Ii(t)
Residue = Residue - IMFk
k = k+1
end
Huang’s “Sifting Process”.
22
10 20 30 40 50 60 70 80 90 100 110 120
-1.5
-1
-0.5
0
0.5
1
1.5
IMF 1; iteration 1
10 20 30 40 50 60 70 80 90 100 110 120
-1.5
-1
-0.5
0
0.5
1
1.5
residue
Residue = s(t)
I1(t) = Residue
i = 1
k = 1
while Residue not equal zero or not
monotone
while Ii has non-negligible local
mean
U(t) = spline through local
maxima of Ii
L(t) = spline through local
minima of IiAv(t) = 1/2 (U(t) + L(t))
Ii(t) = Ii(t) - Av(t)
i = i + 1
end
IMFk(t) = Ii(t)
Residue = Residue - IMFk
k = k+1
end
Huang’s “Sifting Process”.
23
10 20 30 40 50 60 70 80 90 100 110 120
-1.5
-1
-0.5
0
0.5
1
1.5
IMF 1; iteration 1
10 20 30 40 50 60 70 80 90 100 110 120
-1.5
-1
-0.5
0
0.5
1
1.5
residue
Residue = s(t)
I1(t) = Residue
i = 1
k = 1
while Residue not equal zero or not
monotone
while Ii has non-negligible local
mean
U(t) = spline through local
maxima of Ii
L(t) = spline through local minima
of IiAv(t) = 1/2 (U(t) + L(t))
Ii(t) = Ii(t) - Av(t)
i = i + 1
end
IMFk(t) = Ii(t)
Residue = Residue - IMFk
k = k+1
end
Huang’s “Sifting Process”.
24
10 20 30 40 50 60 70 80 90 100 110 120
-1.5
-1
-0.5
0
0.5
1
1.5
IMF 1; iteration 1
10 20 30 40 50 60 70 80 90 100 110 120
-1.5
-1
-0.5
0
0.5
1
1.5
residue
Residue = s(t)
I1(t) = Residue
i = 1
k = 1
while Residue not equal zero or not
monotone
while Ii has non-negligible local
mean
U(t) = spline through local maxima
of IiL(t) = spline through local
minima of Ii
Av(t) = 1/2 (U(t) + L(t))
Ii(t) = Ii(t) - Av(t)
i = i + 1
end
IMFk(t) = Ii(t)
Residue = Residue - IMFk
k = k+1
end
Huang’s “Sifting Process”.
25
10 20 30 40 50 60 70 80 90 100 110 120
-1.5
-1
-0.5
0
0.5
1
1.5
IMF 1; iteration 1
10 20 30 40 50 60 70 80 90 100 110 120
-1.5
-1
-0.5
0
0.5
1
1.5
residue
Residue = s(t)
I1(t) = Residue
i = 1
k = 1
while Residue not equal zero or not
monotone
while Ii has non-negligible local
mean
U(t) = spline through local
maxima of IiL(t) = spline through local
minima of Ii
Av(t) = 1/2 (U(t) + L(t))
Ii(t) = Ii(t) - Av(t)
i = i + 1
end
IMFk(t) = Ii(t)
Residue = Residue - IMFk
k = k+1
end
Huang’s “Sifting Process”.
26
10 20 30 40 50 60 70 80 90 100 110 120
-1.5
-1
-0.5
0
0.5
1
1.5
IMF 1; iteration 1
10 20 30 40 50 60 70 80 90 100 110 120
-1.5
-1
-0.5
0
0.5
1
1.5
residue
Residue = s(t)
I1(t) = Residue
i = 1
k = 1
while Residue not equal zero or
not monotone
while Ii has non-negligible
local mean
U(t) = spline through local
maxima of IiL(t) = spline through local
minima of IiAv(t) = 1/2 (U(t) + L(t))
Ii(t) = Ii(t) - Av(t)
i = i + 1
end
IMFk(t) = Ii(t)
Residue = Residue - IMFk
k = k+1
end
Huang’s “Sifting Process”.
27
10 20 30 40 50 60 70 80 90 100 110 120
-1.5
-1
-0.5
0
0.5
1
1.5
IMF 1; iteration 1
10 20 30 40 50 60 70 80 90 100 110 120
-1.5
-1
-0.5
0
0.5
1
1.5
residue
Residue = s(t)
I1(t) = Residue
i = 1
k = 1
while Residue not equal zero or
not monotone
while Ii has non-negligible
local mean
U(t) = spline through local
maxima of IiL(t) = spline through local
minima of IiAv(t) = 1/2 (U(t) + L(t))
Ii(t) = Ii(t) - Av(t)
(“residue”-->)
i = i + 1
end
IMFk(t) = Ii(t)
Residue = Residue - IMFk
k = k+1
end
Huang’s “Sifting Process”.
28
10 20 30 40 50 60 70 80 90 100 110 120
-1.5
-1
-0.5
0
0.5
1
1.5
IMF 1; iteration 2
10 20 30 40 50 60 70 80 90 100 110 120
-1.5
-1
-0.5
0
0.5
1
1.5
residue
Residue = s(t)
I1(t) = Residue
i = 1
k = 1
while Residue not equal zero or
not monotone
while Ii has non-negligible
local mean
U(t) = spline through local
maxima of IiL(t) = spline through local
minima of IiAv(t) = 1/2 (U(t) + L(t))
Ii(t) = Ii(t) - Av(t)
i = i + 1
end
IMFk(t) = Ii(t)
Residue = Residue - IMFk
k = k+1
end
Huang’s “Sifting Process”.
29
10 20 30 40 50 60 70 80 90 100 110 120
-1.5
-1
-0.5
0
0.5
1
1.5
IMF 1; iteration 2
10 20 30 40 50 60 70 80 90 100 110 120
-1.5
-1
-0.5
0
0.5
1
1.5
residue
Residue = s(t)
I1(t) = Residue
i = 1
k = 1
while Residue not equal zero or not
monotone
while Ii has non-negligible local
mean
U(t) = spline through local
maxima of Ii
L(t) = spline through local minima of
IiAv(t) = 1/2 (U(t) + L(t))
Ii(t) = Ii(t) - Av(t)
i = i + 1
end
IMFk(t) = Ii(t)
Residue = Residue - IMFk
k = k+1
end
Huang’s “Sifting Process”.
30
10 20 30 40 50 60 70 80 90 100 110 120
-1.5
-1
-0.5
0
0.5
1
1.5
IMF 1; iteration 2
10 20 30 40 50 60 70 80 90 100 110 120
-1.5
-1
-0.5
0
0.5
1
1.5
residue
Residue = s(t)
I1(t) = Residue
i = 1
k = 1
while Residue not equal zero or not
monotone
while Ii has non-negligible local
mean
U(t) = spline through local
maxima of Ii
L(t) = spline through local
minima of IiAv(t) = 1/2 (U(t) + L(t))
Ii(t) = Ii(t) - Av(t)
i = i + 1
end
IMFk(t) = Ii(t)
Residue = Residue - IMFk
k = k+1
end
Huang’s “Sifting Process”.
31
10 20 30 40 50 60 70 80 90 100 110 120
-1.5
-1
-0.5
0
0.5
1
1.5
IMF 1; iteration 2
10 20 30 40 50 60 70 80 90 100 110 120
-1.5
-1
-0.5
0
0.5
1
1.5
residue
Residue = s(t)
I1(t) = Residue
i = 1
k = 1
while Residue not equal zero or
not monotone
while Ii has non-negligible local
mean
U(t) = spline through local
maxima of IiL(t) = spline through local
minima of Ii
Av(t) = 1/2 (U(t) + L(t))
Ii(t) = Ii(t) - Av(t)
i = i + 1
end
IMFk(t) = Ii(t)
Residue = Residue - IMFk
k = k+1
end
Huang’s “Sifting Process”.
32
10 20 30 40 50 60 70 80 90 100 110 120
-1.5
-1
-0.5
0
0.5
1
1.5
IMF 1; iteration 2
10 20 30 40 50 60 70 80 90 100 110 120
-1.5
-1
-0.5
0
0.5
1
1.5
residue
Residue = s(t)
I1(t) = Residue
i = 1
k = 1
while Residue not equal zero or not
monotone
while Ii has non-negligible local
mean
U(t) = spline through local
maxima of IiL(t) = spline through local
minima of Ii
Av(t) = 1/2 (U(t) + L(t))
Ii(t) = Ii(t) - Av(t)
i = i + 1
end
IMFk(t) = Ii(t)
Residue = Residue - IMFk
k = k+1
end
Huang’s “Sifting Process”.
33
10 20 30 40 50 60 70 80 90 100 110 120
-1.5
-1
-0.5
0
0.5
1
1.5
IMF 1; iteration 2
10 20 30 40 50 60 70 80 90 100 110 120
-1.5
-1
-0.5
0
0.5
1
1.5
residue
Residue = s(t)
I1(t) = Residue
i = 1
k = 1
while Residue not equal zero or
not monotone
while Ii has non-negligible
local mean
U(t) = spline through local
maxima of IiL(t) = spline through local
minima of IiAv(t) = 1/2 (U(t) + L(t))
Ii(t) = Ii(t) - Av(t)
i = i + 1
end
IMFk(t) = Ii(t)
Residue = Residue - IMFk
k = k+1
end
Huang’s “Sifting Process”.
34
10 20 30 40 50 60 70 80 90 100 110 120
-1.5
-1
-0.5
0
0.5
1
1.5
IMF 1; iteration 2
10 20 30 40 50 60 70 80 90 100 110 120
-1
-0.5
0
0.5
1
residue
Residue = s(t)
I1(t) = Residue
i = 1
k = 1
while Residue not equal zero or
not monotone
while Ii has non-negligible
local mean
U(t) = spline through local
maxima of IiL(t) = spline through local
minima of IiAv(t) = 1/2 (U(t) + L(t))
Ii(t) = Ii(t) - Av(t)
(“residue”-->)
i = i + 1
end
IMFk(t) = Ii(t)
Residue = Residue - IMFk
k = k+1
end
Huang’s “Sifting Process”.
35
10 20 30 40 50 60 70 80 90 100 110 120
-1
-0.5
0
0.5
1
IMF 1; iteration 3
10 20 30 40 50 60 70 80 90 100 110 120
-1
-0.5
0
0.5
1
residue
Residue = s(t)
I1(t) = Residue
i = 1
k = 1
while Residue not equal zero or not
monotone
while Ii has non-negligible local
mean
U(t) = spline through local
maxima of Ii
L(t) = spline through local
minima of Ii
Av(t) = 1/2 (U(t) + L(t))
Ii(t) = Ii(t) - Av(t)
i = i + 1
end
IMFk(t) = Ii(t)
Residue = Residue - IMFk
k = k+1
end
Huang’s “Sifting Process”.
36
10 20 30 40 50 60 70 80 90 100 110 120
-1
-0.5
0
0.5
1
IMF 1; iteration 4
10 20 30 40 50 60 70 80 90 100 110 120
-1
-0.5
0
0.5
1
residue
Residue = s(t)
I1(t) = Residue
i = 1
k = 1
while Residue not equal zero or not
monotone
while Ii has non-negligible local mean
U(t) = spline through local
maxima of Ii
L(t) = spline through local
minima of Ii
Av(t) = 1/2 (U(t) + L(t))
Ii(t) = Ii(t) - Av(t)
i = i + 1
end
IMFk(t) = Ii(t)
Residue = Residue - IMFk
k = k+1
end
Huang’s “Sifting Process”.
37
10 20 30 40 50 60 70 80 90 100 110 120
-1
-0.5
0
0.5
1
IMF 1; iteration 5
10 20 30 40 50 60 70 80 90 100 110 120
-1
-0.5
0
0.5
1
residue
Residue = s(t)
I1(t) = Residue
i = 1
k = 1
while Residue not equal zero or not
monotone
while Ii has non-negligible local
mean
U(t) = spline through local
maxima of Ii
L(t) = spline through local
minima of Ii
Av(t) = 1/2 (U(t) + L(t))
Ii(t) = Ii(t) - Av(t)
i = i + 1
end
IMFk(t) = Ii(t)
Residue = Residue - IMFk
k = k+1
end
Huang’s “Sifting Process”.
38
10 20 30 40 50 60 70 80 90 100 110 120
-1
-0.5
0
0.5
1
IMF 1; iteration 6
10 20 30 40 50 60 70 80 90 100 110 120
-1
-0.5
0
0.5
1
residue
Residue = s(t)
I1(t) = Residue
i = 1
k = 1
while Residue not equal zero or not
monotone
while Ii has non-negligible local
mean
U(t) = spline through local
maxima of Ii
L(t) = spline through local
minima of Ii
Av(t) = 1/2 (U(t) + L(t))
Ii(t) = Ii(t) - Av(t)
i = i + 1
end
IMFk(t) = Ii(t)
Residue = Residue - IMFk
k = k+1
end
Huang’s “Sifting Process”.
39
10 20 30 40 50 60 70 80 90 100 110 120
-1
-0.5
0
0.5
1
IMF 1; iteration 7
10 20 30 40 50 60 70 80 90 100 110 120
-1
-0.5
0
0.5
1
residue
Residue = s(t)
I1(t) = Residue
i = 1
k = 1
while Residue not equal zero or not
monotone
while Ii has non-negligible local
mean
U(t) = spline through local
maxima of Ii
L(t) = spline through local
minima of Ii
Av(t) = 1/2 (U(t) + L(t))
Ii(t) = Ii(t) - Av(t)
i = i + 1
end
IMFk(t) = Ii(t)
Residue = Residue - IMFk
k = k+1
end
Huang’s “Sifting Process”.
40
10 20 30 40 50 60 70 80 90 100 110 120
-1
-0.5
0
0.5
1
IMF 1; iteration 8
10 20 30 40 50 60 70 80 90 100 110 120
-1
-0.5
0
0.5
1
residue
Residue = s(t)
I1(t) = Residue
i = 1
k = 1
while Residue not equal zero or not
monotone
while Ii has non-negligible local
mean
U(t) = spline through local
maxima of Ii
L(t) = spline through local
minima of Ii
Av(t) = 1/2 (U(t) + L(t))
Ii(t) = Ii(t) - Av(t)
i = i + 1
end
IMFk(t) = Ii(t)
Residue = Residue - IMFk
k = k+1
end
Huang’s “Sifting Process”.
41
10 20 30 40 50 60 70 80 90 100 110 120
-1
-0.5
0
0.5
1
IMF 2; iteration 0
10 20 30 40 50 60 70 80 90 100 110 120
-1
-0.5
0
0.5
1
residue
Residue = s(t)
I1(t) = Residue
i = 1
k = 1
while Residue not equal zero or not
monotone
while Ii has non-negligible local
mean
U(t) = spline through local
maxima of IiL(t) = spline through local
minima of IiAv(t) = 1/2 (U(t) + L(t))
Ii(t) = Ii(t) - Av(t)
i = i + 1
end
IMFk(t) = Ii(t)
Residue = Residue - IMFk
k = k+1
end
Huang’s “Sifting Process”.
42
An example:-(puts emphasis on the potentially “nonharmonic” nature of EMD)
the signal is composed of 1. a “high frequency” triangular waveform
whose amplitude is slowly (linearly) growing.
2. a “middle frequency”sine wave whose amplitude is quickly (linearly) decaying
3. a “low frequency” triangular waveform
Both linear and nonlinear oscillations are effectively identified and separated by EMD. whereas any “harmonic” analysis (Fourier, wavelets, . . . ) would end up in the same context with a much less compact and physically less meaningful decomposition.
43
No analytic definition — The decomposition is only defined as the output of an algorithm
Some remarks on Huang’s EMD
Rationale — Intuitive, simple, local and fully data-driven.
Still lacks from solid theoretical grounds
44
Acoustics, noise, and vibration: Machine vibration analysis, Speech/sound analysis , Biomedical signal analysis.
Environmental: Oceanography, Earthquake engineering , Water and wind dynamics.
Industrial: Machine monitoring and failure prediction .
Applications of EMD
45
References:- P. FLANDRIN, P. GONCALVES, 2004 :
"Empirical Mode Decompositions as Data-Driven Wavelet-Like Expansions," Int. J. of Wavelets, Multires. and Info. Proc., Vol. 2, No. 4, pp. 477-496.
ON EMPIRICAL MODE DECOMPOSITION AND ITS ALGORITHMSGabriel Rilling∗, Patrick Flandrin∗ and Paulo Gon¸calv`es∗∗∗Laboratoire de Physique (UMR CNRS 5672), ´Ecole Normale Sup´erieure de Lyon 46, all´ee d’Italie 69364 Lyon Cedex 07, France. The analysis of the Empirical Mode Decomposition MethodVesselin Vatchev, USC ,November 20, 2002
http://perso.ens lyon.fr/patrick.flandrin/emd.html