Empirical Mode Decomposition Method Based on Wavelet with Translation Invariance

Abstract

For the mode mixing problem caused by intermittency signal in empirical mode decomposition (EMD), a novel filtering method is proposed in this paper. In this new method, the original data is pretreated by using wavelet denoising method to avoid the mode mixture in the subsequent EMD procedure. Because traditional wavelet threshold denoising may exhibit pseudo-Gibbs phenomena in the neighborhood of discontinuities, we make use of translation invariance algorithm to suppress the artifacts. Then the processed signal is decomposed into intrinsic mode functions (IMFs) by EMD. The numerical results show that the proposed method is able to effectively avoid the mode mixture and retain the useful information.

Full text

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Hindawi Publishing Corporation
EURASIP Journal on Advances in Signal Processing
Volume 2008, Article ID 526038, 6 pages
doi:10.1155/2008/526038
ResearchArticle
Empirical Mode Decomposition Method Based on
Wavelet with Translation Invariance
Qin Pinle,1,2Lin Yan,1,2and Chen Ming1
1School of Electrical and Information Engineering, Dalian University of Technology, Dalian 116024, Liaoning, China
2Ship CAD Engineering Center, Dalian University of Technology, Dalian 116024, Liaoning, China
Correspondence should be addressed to Qin Pinle, qpl001@sohu.com
Received 20 August 2007; Revised 11 February 2008; Accepted 10 April 2008
Recommended by Nii Attoh-Okine
For the mode mixing problem caused by intermittency signal in empirical mode decomposition (EMD), a novel filtering method
is proposed in this paper. In this new method, the original data is pretreated by using wavelet denoising method to avoid the
mode mixture in the subsequent EMD procedure. Because traditional wavelet threshold denoising may exhibit pseudo-Gibbs
phenomena in the neighborhood of discontinuities, we make use of translation invariance algorithm to suppress the artifacts.
Then the processed signal is decomposed into intrinsic mode functions (IMFs) by EMD. The numerical results show that the
proposed method is able to effectively avoid the mode mixture and retain the useful information.
Copyright © 2008 Qin Pinle et al. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1.INTRODUCTION
A new nonlinear technique, empirical mode decomposition
(EMD), has recently been more and more popular as a new
tool for time-frequency analysis method [1]. The essence of
EMD is to decompose time-varying data series into a finite
setoffunctionsnamedintrinsicmodefunctions(IMFs).The
extracted IMFs represent the local character of original data.
Furthermore, coupled with the Hilbert transform applied to
the IMFs, this decomposition method can obtain instanta-
neous frequency and instantaneous amplitude. This proce-
dure is called Hilbert-Huang transform (HHT). Despite the
success over the past few years of this analysis tool [2–6], it
still has some sections to improve. Simulations showed that
straightforward application of EMD method may run into
mode mixing when the data contain intermittency, the IMFs
will lose intrinsic physics sense. We should find a suitable
way to eliminate the mode mixing. To solve this problem,
a criterion based on the period length was introduced to
separate the waves of different periods into different modes
by Huang et al. [7]. But the detailed manipulation had not
been presented. Zhao [8] made use of three corresponding
characteristics for abnormal signal between the original data
and the first IMF to determine the start and end positions
of an abnormal signal. Then, the abnormal signal is removed
directly.Butitissuitableonlyfortheshortintervalabnormal
signal. Li et al. [9] used wavelet to avoid the mode mixing,
but he did not take into account the effect of artifacts caused
by wavelet.
In this paper, in order to overcome mode mixing, we
firstly combine wavelet transform and translation invariance
algorithm,whichcansuppresstheartifactscausedbywavelet
transform, to process original signal. Then, we execute
empirical mode decomposition to the processed signal. In
this way, we can eliminate mode mixing phenomenon to
obtain excellent effect. Finally, in order to illustrate the
effectiveness of the proposed method, the simulations and
real data analysis are shown.
2.EMPIRICAL MODE DECOMPOSITION
The empirical mode decomposition (EMD) technique has
been developed recently with a view to analyze time-
frequency distribution of nonlinear and nonstationary data.
It is an adaptive decomposition with which any complicated
signal can be decomposed into its intrinsic mode functions
(IMFs). IMFs satisfy the following two constraints.
(i) In the whole signal segment, the number of extrema
(maximumandminimumpointsofdataset)andthenumber
of zero crossing must be either equal or differ at most by one.

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2EURASIP Journal on Advances in Signal Processing
Signal
Imf1
Imf2
Res.
(a)
Signal
Imf1
Imf2
Imf3
Res.
(b)
Figure 1: (a) The decomposed results of formula (2) and (b) the decomposed results of signal with intermittency signal.
(ii) At any point, the mean value of the envelope defined
by the local maxima and the envelope defined by the local
minima is zero.
In practice, most of the signals may involve more than
one oscillatory mode, that is, the signal has more than one
instantaneous frequency at a time locally. Assumed that any
data consist of different simple IMFs, EMD is developed to
decomposea signalintoIMFcomponentsandeveryIMFhas
a unique local frequency. Given a time series data x(t), it can
be decomposed by EMD as follows [1, 10].
(1) Identify all the maxima and minima of x(t).
(2) Generate its upper and lower envelopes, xup(t) and
xlow(t), with cubic spline interpolation.
(3) Compute the local mean m(t) = (xup(t)+xlow(t))/2.
(4) Extract the detail, g(t) = x(t) − m(t).
(5) Check whether g(t) is an IMF or not;
(5.1) if g(t) is an IMF according to the definition
of IMF, extract IMF and replace x(t) with the
residual r(t) = x(t) −g(t),
(5.2) if g(t) is not an IMF, further sifting is needed,
and replace x(t) with g(t).
(6) repeat steps (1–5) until the residual satisfies some
stopping criterion.
The sifting process will be continued until no more IMFs
can be extracted. At the end of the decomposition, the signal
x(t) is represented as follows:
x(t) =
N?
j=1
cj(t)+rN(t),(1)
where N is the number of IMFs, rN(t) is the residue which
is a constant, a monotonic, or a function with only maxima
and one minima from which no more IMF can be derived,
and cjdenotes IMF.
We can apply above EMD procedure to decompose the
time series into set of IMFs and a residue. By applying the
Hilbert transform to each IMF we can farther analyze the
signal and calculate the instantaneous frequency of each
transformed IMF. The whole process is called Hilbert-Huang
transform (HHT) [1].
3.EFFECT OF INTERMITTENCY POINT TO EMD
The EMD method has been applied widely in many areas,
which shows that it has good effectiveness. Yet straight-
forward application of the sifting method may run into
difficulties. Especiallytheoriginal datacontainintermittency
which will cause mode mixing, that is, the first IMF will
contain the information of intermittency signal so that it
could not exhibit normal frequency process. Once the first
IMF caused mixing phenomenon, the subsequent IMFs will
be influenced.
Let us consider the data s(t) given in formula (2).
Figure 1(a) shows the decomposed results of s(t) with
application of the straightforward EMD, and Figure 1(b)
showsthedecomposedresultsofs(t)includingintermittency
signal:
s(t) = sin(2p ×5t)+2cos(2p ×10t)+3.
In Figure 1(b), the first IMF includes the frequency of
intermittency signal, that is, mode mixing is caused in the
part of intermittency signal. As a result, the subsequent IMFs
also contain seriously mixed modes. To explain it more, root
mean square error (RMSE), which is expressed as formula
(3), is adopted as an evaluation criterion:
???? x(t) −x(t)?2/T,
where ? x(t) denotes decomposed IMF data, x(t) denotes the
The RMSE of IMFs can be summarized in Table 1. From
Table 1, we can find that errors of IMF components and
real value become larger due to intermittent signal. As the
mode mixing caused by intermittency is inevitable, it is more
worthwhile to explore a method to solve the problem.
(2)
RMSE =
(3)
real signal data, and T is the length of time series.
4.THE SOLUTIONTO INTERMITTENCY PROBLEM
In the last few years, wavelet transform has become a
well-accepted time-frequency analysis tool, there has been

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Qin Pinle et al.3
Table 1: RMSE of EMD of simulation signal.
IMF1
0.0064
0.3752
IMF2
0.0534
1.1341
Residue
0.0720
1.021
Normal signal
Abnormal signal
Signal
Imf1
Imf2
Imf3
Imf4
Res.
Figure 2: EMD result of signal after tradition doing wavelet
denoising.
considerable interest in the use of wavelet transforms for
removing noise from signals. One method has been the use
of transform-based threshold, working in three steps:
(i) transformthenoisydataintoanwaveletdomain,and
get a group of wavelet coefficients,
(ii) apply soft or hard threshold to the resulting coeffi-
cients, thereby suppressing those coefficients smaller
than certain amplitude, then obtain a group of
estimate coefficients, and
(iii) transform back into the original domain.
Wavelet transform can detect and characterize singulari-
ties in signals so that it offers criterion for the classification
and identification of signal. Now, there are some studies
about comparing wavelet with EMD method [11, 12].
We attempt to adopt the wavelet denoising to eliminate
mode mixing, but simulations show that denoising with
the traditional wavelet transform can exhibit pseudo-Gibbs
phenomena in the neighborhood of discontinuities, which
still causes mode mixing. To make our meaning clear, with
application of the straightforward wavelet denoising in Haar
basis to Figure 1(b), and then using EMD method, we will
obtain the components as shown in Figure 2, in which the
first two IMF components contain seriously mixed modes.
It is evident that the pseudo-Gibbs oscillations caused by
wavelet denoising in the vicinity of discontinuities may run
into mode mixing. One method to suppress pseudo-Gibbs
phenomena is called translation invariant wavelet transform
by Coifman and Donoho [13].
4.1. Waveletthresholddenoisingbasedon
translationinvariance
In the neighborhood of discontinuities, traditional wavelet
denoising can exhibit pseudo-Gibbs phenomena. An impor-
tant observation about the phenomena is that the size
of pseudo-Gibbs depends mainly on the location of a
discontinuityinthesignal.Forexample,whenusingtheHaar
wavelets as basis, a discontinuity located at n/2 will not give
pseudo-Gibbs oscillations; a discontinuity near n/3 will lead
to significant pseudo-Gibbs oscillations. The essence reason
is the misalignment between the signal data and the basis
[14, 15].
A possible way to correct the misalignment between the
data and the basis is to forcibly shift the data so that the
discontinuities change positions, the shifted signal will not
exhibit the pseudo-Gibbs phenomena, and after denoising
the data can be shifted back. Unfortunately, we do not know
the location of the discontinuity. One method solving this
situation is optimization: develops a measure of artifacts and
minimizes it by a proper choice of the shift, but there is
no guarantee that this will always be the case. If the signal
has several discontinuities, they may interfere with each
other, that is, the best shift for one discontinuity may also
be the worst for another discontinuity. Another reasonable
approach is called translation invariant algorithm, which is
to apply a range of shifts, denoise the shifted data by wavelet
threshold and average the several results, then produce a
reconstruction subject. Consequently, the shift dependence
of wavelet basis is eliminated. This method can effectively
suppress the artifacts so that denoised signal is smoother and
has better approximation to original signal.
For a signal xt(0 ≤ t < n), Shdenotes the circulant shift
by h. The Sh(x)tcan be specifically written as
Sh(x)t= x(t+h) mod n.
(4)
The operator is unitary, and hence invertible:
?Sh
?−1= S−h.
(5)
T represents the process of wavelet transform and
denoise based on threshold, the process of eliminating
oscillation by translation is shown as follows:
?T?Sh(x)??.
Then apply a range of shifts, so an average over the several
results is obtained. For time shifts, we consider a range H of
shifts and set
?S−h
or in words in order to compare the efficiency with [8], we
increased the content. We can draw a conclusion that the
efficiency is better than [8] from [13].
The method can be calculated rapidly in nlog(n) time
[13].
In wavelet transform, how to choose desirable wavelet
basis is very difficult. Unsuitable wavelet basis function
maybe reduces denoising efficiency. Fortunately, during
translationinvariancedenoising,abundantsimulationsshow
that when the signal includes intermittency Haar basis can
eliminate primely the pseudo-Gibbs in the neighborhood
of discontinuities [16, 17]. For comparison, in this paper,
x?= S−h
(6)
x?= AVEh∈H
?T?Sh(x)???,(7)

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4EURASIP Journal on Advances in Signal Processing
Original signal
−10
0
10
0100 200 300400500 600700 800900 1000
(a)
Signal with noise
−20
0
20
0 100 200300 400 500600 700 800900 1000
(b)
Threshold filter
−10
0
10
0 100 200300 400 500600700 800900 1000
(c)
TA-threshold filter
−10
0
10
0 100200300400 500 600700 800900 1000
(d)
Figure 3: Comparison of two denoising methods.
tradition denoising method and the denoising method based
on translation invariance algorithm all use Haar as the basis
function.
In order to evaluate the effectiveness of the approach, we
adopt random square-wave as experimental signal to com-
pare traditional threshold algorithm with wavelet transform
based on translation invariance algorithm. Both methods
decompose the signal with Haar wavelet basis to three layers
and use soft threshold denoising which can be defined by
[17]
?sgn?wj,k
0,
wj,k=
????wj,k
??− λ?,
??wj,k
??≥ λ,
??wj,k
??< λ,
(8)
where sgn(•) is sign function, one method of choosing λ is
formula (9):
?
δ ≈ Mx/0.6745,Mxrepresentstheabsolutemedianestimated
on the first scale.
Figure 3 shows that comparison between traditional soft
threshold and soft threshold based on translation invariance.
Here, we average over all n circulant shifts H = Hn = {h :
0 ≤ h < n}, which are called fully translation-invariant [12].
A benefit of the fully translation-invariant approach is that
there are no arbitrary parameters to set: one does not have to
decide whether to average over 10 or 20 shifts.
In Figure 3, it displays that denoising method with
translation invariance is better than traditional threshold
denoising method. The Pseudo-Gibbs phenomenon is elim-
inated effectively and the signal curve is smoother.
λ = δ
2logN,(9)
Table 2: Correlation coefficients of two results with original signal.
Traditional threshold
0.8271
TI-threshold
0.9878
R
Table 3: RMSE of proposed method.
IMF2
0.0203
IMF3
0.0981
Residue
0.0954RMSE
Using correlation coefficient (R) as an evaluation crite-
rion, we compare the two methods with original signal in
Table 2. From the data, also we can see that fully translation-
invariant threshold is better than traditional threshold when
the data includes discontinuity points.
So we can draw a conclusion that TI-threshold wavelet
can better solve the problems which the intermittency signal
effects on wavelet transform.
4.2. EMDwithtranslationinvariancewavelet
transform
Intermittency signal has the characteristic of sharp variation,
so we firstly use wavelet denoising based on translation
invariance to pretreat original signal which will eliminate the
mode mixing caused by discontinuities. Then, we make use
ofEMDtoextracttheIMFcomponents;asetofnewmodesis
obtained. By this way, we can guarantee the validity of EMD
method.
The flowchart of the proposed method is shown in
Figure 4.
Figure 5 shows the processing result of signal in
Figure 1(b) with the proposed method. We can see that the
mode mixing phenomenon is eliminated effectively.
Table 3 shows the RMSE using proposed method. From
the above analysis, it is shown that the denoised signal can be
better fitting the real signal by using translation invariance
wavelet denoise, which could not blur out important signal
features.Andthedecompositionnolongerhasmodemixing.
Also, the IMFs obtained are closer to real value.
5. APPLICATION TO REAL TEMPERATURE DATA
In order to validate the feasibility of the proposed method,
we adopt the data with Zhao [8] to analyze the result.
In Figure 6, the top curve is the original signal which
is 1000hPa monthly averaged air temperature from 1958–
1996 in Barrow, AK, USA. There are persistent several days
of high or low temperature which do not exist in other
months and come into being several local maxima which
will produce frequency mode mixing by EMD method.
We can easily see from Figure 4 that abnormality data not
only affects the results of high-frequency portion but also
influences the signal of multiyear variation, which attributes
to effect of intermittency for whole IMFs in empirical mode
decomposition. Intermittency signal may expand to every
IMF in the course of EMD, which brings about whole

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Qin Pinle et al.5
Start
Signal
Shift data
Wavelet
transform
De-noise
Wavelet reverse
transform
Unshift data
Achieve cycle time
N
Y
Average result
x(t) = r(t)
Local maxima and minima extraction
Upper and lower envelope fits by
spline interpolation
Compute mean envelope m(t)
x(t) = r
g(t) = x(t) −m(t)
x(t) = g(t)
N
Y
g(t) achieves criterion
n = n+1, c(n) = (t), r = r −c(n)
r or c(n) achieves criterion
N
Y
End
Figure 4: The flowchart of the proposed method.
Signal
Imf1
Imf2
Imf3
Imf4
Figure 5: The EMD result of the proposed method.
result distortion. So, it is necessary to pretreat original data
contained intermittency before processing data.
Figure 7 displays the results of the application of pro-
posed method to the original signal data. The original data
pretreat by translation invariance wavelet, the abnormal dis-
turbance could be prevented efficiently. Then, decomposing
the new dataset by EMD method again, a set of IMFs is
obtained, which has a more reasonable physical significance.
In this way, we can guarantee the validity of EMD method.
In order to further research the signal features of
exceptional temperature, we can remove the residue (res. in
Figure 7) and climatic period changing (imf2 in Figure 7),
and then restructure the signal.
6.CONCLUSION
In this paper, we analyze the effect of intermittency to EMD
method and point out that the signal with intermittency
will produce mode mixing phenomenon by directly using
EMD approach. Wavelet based on threshold method is an
appropriate method for multiscale analysis signal, but it will
come into being pseudo-Gibbs phenomena on intermittent
points which will affect empirical mode decomposition. So,
we adopt translation invariance algorithm to eliminate the
artifacts and then proceed to empirical mode decomposition
to get IMF components which have genuine physics sense.
Theoretical analysis and the given example show that.
(1) The proposed method, which combines empirical
mode decomposition and wavelet denoising based on trans-
lation invariance algorithm, effectively eliminates the mode
mixing caused by intermittency.
(2) Compared with [8], the efficiency of method remov-
ing the mode mixing in our paper is O(nlogn) [13].
Comparing the reference [8], the method is better.