[IEEE 2010 3rd International Symposium on Systems and Control in Aeronautics and Astronautics...

Deal with boundary effect of EMD method based on ImprovedCharacteristic Wave Extending

Jun Wang, Xiu-Feng Zhong, and Xi-Yuan Peng

Abstract— Empirical Mode Decomposition (EMD) is a newway to process the non-stationary and nonlinear data. But theedge effect or boundary effect appears when spline interpolationis used to get two envelopes of the data. A novel methodbased on improved characteristic wave algorithm is proposedto prolong the data series and get extra-extremes to deal withthe edge effect. Two simulated signals are decomposed bythree methods for comparison: ‘Extending Method with Char-acteristic Wave’, ‘Extending Method with Mirror Imaging’,and ‘Extending Method with Improved Characteristic Wave’.Experimental results show that the proposed method restrainsthe edge effects effectively.

I. INTRODUCTION

The natural physical processes are mostly nonlinear ornon-stationary, so there are very limited options in dataanalysis methods that can correctly handle data from suchprocess. The available methods are either for linear but non-stationary processes such as the Wavelet analysis, Wagner-Ville, short time Fourier spectrograms and etc., or for non-linear but stationary and statistically deterministic processessuch as time-delayed imbedded methods. The aforemen-tioned methods such as wavelet have been applied to analyzethe non-stationary signal successfully, but will generate somefake spectrum and spectral leakage to high-frequency, so weurgently need to new approaches to examine data from realworld nonlinear and non-stationary stochastic processes orsignals.

A novel signal processing method for analyzing the non-linear and non-stationary stochastic signal named empiricalmode decomposition (EMD) was proposed in 1998 by Nor-den E. Huang [1]. Unlike the traditional signal processingmethods, the EMD method thinks that any signals are com-posed with a series of intrinsic mode functions (IMF). TheIMF highlights the various components of the local featuresof data and can more accurately grasp the characteristics ofthe original data information.

In the decomposition process of EMD, it is needed tocalculate the mean of the upper and lower envelopes thatare obtained with the local maxima and minima by cubicspline interpolation. But unfortunately, the endpoint of thesignal may not be the local maxima or minima, so using

This work was supported by the Natural Science Foundation of Guang-dong Province of China(No.9451503101003263), Educational Commissionof Guangdong Province of China and the Youth Foundation of ShantouUniversity.

J. Wang and X.F. Zhong are with the Department of Electronics Engineer-ing, Shantou University, No.243 Daxue Road, Shantou, Guangdong, 515063,People’s Republic of China. Email: wangjun@stu.edu.cn

X.Y. Peng is with the Auto-testing and Control Laboratory, HarbinInstitute of Technology, Harbin, Heilongjiang, P. R. China.

the endpoints to do cubic spline interpolation will cause theinterpolation of the two endpoints in the signal appearingrelatively large swing. The swing may be spread to themiddle segment of the signal and destruct the characteristicsof the entire data [2]. This is called the endpoint or boundaryeffect problem. Huang et al proposed the mirror imageextending method (MM) to resolve the problem [3]. Theadvantage of MM method is the easy to process and has lowcomputational complexity. The disadvantage of MM methodis that its performance of boundary effect processing willdegrade greatly when the endpoints remote from the realcase of extreme points. Zhong proposed the characteristicwave extending method (CW) for processing boundary effect[4]. Deng et al proposed the neural network based extendingmethod (NN) for processing the boundary effect. The NNmethod resolve the boundary effect problem very well underthe conditions that the noise in the original signal is relativelower and the mean value curve has good symmetry [5].But NN method has the shortages such as high computingcomplexity, bad real-time property of decomposition, andrequirement that the signal changes can not too complicated.[6] proposed the polynomial fitting based extending method(PT) and [7] proposed the AR model based extending methodfor processing the boundary effect. According to [3], perfor-mance of the MM method is better than those of the NN andPT methods. But the MM and the CW methods are sensitiveto the phase of the endpoints. Different phase of the endpointwill cause different boundary effect. To resolve this problem,we proposed an improved characteristic wave extendingmethod for processing the boundary effect in the EMD.Experimental results showed that our proposed method hadbetter performance than the MM and CW method and wasless sensitive to phase of endpoints

II. EMPIRICAL MODE DECOMPOSITION AND BOUNDARYEFFECT

The EMD will decompose the data into a collectionof IMF defined as any function satisfying the followingconditions or assumptions: (a) in the whole data set, thenumber of extrema and the number of zero-crossings musteither equal or differ at most by one, and (b) at any point,the mean value of the envelope defined by the local maximaand the envelope defined by the local minima is zero [1].Based on these assumptions, the process of empirical modedecomposition of any signal x(t) can be described as follows.Step 1: determine all local maxima and minima of the signalx(t). The upper envelope can be obtained by cubic splineinterpolating with all local maxima. The lower envelope can

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also be obtained by cubic spline interpolating with all localminima. So the signal x(t) will be enveloped with upper andlower envelopes.Step 2: the mean value function m1 can be computed asEq.(1).

m1 = UpperEnvelope(x(t))−LowerEnvelope(x(t)) (1)

And the signal y1(t) can be computed as Eq.(2).

y1(t) = x(t)−m1 (2)

Step 3: Determine whether y1(t) an IMF or not. Repeat thestep 1 and step 2 if the y1(t) is not satisfied with assumptionof IMF. Until the y1(t) is satisfied with the assumption ofIMF, then c1(t) = y1(t) is the first component sifted out thatrepresent the component with the highest frequency band.Step 4: The differential signal r1(t) can be obtained byfiltering c1(t) out.

r1(t) = x(t)− c1(t) (3)

Step 5: Regard the r1(t) as the new original signal, repeatthe Step 1 to Step 3, and the second IMF component can begot. Repeating the aforementioned steps with n times, wecan get n IMF components.

After a series of decomposition, the signal x(t) can berepresented with n IMF components and a residual rn(t)i.e.

x(t) =n∑

i=1

ci(t) + rn(t) (4)

where ci is appeared from highest frequency band to lowestfrequency band, the c1 represents the component with highestfrequency band, cn represents the component with lowestfrequency band, and the residual rn(t) is a monotone series.

In fact, the EMD process can be viewed as a siftingprocess, in which the superposition of the modal waveformsis eliminated and the contour of the waveform becomes muchmore symmetrical. Firstly, EMD method sifts the modalwaveform with little time scale out. And then it sifts themodal waveform with relative higher time scale out. At last,the modal waveform with largest time scale is sifted out. Sothe EMD method can be viewed as a high frequency passedfilter bank.

In the EMD, the cubic spline interpolation is frequentlyused to obtain the upper and lower envelopes to get themean curve of these two envelopes. When the endpointsare not the extreme points, the decomposed signals will bedistorted by directly doing cubic spline interpolating with theendpoints. This problem is called the boundary effect. To thehigh frequency components, the boundary effect happened inthe neighborhood of the endpoints for its small time scale.To the low frequency components, the boundary effect willbe propagated from the boundary to the inner of the signals.The more short the length of the signals, the more serious theboundary effect will be. The IMFs decomposed from signalby EMD will lose the physical meaning when the lengthof signal is too short and dealt without any boundary effectprocessing methods.

III. IMPROVED CHARACTERISTIC WAVE EXTENDINGBASED BOUNDARY EFFECT PROCESSING METHOD

The characteristic wave extending method for boundaryeffect processing just takes a sin wave whose circle equalsto the mean circle of three circle characteristic or sin waveadjacent to the endpoints as the extending part of the series,as Fig.1(a). Assumed the endpoint is the maximum point asshowed in the Fig.1(a), when the distance between the lastextreme point and the endpoint is less than the 1/4 circleof the characteristic wave, the extra extreme point obtainedby extending the series or signal with the characteristicwave will arrive before the real extreme point of the signaldoes. The extra extreme points obtained by extending withcharacteristic wave will be larger (for the local maxima) orless (for the local minima) than the real extreme point. So theenvelopes gotten by cubic spline interpolation with the localextreme points and the extra extreme points can not wrap upthe real series of the signal after the endpoints. Under thiscondition, serious boundary effect will be caused.

In this paper, an improved characteristic wave extend-ing method is proposed for boundary effect processing, asshowed in Fig.1 (b). When the distance between the lastextreme point and the endpoint is less than the 1/8 circleof characteristic wave, the series for extending is shifted1/4 circle of characteristic wave firstly and then attached tothe original signal as the extended series. When the distancebetween the last extreme point and the endpoint is larger thanthe 1/8 circle of characteristic wave and less than 1/4 circleof characteristic wave, the series for extending is shifted 1/8circle of characteristic wave firstly and then attached to theoriginal signal as the extended series. When the distancebetween the last extreme point and the endpoint is larger than1/4 circle of characteristic wave, the series for extending isattached to the original signal directly.

Furthermore, the improved characteristic wave methoddoes time scaling for the characteristic wave. This operationwill alleviate the changing intensity of envelopes nearing theendpoint, which reduces the boundary effect further. In theimproved characteristic wave extending method, there areonly some simple operations to be needed such as simplediscriminating, shifting and scaling of the basic characteristicwave and so on, which make the computational complexityof the improved characteristic wave extending method to bevery low and easy to be implemented for processing theboundary effect.

IV. EXPERIMENTAL RESULTS

Two simulation signals are used to compare the perfor-mance of boundary effect processing techniques. The signalone is a nonlinear signal defined as

x(t) =

sin(0.1πt), 0 ≤ t ≤ 34.1sin(0.4πt), 34.2 ≤ t ≤ 68.2sin(0.7πt), 68.3 ≤ t ≤ 102.3

(5)

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Fig. 1. Principle comparisions with the Characteristic Wave method andthe Improved Characteristic Wave method caption.

The signal two is a linear signal defined as

x(t) = sin

(110

πt

)+ sin

(225

πt))

, t = 0, 1, 2, ..., 511

(6)Two above signals both satisfy the conditions of IntrinsicMode Functions (IMF), and can be decomposed into twocomponents (first IMF component and a residue), so theresidue is caused by boundary effect. To measure how greatboundary effect is caused, two evaluating criterions are usedi.e. maximum absolute error (MAE1) and mean absoluteerror (MAE2). To show performance of the novel method, thecharacteristic wave (CW) extending method and the mirrorimage (MM) extending method are used for comparison.

The residues of the signal 1 with EMD algorithms basedon three different boundary effect processing methods areshowed in Figure 2, where the horizontal axis representstime and vertical axis represents the amplitude of the residue.The residues of the signal 2 with EMD algorithms basedon three different boundary effect processing methods areshowed in Figure 3, where the horizontal axis represents timeand vertical axis represents the amplitude of the residue.

The front endpoint phases of two signals are both equalto kπ/2. From Figure 2(a) and Figure 3(a), we can find thatthe residues are very small. That is to say, the boundaryeffect happened in this case is not serious or very mild. Thephases of the back endpoints of two signals are not equal tothe kπ/2. From Figure 2(b) and Figure 3(b), we can find thatthe residues with the characteristic wave extending methodand the mirror image extending method are much large thanthe residue with the improved characteristic wave extendingmethod. All of these indicated that the characteristic waveextending method and the mirror image extending methodfor processing the boundary effect are both sensitive to thephases of the endpoints. When the phases of the endpointsare not adequate, these methods will do poorly boundaryeffect processing. But our proposed method is relatively lesssensitive to the phases of the endpoints and much qualifiedfor boundary effect processing.

Table 1 gives the MAE1 and MAE2 of two simulationsignals with three different methods. From Table 1, we alsosee that the boundary effects caused by our method are lessthan by the other two methods. This also proved that theproposed method much qualified for doing the boundaryeffect processing than other two methods.

(a) The residue waveform of thefront endpoint.

(b) The residue waveform of theback endpoint.

Fig. 2. EMD risidues of signal 1 with three boundary effect processingmethods, where CW represents the characteristic wave extending method,MM represents the mirror image extending method, and ICW represents theimproved characteristic wave extending

(a) The residue waveform of thefront endpoint.

(b) The residue waveform of theback endpoint.

Fig. 3. EMD risidues of signal 2 with three boundary effect processingmethods, where CW represents the characteristic wave extending method,MM represents the mirror image extending method, and ICW represents theimproved characteristic wave extending method.

TABLE IMAE1 AND MAE2 COMPARISON OF TWO SIMULATION

SIGNALS WITH THREE METHODS

Method Signal 1 Signal 2MAE 1 MAE 2 MAE 1 MAE 2

CW 0.1636 0.0027 0.3333 0.0022MM 0.3388 0.0063 0.5984 0.0038ICW 0.0388 0.0009 0.0624 0.0019

V. CONCLUSION

In this paper, we proposed an improved characteristicwave extending method for processing the boundary effect inthe empirical mode decomposition. The experimental resultsshowed that the boundary effect caused by our method wasmuch less than the other two widely used methods and ourmethod was much less sensitive or more robust to the phasesof the endpoints. Furthermore, only simple operations are

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needed to do boundary effect processing such as shifting,scaling and so on. That is to say, our method has the verylow computing complexity. So our method is more qualifiedfor doing boundary effect processing.

REFERENCES

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[2] H. Li, X. Wu, Y. Ge. “Dealing with end issue of EMD method”.Electric Power Automation Equipment, Vol.25,2005, pp.47-49.

[3] N. Huang and N.O. Attoh-Okine. The Hilbert-Huang Transform inEngineering. Taylor & Francis, 2005.

[4] Y. Zhong, “Research on the Local-instantaneous Signal Analysis The-ory of the Hilbert-Huang Transform”. Ph.D dissertation of ChongqingUniversity, 2002.

[5] Y. Deng, W.Wang, C.Qian, Z.Wang, and D.Dai. “Processing of theboundary problems in EMD and Hilbert Transform”. Chinese ScienceBulletin, Vol.46, 2001, pp:257–263.

[6] H. Liu, M. Zhang, J. Cheng. “Dealing with the End Issue of EMDBased on Polynomial Fitting Al gorithm”. Computer Engineering andAmplications, Vol.16, 2004, pp.84–86.

[7] Y. Zhang, J. Liang, Y. Hu.“Applying AR Model to resolve the bound-ary effect problem in EMD method” Progress in Natural Science,Vol.13, 2003,pp.1054-1059.

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