Proceedings of the 2012 International Conference on Wavelet Analysis and Pattern Recognition, Xian, 15-17 July, 2012

NONSTATIONARY SIGNAL ANALYSIS BASED ON EMD AND EXTREMUMPOINTS

JIAN-JIA PANt, YUAN-YAN TANG2

'Department ofComputer Science, Hong Kong Baptist University, Hong Kong2Department ofComputer and Information Science, The University ofMacao, Macao

E-MAIL: jjpan@comp.hkbu.edu.hk.yytang@umac.mo

Abstract:Empirical mode decomposition (EMD) is a data driven

processing algorithm, which has no predetermined filter. It isable to perfectly analyze the nonlinear and nonstationary signals.In EMD decomposition processing, the envelopes are computedby spline interpolation, which is time-consuming. In this work,flrstly, we proposed a boundary extending method based onlinear prediction and boundary extrema points adjusting, whichreduce the end effects problem. And then, based on the straightline method, we proposed just using the extrema points to detectthe extrema information about the signal, which is ExtremaPoints Empirical Mode Decomposition (EPEMD). By using theextrema points information, a fast and distinct frequencychange detection method is proposed.

Keywords:EMD; Time-frequency analysis; Boundary extending;

Extrema points

1. Introduction

The combination of the well-known Hilbert spectralanalysis (HAS) and the recently developed empirical modedecomposition (EMD) designated as the Hilbert-Huangtransform (HHT) by Huang [1], represents a paradigm shift ofdata analysis methodology. The HHT is designed specificallyfor analyzing nonlinear and nonstationary data. The key partof HHT is EMD with which any complicated data set can bedecomposed into a fmite and often small number of intrinsicmode functions (IMFs) [4].

Nonstationary signals have statistical properties that varyas a function of time and should be analyzed differently thanstationary data. Rather than assuming that a signal is a linearcombination of predetermined basis functions, the data areinstead thought of as a superposition of fast oscillations ontoslow oscillations. EMD identifies those oscillations that areintrinsically present in the signal and produces adecomposition using these modes as the expansion basis.

Since Huang proposed HHT and EMD, there have beensome improvements and applications. In practice, the EMD

978·1-4673·1535·7/121$31.00 ©2012 IEEE

has met several unsettled problems, such as boundaryextension, curve fitting, stop criteria and so on. One key topicis the boundary extending problem or end effects problem,and some solutions such as mirror extending, neural networktraining or AR model have been proposed [8]. In this paper,we proposed using linear prediction combined with boundaryextrema points information to extend the signal, whichreduces the end effects.

Then, based on the straight linear method, we proposejust using the extrema points to detect the frequency changedinformation of the signal, which is Extrema Points EmpiricalMode Decomposition (EPEMD). The detected pointsidentified by EPEMD are clear and distinct such that thefrequency changed location is present. EPEMD also offerscomputational efficiency.

2. Empirical mode decomposition (EMD)

EMD is first proposed by Huang et al. [1] for theprocessing of non-stationary functions. The tool decomposessignals into components called Intrinsic Mode Functions(IMFs) satisfying the following two conditions: (a).Thenumbers of extrema and zero-crossings must either equal ordiffer at most by one; (b).At any point, the mean value of theenvelope defmed by the local maxima and the envelope by thelocal minima is zero.

Huang [1] proposed an algorithm called 'sifting' toextract IMFs from the original signalf(t) as follows:

(1) For any given data, f(t), we identify all the localextrema.

(2) Separately connect all the maxima and minima withnatural cubic spline lines to form the upper u(t), and lower l(t),envelopes.

(3) Find the mean of the envelopes by m(t) = [u(t) +l(t)JI2.

(4) Take the difference between the data and the mean asthe proto-IMP, h(t) =f(t) - m(t).

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Proceedings of the 2012 International Conference on Wavelet Analysis and Pattern Recognition, Xian, 15-17 July, 2012

(5) Check the proto-IMF against the definition of IMFand the stoppage criterion to determine if it is an IMF.

(6) If the proto-IMF does not satisfy the defmition, repeatstep 1 to 5 on h(t) as many time as needed till it satisfies thedefmition.

(7) If the proto-IMF does satisfy the defmition, assign theproto-IMF as an IMF component, I(t).

(8) Repeat the operation step 1 to 7 on the residue, r(t) =

f(t)-I(t), as the data.(9) The operation ends when the residue contains no

more than one extremum.After then, the original signal is decomposed into its

IMFs and residue:

in which, N is the signal length, I is a select parameter todecide the length ofboundary part, normally, 1=3--5;

3. Extend the signal by linear prediction of the boundarypart, the prediction part of I1eft is used as the left extend part,and the prediction part of Iright is used as the right extend part.In experiment, p=4.

A simulation nonstationary signal is used as an example:Signal 1:

x=O:O.OOl:0.255;

z=cos(401rx)cos(40Dxx)+cos(201rX)cos(2001CX);

T

f(t) = L1i(t)+rT(t)i=l

(1)

Where Ii(t), i=1, ...,Tare IMFs and r~t) is the residue.In the EMD processing, some different stoppage criteria

were proposed. Follow Huang's recently commend [3], weuse the S-number criterion as the stoppage criteria in ourexperiment.

3. Extending based on linear prediction and end pointsadjusting

Linear prediction is a normally extending method insignal processing, which was adopted to many researches. Inprevious EMD end effect processing, linear prediction has notbeen used. In this paper, we firstly used the linear predictionas the boundary extending methods, and then the boundarypoints are adjusted by the extrema points information.

LPC determines the coefficients of a forward linearpredictor by minimizing the prediction error in the leastsquares sense. It has applications in filter design and speechcoding. LPC fmds the coefficients of a p -order linearpredictor (FIR filter) that predicts the current value of the real­valued time series x based on past samples.x(n) = -a(2)x(n -1) - a(3)x(n - 2) - ... - a(p + l)x(n - p)

(2)

P is the order of the prediction filter polynomial, a = [1a(2) ... a(p+l)].. The length ofp must be less than or equal tothe length ofx.

The intact algorithm is as follows:1. Detect the maximum and minimum points of signal

f(t); The maximum point set is (fmax(i)}, the minimum pointset is (fminO)};

2. Detect the boundary part of signal:I1eft=f(1 :tletJ ' tleft=max(fmax(l), fmin(l)); (3)

Iright=f(tright:N) , tright=min(fmax(N-l), fmin(N-l)); (4)

Figure 1. Original signal and the extend result with different methods.

Figure 2. IMFS decomposed by EMD, compared methods and proposedmethod.

Figure 1 showed the original signal and the extend result.The extended part in original signal is set to zero. Figure2showed the decomposed last IMF of signal 1. As Figure 2showed, LPC extending method achieved the best end effectsreducing. No extend EMD has a serious end effect.

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Proceedings of the 2012 International Conference on Wavelet Analysis and Pattern Recognition, Xian, 15-17 July, 2012

Further, based on the boundary extreme points and Wu'work [7], we further proposed a new end point processingmethod. Suppose that we have two maxima max(l) andmax(2), two minima min(l) and min(2) that are closest to anend, as shown in Figure 3. In this case, min(l) is on the left ofmax(I), and the opposition case is the same, just use maxinstead of min. We linearly connect the min points, computethe slope (min(I),min(2)), and then compute the min pointsmin(max(I)) in the max(I)'s location. The mean point isgivem by mean(1)=(max(1)+min(max(1)))/2. Based on theslope (min(I),min(2)), linearly extend straight line mean (1) tothe end to fmd mean_end. If mean_end is larger than the end­point value end(1) of the signal, we consider mean_end as anew maximum points for the upper envelope fitting and end(l)as a new minimum points for the lower envelope fitting.Otherwise, we consider end(l) as a new maximum points forthe upper envelope fitting and mean_end as a new minimumpoints for the lower envelope fitting. In this case, mean_end issmaller than end(I).

2_-----.-----r--_r__-----.-----r----r----r--------.--~-_r__-----.

end(l)

1.5

Further, based on the end points adjusting method, allmethods' decomposition's MAD and reconstruction errors areimproved, especially for the original EMD methods.

Figure 4. IMFS decomposed by EMD, compared methods and proposedLPC method, combined with extrema points adjusted

Figure 3. End points adjusting based on extrema mean information

4. Extrema Points Empirical Mode Decomposition

The nonstationary signal frequency is changed over time,which is also called time-varying signal. To analyze thissignal, it should detect a period of time's frequencyinformation or a period of frequency's time information,Fourier transform can't be used to detect those nonstationarysignals's frequency detection. In this part, we proposed to usethe extrema point's information to detect some usefulinformation for nonstationary signals analysis.

In EMD, the extrema points information are veryimportant, but in the previous works, the extrema points arejust used to build the envelopes. After the interpolation, theextrema points are not used again. We propose that theextrema point's information is more useful.

A simulation nonstationary time-varying signal is usedfor an example:

Signal 2:

x=1:1:128;

zl =sin(0.241rX);

x=129:1:256;

z2=0.15+cos(0.31rX)+0.15cos(0. 61CX);

The example signal is shown in Figure 5.

max(2)

\8 9 10 11 126 74 5

I

__ '-'mean..,...,. - --- I

__~~ --- --- -- min(2)".--

min(l) min(max(l))

max(l)

....----

2 3

o

0.5

_1.51--.1..---'---....1.....--1--.1..----'---....1.....--1--.1..----'---'1

-0.5

Figure 4 shows the decomposed IMF5 of signal 1 withextrema points adjusted. As Figure 4 shows, LPC extendingmethod also achieves the best end effects reducing. Combinedwith extrema points adjusted, all methods' end effects areimproved, especially for the original EMD methods. Ourmethod is also better than Wu's method [7], but the result isnot shown here.

Table 1 demonstrates the mean absolute deviation (MAD)ofIMF5, extending computed time and reconstruction error ofthe methods mentioned above. For MAD and reconstructionerror, the LPC extending method achieves the best result in allcases. Its computation time is longer than mirror extendmethod, but is much faster than AMRA extend method.

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Proceedings of the 2012 International Conference on Wavelet Analysis and Pattern Recognition, Xian, 15-17 July, 2012

TABLE 1.PERFORMANCES COMPARISON OF DIFFERENT EMD METHODS OF SIGNAL 1

Methods EMD Mirror extend AMRA extend Linear predictionextend

MADofIMF5 0.3019 0.0674 0.1259 0.0589

MAD ofIMF5 with end 0.0801 0.0626 0.0981 0.0561extrema adjusted

Reconstruction Error 1.4683e-14 1.4128e-14 1.6126e-14 1.2670e-14

Reconstruction Error with end 7.3275e-15 5.4956e-15 1.3180e-14 4.8956e-15extrema adjusted

Computed time (seconds) None 0.002788 10.756900 0.022366

(6).Repeat the operation step 1 to 5 with N time. All thepoints value and their local information are recorded in theset {fmax(i)} ,{fminlj)},{i,j}. In our experimental, N=10.

In every minimum point fminlj)) location, compute theupper envelop point value umaxO)

umaxO) = fmax(i-1)+ Gfmax(i-1)*0-(i-1)) (8)

(4).The extrema mean points' value is

mean(i) = Umax(i)+ lmin(i)}/2; (9)

meantj)> UminO)+ umaxO)}/2; (10)

(5).The new extrema mean points set is:

We propose a new method to compute the extremapoints of signal, and just detect the extrema points' locationand value.

Extrema Points Empirical Mode Decomposition(EPEMD):(1).For any given data,f(t), we identify all the local extrema.The maximum point set is (fmax(i)}, the minimum point setis (fminO)};

(2).Compute the gradient of each neighbor points in {fmax(i)}and {fminO)} , which are {Gfmax(i)}and {GfminO)};

Gfmax (i)= Umax(i)- fmax(i-1)}/[x(i)-x(i-1)} (5)

Gfmin (i)= Uminlj)- fminlj-1)}/[xlj)-xlj-1)} (6)

(3).In every maximum point fmmli) location, compute thelower envelop point value lmin(i)

lmin(i) = fminlj-1) + GfminO-1)*(i-0-1))

hmax(i)=fmax(i)-mean(i);

hminlj)=fmin lj)-meanO);

(7)

(11)

(12)

Figure S. the Signal 2

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Figure 6. the maximum points set by EPEMD

Proceedings of the 2012 International Conference on Wavelet Analysis and Pattern Recognition, Xian, 15-17 July, 2012

Figure6 shows maximum points set of signal 2 byEPEMD. Figure7 shows minimum points set of signal 2 byEPEMD. From them, we can fmd that:

Figure7. the minimum points set by EPEMD

1). The normal values of extrema points set are about 1and -1, which are the amplitude value of the original signal;

2). In the neighborhood period of the frequencychanged points, the value ofextrema points are much largerthan other periods.

3). The max positive points in Figure6 is in locationi=110, The min negative points in Figure 6 is in locationi=119; The min negative points in Figure7 is in locationj=123, The max positive points in Figure7 is in locationj =115. These points' locations present the locationinformation ofthe frequency changed points.

To confrrm the decomposition by EPEMD, theproposed method is used to decompose another simulatedsignal.

Signal 3:

x1=0:0.1:34.1;

z1=sin(0.111X1);

X2 =34.2:O. 1:68.2;

z2=sin(0.411X~;

x3=68.3:0.1:102.3;

z3=sin(0. 711X3);

This signal is a nonlinear and nonstationary, time-varyfrequency signal, which is shown in Figure 8. In fact, thissignal itself satisfies the definition of IMF, and is verysimilar to an IMF. The EMD method can't detect anysignificant components from this signal. But by use of

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EPEMD, we can detect the frequency changed locationinformation and some others information as discussedbelow.. The detection result of the simulation signal isshown in Figures 9 and 10.

As in the Signal 2, some similar results can be foundfrom these figures:

1) The normal values of extrema points set are about 1and -1, which are the amplitude values of the originalsignal;

2). In the neighborhood period of the frequencychanged points, the value ofextrema points are much largerthan other periods.

3). In different parts of the signal, the frequency of theextrema points is different, which directly present thefrequencies of the signal in different parts. In the formerpart of the signal, the extrema points are sparse, whichpresent low frequencies. In the latter part of the signal, theextrema points are dense, which present high frequencies.

4). The max positive points in Figure9 is in locationi=251, The min negative points in Figure9 is in locationi=251; The second max positive points in Figure9 is inlocation i=664, The second min negative points in Figure9is in location i =613;

5) The max positive points in Figurel0 is in locationj=342, The min negative points in Figurel0 is in locationj=638; The second max positive points in Figurel0 is inlocationj=683, The second min negative points in Figure10is in location j=388. Please note that the location of maxpositive points and second max positive points in Figurel0is the exact location of frequency change points in Signal 3.

Furthermore, the EPEMD's computation is very fast.The experiment computer is Inter®, Core(TM)2, CPU,Q6600 @2.40GHz. The simulation platform is MATLABR2011a. For signal 2, the computed time is 2.202ms, andfor signal 3, the computed time is 12.876ms.

Figure 8. the Signal 3

Proceedings of the 2012 International Conference on Wavelet Analysis and Pattern Recognition, Xian, 15-17 July, 2012

points, the values of extrema points are much larger thanother periods. The location of extrema points presents thelocation ofsignal frequency change points.

In this paper, we emphasize the importance ofextramapoint information. Our work also showed that bycombining with the extrama point information, one can getbetter result both in boundary extend and frequencychanged points' detection. In the future work, we willfurther investigate other aspects of the proposed method.

References

Figure 9. the maximum points set by EPEMD

FigurelO. the minimum points set by EPEMD

5. Summary

In this paper, we firstly propose using the linearprediction combined with boundary extrema pointsinformation to extend the signal, which reduce the endeffects in EMD sifting processing. Experiments show thatthe MAD and reconstruction error of the proposed methodis better than other extending methods.

We also propose a new and fast nonstationary signalanalysis method Extrema Points Empirical ModeDecomposition (EPEMD), which is just following theextrema point information in signal. The extrema point'slocation and value information are used to build an extremapoints sets, which present some interesting result fornonstationary signals analysis. The normal value of thedetected extrema points set is the amplitude value of theoriginal signal, even if the signal is jumped, the extremapoints set value remains unchanged. On the other hand,near the neighborhood period of the frequency changed

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