Improving the empirical mode decomposition method

This article was downloaded by: [Juan J. Trujillo]On: 09 July 2011, At: 04:40Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registeredoffice: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK

Applicable AnalysisPublication details, including instructions for authors andsubscription information:http://www.tandfonline.com/loi/gapa20

Improving the empirical mode

decomposition methodJose L. Sanchez

a & Juan J. Trujillo

b

a Departamento de Ingeniería de Sistemas y Automática y

Arquitectura y Tecnología de Computadores, Universidad de LaLaguna, San Cristobal de La Laguna, Spainb Departamento de Análisis Matemático, Universidad de La

Laguna, San Cristobal de La Laguna, Spain

Available online: 08 Jul 2010

To cite this article: Jose L. Sanchez & Juan J. Trujillo (2011): Improving the empirical modedecomposition method, Applicable Analysis, 90:3-4, 689-713

To link to this article: http://dx.doi.org/10.1080/00036810903569531

PLEASE SCROLL DOWN FOR ARTICLE

Full terms and conditions of use: http://www.tandfonline.com/page/terms-and-conditions

This article may be used for research, teaching and private study purposes. Anysubstantial or systematic reproduction, re-distribution, re-selling, loan, sub-licensing,systematic supply or distribution in any form to anyone is expressly forbidden.

The publisher does not give any warranty express or implied or make any representationthat the contents will be complete or accurate or up to date. The accuracy of anyinstructions, formulae and drug doses should be independently verified with primarysources. The publisher shall not be liable for any loss, actions, claims, proceedings,demand or costs or damages whatsoever or howsoever caused arising directly orindirectly in connection with or arising out of the use of this material.

Applicable AnalysisVol. 90, Nos. 3–4, March–April 2011, 689–713

Improving the empirical mode decomposition methodyJose L. Sancheza and Juan J. Trujillob*

aDepartamento de Ingenierıa de Sistemas y Automatica y Arquitectura y Tecnologıa deComputadores, Universidad de La Laguna, San Cristobal de La Laguna, Spain;

bDepartamento de Analisis Matematico, Universidad de La Laguna, San Cristobal deLa Laguna, Spain

Communicated by R.P. Gilbert

(Received 19 November 2009; final version received 17 December 2009)

In recent years, the analysis of non-stationary signals has taken on greatimportance since the introduction of empirical mode decomposition(EMD) by Huang in 1998. The algorithm is based entirely on a discretealgorithm, meaning that, for now, no clear analytical interpretation exists.In this article, we study several aspects of the EMD algorithm in order toimprove the decomposition. On the one hand, the sampling frequency mustbe optimized so as to maximize the similarity between the discrete andcontinuous signals, minimizing the computational cost required to applyanalysis methods for non-linear and non-stationary signals. On the otherhand, a solution to border effect which gives good results for signals ofapproximately constant, growing or decreasing amplitude near the bordersis provided. Moreover, the stopping criteria must be modified to limit theamplitudes allowed to IMF. Some examples are shown at the end.

Keywords: empirical mode decomposition; nonlinear process;nonstationary

AMS Subject Classifications: 42C99; 94A12; 92C55

1. Introduction

Methods for analysing linear and stationary signals have been in use for many yearsin signal processing, though the results, in general, were only sufficiently good if thesignal complied with the method’s requirements. The signals that exist in the realword, however, frequently contain transient components and significantnon-linearities that cannot be accounted well with classical methods (in particular,the Fourier transform). The analysis of non-linear and non-stationary signals [1] is ofparamount importance in vibration analysis [2], biomedicine (the electroencephalo-gram (EEG) and electrocardiogram (ECG) [3–5]) and in other fields, such asgeophysics [6], imaging [7], structural safety [8] and power systems [9–12]. If the

*Corresponding author. Email: [email protected] to Professor Paul Leo Butzer on the occasion of his 80th birthday.

ISSN 0003–6811 print/ISSN 1563–504X online

! 2011 Taylor & FrancisDOI: 10.1080/00036810903569531http://www.informaworld.com

Dow

nlo

aded

by [

Juan

J. T

ruji

llo]

at 0

4:4

0 0

9 J

uly

2011

signal’s properties change over time then so must be its spectrum. It is thereforenecessary that some other type of method be found that allows for the spectralparameters of the signal to be calculated, or for its decomposition in the time domaininto other signals that explain the spectral qualities of the original signal.

Most spectral analysis methods normally employed to handle non-deterministicsignals are only valid when they are applied to stationary processes (those whosestatistical properties are constant over time). It is difficult to conceive of processesfitting this description in the real world [13]. The most that can be expected is thatthey do not stray too far from stationarity so the resulting analysis is not invalid. Inthese cases it is necessary to introduce the concept of time-dependent spectrum(energy distribution that changes from one instant in time to the next). Variousframeworks have been proposed [14,15] to deal with the questions of instantaneousspectrum, instantaneous frequency and time-variable spectrum. The central compo-nent of these frameworks is generally a transform of the signal (or distribution), thatis, the original signal is mapped onto a two-dimensional space, with one dimensionbeing associated with time and the other to the frequency (time-frequency graph).Within these methods, the best known are probably the short-time Fourier transform(STFT) and the wavelet transform. Fano [16] and Schroeder and Atal [17] developeda rigorous STFT for analogue signals. The technique was not applied to digitalsignals until the digital spectrogram [18] was introduced in 1970. The waveletconcept, on the other hand, was introduced by Haar in 1909 [19], though it wasdeveloped mostly starting in the 1980s with the work of Daubechies [20] amongothers. The greatest drawbacks of STFT are as follows: (a) it still requires that thesignal within the moving window be stationary, which complicates the transientanalysis and (b) the use of equally-sized windows regardless of the frequencyprevents taking advantage of the fact that a high-frequency signal has many moreperiods than a low-frequency signal in the same time window. The wavelet transformtried to remedy this shortfall. The choice of a base wavelet function and thepreliminary determination of the scale, however, continue to limit the utility of thismethod for strongly non-linear or non-stationary signals, as evidenced by variousexamples in the literature [21,22].

The determination of a signal’s instantaneous frequency requires that the signalbe monocomponent, meaning it is not the sum of other, simpler, signals. Thus, thefirst step in being able to analyse a signal’s instantaneous frequency to obtain adecomposition of the said signal, such that each of its components has just a singlefrequency at each moment in time. The non-linear energy operator and the Hilbertspectral analysis are the usual tools for determining the instantaneous frequenciesand amplitudes.

Huang [1] devised a novel approach for the analysis of non-stationary andnon-linear signals. The method is simple and makes no pre-assumptions about thetype of signal or its stationarity. The method is self adaptive, though the techniqueexhibits, among others, the difficulty of being defined essentially by a discretealgorithm for which no analytical formulation currently exists that allows for atheoretical interpretation.

The Shannon sampling theorem states that a band-limited continuous signal x(t)with a cutoff frequency fc can be recovered from its samples x(nT ) using a kernel sincinterpolator if the sampling frequency is greater than twice the highest fre-quency present in the signal (Nyquist frequency), which is quite costly from a

690 J.L. Sanchez and J.J. Trujillo

Dow

nlo

aded

by [

Juan

J. T

ruji

llo]

at 0

4:4

0 0

9 J

uly

2011

computational standpoint. By itself, this theorem does not allow for the use of adiscrete signal in place of a continuous one. In practice, it is normal to substitute thecontinuous and discrete signals without having done the said reconstruction andintroducing distortions in the amplitude of the sampled signal that are unrelated toany underlying amplitude modulation phenomena that may exist in the continuoussignal. As the signal frequency increases, so does the distortion, since there will befewer samples for each signal period. If the sampling frequency fs is sufficiently high,the errors introduced by this substitution are manageable in practice. This leads tousing oversampling to reduce the amplitude distortions of the discrete versionswithout a clear criterion for setting fs.

The discrete methods for the analysis of non-linear and non-stationary signals infinite intervals should be based on very good approximations of the continuoussignals if meaningful results are to be obtained. The correct sampling frequencyshould be the minimum possible which guarantees that the sampled signal can beused instead of the continuous one. In the case of empirical mode decomposition(EMD), having the amplitude of the sampled signal possess a great variability rangeversus the continuous signal hampers the method severely because the first step in theEMD consists of estimating the signal envelope, which is conditioned by its extremaand, thus, by its amplitude. If there are large amplitude deviations between bothmentioned signals, the envelopes returned by the method will be erroneous, leadingto decompositions whose interpretation lacks a clear physical meaning [23].

The main objective of this article is to address the sampling problem describedabove, which will allow the performance of Huang’s original algorithm, and byextension, of any modification thereof, to be significantly improved. First, we willpropose a minimum boundary for the sampling frequency, based on the range of theenvelope. Continuous sinusoidal signals have a constant envelope. However, discretesinusoidal signals envelopes are modified by the discretization errors. From period toperiod the maxima position does not correspond to the position of the maxima inthe continuous signal, producing amplitude smaller than the correct one. So, theenvelope will vary from 1 in those samples at the correct maxima position to a givenvalue smaller than one that corresponds to the maximum deviation from the correctmaxima position. The worst case is at half the sampling frequency in which theenvelope amplitude is of the same size than the sinusoidal amplitude. On the otherhand, for an empirical decomposition method to work correctly, it is necessary totake into account other considerations. A border effect usually appears due to a badadjust of the spline near the ends. The inner and outer criteria are critical forobtained decomposition. A bad election would produce a meaningless decomposi-tion. Energy must also be taken into account.

The sections in this article are divided as follows. First, the methods used aredescribed. An explanation of the experiments conducted and of their results isdiscussed next. Finally, the conclusions are presented.

2. Methods

The method being considered in this article is Huang’s original EMD [1]. We willtherefore start in Section 2.1 by explaining this method and some of the drawbacksassociated with its implementation. The improvements introduced in this article are

Applicable Analysis 691

Dow

nlo

aded

by [

Juan

J. T

ruji

llo]

at 0

4:4

0 0

9 J

uly

2011

the criterion for selecting the sampling frequency (Section 2.2), some energyconsiderations (Section 2.3), extrapolation to solve the border effect (Section 2.4)and inner and outer stopping criteria (Sections 2.5 and 2.6).

2.1. EMD algorithm

The purpose of the EMD method is to decompose a signal into a series of signals ofvariable amplitude and frequency that can be considered as amplitude and frequencymodulated signals. It is an iterative method which, save for the spline determination,has a polynomial algorithmic complexity. Given a discrete signal x[n], the EMDalgorithm utilizes a sifting method that can be summarized by the following steps:

. Verify whether the signal to be analysed satisfies the outer stopping criteria,for example, that the signal is not monotonic.

. Identify all the extrema of signal x[n].

. Compute the interpolating signals between maxima on the one hand andminima on the other that will correspond to upper envelope s1[n] and lowerenvelope s2[n], respectively, of the signal in question.

. Compute the mean of both envelopes:

M!n" # $s1!n" % s2!n"&2

: $1&

. Subtract the mean of both envelopes from the signal to obtain (which wewill call test signal in our method):

Sjfx!n"g # x!n" 'M!n": $2&

The process in Steps 2–5 (inner loop) is iterated over successive Sj{x[n]} and isstopped when a certain criterion is satisfied (e.g. that the difference between signalsSj{x[n]} and Sj'1{x[n]} is very small). IMF1#Sj{x[n]} is then referred to as the firstdetail component and residue1[n]# x[n]' IMF1 as the first residue. This criterionwill be named as inner stopping criterion in our method. The original signal x[n] isreplaced by residue1[n] and the process is iterated over Steps 1–6 (outer loop),obtaining the second IMF (IMF2) and the corresponding residual: residue2[n]#x[n]' IMF1' IMF2. The whole process (outer loop) is iterated as many times asnecessary, with each iteration of the outer loop yielding a new IMF. As a result,qIMF components are obtained such that x[n]# IMF1[n]% IMF2[n]% ( ( ( %IMFq[n]% residueq[n]. The process ends when a given criterion is satisfied (namedouter stopping criterion in our method). Since the number of extrema of a digitalsignal is finite, as a consequence of both the election of a finite working interval, andthat no accumulation points exists in the interval, the number of IMFs will also befinite if the stopping criterion of the inner loop is based in signal extrema.

Some authors have tried to offer improvements to the original EMD method[22–26] given the admitted problems of the Huang algorithm, such as: (1) sampling:since the algorithm operates on digital signals, the maxima and minima must becorrectly identified. A prerequisite to do this involves oversampling the signal inexcess of the limit given by Nyquist so as to guarantee the correct positioning of theextrema. (2) Envelope determination: the envelopes are determined based on curves

692 J.L. Sanchez and J.J. Trujillo

Dow

nlo

aded

by [

Juan

J. T

ruji

llo]

at 0

4:4

0 0

9 J

uly

2011

interpolated between the extrema. Normally cubic splines are used. However,another result could be obtained by changing the interpolator. (3) Boundaryconditions: the assumptions for the boundary values of the splines are crucial foravoiding propagation errors in the various IMFs. (4) Stopping criteria: the siftingalgorithm (iterative subtraction of the mean) requires two stopping criteria, one todetermine that one iteration is complete and an IMF has been obtained, and anotherto determine when to finish the signal decomposition. (5) Frequency resolution: theoriginal EMD method fails when attempting to separate modes with frequencieswhose range is less than an octave.

In this article, we present several improvements to the method for the optimalselection of the sampling frequency, border effects and stopping criteria.

2.2. Method proposed for selecting the sampling frequency

This problem has already been addressed by various researchers. Among otherauthors: (1) Rilling [24] states that the absolute minimum sampling period must beat least half the distance between extrema or, put another way, twice the Nyquistfrequency. (2) Stevenson [25] sets his sampling limit to five times the Nyquistfrequency so as to improve EMD performance. This means a fivefold increase in thenumber of signal points, which affects its computational cost significantly. (3) Rato[23] proposes a modification of the algorithm which efficiently addresses thedrawbacks mentioned above. With regard to the sampling, however, he does notfocus on the signal, but rather on its envelope, since the first step in the algorithmis the determination of the upper and lower envelopes of the signal. In order tocorrectly locate the maxima through which the envelope passes, it is necessary todetermine its position as accurately as possible. To do this, he interpolates the signalbetween the neighbour immediately before and after each extreme so as to improvethe localization and individualization of extrema. The classical interpolation forband-limited signals relies on the use of a computationally expensive kernel sinc.Rato proposes the use of a parabolic interpolation as an intermediate solutionbetween the non-use of interpolation and the use of a sinc-based interpolation.

Although data is usually obtained from continuous time process, the EMDalgorithm works on digital signals. As we have mentioned before, a digital signalcannot be used instead of the continuous one if it has not be interpolated with akernel sync to reduce the discretization errors. On a digital signal, as the samplingfrequency increases, the digital version is more similar to the continuous one, becauseof the reduction in the sampling period. That makes some authors to recommendoversampling to work with EMD method. However, it is under discussion that howmuch oversampling must be applied as a big increase in the sampling frequency alsoproduces a big increase in the number of samples and in the computation time.So, one important question is the election of the correct sampling frequency. Thesampling theorem for Fourier Transform states that the sampling frequency must beat least two times the maximum frequency for a band limited signal. However,modulated signals, particularly if they are frequency modulated, cannot beconsidered band limited. A new rule must be stated for the election of the correctsampling frequency. In that case we will not make the supposition that the signal isband limited but that the signal has most of its energy concentrated below a given

Applicable Analysis 693

Dow

nlo

aded

by [

Juan

J. T

ruji

llo]

at 0

4:4

0 0

9 J

uly

2011

frequency fc, being the amplitude of higher frequencies smaller than a giventhreshold d, as shown in Figure 1.

In practice, it is not necessary to obtain that cutoff frequency for every signal.Instead, it must be obtained from a set of control signals representing the sameprocess. The cutoff frequency must verify the condition for all the control signals. Ifthe control signals are not correctly selected, the condition could not be held forsome experimental data.

It is well known that the sifting procedure of the EMD algorithm removes thedissymmetry between upper and lower envelopes to transform the original signal intoan AM signal (whose envelopes are symmetric). If we consider an AM continuoustime signal, it is clear than its envelopes will be symmetric. However, can we considersymmetric the envelopes of its digital version? Discretization errors make theextrema position to be not located at the same position in both the discrete andcontinuous signals. So the amplitudes of the extrema will differ in both cases. Thesimpler example is in the case of sinusoidal signals. They have constant amplitudeenvelopes in the continuous case. However, discrete sinusoidal signals haveenvelopes which are oscillating as the extrema position varies from the correctposition (maximum envelope amplitude) to the most deviated from the continuouscase (minimum envelope amplitude).

In Figure 2 it can be observed that the envelope range increases with thefrequency. Figure 2(a) shows a sinusoid with a low range of variation in theamplitude of its envelope as the frequency is very low compared with fs/2. Figure 2(b)shows a higher range in the oscillation of the envelopes for a frequency near fs/4.For frequencies bigger to fs/4, an amplitude modulation effect between the signalfrequency and the sampling frequency appears so that the envelopes have widerranges. Figure 2(c) shows a sinusoid of 0.36Hz which is bigger than fs/4 andFigure 2(d) shows a sinusoid of 0.49Hz which is near to fs s/2. It can be appreciated

Figure 1. Fourier spectrum of a signal whose energy is concentrated below a given frequencyfc. From that frequency in advance the amplitudes of the spectrum are smaller than thethreshold d.

694 J.L. Sanchez and J.J. Trujillo

Dow

nlo

aded

by [

Juan

J. T

ruji

llo]

at 0

4:4

0 0

9 J

uly

2011

that the number of periods in the envelope diminishes as the frequency approaches tofs s/2 from fs/4 and increases from 0 to fs/4. When fs4 fs/4, the envelope range is nearor above to 30% of the signal’s amplitude while it is supposed to be zero in thecontinuous case.

In Figure 3 it can be observed how the range of the envelope increases withfrequency. The blue line represents the case in which fs! 2 * fmax wherefmax! 0.49Hz. It is remarkable at the valleys at 0.25Hz ( fs/4), 0.20Hz ( fs/5),0.125Hz ( fs/8) and 0.10Hz ( fs/10). If we increase the sampling frequency to thedouble, maintaining the same range of interest, the maximum deviation (green line) isthe one that reaches the first curve just before the valley at 0.25Hz. That means thatthe new maximum amplitude is only a 28% of the first one. If we increase thesampling frequency three times, the maximum deviation is the same that occurs inthe blue line for 0.166Hz, which is approximately 13% of the initial maximumamplitude. It can be observed that a higher increase in the sampling frequency doesnot represent much higher contribution to the reduction of the range of theamplitude. This effect is due to the fact that the curve of the amplitude of theenvelope has an exponential shape, growing very slowly at the beginning.

Taking into account this and Figure 1, we can conclude that a good election forthreshold d is 0.13 of the maximum spectrum amplitude. That makes discretizationerrors of the higher amplitudes components to be as low as possible. In other case thediscretization error for the higher amplitude components will be too big resulting in

Figure 2. Several sinusoidal signals and its envelopes. fs! 1Hz in all of them. (a) f! 0.6Hz,(b) f! 0.24Hz, (c) f! 0.36Hz and (d) f! 0.49Hz.

Applicable Analysis 695

Dow

nlo

aded

by [

Juan

J. T

ruji

llo]

at 0

4:4

0 0

9 J

uly

2011

extra IMF’s in the decomposition. In real signals most of the energy is concentratedin the low frequencies, so this condition is possible to be satisfied by real worldsignals.

2.3. Energy and the EMD

Many works have been done trying to apply the EMD to real signals and to finda physical interpretation of each component. A lot of effort has been done aboutthe definition of instantaneous frequency to be meaningful. However, energeticdescription has not been incorporated to the EMD, except for the interpretation ofthe instantaneous amplitude and for the stopping criteria in [23]. In this article wewill use the energy to check the quality of the decomposition (Figure 4). Averageenergy will be used as it has a smaller dynamic range than the total energy. Theaverage Energy of a signal of length N is defined as

AE ! 1

N

XN

i!1

x"i #2 i ! 1, . . . ,N: $3%

The decomposition acts over an original signal (OS) with an average energyAEOS, producing a set of IMF, each of one has an average energy AEIMF( j)j! 1, . . . ,M, where M is the number of IMF obtained by the decomposition. Thesum of all the components will produce a synthesized signal with an average energyAESIMF and the sum of the average energies of each IMF will have a total value ofAETIMF.

The first basic principle for a decomposition to be valid is that the sum of all theIMF will reconstruct the original signal. That implies that the energy of the sum ofthe IMF (AESIMF) will be equal to the original signal energy (AEOS).

Figure 3. Range of the envelope for sinusoids with frequencies from f! 0 to f! 0.49Hz fordifferent sampling frequencies ranging from fs! 1 to fs! 5Hz. The observed frequency rangeis constant between 0 and 0.5Hz.

696 J.L. Sanchez and J.J. Trujillo

Dow

nlo

aded

by [

Juan

J. T

ruji

llo]

at 0

4:4

0 0

9 J

uly

2011

The second basic principle is that the sum of the energies of all the IMF will alsobe equal to the energy of the original signal.

If the second principle is not satisfied, several situations can occur: (a) new energyis created in the decomposition by the effect of components which are practicallycancelled when they are added. That kind of decomposition cannot be consideredmeaningful physically. (b) Due to border effects the synthesized signal cannotreproduce the original signals near the borders, so some energy could be created orlost depending on the spline adaptation to the envelope. (c) If discretization errorsare important, some false IMF are created with no physical meaning at all. Thedecomposition will be better if the difference between AETIMF and AEOS is assmall as possible.

2.4. Border effect

A very well-known problem with the EMD algorithm is related to boundaryconditions. If the splines are poorly related to the true envelope, an error thatappears in the first IMF can be propagated to the following IMF, moving itselftowards the centre as the frequency band becomes narrower. As it does notremain in a fixed position, no truncation of the signal can be done to eliminatesuch error. The importance of border effects is that some aliased IMF areproduced just to compensate it, while they do not belong to the continuous signalat all. To minimize the number of IMF, it is necessary to control the bordereffect.

Different authors have considered this problem. Rilling [27] had obtainedgood results mirrorizing the extrema close to the edges. Rato [23] locate the first

Original signal x(n)

Average energy AEOS

IMF1 average energy

AEIMF1

IMF2 average energy

AEIMF2

Synthesized signal

y(n)= IMF1+IMF2

Average energy AESIMF

Sum average energy IMF’s

AETIMF=AEIMF1 +AEIMF2

Figure 4. Energetic description of EMD decomposition.

Applicable Analysis 697

Dow

nlo

aded

by [

Juan

J. T

ruji

llo]

at 0

4:4

0 0

9 J

uly

2011

maxima and minima of the signal and replicate them at the mirror positions inthe left side of the signal. The same procedure is done with the last maxima andminima. However, the risk in the extrapolation of a signal grows with thedistance from the last signal point. This procedure can fail if the extremes followan increasing or decreasing trend as the repetition will force to smalleramplitudes.

Our solution is related to Rato, in the sense of considering only the extrema andnot the signal. A line will be fitted between the two first maxima and the two firstminima. The independent term will give us the value of the fitted line in zeroposition. Equally, a line will be fitted between the last two maxima and minima. Theline fitted up to point N (length of the signal) will be extrapolated to point N! 1. Incase a change in the trend of the signal appears in the very beginning or end of thesignal, the amplitude of the signal will differ from the extrapolated line. If theamplitude on the edges is bigger than the maximum of the peaks of the upperenvelope or lower than the minimum of the peaks of the lower envelope, then thevalues of the splines at 0 and N! 1 are corrected. The correction implies to add orsubtract the difference between the first (last) value of the signal and the spline fittedone. With this procedure no suppositions are made about points at distance biggerthan one sample in the signal.

Figure 5 shows the envelopes of a sinuosoidal signal with f" 0.037 and n" [0:199]( fs" 1). Huang’s original method ties the envelopes to zero at both extremes wheninterpolation is done fixing the values of the splines at 0 and N! 1 (where N is thelength of the signal) to zero. As a result, big overshoot of the envelopes can occurnear both edges. The mean of the envelopes will differ from the true mean of theenvelopes originating false components. Figure 6 shows the envelopes of the samesinusoidal signal calculated with our method. In that case, a perfect reconstruction of

Figure 5. Envelopes for a sinusoidal signal with fade out by the original Huang method. Thesplines are tied to 0 at the borders.

698 J.L. Sanchez and J.J. Trujillo

Dow

nlo

aded

by [

Juan

J. T

ruji

llo]

at 0

4:4

0 0

9 J

uly

2011

the envelopes is achieved and no false components are generated because of thoseborder errors.

2.5. Inner stopping criteria

Huang focus its inner stopping criteria just on the consideration of change betweenthe signal obtained on a given iteration and the one obtained in the previousiteration. If they are very similar, the loop stops. Arbitrarily, the threshold is set onsd! 0.1. However, there is no justification of this value except that it means bothsignals are very similar.

On the other hand, Rato [23] defines a resolution factor by the ratio between theenergy of the signal at the beginning of the sifting, x(t), and the energy of average ofthe envelopes, m(t). If this ratio grows above the allowed resolution, then the IMFcomputation must stop. This ratio is expressed in dB. Although this is a betterstopping criteria than the Huang’s one with more physical meaning, the drawback isthe convergence time to achieve a given value. High values of the resolution factor,such as 40 dB, could require hundred of iterations which make the time to get thesolution too long. However, we will use this criterion as a limit for the decomposition.

Before exposing our stopping criteria we will define some previous terms: As wehave mentioned before, a continuous signal and its discrete version could be verydifferent if no reconstruction has been done. While for a continuous signal thatsatisfies the requirements of an IMF, it is clear that its envelopes are absolutelysymmetric, a discrete version of an IMF has discretization errors which results innone perfectly symmetric envelopes. To measure the error in the symmetry of bothenvelopes, we define

Esim ! mean"jjs1j# js2jj$: "4$

Figure 6. Envelopes for a sinusoidal signal with fade out in our method.

Applicable Analysis 699

Dow

nlo

aded

by [

Juan

J. T

ruji

llo]

at 0

4:4

0 0

9 J

uly

2011

The default value is 2e–3 as it represents a very small error in the symmetry.DbqEsim is defined as the ratio between the Esim value in the actual iteration andthe Esim value in the previous iteration expressed in dB.

DbqEsim ! 20 log 10Esim

Esim"k# 1$

! ", "5$

where k represents the iteration. Normally Esim grows very slowly with iterationsso DbqEsim tends to a small value. As DbqEsim takes bigger values in the firstiterations, a threshold can be put so that if its value is smaller than the threshold, thecomputation must stop. DbqResol is defined in [23]. However we will not use suchparameter as an objective to reach but as a limit in the number of iterations.

It is well known that the sifting procedure is an iterative way of removing thedissymmetry between the upper and lower envelopes [23]. Our stopping criterion iscomposed of three conditions with a logical AND between them. The first conditionis related to symmetry. The inner loop must be stopped when the error in thesymmetry reaches a user defined threshold value (TH1). We recommend a value ofTH1! 2e–3. The obtained IMF will have the envelopes much more symmetric thanif the Huang criterion is applied, as generally it takes some steps more than Huang toobtain the IMF. The second condition is related to the rate of change in the Esimparameter. If Esim grows very slowly, DbqEsim reaches very low values. Putting athreshold to DbqEsim, we satisfy two purposes:

The iteration must follow until the threshold is reached. If DbqEsim is set to#3Db, as soon as the Esim parameter will change at a lower rate, the iteration willbe finished. Sometimes, when long iterations are done, discretization errors appear inthe position of extrema producing a bigger discretization error than in the previousstage. In that case DbqEsim will be positive and of course bigger than the threshold,so the computation will also be finished retaining as solution the previous signalbefore the error appears. The third condition acts as a limit in the number ofiterations. In some occasions DbqEsim takes nearly the same value during a lot ofiterations. It will have the effect of a never-ending loop if the Esim value do notreaches its threshold before. As in every iteration the DbqResol increases, athreshold can avoid an excessive number of iterations.

In summary, our stopping criterion is

Esim4TH1 & DbqEsim4TH2 & DbqResol5TH3: "6$As the loop will be finished as soon as one of the conditions will be satisfied, we

are dealing with an optimal algorithm in the sense of obtaining parameters as close tothe thresholds as possible with as few iterations as possible.

2.6. Outer stopping criteria

Huang considers the monotony of the obtained signal as the only one requirement tostop the procedure for obtaining IMF. That tends to produce too many IMF, even ifthey do not exist. For example, if we know that our signal is composed by a numberN of pure IMF, the number of obtained IMF must be no bigger than N. However,energy will be used in our algorithm as a second condition. The decomposition willbe finished as soon as AETIMF will become bigger than AEOS. If on iteration NAETIMF5AEOS and on iteration N% 1 AETIMF4AEOS, a correction stage

700 J.L. Sanchez and J.J. Trujillo

Dow

nlo

aded

by [

Juan

J. T

ruji

llo]

at 0

4:4

0 0

9 J

uly

2011

is necessary. The obtained IMFs for N and N! 1 will be combined at stage N againto keep AETIMF as close as possible to AEOS.

2.7. EMD decomposition

The EMD decomposition as given by Huang consists of only IMF components whosenumber of extrema is decreasing on every step until reaching a signal with only oneextrema. This is a very mathematical description that does not give information abouttheir physical content. In our case, our decomposition has the next components:(a) the IMF detected with an important energy compared with the original signal.(b) An IMF whose energy is below the threshold, established to consider it as an IMF.That component usually is very small and even can be consider a residual. (c) Themean of the signal is added as an IMF as its envelopes are symmetrical (equal to themean) because it has an important component of energy if it is different from zero.The residual itself is obtained after subtracting the whole set of IMF to the originalsignal. It is a zero mean signal in order to have the minimum energy as possible. Thissecond residual only has importance if minor edge errors remain after decomposition.Normally the residual will be the first IMFwhich satisfies the condition to be residual.

3. Experimental results

In this section the different experiments carried out are shown.

3.1. Sinusoidal signal

Sinusoidal signals are the simpler IMF that can be defined (apart from a continuousone). Its envelopes are straight parallel lines. A good decomposition must show onlyone component.

AETIMF is bigger than AESIMF by more than two times (Table 1) due to thegreat number of components obtained (Figure 7). This amount of extra energy canonly be explained by the use of components that cancel between them. It can beobserved in Figure 8 that our method obtains only two components (Table 2) and thesecond is due to discretization errors. In that case AETIMF is of the same order ofmagnitude than AEOS (slightly bigger).

Figure 9, sampled at 3Hz shows a second component with much smalleramplitude (Table 3) than when sampled at 1Hz. That means such component is dueto discretization errors as its energy diminish two orders of magnitude just due tochanging the sampling frequency. Moreover, in both cases the second componentsatisfies the condition for being a residue. So, we can consider that our algorithmextracts correctly the IMF.

3.2. Adding two IMFs

To illustrate the differences between our algorithm and others, a signal composed bya sum of two IMF is used.

X1 " sin#2!f0n$, X2 " 2 % j sin#2!f1n$j& 1, X " X1! X2, n " 0, 1, . . . , 99:

Applicable Analysis 701

Dow

nlo

aded

by [

Juan

J. T

ruji

llo]

at 0

4:4

0 0

9 J

uly

2011

This signal has been used by Rato [23] to compare its method with the Rillingroutine. In both cases six IMF are produced while two must be the correct numberaccording to Rato. Our decomposition gives us a total of only 2 components(Figure 10) with the next energy information. Table 4 shows a quantitativedescription of the energy of obtained IMF. Only two components are obtained for asignal composed of two IMF’s.

Figure 7. Original Huang’s decomposition of a sinusoidal signal with f! 0.11 and n! [0:199].The upper component is the original signal.

Table 1. Energy of Huang’s decomposition for fs! 1Hz.

IMF AEOS AEIMF AETIMF AESIMF

1 0.292 0.292 0.292 0.2922 0.292 0.121 0.413 0.4093 0.292 0.235 0.648 0.3204 0.292 0.028 0.676 0.2925 0.292 6.6e" 5 0.676 0.292

Table 2. Energy of our method decomposition for fs! 1Hz.

IMF AEOS AEIMF AETIMF AESIMF

1 0.292 0.292 0.292 0.2922 0.292 2e" 4 0.292 0.292

702 J.L. Sanchez and J.J. Trujillo

Dow

nlo

aded

by [

Juan

J. T

ruji

llo]

at 0

4:4

0 0

9 J

uly

2011

Figure 9. Decomposition of a sinusoidal signal with f! 0.11 and n! [0:1/3:199] ( fs! 3Hz).Upper graphic shows the original signal. The upper component is the original signal.

Figure 8. Decomposition by our method of the sinusoidal signal with f! 0.11 and n! [0:199]( fs! 1Hz). The upper component is the original signal.

Applicable Analysis 703

Dow

nlo

aded

by [

Juan

J. T

ruji

llo]

at 0

4:4

0 0

9 J

uly

2011

3.3. The chirp signal

The chirp signal is a good test for the algorithm. On the one hand, it satisfies thedefinition of an IMF because the continuous version has constant amplitude.Moreover, frequency ranges in a wide range so that constant amplitude for thediscrete signal is not obtained because of the discretization errors. This signal is also

Figure 10. Decomposition of the signal X1!X2 using our algorithm. The upper component isthe original signal.

Table 3. Energy of our method decomposition for fs" 3Hz.

IMF AEOS AEIMF AETIMF AESIMF

1 0.292 0.292 0.292 0.2922 0.292 3e# 6 0.292 0.292

Table 4. Energy description of the EMD decomposition.

IMF AEOS AEIMF AETIMF AESIMF

1 0.954 0.363 0.363 0.3622 0.954 0.588 0.951 0.954

704 J.L. Sanchez and J.J. Trujillo

Dow

nlo

aded

by [

Juan

J. T

ruji

llo]

at 0

4:4

0 0

9 J

uly

2011

used by Rato [23] with the result of being unmodified by the algorithm, whilethe Rilling algorithm obtains 10 IMFs. The chirp signal is calculated fromcos(2pin2/4507! 2pin/213), with 05" n5" 999 ( fs" 1Hz). This is a problemhighly dependent of the sampling frequency.

Figure 11(c) shows that the 13% level of the maximum amplitude of the spectrumis very near to fs/2. Increasing the sampling frequency by 3 from the thresholdfrequency, we will increase the sampling frequency three times just by doing n" 0:1/3:999. We now observe that the low-pass part of the spectrum is far away of the fs/2limit (Figure 11(d)). Correspondingly, the Figure 11(b) is much more approximate toa constant level IMF than the first one (Figures 11(a) and 12). In Figure 13important discretization errors in the envelope can be observed.

Figure 13 shows how the second IMF corresponds practically to thediscretization errors showed in Figure 12. From the energy description in Table 5,it can be observed that the important quantity of energy is generated on thedecomposition (AETIMF42 AEOS). That is due to the generation of componentswhich cancels between them. With our method, only two components are generatedto decompose the chirp signal (Table 6). From the energy description it can beobserved that the second component satisfies the condition for being residue (lessthan 1e–2 the energy of the original signal), so we can consider the chirp restsundecomposed by our algorithm as it happens in Rato [23]. Figure 14 shows that thisresidue is due to discretization errors in the higher frequencies.

Figure 11. Chirp for (a) fs" 1 and (b) for fs" 3, (c) normalized Abs(fft(x)) for fs" 1(500 samples), (d) normalized abs(fft(y)) (1500 samples).

Applicable Analysis 705

Dow

nlo

aded

by [

Juan

J. T

ruji

llo]

at 0

4:4

0 0

9 J

uly

2011

3.4. EEG signal

We will check our algorithm with an EEG data set. This data has been downloadedfrom project BCI [28]. This is the subject of a 21-year-old, right-handed male with noknown medical conditions. The EEG consists of actual random movements of left

Figure 13. Decomposition by Huang’s original method of the chirp signal. The uppercomponent is the original signal.

Figure 12. First iteration to obtain the IMF by Huang’s method.

706 J.L. Sanchez and J.J. Trujillo

Dow

nlo

aded

by [

Juan

J. T

ruji

llo]

at 0

4:4

0 0

9 J

uly

2011

Figure 14. Decomposition of the chirp signal by our method. The upper component is theoriginal signal.

Table 5. Energy description of the chirp decomposition by Huang’smethod with fs! 1.

IMF AEOS AEIMF AETIMF AESIMF

1 0.501 0.508 0.508 0.5082 0.501 0.005 0.514 0.5033 0.501 9e" 4 0.515 0.5024 0.501 4e" 4 0.515 0.5025 0.501 0.007 0.523 0.5096 0.501 0.331 0.853 0.8157 0.501 0.209 1.063 0.5168 0.501 0.014 1.077 0.501

Table 6. Energy description of the chirp decomposition by our methodwith fs! 1.

IMF AEOS AEIMF AETIMF AESIMF

1 0.502 0.509 0.509 0.5022 0.502 0.009 0.519 0.502

Applicable Analysis 707

Dow

nlo

aded

by [

Juan

J. T

ruji

llo]

at 0

4:4

0 0

9 J

uly

2011

and right hand recorded with eyes closed. Each row represents one electrode. Theorder of the electrodes is FP1 FP2 F3 F4 C3 C4 P3 P4 O1 O2 F7 F8 T3 T4 T5 T6 FZCZ PZ. The recording was done at 500Hz using Neurofax EEG System which usesa daisy chain montage. The data was exported with a common reference usingEemagine EEG. AC Lines in this country work at 50Hz. The signal analysed is FZleft because it is one of the more complex signals in the register. In the first place, wewill observe the result from the original Huang’s algorithm. Twelve IMFs areobtained, most of them requiring only 2 or 3 steps while the 6th requires 5 steps to becalculated.

Figure 15 shows a big end effect in the right side due to the original signal is veryfar from zero at this edge. This creates an important energy in the 6th IMF andpropagates in the rest to the centre of the signal.

The energetic description of the decomposition is shown in Tables 7 and 8. It canbe observed that although AESIMF equals AEOS, AETIMF is considerably higher

Figure 15. Seven first IMF of the Decomposition by Huang’s original method of the EEGsignal. The upper component is the original signal.

Table 7. Global energy parameters of Huang’s decomposition forthe EEG signal.

AEOS AETIMF AESIMF

3.4276e! 003 1.3429e! 004 3.4276e! 003

708 J.L. Sanchez and J.J. Trujillo

Dow

nlo

aded

by [

Juan

J. T

ruji

llo]

at 0

4:4

0 0

9 J

uly

2011

(one order of magnitude), which means a lot of energy has been created due to theedge effect mainly. This means not all of them can be considered to have a physicalmeaning.

In the second place, we will observe the results from Rato [23] algorithm(Tables 9 and 10). While Huang tries not to iterate too much to obtain the IMFs,Rato goes one step beyond and parameterizes the evolution not in term of the wellknow sd Huang’s parameter but in terms of the rate of energy among the signal andthe mean of the envelopes (expressed in dB). Better IMFs are obtained as the meanof the envelopes approach to zero. With this scheme, Rato algorithm with a qResolparameter of 10 dB decomposes this signal in nine components. The first question tobe solved with this algorithm is the parameter election. While it is recommendeda value between 40 and 60, a first trial with 40 dB was done resulting in badconvergence time. After 175 iterations, dBqResol only achieves to 26 dB. This meanswe must relax the stopping criteria. If we reduce the qResol parameter to 25 dB, we

Table 10. Detailed energy of Rato’s decomposition for theEEG signal.

IMF AEIMF IMF AESIMF

!103

1 0.024 7 0.022 0.117 8 0.6433 0.009 9 0.9434 0.006 10 1.4795 0.004 11 1.1956 0.003 12 1.449

Table 8. Detailed energy of Huang’s decomposition for theEEG signal.

IMF AEIMF IMF AESIMF

!103

1 0.117 7 3.0922 0.066 8 4.9733 0.015 9 3.8234 0.006 10 0.8405 0.005 11 0.3796 0.109 12 0.775

Table 9. Global energy parameters of Rato’s decompositionfor the EEG signal.

AEOS AETIMF AESIMF

3.4276e" 003 5.8960e" 003 3.4276e" 003

Applicable Analysis 709

Dow

nlo

aded

by [

Juan

J. T

ruji

llo]

at 0

4:4

0 0

9 J

uly

2011

obtain 11 IMF of which the first requires 72 iterations while the rest requires about10 iterations. From Figure 16 it can be observed that no big end effects are shown,meaning that the Rato’s method is a great improvement from the Huang’s one.While AESIMF equals AEOS as it was expected, AETIMF is 1.7 times bigger thanAEOS. This means the components are not completely independent and some extraenergy is created (Tables 11 and 12).

In Figure 17 the decomposition by our method can be observed. The first fact isthat it has only nine components versus 12 of the other methods. That is due to thesecond outer loop stopping criteria, the ATIMF condition. That is, the decompo-sition must satisfy the second criteria exposed in the introduction: Not only the sumof the components must have the same energy than the original signal, but also the

Figure 16. First 8th IMF of the Rato’s algorithm with qResol! 25 dB. The upper componentis the original signal.

Table 11. Global energy parameters of our algorithm decom-position for the EEG signal.

AEOS AETIMF AESIMF

3.4276e" 003 4.6908e" 003 3.4276e" 003

710 J.L. Sanchez and J.J. Trujillo

Dow

nlo

aded

by [

Juan

J. T

ruji

llo]

at 0

4:4

0 0

9 J

uly

2011

Figure 17. Decomposition of the EEG signal by our method. The upper one is the originalsignal.

Table 12. Detailed energy of our algorithm decomposition forthe EEG signal.

IMF AEIMF IMF AEIMF

!103

1 0.024 7 0.022 0.117 8 0.6433 0.009 9 0.9434 0.006 10 1.4795 0.004 11 1.1956 0.003 12 1.449

Table 13. Global energy parameters of our algorithm decom-position for the EEG signal.

AEOS AETIMF AESIMF

3.4276e" 003 3.601e3 3.4276e" 003

Applicable Analysis 711

Dow

nlo

aded

by [

Juan

J. T

ruji

llo]

at 0

4:4

0 0

9 J

uly

2011

sum of the energies of each component must be approximately equal to the energyof the original signal (some energy can be created or destroyed by the iterativeprocedure and by the spline interpolation). Further decomposition of the lowfrequencies takes as a consequence an ATIMF energy much higher than the AEOS,as it can be observed in Table 11. From Tables 13 and 14 it can be observed thatAETIMF in our decomposition is very similar to AEOS, so our decomposition is themore physically meaningful of the three.

References

[1] N.E. Huang, Z. Shen, S.R. Long, C. Manli, H.H. Shih, Q. Zheng, N. Yen, C.C. Tung,and H.H. Liu, The empirical mode decomposition and Hilbert spectrum for nonlinear andnonstationary time series analysis, Proc. R. Soc., London, A 454 (1998), pp. 903–995.

[2] W. Yang, Interpretation of mechanical signals using an improved Hilbert–Huang transform,Mech. Syst. Signal Pr. 22 (2008), pp. 1061–1071.

[3] R. Agarwal and J. Gotman, Adaptive segmentation of electroencephalographic data usinga nonlinear energy operator, ISCAS’99, Orlando, 1999.

[4] N. Martins and A. Rosa, EEG non-stationary spectrum analysis and feature extraction,IEEE International Conference on Systems, Man and Cybernatics: Information,Intelligence and Systems, Beijing, 1996.

[5] H. Liang, Q. Lin, and J.D. Chenv, Application of the empirical mode decomposition to theanalysis of esophageal manometric data in gastroesophageal reflux disease, IEEE Trans.Biomed. Eng. 52 (2005), pp. 1692–1701.

[6] N.E. Huang, Z. Shen, and S.R. Long, A new view of nonlinear water waves: The HilbertSpectrum 1, Annu. Rev. Fluid Mech. 31 (1999), pp. 417–457.

[7] J. Nunes, Y. Bouaoune, E. Delechelle, O. Niang, and P. Bunel, Image analysis bybidimensional empirical mode decomposition, Image Vision Comput. 21 (2003),pp. 1019–1026.

[8] N.E. Huang and S.S. Shen, Hilbert–Huang Transform and its Applications,Interdisciplinary Mathematical Sciences, Vol. 5, World Scientific, New Jersey, 2005.

[9] M.A. Andrade, A.R. Messina, C.A. Rivera, and D. Olguin, Identification of instantaneousattributes of torsional shaft signals using the Hilbert transform, IEEE Trans. Power Syst. 19(2004), pp. 1422–1429.

[10] A.R. Messina and V. Vittal, Nonlinear, non-stationary analysis of interarea oscillations viaHilbert spectral analysis, IEEE Trans. Power Syst. 21 (2006), pp. 1234–1241.

Table 14. Detailed energy of our algorithm decomposition forthe EEG signal.

IMF AEIMF IMF AESIMF

!103

1 0.101 7 0.5212 0.070 8 0.3153 0.014 9 2.56284 0.0075 0.0046 0.004

712 J.L. Sanchez and J.J. Trujillo

Dow

nlo

aded

by [

Juan

J. T

ruji

llo]

at 0

4:4

0 0

9 J

uly

2011

[11] A.R. Messina, V. Vittal, D. Ruiz-vega, and G. Enrv, Interpretation and visualization ofwide-area PMU measurements using Hilbert analysis, IEEE Trans. Power Syst. 21 (2006),pp. 1763–1771.

[12] Z. Lu, J. Smith, Q. Wu, and J. Fitch, Empirical mode decomposition for power qualitymonitoring, IEEE/PES. Transmission and Distribution Conference, Dalian Xinghai,China, 2005.

[13] M.B. Priestley, Spectral Analysis and Time Series, Academic Press, London; New York,1981.

[14] E. Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev. 40(1932), pp. 0749–0759.

[15] A.W. Rihaczek, Signal energy distribution in time and frequency, IEEE Trans. Info.Theory 14 (1968), pp. 369–374.

[16] R. Fano, Short time autocorrelation functions and power spectra, J. Acoustical Soc. Amer.22 (1950), pp. 546–550.

[17] M. Schroeder and B. Atal, Generalized short-time power spectra and autocorrelationfunctions, J. Acoustical Soc. Amer. 34 (1962), pp. 1679–1683.

[18] A. Oppenheim, Speech spectrograms using the fast Fourier transform, IEEE Spectr. 7(1970), pp. 57–62.

[19] A. Haar, Zur Theorie der orthogonalen Funktionsysteme (Zweite Mitteilung), Abh.Geschichte Math. 71 (1912), pp. 38–53.

[20] I. Daubechies, Ten Lectures on Wavelets, SIAM, Philadelphia, PA, USA, 1992.[21] G. Rilling and P. Flandrin, One or two frequencies? The empirical mode decomposition

answers, IEEE Trans. Signal Process. 56 (2008), pp. 85–95.[22] C. Junsheng, Y. Dejie, and Y. Yu, Research on the intrinsic mode function (IMF) criterion

in EMD method, Mech. Syst. Signal Pr. 20 (2006), pp. 817–824.[23] R. Rato, M. Ortigueira, and A. Batista, On the HHT, its problems, and some solutions,

Mech. Syst. Signal Pr. 22 (2008), pp. 1374–1394.[24] G. Rilling and P. Flandrin, On the influence of sampling on the empirical mode

decomposition, IEEE ICASP, Tolousse, France, 2006.[25] N. Stevenson, M. Mesbah, and B. Boashash, A sampling limit for the empirical mode

decomposition, ISSPA-05, Sydney, Australia, 2005.[26] N. Senroy, S. Suryanarayanan, and P.F. Ribeiro, An Improved Hilbert–Huang method for

analysis of time-varying waveforms in power quality, IEEE Trans. Pow. Syst. 22 (2007),pp. 1843–1850.

[27] G. Rilling, P. Flandrin, and P. Gonzalves, On empirical mode decomposition and itsalgorithms, IEEE-EURASIP Workshop on Nonlinear Signal and Image ProcessingNSIP-03, Grado, Italy, 2003.

[28] Brain Computer Interface Research at NUST, Pakistan, Project BCI – EEG motoractivity data set, available from: http://sites.google.com/site/projectbci/.

Applicable Analysis 713

Dow

nlo

aded

by [

Juan

J. T

ruji

llo]

at 0

4:4

0 0

9 J

uly

2011

Dow

nlo

aded

by [

Juan

J. T

ruji

llo]

at 0

4:4

0 0

9 J

uly

2011