AN ENHANCED EMD ALGORITHM FOR ECG SIGNAL PROCESSING

Foteini Agrafioti, Student Member, IEEE and Dimitrios Hatzinakos, Senior Member, IEEE

The Edward S. Rogers Sr. Department of Electrical and Computer Engineering,

University of Toronto,

10 King’s College Road, Toronto, ON, Canada, M5S 3G4

{foteini,dimitris}@comm.utoronto.ca

ABSTRACT

The Empirical Mode Decomposition (EMD) is becomingincreasingly popular for the multi-scale analysis of signals.However, the data-driven and adaptive nature of the EMDraises concerns regarding the uniqueness of the decomposi-tion as well as the extend to which oscillatory modes can bemixed across different IMFs. This paper proposes a solutionto this problem for the analysis of ECG signals. The bivariateextension of the decomposition (BEMD) is used as the basisof an analysis in which a synthetic ECG signal of idealizedwaveform guides the decomposition of an input ECG seg-ment. Essentially, this work provides the necessary groundfor the deployment of signal processing algorithms on theECG signal using a more robust EMD analysis.

Index Terms— Electrocardiogram, bivariate empiricalmode decomposition, intrinsic mode function

1. INTRODUCTION

The purpose of pattern recognition is to extract informationthat is originally hidden in the data. Multi-scale analysisis one way to address this problem. For one dimensionalphysiological signals such as the electrocardiogram (ECG),phonocardiogram (PPG), blood volume pressure (BVP), pho-toplethysmogram (PPG) and other that are widely studied indiagnostics, typical approaches include the Fourier transform,the spectogram, the cosine transform, the Wigner-Ville dis-tribution and wavelet analysis. Although all these approachesare well established both theoretically and in practice, theyare all employed on linearity and stationarity assumptions.However, this is not the case with most biosignals as thephysical process that generates them is neither stationary norlinear.

This gap in digital signal processing was emphasized byHuang et al.’s work [1], and the Empirical Mode Decompo-sition (EMD) was proposed as an alternative solution. Sincethen, a significant amount of work has been reported on theapplications of EMD along with several variations of the orig-inal algorithm [2, 3, 4, 5, 6]. The essence of EMD is that

it decomposes signals in a data-driven and adaptive mannerwithout linearity or stationarity restrictions.

For EMD, every signal is the superposition of fast oscil-lations over slow oscillations [2]. Despite the fact that manyreal life signals do not exhibit oscillations naturally (for exam-ple images), rapidly oscillatory components can be extractedto describe the finest high frequency characteristics of the sig-nal. Similarly, slow oscillations can be extracted to describelow frequency underlying phenomena that may exist in thesignal. In EMD terminology, the first are low order Intrin-sic Mode Functions (IMFs) while the latter are higher orderIMFs. It is important to note, that EMD can detect and extractthese components without prior information on the morpho-logical properties of the signal.

However, adaptivity is a mixed blessing. The benefit, asmentioned before, is that any signal can be processed witha decomposition that explores its intrinsic properties. Thedrawback of EMD is that it cannot be analytically defined.This causes uniqueness and mode mixing problems that limitits applicability. Since the decomposition depends on the sig-nal itself, the number of IMFs in which it will result in, isunpredictable. In addition, similar oscillatory modes may bepresented across different IMFs. This uncertainty poses greatthreats to the automatic deployment of EMD in signal pro-cessing. For instance, it is difficult to establish correspondingIMFs across different instances of the same signal (for exam-ple consecutive PPG recordings). This problem may be prac-tically addressed by forcing the decomposition to stop once apredefined number of IMFs is reached, however this does notsolve the mode mixing problem nor does it guarantee that theresulting IMFs will have physical meaning.

We advocate that this problem can be addressed by takinginto account the particular morphology of the signal in-hand.Although it is difficult to predict the appearance of a stochas-tic signal, one can take explore its general structure to en-hance the empirical decomposition. This work is interested inthe ECG signal, as it is widely examined by both the medical[7, 8, 9] and lately by the biometrics [10, 11, 12] commu-nities. A solution based on the Bivariate EMD (BEMD) [2]extension of the algorithm is presented.

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Fig. 1. Main components of an ECG heart beat.

The remaining of this paper is organized as follows. Sec-tion 2 provides an overview of the ECG from a signal process-ing point of view while Section 3 is a brief introduction to theEMD algorithm. The proposed decomposition is presentedin Section 4 and some experimental results are provided inSection 5.

2. THE ECG WAVEFORM

The ECG reflects the cardiac electrical potential over time.From a signal processing perspective, the ECG is a non pe-riodic but highly repetitive signal that is composed of threemain waves, as shown in Figure 1, typically referred to asthe P wave, QRS complex and the T wave. The P wave hasusually positive polarity and a duration of approximately 120ms. This wave mainly reflects the depolarization of the rightand left atria. The QRS complex describes the depolarizationof right and left ventricles. In normal rhythms, its durationvaries between 70-110 ms. Finally, the T wave reflects a de-polarization of the ventricles and is usually observed about300 ms after the QRS complex. However, its exact positionrelies on the heart rate and appears closer to the QRS complexat rapid rhythms [13].

The spectral characteristics of ECG waves are central tothe application of signal processing algorithms. A healthy Pwave is considered to contribute to the low frequency com-ponents at about 10-15 Hz. On the other hand, a QRS com-plex has a spectrum of comparably high frequencies due to itssteep slopes. The spectral content of this complex is usuallyfound in the 10-40 Hz band.

3. THE EMPIRICAL MODE DECOMPOSITION

This section provides an overview of the EMD algorithm, pro-posed by Huang et al. [1], in order to then extend it to the bi-variate case, and present the proposed solution. EMD decom-poses a signal adaptively into a number of IMFs. Each IMFdescribes a disctint oscillation and has the following charac-teristics [1]:

1. The number of extrema and the number of zero cross-ings must be equal or differ at most by one.

2. The mean of the envelopes that are defined by the max-ima and the minima is zero at every time instance.

EMD seeks for oscillations through a sifting process.Once an IMF is found, it is removed from the signal and thealgorithm iterates on the residual in order to find more oscil-latory modes. Fast oscillations (high frequency) are detectedfirst.

Given a signal 𝑥(𝑡), EMD operates as follows:

1. Detects local maxima 𝑥𝑚𝑎𝑥(𝑖) and minima 𝑥𝑚𝑖𝑛(𝑗) of𝑥(𝑡).

2. Interpolates among 𝑥𝑚𝑎𝑥(𝑖) to get an upper envelope𝑥𝑢𝑝(𝑡), and 𝑥𝑙𝑜𝑤(𝑡) for minima respectively.

3. Computes the average of the two envelopes 𝑚(𝑡) =𝑥𝑢𝑝(𝑡)+𝑥𝑙𝑜𝑤(𝑡)

2 .

4. Subtracts from the original signal 𝑢(𝑡) = 𝑥(𝑡)−𝑚(𝑡).

5. Iterates for the residual: 𝑥(𝑡) = 𝑢(𝑡).

This process is terminated when 𝑢(𝑡) meets the IMF cri-teria. If it does, 𝑢(𝑡) describes an underlying oscillation of𝑥(𝑡), refereed herein as 𝑑(𝑡). EMD continues with sifting onthe residual 𝑟(𝑡) = 𝑥(𝑡) − 𝑑(𝑡) i.e., on a signal stripped of afast oscillation. The original signal can then be expressed as:

𝑥(𝑡) =

𝑁−1∑

𝑖=1

𝑑𝑖(𝑡) + 𝑟(𝑡) (1)

where 𝑑𝑖(𝑡) denotes the 𝑖𝑡ℎ extracted IMF and 𝑟(𝑡) thefinal residual. Note that by definition, 𝑟(𝑡) is not an IMF.

BEMD was proposed by Rilling et al. [2] as a way todecompose complex signals with the EMD. The BEMD algo-rithm decomposes naturally bivariate signals in a consistentway i.e., by examining the real and imaginary parts simulta-neously. Other complex solutions [5] risk a physically mean-ingless decomposition because the complex components aretreated independently. In [2], the oscillatory rationale of theEMD is directly translated to the bivariate case. Because ofthe consistency in the analysis of the real and the imaginaryparts, the BEMD has been suggested for signal separation (de-trending) in filtering applications [14, 15].

4. PROPOSED DECOMPOSITION

In this work, the BEMD is used on ECG signals in orderto overcome the drawbacks of the data-driven EMD. As ex-plained in Section 1, the EMD suffers from uniqueness issuesi.e., the number and type of IMFs at which the decompositionwill result in, is uncertain, even for signals of similar statis-tics. Unless properly treated, this poses great restrictions to itsusability, as it renders comparisons among different ECG seg-ments meaningless. Furthermore, predetermining the numberof IMFs (by forcing the decomposition to stop) beats the pur-pose of EMD as the analysis will no longer be adaptive.

3a) Real ECG segment

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Fig. 2. Example of ECG synthesis. a) Real ECG segment, b)Generated synthetic ECG. The two signals are synchronizedat the main waves.

What is needed, is a way for the decomposition to focuson oscillations that are directly linked to the inherent char-acteristics of the signal in-hand, and to ignore random oscil-lations that may be presented due to noise. We address thisproblem by decomposing a real ECG signal together with asynthetic, in a framework that allows the latter to act as therule of decomposition i.e., to determine which type of IMFsare important from on the input side.

The proposed framework encompasses two steps, namelya) ECG synthesis and b) driven BEMD analysis. Given anECG segment, 𝑥𝐼(𝑡), the first step is to synthesize a syn-chronous ECG segment, 𝑥𝑆(𝑡) in order to decompose themsimultaneously at the second step.

4.1. ECG Synthesis

The goal of this step is to design a synthetic ECG signal,𝑥𝑆(𝑡), that is synchronized at the main waves with the inputECG signal (𝑥𝐼(𝑡)) that is to be decomposed. Synchroniza-tion is necessary because both the EMD and BEMD algo-rithms operate in the time domain. The purpose if to decom-pose the synthetic signal simultaneously with 𝑥𝐼(𝑡).

𝑥𝐼(𝑛) is first filtered, to remove major noise artifacts(baseline wander, powerline interferences etc). A Butter-worth bandpass filter or order 4, with cutoffs at 0.5𝐻𝑧 and40𝐻𝑧 is used based on empirical results. The signal is thendelineated, i.e., fiducial points such as the ones shown inFigure 1 are detected. The QRS complex is detected using thealgorithm described in [16], and surrounding waves are lo-calized with empirical rules (the P wave’s healthy duration isapproximately 120𝑚𝑠𝑒𝑐, and T wave extends about 300𝑚𝑠𝑒𝑐after the QRS complex [13]).

The position of the fiducial points on 𝑥𝐼(𝑛), guide thegeneration of a synthetic ECG signal 𝑥𝑆(𝑛). Synthesis is per-formed using a dynamic model of three coupled differentialequations, proposed by McSharry et al. [17]. A heart beatis modeled as a a circular movement of a trajectory in a 3Dspace, where a cycle completion corresponds to one repeti-

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Fig. 3. Simultaneous decomposition of real and a syntheticECG signal using the BEMD.

tion. Because of the circular movement, the locations of thefiducial points 𝑃 , 𝑄, 𝑅, 𝑆, 𝑇 , are transformed to the respec-tive angles 𝜃𝑃 , 𝜃𝑄, 𝜃𝑅, 𝜃𝑆 , 𝜃𝑇 within the unit circle. Thegenerated synthetic ECG, 𝑥𝑆(𝑛), is an idealized, noise free,representation of an ECG signal. Figure 2 shows an exampleof an input ECG segment and the corresponding synthetic thatwas generated.

4.2. Driven BEMD

In this step, the synthetic signal 𝑥𝑆(𝑡) is decomposed si-multaneously with the input one, 𝑥𝐼(𝑡), using the BEMD.In essence, the difference between EMD and BEMD is thatwhere EMD sifting builds envelopes around 𝑥(𝑡), BEMDbuilds 3D cubes that surround the complex function 𝑥𝑐(𝑡)[2]. Thus, the analysis is performed simultaneously for thereal and imaginary components, and results in the same num-ber of IMFs for both:

𝑥𝑐(𝑡) =𝑁−1∑

𝑖=1

𝑑𝑐𝑖(𝑡) + 𝑟𝑐(𝑡) (2)

where 𝑑𝑐𝑖(𝑡) denotes a complex IMF and 𝑟𝑐(𝑡) the com-plex residual. A complex signal is formed using 𝑥𝐼(𝑡) and𝑥𝑆(𝑡) for the real and imaginary parts respectively:

𝑥𝑐(𝑡) = 𝑥𝐼(𝑡) + 𝑗𝑥𝑆(𝑡) (3)

By applying BEMD on 𝑥𝑐(𝑡) we get

𝑥𝑐(𝑡) =

𝑁∑

𝑖=1

𝑅𝑒{𝑑𝑖(𝑡)}+ 𝑗

𝑁∑

𝑖=1

𝐼𝑚{𝑑𝑖(𝑡)} (4)

where the residual has been included in the summationfor simplicity. A BEMD decomposition example is depictedin Figure 3. Since the synthetic ECG has an idealized wave-form, the presence of oscillatory activity on the imaginary

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Fig. 4. EMD decomposition of a synthetic ECG signal. The first three IMFs are quasi periodic.

side guarantees that the corresponding mode is present on thereal side. When the input ECG is contaminated with highfrequency noise, low order IMFs on the real part will exhibitstrong (but physiologically random) oscillations while almostzero activity will exist on the imaginary side,. The last obser-vation makes noisy IMFs easily detectable.

5. ILLUSTRATION

It has been observed that when analyzing the ECG signal withthe EMD, the first three IMFs tend to preserve informationfrom the QRS complex [18]. The same is true for the BEMD.

In the absence of noise, the first IMF typically exhibitsthree distinct oscillations (tricomponent). Once this oscilla-tion is removed from the signal and a sifting process is com-pleted, the subsequent IMF exhibits two oscillations (bicom-ponent) and similarly the third is monocomponent. Figure 4shows an example of a synthetic ECG’s IMFs that illustratethis property. The IMFs of order higher than three are notquasi-periodic and the oscillation magnitude is near zero.

However, due to noise the clear oscillatory structure ofFigure 4 is not always apparent when decomposing real ECGsignals. This is because the decomposition is driven by thenoise as well. In the proposed extension for the BEMD algo-rithm, since the a noise-free synthetic ECG (imaginary part)is decomposed together with the input segment (real part), os-cillations due to noise on the real side will result in near zeroIMF activity on the imaginary side, which renders these IMFseasily detectable. In addition, by driving the BEMD decom-position using an idealized waveform, the IMFs will conformbetter to the expected structure i.e., IMF 1 to be a tricompo-nent function, IMF 2 a bicomponent and so on.

Figure 5 shows a comparison between the univariate EMD

and the driven BEMD decompositions for the same ECG seg-ment. Even though there is no theoretical guarantee foruniqueness, the proposed driven BEMD algorithm bypassesthis inadequacy by ensuring that the three most substantialIMFs for ECG analysis will be present in the decomposition,without mode mixing.

The purpose of this analysis is to examine the consistencyof the IMFs across different ECG recordings. To quantifythis performance we define a measure of oscillatory activityas follows. For every IMF we measure the time between con-secutive extrema interchange which is essentially a measureof local oscillation. The assumption is that that there will bethree dominant oscillation types within the first IMF of anECG, two in the second and one in the third.

Even though some variability is expected among differentECG recordings (ECG is affected by both physical and psy-chological activity), it is anticipated that with the proposedBEMD decomposition the dominant oscillations at every IMFlevel will exhibit small variance i.e., they will not be affectedby random noise oscillations or by mode mixing. Thereforethe standard deviation of the dominant oscillations within ev-ery IMF is used in this work as a measure of stability for theproposed decomposition.

The driven BEMD was tested on ECG recordings from44 individuals. The collection took place at the Affect andCognition Laboratory at the University of Toronto. For ev-ery subject, 50 recordings are available (5 sec each) whichmakes a total of 2200 ECG segments. The BIOPAC MP150 system was used for the collection, and the signals weredigitized at 1KHz. Across all recordings, the standard de-viations of the three most dominant oscillations in IMF1, obtained with the proposed BEMD decomposition, are{0.0820, 0.0593, 0.0072}. When the same measure is esti-

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Fig. 5. Comparison of Univariate and driven Bivariate EMD decomposition on the same ECG signal. For the BEMD case theIMFs exhibit less mode mixing as well as the oscillation structure follows the properties of ECG decomposition in the absenceof noise i.e., IMF 1 is tricomponent, IMF 2 is bicomponent and IMF 3 is monocomponent.

mated for a decomposition using the typical EMD algorithm,the oscillations of the first IMF are not as consistent, witha standard deviation of {0.1652, 0.1034, 0.0453}. Similarly,for the second IMF, the deviation of the two dominant IMFsof the proposed method is {0.0871, 0.0268} while for theunivariate EMD it is {0.1320, 0.0569}. For the third IMFthere should be only one dominant oscillation, the deviationof which using the driven BEMD and the univariate EMDare 0.0283 and 0.0835 respectively. These measurementsindicate that the proposed driven BEMD algorithm managesto provide a more consistent decompositions for the ECGsignal.

6. CONCLUSION

This work dealt with the problem of ECG analysis using theEMD algorithm. Despite the advantages of the data-drivennature of the decomposition, the EMD is considerably vulner-able to noise. This practically leads to uniqueness and modemixing issues. We advocate that this problem can only be ad-dressed by taking into consideration the morphology of thesignal on which the method is applied.

In particular, this work provides a standard for the deploy-ment of the EMD algorithm on ECG signals. Instead of risk-ing a univariate decomposition that may lead to incomparableIMFs across different ECG segments, the bivariate extension

of the algorithm is proposed as promising tool for the simul-taneous decomposition of the ECG signal with a syntheticequivalent. The synthetic ECG segment has a noise-free ide-alized waveform that acts as the rule of the decompositionprocess. This driven BEMD decomposition results in robustmode functions, with respect to the oscillatory functions thatare conveyed.

7. ACKNOWLEDGEMENTS

This work has been supported by the Natural Sciences andEngineering Research Council of Canada (NSERC).

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