Combination of the CEEM Decomposition with Adaptive Noise and Periodogram Technique for ECG Signals Analysis

The electrocardiogram (ECG) signal is a fundamental tool for patient treatment, especially in the cardiology domain, due to the high mortality rate of heart diseases. The main objective of this paper is to present the most optimal techniques that can link the processing and analysis of ECG signals. This work is divided into two steps. In the first one, we propose a comparison between some denoising techniques that can reduce noise affecting the ECG signals; these techniques are the empirical mode decomposition (EMD), the ensemble empirical mode decomposition (EEMD), and the complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN). In the second one, we make a comparison of three time-frequency techniques: the Choi-Williams (CW), the periodogram (PE), and the smoothed pseudo Wigner-Ville (SPWV). Firstly, the obtained results illustrate the effectiveness of the CEEMDAN in reducing noise that interferes with ECG signals compared to other denoising methods. Secondly, they show that the periodogram time-frequency technique gives a good detection and localization of the main components in the time-frequency plan of ECG signals. This work proves the utility of the combination of the periodogram and CEEMDAN techniques in analyzing the ECG signals. to analyze these biomedical signals. The obtained results show that, in the first part, the CEEMDAN presents a high effectiveness in the noise elimination and, in the second one, the periodogram provides the best solution for analyzing ECG signals. We conclude that a combination of the CEEMDAN denoising method and the PE time-frequency technique can be a good issue in analyzing the ECG signals.


Introduction
The heart function can be obtained by storing the voltage variations which occur on some parts of the human body surface [1][2][3]. The electrocardiogram (ECG) is the record of those voltage variations over time. This biomedical signal presents a fundamental tool used in cardiology to detect cardiac diseases. The normal ECG signal is characterized by a sequence of some well-defined components as P wave, QRS complex, and T wave [1][2][3]. ECG signals are most of the time contaminated by different noise sources, like power-line interference, baseline wander, muscle noise and motion artifact, and other noises, which in different cases make the 2. Theory background 2.1 Denoising techniques used

Empirical mode decomposition (EMD)
Huang et al. had defined a tool named EMD to decompose adaptively a signal in a set of AM-FM components [5]. No mathematical foundations or analytical expressions have been proposed for the technique theoretical study. In various domains, such as biomedicine, acoustics, seismology, or study of climate phenomena, the EMD has been used successfully in several works to treat real data [20,21]. These studies had provided satisfaction and good results in signal processing, especially for nonstationary ones. A nonstationary signal is decomposed adaptively by the EMD technique into a sum of functions oscillatory band-limited d(t). These functions, called intrinsic mode functions IMF J (t), oscillate around zero. The intrinsic mode functions can express the signal x(t) by the following expression: where r(t) is the low-frequency residue. Two conditions must be satisfied by each IMF J (t): • The zero crossings and extreme signal numbers must be equal all over the analyzed signal.
• The envelope average defined by signal local extreme must be equal to 0 at any point. On the one hand, the low-oscillation components are represented by the higher-order IMF J (t), and on the other hand, the fast ones are presented by lower-order IMF J (t). The IMF J (t) number is variable for different decomposed signals and depends on the signal spectral content. The technical aspects of the EMD implementation are decomposed on five steps given by the following algorithm [5]: Step 1: Extraction of the signal x(t) extreme.
Step 2: By the maximum interpolation (resp. minima), an upper envelope e max (t) (resp. lower e min (t)) is deduced.
Step 3: The half envelope sum is defined as a local average m(t) by the following expression: Step 4: Deduction of d J (t) = IMF J (t), a local detail by Step 5: The expression (1) gives the iteration. The high frequency terms are contained in the first IMF, which also involves the following terms of decreasing frequency up to forwarding only a low-frequency residue.

Ensemble empirical mode decomposition (EEMD)
The ensemble empirical mode decomposition (EEMD) method was proposed to surpass the mode mixing disadvantage which exists in EMD technique [22]. By repeating the processes of decomposition, the EMD provides all solutions giving the true IMF.
The following steps give the EEMD method algorithm: Step 1: The analyzed signal is added with a predefined amplitude white noise.
Step 2: The resulted signal is decomposed by using the EMD method.
Step 3: The above signal decomposition is repeated with different fixed amplitude white noises.
Step 4: Calculation of the final results is equal to the ensemble means of the decomposition results.
As finite number of intrinsic mode functions (IMFs) and a residue, the signal x (k) is decomposed: where n defines the IMF number, c _ i is the i-th IMF which is the corresponding IMF ensemble mean resulted from all of the decomposition processes, and r _ is the residue mean obtained from all processes of the decomposition. Although the mode mixing effect is mitigated by the EEMD method, if the ensemble number is small, some noise will continue to exist in the corresponding IMF(s). To deal with this problem and assure a noise-free IMF, a CEEMD algorithm [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23] is defined by the following steps: Step 1: The target signal x(t) is added by positive and negative white noise ε + , À (t) in order to create two new signals x + (t) and x À (t): Step 2: Step 1 is repeated, and by using the EMD algorithm, each of the new signals x + (t) and x À (t) is decomposed.
Step 3: For the x + (t) and x À (t) data sets, two IMF sets are obtained; (4) by averaging the IMF i k in Eq. (11), the decomposed result is calculated, where IMF i k defines the i-th IMF of the k-th iteration; On the one hand, using EEMD overcomes the EMD mode mixing problem, but on the other hand, this technique presents a problem. The number of iterations required in EEMD process is higher, added to the residual noise remaining in the reconstructed signal. So we have a new method called CEEMD; this technique provides an exact reconstruction of the original signal and gives better separation of modes with low computation cost. In this method the first mode IMF 1 n ð Þ is obtained in the same way as in EEMD. It's computed over an ensemble of r 1 (n) plus different realizations of a given noise obtaining IMF 2 n ð Þ by averaging. Here E j [.] operator provides j th mode obtained by EMD. w i is the white noise [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24].
The steps of CEEMDAN decomposition are as follows: Step 1: Decompose x n ð Þ þ ε 0 w i n ð Þ to obtain the first mode by using: where w 0 is the added white noise amplitude and ε t ð Þ is the white noise with unit variance.
Step 2: Compute the difference signal: Step 3: Decompose r 1 n ð Þ þ w 1 E 1 ε i n ð Þ À Á , to obtain the first mode, and define the second mode by For k = 2, …, K, calculate the k-th residue and obtain the first mode. Define the (k + 1)-th mode as follows: where E 1 is a function to extract the j th IMF decomposed by EMD.
Step 4: Continue this process until residue is no longer feasible. Final residue So the given signal can be expressed as A quantitatively comparison of these three filtering methods' performance will be made based on two metrics: mean square error (MSE) and percent root mean square difference (PRD). The MSE and PRD are used to evaluate the quality of the information which is preserved in the denoised ECG signal. The MSE and the PRD are computed as follows: where x(n) is the original ECG signal, x n ð Þ denotes the reconstruction of the ECG signal, and N is the number of ECG samples used.

Time-frequency techniques
In time-varying spectral analysis, time-frequency techniques have found a large number of application [8][9][10][11][12][13][14][15]. There is no single time-frequency representation due to the signal energy distribution which joints time and frequency coordinates. There are many time-frequency techniques and many ways to define them. The most popular time-frequency representation class is called the quadratic or Cohen (1989) class. The Choi-Williams (CW), the periodgram (PE), and the smoothed pseudo Wigner-Ville (SPWV) techniques were chosen from the different time-frequency techniques belonging to this representation time-frequency set, due to its interesting properties.

Choi-Williams distribution (CWD)
The Choi-Williams distribution CWD(t, f) was a significant step in the field of time-frequency analysis where it opened the way for optimizing resolution with cross-term reduction [25]: where and ϕ θ; τ ð Þ ¼ e θ 2 τ 2 σ . The smoothing of the distribution is controlled by the constant σ. If σ ! ∞, the Choi-Williams distribution (CWD) will simply converge to the Wigner-Ville distribution, as the kernel goes to 1.

Periodogram technique
The minimum variance estimator, named Capon estimator (CA), does not fix a model on the signal. At each frequency f, this method seeks a matched filter whose response is 1 for the frequency f and 0 everywhere else [26]: where CA(n, f) means the filter Capon output power. By the discrete signal x(n) sampled at the period t e , this filter is excited; a(n, f) = (a 0 , …, a p ) is the filter impulse response at frequency n; R x [n] = E{x[n]x T [n]} is the crossed x(n) autocorrelation matrix of dimension (p + 1)*(p + 1); x[n] = (x(n À p), …, x(n)) is the selected signal at time n; Z H f (1, e 2iπft e , …, e 2iπft e p ) is the steering vector; (p + 1) is the number of filter coefficient and the exponent H for conjugate transpose and the superscript T for transpose. The periodogram (PE) is the derivate of the Capon (CA) technique. The spectral estimator of this method is defined by the following equation [26]: By sliding windows, the PE technique can be used. Theoretical criterion does not exist for selecting window duration and filter order. The parametric technique frequency response presents different properties according to the signal characteristics. The time-frequency resolution depends principally of the window choice. Usually, the PE estimator gives a better frequency resolution.

Smoothed pseudo Wigner-Ville technique (SPWV)
The Cohen class exhibits most nonparametric time-frequency techniques [16,17]. The smoothed pseudo Wigner-Ville technique belongs in particular to this class [16,17]. To overcome the major weakness of the Wigner-Ville time-frequency representation, which is the covering of frequential components, the SPWV has been proposed between the different existing nonparametric time-frequency techniques; for that, the analytical signal x a (t) replaces the real signal x(t). The following expression defines this signal: where i 2 = À1, H{x(t)} is the Hilbert transform of the signal with real values, x(t).
Expression (20) defines the analytical signal x a (t) spectrum, F a (k): where X(k) represents the original signal x(t) Fourier transform and N is the point number.
The function W x (t, f) is the Wigner-Ville distribution related to a signal x(t), of finished energy. This distribution depends on the temporal (t) and frequential (f) parameters. The following expression defines this distribution [16,17]: where x* a (t) indicates the complex conjugate of x a (t).
The SPWV is used principally to decrease the problem of the interference terms happening between the inner components that existed in Wigner-Ville image. The time-frequency image visibility is reduced by these terms [13,14]. The SPWV technique is applied by using two smoothing windows h(t) and g(t). The utility of these smoothing windows entered into the definition of the Wigner-Ville technique is to guarantee an interference separate control both in time (g) and in frequency (h). This representation is defined by the following expression [16,17]: where h(t) is a smoothing frequential window and g(t) is a smoothing temporal window.
We compare also the performance of these three time-frequency techniques by using the same metrics that were used in the filtering method comparison.

Biomedical signals
Electrocardiogram signals allow to represent the human heart state. ECG signal is a fundamental tool commonly used in the heart medical domain to treat patients suffering from cardiac diseases. By measuring the potential difference between electrodes posed in well-known places in the patient skin, these signals are usually obtained. The ECG signal can be single channel or multichannel depending on how many electrodes are used, one or several. Important knowledge is obtained by cardiologists about the patient's heart function only by analyzing a minute feature of these signals.
The ECG signal has a well-defined P, QRS, and T signatures that represent each heartbeat. The duration, shape, and amplitude of these waves are considered as major features in time domain analysis.
Changes in the normal rhythmicity of a human heart may result in different cardiac arrhythmias, which may be immediately fatal or cause irreparable damage to the heart when sustained over a long period of time.
The following subsections present the different normal and abnormal ECG signals chosen for this study. These data were obtained from [19]. Figure 1 shows the time domain of a normal ECG signal. The sampling frequency for this normal ECG signal was 128 samples/s and the signal length 8 s. Figure 2 shows a length of 4 s of an abnormal atrial fibrillation ECG signal obtained from a patient with malignant ventricular arrhythmia. The sampling frequency for this signal was 250 samples/s.

Atrial fibrillation ECG
The atrial rate exceeds 350 beats per minute in this type of arrhythmias. This arrhythmia occurs due to an uncoordinated activation and contraction of different parts of the atrial which leads to ineffective pumping of blood into the ventricles. This abnormal signal presents a misalignment of the third QRS complex. Figure 4 shows a length of 4 s of the time domain ECG signal obtained from a patient with malignant ventricular arrhythmia. The sampling frequency for this signal was 250 samples/s. The depolarization wave spreads through the ventricles by an irregular and therefore slower pathway. The QRS complex is thus wide and abnormal. Repolarization pathways are also different, causing the T wave to have an unusual morphology.   Figure 5 shows the time domain ECG signal of a patient with supraventricular arrhythmia. The sampling frequency for this abnormal ECG signal was 128 samples/ s and the signal length 8 s. The shape of the QRS complex in this signal is abnormal at the QR part.  These normal and abnormal ECG signals were corrupted with noise CN generated by the following Eq. [4]:

Supraventricular ECG
where BW is the baseline wander noise, EM is the electromyogram noise, and MA is the motion artifact. Wbw, wem, and wma define the added noise percentage of baseline wander, electromyogram noise, and motion artifact noises, respectively. These parameters have been chosen with the following values wbw = 2, wem = 2, and wma = 5, which signified that the predominant noise in the noisy ECG signal is the motion artifact.

Denoising methods
To evaluate the performance of the three denoising methods, EMD, EEMD, and CEEMDAN, a set of normal and abnormal ECG signals with different shapes were chosen. Before applying the proposed denoising methods, the ECG signals were corrupted with different values of signal-to-noise ratio (SNR); these values are from À5 dB to 20 dB with a 5 dB step.
Tables 1-5 report the performance of the denoising methods for the five ECG recordings. These tables present the obtained results of the mean square error (MSE), the root mean square error (RMSE), and the percent root mean square difference (PRD) for the following ECG signals, respectively, a normal ECG, an atrial fibrillation ECG, a ventricular tachyarrhythmia ECG, a malignant ventricular arrhythmia ECG, and a supraventricular arrhythmia ECG.  considered ECG signals: normal ECG, atrial fibrillation ECG, ventricular tachyarrhythmia ECG, malignant ventricular arrhythmia ECG, and supraventricular arrhythmia ECG, respectively, at a SNR interval varying from À5 to 20 dB. Figures 11-15 are presenting the obtained PRD results of the three denoising methods (EMD, EEMD, and CEEMDAN) to all the chosen ECG signals, normal ECG, atrial fibrillation ECG, ventricular tachyarrhythmia ECG, malignant ventricular arrhythmia ECG, and supraventricular arrhythmia ECG, respectively, at a SNR interval varying from À5 to 20 dB.  Table 3. MSE, RMSE, and PRD of the ventricular tachyarrhythmia ECG signal.

Time-frequency techniques
To compare the performance of the three chosen time-frequency techniques, Choi-Williams (CW), periodogram (PE), and smoothed pseudo Wigner-Ville (SPWV), we applied these time-frequency methods to ECG signals presented in  Table 5. MSE, RMSE, and PRD of the supraventricular arrhythmia ECG signal.          Section 2.3. Before applying the time-frequency techniques, these ECG signals were corrupted with different values of signal-to-noise ratio, varying from À5 to 20 dB with a 5 dB step. Tables 6-10 report the obtained results of the mean square error (MSE), the root mean square error (RMSE), and the percent root mean square difference (PRD) after applying the three time-frequency methods (CW, PE, and SPWV) to the ECG signals, normal ECG, atrial fibrillation ECG, ventricular tachyarrhythmia ECG, malignant ventricular arrhythmia ECG, and supraventricular arrhythmia ECG, respectively.  Table 10. MSE, RMSE, and PRD of the supraventricular arrhythmia ECG signal.

Discussion
The study was divided to two separate steps. The first part involved a comparison between three denoising methods, empirical mode decomposition (EMD) and its two variants ensemble empirical mode decomposition (EEMD) and complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN). We note from the obtained results (Tables 1-5 and Figures 6-15) that the EEMD and the CEEMDAN methods for the most selected ECG signals for the different signalto-noise ratio (SNR) values yield the smallest values of MSE, RMSE, and PRD compared to those obtained by the EMD method. Despite the EEMD method providing a slight advantage than CEEMDAN for a few degrees of SNR in some ECG signals, we concluded that the most optimal technique for denoising this type of biomedical signals is CEEMDAN, especially for the large number of iterations required in EEMD process.
The second part of study presents a comparison of the different results of the three time-frequency techniques Choi-Williams (CW), periodogram (PE), and smoothed pseudo Wigner-Ville (SPWV). These time-frequency techniques were applied to normal and abnormal ECG signals with different degrees of SNR varying from À5 dB to 20 dB. We note that the PE technique provides the best results; it furnishes the smallest values of MSE, RMSE, and PRD than those obtained by the two other techniques, CW and SPWV.
After these two steps, we concluded that a combination of the two techniques, CEEMDAN denoising method and PE time-frequency technique, would be ideal for the ECG signal analysis. The CEEMDAN method will be reserved for the pretreatment phase to filter the noise, and in the second phase, the PE technique will be applied to supply the evolution of the ECG signal fequential components over the time in order to provide a good diagnosis.

Conclusion
The work purpose was to conduct two comparative studies to determine the best techniques for ECG signal processing. The first one focused on the comparison between techniques aimed at preprocessing ECG signals, namely, denoising methods. The second one was to compare some time-frequency techniques that are intended to analyze these biomedical signals. The obtained results show that, in the first part, the CEEMDAN presents a high effectiveness in the noise elimination and, in the second one, the periodogram provides the best solution for analyzing ECG signals. We conclude that a combination of the CEEMDAN denoising method and the PE time-frequency technique can be a good issue in analyzing the ECG signals.