According to Abeysekera et al., the QRS complex clearly resembles a ’slightly’ distorted sine wave [14] . Based on this assumption, we undertook to give an estimation of the RMS of the electrocardiogram since an ECG is nothing but a periodical repetition of QRS complexes. Let V ( t ) = V max sin ( ω t + φ ) a sine wave representing an electrocardiogram, with ω = 2 π / T its pulsation, T the period, and V_max the maximum amplitude which is nothing but the highest R-peak. The root means square of such kind of curve is given by:

RMS 2 = 1 T ∫ t 0 t 0 + T V 2 ( t ) d t = 1 T ∫ t 0 t 0 + T V max 2 sin 2 ( ω t + φ ) d t = V max 2 T ∫ t 0 t 0 + T sin 2 ( ω t + φ ) d t = V max 2 T ∫ t 0 t 0 + T 1 − cos ( 2 ω t + 2 φ ) 2 d t = V max 2 2 T ∫ t 0 t 0 + T d t − V max 2 2 T ∫ t 0 t 0 + T cos ( 2 ω t + 2 φ ) d t

RMS 2 = V max 2 2

Therefore: RMS = V max 2 .

Throughout this article, we are going to consider this expression of the root means square. The key idea behind the RMS shifting formalism is to remove the amount of the RMS in the original signal to keep only the R-peaks in the positive part of the shifted signal. Then, the identification of R-peaks can be performed by the detection of all the zeros of the shifted signal. However, owing to the baseline wander cancellation and the denoising subroutines applied on the original electrocardiogram signal, the standard deviation of the R-peaks is naturally affected. Accordingly, instead of removing the previously calculated RMS, we set out to remove an amount proportional to this RMS value to account for the decay of the standard deviation.

Let V₀(t) be the original signal and V_s(t) be the shifted signal. Hence, the expression of the shifted signal with respect to the original signal is given by:

V s ( t ) = V 0 ( t ) − V max λ 2 ,     with   λ ∈ ℕ \ { 0 } . (1)

We can get empirically the optimal value of the coefficient in the denominator λ = 4. Using the equation above, we have shifted all the P and T waves in the negative part along the y-axis and it remains only the R-peaks in the positive part. Thus, we can detect the zeros of the shifted signal using the intermediate value theorem. This theorem is stated as follows:

Theorem (Intermediate value theorem). Let f be a continuous function on [a, b] and s be a number with: f(a) < s < f(b). Then there exists some x between a and b such that f(x) = s.

In our case, f stands for the shifted electrocardiogram, and s = 0 since we are investigating the roots of the shifted signal. Actually, this formulation of the intermediate value theorem is not suitable for our use case. To make this theorem useful for our algorithm, we have been interested in its corollary rather than its previous formulation. The used corollary is known as Bolzano’s theorem:

Corollary (Bolzano’s theorem). If a continuous function has values of opposite signs inside an interval, then it has a root in that interval.

Based on Bolzano’s theorem, we inferred the following criterion for zero detection in the shifted signal expression as follow: ∀ t 0 , V s ( t 0 ) . V s ( t 0 + Δ t ) ≤ 0 ⇒ root detection. Here, ∆t represents the time-marching step of our algorithm.

After detecting all the zeros of the shifted signal, the presence of R-peaks is ensured by Rolle’s theorem. This theorem is stated below:

Lemma (Rolle’s theorem). If real-valued function f is continuous on a closed interval [a, b], differentiable on the open interval ]a,b[, and f(a) = f(b), then there exists at least one c in the open interval ]a,b[ such that f'(c) = 0.

In our use case, c is nothing but the position of an R-peak in the time axis. Let N_root be the total number of roots detected in the shifted signal. Using Rolle’s theorem, we can prove that the amount of QRS complexes is given by:

N Q R S = { N r o o t 2                   if   N r o o t   isanevennumber 1 − N r o o t 2         if   N r o o t   isanoddnumber (2)

Finally, if the total duration of the raw data acquisition experiment is noted d in seconds, we can obviously infer the heart rate (HR) using the following formula:

HR = N Q R S d × 60 (3)

It is really important to notice that the RSM shifting scheme should be preceded by pre-processing algorithms to ensure a good result. After recording the raw data, we need to cancel the baseline drift and denoise the signal. Then centralize the obtained signal before performing the RMS shifting scheme. The block diagram of our proposed algorithm is given in Figure 3.

A recording of the body surface potentials caused by the electrical activity of the heart is called an electrocardiogram (ECG). The ECG recording can be used by clinicians to diagnose further and assess a patient’s overall health and heart condition.

Three main sources of pre-recorded ECG data including biomedical databases (such as the MIT-BIH Arrhythmia Database), ECG simulators, and real-time

ECG data gathering are where the majority of the ECG signals utilized in the creation and testing of biomedical signal processing algorithms come from.

In this paper, to verify our model, we used a sample of the MIT-BIH Arrhythmia Database stored in Matlab. One collection of genuine ECG data was recorded and came from a healthy volunteer with an average heart rate of 82 beats per minute (bpm). Before being fed to the 12-bit ADC, this ECG data was pre-filtered and boosted by the analog front end. This signal has the following features: a sampling frequency of 360 Hz, an additive uniformly distributed measurement noise of 0.005 mV, and a standard deviation of heart rate of 1 bpm [15] . The scheme of the data acquisition simulation implemented in Simulink is depicted in Figure 4.

The electrocardiogram signal processing block details are given in Figure 5. This describes the different stages of the well-known Pan-Tompinks algorithm. We will give further explanation about this model in the next section.

The raw data displayed by the scope are depicted in Figure 6. It’s a sample of the electrocardiogram signal recorded for 10 seconds. On the same graph, we can see below the waveform of the filtered ECG. Especially, the P and T waves have been strongly attenuated. Consequently, the filtering makes it possible to highlight the R-peaks of the electrocardiogram signal.

Furthermore, we can also see the evolution of the heart rate versus time in Figure 7, and the variation of the adaptive threshold implemented in the Pan-Tompinks algorithm.

While recording the heart’s electrical activity, the electrocardiogram signal is

corrupted by numerous phenomena such as muscle activity noise or artifacts due to electrode motion. Therefore, considerable attention should be paid to applying algorithms for the purpose of removing baseline wander and powerline interference present in the ECG raw data. To reduce variations in beat morphology that do not have a cardiac cause, baseline wander must be eliminated. We can see below the electrocardiogram raw data extracted from Simulink in Figure 8(a), and the waveform after applying the baseline wander removal

subroutine in Figure 8(b).

Electromagnetic fields caused by power supply represent also a common noise source that introduces spurious waveforms in the electrocardiogram raw data. This type of noise is known as powerline interference. To reduce the effect of such kind of noise we undertook to denoise the ECG raw data using the wavelet transform. The signal processing software Matlab provides a graphical user interface (GUI) to denoise a one, two, or three-dimensional signal through the wavelet transform. This GUI is called wavelet Analyzer toolbox [16] . In our use case, the electrocardiogram was a one-dimensional time-domain signal. In Table 1, we can find the parameters used in our case and a screenshot of the wavelet Analyzer Toolbox in Figure 9.

Finally, we applied the RMS shifting scheme onto the resulting denoised signal to detect the R-peaks. RMS shifting algorithm has been performed after the centralization of the denoised signal. All the peaks detected are pinpointed in the graph shown in Figure 10.

Furthermore, our algorithm indicates that 54 R-peaks have been detected. Since the total duration of the recording experiment is d = 39.72   s , thus we can readily infer the value of the heartbeat rate of this sample:

HR = N Q R S d × 60 = 54 39.72 × 60 = 81.57   bpm .

This result is in agreement with the features of this ECG signal described in

the previous section. Indeed, according to Matlab documentation, the sample of the ECG signal provided has an average heart rate of 82 bpm with a standard deviation of 1 bpm. Therefore, our algorithm accurately estimated the heart rate value for this sample. Additionally, to verify and confirm again the R-peaks detected by our algorithm, we undertook to use another QRS complex detection algorithm. We used the well-known Pan-Tompinks algorithm implemented in Matlab to accomplish this task. In the next section, we will outline this algorithm and show its effects on the ECG signal.

Simulink has been used to create a real-time QRS detection algorithm on the presumption that the input ECG signal’s sampling frequency is always 200 Hz (or 200 samples/s) [15] .

The recorded real ECG data, on the other hand, can have various sampling frequencies ranging from 200 Hz to 1000 Hz, for example, 360 Hz in our use case. A sample rate converter block is used to convert the sample rate to 200 Hz in order to bridge the disparate sampling frequencies. To make sure the duration of the incoming ECG signal is greater than the sample-rate converter block’s computed decimation factor, a buffer block is inserted as depicted in Figure 4. To provide a windowed estimate of the energy in the QRS frequency region, the ECG signal is filtered. These steps comprise the filtering process: FIR bandpass filter with cut-off frequencies of 5 to 26 Hz, taking the derivative of the bandpass filtered signal, taking the absolute value of the signal, and averaging the absolute value over an 80 ms window [15] . All of these stages are presented in the ECG signal processing block details (Figure 5). The effects of each pre-processing algorithm on the signal are depicted in Figure 11.

Simulink has been used to create a real-time QRS detection algorithm on the presumption that the input ECG signal’s sampling frequency is always 200 Hz (or 200 samples/s) [15] .

Real-time peak identification of the filtered ECG signal is performed by the QRS detection block (Figure 5). This block implements the well-known Pan-Tompinks algorithm. Based on the mean estimation of the average noise peak and the average QRS peak, the detection threshold is automatically modified. Depending on whether the observed peak is above the threshold, it is either categorized as a QRS complex or as noise. The PIC-based QRS detector used in that block is referenced by the following QRS detection rules: Ignore all peaks that precede or follow larger peaks by less than 196 ms (306 bpm); if a peak occurs, check to see whether the raw signal contains both positive and negative slopes if true report a peak is found, otherwise, the peak represents a baseline shift; if the peak is larger than the detection threshold, classify it as a QRS complex, otherwise, classify it as noise.

If no QRS has been detected within 1.5 R-to-R intervals, but there is a peak that was larger than half the detection threshold, and that peak followed the preceding detection by at least 360 ms, classify that peak as a QRS complex [3] [15] . Further details on Pan-Tompinks algorithm and its behavior in noisy signals are presented in [17] . After following these rules, we got the results shown in Figure 12.

As well as our proposed method, the Pan-Tompinks algorithm also detected 54 R-peaks. The peaks detected overlap perfectly with those detected by our

proposed method. This result proves that the peaks detected using the RMS shifting scheme are reliable.

As well as our proposed method, the Pan-Tompinks algorithm also detected 54 R-peaks. The peaks detected overlap perfectly with those detected by our proposed method. This result proves that the peaks detected using the RMS shifting scheme are reliable.

However, let us mention that the time duration in the Pan-Tompinks graph is different from that used in our scheme. This is due to the difference in the sampling frequencies used in the two schemes. In our case, we kept the original sampling frequency of 360 Hz whereas in the Pan-Tompinks case, a sample rate converter block has been used to reduce the sampling frequency to 200 Hz. The modification of the sampling frequency affects the time vector since its size remains the same in the two cases. Indeed, the time step value of the time axis is nothing but the inverse of the sampling frequency.

In the last section, we will use the RMS shifting algorithm to estimate the heart rate of 9 healthy volunteers. The results obtained will be compared with those obtained by the conventional method of calculating the heart rate based on the spectral analysis of the ECG signal.

In this work, we particularly focused on the electrocardiogram data. For each volunteer, we collected a sample with an acquisition time of one minute ranging from 100 s to 160 s. In order to assess the accuracy of our proposed method, we first set out to detect R peaks using both our algorithm and the Pan-Tompinks algorithm implemented in Matlab. The results of the two algorithms are shown in Table 2. The results clearly show that for all cases, both algorithms found the same number of peaks. This implies that the two algorithms give the same heart rate values for all cases since the heart rate only depends on the number of peaks detected and the acquisition time.

Therefore, for further assessment of our algorithm, we undertook to compare the heart rate that had been calculated using our proposed method and the conventional method. The conventional method consists of the spectral analysis of the signal. Let s(t) be a signal, its spectrum is given by the Fourier transform: In this work, we particularly focused on the electrocardiogram data. For each volunteer, we collected a sample with an acquisition time of one minute ranging from 100 s to 160 s. In order to assess the accuracy of our proposed method, we first set out to detect R peaks using both our algorithm and the Pan-Tompinks

algorithm implemented in Matlab. The results of the two algorithms are shown in Table 2. The results clearly show that for all cases, both algorithms found the same number of peaks. This implies that the two algorithms give the same heart rate values for all cases since the heart rate only depends on the number of peaks detected and the acquisition time.

Therefore, for further assessment of our algorithm, we undertook to compare the heart rate that had been calculated using our proposed method and the conventional method. The conventional method consists of the spectral analysis of the signal. Let s(t) a signal, its spectrum is given by the Fourier transform:

S ( f ) = F T { s ( t ) } = ∫ − ∞ ∞ s ( t ) e − j 2 π f t d t (4)

To be able to utilize the Fourier transform while working with a digital signal, the digital Fourier Transform (DFT) should be used. The fast Fourier transform (FFT) is an efficient and fast way of evaluating the DFT of a signal. After performing the FFT algorithm, the highest peak in the frequency domain is nothing but the heart rate.

We applied the conventional method and the RMS shifting scheme to the dataset and the comparison of the results of the two algorithms is shown in Figure 16.

The results show that for the two first cases, the values found by the proposed method are exactly the same as those found through the conventional method. However, there is a difference of 1 bpm from the third volunteer to the eighth one, whereas we have a difference of 2 bpm in the last case. This difference can be justified by the fact that the conventional method doesn’t account for the

powerline interface, the baseline wander, electrode motion noise, or other artifacts that can make the signal noisy.

On the other hand, we also performed the calculation of the correlation coefficient (r) and the root mean square error (RMSE) between these two sequences of data to assess their statistical relationship:

RMSE = 1 N ∑ i ‖ x i − y i ‖ 2 (5)

r = ∑ ​ ( x i − x ¯ ) ( y i − y ¯ ) ∑ ​ ( x i − x ¯ ) 2 ∑ ​ ( y i − y ¯ ) 2 (6)

where x i and y i are respectively the heartbeat rate of the ith volunteer calculated through our proposed method and the conventional method, and x ¯ and y ¯ are their respective average. After performing the calculation of the correlation coefficient, we found r = 0.9938 (Figure 16). Thus, the results of the two methods have a strong linear relationship.

The Root Mean Square Error (RMSE) is a statistical coefficient that calculates the amount of error between two data sets. In other words, it contrasts a value that was anticipated with one that was observed or known. The anticipated and observed values are closer together when the RMSE is small. As shown in Figure 16, we found the value of RMSE = 1.0541 bpm. Therefore, this means that there is only 1 bpm difference between the heart rate values estimated by the two methods.

Finally, we have also calculated the averages of the absolute and relative errors of our proposed methods with respect to the conventional method. The outcomes are displayed in the graph in Figure 17. In both cases, the error values are very

low. We found an absolute error average of 0.89 bpm and a relative error average of 1.3%. Consequently, all these indicators confirm the accuracy and reliability of our algorithm for heart rate estimation for electrocardiography applications.

In order to assess the performance of our algorithm, we set out to measure the CPU time of our proposed method, the Pan-Tompinks algorithm, and the conventional method. Then, we compared them. CPU time, also known as process time, is the length of time a central processing unit (CPU) was employed to carry out an operating system’s or computer program’s instructions [20] . The overall empirical effectiveness of two functionally equivalent algorithms is measured using the CPU time. The algorithm with the lowest CPU time is the most efficient in terms of execution speed.

We compared the three algorithms with different data sizes. The first experiment was done using data collected over a period of 1 minute. Since the sampling frequency is 1000 Hz, hence, the size of the input data is 60,000 samples. We carried out a total of 10 experiments. When we went from an experience of rank n to an experience of rank n + 1 we added 60,000 additional values. The objective was to test the speed of the algorithms with increasingly large data sizes.

The algorithms were executed on an INTEL Core i7-9700 type processor, with a clock frequency of 3 GHz (8 CPUs) and a 64-bit architecture. The RAM memory size was 8 GB. The results are shown in Figure 18. It can be clearly seen that our proposed method has a shorter execution cycle than the other two methods. We also note that for our algorithm, the CPU time remains almost constant

while the size of the data increases, whereas the CUP time of the other algorithms increases when the size of the data increases. Also, by analyzing the CPU time averages, we realize that our algorithm is twice as fast as the conventional method.

The parameters that we have studied show that our algorithm presents quite reliable performances, an execution speed twice superior to the conventional method based on spectral analysis and its quasi-insensitivity to the mass of input data. The last two properties make our algorithm a good alternative to be implemented in systems with low resources like connected objects. However, the speed of this algorithm can still be improved by integrating parallelism. The main idea is to split the main program into independent subroutines and to execute these subroutines in parallel on different processor cores.