ECG SIGNAL QUANTITATIVE ANALYSIS BASED ON EXTREMUM ENERGY DECOMPOSITION METHOD

1. Introduction

Physiological signals are generated by the interaction of multiple living systems. The acting time and intensity of different systems are different, leading to the complexity of physiological signals in time and space.^1,2,3,4,5 ECG signal quantitative analysis plays a significant role in clinical medicine and biomedical engineering.^6,7,8 ECG analysis is essential for the diagnosis of various cardiac conditions, including arrhythmias, ischemic events, conduction abnormalities, and structural heart diseases. By applying various mathematical and statistical methods, quantitative features related to cardiac electrical activity can be extracted from ECG signals, providing effective means for the diagnosis and monitoring of heart diseases. In the quantitative analysis of ECG signals, signal preprocessing is performed as a first step, including noise removal, baseline correction, filtering, and artifact detection, to ensure the accuracy and reliability of the signal quality.^9,10,11 In the latest references, Li et al.¹² proposed a baseline correction algorithm for electrocardiogram (ECG) signals based on empirical wavelet transform and piecewise polynomial fitting theory. The algorithm utilizes empirical wavelet transform to adaptively segment the spectrum of ECG signals. Within these segmented intervals, suitable wavelet windows are constructed to extract empirically mode components with compact support. These empirical mode components, after removing the baseline drift, are then subjected to piecewise polynomial fitting to eliminate any remaining baseline drift. Giorgio et al.¹³ proposed a quantitative ECG analysis approach based on principal component analysis, which was applied on the 12-lead ECGs, and six RQA-based features were extracted from the most significant principal component scores.

Subsequently, feature extraction techniques are employed to extract quantitative features such as waveform morphology, heart rate variability, and timing intervals that are relevant to cardiac function. These features reflect the cardiac electrophysiological activity and provide important information for the evaluation and diagnosis of heart diseases. Following the extraction of quantitative features, classification and pattern recognition algorithms are commonly utilized to classify ECG signals into normal or abnormal states. Machine learning techniques such as support vector machines, neural networks, and deep learning are widely applied in this field, enabling automated identification and classification of cardiac diseases, thereby improving the accuracy and efficiency of diagnosis. In the latest references, Diao¹⁴ proposed heart rate variability analysis method utilizing the local mean algorithm that can adaptively decompose signals into a series of amplitude-modulated and frequency-modulated components for the assessment of ECG quality. Su et al.¹⁵ proposed a method based on adaptive differential double thresholding method and distribution to achieve quantitative analysis. The references above have analyzed quantitative analysis methods for electrocardiographic signals from different steps and perspectives.

Based on the above, an extremum energy decomposition (EED) method^16,17 is proposed in this paper for quantitative analysis of ECG signals. The original signal is decomposed into several components, i.e., extremum component function, then calculating the energy of each component to obtain its energy distribution. The EED proposed in this paper has two advantages: (1) The data length remains consistent across all scales during extremum decomposition, thereby avoiding and reduction in data length, which makes it suitable for short-term data analysis; (2) the energy analysis of multi-scale components is relatively resistant to the affluence of noise.

2. EED Method

The EED method is a signal processing technique used to analyze and decompose complex signals, often encountered in applications like speech and image processing. In the beginning, the concept of extremum modal function is presented. This concept pertains to a category of signal featuring a singular frequency which simultaneously fulfills the subsequent two conditions.

(1) Throughout the entire data sequence, the number of extreme points (including both maximum and minimum) and the number of zero crossing must be either equal or at most one difference.

(2) At any time, the mean value of the upper envelope, constructed by using local maximum point, and the lower envelope, formed by local minimum point, is maintained at zero. This signifies that the local upper and lower envelope exhibit local symmetric with respect to the time axis.

For the above two conditions, condition (1) is similar to the requirements of the Gaussian normal stationary process for traditional narrowband, condition (2) ensures that the instantaneous frequency calculated by extremum modal function has physical significance.

The steps of the EED method are as follows:

(1)	For a given time sequences $x(t)$, find out all the local extreme points. Then, connect all the maximum and minimum points using spline curves to form the upper and lower envelope lines, respectively. Obtain the mean value of upper and lower envelope, denoted as $m(t)$, as it is shown in Fig. 1.
(2)	The original signal $x(t)$ minus the mean envelope signal $m(t)$ gets $h_1(t)=x(t)-m(t)$. Then, determine whether $h_1(t)$ meets the judgment criteria of extremum modal function. If not, take $h_1(t)$ as the original signal and repeat step (1) until $h_k(t)$ satisfies the two conditions of extremum modal function. At this time, taking $c_1(t)=h_k(t)$, as the first extremum modal function component.
(3)	The original sequence $x(t)$ minus the first extremum modal function component $c_1(t)$ to get residual $x_1(t)=x(t)-c_1(t)$, which is taken as new original sequence, and repeat steps (1) and (2) to get the second, third, …, n extremum modal function components and residual $r_n(t)$. Then, $x(t)$ is decomposed into n extremum modal function components and a residual. $$x(t)=\sum _{i=1}^n{c}_i(t)+r_n(t).$$(1)

It is difficult for $h_i(t)$ obtained in step (2) to fully satisfy the two conditions of extremum modal function. Typically, the screening process ends when $h_i(t)$ meets the following criteria :

where $\epsilon$ denotes the screening threshold, usually between 0.2 and 0.3.

There are two termination conditions for extremum modal decomposition. The first one is when the final extremum modal function component $c_n(t)$ or residual $r_n(t)$ falls below the preset threshold; the second one is when the residual signal $r_n(t)$ becomes a monotone signal, from which extremum modal function signals cannot be extracted.

The termination criteria of extremum modal function should be moderate. Excessively stringent conditions can result in the loss of significance in the last several extremum modal function component, while overly lenient conditions may lead to the loss of useful components. In practical applications, the decomposition levels of extremum modal function component can be determined based on experience. The calculation process will be terminated when decomposition level is satisfied.

The schematic diagram of extremum modal decomposition is shown in Fig. 1.

(4) The n extremum modal function components decomposed from the original signal represent the components of the original signal in different frequency bands. Subsequently, the energy of each component is calculated :

Each energy value is normalized to obtain the normalized energy distribution vector. where $E={\sum }_{i=1}^n{E}_i$, the first component $p_1$ denotes the energy of the highest band, representing the energy distribution proportion of signal within the range of the highest frequency band. The last component $p_n$ describes the energy distribution proportion of signal within the range of the lowest frequency band.

Generally, for a sequence of N points, it is possible to decompose a maximum of n components.

In other words, when n effective components have been decomposed, at least, the following condition should be satisfied : The above numbers of sequence points are required.

The calculation flow chart of EED is shown in Fig. 2.

3. ECG Signal Analysis Based on EED Method

3.1. ECG signal analysis based on EED method

The ECG signal analysis utilizing EED presents a novel approach to quantify cardiac activity. By extracting extremum modal function components, the method offers insights into frequency bands and abnormality detection. This technique holds potential for enhancing objective cardiac health assessment, advancing the field’s diagnostic capabilities. The details of this method are described as follows:

(1)	Data preparation

The initial step involves obtaining raw ECG signal data from the nsrdb database, which is sourced from healthy individuals’ normal sinus rhythms on PhysioNet.

(2)	Preprocessing

To enhance the signal quality, a zero-phase Finite Impulse Response (FIR) low-pass filter with a cutoff frequency of 40$$Hz^6,7 is applied. This filtering process effectively removes high-frequency noise, preserving ECG energy within the significant 0–40$$Hz range.

Additionally, baseline drift is mitigated through the utilization of a median filter, which eliminates fluctuations stemming from drift.

(3)	Extremum Points Detection

To enhance the signal quality, a zero-phase FIR low-pass filter with a cutoff frequency of 40$$Hz is applied. This filtering process effectively removes high-frequency noise, preserving ECG energy within the significant 0–40$$Hz range.

Additionally, baseline drift is mitigated through the utilization of a median filter, which eliminates fluctuations stemming from drift.

(4)	Energy Decomposition

The starting point for energy decomposition is the extremum point with the highest energy. A sequential iteration through the remaining extremum points is conducted, prioritizing them based on descending energy.

During this process, extremum modal function components are constructed by connecting the energies of adjacent extremum points while maintaining the order of these points.

(5)	Residual Component extraction

The residual component of the signal is extracted, capturing the signal information that is not encapsulated by the previously identified extremum modal function components.

The data length is 8$$s, and the EED decomposition levels are set as $n=8$. The results are shown in Fig. 3.

The x-coordinate represents time, the y-ordinate represents amplitude. $x(t)$ is the original signal, $c_1$ is the first component of extremum modal function, $c_2$ is the second component, and so on, and $c_8$ is the eighth component. $r_n$ is residual.

Based on waveform characteristics, the ECG signal can be divided into three wave groups: P-, QRS-, and T-wave. By observing the wave frequency of the signal in Fig. 3, the corresponding physical meaning for each component can be concluded. Notably, $c_1$ and $c_2$ exhibit the highest frequency, mainly representing the decomposition components of QRS-wave group characterized by the highest frequency. The frequency component corresponding to the P-wave starts increasing from $c_3$. From $c_4$ onwards, the decomposition component of T-wave is added, resulting from the superposition of components of P-, QRS-, and T-wave. Meanwhile, $c_5$ reflects the superposition of low-frequency components of P-, QRS-, and T-waves. $c_6$ describes the cardiac cycle, representing the rhythm of heart beating. On the other hand, $c_7$ and $c_8$ delineate the physiological adjustment rhythm of the heart on a larger time scale, illustrating the long-term rhythm of the heart. Notably, the component with the highest frequency exhibits more pronounced amplitude and greater energy, whereas, those with the lowest frequency display lower amplitude and lower energy.

$c_1$ was analyzed to obtain the spectrum diagram shown in Fig. 4. The examination reveals that the central frequency of $c_1$ is approximately 20$$Hz, with the primary frequency concentration ranging from 15$$Hz to 25$$Hz. Similarly, spectrum analysis was conducted on signals at other scales to obtain their center frequency, as documented in Table 1.

Related studies indicate that the spectrum range of P-wave spans from 0$$Hz to 18$$Hz (± 3$$Hz), with energy mainly concentrated between 5$$Hz and 12$$Hz. Similarly, the QRS complex covers a spectrum range from 0$$Hz to 37$$Hz (± 5$$Hz), with energy primarily concentrated between 6$$Hz and 18$$Hz. As for the T-wave, its spectrum range extends from 0$$Hz to 8$$Hz (± 2$$Hz), and energy is chiefly concentrated within the range of 0–8$$Hz. A comparison with the information in Table 1 yields the conclusion that the frequency band of QRS group mainly contains $c_1$ and $c_2$ components. The P-wave mainly includes $c_3$ and $c_4$ components, and T-wave mainly contains $c_4$–$c_8$ components. It is worth noting that the inclusion mentioned does not mean that each component is only determined by a specific ECG group (P, QRS, T), or that each ECG group is only contained in a specific component. The above-mentioned relationship between wave groups and energy represents a significant correlation, but it is not exhaustive. For example, components $c_5$–$c_8$, representing low-frequency components, result from superposition of low frequencies from each ECG group rather than a single specific ECG group. The results analyzed here are basically consistent with those observed in the EED decomposition waveforms in Fig. 3. This correspondence indicates that the extremum modal function component of ECG can capture certain fluctuations of ECG wave group and reveal the fluctuation pattern of the ECG across various levels. In contrast to frequency domain analysis method, the EED method can directly present ECG fluctuations at all levels, providing a remarkably intuitive representation.

3.2. EED analysis of ECG in healthy individuals and congestive heart failure patients

EED analysis was applied to analyze the energy distribution of ECG in both healthy individuals and patients with congestive heart failure (CHF) across various levels. The data utilized are sourced from the Normal Sinus Rhythm Database (NSRDB) and the Congestive Heart Failure Database (CHFDB) available on physionet’s normal sinus database,^8,9,10,11 where NSRDB database consists of records from 18 healthy individuals (age: $34.3\pm 8.4$), while the CHFDB database comprises data from 15 patients with CHF (age: $58.8\pm 9.1$). Due to the disparate sampling rates between NSRDB and CHFDB- 128$$Hz and 50$$Hz, respectively, it is necessary to resample both datasets to a uniform value before conducting EED analysis. This is because the ECG signal with different sampling rates would obtain different levels through EED decomposition. Consequently, the corresponding frequencies of components at the same level would diverge, rendering the inter-group comparison within identical frequency ranges unfeasible. Therefore, we resample the ECG data from CHFDB to 128$$Hz, consistent with that of NSRDB. The experimental data length is 10$$s, and the maximum scale n is set to 8. The results of the EED analysis are depicted in Fig. 5.

The x-coordinate represents the component level, while the y-coordinate represents the normalized energy value. The curve describes the mean value, and the error bar shows the standard deviation.

As evident from Fig. 5, at level 1, the energy of healthy individuals surpasses that of CHF patients. With the increase in level, the energy of healthy individuals gradually declines. In contrast, CHF patients exhibit a gradual increment in energy from level 1 to level 3, reaching its peak at level 3. Beyond level 3, the energy gradually diminishes with the escalating level. The energy distribution of healthy individuals predominantly within the lower levels (level 1–4) signifies a focus on a higher frequency range. This observation suggests that healthy individuals possess a more robust capacity for short-term heart regulation. Conversely, the lower-level energy of CHF patients is relatively reduced, manifesting a reduction in short-term regulatory ability, and a greater concentration of power in the moderate level. Notably, at level 6, which reflects heart rhythm, CHF patients exhibit higher energy compared to healthy individuals. This indicates that CHF patients allocate a greater proportion of energy towards regulating heart rhythm.

To further verify the energy distribution of the two groups, we conducted spectral analysis on each group. The frequency domain analysis results of a healthy individual and a CHF patient are shown in Fig. 6. It is evident that the energy distribution of healthy individuals spans the range from 0$$Hz to 40$$Hz, with a notable emphasis on high-frequency energy above 20$$Hz. Conversely, the energy of CHF patients is predominantly below 20$$Hz, and there is a significant reduction in high-frequency energy segment.

A significance test, using students’ t-test, was conducted on the EED analysis results of two groups. The results are listed in Table 2 (when $p<0.05$, it is generally regarded as statistically significant, and significant results are presented in bold.). The outcomes illustrate notable disparities between the two groups at levels 1, 2, 3, 5, and 6. The CHF patients present reduced energy at the low levels of decomposition, implying that the disease causes a decline in the short-term regulating ability of the heart. Conversely, healthy individuals display higher energy at lower stages of decomposition, indicating an improved short-term regulatory capacity of the heart and better adaptability to variations in the external and physical environment.

4. EED for the Quantitative Analysis of ECG Signals

To further investigate the effects of age and disease on ECG energy distribution, we added the PTPDB database of physionet^8,9,10,11 for analysis, which contains 549 sets of data from 297 individuals, including healthy individuals and patients with multiple heart conditions (myocardial infarction, cardiomyopathy, bundle branch block, etc.). Several groups with a large number of medical records were selected for analysis, including 18 healthy younger individuals (individuals younger than 36 years old, age $28.3\pm 4.7$) and seven healthy old individuals (individuals older than 59 years old, age $69.4\pm 5.4$), 148 patients ($60.5\pm 11.0$) suffering myocardial infarction (MI) and 15 patients ($59.7\pm 14.4$) with cardiomyopathy (CM), as well as CHF patients (age $60.5\pm 11$) from CHFDB database. Due to the difference in sampling rate, all data were initially resampled to 128$$Hz. Subsequently, a preprocessing step involved 40$$Hz low-pass filtering and the removal of baseline drift. The results of the EED analysis are shown in Fig. 7.

As shown in Fig. 7, at level 1, the energy values gradually decline, transitioning from healthy younger individuals to elderly individuals, and further to various disease conditions. The energy curves for both healthy younger individuals and healthy old individuals demonstrate a tendency to plateau at lower level of energy, specifically, at level 1, level 2, and level 3. However, at level 4, the energy spikes significantly. The energy of the other three kinds of diseases increases at levels 1–3, reaching the maximum at level 3, after which it gradually decreases. Between levels 1 and 3, the energy trends of multiple groups exhibit a complete reversal, resulting in their curves intersecting. The energy distribution curve for patients with heart disease forms an inverted “V” shape overall, and the inverted “V” phenomenon becomes more pronounced with the severity of the disease. Heart disease significantly reduces ECG extremum energy at lower levels, indicating a reduction in the heart’s short-term regulatory capability and its adaptability to the environment. Concurrently, patients with heart disease have a higher energy distribution at the levels (levels 5 and 6) governing heart rhythm. This suggests that, for patients with heart disease, regulating heart rhythm might hold increased significance. Furthermore, the mean energy of elderly individuals is slightly lower than that of younger individuals at scale 1. However, the decreasing trend is not obvious, indicating that age does not significantly affect the distribution of ECG extremum energy. In essence, advanced age does not necessarily correlate with diminished cardiac function; elderly individuals could still maintain a commendable level of cardiac function.

The normalized energy distribution vector $p_i$ of ECG signals from healthy younger individuals and the normalized energy distribution vector $p_i$ of ECG signals for healthy elderly individuals were significantly detected, resulting in the computation of probability value $p_i$ for each extremum modal function component. If the value exceeds 0.05, it indicates that there is no significant difference between the two groups, as illustrated in Table 3.

The normalized energy distribution vector $p_i$ of ECG signal from healthy elderly individuals and the normalized energy distribution vector $p_i$ of ECG signal from patients with MI, CM, and CHF were significantly detected to obtain the probability value $p_i$ for each extremum modal function component. When the probability value $p_i$ falls below 0.05, it indicates that there is a statistically significant difference between the two groups, as presented in Table 4.

The normalized energy distribution vector $p_i$ of ECG signals from healthy younger individuals and patients with MI, CM, and CHF were significantly detected to obtain the probability value $p_i$ for each extremum modal function component. When the probability value $p_i$ falls below 0.05, it indicates that there is a statistically significant difference between the two groups, as shown in Table 5.

The normalized energy distribution vector $p_i$ from patients with MI, CM, and CHF was significantly detected to obtain the probability value $p_i$ of each extreme modal function component. When the probability value $p_i$ falls below 0.05, it indicates that there is a statistically significant difference between the two groups, as presented in Table 6.

The information presented in Table 6 illustrates the following:

(1)	There is no significant difference between healthy younger individuals and healthy elderly individuals at any energy level.
(2)	At levels 1, 5, and 6, a clear distinction is evident between healthy elderly individuals and patients with three kinds of heart disease.
(3)	Healthy younger individuals and patients with three kinds of heart disease are well distinguished at levels 1, 3, 4, 5, and 6, especially at levels 1 and 3. Notably, significant differences are apparent for patients with three kinds of heart disease. When compared to the contrast between elderly individuals and patients with three kinds of heart disease in Table 3, the difference between patients with heart disease and younger individuals is more pronounced.
(4)	Variations are evident among the three diseases, reflecting the distinct characteristics of heart damage associated with each condition.

These results demonstrate that cardiac disease has a notable impact on reducing the energy of ECG signals, particularly at high-frequency levels (Level 1). This reduction signifies a decline in the heart’s short-term regulation capacity. Additionally, the energy distribution curve shows an inverted “V” shape, with the degree of severity of this disease correlating with the prominence of this pattern. In addition, at levels 5 and 6, which represent the heart beating rhythm, patients with heart disease are well distinguished from healthy individuals. In these levels, the peak energy of patients with heart disease surpasses that of healthy individuals. This indicates that maintaining control over heart rhythm holds greater significance for patients with heart disease compared to those who are healthy.

By quantifying the normalized energies at three levels (levels 1, 2, and 3), we discovered a significant reduction in the extremum energy of ECG signals at high-frequency levels (level 1) due to heart disease. This reduction shows a decline in the short-term regulation of the heart. The quantitative analysis of these three levels (levels 1, 5, and 6) can provide a reference for clinical diagnosis.

As shown in Fig. 7, the graph’s pattern takes the normalized energy value of level “3” as the vertex to form an inverted “V” shape. This shape is constituted by lines connecting energy points from levels 1, 2, and 3 on the left-hand side, and energy points at levels 3, 4, 5, 6, 7, and 8 on the right-hand side. The slope values of the energy points at levels 2 and 3 are taken as a reference for quantitative analysis to assess the degree of abnormality in the ECG signal. A higher slope value indicates a more severe of abnormality.

The left slope K of inverted “V” shape is calculated as follows: $K1=0.08$ for patients with MI, $K2=0.17$ for patients with CM, and $K3=0.22$ for patients with CHF. The right slope K of inverted “V” shape is calculated as follows: $K1=-0.05$ for patients with MI, $K2=-0.11$ for patients with CM, and $K3=-0.13$ for patients with CHF.

5. Conclusion

This paper employs EED to analyze ECG signals. Through this method, the original signal is decomposed into distinct components, specifically extremum component function, and the energy of each component is calculated to obtain its energy distributions. Subsequently, the energy of each component is computed to establish its energy distribution. According to the fluctuation rule of biomedical signals, the technology in this paper can not only decompose the signals into different time levels from high frequency to low frequency, but also divide the abnormal degree of signal into several levels. Higher levels denote a more pronounced degree of signal abnormality. The utilizations of extremum value decomposition ensures consistent data length at all levels. Consequently, this approach avoids data length reduction, rendering it suitable for short-term data analysis. In essence, accurate results can be obtained from a limited dataset.

The ECG signal quantitative analysis method introduced in this paper, which is based on EED, introduces a novel approach for quantitatively assessing cardiac electrical activity. However, there remain several aspects that warrant further exploration and improvement. In the future, we will focus on optimizing and refining the EED method to enhance the accuracy and stability of the decomposition process. This entails exploring alternative decomposition algorithms and incorporating additional signal processing techniques to achieve superior signal decomposition results. The insights and outcomes derived from this research contribute new dimensions to cardiac electrophysiology and hold notable significance.

Ethical Compliance

Research experiments conducted in this paper with animals or humans were approved by the Ethical Committee and responsible authorities of our research organization(s) following all guidelines, regulations, and legal, and ethical standards as required for humans or animals.

Conflicts of Interest

The authors declare no conflict of interest.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (No. 82004498); the University Philosophy and Social Science Project of Jiangsu Province (No. 2022SJYB0316); and the Life and Health Technology Special Project of Nanjing (No. 202205055).

ORCID

Yanyu Zhou  https://orcid.org/0009-0008-5807-9109

Yihua Song  https://orcid.org/0009-0000-5431-2163

Kankan She  https://orcid.org/0000-0002-4805-1064

Xinxia Li  https://orcid.org/0009-0009-0443-1215

Yu Hu  https://orcid.org/0000-0003-3060-8095

Xinbao Ning  https://orcid.org/0000-0002-9193-3118