The integrated approach centers on an adaptive filter using the LMS algorithm to separate LF and HF components by modeling the relationship between the RR interval signal and the respiratory tidal volume signal.

$W=\left[w_0,w_1,\cdots ,w_{N-1}\right]$ (1)

which adapt to minimize the MSE between the predicted high-frequency component $y\left(k\right)$ and the actual observed RR interval $RR\left(k\right)$, where the filter output at discrete time k is given by:

$y\left(k\right)={\displaystyle {\sum }_{i=0}^{N-1}{w}_i\left(k\right)V_t\left(k_i\right)}$ (2)

Here, $V_t\left(k\right)$ is the respiratory tidal volume reference signal at time k.

$z\left(k\right)=RR\left(k\right)y\left(k\right)$ (3)

The LMS algorithm updates the filter weights iteratively to minimize the expected squared error, using the gradient descent approach:

$w_i\left(k+1\right)=w_i\left(k\right)+z\left(k\right)V_t\left(k_i\right)$ (4)

$s\left(t\right)=C+A\sin \left(2\pi f_{h_t}+{\alpha }_h\right)+B\sin \left(2\pi f_{l_t}+{\alpha }_l\right)$ (5)

Then tidal volume $V_t\left(k\right)$:

$V_t\left(k\right)=D\sin \left(2\pi f_{h_t}\right)+E$ (6)

where, A and B are the amplitudes of HF and LF components and C is the mean RR interval, while $f_{h_t}$ and $f_{l_t}$ are the high and low frequencies of respective components and α_h and α_l are the phases of HF and LF components, with D and E as amplitude and offset of tidal volume signal These signals are sampled with 1 ms resolution and then processed to a 5 Hz uniform sample rate for filtering [9] .

This approach was developed to generate precise artificial heart rate variability (HRV) time series by mathematically modeling simplified physiological processes of the human body using MATLAB tools. It represents a significant advancement over existing methods that primarily replicate statistical features of recorded data but often overlook the underlying biophysical mechanisms governing heart rate variability, which many studies fail to address [10] . To prevent phase distortion artifacts common with filtering physiological signals, a zero-phase filtering technique is integrated with the adaptive filtering. The zero-phase filtering is achieved by:

This two-pass filtering cancels phase distortion while maintaining the frequency response, thus preserving the shape and timing relationships in the heartbeat signal. Outcomes of the Mathematical Application:

1) The adaptive LMS filter successfully separates LF and HF components by using respiratory signal reference.

2) Zero-phase filtering ensures no phase shifts distort the filtered signals.

3) The algorithm converges over time, requiring tuning of filter length (typically N = 20) for optimal performance.

4) Simulated and real data analyses show improved spectral separation for cardiac autonomic control measurements.

This section synthesizes the core mathematical framework and adaptive filtering theory applied to heartbeat signal processing in the paper, detailing the LMS algorithm, signal models, zero-phase filtering, and their integration for physiological signal enhancement [11] [12] . Additionally, changing the seed of the random number generator produces different realizations of the random phases, ensuring that each new time series maintains the same temporal and spectral characteristics as the original data exactly as in Figure 1 shown:

In this integrated framework, noise models and MATLAB simulations are designed to realistically replicate the distortions commonly encountered in various practical ECG acquisition scenarios, which include:

1) Power-line Interference (PLI): Spectral noise caused by electrical power systems, usually modeled as narrow-band sinusoidal interference at 50 or 60 Hz and harmonics.

2) Baseline Wander (BW): Low-frequency noise caused by respiration or body movements, modeled as slow varying trends or wandering baselines in the ECG waveform.

3) Electrocardiogram (EMG) Noise: Muscle activity noise with variable amplitude and frequency components, often modeled as band-limited stochastic or Gaussian noise to simulate muscle contractions contaminating the ECG.

4) Motion Artifacts: Noise introduced from patient movement or electrode cable motion, simulated as transient, irregular bursts or shifts, posing challenges due to their non-stationary, dynamic nature.

5) Electrode Contact Noise: Disturbances caused by electrode displacement or wearing, introduced as intermittent signal drops or spikes.

6) Simulation and Environmental Conditions: The data simulation uses short ECG segments (e.g., 10-second clips) often overlapping by 50 percent to avoid information loss and abrupt transitions [17] .

Different random noise types are superimposed on clean ECG signals under controlled or semi-supervised settings, replicating stable low-noise environments and highly dynamic, noisy environments respectively. Simulation models incorporate realistic noise amplitude and temporal characteristics to stress-test the adaptive LMS filter’s ability to track and cancel non-stationary noise. Hybrid simulation environments combine multiple noise sources to test robustness across varied clinical and ambulatory conditions, where using these noise models and simulations enables realistic evaluation of the framework’s effectiveness in improving the resulting ECG signal quality by increasing SNR and minimizing MSE, while preserving diagnostically important features in the ECG waveforms [18] . This type of noise modeling and environmental simulation reflects the requirements for practical world-time ECG monitoring systems to handle diverse noise challenges encountered in real-world conditions [19] .

The primary performance metrics for evaluation focus on noise attenuation and signal fidelity, and most crucial of these metrics are:

In various studies, combinations of these metrics show significant improvements in SNR (which often 30 - 60 percent), reduced MSE, and good percentage for correlation coefficients, demonstrating their effectiveness in realistic noisy conditions for reliable ECG monitoring and analysis. These metrics collectively ensure the framework not only removes the unwanted noise but also maintaining the diagnostic features of the ECG signal critical to be as downstream clinical interpretation and automated analysis [22] .

The integrated framework that combines ZP and adaptive LMS filters demonstrates strong noise reduction performance for enhancing ECG signals:

Adaptive LMS filters dynamically update their filter coefficients in real time based on the error between the noisy input and the desired signal, enabling the framework to effectively track and reduce time-varying noise components such as emotion artifacts, muscle (EMG) noise, and electrode interference (EI). While ZP filtering efficiently removes stationary and repetitive noise like power-line interference and baseline wander without phase distortion, where the adaptive LMS filter complements this by continuously adjusting to changing noise characteristics, resulting in superior overall noise suppression and signal fidelity in dynamic environments. Experimental results, including those from real-world ECG datasets, demonstrate that adaptive LMS filtering outperforms fixed filters in reducing noise metrics such as SNR and MSE under non-stationary conditions [26] .

The Key points on noise reduction effectiveness across diverse noise scenarios include performance highlights under varying noise conditions:

This integrated framework demonstrates strong noise reduction capabilities, as clearly shown through a simulation that closely replicates real-world conditions using MATLAB. Figure 2 presents the ideal human heart rate function, which is then approximated by adding noise—illustrated in Figure 3 —to create a more realistic simulation. The noisy signal is subsequently processed by the filters of the Hybrid System, shown in Figure 4 , which is the main focus of our study.

The enhancement of noise performance in communication systems can be effectively evaluated using key signal quality metrics such as:

1) Signal-to-Noise Ratio (SNR): The Hybrid Filtering Framework aims to maximize SNR by minimizing noise components while preserving signal integrity. ZP filtering ensures no phase distortion, which is critical in the resulted signals, and LMS filtering adaptively suppresses time-varying noise.

2) Mean Squared Error (MSE): MSE serves as an objective measure of the filter’s accuracy in noise reduction. The Hybrid scheme reduces MSE significantly compared to standalone filtering techniques, indicating improved signal reconstruction and reduced distortion.

3) Correlation Coefficient (CC): Higher CC values indicate that the hybrid framework effectively retains the key features of the ideal signal while eliminating unwanted noise. This is essential for preserving the similarity between the desired and resulting signals, ensuring a more accurate determination of the convergence ratio [24] [25] . Table 1 presents a comparison between the noisy signal and the signal filtered through the hybrid filtering system, accompanied by an illustration of this relationship in the corresponding Figure 5 .