## 4 Development of Proposed Method
### Proposed Method
#### 4.2.5 Signal Reconstruction
This notable superiority can be attributed to the implementation of a robust multistage procedure, which is adept at recovering signals in the presence of noisy modes. The robustness of the proposed method is demonstrated by its consistent performance across all input signals, even those with varying noise characteristics. This practical validation strongly supports the applicability of the proposed method. The effectiveness of the multistage procedure in signal recovery despite the presence of noise is highlighted, leading to the conclusion that the proposed method is more reliable and accurate than the existing methods. Its ability to mitigate the impact of noise on signal recovery further enhances its superiority. The presented results primarily aim to introduce and validate the efficacy of the proposed VMD-based denoising method. The results presented in this subsection clearly demonstrate the effectiveness of the proposed method in cleaning ECG signals tainted by thermal noise. However, the inclusion of arrhythmic events has the potential to broaden the study's scope. Therefore, a preliminary experiment has been conducted to explore this potential avenue for future research. The same database has been utilized, and record 203 has been selected for its severe noise contamination, where arrhythmic events cause beat-by-beat morphological variations. Figure 13 illustrates the outcomes, with the top panel displaying the original ECG signal and the bottom panel showcasing the denoised result using the proposed VMD-based method. Though the ground truth signal is unavailable, visual inspection shows that the denoised signal appears noticeably cleaner and less noisy than the original noisy signal. The method successfully restores critical features, such as peaks and highly variable segments of the signal. Despite subtle variations, the overall quality of the denoised signal remains uncompromised. These findings affirm the practicality of the proposed method in mitigating the sensor noise while retaining the previously concealed details. Figure 11: Plot of “ECG100” signal (red) along with its noise-contaminated version (black). Figure 12: Results for the proposed VMD-based method for denoising a few ECG signals taken from the freely available Physionet MIT-BIH arrhythmia database: (**a**,**b**) results for “ECG100” and a zoomed version, respectively; (**c**,**d**) results for “ECG103” and a zoomed version, respectively; (**e**,**f**) results for “ECG105” and a zoomed version, respectively; (**g**,**h**) results for “ECG115” and a zoomed version, respectively; and (**i**,**j**) results for “ECG215” and a zoomed version, respectively. ## 7 Conclusions

In this paper, a new signal-denoising method using VMD with DFA-based thresholding and SWT has been developed. The proposed method separates signal and noise components into different groups of BLIMFs by taking advantage of VMD's strength in handling noise and mode mixing. The adaptive thresholding (based on DFA scaling exponents) allows the method to adjust the threshold value according to the characteristics of each mode, which helps in identifying and separating noise more effectively. Using SWT after VMD helps keep shift invariance, which improves the ability to remove noise without affecting the main signal. The results from numerical experiments show that the proposed method performs better than traditional methods in terms of denoising quality. The performance has been evaluated using metrics such as NMSE, MI, and SNR improvement. The analysis of computational complexity shows that the proposed method exhibits a similar computational demand to other VMD-based methods while providing better denoising results. These findings show that the proposed method is effective for both synthetic and real-world signals, especially for complex signals like ECG data. Future work could focus on adapting the proposed method to work well with non-Gaussian noise, especially impulsive noise.