## 4 Development of Proposed Method
### Proposed Method
#### 4.2.5 Signal Reconstruction
VMD can separate different frequency components, and using DFA-based adaptive thresholding can help detect and reduce impulsive noise. More research could involve creating a better mode classification system by using

Figure 13: Performance of the proposed method for an ECG signal with arrhythmic events causing beat-by-beat morphological variations: (**top**) original signal and (**bottom**) denoised result. higher-order statistics or machine learning. Further work should also consider applying the method to other types of real-world signals beyond ECG, such as radar and audio signals, to demonstrate its versatility. Using recent developments in deep learning for mode selection and adaptive thresholding could also improve the method's robustness and adaptability. Another possible direction is to combine the proposed method with adaptive filtering techniques like the RLS algorithm [31; 32]. This hybrid approach could be useful in non-stationary environments, where adaptive filtering could use the pre-processed signal from the proposed (VMD-DFA-SWT) method to improve the tracking of the changing signal characteristics. Finally, making the computational implementation more efficient by using parallel processing and GPU acceleration could solve scalability problems and make the method more suitable for large-scale applications. Conceptualization, methodology, software, validation, and formal analysis, F.K. ; resources, M.T.A. ; writing--original draft preparation, F.K. ; writing--review and editing, M.T.A. ; visualization, F.K. and M.T.A. ; supervision, M.T.A. All authors have read and agreed to the published version of the manuscript. This research received no external funding. The study did not require ethical approval. Not applicable. The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding authors. The authors declare no conflicts of interest. The following abbreviations are used in this manuscript: \begin{tabular}{l l} DFA & Detrended Fluctuation Analysis \\ SWT & Stationary Wavelet Transform \\ PDF & probability density function \\ RMS & Root-Mean-Square \\ wGn & white Gaussian noise \\ LiDAR & Light Detection And Ranging \\ EEG & Electro-Energhola-Gram \\ ECG & Electro-Cardi-Gram \\ EMD & Empirical Mode Decomposition \\ EEMD & Ensemble EMD \\ IMFs & Intrinsic Mode Functions \\ VMD & Variational Mode Decomposition \\ BLIMFs & Band-Limited IMFs \\ EMD-DFA & Method based on EMD combined with DFA \\ EMD-SWT3 & Method based on combining EMD and 3-level SWT \\ MI & Mutual Information \\ SC & Stringent Correlation \\ EMD-SC & Method based on EMD combined with SC \\ EMD-MI & Method based on EMD combined with MI \\ VMD-DFA & Method based on VMD combined with DFA \\ DWT & Discrete Wavelet Transform \\ VMD-DFA-DWT & VMD-DFA incorporating DWT \\ ADMM & Alternating Direction Method of Multipliers \\ \end{tabular} 

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