## 4 Development of Proposed Method
### Proposed Method
#### 4.2.5 Signal Reconstruction
However, in the context of signal reconstruction, this estimation process can introduce a bias or inaccuracy in amplitude estimation. These estimation errors can lead to disparities between the reconstructed and original signals, manifesting as differences in their amplitudes. In the case of EMD-SWT3, wavelet decomposition and reconstruction can impact the signal's amplitudes. During wavelet decomposition, the signal is divided into different frequency components, and during reconstruction, these components are combined to reconstruct the signal. However, this process can introduce artifacts and alterations, affecting the amplitudes of the reconstructed signal. Certain frequency components may be amplified or attenuated during the reconstruction, leading to discrepancies in the amplitudes of the reconstructed signal compared to the original noiseless signal. As shown in Figure 7e, the previous EMD-based method generates an estimate of the original signal that is highly accurate and nearly devoid of artifacts. This is accomplished through the utilization of adaptive threshold estimation, which significantly enhances the denoising process by effectively distinguishing between noise and signal components. As a result, the reconstruction of the denoised signal based on the thresholded modes successfully avoids the artifacts that are typically observed in the results obtained from other methods considered in this paper. Overall, while various methods have been employed to denoise signals, their effectiveness varies significantly. The presence of artifacts in denoised signals still remains a significant challenge, affecting the overall quality of the signal. ### Performance Comparison of VMD-Based Methods for Test Signals

Figure 8 shows BLIMFs of the decomposed Bumps signal contaminated with 10 dB SNR. Here, 10 levels of decomposition have been performed for VMD in order to ensure compatibility with the EMD-based methods. The high-order modes correspond to fast oscillations, and the decomposition starts from the coarsest level. Therefore, to remove noise, the noise-dominant modes should be separated into several last BLIMFs, unlike in the case of IMFs. Figure 9 shows the variation in the scaling exponent \(\alpha\) versus the index of BLIMFs (shown in Figure 8), which shows that \(\alpha\) decreases for noise-dominant BLIMFs, highlighting the distinction between noise-dominant and signal-dominant modes. In the existing VMD-DFA method [18], a fixed threshold of \(\alpha\geq 0.75\) is used (see the dashed black horizontal line in Figure 9) to select the modes for reconstruction. As shown, BLIMFs 1-4 have scaling exponents greater than or equal to 0.75, indicating that these modes are used in signal reconstruction. However, this fixed threshold of 0.75 may be inadequate in distinguishing noise-dominant modes in certain scenarios, particularly in the presence of mode mixing. To address mode mixing and improve the denoising process, the VMD-DFA-DWT [19] approach uses a lower fixed threshold of \(\alpha<0.25\), represented by the dotted horizontal line in Figure 9. As shown, BLIMFs 6-10 are identified as noise-dominant modes and further processed using wavelet-based windowing, while BLIMFs 1-5 are retained for reconstruction without modification. The wavelet-based processing is used further to preserve useful information contained in the modes instead of completely discarding them. The \(db8\) wavelet function is used here, and the level of decomposition is determined by

Figure 7: Denoised signal (in black) for various EMD-based methods along with the original signal (in red): (**a**) EMD-SC [10], (**b**) EMD-MI [14], (**c**) EMD-DFA [11], (**d**) EMD-SWT3 [13], and (**e**) previous method [21]. Shannon entropy. The decomposition process continues until the entropy of the detail coefficient (\(CD_{k}\)) exceeds that of the approximation coefficient (\(CA_{k}\)).