## 4 Development of Proposed Method
### Proposed Method
#### 4.2.5 Signal Reconstruction
In the EMD-SC algorithm [10], the subsequent correlation calculation, which includes linear operations like signal comparison, is influenced by the number of IMFs. Furthermore, fundamental arithmetic and comparison operations come to the forefront in the thresholding and de noising phase. Taking everything into account, the overall complexity of EMD-SC [10] can be expressed as \(\mathcal{O}\{(\frac{m}{2}+2)N\log_{2}N+(\frac{m}{2}+1)N\}\). In EMD-MI [14], the calculation of correlations involves basic operations such as comparing signals, the assessment of MI, and the application of thresholds employing a modified forward selection method. Since the second part of the method consists of the initial separation of high and low-frequency components, this step approximates to \(m/2\) iterations. For each high-frequency IMF, wavelet denoising typically has a complexity that depends on the specific implementation but can be approximated as \(\mathcal{O}\{N\log_{2}N\}\) or slightly higher. Therefore, the complexity of this part is approximately \(\mathcal{O}\{(\frac{m}{2}+1)N\log_{2}N\}\). Combining these complexities, the overall computational complexity of the EMD-MI algorithm [14] can be estimated as \(\mathcal{O}\{(\frac{m}{2}+2)N\log_{2}N+(\frac{m}{2}+1)N\}\). The EMD-DFA method in [11] embarks on a process involving IMF iteration and scaling exponent calculation. The overall computational complexity of EMD-DFA approximates to \(\mathcal{O}\{N\log_{2}N+(m+1)(13N+1)\}\), where the first term arises from the EMD step, and the second to the DFA calculation. The latter part consists of the fixed region for the scaling exponent computation, which lies within a fixed range (see [21; 26] for further details). The reconstruction part involves iterating over the scaling exponent's array to identify which IMFs to include in the final signal reconstruction. In EMD-SWT3 [13], the inherent computational complexity of the SWT function adheres to \(\mathcal{O}\{N\log_{2}N\}\)[27]. Consequently, the denoising algorithm's combined complexity appears to be \(\mathcal{O}\{N\log_{2}N+N\log_{2}N\}\). In the previous method [21], the inclusion of the embedded DFA step within the mode loop (while performing EMD) results in a computational complexity of approximately \(\mathcal{O}\{13N(m+1)\}\). Following this, the identification of breakpoints and the separation of high-frequency and low-frequency IMF clusters contribute to the overall computational load of the SWT process. In summary, the total complexity of the previous method [21] can be approximated as \(\mathcal{O}\{N\log_{2}N+N\log_{2}N+(m+1)(13N+1)\}\). In the case of VMD, the signal decomposition into \(B\) BLIMFs is executed iteratively, and the corresponding computational complexity can be expressed as \(\mathcal{O}\{BN\log_{2}N\}\)[28]. In the case of the VMD-DFA method [18], the overall computational complexity can be framed as \(\mathcal{O}\{BN\log_{2}N+(m+1)(13N+1)\}\). For the VMD-DFA-DWT method [19], the DWT complexity aligns with \(\mathcal{O}\{N\}\), and hence, the overall computational complexity approximates to \(\mathcal{O}\{BN\log_{2}N+(m+1)(13N+1)+N\}\). Finally, the computational complexity of the proposed VMD-based method (involving VMD, DFA, and SWT) can be approximated as \(\mathcal{O}\{BN\log_{2}N+N\log_{2}N+(m+1)(13N+1)\}\). The last column in Table 1 presents a computational complexity comparison for a typical case with parameter values selected to match those used in the experiments conducted in this paper (\(N=2^{11}\) for signal length, and 10 modes for EMD and VMD). It is observed that the computational complexities of EMD-SC [10], EMD-SWT3 [13] and EMD-MI [14] are about 10 times lower than the rest of the methods considered in this paper. These methods would appear as suitable choices when the computational resources are constrained. However, it is worth mentioning that the denoising performance (as detailed in the next section) of these methods appears far inferior to that of the rest of the