## 4 Development of Proposed Method
### Proposed Method
#### 4.2.5 Signal Reconstruction
\begin{table}
\begin{tabular}{l l l} \hline \hline
**Denoising Method** & **Operation Count** & **Example** \\ \hline EMD-SC [10] & \(\mathcal{O}\{N\log_{2}N+m(N+1)\}\) & \(2.727\times 10^{4}\) \\ EMD-MI [14] & \(\mathcal{O}\{(\frac{m}{2}+2)N\log_{2}N+(\frac{m}{2}+1)N\}\) & \(5.976\times 10^{4}\) \\ EMD-DFA [11] & \(\mathcal{O}\{N\log_{2}N+(m+1)(13N+1)\}\) & \(2.997\times 10^{5}\) \\ EMD-SWT3 [13] & \(\mathcal{O}\{N\log_{2}N+N\log_{2}N\}\) & \(2.931\times 10^{4}\) \\ Previous Method [21] & \(\mathcal{O}\{N\log_{2}N+N\log_{2}N+(m+1)(13N+1)\}\) & \(3.222\times 10^{5}\) \\ VMD-DFA [18] & \(\mathcal{O}\{BN\log_{2}N+(m+1)(13N+1)\}\) & \(3.607\times 10^{5}\) \\ VMD-DFA-DWT [19] & \(\mathcal{O}\{BN\log_{2}N+(m+1)(13N+1)+N\}\) & \(3.624\times 10^{5}\) \\ Proposed Method & \(\mathcal{O}\{BN\log_{2}N+N\log_{2}N+(m+1)(13N+1)\}\) & \(3.832\times 10^{5}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summary of computational complexity analysis for various denoising methods discussed in this paper. (The parameters used in Example: \(N=2^{11}\), \(m=10\), \(B=10\)). methods. The computational complexities of the previous method [21] and the proposed method appear to be of a similar order as those of the EMD-DFA [11], VMD-DFA [18], and VMD-DFA-DWT [19], yet the previous method outperforms the rest of the EMD-based methods and the proposed VMD-based method appears as the best-performing denoising method among the methods considered in this paper (for example, see Tables 3 and 4 in the next section on the simulation results). ## 6 Simulation Results

This section presents the results of extensive computer simulations carried out to demonstrate the efficacy of the proposed method in comparison with the related state-of-the-art methods. The following methods have been considered for performance comparison. * EMD-SC [10]: A correlation-based method is used to identify and eliminate replicated modes, under the assumption that noise is distributed among all IMFs. * EMD-MI [14]: Conditional MI is used to detect the noise-dominant and signal-dominant cutoff and applies SURE for denoising only on the noise-dominant group of IMFs derived from the original signal using mutual dependency measurements. * EMD-DFA [11]: Utilizes DFA to remove trends from non-stationary time series and to identify scaling properties and long-term correlations by generating a fluctuation graph. * EMD-SWT3 [13]: Employs three-level SWT to remove noise and artifacts from signals by first decomposing them into IMFs and then applying SWT to extract the detailed coefficients, which are used to reconstruct the denoised signal using inverse wavelet transform. * Previous method [21]: A previous EMD-based method for which preliminary results have been presented at a conference [21]. * VMD-DFA [18]: An approach similar to EMD-DFA [11], with EMD being replaced with VMD. * VMD-DFA-DWT [19]: Employs VMD and utilizes DFA for identifying important modes to reconstruct the filtered signal while using DWT to suppress noise in the selected BLIMFs. * Proposed method: The VMD-based proposed method presented in this paper. ### Performance Comparison of EMD-Based Methods for Test Signals

The dataset for synthetic signals is shown in Figure 4 with five test signals: (a) Piecewise-regular, (b) Doppler, (c) HeavySine, (d) Blocks, and (e) Bumps. As a typical example, the decomposed IMFs for the Bumps signal contaminated with 10 dB SNR (given in Figure 4e) are displayed as an example in Figure 5. As shown in Figure 5, the first few IMFs are found to capture the majority of the noise; nevertheless, they may also include some true information. In a nutshell, the low-order IMFs in EMD correspond to noise-dominant modes, while the high-order IMFs correspond to signal-dominant modes. For EMD-SC, a threshold is calculated based on the greatest correlation coefficient \(\rho\) such that

\[\theta=\frac{\max(\rho)}{10\times\max(\rho)-3}. \tag{18}\]

Only IMFs with correlation coefficients \(\rho\) larger than the stipulated threshold are kept.