## 4 Development of Proposed Method
### Proposed Method
#### 4.2.5 Signal Reconstruction
Table 2 shows the correlation coefficient with the original signal of each IMF. It should be emphasized that the existence of noise tends to muddle the results, since the initial few IMFs may contain meaningful information, but the EMD-SC method has removed the majority of IMFs. EMD-MI is another thresholding-based method where the separation of IMFs is carried out as follows. * The initial partition index (IPI) is \(\left[\frac{NM}{2}\right]\), where \(NM\) specifies the number of modes and \(\left[\cdot\right]\) signifies the integer portion. * The first signal-dominant group is the sum of all IMFs with indexes greater than IPI. The remaining IMFs are possible modes for the noise-dominant group. There are 10 IMFs, and the IPI for the Piecewise-regular signal is 5. As a result, IMFs with indexes ranging from 1 to 5 are perspective modes for the noise-dominant group, while the sum of IMFs with indexes ranging from 6 to 10 is referred to as the initial signal-dominant group. The modified forward selection strategy [14] is as follows. 1. Compute the MI between the original noisy signal and the initial signal-dominant group (i.e., the low-frequency group formed from the sum of all IMFs with an index greater than or equal to the IPI). \begin{table}
\begin{tabular}{c c c} \hline \hline
**IMFs** &
\begin{tabular}{c} **Correlation** \\ **Coefficient** \\ \end{tabular} & **Threshold** & **Retained IMFs** \\ \hline
1 & 0.019 & - \\
2 & 0.023 & - \\
3 & 0.054 & - \\
4 & 0.101 & - \\
5 & 0.235 & 0.213 & 5 \\
6 & 0.317 & 6 \\
7 & 0.433 & 7 \\
8 & 0.565 & 8 \\
9 & 0.479 & 9 \\
10 & 0.189 & - \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Correlation coefficient between each IMF and the original signal obtained from Bumps signal corrupted with 10 dB SNR for EMD-SC [10] method. Figure 4: Test signals used in the computer simulations: (**a**) Piecewise-regular, (**b**) Doppler, (**c**) Heavy-Sine, (**d**) Blocks, and (**e**) Bumps. 2. Iteratively add each remaining IMF, starting from the IMF at the IPI. For each new signal formed by adding one IMF at a time, calculate the MI between this new signal and the original noisy signal. This process continues until all IMFs have been added. 3. The final partition index is identified as the IMF index where the MI between the newly formed signal and the original noisy signal is at its minimum. 4. The final noise-dominant group is formed by the sum of all IMFs with indices less than the final partition index, while the final signal-dominant group is formed by the sum of all IMFs with indices equal to or greater than the final partition index. Figure 5: IMFs obtained from EMD of the Bumps signal with 10 dB SNR. Figure 6 plots the scaling exponent \(\alpha\) obtained for the respective IMFs as shown in Figure 5. It is observed that the scaling exponent \(\alpha\) is very small for initial (noise-dominant) IMFs and increases for the later (signal-dominant) IMFs. Consequently, for EMD-based denoising, the last modes are selected to reconstruct the filtered signal. The denoised signal for EMD-DFA is

\[\hat{\mathbf{s}}(n)=\sum_{i=3}^{10}\mathbf{d}_{i}(n). \tag{19}\]

In EMD-SWT3 [13], following the EMD decomposition process, the initial IMF is deducted from the noisy original signal. Subsequently, the signal undergoes processing via the SWT filter at a three-level depth to further decrease noise and enhance the SNR. The detailed coefficients of the SWT, which provide additional data, are extracted and employed in the reconstruction process via the inverse wavelet transform. The EMD-DFA [11] method employs fixed thresholds (see dotted and dashed black lines for \(\theta=0.25\) and \(\theta=0.75\), respectively, in Figure 6), selecting IMFs between \(0.25\leq\alpha\leq 0.75\) to identify which IMFs are signal-dominant and good candidates for reconstruction. However, fixed thresholds inherently suffer from a lack of adaptability as they cannot account for the variability in signal characteristics. This might lead to retaining noisy IMFs or discarding important signal content.