## 4 Development of Proposed Method
### Proposed Method
#### 4.2.5 Signal Reconstruction
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **SNR** & & \multicolumn{2}{c}{**VMD-DFA**} & \multicolumn{2}{c}{**VMD-DFA-**} & \multicolumn{1}{c}{**Proposed Method**} \\ \hline \multirow{4}{*}{Piecewise-regular} & \multirow{2}{*}{0} & Mean & 3.623 & 3.368 & **4.001** \\  & & SD & 0.0712 & 0.0735 & 0.0756 \\  & & Median & 3.644 & 3.691 & 4.009 \\ \cline{2-6}  & \multirow{2}{*}{10} & Mean & 4.069 & 4.056 & **4.369** \\  & & SD & 0.0496 & 0.0570 & 0.0614 \\  & & Median & 4.075 & 4.063 & 4.378 \\ \cline{2-6}  & \multirow{2}{*}{20} & Mean & 4.348 & 4.424 & **4.801** \\  & & SD & 0.0442 & 0.0471 & 0.0493 \\  & & Median & 4.334 & 4.432 & 4.814 \\ \hline \multirow{4}{*}{Doppler} & \multirow{2}{*}{0} & Mean & 5.362 & 4.506 & **5.864** \\  & & SD & 0.0848 & 0.0795 & 0.0564 \\  & & Median & 5.342 & 4.502 & 5.855 \\ \cline{2-6}  & \multirow{2}{*}{10} & Mean & 5.646 & 4.914 & **6.257** \\  & & SD & 0.0536 & 0.0477 & 0.0474 \\  & & Median & 5.659 & 4.907 & 6.267 \\ \cline{2-6}  & \multirow{2}{*}{20} & Mean & 5.867 & 5.317 & **6.731** \\  & & SD & 0.0451 & 0.0266 & 0.0276 \\  & & Median & 5.858 & 5.314 & 6.737 \\ \hline \multirow{4}{*}{HeavySine} & \multirow{2}{*}{0} & Mean & 5.359 & 4.465 & **6.999** \\  & & SD & 0.0807 & 0.0571 & 0.0533 \\  & & Median & 5.360 & 4.459 & 6.995 \\ \cline{2-6}  & \multirow{2}{*}{10} & Mean & 5.673 & 4.918 & **6.675** \\  & & SD & 0.0308 & 0.0359 & 0.0551 \\  & & Median & 5.675 & 4.908 & 6.732 \\ \cline{2-6}  & \multirow{2}{*}{20} & Mean & 5.905 & 5.389 & **6.862** \\  & & SD & 0.0676 & 0.0281 & 0.0386 \\  & & Median & 5.918 & 5.388 & 6.867 \\ \hline \multirow{4}{*}{Blocks} & \multirow{2}{*}{0} & Mean & 1.306 & 2.759 & **2.795** \\  & & SD & 0.0950 & 0.0308 & 0.0856 \\  & & Median & 1.300 & 2.759 & 2.802 \\ \cline{2-6}  & \multirow{2}{*}{10} & Mean & 2.304 & 2.935 & **3.498** \\  & & SD & 0.0472 & 0.0352 & 0.0223 \\  & & Median & 2.322 & 2.926 & 3.499 \\ \cline{2-6}  & \multirow{2}{*}{20} & Mean & 2.547 & 3.160 & **3.623** \\  & & SD & 0.0186 & 0.0344 & 0.0274 \\  & & Median & 2.544 & 3.162 & 3.616 \\ \hline \multirow{4}{*}{Bumps} & \multirow{2}{*}{0} & Mean & 2.756 & 2.745 & **2.952** \\  & & SD & 0.0411 & 0.0339 & 0.0652 \\  & & Median & 2.763 & 2.740 & 2.928 \\ \cline{2-6}  & \multirow{2}{*}{10} & Mean & 2.938 & 2.913 & **3.208** \\  & & SD & 0.0418 & 0.0247 & 0.0508 \\  & & Median & 2.939 & 2.908 & 3.207 \\ \cline{2-6}  & \multirow{2}{*}{20} & Mean & 3.149 & 3.037 & **3.562** \\  & & SD & 0.0411 & 0.0833 & 0.0961 \\  & & Median & 3.151 & 3.012 & 3.555 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Quantitative results for test signals: MI for various VMD-based denoising methods for different input SNR levels. with 0 dB SNR, the performance of the previous method is comparable to that achieved by the EMD-DFA method [11]. The corresponding NMSE results for VMD-based methods are presented in Table 4. It is noticed that the proposed VMD-based method (as presented in this paper) exhibits significantly improved performance compared with that achieved by the rest of the methods for all experiments except for the Blocks signal with 20 dB SNR. Even in this situation, the NMSE performance of the proposed VMD-based method is comparable to that achieved by the VMD-DFA-DWT method [19]. By comparing the results presented in Tables 3 and 4, it is observed that, in almost all experiments, the proposed VMD-based method and previous EMD-based method [21] give better performance as compared with that achieved for the other methods, with the proposed method appearing as the best performing method (there are only a few exceptions, as discussed earlier). This underscores the effectiveness of the proposed method, which is based on the strong architecture of VMD. The overall performance comparison from the viewpoint of MI (16) for EMD-based methods is presented in Table 5, which shows the effectiveness of the previous method [21] in comparison with the rest of the EMD-based methods considered in this paper.