# Research on a Noise Reduction Method Based on Multi-Resolution Singular Value Decomposition
## 3 Analysis
### The Optimal Model of MRSVD
Comparing the peak SNR and minimum MSE of three decomposition types, it can be seen that it has the best decomposition effect when N = 2. Figure 4: The SNR and MSE in the three-division method. (a) x1, (b) x2. Due to space limitations, the SNR and MSE obtained by adding the white noise with the SNR is respectively \(-5\), \(5\) and \(20\) db are given in the tables below. Table 1 lists the SNR under different noise reduction models, Table 2 shows the MSE values, and Table 3 shows the optimal decomposition layers. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**SNR/db**} & \multicolumn{4}{c}{x1} & \multicolumn{4}{c}{x2} \\ \cline{2-9}  & \(\mathbf{-5}\) & \(\mathbf{1}\) & \(\mathbf{5}\) & \(\mathbf{20}\) & \(\mathbf{-5}\) & \(\mathbf{1}\) & \(\mathbf{5}\) & \(\mathbf{20}\) \\ \hline Two-division & 9.182 & 12.621 & 14.210 & 26.718 & 7.129 & 12.526 & 15.618 & 26.458 \\ Three-division(N = 1) & 0.752 & 6.253 & 8.071 & 7.107 & 2.445 & 7.375 & 8.031 & 6.997 \\ Three-division(N = 2) & \(-\)2.186 & 5.734 & 7.361 & 5.585 & 1.762 & 7.322 & 8.039 & 5.867 \\ Four-division(N = 1) & 8.396 & 11.735 & 13.374 & 26.135 & 7.597 & 11.662 & 14.388 & 22.725 \\ Four-division(N = 2) & 5.501 & 12.611 & 14.125 & 30.321 & 7.012 & 12.523 & 15.606 & 30.385 \\ Four-division(N = 3) & 7.453 & 8.255 & 14.212 & 29.712 & 6.821 & 12.463 & 15.528 & 29.263 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The SNR of different noise reduction models. Figure 5: The SNR and MSE in the four-division method. (**a**) x1, (**b**) x2. By observing Tables 1-3, it is usually satisfied (\(-5\) db, \(1\) db, \(5\) db) that:  
Maximum value of SNR: two-division \(>\) four-division \(>\) three-division  
Minimum value of MSE: two-division \(<\) four-division \(<\) three-division  
When the noise power is much smaller than the original signal (\(20\) db), the following is satisfied:  
Maximum value of SNR: four-division (N = \(2\)) \(>\) two-division \(>\) three-division  
Minimum value of MSE: four-division (N = \(2\)) \(<\) two-division \(<\) three-division  
The noise reduction effect of the three-division method is inferior to the two-division method and the four-division method, because the power leakage of the three-division method is more serious when it is decomposed. In fact, when the number of division of decomposition method is even, the effect is always better than that is odd. The SNR of the four-division method is slightly higher than the two-division method in the case of low noise power, but it has obvious drawbacks. Its decomposition layer is obviously much higher than the two-division method, and the noise reduction efficiency is significantly lower than that of the two-division method. Before the peak SNR is reached, the SNR of the two-division method is higher than the four-division under the same decomposition layer. When the initial SNR is \(20\) db, the power of the original signal is about \(100\) times that of the noise. At this time, the influence on the signal by increasing the SNR is very small, and the rate of noise reduction should be taken as the main consideration. The optimal decomposition layers of signal \(x1\) and \(x2\) decomposed by the two-division method is seven and one layers, respectively, while for the four-division method, it is \(1755\) and \(15\) layers. In this case, the noise reduction speed of the two-division method is much higher than that of the four-division method. For the multi-division method of L \(>\) 2 (L is even), there is also the same problem as the four-division method, and the larger the L, the lower the noise reduction speed, which obtained a better SNR and MSE by sacrificing the speed of noise reduction. In actual engineering applications,