# Research on a Noise Reduction Method Based on Multi-Resolution Singular Value Decomposition
## 3 Analysis
### The Optimal Model of MRSVD  
Using different parameters, MRSVD will have different forms, and not all noise reduction models have good effects. Therefore, it is necessary to make a discussion on various models to select the optimal model. In this paper, the different values of the decomposition method L, the decomposition type N and the decomposition layer M are used as variables to draw the relationship curve between two evaluation indicators (SNR, MSE) and the number of decomposition layers in the two-division method, the three-division method, and the four-division under different decomposition types. The optimal model of MRSVD is selected by the specific analysis on the change trend of SNR curve and the MSE curve. Figure 3 is the SNR and MSE curves in the two-division method at an initial SNR of 1 db, where (a) is the curve for \(x1\) and (b) is the curve for \(x2\). It can be seen from Figure 3a,b that \(x1\) and \(x2\) reach the peak SNR at M = 48 and 9, respectively, and the value of MSE just reaches the minimum value. By observing the curve, it can be seen that the initial growth of the SNR curve is faster, then slows down gradually, and begins to decrease after reaching the peak SNR. This is because most of the noise signal is separated into the detail signal at the beginning, and the original signal is basically retained as an approximation signal, so the SNR rises very quickly. When the SNR becomes large, the noise power is already quite small. At this time, the separated noise will inevitably be mixed with some of the original signals, resulting in a decrease in SNR. This is consistent with the principle of noise reduction. An increase on the MSE value indicates that the amplitude of the original signal \(x(n)\) is already bigger than the noise-reduced signal \(A_{M}\), and the original signal has been excessively separated into the detail signal. If the decomposition continues, the signal distortion will become more and more serious, and this should be avoided during the noise reduction process. Figure 4 is the SNR and MSE curve in the three-division method at an initial SNR of 1 db, where (a) is the curve for \(x1\) and (b) is the curve for \(x2\). Figure 3: The SNR and MSE in the two-division method. (a) \(X_{1}\), (b) \(X_{2}\). It can be seen from Figure 4a,b that the three-division method has two types of decomposition: N = 1 and N = 2. For the three-division method, regardless of the value of decomposition type N of the signal x1 or x2, its SNR reaches the maximum value when M = 1. The SNR curve shows a downward trend and a negative value occurs. If the SNR is negative, then the original signal power is less than the noise power, it shows that the three-division method separates too much of the original signal, which is seen noise wrongly, into the detail signal, that is, the power of the original signal is leaked into the noise, and the signal is severely distorted. It is observed that the value of the MSE increases while the SNR decreases, indicating that the power of the original signal is getting smaller and smaller, which also means that the accuracy of the reconstructed signal after decomposition is gradually degraded as the number of decomposition layers increases. Figure 5 is the SNR and MSE curve in the four-division method at an initial SNR of 1 db, where (a) is the curve for x1 and (b) is the curve for x2. It can be seen from Figure 5a,b that the four-division method has three decomposition types. When N = 1, the SNR of signal x1 increases first and then decreases gradually, and the SNR of x2 has the trend of decreasing, reaching maximum at M = 9 and M = 2 respectively; while N = 2 and N = 3, the SNR has been increasing until M = 500, and the curve becomes very gentle after the number of decomposition layer reach to 500, which indicates that the larger the value of N, the less noise is separated from each layer, and the lower the efficiency of noise reduction.