## 4 Model Testing of Actual Signal Denoising
### Signal Denoising Processing  
A further analysis requires denoising the obtained vibration acceleration signals. Due to space constraints, this section presents results using the z-direction acceleration of the sensor at the front end of the lower arm when the ecological discharge is at 20% of opening. Acceleration time-history data from each channel show that the measured vibration signals contain significant noise, with multiple spikes in the curves.  
When denoising the measured vibration signals using the method described in this paper, the VMD parameters are first adaptively optimized based on the MVO algorithm. The MVO-VMD-MPE-WTD iterative optimization curve is shown in Figure 4. From Figure 4, it can be determined that the optimal number of VMD modes is \(K=12\) and the secondary penalty factor is \(\alpha=2000\).  
The MVO-optimized VMD decomposition results are shown in Figure 5.  
Figure 4: MVO-VMD-MPE-WTD convergence curve: (**a**) MVO-VMD adaptive curve; (**b**) VMD convergence curve.  
Figure 5: MVO-VMD-MPE-WTD decomposition results.  
Based on the VMD decomposition results, the original signal is decomposed into 12 IMF components. The MPE values for each IMF component are calculated and displayed in Figure 6.  
According to Figure 6, among the 12 IMF components obtained from VMD decomposition, the MPE values of the 1 st, 4 th, 7 th, and 12 th components are less than 0.6 and are considered pure components. The remaining IMF components are regarded as noise and need to be denoised using wavelet thresholding. The db4 wavelet is used as the basis function, and a hard threshold is applied. After denoising the noise components, they are then combined with the pure components and reconstructed to obtain the denoised vibration signal. The comparison of the signal before and after denoising is shown in Figure 7.  
In practical engineering, the power of both the useful signal and noise in the measured signal is unknown, making it impossible to assess denoising performance using _SNR_. Therefore, this paper introduces the \(d_{n\textit{SNR}}\)[23] to quantitatively evaluate the quality of denoising, with the calculation formula as follows:  
\[d_{n\textit{SNR}}=10\ \log_{10}\bigl{(}P_{s}/P_{g}\bigr{)} \tag{17}\]  
Figure 6: MPE of each IMF component.  
Figure 7: Comparison of signals on pre-denoising and post-denoising.  
In the equation, \(P_{s}\) represents the power of the noisy signal; \(P_{g}\) represents the power of the denoised signal.  
When denoising the measured signal, three approaches are used: Experimental Group A only uses WTD for denoising; Group B discards noise components directly after MVO-VMD-MPE decomposition; and the control group uses the method described in this paper for denoising. The \(d_{nSNR}\) results for each method are shown in Table 2:  
According to the calculation results, the MVO-VMD-MPE-WTD method achieves the lowest dnSNR value of 1.14, demonstrating superior performance compared to the other two denoising methods. The combination of MVO-optimized VMD decomposition and WTD offers the best denoising performance. Based on \(d_{nSNR}\), the method proposed in this paper results in the smallest error and the most effective denoising of the measured signal from the spillway radial gate.