## 3 Artificial Simulation Signal Denoising  
To evaluate the performance of the denoising algorithm proposed in this paper, we first verify the effectiveness of the denoising method (MVO-VMD-MPE-WTD) using artificially simulated signals. Let  
\[\left\{ \begin{array}{l}Y=X_{1}+X_{2}+X_{3}+X_{noise}\\ X_{1}=\cos(\pi t)\\ X_{2}=2\sin(2\pi t)\\ X_{3}=4\cos(8\pi t)\\ X_{noise}=2 randn(N,1)\end{array} \right. \tag{12}\]  
In the equation, \(X_{1}\), \(X_{2}\), and \(X_{3}\) are three pure signals with different frequencies; \(X_{noise}\) is a random white noise signal; and \(Y\) is the noisy simulation signal. The waveforms of the pure signals, noisy signals, and denoised signals obtained using the proposed denoising method are compared as shown in Figure 2.  
From the comparison of the denoising results, it is clear that after noise is added to the original signal, its trend becomes irregular. Noise completely obscures the original signal, which makes it difficult to distinguish the original waveform. After applying the proposed denoising method, the denoised signal closely matches the waveform of the original signal. To quantitatively compare the effectiveness of the denoising algorithm, we analyze the signal-to-noise ratio (\(SNR\)) and mean squared error (\(MSE\)) before and after denoising. \(SNR\) is a crucial indicator of signal quality; a higher \(SNR\) means less noise in the signal. \(MSE\) reflects the average squared error between the original and denoised signals; a lower \(MSE\) indicates better denoising performance [20; 21; 22]. To further verify the accuracy of the \(SNR\) evaluation trend, both frequency-domain \(SNR\) and time-domain \(SNR\) are introduced. The calculation formulas for frequency-domain \(SNR\), time-domain \(SNR\), and \(MSE\) are as follows:  
\[SNR_{time}=10\ \log_{10}\left(\frac{\sum_{n=1}^{N}s[n]^{2}}{\sum_{n=1}^{N}n[n]^{2 }}\right) \tag{13}\]  
\[SNR_{freq}=10\ \log_{10}\left(\frac{\sum_{n=1}^{N}\left|X[k]\right|^{2}}{\sum_{n=1}^{N }\left|N[k]\right|^{2}}\right) \tag{14}\]  
\[MSE=\frac{1}{N}{\sum_{i=1}^{N}}(x_{i}-\hat{x}_{i})^{2} \tag{15}\]  
Figure 2: Comparison of denoising results for artificial signals. In these equations, \(SNR_{time}\) represents the signal-to-noise ratio in the time domain, and \(SNR_{freq}\) represents the signal-to-noise ratio in the frequency domain. \(X[k]\) denotes the FFT (Fast Fourier Transform) result of the signal at the \(k\)-th frequency component, and \(N[k]\) represents the FFT result of the noise at the \(k\)-th frequency component. \(s[n]\) is the signal value at the \(n\)-th time point, and \(n[n]\) is the noise value at the \(n\)-th time point. Using the proposed method for denoising artificial signals, it is found that the optimal number of VMD decomposition layers is 12. To further highlight the effectiveness of this algorithm, instead of optimizing the VMD decomposition layers with MVO, we manually set the VMD decomposition layers to 5-11 and 13-14, resulting in the VMD-MPE-WTD algorithm (\(K=5\), 6, 7, 8, 9, 10, 11, 13, 14). Additionally, we include a comparison with wavelet thresholding denoising (WTD) to emphasize the benefits of joint decomposition and denoising. The results of various denoising methods are compared in Table 1.  
Based on the calculations, it is observed that the time-domain \(SNR\) of the original noisy signal is 5.7327, while the frequency-domain \(SNR\) is 5.6694. All denoising methods improve the \(SNR\) of the noisy signals to some extent. The joint denoising method shows a significant enhancement in performance compared to the single WTD denoising method. There is no clear correlation between the number of decomposition layers and denoising effectiveness, since an increase in the number of layers may either improve or worsen the denoising results. However, among all results, the decomposition with 12 layers optimized by MVO provides the best results.