## 2 Principles and Methods
### Noise Suppression Method Based on PSO-Optimized VMD  
From the above theory of the VMD algorithm, it is evident that setting different penalty factors K and the number of decomposition modes \(\alpha\) affects the solution performance of \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(\alpha\) & \multicolumn{3}{c}{**Center Frequency (kHz)**} & \multicolumn{3}{c}{**NCC**} & **SNR** \\ \hline
1000 & 0.0013 & 0.1357 & 0.2846 & 0.4206 & 0.934 & 8.856 \\
1500 & 0.0013 & 0.1344 & 0.2795 & 0.4165 & 0.942 & 9.403 \\
2000 & 0.0013 & 0.1343 & 0.2769 & 0.4152 & 0.948 & 9.835 \\
2500 & 0.0013 & 0.1346 & 0.2755 & 0.4151 & 0.946 & 9.726 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of the VMD denoising performance of simulated impulse noise signals with different \(\alpha\) values. the VMD algorithm. During each particle position update, the fitness value must be calculated once, and updates are performed by comparing it with the fitness value of the new particle. Typically, the selection of these two parameters relies on the experience of the data processing personnel [32]. To reduce reliance on human experience, this paper proposes the use of the PSO algorithm to find the optimal parameter combination for the VMD algorithm through adaptive parameter selection. It is important to note that setting an appropriate fitness function significantly influences the optimization results when using the PSO algorithm to determine the optimal parameter combination for the VMD algorithm. In this regard, the following analysis is provided: In the field of signal analysis, envelope entropy is used to reflect the sparsity characteristics of a signal. If the signal contains a large amount of information, its sparsity is strong, resulting in a lower envelope entropy value. Conversely, if the signal contains little information, its sparsity is weak, leading to a higher envelope entropy value [33]. The envelope entropy Ep was calculated using the following formula:  
\[\left\{ \begin{array}{l}p_{i}=\frac{a(t)}{\sum\limits_{i=1}^{N}a(t)}\\ E_{p}=-\sum\limits_{i=1}^{N}p_{i}\text{lg}p_{i}\end{array} \right. \tag{11}\]  
where \(\alpha(t)\) represents the envelope signal obtained by applying the Hilbert-Huang transform to the original signal \(x(t)\); \(p(t)\) is the result obtained by normalizing the envelope signal \(\alpha(t)\); \(N\) is the length of the time series of the signal, where \(t=1,2,\ldots,N\); and \(E_{p}\) is the calculated envelope entropy. Under the parameter combinations of the penalty factor K and decomposition mode \(\alpha\), the decomposition of strong cultural noise signals using the VMD algorithm yields K IMFs. If a particular IMF contains a significant amount of noise, the calculated envelope entropy Ep will be large. Conversely, if an IMF contains little noise, the calculated envelope entropy Ep will be small. This indicates that the IMF corresponding to the minimum entropy value Ep contains a wealth of useful information. Therefore, this paper used the minimum envelope entropy Ep obtained from K IMFs as the local minimum entropy minEp. Throughout the optimization process of the PSO algorithm, the local minimum entropy minEp serves as the fitness function. The noise suppression process for optimizing the VMD parameter combinations using PSO is as follows:  
1. Initialize PSO Algorithm Parameters. Set the parameters for the PSO algorithm, including the maximum number of iterations Gmax and population size M, and calculate the initial fitness values. 2. Initialize Particle Population. Treat the influencing parameter combination [K, \(\alpha\)] as the position of the particles. To construct the particle swarm, randomly generate a certain number of influencing parameter combinations that serve as the initial positions for the particles, along with randomly initializing the velocity of each particle. 3. Determine the Termination Conditions. During the iterative optimization of the particle swarm, compare the fitness values to determine when to end the search process.