## 2 Principles and Methods
### Impact and Analysis of Initial Parameters  
In the decomposition process of the VMD algorithm, the decomposition scale K directly affects the accuracy of the results. Thus, selecting an appropriate K value can effectively eliminate modal aliasing. Therefore, using the center frequency method, the initial parameter convergence tolerance \(\epsilon\) is set to \(1\times 10^{-7}\), and the penalty factor \(\alpha\) is set to an empirical value of 2000. Taking the simulated impulse noise signal as an example, Table 1 shows the denoising performance of the VMD algorithm for the simulated impulse noise signals under different K values. Here, the center frequency corresponds to the intrinsic frequency of each modal component, whereas NCC and SNR are indicators for a comparative analysis between the profile characteristics of strong cultural noise obtained after VMD and the simulated strong cultural noise signals. The denoising performance analysis results for the other three types of simulated strong cultural noise signals under different K values were consistent with the above findings.  
As shown in Table 1, for the four types of simulated strong cultural noise, when \(\alpha\) remains constant, both NCC and SNR begin to decline starting from K = 5. At K = 4, the NCC and SNR of the four types of strong cultural noise signals reach their maximum values. Additionally, the corresponding center frequencies indicate that the modes effectively avoid aliasing [30].  
The penalty factor \(\alpha\) also affects the accuracy of the signal decomposition. Similarly, following the previous analysis, the decomposition scale K was set to an empirical value of 4. Taking the simulated impulse noise signal as an example, Table 2 displays the denoising performance of the VMD algorithm for the simulated impulse noise signals under different \(\alpha\) values.  
\begin{table}
\begin{tabular}{c c c c c c} \hline K & \multicolumn{3}{c}{**Center Frequency (kHz)**} & **NCC** & **SNR** \\ \hline
2 & 0.0014 & 0.3152 & & 0.947 & 9.752 \\
3 & 0.0014 & 0.1405 & 0.3397 & & 0.947 & 9.777 \\
4 & 0.0013 & 0.1343 & 0.2769 & 0.4152 & 0.948 & 9.835 \\
5 & 0.0011 & 0.0547 & 0.1805 & 0.3899 & 0.4224 & 0.848 & 8.809 \\ \hline \end{tabular}
\end{table}
Table 1: Comparison of the VMD denoising performance of simulated impulse noise signals with different K values.  
From the above analysis, it is evident that the selection of the decomposition scale K and penalty factor \(\alpha\) significantly impacts the decomposition results of the VMD algorithm. When K is too large, it may result in over-decomposition and the appearance of spurious components. Conversely, when K is too small, the effective frequency components of the signal may not be fully decomposed. An improper selection of \(\alpha\) can also lead to over-decomposition or under-decomposition, whereas a larger number of iterations can affect the computational efficiency. The selection of these two parameters is typically irregular and can influence each other. To determine appropriate values for K and \(\alpha\), this paper proposes the use of the PSO algorithm to optimize the parameters in the VMD algorithm, thereby obtaining the optimal parameter combination for K and \(\alpha\) to enhance noise suppression capabilities.