## 4 Simulation
### Discussion of Results
#### 4.2.2 Method Performance Discussion  
In this section, the effects of source frequency, SNR, and microphone array on the performance of the sound source identification results are analyzed using the same simulation conditions as in Section 4.2.1 and compared with the conventional methods SBL, OMP, FISTA, off-gird-GL\({}_{1}\), and RPCA-SBL. The source frequency ranges from 100 to 1000 Hz with an interval of 100 Hz, and the SNR ranges from 10 to 40 dB with an interval of 5 dB. For the OMP method, the sparsity is assumed to be five here. 1. Influence of source frequency  
The RMSEL and RMSEM of the sound source identification results for different methods at multiple frequencies are shown in Figure 5, and the average values are shown in Table 1. It should be noted that the error comparison figures below are shown using the log vertical axis to demonstrate the differences. Figure 4: Analysis of the impact of decay factor \(\delta\) on sound source identification results. (**a**) Impact on RMSEL and RMSEM. (**b**) Impact on the number of iterations and calculation efficiency. As shown in Figure 5a, among all methods, OMP shows the most significant localization error, followed by FISTA. Off-grid-GL\({}_{1}\) performs poorly for low-frequency sources, SBL and RPCA-SBL perform similarly, while VG-SBL shows the smallest error. For the minimum frequency of 100 Hz, the errors of all methods increase significantly, while for source frequencies of 200 Hz and above, the performance of all methods improves. The reason for this phenomenon can be explained from the following two angles. On the one hand, the low-frequency sound source signal has a longer wavelength, the effective sampling volume is small, and the truncation effect inherent in digital signal processing is serious. On the other hand, the condition number of the transfer matrix increases significantly as the frequency decreases, making low-frequency sound source identification quite challenging at such a long measurement distance. To mitigate these errors, a possible approach could be to optimize the array configuration or improve the signal processing algorithms, for example, using a microphone array with a larger aperture [57; 58] or further proposing a sound source identification algorithm with higher-resolution performance [59; 60; 61]. In addition, the method can also be used to extend the lower limit of the working frequency by utilizing asynchronous measurement techniques [62; 63; 64]. However, implementing these improvements might present certain challenges, such as practical limitations in the experimental setup or increased complexity in data processing. Therefore, how to further improve the recognition resolution of low-frequency sound sources is the next key research direction. The resolution performance of FISTA and OMP methods for low-frequency sound sources is insufficient. Thus, there is a significant discrepancy between the identified sound sources and their actual locations. For the off-grid-GL\({}_{1}\) method, since the \(L_{1}\) norm is prone to additional bias and improper group sparsity penalty, multiple interference sources exist near the real sound source, and source positioning is inaccurate. Owing to the iterative estimation of the likelihood and a priori hyperparameters, SBL and RPCA-SBL provide \begin{table}
\begin{tabular}{c c c} \hline \hline
**Method** & **Average RMSEL (m)** & **Average RMSEM (dB)** \\ \hline SBL & 0.16 & 4.82 \\ OMP & 0.32 & 5.42 \\ FISTA & 0.20 & 27.11 \\ Off-grid-GL\({}_{1}\) & 0.20 & 6.94 \\ RPCA-SBL & 0.18 & 4.38 \\ VG-SBL & 0.09 & 2.45 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Average RMSEL (m) and RMSEM (dB) between 100 and 1000 Hz, SNR = 20 dB. Figure 5: Identification error statistics of the methods at different frequencies for the SNR of 20 dB. (**a**) RMSEL. (**b**) RMSEM. the most robust and accurate results among the compared methods. Since RPCA mainly removes ghost noise in cloud images, it has little effect on improving the accuracy of location recognition.