## 3 Simulation Results
### Performance Evaluation of the Localization Results  
Table 1 shows the errors in the absolute distance of localization for different simulation methods. A maximal error of 0.79 m occurs in the DAS beamforming map after removing \(s_{1}\) and \(s_{2}\) without DNNs. In accordance with the localization results shown in the figures, the DNN-based beamforming method provides a smaller error, and the localization results can be observed to be accurate.  
Figure 8: DAS Beamforming maps after removing \(s_{1}\) and \(s_{2}\). (**a**) Without DNNs; (**b**) with DNNs.  
Figure 7: DAS Beamforming maps after removing \(s_{1}\). (**a**) Without DNNs; (**b**) with DNNs.  
It is worth mentioning that the virtual measurement signal could not be constructed perfectly in the time domain. This means that the residual error after removing the dominant sound source could still be comparable in the upgraded sound area. Therefore, the location of \(s_{1}\) can still be observed in Figure 9b,c. The same conclusion can be drawn from the signal reconstruction section.  
As the interference in the beamforming maps is usually generated by two other sound sources instead of side-lobes, in this case, the common MSR may lead to an inaccurate evaluation to describe the localization performance. Therefore, a parameter called the main-to-second-lobe level (MSEL) was defined and utilized in this study:  
\[\text{MSEL}=20\log_{10}\!\left(\frac{L_{m}}{L_{s}}\right), \tag{16}\]  
where \(L_{m}\) is the height of the main lobe, and \(L_{s}\) is the height of the second lobe generated by other sources. Figure 10a shows the MSEL results of the three sources using different approaches. First, the MSEL of the beamforming maps of the third source obtained by DAS without DNNs is negative, which means that this method fails to locate the third sound source, as shown in Figure 8a. The DNNs could significantly improve the MSEL in the beamforming maps after removing \(s_{1}\) and \(s_{2}\). Second, the MSEL of the BW-MUSIC method is not always higher than that of the DAS method, even though the BW-MUSIC method could provide distinct beamforming maps.  
The effective SNR ranges of the different approaches were evaluated by MSEL as well. Since no additional noise was added in this case, the decay rate between the two adjacent sound sources was utilized to describe the SNR. For example, assuming that the primary amplitudes of the three sound sources were \(A_{1}\), \(A_{2}\), and \(A_{3}\), the actual amplitudes of the three sound sources in the simulation would be set to \(A_{1}\), \(0.8A_{2}\), and \(0.64A_{3}\) when the decay rate is 80%. Figure 10b shows the MSEL results of the beamforming maps after removing \(s_{1}\) and \(s_{2}\), which vary with the decay rate. Apparently, the DAS without DNNs method could not locate the third sound source at all the SNR ranges, while the other two approaches failed when the decay rate was lower than 20%. According to the curves shown in Figure 10b, a decay rate higher than 40% could be a reasonably effective SNR range  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Sound Source** & **Localization Method** & **Error (m)** \\ \hline \multirow{2}{*}{\(s_{1}\)} & DAS & 0 \\  & BW-MUSIC & 0 \\ \hline \multirow{2}{*}{\(s_{2}\)} & DAS without DNNs & 0.01 \\  & DAS with DNNs & 0 \\  & BW-MUSIC with DNNs & 0 \\ \hline \multirow{2}{*}{\(s_{3}\)} & DAS without DNNs & 0.79 \\  & DAS with DNNs & 0 \\ \cline{1-1}  & BW-MUSIC with DNNs & 0 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Localization errors of the simulations.  
Figure 9: Beamforming maps obtained with the BW-MUSIC with DNNs. (**a**) \(s_{1}\); (**b**) \(s_{2}\); (**c**) \(s_{3}\).  
for the DAS with the DNNs method. For BW-MUSIC with DNNs, the criterion should be higher than 60%.