# A Forward\(-\)Backward Splitting Equivalent Source Method Based on S\(-\)Difference
## 3 Numerical Simulations
### Coherent Sound Source Simulation  
This section analyzes the reconstruction performance of the four regularization methods mentioned above for coherent sound sources. The simulation settings for both coherent and single sound sources are the same. Sound sources are symmetrically distributed about the Y\(-\)axis and are separated by 0.4m along the X\(-\)axis, then the sources are located at the following coordinates: (0.2, 0, 0) m and (\(-\)0.2, 0, 0) m.  
Figure 6 shows the reconstructed sound pressure amplitude of the coherent sources in the middle row of the reconstruction plane. At the source frequency of 200 Hz, the peak value of the sound pressure amplitude of the coherent sound source of the intermediate row is close to the theoretical value, which can accurately reconstruct the sound field. At the frequency of 2500 Hz, the sound pressure amplitude calculated by WBH, ISTA, and FBS is almost identical to the theoretical value, which can achieve sound field reconstruction.  
The sound field reconstruction results of coherent sound sources at 200 Hz and 2500 Hz are shown in Figure 7. At the coherent source frequency of 200 Hz, the two real sources in the Tikhonov regularization method are covered by larger hot zones and cannot be accurately identified; WBH identifies the number and position of the sources incorrectly. In contrast, ISTA and FBS can clearly identify the two sources, and the hot zone generated by FBS is more concentrated at the source position. At a frequency of 2500 Hz, the hot zones of these five algorithms can clearly indicate the positions of the given sources and perform sound field reconstruction. The identification effects of ISTA and FBS are relatively close, and the sound pressure level graph fits well with the theoretical sound pressure level graph in hot zone size.  
The reconstruction error of the coherent sources is shown in Figure 8. It can be seen that the reconstruction error value of Tikhonov regularization in the full frequency band is mainly concentrated in the range of 40\(-\)50%; the reconstruction fails and the algorithm is no longer applicable. There exists a conversion frequency for the WBH algorithm, and according to the analysis of the curve direction, it is concluded that the conversion frequency is close to 1000 Hz; when the computation frequency is higher than the critical value, the overall error is kept under 15%, showing good reconstruction performance. In contrast, the frequency range of the ISTA and FBS algorithms is widened, and FBS is able to reconstruct the sound field in the whole frequency band from 200 to 2500 Hz. The frequency range is broadened, and the error value of FBS in reconstructing the sound field in the whole frequency band from 200 to 2500 Hz is kept below 10%, which shows that the algorithm can reconstruct the sound field stably and accurately. This shows that the algorithm can reconstruct the sound field stably and accurately.  
Figure 6: Reconstructed sound pressure amplitude of a single sound source along the middle row of the reconstructed surface.  
Figure 7: Calculated sound pressure level diagram of 200 Hz and 2500 Hz coherent sound sources.