# A Forward\(-\)Backward Splitting Equivalent Source Method Based on S\(-\)Difference
## 3 Numerical Simulations
### Single Source Simulation  
The purpose of the single-source simulation is to compare four algorithms--Tikhonov regularization, WBH, ISTA and FBS--for sound field reconstruction analysis under a single source. The coordinates of the sound source were (0, 0, 0,) m, and the distance from the holographic surface was set to 0.1 m. Random noise was added to the microphone sound pressure data at a level of 15 dB below the average sound pressure of the microphone, while the dynamic display range was set to 15 dB.  
Figure 3 shows the sound pressure level diagrams of the theoretical values on the reconstruction plane, as well as the results calculated by Tikhonov regularization, WBH, ISTA, and FBS algorithms for a single sound source at 200 Hz and 2500 Hz. At 200 Hz, Tikhonov regularization produces a larger hot zone at the sound source location, while WBH, ISTA, and FBS algorithms gradually produce a hot zone that tends toward the center of the sound source, with good recognition performance. At 2500 Hz, all four algorithms  
produce a concentrated hot zone, and can accurately identify the sound source. The sound pressure cloud maps of several algorithms show that the range of the hot zone is reduced compared with that of 200 Hz, and the sound source identification resolution is improved.  
Figure 2: Schematic diagram of sound source layout.  
Figure 4 shows the reconstructed sound pressure amplitude of a single sound source in the middle row of the reconstruction plane. At 200 Hz, all algorithms except for FBS have a reconstructed sound pressure amplitude far from the theoretical value, but FBS maintains its high reconstruction accuracy at low frequencies. At 2500 Hz, the amplitude curve of  
Figure 3: The calculated sound pressure level of single sound source at 200 Hz and 2500 Hz.  
the reconstructed sound pressure in the middle plane calculated by WBH, ISTA, and FBS algorithms matches the theoretical curve, and all algorithms can achieve high\(-\)precision sound field reconstruction. The amplitude plot shows that the reconstructed sound pressure peaks of the FBS algorithm are closer to the theoretical sound pressure values, and the reconstruction accuracy is higher than that of the FBS algorithm. The amplitude plot shows that the reconstructed sound pressure peaks of the FBS algorithm are closer to the theoretical sound pressure values, which maintains a high reconstruction accuracy. When the source frequency is 2500 Hz, except for the traditional Tikhonov regularization, which still fails to reconstruct the sound field, the other four algorithms can reconstruct the sound field.  
According to Equation (15), Figure 5 shows the reconstruction error curves for a single source in the entire analyzed frequency band from 200 to 2500 Hz, with the frequency sampling points of each curve spaced 50 Hz apart, and the error values averaged over 10 calculations. Overall, WBH, ISTA, and FBS algorithms can reconstruct the sound field more stably than the Tikhonov regularization method. Except for the traditional Tikhonov regularization, the remaining three algorithms maintain more satisfactory reconstruction performance throughout the analyzed frequency band. The three algorithms, WBH, ISTA, and FBS, have good reconstruction accuracy in the middle\(-\)frequency and high\(-\)frequency bands from 1000 Hz to 2500 Hz, and the reconstruction errors are kept below 20%. Compared with WBH and ISTA, the FBS algorithm maintains higher reconstruction accuracy in the low\(-\)frequency band from 200 to 500 Hz, and the reconstruction errors in the whole frequency range are kept below 10%, which broadens the frequency range and ensures the best sound field reconstruction performance.  
Figure 4: Reconstructed sound pressure amplitude of a single sound source along the middle row of the reconstructed surface.  
Figure 5: Single source reconstruction errors between Tikhonov, WBH, ISTA, and FBS.