# A Forward\(-\)Backward Splitting Equivalent Source Method Based on S\(-\)Difference
## 4 Experimental Verifications
### Experimental Treatment  
In the experiment, let the time-domain signal measured by the microphones in the measurement array be \(P_{h}(t)\). The data processing process starts with obtaining the complex sound pressure at each microphone position in the measurement array. The equation for solving the complex sound pressure is as follows:  
\[A(\omega)=\sqrt{s_{h,h}(\omega)} \tag{16}\]  
In Equation (16), \(s_{h,h}(\omega)\) is the self-spectrum of the time-domain data \(P_{h}(t)\) at the array measurement points.  
The phase is solved as follows:  
\[\theta(\omega)=arg(s_{h,h}(\omega)) \tag{17}\]  
In Equation (17), arg is the function that takes the phase. According to Equations (16) and (17), the complex sound pressure at the location of the microphone measurement point corresponding to the location of each microphone measurement point in the array can be found.  
According to the amplitude and phase of the sound pressure at each measurement point location on the holographic surface, the complex sound pressure at the corresponding location of each microphone can be derived. The reconstruction of the sound field formed by single and coherent sources is performed using the equivalent source method and the s\(-\)differential forward\(-\)backward splitting (FBS) algorithm, and the reconstructed sound field is shown in Figures 11 and 12. The sound field formed by the single source and coherent sources is reconstructed using the equivalent source method and the s\(-\)differential forward\(-\)backward split (FBS) algorithm.  
Figure 11 is the reconstructed sound pressure level diagram of the single sound source experiment. At the single sound source frequency of 200 Hz, the Tikhonov regularization method produced a large hot zone around the center, resulting in poor reconstruction performance. Although the WBH and ISTA algorithms accurately identified the single source location, a significant hot zone was still visible around the origin coordinates. Only the FBS algorithm produced a smaller hot zone and achieved accurate sound field reconstruction. At the single sound source frequency of 2500 Hz,  
Figure 10: Experimental conditions.  
by these four regularization algorithms were concentrated at the sound source location, which was consistent with the simulation results, and the reconstruction effect of FBS was the best.  
Figure 12 shows the reconstructed sound pressure level diagram of the coherent sound source experiment. It can be seen from the figure below that when the frequency of the coherent sound source is 200 Hz, the sound pressure level cloud map is consistent with the theoretical sound pressure level cloud map, and the hot zone displayed by FBS accurately identified the source location, although the sound pressure amplitude was affected by external experimental conditions. When the frequency of the coherent sound source is 2500 Hz, although the hot zones produced by all four algorithms could deduce the approximate source location, only the FBS two\(-\)source hot zone image was close, maintaining high reconstruction accuracy.  
Figure 11: Reconstructed sound pressure level diagram of 200 Hz and 2500 Hz single sound source.