# Particle-Velocity-Based Mixed-Source Sound Field Translation for Binaural Reproduction
## 5 Simulation Study
### Simulation Results
#### 5.2.1 Reproduced Pressure Field  
In terms of the single source scenario, Figure 2 shows the true pressure field calculated by (33), and the recorded pressure field calculated using the theoretical pressure coefficients in (34). The black circle denotes the receiver region. The receiver can record the sound field within the receiver region. However, truncation error degrades the accuracy of the pressure field beyond the receiver region, which is the constraint in Section 2 that we wish to relax through the proposed method. To compare the proposed method with the sound-pressure-based expansion, we first show the results of the sound-pressure-based expansion for the closed-form solution and the sparse solution in Figure 3. We observe that the closed-form solution can guarantee the reproduction accuracy within the receiver region. However, for a position away from the receiver region, the mismatch percentage at the position quickly increases to a value close to 100 %, which is similar (but not exact)to the recorded pressure field and is not good enough for translation. The sparse solution improves the performance and reconstructs the pressure field accurately around the receiver region, which allows for the listener to move to the position exterior to the receiver region. Figure 4 shows the reconstructed pressure field and the error field controlled by the least squares solution and the sparse solution using the particle-velocity-based expansion, and these two solutions have the exactly same pressure field mismatch, which means that the least squares solution of the particle-velocity-based expansion has the same performance as the closed-form solution of the sound-pressure-based expansion. However, for the sparse solution of the particle-velocity-based expansion, it provides a larger area with 0 % pressure field mismatch around the receiver region compared with the sparse solution of the sound-pressure-based expansion, which allows for the listener to translate further away from the original receiver region. As for the multiple sources scenario, the true pressure field and recorded pressure field are given in Figure 5. The pressure field difference between them also increases gradually (i.e., the value of the mismatch gradually increases from 0% to 100%) as the  
Figure 4: The reconstructed pressure field and the error field controlled by the least squares solution (V-LSS) and the sparse solution (V-SS) using the particle-velocity-based expansion. (**a**) Reconstructed pressure field controlled by V-LSS; (**b**) error field controlled by V-LSS; (**c**) reconstructed pressure field controlled by V-SS; (**d**) error field controlled by V-SS. Figure 3: The reconstructedpressure field and the error field controlled by the closed-form solution (P-CFS) and the sparse solution (P-SS) using the sound-pressure-based expansion. (**a**) Reconstructed pressure field controlled by P-CFS; (**b**) error field controlled by P-CFS; (**c**) reconstructed pressure field controlled by P-SS; (**d**) error field controlled by P-SS. Figure 2: The truepressure field and recorded pressure field at 1500 Hz. The black circle denotes the receiver region. (**a**) the true pressure field; (**b**) the recorded pressure field. observation point moves away from the target region due to the truncation error. Similar to the evaluation for the single source scenario, we first show the reconstructed pressure fields controlled by the closed-form solution and the sparse solution based on the sound-pressure-based expansion in the multiple sources scenario. The results with the error fields are shown in Figure 6. It shows that the closed-form solution still can only guarantee accurate reproduction around the target region, whereas the sparse solution can provide greater translation area. However, the area with 0% pressure field mismatch is much smaller than that in the single source scenario. Figure 7 shows the results of the least squares solution and the sparse solution based on the particle-velocity-based expansion.