## 3 Evaluation Experiments
### Objective Evaluation Experiments on Sound Pressure Distribution  
In this experiment, we measured the sound pressure distribution of the sound fields synthesized with the comparison patterns. A band-limited white noise of 0-8 kHz was used  
Figure 4: Experimental arrangement of the evaluation experiments.  
Figure 5: Loudspeaker array used in the experiments.  
as the target signal. The sound pressure was calculated from the measured microphone signals as  
\[p(\mathbf{r})=10\log_{10}\left(\frac{1}{N}\sum_{n=0}^{N-1}\sigma_{\mathbf{r}}^{2}[n]\right), \tag{27}\]  
where \(\sigma_{\mathbf{r}}^{2}[n]\) denotes the audible signal observed at \(\mathbf{r}\). When PALs were used, a low-pass filter was applied to the signal before calculating the sound pressure to remove the inaudible band. The microphones were set in the range of \(x=-2.0\)-\(2.0\) m, \(y=0.2\)-\(3.0\) m, with intervals of \(0.1\) m in both directions.  
To evaluate the sound pressure distribution, we focused on the sound pressure transition along the \(y\)-axis. The results are shown in Figure 6. The gray zones in Figure 6 indicate the positions less than \(0.3\) m away from the real sound source in 'Real' cases. For 'Conv. Hybrid', the results overlap with the results of 'Baseline' in Figure 6a,b and overlap with the results of 'Conv. PAL' in Figure 6c,d. Here, we calculated the slope of the sound pressure, which can be defined as  
\[\Delta p(y)=p((0,y))-p((0,y-\Delta y)), \tag{28}\]  
where \(\Delta y\) (\(\Delta y=0.1\) m) is the interval of observation points in the \(y\)-axis direction. The error in the sound pressure slope (ESPS) is defined as the root mean square error for the slope of the sound pressure, which can be given as  
\[\text{ESPS}_{\text{pt}}=\sqrt{\frac{1}{N_{y}}\sum_{y\in Y}\bigl{(}\Delta p_{ \text{pt}}(y)-\Delta p_{\text{Real}}(y)\bigr{)}^{2}}, \tag{29}\]  
where 'pt' in the subscript denotes the comparison pattern (e.g., 'Baseline'), \(N_{y}\) denotes the number of observation points, and \(\mathbf{Y}\) denotes the group of \(y\) for evaluation. A smaller value of ESPS suggests a better reproduction of the sound pressure distribution. The sound pressure transition of the 'Real' in the gray zones is impossible to reproduce physically utilizing secondary sound sources at \(y=0\). Therefore, we only used the results outside the gray zones for evaluation. For example, regarding \(r_{\text{S}}=(0.0,\,0.5)\), the observation points at \(y=0.2,\,0.3,\,\cdots,\,0.8\) m were excluded from the evaluation. In this case, we have \(\mathbf{Y}=\{0.9,\,1.0,\,\cdots\,3.0\}\) and \(N_{y}=22\).  
Figure 6: Results concerning sound pressure transition along \(y\)-axis.  
The results for the ESPS are given in Table 3. Values in bold font indicate the best performance among the comparison patterns. It can be derived from Table 3 that the proposed method achieves the least error of sound pressure slope in total. Moreover, the error does not change significantly when the VSS position changes; therefore, it can be concluded that the performance of the proposed method is robust against the position of VSSs.  
The results regarding the sound pressure distribution for \(r_{\text{S}}\) = (0.0, 1.0) and \(r_{\text{S}}\) = (0.0, 2.0) are shown in Figure 7 and Figure 8, respectively. The black crosses in these figures indicate the positions of the VSS. In the cases of 'Baseline', 'Conv. PAL', and 'Conv. Hybrid', the tendency of sound pressure is confirmed to be the same as that in previous studies [22; 24]. In the cases of 'Prop.(8.8)' and 'Prop.(16.16)', it can be noticed that the sound pressure distribution for the proposed method has taken a balance between 'Baseline' and 'Conv. PAL'.