## 4 Evaluation Experiment
### Experiment 1: Evaluation of Sound Image Localization Accuracy  
In this experiment, the sound image localization accuracy was evaluated on the basis of [18]. The evaluation method [18] utilizes the head-related transfer function (HRTF) database and the direction of the sound image \((\hat{\theta},\hat{\phi})\) can be estimated accurately. However, the distance between the recorded position and sound image cannot be obtained directly by this evaluation method. Hence, we estimated the position of the sound image \((\hat{r},\hat{\theta},\hat{\phi})\) by using the image's estimated direction and the recorded sounds at five positions shown in Table 5. The evaluation procedure is shown as follows:  
1. Calculation of inter-aural level difference (ILD) and inter-aural phase difference (IPD) The ILD and IPD were calculated for each recorded sound at positions A to E shown in Table 5 [19]. Here, the ILD and IPD are known as factors for the sound localization of humans. The ILD and IPD can be calculated by: \[\mathrm{ILD}_{\Omega}(\omega) =20\log_{10}\left|\frac{C_{\Omega}(\omega)}{P_{\Omega}(\omega)} \right|,\] (20) \[\mathrm{IPD}_{\Omega}(\omega) =\tan^{-1}\biggl{(}\frac{\mathrm{Im}(C_{\Omega}(\omega))}{ \mathrm{Re}(P_{\Omega}(\omega))}\biggr{)},\] (21) where \(C_{\Omega}(\omega)\) is the cross spectrum between the signals obtained at the left and right ears of the dummy head microphone, \(P_{\Omega}(\omega)\) is the power spectrum of the signal obtained at the left ear of the dummy head microphone, and \(\mathrm{Re}(\cdot)\) and \(\mathrm{Im}(\cdot)\) represent the real and imaginary parts of the complex value, respectively. \(\Omega\in\{\mathrm{A,B,C,D,E}\}\) is the index for the recording position. In addition, \(\mathrm{ILD}_{\mathrm{HRTF}}(\omega,\theta,\phi)\) and \(\mathrm{IPD}_{\mathrm{HRTF}}(\omega,\theta,\phi)\) were calculated by using HRTF in the CIPIC database [20]. \(\mathrm{ILD}_{\mathrm{HRTF}}(\omega,\theta,\phi)\) and \(\mathrm{IPD}_{\mathrm{HRTF}}(\omega,\theta,\phi)\) represent the value for which the sound image perfectly localizes at the desired position. 2. Calculation of the differences of the ILD and IPD between recorded sound and HRTF database. The differences of the ILD and IPD between the recorded sound and HRTF database were calculated. If the differences are small, it can be said that the sound image localizes at the direction \((\theta,\phi)\). The difference between the inter-aural time difference (ITD) and IPD are defined as: \[E_{\Omega,\mathrm{ILD}}(\omega,\theta,\phi) =|\mathrm{ILD}_{\Omega}(\omega)-\mathrm{ILD}_{\mathrm{HRTF}}( \omega,\theta,\phi)|,\] (22) \[E_{\Omega,\mathrm{IPD}}(\omega,\theta,\phi) =|\mathrm{IPD}_{\Omega}(\omega)-\mathrm{IPD}_{\mathrm{HRTF}}( \omega,\theta,\phi)|.\] (23) \begin{table}
\begin{tabular}{c c} \hline A & \((0.5,90^{\circ},180^{\circ})\) \\ B & \((0.5,90^{\circ},225^{\circ})\) \\ C & \((0.5,90^{\circ},270^{\circ})\) \\ D & \((0.5,90^{\circ},315^{\circ})\) \\ E & \((0.5,0^{\circ},90^{\circ})\) \\ \hline \end{tabular}
\end{table} 
Table 5: Position of the dummy head microphone \((r,\theta,\phi)\). Then, \(E_{\Omega,\text{ILD}}(\omega,\theta,\phi)\) and \(E_{\Omega,\text{IPD}}(\omega,\theta,\phi)\) were combined as the following form. \[E_{\Omega}(\omega,\theta,\phi) =\beta(\omega)E_{\Omega,\text{IPD}}(\omega,\theta,\phi)+(1-\beta( \omega))E_{\Omega,\text{ILD}}(\omega,\theta,\phi), \tag{24}\] \[\beta(\omega) = \begin{cases}1&(\omega\leq\omega_{\text{L}})\\ 1-\frac{\omega^{\prime}-\omega_{\text{L}}}{\omega_{\text{H}}-\omega_{\text{L} }}&(\omega_{\text{L}}<\omega<\omega_{\text{H}})\,,\\ 0&(\omega\geq\omega_{\text{H}})\end{cases}  \tag{25}\]  
where \(\beta(\omega)\) is the weighting function that controls the ratio between \(E_{\Omega,\text{IPD}}(\omega,\theta,\phi)\) and \(E_{\Omega,\text{ILD}}(\omega,\theta,\phi)\) in Equation (24). \(\beta(\omega)\) has a characteristic shown in Figure 9. The reason for using \(\beta(\omega)\) is that the IPD affects the sound localization of humans below 1500 Hz and is dominant above 1500 Hz [21].