## 5 Evaluation of the Auralization System
### Monaural Analysis  
A comparison of the predicted omnidirectional frequency responses at different receiver positions was carried out. This was performed by rendering the sound field in real time using the auralization system developed in Max. The omnidirectional frequency responses from the auralization system were obtained by adding all of the directional impulse responses corresponding to the different \(L\) plane waves used to represent the field and recording the total output after the rotation stage. This information is compared to omnidirectional frequency responses that were synthesized individually at the receiver locations. These omnidirectional references do not use the directional information of the plane wave expansion. They correspond to the frequency response of omnidirectional receivers obtained directly from the FE solution. The use of the numerical information as a reference is due to the lack of measurements and spatial information across the enclosure to evaluate different positions. Five receivers' positions were selected as shown in Figure 19. The central point of the expansion corresponds to Location 01. The predicted frequency responses in narrow band and in 1/3 octave band resolution for each receiver are illustrated in Figures 20-24. The vertical cyan line indicates the cut-off frequency (355 Hz) of the low-pass filter, and the vertical black line indicates the maximum frequency below which the translation is expected to provide accurate results. The maximum frequency was estimated by solving Equation (14) for \(R\). This was done taking into account the distance between  
Figure 18: Room model created in Unity. Exterior (**a**) and interior (**b**) view of enclosure. the central point of the expansion and each receiver's position. The mean error displayed in the figures was selected as a metric, and it is defined as:  
\[\mathrm{ME}(\mathrm{dB})=\frac{1}{n}\sum_{i=1}^{n}\left|10\log_{10}(|\widetilde{ p_{i}}|^{2})-10\log_{10}(|p_{i}|^{2})\right|, \tag{22}\]  
in which \(n\) is the number of \(1/3\) octave frequency bands and \(|\widetilde{p_{i}}|^{2}\) and \(|p_{i}|^{2}\) are the predicted and reference energy of the acoustic pressure in the \(1/3\) octave band \(i\), respectively. This error is based on an equal contribution from all of the \(1/3\) octave bands, being analogous to a model in which pink noise is used as the input signal. It was created to provide insight into how dissimilar on average the reconstructed field is from the reference one. A summary of the mean errors according to the receiver location is presented in Table 2.  
Figure 19: Listener positions selected to evaluate the auralization system. Figure 20: Comparison of full (**a**) and \(1/3\) octave band (**b**) frequency responses. PWE and omnidirectional room impulse response (FEM) at reference Position 1. The results indicate that the changes in the modal response of the enclosure are correctly predicted by the translation operator. Furthermore, the figures show good accuracy in the sound field reconstruction up to the frequencies predicted by Equation (14) as long as these frequencies are below the cut-off frequency. The differences found at this frequency range may arise from three causes: the implementation of integer delays in Max, the numerical accuracy used by Max to perform mathematical operations (summing the directional impulse responses) and the application of regularization in the inverse problem, which decreases the matching between the radius of validity and the effective area where the reconstruction is accurate. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Receiver** & **Distance (m)** & **Frequency (Hz)** & **ME (dB)** \\ \hline
2 & 0.5 & \(\approx\)562 & 1.5 \\
3 & 0.5 & \(\approx\)562 & 1.7 \\
4 & 1 & \(\approx\)281 & 1.9 \\
5 & 1.5 & \(\approx\)187 & 2.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Mean errors at different receiver locations.