## 5 Conclusions  
In this paper, we proposed a simulation method for demodulated PAL sound. This method is based on the sequential ray-tracing method and is extended to express the self-demodulation phenomenon in the sound reproduction; therefore, it can simulate the distribution of the demodulated sound of a PAL with higher accuracy than can other geometry-based simulation methods and has much lower calculation complexity than FDTD-based methods. The experimental results demonstrated the effectiveness of the proposed method with a quantitative evaluation.  
In the future, we plan to further improve the accuracy of the proposed method by investigating a feasible approach to estimate the simulation parameters with higher accuracy. The simulation of the proposed method is in three-dimensional space; however, it is difficult to show the results in three-dimensional space. Therefore, we also plan to develop a user-friendly interface to show the simulation results.  
Conceptualization, M.S. and T.N.; methodology, Y.G. and M.N.; software, M.S.; validation, Y.G., M.N. and T.N.; formal analysis, Y.G.; investigation, Y.G. and M.S.; resources, T.N.; data curation, M.S.; writing--original draft preparation, Y.G.; writing--review and editing, Y.G., M.N. and T.N.; visualization, M.S. and Y.G.; supervision, M.N. and T.N.; project administration, T.N.; and funding acquisition, Y.G., M.N. and T.N. All authors have read and agreed to the published version of the manuscript.  
This work was partly supported by Ritsumeikan University R-GIRO, RARA, and ARC, as well as JSPS KAKENHI Grant Numbers JP21H03488, JP21H04427, JP23H03425, JP23K21691, JP23K28115, and JP24K20803.  
\begin{table}
\begin{tabular}{l c c c c c c c} \hline \hline \multirow{2}{*}{**Evaluation Area**} & \multicolumn{6}{c}{**Demodulated Frequency \(f_{\text{d}}\) (kHz)**} \\ \cline{2-7}  & **1** & **2** & **3** & **4** & **5** & **6** & **7** \\ \hline Points for parameter determination & 1.94 & 0.87 & 0.83 & 0.81 & 0.93 & 0.65 & 0.52 \\ \hline Points for evaluation (without reflection) & 2.15 & 1.14 & 1.39 & 1.13 & 1.16 & 0.98 & 1.26 \\ \hline Points for evaluation (with reflection) & 2.48 & 1.89 & 2.28 & 2.94 & 4.23 & 4.25 & 4.95 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Root mean square error \(E_{\text{rms}}\) for the sound pressure level of demodulated sound between the simulation and measurement results (dB).  
\begin{table}
\begin{tabular}{l c c c c c c c} \hline \hline \multirow{2}{*}{**Evaluation Area**} & \multicolumn{6}{c}{**Demodulated Frequency \(f_{\text{d}}\) (kHz)**} \\ \cline{2-7}  & **1** & **2** & **3** & **4** & **5** & **6** & **7** \\ \hline Points for parameter determination & 1.67 & 0.64 & 0.62 & 0.59 & 0.73 & 0.46 & 0.42 \\ \hline Points for evaluation (without reflection) & 1.84 & 0.81 & 0.90 & 0.75 & 0.90 & 0.63 & 0.71 \\ \hline Points for evaluation (with reflection) & 1.93 & 1.26 & 1.90 & 2.22 & 3.13 & 3.22 & 3.67 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Average absolute error \(E_{\text{abs}}\) for the sound pressure level of demodulated sound between the simulation and measurement results (dB).  
Data Availability Statement: The raw data supporting the conclusions of this article will be made available by the authors on request.  
The authors declare no conflicts of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.