# Discrete Environment-Driven GPU-Based Ray Launching: Validation and Applications
## 3 Validation
### Validation of the DED-RL Algorithm  
As in [6] for the validation of a standard image RT algorithm, a subset of 18 sites in the San Francisco environment was selected in order to perform a comparative assessment of the overall performance of the new DED-RL algorithm vs. the traditional RT algorithm.  
As one of the main sources of error in DED-RL is environment discretization, prediction accuracy has been analyzed for different tile sizes. RL simulations have been performed with the parameters shown in Table 1. As the DED-RL is discrete and allows the computation of the field only on discrete points upon surfaces (e.g., building walls and ground), the predicted field value on the closest ground tile has been considered for each drive-test measurement location.  
Results in term of prediction error (mean value and standard deviation) as a function of the tile size in the San Francisco scenario are shown in Figure 4.  
Prediction error \"\(Err_{i}\)\" is computed as the predicted RSSI minus the measured RSSI for the generic, i-th test point. Then, mean error, \(\mu_{E}\) and standard deviation of the error, \(\sigma_{E}\) are computed for the \(N_{RX}\) locations covered by each cell site in the following way:  
\[\begin{array}{l}\mu_{E}=\frac{\sum\limits_{i=1}^{N_{RX}}Err_{i}}{\sum \limits_{i=1}^{N_{RX}}}\ \sigma_{E}=\sqrt{\frac{\sum\limits_{i=1}^{N_{RX}}\left(Err_{i}-\mu_{E} \right)^{2}}{N_{RX}}}\\ \text{where}\ {Err_{i}}=\left(RSSI_{sim}\right)_{i}-\left(RSSI_{meas}\right)_{i} \end{array} \tag{3}\]  
\begin{table}
\begin{tabular}{l l} \hline \hline
**Parameter** & **Value** \\ \hline Max. Number of interactions & 5 bounces: 5 reflections, 2 diffractions, 1 scattering \\ \hline Max. Number of combined interactions & 3 among reflections and diffractions, 3 among diffractions and scattering \\ \hline Over-Rooftop (ORT) diffraction enabled & Enabled \\ \hline Combination of single reflection and scattering & Enabled 1 \\ with ORT & \\ \hline Attenuation from vegetation & Enabled 1 \\ \hline Transmission through the closest building to BS & Enabled 1 \\ \hline Diffuse scattering coefficient (S) & 0.4 \\ \hline Wall relative permittivity & 5 \\ \hline Wall conductivity & 0.01 S/m \\ \hline Scattering coefficient & 0.5 \\ \hline Specific Attenuation through vegetation & 0.04 dB/m \\ \hline \hline \end{tabular} \({}^{\dagger}\) Only for some selected sites.  
\end{table}
Table 1: Main settings for ray launching simulations.  
Global error statistics are finally derived computing the mean and the standard deviation of \(\mu_{E}\) and \(\sigma_{E}\) for all the \(N_{BS}\) cell sites:  
\[\overline{\mu}_{E}=\frac{\sum\limits_{j=1}^{N_{BS}}\mu_{E_{j}}}{N_{BS}}\ \ \sigma_{\mu_{E}}=\sqrt{\frac{\sum\limits_{j=1}^{N_{BS}}\left(\mu_{E_{j}}-\mu_{ \mu_{E}}\right)^{2}}{N_{BS}}} \tag{4}\]  
\[\overline{\sigma}_{E}=\frac{\sum\limits_{j=1}^{N_{BS}}\sigma_{E_{j}}}{N_{BS}} \sigma_{\sigma_{E}}=\sqrt{\frac{\sum\limits_{j=1}^{N_{BS}}\left(\sigma_{E_{j}} -\mu_{\sigma_{E}}\right)^{2}}{N_{BS}}} \tag{5}\]  
These global statistics are shown more in detail for all the considered environments (San Francisco, San Jose, Atlanta) in the next sections.  
Figure 4 shows that overall performance for San Francisco slightly improves when reducing the tile size, from very large tiles (15 \(\times\) 15 m\({}^{2}\)), down to 5 \(\times\) 5 m\({}^{2}\): in particular, the global mean error (\(\overline{\mu}_{E}\)) becomes very close to zero, while the average error standard deviation (\(\overline{\sigma}_{E}\)) reduces from 10 dB for the largest tiles, down to 9 dB, which is even better than what was obtained in [6]. This means that it is not necessary to consider very small tiles in order to achieve a good performance: this point is very important, as the amount of memory occupation (i.e., the size of the visibility matrix) and the computation time rapidly increase when reducing the tile size.