## 4 Results
Interestingly, despite the capture rate increases with \(\sigma\), the FSR exhibits a diminishing trend with \(\sigma\). This can be explained by considering nodes powering failure. Specifically, a higher fading spread on the forward link affects the nodes powering probability, which is dominant compared to the increment of capture rate. Consequently, the FRS is generally affected negatively by the fading spread. As expected, increasing the number of deployed receivers substantially increases the FSR due to a higher diversity gain. Observe that the accuracy of the proposed approximation approach is very high for all the considered values of \(\sigma\). This confirms that the log-normal approximation can be safely applied for the derivation of capture probabilities. Indeed, although a low accuracy is expected for \(\sigma>4\), this only occurs if the target is the whole PDF of the weighed sum of log-normal components. Nevertheless, capture probabilities are defined in terms of the tail distribution of the sum, thus, the proposed approach maintains a good accuracy within a wide range of \(\sigma\) values. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Value** & **Unit** \\ \hline \(P_{\text{tx}}\) & \(33\) & dBm \\ \(\gamma_{h}\) & \(-20\) & dBm \\ \(\gamma_{d}\) & \(-80\) & dBm \\ \(f\) & \(915\) & Mhz \\ \(k_{0}\) & \(2\) & - \\ \(\gamma_{\text{SIR}}\) & \(6\) & dB \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Main simulation parameters. It is expected that the illuminator-receivers distance \(d_{\rm IR}\) has a substantial impact on the system performance. Indeed, the distance between the receivers and the center of the illumination area affects the SIR at the generic receiver and, consequently the capture occurrence. Moreover, \(d_{\rm IR}\) also affect the single reply detection probability defined in Equation (5). The FSR as a function of \(d_{\rm IR}\) is illustrated in in Figure 6, for different values of the fading spread \(\sigma\) and considering different number of deployed receivers. Figure 5(a) shows the expected FSR for a single deployed receiver. It can be seen that best performance is obtained when the receiver is located within the illumination area and close to the illuminator, especially for \(\sigma=6\) dB. As \(d_{\rm IR}\) increases the expected FSR diminishes since the spatial configuration is less favorable for capture diversity and the miss-detection probability increases. The case with two antipodal receivers is analyzed in Figure 5(b), where the FSR is clearly higher with respect to the case with \(M=1\) due to the increased capture diversity gain. The FSR curve also exhibits an evident maximum when the receivers are placed at a distance that is approximately half of the illumination area radius. We remark that the presence of the maximum is mainly due to the spatial symmetry of the considered network geometry. Specifically, given the fixed dimension of the illumination area, when the receivers are either too close or too far from the illuminator, the nodes distribution is unfavorable to the capture effect. Interestingly, as in the previous case, performance diminishes as the illuminator-reader distance increases. Nevertheless, at a given distance, larger fading spread yields better performance since the higher capture chance better compensates performance loss (i.e., packets miss-detection). Note also that the distance that returns the maximum performance does not depend on the fading severity. A similar trend is observed when \(M=10\) in Figure 5(c), where a substantial increment of the FSR is observed due to the additional gain in diversity. Moreover, the effect of the spatial symmetry is exacerbated in this scenario. Indeed, we clearly observe two intersection points that dictate an inversion trend of the FSR with respect to \(\sigma\).