## 4 Results
Again, this effect can be explained by first observing that the illumination area is of fixed size, and thus placing receivers either too close or too far from the illuminator results in lower FSR because of bad capture configurations and higher packets miss-detection. On the other hand, capture effect becomes dominating at the intersection points for higher values of the fading spread, thus yielding an increasing FSR as \(\sigma\) increases. Figure 5: Impact of fading spread \(\sigma\) for different number of deployed receivers. The considered scenario consists of \(K=50\) backscattering nodes and the receivers are located at the border of the illumination area at a distance of \(d_{IR}=10\) m from the illuminator. (**a**) Capture rate. (**b**) Frame success rate. To analyze how the number of deployed receivers contributes to the diversity gain, we derive the expected CR and the expected FSR as a function of \(M\), for different values of the fading spread and a fixed illuminator-receivers distance of \(d_{\text{IR}}=6\) m, which has been shown to provide close-to optimum performance in the considered settings. It can be seen from Figure 7a that CR rapidly reaches its maximum as \(M\) increases. Note that, when \(\sigma=2\) dB, CR does not saturate to 1, meaning that increasing the number of receivers to much does not guarantee capture occurrence in all multiple transmission slots. As expected, the increasing trend with \(M\) is also visible for the FSR in Figure 7b. Observe that the saturation effect is less pronounced when referring to the FSR, suggesting that by increasing the number of deployed receivers, the average FSR eventually approaches to 1. This is due to the fact that, during multiple transmission slots, multiple packets reception can occur thanks to capture diversity, therefore a higher \(M\) produces a higher average number of simultaneously detected packets. Nevertheless, the average number of simultaneously detected packets is also affected by the network topology and, more precisely, by the spatial configuration of the nodes. Indeed, capture diversity depends on the combination of scenario geometry and channel contention mechanism, which randomly selects a subset of transmitting nodes in a slot. To better highlight this aspect, we present the CR and the FSR for a different nodes spatial dissemination in Figure 8a,b, respectively. Specifically, nodes are uniformly distributed on a horizontal line at the center of the illumination area. It can be seen that, in this scenario, the FSR exhibits an evident saturation, especially for \(\sigma=2\) dB. In this situation, there is no benefit in increasing \(M\) to much, and satisfactory performance are already obtained for \(M=2\). Figure 6: Expected frame success rate as a function of illuminator-receivers distance \(d_{IR}\) for different values of \(\sigma\). (**a**) \(M=1\). (**b**) \(M=2\). (**c**) \(M=10\). All the results derived above assume that the frame length is set to \(L=K\), which corresponds to the theoretical optimal frame length. Nevertheless, due to capture effect, multiple transmissions does not necessarily result in collisions, and, in the considered multi-static setting, multiple packets detection may occur due to capture diversity. Therefore, the actual optimal frame length differs from the theoretical optimum and is shorter than \(K\). Observe also that fading and network topology affect the capture probability and, more generally, the capture distribution at different receivers. Consequently, the optimal frame length depends on different system parameters, non-trivially. To unveil the impact of such dependence, we consider a network scenario with \(K=10\) backscattering nodes, where receivers are located at the border of the illumination area at a distance of \(d_{IR}=10\) m. Figure 9 shows the expected FSR as a function of \(L\) for different values of \(M\) and \(\sigma=2\) dB (Figure 9a), \(\sigma=4\) dB (Figure 9b), and \(\sigma=6\) dB (Figure 9c).