## 4 Results  
In order to analyze the performance of the considered multi-static backscattering communication system and to validate the proposed approximation method we consider the expected FSR and the expected CR as defined in Equations (11) and (28), respectively. Specifically, we study the complex and often counter-intuitive impact of relevant system parameters, such as the number of disseminated receivers \(M\), the network topology, and the fading spread \(\sigma\) on the FSR and the CR. Moreover, we also analyze the impact of the frame length \(L\) on the optimal system performance. We considered a reference network scenario where backscattering nodes are disseminated within a circle of radius \(d_{I}\), and the illuminator is placed at the center of the circle, which we refer to as the illumination area. Unless otherwise stated, we assume that the \(N=50\) nodes are located at random positions within the circle, and the FSR is obtained by averaging over 100 different realizations of nodes spatial dissemination. The receivers are placed at fixed positions uniformly dislocated on a circle of radius \(d_{IR}\) in the illumination area. The illuminator transmit power \(P_{\text{tx}}\), the nodes sensitivity threshold \(\gamma_{h}\), the receivers' sensitivity threshold \(\gamma_{d}\), the SIR threshold \(\gamma_{\text{SIR}}\), the operative frequency \(f\) and the path-loss  
Figure 4: Example of capture distributions for \(\sigma=6\) dB in a scenario with \(K=50\) nodes, \(M=2\) receivers and 3 randomly selected transmitting nodes in a slot. (**a**) Receiver 1. (**b**) Receiver 2.  
exponent \(k_{0}\) are fixed and the relative values are summarized in Table 1. We observe that the values considered for the aforementioned parameters are compliant with realistic passive backscatters. Specifically, the illuminator transmit power and the operative frequency are set to standard values for passive RFID systems [50]. Similarly, the nodes' sensitivity threshold is set to a reference value of commercial passive tags and the receivers' sensitivity threshold is compliant with commercial RFID readers. We further assume that all receivers and all backscattering nodes have the same characteristics, thus \(\gamma_{d}\) is the same for all receivers and \(\gamma_{h}\) is the same for all the nodes. Finally, the SIR threshold is set to a value commonly assumed in several capture effect models [19]. Although the considered parameters are representative of state-of-the-art backscattering systems, we remark that the proposed framework easily allows to consider different parameters' sets, which are not addressed in this study. To obtain the FSR and the CR we enforce the approximation approach described in Section 3. Unless otherwise stated, the frame length \(L\) is set equal to the number of nodes \(K\), which corresponds to the theoretical optimal frame length of FSA in absence of propagation phenomena. Furthermore, to validate the accuracy of the proposed method, we compare the obtained results with Monte Carlo simulation. Specifically, the simulated FSR and CR are obtained by averaging over 1000 Monte Carlo trials. The impact of fading spread \(\sigma\) is illustrated in Figure 5 for different numbers+ of receivers, specifically \(M=1,2,10\). The illumination area radius is set to \(d_{I}=10\) m and the receivers are placed at the border of the illumination area (i.e., \(d_{IR}=10\) m). The expected capture rate is depicted in Figure 5a. Firstly, we notice that higher fading spread values are beneficial in terms of capture rate. This is due to the fact that a larger fading spread \(\sigma\) induces a larger spread in the capture distribution, which increases chance of capture. Secondly, a higher number of deployed receivers substantially increases the capture rate, as expected. Observe that when \(M\) is set to 10, the capture rate approaches 1 as \(\sigma\) increases, meaning that any multiple transmission yields at least a correctly detected data packet. The expected FSR is shown in Figure 5b.