## 3 Multiple Access Interference Characterization and Capture Diversity
### Capture Diversity  
For any multiple transmission slot, it is possible to associate a capture distribution to each receiver. Formally, given a generic receiver \(m\) and the interferes set \(\mathcal{I}\), the capture distribution is given by the capture probabilities \(p_{c,k}^{(m)}\), with \(k{\in}\mathcal{I}\), including the collision probability \(p_{\text{coll}}^{(m)}\). Since backscattered signals are received at different spatial locations, and thus experiment different attenuation and fading realizations, the SIR associated to a generic node \(k\) differs at different receivers. This implies that the capture probabilities \(p_{c,k}^{(m)}\) distribute differently from one receiver to another. Therefore, data packet from different nodes can be captured simultaneously at different receivers, yielding multiple packet reception during multiple transmission slots. We refer to this as capture diversity. To illustrate capture diversity, we consider an example where a multiple transmission slot occurs in presence of two spatially dislocated receivers. We assume a set of 50 backscattering nodes disseminated at random within a circular area of radius 5 m around the illuminator. The receivers are placed at antipodal positions with respect to the illuminator, at the border of the circular area. We assume that three randomly selected nodes transmit simultaneously in a slot and we compute the approximated capture distributions relying on Equation (22). The capture distributions associated to the two receivers for \(\sigma=2\) dB are depicted in Figure 3a,b, respectively. To show the accuracy of the proposed approximation we also derive the capture distribution by means of Monte Carlo simulations, which converge to the actual value of the capture probabilities. Observe that the second node's packet has a good chance of being captured by the first receiver, whereas, the first node's packet is likely captured at the second receiver. Thus, the probability of successfully receiving two data packets simultaneously in the same slot is substantially high. Note also that the accuracy of the approximation method is very good for the selected value of \(\sigma\). A second example is considered, where the scenario is kept fixed and \(\sigma\) is set to 6 dB. The approximated and simulated capture distributions associated to the first and second receiver are reported in Figure 4a,b, respectively. In this case, the higher fading spread reflects on the spread of the capture distributions. Indeed, the first receiver have now a non zero probability of capturing either the second node's packet or the third node's packet. The second receiver, instead, has a good chance of capturing the first node's data packet. Note that the larger value of \(\sigma\) also affects the accuracy of the proposed approximation method. However, although a lower accuracy is associated to higher value of the fading spread, the approximation still perform well. Figure 3: Example of capture distributions for \(\sigma=2\) 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. In order to quantify the impact of capture diversity, we define the Capture Rate (CR) as the probability of having capture events in multiple transmission slots. Formally, conditioned on a specific set of active nodes in a frame, the CR can be expressed as  
\[CR|\mathcal{P}_{\mathcal{A}}=\frac{\sum_{i=1}^{L}\phi_{i}|\mathcal{P}_{\mathcal{ A}}}{n_{\text{mr}}}, \tag{27}\]  
where \(\phi_{i}\) is a binary flag that is high if capture occurs in a slot and \(n_{\text{mr}}\) is the number of multiple transmission slots in a frame. Then, the expected capture rate can be obtained as  
\[C=\sum_{\mathcal{A}\subset\mathcal{K}}E[CR|\mathcal{P}_{\mathcal{A}}]\Pr( \mathcal{P}_{\mathcal{A}}). \tag{28}\]  
We remark that capture diversity differs from conventional micro-diversity concept, where redundancy is exploited by means of signal combining approaches that aim at maximizing detection performance.