## 3 Multiple Access Interference Characterization and Capture Diversity  
In order to study the FSR for the considered multi-static system, a crucial step is the statistical characterization of the interference that allows to determine the capture probability defined in Equation (7). To this aim, we consider a multiple transmission slot with a generic set of transmitting nodes \(\mathcal{I}\), with \(|\mathcal{I}|>1\). Conditioned on the event \(\mathcal{P}_{\mathcal{A}}\), the capture probability at the \(m\)-th receiver for the \(k\)-th node is  
\[ \begin{split} p_{c,k}^{(m)}&=\Pr\left(\frac{ \mathsf{P}_{b,k}^{(m)}}{\sum_{j\in\mathcal{I}}\mathsf{P}_{b,k}^{(m)}}{\sum_{j \in\mathcal{I}}\mathsf{P}_{b,k}^{(m)}}{<}\frac{1}{\gamma_{\text{SIR}}},\mathsf{ P}_{fj}{>}\gamma_{h},\forall i\in\mathcal{A}\right)\\ &=\frac{\Pr\left(\frac{\mathsf{Y}^{j\
eq k}\mathsf{P}_{b,k}^{(m)}} {\sum_{j\in\mathcal{I}}\mathsf{P}_{b,k}^{(m)}}{<}\frac{1}{\gamma_{\text{SIR}} },\mathsf{P}_{fj}{>}\gamma_{h},\forall i\in\mathcal{I}\right)}{\Pr\left( \mathsf{P}_{f,j}{>}\gamma_{h},\forall i\in\mathcal{I}\right)}\\ &=\frac{\Pr\left(\sum_{j\in\mathcal{I}}\frac{D_{m,j}}{D_{m,k}} \mathsf{c}\mathsf{Y}_{j}^{(m)}{-}\mathsf{Y}_{k}^{(m)}{<}\frac{1}{\gamma_{ \text{SIR}}},\mathsf{S}_{f,j}{>}\widehat{\gamma}_{h,i},\forall i\in\mathcal{I} \right)}{\prod_{i\in\mathcal{I}}[1-F_{\mathsf{S}}(\widehat{\gamma}_{h,i})]}. \end{split}  \tag{12}\]  
The derivation of the above probability requires the statistics of the random variable  
\[\mathsf{X}_{k}^{(m)}{=}\sum_{j\in\mathcal{I}}\frac{D_{m,j}}{D_{m,k}}\mathsf{c }\mathsf{Y}_{j}^{(m)}{-}\mathsf{Y}_{k}^{(m)}, \tag{13}\]  
with \(k\in\mathcal{I}\), which represents the inverse of the SIR. However, the random variable \(\mathsf{X}_{k}^{(m)}\) is given as a weighted sum of log-normal random variables, which has an unknown distribution (i.e., the PDF of \(\mathsf{X}_{k}^{(m)}\) does not exist in closed form). Therefore, we approximate the inverse SIR with a log-normal random variable as  
\[\mathsf{X}_{k}^{(m)}{\approx}\varepsilon^{\mathsf{T}_{k}^{(m)}}, \tag{14}\]  
where we defined \(\hat{\gamma}=-\ln\gamma_{\text{SIR}}\). To compute the approximated capture probability in Equation (22) we first need to determine the covariance between \(\Upsilon_{k}^{(m)}\) and \(\mathsf{S}_{f,i}\) for all \(i\in\mathcal{I}\), which we define as  
\[c_{\Upsilon_{k}^{(m)},i}{=}Cov(\Upsilon_{k}^{(r)},\mathsf{S}_{f,i}). \tag{23}\]  
The conditional probability in Equation (12) can be expressed by deriving the correlations  
\[\delta_{\mathsf{X}_{k}^{(m)},i} \triangleq E[\mathsf{X}_{k}^{(m)}e^{\mathsf{S}_{f,i}}] \tag{24}\] \[=E\bigg{[}\sum_{j\in\mathcal{I}}^{j\
eq k}\frac{D_{m,j}}{D_{m,k}} \mathsf{Y}_{j}^{(m)}{-}\mathsf{Y}_{k}^{(m)}e^{\mathsf{S}_{f,i}}\bigg{]}\] \[=\frac{1}{D_{m,k}}\sum_{j\in\mathcal{I}}^{j\
eq k}D_{m,j}E\bigg{[} e^{\mathsf{Y}_{j}^{(m)}-\mathsf{Y}_{k}^{(m)}+\mathsf{S}_{f,j}}\bigg{]}\] \[= \begin{cases}\frac{\xi_{k}^{2}e^{2}}{D_{m,k}}\sum_{j\in\mathcal{ I}}^{j\
eq k}D_{m,j}&k=i\\ \frac{\xi_{k}^{2}e^{2}}{D_{m,k}}\sum_{j\in\mathcal{I}}^{j\
eq k,j\
eq i}D_{m,j}+ \frac{D_{m,j}}{D_{m,k}}e^{2}e^{2}&k\
eq i.\end{cases} \]  
Being  
\[E[e^{\Upsilon_{k}^{(m)}}e^{\mathsf{S}_{f,i}}]=E[e^{\Upsilon_{k}^{(m)}}]E[e^{ \mathsf{S}_{f,i}}]e^{c_{\Upsilon_{k}^{(m)},i}}, \tag{25}\]  
with \(i\in\mathcal{I}\), the covariance between \(\Upsilon_{k}^{(m)}\) and the fading component on the \(i\)-th forward link is thus obtained as  
\[c_{\Upsilon_{k}^{(m)},i}=\ln\frac{\delta_{\mathsf{X}_{k}^{(m)},i}}{e^{\eta_{ \Upsilon_{k}^{(m)}+\frac{1}{2}}(\sigma_{\Upsilon_{k}^{(m)}+\sigma^{2}}^{2})}}. \tag{26}\]  
The approximated capture probability in Equation (22) can be expressed in terms of multi-variate Gaussian CDF and can be efficiently evaluated by means of standard numeric integration approaches [47]. The proposed approximation is an extension of the generalized method proposed in [14], which is derived from the Fenton-Wilkinson approach.