## 3 Random Antenna Selection (RAS)
### Outage Probability  
At user \(U_{1}\), the outage probability is defined via the event this user cannot detect its own signal, i.e., \(R_{1}\) falls below a data rate threshold \(R_{1}\) (\(\rm{bps/Hz}\)). Subsequently, the outage probability at this user while being assisted by the \(k\)-th antenna is given by  
\[\begin{array}{ll}OP_{1}^{RAS}(k)&=\Pr\left(R_{1}<\bar{R}_{1}\right)\\ &=\Pr\left(\frac{X_{1,k}}{Z_{1}}<\frac{\theta_{1}}{\theta_{1}-\theta_{1} \theta_{2}}\right),\ \bar{R}_{1}<\frac{\theta_{1}}{\theta_{2}}.\end{array} \tag{7}\]  
in which \(\theta_{n}=2^{R_{n}}-1\). At user \(U_{2}\), the outage probability is defined via the events (i) this user cannot decode \(x_{1}\) or (ii) this user correctly detects \(x_{1}\) but cannot decode \(x_{2}\) after successful SIC. Hence, this probability can be formulated as  
\[\begin{array}{ll}OP_{2}^{RAS}(k)&=\Pr\left(R_{1\to 2}(k)<\bar{R}_{1} \right)+\Pr\left(R_{1\to 2}(k)>\bar{R}_{1},R_{2}(k)<\bar{R}_{2}\right)\\ &=\Pr\left(R_{1\to 2}(k)<\bar{R}_{1}\text{ or }R_{2}(k)<\bar{R}_{2}\right)\\ &=\Pr\left(\frac{X_{2,k}}{Z_{2}}<\max\left(\frac{\theta_{1}}{\theta_{1}- \theta_{1}\theta_{2}},\frac{\theta_{2}}{\theta_{2}}\right)\right),\ \bar{R}_{1}<\frac{\theta_{1}}{\theta_{2}}.\end{array} \tag{8}\]  
Let \(\tau_{1}=\frac{\theta_{1}}{\theta_{1}-\theta_{1}\theta_{2}}\), \(\tau_{2}=\max\left(\frac{\theta_{1}}{\theta_{1}-\theta_{1}\theta_{2}},\frac{ \theta_{2}}{\theta_{2}}\right)\) and \(G_{n}^{(k)}=X_{n,k}/Z_{n}\) be the ordered channel fading gain normalized by the aggregated interference. It can be seen from (8) and (11) that \(OP_{n}^{(k)}=F_{G_{n}^{(k)}}(\tau_{n})\), where \(F_{G_{n}^{(k)}}(\gamma)\) is the CDF of \(G_{n}^{(k)}\). By use of order statistics, \(F_{G_{1}^{(k)}}(\gamma)\) and \(F_{G_{2}^{(k)}}(\gamma)\) are given by  
\[F_{G_{1}^{(k)}}(\gamma)=\Pr\bigg{(}\min\left(\bar{G}_{1}^{(k)}, \bar{G}_{2}^{(k)}\right)<\gamma\bigg{)}=1-\bigg{(}1-F_{\bar{G}_{1}^{(k)}}( \gamma)\bigg{)}\bigg{(}1-F_{\bar{G}_{2}^{(k)}}(\gamma)\bigg{)}, \tag{9}\] \[F_{G_{2}^{(k)}}(\gamma)=\Pr\bigg{(}\max\left(\bar{G}_{1}^{(k)}, \bar{G}_{2}^{(k)}\right)<\gamma\bigg{)}=F_{\bar{G}_{1}^{(k)}}(\gamma)F_{\bar{G }_{2}^{(k)}}(\gamma), \tag{10}\]  
respectively, in which \(\bar{G}_{n}^{(k)}\)'s specify the unordered ones with \(F_{\bar{G}_{n}^{(k)}}(\gamma)\) being the corresponding CDF. Since \(X_{n,k}\)'s are independent and identically distributed \(\forall k\), thus \(F_{\bar{G}_{n}^{(1)}}(\gamma)=F_{\bar{G}_{n}^{(2)}}(\gamma)=\cdots=F_{\bar{G} _{n}^{(N)}}(\gamma)\) which in turns obtains \(F_{G_{n}^{(1)}}(\gamma)=F_{G_{n}^{(2)}}(\gamma)=\cdots=F_{G_{n}^{(N)}}(\gamma)\). For convenience, let \(F_{\bar{G}_{n}}(\gamma)=F_{\bar{G}_{n}^{(k)}}(\gamma)\) and \(F_{G_{n}}(\gamma)=F_{G_{n}^{(k)}}(\gamma)\), \(\forall k\). The CDF of the ordered \(G^{k}\) is given in the following proposition.  
**Proposition 1**.: _The CDF of the unordered \(\bar{G}_{n}^{(k)}\) is given by_  
\[F_{\bar{G}_{n}^{(k)}}(\gamma)= F_{\bar{G}_{n}}(\gamma)=1-\exp\bigg{(}-\underbrace{\pi\lambda_{n}}_{ \Gamma(a_{n})}\Gamma\left(\alpha_{n}+\frac{2}{\eta}\right)\Gamma\left(1-\frac{ 2}{\eta}\right)\left(\frac{\gamma}{\mu_{n}\alpha_{n}}\right)^{\frac{2}{\eta}} \bigg{)},\ \gamma>0. \tag{11}\]  
_in which \(\mu_{n}=\gamma/d_{n}^{\eta}\) and \(\alpha_{n}\) specifies the shape parameter of \(|g_{n}^{(l)}|^{2}\), \(\forall l\in\Phi_{n}\)._  
Proof.: See Appendix A.  
Substituting (11) into (7) and (8), one can obtain the analytical forms for the outage probability at \(U_{1}\) and \(U_{2}\), respectively. Please note that both \(OP_{1}^{RAS}(k)\) and \(OP_{2}^{RAS}(k)\) become one (\(\forall k\)) when \(\bar{R}_{1}\geq\theta_{1}/\theta_{2}\).