## 4 Best Antenna Selection Mode
### Best Antenna Serving \(U_{2}\) (Bas-2)  
In this scheme, the source node selects the best antenna for maximal performance at \(U_{2}\). This strategy can optimize the performance of the system in the scenario where the performance at \(U_{1}\) cannot be guaranteed due to high interference level and/or poor channel condition. Subsequently, the source prefers a suitable antenna to serve \(U_{2}\). The selected antenna, denoted by \(k_{2}^{*}\), can be formulated as  
\[k_{2}^{*}=\arg\ \max_{k=1,2,\ldots,K}\ G_{2}^{(k)}. \tag{28}\]  
It then immediately follows that \(F_{G_{1}}^{k_{2}}(\gamma)=F_{G_{1}}(\gamma)\), thus the outage probability and the Ergodic Capacity at \(U_{1}\) in this scheme is similar to those in \(RAS\), i.e., \(OP_{1}^{BAS-2}=OP_{1}^{RAS}\) and \(E_{x_{1}}^{BAS-2}=E_{x_{1}}^{RAS}\). In addition, the ordered random variables \(G_{2}^{(k_{2}^{*})}\) can be re-expressed as  
\[G_{2}^{(k_{2}^{*})}= \max(\max(\tilde{G}_{1}^{(1)},\tilde{G}_{2}^{(1)}),\max(\tilde{G} _{1}^{(2)},\tilde{G}_{2}^{(2)}),\ldots,\max(\tilde{G}_{1}^{(K)},\tilde{G}_{2} ^{(K)}) \tag{29}\] \[= \max\left(\frac{\max((X_{1,k})_{k=1}^{K})}{I_{1}},\frac{\max((X_{ 2,k})_{k=1}^{K})}{I_{2}}\right), \tag{30}\]  
in which the last equality is obtained by using the definition of \(\tilde{G}_{n}^{(k)}\) and then performing some necessary algebraic transforms. The CDF of \(\max((X_{1,k})_{k=1}^{K})\), denoted by \(F_{M_{n}}(\gamma)\), can be given by  
\[F_{M_{n}}(\gamma)= \left(1-\exp\left(-\frac{\gamma}{\mu_{n}}\right)\ \right)^{K},\ \gamma>0. \tag{31}\]  
By use of binomial theorem, one can obtain  
\[F_{M_{n}}(\gamma)=\sum_{k_{n}=0}^{K}{K\choose k_{n}}(-1)^{k_{n}}\exp\left(- \frac{k_{n}\gamma}{\mu_{n}}\right),\ \gamma>0. \tag{32}\]  
To perform the remaining analysis, the CDF of \(G_{2}^{(k_{2}^{*})}\) need be computed by  
\[F_{G_{2}}^{(k_{2}^{*})}(\gamma)= 1+\sum_{\begin{subarray}{c}k_{1},k_{2}=0\\ k_{1}+k_{2}\
eq 0\end{subarray}}^{K}{K\choose k_{1}}{K\choose k_{2}}(-1)^{k_{1}+k_{2}} \exp\bigg{(}-C_{1}\left(\frac{k_{1}\gamma}{\mu_{1}\alpha_{1}}\right)^{\frac{2} {\eta}}-C_{2}\left(\frac{k_{2}\gamma}{\mu_{2}\alpha_{2}}\right)^{\frac{2}{\eta }}\bigg{)}. \tag{33}\]  
Substituting (32) into (13) yields similar integrals as in (16), thus by use of (16), one can obtain the closed-form of the Ergodic Capacity as \(U_{2}\) for \(BAS-2\) scheme as  
\[\begin{split} E_{x_{2}}^{BAS-2}=&\frac{1}{\mu\alpha }\sum_{\begin{subarray}{c}k_{1},k_{2}=0\\ k_{1}+k_{2}\
eq 0\end{subarray}}^{K}{K\choose k_{1}}(k_{2})(-1)^{k_{1}+k_{2}-1}\\ &\times H_{1,2}^{2,1}\bigg{(}C_{1}\left(\frac{k_{1}}{\mu_{1}\alpha _{1}\alpha_{2}}\right)^{\frac{2}{\eta}}+C_{2}\left(\frac{k_{2}}{\mu_{2}\alpha_ {2}\alpha_{2}}\right)^{\frac{2}{\eta}}\left|\begin{array}{c}(0,2\eta^{-1}) \\ (0,1),(0,2\eta^{-1})\end{array}\right.\right).\end{split} \tag{34}\]The optimal sum Ergodic Capacity can be obtained via the following antenna selection strategy  
\[k^{opt.}=\arg\max_{k=1,2,\ldots,K}\bigg{(}\log_{2}\bigg{(}1+\frac{\theta_{1}X_{1, k}}{\theta_{2}X_{1,k}+Z_{1}}\bigg{)}+\bigg{(}\log_{2}\bigg{(}1+\frac{\theta_{2}X_{2, k}}{Z_{2}}\bigg{)}\bigg{)}, \tag{35}\]  
in which \(k^{opt.}\) can be effectively achieved by simple search methods, e.g., exhaustive search.  
For ease of reference, we provide Table 1 as follows.