## 3 System Model  
Figure 1 presents the system model, which includes a mobile source (MS) node with \(N_{\rm t}\) antennas, a mobile destination (MD) node with \(N_{\rm r}\) antennas, and \(L\) mobile relay (MR) nodes. The nodes operate in half-duplex mode. The \(L\)MR nodes utilize their individual uplink channel state information (CSI) to select the best MR. The best MR node participates in collaboration. The MD node calculates the received signal-to-noise ratio (SNR) from the best MR node. The MD node orders the received SNR from \(N_{\rm t}\) antennas, and then feedbacks the selected antenna to the MS node. The channel coefficient \(h=h_{\rm k}\), \(k\)\(\in\)\(\{\)SD\(ij\), SR\(il\), RD\(|j\}\), \(i\)\(\in\)(1,\(N_{\rm t}\)), \(j\)\(\in\)(1,\(N_{\rm r}\)), \(l\)\(\in\)(1,\(L\)). The amplitude of \(h\) follows \(N\)-Nakagami distribution [19]. The total power is \(E\). In the first timeslot, MS\({}_{i}\) transmits the signal \(a\). MD\({}_{j}\) and MR\({}_{l}\) receive the signals \(r_{\rm SDj}\) and \(r_{\rm SRi}\) as:  
\[r_{\rm SDj}=\sqrt{KEh_{\rm SDj}}a+n_{\rm SDj} \tag{1}\]  
\[r_{\rm SRi}=\sqrt{G_{\rm SRi}KEh_{\rm SRi}}a+n_{\rm SRi} \tag{2}\]  
where \(G_{\rm SDj}\) = 1 is the relative gain of MS\(\rightarrow\)MD, \(G_{\rm SRi}\) is the relative gain of MS\({}_{i}\)\(\rightarrow\)MR\({}_{l}\). \(n_{\rm SRi}\) and \(n_{\rm SDj}\) have mean 0 and variance \(N_{0}\)/2. \(K\) is the power allocation parameter, which controls power distribution between the MS and MR. For MS, the power is \(KE\). For MR, the power is \((1\)\(-\)\(K)E\), \(K\)\(\in\)(0,1). In the second timeslot, with AF method, MD\({}_{j}\) receives the signal as:  
\[r_{\rm RDj}=\sqrt{c_{ij}}El_{\rm SRi}ln_{\rm RDj}a+n_{\rm RDj} \tag{3}\]  
where \(G_{\rm RDj}\) is the relative gain of MR\({}_{l}\)\(\rightarrow\)MD\({}_{j}\), \(n_{\rm RDj}\) has mean 0 and variance \(N_{0}\)/2. \(c_{ij}\) is given as [39]:  
\[c_{ilj}=\frac{K(1-K)G_{\rm SRi}G_{\rm RDj}E/N_{0}}{1+K_{\rm GRi}ll|t_{\rm SRi} |^{2}E/N_{0}+(1-K)G_{\rm RDj}|\big{|}h_{\rm RDj}\big{|}^{2}E/N_{0}} \tag{4}\]  
where \(N_{0}\) is noise power. Figure 1: The system model. MD\({}_{j}\) calculates the received signal-to-noise ratio (SNR) as:  
\[\gamma_{ij}=\max(\gamma_{\text{SD}ij},\gamma_{\text{SRD}ij}) \tag{5}\]  
\[\gamma_{\text{SD}ij}=\frac{K\big{|}h_{\text{SD}ij}\big{|}^{2}E}{N_{0}}=K\big{|}h _{\text{SD}ij}\big{|}^{2}\overline{\gamma} \tag{6}\]  
\[\gamma_{\text{SRD}ij}=\max\limits_{1\leq I\leq L}(\frac{\gamma_{\text{SR}il} \gamma_{\text{RD}lj}}{1+\gamma_{\text{SR}il}+\gamma_{\text{RD}lj}}) \tag{7}\]  
where \(\gamma_{\text{SRD}ij}\) is the SNR of MS\(\rightarrow\)MR\(\rightarrow\)MD link\({}_{j}\)\(\gamma_{\text{SR}il}\) is the SNR of MS\(\rightarrow\)MD link\({}_{j}\)\(\gamma_{\text{SR}il}\) is the SNR of MS\({}_{i}\)\(\rightarrow\)MR\({}_{l}\) link, and\(\gamma_{\text{RD}lj}\) is the SNR of MR\({}_{l}\)\(\rightarrow\)MD\({}_{j}\) link. \[\gamma_{\text{SR}il}=\frac{G_{\text{SR}il}K\big{|}h_{\text{SR}il}\big{|}^{2}E}{ N_{0}}=G_{\text{SR}il}K\big{|}h_{\text{SR}il}\big{|}^{2}\overline{\gamma} \tag{8}\]  
\[\gamma_{\text{RD}lj}=\frac{(1-K)G_{\text{RD}lj}\big{|}h_{\text{RD}lj}\big{|}^{ 2}E}{N_{0}}=(1-K)G_{\text{RD}lj}\big{|}h_{\text{RD}lj}\big{|}^{2}\overline{\gamma} \tag{9}\]  
It is difficult to obtain the closed-form solution to \(\gamma_{\text{SRD}ij}\). With the help of [40,41], we obtain an upper bound of \(\gamma_{\text{SRD}ij}\) as:  
\[r_{\rm SRi}=\sqrt{G_{\rm SRi}KEh_{\rm SRi}}a+n_{\rm SRi} \tag{2}\]0  
MD calculates the received SNR as:  
\[r_{\rm SRi}=\sqrt{G_{\rm SRi}KEh_{\rm SRi}}a+n_{\rm SRi} \tag{2}\]1  
where \(\gamma_{ij}=\max(\gamma_{\text{SD}ij},\gamma_{\text{up}ij})\). We select \(g\) of TAS scheme as:  
\[r_{\rm SRi}=\sqrt{G_{\rm SRi}KEh_{\rm SRi}}a+n_{\rm SRi} \tag{2}\]2  
**4. The OP of Optimal TAS Scheme**  
We obtain the OP as:  
\[