## 3 Performance Analysis
### Asymptotic OP  
Recalling the fact in Equation (20), when \(\overline{\gamma}_{SR}\) is large enough, it can be presented as:  
\[F_{\gamma_{SR}}\left(x\right)\approx\frac{\alpha_{SR}x}{\tilde{\gamma}_{SR}}. \tag{31}\]Taking Equations (23) and (31) into Equation (25), we could have Equation (32), which is shown at the top of this page.  
\[\Pr\left(\gamma_{R}\leq\gamma_{0}\right)\approx\sum\limits_{\xi_{1}=0}^{m_{SP}-1} \cdots\sum\limits_{\xi_{M}=0}^{m_{SP}-1}\frac{\alpha_{SR}\Xi\left(M\right)}{ \tilde{\gamma}_{SR}}\left[\frac{C_{1}\Lambda_{SP}!}{\Delta_{SP}^{\Lambda_{SP}+1 }}+\frac{C_{2}\left(\Lambda_{SP}-1\right)!}{\Delta_{SP}^{\Lambda_{SP}}}\right]. \tag{32}\]  
Utilizing the similar method and supposing the identical channel parameters, the CDF of \(\gamma_{RD}\) at high SNRs is given by:  
\[F_{\gamma_{RD}}\left(x\right)\approx\frac{x}{\tilde{\gamma}_{RD}}. \tag{33}\]  
Then, substituting Equations (13) and (33) into Equation (27), we can obtain:  
\[\Pr\left(\gamma_{D}\leq\gamma_{0}\right)\approx\sum\limits_{i=1}^{\rho\left( \text{A}_{RP}\right)}\sum\limits_{j=1}^{\tau_{\left(\text{A}_{RP}\right)}} \frac{\chi_{i,j}\left(\text{A}_{RP}\right)}{\tilde{\gamma}_{RD}}\left(D_{1}j \mu_{\left\langle i\right\rangle}+D_{2}\right). \tag{34}\]  
Furthermore, taking Equations (32) and (34) into Equation (24), the asymptotic expression of OP can be given by:  
\[\begin{split}& P_{out}^{\infty}\left(\gamma_{0}\right)\approx\\ &\begin{cases}\sum\limits_{\xi_{1}=0}^{m_{SP}-1}\cdots\sum\limits _{\xi_{M}=0}^{m_{SP}-1}\frac{\alpha_{SR}\Xi\left(M\right)}{\tilde{\gamma}_{SR}} \left[\frac{C_{1}\Lambda_{SP}!}{\Delta_{SP}^{\Lambda_{SP}+1}}+\frac{C_{2}( \Lambda_{SP}-1)!}{\Delta_{SP}^{\Lambda_{SP}}}\right]+\sum\limits_{i=1}^{\rho \left(\text{A}_{RP}\right)}\sum\limits_{j=1}^{\tau_{\left(\text{A}_{RP} \right)}}\frac{\chi_{i,j}\left(\text{A}_{RP}\right)}{\tilde{\gamma}_{RD}}\left( D_{1}j\mu_{\left\langle i\right\rangle}+D_{2}\right)\\ -\sum\limits_{\xi_{1}=0}^{m_{SP}-1}\cdots\sum\limits_{\xi_{M}=0}^{m_{SP}-1} \frac{\alpha_{SR}\Xi\left(M\right)}{\tilde{\gamma}_{SR}}\left[\frac{C_{1} \Lambda_{SP}!}{\Delta_{SP}^{\Lambda_{SP}+1}}+\frac{C_{2}(\Lambda_{SP}-1)!}{ \Delta_{SP}^{\Lambda_{SP}}}\right]\\ \times\left[\sum\limits_{i=1}^{\rho\left(\text{A}_{RP}\right)}\sum\limits_{j =1}^{\tau_{\left(\text{A}_{RP}\right)}}\frac{\chi_{i,j}\left(\text{A}_{RP} \right)}{\tilde{\gamma}_{RD}}\left(D_{1}j\mu_{\left\langle i\right\rangle}+D_{ 2}\right)\right],\gamma_{0}<\min\left(\frac{1}{\tilde{k}_{1}^{2}},\frac{1}{ \tilde{k}_{2}^{2}}\right)\\ 1,\gamma_{0}\geq\min\left(\frac{1}{\tilde{k}_{1}^{2}},\frac{1}{\tilde{k}_{2}^{ 2}}\right).\end{cases}\end{split} \tag{35}\]  
Finally, denoting \(\tilde{\gamma}_{SR}=\tilde{\gamma}_{SP}=\tilde{\gamma}_{RD}=\tilde{\gamma}_{ RP}=\tilde{\gamma}\) and ignoring the higher order terms, we can get:  
\[P_{out}^{\infty}\left(\gamma_{0}\right)=\Phi\bigg{(}\frac{1}{\tilde{\gamma}} \bigg{)}^{\Theta}, \tag{36}\]  
where the diversity order \(\Theta\) and coding gain \(\Phi\) can be, respectively, derived as:  
\[\Theta=\begin{cases}\Theta_{1}=1+\sum\limits_{\xi_{1}=0}^{m_{SP}-1}\cdots\sum \limits_{\xi_{M}=0}^{m_{SP}-1}\Lambda_{SP},\Phi_{1}>\Phi_{2},\\ \Theta_{2}=1,\Phi_{1}\leq\Phi_{2},\end{cases} \tag{37}\]  
\[\Phi=\begin{cases}\Phi_{1}=\sum\limits_{\xi_{1}=0}^{m_{SP}-1}\cdots\sum\limits _{\xi_{M}=0}^{m_{SP}-1}\pi_{SR}\Xi\left(M\right)\\ \times\left[\frac{C_{1}\Lambda_{SP}!}{\left(\tilde{\beta}_{SP}-\delta_{SP} \right)^{\Lambda_{SP}+1}\tilde{\gamma}_{SP}}+\frac{C_{2}(\Lambda_{SP}-1)!}{ \left(\tilde{\beta}_{SP}-\delta_{SP}\right)^{\Lambda_{SP}}}\right],\\ \Phi_{2}=\frac{D_{2}}{\tilde{\gamma}_{RD}}.\end{cases} \tag{38}\]