## 4 High SNR Analysis and Secrecy Throughput
### High SNR Analysis
\begin{split}\Delta_{2}&=\sum_{j=0}^{N_{R}N_{B}} \left(\begin{array}{c}N_{R}N_{B}\\ j\end{array}\right)\left(2^{R_{S}}-1\right)^{N_{R}N_{B}-j}\!2^{R_{S}j}\!\! \left(\frac{\overline{\gamma}_{E2}}{\overline{\gamma}_{I}}\right)^{j}\!\Gamma \left(j+1\right)\left(\frac{1}{N_{R}!}\right)^{N_{B}}\\ &\times\left[\frac{1}{\overline{\gamma}_{I}}\Psi\left(j+1,j+4-N_{ R},\frac{1}{\overline{\gamma}_{I}}\right)+\left(N_{B}-2\right)\ \Psi\left(j+1,j+3-N_{B},\frac{1}{\overline{\gamma}_{I}}\right)\right]\\ &\times\left[\left(1-\exp\left(-\frac{\mu}{\lambda_{RP}}\right) \right)\right.+\!\left(\frac{\lambda_{RP}}{\mu}\right)^{N_{R}N_{B}}\!\!\Gamma \left(N_{R}N_{B}+1,\frac{\mu}{\lambda_{RP}}\right)\right]\left(\frac{1}{ \epsilon}\right)^{N_{R}N_{B}}.\end{split}  \tag{52}\]Proof of Corollary 2.: See Appendix D.  
**Remark 1**. : _The secrecy diversity gain of the MRC/MRT-SC and MRC-ZFB/MRT-SC-ZFB schemes is \(\min\left(N_{R},N_{R}N_{B}\right)\) under Scenario I, which is only determined by the number of \(N_{R}\) and \(N_{B}\) of Relay and Bob, respectively. Additionally, the quality of the main channel and wiretap channel influence the secrecy performance of two scenarios through the coding gain, i.e., \(G=\Delta_{\phi}{}^{-1/\min\left(N_{R},N_{R}N_{B}\right)}\), in which \(\phi\in\)(MRC/MRT-SC, MRC-ZFB/MRT-SC-ZFB), respectively. We found that the more multiple antennas of Relay and Bob, the better system performance._  
1.2 Scenario Ii: \(\overline{\gamma}_{Bj}\rightarrow\infty\) and \(\overline{\gamma}_{Ej}\rightarrow\infty\)  
Now, the approximated SOP of the considered CRNs is analyzed in this section. _(1) MRC/MRT-SC scenario:_  
**Corollary 3**. : _The asymptotic SOP of the MRC/MRT-SC scenario under \(\overline{\gamma}_{B1}\rightarrow\infty\) and \(\overline{\gamma}_{E1}\rightarrow\infty\) can be expressed as_  
\[p_{out}^{MRC/MRT-SC}\left(R_{S}\right)\approx 1-\left(1-F_{\gamma_{1}^{MRC}} \left(R_{S}\right)\right)\left(1-F_{\gamma_{1}^{MRT-SC}}\left(R_{S}\right) \right), \tag{53}\]  
_where \(F_{\gamma_{1}^{MRC}}\left(R_{S}\right)\) and \(F_{\gamma_{2}^{MRT-SC}}\left(R_{S}\right)\) being expressed as_  
\[F_{\gamma_{1}^{MRC}}\left(R_{S}\right)\approx 1-\sum_{k=0}^{N_{B}-1}\frac{ \left(2^{R_{S}}\right)^{k}}{\left(\overline{\gamma}_{B1}\right)^{k}\bar{ \gamma}_{E1}}\left(\frac{\bar{\gamma}_{B1}\bar{\gamma}_{E1}}{2^{R_{S}}\bar{ \gamma}_{E1}+\bar{\gamma}_{B1}}\right)^{k+1}\left[\left(1-\exp\left(-\frac{ \mu}{\lambda_{AP}}\right)\right)+\Gamma\left(1,\frac{\mu}{\lambda_{AP}}\right) \right], \tag{54}\]  
_and_  
\[F_{\gamma_{2}^{MRT-SC}}\left(R_{S}\right) \approx\sum_{l=0}^{N_{B}}\left( \begin{array}{c}N_{B}\\ i\end{array} \right)(-1)^{i}\Theta_{N_{R}i}2^{R_{S}\phi}\phi!\left(\frac{1}{ \varepsilon_{TB1}}\right)^{\phi}\!\left(\frac{i2^{R_{S}}\eta\bar{\gamma}_{E1} +\varepsilon\bar{\gamma}_{B1}}\right)^{-\left(\phi+1\right)}\frac{1}{\eta\bar {\gamma}_{E1}} \tag{55}\] \[\times\left[\left(1-\exp\left(-\frac{\mu}{\lambda_{RP}}\right) \right)+\frac{\mu}{\lambda_{RP}}\Gamma\left(1,\frac{\mu}{\lambda_{RP}}\right) \ \left(\frac{i\left(2^{R_{S}}-1\right)\lambda_{RP}+\mu\bar{\gamma}_{B1}}{ \varepsilon\bar{\gamma}_{B1}\lambda_{RP}}\right)^{-1}\right].\]  
Proof. : By using (32) and (36), when \(\overline{\gamma}_{B1}\rightarrow\infty\) and \(\overline{\gamma}_{E1}\rightarrow\infty\), \(F_{\gamma_{1}^{MRC}}\left(R_{S}\right)\) and \(F_{\gamma_{2}^{MRT-SC}}\left(R_{S}\right)\) can be easily derived, respectively. Thus, substituting (54) and (55) into (53), the final result of MRC/MRT-SC scenario can be easily derived. _(2) MRC-ZFB/MRT-SC-ZFB scheme:_  
**Corollary 4**.