## 3 Secrecy Performance Analysis
### MRC/MRT-SC with HD Scenario  
For MRC phase, observing from (1), (3) and [11], the conditional CDF of \(\gamma_{AR,1}\) and the conditional PDF of \(\gamma_{AE,1}\), are respectively written as  
\[F_{\gamma AR,1}\left(x|G\right)=1-\exp\left(-\frac{\sigma_{R}^{2}}{P_{S}\lambda _{AR}}x\right)\sum_{k=0}^{N_{R}-1}\frac{1}{k!}\Bigg{(}\frac{\sigma_{R}^{2}x}{P_ {S}\lambda_{AR}}\Bigg{)}^{k}, \tag{28}\]  
and  
\[f_{\gamma_{AE,1}}\left(y|G\right)=\frac{\sigma_{E}^{2}}{P_{S}\lambda_{AE}}\exp \left(-\frac{\sigma_{E}^{2}}{P_{S}\lambda_{AE}}y\right). \tag{29}\]  
Then, substituting (28) and (29) into (15), the conditional \(F_{\gamma AR,1}\left(R_{S}|G\right)\) is given by  
\[ \begin{split} F_{\gamma AR,1}\left(R_{S}|G\right)& =\int_{0}^{\infty}F_{\gamma AR,1}\left(2^{R_{S}}\left(1+y\right)-1 |G\right)f_{\gamma_{AE,1}}\left(y|G\right)dy\\ &=1-\exp\left(-\frac{\sigma_{R}^{2}\left(2^{R_{S}}-1\right)}{P_{ S}\lambda_{AR}}\right)\sum_{k=0}^{N_{R}-1}\frac{1}{k!}\Bigg{(}\frac{\sigma_{R}^{2}}{P _{S}\lambda_{AR}}\Bigg{)}^{k}\frac{\sigma_{E}^{2}}{P_{S}\lambda_{AE}}\\ &\times\sum_{i=0}^{k}\left(\begin{array}{c}k\\ i\end{array}\right)\!\left(2^{R_{S}}\right)^{i}\!\left(2^{R_{S}}-1\right)^{k-i }!\Bigg{(}\frac{P_{S}\lambda_{AR}P_{S}\lambda_{AE}}{\sigma_{R}^{2}\left(2^{R_ {S}}P_{S}\lambda_{AE}+P_{S}\lambda_{AR}\right)}\Bigg{)}^{i+1}.\end{split}  \tag{30}\]  
The random variable \(G=\left|h_{AP}\right|^{2}\) obeys the exponential distribution, and the PDF expression of \(G\) is given by  
\[f_{G}\left(g\right)=\frac{1}{\lambda_{AP}}\exp\left(-\frac{1}{\lambda_{AP}}g \right). \tag{31}\]  
Thus, by using (30) and (31), we can write the CDF of \(\gamma_{AR,1}\) for the MRC operation with the HD mechanism in the following theorem. **Theorem 1**. : _By using MRC scheme we can calculate the CDF of \(\gamma_{AR,1}\), which can be derived as_  
\[ \begin{split} F_{\gamma_{1}^{\text{ MRC}}}\left(R_{S}\right)& =1-\sum_{k=0}^{N_{R}-1}\frac{1}{k!}\frac{1}{\left(\gamma_{R}\right)^{k}\!\gamma _{E}}\sum_{i=0}^{k}\left(\begin{array}{c}k\\ i\end{array}\right)\!\left(2^{R_{S}}-1\right)^{k-i}\!\left(2^{R_{S}}\right)^{ i}\!i\!\left(\frac{\gamma_{B}\sigma_{E}}{2^{R_{S}}\bar{\gamma}_{E}+\bar{ \gamma}_{B}}\right)^{i+1}\\ &\times\left[\left(1-\exp\left(-\frac{\mu}{\lambda_{AP}}\right) \right)\exp\left(-\frac{2^{R_{S}}-1}{\bar{\gamma}_{B}}\right)+\frac{\mu}{ \lambda_{AP}}\!\left(\frac{\bar{\gamma}_{B}\lambda_{AP}}{\left(2^{R_{S}}-1 \right)\lambda_{AP}+\bar{\gamma}_{B}\mu}\right)^{k-i+1}\\ &\times\left.\Gamma\left(k-i+1,\frac{\left(2^{R_{S}}-1\right)\lambda_{AP}+ \mu\bar{\gamma}_{B}}{\bar{\gamma}_{B}\lambda_{AP}}\right)\right],\end{split}  \tag{32}\]  
_where \(\Gamma\left(\cdot,\cdot\right)\) is the incomplete upper gamma function, as defined in (Equation (8.350.2), [41])._  
Proof of Theorem 1.: See Appendix A. For MRT-SC phase, observing from (4), (5), [12; 42], the conditional CDF of \(\gamma_{RB,1}\) and the conditional PDF of \(\gamma_{RE,1}\), are respectively given by  
\[F_{\gamma_{RB,1}}\left(y|G_{1}\right)\,=\sum_{i=0}^{N_{R}}\Bigg{(} \begin{array} []{c}N_{B}\\ i\end{array} \Bigg{)}(-1)^{i}\exp\left(-\frac{i\sigma_{R}^{2}y}{P_{R}\lambda_{ RB}}\right)\Theta_{N_{R},i}\Bigg{(}\frac{\sigma_{B}^{2}y}{P_{R}\lambda_{RB}} \Bigg{)}^{\phi}, \tag{33}\]  
and  
\[f_{\gamma_{RE,1}}\left(y|G_{1}\right)=\frac{\sigma_{E}^{2}}{P_{R}\lambda_{RE}} \exp\left(-\frac{\sigma_{E}^{2}}{P_{R}\lambda_{RE}}y\right), \tag{34}\]where \(\Theta_{N_{R,i}}=\sum\limits_{n_{1}=0}^{q}\sum\limits_{n_{2}=0}^{n_{1}}\cdots \sum\limits_{n_{N_{R}-1}=0}^{n_{N_{R}-2}}\frac{q!}{n^{N_{R}-1}! }\prod\limits_{i =1}^{N_{R}-1}\frac{(i!)^{n_{i+1}-n_{i}}!}{(n_{i-1}-n_{i})! }\), with \(n_{0}=q\), \(n_{N_{R}}=0\), and \(\phi=\sum\limits_{q_{0}=1}^{N_{R}-1}n_{q_{0}}\). Then, substituting (33) and (34) into (15), the conditional \(F_{7 RB,1}\left(R_{S}|G_{1}\right)\) is written as  
\[