## 3 Secrecy Performance Analysis
### MRC-ZFB/MRT-SC-ZFB with FD Scenario  
For MRC-ZFB phase, observing from (8) and (9), and in the presence of [12], the conditional CDF of \(\gamma_{AR,2}\) can be written as  
\[F_{\gamma_{AR,2}}\left(x|G\right)=1-\exp\left(-\frac{\sigma_{R}^{2}}{P_{S} \lambda_{AR}}x\right)\sum\limits_{k=0}^{N_{R}-1}\frac{1}{k!}\left(\frac{\sigma_{ R}^{2}x}{P_{S}\lambda_{AR}}\right)^{k}. \tag{37}\]  
Lemma 3.2: _The conditional PDF of \(\gamma_{AE,2}\) is given by_  
\[ \begin{split} f_{\gamma_{AE,2}}\left(y|G\right)&= \frac{\sigma_{E}^{2}}{P_{S}\lambda_{AE}}\left(\frac{P_{S}\lambda_{AE}}{P_{I} \lambda_{JE}y+P_{S}\lambda_{AE}}\right)^{N_{B}-2}\exp\left(-\frac{y\sigma_{E}^ {2}}{P_{S}\lambda_{AE}}\right)+\exp\left(-\frac{\sigma_{E}^{2}y}{P_{S}\lambda_{ AE}}\right)\\ &\times\frac{\left(N_{B}-2\right)P_{I}\lambda_{JE}\left(P_{S} \lambda_{AE}\right)^{N_{B}-2}}{\left(P_{I}\lambda_{JE}y+P_{S}\lambda_{AE} \right)^{N_{B}-1}}.\end{split}  \tag{38}\]  
Proof of Lemma 3.2: See Appendix B. Then, substituting (37) and (38) into (15), the conditional \(F_{\gamma AR,2}\left(R_{S}|G\right)\) is derived as  
\[ \begin{split}& F_{\gamma AR,2}\left(R_{S}|G\right)=\int_{0}^{ \infty}F_{\gamma AR,2}\left(2^{R_{S}}\left(1+y\right)-1\right)f_{\gamma AR,2} \left(y\right)dy\\ &=1-\underbrace{\int_{0}^{\infty}\exp\left(-\frac{\sigma_{R}^{2 }\left(2^{R_{S}}\left(1+y\right)-1\right)}{P_{S}\lambda_{AR}}\right)\sum_{k=0 }^{N_{R}-1}\frac{1}{k!}\left(\frac{\sigma_{R}^{2}\left(2^{R_{S}}\left(1+y \right)-1\right)}{P_{S}\lambda_{AR}}\right)^{k}\frac{\sigma_{I}^{2}}{P_{S} \lambda_{AE}}\exp\left(-\frac{y\sigma_{R}^{2}}{P_{S}\lambda_{AE}}\right)\left( \frac{P_{S}\lambda_{AE}}{P_{J}xy+P_{S}\lambda_{AE}}\right)^{N_{R}-2}dy}\\ &-\underbrace{\int_{0}^{\infty}\exp\left(-\frac{\sigma_{R}^{2} \left(2^{R_{S}}\left(1+y\right)-1\right)}{P_{S}\lambda_{AR}}\right)\sum_{k=0}^ {N_{R}-1}\frac{1}{k!}\left(\frac{\sigma_{R}^{2}\left(2^{R_{S}}\left(1+y \right)-1\right)}{P_{S}\lambda_{AR}}\right)^{k}\exp\left(-\frac{y\sigma_{R}^{ 2}}{P_{S}\lambda_{AE}}\right)\frac{\left(N_{R}-2\right)P\lambda_{IE}\left(P_ {S}\lambda_{AE}\right)^{N_{R}-2}}{\left(p_{J}xy+P_{S}\lambda_{AE}\right)^{N_{ R}-1}}dy}.\end{split}  \tag{39}\]  
Then, we can operate several mathematical manipulations. By making use of (Equation (9.211.4), [41]), K1 and K2 can be rewritten as  
\[ \begin{split}\text{K1}&=\exp\left(-\frac{\sigma_{R}^ {2}\left(2^{R_{S}}-1\right)}{P_{S}\lambda_{AR}}\right)\sum_{k=0}^{N_{R}-1} \frac{1}{k!}\left(\frac{\sigma_{R}^{2}}{P_{S}\lambda_{AR}}\right)\sum_{i=0}^ {k}\left(\begin{array}{c}k\\ i\end{array}\right)\left(2^{R_{S}}-1\right)^{k-i}2^{iR_{S}}\frac{\sigma_{E}^{2 }}{P_{S}\lambda_{AE}}\\ &\times\left(\frac{P_{S}\lambda_{AE}}{P_{J}\lambda_{IE}}\right)^{i }\Psi\left(i+1,4+i-N_{B},\frac{2^{R_{S}}P_{S}\lambda_{AE}+P_{S}\lambda_{AR}}{ P_{S}\lambda_{AR}P_{J}\lambda_{IE}}\right)\Gamma\left(i+1\right),\end{split}  \tag{40}\]  
and  
\[ \begin{split}\text{K2}&=\exp\left(-\frac{\sigma_{R}^ {2}\left(2^{R_{S}}-1\right)}{P_{S}\lambda_{AR}}\right)\sum_{k=0}^{N_{R}-1} \frac{1}{k!}\left(\frac{\sigma_{R}^{2}}{P_{S}\lambda_{AR}}\right)^{k}\sum_{i=0 }^{k}\left(\begin{array}{c}k\\ i\end{array}\right)\left(2^{R_{S}}-1\right)^{k-i}\left(N_{B}-2\right)\\ &\times\left(\frac{P_{S}\lambda_{AE}}{P_{J}\lambda_{IE}}\right)^{i }\Psi\left(i+1,3+i-N_{B},\frac{2^{R_{S}}P_{S}\lambda_{AE}+P_{S}\lambda_{AR}}{ P_{S}\lambda_{AR}P_{J}\lambda_{IE}}\right)2^{iR_{S}}\Gamma\left(i+1\right),\end{split}  \tag{41}\]  
where \(\Psi\left(\cdot,\cdot;\cdot\right)\) being the confluent hypergeometric function of second kind. Thus, the conditional \(F_{AR,2}\left(R_{S}|G\right)\) can be rewritten as  
\[F_{AR,2}\left(R_{S}|G\right)=1-\text{K}_{1}-\text{K}_{2}. \tag{42}\]  
Hence, the \(F_{\gamma_{1}^{MRC-ZFB}}\left(R_{S}\right)\) can be derived as  
\[F_{\gamma_{1}^{MRC-ZFB}}\left(R_{S}\right)=\int_{0}^{\infty}\left[1-\text{K1} \right.\left.-\text{K2}\right]f_{G}\left(g\right)dg. \tag{43}\]  
Now, substituting (31) into (43), we can write the CDF of \(\gamma_{AR,2}\) for the MRC-ZFB scheme with the FD mechanism in the following theorem.