## 3 Secrecy Performance Analysis  
In this part, we study the SOP of the secure transmission in the dual-hop cognitive wiretap networks with FD receiver under two different scenarios. The SOP, defined as the probability of the secrecy capacity, \(C_{S}\), is less than a given threshold, \(R_{S}\). According to RaF protocol [20,40], we only solve the SOP of each independent phase. Mathematically, given by  
\[P_{out}\left(R_{S}\right)=\Pr\left(C_{S,k}<R_{S}\right)=\int_{0}^{\infty}F_{ \gamma_{RB}}\left(2^{R_{S}}\left(1+y\right)-1\right)f_{\gamma_{TE}}\left(y \right)dy. \tag{15}\]  
where \(C_{S,k}\) and \(R_{S}\) represent instantaneous secrecy rate and a predetermined threshold, respectively. Furthermore, the outage probability of \(P_{out}\left(R_{S}\right)\) is expressed as  
\[P_{out}\left(R_{S}\right)=1-\Pr\left\{C_{S,1}^{\zeta}>R_{S}\right\}\Pr\left\{ C_{S,2}^{\zeta}>R_{S}\right\}, \tag{16}\]  
where \(C_{S,1}^{\zeta}\) and \(C_{S,2}^{\zeta}\) refer to the instantaneous secrecy rate of the first phase and the second phase, respectively, \(\zeta\) \(\in\)(MRC, MRC-ZFB) in the first phase, \(\zeta\) \(\in\)(MRT-SC,MRT-SC-ZFB) in the second phase. The instantaneous SOP of \(C_{S,1}^{MRC}\) and \(C_{S,2}^{MRT-SC}\) in MRC/MRT-SC scheme are written as  
\[C_{S,1}^{MRC}=\log_{2}\frac{1+\gamma_{AR,1}}{1+\gamma_{AE,1}}=\log_{2}\frac{ 1+\min\left(\frac{Q}{\left|h_{AP}\right|^{2}},P_{t}\right)\frac{\left|h_{AR} \right|^{2}}{\sigma_{E}^{2}}}{1+\min\left(\frac{Q}{\left|h_{AP}\right|^{2}}, P_{t}\right)\frac{\left|h_{AE}\right|^{2}}{\sigma_{E}^{2}}}, \tag{17}\]  
and  
\[C_{S,2}^{MRT-SC}=\log_{2}\frac{1+\gamma_{RB,1}}{1+\gamma_{RE,1}}=\log_{2} \frac{1+\min\left(\frac{Q}{\left|h_{RP}\right|^{2}},P_{t}\right)\frac{\max \left|h_{RB}\right|^{2}}{\sigma_{E}^{2}}}{1+\min\left(\frac{Q}{\left|h_{RP} \right|^{2}},P_{t}\right)\frac{\left|h_{RE}\right|^{2}}{\sigma_{E}^{2}}}. \tag{18}\]  
The instantaneous SOP of \(C_{S,1}^{MRC-ZFB}\) and \(C_{S,2}^{MRT-SC-ZFB}\) in MRC-ZFB/MRT-SC-ZFB scheme are formulated as  
\[C_{S,1}^{MRC-ZFB}=\log_{2}\frac{1+\gamma_{AR,2}}{1+\gamma_{AE,2}}=\log_{2} \frac{1+\min\left(\frac{Q}{\left|h_{AP}\right|^{2}},P_{t}\right)\frac{\left|h_ {AR}\right|^{2}}{\sigma_{E}^{2}}}{1+\min\left(\frac{Q}{\left|h_{AP}\right|^{2} },P_{t}\right)\frac{\left|h_{AR}\right|^{2}}{\left|h_{SR}^{2}\left|h_{RF} \right|^{2}+\sigma_{E}^{2}}}, \tag{19}\]and  
\[C_{S,2}^{MRT-SC-ZFB}=\log_{2}\frac{1+\gamma_{RB,2}}{1+\gamma_{RE,2}}=\log_{2}\frac{ 1+\min\left(\frac{Q}{|h_{RP}|^{2}},P_{t}\right)\frac{\max\limits_{i\in N_{B}} \left\|\mathbf{h}_{RBi}\right\|^{2}}{\sigma_{B}^{2}}}{1+\min\left(\frac{Q}{|h_ {RP}|^{2}},P_{t}\right)\frac{\left\|\mathbf{h}_{RE}\right\|^{2}}{P_{j}\left| \mathbf{h}_{BE}^{\dagger}\mathbf{w}_{ZF1}\right|^{2}+\sigma_{E}^{2}}}. \tag{20}\]  
In order to solve (16), \(P_{out}\left(R_{S}\right)\) needs to use (17), (18), (19) and (20). We have  
\[\Pr\left\{C_{S,1}^{\zeta}>R_{S}\right\}= 1-\Pr\left(\gamma_{1}^{\zeta}\leq 2^{R_{S}}\right)=1-F_{ \gamma_{1}^{\zeta}}\left(R_{S}\right), \tag{21}\]  
and  
\[\Pr\left\{C_{S,2}^{\zeta}>R_{S}\right\}= 1-\Pr\left(\gamma_{2}^{\zeta}\leq 2^{R_{S}}\right)=1-F_{ \gamma_{2}^{\zeta}}\left(R_{S}\right), \tag{22}\]  
where \(\Pr\left(\gamma_{1}^{\zeta}\leq 2^{R_{S}}\right)\) and \(\Pr\left(\gamma_{2}^{\zeta}\leq 2^{R_{S}}\right)\) are the SOP of the first phase and second phase of each scenario, respectively.