# Security Performance Analysis of Full-Duplex UAV Assisted Relay System Based on SWIPT Technology
## 3 Performance Analysis
### Probability of Safety Interruption  
The secrecy capacity of the system is defined as the difference between the legitimate channel capacity and the eavesdropping channel capacity of the system, \(C_{s}=max\{C_{AB}-C_{E},0\}\), and \(R_{s}^{*}\) is the threshold of the secrecy transmission rate for the whole information transmission process. When \(C_{s}>R_{s}^{*}\), it indicates that the system is securely connected and the signal can be securely transmitted; when \(C_{s}\leq R_{s}^{*}\), it indicates that the secure transmission is interrupted. \(C_{E}\) is the instantaneous capacity of the eavesdropping link, \(C_{E}=C_{AB}-C_{S}\), defined as \(C_{E}=\frac{1}{2}log_{2}(1+\gamma_{E})\). When \(C_{E}\) greater than the rate difference \(R_{E}=R_{t}-R_{s}\), the message is intercepted and the node E has access to the transmitted confidential information, again indicating an interruption of the secure transmission.  
**Theorem 2**: _Assuming \(R_{e}\) as the threshold for secrecy transmission, let \(\delta_{e}\triangleq 2^{2R_{e}}-1\), the lower bound of the system's secrecy outage probability in an aerial drone eavesdropping scenario is denoted by:_  
\[\begin{array}{l}P_{{\it{\it{\it{\it{\it{\it{\it{\it{\it{\it{\it{\it{ \it{\it{\it{\it{\it{\it{\itit{\itit{\ititit{ \ititit{ \ititit{ \itit{ \it{          }}}}}}}}}}}}}}} {\left.\right.\end} \tag{17}\]  
\[\begin{array}{l}D_{e1}=P_{r}\Big{\{}\gamma_{E}^{(1)}\leq\delta_{e}\Big{\}}= 1-{\it exp}[-K_{ae}-K_{ue}-a_{2}(1+K_{ae})]\cdot\\ \sum\
olimits_{d_{1}=0}^{D_{1}}\sum\
olimits_{u_{1}=0}^{d_{1}}\sum\
olimits_{ s_{1}=0}^{u_{1}}\sum\
olimits_{r_{1}=0}^{R_{1}}\frac{\Gamma(D_{1}+d_{1})D_{1}^{ 1-2d_{1}}\Gamma(R_{1}+r_{1})R_{1}^{1-2r}K_{au}^{d_{1}}(1+K_{au})^{r_{1}+K_{ au}^{d_{1}}(1+K_{au})^{u}}}{\Gamma(D_{1}-d_{1}+1)\Gamma(d_{1}+1)\Gamma(u_{1}-s_{1}+1) \Gamma(s_{1}+1)\Gamma^{2}(r_{1}+1)\Gamma(R_{1}-r_{1}+1)}a_{2}{}^{u_{1}-s_{1}} \cdot\\ a_{1}{}^{s_{1}}\cdot(1+K_{ue}+(1+K_{ae})a_{1})^{-\gamma}\cdot\Gamma(v)\\ \end{array} \tag{18}\]  
_where \(a_{1}=\frac{\delta_{e}P_{u}L_{\omega}}{P_{s}L_{\omega}}\), \(a_{2}=\frac{\delta_{e}N_{0}}{P_{s}L_{\omega}}\), \(v=s_{1}+r_{1}+1\)._\[\begin{array}{l}D_{\ell 2}=P_{r}\Big{\{}\gamma_{E}^{(2)}\leq\delta_{e}\Big{\}}=1-exp[-K_{ au}-K_{ue}-K_{be}-c_{3}(1+K_{au})].\\ \sum_{d_{2}=0}^{D_{2}}\sum_{u_{2}=0}^{d_{2}}\sum_{u_{2}=0}^{u_{2}}\sum_{u_{2}=0}^{ R_{2}}\sum_{u_{2}=0}^{R_{2}}\sum_{u_{2}=0}^{R_{3}}\frac{\Gamma(D_{2}+d_{2})D_{2} ^{1-4}\lambda_{2}}{(D_{2}-d_{2}+1)\Gamma(\lambda_{2}-s_{2}+1)\Gamma(s_{2}+1)}. \Big{(}\frac{c_{1}}{c_{2}}\Big{)}^{s_{2}}\cdot c_{3}u_{2}-s_{2}.\\ \Gamma(v_{1})\cdot k_{ai}^{2}k_{ue}^{r_{2}}K_{be}^{r_{3}}\cdot(1+K_{be})^{r_{3} +1}(1+K_{ue})^{r_{2}}(1+K_{au})^{s_{2}}(1+K_{ue})^{s_{2}-r_{2}}.\\ \frac{\Gamma(R_{2}+r_{2})R_{2}^{s_{1}-s_{2}}}{(r_{2}+1)\Gamma(R_{2}-r_{2}+1)} \cdot\frac{\Gamma(R_{3}+r_{3})R_{3}^{1-2}\lambda_{3}}{(r_{3}+1)\Gamma(R_{3}-r_ {3}+1)}.\bigg{\{}\Big{[}(1+K_{be})-(1+K_{au})\cdot\Big{(}\frac{c_{1}}{c_{2}} \Big{)}\Big{]}^{-(s_{2}+r_{3}+1)}.\\ \Gamma(s_{2}+r_{3}+1)+\tau(1+K_{au})\cdot\Big{(}\frac{c_{1}}{c_{2}}\Big{)}\cdot \Big{[}(1+K_{be})-(1+K_{au})\cdot\Big{(}\frac{c_{1}}{c_{2}}\Big{)}\Big{]}^{-(s _{2}+r_{3}+2)}.\\ \Gamma(s_{2}+r_{3}+2)\big{\}}\end{array} \tag{19}\]  
where \(c_{1}=\delta_{e}[P_{b}L_{be}(1-\beta)+(1-\beta)N_{0}],c_{2}=\eta\beta(1-\beta )P_{a}L_{ae}L_{ue},c_{3}=\delta_{e}\Big{[}\frac{P_{c}S_{u}}{P_{a}^{4}L_{m}}+ \frac{(1-\beta+\xi)N_{0}}{(1-\beta)P_{a}^{4}L_{m}}\Big{]}\), \(\tau=v_{2}(1+K_{ue})^{-1}-1\).  
Since \(max\big{\{}a,b\big{\}}\leq a+b\), a lower bound on the secrecy outage probability can be derived after a series of complex calculations as shown in Equations (17)-(19). Where \(D_{1},D_{2},R_{1},R_{2},R_{3}\) are integers of definite precision, \(\Gamma(\cdot)\) is the gamma function, and \(\gamma_{E}^{(1)}\) and \(\gamma_{E}^{(2)}\) are given by Equations (14) and (15).  
Proof of Theorem 2.: The process is detailed in Appendix B. \(\square\)