# Security Performance Analysis of Full-Duplex UAV Assisted Relay System Based on SWIPT Technology
## Appendix B Safety Outage Probability (SOP) Derivation Procedure:  
From Equation (17), the derivation of \(P_{sop}\) requires the separate computation of the two parts \(D_{e1}\) and \(D_{e2}\), so:  
\[P_{sop}=1-D_{e1}\cdot D_{e2} \tag{18}\]  
Defining \(V=S_{ae}\), \(W=S_{ue}\), it follows:  
\[D_{e1}=P_{r}\bigg{\{}\frac{P_{a}L_{ae}V}{P_{b}L_{ae}W+N_{0}}\leq\delta_{e}\bigg{\}} =P_{r}\bigg{\{}V\leq\frac{\delta_{e}P_{a}L_{ae}}{P_{b}L_{ae}}W+\frac{\delta_{e} N_{0}}{P_{a}L_{ae}}\bigg{\}}=E_{w}\bigg{\{}F_{w|w}\bigg{(}\frac{\delta_{e}P_{a}L_{ ae}}{P_{b}L_{ae}}w+\frac{\delta_{e}N_{0}}{P_{a}L_{ae}}\bigg{)}\bigg{\}} \tag{19}\]  
Defining \(a_{1}=\frac{\delta_{e}P_{a}L_{ae}}{P_{a}L_{ae}}\), \(a_{2}=\frac{\delta_{e}N_{0}}{P_{a}L_{ae}}\), Equation (19) reduces to:\[ \begin{split} D_{e1}=E_{w}\Big{\{}F_{v|w}(a_{1}w+a_{2})\Big{\}}=\int_{ 0}^{\infty}F_{ae}(a_{1}w+a_{2})f_{ae}(w)dw\\ =\int_{0}^{\infty}\Bigl{[}1-\Omega\Bigl{(}\sqrt{2K_{ae}},\sqrt{2(1+ K_{ae})(a_{1}w+a_{2})}\Bigr{)}\Bigr{]}f_{ae}(w)dw\\ =\int_{0}^{\infty}f_{ae}(w)dw-\int_{0}^{\infty}Q\Bigl{(}\sqrt{2K_{ae }},\sqrt{2(1+K_{ae})(a_{1}w+a_{2})}\Bigr{)}\cdot f_{ae}(w)dw\end{split}  \tag{10}\]  
Since the derivation of Equation (10) is similar to that of Theorem 1, the derivation will not be repeated, and the expression for \(D_{e1}\) is finally obtained according to (Equation (3.481.4) [24]) and is given in Equation (18), where \(v=s_{1}+r_{1}+1\). Since \(X=S_{au},W=S_{ue}\), here \(Z=S_{be}\) is defined, and the expression (15) for \(\gamma_{E}^{(2)}\) is obtained by substituting it into Equation (19):  
\[D_{e2}=P_{r}\Big{\{}\gamma_{E}^{(2)}\leq\delta_{e}\Big{\}}=P_{r}\bigg{\{}X \leq\frac{\delta_{e}P_{u}S_{u}}{P_{a}L_{au}}+\frac{N_{0}(1-\beta+\zeta)\delta_ {e}}{(1-\beta)P_{a}L_{au}}+\frac{\delta_{e}P_{b}L_{be}Z+N_{0}\delta_{e}}{\eta \beta P_{a}L_{au}L_{ue}W}\bigg{\}} \tag{11}\]  
Defining \(c_{1}=\delta_{e}P_{b}L_{be}c_{2}=N_{0}\delta_{e}c_{3}=\eta\beta P_{a}L_{au}L_ {ue}c_{4}=\delta_{e}\Bigl{[}\frac{P_{u}S_{u}}{P_{a}L_{au}}+\frac{(1-\beta+ \zeta)N_{0}}{(1-\beta)P_{a}L_{au}}\Bigr{]}\), Equation (11) is further simplified to obtain:  
\[D_{e2}=E_{z}\bigg{\{}E_{w|z}\bigg{\{}F_{x|w,z}\bigg{(}\frac{c_{1}z+c_{2}}{c_{ 3}w}+c_{4}\bigg{)}\bigg{\}}\bigg{\}}=1-\int_{0}^{\infty}\psi_{x}(z)f_{z}(z)dz \tag{12}\]  
where \(\psi_{x}(z)\) is represented by Equation (13). \[ \begin{split}&\psi_{x}(z)=\int_{0}^{\infty}Q\Biggl{(}\sqrt{2K_{ au}},\sqrt{2(1+K_{au})\bigg{(}\frac{c_{1}z+c_{2}}{c_{3}w}+c_{4}\bigg{)}}f_{ae}(w)dw\\ &=\sum\limits_{d_{2}=0}^{D_{2}}\alpha_{u2}^{d_{2}}\sum\limits_{ s_{2}=0}^{u_{2}}\sum\limits_{s_{3}=0}^{u_{2}}\frac{\Gamma(D_{2}+d_{2})D_{2} ^{1-2d_{2}}}{\Gamma(D_{2}-d_{2}+1)\Gamma(d_{2}+1)\Gamma(u_{2}-s_{2}+1)\Gamma( s_{2}-s_{3}+1)\Gamma(s_{3}+1)}\\ &\cdot exp\bigl{(}-K_{au}-K_{ue}-(1+K_{au})c_{4}\bigr{)}\cdot c_{ 1}s_{3}c_{2}s_{3}c_{3}s_{2}s_{3}c_{4}s_{4}s_{4}s_{2}s_{2}\cdots 1\int_{0}^{\infty}w^{-s_{2}}\\ &\cdot exp\Bigl{[}-(1+K_{ue})w-(1+K_{au})\bigg{(}\frac{c_{1}z+c _{2}}{c_{3}w}\bigg{)}\cdot w^{-1}\Bigr{]}\cdot l_{0}\Bigl{(}2\sqrt{K_{ue}(K_{ ue}+1)w}\Bigr{)}dw\end{split}  \tag{13}\]  
Let the integral in Equation (13) be \(DD\), and Equation (14) is obtained by simplification according to the Taylor series formula and the equation (Equation (3.381.4) [24]). \[ \begin{split}& DD=\int_{0}^{\infty}w^{-s_{2}}\cdot exp\Bigl{[}-(1+K_ {au})w-(1+K_{au})\bigg{(}\frac{c_{1}z+c_{2}}{c_{3}w}\bigg{)}\cdot w^{-1}\Bigr{]} \cdot l_{0}\Bigl{(}2\sqrt{K_{ue}(K_{ue}+1)w}\Bigr{)}dw\\ &=\sum\limits_{r_{2}=0}^{R_{1}}\frac{\Gamma(R_{2}+r_{2})R_{2}^{1- r_{2}}K_{ue}^{r_{2}}\cdot(1+K_{ae}^{r_{2}})^{r_{2}}\cdot\Gamma^{\Omega}}{ \Gamma^{2}(r_{2}+1)\Gamma(R_{2}-r_{2}+1)}\left[(1-\Omega)(1+K_{au})^{-r_{1}} \cdot\Gamma(v_{1})+\Omega(1+K_{au})^{-r_{2}}\right.\\ &\cdot\Gamma(v_{2})\Bigr{]}\end{split}  \tag{14}\]  
where \(\Omega=(1+K_{au})\Big{(}\frac{c_{1}}{c_{3}}z+\frac{c_{2}}{c_{3}}\Big{)}\). Substituting Equations (13) and (14) into Equation (12) and further simplifying according to equation (Equation (3.381.4) [24]), the final expression for SOP can be obtained as shown in Equation (17).