# Outage Performance of SWIPT-D2D-Based Hybrid Satellite-Terrestrial Networks
## 3 Performance Analysis
### PS-D2D Scheme
#### 3.1.2 Outage Probability of U\({}_{2}\)  
Firstly, we assume that the target rate at \(U_{2}\) is \(R_{p}\) with only the satellite-terrestrial direct link. Thus, the OP of \(U_{2}\) for the satellite-terrestrial direct link is as follows:  
\[\mathrm{P}_{\mathrm{out,PS}}^{\mathrm{su2}}(R_{p})=\mathrm{Pr}\big{[}\log_{2}(1 +\Lambda_{su2,ps})<R_{p}\big{]}=F_{\Lambda_{su2,ps}}(\gamma^{\prime}{}_{p}). \tag{33}\]  
Substituting Equation (22) into Equation (33), we can compute the OP of \(U_{2}\) for the satellite-terrestrial direct link. Secondly, considering that the terrestrial relay link of \(U_{2}\) originates from \(U_{1}\) with the SNR of \(\Lambda_{u1u2,ps}\) in Equation (10), the OP of \(U_{2}\) for the terrestrial relay link is the CDF of \(\Lambda_{u1u2,ps}\). Making use of the PDF of the gamma random variable in reference [24] for Equation (10), we can obtain the PDF of \(\Lambda_{u1u2,ps}\) as follows:  
\[ \begin{split} f_{\Lambda_{u1u2,ps}}(x)&=\int\limits _{0}^{\infty}\frac{1}{\omega(1-\rho)y}f_{\eta_{r}|S_{rn1}|^{2}}\bigg{(}\frac{x }{\omega y-\omega y\rho}\bigg{)}f_{|S_{su2}|^{2}}(y)\mathrm{d}y\\ &=\frac{2(\chi_{1})}{\Gamma(m_{ru1})\Gamma(m_{u1u2})}x\frac{m_{ ru1}+m_{u1u2}}{2}-\chi_{m_{ru1u2}-m_{ru1}}(2\sqrt{\chi_{1}x}),\end{split}  \tag{34}\]  
where \(\chi_{1}=\frac{m_{ru1u2,ps}}{\omega(1-\rho)\Omega_{ru1u2}\Omega_{ru1u2}\eta_{r}}\). Making use of [PERSON]'s G-function in reference [25], we can rewrite the PDF of \(\Lambda_{u1u2,ps}\) as  
\[f_{\Lambda_{u1u2,ps}}(x)=\frac{\chi_{1}}{\Gamma(m_{ru1})\Gamma(m_{u1u2})}G_{0,2}^{2,0}\Bigg{[}\chi_{1}x\Bigg{]} \begin{array}{c}-\\ -\\ \frac{m_{u1u2}-m_{ru1}}{2},-\frac{m_{u1u2}-m_{ru1}}{2}\end{array} \Bigg{]}, \tag{35}\]  
where \(G_{p,q}^{m,u}\left[x\Bigg{|} \begin{array}{c}a_{1},\cdots,a_{p}\\ b_{1},\cdots,b_{q}\end{array} \right]\) is the [PERSON]'s G-function. Making use of the integral formula of [PERSON]'s G-function in reference [26], we can obtain the corresponding CDF of \(\Lambda_{u1u2,ps}\) as follows:  
\[F_{\Lambda_{u1u2,ps}}(x)=\frac{\chi_{1}x}{\Gamma(m_{ru1})\Gamma(m_{u1u2})}G_{1,3}^{2,1}\Bigg{[}\chi_{1}x\Bigg{]} \begin{array}{c}0,\\ \frac{m_{u1u2}-m_{ru1}}{2},-\frac{m_{ru1u2}-m_{ru1}}{2},-1\end{array} \Bigg{]}. \tag{36}\]  
Thirdly, utilizing Equations (9) and (10) for MRC, we have the OP of \(U_{2}\) for the PS-D2D scheme as follows:  
\[ \begin{split}\mathrm{P}_{\mathrm{out,PS}}^{\mathrm{u2}}(R_{p})& =\mathrm{Pr}\bigg{[}\log_{2}(1+\Lambda_{su2,ps})<R_{p},\frac{1}{3}\log_{2}( \Lambda_{su2,ps}+\Lambda_{u1u2,ps})<R_{p}\bigg{]}\\ &=\mathrm{Pr}\Big{[}\Lambda_{su2,ps}<\gamma^{\prime}{}_{p^{\prime}} \Lambda_{su2,ps}+\Lambda_{u1u2,ps}<\gamma_{p}\Big{]}\\ &=\mathrm{Pr}\Big{[}\Lambda_{su2,ps}<\mathrm{min}(\gamma_{p}-\Lambda_{u1u2,ps},\gamma^{\prime}{}_{p})\Big{]}\\ &=\phi_{3}+\phi_{4},\end{split}  \tag{37}\]  
where \(\phi_{3}\) denotes the probability of case 3, and \(\phi_{4}\) denotes the probability of case 4. The probability of case 3 is as follows:  
\[\phi_{3}=\mathrm{Pr}\Big{[}\Lambda_{su2,ps}<\gamma^{\prime}{}_{p},\gamma^{ \prime}{}_{p}<\gamma_{p}-\Lambda_{su2,ps}\Big{]}=F_{\Lambda_{su2,ps}}(\gamma^ {\prime}{}_{p})F_{\Lambda_{su2,ps}}(\gamma_{p}-\gamma^{\prime}{}_{p}). \tag{38}\]Substituting Equations (33) and (36) into Equation (38), we can obtain \(\phi_{3}\). The probability of case 4 is as follows:  
\[ \begin{split}\phi_{4}&=\Pr\Bigl{[}\Lambda_{su2,ps}< \gamma_{p}-\Lambda_{u1u2,ps},\gamma^{\prime}{}_{p}\geq\gamma_{p}-\Lambda_{u1u2, ps}\Bigr{]}\\ &=\Pr\Bigl{[}\Lambda_{su2,ps}<\gamma_{p}-\Lambda_{u1u2,ps}, \Lambda_{u1u2,ps}\geq\gamma_{p}-\gamma^{\prime}{}_{p}\Bigr{]}\\ &=\int_{0}^{\gamma_{p}}\int_{\gamma_{p}-\gamma^{\prime}{}_{p}}^{ \gamma_{p}-y}f_{\Lambda_{u1u2,ps}}(x)f_{\Lambda_{su2,ps}}(y)\mathrm{d}x\mathrm{ d}y\\ &=\int_{0}^{\gamma_{p}}F_{\Lambda_{u1u2,ps}}(\gamma_{p}-y)f_{ \Lambda_{su2,ps}}(y)\mathrm{d}y-\int_{0}^{\gamma_{p}}F_{\Lambda_{u1u2,ps}}( \gamma_{p}-\gamma^{\prime}{}_{p})f_{\Lambda_{su2,ps}}(y)\mathrm{d}y.\end{split}