# Outage Performance of SWIPT-D2D-Based Hybrid Satellite-Terrestrial Networks
## 3 Performance Analysis
### PS-D2D Scheme
#### 3.1.1 Outage Probability of U\({}_{1}\)  
Firstly, we assume that the target rate at \(U_{1}\) is \(R_{p}\) with only the satellite-terrestrial direct link. Thus, the OP of \(U_{1}\) for the satellite-terrestrial direct link is as follows:  
\[\mathrm{P}_{\mathrm{out,PS}}^{\mathrm{sul}}(R_{p})=\Pr\bigl{[}\log_{2}(1+ \Lambda_{su1,ps})<R_{p}\bigr{]}=F_{\Lambda_{su1,ps}}(\gamma^{\prime}_{p}), \tag{25}\]  
where \(\gamma^{\prime}_{p}=2^{R_{p}}-1\). Substituting Equation (22) into Equation (25), we can compute the OP of \(U_{1}\) for the satellite-terrestrial direct link. Secondly, we assume that the target rate at \(U_{1}\) is \(R_{p}\) with only the terrestrial relay link. Thus, the OP of \(U_{1}\) for the terrestrial relay link is as follows:  
\[\mathrm{P}_{\mathrm{out,PS}}^{\mathrm{sul}}(R_{p})=\Pr\Bigl{[}\frac{1}{3}\log_ {2}(1+\Lambda_{su1,ps})<R_{p}\Bigr{]}=F_{\Lambda_{su1,ps}}(\gamma_{p}), \tag{26}\]  
where \(\gamma_{p}=2^{3R_{p}}-1\), and \(1/3\) results from the three time slots to complete the transmission process from \(S\) to \(U_{1}\). Substituting Equation (6) into Equation (26), we can present the CDF of \(\Lambda_{su1,ps}\) as follows:\[F_{\Lambda_{svt,ps}}(x)=\Pr\left[\Lambda_{sr,ps}<\frac{x(\rho^{\prime}\Lambda_{svt,ps}+1)}{\rho^{\prime}\Lambda_{svt,ps}-x}\right]=1-\int_{\frac{x}{\rho^{\prime} }}^{\infty}\overline{F}_{\Lambda_{svt,ps}}\left(\frac{\rho^{\prime}xy+x}{\rho^ {\prime}y-x}\right)f_{\Lambda_{svt,ps}}(y)\mathrm{d}y, \tag{27}\]  
where \(\overline{F}_{\Lambda_{svt,ps}}(\cdot)=1-F_{\Lambda_{svt,ps}}(\cdot)\) denotes the complementary CDF of \(\Lambda_{sr,ps}\). Substituting Equations (22) and (23) into Equation (27), and simplifying using binomial expansion in reference [22], we can obtain the CDF of \(\Lambda_{svt1,ps}\) as follows:  
\[ \begin{split} F_{\Lambda_{svt1,ps}}(x)=1-2\alpha_{r}\sum\limits_ {k=0}^{m_{rt}-1}\sum\limits_{l=0}^{x}\sum\limits_{m=0}^{l}\sum\limits_{ \begin{subarray}{c}s=0\\ \end{subarray}}^{l}\frac{m+m_{\mathrm{cb}s}-1}{\sum\limits_{\begin{subarray}{ c}s=0\\ \end{subarray}}^{(m+m_{\mathrm{cb}s}-1)}}\frac{\left(-1\right)^{\kappa}\left(1-m_{sr} \right)_{\kappa}\delta_{r}^{\kappa}}{\eta_{s}}\frac{\left(\frac{\rho_{r}- \delta_{r}}{\eta_{s}}\right)^{-\left(\kappa+1-l\right)}}{\left(\eta_{s}\right) ^{\kappa+1}\kappa!t!}\left(\frac{\beta_{r}-\delta_{r}}{\eta_{s}}\right)^{- \left(\kappa+1-l\right)}\\ \times e^{-\left(\frac{\beta_{r}-\delta_{r}}{\eta_{s}}\right)^{ \kappa}}+\frac{m_{r1}}{\rho^{\prime}\Omega_{r1}\eta_{r}}^{x}\left(\frac{\beta _{r}-\delta_{r}}{\eta_{s}}\frac{\Omega_{r1}\eta_{r}(x^{2}+\rho^{\prime}x)}{m_ {r1}}\right)\frac{g-l+1}{2}x^{l}\\ \times\mathcal{K}_{g-l+1}(\frac{2}{\rho^{\prime}}\sqrt{\frac{\beta_{r}- \delta_{r}}{\eta_{s}}}\frac{m_{r1}\left(x^{2}+\rho^{\prime}x\right)}{\rho^{ \prime}\Omega_{r1}\eta_{r}}),\end{split}  \tag{28}\]  
where \(\mathcal{K}_{\varphi}(\cdot)\) is the modified Bessel function of the second kind. After substituting Equation (28) into Equation (26) with the threshold data rate \(\gamma_{p}\), we can obtain the OP of \(U_{1}\) for the terrestrial relay link. Thirdly, utilizing Equations (5) and (6) for MRC, we have the OP of \(U_{1}\) for PS-D2D scheme as follows:  
\[ \begin{split}\mathrm{Pu}_{\mathrm{out,PS}}^{\mathrm{u1}}(R_{p})& =\Pr\left[\log_{2}(1+\Lambda_{svt1,ps})<R_{p},\frac{1}{3}\log_{2}( \Lambda_{svt1,ps}+\Lambda_{svt1,ps})<R_{p}\right]\\ &=\Pr\left[\Lambda_{svt1,ps}<\gamma^{\prime}_{p^{\prime}}\Lambda_ {svt1,ps}+\Lambda_{svt1,ps}<\gamma_{p}\right]\\ &=\Pr\left[\Lambda_{svt1,ps}<\min(\gamma_{p}-\Lambda_{svt1,ps}, \gamma^{\prime}_{p})\right]\\ &=\phi_{1}+\phi_{2},\end{split}  \tag{29}\]  
where \(\phi_{1}\) denotes the probability of case 1, and \(\phi_{2}\) denotes the probability of case 2.