# Outage Performance of SWIPT-D2D-Based Hybrid Satellite-Terrestrial Networks
## 3 Performance Analysis
### Ts-D2d Scheme
#### 3.2.1 Outage Probability of \(\mathrm{U}_{1}\)  
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:  
\[ \begin{split}\mathrm{P}^{\mathrm{s}nl}_{\mathrm{out,TS}}(R_{p}) =\Pr\Bigl{[}\log_{2}(1+\Lambda_{su1,ts})<R_{p}\Bigr{]}=F_{\Lambda_{su1,ts}}( \gamma^{\prime}{}_{p}).\end{split}  \tag{41}\]  
Substituting Equation (22) into Equation (41), 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:  
\[ \begin{split}\mathrm{P}^{\mathrm{s}nl}_{\mathrm{out,TS}}(R_{p}) =\Pr\Bigl{[}\frac{1-\tau}{3}\log_{2}(1+\Lambda_{srul,ts})<R_{p}\Bigr{]}=F_{ \Lambda_{su1,ts}}(\gamma_{p,ts}),\end{split}  \tag{42}\]  
where \(\gamma_{p,ts}=2^{(3R_{p})/(1-\tau)}-1\). Substituting Equation (14) into Equation (42), we can present the CDF of \(\Lambda_{srul,ts}\) as follows:  
\[ \begin{split} F_{\Lambda_{srul,ts}}(x)=\Pr\biggl{[}\Lambda_{sr, ts}<\frac{x(\Lambda_{ru1,ts}+1)}{\Lambda_{ru1,ts}-x}\biggr{]}=1-\int_{x}^{ \infty}\overline{F}_{\Lambda_{sr,ts}}\biggl{(}\frac{xy+x}{y-x}\biggr{)}f_{ \Lambda_{ru1,ts}}(y)\mathrm{d}y,\end{split}  \tag{43}\]  
where \(\overline{F}_{\Lambda_{sr,ts}}(\cdot)=1-F_{\Lambda_{sr,ts}}(\cdot)\) denotes the complementary CDF of \(\Lambda_{sr,ts}\). Substituting Equations (22) and (23) into Equation (43), and simplifying using binomial expansion, we can obtain the CDF of \(\Lambda_{srul,ts}\) as follows:\[ \begin{split} F_{\Lambda_{\text{srnl,ls}}}(x)&=1-2\alpha _{r}\sum\limits_{\kappa=0}^{m_{r}-1}\sum\limits_{l=0}^{\kappa}\sum\limits_{m=0}^ {l}\sum\limits_{\beta=0}^{m+m_{r}-1}\frac{(-1)^{\kappa}(1-m_{sr})_{\kappa}g_{r} ^{\kappa}}{(\eta_{s})^{\kappa+1}\kappa l!}(\frac{\beta_{r}-\delta_{r}}{\eta_{s }})^{-(\kappa+1-l)}\\ &\times\left(\frac{m_{r\mu 1}}{\Omega_{r\mu 1}\eta_{r}}\right)^{m_{r \mu 1}}\frac{x^{m+m_{r}-1-g}}{\Gamma(m_{r\mu 1})}\left(\frac{m}{m}\right)\left( \frac{m+m_{r\mu 1}-1}{g}\right)\\ &\times e^{-(\frac{\beta_{r}-\delta_{r}}{\eta_{s}}+\frac{m_{r\mu 1 }}{\Omega_{r\mu 1}\eta_{r}})^{x}}\left(\frac{\beta_{r}-\delta_{r}}{\eta_{s}} \frac{\Omega_{r\mu 1}\eta_{r}(x^{2}+x)}{m_{r\mu 1}}\right)\frac{g-l+1}{2}x^{l}\\ &\times\mathcal{K}_{g-l+1}(2\sqrt{\frac{\beta_{r}-\delta_{r}}{ \eta_{s}}}\frac{m_{r\mu 1}(x^{2}+x)}{\Omega_{r\mu 1}\eta_{r}}).\end{split}  \tag{44}\]  
After substituting Equation (44) into Equation (42) with the threshold data rate \(\gamma_{p,ls}\), we can obtain the OP of \(U_{1}\) for the terrestrial relay link. Thirdly, utilizing Equations (13) and (14) for MRC, we have the OP of \(U_{1}\) for TS-D2D scheme as follows:  
\[ \begin{split}\text{P}_{\text{out,TS}}^{\text{u1}}(R_{p})& =\text{Pr}\left[\log_{2}(1+\Lambda_{\text{su1,ts}})<R_{p},\frac{1- \tau}{3}\log_{2}(\Lambda_{\text{su1,ts}}+\Lambda_{\text{su1,ts}})<R_{p}\right] \\ &=\text{Pr}\left[\Lambda_{\text{su1,ts}}<\gamma^{\prime}_{p^{ \prime}}\Lambda_{\text{su1,ts}}+\Lambda_{\text{su1,ts}}<\gamma_{p,ts}\right] \\ &=\text{Pr}\left[\Lambda_{\text{su1,ts}}<\text{min}(\gamma_{p,ts} -\Lambda_{\text{su1,ts}},\gamma^{\prime}_{p})\right]\\ &=\varphi_{1}+\varphi_{2},\end{split}  \tag{45}\]  
where \(\varphi_{1}\) denotes the probability of case 1, and \(\varphi_{2}\) denotes the probability of case 2. The probability of case 1 is given by  
\[\varphi_{1}=\text{Pr}\left[\Lambda_{\text{su1,ts}}<\gamma^{\prime}_{p^{\prime} },\gamma^{\prime}_{p}<\gamma_{p,ts}-\Lambda_{\text{su1,ts}}\right]=F_{\Lambda_ {\text{su1,ts}}}(\gamma^{\prime}_{p})F_{\Lambda_{\text{su1,bs}}}(\gamma_{p,ts} -\gamma^{\prime}_{p}). \tag{46}\]  
Substituting Equations (41) and (42) into Equation (46), we can obtain \(\varphi_{1}\). The probability of case 2 is given by  
\[