# High Reliable Uplink Transmission Methods in GEO-LEO Heterogeneous Satellite Network
## 3 Performance Evaluation
### Outage Probability of NOMA Method  
In downlink NOMA, which has been widely discussed, the weak user's signal should be firstly decoded at the strong user side since the power allocated to the weak user's signal is always high. However, as the signals of beam-center and beam-edge users may experience different channel attenuation in uplink SATCOM, the signal of beam-center user is considered to be demodulated first at the satellite, which is shown in Figure 2.  
Provided a demodulation threshold \(T_{1}\), the outage probability of \(U_{1}\) can be defined as \(Pr_{1}=\mathbb{P}(SINR_{1}<T_{1})\). Let \(\beta=\frac{P}{\sigma^{2}}\). Then, we have the following theorem. **Theorem 1**. : _The outage probability of beam-center user in the uplink LEO NOMA is provided by_  
\[Pr_{1}=\mathbb{P}(SINR_{1}<T_{1})=\frac{1}{\kappa_{2,l}}\sum_{i=1}^{n}\omega_{ i}F(\frac{x_{i}}{\kappa_{2,l}}), \tag{10}\]  
_where \(F(x)=\varepsilon_{1,l}\sum_{k=0}^{\infty}\frac{(m_{1,l})x_{1,l}^{k}}{(k! )^{2}x _{1,l}^{k+1}}\sum_{l=0}^{\infty}\frac{(-1)^{l}}{t(k+1+t)}\Big{(}\frac{T_{1}(k \kappa_{2}\beta+1)}{K_{1}(1-\alpha)\beta}\kappa_{1,l}\Big{)}^{k+1+t} \varepsilon_{2,l1}F_{1}(m_{2,l};1;\
u_{2,l}x)\). Here, \(K_{1}=l^{2}(d_{l})G_{1}G_{l}(\theta_{1})\) and \(K_{2}=l^{2}(d_{l})G_{2}G_{l}(\theta_{2})\), \(x_{i}\) and \(\omega_{i}\) denote the Gauss-Laguerre quadrature nodes and weights over \([0,+\infty]\)[25]._  
Proof. : According to (4), the outage probability of \(U_{1}\) can be calculated as  
\[ \begin{array}{ll}Pr_{1}&=\mathbb{P}(SINR_{1}<T_{1})=\Pr(\frac{|h_{1}|^{2}( 1-\alpha)P}{|h_{2}|^{2}(\alpha P+\sigma^{2}}<T_{1})\\ &=\Pr(|h_{1}|^{2}<\frac{T_{1}(|h_{2}|^{2}\alpha P+\sigma^{2})}{(1-\alpha)P}) \\ &=\Pr(|\rho_{1,l}|^{2}<\frac{T_{1}(|\rho_{2}|^{2}K_{2}\alpha\beta+1)}{K_{1}(1 -\alpha)\beta}).\end{array}  \tag{11}\]  
Since Shadowed Rician channel model is assumed in this paper, (11) can be further expressed as  
\[ \begin{array}{ll}Pr_{1}&=\int_{0}^{\infty}\int_{0}^{\frac{T_{1}(|\rho_{2}|^ {2}K_{2}\alpha\beta+1)}{K_{1}(1-\alpha)\beta}}f_{|\rho_{1}|^{2}}(x)dxf_{|\rho _{2}|^{2}}(y)dy\\ &=\int_{0}^{\infty}\varepsilon_{1,l}\sum_{k=0}^{\infty}\frac{(m_{1,l})x_{1,l}^ {k+1}}{(k!)^{2}x_{1,l}^{k+1}}\gamma(k+1,\frac{T_{1}(yk_{2}\alpha\beta+1)}{K_ {1}(1-\alpha)\beta}\kappa_{1,l})f_{|\rho_{2}|^{2}}(y)dy\\ &=\int_{0}^{\infty}\varepsilon_{1,l}\sum_{k=0}^{\infty}\frac{(m_{1,l})x_{1,l} ^{k+1}}{(k!)^{2}x_{1,l}^{k+1}}\gamma(k+1,\frac{T_{1}(yk_{2}\alpha\beta+1)}{K_ {1}(1-\alpha)\beta}\kappa_{1,l})\varepsilon_{2,l}e^{-\kappa_{2,l}y}{}_{1}F_{1} (m_{2,l};1;\
u_{2,l}y)dy.\end{array}  \tag{12}\]  
Motivated by Equation (8.354.1) in [24], Equation (12) can be further derived as  
Figure 2: The receiver structure of the satellite. \[ \begin{array}{ll}Pr_{1}&=\int_{0}^{\infty}\varepsilon_{1,J}\sum_{k=0}^{\infty} \frac{(m_{1,J})\kappa_{1,J}^{k}}{(k!)^{2}\kappa_{1,J}^{k+1}}\sum_{l=0}^{\infty} \frac{(-1)^{l}}{l!(k+1+l)}\Big{(}\frac{T_{1}(y\kappa_{2}\alpha\beta+1)}{K_{1}(1 -\alpha)\beta}\kappa_{1,J}\Big{)}^{k+1+t}\varepsilon_{2,l}e^{-\kappa_{2,l}y_{ 1}}F_{1}(m_{2,l};1;\
u_{2,l}y)dy\\ &=\int_{0}^{\infty}e^{-\kappa_{2,l}y}\underbrace{\varepsilon_{1,l}\sum_{k=0}^{ \infty}\frac{(m_{1,J})\kappa_{1,J}^{k}}{(k!)^{2}\kappa_{1,J}^{k+1}}\sum_{l=0}^ {\infty}\frac{(-1)^{l}}{t!(k+1+t)}\bigg{(}\frac{T_{1}(yK_{2}\alpha\beta+1)}{K_ {1}(1-\alpha)\beta}\kappa_{1,J}\bigg{)}^{k+1+t}\varepsilon_{2,l1}F_{1}(m_{2,l };1;\
u_{2,l}y)}_{F(y)}dy\\ &\stackrel{{(a)}}{{=}}\frac{1}{\kappa_{2,J}}\sum_{i=1}^{n}\omega_ {i}F_{(\frac{x_{i}}{\kappa_{2,J}})},\end{array}  \tag{13}\]  
where \((a)\) follows Gauss-Laguerre quadrature. The theorem is proved. To obtain the information of \(U_{2}\), the satellite should correctly decode both \(x_{1}\) and \(x_{2}\). In this case, outage probability of \(U_{2}\) happens when the satellite failed to demodulate \(x_{1}\) or \(x_{2}\), i.e., \(Pr_{2}=\mathbb{P}(SINR_{1}<T_{1}\;or\;SNR_{2}<T_{2})\). Then, we can derive the following theorem for the beam-edge user. **Theorem 2**. : _The outage probability of the beam-edge user can be calculated as_  
\[