# Performance Evaluation of a Full-Duplex Relaying-Enabled Satellite Sensor Network
## 3 Performance Evaluation
### Ergodic Capacity
Due to the independent nature of \(\tilde{\gamma}_{1}\) and \(\gamma_{2}\), we have the following property of \(M_{\gamma_{3}}\left(s\right)\) as  
\[M_{\gamma_{3}}\left(s\right)= M_{\tilde{\gamma}_{1}}\left(s\right)M_{\gamma_{2}}\left(s\right), \tag{34}\]  
and then the \(M_{\gamma_{3}}^{\left(1\right)}\left(s\right)\) can be expressed as  
\[M_{\gamma_{3}}^{\left(1\right)}\left(s\right) =\left[M_{\tilde{\gamma}_{1}}\left(s\right)M_{\gamma_{2}}\left(s \right)\right]^{\left(1\right)}\] \[=M_{\tilde{\gamma}_{1}}^{\left(1\right)}\left(s\right)M_{\gamma_{ 2}}\left(s\right)+M_{\tilde{\gamma}_{1}}\left(s\right)M_{\gamma_{2}}^{\left(1 \right)}\left(s\right). \tag{35}\]  
According to the definition of MGF, \(M_{\tilde{\gamma}_{1}}\left(s\right)\) and \(M_{\gamma_{2}}\left(s\right)\) can be derived as  
\[M_{\tilde{\gamma}_{1}}\left(s\right)=\alpha_{1}\sum_{k_{1}=0}^{m_{1}-1}\Lambda \left(k_{1},\tilde{\gamma}_{1}\right)\frac{k_{1}!}{\left(s+\mathcal{AC} \right)^{k_{1}+1}}, \tag{36}\]  
and  
\[M_{\gamma_{2}}\left(s\right)=\alpha_{2}\sum_{k_{2}=0}^{m_{2}-1}\Lambda\left(k _{2},\tilde{\gamma}_{2}\right)\frac{k_{2}!}{\left(s+\mathcal{B}\right)^{k_{2} +1}}. \tag{37}\]Then, by differentiating (33) and (34) with respect to \(s\), the analytical results of \(M_{\tilde{\gamma}_{1}}^{\left(1\right)}\left(s\right)\) and \(M_{\gamma_{2}}^{\left(1\right)}\left(s\right)\) can be obtained as  
\[M_{\tilde{\gamma}_{1}}^{\left(1\right)}\left(s\right) = -\alpha_{1}\sum_{k_{1}=0}^{m_{1}-1}\Lambda\left(k_{1},\tilde{ \gamma}_{1}\right)\frac{\left(k_{1}+1\right)!}{\left(s+\mathcal{A}C\right)^{k _{1}+2}}, \tag{38}\]  
and  
\[M_{\gamma_{2}}^{\left(1\right)}\left(s\right) = -\alpha_{2}\sum_{k_{2}=0}^{m_{2}-1}\Lambda\left(k_{2},\tilde{ \gamma}_{2}\right)\frac{\left(k_{2}+1\right)!}{\left(s+\mathcal{B}\right)^{k _{2}+2}}. \tag{39}\]  
Subsequently, by combining (33) and (35), \(C_{3}\) can be further expressed as  
\[C_{3} = \frac{1}{\ln 2}\left[\int_{0}^{\infty}Ei\left(-s\right)M_{ \tilde{\gamma}_{1}}^{\left(1\right)}\left(s\right)M_{\gamma_{2}}\left(s \right)ds\right. \tag{40}\] \[\left.+\int_{0}^{\infty}Ei\left(-s\right)M_{\tilde{\gamma}_{1}} \left(s\right)M_{\gamma_{2}}^{\left(1\right)}\left(s\right)ds\right].\]  
\[M_{\gamma_{3}}\left(s\right)= M_{\tilde{\gamma}_{1}}\left(s\right)M_{\gamma_{2}}\left(s\right), \tag{34}\]0 \[\left.\times\underbrace{\int_{0}^{\infty}\mathrm{G}_{1,1}^{1,1} \left[\left(\tilde{\gamma}_{1}+1\right)\mathcal{A}s\left| \begin{array}{c}-k _{1}-1\\ 0\end{array} \right. }_{I_{1}}\right]\mathrm{G}_{1,1}^{1,1}\left[\mathcal{A}x \left| \begin{array}{c}-k_{2}\\ 0\end{array} \right. }_{I_{2}}\right]\mathrm{G}_{1,2}^{2,0}\left[s\left| \begin{array} []{c}1\\ 0,0\end{array} \right.\right]ds\] \[+ \underbrace{\frac{1}{\mathcal{A}^{k_{2}+1}}\int_{0}^{\infty} \mathrm{G}_{1,1}^{1,1}\left[\left(\tilde{\gamma}_{1}+1\right)\mathcal{A}s \left| \begin{array}{c}-k_{1}\\ 0\end{array} \right. }_{I_{2}}\right]\mathrm{G}_{1,1}^{1,1}\left[\mathcal{A}x \left| \begin{array}{c}-k_{2}-1\\ 0\end{array} \right. }_{I_{2}}\right]\mathrm{G}_{1,2}^{2,0}\left[s\left| \begin{array} []{c}1\\ 0,0\end{array} \right.\right]ds\]  
Then, substituting (36)-(39) into (40) yields (41). To solve the associated multiple integrals, we then apply the following Meijer-G function representations  
\[\left(1+\alpha x\right)^{\beta} = \frac{1}{\Gamma\left(-\beta\right)}\mathrm{G}_{1,1}^{1,1}\left[ \alpha x\left| \begin{array}{c}\beta+1\\ 0\end{array} \right.\right], \tag{42}\]  
and  
\[Ei\left(-s\right) = -\mathrm{G}_{1,2}^{2,0}\left[s\left| \begin{array}{c}1\\ 0,0\end{array} \right.\right], \tag{43}\]  
along with ([32], Equation (3.1)), and derive the closed-form expression of \(C_{3}\) as (44). Finally, by substituting (31), (32) and (40) into (26), one can directly calculate the ergodic capacity of the FD satellite relaying system as (45). Figure 3 depicts the outage probability of FD satellite relaying system for different off-boresight beam angles \(\varphi_{1}=\varphi_{2}=\varphi\) and elevation angles \(\theta_{1}=\theta_{1}=\theta\).