# Performance Evaluation of a Full-Duplex Relaying-Enabled Satellite Sensor Network
## 3 Performance Evaluation
### Outage Probability  
The quality of satellite services can be determined by using the outage probability, which is commonly defined as the probability that the received SINR falls below a predefined threshold \(\gamma_{th}\), namely [28]  
\[P_{out}=\Pr\left(\gamma\leq\gamma_{th}\right)=F_{\gamma}\left(\gamma_{th} \right), \tag{19}\]  
where \(F_{\gamma}\left(\cdot\right)\) denotes the cumulative distribution function (CDF) of \(\gamma\). From (9), \(F_{\gamma}\left(x\right)\) can be expressed as  
\[F_{\gamma}\left(x\right)=\int_{0}^{\left(\gamma_{l}+1\right) \gamma_{th}}f_{\gamma_{1}}\left(y\right)dy\int_{0}^{\infty}f_{\gamma_{2}}\left( x\right)dx\] \[+\int_{\left(\gamma_{l}+1\right)\gamma_{th}}^{\infty}F_{\gamma_{ 2}}\left(\frac{\left(y+\bar{\gamma}_{l}+1\right)\gamma_{th}}{y-\left(\bar{ \gamma}_{l}+1\right)\gamma_{th}}\right)f_{\gamma_{1}}\left(y\right)dy. \tag{20}\]  
For simplicity, by assuming the Nakagami parameter \(m_{i}\) in (13) takes on integer values [21; 23], and defining \(\bar{\gamma}_{i}=\frac{P_{i}C_{i}}{\sigma_{i}^{2}}\), \(f_{\gamma_{i}}\left(x\right)\) can be simplified as  
\[f_{\gamma_{i}}\left(x\right)=\alpha_{i}\exp\left(-\frac{\left(\beta_{i}-\delta _{i}\right)}{\bar{\gamma}_{i}}x\right)\sum_{k_{i}=0}^{m_{i}-1}\Lambda\left(k_{ i},\bar{\gamma}_{i}\right)x^{k_{i}}, \tag{21}\]  
with  
\[\Lambda\left(k_{i},\bar{\gamma}_{i}\right)=\frac{\left(-1\right)^{k_{i}}\left( 1-m_{i}\right)_{k}\delta_{i}^{k_{i}}}{\left(k_{i}\right)^{2}\bar{\gamma}_{i}^ {k_{i}+1}}.\]  
Then, by applying ([27], Equation (3.351.2)) along with integration computation, \(F_{\gamma_{l}}\left(x\right)\) can be obtained as  
\[F_{\gamma_{l}}\left(x\right) =\int_{0}^{x}f_{\gamma_{l}}\left(y\right)dy\] \[=1-\alpha_{i}\exp\left(-\frac{\left(\beta_{i}-\delta_{i}\right)x} {\bar{\gamma}_{i}}\right)\sum_{k_{i}=0}^{m_{i}-1}\sum_{l_{i}=0}^{k_{i}}\Xi \left(k_{i},l_{i},\bar{\gamma}_{i}\right)x^{l_{i}}, \tag{22}\]\[F_{\gamma}\left(x\right) =1-\alpha_{1}\alpha_{2}\sum_{k_{1}=0}^{m_{1}-1}\Lambda\left(k_{1}, \tilde{\gamma}_{1}\right)\sum_{k_{2}=0}^{m_{2}-1}\sum_{j_{2}=0}^{k_{2}}\Xi \left(k_{2},l_{2},\tilde{\gamma}_{2}\right)\] \[\times\int_{\left(\gamma_{t}+1\right)\gamma_{th}}^{\infty}y^{k_{1 }}\left(\frac{\left(y+\tilde{\gamma}_{1}+1\right)\gamma_{th}}{y-\left(\tilde{ \gamma}_{t}+1\right)\gamma_{th}}\right)^{l_{2}}\exp\left(-\frac{\left(\beta_{ 2}-\delta_{2}\right)}{\tilde{\gamma}_{2}}\frac{\left(y+\tilde{\gamma}_{1}+1 \right)\gamma_{th}}{y-\left(\tilde{\gamma}_{t}+1\right)\gamma_{th}}-\frac{ \left(\beta_{1}-\delta_{1}\right)}{\tilde{\gamma}_{1}}y\right)dy, \tag{23}\]  
\[F_{\gamma}\left(x\right) =1-2\alpha_{1}\alpha_{2}\sum_{k_{1}=0}^{m_{1}-1}\Lambda\left(k_{ 1},\tilde{\gamma}_{1}\right)\sum_{k_{2}=0}^{m_{2}-1}\sum_{l_{2}=0}^{k_{2}}\Xi \left(k_{2},l_{2},\tilde{\gamma}_{2}\right)\exp\left(-\left(\mathcal{B}+ \mathcal{AC}\right)\gamma_{th}\right)\] \[\times\sum_{i=0}^{k_{1}}\left( \begin{array}{c}k_{1}\\ i\end{array} \right)\sum_{j=0}^{l_{2}}\left( \begin{array}{c}l_{2}\\ j\end{array} \right)\frac{\mathcal{C}^{k_{1}-i+j}\mathcal{D}^{j}}{\gamma_{th}^ {i-k_{1}-l_{2}}}\left(\frac{\mathcal{BCD}\gamma_{th}}{\mathcal{A}}\right)^{ \frac{i-j+1}{2}}K_{i-j+1}\left(2\sqrt{\mathcal{ABCDD}\gamma_{th}}\right), \tag{24}\]  
with  
\[\Xi\left(k_{i^{\prime}},l_{i^{\prime}},\tilde{\gamma}_{i}\right)=\frac{\left( -1\right)^{k_{i}}\left(1-m_{i}\right)_{k_{j}}\delta_{i}^{k_{i}}}{k_{i}!l_{i}! \left(\beta_{i}-\delta_{i}\right)^{k_{i}-l_{i}+1}\tilde{\gamma}_{i}^{l_{i}}}.\]  
Substituting (21) and (22) into (20), we get (23) as shown on the top of next page. Then, by letting \(z=y-\left(\tilde{\gamma}_{I}+1\right)\gamma_{th}\), and \(\mathcal{A}=\frac{\left(\beta_{1}-\delta_{1}\right)}{\tilde{\gamma}_{1}}\), \(\mathcal{B}=\frac{\left(\beta_{2}-\delta_{2}\right)}{\tilde{\gamma}_{2}}\), \(\mathcal{C}=\tilde{\gamma}_{I}+1\), \(\mathcal{D}=\gamma_{th}+1\) and applying ((27], Equation (3.471.9)), the closed-form expression of \(F_{\gamma}\left(x\right)\) can be derived as (21).