# Performance Evaluation of a Full-Duplex Relaying-Enabled Satellite Sensor Network
## 3 Performance Evaluation
### Satellite Channel Model  
By considering the on-board beam gain, pathloss, channel fading and ES antenna gain, \(h_{i},\left(i\in\{1,2\}\right)\) can be written as [22]  
\[h_{i}=G_{i}^{\frac{1}{2}}g_{i}, \tag{13}\]  
with [23] \begin{table}
\begin{tabular}{c c} \hline \hline
**Error Probability \(\varepsilon\)** & **Cancellation Performance** \\ \hline
0.5\% & 46 dB \\
1\% & 40 dB \\
2\% & 34 dB \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Echo residual interference. \[G_{i}=LG_{\text{U}}G_{\text{S, max}}\left(\frac{I_{1}\left(u_{i}\right)}{2u_{i}}+36 \frac{I_{3}\left(u_{i}\right)}{u_{i}^{3}}\right)^{2}, \tag{14}\]  
where the above parameters are defined below [24]:  
* \(L\): Free space loss between for the uplink and downlink computed as \[L=\left(\frac{c}{4\pi df_{c}}\right)^{2},\] where \(c\) is the propagation speed, \(f_{c}\) the frequency and \(d\) the distance, and \(d\) = 35,786 km. * \(G_{\text{U}}\): Antenna gain at the ES-\(i\), (\(i\in\{1,2\}\)). * \(G_{\text{S, max}}\): Maximum beam gain at the on-board antenna boresight. * \(u_{i}\): Auxiliary variable in determining the on-board beam gain factor for a given a ES's position, which is defined as [25] \[u_{i}=2.07123\frac{\sin\varphi_{i}}{\sin\varphi_{\text{3 dB}}},\] (15) where \(\varphi_{i}\) and \(\varphi_{\text{3 dB}}\) represents the beam angle with respect to the beam center and 3-dB angle, respectively. * \(g_{i}\): Fading channel coefficient of the satellite links. Among the different atmosphere effects, rain attenuation is regarded as the major impairment which is commonly described as a log-normal distribution. However, in a practical scenario of the existing literature, the application of log-normal distribution in modeling the shadowing fading would lead to a quite complicated expressions for characterizing the key merits of both the first- and second-order statistical properties [26]. On the other hand, the shadowed-Rician model proposed originally in [26], which adopts the gamma distribution to approximate the log-normal distribution, can result in a simpler form for channel statistics with the similar performance for practical cases. As can be found in the existing literature, the shadowed-Rician distribution can be applied in difference frequency bands, including UHF-band, L-band, Ku-band, Ka-band and etc [7; 8; 9]. Under this situation, this paper has employed an alternative approach for atmosphere and weather effects according to the existing references, which can be applied to both the fixed and mobile terminals operating in various propagation environments [7; 9; 23; 24; 25]. Accordingly, the probability density function (PDF) of the channel gain \(\left|g_{i}\right|^{2}\) can be given by [26] \[f_{\left|g_{i}\right|^{2}}\left(x\right)=\alpha_{i}\exp\left(-\beta_{i}x \right){}_{1}F_{1}\left(m_{i};1;\delta_{i}x\right),\] (16) where \({}_{1}F_{1}\left(a;b;c\right)\) denotes the confluent hypergeometric function ([27], Equation (9.210.1)). Based on the results in [26], the related parameters \(\alpha_{i}\), \(\beta_{i}\) and \(\delta_{i}\) can be, respectively, calculated by the following identities \[\alpha_{i}=2b_{i}m_{i}/(2b_{i}m_{i}+\Omega_{i})^{m_{i}}\Big{/}2b_{ i},\] (17a) \[\beta_{i}=1/2b_{i},\] (17b) \[\delta_{i}=\Omega_{i}/2b_{i}\left(2b_{i}m_{i}+\Omega_{i}\right),\] (17c) where \[\Omega_{i}\] denotes the average power in terms of the LoS component, \[2b_{i}\] stands for the multipath average power, and \[G_{i}=LG_{\text{U}}G_{\text{S, max}}\left(\frac{I_{1}\left(u_{i}\right)}{2u_{i}}+36 \frac{I_{3}\left(u_{i}\right)}{u_{i}^{3}}\right)^{2}, \tag{14}\]0 represents the Nakagami-\(m\) fading severity.