## 3 Performance Analysis
### Statistical Properties of Channels  
In this paper, it is assumed that all terrestrial links experience the independent identically distributed (i.i.d) Rayleigh fading. With the help of Equation (1), the probability distribution function (PDF) and cumulative distribution function (CDF) of \(|h_{Rn}|^{2}\) can be obtained as, respectively [9]:  
\[f_{\left|h_{Rn}\right|^{2}}(x)=N\left(1-e^{-x}\right)^{N-1}e^{-x}=N\sum_{k=0}^ {N-1}\left( \begin{array}{c}N-1\\ k\end{array} \right)e^{-(k+1)x}(-1)^{k}, \tag{23}\]  
\[F_{\left|h_{Rn}\right|^{2}}(x)=\left(1-e^{-x}\right)^{N}=\sum_{k=0}^{N}\left( \begin{array}{c}N\\ k\end{array} \right)e^{-kx}(-1)^{k}. \tag{24}\]  
Moreover, it is inevitable to use multi-beam technology to improve the spectral efficiency in SatCom. Therefore, this factor must be considered in the channel model. For geosynchronous earth orbit (GEO) satellites, multiple beams are usually generated by array reflectors, which have a higher efficiency than direct radiation arrays. In this case, the radiation pattern of each beam is a constant, which significantly reduces the need for on-satellite processing capacity [34]. Furthermore, time division multiple access (TDMA) technology is applied to ensure that only one terrestrial user is accessed for the given time in the system. Furthermore, the channel coefficient \(h_{SR}\) between the satellite and terrestrial relay can be shown as:  
\[h_{SR}=C_{SR}g_{SR}, \tag{25}\]  
where \(g_{SR}\) represents the SR coefficient, and \(C_{SR}\) denotes the radio frequency loss, which consists of free space loss and antenna gain and is expressed as:  
\[C_{SR}=\frac{\lambda}{4\pi}\frac{\sqrt{G_{S}G_{R}}}{\sqrt{s^{2}+R_{0}}^{2}}, \tag{26}\]  
where \(\lambda\) is the carrier wavelength, \(s\) denotes the distance between terrestrial relay and satellite beam center, and \(R_{0}\approx 35,786\) km represents the radius of geosynchronous earth orbit. Moreover, \(G_{S}\) is the beam gain of the satellite, and \(G_{R}\) denotes the antenna gain of terrestrial relay. Besides, the antenna gain for terrestrial relay with parabolic antenna can be shown as [35]:  
\[G_{R}(dB)\simeq\left\{ \begin{array}{c}G_{MR},0^{\circ}<\psi<1^{\circ}\\ 32-25\log\psi,1^{\circ}\leq\psi<48^{\circ}\\ -10,48^{\circ}<\psi\leq 180^{\circ}\end{array} \right., \tag{27}\]  
where \(G_{MR}\) is the upper limit of the antenna gain, and \(\psi\) represents the deflection angle of the satellite. As for \(G_{S}\), \(\vartheta\) is defined as the angle between the terrestrial relay and the center of the satellite beam, and \(\bar{\vartheta}\) represents the 3 dB angle of the satellite beam. Furthermore, the beam gain of the satellite can be given by [36]:  
\[G_{S}(dB)\simeq G_{MS}\bigg{(}\frac{I_{1}(\tau)}{2\tau}+36\frac{I_{3}(\tau)}{ \tau^{3}}\bigg{)}^{2}, \tag{28}\]  
where \(G_{MS}\) represents the maximal beam gain, \(\tau=2.07123\sin\vartheta/\sin\bar{\vartheta}\), \(I_{1}(x)\) and \(I_{3}(x)\) are the first-kind bessel function of order 1 and 3, respectively. Ideally, to obtain the best system performance, generally \(\vartheta\to 0\). Therefore, we can get \(G_{S}=G_{MS}\), and further, \(h_{SR}=C_{SR}^{Max}g_{SR}\). From [37], the PDF of \(\left|h_{SR_{m}}\right|^{2}=\left|C_{SR_{m}}^{Max}g_{SR_{m}}\right|^{2}\) is given by  
\[f_{\left|h_{SR_{m}}\right|^{2}}(x)=\alpha_{SR_{m}}e^{-\beta_{SR_{m}}x}{}_{1}F_ {1}(m_{SR_{m}};1;\delta_{SR_{m}}x), \tag{29}\]  
where \(\alpha_{SR_{m}}=\frac{1}{2b_{SR_{m}}m_{SR_{m}}}\bigg{(}\frac{2b_{SR_{m}}m_{SR_{ m}}}{\left(2b_{SR_{m}}m_{SR_{m}}+\Omega_{SR_{m}}\right)}\bigg{)}^{m_{SR_{m}}}\), \(\delta_{SR_{m}}=\frac{\Omega_{SR_{m}}}{\left[2b_{SR_{m}}\left(2b_{SR_{m}}m_{SR_{ m}}+\Omega_{SR_{m}}\right)\right]}\), and \(\beta_{SR_{m}}=\frac{1}{2b_{SR_{m}}}\). Besides, \(m_{SR_{m}}\) represents the Nakagami-m parameter, \(\Omega_{SR_{m}}\) denotes average power of line of sight (LOS) link and \(b_{SR_{m}}\) represents half average power of the multi-path.