## 2 System Model
### Propagation Model  
Utilizing the basic large-scale propagation characteristics of the satellite-terrestrial channel, the channel coefficient \(h_{k}\) linking the satellite with the UT can be represented by  
\[h_{k}=\left(\frac{c_{0}}{4\pi f_{c}d_{k}(\hat{\vartheta})}\right)\left[\frac {G^{\text{UT}}(\varphi)G_{k}^{\text{Sat}}(\theta)}{\kappa BT}\right]^{1/2} \widetilde{h}, \tag{8}\]  
where \(f_{c}\) represents the carrier frequency; \(c_{0}\) signifies the light speed; \(\kappa\) stands for Boltzmann constant; \(B\) represents the bandwidth of the signal; \(T\) corresponds to the noise temperature of the receiver. Furthermore, \(G^{\text{UT}}(\varphi)\) denotes the antenna gain of the UT with respect to the off-axis angle \(\varphi\) and \(G_{k}^{\text{Sat}}(\theta)\) refers to the antenna gain of a satellite with regard to the off-axis angle \(\theta\). \(\widetilde{h}\) represents the coefficient of the small-scale fading and follows a shadowed Rician (SR) distribution. The probability density function (PDF) of \(\widetilde{h}\) is represented by the following:  
\[f_{|h|^{2}}(x)=\left(\frac{2 np_{c}}{2 np_{c}+V}\right)^{n}\frac{1}{2p_{c}}\exp \left(-\frac{x}{2p_{c}}\right)_{1}F_{1}\bigg{(}n,1,\frac{Vx}{2p_{c}(2 np_{c}+V )}\bigg{)}. \tag{9}\]  
In the above, \({}_{1}F_{1}(\cdot,\cdot,\cdot)\) represents the hypergeometric function, while \(n\), \(p_{c}\), and \(V\) denote the Nakagami fading coefficient, half-power of the scattering component, and half-power of the direct path, respectively. According to ITU-R. S. 672-4 [21], the antenna pattern of the GEO satellite is given by  
\[G_{\text{GEO}}^{\text{Sat}}(\theta)=\left\{\begin{array}{ccc}G_{\text{max} }&0<\theta\leq\theta_{3\text{dB}}\\ G_{\text{max}}-3(\theta/\theta_{3\text{dB}})^{2}&\theta_{3\text{dB}}<\theta \leq Y_{1}\theta_{3\text{dB}}\\ G_{\text{max}}+L_{s}&Y_{1}\theta_{3\text{dB}}<\theta\leq Y_{2}\theta_{3\text{dB }}\\ G_{\text{max}}+L_{s}+20-25\log(\theta/\theta_{3\text{dB}})&Y_{2}\theta_{3\text {dB}}<\theta\leq\theta_{1}\\ 0&\theta_{1}<\theta\end{array}\right.,\quad(\text{dBi}) \tag{10}\]  
where \(\theta_{3\text{dB}}\) is the angle corresponding to the half of the maximum gain, \(L_{s}=-25\), \(Y_{1}=2.887\), and \(Y_{2}=6.32\). \(G_{\text{max}}=10\log(\eta(\pi Df/c)^{2})\) is the antenna maximum gain,where \(\eta\) and \(D\) are the antenna efficiency and the antenna diameter, respectively. The antenna pattern of the NGEO satellite is obtained from ITU-R. S. 1528 [22] as  
\[G_{\text{NGEO}}^{\text{Sat}}(\theta)=\left\{\begin{array}{cc}G_{\text{max}}&0< \theta\leq\theta_{\text{3 dB}}\\ G_{\text{max}}-3(\theta/\theta_{\text{3 dB}})^{2}&\theta_{\text{3 dB}}<\theta \leq Y_{3}\\ G_{\text{max}}+L_{\text{s}}-25\log(\theta/\theta_{\text{3 dB}})&Y_{3}<\theta \leq Y_{4}\\ L_{F}&Y_{4}<\theta\end{array}\right.,\ \ (\text{dBi}). \tag{11}\]  
In the above, \(Y_{3}=\theta_{\text{3 dB}}(-L_{\text{s}}/3)^{1/2}\), \(Y_{4}=Y_{3}10^{0.04(G_{\text{max}}+L_{\text{s}}+L_{F})}\) and \(L_{\text{F}}=5\). Based on ITU-R. S. 465-6 [23] and ITU-R. S. 1428 [24], the antenna pattern of GEO UT and NGEO UT are, respectively, given by  
\[G_{\text{GEO}}^{\text{UT}}(\theta)=\left\{\begin{array}{cc}G_{\text{max}}&0< \theta\leq\theta_{t}\\ 32-25\log\theta&\theta_{t}<\theta\leq 48^{\circ}\\ -10&48^{\circ}<\theta\end{array}\right.,\ \ (\text{dBi}). \tag{12}\]  
\[G_{\text{NGEO}}^{\text{UT}}(\theta)=\left\{\begin{array}{cc}G_{\text{max}}-2.5\times 10^{-3}\left(\frac{D}{D}\right)^{2}&0<\theta\leq\theta_{m}\\ G_{\text{max}}-25\log\left(95\frac{\lambda}{D}\right)&\theta_{m}<\theta\leq 95 \frac{\lambda}{D}\\ G_{\text{max}}-25\log\theta&95\frac{\lambda}{D}<\theta\leq 33.1^{\circ}\\ -9&33.1^{\circ}<\theta\leq 80^{\circ}\\ -4&80^{\circ}<\theta\leq 120^{\circ}\\ -9&120^{\circ}<\theta\leq 180^{\circ}\end{array}\right.,\ \ (\text{dBi}). \tag{13}\]  
where \(\theta_{t}=\text{max}\big{(}2^{\circ},144(D/\lambda)^{-1.09}\big{)}\) and \(\theta_{m}=20\lambda/D\sqrt{G_{\text{max}}-\left(29-25\log\left(95\frac{ \lambda}{D}\right)\right)}\) are the threshold parameters; \(\lambda\) is the wavelength.