## 3 Performance Analysis
### Preliminary Results
#### 3.1.2 Satellite Channel Model  
In modern satellite communications, multibeam technology is widely used to increase the spectral efficiency, which should be taken into account in modeling the satellite channel. For a geosynchronous Earth orbit (GEO) satellite, multiple beams are often generated through array-fed reflectors, which is more efficient than direct radiating arrays. In this case, the radiation pattern of each beam is fixed, so that the on-board precessing can be significantly reduced [28]. Furthermore, the time division multiple access (TDMA) scheme is adopted so that there is only one Earth station (ES) scheduled within each beam at any given time. Next, the channel coefficient \(\widetilde{h}_{SJ}\) between the ES and the \(k\)-th on-board beam for the downlink is given by:  
\[\widetilde{h}_{SJ}=C_{SJ}g_{SJ},\left(J\in\left\{R,P_{i}\right\} \right), \tag{14}\]  
where \(g_{SJ}\) represents the complex coefficient of the satellite channel and \(C_{SJ}\) denotes the radio propagation loss including the effects of free space loss (FSL) and the antenna pattern, which is described as:  
\[C_{SJ}=\frac{\lambda}{4\pi}\frac{\sqrt{G_{SJ}G_{ES}}}{\sqrt{d^{2}+d_{0}^{2}}}, \tag{15}\]  
where \(\lambda\) denotes the carrier wavelength, \(d\) is the distance between the ES and the center of the \(k\)-th center beam and \(d_{0}\approx\) 35,786 km is the height of a GEO satellite. Besides, \(G_{ES}\) is the antenna gain of the ES and \(G_{SJ}\) is the \(k\)-th satellite on-board beam gain. According to [29], the antenna gain for the ES with parabolic antenna can be approximately expressed as:  
\[G_{ES}\left(dB\right)\simeq \begin{cases}\overline{G}_{\max},&for\ 0^{\circ}< \beta<1^{\circ}\\ 32-25\log\beta,for\ 1^{\circ}<\beta<48^{\circ}\\ -10,&for\ 48^{\circ}<\beta\leq 180^{\circ},\end{cases}  \tag{16}\]where \(\overline{G}_{\max}\) is the maximum beam gain at the boresight and \(\beta\) the off-boresight angle. As for \(G_{SJ}\), by defining \(\theta_{k}\) as the angle between the ES position and the \(k\)-th beam center with respect to the satellite and \(\overline{\theta}_{k}\) as the 3-dB angle of the \(k\)-th on-board beam, the antenna gain from the \(k\)-th beam to the ES is approximated by [30]:  
\[G_{SJ}\simeq G_{\max}\Bigg{(}\frac{I_{1}\left(u_{k}\right)}{2u_{k}}+36\frac{I_ {3}\left(u_{k}\right)}{u_{k}^{3}}\Bigg{)}^{2}, \tag{17}\]  
where \(G_{max}\) denotes the maximal beam gain, \(u_{k}=2.07123\sin\theta_{k}/\sin\overline{\theta}_{k}\) and \(J_{1}\) and \(J_{3}\) denote the first-kind Bessel function of order one and three, respectively. In order to obtain the maximum beam gain, hence \(\theta_{k}\to 0\), as a result of \(G_{SJ}\approx G_{\max}\). On this foundation, we have \(\widetilde{h}_{SJ}=C_{SJ}^{\max}g_{SJ}\) (\(C_{SJ}^{max}\) can be derived by submitting (17) and (16) into (15) when \(\theta_{k}\to 0\), which is given by \(C_{SJ}^{\max}\approx\frac{\lambda}{4\pi}\frac{\sqrt{G_{\max}G_{FS}}}{\sqrt{d^ {2}+d_{0}^{2}}}\). As for the complex random shadowing \(g_{SJ}\), besides the mathematical models, including Loo, Barts-Stutzman and Karasawa, the SR channel proposed in [31] is the commonly-used channel model for LMS communication [3; 32; 33; 4]. According to [31], the complex random shadowing of \(g_{SJ}\) undergoing the SR model can be expressed as:  
\[g_{SJ}=A\exp\left(j\theta\right)+Z\exp\left(j\psi\right), \tag{18}\]  
where \(A\) and \(Z\) denote the amplitudes of the scattering and the LOS components, which are the independent stationary random processes following Rayleigh and Nakagami-\(m\) distributions, respectively. Besides, \(\theta\) is a stationary random phase uniformly distributed over \([0,2\pi)\), and \(\psi\) is the deterministic phase of the LOS component.