## 2 System Model  
### Description of the SATCOM Star Network  
The SATCOM star network considered in this study is illustrated in Figure 1. The HS broadcasts to the \(M\) UTs and collects information from the UTs in backward links. Different from the full-duplex in [24], we apply frequency division duplexing in the uplink and downlink which could help to avoid self-interference. Specifically, the system works in transparent mode, which indicates that the satellite amplifies signals without digital processing [25]. Considering the mobility of the UTs, we assume that uniform planar arrays (UPAs) are deployed at the satellite and UTs while the HS is equipped with a reflector antenna (since the HS appears as a fixed satellite station which does not have constraints on antenna weight, we consider that a reflector antenna is equipped at the HS). In the star network, the satellite generates two beams (called feeder beam and service beam) which could cover the HS and UTs separately.  
For notational convenience, \(yz\)-plane (note that the results of the paper could still hold for other orientation UPAs) UPAs with \(N_{y}^{S}\times N_{z}^{S}\) and \(N_{y}^{U}\times N_{z}^{U}\) antenna elements are employed at the satellite and UTs. Let \(\phi\) (\(\theta\)) and \(\omega\) (\(v\)) denote the azimuth (elevation) angles of arrival at the satellite and UTs, respectively. The azimuth (elevation) angles of departure are denoted by \(\phi^{{}^{\prime}}\) (\(\theta^{{}^{\prime}}\)) and \(\omega^{{}^{\prime}}\) (\(v^{{}^{\prime}}\)), correspondingly. Then, the receive array response vectors at the satellite and UTs can be derived as  
Figure 1: The SATCOM star network.  
\[\mathbf{a}(\phi,\theta) =\frac{1}{\sqrt{N_{y}^{S}N_{z}^{S}}}[1,\dots,e^{ik\mu(m\sin(\phi)\sin( \theta)+n\cos(\theta))},\dots,e^{ik\mu((N_{y}^{S}-1)\sin(\phi)\sin(\theta)+(N_{z} ^{S}-1)\cos(\theta))}]^{\mathrm{T}} \tag{1}\]  
and  
\[\mathbf{b}(\omega,v) =\frac{1}{\sqrt{N_{y}^{U}N_{z}^{U}}}[1,\dots,e^{ik\mu(m\sin( \omega)\sin(v)+n\cos(v))},\dots,e^{ik\mu((N_{y}^{U}-1)\sin(\omega)\sin(v)+(N_{z }^{U}-1)\cos(v))}]^{\mathrm{T}}, \tag{2}\]  
where \(k=2\pi/\lambda\), and \(\mu=\lambda/2\) denotes the interelement spacing. Here, \(\lambda\) is the carrier wavelength. The transmit array response vectors can be denoted as \(\mathbf{a}(\phi^{{}^{\prime}},\theta^{{}^{\prime}})\) and \(\mathbf{b}(\omega^{{}^{\prime}},v^{{}^{\prime}})\), correspondingly. We denote \(G_{t,0}\) and \(G_{r,0}\) as transmit and receive antenna gains at the HS, respectively. The normal-direction gains of transmit (receive) beams at the satellite and UT UPAs are represented by \(E_{t}\) (\(E_{r}\)) and \(G_{t}\) (\(G_{r}\)), respectively. For the reason of the long travel distance of space-to-ground links, signals may mainly experience large-scale fading, while small-scale fading only exists in the downlink. Let \(l(d)=\sqrt{(\lambda/(4\pi d))^{2}}\) denote the large-scale channel coefficient, where \(d\) is the orbit height. The uplink and downlink channels between the HS and satellite can be modeled as  
\[\mathbf{g}_{0}=\mathbf{a}(\phi_{0},\theta_{0})\sqrt{E_{r}}l(d)\sqrt{G_{t,0}} \tag{3}\]  
and  
\[\mathbf{h}_{0}^{\mathrm{H}}=\rho_{0}\sqrt{G_{r,0}}l(d)\sqrt{E_{t}}\mathbf{a}^ {\mathrm{H}}(\phi_{0}^{{}^{\prime}},\theta_{0}^{{}^{\prime}}), \tag{4}\]  
respectively.  
Similarly, the uplink and downlink channels between the \(i\)-th UT and satellite can be given as  
\[\mathbf{G}_{i}=\mathbf{a}(\phi_{i},\theta_{i})\sqrt{E_{r}}l(d)\sqrt{G_{t}} \mathbf{b}^{\mathrm{H}}(\omega_{i}^{{}^{\prime}},\upsilon_{i}^{{}^{\prime}}) \tag{5}\]  
and  
\[\mathbf{H}_{i}=\rho_{i}\mathbf{b}(\omega_{i},\upsilon_{i})\sqrt{G_{r}}l(d) \sqrt{E_{t}}\mathbf{a}^{\mathrm{H}}(\phi_{i}^{{}^{\prime}},\theta_{i}^{{}^{ \prime}}), \tag{6}\]  
respectively. Here, \(\rho_{i}\sim\mathcal{CN}(0,1)\), \(i\in\{0,1,\dots,M\}\), denotes the small-scale fading coefficient.