## 5 Simulation Results  
In this section, we conduct comprehensive simulations in terms of the achievable rate ratio, ergodic rate, and outage probability. We consider a practical RIS-assisted SatCom system in which a low earth orbit (LEO) satellite serves a cluster of seen UEs via seven RISs (i.e., one RIS per UE and \(K=7\)). The parameters used for the simulations are detailed in Table 1. Statistical averaging is performed over a considerable number of independent runs for all simulations. Here, the label \"Estimation\" represents the estimated result obtained from our analyses, whereas the label \"Simulation\" represents the simulated results within the practical RIS-assisted SatCom system. In Table 1, the transmission power for UE \(k\) is defined as \(P_{k}\); thus, we have \(\rho_{k}=P_{k}/N_{0}\), where \(N_{0}\) is the noise power and equal power allocation is assumed for all UEs (i.e., \(P_{1}=P_{2}=\cdots=P_{K}\)). In Figure 2, Lemma 1 is verified when the optimal achievable rate and its upper bound, respectively, follow (20) and (21). As shown in Figure 2, as \(N\) increases, the achievable rate ratio obtained using the proposed scheme approaches 1, confirming the optimality of the proposed scheme, as proven in Lemma 1. Also, the rate ratio of the optimal performance is larger than that of the proposed scheme. In particular, the proposed scheme achieves approximately 89% of the upper bound, while the optimal scheme achieves approximately 90% of the upper bound when \(N=64\). However, the optimal performance is only achievable using the full CSI (i.e., \(f_{k}\)), which is an impractical assumption in the considered RIS-assisted SatCom system. By contrast, the proposed scheme only requires partial CSI (i.e., \(\bar{f}_{k}\)), which is deterministic LoS channel information and is achievable at the ES using the GNSSs. In Figures 3 and 4, the asymptotic ergodic sum-rate of the proposed scheme analyzed in (31) is verified. In these figures, the label \"Random\" represents the case of the random passive beamforming, in which \(\theta_{k,n}\) is determined randomly. Both figures show that the proposed scheme utilizes partial CSI, but it can achieve a performance of optimal one, which is operated based on full CSI. Evidently, the asymptotic ergodic sum-rates derived from (31) align with the simulation results across the entire range of \(\rho_{k}\) and \(N\). In Figures 3 and 4, the random passive beamforming technique requires no CSI; however, its ergodic sum-rate is much lower than that of the proposed scheme and the optimal case. As shown \begin{table}
\begin{tabular}{l c} \hline \hline
**Parameter** & **Value** \\ \hline Orbit & LEO (2520 km) \\ Carrier frequency & 2 GHz (L-band) \\ Bandwidth per beam, \(P_{k}\) & 20 MHz, 50 dBW \\ \(K\), \(G_{\max}\), \(\phi_{3\text{dB}}\) & 7, 49.5 dB, \(32^{\circ}\) \\ UE antenna gain & 38.16 dB \\ Boltzmann’s constant & \(1.38\times 10^{-23}\) J/K \\ Noise temperature at UE & 207 K \\ RIS element layout & 2D square lattice (spacing 0.01 m) \\ Average rain attenuation & 3 dB (\(\eta_{k}=-1.5617,\sigma_{k}=1,\forall k\)) \\ \(\kappa_{k}\), \(d_{k}^{\text{v}}\), \(d_{k}^{\text{h}}\), \(\forall k\) & 10, 1 km, 0.1 km \\ \(\eta_{\text{L}}\), \(\eta_{\text{NL}}\), \(c_{1}\), \(c_{2}\) & \(1,20,15,0.4\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Simulation parameters. Figure 2: Performance comparison of the achievable rate ratios as a function of \(N\). in (31), the ergodic sum-rate increases with \(\mathcal{O}(\log N)\) and \(\mathcal{O}(\log\rho_{k})\) as \(N\) and \(\rho_{k}\) increase, respectively. Accordingly, the ergodic sum-rates linearly increase with increasing \(\rho_{k}\) in the dBW scale, as shown in Figure 3, and they increase following the logarithm shape as \(N\) increases, as shown in Figure 4, verifying (31). In Figure 5, the optimality of the proposed scheme in the multicast system is verified.