# Performance Analysis of RIS-Assisted SatComs Based on a ZFBF and Co-Phasing Scheme
## 4 Simulation Results and Analysis  
We conduct extensive simulations to evaluate the performance of a practical RIS-assisted SatCom system, considering a limited number of reflecting elements \(N\). It is assumed that a low equatorial orbit (LEO) satellite serves a cluster of 7 UEs through 7 RISs (i.e., \(K=7\)). The noise power at each UE is given by \(\kappa_{\text{B}}TB\), where \(\kappa_{\text{B}}\), \(T\), and \(B\) are the Boltzmann constant, noise temperature at each UE, and bandwidth per beam, respectively. To ensure statistical reliability, all simulations are averaged over a large number of independent runs, and the simulation parameters used are provided in Table 1 based on [17].  
\begin{table}
\begin{tabular}{l c} \hline \hline
**Parameter** & **Value** \\ \hline Orbit & LEO (\(3.6\times 700\) km) \\ \hline Distance between UAV and UE & 1 km \\ \hline Carrier frequency & 2 GHz (L-band) \\ \hline \(K\), \(G_{\text{max}}\), \(\phi_{\text{3 dB}}\) & 7, 49.5 dB, 3.2\({}^{\circ}\) \\ \hline Rician factor \((\kappa_{j,k},\forall i,k)\) & 10 dB \\ \hline \hline \end{tabular}
\end{table}
Table 1: Simulation parameters.  
In Figure 2, the asymptotic ergodic capacity of the proposed scheme analyzed in (20) is verified. In this figure, \"MRT without IBI and IRI\" shows an ideal performance representing the case where the ES employs the maximum ratio transmission (MRT) to generate beamforming weights \(\mathbf{w}_{k}=\mathbf{h}_{k,k}^{\mathrm{H}}/\left(K\middle\|\mathbf{h}_{k,k}\right\|\right)\) that align the transmitted signals towards the intended UE. On the other hand, \"random phase shift\" represents the scenario of random passive beamforming, where the phase shifts \(\theta_{k,n},\forall k\), and \(n\) are randomly determined. The figure demonstrates that the asymptotic ergodic capacity derived from (20) aligns well with the simulation results as the value of \(N\) increases. The proposed deterministic ZFBF combined with the co-phasing technique utilizes partial CSI but achieves a performance close to that obtained using full CSI. Moreover, when \(N=64\), the proposed deterministic beamforming achieves approximately 92.8% of the ideal performance, and it increases as \(N\) increases, further confirming the asymptotic optimality of the proposed scheme.  
In Figure 3, the outage probability analyzed in (23) is verified. The figure compares the outage probabilities obtained from the simulations for different values of \(\epsilon\), specifically for \(\epsilon=0.90,0.85\), and \(0.80\). As shown in Figure 3, the outage probabilities tend to decrease  
\begin{table}
\begin{tabular}{l c} \hline \hline
**Parameter** & **Value** \\ \hline UE antenna gain & 38.16 dB \\ \hline Transmit power & 50 dBW \\ \hline [PERSON]’s constant (\(\kappa_{\mathrm{B}}\)) & \(1.38\times 10^{-23}\) J/m \\ \hline Bandwidth per beam (\(B\)) & 20 MHz \\ \hline Noise temperature (\(T\)) & 300 K \\ \hline RIS element layout & 2D square lattice \\ \hline RIS element spacing & 0.01 m \\ \hline Average rain attenuation (\(\forall k\)) & 3 dB (\(\eta_{k}=-1.5617,\sigma_{k}=1\)) \\ \hline \hline \end{tabular}
\end{table}
Table 1: _Cont._  
Figure 2: Performance comparison of the ergodic sum capacities for \(N\).  
towards zero as \(N\) increases. This decreasing trend in the outage probability indicates improved system performance and enhanced reliability as the RIS is able to mitigate potential communication outages. Furthermore, Figure 3 demonstrates that the proposed deterministic ZFBF combined with the co-phasing technique utilizes partial CSI while achieving outage performance close to that obtained using full CSI. In conclusion, this result provides empirical evidence that supports the notion of the RIS as a viable solution for ensuring reliable SatCom in the presence of a large number of RIS elements.