# Performance Analysis of RIS-Assisted SatComs Based on a ZFBF and Co-Phasing Scheme
## 3 Active Beamforming and Phase Shift Design
### ZFBF and Co-Phasing Technique Using Partial CSI  
Acquiring complete CSI at the ES results in substantial signaling overhead, leading to significant latency. Additionally, due to the fully passive nature of the RIS, the ES cannot accurately estimate RIS-related channels such as \(\mathbf{D}_{k}\) and \(\mathbf{f}_{k,k}\). In this regard, we propose a deterministic ZFBF and co-phasing technique that utilizes partial CSI. Based on (13) and (14), we can observe that \(\mathbf{h}_{k,k}\bar{\mathbf{t}}_{i}=0,\forall i\
eq k\). Motivated by this observation, we propose a ZFBF vector as \(\mathbf{w}_{k}=\bar{\mathbf{t}}_{k}/(K\|\bar{\mathbf{t}}_{k}\|)\), which can perfectly eliminate the IBI as follows:  
\[ \begin{split}&\sum_{i\
eq k}^{K}\sqrt{\rho_{i}}\mathbf{h}_{k,k}\mathbf{w}_{i }x_{i}\\ &=\sum_{i\
eq k}^{K}\sqrt{\rho_{i}\bar{\xi}_{k}}|c_{k}|\exp(j\psi_ {k})\sum_{n}\left|f_{k,k,n}\right|\frac{\mathbf{a}_{k}^{\mathrm{T}}\bar{\mathbf{t}}_{i }}{K\|\bar{\mathbf{t}}_{i}\|}x_{i}=0.\end{split} \]To design the proposed ZFBF vector \(\mathbf{w}_{k}=\bar{\mathbf{t}}_{k}/(K\|\bar{\mathbf{t}}_{k}\|)\), the ES requires access to \(\mathbf{a}_{k}\) for all \(k\), which consists of the deterministic beam pattern coefficients. These coefficients are deterministic information and can be accurately estimated using location-aware channel estimation based on global navigation satellite systems (GNSSs) [18]. Also, to maximize the desired signal power using partial CSI, the phase shifts at the RIS can be designed such that the reflected \(N\) signals from \(N\) LoS paths are aligned in phase as follows: \(\theta_{k,n}=-\angle\bar{f}_{k,k,n}^{*},\forall k,n.\) Similarly, in (16), as \(N\) increases, both the IRI and the noise become negligible compared to the desired signal as follows:  
\[ \begin{split}& y_{k}=\frac{\sqrt{\rho_{k}\xi_{k}}|c_{k}|e^{i\psi_{k}} \sum_{n}\left(\tilde{\beta}_{k,k}+\tilde{\beta}_{k,k}\bar{f}_{k,k,n}^{*}e^{-j \angle\bar{f}_{k,k,n}^{*}}\right)}{K\|\bar{\mathbf{t}}_{k}\|}x_{k}\\ &+\sum_{i\
eq k}^{K}\frac{\sqrt{\rho_{i}\xi_{i}}|c_{i}|e^{i\psi_{ k}}\sum_{n}\left[f_{i,k,n}\right]e^{i\tilde{\vartheta}_{k,n}}}{K\|\bar{\mathbf{t}}_{i }\|}x_{i}+n_{k}\\ &\xrightarrow[(b)]{\text{a.s.}}\frac{N\sqrt{\rho_{k}\xi_{k}}|c_{ k}|e^{i\tilde{\vartheta}_{k}}\left(\tilde{\beta}_{k,k}+\tilde{\beta}_{k,k} \text{E}\big{[}\bar{f}_{k,k,n}^{*}e^{-j\angle\bar{f}_{k,k,n}^{*}}\big{]} \right)}{K\|\bar{\mathbf{t}}_{k}\|}x_{k}\\ &=\frac{N\sqrt{\rho_{k}\xi_{k}}|c_{k}|e^{i\tilde{\vartheta}_{k}} \tilde{\beta}_{k,k}}{K\|\bar{\mathbf{t}}_{k}\|}x_{k},\end{split}  \tag{18}\]  
where \(\tilde{f}_{k,k,n}\) is the \(n\)th element of \(\bar{\mathbf{f}}_{k,k}\) and \(\tilde{\vartheta}_{k,n}=\angle f_{i,k,n}^{*}-\angle\bar{f}_{i,i,n}^{*}\). In (18), \((b)\) arises from the fact that \(\text{E}\big{[}\bar{f}_{k,k,n}^{*}e^{-j\angle\bar{f}_{k,k,n}^{*}}\big{]}= \text{E}\big{[}\bar{f}_{k,k,n}^{*}\big{]}\cdot\text{E}\big{[}e^{-j\angle\bar{ f}_{k,k,n}^{*}}\big{]}=0\). Thus, to design the proposed RIS phase shift matrix, the ES only requires \(\bar{\mathbf{f}}_{k,k}\), which consists of the deterministic LoS components, which can also be accurately estimated using location-aware channel estimation based on GNSSs. Using the proposed deterministic ZFBF and co-phasing technique, we can obtain the asymptotic SINR at UE \(k\) as follows:  
\[\gamma_{k}^{\text{part}}=\frac{N^{2}\rho_{k}\xi_{k}|c_{k}|^{2}\bar{\beta}_{k, k}^{2}}{K^{2}\|\bar{\mathbf{t}}_{k}\|^{2}}. \tag{19}\]  
From (19), we can observe that as \(N\) increases, the received SINR also increases with \(\mathcal{O}\big{(}N^{2}\big{)}\), as seen in the case of the ZFBF with co-phasing approach using full CSI. Furthermore, in practical A2G channel models, the average and maximum values of the Rician factor measured in [23] are 10 dB and 20 dB, respectively. As a result, \(\text{E}\big{[}\big{|}f_{k,k,n}\big{|}\big{]}\) is approximately similar to \(\tilde{\beta}_{k,k}|\text{exp}(-j2\pi d_{k,k,n}/\lambda)\big{|}=\tilde{\beta }_{k,k}\), leading to \(\gamma_{k}^{\text{part}}\approx\gamma_{k}^{\text{full}}\).