# 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 Full CSI  
Considering the ZFBF at the ES and the co-phasing technique at the RIS, the effective channel between the satellite and UE \(k\) via RIS \(k\) and the corresponding ZFBF vector for UE \(k\) can be rewritten, respectively, as follows:  
\[\mathbf{h}_{k,k}=\sqrt{\xi_{k}}|c_{k}|\exp(\jmath\psi_{k})\mathbf{a}_{k}^{\mathrm{T}} \sum\limits_{n}|f_{k,k,n}|, \tag{13}\]\[\mathbf{w}_{k}=\frac{\bar{\mathbf{t}}_{k}}{K\exp(j\psi_{k})\|\bar{\mathbf{t}}_{k}\|}. \tag{14}\]  
Here, \(\bar{\mathbf{t}}_{k}\) is defined as the \(k\)th column vector of \(\bar{\mathbf{T}}\), where \(\bar{\mathbf{T}}=\bar{\mathbf{H}}^{-1}\) and \(\bar{\mathbf{H}}=\left[a_{1}^{\mathrm{T}},a_{2}^{\mathrm{T}},\cdots,a_{K}^{ \mathrm{T}}\right]^{\mathrm{T}}\). Then, the received signal at UE \(k\) can be obtained as:  
\[\begin{split} y_{k}&=\frac{\sqrt{\rho_{k}\xi_{k}}| c_{k}|\sum_{n}\left|f_{k,k,n}\right|}{K\|\bar{\mathbf{t}}_{k}\|}x_{k}\\ &+\sum_{i\
eq k}^{K}\frac{\sqrt{\rho_{i}\xi_{i}}|c_{i}|\sum_{n} \left|f_{i,k,n}\right|e^{i\theta_{i,k,n}}}{K\|\bar{\mathbf{t}}_{i}\|}x_{i}+n_{k}, \end{split} \tag{15}\]  
where \(\theta_{i,k,n}=\angle f_{i,k,n}^{*}-\angle f_{i,k,n}^{*}\). The second term in the right-hand side of (15) represents the residual IRI generated by the reflected signal from other RISs. As \(N\) increases, the residual IRI and the noise become negligible compared to the desired signal as follows:  
\[y_{k}\xrightarrow[(a)]{\mathrm{a.s.}}\frac{\sqrt{\rho_{k}\xi_{k}}|c_{k}|\sum_ {n}\left|f_{k,k,n}\right|}{K\|\bar{\mathbf{t}}_{k}\|}x_{k}. \tag{16}\]  
In (16), \((a)\) results from the following two observations: (i) as \(N\) increases, \(\sum_{n}\left|f_{k,k,n}\right|\) increases with \(\mathcal{O}(N)\), while (ii) \(\sum_{n}\left|f_{i,k,n}\right|e^{i\theta_{i,k,n}}\) and \(n_{k}\) do not increase because the values of \(\theta_{i,k,n},\forall n\) are different from each other. Hence, as \(N\) increases, we can determine the asymptotic SINR at UE \(k\) for ZFBF with the co-phasing approach as follows:  
\[\begin{split}\gamma_{k}^{\mathrm{full}}&=\frac{N^{ 2}\rho_{k}\xi_{k}|c_{k}|^{2}\left(\sum_{n}\left|f_{k,k,n}\right|/N\right)^{2} }{K^{2}\|\bar{\mathbf{t}}_{k}\|^{2}}\\ &\xrightarrow[]{\mathrm{a.s.}}\frac{N^{2}\rho_{k}\xi_{k}|c_{k}|^ {2}\left[\left|f_{k,k,n}\right|\right]^{2}}{K^{2}\|\bar{\mathbf{t}}_{k}\|^{2}}. \end{split} \tag{17}\]  
In (17), \(\mathrm{E}\left[\left|f_{k,k,n}\right|\right]\) represents the expectation of the random variable \(\left|f_{k,k,n}\right|\) for \(n\), where \(n\) ranges from \(1\) to \(N\). From (17), it can be observed that, as \(N\) increases, the received SINR increases with \(\mathcal{O}\left(N^{2}\right)\), and this is equivalent to the asymptotic SNR of an ideal RIS-assisted cellular network, as demonstrated in [22]. While the implementation of the ZFBF and the co-phasing technique may not be optimal, they have the potential to achieve optimal performance in the asymptotic regime. However, a general ZFBF requires complete CSI of \(\mathbf{h}_{k,k}\), and the co-phasing technique requires complete CSI of \(\mathbf{f}_{k,k}\). Acquiring complete CSI at the ES leads to a considerable increase in signaling overhead and introduces significant latency due to extended round-trip delays, particularly in SatCom systems. Therefore, in the next section, we propose a deterministic ZFBF and co-phasing technique that utilizes partial CSI.