## 3 Asymptotic Performance Analysis in Unicast System  
We analyze the asymptotic data rate considering the practical SatCom system according to the following steps. Considering the achievable rate \(\tilde{R}_{k}\), we first analyze the upper bound of \(\tilde{R}_{k}\) as \(N\) increases to infinity. We then propose a practical passive beamformer and verify the asymptotic optimality of the proposed scheme. We finally derive the mean and variance of the achievable data rate of the proposed scheme. From (19), the achievable rate of UE \(k\), \(\tilde{R}_{k}\), and its upper bound, \(\hat{R}_{k}\), are obtained as  
\[\tilde{R}_{k} =\log\bigg{(}1+\rho_{k}\xi_{k}^{2}a_{k}|c_{k}|^{2}\Big{(}\sum_{n= 1}^{N}\left|f_{k,n}\right|\Big{)}^{2}\bigg{)} \tag{20}\] \[\underset{(a)}{\leq}\log\left(1+N^{2}\rho_{k}\xi_{k}^{2}a_{k}|c_ {k}|^{2}\Bigg{(}\tilde{\beta}_{k}+\tilde{\beta}_{k}\frac{\sum_{n}\left|\tilde{ f}_{k,n}\right|}{N}\Bigg{)}^{2}\right)\] \[\overset{\Delta}{=}\tilde{R}_{k}, \tag{21}\]  
where \(\tilde{f}_{k,n}\) is the \(n\)-th element of a vector, \(\tilde{f}_{k}\), and (a) is obtained from the following inequality: \(\sum_{n}\left|f_{k,n}\right|\leq\sum_{n}\tilde{\beta}_{k}\left|\tilde{f}_{k,n} \right|+\sum_{n}\tilde{\beta}_{k}\left|\tilde{f}_{k,n}\right|\), where \(\left|\tilde{f}_{k,n}\right|=1\) from (7). As \(N\) increases to infinity, we derive the asymptotic upper bound of \(\tilde{R}_{k}\) as follows:  
\[\hat{R}_{k} \underset{(b)}{\leq}\log\Big{(}1+N^{2}\rho_{k}\xi_{k}^{2}a_{k}|c _{k}|^{2}\big{(}\tilde{\beta}_{k}+\tilde{\beta}_{k}\big{)}^{2}\Big{)}\] \[\approx\log\Big{(}N^{2}\rho_{k}\xi_{k}^{2}a_{k}|c_{k}|^{2}\tilde{ \beta}_{k}^{2}\Big{)}+2\log\left(1+\sqrt{\frac{1}{\kappa_{k}}}\right)\] \[\underset{(c)}{\approx}\log\Big{(}N^{2}\rho_{k}\xi_{k}^{2}a_{k}|c _{k}|^{2}\tilde{\beta}_{k}^{2}\Big{)}, \tag{22}\]  
where (b) results from [PERSON]'s inequality: \(\left(\sum_{n}\left|\tilde{f}_{k,n}\right|/N\right)^{2}\leq\sum_{n}\left| \tilde{f}_{k,n}\right|^{2}/N\). Here, the sample mean of \(\left|\tilde{f}_{k,n}\right|^{2}\) can be obtained from the asymptotic channel hardening effect [26], under the assumption that the spectral norm of \(\tilde{G}_{k}\) is bounded as  
\[\frac{\sum_{n=1}^{N}\left|\tilde{f}_{k,n}\right|^{2}}{N}\to 1, \tag{23}\]  
almost surely as \(N\rightarrow\infty\). Also, (c) results from a large \(N\) and a small \(\sqrt{1/\kappa_{k}}\) because the Rician factor, \(\kappa_{k}\), will be a large value (in the A2G channel model, the maximum value and the average value of the Rician factor introduced in [27] are 20 dB and 10 dB, respectively), considering a practical A2G channel model [27; 28]. We can observe from (22) that, as \(N\) increases, the upper bound of the SNR increases with \(\mathcal{O}\big{(}N^{2}\big{)}\), and the upper bound of the achievable rate increases with \(\mathcal{O}(\log N)\). Note that an ES can leverage global navigation satellite systems (GNSSs) to accurately estimate the three-dimensional positions of UAVs and UEs [23]. Thus, the ES can estimate the deterministic LoS components, denoted as \(\mathbf{\tilde{f}}_{k}\), by using the distance between RIS \(k\) and UE \(k\), which is partial CSI; however, this is the only information available at the ES side. Hence, we design a suboptimal passive beamformer exclusively based on \(\mathbf{\tilde{f}}_{k}\), as follows:  
\[\bar{\theta}_{k,n}=-\angle\tilde{f}_{k,n}^{*}. \tag{24}\]  
Using (24), the achievable rate is given as follows:  
\[\bar{R}_{k}=\log\left(1+N^{2}\rho_{k}\xi_{k}^{2}a_{k}|c_{k}|^{2} \middle|\bar{\beta}_{k}+\frac{\sum_{n}\bar{\beta}_{k}\tilde{f}_{k,n}e^{i\bar{ \theta}_{k,n}}}{N}\right|^{2}\right). \tag{25}\]  
Then, we derive the following lemma from the asymptotic analyses. **Lemma 1**. : _On the basis of the scaling law for \(N\), the rate ratio between the achievable rate of the proposed passive beamformer and its performance bound becomes one (i.e., \(\frac{\hat{R}_{k}}{\hat{R}_{k}}=1\)) as \(N\) increases to infinity._  
Proof.