## 3 Asymptotic Performance Analysis in Unicast System
: In (25), the term \(\sum_{n}\tilde{f}_{k,n}e^{i\bar{\theta}_{k,n}}\) represents the sum of \(N\) samples, where each sample is an individual random variable whose mean value is equal to zero because \(\mathrm{E}\left[\tilde{f}_{k,n}\right]=0,\forall n\). Hence, as \(N\) increases, \(\sum_{n}\tilde{f}_{k,n}e^{i\bar{\theta}_{k,n}}/N\) approaches zero, and we can then approximate \(\bar{R}_{k}\) as follows:  
\[\bar{R}_{k} \approx\log\left(N^{2}\rho_{k}\xi_{k}^{2}a_{k}|c_{k}|^{2} \middle|\bar{\beta}_{k}+\bar{\beta}_{k}\frac{\sum_{n}\tilde{f}_{k,n}e^{i\bar{ \theta}_{k,n}}}{N}\right|^{2}\right)\] \[\underset{(a)}{\leq}\log\left(1+N^{2}\rho_{k}\xi_{k}^{2}a_{k}|c_ {k}|^{2}\Bigg{(}\tilde{\beta}_{k}+\tilde{\beta}_{k}\frac{\sum_{n}\left|\tilde{ f}_{k,n}\right|}{N}\Bigg{)}^{2}\right)\]0  
where \(\dot{R}_{k}\) is defined as the asymptotic achievable rate for a large \(N\). From (22) and (26), the rate ratio between \(\bar{R}_{k}\) and \(\bar{R}_{k}\) becomes one as \(N\) increases to infinity, which completes the proof. Lemma 1 shows that the channel fading of an RIS-assisted SatCom system exclusively depends on the rain fading channel gain, \(|c_{k}|\), and the impact of the Rician fading on the achievable rate becomes negligible as \(N\) increases. Lemma 1 also proves that the proposed passive beamformer achieves the upper bound of the achievable rate and asymptotically achieves the SNR order of \(\mathcal{O}\left(N^{2}\right)\). Moreover, since \(\bar{R}_{k}\leq\bar{R}_{k}\) from (21), it is proven that the proposed passive beamformer achieves the optimal performance, \(\bar{R}_{k}\), which is only achievable using the full CSI of \(f_{k}\) when \(\theta_{k,n}=-\angle\tilde{f}_{k,n}^{*}\). Note that the proposed \(\bar{\theta}_{k,n}\) can be readily obtained from the estimated value of \(d_{k,n}^{\text{ru}}\), leading to a low computational complexity (i.e., constant complexity) to calculate all phase shift values of the RIS. Additionally, the proposed scheme exclusively relies on deterministic LoS channel information (i.e., large-scale information), which exhibits slower variation compared to NLoS channels. Consequently, the proposed scheme does not require frequent feedback for CSI and location information, effectively reducing feedback/signaling overhead and rendering the proposed scheme practically feasible. Next, we analyze the mean (i.e., ergodic achievable rate) and variance of the asymptotic achievable rate for the proposed passive beamformer. From (26), the asymptotic achievable rate, \(\bar{R}_{k}\), can be obtained using (13) as follows:  
\[\underset{(a)}{\leq}\log\left(1+N^{2}\rho_{k}\xi_{k}^{2}a_{k}|c_ {k}|^{2}\Bigg{(}\tilde{\beta}_{k}+\tilde{\beta}_{k}\frac{\sum_{n}\left|\tilde{ f}_{k,n}\right|}{N}\Bigg{)}^{2}\right)\]1 \[\underset{(a)}{\leq}\log\left(1+N^{2}\rho_{k}\xi_{k}^{2}a_{k}|c_ {k}|^{2}\Bigg{(}\tilde{\beta}_{k}+\tilde{\beta}_{k}\frac{\sum_{n}\left|\tilde{ f}_{k,n}\right|}{N}\Bigg{)}^{2}\right)\]2as \(N\) increases, where \(z_{1}\sim\mathcal{LN}\big{(}\mu_{k},\sigma_{k}^{2}\big{)}\) and \(\mu_{k}=\eta_{k}-\log(\log 10)+\log 20\)[25]. As \(N\) increases, the mean and variance of the asymptotic achievable rate are obtained, respectively, as follows:  
\[\underset{(a)}{\leq}\log\left(1+N^{2}\rho_{k}\xi_{k}^{2}a_{k}|c_ {k}|^{2}\Bigg{(}\tilde{\beta}_{k}+\tilde{\beta}_{k}\frac{\sum_{n}\left|\tilde{ f}_{k,n}\right|}{N}\Bigg{)}^{2}\right)\]3 \[\underset{(a)}{\leq}\log\left(1+N^{2}\rho_{k}\xi_{k}^{2}a_{k}|c_ {k}|^{2}\Bigg{(}\tilde{\beta}_{k}+\tilde{\beta}_{k}\frac{\sum_{n}\left|\tilde{ f}_{k,n}\right|}{N}\Bigg{)}^{2}\right)\]4  
Then, the ergodic sum-rate is asymptotically derived as  
\[\underset{(a)}{\leq}\log\left(1+N^{2}\rho_{k}\xi_{k}^{2}a_{k}|c_ {k}|^{2}\Bigg{(}\tilde{\beta}_{k}+\tilde{\beta}_{k}\frac{\sum_{n}\left|\tilde{ f}_{k,n}\right|}{N}\Bigg{)}^{2}\right)\]5  
when \(\mu_{k}=\mu_{0}\) and \(\sigma_{k}=\sigma_{0},\forall k\).