# Performance Analysis of RIS-Assisted SatComs Based on a ZFBF and Co-Phasing Scheme
## 3 Active Beamforming and Phase Shift Design
### Asymptotic Performance Analysis  
Next, we conduct an asymptotic analysis of the mean of the capacity (i.e., ergodic capacity) for the proposed deterministic beamforming. From (19), the ergodic capacity can be derived asymptotically for a large \(N\) as follows:  
\[\begin{split}\mu_{R_{k}}&=\text{E}\left[\log_{2} \left(1+\frac{N^{2}\rho_{k}\xi_{k}|c_{k}|^{2}\bar{\beta}_{k,k}^{2}}{K^{2}\| \bar{\mathbf{t}}_{k}\|^{2}}\right)\right]\\ &\approx\log_{2}\left(\frac{N^{2}\rho_{k}\xi_{k}\tilde{\beta}_{k, k}^{2}}{K^{2}\|\bar{\mathbf{t}}_{k}\|^{2}}\right)+2\text{E}[\log_{2}|c_{k}|]\\ &\underset{(c)}{=}\log_{2}\left(\frac{N^{2}\rho_{k}\xi_{k}\tilde{ \beta}_{k,k}^{2}}{K^{2}\|\bar{\mathbf{t}}_{k}\|^{2}}\right)-\frac{\log_{2}10}{10}e^ {\mu_{k}+\frac{\sigma_{k}^{2}}{K}},\end{split} \tag{20}\]where \(\mu_{k}=\eta_{k}-\log(\log 10)+\log 20\)[18]. In (20), (\(c\)) arises from the characteristic of the double-log-normal distribution of \(|c_{k}|\sim\mathcal{L}^{2}\mathcal{N}\big{(}\eta_{k},\sigma_{k}^{2}\big{)}\) and a large \(N\). Similarly, the variance of \(R_{k}\) can be obtained by the following:  
\[\sigma_{R_{k}}^{2} =\text{Var}\left[\log_{2}\left(1+\frac{N^{2}\rho_{k}\xi_{k}|c_{k} |^{2}\bar{\beta}_{k,k}^{2}}{K^{2}\|\bar{\mathbf{\imath}}_{k}\|^{2}}\right)\right]\] \[\underset{(d)}{\approx}4\text{Var}[\log_{2}|c_{k}|] \tag{21}\] \[=\left(\frac{\log_{2}10}{10}\right)^{2}\Big{(}e^{\sigma_{k}^{2}}- 1\Big{)}e^{2\mu_{k}+\sigma_{k}^{2}},\]  
where \((d)\) results from a large \(N\) approximation. Given that the values of \(\rho_{k}\), \(\xi_{k}\), \(\bar{\beta}_{k,k}\), \(\mu_{k}\), \(\sigma_{k}\), and \(\|\bar{\mathbf{\imath}}_{k}\|\) are deterministic, it is possible to obtain \(\mu_{R_{k}}\) and \(\sigma_{R_{k}}^{2}\) deterministically. Consequently, we can assess the performance of an RIS-assisted SatCom system in terms of ergodic capacity without the need for extensive simulations.  
Based on the scaling law for \(N\), we can observe from (20) that the ergodic capacity increases with \(\mathcal{O}(\log N)\), while (21) shows that \(\sigma_{R_{k}}^{2}\) remains constant regardless of \(N\). Therefore, we can characterize the outage probability, defined as the probability that \(R_{k}\) falls below a target value \(R_{0}\), as follows:  
\[P_{\text{o}}=\text{Pr}(R_{k}<R_{0})=\text{Pr}\big{(}R_{k}-\mu_{R_{k}}<R_{0}- \mu_{R_{k}}\big{)}. \tag{22}\]  
Consider a target value \(R_{0}=\epsilon\cdot\mu_{R_{k}}\) where \(0<\epsilon\leq 1\). Then, we can obtain the outage probability as follows:  
\[P_{\text{o}}=\text{Pr}\big{(}R_{k}-\mu_{R_{k}}<(\epsilon-1)\mu_{R_{k}}\big{)}. \tag{23}\]  
From (20), we can observe that  
\[R_{k}-\mu_{R_{k}}\approx 2\log_{2}|c_{k}|+\frac{\log_{2}10}{10}e^{\mu_{k}+ \frac{\sigma_{k}^{2}}{2}}, \tag{24}\]  
and thus, a random variable \(R_{k}-\mu_{R_{k}}\) is unrelated to \(N\). Then, the probability density function (PDF) of the random variable \(R_{k}-\mu_{R_{k}}\) is not affected by the increase in \(N\), while \(\mu_{R_{k}}\) increases with \(\mathcal{O}(\log N)\). Hence, as \(N\) increases, the PDF of \(R_{k}-\mu_{R_{k}}\) remains unchanged, and \((\epsilon-1)\mu_{R_{k}}\) decreases, resulting in a decreasing \(P_{\text{o}}\). Hence, with the increase in \(N\), the outage probability decreases, thereby verifying the ability of an RIS to offer reliable SatCom services for a large \(N\) regime.