# RIS-Assisted Fixed NOMA: Outage Probability Analysis and Transmit Power Optimization
## 3 Outage Probability Analysis of RIS-SR-FNOMA  
The conceptual diagram of RIS-SR-FNOMA is illustrated in Figure 1. In this system, RIS with \(N\) reflecting elements acts as a smart reflector. The RIS elements are distributed evenly across the users. RIS is placed sufficiently far from the BS and the users. This results in a two-hop communication system. The BS, NU, and FU are considered to have a single antenna. In a two-user scenario, FU performs direct decoding while dealing with the NUsignal as interference. The NU decodes the FU signal first, and the influence of that signal is eliminated through SIC. The resulting signal is then used to decode the NU signal. The superposition-coded symbol at the BS is given by [27]  
\[x_{s}=\sqrt{\alpha_{NUI}P_{s}}\ x_{NUI}+\sqrt{\alpha_{FU}P_{s}}\ x_{FU}, \tag{1}\]  
where \(\alpha_{NUI}P_{s}\) and \(\alpha_{FU}P_{s}\) are the powers allocated to NU and FU, respectively. In FNOMA, \(\alpha_{NUI}<\alpha_{FU}\) and \(\alpha_{NUI}+\alpha_{FU}=1\). \(x_{NUI}\) and \(x_{FU}\) are the transmit symbols corresponding to NU and FU, respectively. \(P_{s}\) is the average transmit symbol power. The channel condition between BS to RIS and RIS to UE are modeled as \(h_{i}=\alpha_{i}e^{i\theta_{i}},h_{i}\sim C\mathcal{N}(0,1)\), \(i=1,2,\ldots,N\) and \(g_{i}=\beta_{i}e^{i\psi_{i}},g_{i}\sim C\mathcal{N}(0,1)\), \(i=1,2,\ldots,N\) respectively. Here, \(\alpha_{i}=\sqrt{(Re(h_{i})^{2}+Im(h_{i})^{2})}\), \(\theta_{i}=tan^{-1}\Big{(}\frac{Im(h_{i})}{Re(h_{i})}\Big{)}\), \(\beta_{i}=\sqrt{(Re(g_{i})^{2}+Im(g_{i})^{2})}\) and \(\psi_{i}=tan^{-1}\Big{(}\frac{Im(g_{i})}{Re(g_{i})}\Big{)}\). The passive RIS introduces phase shift \(\phi_{i}\) at each reflecting element, and it is given by  
\[\phi_{i}=-\theta_{i}-\psi_{i},i=1,2,\ldots,N. \tag{2}\]  
The signal received at the FU is [42]  
\[r_{FU}=\left(\sum_{i=\frac{N}{2}+1}^{N}g_{i}h_{i}e^{i\phi_{i}}\right)x_{s}+n_ {FU}, \tag{3}\]  
where \(n_{FU}\) is the additive white Gaussian noise (AWGN) at FU and \(n_{FU}\in C\mathcal{N}(0,N_{0})\). \(N_{0}\) is the noise variance. Substituting the channel conditions, the received signal at the FU is  
\[r_{FU}=\left(\sum_{i=\frac{N}{2}+1}^{N}\alpha_{i}\beta_{i}\right)x_{s}+n_{FU}. \tag{4}\]  
Let \(B=\sum_{i=\frac{N}{2}+1}^{N}\alpha_{i}\beta_{i}\) be the cumulative effect of the dual-hop channel for the FU. Substituting \(B\) and (1) in (4), the received signal at FU is  
\[r_{FU}=B\ \sqrt{\alpha_{NUI}P_{s}}\ x_{NUI}+B\ \sqrt{\alpha_{FU}P_{s}}\ x_{FU}+n_{FU}. \tag{5}\]  
Since the cumulative dual-hop channel for the FU is weaker and \(\alpha_{NUI}\) is smaller, the first term in (5) is not more dominant than the second term. Hence, by treating the NU  
Figure 1: The conceptual diagram of RIS-SR-FNOMA system. signal as interference, the FU signal is detected. The SINR for decoding FU signal, while treating NU signal as interference, is  
\[\gamma_{FUI}^{x_{FUI}}=\frac{B^{2}\alpha_{FUI}P_{s}}{B^{2}\alpha_{NUI}P_{s}+N_{0}}. \tag{6}\]  
At the desired data rate \(\tilde{R}_{FUI}\), the outage for FU occurs when  
\[\log_{2}\left(1+\gamma_{FUI}^{x_{FUI}}\right)\ \leq\ \tilde{R}_{FUI}. \tag{7}\]  
Substituting (6) in (7), the outage probability at the FU is defined as  
\[P_{FUI}^{o,SR}=P\bigg{[}\log_{2}\left(1+\frac{B^{2}\alpha_{FUI}P_{s}}{B^{2} \alpha_{NUI}P_{s}+N_{0}}\right)\leq\tilde{R}_{FUI}\bigg{]}. \tag{8}\]  
Simplifying (8) gives  
\[x_{s}=\sqrt{\alpha_{NUI}P_{s}}\ x_{NUI}+\sqrt{\alpha_{FU}P_{s}}\ x_{FU}, \tag{1}\]0  
where \(R_{FUI}=2^{\tilde{R}_{FUI}}-1\). Since \(\alpha_{i}\) and \(\beta_{i}\) are positive real, \(B\) is always positive. Hence, Equation (9) can be presented as  
\[x_{s}=\sqrt{\alpha_{NUI}P_{s}}\ x_{NUI}+\sqrt{\alpha_{FU}P_{s}}\ x_{FU}, \tag{1}\]1  
As the number of reflecting elements allocated to FU (\(N_{FUI}\)) is larger, according to the central limit theorem (CLT), \(B\) follows the Gaussian probability density function \(f_{B}(b)\) with mean \(m_{B}=\frac{N_{FUI}\pi}{4}\) and variance \(\sigma_{B}^{2}=N_{FUI}\left[1-\frac{\pi^{2}}{16}\right]\).