# RIS-Assisted Fixed NOMA: Outage Probability Analysis and Transmit Power Optimization
## 3 Outage Probability Analysis of RIS-SR-FNOMA
The outage probability at FU is  
\[x_{s}=\sqrt{\alpha_{NUI}P_{s}}\ x_{NUI}+\sqrt{\alpha_{FU}P_{s}}\ x_{FU}, \tag{1}\]2  
where \(y_{FUI}=\sqrt{\frac{R_{FUI}N_{0}}{(\alpha_{FUI}-\alpha_{NUI}R_{FUI})P_{s}}}\). Substituting \(f_{B}(b)\) and integrating within the limit yields the outage probability as  
\[x_{s}=\sqrt{\alpha_{NUI}P_{s}}\ x_{NUI}+\sqrt{\alpha_{FU}P_{s}}\ x_{FU}, \tag{1}\]3  
where \(erf(. )\) is the error function. The signal received by the NU is  
\[x_{s}=\sqrt{\alpha_{NUI}P_{s}}\ x_{NUI}+\sqrt{\alpha_{FU}P_{s}}\ x_{FU}, \tag{1}\]4  
where \(n_{NUI}\) is the AWGN at the NU and \(n_{NUI}\in\mathcal{CN}(0,N_{0})\). Substituting channel conditions and (2) in (13) gives  
\[x_{s}=\sqrt{\alpha_{NUI}P_{s}}\ x_{NUI}+\sqrt{\alpha_{FU}P_{s}}\ x_{FU}, \tag{1}\]5  
Let \(A=\sum_{i=1}^{N}\alpha_{i}\beta_{i}\) be the cumulative effect of the dual-hop channel for the NU. Substituting \(A\) and (1) in (14) gives  
\[x_{s}=\sqrt{\alpha_{NUI}P_{s}}\ x_{NUI}+\sqrt{\alpha_{FU}P_{s}}\ x_{FU}, \tag{1}\]6Since \(A\) and \(\alpha_{FU}\) are higher, the second term in (15) is more dominant than the first term. Hence, the FU signal is detected first by the NU. The SINR of decoding the FU signal at the NU is  
\[x_{s}=\sqrt{\alpha_{NUI}P_{s}}\ x_{NUI}+\sqrt{\alpha_{FU}P_{s}}\ x_{FU}, \tag{1}\]7  
After decoding \(x_{FU}\), its effect is removed from \(r_{NUI}\) using SIC. \[x_{s}=\sqrt{\alpha_{NUI}P_{s}}\ x_{NUI}+\sqrt{\alpha_{FU}P_{s}}\ x_{FU}, \tag{1}\]8  
Then, the NU signal is detected. The SNR for decoding \(x_{NUI}\) at NU is given by  
\[x_{s}=\sqrt{\alpha_{NUI}P_{s}}\ x_{NUI}+\sqrt{\alpha_{FU}P_{s}}\ x_{FU}, \tag{1}\]9  
The decoding of the FU signal fails at NU when  
\[\phi_{i}=-\theta_{i}-\psi_{i},i=1,2,\ldots,N. \tag{2}\]0  
Substituting (16) in (19), the outage probability is given by  
\[\phi_{i}=-\theta_{i}-\psi_{i},i=1,2,\ldots,N. \tag{2}\]1  
Simplifying (20) results in  
\[\phi_{i}=-\theta_{i}-\psi_{i},i=1,2,\ldots,N. \tag{2}\]2  
Since \(\alpha_{i}\) and \(\beta_{i}\) are positive real, \(A\) is always positive. Hence, Equation (21) can be written as  
\[\phi_{i}=-\theta_{i}-\psi_{i},i=1,2,\ldots,N. \tag{2}\]3  
As the number of reflecting elements allocated to NU is larger, according to CLT, \(A\) follows Gaussian distribution \(f_{A}(a)\) with mean \(m_{A}=\frac{N_{NUI}\pi}{4}\) and variance \(\sigma_{A}^{2}=N_{NUI}\left[1-\frac{\pi^{2}}{16}\right]\). At the desired rate of \(\bar{R}_{NUI}\) at NU, the decoding of the NU signal fails at NU when  
\[\phi_{i}=-\theta_{i}-\psi_{i},i=1,2,\ldots,N. \tag{2}\]4  
Substituting (18) in (23) gives  
\[\phi_{i}=-\theta_{i}-\psi_{i},i=1,2,\ldots,N. \tag{2}\]5  
Simplifying (24), the outage probability at NU is  
\[\phi_{i}=-\theta_{i}-\psi_{i},i=1,2,\ldots,N. \tag{2}\]6  
where \(R_{NUI}=2^{\bar{R}_{NUI}}-1\). By combining (22) and (25), the overall condition for which decoding of NU signal fails at NU is  
\[\phi_{i}=-\theta_{i}-\psi_{i},i=1,2,\ldots,N. \tag{2}\]7The outage probability of NU is  
\[\phi_{i}=-\theta_{i}-\psi_{i},i=1,2,\ldots,N. \tag{2}\]8  
where  
\[\phi_{i}=-\theta_{i}-\psi_{i},i=1,2,\ldots,N. \tag{2}\]9  
After mathematical simplification, the outage probability \(P_{NU}^{o,SR}\) at the NU is given by  
\[r_{FU}=\left(\sum_{i=\frac{N}{2}+1}^{N}g_{i}h_{i}e^{i\phi_{i}}\right)x_{s}+n_ {FU}, \tag{3}\]0