# RIS-Assisted Fixed NOMA: Outage Probability Analysis and Transmit Power Optimization
## 5 Transmit Power Optimization for Sum Capacity Maximization  
In this section, optimal power allocation strategies for obtaining maximum sum capacity in RIS-SR-FNOMA and RIS-AP-NOMA systems are discussed in detail. The sum capacity of downlink RIS-SR-FNOMA for a two-user scenario is given by  
\[C_{N\bar{U}}+C_{F\bar{U}}=\log_{2}\big{(}1+\tau_{N\bar{U}}^{x_{N\bar{U}}}\big{)} \ +\log_{2}\big{(}1+\gamma_{F\bar{U}}^{x_{F\bar{U}}}\big{)}\, \tag{37}\]  
where \(C_{N\bar{U}}\) and \(C_{F\bar{U}}\) are the capacities of NU and FU, respectively. The optimization problem for determining the power allocation factor \(\alpha_{N\bar{U}}\) and \(\alpha_{F\bar{U}}\) of both NU and FU is formulated as maximizing the sum capacity while satisfying the quality-of-service (QoS) constraints at both NU and FU. It is defined as  
\[\max_{\alpha_{N\bar{U}},\alpha_{F\bar{U}}}\qquad C_{N\bar{U}}+C_{F\bar{U}}. \tag{38}\]  
Subject to  
\[\log_{2}\big{(}1+\gamma_{N\bar{U}}^{x_{N\bar{U}}}\big{)}\ \geq\ \bar{R}_{N\bar{U}}. \tag{39}\]  
\[\log_{2}\big{(}1+\gamma_{F\bar{U}}^{x_{F\bar{U}}}\big{)}\ \geq\ \bar{R}_{F\bar{U}}. \tag{40}\]  
\[\alpha_{N\bar{U}}+\alpha_{F\bar{U}}=1. \tag{41}\]  
Substituting (6) and (18) in (37) gives  
\[C_{N\bar{U}}+C_{F\bar{U}}=\log_{2}\!\left(1+\frac{A^{2}\alpha_{N\bar{Simplifying (42), it results in  
\[C_{NUI}+C_{FU} =\log_{2}\left(\left(1+B^{2}\chi\right)\left(\frac{1+A^{2}\alpha_{ NU}\chi}{1+B^{2}\alpha_{NU}\chi}\right)\right)\,. \tag{43}\]  
where \(\chi=\frac{P_{t}}{N_{0}}\) is the received SNR. The term \(\left(1+B^{2}\chi\right)\) is constant. As the term \(\left(\frac{1+A^{2}\alpha_{NU}\chi}{1+B^{2}\alpha_{NU}\chi}\right)\) is a monotonically increasing function of \(\alpha_{NU}\), the maximum sum capacity can be achieved by assigning a maximum possible value for \(\alpha_{NU}\). By simplifying (43), the condition for \(\alpha_{NU}\), which satisfies the QoS constraints, is given by  
\[\underbrace{\frac{R_{NU}}{A^{2}\chi}}_{\alpha_{NU}^{min}}\leq\alpha_{NU}\leq \underbrace{\frac{B^{2}\chi-R_{FU}}{B^{2}\chi(1+R_{FU})}}_{\alpha_{NU}^{max}}. \tag{44}\]  
Any value between \(\alpha_{NU}^{min}\) and \(\alpha_{NU}^{max}\) is sub-optimal for \(\alpha_{NU}\). In simulations, the average of \(\alpha_{NU}^{min}\) and \(\alpha_{NU}^{max}\) is assumed to be \(\alpha_{NU}\). The sub-optimal power assigned to NU is  
\[\alpha_{NU}^{sub-opt}=\frac{\left(\alpha_{NU}^{min}+\alpha_{NU}^{max}\right) }{2}. \tag{45}\]  
The balance power is assigned to the FU:  
\[\alpha_{FU}^{sub-opt}=1-\alpha_{NU}^{sub-opt}. \tag{46}\]  
To find the optimal value of \(\alpha_{NU}\), the first threshold \(\alpha_{NU}^{min}\) should be less than the second threshold \(\alpha_{NU}^{max}\)  
\[\frac{R_{NU}}{A^{2}\chi}\leq\frac{B^{2}\chi-R_{FU}}{B^{2}\chi(1+R_{FU})}. \tag{47}\]  
By solving (47), the minimal SNR demand to satisfy the QoS constraints of users is attained. This is called a feasible region:  
\[\chi\geq\frac{R_{NU}}{A^{2}}+\frac{R_{NU}R_{FU}}{A^{2}}+\frac{R_{FU}}{B^{2}}. \tag{48}\]  
If the condition in (48) is not met, then it is not possible to choose the value of \(\alpha_{NU}\). The largest value of \(\alpha_{NU}\) is the optimal value of \(\alpha_{NU}\), which maximizes the sum capacity:  
\[\alpha_{NU}^{opt}=\frac{B^{2}\chi-R_{FU}}{B^{2}\chi(1+R_{FU})}. \tag{49}\]  
The balance power is assigned to FU, which is  
\[\alpha_{FU}^{opt}=1-\alpha_{NU}^{opt}. \tag{50}\]  
In a similar way, the optimal powers to be assigned for the users of RIS-AP-FNOMA are derived. The optimal power assigned to the NU of RIS-AP-FNOMA system is given by  
\[\alpha_{NU}^{opt}=\frac{C^{2}\chi-R_{FU}}{C^{2}\chi_{s}(1+R_{FU})}. \tag{51}\]  
The balance power is assigned to the FU as in (50). The minimal SNR demand to meet the QoS constraints of the users of RIS-AP-FNOMA is given by  
\[\chi\geq\frac{R_{NU}}{D^{2}}+\frac{R_{NU}R_{FU}}{D^{2}}+\frac{R_{FU}}{C^{2}}. \tag{52}\]