## 3 System Model
### Fair PA Analysis  
The fair PA in SISO-NOMA downlink communication is crucial to ensure that all system users (\(U_{1}\), \(U_{2}\),..., \(U_{N}\)) have a guaranteed QoS and to maximise the sum rate. Thus, fair PA promotes user fairness, balances interference (undesired signal components) and enhances spectral efficiency. Moreover, the PA coefficients (\(\alpha_{1}\), \(\alpha_{2}\),..., \(\alpha_{N}\)) are directly dependent on channel conditions. Without considering the channel conditions, the fixed PA coefficients are used, where the outage probabilities of the users are higher with a lower sum rate (\(\text{bps}/\text{Hz}\)). Considering the CSI, the PA coefficients can be fairly optimised to improve the system performance. The fair PA focuses on the far user (User 1 in our case), as the far user is weak and located relatively far away from the BS compared to other users. To better evaluate the SISO-NOMA scheme, let us consider two users (User 1--far user, and User 2--near user) presented in the system. From Equation (12), the achievable sum rate for Users 1 and 2, respectively, can be given as  
\[R_{U_{1}}=\log_{2}\biggl{(}1+\frac{P_{T}\alpha_{1}|h_{1}|^{2}}{P_{T}\alpha_{2}| h_{1}|^{2}+\sigma^{2}}\biggr{)} \tag{15}\]  
\[R_{U_{2}}=\log_{2}\biggl{(}1+\frac{P_{T}\alpha_{2}|h_{2}|^{2}}{\sigma^{2}} \biggr{)} \tag{16}\]  
Hence, in this case, the PA coefficients (\(\alpha_{1}\) and \(\alpha_{2}\)) are designed/derived by setting up the target rate (\(R_{T}\)) for the far user (User 1), which is less than or equal to the sum rate, i.e., \(R_{T}\leq R_{U_{1}}\), from Equation (15). Once the \(R_{T}\) for User 1 is met, the optimised PA coefficients can be derived instead of having a fixed PA. Therefore, Equation (15) becomes  
\[R_{T}=\log_{2}\biggl{(}1+\frac{P_{T}\alpha_{1}|h_{1}|^{2}}{P_{T}\alpha_{2}|h_{ 1}|^{2}+\sigma^{2}}\biggr{)} \tag{17}\]  
\[2^{R_{T}}-1=\frac{P_{T}\alpha_{1}|h_{1}|^{2}}{P_{T}\alpha_{2}|h_{1}|^{2}+ \sigma^{2}}\]  
Let us consider \(2^{R_{T}}-1=\beta\)  
\[\beta=\frac{P_{T}\alpha_{1}|h_{1}|^{2}}{P_{T}\alpha_{2}|h_{1}|^{2}+\sigma^{2}}\]  
\[\beta P_{T}\alpha_{2}|h_{1}|^{2}+\beta\sigma^{2}=P_{T}\alpha_{1}|h_{1}|^{2} \tag{18}\]  
We know that the sum of the PA coefficients is equal to 1 as we have two users (User 1 and User 2, having PA coefficients of \(\alpha_{1}\) and \(\alpha_{2}\), respectively), \(\alpha_{1}+\alpha_{2}=1\). Therefore, Equation (18) implies  
\[\alpha_{1}=\frac{\beta(P_{T}|h_{1}|^{2})+\sigma^{2}}{(1+\beta)P_{T}|h_{1}|^{2} };\alpha_{1}<1 \tag{19}\]  
Hence, Equation (19) can be also written as  
\[\alpha_{1}=\min\biggl{(}\frac{\beta(P_{T}|h_{1}|^{2})+\sigma^{2}}{(1+\beta)P_ {T}|h_{1}|^{2}},1\biggr{)}\]  
Once the PA coefficient (\(\alpha_{1}\)) of a far user (User 1) is calculated, the PA coefficient (\(\alpha_{2}\)) of a near user (User 2) can be calculated using  
\[\alpha_{2}=1-\alpha_{1} \tag{20}\]  
In this case, if the calculation for the term \(\left(\frac{\beta(P_{T}|h_{1}|^{2})+\sigma^{2}}{(1+\beta)P_{T}|h_{1}|^{2}} \right)>1\) (let us consider 20), then \(\alpha_{1}=\min\biggl{(}\frac{\beta(P_{T}|h_{1}|^{2})+\sigma^{2}}{(1+\beta)P_ {T}|h_{1}|^{2}},1\biggr{)}=1\), which does not satisfy the condition \(\alpha_{1}+\alpha_{2}=1\). If \(\alpha_{1}\) is designed to have 20, then User 1 will meet the target rate (\(R_{T}\)). But, if \(\alpha_{1}<20\), User 1 will not meet \(R_{T}\), leading to an outage condition. Moreover, \(\alpha_{1}\) cannot hold the value 20, which violates the condition \(\alpha_{1}+\alpha_{2}=1\). Consequently, even if User 1's \(\alpha_{1}\) holds the value 1 (not meeting the considered condition \(\alpha_{1}=20\)), User 1 will be in an outage situation. In contrast, if \(\alpha_{1}=1\), then \(\alpha_{2}=1-\alpha_{1}\) will be 0. This infers that User 2 will not have any PA coefficient. Hence, User 2 will also be in an outage condition. The solution for this outage problem of Users 1 and 2 can be resolved by setting up \(\alpha_{1}\) as 0 (no PA) and \(\alpha_{2}\) as 1 (entire PA).