## 4 Performance Analysis
### Maximum SSE Through Power Fraction Optimization  
The SSE measures the overall capacity of the system and facilitates performance evaluation. The SSE is calculated as the sum of the SE of the individual pieces of UE within the system. PA factors can be optimized for maximizing SSE by dynamically distributing power among the UE based on their channel conditions during direct transmission. In the case of cooperative relaying, there is no power splitting and, therefore, no need for PA optimization. The SSE is given by  
\[C_{p}^{i}=I_{d-s,p}^{i}+max\Big{\{}I_{d-w,p}^{i}\,\ \Gamma_{c-w,p}^{i}\Big{\}} \tag{33}\]  
Providing optimal power for the UE during direct transmission is crucial, particularly for the SUE, as they need to manage a complex SIC process and engage in cooperative communication. To ensure fairness, the SUE receives a lower power fraction. Furthermore, while decoding, the SUE must meet the QoS requirements of the WUE. Consequently, power fractions should be optimized to maximize SSE. The condition for the UE to achieve maximum SSE while satisfying the individual QoS requirements of each piece of UE is outlined as follows [4]:  
\[\max_{\mu_{si}}\ \mu_{ai}\ C_{p}^{i} \tag{34}\]  
Subject to  
\[\Gamma_{d-w,p}^{i}\geq\hat{R}_{w,p}^{i}\] \[\Gamma_{d-s,p}^{i}\geq\hat{R}_{s,p}^{i}\] \[\mu_{si}+\ \mu_{ai}=1\] \[0\ <\mu_{si}\ \leq\ \mu_{ai}<1\]  
The condition for the SUE with no outage is given by \(\Gamma_{d-s,p}^{i}\geq\hat{R}_{s,p}^{i}\). Substituting (3) and solving for \(\mu_{si}\) gives  
\[\mu_{si}\geq\mathcal{A}_{s,p}^{min}=\frac{\varrho_{s,p}^{i}\Big{(}\Big{(} \varepsilon+\left[k_{g}^{2}+k_{g}^{2}\right]\Big{)}G_{s}\rho+1\Big{)}}{\Big{(} \varepsilon\rho_{s,p}^{i}+1\Big{)}G_{s}\rho} \tag{35}\]  
where \(\varrho_{s,p}^{i}\)is the threshold SNR of the SUE. The condition for the WUE experiencing no outage is \(\Gamma_{d-w,p}^{i}\geq\hat{R}_{w,p}^{i}\). Solving this condition gives  
\[\mu_{si}\leq\mathcal{A}_{s,p}^{max}=\frac{G_{w}\rho-\varrho_{w,p}^{i}\Big{(}G _{w}\rho\Big{(}\left[k_{g}^{2}+k_{w}^{2}\right]+1\Big{)}\Big{)}}{G_{w}\rho \Big{(}\varrho_{w,p}^{i}+1\Big{)}} \tag{36}\]  
where \(\varrho_{w,p}^{i}\)is threshold SNR of the WUE. Combining (35) and (36) gives,  
\[\mathcal{A}_{s,p}^{min}\leq\mu_{si}\leq\mathcal{A}_{s,p}^{max} \tag{37}\]  
In this optimization problem, both paired pieces of UE must meet their minimum QoS requirements while maximizing the SSE. In a NOMA system, the WUE is generally assigned the majority of the power, as the SUE benefits from better channel conditions and thus requires less power. However, this PA strategy may fail to satisfy the QoS requirements of both pieces of UE. To ensure that the SUE meets its minimum QoS requirements, the upper limit, as formulated in (37), is allocated accordingly.  
\[\mu_{si}^{opt}=\mathcal{A}_{s,p}^{max} \tag{38}\]  
As \(\mu_{si}+\ \mu_{ai}=1\), the remaining power is allocated to the WUE, which is given by  
\[\mu_{wi}^{opt}=1-\mu_{si}^{opt} \tag{39}\]The feasibility region for the optimal power fractions allocated for the SUE and the WUE can be obtained by the condition  
\[\mathcal{A}_{s,p}^{min}\leq\mathcal{A}_{s,p}^{max} \tag{40}\]  
From (40), the minimum SNR requirement is given by  
\[\rho\geq\frac{\rho_{s,p}^{i}G_{w}\left(\rho_{w,p}^{i}+1\right)+\rho_{w,p}^{i}G_ {s}\left(\rho_{s,p}^{i}+1\right)}{G_{w}G_{s}\left(1-\left(\varepsilon+\left[k_{ g}^{2}+k_{w}^{2}\right]\right)\rho_{s,p}^{i}\rho_{w,p}^{i}-\left[k_{g}^{2}+k_{ g}^{2}\right]\left(\rho_{s,p}^{i}+\rho_{w,p}^{i}+\rho_{s,p}^{i}\rho_{w,p}^{i} \right)\right)} \tag{41}\]