# Efficient Resource Management for Sum Capacity Maximization in 5G NOMA Systems
## 3 Resource Management With Fixed Power Control
### User Minimum Capacity Requirement Based Sub-Channel Algorithm
We first calculate the minimum transmit power to satisfy the minimum capacity requirement of \(CUl_{i}\) and \(CUl_{j}\) on \(SC_{m}\), which can be derived as:  
\[p_{i,m}^{*}=\frac{(2^{\bar{C}_{i,m}^{\min}}-1)n_{0}}{|h_{i,m}|^{2}}, \tag{21}\]  
\[p_{j,m}^{*}=\frac{(2^{\bar{C}_{j,m}^{\min}}-1)(p_{i,m}^{*}|h_{j,m}|^{2}+n_{0}) }{|h_{j,m}|^{2}}. \tag{22}\]  
The remaining power after capacity requirements of \(CUl_{i}\) and \(CUl_{j}\) on \(SC_{m}\) which is denoted as \(P_{r}\) and can be given by  
\[P_{r}=P_{m}-(p_{i,m}^{*}+p_{j,m}^{*}), \tag{23}\]  
In this article, our aim is to optimize the remaining power \(P_{r}\) among \(CUl_{i}\) and \(CUl_{j}\) on \(SC_{m}\). With \(p_{i,m}^{*}\) and \(p_{j,m}^{*}\), the sum capacity problem (P3) can be modified as:  
\[\text{(P4)} \tag{24}\] \[\text{s.t.} \log_{2}\left(1+\frac{p_{i,m}|h_{i,m}|^{2}}{n_{0}}\right)\geq \bar{C}_{i,m}^{\min}, \tag{16}\]0 (25) \[\text{s.t.} \log_{2}\left(1+\frac{p_{i,m}|h_{i,m}|^{2}}{n_{0}}\right)\geq \bar{C}_{i,m}^{\min}, \tag{16}\]1 (26) \[\text{s.t.} \log_{2}\left(1+\frac{p_{i,m}|h_{i,m}|^{2}}{n_{0}}\right)\geq \bar{C}_{i,m}^{\min}, \tag{16}\]2  
where (25) in (P4) can be satisfied with equality by utilizing the NOMA downlink framework [10]. Now let \(\alpha\) be the fraction of power allocation to \(CUl_{i}\) over sub-channel \(m\), then the transmit power of \(CUl_{i}\) and \(CUl_{j}\) on \(SC_{m}\) can be respectively expressed as \(p_{i,m}=\alpha P_{r}\) and \(P_{j,m}=(1-\alpha)P_{r}\). Note that the value of \(\alpha\) lies between 0 and 1. By introducing \(\alpha\), the optimization problem (P4) can be further simplified as (P5):  
\[\text{s.t.} \log_{2}\left(1+\frac{p_{i,m}|h_{i,m}|^{2}}{n_{0}}\right)\geq \bar{C}_{i,m}^{\min}, \tag{16}\]3 (28) s.t. \[\text{s.t.} \log_{2}\left(1+\frac{p_{i,m}|h_{i,m}|^{2}}{n_{0}}\right)\geq \bar{C}_{i,m}^{\min}, \tag{16}\]4 \[\text{s.t.} \log_{2}\left(1+\frac{p_{i,m}|h_{i,m}|^{2}}{n_{0}}\right)\geq \bar{C}_{i,m}^{\min}, \tag{16}\]5  
where the last constraint in (30) ensures that the fraction power of \(CU_{i}\) on \(SC_{m}\) should be less than or equal to 0.5. The problem (P5) is a standard convex optimization problem and dual method can be employed to obtain the efficient solution [11]. The corresponding dual problem is  
\[\text{s.t.} \log_{2}\left(1+\frac{p_{i,m}|h_{i,m}|^{2}}{n_{0}}\right)\geq \bar{C}_{i,m}^{\min}, \tag{16}\]6  
where \(D(\lambda,\eta)\) is called the dual function and can be given by  
\[\text{s.t.} \log_{2}\left(1+\frac{p_{i,m}|h_{i,m}|^{2}}{n_{0}}\right)\geq \bar{C}_{i,m}^{\min}, \tag{16}\]7  
The Lagrangian function of optimization problem (P5) can be defined as:  
\[ \begin{split}\mathcal{L}(\alpha,\lambda,\eta)=&\log _{2}\left(1+\frac{(\alpha P_{r}+p_{i,m}^{*})|h_{i,m}|^{2}}{n_{0}}\right)+\log_ {2}\left(1+\frac{((1-\alpha)P_{r}+p_{j,m}^{*})|h_{j,m}|^{2}}{|h_{j,m}|^{2}( \alpha P_{r}+p_{i,m}^{*})+n_{0}}\right)\\ &-\lambda(2\alpha P_{r}+p_{i,m}^{*}-P_{r}-p_{j,m}^{*})-\eta( \alpha-0.5),\end{split}  \tag{33}\]  
Under high signal to noise ratio (SNR) approximation, from the properties of log, we obtain  
\[ \begin{split}\mathcal{L}(\alpha,\lambda,\eta)=&\log _{2}((\alpha P_{r}+p_{i,m}^{*})|h_{i,m}|^{2})-\log_{2}(n_{0})+\log_{2}(|h_{j,m} |^{2}((1-\alpha)P_{r}+p_{j,m}^{*}))\\ &-\log_{2}(|h_{j,m}|^{2}(\alpha P_{r}+p_{i,m}^{*})+n_{0})-\lambda (2\alpha P_{r}+p_{i,m}^{*}-P_{r}-p_{j,m}^{*})-\eta(\alpha-0.5).\end{split}  \tag{34}\]  
Now differentiating by \(\alpha\) as \(\frac{\partial\mathcal{L}(\alpha,\lambda,\eta)}{\partial\alpha}\), we obtain  
\[ \begin{split}\frac{\partial\mathcal{L}(\alpha,\lambda,\eta)}{ \partial\alpha}=&\frac{1}{\ln_{2}}\bigg{(}\frac{P_{r}|h_{i,m}|^{2} }{|h_{i,m}|^{2}(\alpha P_{r}+p_{i,m}^{*})}\bigg{)}+\frac{1}{\ln_{2}}\bigg{(} \frac{-P_{r}|h_{j,m}|^{2}}{|h_{j,m}|^{2}((1-\alpha)P_{r}+p_{j,m}^{*})}\bigg{)} \\ &-\frac{1}{\ln_{2}}\bigg{(}\frac{P_{r}|h_{j,m}|^{2}}{|h_{j,m}|^{2} (\alpha P_{r}+p_{i,m}^{*})+n_{0}}\bigg{)}-2\lambda P_{r}-\eta,\end{split}