# Efficient Resource Management for Sum Capacity Maximization in 5G NOMA Systems
## 2 System Model and Problem Formulation  
As depicted in Figure 1, a NOMA network exploiting power multiplexing in downlink transmission is considered, where a single BS which is located at the center of the cell and communicates with \(K\) different users using NOMA protocol. The \(K\) is an even integer and we assume that \(\mathcal{K}=\{1,2,3,\ldots,K\}\) denotes the set of \(K\) cellular users. We divide the total spectrum of the system into \(M\) sub-channels uniformly, where \(\mathcal{M}=\{1,2,3,\ldots,M\}\) denotes the set of \(M\) sub-channels. This network considers single antenna configuration for all its associated nodes. The information of all users is assumed to be available at the source. In addition, we consider independent and identically distributed (i.i.d) Rayleigh fading channels between BS and users. According to the NOMA protocol, we assume that a sub-channel can accommodate more than one user at one time. However, to reduce the SIC complexity at the users, we restrict a sub-channel to accommodate no more than \(U_{\text{max}}\) users at one time. This condition also helps to minimize the interference among NOMA users which operate on the same sub-channel. We denote the transmit power of cellular user \(CU_{j}\in\mathcal{K}\) on sub-channel \(SC_{m}\in\mathcal{M}\) by \(p_{j,m}\) while the channel gain between BS and \(CU_{j}\) by \(h_{j,m}\), respectively. The transmit power should satisfy \(\sum\limits_{j=1}^{K}p_{j,m}\leq P_{BS}\), where \(P_{BS}\) is the total power of BS. If \(U_{m}\) denotes the set of users on \(SC_{m}\) and \(\omega_{i,m}\) denotes the signal of \(CU_{i}\in U_{m}\) on \(SC_{m}\). Then, the received signal at \(CU_{j}\) on \(SC_{m}\) can be given by  
\[y_{j,m}=h_{j,m}\sum\limits_{i\in U_{m}}\sqrt{p_{i,m}}\omega_{i,m}+x_{j,m}, \tag{1}\]  
where \(x_{j,m}\) in (1) is the additive white Gaussian noise of \(CU_{j}\) on \(SC_{m}\) with zero mean and variance \(n_{0}\). As the \(SC_{m}\) can accommodate \(U_{m}\) users, the \(CU_{j}\) in \(U_{m}\) can receives interference from any other \(CU_{i}\) if \(h_{j,m}/n_{0}<h_{i,m}/n_{0}\). However, \(U_{j}\) will apply SIC to the users in \(U_{m}\) which have weak channel gains. The achievable capacity of \(CU_{j}\) on \(SC_{m}\) can be given as  
\[C_{j,m}=\log_{2}\left(1+Y_{j,m}\right), \tag{2}\]  
where \(Y_{j,m}\) in (2) is the SINR of \(CU_{j}\) on \(SC_{m}\) and can be defined as  
\[Y_{j,m}=\frac{p_{j,m}|h_{j,m}|^{2}}{|h_{j,m}|^{2}\sum\limits_{i=j+1}^{m}p_{i, m}+n_{0}}. \tag{3}\]  
where the denominator term \(|h_{j,m}|^{2}\sum\limits_{i=j+1}p_{i,m}\) in (3) denotes the interference of other users in \(U_{m}\) which have strong channel gains than \(U_{j}\). On the account of sub-channel allocation to the users, we define a \(M\times K\) sub-channel matrix \(H\), where \(\beta_{j,m}\in\{0,1\}\) denotes whether \(SC_{m}\) is assigned to \(CU_{j}\). In this paper, we evaluate the network performance by sum capacity of the users which is given by  
\[C_{\text{sum}}=\sum\limits_{j=1}^{K}\sum\limits_{m=1}^{M}\beta_{j,m}C_{j,m}. \tag{4}\]  
Our objective is to optimize \(\{\beta_{j,m},p_{j,m}\}\) in each time slot to enhance the sum capacity of the NOMA network subject to guarantee the minimum user capacity. To do so, we need to investigate the resource optimization problem which can be formed as:(P1) \[\underset{\beta_{j,m}P_{j,m}}{\text{maximize}}\bigg{\{}C_{\text{sum} }=\sum_{j=1}^{K}\sum_{m=1}^{M}\beta_{j,m}C_{j,m}\bigg{\}}\] (5) s.t. \[\sum_{m=1}^{M}\beta_{j,m}\leq 1,\ \forall j,\] (6) \[\sum_{j=1}^{K}\beta_{j,m}\leq U_{\text{max}},\ \forall m,\] (7) \[C_{j,m}\geq\bar{C}_{\text{min}},\ \forall j,\forall m,\] (8) \[\sum_{j=1}^{K}\sum_{m=1}^{M}\beta_{j,m}p_{j,m}\leq P_{BS},\] (9) \[p_{j,m}\geq 0,\ \forall j,\forall m.\] (10)  
where (5) denotes the objective function for sum capacity maximization. Constraint in (6) ensures that \(CU_{j}\) has access to no more than one sub-channel at one time. Constraint in (7) shows that a sub-channel can accommodates no more than \(U_{\text{max}}\) users at one time.