# Energy-Efficient Resource Allocation for Downlink Non-Orthogonal Multiple Access Systems
## 2 System Model and Problem Formulation  
### System Model  
We consider a single-cell multiuser NOMA downlink system in which the base station is located in the center of the cell, and the users are distributed randomly. The system model diagram is shown in Figure 1. In this system, \(M\) users are allocated to \(N\) subchannels, and the maximum number of users multiplexed per subchannel is \(M_{n}=M/N\). The total bandwidth of the system is equally divided into orthogonal subchannels; that is, the bandwidth of each subchannel is \(B_{s}=B/N\). Additionally, the \(m\)th user is denoted as \(U_{m}\), while the \(n\)th subchannel is denoted as \(SC_{n}\). \(U_{m,n}\) represents the \(m\)th user on \(SC_{n}\). In this NOMA system, we assume that the channel state information can be obtained by channel estimation, and the superposition information sent to each subchannel is independent. Then the transmitted superposed signal by the BS on \(SC_{n}\) is  
\[x_{n}=\sum\limits_{m=1}^{M_{n}}\sqrt{p_{m,n}}x_{m,n} \tag{1}\]  
where \(p_{m,n}\) is the power allocated to \(U_{m,n}\), \(x_{m,n}\) is the signal sent by user \(U_{m}\) on \(SC_{n}\). \(p_{n}=\sum\limits_{m=1}^{M_{n}}p_{m,n}\) indicates the transmit power on the subchannel \(n\). When the superimposed signal is transmitted to the receiving end through the channel, the received signal of \(U_{m,n}\) is given by  
\[y_{m,n}=h_{m,n}x_{n}+z_{m,n}=\sqrt{p_{m,n}}h_{m,n}x_{m,n}+h_{m,n}\sum\limits_{ i=1,j\
eq m}^{M_{n}}\sqrt{p_{i,n}}x_{i,n}+z_{m,n} \tag{2}\]  
where \(h_{m,n}\) is the channel gain from the base station to \(U_{m,n}\). The additive white Gaussian noise (AWGN) is represented by \(z_{m,n}\) with zero mean and \(\sigma_{n}^{2}\) variance, that is, \(z_{m,n}\sim\mathcal{CN}\big{(}0,\sigma_{n}^{2}\big{)}\).  
Since NOMA multiplexes several users on the same sub-channel, multiple users are not orthogonal to each other. Therefore, co-channel interference between users is inevitable. In this case, each user eliminates interference from other users' signals through SIC at the receiving end. Due to the users' fairness, the base station allocates less power to users with good channel conditions and more power to users with poor channel conditions. The SIC performs data decisions on a plurality of users according to the user power in the received signal. In this process, once the SIC demodulates a user, the interference caused by the signal is subtracted. This operation is generally performed according to the increasing order of channel response normalized by noise (CRNN). In this system, we assume that  
Figure 1: Downlink multicarrier NOMA system model.  
\(|H_{1,n}|\geq|H_{2,n}|\geq\cdots\geq|H_{m,n}|\geq|H_{m+1,n}|\geq\cdots\geq|H_{m,n}|\) and \(H_{m,n}=|h_{m,n}|^{2}/\sigma_{n}^{2}\). After SIC demodulation, in accordance with [PERSON]'s capacity law, the swallow volume of \(U_{m,n}\) can be written as  
\[R_{m,n}=B_{s}\text{log}_{2}\Bigg{(}1+\frac{p_{m,n}H_{m,n}}{1+H_{m,n}\sum\limits _{i=1}^{m-1}p_{i,n}}\Bigg{)}=B_{s}\text{log}_{2}(1+SINR_{m,n}) \tag{3}\]  
Then, the sum rate of the \(\text{SC}_{n}\) can be written as  
\[R_{n}=B_{s}\sum\limits_{i=1}^{M_{n}}\text{log}_{2}(1+\text{SINR}_{i,n}) \tag{4}\]