# Power Allocation and User Grouping for NOMA Downlink Systems
## 2 System Model  
We consider the downlink of a NOMA system [22]. Figure 1 shows the concept of the system. We assume that there is a BS and \(M\) users in a cell. The BS is located in the center of the cell, and \(M\) users are randomly distributed. The BS and users are equipped with a single antenna. The total bandwidth of the system, \(B_{0}\), is equally distributed to \(N\) subchannels and the bandwidth of each subchannel is \(B=B_{0}/N\). \(P_{T}\) denotes the total transmit power. If the number of users on a subchannel is \(M_{n}(n=1,2,\cdots,N)\), \(M=\sum_{n=1}^{N}M_{n}\) and the power of the \(m\)-th user on the \(n\)-th subchannel is \(p_{n,m}(m=1,2,\cdots,M_{n})\). The BS sends a signal with superposition-coded symbols on the subchannel \(n\) as follows:  
\[x_{n}=\sum_{m=1}^{M_{n}}\sqrt{p_{n,m}}\cdot x_{n,m} \tag{1}\]  
where \(x_{n,m}\) is the transmitted symbol of the \(m\)-th user on the subchannel \(n\).  
The received signal at the user \(m\) is expressed as follows:  
\[y_{n,m}=h_{n,m}\cdot x_{n}+w_{n,m}=h_{n,m}\sum_{i=1}^{M_{n}}\sqrt{p_{n,i}} \cdot x_{n,i}+w_{n,m} \tag{2}\]  
where \(h_{n,m}\) represents the channel coefficient from the BS to the user \(m\), including path loss and Rayleigh fading. \(w_{n,m}\) denotes additive white Gaussian noise with mean \(0\) and variance \(\sigma_{n}^{2}\), where \(w_{n,m}\sim CN\big{(}0,\sigma_{n}^{2}\big{)}\). If ee expand Equation (2) further:  
\[y_{n,m}=h_{n,m}\sqrt{p_{n,m}}\cdot x_{n,m}+h_{n,m}\sum_{i=1,j\
eq m}^{M_{n}} \sqrt{p_{n,i}}\cdot x_{n,i}+w_{n,m}, \tag{3}\]  
we can see that the received signal includes its useful signal and other interference signals. \(h_{n,m}\sum_{i=1,j\
eq m}^{M_{n}}\sqrt{p_{n,i}}\cdot x_{n,i}\) is the inter-user interference signal on the subcarrier \(n\).  
According to SIC technology, a user signal with a high power will be demodulated first, followed by a user signal with a low power. The order of power allocated to users is \(p_{1}<p_{2}<\cdots<p_{M_{n}}\). Therefore, the user \(m\) can decode and successfully remove the inter-user interference from users \(m+1,\cdots,M_{n}\). However, the signal power from users \(1\) to \(m-1\), who have better channel conditions than the user \(m\), will be considered intra-cell interference. Then, the receiver is processed by SIC and according to the Shannon theorem, the rate of the user \(m\) can be expressed as:  
\[R_{n,m}=B\log_{2}\!\left(1+\frac{\left|h_{n,m}\right|^{2}p_{n,m}}{\left|h_{n,m }\right|^{2}\sum_{i=1}^{m-1}p_{n,i}+\sigma_{n}^{2}}\right) \tag{4}\]  
Consequently, the sum rate can be expressed as:  
\[R_{n}=\sum_{m=1}^{M_{n}}R_{n,m} \tag{5}\]  
Figure 1: The downlink of a NOMA network.  
From (4) and (5) we can see that different user combination methods and power allocation results directly impact the total system rate. Therefore, determining the multiplied users and power to reach the system's optimal performance is very important. In this paper, we study how to maximize the sum rate of the NOMA system. The total transmit power and minimum transmission rate are the limiting conditions. The maximum system sum rate is modeled as below:  
\[\max_{P_{n,m}}\sum_{n}^{N}\sum_{m=1}^{M_{n}}R_{n,m} \tag{6}\]  
\[\text{s.t.}\left\{\begin{array}{l}C1:\sum_{n=1}^{N}\sum_{m=1}^{M_{n}}p_{n,m} \leq P_{T}\\ C2:p_{n,m}\geq 0\\ C3:R_{n,m}\geq R_{\text{min}}\end{array}\right. \tag{7}\]  
where \(R_{\text{min}}\) is the minimum data rate. The maximum power \(P_{T}\) limit of the BS is guaranteed according to \(C1\). \(C2\) indicates that the power allocated to users cannot be less than 0. The user's actual rate cannot be lower than the minimum rate to guarantee the user's quality of service in \(C3\).  
The above problem is a non-convex optimization problem; therefore, it is difficult to solve the optimal global solution directly. Therefore, we adopt a step-by-step optimization to find a suboptimal solution. Users are grouped first and then power is allocated.