# Power Allocation and User Grouping for NOMA Downlink Systems
## 1 Introduction
### Related Work  
In the research into user grouping strategies, the best scheme is the exhaustive search method. An algorithm traverses all the user grouping methods and selects the user combination that achieves the best system performance. However, the complexity of this method increases exponentially, so it is not used in practical systems. In [8], a predefined threshold method was proposed which compares the channel gain of users with the threshold value. The users whose channel gain is higher than the threshold are grouped together. The users whose channel gain is lower than the threshold are grouped into another group. One user is selected from each of the two groups to form a NOMA group. However, the performance of this method is affected by the threshold selection, and the threshold selection is not mentioned. In [9], the authors provided a user grouping method based on maximum channel gain difference. Firstly, all users are sorted according to the channel gain. Then, the users with the highest and the lowest channel gain are paired into the first NOMA group. The users with the second highest and second lowest channel gain are paired into the second group. However, as the grouping process progresses, the channel gain difference between users decreases gradually, which aggravates the interference between users, leading to performance degradation. In [10], the authors proposed a method according to matching theory, which considers the process of determining two users on a subcarrier as a two-sided matching problem between users and subchannels. A downlink NOMA system was considered in [11]. In the case of only two users, the optimal power allocation of users under the minimum rate constraint is obtained first. Then, under the obtaining power, the user grouping problem is analyzed in the case of four users to optimize the sum rate. Finally, the closed-form expression of the optimal user pairing in a general case is given. However, this method requires obtaining the power of each user first. Moreover, heuristic algorithms such as the hill climbing pairing algorithm [12], the simulated annealing algorithm [12; 13], the Hungarian algorithm [14], the particle swarm optimization algorithm [15] and the artificial neural network approach [16] have also been applied to user pairing problems. However, heuristic algorithms are unstable and need iterative calculations to obtain the optimal solution, which has high complexity. The wireless communication system needs to complete the resource allocation in milliseconds; therefore, user grouping methods require low complexity and ensured system performance. In the research into power allocation, the full space search algorithm can lead to the best performance of NOMA systems. It searches the entire power allocation space according to the minimum search interval and obtains the optimal power allocation scheme by traversing all power allocation situations. However, the complexity of this algorithm is very high, so it is generally not used in practice. In [1], fixed power allocation (FPA) was proposed to allocate power to users according to a fixed factor without considering the channel conditions. The algorithm complexity is very low, but the system performance is also poor. In [4], a suboptimal fractional transmit power allocation (FTPA) algorithm was adopted which allocates power according to the channel conditions. Moreover, the computational complexity is low and a better system performance can be achieved. However, this algorithm is a local optimization algorithm. In [9], by applying the Karush-Kuhn-Tucker (KKT) condition, a closed-form solution was deduced to optimize the system throughput. The optimal global solution is obtained by mathematical theory by establishing an optimization problem with inequality constraints. In [17], the authors optimized the weighted sum rate and a suboptimal solution was obtained by step-by-step optimization. First of all, the total power is equally distributed to each subcarrier.