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We evaluate a downlink non-orthogonal multiple access (NOMA)-enable UAV-aided communication system to address the demand of spectrum usage of unmanned aerial vehicles (UAVs). In this paper, multiple NOMA users are served by an UAV to improve the effective spectrum usage. Over Nakagami- $m$ fading model, the performance of the system was investigated based on the analysis of outage probabilities (OPs) of the NOMA users, the ergodic rate, and symbol error rate of the system under two scenarios, i.e., perfect successive interference cancellation (pSIC) and imperfect successive interference cancellation (ipSIC). Additionally, the effects of the system parameters such as the transmit power and altitude of the UAV, the coefficients of channel model on the system performance were studied. The results demonstrate that the performance of the NOMA-based system is better compared with that of the conventional orthogonal multiple access (OMA)-based system in terms of OP, throughput and ergodic rate. Considering outage probabilities at the users, the system in the ipSIC case achieves the same performance as the pSIC case at a low transmit power of the UAV. In addition, an increase in the height of the UAV decreases the ergodic capacity of each user.