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This paper considers a two-user downlink transmission in reconfigurable intelligent surface (RIS)-aided network over fading channels. Particularly, by employing user grouping and fixed power allocation scheme to enable non-orthogonal multiple access (NOMA) approach, RIS-aided network can significant benefit from NOMA. To highlight the advantages of RIS, we compare the system performance of NOMA-RIS and traditional orthogonal multiple access (OMA) based RIS. Main practical circumstances are carefully analyzed, such as, RIS with direct link, system without RIS, and imperfect phase shifts. More specifically, we consider a RIS-aided downlink network, where the base station communicates with a group of two users under assistance of RIS, which acts equivalently as relay. As key expectation, the RIS is efficiently designed to improve the performance of users. To evaluate the system performance, two main system performance metrics including outage probability and average capacity are studied by deriving new closed-form expressions. The goal is to find out which system parameters need to be adjusted to achieve the expected performance. The numerical results reveal that: i) <italic>the outage probability and average capacity of considered NOMA-RIS aided wireless system outperforms the conventional NOMA network over fading channels</italic>; ii) <italic>with different power allocation factors assigned to users, the performance gap among two users can be adjusted to guarantee the fairness characteristic</italic>; iii) <italic>the number of reflecting elements in RIS has significant impact on the system performance of the considered NOMA-RIS</italic> system, which shows advantage of both RIS and NOMA compared with conventional OMA system.

Enabling User Grouping and Fixed Power Allocation Scheme for Reconfigurable Intelligent Surfaces-Aided Wireless Systems