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Reconfigurable intelligent surface (RIS) is a revolutionary technology to achieve spectrum-, energy-, and cost-efficient wireless networks. This paper considers an RISassisted downlink non-orthogonal-multiple-access (NOMA) system. To optimize the rate performance and ensure user fairness, we maximize the minimum decoding signal-to-interference-plusnoise-ratio (equivalently the rate) of all users, by jointly optimizing the (active) transmit beamforming at the base station (BS) and the phase shifts (i.e., passive beamforming) at the RIS. A combined-channel-strength based user-ordering scheme for NOMA decoding is first proposed to decouple the user-ordering design and the joint beamforming design. Efficient algorithms are further proposed to solve the non-convex problem, by leveraging the block coordinated descent and semidefinite relaxation (SDR) techniques. For the single-antenna BS setup, the optimal power allocation at the BS and the asymptotically optimal phase shifts at the RIS are obtained in closed forms. For the multiple-antenna BS setup, it is shown that the rank of the SDR solution of the transmit beamforming design is upper bounded by two. Also, the proposed algorithms are analyzed in terms of convergence and complexity. Simulation results show that the RIS-assisted NOMA system can enhance the rate performance significantly, compared to traditional NOMA without RIS and traditional orthogonal multiple access with/without RIS.

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