  Intelligent reflecting surface (IRS) is a new and disruptive technology to
achieve spectrum- and energy-efficient as well as cost-efficient wireless
networks. This paper considers an IRS-assisted downlink
non-orthogonal-multiple-access (NOMA) system. To optimize the rate performance
and ensure user fairness, we maximize the minimum decoding
signal-to-interference-plus-noise-ratio (i.e., 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 IRS. A
combined-channel-strength based user ordering scheme is first proposed to
decouple the user-ordering design and the joint beamforming design. Efficient
algorithms are further proposed to solve the formulated non-convex problem for
the cases of a single-antenna BS and a multi-antenna BS, respectively, by
leveraging the block coordinated decent and semidefinite relaxation (SDR)
techniques. For the single-antenna BS case, the optimal solution for the power
allocation at the BS and the asymptotically optimal solution for the phase
shifts at the IRS are obtained in closed forms. For the multi-antenna BS case,
it is shown that the rank of the SDR solution to the transmit beamforming
design is upper bounded by two. Also, the convergence proof and the complexity
analysis are given for the proposed algorithms. Simulation results show that
the IRS-assisted downlink NOMA system can enhance the rate performance
significantly, compared to traditional NOMA without IRS and traditional
orthogonal multiple access with/without IRS. In addition, numerical results
demonstrate that the rate degradation due to the IRS's finite phase resolution
is slight, and good rate fairness among users can be always guaranteed.
