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The reconfigurable intelligent surface (RIS), which consists of a large number of passive and low-cost reflecting elements, has been recognized as a revolutionary technology to enhance the performance of future wireless networks. This paper considers an RIS assisted multicast transmission, where a base station (BS) with multiple-antenna multicasts common message to multiple single-antenna mobile users (MUs) under the assistance of an RIS. An equivalent channel model for the considered multicast transmission is analyzed, and then an optimization problem for the corresponding channel capacity is formulated to obtain the optimal covariance matrix and phase shifts. In order to solve the above non-convex and non-differentiable problem, this paper first exploits the gradient descent method and alternating optimization, to approach the locally optimal solution for any number of MUs. Then, this paper considers a special case, which can obtain the global optimal solution, and shows the sufficient and necessary condition for this special case. Finally, the order growth of the maximal capacity is obtained when the numbers of the reflecting elements, the BS antennas, and the MUs go to infinity.

Capacity Characterization for Reconfigurable Intelligent Surfaces Assisted Multiple-Antenna Multicast