Multiple Antenna Multicast Transmission Assisted by Reconfigurable Intelligent Surfaces

In order to improve the multiple antenna multicasting in the obstacles environment, a reconfigurable intelligent surface (RIS), which consists of a large number of reconfigurable reflecting elements each being able to reflect the received signals with phase shifts, is set up to assist multicast transmission. Since RIS hardly consumes any energy, it is more environmental-friendly than conventional relaying mode. This paper considers a multicast transmission assisted by the RIS, i.e., a multiple-antennas base station (BS) sends the common messages to $K$ single-antenna mobile users (MUs), where an RIS is deployed as an amplify-and-forward no-power relay to assist this transmission. An equivalent channel model for the considered multicast system is analyzed, and then a capacity maximization problem is formulated to obtain the optimal covariance matrix of the transmitted symbol vector and phase shifts of RIS, which is a non-convex non-differentiable problem. First, we consider this problem for a special scenario, i.e., $K=2$, which owns three differentiable cases, and the optimal solutions for the threes cases are obtained by KKT conditions and a proposed numerical algorithm, respectively. Then, for $K>2$, since the maximization problem is non-differentiable, this paper reformulates this problem as a differentiable problem, and proposes two numerical algorithms, i.e., subgradient and gradient descent methods, to approach the optimal solution. Finally, in oder to more intuitively comprehend the performances of RIS in multicast transmission, the order growth of the maximum capacity of the considered multicast system is obtained in the scenarios that the numbers of reflecting elements, BS antennas, and MUs go to infinity.