Capacity region of the multiantenna Gaussian broadcast channel with analog TX-RX beamforming

In this paper we characterize the capacity boundary of the two-user multiple-input multiple-output (MIMO) broadcast channel (BC) when the multiantenna terminals use analog beamforming at both sides of the link. Basically, the problem reduces to finding the optimal transmit direction and the optimal power allocation between users to operate at any point on the boundary. We show that both optimization problems can be solved in closed form. First, exploiting the fact that any Pareto optimal pair of rates must also be Pareto optimal of a convex region defined by the channel energies, the optimal transmit direction is shown to be the principal eigenvector of a matrix. Second, the optimal power allocation is obtained by exploiting again the Pareto optimality of the sought pair of rates. Although this paper focuses on the BC, the obtained results also serve to characterize the dual multiple-access channel (MAC). Moreover, this explicit characterization of the boundary can be extended to a $K$-user system for $K>2$. Numerical examples are provided to illustrate the results of the paper, and to show the enlargement of the capacity region achieved by analog beamforming in comparison to a single-input single-output (SISO) BC.