Closed-Form Sum-MSE Minimization for the Two-User Gaussian MIMO Broadcast Channel

We study the Gaussian multiple-input, multiple-output broadcast channel, where a base station with N_T antennas transmits K independent messages to K users, each having a single receive antenna. The messages consist of independent, identically distributed Gaussian random variables and we study linear transmission with an end-to-end distortion criterion. By using an already established uplink/downlink duality and a recently discovered special relation between beamforming vectors and channel vectors, we present a closed-form expression for the optimal power allocation in the two-user case. We also outline an iterative algorithm that finds the optimal power allocation for an arbitrary number of users.