Downlink beamforming with per-antenna power constraints

This paper addresses the problem of linear beamformer design for a multiuser downlink channel in which the base-station is equipped with multiple antennas and the remote users are equipped with a single-antenna each. The design objective is to minimize the maximum transmit power on each transmit antenna while satisfying an SINR constraint for each user. This per-antenna power optimization is more realistic than the usual sum-power criterion because in practice each transmit antenna has its own power amplifier. The main contribution of this paper is a uplink-downlink duality result for this new setting. We prove that the optimal downlink transmit beamformers with per-antenna power constraints are identical to the optimal receive beamformers of a dual uplink channel with a uncertain noise. This mirrors our previous result on the capacity region duality with per-antenna power constraints, and it extends the previously known beam-forming duality for sum-power constrained channels. The per-antenna beamforming duality derived in this paper is based on a new interpretation of beamforming duality based on Lagrangian duality. Duality is useful in practice because the uplink problem is typically easier to solve. In the final part of this paper, we propose efficient numerical algorithms to solve the per-antenna problem.