Distributionally robust chance-constrained transmit beamforming for multiuser MISO downlink

Abstract

This paper considers robust transmit beamforming for multiuser multi-input single-output (MISO) downlink transmission, where imperfect channel state information (CSI) is assumed at the base station (BS). The imperfect CSI is captured by a moment-based random error model, in which the BS knows only the mean and covariance of each CSI error, but not the exact distribution. Under this error model, we formulate a distributionally robust beamforming (DRB) problem, in which the total transmit power at the BS is to be minimized, while each user's SINR outage probability, evaluated w.r.t. any distribution with the given mean and covariance, is kept below a given threshold. The DRB problem is a semi-infinite chance-constrained problem. By employing recent results in distributionally robust optimization, we show that the DRB problem admits an explicit conic reformulation, which can be conveniently turned into a convex optimization problem after semidefinite relaxation (SDR). We also consider the case where the mean and covariance are not perfectly known. We show that the resulting DRB problem still admits a conic reformulation and can be approximately solved using SDR. The robustness of the proposed designs are demonstrated by numerical simulations.