Unraveling the Rank-One Solution Mystery of Robust MISO Downlink Transmit Optimization: A Verifiable Sufficient Condition via a New Duality Result

摘要

This paper concentrates on a robust transmit optimization problem for themultiuser multi-input single-output (MISO) downlink scenario and underinaccurate channel state information (CSI). This robust problem deals with ageneral-rank transmit covariance design, and it follows a safe rate-constrainedformulation under spherically bounded CSI uncertainties. Curiously, simulationresults in previous works suggested that the robust problem admits rank-oneoptimal transmit covariances in most cases. Such a numerical finding isappealing because transmission with rank-one covariances can be easily realizedby single-stream transmit beamforming. This gives rise to a fundamentallyimportant question, namely, whether we can theoretically identify conditionsunder which the robust problem admits a rank-one solution. In this paper, weidentify one such condition. Simply speaking, we show that the robust problemis guaranteed to admit a rank-one solution if the CSI uncertainties are not toolarge and the multiuser channel is not too poorly conditioned. To establish theaforementioned condition, we develop a novel duality framework, through whichan intimate relationship between the robust problem and a related maximinproblem is revealed. Our condition involves only a simple expression withrespect to the multiuser channel and other system parameters. In particular,unlike other sufficient rank-one conditions that have appeared in theliterature, ours is verifiable. The application of our analysis framework toseveral other CSI uncertainty models is also discussed.