On the Robustness of Transmit Beamforming

Beamforming is a simple transmit strategy that uses only one eigen-direction in multiple-input multiple-output channels. This simplicity makes beamforming a competitive strategy in practice, but at the same time poses a doubt on the sensitivity of beamforming to the imperfectness of the channel state information at the transmitter (CSIT). This paper studies beamforming from the perspective of worst-case robustness. We show that beamforming can achieve the maximum received signal-to-noise ratio (SNR) or guarantees a given received SNR with the minimum transmit power, in the worst channel within an elliptical uncertainty region defined by the weighted spectral norm. This result further implies that beamforming has the ability to combat against the imperfectness of CSIT, especially for small channel dimensions or small channel uncertainty.