Under-determined Training and Estimation for Distributed Transmit Beamforming Systems

Distributed transmit beamforming (DTB) can significantly boost the signal-to-noise ratio (SNR) of a wireless communication system. To realize the benefits of DTB, generating and feeding back beamforming vector are very challenging tasks. Existing schemes have either enormous overhead or weak robustness in noisy channels. In this paper, we investigate the design of training sequences and beamforming vector estimators in DTB systems. We consider an under-determined case, where the length of training sequence N sent from each node is smaller than the number of source nodes M. We derive the optimal estimation of the beamforming vector that maximizes the beamforming gain and show that it can be well approximated as the linear minimum mean square error (LMMSE) estimator. Based on the LMMSE estimator, we investigate the optimal design of training sequences and propose efficient DTB schemes. We analytically show that these schemes can achieve approximately N times increased SNR in uncorrelated channels, and even higher gain in correlated ones. We also propose a concatenated training scheme which optimally combines the training signals over multiple frames to obtain the beamforming vector. Simulation results demonstrate that the proposed DTB schemes can yield significant gains even at very low SNRs, with total feedback bits much less than those required in the existing schemes.