This paper considers the beamforming design for a multiuser multiple-input single-output downlink with an arbitrary number of (context specific) shaping constraints. In this setup, the state-of-the-art beamforming schemes cannot attain the well known performance bound promised by the semidefinite program (SDP) relaxation technique. To close the gap, we propose a redundant-signal embedded linear beamforming (REEL-BF) scheme, where each user is assigned with one information beamformer and several shaping beamformers. It is shown that the proposed REEL-BF scheme can perform general rank-  $K$   beamforming for user symbols in a low-complexity and structured manner. In addition, sufficient conditions are derived to guarantee that the REEL-BF scheme always achieves the SDP bound for linear beamforming schemes. Based on such conditions, an efficient algorithm is then developed to obtain the optimal REEL-BF solution in polynomial time. Numerical results demonstrate that the proposed scheme enjoys substantial performance gains over the existing alternatives.
['Feng Wang', 'Chongbin Xu', 'Yongwei Huang', 'Xin Wang', 'Xiqi Gao']
REEL-BF Design: Achieving the SDP Bound for Downlink Beamforming With Arbitrary Shaping Constraints