Solution of the multiuser downlink beamforming problem with individual SINR constraints

We address the problem of joint downlink beamforming in a power-controlled network, where independent data streams are to be transmitted from a multi-antenna base station to several decentralized single-antenna terminals. The total transmit power is limited and channel information (possibly statistical) is available at the transmitter. The design goal: jointly adjust the beamformers and transmission powers according to individual SINR requirements. In this context, there are two closely related optimization problems. P1: maximize the jointly achievable SINR margin under a total power constraint. P2: minimize the total transmission power while satisfying a set of SINR constraints. In this paper, both problems are solved within a unified analytical framework. Problem P1 is solved by minimizing the maximal eigenvalue of an extended cross-talk matrix. The solution provides a necessary and sufficient condition for the feasibility of the SINR requirements. Problem P2 is a variation of problem P1. An important step in our analysis is to show that the global optimum of the downlink beamforming problem is equivalently obtained from solving a dual uplink problem, which has an easier-to-handle analytical structure. Then, we make use of the special structure of the extended cross-talk matrix to develop a rapidly converging iterative algorithm. The optimality and global convergence of the algorithm is proven and stopping criteria are given.