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We consider the problem of transmit downlink beamforming for wireless transmission in the context of certain broadcasting or multicasting applications wherein channel state information (CSI) is available at the transmitter, and a common message is to be transmitted to the users. Unlike the usual "blind" isotropic broadcasting scenario, the availability of CSI allows transmit optimization. We adopt a minimum transmission power criterion, subject to prescribed minimum received signal-to-noise ratio (SNR) at each of the intended receivers. We also consider a related max-min SNR "fair" problem formulation subject to a transmit power constraint. The basic problem is non-convex and thus difficult to solve; however, we show that a suitable reformulation allows the application of semidefinite relaxation (SDR) techniques. SDR yields a (generally approximate) solution, but in many cases our solution is optimal, and in most cases it is within 3-4 dB from the optimal solution, which is often good enough in our intended applications. While the focus of the paper is on a wireless communication scenario, we also discuss related problems in downstream preceding for broadcasting in digital subscriber line systems.
Sidiropoulos et al. (Fri,) studied this question.