Feasibility test and globally optimal beamformer design in the satellite downlink based on instantaneous and ergodic rates

We focus on the globally optimal linear beamformer design based on quality-of-service (QoS) power minimization and balancing in the satellite downlink (DL) with perfect and statistical channel state information (CSI) users. Contrary to the usual rate requirements for the perfect CSI users, we consider ergodic rate requirements for the statistical CSI users. Assuming that the satellite works at Ka-band, we model the fading channels as zero-mean Gaussian vectors with rank-one covariance matrices. This leads to tractable ergodic mutual information expressions that enable to formulate a simple necessary and sufficient feasibility test for the power minimization under minimum rate constraints. Moreover, we are able to solve the power minimization with a globally optimal branch-and-bound (BB) method via properly reformulating the optimization problem. This BB formulation can also be applied to the rate balancing problem. The results serve as a benchmark for other optimization methods.