Bounds on optimal power minimization and rate balancing in the satellite downlink

We focus on the design of linear beamforming based on quality-of-service (QoS) power minimization and rate balancing in the downlink (DL) of a satellite communication system with mobile and static users. Since only the rank-one covariance matrices of the channels to mobile users are known, we introduce average rate requirements for these users contrary to the perfect channel state information (CSI) rate requirements for the static users. Due to the structure of the ergodic rates, we cannot directly resort to the optimization techniques for the purely complete CSI vector broadcast channel (BC). Therefore, we propose two suitable lower and upper bounds on the ergodic rates. Incorporating these bounds in the optimization problems, instead of the ergodic expressions, the usual power minimization algorithms can be applied. This allows the computation of bounds on the optima of the considered problems. Furthermore, employing an additional power allocation on the obtained solutions, we achieve outcomes that reside close to the expected optima.