The geometry of the MIMO broadcast channel rate region under linear filtering at high SNR

We present a high SNR analysis of the rate region geometry when linear filtering is applied in the MIMO broadcast channel and time sharing is not considered. In particular, the set of weakly Pareto optimal points of the asymptotic rate region is derived. While all different permutations of user sorting have to be investigated for the complete rate region when nonlinear dirty paper coding is applied, we show that the analogon to different sorting orders is the number of active streams per user when linear filtering is applied. Furthermore, we reveal that only a small fraction of weakly Pareto optimal points is in charge of maximizing an asymptotic weighted sum rate and points belonging to that fraction are all obtained when every user applies full multiplexing. In contrast, switching off some data streams may be optimal when it comes to balancing or maximizing a user´s rate while keeping the other users´ rates constant. Many of the derived results are not only applicable to systems with enough antennas at the base station, but also to configurations with too few degrees of freedom, which so far have never been discussed in any asymptotic analysis before.