A Complete Description of the QoS Feasibility Region in the Vector Broadcast Channel

We characterize the complete quality-of-service (QoS) feasibility region and present a simple feasibility test for given QoS requirements in the Gaussian vector broadcast channel. While most contributions in the literature recast the QoS constraints into requirements for the signal-to-interference-and-noise ratios (SINRs), we convert them into upper bounds for the minimum mean square errors (MMSEs) instead and test feasibility in the MMSE domain. Our main contribution is a complete description of the feasible MMSE region. Its closure is shown to be a polytope and we find the complete set of its bounding half-spaces noniteratively after a finite number of steps. The polytope can easily be converted into any other QoS domain like SINR or rate. However, the simple geometry of the MMSE domain is lost in other domains. Once the bounding half-spaces are determined, any target MMSE tuple can quickly be checked for feasibility by verifying its membership to the interior of the polytope. For nondegenerate channels, the only relevant bounding half-space is essentially the lower bound on the sum mean square error. No further computations are necessary contrary to existing feasibility checks which iteratively solve eigenproblems in an alternating optimization framework for every single QoS requirement to test. For two particular user/antenna configurations, we find a noniterative closed form solution for the optimum power allocation of the signal-to-interference ratio (SIR) balancing.