On the Convexity of the MSE Region of Single-Antenna Users

TL;DR

This paper proves the convexity of the MSE region for two single-antenna users in a multi-antenna system, but shows that this convexity does not hold for more than two users, refining the understanding of MSE region geometry.

Contribution

It establishes the convexity of the MSE region for two single-antenna users and demonstrates non-convexity for more than two users, correcting previous assumptions.

Findings

Convex MSE region for two single-antenna users.

Non-convex MSE region for more than two users.

Extension and correction of prior MSE region geometry notions.

Abstract

We prove convexity of the sum-power constrained mean square error (MSE) region in case of two single-antenna users communicating with a multi-antenna base station. Due to the MSE duality this holds both for the vector broadcast channel and the dual multiple access channel. Increasing the number of users to more than two, we show by means of a simple counter-example that the resulting MSE region is not necessarily convex any longer, even under the assumption of single-antenna users. In conjunction with our former observation that the two user MSE region is not necessarily convex for two multi-antenna users, this extends and corrects the hitherto existing notion of the MSE region geometry.