On the MSE-duality of the broadcast channel and the multiple access channel

We present a mean-square-error (MSE) duality between the broadcast channel and the multiple access channel for multiantenna users communicating with a single base station. We introduce three levels of the duality which allow for a problem specific customization with different computational complexities and resolutions. The first level preserves the sum-MSE during the conversion from the uplink to the downlink and vice versa, whereas the second level not only keeps the sum-MSE constant but also ensures the preservation of the individual users' MSEs. The third level involves the finest resolution and preserves the individual streams' MSEs and the individual streams' signal-to-interference-and-noise ratio (SINR) simultaneously. In contrast with hitherto existing MSE-dualities, the proposed sort of duality features a lower complexity since no MSE computation detouring is necessary during the conversion to the dual domain and is capable of handling all combinations of active and passive transmitters and receivers. Moreover, we show how two of these three dualities can be exploited to solve the unweighted total sum-MSE minimization problem and the weighted sum-MSE minimization in the broadcast channel in an efficient way by revealing the hidden convexity in the first case and drastically reducing the computational complexity in the latter case.