Average sum MSE minimization in the multi-user downlink with multiple power constraints

We consider a sum mean square error (SMSE) based transceiver design for the multi-user downlink with multiple linear transmit power constraints. Since the multi-antenna transmitter has only imperfect channel state information (CSI), the average SMSE is minimized via an alternating optimization (AO). For fixed equalizers, the average SMSE minimizing precoders are found via an uplink-downlink SMSE duality. The precoder update therewith transforms to an uplink max-min average SMSE problem, i.e., a minimum mean square error (MMSE) equalizer design and an outer worst-case noise search. This dual problem is optimally solved, e.g., by a gradient projection approach, and strong duality is shown.