Massive MIMO Channels Under the Joint Power Constraints

The capacity and optimal signaling for a fixed Gaussian MIMO channel under the total power constraint (TPC) are well-known. When per-antenna power constraint (PAC) are added to the TPC, the problem becomes significantly more complicated and no closed-form solution is known in the general case, while some special cases (MISO and full-rank MIMO) have been solved. Here, we consider a massive MIMO Gaussian channel under the joint (TPC+PAC) power constraints and obtain a closed-form solution for its optimal covariance matrix under the favorable propagation conditions. The impact of joint constraints on the optimal signaling and capacity is discussed. In particular, while optimal signaling under the joint constraints has some similarities to the standard WF, it also reveals a number of significant differences. A water-filling-type interpretation of the obtained solution is given. The number of active streams as well as the number of active PACs are determined. A closed-form solution for the optimal dual variable under the joint constraints is given.