Optimal rank-constrained transmission for MIMO under per-group power constraints

In this paper, we determine the optimal rank-constrained transmission strategy for the Multiple-Input Multiple-Output (MIMO) channel under Per-group Power constraints (PGPC). PGPC includes sum power constraint and per-antenna power constraints as special cases. We propose a Projected Factored Gradient Descent (PFGD) algorithm to obtain optimal rank-constrained strategy. If the rank constraint is greater than or equal to the rank of the unconstrained optimal covariance matrix, then the rank-constrained capacity coincides with the MIMO capacity under PGPC. The study of rank-constrained transmission under PGPC is important in the context of mmWave systems where the number of antennas could be large, but the number of streams is limited by the rank of the channel and the complexity of implementation of spatial multiplexing with a large number of streams. The proposed algorithm has lower complexity compared to standard approaches for semi-definite programs especially for low rank transmission. Numerical results are shown to study the behavior of the resulting rank-constrained capacity in the context of some existing mmWave channel models and to illustrate the convergence of the algorithm. For a special case where the channel has full column rank and the optimal covariance matrix is also full rank, we determine the MIMO capacity under PGPC analytically.