The Capacity and Optimal Signaling for Gaussian MIMO Channels Under Interference Constraints

Gaussian MIMO channel under the joint transmit (Tx) and interference power constraints (TPC and IPC) is studied. A closed-form solution for the optimal Tx covariance matrix in the general case is obtained using the KKT-based approach, up to dual variables. A number of more explicit closed-form solutions are given with optimal dual variables, including full-rank and rank-1 (beamforming) cases as well as the case of identical eigenvectors (typical for massive MIMO settings), which differer significantly from the standard water-filling solution. A “whitening” filter is shown to be an important part of optimal precoding under interference constraints. Sufficient and necessary conditions for each constraint to be redundant are given. Capacity scaling with the SNR is shown to be determined by a natural linear-algebraic structure of sub-spaces induced by channel matrices of multiple users. A number of unusual properties of optimal Tx covariance matrix under the joint TPC and IPC are pointed out and a bound on its rank is established. An interplay between the TPC and IPC is studied, including the transition from power-limited to interference-limited regimes as the Tx power increases. While closed-from solutions for optimal dual variables are given in some special cases, an iterative bisection algorithm (IBA) is proposed to find optimal dual variables in the general case and its convergence is proved for some special cases. Numerical experiments illustrate its efficient performance. Bounds for the optimal dual variables are given.