Capacity for deterministic MIMO channels under rank constraint: A textbook derivation

This paper provides a text-book derivation for the capacity of the deterministic multiple-input multiple-output (MIMO) channels with a rank-constrained source covariance matrix. We prove that the optimal power allocation aligns with the water-filling interpretation but only over the best K eigen-modes, where K is the upper bound of the source covariance rank. Moreover, we prove that the capacity is an increasing and concave function of the total power under the considered scenario.