The Capacity of a MIMO Ricean Channel Is Monotonic in the Singular Values of the Mean

We consider a discrete-time memoryless Multiple-Input Multiple-Output (MIMO) fading channel where the fading matrix can be written as the sum of a deterministic (line-of-sight) matrix D and a random matrix H̃ whose entries are IID zero-mean unit-variance complex circularly-symmetric Gaussian random variables. It is demonstrated that if the realization of the fading matrix is known at the receiver but not at the transmitter, then the capacity of this channel under an average power constraint is monotonically non-decreasing in the singular values of D. This complements a recent result of Kim and Lapidoth [1] demonstrating the monotonicity of the mutual information corresponding to isotropically distributed Gaussian input vectors. We also show that the optimal covariance matrix of the Gaussian input vector has the same eigenvectors as D†D.