Optimizing the Input Covariance for MIMO Channels

A multivariate Gaussian input density achieves capacity for multiple-input multiple-output flat fading channels with additive white Gaussian noise and perfect receiver channel state information. Capacity computation for these channels reduces to the problem of finding the best input covariance matrix, and is in general a convex semi-definite program. This paper presents Kuhn-Tucker optimality conditions for this problem, and iterative algorithms for determination of the optimal covariance matrix. For channels with a separable covariance structure, the optimal eigenvectors are known, and only the eigenvalues need to be determined. In the completely general case, both the eigenvectors and the eigenvalues must be found by numerical solutions. Large systems asymptotics are also considered.