Derivatives of Mutual Information in Gaussian Vector Channels with Applications

In this paper, derivatives of mutual information for a general linear Gaussian vector channel are considered. We consider two applications. First, it is shown how the corresponding gradient relates to the minimum mean squared error (MMSE) estimator and its error matrix. Secondly, we determine the directional derivative of mutual information and use this geometrically intuitive concept to characterize the capacity-achieving input distribution of the above channel subject to certain power constraints. The well-known water-filling solution is revisited and obtained as a special case. Also for shaping constraints on the maximum and the Euclidean norm of mean powers explicit solutions are derived. Moreover, uncorrected sum power constraints are considered. The optimum input can here always be achieved by linear precoding.