  In this paper we consider the computation of channel capacity for ergodic
multiple-input multiple-output channels with additive white Gaussian noise. Two
scenarios are considered. Firstly, a time-varying channel is considered in
which both the transmitter and the receiver have knowledge of the channel
realization. The optimal transmission strategy is water-filling over space and
time. It is shown that this may be achieved in a causal, indeed instantaneous
fashion. In the second scenario, only the receiver has perfect knowledge of the
channel realization, while the transmitter has knowledge of the channel gain
probability law. In this case we determine an optimality condition on the input
covariance for ergodic Gaussian vector channels with arbitrary channel
distribution under the condition that the channel gains are independent of the
transmit signal. Using this optimality condition, we find an iterative
algorithm for numerical computation of optimal input covariance matrices.
Applications to correlated Rayleigh and Ricean channels are given.
