Symmetry and Asymmetry of MIMO Fading Channels

We consider ergodic coherent MIMO channels, and characterize the optimal input distribution under general distributions of the fading matrix. We describe how symmetries in the fading matrix distribution and the constraint set, described naturally with group structures, imply the symmetric properties of the optimal input distribution. We use this approach to give a general condition under which the white Gaussian input is optimal, and to compute the optimal input for MISO Ricean channel. In contrast, we also investigate the Kronecker model, in which case we show how an asymmetric structure in the problem is also preserved in the optimal input, leading to a new waterfilling solution.