The Algebra of MIMO Channels

We consider ergodic coherent MIMO channels. We characterize the optimal input distribution for various fading matrix distributions. First, we describe how symmetries in the fading matrix distribution are preserved as symmetries in the optimal input covariance and thus yield to specification of the optimal input. We will see that group structures and notion of commutant appear as key elements. Second, we investigate the Kronecker model, in this case we will show how an asymmetric structure in the problem is also preserved in the optimal input structure. Notations: We define the following subsets of the n× n complex matrices Mn(C), n ≥ 1: H(n): the hermitian matrices, H+(n): the hermitian positive semidefinite matrices, H +(n): the hermitian positive definite matrices, U(n): the unitary matrices, S(n): the symmetric group of all permutations matrices, C(n): the group of cyclic permutations matrices, Z2(n): the group of diagonal matrices with {−1,+1} elements.