Capacity of compound MIMO Gaussian channels with additive uncertainty

This paper considers reliable communications over a multiple-input multiple-output (MIMO) Gaussian channel, where the channel matrix is within a bounded channel uncertainty region around a nominal channel matrix, i.e., an instance of the compound MIMO Gaussian channel. We study the optimal transmit covariance design to achieve the capacity of compound MIMO Gaussian channels, where the channel uncertainty region is characterized by the spectral norm. This design problem is a challenging non-convex optimization problem. However, in this paper, we reveal that this design problem has a hidden convexity property, and hence it can be simplified as a convex optimization problem. Towards this goal, we first prove that the optimal transmit design is to diagonalize the nominal channel, and then show that the duality gap between the capacity of the compound MIMO Gaussian channel and the minimal channel capacity is zero, which proves the conjecture of Loyka and Charalambous (IEEE Trans. Inf. Theory, vol. 58, no. 4, pp. 2048-2063, 2012). The key tools for showing these results are a novel matrix determinant inequality and some unitarily invariant properties.