On the Compound Capacity of a Class of MIMO Channels Subject to Normed Uncertainty

The compound capacity of uncertain multiple-input multiple-output channels is considered, when the channel is modeled by a class described by a (known) nominal channel and a constrained-norm (unknown) uncertainty. Within this framework, two types of classes are investigated with additive and multiplicative uncertainties subject to a spectral norm constraint, using the singular value decomposition and related singular value inequalities as the main tools. The compound capacity is a maxmin mutual information, representing the capacity of the class, in which the minimization is done over the class of channels while the maximization is done over the transmit covariance. Closed-form solutions for the compound capacity of the classes are obtained and several properties related to transmit and receive eigenvectors are presented. It is shown that, under certain conditions, the compound capacity of the class is equal to the worst-case channel capacity, thus establishing a saddle-point property. Explicit closed-form solutions are given for the worst-case channel uncertainty and the capacity-achieving transmit covariance matrix: the best transmission strategy achieving the compound capacity is a multiple beamforming on the nominal (known) channel eigenmodes with the beam power distribution via the water filling at a degraded SNR. As the uncertainty increases, fewer eigenmodes are used until only the strongest one remains active so that transmit beamforming is an optimal robust transmission strategy in this large-uncertainty regime, for which explicit conditions are given. Using these results, upper and lower bounds of the compound capacity are constructed for other bounded uncertainties and some generic properties are pointed out. The results are extended to compound multiple-access and broadcast channels. In all considered cases, the price to pay for channel uncertainty is an SNR loss (or, equivalently, the nominal channel degradation) commensurate with the uncertainty set radius measured by the spectral norm and the optimal signaling strategy is the transmission on the degraded nominal channel.