Information Theoretic Bounds for Compound MIMO Gaussian Channels

In this paper, achievable rates for compound Gaussian multiple-input–multiple-output (MIMO) channels are derived. Two types of channels, modeled in the frequency domain, are considered when: 1) the channel frequency response matrix $H$ belongs to a subset of $H^{\infty}$ normed linear space, and 2) the power spectral density (PSD) matrix of the Gaussian noise belongs to a subset of $L_1$ space. The achievable rates of these two compound channels are related to the maximin of the mutual information rate. The minimum is with respect to the set of all possible $H$ matrices or all possible PSD matrices of the noise. The maximum is with respect to all possible PSD matrices of the transmitted signal with bounded power. For the compound channel modeled by the set of $H$ matrices, it is shown, under certain conditions, that the code for the worst case channel can be used for the whole class of channels. For the same model, the water-filling argument implies that the larger the set of matrices $H$, the smaller the bandwidth of the transmitted signal will be. For the second compound channel, the explicit relation between the maximizing PSD matrix of the transmitted signal and the minimizing PSD matrix of the noise is found. Two PSD matrices are related through a Riccati equation, which is always present in Kalman filtering and liner-quadratic Gaussian control problems.