Information Theoretic Bounds for Compound MIMO Gaussian Channels

Abstract

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> 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 <emphasis emphasistype="italic"><formula formulatype="inline"><tex Notation="TeX">$H$</tex> </formula></emphasis> belongs to a subset of <emphasis emphasistype="italic"><formula formulatype="inline"><tex Notation="TeX">$H^{\infty}$</tex></formula></emphasis> normed linear space, and 2) the power spectral density (PSD) matrix of the Gaussian noise belongs to a subset of <emphasis emphasistype="italic"><formula formulatype="inline"> <tex Notation="TeX">$L_1$</tex></formula></emphasis> 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 <emphasis emphasistype="italic"><formula formulatype="inline"><tex Notation="TeX">$H$</tex> </formula></emphasis> 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 <emphasis emphasistype="italic"><formula formulatype="inline"><tex Notation="TeX">$H$</tex> </formula></emphasis> 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 <emphasis emphasistype="italic"><formula formulatype="inline"> <tex Notation="TeX">$H$</tex></formula></emphasis>, 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. </para>