Communication subject to normed channel uncertainties

Abstract

The transmission of information over a communication channel vastly depends on the level of knowledge that a transmitter and a receiver have about the channel and the interference. The transmission of information subject to insufficient knowledge of communication environment is called communication subject to uncertainties. The goal of this thesis is twofold: (1) To introduce new models for uncertain communication channels; (2) To define, compute, and analyze the performance of communication systems subject to introduced uncertainties from an information theoretic point of view. Various communication scenarios of compound single-input single-output and multiple-input multiple-output Gaussian channels are considered. There are three main contributions of the thesis: (1) The modeling of the channel and the noise uncertainties using Hinfinity and L1 normed liner spaces in frequency domain; (2) In the case of single-input single-output channels, the channel uncertainty is modeled as a subset of H infinity space; while the noise uncertainty is modeled either by a subset of Hinfinity space or by a subset of L1 space. Explicit formulas for the channel capacities, called robust capacities, and the optimal transmitted powers in the form of new water-filling formulas, are derived that explicitly depend on the sizes of the uncertainty sets. Moreover, when the noise uncertainty is modeled by a subset of L1 space, the capacity formula has a game theoretical interpretation, where the transmitter tries to maximize the mutual information, while the noise tries to minimize it. It is shown that a saddle point exists and that the optimal PSD of the transmitter is proportional to the optimal PSD of the noise; (3) In the case of multiple input multiple-output channels, two problems are considered. When the channel uncertainty is described by a subset of Hinfinity space, it is found that the transmission over the strongest singular value of the nominal channel frequency response matrix, representing the partial channel knowledge, is optimal for a large uncertainty set. When the noise uncertainty is described by a subset of L1 space, the optimal power spectral density matrix of the noise is proportional to the optimal power spectral density matrix of the transmitted signal.