Robust capacity of a Gaussian noise channel with channel and noise uncertainty

This paper is concerned with the problem of defining, and computing the capacity of a continuous-time additive Gaussian noise communication channel when the true frequency response of the channel, and the power spectral density of the noise are not perfectly known, and the transmitted signal is a wide-sense stationary process constrained in power. The uncertainties of a true channel frequency response and power spectral density of the noise are described by weighted balls in the H_∞ space. In that way, two sets are defined that describe the set of all possible channel frequency responses, and the set of all possible power spectral densities of the noise. The ball radii depend on the degree of uncertainty that one has about the true channel frequency response, and power spectral density of the noise. The channel capacity is defined as the max-min-min of a mutual information rate between transmitted, and received signals, where the first minimum is taken over the set of all possible noises, the second minimum is taken over the set of all possible channel frequency responses, and maximum is over the set of all possible power spectral densities of transmitted signal with constrained power. It is shown that such defined channel capacity, called robust capacity, is equal to the operational capacity that represents the theoretical maximum of all attainable rates over a given channel.