Capacity of Gaussian channels with noise uncertainty

The problem addressed is that of defining, and computing, the capacity of a communication channel when the additive noise statistic is not fully known. The communication channel is specified as a continuous time channel with a known transfer function, where the transmitted signal is constrained in power, and an additive Gaussian noise channel is assumed. The power spectral density of the noise, although unknown, belongs to a known set defined through the uncertainty of the filter that shapes the power spectral density of the noise. The channel capacity is defined as the max-min of the mutual information rate between the transmitted and received signals, where the infimum is taken over the set of all possible power spectral densities of the noise, and the supremum is taken over all power spectral densities of the transmitted signal with constrained power. It is shown that the so defined channel capacity is equal to the operational capacity that represents the supremum of all attainable rates over a given channel.