Robust capacity of white Gaussian noise channels with uncertainty

Abstract

This paper concerns the problem of defining, and computing the channel capacity of a continuous time additive white Gaussian noise channel when the true frequency response of the channel is not completely known to the transmitter, and receiver, and when a transmitted signal is a wide sense stationary process constrained in power. To represent the uncertainty of a true frequency response two basic uncertainty models are used that are borrowed from the control theory, additive, and multiplicative. Here, the true frequency response although unknown, belongs to a ball in a normed linear space. The radius of the ball is a function of frequency, and it depends on the size of the uncertainty. The channel capacity, called robust capacity is defined as a max-min of the mutual information rate, where the maximum is over all power spectral densities of the input signal with constrained power, and minimum is over the uncertainty set of frequency response. The robust capacity formula is explicitly computed describing how the channel uncertainty reduces the capacity. The water-filling formula is derived showing how the optimal transmitted power changes with uncertainty. At the end, it is shown that a channel coding theorem, and its converse under certain conditions imposed on the uncertainty set hold for the robust maximin capacity.