Gaussian arbitrarily varying channels

Abstract

The {\em arbitrarily varying channel} (AVC) can be interpreted as a model of a channel jammed by an intelligent and unpredictable adversary. We investigate the asymptotic reliability of optimal random block codes on Gaussian arbitrarily varying channels (GAVC's). A GAVC is a discrete-time memoryless Gaussian channel with input power constraint <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P_{T}</tex> and noise power <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N_{e}</tex> , which is further corrupted by an additive "jamming signal." The statistics of this signal are unknown and may be arbitrary, except that they are subject to a power constraint <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P_{J}</tex> . We distinguish between two types of power constraints: {\em peak} and {\em average.} For peak constraints on the input power and the jamming power we show that the GAVC has a random coding capacity. For the remaining cases in which either the transmitter or the jammer or both are subject to average power constraints, no capacities exist and only <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\lambda</tex> -capacities are found. The asymptotic error probability suffered by optimal random codes in these cases is determined. Our results suggest that if the jammer is subject only to an average power constraint, reliable communication is impossible at any positive code rate.