The Gaussian test channel with an intelligent jammer

The problem of transmitting a sequence of identically distributed independent Gaussian random variables through a Gaussian memoryless channel with a given input power constraint, in the presence of an intelligent jammer, is considered. The jammer taps the channel and feeds back a signal, at a given energy level, for the purpose of jamming the transmitting sequence. Under a square-difference distortion measure which is to be maximized by the jammer and to be minimized by the transmitter and the receiver, this correspondence obtains the complete set of optimal (saddle-point) policies. The solution is essentially unique, and it is structurally different in three different regions in the parameter space, which are determined by the signal-to-noise ratios and relative magnitudes of the noise variances. The best (maximin) policy of the jammer is either to choose a linear function of the measurement he receives through channel-tapping, or to choose, in addition (and additively), an independent Gaussian noise sequence, depending on the region where the parameters lie. The optimal (minimax) policy of the transmitter is to amplify the input sequence to the given power level by a linear transformation, and that of the receiver is to use a Bayes estimator.