Solutions to a class of minimax decision problems arising in communication systems

In this paper, we consider a class of minimax decision problems which arise in the transmission of a Gaussian vector message over a vector channel with partially unknown statistical description. The statistically unknown part of the channel is modelled as one which is controlled by a jammer who can corrupt the transmitted message by sending noise which may be correlated with the original message under a given power constraint. Under two types of structural assumptions on the transmitter (encoder), the problem is posed as one in which the optimum decision rules at the encoder and the decoder jointly minimize a square distortion measure at the output, under worst possible choices for the jamming noise. It is shown that a saddle-point solution exists when the linear encoder structure is of the mixed type, whereas it does not exist when it is restricted to be deterministic. In the former case, explicit expressions for the saddle-point solution have been presented, whereas in the latter case minimax and maximin solutions have been obtained. An important feature of the saddle-point solution is that it depends on two integer-valued parameters, one of which determines (in a new rotated coordinate system) the number of components of the message vector to be transmitted through the channel, and the second one determines the number of channels that the jammer actually jams. Some worked out numerical examples complement the theoretical results.