Kalman meets Shannon

We consider the problem of communicating the state of a dynamical system via a Shannon Gaussian channel with a given power constraint and no feedback. The transmitter observes a possibly noisy measurement of the state. These measurements are then used to encode the message to be transmitted over a noisy Gaussian channel, where a power constraint is imposed on the transmitted message. The receiver, which acts as both a decoder and estimator, observes the noisy measurement of the channel output and makes an optimal estimate of the state of the dynamical system in the minimum mean square sense. Thus, we get a mixed problem of Shannon’s source-channel coding problem and a sort of Kalman filtering problem. We show that optimal encoders and decoders are linear filters with a finite memory and we give explicitly the state space realization of the optimal filters. We also present the solution of the case where the transmitter has access to noisy measurements of the state where we derive a separation principle for this communication scheme. Finally, we give necessary and sufficient conditions for the existence of a stationary solution.