Optimal Communication of States of Dynamical Systems over Gaussian Channels with Noisy Feedback: The Scalar Case
We consider the problem of communicating the state of a dynamical system via a Shannon Gaussian channel. 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. Noisy feedback from the receiver to the transmitter is present. The transmitter observes the noise-corrupted feedback message from the receiver together with a possibly noisy measurement of the state the dynamical system. These measurements are then used to encode the message to be transmitted over a noisy Gaussian channel, where a per symbol power constraint is imposed on the transmitted message. Thus, we get a mixed problem of Shannon's source-channel coding problem and a sort of Kalman filtering problem. In particular, we consider two feedback instances, one being feedback of receiver measurements and the second being the receiver's state estimates. We show that optimal encoders and decoders are linear filters with a finite memory and we give explicitly the state space realizations of the optimal filters. For the case where the transmitter has access to noisy measurements of the state, we derive a separation principle for the optimal communication scheme. Furthermore, we investigate the presence of noiseless feedback or no feedback from the receiver to the transmitter. Necessary and sufficient conditions for the existence of a stationary solution are also given for the feedback cases considered.