  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. The transmitter observes a possibly noisy measurement of the
state of the dynamical system. These measurements are then used to encode the
message to be transmitted over a noisy Gaussian channel, where a per sample
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. We first consider the problem of communication with full
state measurements at the transmitter and show that optimal linear encoders
don't need to have memory and the optimal linear decoders have an order of at
most that of the state dimension. We also give explicitly the structure of the
optimal linear 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, where the transmitter needs a filter with an order of at
most the dimension of the state of the dynamical system. The results are
derived for first order linear dynamical systems, but may be extended to MIMO
systems with arbitrary order.
