Estimation with communication cost over a finite alphabet - a transport theory approach

Consider a stochastic process, a pre-processor that accepts causal measurements of the process and a state estimator. The pre-processor and the estimator are not co-located and each time the pre-processor can send a symbol to the estimator from a finite alphabet. We seek the pro-processor and the estimator that jointly minimize a cost combining two terms: the expected squared state estimation error and a communication cost, under a constraint for the communication cost. The communication cost comes from the fact that each transmission of a symbol incurs a cost. In this paper we find what is the optimal encoding at the pre-processor side and which is the optimal estimate.