The Gaussian CEO problem for a scalar source with memory: A necessary condition

In wireless sensor networks, many monitoring problems can be cast in the form of distributed data collection and centralized data fusion. If the data links from the sensor nodes to the fusion center have limited capacity, there is a tradeoff between estimation precision and transmission rate. This kind of decentralized estimation system is equivalent to the so-called indirect multiterminal source coding problem, and the Berger-Tung inner bound is the best known achievable rate region boundary. In this paper, we attempt to evaluate the Berger-Tung achievable sum rate for a Gaussian scalar source with arbitrary memory, and we find an analytic necessary condition which the solution to the sum rate problem must satisfy. First, we describe the system model and discuss how the arbitrary memory problem can be tackled by utilizing known results for the vector central estimation officer (CEO) problem. Then we formulate the sum rate calculation as a variational calculus problem with a distortion constraint, and show how to find the necessary condition. Based on our analytic form of the condition, we discuss how to compute the rate-distortion curve and an example is given. We also note that the solution is compatible with previous findings in rate distortion theory, which supports the validity of the necessary condition.