Gaussian Broadcast Channel with State Estimation

A state amplification problem, where the common additive state to a Gaussian broadcast channel (BC) is to be estimated at both receivers, is considered. The state process, known non-causally at the encoder, is assumed to be independent and identically distributed (i.i.d.) Gaussian. Both receivers must estimate the state process to within acceptable squared error distortion limits. In addition to the state estimation, our setting also requires message transmission to the stronger receiver at a given rate. We are interested in the optimal trade-offs between the distortions incurred at the receivers when a message at a given rate is to be delivered from the encoder to the strong receiver. A complete characterization of the rate-distortion trade-off region is presented. Our result differs from a recent result where an additional common reconstruction constraint was imposed on the state estimates in the same setting, and it was observed that allowing the weak user to decode part of the private message to the stronger user helps the distortion trade-offs.