Gaussian degraded relay channel with lossy state reconstruction

We consider a state reconstruction problem where the additive state process to a Gaussian degraded relay channel (GRC) is to be reconstructed at the receiver subject to some distortion/fidelity constraints. The state process we consider is Gaussian, independent and identically distributed (i.i.d.) and known a priori at the encoder as well as the relay. The receiver has to perform a lossy estimation of the state process, to within some prescribed distortion tolerance under a squared error distortion criterion. The setting also involves communicating messages across to the receiver, in addition to the estimation. Naturally, there is a tension between communication rate and state reconstruction fidelity. We are interested in the optimal trade-off between the distortion incurred at the receiver versus the rate at which messages can be communicated. We provide a complete characterization of the optimal rate versus distortion performance. The main contribution of the paper is a characterization of the fundamental limits of achievable rate versus distortion pairs in the Gaussian degraded relay model under consideration.