State-Dependent Gaussian Broadcast Channel with Common State Reconstructions

A common reconstruction (CR) problem, where the common additive state to a Gaussian broadcast channel (BC) is to be estimated at two receivers, is considered. The state process is assumed to be IID Gaussian, and known non-causally at the encoder. Each receiver has to make separate estimates of the state-process, with the CR constraints that the individual receiver's estimate should match a corresponding estimate at the transmitter. We study the trade-offs between the two distortions and a private rate to the strong receiver. We compute inner and outer bounds which are numerically shown to characterize the optimal performance in several regimes of interest. Interestingly, it is observed that allowing the weak user to decode part of the private message to the stronger user helps the distortion trade-offs, even though the objective concerns only a private rate to the strong user. Also, as a special case of our BC results, we show that Gaussian auxiliaries are optimal for a single user Gaussian CR problem.