Broadcast Correlated Gaussians: The Vector-Scalar Case

The problem of sending a set of correlated Gaussian sources over a bandwidth-matched two-user scalar Gaussian broadcast channel is studied in this work, where the strong receiver wishes to reconstruct several source components (i.e., a vector source) under a distortion covariance matrix constraint and the weak receiver wishes to reconstruct a single source component (i.e., a scalar source) under the mean squared error distortion constraint. We provide a complete characterization of the optimal tradeoff between the transmit power and the achievable reconstruction distortion pair for this problem. The converse part is based on a new bounding technique which involves the introduction of an appropriate remote source. The forward part is based on a hybrid scheme where the digital portion uses dirty paper channel code and Wyner-Ziv source code. This scheme is different from the optimal scheme proposed by Tian et al. in a recent work for the scalar-scalar case, which implies that the optimal scheme for the scalar-scalar case is in fact not unique.