  We provide a complete characterization of the achievable distortion region
for the problem of sending a bivariate Gaussian source over bandwidth-matched
Gaussian broadcast channels, where each receiver is interested in only one
component of the source. This setting naturally generalizes the simple single
Gaussian source bandwidth-matched broadcast problem for which the uncoded
scheme is known to be optimal. We show that a hybrid scheme can achieve the
optimum for the bivariate case, but neither an uncoded scheme alone nor a
separation-based scheme alone is sufficient. We further show that in this joint
source channel coding setting, the Gaussian setting is the worst scenario among
the sources and channel noises with the same covariances.
