Broadcasting correlated Gaussians

We consider a one-to-two Gaussian broadcasting problem where the transmitter observes a memoryless bi-variate Gaussian source and each receiver wishes to estimate one of the source components. The transmitter describes the source pair by means of an average-power-constrained signal and each receiver observes this signal corrupted by a different additive white Gaussian noise. From its respective observation, Receiver 1 wishes to estimate the first source component and Receiver 2 wishes to estimate the second. We seek to characterize the pairs of expected squared-error distortions that are simultaneously achievable at the two receivers. Our result is that below a certain SNR-threshold an ldquouncoded schemerdquo that sends a linear combination of the source components is optimal. We present a lower bound on this threshold in terms of the source correlation and the distortion at the receiver with weaker channel noise.