Sending a Bivariate Gaussian Over a Gaussian MAC

We study the power-versus-distortion tradeoff for the distributed transmission of a memoryless bivariate Gaussian source over a two-to-one average-power limited Gaussian multiple-access channel. In this problem, each of two separate transmitters observes a different component of a memoryless bivariate Gaussian source. The two transmitters then describe their source component to a common receiver via an average-power constrained Gaussian multiple-access channel. From the output of the multiple-access channel, the receiver wishes to reconstruct each source component with the least possible expected squared-error distortion. Our interest is in characterizing the distortion pairs that are simultaneously achievable on the two source components. We focus on the "equal bandwidth" case, where the source rate in source-symbols per second is equal to the channel rate in channel-uses per second. We present sufficient conditions and necessary conditions for the achievability of a distortion pair. These conditions are expressed as a function of the channel signal-to-noise ratio (SNR) and of the source correlation. In several cases, the necessary conditions and sufficient conditions are shown to agree. In particular, we show that if the channel SNR is below a certain threshold, then an uncoded transmission scheme is optimal. Moreover, we introduce a "source-channel vector-quantizer" scheme which is asymptotically optimal as the SNR tends to infinity.