Optimal Distributed Linear Transceivers for Sending Independently Corrupted Copies of a Colored Source Over the Gaussian MAC

We consider distributed linear transceivers for sending a second-order wide-sense stationary process observed by two noisy sensors over a Gaussian multiple-access channel (MAC). We derive the minimum mean-square error (MSE) distributed linear transceiver. The optimal linear transmitter exploits bandwidth expansion by repeating transmission and the transmitters at the two sensors are the same except for a constant factor. When the source is white, encoded transmission is the best linear code for any SNR. But for a colored source, whitening transmit filter is sub-optimal. In high SNR regime, the magnitude response of the optimal transmission filter is inversely proportional to fourth-root of the power spectrum of the process (while that for the whitening filler is inversely proportional to the square-root of the spectrum). In the special case of n single sensor with Gaussian source, we also quantify the performance loss of linear source-channel codes with respect to the Shannon limit.