Distributed source coding system for correlated Gaussian observations

We consider the distributed source coding system for L correlated Gaussian remote sources X_i, i = 1, 2,···, L, where X_i, i = 1, 2, ·, L are L correlated Gaussian random variables. We deal with the case where each of L distributed encoders can not directly observe X_i but its noisy version Y_i = X_i +N_i. Here N_i, i = 1,2,···, L are independent additive L Gaussian noises also independent of X_i, i = 1, 2,···, L. On this coding system the determination problem of the rate distortion region remains open. In our previous works, we derived explicit outer and inner bounds of the rate distortion region and explicit sufficient condition for those two to match. In this paper we derive a stronger sufficient condition for the inner and outer bound to match. We further study the sum rate part of the rate distortion region when the correlation has some symmetrical property and derive a new lower bound of the sum rate part. We further derive a sufficient condition for this lower bound to be tight. The derived sufficient condition depends only on the correlation property of the sources and their observations.