Structural Properties of Test Channels of the RDF for Gaussian Multivariate Distributed Sources

In this paper, we consider Wyner's [1] lossy distributed source coding problem of Fig. 1, for multivariate Gaussian distributed sources with square-error fidelity, when i) the side information, random variable (RV) Y, is available at the decoder only, and ii) Y is available at both the encoder and the decoder. We derive structural properties of optimal test channels that achieve the operational information rate distortion functions (RDFs) for i) and ii), i.e., (R) over bar (d) and R-X vertical bar Y (d), respectively. We also show there is no loss of compression if the side information Y is only available at the decoder, i.e., (R) over bar (d) = R-X vertical bar Y (d). This paper also points out that the two test channels of the RDF of the distributed remote source coding problem given in [2] and [3], do not generate Wyner's value of the RDF of scalar RVs, as one would expect; this observation implies that the results in [2, Theorem 4] and [3, Theorem 3A] should he read with caution.