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This paper focuses on the structural properties of test channels, of Wyner'soperational information rate distortion function (RDF),$\overline{R}(\Delta_X)$, of a tuple of multivariate correlated, jointlyindependent and identically distributed Gaussian random variables (RVs),$\{X_t, Y_t\}_{t=1}^\infty$, $X_t: \Omega \rightarrow {\mathbb R}^{n_x}$, $Y_t:\Omega \rightarrow {\mathbb R}^{n_y}$, with average mean-square error at thedecoder, $\frac{1}{n} {\bf E}\sum_{t=1}^n||X_t - \widehat{X}_t||^2\leq\Delta_X$, when $\{Y_t\}_{t=1}^\infty$ is the side information available to thedecoder only. We construct optimal test channel realizations, which achieve theinformational RDF, $\overline{R}(\Delta_X) \triangleq\inf_{{\cal M}(\Delta_X)}I(X;Z|Y)$, where ${\cal M}(\Delta_X)$ is the set of auxiliary RVs $Z$ suchthat, ${\bf P}_{Z|X,Y}={\bf P}_{Z|X}$, $\widehat{X}=f(Y,Z)$, and ${\bfE}\{||X-\widehat{X}||^2\}\leq \Delta_X$. We show the fundamental structuralproperties: (1) Optimal test channel realizations that achieve the RDF,$\overline{R}(\Delta_X)$, satisfy conditional independence, $  {\bf P}_{X|\widehat{X}, Y, Z}={\bf P}_{X|\widehat{X},Y}={\bfP}_{X|\widehat{X}}, \hspace{.2in} {\bf E}\Big\{X\Big|\widehat{X}, Y,Z\Big\}={\bf E}\Big\{X\Big|\widehat{X}\Big\}=\widehat{X} $ and (2) similarlyfor the conditional RDF, ${R}_{X|Y}(\Delta_X) \triangleq \inf_{{\bfP}_{\widehat{X}|X,Y}:{\bf E}\{||X-\widehat{X}||^2\} \leq \Delta_X} I(X;\widehat{X}|Y)$, when $\{Y_t\}_{t=1}^\infty$ is available to both the encoderand decoder, and the equality $\overline{R}(\Delta_X)={R}_{X|Y}(\Delta_X)$.

Structural Properties of Optimal Test Channels for Distributed Source  Coding with Decoder Side Information for Multivariate Gaussian Sources with  Square-Error Fidelity