This paper focuses on the structural properties of test channels, of Wyner's operational information rate distortion function (RDF), $\overline{ R}(\Delta_X)$, of a tuple of multivariate correlated, jointly independent 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 the decoder, $\frac{ 1}{ n} { \bf E}\sum_{ t=1}^n||X_t - \widehat{ X}_t||^2\leq \Delta_X$, when $\{ Y_t\}_{ t=1}^\infty$ is the side information available to the decoder only. We construct optimal test channel realizations, which achieve the informational 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 $Z$ such that, ${ \bf P}_{ Z|X,Y}={ \bf P}_{ Z|X}$, $\widehat{ X}=f(Y,Z)$, and ${ \bf E}\{ ||X-\widehat{ X}||^2\}\leq \Delta_X$. We show the fundamental structural properties: (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}={ \bf P}_{ 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) similarly for the conditional RDF, ${ R}_{ X|Y}(\Delta_X) \triangleq \inf_{ { \bf P}_{ \widehat{ X}|X,Y}:{ \bf E}\{ ||X-\widehat{ X}||^2\} \leq \Delta_X} I(X; \widehat{ X}|Y)$, when $\{ Y_t\}_{ t=1}^\infty$ is available to both the encoder and decoder, and the equality $\overline{ R}(\Delta_X)={ R}_{ X|Y}(\Delta_X)$.
Categories: cs.IT math.IT