Gaussian codes and Shannon bounds for multiple descriptions

A pair of well-known inequalities, due to Shannon, upper/lower bound the rate-distortion function of a real source by the rate-distortion function of the Gaussian source with the same variance/entropy. We extend these bounds to multiple descriptions, a problem for which a general "single-letter" solution is not known. We show that the set D/sub X/(R/sub 1/, R/sub 2/) of achievable marginal (d/sub 1/, d/sub 2/) and central (d/sub 0/) mean-squared errors in decoding X from two descriptions at rates R/sub 1/ and R/sub 2/ satisfies D*(/spl sigma//sub x//sup 2/, R/sub 1/, R/sub 2/)/spl sube/D/sub X/(R/sub 1/, R/sub 2/)/spl sube/D*(P/sub x/, R/sub 1/, R/sub 2/) where /spl sigma//sub x//sup 2/ and P/sub x/ are the variance and the entropy-power of X, respectively, and D*(/spl sigma//sup 2/, R/sub 1/, R/sub 2/) is the multiple description distortion region for a Gaussian source with variance /spl sigma//sup 2/ found by Ozarow (1980). We further show that like in the single description case, a Gaussian random code achieves the outer bound in the limit as d/sub 1/, d/sub 2//spl rarr/0, thus the outer bound is asymptotically tight at high resolution conditions.