Gaussian codes and Shannon bounds for multiple descriptions

A well known inequality due to Shannon, upper / lower bounds 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 X (R 1 ; R 2) of achievable marginal (d 1 ; d 2) and central (d 0) mean squared errors in decoding X from two descriptions at rates R 1 and R 2 satisses where 2 x and P x are the variance and the entropy-power of X, respectively, and D (2 ; R 1 ; R 2) is the distortion region for a Gaussian source with variance 2 found by Ozarow. We further show that like in the single description case, a Gaussian random code achieves the outer bound in the limit as d 1 ; d 2 ! 0, thus the outer bound is asymptotically tight at high resolution conditions.