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_X(R₁, R₂) of achievable marginal (d₁, d₂) and central (d₀) mean-squared errors in decoding X from two descriptions at rates R₁ and R₂ satisfies D*(σ_x², R₁, R₂)⊆D _X(R₁, R₂)⊆D*(P_x, R₁, R₂) where σ_x² and P_x are the variance and the entropy-power of X, respectively, and D*(σ², R₁, R₂) is the multiple description distortion region for a Gaussian source with variance σ² 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 d₁, d₂→0, thus the outer bound is asymptotically tight at high resolution conditions