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.
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机译：归因于香农的一对众所周知的不等式，用相同的方差/熵的高斯源的速率失真函数将实际源的速率失真函数上/下限。我们将这些界限扩展到多个描述，对于这个问题，一般的“单字母”解决方案是未知的。我们表明，可实现的边际（d / sub 1 /，d / sub 2 /）和中心（d / sub 0 /）的集合D / sub X /（R / sub 1 /，R / sub 2 /）均值-从两个描述以速率R / sub 1 /和R / sub 2 /解码X的平方误差满足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 /）其中/ spl sigma // sub x // sup 2 /和P / sub x /分别是X和D *（/ spl sigma // sup 2 /，R / sub 1 /，R / sub 2 /）是由Ozarow（1980）找到的，方差为/ spl sigma // sup 2 /的高斯源的多描述失真区域。我们进一步证明，与在单个描述情况下一样，高斯随机代码在极限中达到了d / sub 1 /，d / sub 2 // spl rarr / 0的外边界，因此该外边界在高分辨率下渐近紧条件。
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