An encoder whose input is a binary equiprobable memoryless source produces one output of rateR_{1}and another of rateR_{2}. LetD_{1}, D_{2}, and D_{0}, respectively, denote the average error frequencies with which the source data can be reproduced on the basis of the encoder output of rateR_{l}only, the encoder output of rateR_{2}only, and both encoder outputs. The two-descriptions problem is to determine the regionRof all quintuples(R_{1}, R_{2}, D_{1}, D_{2}, D_{0})that are achievable in thc usual Shannon sense. LetR(D)=1+D log_{2} D+(1-D) log_{2}(1-D)denote the error frequency rate-distortion function of the source. The "no excess rate case" prevails whenR_{1} + R_{2} = R(D_{0}), and the "excess rate case" whenR_{1} + R_{2} > R(D_{0}). Denote the section ofRat(R_{1}, R_{2}, D_{0})byD(R_{1} R_{2}, D_{0}) ={(D_{1},D_{2}): (R_{1}, R_{2}, D_{1},D_{2},D_{0}) in R}. In the no excess rate case we show that a portion of the boundary ofD(R_{1}, R_{2}, D_{0})coincides with the curve(frac{1}{2} + D_{1}-2D_{0})(frac_{1}_{2} + D_{2}-2D_{0})= frac{1}{2}(1-2D_{0})^{2}. This curve is an extension of Witsenhausen's hyperbola bound to the caseD_{0} > 0. It follows that the projection ofRonto the(D_{1}, D_{2})-plane at fixedD_{0}consists of allD_{1} geq D_{0}andD_{2} geq D_{0}that lie on or above this hyperbola. In the excess rate case we show by counterexample that the achievable region of El Gamal and Cover is not tight.

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Information Theory, IEEE Transactions on  (Volume:33 ,  Issue: 4 )

Jul 1987