مرکز منطقه ای اطلاع رساني علوم و فناوري - New results in binary multiple descriptions

Abstract :

An encoder whose input is a binary equiprobable memoryless source produces one output of rate $R_{1}$ and another of rate $R_{2}$ . Let $D_{1}, D_{2}, and D_{0}$ , respectively, denote the average error frequencies with which the source data can be reproduced on the basis of the encoder output of rate $R_{l}$ only, the encoder output of rate $R_{2}$ only, and both encoder outputs. The two-descriptions problem is to determine the region $R$ of all quintuples $(R_{1}, R_{2}, D_{1}, D_{2}, D_{0})$ that are achievable in thc usual Shannon sense. Let $R(D)=1+D \\log _{2} D+(1-D) \\log _{2}(1-D)$ denote the error frequency rate-distortion function of the source. The "no excess rate case" prevails when $R_{1} + R_{2} = R(D_{0})$ , and the "excess rate case" when $R_{1} + R_{2} > R(D_{0})$ . Denote the section of $R$ at $(R_{1}, R_{2}, D_{0})$ by $D(R_{1} R_{2}, D_{0}) ={(D_{1},D_{2}): (R_{1}, R_{2}, D_{1},D_{2},D_{0}) \\in R}$ . In the no excess rate case we show that a portion of the boundary of $D(R_{1}, R_{2}, D_{0})$ 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 case $D_{0} > 0$ . It follows that the projection of $R$ onto the $(D_{1}, D_{2})$ -plane at fixed $D_{0}$ consists of all $D_{1} \\geq D_{0}$ and $D_{2} \\geq D_{0}$ 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.