New results in binary multiple descriptions

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.