Achievable rates for multiple descriptions
Abbas El Gamal, Thomas M. Cover · IEEE Transactions on Information Theory · 1982
Consider a sequence of independent identically distributed (i.i.d.) random variablesX_{l},X_{2}, \cdots, X_{n}and a distortion measured(X_{i},X̂_{i})on the estimatesX̂_{i}ofX_{i}. Two descriptionsi(X)\in \{1,2, \cdots ,2^{nR_{1}\}andj(X)\in \{1,2, \cdots,2^{nR_{2}\}are given of the sequenceX=(X_{1}, X_{2}, \cdots ,X_{n}). From these two descriptions, three estimates(i(X)), X2(j(X)), and\hat{X}_{O}(i(X),j(X))are formed, with resulting expected distortionsE \frac{1/n} \sum^{n}_{k=1} d(X_{k}, \hat{X}_{mk})=D_{m}, m=0,1,2.We find that the distortion constraintsD_{0}, D_{1}, D_{2}are achievable if there exists a probability mass distributionp(x)p(\hat{x}_{1},\hat{x}_{2},\hat{x}_{0}|x)withEd(X,\hat{x}_{m})\leq D_{m}such thatR_{1}>I(X;\hat{X}_{1}),R_{2}>I(X;\hat{X}_{2}),whereI(\cdot)denotes Shannon mutual information. These rates are shown to be optimal for deterministic distortion measures.