Achievable rates for multiple descriptions

Abstract

Consider a sequence of independent identically distributed (i.i.d.) random variables <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X_{l},X_{2}, \cdots, X_{n}</tex> and a distortion measure <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d(X_{i},X̂_{i})</tex> on the estimates <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X̂_{i}</tex> of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X_{i}</tex> . Two descriptions <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i(X)\in \{1,2, \cdots ,2^{nR_{1}\}</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">j(X)\in \{1,2, \cdots,2^{nR_{2}\}</tex> are given of the sequence <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X=(X_{1}, X_{2}, \cdots ,X_{n})</tex> . From these two descriptions, three estimates <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(i(X)), X2(j(X))</tex> , and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\hat{X}_{O}(i(X),j(X))</tex> are formed, with resulting expected distortions <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E \frac{1/n} \sum^{n}_{k=1} d(X_{k}, \hat{X}_{mk})=D_{m}, m=0,1,2.</tex> We find that the distortion constraints <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D_{0}, D_{1}, D_{2}</tex> are achievable if there exists a probability mass distribution <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p(x)p(\hat{x}_{1},\hat{x}_{2},\hat{x}_{0}|x)</tex> with <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Ed(X,\hat{x}_{m})\leq D_{m}</tex> such that <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R_{1}&gt;I(X;\hat{X}_{1}),</tex> <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R_{2}&gt;I(X;\hat{X}_{2}),</tex> where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">I(\cdot)</tex> denotes Shannon mutual information. These rates are shown to be optimal for deterministic distortion measures.

Keywords

Affiliated Institutions

• Stanford University US

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