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The rate-distortion function for source coding with side information at the decoder

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Let{(X_{k}, Y_{k}) }^{ infty}_{k=1}be a sequence of independent drawings of a pair of dependent random variablesX, Y. Let us say thatXtakes values in the finite setcal X. It is desired to encode the sequence{X_{k}}in blocks of length n into a binary stream of rateR, which can in turn be decoded as a sequence{ hat{X}_{k} }, wherehat{X}_{k} in hat{ cal X}, the reproduction alphabet. The average distortion level is(1/n) sum^{n}_{k=1} E[D(X_{k},hat{X}_{k})], whereD(x,hat{x}) geq 0, x in {cal X}, hat{x} in hat{ cal X}, is a preassigned distortion measure. The special assumption made here is that the decoder has access to the side information{Y_{k}}. In this paper we determine the quantityR ast (d), defined as the infimum ofratesRsuch that (withvarepsilon > 0arbitrarily small and with suitably largen)communication is possible in the above setting at an average distortion level (as defined above) not exceedingd + varepsilon. The main result is thatR ast (d) = inf [I(X;Z) - I(Y;Z)], where the infimum is with respect to all auxiliary random variablesZ(which take values in a finite setcal Z) that satisfy: i)Y,Zconditionally independent givenX; ii) there exists a functionf: {cal Y} times {cal Z} rightarrow hat{ cal X}, such thatE[D(X,f(Y,Z))] leq d. LetR_{X | Y}(d)be the rate-distortion function which results when the encoder as well as the decoder has access to the side information{ Y_{k} }. In nearly all cases it is shown that whend > 0thenR ast(d) > R_{X|Y} (d), so that knowledge of the side information at the encoder permits transmission of the{X_{k}}at a given distortion level using a smaller transmission rate. This is in contrast to the situation treated by Slepian and Wolf [5] where, for arbitrarily accurate reproduction of{X_{k}}, i.e.,d = varepsilonfor anyvarepsilon >0, knowledge of the side information at the- encoder does not allow a reduction of the transmission rate.

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Information Theory, IEEE Transactions on  (Volume:22 ,  Issue: 1 )