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1、IEEETRANSACTIONSONINFORMATIONTHEORY,VOL.IT-22,NO.1,JANUARY19761TheRate-DistortionFunctionforSourceCodingwithSideInformationattheDecoderAAROND.WYNER,FELLOW,IEEE,ANDJACOBZIV,FELLOW,IEEEAbstract-Let{(X,,Y,J}r=1beasequenceofindependentdrawingsofwhichtakevaluesinthefi
2、nitereproductionalphabet.!?.apairofdependentrandomvariablesX,Y.LetussaythatXtakesvaluesTheencodinganddecodingisdoneinblocksoflengthn,inthefiniteset6.Itisdesiredtoencodethesequence{X,}inblocksandthefidelitycriterionistheexpectationofoflengthnintoabinarystream*ofra
3、teR,whichcaninturnbedecodedasasequence{2k},wherezkE%,thereproductionalphabet.Theaveragedistorjionlevelis(l/n)cl=1E[D(X,,z&,whereD(x,$20,xEI,2EJ,isapre-assigneddistortionmeasure.Thespecialassumptionmadehereisthatthedecoderhasaccesstothesideinformation{Yk}.whereD(x
4、,R)20,xE%‘,2E@‘,isagivendistortionfunc-InthispaperwedeterminethequantityR*(d).definedastheinfimumofratesRsuchthat(withE>0arbitrarilysmallandwithsuitablylargen)tion.IfswitchAand/orBisclosedthentheencoderand/orcommunicationispossibleintheabovesettingatanaveragedist
5、ortiondecoder,respectively,areassumedtohaveknowledgeoflevel(asdefinedabove)notexceedingd+E.Themainresultisthatthesideinformationsequence{Y,}.IfswitchAand/orBisR*(d)=inf[Z(X,Z)-Z(Y,Z)],wheretheinfimumiswithrespecttoallopen,thenthesideinformationisnotavailabletothe
6、auxiliaryrandomvariablesZ(whichtakevaluesinafiniteset3)thatencoderand/ordecoder,respectively.satisfy:i)Y,ZconditiofallyindependentgivenX;ii)thereexistsafunctionf:“YxE+.%,suchthatE[D(X,f(Y,Z))]5d.Nowconsiderthefollowingcases:LetRx,y(d)betherate-distortionfunctionw
7、hichresultswhenthei)switchesAandBareopen,i.e.,thereisnoavailableencoderaswellasthedecoderhasaccesstothesideinformation{Y,}.sideinformation;Innearlyallcasesitisshownthatwhend>0thenR*(d)>Rx,y(d),sothatknowledgeofthesideinformationattheencoderpermitstrans-ii)switche
8、sAandBareclosed,i.e.,boththeencodermissionofthe{X,}atagivendistortionlevelusingasmallertransmissionandthedecoderhaveaccesstothesideinformationrate.Thisisincont
