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1、AppendixBInformationtheoryfromfirstprinciplesThisappendixdiscussestheinformationtheorybehindthecapacityexpres-sionsusedinthebook.Section8.3.4istheonlypartofthebookthatsupposesanunderstandingofthematerialinthisappendix.Morein-depthandbroaderexpositionsof
2、informationtheorycanbefoundinstandardtextssuchas[26]and[43].B.1DiscretememorylesschannelsAlthoughthetransmittedandreceivedsignalsarecontinuous-valuedinmostofthechannelsweconsideredinthisbook,theheartofthecommunicationproblemisdiscreteinnature:thetransmi
3、ttersendsoneoutofafinitenum-berofcodewordsandthereceiverwouldliketofigureoutwhichcodewordistransmitted.Thus,tofocusontheessenceoftheproblem,wefirstcon-siderchannelswithdiscreteinputandoutput,so-calleddiscretememorylesschannels(DMCs).Boththeinputxmandt
4、heoutputymofaDMClieinfinitesetsandrespectively.(Thesesetsarecalledtheinputandoutputalphabetsofthechannelrespectively.)Thestatisticsofthechannelaredescribedbyconditionalprobabilitiespjii∈j∈.Thesearealsocalledtransitionprob-abilities.Givenanin
5、putsequencex=x1xN,theprobabilityofobservinganoutputsequencey=y1yNisgivenby1Npyx=pymxm(B.1)m=1Theinterpretationisthatthechannelnoisecorruptstheinputsymbolsindependently(hencethetermmemoryless).1Thisformulaisonlyvalidwhent
6、hereisnofeedbackfromthereceivertothetransmitter,i.e.,theinputisnotafunctionofpastoutputs.Thisweassumethroughout.516517B.1DiscretememorylesschannelsExampleB.1BinarysymmetricchannelThebinarysymmetricchannelhasbinaryinputandbinaryoutput==01.Thetrans
7、itionprobabilitiesarep01=p10=p00=p11=1?.A0anda1arebothflippedwithprobability.SeeFigureB.1(a).ExampleB.2BinaryerasurechannelThebinaryerasurechannelhasbinaryinputandternaryoutput=01=01e.Thetransitionprobabilitiesarep00=p11=1
8、?pe0=pe1=.Here,symbolscannotbeflippedbutcanbeerased.SeeFigureB.1(b).AnabstractionofthecommunicationsystemisshowninFigureB.2.Thesenderhasoneoutofseveralequallylikelymessagesitwantstotransmittothereceiver.Toconveytheinformat
