Fundamentals_Wireless_Communication_AppendixB

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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).Boththeinputx
mandt

4、heoutputy
mofaDMClieinfinitesets
andrespectively.(Thesesetsarecalledtheinputandoutputalphabetsofthechannelrespectively.)Thestatisticsofthechannelaredescribedbyconditionalprobabilitiespj
ii∈
j∈.Thesearealsocalledtransitionprob-abilities.Givenanin

5、putsequencex=x
1x
N,theprobabilityofobservinganoutputsequencey=y
1y
Nisgivenby1
Npy
x=py
m
x
m(B.1)m=1Theinterpretationisthatthechannelnoisecorruptstheinputsymbolsindependently(hencethetermmemoryless).1Thisformulaisonlyvalidwhent

6、hereisnofeedbackfromthereceivertothetransmitter,i.e.,theinputisnotafunctionofpastoutputs.Thisweassumethroughout.516517B.1DiscretememorylesschannelsExampleB.1BinarysymmetricchannelThebinarysymmetricchannelhasbinaryinputandbinaryoutput
==01.Thetrans

7、itionprobabilitiesarep0
1=p1
0=p0
0=p1
1=1−.A0anda1arebothflippedwithprobability.SeeFigureB.1(a).ExampleB.2BinaryerasurechannelThebinaryerasurechannelhasbinaryinputandternaryoutput
=01=01e.Thetransitionprobabilitiesarep0
0=p1
1=1

8、−pe
0=pe
1=.Here,symbolscannotbeflippedbutcanbeerased.SeeFigureB.1(b).AnabstractionofthecommunicationsystemisshowninFigureB.2.Thesenderhasoneoutofseveralequallylikelymessagesitwantstotransmittothereceiver.Toconveytheinformat