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1、ReprintedwithcorrectionsfromTheBellSystemTechnicalJournal,Vol.27,pp.379–423,623–656,July,October,1948.AMathematicalTheoryofCommunicationByC.E.SHANNONINTRODUCTIONHErecentdevelopmentofvariousmethodsofmodulationsuchasPCMandPPMwhichexchangebandwidthforsignal-to-noiser
2、atiohasintensi?edtheinterestinageneraltheoryofcommunication.ATbasisforsuchatheoryiscontainedintheimportantpapersofNyquist1andHartley2onthissubject.Inthepresentpaperwewillextendthetheorytoincludeanumberofnewfactors,inparticulartheeffectofnoiseinthechannel,andthesav
3、ingspossibleduetothestatisticalstructureoftheoriginalmessageandduetothenatureofthe?naldestinationoftheinformation.Thefundamentalproblemofcommunicationisthatofreproducingatonepointeitherexactlyorap-proximatelyamessageselectedatanotherpoint.Frequentlythemessageshave
4、meaning;thatistheyrefertoorarecorrelatedaccordingtosomesystemwithcertainphysicalorconceptualentities.Thesesemanticaspectsofcommunicationareirrelevanttotheengineeringproblem.Thesigni?cantaspectisthattheactualmessageisoneselectedfromasetofpossiblemessages.Thesystemm
5、ustbedesignedtooperateforeachpossibleselection,notjusttheonewhichwillactuallybechosensincethisisunknownatthetimeofdesign.Ifthenumberofmessagesinthesetis?nitethenthisnumberoranymonotonicfunctionofthisnumbercanberegardedasameasureoftheinformationproducedwhenonemessa
6、geischosenfromtheset,allchoicesbeingequallylikely.AswaspointedoutbyHartleythemostnaturalchoiceisthelogarithmicfunction.Althoughthisde?nitionmustbegeneralizedconsiderablywhenweconsiderthein?uenceofthestatisticsofthemessageandwhenwehaveacontinuousrangeofmessages,wew
7、illinallcasesuseanessentiallylogarithmicmeasure.Thelogarithmicmeasureismoreconvenientforvariousreasons:1.Itispracticallymoreuseful.Parametersofengineeringimportancesuchastime,bandwidth,numberofrelays,etc.,tendtovarylinearlywiththelogarithmofthenumberofpossibilitie
8、s.Forexample,addingonerelaytoagroupdoublesthenumberofpossiblestatesoftherelays.Itadds1tothebase2logarithmofthisnumber.Doublingthetimeroughlysquaresthenu
