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1、ReprintedwithcorrectionsfromTheBellSystemTechnicalJournal,Vol.27,pp.379–423,623–656,July,October,1948.AMathematicalTheoryofCommunicationByC.E.SHANNONINTRODUCTIONHErecentdevelopmentofvariousmethodsofmodulationsuchasPCMandPPMwhichexchangebandwidthforsignal-to-noiseratiohasint
2、ensifiedtheinterestinageneraltheoryofcommunication.ATbasisforsuchatheoryiscontainedintheimportantpapersofNyquist1andHartley2onthissubject.Inthepresentpaperwewillextendthetheorytoincludeanumberofnewfactors,inparticulartheeffectofnoiseinthechannel,andthesavingspossibleduetothe
3、statisticalstructureoftheoriginalmessageandduetothenatureofthefinaldestinationoftheinformation.Thefundamentalproblemofcommunicationisthatofreproducingatonepointeitherexactlyorap-proximatelyamessageselectedatanotherpoint.Frequentlythemessageshavemeaning;thatistheyrefertoorare
4、correlatedaccordingtosomesystemwithcertainphysicalorconceptualentities.Thesesemanticaspectsofcommunicationareirrelevanttotheengineeringproblem.Thesignificantaspectisthattheactualmessageisoneselectedfromasetofpossiblemessages.Thesystemmustbedesignedtooperateforeachpossiblesel
5、ection,notjusttheonewhichwillactuallybechosensincethisisunknownatthetimeofdesign.Ifthenumberofmessagesinthesetisfinitethenthisnumberoranymonotonicfunctionofthisnumbercanberegardedasameasureoftheinformationproducedwhenonemessageischosenfromtheset,allchoicesbeingequallylikely.
6、AswaspointedoutbyHartleythemostnaturalchoiceisthelogarithmicfunction.Althoughthisdefinitionmustbegeneralizedconsiderablywhenweconsidertheinfluenceofthestatisticsofthemessageandwhenwehaveacontinuousrangeofmessages,wewillinallcasesuseanessentiallylogarithmicmeasure.Thelogarithm
7、icmeasureismoreconvenientforvariousreasons:1.Itispracticallymoreuseful.Parametersofengineeringimportancesuchastime,bandwidth,numberofrelays,etc.,tendtovarylinearlywiththelogarithmofthenumberofpossibilities.Forexample,addingonerelaytoagroupdoublesthenumberofpossiblestatesoft
8、herelays.Itadds1tothebase2logarithmofthisnumber.Doublingthetimeroughlysquaresthenu
