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1、ReprintedwithcorrectionsfromTheBellSystemTechnicalJournal,Vol.27,pp.379–423,623–656,July,October,1948.AMathematicalTheoryofCommunicationByC.E.SHANNONINTRODUCTIONHErecentdevelopmentofvariousmethodsofmodulationsuchasPCMandPPMwhichexchangebandwidthforsignal-to-noiseratiohasintens
2、i?edtheinterestinageneraltheoryofcommunication.ATbasisforsuchatheoryiscontainedintheimportantpapersofNyquist1andHartley2onthissubject.Inthepresentpaperwewillextendthetheorytoincludeanumberofnewfactors,inparticulartheeffectofnoiseinthechannel,andthesavingspossibleduetothestatis
3、ticalstructureoftheoriginalmessageandduetothenatureofthe?naldestinationoftheinformation.Thefundamentalproblemofcommunicationisthatofreproducingatonepointeitherexactlyorap-proximatelyamessageselectedatanotherpoint.Frequentlythemessageshavemeaning;thatistheyrefertoorarecorrelate
4、daccordingtosomesystemwithcertainphysicalorconceptualentities.Thesesemanticaspectsofcommunicationareirrelevanttotheengineeringproblem.Thesigni?cantaspectisthattheactualmessageisoneselectedfromasetofpossiblemessages.Thesystemmustbedesignedtooperateforeachpossibleselection,notju
5、sttheonewhichwillactuallybechosensincethisisunknownatthetimeofdesign.Ifthenumberofmessagesinthesetis?nitethenthisnumberoranymonotonicfunctionofthisnumbercanberegardedasameasureoftheinformationproducedwhenonemessageischosenfromtheset,allchoicesbeingequallylikely.Aswaspointedout
6、byHartleythemostnaturalchoiceisthelogarithmicfunction.Althoughthisde?nitionmustbegeneralizedconsiderablywhenweconsiderthein?uenceofthestatisticsofthemessageandwhenwehaveacontinuousrangeofmessages,wewillinallcasesuseanessentiallylogarithmicmeasure.Thelogarithmicmeasureismorecon
7、venientforvariousreasons:1.Itispracticallymoreuseful.Parametersofengineeringimportancesuchastime,bandwidth,numberofrelays,etc.,tendtovarylinearlywiththelogarithmofthenumberofpossibilities.Forexample,addingonerelaytoagroupdoublesthenumberofpossiblestatesoftherelays.Itadds1tothe
8、base2logarithmofthisnumber.Doublingthetimeroughlysquaresthenu
