Nyquist–Shannon sampling theorem(奈奎斯特-香农采样定理)

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1、Nyquist–ShannonsamplingtheoremX(ƒ)1IntroductionSamplingistheprocessofconvertingasignal(forex-ample,afunctionofcontinuoustimeand/orspace)intoanumericsequence(afunctionofdiscretetimeand/orspace).Shannon’sversionofthetheoremstates:[1]Ifafunctionx(t)containsnofrequencieshighe

2、rthanB−BBƒhertz,itiscompletelydeterminedbygivingitsordinatesataseriesofpointsspaced1/(2B)secondsapart.Fig.1:MagnitudeoftheFouriertransformofabandlimitedAsufficientsample-rateistherefore2Bsamples/second,functionoranythinglarger.Equivalently,foragivensampleratef⛿,perfectrecon

3、structionisguaranteedpossibleforabandlimitB≤f⛿/2.Inthefieldofdigitalsignalprocessing,thesamplingthe-Whenthebandlimitistoohigh(orthereisnoban-oremisafundamentalbridgebetweencontinuous-timedlimit),thereconstructionexhibitsimperfectionsknownsignals(oftencalled“analogsignals”)

4、anddiscrete-timeasaliasing.Modernstatementsofthetheoremaresome-signals(oftencalled“digitalsignals”).Itestablishesatimescarefultoexplicitlystatethatx(t)mustcontainsufficientconditionbetweenasignal’sbandwidthandthenosinusoidalcomponentatexactlyfrequencyB,orthatsampleratethatp

5、ermitsadiscretesequenceofsamplesBmustbestrictlylessthan½thesamplerate.Thetocapturealltheinformationfromthecontinuous-timetwothresholds,2Bandf⛿/2arerespectivelycalledthesignal.NyquistrateandNyquistfrequency.Andrespectively,Strictlyspeaking,thetheoremonlyappliestoaclassofth

6、eyareattributesofx(t)andofthesamplingequipment.mathematicalfunctionshavingaFouriertransformthatisTheconditiondescribedbytheseinequalitiesiscalledzerooutsideofafiniteregionoffrequencies(seeFig1).theNyquistcriterion,orsometimestheRaabecondi-Intuitivelyweexpectthatwhenoneredu

7、cesacontinuoustion.Thetheoremisalsoapplicabletofunctionsofotherfunctiontoadiscretesequenceandinterpolatesbacktodomains,suchasspace,inthecaseofadigitizedimage.acontinuousfunction,thefidelityoftheresultdependsTheonlychange,inthecaseofotherdomains,istheunitsonthedensity(orsam

8、plerate)oftheoriginalsamples.ofmeasureappliedtot,f⛿,andB.Thesamplingtheoremintroducestheconcepto