6 Modulation and Sampling.docx
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6 Modulation and Sampling.docx
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6ModulationandSampling
6.ModulationandSampling
6.1Introductionmodulation
Inthislessonweinvestigateanumberofsignaltechniqueswhichfallunderthegeneralheadingof‘modulation’processes.Theterm‘modulation’impliesthevariationoradjustmentofapropertyofonesignal,generallyreferredtoasthe‘carrier’,byasecondwaveformknownasthe‘modulationsignal’.Themodulatedcarriercontains(atleastinprinciple)completeinformationaboutthemodulatingsignal,andmaythereforebeusedasasubstituteforit.Figure6.1showspartofatypicalmodulationsignal
andillustratesfourwaysinwhichitmightbemadetomodulateacarrier.Intwocases,thecarrierisassumedtobearelativelyhighfrequencysinusoid,theamplitudeorfrequencyofwhichismodulatedinsympathywiththeinstantaneousvalueofthesignal
;andintheothertwoexamples,
ismadetomodulateeitherthewidthortheheightofsuccessivepulsesinapulse-train.
Figure6.1
Figure6.1Fourtypesofmodulation
Therearevariousreasonswhyamodulatedcarriermightbeusedasasubstituteforthemodulatingsignalitself.Perhapsthebestknownistheuseofsuchacarrierforradiotransmission;thecarrier,beingofhighfrequency,maybeefficientlyradiatedfromandreceivedbysuitableaerials,whereastherelativelylowfrequencymodulatingsignal(whichmight,forexample,representmusicorspeech)wouldrequireenormousaerialsifitwastobetransmitteddirectly.Inmagnetictaperecording,itistechnicallydifficulttorecordverylowfrequencysignals(saybelowafewhertz):
butifsuchsignalsareusedtomodulateeithertheamplitudeorfrequencyofarelativelyhigh-frequencycarrier,theymayeasilyberecorded.Finally,acontinuoussignalmaybeconvertedintosampled-dataformbycausingittomodulatetheamplitudeofatrainofnarrow(Dirac)pulse;thesampled-datasignalmaythenbestoredoranalyzedbyadigitalcomputer,ortransmittedasasetofnumericalvaluesratherthanasacontinuouswaveform.
Astheabovediscussionimplies,oneofthemajoreffectsofmodulationistoshiftthespectrumofthemodulatingsignal-oftenfromanessentiallylow-frequencyregiontoahigh-frequencyone.Theintroductionofnewfrequenciesisanessentialpropertyofanyusefulmodulationprocess:
suchaprocessmayneverthereforeberealizedbyalinearsystemwiththepropertyoffrequencypreservation,nordomodulationprocesses,asageneralrule,fitintotheframeworkoftimeandfrequencydescriptionsoflinearsystemdevelopedinthepreviouslesson.Fortunately,however,thetypeofmodulationinwhichasignalcausesvariationsintheinstantaneousamplitudeofacarrierisanotableexception,forreasonswhichwillnowbeoutlined。
Themainpointtonoteabout‘amplitudemodulation’isthatmodulatedcarrierisformedbymultiplyingthemodulatingsignalbythecarrier.Henceif
representsthecarrierpriortomodulation,and
representsthemodulatingsignal,theprocessofamplitudemodulationinvolvesformingtheproduct
;thisgeneratesfrequencycomponentswhicharenottobefoundineither
or
alone.Modulatedcarriers(a)and(d)offigure6.1areexamplesofamplitudemodulation.Insection5.2.3,wesawhowmultiplicationoftwofunctionsoffrequencyisequivalenttoconvolutionoftheirrespectivetimefunctions.Infact,theconvolutiontheoremismoregeneralinscopethanthis;itmerelystatesthatthemultiplicationoftwofunctionsisequivalenttoconvolutionoftheirFouriertransforms,regardlessofwhatthosefunctionsrepresent.Inamplitudemodulation,itisthetwotimefunctionswhicharemultiplied,andtheconvolutiontheoremthereforeleadsustoexpectthatthisisequivalenttoconvolutiontheirindividualfrequencyspectra.Forthesereasons,amplitudemodulationisaratherspecialcase,towhichmuchofourearlierdiscussionofconvolutionandlinearsystemstheoryisrelevant.Wenowlookatthisgeneralconceptinrathermoredetail,andapplyitfirstofalltotheprocessofsignalsampling.
6.2Signalsamplingandreconstitution
6.2.1Thesamplingprocess
Wheneveracontinuoussignalistoberepresentedbyasetofsamples,adecisionmustbemadeaboutthesamplingrate.Ifthesamplingrateistoolow,informationaboutthedetailedfluctuationsofthecontinuouswaveformwillbelost:
andiftoohigh,anunnecessarilylargenumberofsampleswillhavetobestoredorprocessed.Weshallnowshowthatthecluetoanappropriatesamplingrateliesintherelationshipbetweenthespectrumofacontinuoussignalandthatofitssampledversion.
Thefirstpointtomakeaboutthesamplingprocessisthatitisaformofamplitudemodulation:
fromamathematicalpointofview,samplingacontinuoussignalisequivalenttomultiplyingitbyatrainofequally-spacedunitDiracpulses.Wemayconvenientlyconsidereachsampletobeaweightedvalueofthecontinuouswaveformattherelevantinstant,sincethesampledsignalisobtainedbymultiplicationofthecontinuouswaveformandtheDiracpulsetrain,itsspectrummaybefoundbyconvolutingtheirrespectivespectra.
OurfirsttaskisthereforetodefinethespectrumofaninfinitetrainofunitDiracpulses,separatedfromoneanotherbysomesamplinginterval.Forconvenience,letusassumethatoneofthepulsesoccursat
.Suchapulsetrainisastrictlyperiodicwaveform,symmetricalabout
andmustthereforehavealinespectrumcontainingonlycosineterms:
itsfundamentalfrequencywillclearlybe
hertz,or
radians/second.Itremainstodefinetherelativemagnitudesofthevariousharmonics.Actuallythisisquitesimplydone,ifwerecallthatthemagnitudeofanyharmonicterminaperiodicwaveformmaybefoundbymultiplyingthewaveformbyacosine(orsine)ofappropriatefrequency,andintegratingtheproductoveronecompleteperiod.Figure6.2shows,asanexample,partofacosineofthethirdharmonicfrequency(
):
ifwemultiplythisbythepulsetrainwaveformabove,andintegrateovertheperiod
thesiftingpropertyoftheDiracpulseensuresthattheresultwillbesimplyequaltothevalueofthecosinewaveat
whichisunity.Indeedthissameresultappliestoanyotherharmonicfrequency(andforthezero-frequencyterm)becausecosinesoffrequencieshaveavalueofunityat
.Weconcludethatthespectrumofallpulsetrainoffigure6.2(a)containsaninfinitesetofcosineharmonics,allofthesameamplitudeandseparatedby
radians/second.ThisresultemphasizesaspecialpropertyoftheDiracpulsetrain—thatitstime-domainwaveformanditsfrequencyspectrumareidenticalinform.Itisperhapsunsurprisingthatitsspectrumexentdstoinfinitelyhighfrequencies,becauseitmadeupfromDiracpulseswhicharethemselvescompletely‘wideband’;andsincethetime-functionisstrictlyrepetitive,thespectrumcancontainonlydiscreteharmonicfrequencies.
Figure6.2
Thenextproblemistoinvestigatewhathappenswhenthespectrumjustdiscussedisconvolutedwiththatofatypicalcontinuoussignal,forsuchaconvolutionwillyieldthespectrumofthesignal’ssampledversion.Letusdenotethespectrumofthecontinuoussignalby
andthespectrumofthepulsetrainby
.Theresultofconvolutingthesetwofunctionsisafurtherfunction
givenby
(6-1)
where
isanauxiliaryfrequencyvariable.Thisformoftheconvolutionintegralisidenticaltothatusedinsection5.2.3,exceptthattimefunctionsarereplacedbyfrequencyfunctions.Weshouldofcourserememberthat
isgenerallycomplex,whereas
ispurelyreal
sinceitisthespectrumofaneventimefunction.However,theaboveconvolutionintegralissimplertovisualizeifwethinkofboth
and
asrealfunctionsof
;andthegeneralvalidityoftheresultisnotaffectedbythisassumption.Nowaswehavealreadyshownthespectrum
consistsofaseriesofequally-spacedspectrallines,whichmayalsoconvenientlyberepresentedbyDiracfunctions(thatis,frequency-domain‘pulses’).Hence
(6-2)
Thisimportantresultshowsthatthespectrumofasampledsignalisarepeatedversionofthatoftheunderlyingcontinuouswaveform,italsogivesavitalcluetotheminimumsamplingratewhichmaybeusedifthesamplevaluesaretoformanadequatesubstitutefortheoriginalsignal.Figure6.3hasshownspectrum
withsignificantfrequencycomponentsinthefrequencyrange
to
:
samplingcausesthisspectrumtorepeatevery
radians/second.Itisthereforeclearthatif
isgreaterthan
therewillbeoverlapbetweenadjacentrepetitionsof
.Withoutsuchoverlapitwouldbepossible,atleastinprinciple,torecovertheoriginalcontinuouswaveformfromitssampledversionbypassingthelatterthroughalinearfilterwhichtransmittedequallyallcomponentsintherange
to
andrejectedallothers.Butwithsuchoverlap,thespectrumofthesampledsignalisnolongersimplyrelatedtothatoftheoriginal(especiallyintheregionsof
),andnolinearfilteringoperationcouldbeexpectedtorecovertheoriginalfromitssampledversion.Theoverlapwhichariseswhenthesamplingrateistoolowisoftenreferredtoas‘aliasing’.
Figure6.3
Figure6.3(a)thespectrumofatypicalcontinuoussignaland(b)thatofitssampledversionwhenthesamplingintervalisTseconds
Theaboveargumentspecifies,ineffect,aminimumsamplingrateforthefaithfulrepresentationofacontinuoussignalbyasampleset.By‘faithfulrepresentation’weimplythatthesamplevaluescontaincompleteinformationaboutfluctuationsintheoriginalsignal,andthattheycouldthereforebeusedtoreconstitutetheoriginalifrequired.Itisatfirstsightremarkablethatasetofsamples
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