通信原理英文版教學(xué)課件：class two version 2

SignalandNoiseinCommunicationsSystemsIncommunicationsystems,thereceivedwaveformisusuallycategorizedintothedesiredpartcontainingtheinformationandtheextraneousorundesiredpart.Thedesiredpartiscalledthesignal,andtheundesiredpartiscallednoise.Noiseisoneofthemostcriticalandfundamentalconceptsaffectingcommunicationsystems.Theentiresubjectofcommunicationsystemisallaboutmethodstoovercomethedistortingorbadeffectsofnoise.Todoso,understandingrandomvariablesandrandomprocessesbecomesquiteessential.TypicalnoisesourceClassificationofSignalsWhatisSignal?Anyphysicalquantitythatvarieswithtime,spaceoranyotherindependentvariablesiscalledasignal.Itisthecarrierofmessage.Themeaningfuloreffectivecontentincludedinthemessageisinformation.

ClassificationofSignalsThereareseveralwaystocharacterizethesignals:Continuous-timesignalvs.discrete-timesignalContinuousvaluedsignalvs.discrete-valuedsignal

Continuous-timeandcontinuousvalued:analogsignal(speech)Discrete-timeanddiscretevalued:digitalsignal(CD)Discrete-timeandcontinuousvalued:sampledsignalContinuous-timeandDiscretevalued:quantizedsignal

DeterministicSignalsandRandomSignals

EnergySignalsandPowerSignals

Energy:E=s2(t)dt;

Power:Energysignalhasfiniteenergy,butitsaveragepowerequals0.Physicallyrealizablewaveformsareofenergy-typePowersignalhasfiniteaveragepower,longduration,soitsenergyequalsinfinity.

DeterministicSignal:Thevalueatanytimearedeterministicandpredictable.RandomSignal:Indeterministicandunpredictable.However,theyhavestatisticcharacteristics.CharacteristicsofDeterminationSignalinfrequencydomainFrequencyspectrumofpowersignal:lets(t)beaperiodicpowersignal,itsperiodisT0,thenwehave

where0=2/T0=2f0

∵C(jn0)isacomplexfunction,

∴C(jn0)=|Cn|ejnwhere|Cn|－amplitudeofthecomponentwithfrequencynf0

n

－phaseofthecomponentwithfrequencynf0Fourierseriesofsignals(t):

?

【Example2.1】Findthespectrumofaperiodicrectangularwave.

Solution:AssumetheperiodofaperiodicrectangularwaveisT,thewidthis,andtheamplitudeisV,then

Itsfrequencyspectrumis

FrequencyspectrumfigureFrequencyspectraldensityofenergysignals

Letanenergysignalbes(t),thenitsfrequencyspectraldensityis

TheinverseFouriertransformofS()istheoriginalsignal:【Example2.3】Findthefrequencyspectraldensityofarectangularpulse.Solution:LettheexpressionoftherectangularpulsebeThenitsfrequencyspectraldensityisitsFouriertransform:?Frequencyspectrumfigure【Example2.4】Findthewaveformandthefrequencyspectraldensityofasamplefunction.

Solution:Thedefinitionofthesamplefunctionis

thefrequencyspectraldensitySa(t)is:

Fromtheaboveequation,weseethatSa()isagatefunction.【Example2.5】Findtheunitimpulsefunctionanditsfrequencyspectraldensity.

Solution:Unitimpulsefunctionisusuallycalleddfunctiond(t).Itsdefinitionis

Thefrequencyspectraldensityof(t):13DifferencebetweenfrequencyspectraldensityS(f)ofenergysignalandfrequencyspectrumofperiodicpowersignal:S(f)－continuousspectrum；C(jn0)－discreteUnitofS(f):V/Hz；UnitofC(jn0):

【Example2.6】Findthefrequencyspectraldensityofacosinusoidalwave(PowerSignal)withinfinitelength.Solution:Lettheexpressionofacosinusoidalwavebef(t)=cos0t,F()canbewrittenasIntroducingd(t)cangeneralizetheconceptoffrequencyspectraldensitytopowersignals.t0－00(a)波形(a)波形－000EnergyspectraldensityLettheenergyofanenergysignals(t)beE,thentheenergyofthesignalisdecidedbyIfitsfrequencyspectraldensityisS(f),thenfromParseval’stheoremwehavewhere|S(f)|2iscalledenergyspectraldensity.Theaboveequationcanberewrittenas：whereG(f)＝|S(f)|2（J/Hz）isenergyspectraldensity.PropertyofG(f):Sinces(t)isarealfunction,|S(f)|2isanevenfunction,∴ PowerspectraldensityLetthetruncatedsignalofs(t)issT(t)，-T/2<t<T/2,thenTodefinethepowerspectraldensityofthesignalas:

obtainthesignalpower:CharacteristicsofDeterministicSignalintimedomainAutocorrelationfunctionDefinitionoftheautocorrelationfunctionforenergysignal:Definitionoftheautocorrelationfunctionforpowersignal:Characteristics:R()isonlydependenton,butindependentoft.When=0,R()ofenergysignalequalstheenergyofthesignal,andR()ofpowersignalequalstheaveragepowerofthesignalCross-correlationfunctionDefinitionofthecross-correlationfunctionforenergysignal:Definitionofthecross-correlationfunctionforpowersignal:Characteristics:1.R12()isdependenton,andindependentoft.2. Proof:Letx=t+，thenProbability:LetAbeaneventinasamplespaceSTheprobabilityP(A)isarealnumberthatmeasuresthelikelihoodoftheeventAAxiomsofProbability

RandomVariables(r.v.)Forexample,thenumberofcallsreceivedwithinagivenperiodoftimeatthetelephoneexchangeisarandomvariable.Distributionfunctionofrandomvariable(orcumulativedistributionprobability/CDF):Definition:FX(x)=P(X

x)Characteristics:∵P(a<X

b)+P(X

a)=P(X

b),

P(a<X

b)=P(X

b)–P(X

a)，

∴

P(a<X

b)=FX(b)–FX(a) 27Distributionfunctionofdiscreterandomvariable:LetthevaluesofXbe:x1

x2…xixn，theirprobabilitiesarerespectivelyp1,p2,…,pi,…,pn,then

P(X<x1)=0, P(Xxn)=1

∵P(Xxi)=P(X=x1)+P(X=x2)+…+P(X=xi),

∴

Characteristics:

FX(-)=0

FX(+)=1

Ifx1<x2,then

FX(x1)FX(x2)28Distributionfunctionofcontinuousrandomvariable:Whenxiscontinuous,fromthedefinitionofdistributionfunction

FX(x)=P(X

x)

weknowthatFX(x)isacontinuousmonotonicincreasingfunction.29Probabilitydensityofrandomvariable(pdf)ProbabilitydensityofcontinuousrandomvariablepX(x)DefinitionofpX(x):MeaningofpX(x):pX(x)isthederivativeofFX(x),andistheslopeofthecurveofFX(x)P(a<X

b)canbefoundfrompX(x):CharacteristicsofpX(x):

pX(x)031Probabilitydensityofdiscreterandomvariable

Distributionfunctionofdiscreterandomvariablecanbewrittenas:wherepi

－probabilityofx=xi

u(x)

－unitstepfunction Findingthederivativesofthetwosidesoftheaboveequation,weobtainitsprobabilitydensity:

Characteristics: Whenx

xi,px(x)=0 Whenx=xi

,px(x)=JointDistribution34

ExamplesoffrequentlyusedrandomvariablesRandomvariablewithnormaldistribution(alsocalledGaussiandistribution)MostImportantDefinition:Probabilitydensitywhere>0,a=const.Probabilitydensitycurve:JointGaussianRandomVariablesTwo-VariateGaussianPDF37RandomvariablewithuniformdistributionDefinition:probabilitydensity

wherea,bareconstants.Probabilitydensitycurve:bax0pA(x)38RandomvariablewithRayleighdistributionpdfwherea>0,andisaconstant.Probabilitydensitycurve:RandomvariablewithBinarydistributionRandomvariablewithBinomialdistribution42NumericalcharacteristicsofrandomvariableMathematicalexpectationDefinition:forcontinuousrandomvariableCharacteristics:

IfXandYareindependentofeachother,andE(X)andE(Y)exist,then

43VarianceDefinition:

whereVariancecanberewrittenas:Proof:Fordiscretevariable:Forcontinuousvariable:Characteristics:D(C)=0D(X+C)=D(X)，D(CX)=C2D(X)D(X+Y)=D(X)+D(Y)D(X1+X2+…+Xn)=D(X1)+D(X2)+…+D(Xn) 44

MomentDefinition:thek-thmomentofarandomvariableXisk-thoriginmomentisthemomentwhena=0:k-thcentralmomentisthemomentwhen:Characteristics:Thefirstoriginmomentisthemathematicalexpectation:Thesecondcentralmomentisthevariance:RandomProcessArandomprocessisthenaturalextensionofrandomvariableswhendealingwithsignalsAlsoreferredtoasstochasticprocessorrandomsignalVoicessignals,TVsignals,thermalnoisegeneratedbyaraidoreceiverareallexamplesofrandomsignals.46Basicconceptofrandomprocess

X(A,t)－ensembleconsistingofallpossible“realizations”ofaneventAX(Ai,t)－arealizationofeventA,itisadeterminedtimefunctionX(A,tk)－valueofthefunctionatthegiventimetkDenoteforshort:

X(A,t)

X(t)

X(Ai,t)Xi

(t)47Example:receivernoiseNumericalcharacteristicsofrandomprocess:Statisticalmean:Variance:Autocorrelationfunction:50

StationaryrandomprocessDefinitionofstationaryrandomprocess:(strict) Arandomprocesswhosestatisticalcharacteristicsisindependentofthetimeoriginiscalledastationaryrandomprocess.(or,strictstationaryrandomprocess)Definitionofgeneralizedstationaryrandomprocess:(weak)

Therandomprocesswhosemean,varianceandautocorrelationfunctionareindependentofthetimeoriginCharacteristicsofgeneralizedstationaryrandomprocess:

Astrictstationaryrandomprocessmustbeageneralizedstationaryrandomprocess;butageneralizedstationaryrandomprocessisnotalwaysastrictstationaryrandomprocess.51

ErgodicityCharacteristicofergodicity:timeaveragemaybereplacedbystatisticalmean.Forexample,StatisticalmeanofergodicprocessmX：AutocorrelationfunctionofergodicprocessRX():Ifarandomprocesshasergodicity,thenitmustbeastrictstationaryrandomprocess.However,astrictstationaryrandomprocessisnotalwaysergodic.52Ergodicityofstationarycommunicationsystem

Ifthesignalandthenoisearebothergodic,thenFirstoriginmomentmX

=E[X(t)]－D.C.componentofsignalSquareoffirstoriginmomentmX

2－powerofnormalizedD.C.componentofsignalSecondoriginmomentE[X2(t)]－normalizedaveragepowerofsignalSecondcentralmomentX2

－normalizedaveragepowerfoA.C.componentofsignalIfmX

=mX

2=0,thenX2=E[X2(t)]；

53CharacteristicsofpowerspectraldensityReview:powerspectraldensityofdeterministicsignalSimilarly,powerspectraldensityofstationaryrandomprocessequals:Averagepower:54Relationshipbetweenautocorrelationfunction&powerspectraldensityforRandomProcess

PX(f)andR()areapairofFouriertransform.55

【Example2.7】Letabinarysignalbex(t)asshowninthefigure.Itsamplitudeis+aor–a;andthenumberkofitssignchangesintimeintervalTobeysPoissondistribution:where,isaveragenumberofsignchangesofamplitudeinunittime.FinditsautocorrelationfunctionR()andpowerspectraldensityP(f).+a-ax(t)tt0t-56

Solution:Itcanbeseenfromtheabovefigure,x(t)x(t-)hasonlytwopossiblevalues:a2or-a2.Hence,equation canbereducedto

R()=a2

[occurrenceprobabilityofa2]+(-a2)[occurrenceprobabilityof(-a2)] where,theoccurrenceprobabilitycanbecalculatedaccordingtoPoissondistributionP(k). Ifthenumberofsignchangesofx(t)iseveninsecond,then+a2occurs;ifthenumberofsignchangesofx(t)isoddinsecond,then-a2occurs。Therefore

UseinsteadofTinPoissondistribution,thenobtain57

whenthevalueofisnegative,theaboveequationshouldberewrittenas Combiningtheabovetwoequations,finallyobtain: TheP(f)canbeobtainedfromtheFouriertransformoftheR():

CurvesofP(f)andR():58

【Example2.9】Findtheautocorrelationfunctionandthepowerspectraldensityofwhitenoise.

Solution:WhitenoisehasuniformpowerspectraldensityPn(f):

Pn(f)＝n0/2

where,n0:singlesidepowerspectraldensity（W/Hz）

Theautocorrelationfunctionofwhitenoisecanbeobtainedfromitspowerspectraldensity:

thesamplesofwhitenoiseatanytwoadjacentinstants(i.e.,

0)areuncorrelated. Averagepowerof whitenoise: Theaboveequationshowsthattheaveragepowerofwhitenoiseisinfinity.Pn(f)n0/20fRn()n0/2059Powerspectraldensity&autocorrelationfunctionofband-limitedwhitenoisePowerspectraldensityofband-limitedwhitenoise: Ifthebandwidthofawhitenoiseinlimitedintheinterval(-fH,fH),thenwehave

Pn(f)=n0/2, -fH

<f<fH =0, else itsautocorrelationfunctionis:Curves:

Whenband-limitedwhitenoiseissampledaccordingtothesamplingtheorem,eachsampleisanuncorrelatedrandomvariable.

60

GaussianprocessDefinitionOnedimensionalprobabilitydensityofGaussprocess:where,a=E[X(t)]---mean

2=E[X(t)

-a]2---variance

---standarddeviation∵Gaussprocessisastationaryprocess,henceitsprobabilitydensitypX(x,t1)isindependentoft1 i.e.,pX(x,t1)＝pX(x)CurveofpX(x):Multi-dimensionalprobabilitydensityofGaussprocess:63NormaldistributionexpressedbyerrorfunctionwhichcouldbeadopteddirectlyinmatlabDefinitionoferrorfunction:DefinitionofComplementaryerrorfunction:

Expressionofnormaldistribution64頻率近似為fcNarrowbandrandomprocess

BasicconceptofnarrowbandrandomprocessWhatdoesitmeannarrowband? Assumethebandwidthofarandomprocessisf,thecentralfrequencyisfc.Iff<<fc,thentherandomprocessiscalledanarrowbandrandomprocess.WaveformandexpressionofnarrowbandrandomprocessWaveformandspectrum:65Expression

where,aX(t)－randomenvelopeofnarrowbandrandomprocess

X(t)

－randomphaseofnarrowbandrandomprocess

0

－angularfrequencyofsinusoidalwave

Theaboveequationcanberewrittenas: where,

－inphasecomponentofX(t)

－orthogonalcomponentofX(t)

66

CharacteristicsofnarrowbandrandomprocessStatisticalcharacteristicsofXc(t)andXs(t) IfX(t)isastationarynarrowbandGaussianprocesswithzeromean,then

Xc(t)andXs(t)arealsoGaussianprocesses.

Xc(t)andXs(t)haveidenticalvariance,andthevarianceisequaltothevarianceofX(t)Xc

andXsatthesameinstantareuncorrelatedandstatisticallyindependent.StatisticalcharacteristicsofaX(t)andX(t)ProbabilitydensityofaX(t):ProbabilitydensityofX(t):

67SinusoidalwaveplusnarrowbandGaussianprocessExpressionofsinusoidalwaveplusnoise:where,A

－deterministicamplitudeofsinusoidalwave

0

－angularfrequencyofsinusoidalwave

－randomphaseofsinusoidalwave

n(t)－narrowbandGaussiannoiseProbabilitydensityfunctionoftheenvelopeofr(t): where,

2

－varianceofn(t)

I0()－zero-ordermodifiedBesselfunctionpr(x)iscalledgeneralizedRayleighdistribution,orRiciandistribution. WhenA=0,pr(x)becomesRayleighprobabilitydensity68Conditionalprobabilitydensityofthephaseofr(t): where,

－phaseofr(t)includingthephaseofsinusoidalwaveandthephaseofnoise

pr(/)－conditionalprobabilitydensityofthephaseofr(t)undertheconditionofgiverProbabilitydensityofthephaseofr(t):When=0, where,69CurvesofRiciandistributionWhenA/=0, EnvelopeRayleigh distribution PhaseUniform distributionWhenA/is verylarge, Envelopenormal distribution Phaseimpulse functionrRayleighdistributionProbabilitydensityEnveloper(a)ProbabilitydensityoftheenvelopewithRiciandistribution(b)ProbabilitydensityofthephasewithRiciandistributionUniformphasePhaseProbabilitydensityFigure2.9.1Raciandistributioncurves702.10

Signaltransferthroughlinearsystems

2.10.1BasicconceptoflinearsystemsCharacteristicsofthelinearsystemsdiscussedhereHaveapairofinputandapairofoutputPassiveMemorylessTime-invariantCausalityLinear:satisfyingsuperpositionprinciple Ifwheninputisxi(t),outputisyi(t),thenwheninputis theoutputis where,a1anda2arebotharbitraryconstants.71SketchoflinearsystemLinearsystemOutputInputx(t)y(t)X(f)Y(f)h(t)H(f)Figure2.10.1Linearsystemsketcht(t)h(t)t00722.10.2DeterministicsignaltransferthroughlinearsystemsTimedomainanalysismethod Leth(t)－impulseresponseofthesystem

x(t)－inputsignalwaveform

y(t)－outputsignalwaveform thenwehave:Forphysicallyrealizablesystems:

73FrequencydomainanalysismethodAssumeinputisaenergysignal,let

x(t)－inputenergysignal H(f)－Fouriertransformofh(t)

X(f)－Fouriertransformofx(t)

y(t)－outputsignal thenthefrequencyspectraldensityY(f)oftheoutputsignaly(t)ofthesystemis:

y(t)canbefoundfromtheinverseFouriertransformofY(f):74Assume:theinputx(t)isaperiodicpowersignal,then

where, theoutputis:

Iftheinputx(t)isanonperiodicalpowersignal,thenitwillbeprocessedasarandomsignal.0=2/T0T0

－periodofthesignalf0=0/2---fundamentalfrequencyofsignal75【Example2.10】ThereisaRClow-passfilterasshowninFig.2.10.4.Finditsimpulseresponseandtheexpressionofitsoutputsignalwhentheinputisexponentiallyattenuated. Solution:Assumex(t)－inputenergysignal

y(t)－outputenergysignal X(f)－frequencyspectraldensityofx(t)

Y(f)－frequencyspectraldensityofy(t) thenthetransferfunctionofthecircuitis:

圖2.10.4RC濾波器RCx(t)y(t)76

Theimpulseresponseh(t)ofthefilter:

Therelationshipbetweentheoutputandtheinputofthefilter: Assumetheinputx(t)equals: thentheoutputofthefilteris:77Conditionsofdistortionlesstransmission Thereisadistortionlesslineartransmissionsystem,anditsinputisanen