信号与系统英文版课件Chapter 5 to 8

Contents§5.1Representationofdiscrete aperiodicsignals:Thediscrete-time Fouriertransform§5.2TheFTforperiodicsignals§5.1Thediscrete-timeFouriertransformDevelopmentofthediscreteFTAgeneralsequencex[n]0Fromx[n],weconstructaperiodicx[n]forwhichx[n]isoneperiod.§5.1Thediscrete-timeFouriertransform0N-N–AsN,x[n]=x[n]§5.1Thediscrete-timeFouriertransformDefiningthefunctionthen§5.1Thediscrete-timeFouriertransformEq.(5.1)becomes

AsN,0,andasN,,0d,k0.

For,∵X(ejω)·ejn

withperiod2π,theisNinnumber,withintervalofwidth=2/N,∴intervalofhaveawidthof2π.§5.1Thediscrete-timeFouriertransformTherefore,asN→∞,andEq(5.7)becomeswhereThuswehaveanothertransformpair,discreteFTpair,X(e

jω)—thediscreteFT,orspectrumofx[n].

§5.1Thediscrete-timeFouriertransformNotealsothat§5.1Thediscrete-timeFouriertransformThemajordifferencesbetweendiscreteFTwithcontinuousFTare(1)X(ej)isperiodic(withperiod2),whileX(j)isnot.Interestingcorrespondingandduality:

TimesignalCorrespondingakorX(j)orX(ej)

Periodic↔Discrete

Aperiodic↔ContinuousContinuous↔

Aperiodic Discrete↔Periodic§5.1Thediscrete-timeFouriertransformEquation(5.8)involvesanintegrationonlyoverafrequencyinterval(anyintervaloflength2)

whilecontinuous(3)Inthespectrumgraph,lowfrequenciesnearevenmultiplesofandhighfrequenciesnearoddmultiplesof(seePage362:Figure5.3)§5.1Thediscrete-timeFouriertransformExamplesofdiscreteFT Example5.1 Considerthesignal§5.1Thediscrete-timeFouriertransformThemagnitudeandphaseofX(ejω)showninFigurebelow.1023nx[n](0<a<1)1a2aa3|X(ejω)|0π-2π-π2π1/(1-a)1/(1+a)§5.1Thediscrete-timeFouriertransformExample5.2 ThissignalissketchedinFigure.1

Inthiscase,X(ejω)isrealandisillustratedinFigure5.5(b)for0<a<1P(364).§5.1Thediscrete-timeFouriertransformExample5.3ThespectrumseeFigure5.6(b)(P365).

x[n]=1,|n|≤N10,|n|>N110n§5.1Thediscrete-timeFouriertransformExample5.4[n]1§5.1Thediscrete-timeFouriertransformConvergenceissuesassociatedwiththediscreteFTForfiniteduration,forextremelybroadclassofsignalswithinfiniteduration.Eitherifx[n]isabsolutelysummableorhas

finiteenergyEq(5.9)willconverge.§5.2TheFTforperiodicsignalsConsiderthesignal(simplicity)

thisFTmightexpect

20§5.2TheFTforperiodicsignalsTocheck,substitutingEq.(5.18)intoEq.(5.8)SoEq(5.17)and(5.18)isFTpair.§5.2TheFTforperiodicsignalsTherefore,aperiodicsignalitsFTis§5.2TheFTforperiodicsignalsExample5.5FromEq.(5.18)§5.2TheFTforperiodicsignals……1/21/2……Assignments(P400):5.2,5.3Contents§6.1Themagnitude-phaserepresentationofFT§6.2Themagnitude-phaserepresentationofthefrequencyresponseofLTIsystems§6.3Time-domainpropertiesofidealfrequencyselectivefilters§6.4Time-domainandfrequency-domainaspectsofnon-idealfilters（omit）Themagnitude-phaserepresentationofFTContinuousFTDiscreteFT|X(j)|describestheinformationaboutthebasicfrequencycontentofasignal,whileX(j)doesnot.X(j)haveasignificanteffectonthenatureofthesignalandthuscontainasubstantialamountofinformationaboutthesignal.Themagnitude-phaserepresentationofFT

Different1,2,and3,resultingsignalscandiffersignificantly(seefigure6.1inP425),andexamplefromx(t)x(-t)

Ingeneral,changesinthephasefunctionofX(jω)leadtochangesinthetime-domaincharacteristicsofthesignalx(t).Example:Themagnitude-phaserepresentationofLTI|H(jω)|iscommonlyreferredtoasthegainofthesystem.H(j)isreferredtoasthephaseshiftofthesystem.Themagnitude-phaserepresentationofLTILinearandnonlinearphasesystemOutputofsystemisatimeshiftoftheinput

y(t)=x(t-t0)Themagnitude-phaserepresentationofLTIInthediscretecase,theeffectoflinearphaseissimilartothatinthecontinuouscasewhenslopeofthelinearphaseisaninteger.Obtainasignalthatmaylookconsiderablydifferentfromtheinputsignal.(seefigure6.3cinP429)Themagnitude-phaserepresentationofLTIGroupdelay GroupdelayisdefinedasGroupdelayateachfrequencyequalsthenegativeoftheslopeofthephaseatthefrequency.constantTime-domainpropertiesidealfrequencyselectivefilters

Ingeneral,anideallowpassfilterhasafrequencyresponseoftheformForzerophasefora=0e-ja,||c0,||>c1,

||c0,||>cTheimpulseresponse:tTime-domainpropertiesidealfrequencyselectivefilters

Thestepresponse

isSineintegralfunction.TwospecialfeaturesoffunctionSi(y):(1)

Asy,Si(y)/2,waveformhasoscillation,maxaty=.(2)Si(-y)=-Si(y),oddfunction.Time-domainpropertiesidealfrequencyselectivefilters

Time-domainpropertiesidealfrequencyselectivefilters

Fromwaveformofh(t)ands(t),wecansee:(1)

Ifa0,thenh(t)ands(t)aredelayedbya.Theresponsehasdeformation,extentofdeformation1/c,asc,deformationdisappear.

Theideallowpassfilterisanticausal.

(4)

Establishofresponserequiresometimetr—risetime,tr

1/c.Time-domainpropertiesidealfrequencyselectivefilters

Assignment(P489):6.1,6.3Contents§7.1Representationofacontinuoussignalbyitssamples:thesamplingtheorem.§7.2(omit)§7.3Theeffectofundersampling:aliasingIntroduction

Whatisthemeaningofsampling? Representationofacontinuoussignalbyitssamples.Whysampling? Processingdiscretesignalsismoreflexible.Time-divisionmultiplexing（TDM）.RepresentationofacontinuoussignalthesamplingtheoremImpulse-trainsampling 1)Samplingprinciple SamplingprocessisdepictedintheFigurebelow.wherex(t)isthatwewishtosample,andx(t)xp(t)p(t)tx(t)isimpulsestrainRepresentationofacontinuoussignal

ItsperiodTisreferredassamplingperiod,s=2/Tas

samplingfrequency.tx(t)tp(t)-2T2T-TT……txp(t)……-2T2T-TTx(0)(t)x(T)(t-T)Representationofacontinuoussignal

xp(t)isanimpulsestrainwiththeamplitudesofimpulsesequaltosamplesofx(t)atintervalsspacedbyT.Representationofacontinuoussignal

ThatisXp(jω)isaperiodicofωconsistingofasuperpositionofshiftedreplicasofX(jω),scaledby1/T,asillustratedinrightfigurebelow.0P(j)……0XP(j)……s>2ms<2m0XP(j)……RepresentationofacontinuoussignalFromEq(7.6)andfigureabovetoobtain:(1)

Xp(j)isaperiodicfunctionwithperiodofsAslongasmislimitedands2m,eachofXp(j)willcontainalltheinformationaboutX(j),withoutoverlapping.Inthiscase,x(t)canberecoveredexactlyfromxp(t)bymeansofalowpassfilterwiththegainTandacutofffrequencyωc.ωm<ωc<(ωs-ωm)Thisbasicresultsarereferredtoasthesamplingtheorem.Representationofacontinuoussignal2)SamplingTheorem

Letx(t)beaband-limitedsignal

withX(j)=0for>m,thenx(t)isuniquely

determinedbyitssamples

x(nT),n=0,±1,±2,…

Where

Giventhesesamples,wecanreconstructx(t).RepresentationofacontinuoussignalThefrequency2miscommonlyreferredtoastheNyquistrate.Thefrequencym

isoftenreferredtoastheNyquistfrequency.0-ccRepresentationofacontinuoussignalSamplingwithazeroorderholdSamplingprinciplewithazero-hold: impulse-trainsamplingfollowingbyLTIsystemwitharectangularimpulseresponse.t0T1Representationofacontinuoussignaltx(t)txp(t)……-2T2T-TTtx0(t)…Clearly,bandwidthofx0(t)<<bandwidthofxp(t).

andx0(t)x(t)Tx(nT)RepresentationofacontinuoussignalReconstructx(t)fromx0(t)Zero-orderholdfollowingbyareconstructionfilterwithhr(t)[orHr(j)]hr(t）Hr(j)t0T1equivalentH(j)Representationofacontinuoussignal

Thisrequiresthat IfcofH(j)equaltos/2,thenmagnitudeandphaseforHr(j)isshowninfigurebelow.Representationofacontinuoussignalnow,cofH(j)iss/2.§7.3Theeffectofundersampling:aliasingWhen,theindividualtermsinEq(7.6)overlap.Thiseffectisreferredtoasaliasing.Forsimply,consideroriginalsignalisAliasinggiveseffectandconsequences:

Whenaliasingoccurs,theoriginalfrequencyω0

takesontheidentityoflowerfrequency(ωs-ω0).Inthesecase,thereconstructedsignalhaveachangeinthesignofthephase,i.e.aphasereversal.seeFigure7.16(b)and(c)below.§7.3Theeffectofundersampling:aliasingoriginalsignalsamplesreconstructedtsamplests=30Fig.7.16(b)§7.3Theeffectofundersampling:aliasingoriginalsignalsampless=1.50ttsamplesreconstructedphasereversalFig.7.16(c)Assignments(P556:7.1,7.2,7.3,

7.6)Contents§8.0Introduction§8.1Complexexponentialandsinusoidalamplitudemodulation§8.2DemodulationforsinusoidalAM§8.0IntroductionThefunctionofcommunicationsystems:Modulation,Transmit,DemodulationWhymodulation?§8.0IntroductionModulation—Theprocessofcarryingx(t)onc(t).Modulationmethods: AmplitudeModulation(AM) FrequencyModulation(FM) PulseCodeModulation(PCM)Demodulation: Theinverseprocessofmodulation,i.e.theprocessofrecoveringtheoriginalsignalfrommodulatedsignal.ComplexexponentialandsinusoidalAMAmplitudemodulationwithacomplexexponentialcarrierPrincipleLetX(jω),Y(jω),andC(jω)denotingFTofx(t),y(t),andc(t)respectively.ComplexexponentialandsinusoidalAM

Thus,thespectrumofy(t)issimplythatofinput,shiftedinfrequencybyanamountequaltoc.

ComplexexponentialandsinusoidalAMItisclearthatx(t)canberecoveredfromy(t),tjce)t(y)t(xw-