离散时间系统分析课件

1§1-1IntroductionChapter1FundamentalConcepts§1-2Discrete-TimeSignals

§1-3Discrete-timeSystems§1-4BasicPropertiesofDiscrete-timeSystemsProblems2§1-1Introduction

Theconceptsofsignalsandsystemsariseinvirtuallyallareasoftechnology,rangingfromappliancesordevicesfoundinhomestoverysophisticatedengineeringinnovations.Infact,itcanbearguedthatmuchofthedevelopmentofhightechnologyisaresultofadvancementsinthetheoryandtechniquesofsignalsandsystems.Lastterm,inthecourseofsignalsandsystems,wehavestudiedthecontinuous-timesystemanalysis.Inthisterm,wearegoingtostudythediscrete-timesystemanalysis.Letusbeginwiththefundamentalconceptsofthediscrete-timesystemanalysis.3§1-2Discrete-TimeSignals

Definitions:Discrete-timevariable:Thetimevariabletissaidtobeadiscretetimevariableifttakesononlythediscretevaluest=tnforsomerangeofintegervaluesofn.Discrete-timesignals:Adiscrete-timesignalisasignalthatisafunctionofthediscretetimevariabletn,denotedwithx(tn),wherex(t)isacontinuous-timesignal.Theresultingdiscrete-timesignalx(tn)iscalledthesampledversionoftheoriginalcontinuous-timesignalx(t).4Lettn=nT,whereTiscalledthesamplinginterval.IfTisaconstant,thesamplingprocessiscalleduniformsampling,otherwise,nonuniformsampling.Notethatthenonuniformsamplingissometimesusedinapplicationsbutisnotconsideredinthiscourse.So,weoftenusex[n]todenotex(tn),i.e.,

x[n]=x(tn)=x(t)|t=nT=x(nT)Alsonotethatthesquarebracketsinsteadofparenthesesareusedtodenotethediscrete-timesignalx[n].

5Forexample,whereT=1/15,n=0,1,2,…,30.Theplotsofx[n]andx(t)aregiveninFig.1-1Fig.1-16TypicalandSimpleExamplesofDiscrete-TimeSignalsDiscrete-timeUnit-stepFunctionu[n]Definition:Theplotofu[n]isshowninFig.1-2Fig.1-27Discrete-timeUnit-rampFunctionr[n]Definition:

Theplotofr[n]isshowninFig.1-3Fig.1-38UnitPulseDefinition:Theplotofδ[n]isshowninFig.1-4Fig.1-49PeriodicDiscrete-timeSignals

Definition:Adiscrete-timesignalx[n]isperiodicifthereexistsapositiveintegerrsuchthatx[n+r]=x[n]forallintegersnHencex[n]isperiodicifandonlyifthereisapositiveintegersuchthatx[n]repeatsitselfeveryrtimeinstants,whereriscalledtheperiod.Thefundamentalperiodisthesmallestvalueofrforwhichthesignalrepeats.10Forexample,letusexaminetheperiodicityofadiscrete-timesinusoidgivenbyx[n]=Acos(Ωn+θ)ThesignalisperiodicifAcos[Ω(n+r)+θ]=Acos(Ωn+θ)Recallthatthecosinefunctionrepeatsevery2πradians,sothatAcos(Ωn+θ)=Acos(Ωn+2πq+θ)Forallintegersq.Therefore,thesignalAcos(Ωn+θ)isperiodicifandonlyifthereexistsapositiveintegerrsuchthatΩr=2πqforsomeintegerq,orequivalently,thatthediscrete-timefrequencyΩissuchthatΩ=2πq/rforsomepositiveintegersqandr.11Thediscrete-timesinusoidx[n]=Acos(Ωn+θ)isplottedinFig.1-5.ForthecasewhenΩ=π/3andθ=0,whichisplottedinFig.1-5(a),thecorrespondingperiodisfoundtober=2πq/Ω=6qandthefundamentalperiodis6.Fig.1-5(a)12ThecasewhenΩ=1andθ=0isplottedinFig.1-5(b).Notethatinthiscasetheenvelopeofthesignalisperiodicbutthesignalitselfisnotperiodic.Fig.1-5(b)r=2πq/Ω=2πq13Discrete-timeComplexExponentialSignalsDefinition:where,,j=14Discrete-timeRectangularPulseDefinition:whereLisapositiveoddinteger.TheplotofpL[n]isdisplayedinFig.1-6Fig.1-615DigitalSignalsDefinition:Let{a1,a2,…,aN}beasetofNrealnumbers.Adigitalsignalisadiscrete-timesignalwhosevaluesbelongtothefiniteset{a1,a2,…,aN};thatis,ateachtimeinstanttn,x(tn)=x[n]=aiforsomei,where1≤i≤N.Soadigitalsignalcanhaveafinitenumberofdifferentvalues.Asampledcontinuous-timesignalisnotnecessarilyadigitalsignal.Forexample,thesampledunit-rampfunctionr[n]showninFig.1-3isnotadigitalsignalsinceittakesonaninfiniterangeofvalueswhenn=…,–2,–10,1,2,….Abinarysignalisadigitalsignalwhosevaluesareequalto1or0;thatis,x[n]=0or1forn=…,–2,–10,1,2,….Thesampledunit-stepfunctionandtheunit-pulsefunctionarebothexamplesofbinarysignals.16Time-shiftedSignalsGivenadiscrete-timesignalx[n]andapositiveintegerq,thediscrete-timesignalx[n–q]istheq-steprightshiftsofx[n]andx[n+q]istheq-stepleftshiftsofx[n].Forexample,p3[n–2]isthetwo-steprightshiftsofthediscrete-timerectangularpulsep3[n];p3[n+2]isthetwo-stepleftshiftsofp3[n].TheshiftedsignalsareplottedinFig.1-7.(a)(b)Fig.1-7Two-stepshiftsofp3[n]:(a)rightshift;(b)leftshift.17Discrete-timeSignalsdefinedIntervalbyIntervalAsinthecaseofcontinuous-timesignals,discrete-timesignalsaresometimesdefinedintervalbyinterval.Forinstance,x[n]maybespecifiedbywherex1[n],x2[n]andx3[n]arediscrete-timesignalsandn1,n2andn3areintegerswithn1<n2<n3.Withu[n]equaltothediscrete-timeunit-stepfunction,x[n]canbewrittenintheform18§1-3Discrete-timeSystems

Definitionof

Discrete-timeSystems:Adiscrete-timesystemisasystemthattransformsdiscrete-timeinputsintodiscrete-timeoutputs,asshowninFig.1-8.Fig.1-8Discrete-TimeSystemDefinitionofDiscrete-timeSystemAnalysis:Givenadiscrete-timeinputandadiscrete-timesystem,howtosolvethediscrete-timeoutputofthesystemiscalleddiscrete-timesystemanalysis.19ExampleofaDiscrete-timeSystemAsasimpleexampleofadiscrete-timesystem,considerasimplemodelforthebalanceinabankaccountfrommonthtomonth.Specifically,lety[n]denotethebalanceattheendofthen-thmonth,andsupposethaty[n]evolvesfrommonthtomonthaccordingtotheequationy[n]=1.01y[n–1]+x[n],orequivalently,y[n]–1.01y[n–1]=x[n],wherex[n]representsthenetdeposit(i.e.,depositsminuswithdrawals)duringthen-thmonthandtheterm1.01y[n–1]modelsthefactthatweaccrue1%interesteachmonth.20Asasecondexample,considerasimpledigitalsimulationofthedifferentialequationinwhichweresolvetimeintodiscreteintervalsoflength

andapproximatedv(t)/dtatt=n

bythefirstbackwarddifference,i.e.,inthiscase,ifweletv[n]=v(n

)andx[n]=x(n

),weobtainthefollowingdiscrete-timemodelrelatingthesampledsignalsx[n]andv[n]:21§1-4BasicPropertiesofDiscrete-timeSystemsSystemswithandwithoutMemoryDefinition:Asystemissaidtobememorylessifitsoutputforeachvalueoftheindependentvariableatagiventimeisdependentonlyontheinputatthatsametime,otherwisethesystemisonewithmemory.Forexample,thesystemspecifiedbytherelationshipy[n]=(2x[n]–x[n]2)2ismemoryless,asthevalueofy[n]atanyparticulartimedependsonlyonthevalueofx[n]atthattime.Anexampleofadiscrete-timesystemwithmemoryisanaccumulatororsummer:andasecondexampleisadelay:y[n]=x[n–1].22Causality

Definition:Asystemiscausaliftheoutputatanytimedependsonlyonvaluesoftheinputatthepresenttimeandinthepast.Forexample,thesystemsandy[n]=x[n–1]arecausal,butthesystemsdefinedbyy[n]=x[n]–x[n+1]andy[n]=x[–n]arenot.23StabilityDefinition:Astablesystemisoneinwhichboundedinputsleadtoboundedoutputs.Forexample,theaveragingsystemdefinedbyisstable,buttheaccumulatorsystemdefinedbyisnot,becausey[n]growswithoutbound.24TimeInvarianceDefinition:Asystemistimeinvariantifatimeshiftintheinputsignalresultsinanidenticaltimeshiftintheoutputsignal.Forexample,thesystemdefinedbyy[n]=nx[n]isobviouslynottimeinvariant.Asasecondexample,thesystemdefinedbyy[n]=x[2n]representsatimescaling.Thatis,y[n]isatime-compressed(byafactorof2)versionofx[n].Intuitively,anytimeshiftintheinputwillalsobecompressedbyafactorof2,anditisforthisreasonthatthesystemisnottimeinvariant.25LinearityDefinition:Alinearsystemisasystemthatifaninputconsistsoftheweightedsumofseveralsignals,thentheoutputistheweightedsumoftheresponsesofthesystemtoeachofthosesignals.Moreprecisely,lety1[n]betheresponseofadiscrete-timesystemtoaninputx1[n],andlety2[n]beoutputcorrespondingtotheinputx2[n].Thenthesystemislinearif:

Theresponsetox1[n]+x2[n]isy1[n]+y2[n]

Theresponsetoax1[n]isay1[n],whereaisanycomplexconstant.Thefirstofthesetwopropertiesisknownastheadditivityproperty;thesecondisknownasthescalingorhomogeneityproperty26Considerthesystemy[n]=

2x[n]+3.Thissystemisnotlinearbecauseitviolatestheadditivityproperty.Fortwoinputsx1[n]andx2[n],x1[n]

y1[n]=

2x1[n]+3x2[n]

y2[n]=

2x2[n]+3However,theresponsetothex3[n]=x1[n]+x2[n]isy3[n]=

2(x1[n]+x2[n])+3

y1[n]+y2[n]Notethaty[n]=3ifx[n]=0,weseethatthesystemviolatesthe“zero-in/zero-out”property.27Itmayseemsurprisingthatthesystemintheaboveexampleisnonlinear,sincethesystemequationy[n]=

2x[n]+3islinear.Ontheotherhand,asshowninFig.1-9,theoutputofthesystemcanberepresentedasthesumoftheoutputofalinearsystemandanothersignalequaltothezero-inputresponseofthesystem.Forthesystemy[n]=

2x[n]+3,thelinearsystemisyl[n]=

2x[n]andthezero-inputresponseisy0[n]=3.Fig.1-928Problems1.1Sketchthediscrete-timesignalx[n](0≤n≤10)obtainedbysamplingthecontinuous-timesignalx(t)=e–2t

withsamplingintervalT=0.1.1.2Sketchthefollowingdiscrete-timesignals(a)x[n]=u[n]–2u[n–1]+u[n–4](b)x[n]=(n+2)u[n+2]–2u[n]–nu[n–4](c)x[n]=δ[n+1]–δ[n]+u[n+1]–u[n–2]1.3Letx[n]beasignalwithx[n]=0forn<–

2andn>4.foreachsignalgivenbelow,determinethevaluesofnforwhichitisguaranteedtobezero.

(a)x[n–3](b)x[n+4](c)x[–n](d)x[–n+2](e)x[–n–2]291.4Determineifthefollowingdiscrete-timesignalsareperiodic.Ifso,findthefundamentalperiod.(a)x[n]=(b)x[n]=ej7πn(c)x[n]=ej(n/8–π)1.5Consideradiscrete-timesystemwithinputx[n]andoutputy[n].Theinput-outputrelationshipforthissystemisy[n]=x[n]x[n–2](a)Isthesystemmemoryless?(b)Determinetheoutputofthesystemwhentheinputis

Aδ[n],whereAisanyrealorcomplexnumber.301.6Consideradiscrete-timesystemwithinputx[n]andoutputy[n]relatedbywheren0isafinitepositiveinteger.(a)Isthesystemlinear?(b)Isthesystemtime-invariant?1.7Foreachofthefollowinginput-outputrelationships,determinewhetherthecorrespondingsystemislinear,timeinvariantorboth.(a)y[n]=x[n–2]2(b)y[n]=x[n+1]+x[n–1]31§2-1LinearInput/OutputDifferenceEquationswithConstantCoefficients

Chapter2Discrete-TimeSystemAnalysisintheTimeDomain§2-2DiscretizationinTimeofDifferentialEquationsProblems32§2-1LinearInput/OutputDifferenceEquationswithConstantCoefficients

Nowconsidersingle-inputsingle-outputdiscrete-timesystemdefinedbytheinput/outputdifferenceequationwherenistheinteger-valueddiscrete-timeindex,x[n]istheinput,andy[n]istheoutput.Hereitisassumedthatthecoefficientsa1,a2,…,aNandb0,b1,b2,…,bMareconstants.(2.1)33SinceEq.(2.1)isalineardifferenceequationwithconstantcoefficients,thesystemdefinedbytheequationislinear,timeinvariant,andfinitedimensional.TheintegerNin(2.1)istheorderordimensionofthesystem.Also,anydiscrete-timesystemintheformofEq.(2.1)iscausalsincetheoutputy[n]attimendependsonlyonpreviousvaluesoftheoutput

andthecurrentandpreviousvaluesoftheinputx[n].SolutionbyRecursionUnlikelinearinput/outputdifferentialequations,linearinput/outputdifferenceequationscanbesolvedbyadirectnumericalprocedure.Moreprecisely,theoutputy[n]forsomefiniterangeofintegervaluesofncanbecomputedrecursivelyasfollows.First,rewrite(2.1)intheform34(2.2)Thensettingn=0in(2.2)givesy[0]

a1y[

1]

a2y[

2]

···

aNy[

N]+b0x[0]

b1x[

1]

···

bMx[

M]Thustheoutputy[0]attime0isalinearcombinationofy[

1],y[

2],

···,y[

N]andx[0],x[

1],

···,x[

M].Settingn=1in(2.2)givesy[1]

a1y[0]

a2y[

1]

···

aNy[

N+1]+b0x[1]

b1x[0]

···

bMx[

M+1]Soy[1]isalinearcombinationofy[0],y[

1],

···,y[

N+1]andx[1],x[0],

···,x[

M+1].35Ifthisprocessiscontinued,itisclearthatthenextvalueoftheoutputisalinearcombinationoftheNpastvaluesoftheoutputandM+1valuesoftheinput.Ateachstepofthecomputation,itisnecessarytostoreonlyNpastvaluesoftheoutput(plus,ofcourse,theinputvalues).ThisprocessiscalledanNth-orderrecursion.HerethetermrecursionreferstothepropertythatthenextvalueoftheoutputiscomputedfromNpreviousvaluesoftheoutput(plustheinputvalues).Thediscrete-timesystemdefinedby(2.1)[or(2.2)]issometimescalledarecursivediscrete-timesystemorarecursivediscrete-timefiltersinceitsoutputcanbecomputedrecursively.Hereitisassumedthatatleastoneofthecoefficientsaiin(2.1)isnonzero.Ifalltheaiarezero,theinput/outputdifferenceequation(2.1)reducesto36Inthiscase,theoutputatanyfixedtimepointdependsonlyonvaluesoftheinputx[n],andthustheoutputisnotcomputedrecursively.Suchsystemsaresaidtobenonrecursive.Finally,from(2.1)or(2.2)itisclearthatthecomputationoftheoutputresponsey[n]forn≥0requiresthattheNinitialconditionsy[

1],y[

2],

···,y[

N]mustbesatisfied.Inaddition,iftheinputx[n]isnotzeroforn<0,theevaluationof(2.1)or(2.2)alsorequirestheMinitialinputvaluesx[

1],x[

2],

···,x[

M].37Example2.1Considerthediscrete-timesystemgivenbythesecond-orderinput/outputdifferenceequationy[n]

1.5y[n

1]+y[n

2]

2x[n

2](2.3)Write(2.3)intheform(2.2)resultsintheinput/outputequationy[n]

1.5y[n

1]

y[n

2]+2x[n

2](2.4)Nowsupposethattheinputx[n]isthediscrete-timeunit-stepfunctionu[n]andthattheinitialoutputvaluesarey[

2]=2andy[

1]=1.Thussettingn=0in(2.4)givesy[0]

1.5y[

1]

y[

2]+2x[

2]

(1.5)(1)

2+(2)(0)=

0.538Settingn=1in(2.4)givesy[1]

1.5y[0]

y[

1]+2x[

1]

(1.5)(

0.5)

1+(2)(0)=

1.75Continuingtheprocessyieldsy[2]

1.5y[1]

y[0]+2x[0]

(1.5)(

1.75)

0.5+(2)(1)=

0.125y[3]

1.5y[2]

y[1]+2x[1]

(1.5)(

0.125)

1.75+(2)(1)=3.5625andsoon.Insolving(2.1)and(2.2)recursively,theprocessofcomputingtheoutputy[n]canbeginatanytimepointdesired.Inthedevelopmentabove,thefirstvalueoftheoutputthatwascomputedwasy[0].Ifthefirstdesiredvalueistheoutputy[q]attimeq,therecursiveprocessshouldbestartedbysettingn=qin(2.2).Inthiscase,theinitialvaluesoftheoutputthatarerequiredarey[q

1],y[q

2],

···,y[q

N]39CompleteSolutionBysolving(2.1)or(2.2)recursively,itispossibletogenerateanexpressionforthecompletesolution

y[n]resultingfrominitialconditionsandtheapplicationoftheinputx[n].Theprocessisillustratedbyconsideringthefirst-orderlineardifferenceequationy[n]=–

ay[n–1]+bx[n],n=1,2,…(2.5)withtheinitialconditiony[0].First,settingn=1,n=2andn=3in(2.5)givesy[1]=–

ay[0]+bx[1],(2.6)y[2]=–

ay[1]+bx[2],(2.7)y[3]=–

ay[2]+bx[3],(2.8)Insertingtheexpression(2.6)fory[1]into(2.7)givesy[2]=–

a(–

ay[0]+bx[1])+bx[2],

=a2y[0]–abx[1]+bx[2],(2.9)40Insertingtheexpression(2.9)fory[2]into(2.8)yieldsy[3]=–

a(a2y[0]–abx[1]+bx[2])+bx[3],

=–a3y[0]

+a2bx[1]–abx[2]+bx[3],(2.10)Fromthepatternin(2.6),(2.9)and(2.10),itcanbeseenthatforn≥1,Thisequationgivesthecompleteoutputresponsey[n]forn≥1resultingfrominitialconditiony[0]andtheinputx[n]appliedforn≥1.41§2-2DiscretizationinTimeofDifferentialEquationsAsanapplicationofthedifferenceequationframework,inthissectionitisshownthatalinearconstant-coefficientinput/outputdifferentialequationcanbediscretizedintime,resultinginadifferenceequationthatcanbethensolvedbyrecursion.Thisdiscretizationintimeactuallyyieldsadiscrete-timerepresentationofthecontinuous-timesystemdefinedbythegiveninput/outputdifferentialequation.Thedevelopmentbeginswiththefirst-ordercase.42First-OrderCaseConsiderthelineartime-invariantcontinuous-timesystemwiththefirst-orderinput/outputdifferentialequation(2.11)whereaandbareconstants.Eq.(2.11)canbediscretizedintimebysettingt=nT,whereTisafixedpositivenumberandntakesonintegervaluesonly.Thisresultsintheequation(2.12)43Nowthederivativein(2.12)canbeapproximatedbyIfTissuitablesmallandy(t)iscontinuous,theapproximation(2.13)tothederivativedy(t)/dtwillbeaccurate.ThisapproximationiscalledtheEulerapproximationofthederivative.Insertingtheapproximation(2.13)into(2.12)gives(2.13)(2.14)Tobeconsistentwiththenotationthatisbeingusedfordiscrete-timesignals,theinputsignalx(nT)andtheoutputsignaly(nT)willbedenotedbyx[n]andy[n],respectively;thatis,x[n]=x(t)|

t=nTandy[n]=y(t)|

t=nT44Intermsofthisnotation,(2.14)becomesFinally,multiplyingbothsidesof(2.15)byTandreplacingnbyn–1resultsinadiscrete-timeapproximationto(2.11)givenbythefirst-orderinput/outputdifferenceequationy[n]–y[n–1]=–aTy[n–1]+bTx[n–1],ory[n]=(1–aT)y[n–1]+bTx[n–1],(2.16)ThedifferenceequationiscalledtheEulerapproximationofthegiveninput/outputdifferentialequation(2.11)sinceitisbasedontheEulerapproximationofthederivative.(2.15)45Thediscretevaluesy[n]=y(nT)ofthesolutiony(t)to(2.11)canbecomputedbysolvingthedifferenceequation(2.16).Thesolutionof(2.16)withinitialconditiony[0]andwithx[n]=0forallngivenbyy[n]=(1–aT)ny[0],n=0,1,2,…(2.17)Theexactsolutiony(t)to(2.11)withinitialconditiony(0)andzeroinputisgivenby(2.18)Toanalyzetheapproximationerrorbetween(2.17)andtheexactsolution(2.18)ofy(t),sett=nTin(2.18)givesthefollowingexactexpressionfory[n]y[n]=e–anTy[0]=(e–aT)ny[0],n=0,1,2,…(2.19)Further,insertingtheexpansion46fortheexponentialinto(2.19)resultsinthefollowingexactexpressionforthevaluesofy(t)atthetimest=nT:(2.20)Comparing(2.17)and(2.20)showsthat(2.17)isanaccurateapproximationif1–aTisagoodapproximationtotheexponentiale–aT.ThiswillbethecaseifthemagnitudeofaTismuchlessthan1,inwhichcasethemagnitudeofaTwillbemuchsmallerthanthequantity1–aT.47Example2.2RCCircuitConsidertheRCcircuitgiveninFig.2-1.Thecircuithastheinput/outputdifferentialequationwherex(t)isthecurrentappliedtothecircuitandy(t)isthevoltageacrossthecapacitor.(2.21)Fig.2-148Thedifferenceequation(2.22)canbesolvedrecursivelytoyieldapproximatevaluesy[n]ofthevoltageonthecapacitorresultingfrominitialvoltagey[0]=0,inputcurrentx[n]=x(nT)=u(nT)andR=C=1.TherecursioncanbecarriedoutusingtheMATLABprograminthecoursetext.Writing(2.21)intheform(2.11)revealsthatinthiscase,a=1/(RC)andb=1/C.Hence,thediscrete-timerepresentation(2.16)fortheRCcircuitisgivenby(2.22)49Tocomparewiththeexactsolutionof(2.21),theplotsoftheresultingoutput(theunit-stepresponse)fortheapproximationaredisplayedinFig.2-2(a)forT=0.2andFig.2-2(b)forT=0.1alongwiththeexactunit-stepresponsey(t)=(1–e–t)u(t).Obviously,theapproximationerrorinFig.2-2(b)issmallerthanthatinFig.2-2(a)asthesamplingintervalTbecomessmaller.Fig.2-2(a)Fig.2-2(b)50Second-OrderCaseThediscretizationtechniqueforfirst-orderdifferentialequationsdescribedabovecanbegeneralizedtosecond-andhigh-orderdifferentialequations.Inthissecond-ordercasethefollowingapproximationscanbeused:(2.23)(2.24)Combining(2.23)and(2.24)yieldsthefollowingapproximationtothesecondderivative:(2.25)51Theapproximation(2.25)istheEulerapproximationofthesecondderivative.Nowconsideralineartime-invariantcontinuous-timesystemwiththesecond-orderinput/outputdifferentialequation(2.26)Settingt=nTin(2.26)andusingtheapproximations(2.23)and(2.25)resultsinthefollowingtimediscretizationof(2.26):(2.27)52Replacingnbyn–2in(2.27)andmultiplyingbothsidesof(2.27)byT2yieldsthedifferenceequationy[n]+(a1T–2)y[n–1]+(1–a1T+a0T2)y[n–2]=b1Tx[n–1]+(b0T2

–b1T)x[n–2](2.28)Eq.(2.28)isthediscrete-timeapproximationtothesecond-orderinput/outputdifferenceequation(2.26).Thediscretevaluesy(nT)ofthesolutiony(t)to(2.26)canbecomputedbysolvingthedifferenceequation(2.28).Tosolve(2.28),therecursionwillbestartedatn=2sothattheinitialvaluesy[0]=y(0)andy[1]=y(T)arerequired.Theinitialvaluey(T)canbegeneratedbyusingtheapproximation(2.29)wheredenotesthederivativeofy(t).53Solving(2.29)fory(T)givesy[1]=y(T)=y(0)+(2.30)withtheinitialvaluesy[0]andy[1],thesecond-orderdifferenceequation(2.28)canbesolvedusingtheMATLABprograminthecoursetext.54Example2-3SeriesRLCCircuitConsidertheseriesRLCcircuitshowninFig.2-3.Asindicated,theinputx(t)isthevoltageappliedtothecircuitandtheoutputy(t)isthevoltagevC(t)acrossthecapacitor.Wehaveknownthatthedifferentialequationforthecircuitisgivenby(2.31)Fig.2-3SeriesRLCcircuit55Eq.(2.31)isasecond-orderdifferentialequationthatcanbewrittenintheform(2.26)witha1=R/L,a0=1/(LC),b1=0,b0=1/(LC)(2.32)Inserting(2.32)intothediscretizedequation(2.28)yields(2.33)Eq.(2.33)isthedifferenceequationapproximationoftheRLCcircuit.ThevoltagevC(t)acrossthecapacitorwillbecomputedusingthediscretization(2.33)inthecasewhenR=2,L=C=2,vC(0)=1,,andvC(t)=sin(t)u(t).56Tosolvethedifferenceequation(2.33)forn≥2,theinitialconditionsarex[0]=sin(0)=0,x[1]=sin(T),vC[0]=1,andfrom(2.30),wegetNowthesecond-orderdifferenceequation(2.33)canbesolvedusingtheMATLABprogram.TocomparewiththeexactsolutionvC(t)=0.5[(3+t)e–t–cos(t)]u(t)tothedifferentialequation(2.31),theplotsoftheresultingoutputfortheapproximationaredisplayedinFig.2-4(a)forT=0.2andFig.2-4(b)forT=0.1alongwiththeexactsolution.57FromtheplotsitisseenthatthereisasignificanterrorintheapproximationinFig.2-4(a)forT=0.2.Toobtainabetterapproximation,thediscretizationintervalTcanbedecreasedtobe0.1,theresultisshowninFig.2-4(b).Infact,asT

0,theapproximationshouldapproachthetrueresponsevalues.Fig.2-4(a)Fig.2-4(b)58Problems2.1Forthedifferenceequationy[n]+

1.5y[n–1]=x[n],usethemethodofrecursiontocomputey[n]forn=0,1,2,3,whenx[