信号与系统英文版课件Chapter 1

Continuousanddiscretetimesignals

Twobasictypesofsignals:Continuoussignals：Theindependentvariableiscontinuousandsignalsaredefinedforacontinuousvaluesoftheindependentvariable.x(t)tContinuousanddiscretetimesignalsDiscretesignalsx[n]:aredefinedonlyatdiscretetime,indepen-dentvariabletakesononlyadiscretesetofvalues.Note:x[n]isdefinedonlyforintegervaluesoftheindependentvariable.x[n]012345678nContinuousanddiscretetimesignalsForexample:AresistorRwithv(t)andi(t),theinstantaneouspoweris

ThetotalenergyexpendedoverthetimeintervalisContinuousanddiscretetimesignalsandtheaveragepoweroverthisintervalisSimilarlyforanyx(t)oranyx[n],thetotalenergyisdefinedasorWheredenotesthemagnitudeofx.ContinuousanddiscretetimesignalsSimilarly,thetotalpowerisdefinedasOrOverinfinitetimeinterval(or)1n2n(1.6)(1.7)Continuousanddiscretetimesignals

Withthesedefinitions,wecanidentifythreeimportantclassesofsignals:,Energysignal

(2),Powersignal(3),Infiniteenergyandpowersignal(1.8)(1.9)TransformationsofindependentvariableExampleoftransformationsTimeshiftingtx(t-t0)0t0110tx(t)012341x[n]n

-101231x[n+1]nTransformationsofindependentvariableTimereversal:

x[n]x[-n]:

x(t)x(-t)x(1-t)011t-4-3-201nx[-n]-1012341x[n]n01-1t011tTransformationsofindependentvariableTimescalingx(t)x(2t)compressedx(t)x(t/2)stretchedx(t)x(2t)x(t/2)tttTransformationsofindependentvariableLetIfcompressedstretchedreversedshiftedExample1.1Giventhesignalx(t),toillustratex(t+1),x(-t+1),x(3t/2),x(3t/2+1)··0211tx(t)Transformationsofindependentvariablex(-t+1)=x[-(t-1)]01-1t1x(t+1)=x[(t+2/3)]-2/32/30t···1··0211tx(t)-10121tx(t+1)0-21-1tx(-t)··02/34/3tx(3t/2)1x(-t+1)x(3t/2+1)TransformationsofindependentvariableExample:Adiscretesignalx[n]isshowninFigureSketchandlabelfollowingsignals:

(1)x[2n];(2)x[2n+1].2-4-201234x[n]n0.511.5-0.5-1-1.5-2TransformationsofindependentvariableAssignments:P57:1.4,a,eP59:1.21,d,f-2n-1-2-1012x[2n]121.52024-4-2x[n]n0.51-0.5-1-1.5-2n21.5-1.5-1-2-101x[2n+1]12TransformationsofindependentvariablePeriodicsignalsForallvaluesoft,x(t)=x(t+T)(1.11)thenx(t)isperiodicwithperiodT.ThefundamentalperiodTofx(t)isthesmallestpositivevalueofTforwhichEq.(1.11)holds.-T0Ttx(t)……Transformationsofindependentvariablex[n]:x[n]=x[n+N](1.12)

N—somepositiveinteger.

fundamentalperiodisN=3……x[n]0123nTransformationsofindependentvariableEvenandOddSignalsEvensignals:x(-t)=x(t)x[-n]=x[n]Symmetryontheverticalaxis.Oddsignals:x(-t)=-x(t)x[-n]=-x[n]Symmetryontheorigin.x(t)0t0tx(t)TransformationsofindependentvariableAnysignalcanbebrokenintoasumoftwosignals,oneofwhichisevenandoneofwhichisodd.

Transformationsofindependentvariablex[n]…0123n1=-2Ev{x[n]}+-3–2-1Od{x[n]}+t1/2-1/2xo(t)0123-1n1…1/21230n1/2=t1/2xe(t)t1x(t)ExponentialandsinusoidalsignalsContinuouscomplexexponentialandsinusoidalsignalsComplexexponentialwherecandaare,ingeneral,complexnumbers.

(1)Realexponentialsignal(candaarereal)

a>0,ast↑,↑

a<0,ast↑,↓

a=0,isconstant

Exponentialandsinusoidalsignals

(2)Periodiccomplexexponentialandsinusoidalsignals

aisconstrainedpurelyimaginary

itisperiodicandfundamentalperiod

Exponentialandsinusoidalsignals

Asignalcloselyrelatedtotheperiodiccomplexexponentialisthesinusoidalsignal

andprovideimportantexamplesofsignalswithinfinitetotalenergybutfiniteaveragepower.ExponentialandsinusoidalsignalsbutAperiodsignalisapowersignal.Exponentialandsinusoidalsignals

(3)Generalcomplexexponentialsignals

If

Thenr=0,x(t)isperiodic,ReandImaresinusoidal.

r>0andr<0,x(t)isaperiodic.ExponentialandsinusoidalsignalsExample:determinewhetherornoteachofthefollowingsignalsisperiodic.Ifsignalisperiodic,determineitsT0.

(a)periodic(b)nonperiodic(c)periodicExponentialandsinusoidalsignals(d)

IfT01andT02havesmallestcommonmultiple(SCM),thentheSCMisfundamentalperiodT0.ExponentialandsinusoidalsignalsDiscrete–timecomplexexponentialandsinusoidalsignalsAsincontinuous,animportantsignalindiscreteiscomplexexponentialsignalor

sequence:

candaare,ingeneral,complexnumbers.ExponentialandsinusoidalsignalsRealexponentialsignals

(wherecandareal)|a|>1,asn,x[n]exponentially|a|<1,asn,x[n]exponentially

a>0,allvaluesofx[n]havesamesign.

a<0,thenthesignofx[n]alternates.

a=0,x[n]isaconstant.ExponentialandsinusoidalsignalsSinusoidalsignals

Asbefore,Euler’srelationallowsustorelatecomplexexponentialwithsinusoidal.ThensinusoidalsignalscanbepresentedasGeneralcomplexexponentialsignals(omit)

ExponentialandsinusoidalsignalsPeriodicitypropertiesofdiscrete-timecomplexexponentialsWewillseethattherearethreedefinitedifferencebetweenanditscounterpart

(1)Exponentialandsinusoidalsignals

(2)Forthe,thelargerthemagnitudeof0,thehigheristherateofoscillationinthesignal.

Forthe,thelow-frequencyhavevaluesof0near0,2,andanyotherevenmultipleof,whilethehighfrequencyarelocatednear0andotheroddmultiplesof.(3)

ej0tisperiodicforanyvalueof0,whileejk0nmaynotperiodicforany0Exampleisperiodic,T0=2/3,howeverisnonperiodicExponentialandsinusoidalsignals(4)Theperiodof

InorderfortobeperiodicwithN>0,mustorequivalently0N

mustbeamultipleof2,Thatis,theremustbeanintegermsuchthat0N=2morequivalently0/2=m/NExponentialandsinusoidalsignalsIf00and0/2isarationalnumber,ej0nisperiodicandisnotperiodicotherwise.IfNandmhavenofactorsincommon,thenfundamentalperiodisN.∵fundamentalfrequencyis

2/N=0/mfundamentalperiodis

N=m(2/0)whereN>0andmaresomeintegers.ExponentialandsinusoidalsignalsExample:(a)

x[n]=cos(2n/12)periodicN=12(b)

x[n]=cos(8n/31)periodicN=31(c)x[n]=cos(n/6)nonperiodic(d)

x[n]=exp(j(2/3)n)+exp(j(3/4)n)N=24Inthecontinuouscase,alloftheharmonicallyrelatedcomplexexponentialsejk0t,k=0,1,2…,aredistinct.Table1.1Comparisonofej0tand

ej0n

ej0nej0tPeriodicforany0Onlyif0=2m/NforsomeintegersN>0andmDistinctsignalsfordistinct0

Identicalsignalsforvaluesof0separatedbymultiplesof2fundamentalfrequency00mfundamentalperiod2/

0N=m(2/

0)Assignments:1.10,1.11,1.25cf,1.26beTheunitimpulseandunitstepfunctionsThediscreteunitimpulseandunitstepsequenceUnitimpulse(unitsample)isdefinedas0-11n1[n]Unitstepdefinedby01nu[n]…TheunitimpulseandunitstepfunctionsTheimpulseisthefirstdifferenceofthestep.Conversely,thestepistherunningsum

ofunitsample.Theunitimpulseandunitstepfunctions

ThereissamplingpropertymoregenerallyContinuousunitstepandunitimpulsefunctionUnitstepThecontinuousunitstepu(t)istherunningintegralofunitimpulse(t).(t)thefirstderivativeofu(t)1t0u(t)Continuousunitstepandunitimpulsefunction10tAndthefollowedexampleshowedushowwecanget

C1Fi(t)+-u(t)Continuousunitstepandunitimpulsefunction1t0u(t)t0i(t)=du/dtnodefinition1t0u(t)t0du/dt1/00t10tContinuousunitstepandunitimpulsefunction1t0u(t)1t0u(t)t0du/dt1/10tnote:Aswiththe[n],the(t)alsohasaveryimportantsamplingproperty:(t)andu(t)aresingularityfunctions.

Example1.7

Givensignalx(t):-1121234x(t)tContinuousunitstepandunitimpulsefunctionContinuousunitstepandunitimpulsefunctioncalculateandsketchthex’(t);recoverx(t)fromx’(t).

Solution:-1121234x(t)t-3121234x’(t)tContinuousunitstepandunitimpulsefunctionExample(seeproblem1.38)(1)Showthat(2)determine

(1)Consider:t1/(t)t/21/(2t)ContinuousunitstepandunitimpulsefunctionAssignmentsP58:1.13,1.14,1.21(f)ContinuousanddiscretesystemsInputandoutputarecontinuous.Inputandoutputarediscrete.x(t)continuousy(t)x[n]discretey[n]SimpleexamplesofsystemsContinuoussystems—differentialequation.Discretesystems—differenceequation.Example1.10Lety[n]denotethebalanceattheendofthenthmonth,x[n]representsthenetdepositduringthenthmonthandinteresteachmonthis1%,Thentheequationis

y[n]=1.01y[n-1]+x[n]

ory[n]-1.01y[n-1]=x[n]InterconnectionofsystemsSeries(orcascade),parallel,feedbackin

12out

1in2+outseriesparallelInterconnectionofsystemsSeries-parallelin123+4outin+12outfeedbackSystemproperties:memoryormemorylessIfitsoutputisdependentonlyoninputatthatsametime—memoryless.Theconceptofmemorycorrespondstothepresenceinthesystemthatretainsorstoresinformationaboutinputvaluesattimesotherthanthecurrenttime.Examples:Memoryless(identitysystem)Memory

(Accumulatororsummer)

(delay)

(integrator)Systemproperties:memoryormemorylessSystemproperties:invertibilityandinverseAsystemissaidtobeinvertibleifdistinctinputsleadtodistinctoutputs.Ifasystemisinvertible,thenaninversesystemexists,whencascadedwiththeoriginalsystem,yieldanoutputw[n]equaltothex[n].invertiblex[n]systemy[n]inversesystemw[n]=x[n]AnexampleAnotherexample—accumulatorThedifferencebetweentwosuccessiveoutputvaluesispreciselyinputvalue:x(t)y(t)=2x(t)y(t)w(t)=y(t)/2w(t)=x(t)Systemproperties:invertibilityandinverse∴theinversesystemisExamplesofnoninvertiblesystemsare

andand∵

andsamex[n]y[n]w[n]=y[n]-y[n-1]w[n]=x[n]Sy