奥本海姆版信号与系统

Chapter1

SignalsAndSystems崔琳莉ContentsDescriptionofsignalsTransformationsoftheindependentvariableSomebasicsignalsSystemsandtheirmathematicalmodelsBasicsystemsproperties1.1Continuous-Timeand Discrete-TimeSignals1.1.1ExamplesandMathematicalRepresentation(1)AsimpleRCcircuitSourcevoltageVsandCapacitorvoltageVcA.Examples(2)AnautomobileForceffromengineRetardingfrictionalforceρVVelocityV(3)ASpeechSignal(4)APicture(5)vitalstatistics(人口统计)NoteInthisbook,wefocusonourattentiononsignalsinvolvingasingleindependentvariable.Forconvenience,wewillgenerallyrefertotheindependentvariableastime,althoughitmaynotinfactrepresenttimeinspecificapplications.B.Twobasictypesofsignals

t:continuoustimex(t):continuumofvalue1.

Continuous-Timesignal2.

Discrete-Timesignal

n:discretetimex[n]:adiscretesetofvalues(sequence)Example1:1990-2002年的某村农民的年平均收入SamplingExample2:x[n]issampledfromx(t)WhyDT?(1)FunctionRepresentation

Example:x(t)=cos0tx[n]=cos0nx(t)=ej0tx[n]=ej0n(2)GraphicalRepresentation

Example:(Seepagebefore)(3)Sequence-representationfordiscrete-timesignals:

x[n]={-21321–1}orx[n]=(-21321–1)C.RepresentationNote:Sincemanyoftheconceptsassociatedwithcontinuousanddiscretesignalsaresimilar(butnotidentical),wedeveloptheconceptsandtechniquesinparallel.There

aremanyothersignals

classification:Analogvs.DigitalPeriodicvs.AperiodicEvenvs.OddDeterministicvs.Random……1.1.2SignalEnergyandPowerInstantaneouspower:LetR=1Ω,so+R_Energy:t1tt2AveragePower:TotalEnergyAveragePowerDefinition:Continuous-Time:(t1tt2

)Discrete-Time:(n1nn2

)Wewillfrequentlyfinditconvenienttoconsidersignalsthattakeoncomplexvalues.whenTotalEnergyAveragePowerNote:Itisimportanttorememberthattheterms“Power〞and“energy〞areusedhereindependentlyofthequantitiesactuallyarerelatedtophysicalenergy.Withthesedefinitions,wecanidentifythreeimportantclassofsignals——a.finitetotalenergyb.finiteaveragepowerc.infinitetotalenergy,infiniteaveragepowerReadtextbookP71:MATHEMATICALREVIEWHomework：P57--1.21.2.1ExamplesofTransformations1.TimeShiftx(t-t0),x[n-n0]t0<0

Advance1.2TransformationoftheIndependentVariablen0>0

DelayTimeShiftx(t)andx(t-t0),orx[n]andx[n-n0]:TheyareidenticalinshapeIft0>0,x(t-t0)representsadelay

n0>0,x[n-n0]representsadelayIft0<0,x(t-t0)representsanadvance

n0<0,x[n-n0]representsanadvance2.TimeReversalx(-t),x[-n] ——Reflectionofx(t)orx[n]2.TimeReversalx(-t),x[-n] ——Reflectionofx(t)orx[n]amirrorTimeReversalx[n]x[-n]LookingformistakesNote:thedifferencebetweenx(-t)and–x(t)x(t)x(-t)???-x(t)0tx(t)x(t/2)x(2t)3.TimeScalingcompressedstretchx(at)(a>0)TimeScaling x(at)(a>0)

Stretchifa<1Compressedifa>1Howaboutthediscrete-timesignal?x[n]Generally,timescalingonlyforcontinuoustimesignalsx[2n]x[n]x[2n]0123456nThisisalsocalleddecimationofsignals.〔信号的抽取〕x[n/2]x[n]222Example011tx(t)Solution1：Solution2：Solution1：Solution2：011tx(t)01tx(t-1/2)1/23/201tx(3t-1/2)1/61/2011tx(t)01/31tx(3t)01tx(3t-1/2)1/61/2shiftreversalScalingreversalshiftScalingreversalshiftScalingExample f(t)f(1-3t)1.2.2PeriodicSignals

Aperiodicsignalx(t)(orx[n])hasthepropertythatthereisapositivevalueofT(orintegerN)forwhich: x(t)=x(t+T),forallt x[n]=x[n+N],foralln Ifasignalisnotperiodic,itiscalledaperiodicsignal.

ExamplesofperiodicsignalsCT:x(t)=x(t+T)DT:x[n]=x[n+N]PeriodicSignals

Thefundamentalperiod

T0(N0)

ofx(t)(x[n])isthesmallestpositivevalueofT(orN)forwhichtheequationholds.

Note: x(t)=Cisaperiodicsignal,butitsfundamentalperiodisundefined.Examplesofperiodicsignals1.Itisperiodicsignal.ItsperiodisT=16/3.2.

Itisnotperiodic.3.

x(t)isperiodic.ItsperiodisThesmallestmultiplesofT1andT2incommon4.

Itisaperiodic,too.ThereisnothesmallestmultiplesofT1andT2incommon5.

x(t)isaperiodic.6.

ItisperiodicwithperiodN=16.CosπtCos2tcosπt+cos2t1.2.3EvenandOddSignalsNote: Anoddsignalmustnecessarilybe0att=0,orn=0.ie.x(t)=0,orx[n]=0.

Evensignal:

x(-t)=x(t)orx[-n]=x[n]Oddsignal:

x(-t)=-x(t)orx[-n]=-x[n]Even-OddDecomposition—— AnysignalcanbeexpressedasasumofEvenandOddsignals.x(t)=xeven(t)+xodd(t)x[n]=xeven[n]+xodd[n]Exampleoftheeven-odddecompositonExampleoftheeven-odddecompositonHomework：P57--1.91.101.21(a)(b)(c)(d) 1.22(a)(b)(c)(g)1.231.241.3ExponentialandSinusoidalSignals1.3.1Continuous-timeComplexExponentialandSinusoidalSignals Thecontinuous-timecomplexexponentialsignalisoftheform

whereCandaare,ingeneral,complexnumbers.

A.RealExponentialSignalsx(t)=Ceat(C,aarerealvalue)a>0a<0growingdecayingB.PeriodicComplexExponentialand

SinusoidalSignals

x(t)=ej0t x(t)isperiodicforx(t)=x(t+T),anditsfundamentalperiodis .x(t)=Ceat,C=1,a=j0(purelyimaginary)(1)Forej0tif0=0,x(t)=1,thenitisperiodicforanyT0.(2)x(t)=Acos(0t+)0－rad/sf0－HzEuler’sRelation: ej0t=

cos0t+jsin0tandcos0t=(ej0t+

e-j0t)/2sin0t=(ej0t-

e-j0t)/2jWehave

ifcisacomplexnumber,Re{c}denotesitsrealpart;Im{c}denotestheimaginarypart.1<2<3213y=cos2ty=cos5ty=cos10t(3)Harmonicallyrelatedcomplexexponentials

fork=0,k(t)isaconstant; whileforanyothervalueofk,k(t)isperiodicwithfundamentalfrequency|k|0andfundamentalperiodC.GeneralComplexExponentialSignalsx(t)=Ceat,whenC=|C|ej,a=r+j0Sox(t)=|C|ejertej0t=|C|ertej(0t+)

=|C|ertcos(0t+)+j|C|ertsin(0t+)r>0r<01.3.2Discrete-timeComplex ExponentialandSinusoidalSignalsComplexExponentialSignal(sequence):x[n]=Cn whereCandaare,ingeneral,complexnumbers.

orx[n]=CenWhere=e.ConvenientformAnalogoustothecontinuoustimecounterpartA.RealExponentialSignal>10<<1-1<<0<-1x[n]=CnB.SinusoidalSignals

Complexexponential:x[n]=ej0n

=cos0n+jsin0n

Sinusoidalsignal:

x[n]=cos(0n+)

C.GeneralComplexExponentialSignalsIfletCandinpolarformviz. C=|C|ejAnd =||ej0

,thenx[n]=Cn

=|C|||ncos(0n+)+j|C|||nsin(0n+)RealorImaginaryofSignal||<1||>1growingdecaying1.3.3PeriodicityPropertiesofDiscrete-timeComplexExponentialsContinuous-time:ej0t

,

T=2/0Discrete-time:ej0n

,N=?Twopropertiesofcontinuous-timesignalej0t

:

(1)ej0tisperiodicforanyvalueof0.(2)thelagerthemagnitudeof0,thehigheristherateofoscillationinthesignal.PeriodicityPropertiesCalculateperiod:

Bydefinition:ej0n=

ej0(n+N)

thusej0N=1or0N=

2mSoN=

m(2/0)Conditionofperiodicity:0/2isrationalFundamentalperiodFromthesefigures,wecanconlude:Signalsoscillaterapidlywhen0=±,±3,…

(high-frequency);signalsoscillateslowly

when0=0,±2,±4,…(low-frequency)onthemostoccasionswewillusetheintervalHarmonically

relatedcomplexexponentialsNote:

Comparisonofthesignalsej0tandej0n,seeP28Table1.1So,OnlyNdistinctperiodicexponentialsinthesetForExamples:Determinethefollowingequationsfundamentalperiod:(1)T=31/4N=31(2)Itisnotperiod.(3)N1=3,N2=8N=N1×N2=24ThesmallestmultipleofN1andN2

incommonHomework:P61--1.26*1.25(d)(e)(f)1.4TheUnitImpulseandUnitStepFunctions1.4.1TheDiscrete-timeUnitImpulseandUnitStepSequences(1)UnitSample(Impulse):(2)UnitStepFunction:(3)RelationBetweenUnitSampleandUnitSteporrunningsumfirstdifference(4)SamplingPropertyofUnitSampleIllustrationofSampling1.4.2TheContinuous-timeUnitStepandUnitImpulseFunctions(1)UnitStepFunction:(2)UnitImpulseFunction:But, u(t)isdiscontinuousatt=0andconsequentlyisformallynotdifferentiable.So, howcanweget ?firstderivativerunningintegralAnalogoustotherelationshipbetweenu[n]andConsidering:then:When:Thatmeans, hasnodurationbutunitarea.Wecanget:(3)RelationBetweenUnitImpulseandUnitSteprunningintegralfirstderivative(4)SamplingPropertyof(t)(5)Thetransformationof(t)

Proof:

So,

Let

Example

02SignalrepresentationusingstepfunctionsExample

x(t)tt0f(t)SignalrepresentationusingstepfunctionsExample

1-11x(t)t(-1)2-21x1(t)t(-2)(1)1-11x(t)t2-21x1(t)t2-22x2(t)t22x2(t)t1Homework:P571.61.22(e)(f)1.2.3.2-21g(t)tsketch4.sketch1.5Continuous-timeand Discrete-timeSystem(1)Acontinuous-timesystemisasysteminwhichcontinuous-timeinputsignalsareappliedandresultincontinuous-timeoutputsignals.Continuous-timesystem

x(t)y(t)(2)Adiscrete-timesystem－thatis,asystemthattransformsdiscrete-timeinputsintodiscrete-timeoutputs.Discrete-timesystem

x[n]y[n]1.5.1SimpleExampleofsystems1.Example1.8(p39)RCCircuit(system)vs(t)vc(t)FromOhm’slawandWecanget2.Example1.10(p40)Balanceinabankaccountfrommonthtomonth:balancey[n] 〔余额〕netdepositx[n] (净存款〕interest1%soy[n]=y[n-1]+1%y[n-1]+x[n]ory[n]-1.01y[n-1]=x[n]Balanceinbank(system)x[n]y[n]Conclusion:

Themathematicaldescriptionsofsystemsastheprecedingexamplesarethefirst-orderlineardifferentialordifferenceequationofforms:

1.5.2InterconnectionsofSystem Manyrealsystemarebuiltasinterconnectionsofseveralsubsystems.(1)Series(cascade)interconnection(2)Parallelinterconnection(3)Series-Parallelinterconnection(4)Feed-backinterconnection1.6BasicSystemProperties1.6.1SystemswithandwithoutMemoryMemorylesssystem:It’soutputforeachvalueoftheindependentvariableatagiventimeisdependentonlyontheinputatthesametime.Features:Nocapacitor,noconductor,nodelayer.Examples:withmemorywithoutmemoryidentitysystem1.6.2InvertibilityandInverseSystemsNote:(1)Ifsystemisinvertible,thenaninversesystemexists.(2)Aninversesystemcascadedwiththeoriginalsystem,yieldsanoutputequaltotheinput.Invertiblesystem－distinctinputsleadto distinctoutputs.ExamplesofinvertiblesystemsExamplesofnoninvertiblesystems1.6.3Causality Asystemiscausal

Iftheoutputatanytimedependsonlyonvaluesoftheinputatthepresenttimeandinthepast. (nonanticipative不超前)

Note： Forcausalsystem,ifx(t)=0fort<t0,theremustbey(t)=0fort<t0.

Memorylesssystemsarecausal.causalnoncausalExamplesofcausalsystems1.6.4Stability Thestablesystem－Smallinputsleadtoresponsesthatdonnotdiverge. BoundedinputleadtoBoundedoutput(BIBO〕if|x(t)|<M,then|y(t)|<N.(unstablesystem)Examples:S1：S2:S3:(stablesystem)(unstablesystem)BoundedandUnboundedSignalsWhethertheoutputsignalofasystemisboundedorunboundeddeterminesthestabilityofthesystem1.6.5TimeInvariance Asystemiftimeinvariantifthebehaviorandcharacteristicsofthesystemarefixedovertime.Timeinva