课件 信号与系统 奥本海姆

1信号系统的重要性1.是电类本科生的核心专业根底课程〔P6第一段〕，是我们专业根底课程如电路原理、自动控制的母体，电路与自动控制是信号系统在特定应用领域的衍生与开展。2.是电类本科生考研的首选专业课程3.对我们认识与解决人类实践活动大有裨益〔P6第一段末〕2教材的选择上全球信号与系统的经典之作。本课程的学习方法思想上的热爱与重视。适量的习题〔P7第二段〕尽可能与其它课程融会贯穿我的相关姓名：钟俊：3Foreword(p15)Wearepresentedwithaspecificsystemandareinterestedincharacterizingitindetailstounderstandhowitwillrespondtovariousinputs.Ourinterestmybefocusedondesigningsystemstoprocesssignalsinparticularways.Thereisaneedtodesignsystemstoextractspecificpiecesofinformationfromsignals.Wewishtomodifyorcontrolthecharacteristicsofagivensignals.4控制论创始人维纳认为:

信息是人或物体与外部世界交换内容的名称。内容是事物的原形，交换是信息载体[信号]将事物原形[内容]映射到人或物体的感觉器官，人们把这种映射的结果认为获得了信息。通俗地说，信息指人们得到的消息。

信息多种多样、丰富多彩，具体的物理形态也千差万别。语声信息以声压变化来表示；视觉信息以亮度或色彩变化来表示；文字和数据信息以字符串表示；影响物体运动的信息用作用在物体上的外力表示；影响经济运行的信息表现为投资及各产业的统计数据……51.Signalthecarrierofinformation

2.Systemaprocessofsignals,inwhichinputsignalsaretransformedintooutputsignals

6Signal：thecarrierofinformation信号：信息的载体信号是信息的具体物理表现形式，包含了信息的具体内容。总是1个或多个独立变量的函数。

同一信息可以有不同的物理表现形式，因此对应有不同的信号，但这些不同的信号都包含同一个信息。这些不同的信号之间可以相互转换。例如语音信息用声压表示，可用电压或电流信号作为载体；也可以用一组数据(01)信号作载体。对应模拟信号和数字信号，可以AD转换。7aprocessofsignals,inwhichinputsignalsaretransformedintooutputsignals系统：实现输入信号转化为输出信号的处理各种系统：通信系统、导航、定位跟踪、自动化、计算机、Internet、电力系统……消化系统、决策系统；小到1个电阻、细胞或根本粒子，大到人体、全球通信网、宇宙。所有的系统总是对施加于它的信号作出响应，产生出另外的信号。System8System系统的功能表达在什么样的输入信号系统能输出怎样的输出信号System输入信号激励输出信号响应系统和信号密不可分91

SIGNALSANDSYSTEMS信号与系统10Maincontent：Continuous-TimeandDiscrete-TimeSignals〔连续时间与离散时间信号〕TransformationsoftheIndependentVariable〔自变量的变换〕ExponentialandSinusoidalSignals〔指数信号与正弦信号〕TheUnitImpulseandUnitStepFunctions〔单位冲激与单位阶跃函数〕Continuous-TimeandDiscrete-TimeSystems(连续时间与离散时间系统)BasicSystemProperties(根本系统性质)111.1CONTINUOUS-TIMEANDDISCRETE-TIMESIGNALS(p1)

Signals:physicalphenomenaorphysicalquantities,whichchangewithtimeorspace.Signalsarerepresentedmathematicallyasfunctionsofoneormoreindependentvariables.example:x(t)

1.1.1ExamplesandMathematicalRepresentation(p1)(举例与数学表示)Figure1.1(p2)

AsimpleRCcircuit.12AspeechsignalFigure1.3(p2)“Shouldwechase〞这句话的声压随时间变化的波形.13ApictureFigure1.4(p3)一幅黑白(monochrome)照片可用亮度随二维空间变化的函数来表示.14Continuous-timeandDiscrete-timeSignals〔连续时间与离散时间信号〕continuous-timesignals’independentvariableiscontinuous:x(t)(p3)对一切时间t(除有限个不连续点外)都有确定的函数值，这类信号就称为连续时间信号，简称连续信号。discrete-timesignalsaredefinedonlyatdiscretetimes(onlyforintegervaluesoftheindependentvariable):x[n](p4)仅在不连续的瞬间(仅在自变量的整数值上)有确定函数值

15RepresentingSignalsGraphically

0x(t)

tFigure1.7(p5)Graphicalrepresentationsof(a)continuous-timeand(b)discrete-timesignals.(a)-2x[-1]x[0]x[4]-4-3-1012345x[n]

n(b)161.1.2SignalEnergyandPower(p5)〔信号能量与功率〕(p5),(1.1)(p6),(1.2)(p6),(1.3)

Theaveragepoweris

Theinstantaneouspoweris

Thetotalenergyis

Ifv(t)andi(t)are,respectively,thevoltageandcurrentacrossaresistorwithresistanceR,then17SignalEnergyandPower

A.Continuous-TimeSignalInstantaneousPower:Energyovert1t

t2:TotalEnergy:AveragePower:(p6),(1.4)(p6),(1.6)(p7),(1.8)18B.Discrete-TimeSignalInstantaneousPower:Energyovern1n

n2:TotalEnergy:AveragePower:(p6),(1.5)(p6),(1.7)(p6),(1.9)19Withthesedefinitions,wecanidentifythreeimportantclassesofsignals:

A.

FiniteEnergySignal:(P

0)Example：(p7)20B.

FinitePowerSignal:(E)Example：(p7)C.SignalswithneitherfinitetotalenergynorfiniteaveragepowerExample：(p7)211.2

TRANSFORMATIONSOFTHEINDEPENDENTVARIABLE(p7)(自变量的变换)(1)TimeShiftRightshift:x(t-t0)x[n-n0](Delay)Leftshift:x(t+t0)x[n+n0](Advance)当信号经不同路径传输时，所用时间不同，从而产生时移。如电视图像出现的重影是由于信号传输的时移造成。ExamplesofTransformationsoftheIndependentVariable(p8)〔自变量变换举例〕22TimeShift(Example)SignalTransformationFigure1.8(p8)x[n]andx[n-n0]withn0>0.Figure1.9(p9)x(t-t0)andx(t-t0)witht0<0.23(2)TimeReversalx(-t)

orx[-n]:Reflectionofx(t)orx[n]Figure1.10(p9)x[n]andx[-n].Figure1.11(p9)x(t)andx(-t).24(3)TimeScalingx(at)orx[an]

(a>0)Stretchifa<1Compressedifa>1如录像带慢放时，信号被展宽；快放时，信号被压缩；倒放时，那么信号被反褶。25Example1.1c(complementary)Givenasignalx(-t/3+2),showninFig.(a),drawthegraphofx(t).Figure1.12(p9)Continuous-timesignalsrelatedbytimescaling.260

12t(a)x(-t/3+2)1

-2/3-1/301x(t+2)x(t/3+2)

1

-2-1004/35/321x(t)271x(-t/3)-6-5-401x(t/3)045604/35/321x(t)0

12t(a)x(-t/3+2)1281.2.2PeriodicSignals(p11)〔周期信号〕在较长时间内(严格地说，无始无终)每隔一定时间T(或整数N)按相同规律重复变化的信号叫周期信号。Foracontinuous-timesignalx(t)x(t)=x(t+mT),(m=0,+1,-1,+2,-2,……)(p11,1.11)forallvaluesoft.Foradiscrete-timesignalx[n]x[n]=x[n+mN],(m=0,+1,-1,+2,-2,……)(p12,1.12)forallvaluesofn.Inthiscase,wesaythatx(t)(x[n])isperiodicwithFundamentalPeriodT(N).29Examplesofperiodicsignals:sin,cos,etc.withtheirfundamentalperiod(基波周期)N0=3Figure1.14(p12)Acontinuous-timeperiodsignal.Figure1.15(p12)Adiscrete-timeperiodsignal.30Example1.2c(complementary)

Determinethefundamentalperiodofthesignalx(t)=2cos(10πt+1)-sin(4πt-1).Fromtrigonometry,weknowthatthefundamentalperiodofcos(10πt+1)isT1=1/5,andsin(4πt-1)isT2=1/2.Whataboutthefundamentalperiodofx(t)?TheanswerisifthereisarationalT,anditisthelowestcommonmultipleofT1andT2,thenwesaythatx(t)isperiodicwithfundamentalperiodT,orelse,x(t)isaperiodic.Forthex(t)inthisexample,thelowestcommonmultipleof0.2and0.5isunit1,anditisrational,sothatthefundamentalperiodofx(t)is1.311.2.3EvenandOddSignals(p13)(偶信号与奇信号)Evensignal:x(-t)=x(t)(p13,1.14)x[-n]=x[n](p13,1.15)Oddsignal:x(-t)=-x(t)(p13,1.16)x[-n]=-x[n](p13,1.17)32x(t)=Ev{x(t)}+Od{x(t)}or:

evenpartofx(t)

oddpartofx(t)Even-OddDecomposition:(p14,1.18)(p14,1.19)33Example1.3c(complementary)

Even-OddDecomposition341.3EXPONENTIALANDSINUSOIDAL

SIGNALS(p14)

(指数信号与正弦信号)

1.3.1Continuous-timeComplexExponentialandSinusoidalSignals(p15)(连续时间复指数信号与正弦信号)

A.RealExponentialSignals(p15)

x(t)=Ceat(C,aarerealvalue)(p15,1.20)a>０a<０Figure1.19(p15)Continuous-timerealexponentialsignal.35B.PeriodicComplexExponentialandSinusoidalSignals(p16)

(a)x(t)=ej0t(p16,1.21)

(b)x(t)=Acos(0t+)(p16,1.25)

All

x(t)satisfyforx(t)=x(t+T),andT=2/0,sox(t)isperiodic.Euler’sRelation(欧拉关系):

ej0t=

cos0t+jsin0t

(p17,1.26)andcos0t=(ej0t+

e-j0t)/2

sin0t=(ej0t-

e-j0t)/2j

Wealsohave(p17,1.27)36C.GeneralComplexExponentialSignals(p20)

x(t)=Ceat,inwhichC=|C|ej,a=r+j0,so

Forr=0,therealandimaginarypartsofx(t)aresinusoidal;Forr>0(r<0),theycorrespondtosinusoidalsignalsmultipliedbyagrowing(decaying)exponetial.(p20,1.43)37Thedashedcurveistheenvelope(包络)fortheoscillatorcurve.

(a)Growingsinusoidalsignal,，r>0;(b)decayingsinusoid,，r<0.Figure1.23(p21)Continuous-timecomplexexponentialsignals.

381.3.2Discrete-timeComplexExponentialandSinusoidalSignals(p21)(离散时间复指数信号与正弦信号)

A.RealExponentialSignals(p22)

Figure1.24(p23)Discrete-time

RealExponentialSignal

x[n]=Cn:

(a)

>1;(b)0<<1;(c)-1<<0;(d)<-1.39B.SinusoidalSignals(p22)

Discrete-timeComplexexponential:x[n]=ej0n

=cos0n+jsin0n(p20,1.43)Figure1.25(p24)Discrete-timesinusoidalsignal:x[n]=cos(0n+).

40C.GeneralComplexExponentialSignals(p24)

ComplexExponentialSignal:

x[n]=Cn

inwhichC=|C|ej,=||ej0(polarform，极坐标),then

x[n]=|C|||ncos(0n+)+j|C|||nsin(0n+)(p25,1.50)41

(a)Growingsinusoidalsequence(b)DecayingsinusoidalsequenceFigure1.26(p25)ComplexExponentialSignals.

421.3.3PeriodicityPropertiesofDiscrete-timeComplexExponentials(p25)(离散时间复指数序列的周期性质)

Sampling:

Discrete-timesignalshavethreemajordifferencesfromitscontinuous-timepartner.Aretheythesame?No!

Forej0t

，ithastwoproperties:(p26)Thelargerthemagnitudeof0，thehigheristherateofoscillationinthesignal;ej0tisperiodicforanyvalueof0．43A.Fordiscrete-timecomplexexponentialsignals,weneedtoconsiderafrequencyintervalof2.(p26)(在考虑离散时间复指数时，仅需要在某一个2间隔内选择即可〕Thus,ej0nandej(0+

2)narethesamesignals.ej0n不具备随0在数值上的增加而不断增加其振荡速率的特性！当0从０开始增加，其振荡速率愈来愈快，直到0=,到达最大，假设继续增加0，其振荡速率就下降，直到0=2时,又得到与0=0时同样的效果(常数序列).(p26,1.51)(p26,1.52)44Lowestoscillationrate:Highestoscillationrate:Figure1.27(p27)Discrete-timeSinusoidalsequences.45Continuous-time:ej0t

,

T=2/0Discrete-time:ej0n

,N=?Calculateperiod:Bydefinition:ej0n

=

ej0(n+N)(p26,1.53)

thusej0N

=1

(p26,1.54)

or

0N=

2m(p26,1.55)SoN=

2m/0

withintergerN(p28,1.58)Conditionofperiodicity:2/0

isrational.B.Periodicityofej0n

(p26)假设2/0为一有理数，ej0n就是周期的，否那么就不是周期的．46C.Finitenumberofdistinctharmonics(p29)ForaperiodicsignalwithfundamentalperiodofN,ThereareonlyNdistinctperiodicexponentialsfordiscrete-timesignals.IntheContinuous-timecase,alloftheharmonicallyrelatedcomplexexponentialsaredistinct.(p29,1.60)(p30,1.61)471.4THEUNITIMPULSEANDUNITSTEPFUNCTIONS(p30)

(单位冲激与单位阶跃函数)

1.4.1TheDiscrete-TimeUnitImpulseandUnitStepSequences(p30)(离散时间单位脉冲与单位阶跃序列)

UnitSample(Impulse):(p30,1.63)Figure1.28(p30)Discrete-timeunitimpulse

(sample).48UnitStepFunction:(p30,1.64)Figure1.29(p31)Discrete-timeunitstep

sequence.49Relationshipbetweenunitsampleandunitstepsequencetheunitsampleisthefirstdifferenceoftheunitstepsequencetheunitstepsequenceistherunningsumoftheunitsampleorhere(p31,1.65)(p31,1.66)(p31,1.67)50writeanydiscrete-timesignalintermsofdelayedunitsampleasSamplingPropertyofUnitSample(p32,1.68)(p32,1.69)51

UnitStepFunction:1.4.2TheContinuous-TimeUnitStepandUnitImpulseFunctions(p32)

(离散时间单位脉冲与单位阶跃序列)

(p32,1.70)Figure1.32(p33)Continuous-timeunitstep

function.52UnitImpulseFunction:Figure1.35(p34)Continuous-timeunitimpulse.

53RelationBetweenUnitImpulseandUnitStep(p33,1.71)(p33,1.72)54Illustrationsofδ(t)

0ΔuΔ(t)t1

Continuousapproximationtotheunitstep,uΔ(t)

0Δ

1/Δ

δΔ(t)

DerivativeofuΔ(t)δ(t)isthelimitofδΔ(t)asΔ→0.(p34,1.74)

Figure1.33(p33).Figure1.34(p33).55propertiesof

δ(t)Samplingproperty

(p35,1.76)56δ(t)

isaevenfunction:δ(-t)=δ(t)

Why:Scalingproperty:57Example1.4c(complementary)Determineandsketchthefirstderivativeofthesignaldepictedinthefollowingfigure.-1x(t)t01241-1-21012458(1)Definition:

Interconnection(互联)ofComponent,device,subsystem….(Broadestsense广义)Aprocessinwhichsignalscanbetransformed.(Narrowsense狭义)Continuous-timesystem:bothinputsignalsandoutputsignalsarecontinuous-timesignalsDiscrete-timesystem:transformdiscrete-timeinputsintodiscrete-timeoutputs1.5CONTINUOUS-TIMEANDDISCRETE-TIMESYSTEMS(p38)(连续时间与离散时间系统)59(2)RepresentationofSystemPictorialRepresentationContinuous-timesystem

x(t)y(t)Discrete-timesystem

x[n]Y[n]Relationbythenotation

Figure1.41(p38).(p39,1.79)601.5.1SimpleExampleofSystems(p39)

(简单系统举例)

Example1.8(p39)RCCircuitinFigure1.1:Vc(t)Vs(t)RCCircuit(System)

vs(t)vc(t)---LinearConstantCoefficientDifferentialEquation(微分方程)Figure1.1(p2).(p39,1.82)61BalanceinBank(System)

x[n]y[n]Example1.10(p40)Balance(余额)inabankaccountfrommonthtomonth:balance(第n个月末的余额)---y[n]netdeposit(第n个月的净存款)---x[n]interest(利息)---1%soy[n]=y[n-1]+1%y[n-1]+x[n](p40,1.86)ory[n]-1.01y[n-1]=x[n](p41,1.87)621.5.2InterconnectionsofSystem(p41)

(系统的互联)

(1)Series(cascade)interconnection(串联或级联)Figure1.42(a)(p42).63(2)Parallelinterconnection(并联)Figure1.42(b)(p42).64Series-ParallelinterconnectionFigure1.42(c)(p42).65(3)Feedbackinterconnection(反响联结)Figure1.43(p43).66ExampleofFeedbackinterconnectionFigure1.44(p43)(a)Simpleelectricalcircuit;(b)blockdiagram.671.6BASICSYSTEMPROPERTIES(p44)(根本系统性质)1.6.1SystemswithandwithoutMemory(p44)

(记忆系统与无记忆系统)

Memorylesssystem(无记忆系统):Itsoutputisdependentonlyontheinputatthesametime.Features:Nocapacitor,noconductor,nodelayer.Memorysystem(记忆系统):Itsoutputisdependentnotonlyontheinputattimesotherthanthecurrenttime.Features:capacitor,conductor,delayer.68Examplesofmemorylesssystem:

y(t)=Rx(t)(p44,1.91)ory[n]-0.5y[n-1]=2x[n]Examplesofmemorysystem:y[n]=x[n](p44,1.92)691.6.2InvertibilityandInverseSystems(p45)

(可逆性与可逆系统)

Definition:

Asystemissaidtobeinvertible

ifdistinctinputsleadtodistinctoutputs.

Ifsystemisinvertable(可逆),thenaninversesystemexists.Aninversesystemcascaded(级联)withtheoriginalsystem,yieldsanoutputequaltotheinput.y(t)=2x(t)(p45,1.97)70Figure1.45(p46).711.6.3Causality(p46)

(因果性)Definition:AsystemiscausalIftheoutputatanytimedependsonlyonvaluesoftheinputatthepresenttimeandinthepast.Forcausalsystem,ifx(t)=0fort<t0,theremustbey(t)=0fort<t0.(nonanticipative(不可预测的))Memorylesssystemsarecausal.x(t)y(t)t1t272

Example1.12(p47)

Checkingthecausalityoftwosystems.Thefirstsystemisdefinedby

Itdependsonfuturevalue,soitisnotcausal.

Thesecondsystemisdefinedby

Itisimportanttodistinguishcarefullytheeffectsoftheinputfromthoseofanyotherfunctionsusedinthedefinitionofthesystem.Itdependsoncurrentvalue,soitiscausal.(p47,1.105)(p47,1.106)731.6.4Stability(p48)

(稳定性)Definition:

Iftheinputtoastablesystemisbounded(i.e.,ifitsmagnitudedoesnotgrowwithoutbound),thentheoutputmustalsobebounded.Finiteinputleadtofiniteoutput:if|x(t)|<M,then|y(t)|<N.Examples:Stablependulum Motionofautomobile74Example1.13(p49)

Checkthestabilityofthefollowingtwosystems(p49,1.109)Thesystemisunstable.(p50,1.110)(p50,1.111)(p50,1.112)(p50,1.113)Thesystemisstable.75x(t)y(t)x(t-t0)y(t-t0)

1.6.5TimeInvariance(p50)

(时不变性)Definition:

Asystemistimeinvariantifthebehaviorandcharacteristicsofthesystemarefixedovertime.Timeinvariantsystem:Continuous-time:Ifx(t)y(t),thenx(t-t0)

y(t-t0).Discret