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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
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