奥本海姆版信号与系统课件

ContentsnDescription of signalsnTransformations of the independent variablenSome basic signalsnSystems and their mathematical modelsnBasic systems propertiesContentsDescription of signals11.1 Continuous-Time and Discrete-Time Signals1.1.1 Examples and Mathematical Representation(1)A simple RC circuitSource voltage Vs and Capacitor voltage VcA.Examples1.1 Continuous-Time and D2(2)An automobileForce f from engineRetarding frictional force VVelocity V(2)An automobileForce f from3(3)A Speech Signal(3)A Speech Signal4(4)A Picture(4)A Picture5(5)vital statistics(人口统计人口统计)(5)vital statistics(人口统计)6NotenIn this book,we focus on our attention on signals involving a single independent variable.nFor convenience,we will generally refer to the independent variable as time,although it may not in fact represent time in specific applications.NoteIn this book,we focus on 7B.Two basic types of signals t:continuous timex(t):continuum of value1.Continuous-Time signalB.Two basic types of signals82.Discrete-Time signal n:discrete timexn:a discrete set of values(sequence)2.Discrete-Time signal n:9Example1:1990-2019年的某村农民的年平均收入年的某村农民的年平均收入Example1:1990-2019年的某村农民的年平均10SamplingExample2:xn is sampled from x(t)Why DT?SamplingExample2:xn is sam11(1)Function Representation Example:x(t)=cos 0t xn=cos 0n x(t)=ej 0t xn=ej 0n(2)Graphical Representation Example:(See page before)(3)Sequence-representation for discrete-time signals:xn=-2 1 3 2 1 1 or xn=(-2 1 3 2 1 1)C.Representation(1)Function RepresentationC.12Note:nSince many of the concepts associated with continuous and discrete signals are similar(but not identical),we develop the concepts and techniques in parallel.nThere are many other signals classification:nAnalog vs.DigitalnPeriodic vs.Aperiodic nEven vs.OddnDeterministic vs.RandomnNote:Since many of the concept131.1.2 Signal Energy and PowerInstantaneous power:Let R=1,so +R_Energy:t1 t t2Average Power:1.1.2 Signal Energy and PowerI14Total EnergyAverage PowerDefinition:Continuous-Time:(t1 t t2)Discrete-Time:(n1 n n2)We will frequently find it convenient to consider signals that take on complex values.Total EnergyAverage PowerDefin15whenTotal EnergyAverage PowerwhenTotal EnergyAverage Power16Note:vIt is important to remember that the terms“Power”and“energy”are used here independently of the quantities actually are related to physical energy.vWith these definitions,we can identify three important class of signalsNote:It is important to rememb17a.finite total energyb.finite average powerc.infinite total energy,infinite average powerRead textbook P71:MATHEMATICAL REVIEWHomework：P57-1.2a.finite total energyRead tex181.2.1 Examples of Transformations1.Time Shiftx(t-t0),xn-n0t00 DelayTime Shiftn00 DelayTime Shift20 x(t)and x(t-t0),or xn and xn-n0:nThey are identical in shapenIf t00,x(t-t0)represents a delayn00,xn-n0 represents a delaynIf t00,x(t-t0)represents an advance n00)x(t)x(t/2)x(2t)3.Time Scaling26Time Scalingx(at)(a0 )Stretch if a1How about the discrete-time signal?Time Scalingx(at)(a0 )Ho27xnGenerally,time scaling only for continuous time signals x2nxnx2n0 1 2 3 4 5 6nThis is also called decimation of signals.（信号的抽取）信号的抽取）xn/2xn2 2 2xnGenerally,x2nxnx2n028Example0 11tx(t)Solution 1：Solution 2：Example0 11tx(t)Solut29Solution 1：Solution 2：0 11tx(t)01tx(t-1/2)1/2 3/20 1tx(3t-1/2)1/6 1/20 11tx(t)0 1/31tx(3t)0 1tx(3t-1/2)1/6 1/2Solution 1：Solution 2：0 30shiftreversalScalingreversalshiftScalingreversalshiftScalingExamplef(t)f(1-3t)shiftreversalScalingreversalsh311.2.2 Periodic SignalsA periodic signal x(t)(or xn)has the property that there is a positive value of T(or integer N)for which:x(t)=x(t+T),for all txn=xn+N,for all nIf a signal is not periodic,it is called aperiodic signal.1.2.2 Periodic SignalsA perio32Examples of periodic signalsCT:x(t)=x(t+T)DT:xn=xn+NExamples of periodic signalsCT33Periodic SignalsThe fundamental period T0(N0)of x(t)(xn)is the smallest positive value of T(or N)for which the equation holds.Note:x(t)=C is a periodic signal,but its fundamental period is undefined.Periodic SignalsThe fundament34Examples of periodic signals1.It is periodic signal.Its period is T=16/3.2.It is not periodic.3.x(t)is periodic.Its period isThe smallest multiples of T1 and T2 in commonExamples of periodic signals1.354.It is aperiodic,too.There is no the smallest multiples of T1 and T2 in common5.x(t)is aperiodic.6.It is periodic with period N=16.CostCos2tcost+cos2t4.It is aperiodic,too.There361.2.3 Even and Odd SignalsNote:An odd signal must necessarily be 0 at t=0,or n=0.ie.x(t)=0,or xn=0.Even signal:x(-t)=x(t)or x-n=xn Odd signal:x(-t)=-x(t)or x-n=-xn1.2.3 Even and Odd SignalsNote37Even-Odd Decomposition Any signal can be expressed as a sum of Even and Odd signals.x(t)=xeven(t)+xodd(t)xn=xevenn+xoddnEven-Odd Decomposition Any38Example of the even-odd decompositon Example of the even-odd decomp39Example of the even-odd decompositon Homework：P57-1.9 1.10 1.21(a)(b)(c)(d)1.22(a)(b)(c)(g)1.23 1.24Example of the even-odd decomp401.3 Exponential and Sinusoidal Signals1.3.1 Continuous-time Complex Exponential and Sinusoidal SignalsThe continuous-time complex exponential signal is of the formwhere C and a are,in general,complex numbers.1.3 Exponential and Sinusoidal41A.Real Exponential Signalsx(t)=Ceat (C,a are real value)a0a0growingdecayingA.Real Exponential Signalsx(t42B.Periodic Complex Exponential and Sinusoidal Signalsx(t)=ej 0tx(t)is periodic for x(t)=x(t+T),and its fundamental period is .x(t)=Ceat,C=1,a=j 0(purely imaginary)(1)For e j 0tif 0=0,x(t)=1,then it is periodic for any T0.B.Periodic Complex Exponentia43(2)x(t)=Acos(0t+)0rad/s f0Hz(2)x(t)=Acos(0t+)0rad/s 44Eulers Relation:e j 0t =cos 0t+j sin 0t and cos 0t=(e j 0t+e-j 0t)/2 sin 0t=(e j 0t -e-j 0t)/2j We haveif c is a complex number,Rec denotes its real part;Imc denotes the imaginary part.Eulers Relation:e j0t =45 1 2 3 2 1 3y=cos2ty=cos5ty=cos10t1 2 0r0r10 1-1 0 1051B.Sinusoidal Signals Complex exponential:xn=e j 0n =cos 0n+jsin 0n Sinusoidal signal:xn=cos(0n+)B.Sinusoidal Signals Complex 52C.General Complex Exponential SignalsIf let C and in polar formviz.C=|C|ej And =|ej 0,then xn=C n =|C|ncos(0n+)+j|C|nsin(0n+)C.General Complex Exponential53Real or Imaginary of Signal|1growingdecayingReal or Imaginary of Signal|541.3.3 Periodicity Properties of Discrete-time Complex ExponentialsContinuous-time:ej 0t ,T=2/0Discrete-time:ej 0n,N=?Two properties of continuous-time signal ej 0t:(1)ej 0t is periodic for any value of 0.(2)the lager the magnitude of 0,the higher is the rate of oscillation in the signal.1.3.3 Periodicity Properties o55Periodicity Properties Calculate period:By definition:e j 0n=e j 0(n+N)thus e j 0N=1 or 0N=2 m So N=m(2/0)Condition of periodicity:0/2 is rationalFundamental periodPeriodicity Properties Calcula56奥本海姆版信号与系统课件57From these figures,we can conlude:nSignals oscillate rapidly when 0=,3,(high-frequency);nsignals oscillate slowly when 0=0,2,4,(low-frequency)on the most occasions we will use the intervalFrom these figures,we can con58Harmonically related complex exponentialsNote:Comparison of the signals e j 0t and e j 0n,see P28 Table 1.1So,Only N distinct periodic exponentials in the setForHarmonically related complex e59Examples:Determine the following equations fundamental period:(1)T=31/4N=31(2)It is not period.(3)N1=3,N2=8N=N1N2=24The smallest multiple of N1 and N2 in commonHomework:P61-1.26 *1.25(d)(e)(f)Examples:Determine the followi601.4 The Unit Impulse and Unit Step Functions1.4.1 The Discrete-time Unit Impulse and Unit Step Sequences(1)Unit Sample(Impulse):1.4 The Unit Impulse and Unit 61(2)Unit Step Function:(2)Unit Step Function:62(3)Relation Between Unit Sample and Unit Steporrunning sumfirst difference(3)Relation Between Unit Samp63(4)Sampling Property of Unit Sample(4)Sampling Property of Unit 64Illustration of SamplingIllustration of Sampling651.4.2 The Continuous-time Unit Step and Unit Impulse Functions(1)Unit Step Function:1.4.2 The Continuous-time Unit66(2)Unit Impulse Function:But,u(t)is discontinuous at t=0 and consequently is formally not differentiable.So,how can we get?first derivativerunning integralAnalogous to the relationship between un and(2)Unit Impulse Function:But,67Considering:then:When:That means,has no duration but unit area.Considering:then:When:That mea68We can get:We can get:69(3)Relation Between Unit Impulse and Unit Steprunning integralfirst derivative(3)Relation Between Unit Impu70(4)Sampling Property of (t)(4)Sampling Property of(t)71(5)The transformation of (t)Proof:So,Let (5)The transformation of(t)72Example 02Example 0273Signal representation using step functions Example Signal representation using st74x(t)tt0f(t)Signal representation using step functions x(t)tt0f(t)Signal representati75Example 1-1 1 x(t)t(-1)2-2 1 x1(t)t(-2)(1)1-1 1 x(t)t 2-2 1 x1(t)t 2-2 2 x2(t)t 2 2 x2(t)t1Example 1-1 1 x(t)t(-1)2 76Homework:P57 1.6 1.22(e)(f)1.2.3.2-2 1 g(t)tsketch4.sketchHomework:P57 1.6 1.22(e)(f)771.5 Continuous-time and Discrete-time System(1)A continuous-time system is a system in which continuous-time input signals are applied and result in continuous-time output signals.Continuous-time system x(t)y(t)1.5 Continuous-time and Di78(2)A discrete-time systemthat is,a system that transforms discrete-time inputs into discrete-time outputs.Discrete-time system xnyn(2)A discrete-time systemtha791.5.1 Simple Example of systems1.Example 1.8(p39)RC Circuit(system)vs(t)vc(t)From Ohms lawandWe can get1.5.1 Simple Example of system802.Example 1.10(p40)Balance in a bank account from month to month:balance -yn（余额）余额）net deposit -xn(净存款）净存款）interest -1%so yn=yn-1+1%yn-1+xn or yn-1.01yn-1=xnBalance in bank(system)xnyn2.Example 1.10(p40)Balan81Conclusion:The mathematical descriptions of systems as the preceding examples are the first-order linear differential or difference equation of forms:Conclusion:821.5.2 Interconnections of SystemMany real system are built as interconnections of several subsystems.(1)Series(cascade)interconnection1.5.2 Interconnections of Syst83(2)Parallel interconnection(3)Series-Parallel interconnection(2)Parallel interconnection(84(4)Feed-back interconnection(4)Feed-back interconnection851.6 Basic System Properties1.6.1 Systems with and without MemoryMemoryless system:Its output for each value of the independent variable at a given time is dependent only on the input at the same time.Features:No capacitor,no conductor,no delayer.1.6 Basic System Properties1.686Examples:withmemorywithoutmemoryidentity systemExamples:withwithoutidentity s871.6.2 Invertibility and Inverse SystemsNote:(1)If system is invertible,then an inverse system exists.(2)An inverse system cascaded with the original system,yields an output equal to the input.Invertible systemdistinct inputs lead to distinct outputs.1.6.2 Invertibility and Invers88Examples of invertible systemsExamples of noninvertible systemsExamples of invertible systems891.6.3 CausalityA system is causal If the output at any time depends only on values of the input at the present time and in the past.(nonanticipative 不超前不超前)Note：For causal system,if x(t)=0 for tt0,there must be y(t)=0 for tt0.Memoryless systems are causal.1.6.3 CausalityA system is ca90causalnoncausalExamples of causal systemscausalnoncausalExamples of cau911.6.4 StabilityThe stable systemSmall inputs lead to responses that don not diverge.Bounded input lead to Bounded output(BIBO）if|x(t)|M,then|y(t)|0Solution:Then Which is the zero-input response of the systemExample:We want to know y(t),103xiexie!xiexie!谢谢！谢谢！xiexie!谢谢！104xiexie!xiexie!谢谢！谢谢！xiexie!谢谢！105