信号与系统课件(英文版)chapter1

1、Signals and SystemsAlan V.OppenheimAlan S.WillskyS.Hamid NawabTeacher: 湛柏明IntroductionStatus:Signals and Systems is a very important fundamental course, the fundamental theories, concepts and methods established in this course are the foundation of our specialty.IntroductionObjective:Discuss and stu

2、dy the fundamental theories and methods for which the deterministic signals pass through the LTI systems.PreviewChapter 1: Signals and SystemsConstruct the basic concepts about signals and systemsChapter 2: LTI SystemsIntroduce the theories and methods for the time-domain analysis of LTI systems.The

3、 emphasis is the convolution method.Systemx(t)y(t)PreviewChapter 3 and 4: Continuous-Time Fourier Series and TransformAn important tool for signal analysisChapter 6: Time and Frequency Characterization of Signals and SystemsChapter 7: SamplingSampling is a bridge connecting the analog signals and th

4、e digital signalsPreviewChapter 8: Communication SystemsIntroduce the applications of the Fourier TransformChapter 9: The Laplace TransformThe Laplace transform is also an important tool in signal and system analysis.GobackChapter 1Signals and SystemsContents1.1 Continuous-time and discrete-time sig

5、nals1. What is a signal?2. What is a system?3. Continuous-time and discrete-time signals4. How to represent a signal?5. Examples of signals6. SummaryContents1.2 Transformation of the Independent Variable1. Time shifting2. Time reversal3. Time scaling4. Examples5. Periodic signals6. Even and odd sign

6、alsContents1.3, 1.4 Some basic building blocks of signals1. Sinusoidal Signals2. Exponential Signals 3. Unit Impulse and Unit Step FunctionsContents1.5 Continuous-time and discrete-time systems1. Continuous-time and discrete-time systems2. Examples of systems3. Interconnections of systemsContents1.6

7、 The system properties1. Systems with and without Memory2. Invertibility and Inverse Systems3. Causality4. Stability5. Time Invariance6. LinearityExamplesSummaryWhat is a signal?The signals , which are functions of one or more independent variables, contain information about the behavior or nature o

8、f some physical phenomena. The current and voltage in a circuit, the speech signal, etc.GobackWhat is a system?A system is an interconnection of some units, devices and subsystems accomplishing a certain function.The communication systems, the control systems, etc.GobackSystemInput signalOutput sign

9、alTransmit or processx(t)y(t)1.1 Continuous-time andDiscrete-time SignalsDefinitionA signal x(t) is said to be a continuous-time signal or an analog signal if the independent variable is continuous. Otherwise, if the time variable is discrete, the signal is said to be discrete.A continuous-time sign

10、alGobackA discrete-time signalPeriodic Signals where k is integer. T is called the period.1.2.2 Periodic signalsA continuous-time signal is said to be periodic if it satisfiesPeriodic Signals where k and N are integers. N is called the period.A discrete-time signal is said to be periodic if it satis

11、fiesPeriodic SignalsThe smallest period of a periodic signal is called the fundamental period.The reciprocal of the fundamental period is called the fundamental frequency. Fundamental Period and Fundamental FrequencyGobackEven and Odd Signals1.2.3 Even and Odd signalsA real signal can always be expr

12、essed as a sum of two parts: even and odd parts.Goback1.3 Exponential and Sinusoidal SignalsSinusoidal SignalsA : magnitude: phase measured in radians0: angular frequency measured in radians/sGobackBased on the parameters C and a, the signal exhibits different characteristics.1.3 Exponential and Sin

13、usoidal SignalsContinuous-time exponential signalPeriodic complex exponential signalReal exponential signal Harmonic relationExamplesGeneral complex exponential signalLike the continuous-time exponential signal, also based on the parameters C and , the signal exhibits different characteristics.1.3 E

14、xponential and Sinusoidal SignalsA discrete-time exponential signal has the general form:1.3 Exponential and Sinusoidal SignalsA discrete-time exponential signal has the general form:General complex exponential signalReal exponential signalPeriodicitySinusoidal signalGobackharmonic relationExample1.

15、4 The Unit Impulse and Unit Step FunctionThe discrete-time unit impulse and unit step functions1. Definition2. The relationship between the unit impulse and unit step functions3. Sampling property of the impulse1.4 The Unit Impulse and Unit Step FunctionThe continuous-time unit impulse and unit step

16、 functions1. Definition2. The relationship between the unit impulse and unit step functions3. Sampling property of the impulseSignals expressed in terms of the unit stepGobackSummary In this chapter, we have developed a number of basic concepts related to continuous-time and discrete-time signals an

17、d systems.Signals carry informationSignals can be mathematically expressed as functions of one or more independent variables. In our book, we focus on the one-dimensional signal which involves a single time variable.SummarySome times, we often represent a signal in graph.There are some typical signa

18、ls and their characteristics that we must learn by heart: The unit step and unit impulse, the real and complex exponential signals, and sin/cos signals. Using these signals we can build other complicated signals.SummarySystems are an interconnection of subsystems. The physical meanings of systems ar

19、e very broad.In this chapter, we represent systems using these basic ways:Block diagramsMathematical equationsWe have also discussed some basic properties of systems and the ways that how to verify these properties. SummaryThe emphases are: linearity, time-invariant, causality and stability properti

20、es.The systems that satisfy both linearity and time-invariant properties are called to as the LTI systems.The LTI systems are the primary focus in our book, because a large class of nature systems can be characterized by LTI systems.Home workRead the text book from p1 to p56.9, 14, 15, 20, 21, 31 No

21、te: your exercises must be written in English except your name.Examples of Signals1. Voltages and Currents in a circuit:RC circuitvs(t) and vc(t): voltages of source and capacitor; i(t): current in the circuit;Examples of Signals2. Speech signalExamples of Signals3. The stock market indexGobackRepre

22、sentations of SignalsThere are three ways to represent a signal:1. The mathematical function2. Graphic representationRepresentations of Signals3. For a discrete-time signal, we can represent the signal as a sequence of numbers. xn = , 0, 0.1, 0.23, -1.2, 1, 2, GobackSummary for the concept of signal

23、Signals contain information in general. We can use a mathematic function, a graph or a sequence of numbers to represent a signal.Both the function value and the independent variable of continuous-time signals are continuous.Summary for the concept of signal4. The independent variable of a discrete-t

24、ime signal is discrete.5. For some discrete-time signals, the independent variable is inherently discrete, but other discrete-time signals are generated by sampling continuous-time signals.GobackTime shiftingTime shiftingOriginal signal x(t)Time shifted version of x(t)Time shifted versions of x(t)t0

25、 is positiveGobackTime reversalTime reversalOriginal signal x(t)Time reversed version of x(t)GobackTime scalingTime scalingOriginal signal x(t)Time scaled version of x(t)a is an arbitrary real value.When |a|1, x(t) is compressed to x1(t) When |a|1 , x(t) is expanded to x1(t) , as illustrated in the

26、following figure:Time scalingx(t)x(2t)x(0.5t)GobackExamples of Transformations of the independent variableExample 1.1, 1.2, 1.3 Given a signal x(t), find signal x(-3t+1).Solution: Steps: x(t) Time shifting x(t+1) x(t+1) Time reversal x(-t+1) x(-t+1) Time scaling x(-3t+1)Integration for Transformatio

27、n of the Independent VariableGiven a continuous-time signal x(t), the steps to determine the signal x(at+b) are: Determine x(t+b) Time shiftingDetermine x(-t+b) Time reversal if a0, x(t) is growingFor a0, x(t) is decayingFor a=0, x(t) is constantGobackPeriodic Complex Exponential SignalsPeriodic Com

28、plex Exponential SignalsIf a is a purely imaginary number, Let a=j0 , C=1It is periodicLet T be the periodSo Periodic Complex Exponential SignalsThe fundamental frequency is defined byorThe fundamental periodk=0, 1, 2,Periodic Complex Exponential SignalsEulers RelationorPeriodic Complex Exponential

29、SignalsFurther,Periodic Complex Exponential SignalsFor example, a complex exponential signal given byPeriodic Complex Exponential SignalsAnd it will play a central role in much of our treatment of signals and systems, in part because they serve as extremely useful building blocks for many other sign

30、als.Page19Periodic complex exponential is a basic periodic signal which is important both in theory and engineering.GobackHarmonic RelationHarmonic RelationGiven a periodic complex exponential:For signal if its frequency is an integer multiple of 0, i.e., k= k0, then we say that xk(t) is the k-th ha

31、rmonic of x(t).Harmonic RelationUsing a weighted sum of a set of harmonically related complex exponentials, we can construct many other periodic signals:GobackGeneral Complex Exponential SignalsGeneral Complex Exponential SignalsThe most general case of a complex exponential can be expressed in term

32、s of the two cases we have examined so far: the real exponential and the periodic complex exponential. If C and a are complex, then x(t) is a complex exponential.General Complex Exponential SignalsWe often express C in the polar form and express a in the rectangular form.Polar formRectangular formTh

33、en Real part of x(t)Imaginary part of x(t)r0General Complex Exponential SignalsThese growing and decaying sinusoidal signals are plotted in the following figure:The envelopes areGoback1.3.1 Continuous-time Complex ExponentialsExample 1.5 Given a signalExpress x(t) as a product of a single periodic c

34、omplex exponential and a single sinusoid. The answer isGoback1.3.1 Continuous-time Complex ExponentialsSolution:The signal x(t) can be expressed asFrom the Eulers relation, we can getSo the magnitude of x(t) isGoback1.3.2 Discrete-Time Exponential and Sinusoidal SignalsReal Exponential SignalsIf C a

35、nd are real, the signal xn is called the real exponential signal.| |1, 0|1, 01.3.2 Discrete-Time Exponential and Sinusoidal SignalsFor | 1, the signal xn is decaying;| | 0|1, 1 and |1 , Goback1.3.3 Periodicity of Discrete-time Complex Exponentialsm integerPeriodicityNow considerAssume that its perio

36、d is N, thenConclusion:The signal ej0n is periodic if and only if 2/0 is a rational number. 1.3.3 Periodicity of Discrete-time Complex ExponentialsFor 0=0.6 radians0=0.2 radiansnonperiodicperiodic1.3.3 Periodicity of Discrete-time Complex ExponentialsDetermination of the fundamental period For these

37、 periodic signals:Note: Both N and m are positive integers, and have no factor in common.The fundamental frequency is defined byUse the formula:1.3.3 Periodicity of Discrete-time Complex ExponentialsA very interesting phenomenon ofLet 0 be different values respectively: 1.3.3 Periodicity of Discrete

38、-time Complex ExponentialsWe see that the highest rate of oscillation of signal occurs at 0= . 1.3.3 Periodicity of Discrete-time Complex ExponentialsThis implies that the discrete-time complex exponential signals are always periodic signals of with period 2.Because of the periodicity, the signal ex

39、p(j0n) does not have a continually increasing rate of oscillation as 0 is increased in magnitude.Goback1.3.3 Periodicity of Discrete-time Complex ExponentialsThe harmonic relation Given a periodic complex exponentialFor signalIf xkn has a frequency k which is an integer multiple of 0, i.e., k= k0, t

40、hen we say xkn is the k-th harmonic of xn.1.3.3 Periodicity of Discrete-time Complex ExponentialsDefine a set of harmonically related sequences to k=0, 1, 2,It can be seen there are only N distinct periodic exponentials in the set given in kn.GobackThe fundamental period is1.3.3 Periodicity of Discr

41、ete-time Complex ExponentialsExample-1.6 Determine the fundamental period of the discrete-time signalSolution:1.3.3 Periodicity of Discrete-time Complex ExponentialsGoback1.4 The Unit Impulse and Unit Step Function1.4.1 The Discrete-Time Unit Impulse and Unit Step SequencesThe discrete-time unit imp

42、ulse and step sequences are defined byGoback1.4 The Unit Impulse and Unit Step FunctionRelationship between the impulse and the step functions1.The discrete-time impulse is the first difference of the unit step, that is 1.4 The Unit Impulse and Unit Step Function2. The discrete-time unit step is the

43、 running sum of the unit impulse Intervalof summation Intervalof summationFor n0For n0Interval of summation1.4 The Unit Impulse and Unit Step FunctionorFor n0 Intervalof summationFor n0nnGoback1.4 The Unit Impulse and Unit Step FunctionMore generallySampling Property of the discrete-time unit impuls

44、eGoback1.4 The Unit Impulse and Unit Step Function1.4.2 The Continuous-Time Unit Impulse and Unit Step FunctionsThe continuous-time unit step function is defined byNote: the unit step is discontinuous at t=0. so the value of u(t) at t=0 is undefined.1.4 The Unit Impulse and Unit Step FunctionThe uni

45、t impulse function (t) is defined byNote: The unit impulse function (t) has nonzero at t=0, and has zero for all nonzero values of t. And the area under (t) is 1.Goback1.4 The Unit Impulse and Unit Step Function1. The unit impulse (t) is the first derivative of the unit step function u(t)1.4 The Uni

46、t Impulse and Unit Step Function2. The unit step is the running integral of the unit impulseGoback1.4 The Unit Impulse and Unit Step FunctionSampling propertyandGoback1.4 The Unit Impulse and Unit Step FunctionSignals are often defined interval by interval. For example, suppose that x(t) is given by

47、Signals expressed in terms of unit-stepwhere x1(t), x2(t), x3(t) are arbitrary continuous functions of t. 1.4 The Unit Impulse and Unit Step FunctionSuch signals can be expressed analytically in terms of the unit-step function u(t) and time shifts of u(t). 1.4 The Unit Impulse and Unit Step Function

48、Example 1.7 Consider the discontinuous signal x(t) depicted in the figure.It can be expressed in terms of the unit step:1.4 The Unit Impulse and Unit Step FunctionFrom the expression of x(t) in terms of the unit step, we can readily calculate and graph the derivative of x(t) .GobackThe continuous-ti

49、me and discrete-time systemsContinuous-time systemsSystemInputx(t) Outputy(t)Discrete-time systemsSystemInputxn OutputynGoback1.5.1 Simple Examples of Systems1.5.1 Simple Examples of SystemsExample-1.8 Consider the RC circuit in the following figure. Determine the relationship between vc(t) and vs(t

50、).Answer:1.5.1 Simple Examples of SystemsExample-1.10 Consider a simple model for the balance in a bank account from month to month.Let yn denote the balance at the end of the n-th month, and suppose that yn evolves from month to month according to the difference equationxnThe net deposit1.5.1 Simpl

51、e Examples of SystemsHow to represent a system?From the above examples, we see that we describe a system using a mathematical equation called the mathematical model.For a continuous-time system, the mathematical model is a differential equation, and for a discrete-time system, the mathematical model

52、 is a difference equation.Goback1.5.2 Interconnections of Systems1.5.2 Interconnections of SystemsMany real systems are built as interconnections of several subsystems.There are four types of interconnections: series, parallel, series-parallel and feedback interconnections.1.5.2 Interconnections of

53、SystemsSeries (cascade) interconnectionParallel interconnectionSystem1System2System1System21.5.2 Interconnections of SystemsSeries-parallel interconnectionSystem1System3System2System41.5.2 Interconnections of SystemsFeedback interconnectionSystem1System2Goback1.6.1 Systems with and without Memory1.6

54、.1 Systems with and without MemoryDefinition:A system is said to be memoryless if its output for each value of the independent variable at a given time is dependent only on the input at that same time.Consider the system characterized byIt is a memoryless systemA continuous-time system characterized

55、 by1.6.1 Systems with and without MemoryThis system is a memory system .is memoryless.Consider the system described by1.6.1 Systems with and without MemoryA capacitor is an example of a continuous-time system with memory, sinceGoback1.6.2 Invertibility and Inverse Systems1.6.2 Invertibility and Inve

56、rse SystemsDefinition:A system is said to be invertible if distinct inputs lead to distinct outputs.SystemxnynInverse systemxnIdentity system1.6.2 Invertibility and Inverse SystemsAn example of an invertible continuous-time system is w(t)=y(t)/2w(t)=x(t)y(t)=2x(t)x(t)y(t)The inverse system is constr

57、ucted as:1.6.2 Invertibility and Inverse SystemsGiven a system described byIts inverse system is :w(t)=yn-yn-1wn=xnyn=xnx(t)ynCascade the original system with the inverse system we get the identity system:Goback1.6.3 Causality1.6.3 CausalityA system is said to be causal if the output at any time onl

58、y depends on values of the input at the present time and in the past.Page46Thus in a causal system it is impossible to get an output before an input is applied to the system ( assuming no initial energy ).1.6.3 CausalityThe RC circuit is causal, since the capacitor voltage responds only on the past

59、source voltage.But the systems defined byandare not.Why?Goback1.6.4 Stability1.6.4 StabilityStability is another important system property. Informally, a stable system is one in which small input lead to responses that do not diverge.Stable systemUnstable system1.6.4 StabilityConsider the system giv

60、en byWe haveIt can be seen that yn grows without bound. So the system is unstable.1.6.4 StabilityBIBO definitionA system is said to be stable if a bounded input leads to a bounded output.How to check a system whether is stable or not?Goback1.6.5 Time Invariance1.6.5 Time InvarianceConceptually, a sy