信号与系统课件(英文版)chapter1

Signals and SystemsAlan V.OppenheimAlan S.WillskyS.Hamid NawabTeacher: 湛柏明E-mail: ,Introduction,Status: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.,Introduction,Objective:Discuss and study the fundamental theories and methods for which the deterministic signals pass through the LTI systems.,Preview,Chapter 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 emphasis is the convolution method.,x(t),y(t),Preview,Chapter 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 the digital signals,Preview,Chapter 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.,Goback,Chapter 1Signals and Systems,Contents,1.1 Continuous-time and discrete-time signals1. 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. Summary,Contents,1.2 Transformation of the Independent Variable1. Time shifting2. Time reversal3. Time scaling4. Examples5. Periodic signals6. Even and odd signals,Contents,1.3, 1.4 Some basic building blocks of signals1. Sinusoidal Signals2. Exponential Signals 3. Unit Impulse and Unit Step Functions,Contents,1.5 Continuous-time and discrete-time systems1. Continuous-time and discrete-time systems2. Examples of systems3. Interconnections of systems,Contents,1.6 The system properties,1. Systems with and without Memory,2. Invertibility and Inverse Systems,3. Causality,4. Stability,5. Time Invariance,6. Linearity,Examples,Summary,What is a signal?,The signals , which are functions of one or more independent variables, contain information about the behavior or nature of some physical phenomena. The current and voltage in a circuit, the speech signal, etc.,Goback,What 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.,Goback,Transmit or process,x(t),y(t),1.1 Continuous-time andDiscrete-time Signals,DefinitionA 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 signal,Goback,A discrete-time signal,Periodic 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 satisfies,Periodic Signals,where k and N are integers. N is called the period.,A discrete-time signal is said to be periodic if it satisfies,Periodic Signals,The 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 Frequency,Goback,Even and Odd Signals,1.2.3 Even and Odd signalsA real signal can always be expressed as a sum of two parts: even and odd parts.,Goback,1.3 Exponential and Sinusoidal Signals,Sinusoidal Signals,A : magnitude: phase measured in radians0: angular frequency measured in radians/s,Goback,Based on the parameters C and a, the signal exhibits different characteristics.,1.3 Exponential and Sinusoidal Signals,Continuous-time exponential signal,Periodic complex exponential signal,Real exponential signal,Harmonic relation,Examples,General complex exponential signal,Like the continuous-time exponential signal, also based on the parameters C and , the signal exhibits different characteristics.,1.3 Exponential and Sinusoidal Signals,A discrete-time exponential signal has the general form:,1.3 Exponential and Sinusoidal Signals,A discrete-time exponential signal has the general form:,General complex exponential signal,Real exponential signal,Periodicity,Sinusoidal signal,Goback,harmonic relation,Example,1.4 The Unit Impulse and Unit Step Function,The discrete-time unit impulse and unit step functions1. Definition2. The relationship between the unit impulse and unit step functions3. Sampling property of the impulse,1.4 The Unit Impulse and Unit Step Function,The continuous-time unit impulse and unit step functions1. Definition2. The relationship between the unit impulse and unit step functions3. Sampling property of the impulseSignals expressed in terms of the unit step,Goback,Summary,In this chapter, we have developed a number of basic concepts related to continuous-time and discrete-time signals and 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.,Summary,Some times, we often represent a signal in graph.There are some typical signals 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.,Summary,Systems are an interconnection of subsystems. The physical meanings of systems are 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.,Summary,The emphases are: linearity, time-invariant, causality and stability properties.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 work,Read the text book from p1 to p56.9, 14, 15, 20, 21, 31 Note: your exercises must be written in English except your name.,Examples of Signals,1. Voltages and Currents in a circuit:,RC circuit,vs(t) and vc(t): voltages of source and capacitor; i(t): current in the circuit;,Examples of Signals,2. Speech signal,Examples of Signals,3. The stock market index,Goback,Representations of Signals,There are three ways to represent a signal:1. The mathematical function,2. Graphic representation,Representations of Signals,3. 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, ,Goback,Summary for the concept of signal,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 signal,4. The independent variable of a discrete-time 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.,Goback,Time shifting,Time shiftingOriginal signal x(t)Time shifted version of x(t),Time shifted versions of x(t),t0 is positive,Goback,Time reversal,Time reversalOriginal signal x(t)Time reversed version of x(t),Goback,Time scaling,Time 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 following figure:,Time scaling,Goback,Examples of Transformations of the independent variable,Example 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 Transformation of the Independent Variable,Given 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 growing,For a0, x(t) is decaying,For a=0, x(t) is constant,Goback,Periodic Complex Exponential Signals,Periodic Complex Exponential SignalsIf a is a purely imaginary number,Let a=j0 , C=1,It is periodic,Let T be the period,So,Periodic Complex Exponential Signals,The fundamental frequency is defined by,or,The fundamental period,k=0, 1, 2,Periodic Complex Exponential Signals,Eulers Relation,or,Periodic Complex Exponential Signals,Further,Periodic Complex Exponential Signals,For example, a complex exponential signal given by,Periodic Complex Exponential Signals,And 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 signals.-Page19,Periodic complex exponential is a basic periodic signal which is important both in theory and engineering.,Goback,Harmonic Relation,Harmonic 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 harmonic of x(t).,Harmonic Relation,Using a weighted sum of a set of harmonically related complex exponentials, we can construct many other periodic signals:,Goback,General Complex Exponential Signals,General Complex Exponential SignalsThe most general case of a complex exponential can be expressed in terms 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 Signals,We often express C in the polar form and express a in the rectangular form.,Polar form,Rectangular form,Then,Real part of x(t),Imaginary part of x(t),r0,General Complex Exponential Signals,These growing and decaying sinusoidal signals are plotted in the following figure:,The envelopes are,Goback,1.3.1 Continuous-time Complex Exponentials,Example 1.5 Given a signal,Express x(t) as a product of a single periodic complex exponential and a single sinusoid. The answer is,Goback,1.3.1 Continuous-time Complex Exponentials,Solution:,The signal x(t) can be expressed as,From the Eulers relation, we can get,So the magnitude of x(t) is,Goback,1.3.2 Discrete-Time Exponential and Sinusoidal Signals,Real Exponential SignalsIf C and are real, the signal xn is called the real exponential signal.,| |1, 0,|1, 0,1.3.2 Discrete-Time Exponential and Sinusoidal Signals,For | 1, the signal xn is decaying;,| | 0,|1, 1 and |1 ,Goback,1.3.3 Periodicity of Discrete-time Complex Exponentials,m integer,PeriodicityNow consider,Assume that its period is N, then,Conclusion:The signal ej0n is periodic if and only if 2/0 is a rational number.,1.3.3 Periodicity of Discrete-time Complex Exponentials,For,0=0.6 radians,0=0.2 radians,nonperiodic,periodic,1.3.3 Periodicity of Discrete-time Complex Exponentials,Determination of the fundamental period For these periodic signals:,Note: Both N and m are positive integers, and have no factor in common.The fundamental frequency is defined by,Use the formula:,1.3.3 Periodicity of Discrete-time Complex Exponentials,A very interesting phenomenon of,Let 0 be different values respectively:,1.3.3 Periodicity of Discrete-time Complex Exponentials,We see that the highest rate of oscillation of signal occurs at 0= .,1.3.3 Periodicity of Discrete-time Complex Exponentials,This implies that the discrete-time complex exponential signals are always periodic signals of with period 2.Because of the periodicity, the signal exp(j0n) does not have a continually increasing rate of oscillation as 0 is increased in magnitude.,Goback,1.3.3 Periodicity of Discrete-time Complex Exponentials,The harmonic relation Given a periodic complex exponential,For signal,If xkn has a frequency k which is an integer multiple of 0, i.e., k= k0, then we say xkn is the k-th harmonic of xn.,1.3.3 Periodicity of Discrete-time Complex Exponentials,Define 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.,Goback,The fundamental period is,1.3.3 Periodicity of Discrete-time Complex Exponentials,Example-1.6 Determine the fundamental period of the discrete-time signal,Solution:,1.3.3 Periodicity of Discrete-time Complex Exponentials,Goback,1.4 The Unit Impulse and Unit Step Function,1.4.1 The Discrete-Time Unit Impulse and Unit Step SequencesThe discrete-time unit impulse and step sequences are defined by,Goback,1.4 The Unit Impulse and Unit Step Function,Relationship between the impulse and the step functions,1.The discrete-time impulse is the first difference of the unit step, that is,1.4 The Unit Impulse and Unit Step Function,2. The discrete-time unit step is the running sum of the unit impulse,For n0,For n0,1.4 The Unit Impulse and Unit Step Function,or,For n0,For n0,Goback,1.4 The Unit Impulse and Unit Step Function,More generally,Sampling Property of the discrete-time unit impulse,Goback,1.4 The Unit Impulse and Unit Step Function,1.4.2 The Continuous-Time Unit Impulse and Unit Step FunctionsThe continuous-time unit step function is defined by,Note: 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 Function,The unit impulse function (t) is defined by,Note: 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.,Goback,1.4 The Unit Impulse and Unit Step Function,1. The unit impulse (t) is the first derivative of the unit step function u(t),1.4 The Unit Impulse and Unit Step Function,2. The unit step is the running integral of the unit impulse,Goback,1.4 The Unit Impulse and Unit Step Function,Sampling property,and,Goback,1.4 The Unit Impulse and Unit Step Function,Signals are often defined interval by interval. For example, suppose that x(t) is given by,Signals expressed in terms of unit-step,where x1(t), x2(t), x3(t) are arbitrary continuous functions of t.,1.4 The Unit Impulse and Unit Step Function,Such 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,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 Function,From the expression of x(t) in terms of the unit step, we can readily calculate and graph the derivative of x(t) .,Goback,The continuous-time and discrete-time systems,Continuous-time systems,Discrete-time systems,Goback,1.5.1 Simple Examples of Systems,1.5.1 Simple Examples of SystemsExample-1.8 Consider the RC circuit in the following figure. Determine the relationship between vc(t) and vs(t).,Answer:,1.5.1 Simple Examples of Systems,Example-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 equation,xn-The net deposit,1.5.1 Simple Examples of Systems,How 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 is a difference equation.,Goback,1.5.2 Interconnections of Systems,1.5.2 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 Systems,Series (cascade) interconnection,Parallel interconnection,1.5.2 Interconnections of Systems,Series-parallel interconnection,1.5.2 Interconnections of Systems,Feedback interconnection,Goback,1.6.1 Systems with and without Memory,1.6.1 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 by,It is a memoryless system,A continuous-time system characterized by,1.6.1 Systems with and without Memory,This system is a memory system .,is memoryless.,Consider the system described by,1.6.1 Systems with and without Memory,A capacitor is an example of a continuous-time system with memory, since,Goback,1.6.2 Invertibility and Inverse Systems,1.6.2 Invertibility and Inverse SystemsDefinition:A system is said to be invertible if distinct inputs lead to distinct outputs.,xn,1.6.2 Invertibility and Inverse Systems,An example of an invertible continuous-time system is,The inverse system is constructed as:,1.6.2 Invertibility and Inverse Systems,Given a system described by,Its inverse system is :,Cascade the original system with the inverse system we get the identity system:,Goback,1.6.3 Causality,1.6.3 CausalityA system is said to be causal