奥本海姆信号与系统第二版英文课件第一章.ppt

1 Signal and System1.1 Continuous-time and discrete-time signals1.1.1 Examples and Mathematical RepresentationA.Examples(1)A simple RC circuitSource voltage Vs and Capacitor voltage Vc1.Signals and Systems 1 Signal and System(2)An automobileForce f from engineRetarding frictional force VVelocity V 1 Signal and System(3)A Speech Signal 1 Signal and System(4)A Picture 1 Signal and System(5)Vertical Wind Profile 1 Signal and SystemB.Types of Signals(1)Continuous-time Signal 1 Signal and System(2)Discrete-time Signal 1 Signal and SystemC.Representation(1)Function Representation Example:x(t)=cos0t x(t)=ej 0t(2)Graphical Representation Example:(See page before)1 Signal and System1.1.2 Signal Energy and PowerA.Energy(Continuous-time)Instantaneous power:Let R=1,so p(t)=i2(t)=v2(t)=x2(t)1 Signal and SystemEnergy over t1 t t2:Total Energy:Average Power:1 Signal and SystemB.Energy(Discrete-time)Instantaneous power:Energy over n1 n n2:Total Energy:Average Power:1 Signal and SystemC.Finite Energy and Finite Power SignalFinite Energy Signal:Finite Power Signal:(P 0)(E )1 Signal and System1.2 Transformations of the Independent Variable1.2.1 Examples of TransformationsA.Time ShiftRight shift :x(t-t0)xn-n0 (Delay)Left shift :x(t+t0)xn+n0 (Advance)1 Signal and SystemExamples 1 Signal and SystemB.Time Reversalx(-t)or x-n:Reflection of x(t)or xn 1 Signal and SystemC.Time Scalingx(at)(a0 )Stretch if a0Example 1.1 1 Signal and System1.2.2 Periodic SignalsDefinition:There is a posotive value of T which:x(t)=x(t+T),for all t x(t)is periodic with period T.T Fundamental Period For Discrete-time period signal:xn=xn+N for all n N Fundamental Period 1 Signal and SystemExamples of periodic signal 1 Signal and System1.2.3 Even and Odd Signals Even signal:x(-t)=x(t)or x-n=xn Odd signal:x(-t)=-x(t)or x-n=-xnEven-Odd Decomposition:or:1 Signal and SystemExamples 1 Signal and System1.3 Exponential and Sinusoidal signal1.3.1 Continuous-time Complex Exponential and Sinusoidal SignalsA.Real Exponential Signals x(t)=C eat (C,a are real value)1 Signal and SystemB.Periodic Complex Exponential and Sinusoidal Signals (1)x(t)=e j0t (2)x(t)=Acos(0t+)(3)x(t)=e jk0t All x(t)satisfy for x(t)=x(t+T),and T=2/0 So x(t)is periodic.1 Signal and SystemEulers Relation:e j0t =cos0t+sin 0t and cos0t=(e j0t+e-j0t)/2 sin0t =(e j0t -e-j0t)/2 We also have 1 Signal and SystemC.General Complex Exponential Signals x(t)=C e jat,in which C=|C|ej,a=r+j 0 So x(t)=|C|ej eat ej0t =|C|eat ej(0t+)=|C|eat cos(0t+)+j|C|eat sin(0t+)1 Signal and SystemSignal waves 1 Signal and System1.3.2 Discrete-time Complex Exponential and Sinusoidal SignalsComplex Exponential Signal(sequence):xn=C n or xn=C en 1 Signal and SystemA.Real Exponential Signal Real Exponential Signal xn=C n (a)1 (b)01 (c)-10 (d)-1 1 Signal and SystemB.Sinusoidal Signals Complex exponential:xn=e j0n =cos 0n+jsin0n Sinusoidal signal:xn=cos(0n+)1 Signal and SystemC.General Complex Exponential Signals Complex Exponential Signal:xn=C n in which C=|C|ej,=|ej0 (polar form)then xn=|C|ncos(0n+)+j|C|nsin(0n+)1 Signal and SystemReal or Imaginary of Signal 1 Signal and System1.3.3 Periodicity Properties of Discrete-time Complex ExponentialsContinuous-time:e j0t ,T=2/0Discrete-time:e j0n,N=?Calculate period:By definition:e j0n=e j0(n+N)thus e j0N=1 or 0N=2 m So N=2m/0 Condition of periodicity:2/0 is rational 1 Signal and SystemPeriodicity Properties 1 Signal and System1.4 The Unit Impulse and Unit Step Functions1.4.1 The Discrete-time Unit Impulse and Unit Step Sequences(1)Unit Sample(Impulse):1 Signal and SystemUnit Step Function:(2)Relation Between Unit Sample and Unit Stepor 1 Signal and System(3)Sampling Property of Unit Sample 1 Signal and SystemIllustration of Sampling 1 Signal and System1.4.2 The Continuous-time Unit Step and Unit Impulse Functions(1)Unit Step Function:1 Signal and SystemUnit Impulse Function:1 Signal and System(2)Relation Between Unit Impulse and Unit Step 1 Signal and System(3)Sampling Property of(t)Example 1.7 1 Signal and System1.5 Continuous-time and Discrete-time SystemDefinition:(1)Interconnection of Component,device,subsystem.(Broadest sense)(2)A process in which signals can be transformed.(Narrow sense)Representation of System:(1)Relation by the notation 1 Signal and System(2)Pictorial Representation Continuous-time system x(t)y(t)Discrete-time system xnyn 1 Signal and System1.5.1 Simple Example of systemsExample 1.8:RC Circuit in Figure 1.1:Vc(t)Vs(t)RC Circuit(system)vs(t)vc(t)1 Signal and SystemExample 1.10:Balance in a bank account from month to month:balance -yn net deposit -xn interest -1%so yn=yn-1+1%yn-1+xn or yn-1.01yn-1=xnBalance in bank(system)xnyn 1 Signal and System1.5.2 Interconnections of System(1)Series(cascade)interconnection 1 Signal and System(2)Parallel interconnection Series-Parallel interconnection 1 Signal and System(3)Feed-back interconnection 1 Signal and SystemExample of Feed-back interconnection 1 Signal and System1.6 Basic System Properties1.6.1 Systems with and without MemoryMemoryless system:Its output is dependent only on the input at the same time.Features:No capacitor,no conductor,no delayer.Examples of memoryless system:y(t)=C x(t)or yn=C xnExamples of memory system:or yn-0.5yn-1=2xn 1 Signal and System1.6.1 Invertibility and Inverse SystemsDefinition:(1)If system is invertibility,then an inverse system exists.(2)An inverse system cascaded with the original system,yields an output equal to the input.1 Signal and System 1 Signal and System1.6.3 CausalityDefinition:A system is causal If the output at any time depends only on values of the input at the present time and in the past.For causal system,if x(t)=0 for tt0,there must be y(t)=0 for tt0.(nonanticipative)Memoryless systems are causal.1 Signal and Systemx(t)y(t)t1t2 1 Signal and System1.6.4 StabilityDefinition:Small inputs lead to responses that don not diverge.Finite input lead to finite output:if|x(t)|M,then|y(t)|N.Examples:Stable pendulum Motion of automobile 1 Signal and SystemExample 1.13 1 Signal and System1.6.5 Time InvarianceDefinition:Characteristics of the system are fixed over time.Time invariant system:If x(t)y(t),then x(t-t0)y(t-t0).Example 1.14 1.15 1.16 1 Signal and Systemx(t)y(t)x(t-t0)y(t-t0)1 Signal and System1.6.6 LinearityDefinition:The system possesses the important property of superposition:(1)Additivity property:The response to x1(t)+x2(t)is y1(t)+y2(t).(2)Scaling or homogeneity property:The response to ax1(t)is ay1(t).(where a is any complex constant,a0.)1 Signal and SystemLx1(t)x2(t)y1(t)y2(t)a x1(t)x1(t)+x2(t)ax1(t)+bx2(t)a y1(t)y1(t)+y2(t)ay1(t)+by2(t)Represented in block-diagram:Example 1.17 1.18 1 Signal and SystemLTI SystemLTIx(t)y(t)x(t-t0)ax(t)+bx(t-t0)y(t-t0)ay(t)+by(t-t0)Linear and Time-invariant systemProblems:1.14 1.15 1.16 1.17 1.23 1.24 1.31