信号与系统英文版课件：Chap9 Laplace Transform

1、Laplace TransformChapter 9 Lecture 1Signals and Systems Spring 2015Homework 9.2 9.5 9.7 9.8 9.9 9.10 9.13 9.21(a)(b) 9.22(a)(b) 9.28 9.31 9.32 9.33 9.35 9.37(a) 9.45 9.60OutlineDefinition of Laplace transformRegion of convergence for Laplace transformProperties of the Laplace transformAnalysis of CT

2、 LTI systems using the L transfBlock diagram of CT LTI systemsDescription of A LTI Systems : a complex constantH(s): a constant determined by s and h(t)h(t)For general values of the complex variable s, H(s) is referred to as the Laplace transform of the impulse response h(t).When s=j, H(s) becomes t

3、he Fourier transform.Description of A LTI SystemFrom Fourier Transf to Laplace TransfFourier transform of a signal f(t) exists if Dirichlet conditions is satisfied. Dirichlet condition requires the signal f(t) to be absolutely integrable, Absolutely integrable is a stringent constraint that cannot b

4、e satisfied by many signals of interest. For example . Laplace TransformFor arbitrary CT signal f(t), define a new signal via properly choosing , such that the new signal g(t) is absolutely integrable, which ensures the existence of the FT of g(t). whereCan we derive f(t) from its Laplace tran F(s)?

5、let s = + jLaplace transformInverse Laplace TransformSince , we have let s = + jds = jdInverse Laplace transformLaplace TransformLaplace Transform (LT)Inverse Laplace Transform Often use notationLaplace TransformIntuitively, f(t) is decomposed into a sum of weighted complex exponentials est, the mag

6、nitudes of which are Note that must be chosen to meet certain conditions to guarantee that g(t) = f(t)e-t is absolutely integrable.Example 1Compute the Laplace transform of .Solution: s = + jExample 2Compute the Laplace transform of where is a complex number.Solution: s = + jExample 3Compute the Lap

7、lace transform of where is a complex number.Solution: s = + jTwo distinct signals lead to identical algebraic expression for the Laplace transforms. The set of values of s for which the expression is valid is very different in the two examples. In specifying the Laplace transform of a signal, both t

8、he algebraic expression and the range of values of s for which this expression is valid are required.Ex 2Ex 3Region of ConvergenceIn general, the range of values of s for which the integral involved in the Laplace transform converges is referred to as the region of convergence (ROC) of the Laplace t

9、ransform.Since s=+j, the LT of f(t) is the FT of f(t)e-t. Thus the ROC consists of the values of s for which the Fourier transform of f(t)e-t converges.Region of ConvergenceSince the variable s is a complex number, we display the ROC in the complex plane, generally called the s-plane, associated wit

10、h this complex variable. The coordinate axes are Res along the horizontal axis and Ims along the vertical axis. The horizontal and vertical axes are sometimes referred to as the -axis and j-axis, respectively. ExampleZeros and PolesIn general, the Laplace transform F(s) is a rational function and ca

11、n be expressed in the form of The roots of equation are called the zeros of F(s), denoted by z1, z2, , zM.The roots of equation are called the poles of F(s), denoted by p1, p2, , pN.pole-zero plotZeros and PolesEx: has one pole ( p1=a ) and no zero.Ex: has two poles ( p1=1, p2=3 ) and one zero z1=4.

12、The Properties of ROCProperty 1: The ROC of F(s) consists of strips parallel to the j-axis in the s-plane. The ROC of F(s) consists of the values of s=+j for which the Fourier transform of f(t)e-t converges, i.e. the ROC should make f(t)e-t absolutely integrableROC determined by , the real part of s

13、=+j.ExampleDetermine the Laplace transform of Solution: Zeros: Poles: Pole-zero plot & ROC:Property 2: For rational Laplace transforms, the ROC does not contain any poles. Since F(s) equals infinity at a pole, the intergral in does not converge, and hence the ROC cannot contain any poles.Property 3:

14、 If f (t) is of finite duration and absolutely integrable, then the ROC is the entire s-plane. When f (t) has finite duration and is absolutely integrable, sayEx: forthe integral in must be well defined. ROC the entire s-plane Property 4: If the Laplace transform F(s) of f (t) is rational, then if f

15、 (t) is right-sided, the ROC is the region in the s-plane to the right of the rightmost pole. Considers-planeProperty 5: If the Laplace transform F(s) of f (t) is rational, then if f (t) is left-sided, the ROC is the region in the s-plane to the left of the leftmost pole. Considers-planeProperty 6:

16、If the Laplace transform F(s) of f (t) is rational, then if f (t) is two-sided, the ROC is either a strip parallel to the j-axis bounded by two poles or an empty set . s-planeExampleCompute the Laplace transform of Solution: s-planeLaplace TransformChapter 9 Lecture 2Signals and Systems Spring Examp

17、leThe Lapalce transform of f(t) is Determine f(t).Solution: Poles of F(s) are Possible ROC regions:(1) If the ROC is region, i.e. (2) If the ROC is region, i.e. (3) If the ROC is region , i.e.Laplace Transf and Fourier TransfFT:LT:If is in the ROC of , i.e. the ROC contains the j-axis, then Otherwis

18、e, FT dose not exist.ExampleThe Laplace transform of f(t) is If its FT exists, determine f(t).Solution: The poles are Since its FT exists, the j-axis must be in the ROC, so the ROC should be The Properties of Laplace TransformLinearity Time shiftingShifting in S-domainTime scalingConjugationConvolut

19、ionDifferentiation and integration in time-domainDifferentiation in S-domainThe Properties of Laplace TransformThroughout this part, we assumeLinearityEx: linearitythe pole p=-2 disappearsExampleCompute the LT of Solution:linearityThe LT of does not exist.Further, it can be shown that the Laplace tr

20、ansforms of and do not exist.Time ShiftingProof:ExampleDetermine the LT of Solution:linearityShifting in S-DomainProof:Exampleshifting in S-domainlinearityshifting in S-domainTime ScalingProof:Assume the ROC of f(t) is Time ScalingSpecifically, letting givesExampleIf determine the LT ofSolution: shi

21、fting in S-domainscaling in timeConjugationProof:For real signal or thus the coefficients in the function F(s) are real, so that zeros and poles of the LT of a real signal are either real or complex conjugate pairs.ExampleGiven that the LT of a real signal f(t) is F(s), and (1) F(s) has two poles, o

22、ne is (2) F(s) has one zero (3) Determine F(s) and its ROC. Solution: The zeros and poles for a real signal are real or complex conjugate pairsThe third condition impliesSince s = 0 is in the ROC, the ROC isConvolutionProof:letCalculating the system response by LT ExampleAssume ， ， . Compute .Soluti

23、on： Differentiation In Time-domain Proof:Integration In Time-DomainProof:Determine the LT of . Solution：Example: two poles at s = 0two zeros at s = 0ROC = entire s-planeConsistent with Property 3 of the ROCProperty 3: If f (t) is of finite duration and absolutely integrable, then the ROC is the enti

24、re s-plane. Laplace TransformChapter 9 Lecture 3Signals and Systems Spring Differentiation in S-Domain Proof:Determine the LT of .Generally,ExampleSolution:Example Determine the F1(s) and F2(s). Solution:For a causal signal f(t) that contains no impulses or higher order singularities at the origin,

25、one can directly calculate, from the LT, the initial value f(0+) and final value f().Initial- and Final- Value Theoreminitial-value theoremfinal-value theoremThe signal is Determine ExampleSolution:The correctness can be easily verified by observing f(t).Useful TablesProperties of the Laplace transf

26、ormspp. 691, Table 9.1Laplace transform pairspp. 692, Table 9.2Inverse Laplace TransformHow to obtain the original signal using ILT ?By definition, The above integral can be calculated using the residue theorem. Calculation using the above approach can be very lengthy and complicated. Partial Fracti

27、on ExpansionSuppose F(s) can be expressed asAssume M N and denote the poles (can be complex) by p1, p2, pN. Consider two casesall poles are distinct not all poles are distinct If the poles p1, p2, pN are distinctROC on the right of pi :ROC on the left of pi :If some of the poles p1, p2, pN are ident

28、ical, say, p1 is a multiple root with multiplicity KFor i = 2, , N-K+1,To solve for the terms related to p1, we multiply F(s) by (s-p1)KLet LetSimilarlyGenerally ExampleSolution:Given computeExampleGiven , computeSolution: Since we apply polynomial long division to get Let which has polesThe input a

29、nd output of an CT LTI system can be described by its differential equationDefine system functionLTI System Analysis Using the LTDifferential equationSystem functionImpulse responseThe system function is determined by the system poles and zeros up to an unknown factor K. The pole-zero plot can be us

30、ed to describe an LTI system.Pole-Zero Plot of An LTI SystemEquivalent Descriptions of LTI SysDifferential equationSystem functionImpulse responsePole-zero plotExampleThe impulse response of an LTI system is (1) Determine the system function and sketch the pole-zero plot. (2) Determine the different

31、ial equation. Solution: (1) pole , zero Differential equationCausalityFor a causal LTI system, the impulse response is a causal function, i.e. and is thereby right sided. Due to the ROC property, the ROC of a right sided signal is the region in the s-plane to the right of the rightmost pole.The ROC

32、associated with the system function for a causal system is a right-half plane.CausalityFor an anticausal LTI system, the impulse response is an anticausal function, i.e. and is thereby left sided. Due to the ROC property, the ROC of a left sided signal is the region in the s-plane to the left of the

33、 leftmost pole.The ROC associated with the system function for an anticausal system is a left-half plan.InvertibilityThe impulse response hinv(t) of the inverse system of an invertible system h(t) satisfiesThe system function Hinv(s) of the inverse system is the reciprocal of the invertible system H

34、(s).An LTI system function is(1) For possible , discuss the system causality.(2) Find the inverse system and the impulse response of the inverse system. Solution: (1)Causal system Not causalExample(2) The inverse system functionTaking inverse Laplace transformLaplace TransformChapter 9 Lecture 4Sign

35、als and Systems Spring StabilityConsider a system which is a parallel interconn-ection of N subsystems hi(t).The system is stable when all the subsystems hi(t), i=1,N are stable.Subsystem i is stable only when hi(t) satisfies Fourier transform of hi(t) existsROC contains the j-axisROC contains the j

36、-axissystem stableall poles: Re pi 0anticausal system stableFor a causal CT LTI system For a CT LTI system with rational sys funcAn LTI system is(1) Determine the for a causal or an anticausal system.(2) Determine the for a stable system.Solution:Example(1) For a causal systemFor an anticausal syste

37、m(2) For a stable system, the ROC contains the j-axis.The response of an LTI system to the input is Determine , . Judge if the system is stable.Solution:ExampleThe ROC can be one of the two regions, i.e. or (1) ROC:ROC contains the j-axis, so the system is stable.(2) ROC:Since the j-axis is not in t

38、he ROC, the system is unstable.Using LT to Solve CT LTI Response According to the LT convolution propertyExampleGiven a causal LTI system with differential equation and input signal , determine the system response .Solution: Taking LT to the diff eqn yields partial fraction expansions-planeSystem Fr

39、equency Response v.s. Pole and Zero Positions When the ROC of contains the j-axis , the frequency response can be obtained asIn the s-plane, the vectors can be expressed byExampleGiven the system function sketch the system magnitude response and phase response .Solution：Magnitude responsePhase respo

40、nses-planePole-zero plotIf the magnitude of the frequency response is constant and independent of frequency, then the system is referred to as an all-pass system, such that , where is a constant.For every zero/pole of an all-pass system, there is a pole/zero symmetrical to it w.r.t. the j-axis.All-P

41、ass Systems-planeAn LTI system is said to be minimum phase if the system and its inverse are both causal and stable.For a minimum phase system, all the poles and zeros are on the left half of the s-plane.Any causal LTI system can be described as a cascade of a minimum phase system and an all-pass sy

42、stem.Minimum Phase SystemExpress the LTI systemas a cascade of a minimum phase and an all-pass system.Solution: is a minimum phase system.is an all-pass system.ExampleEquivalent Descriptions of CT LTI SysDifferential equationSystem functionImpulse responsePole-zero plotDifferential equationSystem fu

43、nctionImpulse responsePole-zero plotBlock DiagramEquivalent Descriptions of CT LTI SysSystem Function Algebra and Block Diagram RepresentationsSystem function of the parallel interconnection of LTI systems System function of the series interconnection of LTI systems System function of the feed back

44、interconnection of LTI systems AdderBasic Components for Block DiagramMultiplierIntegratorBlock Diagram Representations Consider a first order system i.e.System I: System II: Laplace TransformChapter 9 Lecture 5Signals and Systems Spring b1=0For an Nth order system, ( N M )System II: integrationSyst

45、em I: System II System I Direct-Form Block Diagram Direct-Form Block Diagram Assume H(s) has distinct poles Parallel-Form Block DiagramH(s) has distinct poles , thenSeries-Form Block Diagram (cascade)A casual LTI system has differential equationPlot the direct-, parallel-, and series-form block diagram.Example Solution：Direct-form block diagramParallel-form block diagramSeries-form block diagramSignal Flow Graph Node, branch , and branch gainDirect-form block diagramSignal flow graphParallel-form block diagramSignal flow graphUnilateral Laplace TransformDefine unilateral Lapla