1、1,Chapter VThe Laplace Transform,2,5.1 The Laplace Transform,1. From Fourier Transform to Laplace Transform,3,1. From Fourier Transform to Laplace Transform,4,Above two equations are referred as to Bilateral Laplace Transform peer.,With fixed and vary from - to .,5,2, Region of Convergence (ROC),In

2、general, the range of values of s for which the above integral converges is referred to as the region of convergence (ROC) of the Laplace transform.,6,Example. Determine the Laplace transform of the following causal signal., is real,For causal signals, their Laplace transform are valid if and only i

3、f Res=.,7,For anti-causal signals, their Laplace transform are valid if and only if Res=.,Example. Determine the Laplace transform of the following anti-causal signal., is real,8,Example. Determine the Laplace transform of the following two signal.,ROC is so critical to signals for Laplace Transform

4、.,9,ROC of Casual signals,ROC of Anti-Casual signals,ROCs for Casual and Anti-Casual signals,The Laplace transform of x(t) can be generally expressed in a rational form a ratio of polynomial in the complex variable s,The roots of the numerator polynomial the zeros of X(s); The roots of the denominat

5、or polynomial the poles of X(s); The representation of X(s) through its poles and zeros in the s-plane the pole-zero plot of X(s).,10,Example,11,12,Properties of ROC,Property 1,Property 2,For rational Laplace transforms, the ROC does not contain any poles.,Note. The condition of absolutely integrabi

6、lity depends only on the real part of s.,Property 3,The condition of absolutely integrability,13,Property 4,Properties of ROC,14,Properties of ROC,Property 5,Property 6,A two-sided signal can be divided into the sum of a right-sided and left-sided signal.,15,Properties of ROC,Property 7,If the Lapla

7、ce transform X(s) of x(t) is rational, then its ROC is bounded by poles or extends to infinity. In addition, no poles of X(s) are contained in the ROC.,“A signal with a rational Laplace transform consists of a linear combination of exponentials and the ROC for the transform of individual terms in th

8、is linear combination must have the property.”,16,Property 8,If the Laplace transform X(s) of x(t) is rational, then if x(t) is right sided, the ROC is the region in the s-plane to the right of the rightmost pole. If x(t) is left sided, the ROC is the region in the s-plane to the left of the leftmos

9、t pole.,Properties of ROC,A consequence of Property 4, 5, and 7.,17,If f(t) is of finite duration and is absolutely integrable, the ROC is the whole s-plane.,Example Determine the Laplace transform of the following rectangular impulse.,This rectangular impulse is integrable. No matter what value tak

10、es, we can find,So, its ROC is,18,Example,19,Unilateral Laplace Transform,Unilateral Laplace Transform,Definition of Unilateral Laplace Transform,Usually, common signals have their initial points. Let their initial points as the origin of reference frame. Obviously, f(t)=0 when t0.,20,Laplace Transf

11、orm for Casual Signals,For a casual signal, which can be expressed as f(t)u(t), its Laplace transform is referred as to LTf(t), or F(s) in s-domain, and its Inverse Laplace transform is referred as to ILTF(S) or L-1F(S).,21,Some useful Laplace transform peers,22,5.2 Properties of Laplace Transform,1

12、 Linearity,For constants,(Almost for unilateral Laplace Transform),23,Example Laplace Transform for Unilateral sinusoidal and cosine functions,24,2 Time Scaling,25,3 Time Shifting,26,4 Shifting in the s-domain,27,Example: Determine the Laplace of the attenuated sinusoidal function,According to the p

13、roperty of shifting in the s-domain,28,5 Differentiation in the Time Domain,The ROC of the above is at least,29,Because f(t) has the same order with exponential function, in its region of convergence,30,31,Example: Suppose the Laplace Transform for the current through a inductor is Determine the LT

14、of the voltage function of this inductor.,Here, is the initial value of the current through the inductor.,32,6 Integration in the Time Domain,The ROC of above LT are the intersection of at least.,33,Proof (Additional Contents),34,Proof (Continue) (Additional Contents),35,Example: Suppose Determine t

15、he LT of using the integral of unit step function.,According to the property of integration in the Time Domain,36,7 Convolution Property,37,38,8 Differentiation in the s-Domain,39,40,Example Determine the Laplace Transform of,According to the property of differentiation in the s-domain,41,9 The Init

16、ial- and Final- Value Theorems,The initial-value theorem: Under the constrains that x(t)=0 for t0 and that x(t) contains no impulse or higher order singularities at the origin,42,43,Proof (Additional Contents),44,Proof (Additional Contents),45,5.3 The Inverse Laplace Transform,Looking up reference t

17、able Partial-fraction expansion,46,In general, the Laplace transform of a signal can be expressed as below when it is a rational fraction of s.,in which,assume,in which the order of B(s) is less than the order of A(s).,47,Example,48,1 Looking up reference table,Example Determine the Inverse Laplace

18、transform of,According to reference table,Compared with the corresponding coefficients in the expression of F(s),So that,49,2 Partial-fraction Expansion,In general, the Laplace transform of a signal can be expressed as below when it is a rational fraction of s.,50,(1) F(s) has single poles,If the ro

19、ots of equation A(S)=0 are different and no multiple-order poles, F(s) can be expanded as the following partial-fraction.,in which,51,Example Determine the ILT of,The denominator polynomial of F(s) is,52,(2) F(s) has are conjugated and single poles,If the equation A(s)=0 has a peer of conjugated com

20、plex roots,F(s) can be divided into two parts:,in which,B(s) and A(s) are polynomial about s with real coefficients,53,The ILT is,54,Example Determine the ILT of,The six eigenvalues of the denominator are,55,56,(3) F(s) has multiple-order poles,If A(s)=0 has r-order poles at s=s1, that is s1=s2=sr,w

21、hile the other (n-r) poles, sr+1,sn, are different with s1 Then F(s) can be expressed as,when,57,Both sides of the above equation are multiplied by,The derivative of (*) is,58,The similar method should be performed when determine the ILT of F(s) which has multiple complex poles.,59,60,5.4 Geometric

22、Evaluation of the Fourier Transform from the Pole-Zero Plot,System function Frequency response Frequency response and the pole-zero plot,61,System function,62,Input,Impulse response,Output,System function,63,System function,For a general rational Laplace Transform,64,“Fourier Transform of a signal i

23、s the Laplace transform evaluated on the imaginary axis.”,65,Frequency response,The magnitude and phase vary with . The frequency response can be evaluated by the above method.,66,Example,67,5.5 Analysis and Characterization of LTI Systems Using the Laplace Transform,Causality Stability,68,Causality

24、,The impulse response of a causal LTI system is zero for t0. The ROC associated with the system function for a causal system is a right-half plane. For a system with a rational system function, causality or the system is equivalent to the ROC being the right-half place to the right of the rightmost

25、pole.,69,70,Examples,Casual system,Not Casual system,Stability,The stability of an LTI system is equivalent to its impulse response being absolutely integrable (the Fourier Transform of the impulse response converges) An LTI system is stable if and only if the ROC of its system function H(s) include

26、 the entire j-axis. An causal system with rational system function H(s) is stable if and only is all of the poles of H(s) lie in the left-half of the s-plane.,71,72,Example,If the system is known to be causal,If the system is known to be stable,If the system is known to be anticausal and unstable,73

27、,5.6 Analyze LTI System in s-Domain,Solve differential equations by means of Laplace Transform System Function Systems Diagram in s-Domain Modeling Circuit Systems in s-Domain Relationship between Laplace Transform and Fourier Transform,74,1 Solve differential equations by means of Laplace Transform,75,76,Affected by the initial status,Determin