电子科技大学电子工程学院：《信号与系统》课程教学资源（PPT课件讲稿，英文版）Chapter 9 The Laplace Transform

Chapter g The laplace fransform

1 Chapter 9 The Laplace Transform

Chapter 9 The Laplace transform 59.1 The Laplace Transform st e se H()=。0k=d △ Defining- X(s)=[x(e"dt -Laplace transform 1. The relationship X()=F{x()e} 2

2 Chapter 9 The Laplace Transform §9.1 The Laplace Transform H(s) h(t)e dt −st + − = st e ( ) st H s e Defining X (s) x(t)e dt −st + −   = ——Laplace Transform 1. The relationship ( )  ( )  t X s x t e F − =

Chapter g The Laplace transform 2. Region of Convergence(收敛域) Dirichlet Condition 1:x( andt <OO ROC:对给定的x(),使其拉氏变换存在的对应的 S平面上的区城。 X(ja)=X(s) s=a O=OCROC

3 Chapter 9 The Laplace Transform 2. Region of Convergence（收敛域） Dirichlet Condition 1 : ( )   − + − x t e dt  t ROC：对给定的 ，使其拉氏变换存在的 σ对应的 S平面上的区域。 x(t) ( ) ( ) s j X j X s   = = 0 ROC  = 

Chapter 9 The Laplace transform Example 9.1 =已LL Res>-a s+a pole-zero plot 零极点图 Example 9.2 x() t JO 米 C Res <-a sta pole-zero plot 4

4 Chapter 9 The Laplace Transform Example 9.1 ( ) ( ) at x t e u t − = ( )   1 X s s a Re s a =  − + −a 0  j pole-zero plot 零极点图 Example 9.2 ( ) ( ) at x t e u t − = − − ( )   1 X s s a Re s a =  − + −a  j pole-zero plot

Chapter 9 The Laplace transform e-u(tt Res>=a s+a (-) Re }0 The Fourier transform of u()does not exist. )xo()+ O 5

5 Chapter 9 The Laplace Transform ( ) s a s a e u t a t  − + − ⎯→ Re 1 ( ) s a s a e u t a t  − + − − − ⎯→ Re 1 x(t)⎯→X(s) ;ROC Particularly, ( ) Re  0 1 ⎯→ s  s u t 0  j The Fourier transform of does not exist. u(t) ( ) ( )     j u t F 1 ⎯→ +

Chapter 9 The Laplace transform Example 9.3 ()=3e2l()-2el( 3e2(k 3Rel}>-2 s+2 2e() 2 Res>-1 s+1 3e-2l()-2ev()< X s+1s+2 2-1 R 6

6 Chapter 9 The Laplace Transform Example 9.3 ( ) 3 ( ) 2 ( ) 2 x t e u t e u t − t −t = − ( ) Re  2 2 3 3 2  − + − ⎯→ s s e u t t ( ) Re  1 1 2 2  − + − ⎯→ s s e u t t 3 ( ) 2 ( ) 2 e u t e u t − t −t − ( )( ) 1 2 1 + + − ⎯→ s s s Res −1  j − 2 −1 1

Chapter 9 The Laplace transform Example9.3 x(0=e2u()+e" cos(3t u() x(=e-u(0+e" tu()+e"e 3tu(t) e2u()-2 e/3 1/2 Res> Re-a=-1 S+1-3j 4+3)( 1/2 2 S+1+3j Re)>Rea=-1 2s2+5s+12 X 0 X(s)= (s+2)s2+2+10) Re{s}>-1 1-3 7

7 Chapter 9 The Laplace Transform Example 9.3 ( ) ( ) cos(3 ) ( ) 2 x t e u t e t u t − t −t = + ( ) ( ) ( ) ( ) 2 1 2 2 1 3 3 x t e u t e e u t e e u t − t −t j t −t − j t = + + ( ) Re  2 2 2 1  − + − ⎯→ s s e u t t ( ) ( ) s - j / e u t j t 1 3 1 2 2 1 1 3 + − − ⎯→ Res Re−a= −1 ( ) ( ) s j / e u t j t 1 3 1 2 2 1 1 3 + + − + ⎯→ Res Re−a= −1 ( ) ( )( ) 2 2 10 2 5 12 2 2 + + + + + = s s s s s X s Res −1  j −1−3 j − 2 −1−3 j a

Chapter 9 The Laplace transform Example 9.4 x(=8( 4 2t e ult+=eu 3 5(e>S(le dt=l entire S plane Re 4 4/3 eult Re!s}>-1 s+1 J0 1/3 e ult> Res >2 s-2 O 112 X(s) (+-2R8}>2 (s-1) pole-zero plot σ=0女{Res}>2}F( does not exist. 8

8 Chapter 9 The Laplace Transform Example 9.4 ( ) ( ) ( ) ( ) 3 1 3 4 2 x t t e u t e u t t t = − + −  (t) (t)e dt −st + −  ⎯→  = 1 entire S plane Res − ( ) Re  1 1 4 / 3 3 4  − + − − − ⎯→ s s e u t t ( ) Re  2 2 1/ 3 3 1 2  − ⎯→ s s e u t t ( ) ( ) ( )( ) 1 2 1 2 + − − = s s s X s Res 2  = 0 Res 2 Fx(t) does not exist.  j −1 1 2 pole-zero plot

Chapter g The Laplace transform N( Zeros: N(s)=0 X(s)=D()- Poles:D(s)=0 The direction of signals ROC of X(s) 2. The position of poles 59.2 The Properties of ROC Property 1: The roc of X(s) consists of strips parallel to the jo-axis in the s-plane a<∞- Depends only on o

9 Chapter 9 The Laplace Transform ( ) ( ) ( ) D s N s X s = Poles: D(s) = 0 Zeros: N(s) = 0 1. The direction of signals 2. The position of poles ROC of X(s) §9.2 The Properties of ROC Property 1: The ROC of X(s) consists of strips parallel to the jω-axis in the s-plane ( )   − + − x t e dt  t ——Depends only on σ

Chapter 9 The Laplace transform Property 2: For rational Laplace transforms, the roc does not contain any poles. Property 3: If x(t)is of finite duration and is absolutely integrable, then the roc is the entire s-plane x()=0;tT2 r 2 Ix()d dt T ①whenσ=0 ko<→n =0C ROC

10 Chapter 9 The Laplace Transform Property 2: For rational Laplace transforms, the ROC does not contain any poles. Property 3: If is of finite duration and is absolutely integrable, then the ROC is the entire s-plane. x(t) 2 T t 1 T x(t) ( ) 1 2 x t = 0 ; t T ,t T ( )    x t dt T T 2 1 ( )   −  x t e dt t T T 2 1  ① when  = 0  = 0  ROC  ( )   − + − x t e dt  t