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 9 The Laplace transform Example 9.7 bt XX e x(r=ebiu)+ebu(t) eult< Re b stb 2(-0 Re s <b s-b If b>0 bt -2b b绕 b b b< Res < If b<0, x( has no Laplace transform 16

16 Chapter 9 The Laplace Transform Example 9.7 ( ) b t x t e − = x(t) e u(t) e u( t) b t b t = + − − ( ) s b s b e u t b t  − + − ⎯→ Re 1 ( ) s b s b e u t b t  − − − ⎯→ Re 1 If b>0, 2 2 2 s b b e b t − − ⎯→ − If b≤0, x(t) has no Laplace transform. −b  j b −b  Res b

Chapter 9 The Laplace transform Property 7: If the laplace transform s)ofx()is rational ox() is right sided, Res>0 max ②x() is left sided,Re{s}<σmn Example 9. 8 X(s)= (s+1)s+2) O —米 -2-1 O Res}<2-2<Res}<-1 Res>-1 left sided two sided right sided

17 Chapter 9 The Laplace Transform X(s) x(t) Re  max s x(t) x(t) Re  min s Property 7: If the Laplace transform of is rational, ① is right sided, ② is left sided, Example 9.8 ( ) ( )( ) 1 2 1 + + = s s X s  j − 2 −1  j − 2 −1 Res −2 −2  Res −1  j − 2 −1 Res −1 left sided two sided right sided

Chapter 9 The Laplace transform Basic Laplace Pairs x() X(s) Poles ROC none Re!s}>-∞ s=0 Res>o S 0 Re 6S <0 s=-a Res>=a s+a C Res(<-a s+a 18

18 Chapter 9 The Laplace Transform Basic Laplace Pairs x(t) X(s) Poles ROC  (t) 1 none Res − s 1 u(t) Res 0 −u(−t) Res 0 s 1 e u(t) −at Res −a e u( t) at − − − Res −a s + a 1 s + a 1 s = 0 s = 0 s = −a s = −a

Chapter 9 The Laplace transform 59.3 The Inverse Laplace Transform Ct JO defining X(seds 2 O C ROC Example 9.9 1 Xs S+1X(s+2 Determine the inverse Laplace transform for all possible roc

19 Chapter 9 The Laplace Transform §9.3 The Inverse Laplace Transform   ROC ( ) X (s)e ds j x t st j  j +  −  =  2  1 defining −a 0  j  + j  − j Example 9.9 ( ) ( )( ) 1 2 1 + + = s s X s Determine the inverse Laplace transform for all possible ROC

Chapter 9 The Laplace transform 59.4 Geometric evaluation of the Fourier transform 几何求值 from the pole-Zero plot ∏(o-/B) X(o=m ∏(o-a) =1 J Pole vector: jo-a=AeJ B B Zero vector: j@B =Be 1 B 20

20 Chapter 9 The Laplace Transform §9.4 Geometric evaluation of the Fourier transform 几何求值 from the Pole-Zero plot ( ) ( ) ( ) 1 1 i n i i m i j α j X j M − − =   = =     i  j  i i j − i j −  Pole vector: i j i i j Ae   − = Zero vector: i j i i j B e   −  = Ai Bi  i i