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 }<-a s+a x()<→X():ROC Particular 1()<→-Re{s}>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 ⎯→ +