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 ②σ>0 >T atdt <e <OO o>cROC 3<0 o01<e 02(2>T) (e dt<e o7 lx()o at<e -oz dt<∞ o<OCROC x()=0;t<7,t>72→Re{s}>-∞ 11

11 Chapter 9 The Laplace Transform ②   0 ( ) 2 1 1 2 e e T T T T   − − x(t)e dt t T T 2 1 −    0  ROC ③   0 ( ) 2 1 1 2 e e T T T T   − − x(t)e dt t T T 2 1 −    0  ROC e x(t)dt T T T  −  2 1  1    ( )    − e x t dt T T T 2 1  2 ( ) 1 2 x t = 0 ; t T ,t T Res −

Chapter 9 The Laplace transform Example. x(t)=u(t-u(t-T) Res>o too (t-7) S Re 0 S IT ST ()-u(t-7)< e Re 零极点 2o 抵消 pole: S=0 zero: 1-e=0 e =e/2kr 2kn zeros. k=0.±1· pole-zero plot 12

12 Chapter 9 The Laplace Transform Example x(t) = u(t)−u(t −T) ( ) Re  0 1 ⎯→ s  s u t u(t T) e dt st T − + ⎯→ − s e e s sT T st + − = − = 1 − Res 0 ( ) ( ) 1 s e u t u t T −s T − − − ⎯→ Res − pole: s = 0 zero: 1− = 0 −sT e sT j k e e 2 = − zeros: , 0 1 2 = − k = , T k s j k   j pole-zero plot T j 2 T j 2 − 零极点 抵消

Chapter 9 The Laplace transform Property 4: Ifx() is right sided,x()=0, I<T O CROC口>Res}> O CROC + 三ROC→(ko< → eoat<∞ T g,t oe se cot (>0) + ①71≥0 x(te T x(kal<厂x ②7<0k+ymh fir e x()e O <oO

13 Chapter 9 The Laplace Transform Property 4: If is right sided, x(t)  0  ROC  0  ROC  ( )   − + − x t e dt t  0 ( ) 1 x t = 0 , t T 1  0 ( 0) 1 0   − − e e t  t  t ② T1  0 ① T1  0  ( )   − +  x t e dt t T 0 1  x(t)e dt t T 1 1 − +  x(t)e dt x(t)e dt t t T 1 1 1 0 0  − + −   + finite  ( )   − +  x t e dt t 0 0  Res 0  ROC  ( )   − +  t T x t e 0 1 

Chapter 9 The Laplace transform Property 5: Ifx( is left sided,x(=0, t>T O CROC口Re(s}< R CROC Property 6: Ifx() is two sided, x(t) <t<+ 日a0cROC>a1<Re<a2a x t t 0 0 RQC:Rek<a2∩Re 14

14 Chapter 9 The Laplace Transform Property 5: If is left sided, x(t)  0  ROC ( ) 1 x t = 0 , t T Res 0  ROC Property 6: If is two sided, x(t)  0  ROC x(t), -  t  + 1  Re  2  0 s T0 t x(t) T0 t x (t) L T0 t x (t) R Re 1 ROC: Res 2  s

Chapter 9 The Laplace transform X(s=m x(re"dt △ X(s=Fxlte-otdt X(jo)=X(s)io o=0g ROC 1. The direction of signals ROC of X(s) 2. The position of poles 15

15 Chapter 9 The Laplace Transform ( ) ( ) st X s x t e dt  + − − =  ( )  ( )  t X s F x t e dt   − = ( ) ( ) 0 ROC s j X j X s = = =     1. The direction of signals 2. The position of poles ROC of X(s)