信号与系统Laplace分析.doc

信号与系统Laplace分析h(t)f(t)y(t)H(S) F(S)Y(S)y(t)=f(t)*h(t)Y(S)= F(S) H(s)一. Laplace Transform1. 定义X(s)= -+x(t)e-stdtx(t)=1/(2j)-+X(s)estdsUnilateralX(s)= 0_+x(t)e-stdtx(t)=1/(2j)-+X(s)estdst0x(t)=0 t- (t)1/SReS0e-at1/(S+a)ReS-aeat 1/(S-a) ReSa3.Propertyx(t-t0) e-st0X(S) es0t x(t)X(S-S0) x(t)*h(t)X(S)H(S)dx(t)/dtSX(S)-x(0-) -tx(t)dtX(S)/S+ x(-1) (0-)/S0-tx(t)dtX(S)/S-tx(t)d X(S)/dS x(0+)=SX(S)|s=+x(+)= SX(S)|s=04. 计算a)定义计算法b)性质计算法c)利用常用函数对计算法d)分部式展开（原函数）ex1:f(t)=2*(t)-(t-1) -(t-2)F(S)=?1) 定义计算法F(S)= 0_+f(t)e-stdt=0+1e-stdt+0+2e-stdt= -1/Se-st|0+1 -1/Se-st|0+2= (2-e-s-e-2s)*(1/S)ROC: ReS!=02)性质计算法Ldf(t)/dt=SF(S)-f(0_) = SF(S)df(t)/dt=2*(t) -(t-1) -(t-2)Ldf(t)/dt=2-e- s -e-2 sF(S)=(2-e- s -e-2 s)/S3) 利用常用函数对计算法F(S)=(2-e- s -e-2 s)L(t)=(2-e- s -e-2 s)*(1/S)ex2:f(t)= (1+te-t)(t)F(S)=? ROC:=? F(s)= (1+te-t)(t) =1/s d/ds(1/(s+1) =1/s+1/(s+1) 2 =(s2 +3s+1)/(s(s+1) 2)Causal p1=0 p2=-1;ROC:0Ex3:1)F(s)=( s2+1)/(s(s+2)(s-3)ROC: 32)F(s)=( s2+1)/(s(s+2)(s-3)ROC: 30f(t)=? 1)F(s)=-1/6/s +1/2/(s+2) +2/3/(s-3)f(t)=(-1/6+1/2 e-2t+2/3e3t)(t)2)F(s)=-1/6/s +1/2/(s+2) +2/3/(s-3)f(t)=(-1/6+1/2 e-2t)(t)- 2/3e3t(-t)二系统Laplace分析1系统微分方程分析y(2)(t)+a1y(1)(t)+a0y(t)=b1 (1)f(t)+b0f(t)S2Y(S)-S y(0-)- y(1) (0-)+a1 (SY(S)-y(0-)+a0 Y(S)=b1 SF(S)+b0F(S)Y(S)=?y(t)=?Ex4:y(2)(t)+5y(1)(t)+6y(t)=f(t)y(0-)=2 y(0-) (1)=1 f(t)= e-t(t)y(t)=?S2Y(S)-S y(0-)- y(1) (0-)+5 (SY(S)-y(0-)+6 Y(S)= F(S)Y(S)=(2S2+13S+12)/(S+1)(S+2)(S+3)=1/2/(S+1)+6/(S+2)-9/2/(S+3)y(t)=(1/2e-t+6e-2t-9/2e-3t )(t)2RLC系统分析R:U(S)=I(S) RL: U(S)=SLI(S)-Li(0-)C