1、信号与系统 Laplace 分析f(t) y(t)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)estds t0x(t)=0 t-(t) 1/S ReS0e-at 1/(S+a) ReS-aeat 1/(S-a) ReSa3.Propertyx(t-t0) e-st0X(S)es0t x(t) X(S-S0) x(t)*h(t)
2、 X(S)H(S)dx(t)/dt SX(S)-x(0-) - tx(t)dt X(S)/S+ x(-1) (0-)/S 0-tx(t)dt X(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)RO
3、C: 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
4、;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
5、-)- 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: I(S)=SC U(S)-Cu(0-)KVL KCL 3.系统频率特性H(j)|H(j)|(j)Zeros & Poles Figure:流程图 系统框图
