Chapter 2 LTI Systems 5.卷积的微分、积分性质 ①微分性质 y()=x()*h()=x()*h() y(O)=x(o+hG)=x()+y° ②积分性质 p()=x()*h()=x()*h() y (0)=xm0(+()=x()+nm() m>0 ③推广式 p()=x6()=x()+0()m=0微分 n<0积分7
7 Chapter 2 LTI Systems 5. 卷积的微分、积分性质 ① 微分性质 ② 积分性质 y(t)= x(t)h(t)= x(t)h(t) ( ) ( ) ( ) ( ) ( ) ( ) ( ) y t x t h t x t h (t) n n n = = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 y t x t h t x t h t − − − = = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 m m m y t x t h t x t h t m − − − = = ③ 推广形式 ( ) ( ) ( ) ( ) ( ) ( ) ( ) y t x t h t x t h (t) n n n = = n>0 微分 n<0 积分
Chapter 2 LTI Systems (n+m) Um 特殊地n=1m=1 y(a)=x()*h2()=x()*h() Example 1,0≤t≤2 1,0≤t≤1 h() 0. otherwise 0, otherwise Consider the convolution of the two signals x(t), ht)
8 Chapter 2 LTI Systems ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) y t x t h t x t h (t) n m n m m n = = + 特殊地 n=1 m=-1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 y t x t h t x t h t − − = = Example x(t) = 1, 0 t 2 0, otherwise h(t) = 1, 0 t 1 0, otherwise Consider the convolution of the two signals x(t)，h(t)
Chapter 2 LTI Systems Example Consider the convolution of the two signals x(t), h() x()=h 10
9 Example Consider the convolution of the two signals x(t)，h(t) −1 0 1 1 −1 x(t) = h(t) t Chapter 2 LTI Systems
Chapter 2 LTI Systems 6几种典型系统 x ④恒等系统()=8() ②微分器()=( h(t) x() 3积分器 x ④延迟器M=(t-) x( hl(t) ⑤累加器 hn=uIn ∑
10 Chapter 2 LTI Systems 6 几种典型系统 ① 恒等系统 h(t) = (t) ② 微分器 h(t)=(t) ③ 积分器 h(t)= u(t) ④ 延迟器 ( ) ( ) 0 h t = t − t ⑤ 累加器 hn= un x(t) h(t) x(t) x (t) h(t) x(t) ( ) x (t) −1 h(t) x(t) ( ) 0 x t −t h(t) x(t) hn xk n k =− xn
Chapter 2 LTI Systems 52.3.4 LTI Systems with and without Memory 1. Discrete-time System =kD四 An LTI system without memory 2. Continuous-time System h(t)=ko(t) An LTI system without memory
11 Chapter 2 LTI Systems §2.3.4 LTI Systems with and without Memory 1. Discrete-time System hn= k n An LTI system without memory 2. Continuous-time System h(t) = k (t) An LTI system without memory