Chapter 2 LTI Systems 92.3 Properties of LTI Systems y()=x()*() yln=xin *hn An LtI system is completely characterized by its impulse response h(t/hn Example 2.y ln=0,1 0 otherwse ① LTI system ② Nonlinear System 3 Time-variant System (a)y]=(x[+x[n-1(a)y(4)=c0s(3)x(t) yIn=max (x小]x-(b)y()=ex(t)
2 Chapter 2 LTI Systems §2.3 Properties of LTI Systems x(t) h(t) y(t) = x(t)h(t) hn yn= xnhn xn An LTI system is completely characterized by its impulse response h(t)/h[n] Example 2.9 = 0 1 h n otherwise n = 0 ,1 ① LTI system ② Nonlinear System ( ) ( ) 2 a y n = x n + x n −1 (b) yn= max(xn, xn−1) ③ Time-variant System (a cos 3 ) y t t x t ( ) = ( ) ( ) (b ) ( ) ( ) t y t e x t =
Chapter 2 LTI Systems 92.3. 1 Properties of Convolution Integral and Convolution Sum 1. The Commutative Property(交换律) lt ak h(O)=h()*x() x*小=小小]x口 x()*( h(O)*
3 Chapter 2 LTI Systems §2.3.1 Properties of Convolution Integral and Convolution Sum 1. The Commutative Property (交换律） xnhn= hn xn x(t)h(t) = h(t) x(t) x(t)h(t) h(t) x(t) h(t) x(t) h(t) x(t)
Chapter 2 LTI Systems 2. The Distributive property(分配律) x()*()+h2()}=x()+*h()+x(*h2() x]*{1团+h2[=x小]*h1团+x口小*h四 h( h()+h2() h2(
4 Chapter 2 LTI Systems 2. The Distributive Property (分配律） xn h n h n xn h n xn h n 1 + 2 = 1 + 2 x(t) h (t) h (t) x(t) h (t) x(t) h (t) 1 + 2 = 1 + 2 x(t) y(t) h (t) h (t) 1 + 2 h (t) 1 x(t) h (t) 2 y(t)
Chapter 2 LTI Systems 3. The associative Property(结合律) x()+(G)+()=x(O)*{(0)*1()} x可*h1[吗*h四=x小*{[*h[ x h(0)b2() h()*h Commutative Property x h()h1( h2()+h() y Associative Property
5 Chapter 2 LTI Systems 3. The Associative Property (结合律） xnh1 nh2 n= xnh1 nh2 n x(t)h1 (t)h2 (t) = x(t)h1 (t)h2 (t) h (t) 1 x(t) h (t) 2 y(t) x(t) y(t) h (t) h (t) 1 2 x(t) y(t) h (t) h (t) 2 1 Commutative Property h (t) 1 x(t) h (t) 2 y(t) Associative Property
Chapter 2 LTI Systems 4.含有冲激的卷积 0 x(t*8(0)=x(t) x*=x四 ②y(t)=x(O*h() x(t-4)*h(-t2)=y(-t1-1)
6 Chapter 2 LTI Systems 4. 含有冲激的卷积 ① x(t) (t) = x(t) ② y(t) = x(t)h(t) ( ) ( ) ( ) 1 2 1 2 x t −t h t −t = y t −t −t xn n= xn
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