信号与系统 Lecture 5 Chapter2-1

h [ n ] [ n ] [ n 1]
x[ n ]
h[ n ] x[ n ]
x[ n ]
h[ n ] [ n n 0 ]
h n u n
h[ n ]
x[ n n 0 ]
n
x n
h n

k
x k
10
Chapter 2
1 a 1 a
N
N
BIBO stable <=> -1<a<1
13
Chapter 2

LTI Systems
If h m is not finite, can we find a bounded x that produces an unbounded output?
m
y[n]=?
[n] h[n] [n-k] h[n-k] x[k][n-k] x[k] h[n-k]
y n

k

x k h n k x n h n
k时刻的脉冲在n时刻的响应 系统在n时刻的输出包含所有时刻输入脉冲的影响
Causal system
For any time n0
x[ n ] 0 , n n 0
h n 0 , n 0
y[ n ] 0 , n n0
The system is initial rest. (初始松弛）
Consider a LTI system
Causal Initial Rest (初始松弛）
9
Chapter 2 5 几种典型离散时间系统
x[ n ]
LTI Systems
x[ n ]
h[ n ]
① ②
恒等系统 前向差分器
h[ n ] [ n ]
h [ n ] [ n 1] [ n ]
x[ n ]
h[ n ]
x[ n ]
③ ④ ⑤
后向差分器 延迟器 累加器
LTI
hn L

0 , n

Unit Impulse Response: h[n]
3
Chapter 2
LTI Systems
x n
2. Convolution-Sum （卷积和）
Question:

k

x k n k
x[n] LTI
Solution:
x[ n ]
System
h[ n ]
Inverse System
h1 [ n ]
identity system (恒等系统）
h n h 1 n n
11
Chapter 2 §2.3.6 Causality for LTI Systems LTI系统的因果性
LTI Systems
㈡ 平移 循 环 ㈢ 求乘积
n
x k h n k
x n h n
㈣ 对每一个n求和

k

x k h n k
5
Chapter 2
Example 3 P2.44(b)
LTI Systems
If y[n]=x[n]*h[n], and h[n]≠0 for N0≤n ≤N1, x[n]≠0 for N2≤n≤N3, y[n]≠0 for N4≤n≤N5; Determine N4 and N5 in term of N0, N1, N2, N3. N4=N0+N2, N5=N1+N3; Lh=N1-N0+1, Lx=N3-N2+1, Ly=N5-N4=Lh+Lx-1. ③ 不带进位的普通乘法 ——适用于有限长度序列之间的卷积
C
________
LTI Systems
x n h1 n h 2 n x n h1 n h 2 n
x[ n ]
h1 [ n ]
y[ n ]
h2 [ n ]
x[ n ]
h1 [ n ] h 2 [ n ]
y[ n ]
Commutative Property
12
Chapter 2 §2.3.7 Stability for LTI Systems (稳定性）
LTI Systems
Stable System
h n

n

h n

Absolutely summable （绝对可加）
x[ n m ]h[ m ]

y[ n ]
Signals and Systems
Lecture 5
Convolution Sum and DT-LTI System
1
Chapter 2
LTI Systems
§2.1 Discrete-time LTI Systems : The Convolution Sum （卷积和） §2.1.1 The Representation of Discrete-Time Signals in Terms of impulses

m


m

x[ n m ] h[ m ] M

m

h[ m ]
If
Example:

m
h m
n

y[ n ]
h n a u[n ]
h n

n


n

a u[n ]
n

n0

a
n
lim
1 h n 0
n 0 ,1 otherwise
a y n x n x n 1 2
b y n
max
x n , x n 1
7
Chapter 2
LTI Systems
§2.3.1 Properties of Convolution Integral and Convolution Sum
1. The Commutative Property (交换律）
x n h n h n x n
x[ n ] h[ n ]
x[ n ] h[ n ]
h[ n ] x[ n ]
h[ n ] x[ n ]
2. The Distributive Property (分配律）
x n h1 n h 2 n x n h1 n x n h 2 n
h1 [ n ]
x[ n ]
y[ n ]
x[ n ]
y[ n ]
h1 [ n ] h 2 [ n ]
h2 [ n ]
8
Chapter 2 3. The Associative Property (结合律）
LTI Systems
§2.3.4 LTI Systems with and without Memory
h n k n
An LTI system without memory
§2.3.5 Invertibility of LTI Systems LTI系统的可逆性
x[ n ]
y[ n ]
h[ n ] x[ n ] | h[ n ] | 0

if h [ n ] 0 o th e r w is e
y [0 ]

m
h[ m ] x[ m ]

m
h [m ] h[ m ]
2


m

h[ m ]
14
Chapter 1
Example 4
x n 2 ,1 , 5 h n 3 ,1 , 4 , 2
n 0 ,1 , 2 n 0 ,1 , 2 , 3
Determine
y n x n h n
6
Chapter 2 §2.3 Properties of LTI Systems
x t
x n
LTI Systems
h t
h n
y t x t h t
y n x n h n
LTI系统的特性可由单位冲激响应完全描述 Example 2.9 ① LTI system ② Nonlinear System ③ Time-variant System
4
Chapter 2
LTI Systems
3. 卷积和的计算
① 利用定义计算
Example2(P2.21 a)
x n a u n
n
h n u n
n
x n h n ?
② 图解法
㈠
h n h k
反折
h n k
h k
x[ n ]
y[ n ]
h2 [ n ]
h1 [ n ]
x[ n ]
h 2 [ n ] h1 [ n ]
y[ n ]
4. 含有冲激的卷积 ① x n n x n
Associative Property
② y [ n ] x[ n ] h[ n ]
x [ n n1 ] h [ n n 2 ] y [ n n 1 n 2 ]
LTI Systems
§2.1.2 The Discrete-Time Unit Impulse Responses and the Convolution-Sum Representation of LTI Systems 1. The Unit Impulse Responses 单位冲激响应 x[n]=[n] y[n]=h[n]
Example 1
2 1
3
x n
1
0 1 2

n
xk n k
x n x 1 n 1 x 0 n x 1 n 1
x n

k

x k n k
2Fra Baidu bibliotek
Chapter 2
Chapter 1
x F y
Signals and Systems
LP C
x1[n]= [n]+2[n-1]+3[n-2], y1[n]= 2[n]+ [n-1] TI P x2[n]= 2[n-2]+3[n-3], y2[n]= -[n]+2[n-1] x3[n]= [n-4], y3[n]= [n-3]+2[n-4] + [n-5] If F is Time-Invariance, can F be causal? Can F be linear? x4[n]=x3[n+4]=[n] x1[n]=x4[n], n≤0 but y1[-1] ≠ y4[-1] x1[n]-x2[n+1] = [n] y1[n]-y2[n+1]=[n]+[n+1] x4[n]=x1[n]-x2[n+1], but y4[n] ≠ y1[n]- y2[n+1] y4[n]=y3[n+4]=[n+1]+2[n]+[n-1] So, F can not be causal.
x F y
Signals and Systems
x[n]= [n], y[n]= [n+1]-[n]+2[n-1] If F is linear, is F causal? If F is Time-Invariance, is F causal? x1[n]= [n] x2[n]=0 y1[n]= [n+1]- [n]+ 2 [n-1] y2[n]= 0 x1[n]=x2[n], n≤-1 So, F is not causal. but y1[-1] ≠ y2[-1] If a system F is linear, and its output “proceedes” the input, LP C then F cannnot be causal. x1[n]= [n] x2[n]= [n-1] y1[n]= [n+1]- [n]+ 2 [n-1] y2[n]= [n]- [n-1]+ 2 [n-2] x1[n]=x2[n], n≤-1 So, F is not causal. but y1[-1] ≠ y2[-1] If a system F is time-invariance, and its output “proceedes” 15 TI P C the input, then F cannnot be causal.