1
Chapter 2
Linear Time-invariant Systems
2
Chapter 2 LTI Systems
Example 1 an LTI system
0 2 t
tf1
1
211 tftf
0 1 2 t
ty1
1L
211 tyty
0 2 4 t
1
-1
0 2 4 t
tf2
1 L
ty2
1?
3
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
11011 nxnxnxnx
knkxnx
k



knkx
Example 2
1? 0 1 2
1
2
3nx
n
4
Chapter 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
单位脉冲响应
0,h n L n?
2,Convolution-Sum （卷积和）
knhkxny
k



系统在 n时刻的输出包含所有时刻输入脉冲的影响
k时刻的脉冲在 n时刻的响应
nhnx
5
Chapter 2 LTI Systems
3,卷积和的计算
① 利用定义计算例 2.3nuanx nnunh nhnx
② 图解法
Example 2.4



o t h e r w i s e,0
40,1 n
nx


o t h e r w i s e,0
60,a n nnh 10 a
Determine the output signalny
6
Chapter 2 LTI Systems
Summarizing,we obtain
ny
0
0
64 n
0?n
40 n
a
a n

1
1 1
a
aa nn

1
14
a
aa n

1
74 106 n
10n?
Ly=11 Lx=5 Lh=7
Ly=Lx+Lh-1
7
Chapter 2 LTI Systems
③ 不带进位的普通乘法
—— 适用于因果序列或有限长度序列之间的卷积
Example 3
5,1,2?nx
2,4,1,3?nh
2,1,0?n
3,2,1,0?n
Determinenhnxny
④ 多项式算法（适用于有限长度序列）
利用多项式算法求卷积和的逆运算
8
Chapter 2 LTI Systems
§ 2.2 Continuous-Time LTI Systems,The Convolution Integral
（卷积积分）
§ 2.2.1 The Representation of Continuous-Time Signals
in Terms of impulses
dtxtx —— Sifting Property
§ 2.2.2 The Continuous-Time Unit Impulse Response and the
Convolution Integral Representation of LTI Systems
y t x t h t x h t d
9
Chapter 2 LTI Systems
§ 2.3 卷积的计算
1,由定义计算卷积积分例 2.6 0, atuetx attuthty
2,图解法例 2.7 求下列两信号的卷积
tx
,1 Tt0
,0 其余 t
th
,t Tt 20
,0 其余 t
3,利用卷积积分的运算性质求解
10
Chapter 2 LTI Systems
§ 2.3 Properties of LTI Systems
thtxtythtx
nh
nhnxnynx
LTI系统的特性可由单位冲激响应完全描述
Example 2.9



0
1
nh o t h e r w i s e
1,0?n
① LTI system
② Nonlinear System
21 a nxnxny
1,m a x b nxnxny
③ Time-variant System
a c o s 3y t t x t?
b ty t e x t?
11
Chapter 2 LTI Systems
§ 2.3.1 Properties of Convolution Integral and Convolution Sum
1,The Commutative Property (交换律）
nxnhnhnx
txththtx
thtx?
th
txth
tx
txth?
12
Chapter 2 LTI Systems
2,The Distributive Property (分配律）
nhnxnhnxnhnhnx 2121
thtxthtxththtx 2121
tytx
thth 21?
th1
tx
th2
ty
13
Chapter 2 LTI Systems
3,The Associative Property (结合律）
nhnhnxnhnhnx 2121
ththtxththtx 2121
th1
tx
th2
tytytx
thth 21?
tytx
thth 12?
Commutative
Property
th1
tx
th2
ty
Associative Property
14
Chapter 2 LTI Systems
4,含有冲激的卷积
txttx①
②thtxty
2121 tttytthttx
nxnnx
15
Chapter 2 LTI Systems
5,卷积的微分、积分性质
① 微分性质
② 积分性质
thtxthtxty
thtxthtxty nnn
1 1 1y t x t h t x t h t
0m m my t x t h t x t h t m
③ 推广形式
thtxthtxty nnn
n>0 微分
n<0 积分
16
Chapter 2 LTI Systems
thtxthtxty nmmnmn
特殊地 n=1 m=-1
11 y t x t h t x t h t
Example
tx
,1 20 t
,0 otherwise
th
,1 10 t
,0 otherwise
Consider the convolution of the two signalsthtx,
17
Example
Consider the convolution of the two signalsthtx,
1? 0 1
1
1?
thtx?
t
Chapter 2 LTI Systems
18
Chapter 2 LTI Systems
6 几种典型系统
① 恒等系统tth
② 微分器tth
③ 积分器tuth?
④ 延迟器0ttth
⑤ 累加器nunh?
tx
th
tx
tx?
th
tx
tx 1?
th
tx
0ttx?
th
tx
nh
kxn
k
nx
19
Chapter 2 LTI Systems
§ 2.3.4 LTI Systems with and without Memory
1,Discrete-time System
An LTI system without memory
nknh
2,Continuous-time System
tkth An LTI system without memory
20
Chapter 2 LTI Systems
§ 2.3.5 Invertibility of LTI Systems
LTI系统的可逆性
ty
System
th
tx Inverse System
th1
tx
identity system (恒等系统）
tthth 1
nnhnh 1
21
Chapter 2 LTI Systems
§ 2.3.6 Causality for LTI Systems
LTI系统的因果性
1,Discrete-time System
Causal
system 0,0 nnh
2,Continuous-time System
Causal
system 0,0 tth
22
Chapter 2 LTI Systems
Consider a LTI system
Causal Initial Rest (初始松弛）
For any time t0
0,0 tttx 00,y t t t
The system is initial rest (初始松弛）,
23
Chapter 2 LTI Systems
§ 2.3.7 Stability for LTI Systems (稳定性）
1,Discrete-time System
Stable
System

nh
n
Absolutely summable （绝对可加）nh
2,Continuous-time System
Stable
System dtth
th Absolutely integrable （绝对可积 ）
24
Chapter 2 LTI Systems
§ 2.3.8 The Unit Step Response of an LTI Systems
LTI系统的单位阶跃响应
Discrete-time System
nhnuns
Continuous-time System
thtuts
khns n
k

Unit Step Response
dhts t
Unit Step Response
1 nsnsnh
dt tdsth?
25
Chapter 2 LTI Systems
§ 2,4 Causal LTI Systems described by Differential
and Difference Equations
§ 2,4.1 Linear Constant-coefficient Differential Equations

k
kM
k
kk
kN
k
k dt
txdb
dt
tyda

00
ty
th
tx
一 经典解法
hpy t y t y t
Homogeneous solution
Natural Response
Particular Solution
Forced Response
26
Chapter 2 LTI Systems
输入 特解
C （常数） B （常数）
typtx
nt
1121 nnn BtBtB?
te?
tBe?
tt eBteB 2 1
α不是特征单根
α 是特征单根或t?cos t?sin tBtB s i nco s
21?
27
Chapter 2 LTI Systems
tueeKty tt 235 a
0,551 b 32 teKeKty tt
Example 2.14
txty
dt
tdy 2tuKetx t3?
ty 10 by
(a) The system is initial rest
Determine the output
28
Chapter 2 LTI Systems
由初始状态唯一决定ixc
零输入响应由零初始状态及输入共同决定ifc
零状态响应由初始状态及输入决定ic
自然响应函数形式由输入信号决定受迫响应
tyececty ptλfN
i
tλ
x
N
i
i
i
i
i


1
1
零输入零状态解法
tyecty ptλiN
i
i
1

经典解法三 两种响应分解形式的关系二 零输入、零状态解法
29
Chapter 2 LTI Systems
1,Zero-input response
0,2 tety tx
2,Zero-state response
0,55 32 teKeKty ttf
3,Full response
0,551 32 teKeKty ttf
Example 2.14
txtydt tdy 2tuKetx t3?
10ytyDetermine the output
30
Chapter 2 LTI Systems
§ 2.4.2 Linear Constant-coefficient Difference Equations
线性常系数差分方程
knxbknya M
k
k
N
k
k
00
Nth order
1,Recursive Solution（递推算法）
Example 2.15
nxnyny 121
nKnx(a) The system is initial rest,and the input
a-ynKnx 1 b,?
nyDetermine the output
31
Chapter 2 LTI Systems
2,Typical solution
nynyny ph
Homogeneous solution
Natural Response
Particular Solution
Forced Response
32
Chapter 2 LTI Systems
② Particular solution (Forced Response)
输入 特解
n?
nA?
nn BAn
α不是特征单根
α 是特征单根或
nk 0co s?
nk 0sin?
nBnB 0201 s i nco s
nypnx
2210 nKnKK 2210 nCnCC
0 0 b eK bn0 b Ce bn
33
Chapter 2 LTI Systems
3,零输入、零状态解法
nynyny fx
Zero-input
Response
4,两种响应分解形式的关系
Zero-state
Response
34
Chapter 2 LTI Systems
Example
Consider an LTI system,nxnynyny 2213
; 0,2 nnx n ;2/12,01 yy
Determine the full response ; 0,?nny
Solution
Full Solution
0,23/121221
3
1

n----ny nnnnn
Zero-input response Zero-state response
0,23/121
3
2

n- -ny nnn
Natural response Forced response
35
Chapter 2 LTI Systems
Discrete-time System
Continuous-time System
Causal
system 0,0 tth
Causal
system 0,0 nnh
Stable
System

nh
n
Stable
System dtth
36
Chapter 2 LTI Systems

k
kM
k
kk
kN
k
k dt
txdb
dt
tyda

00
ty
th
tx
hpy t y t y t
Homogeneous Solution
Natural Response
Particular Solution
Forced Response
xfy t y t y t
Zero-input Response Zero-state Response
37
Chapter 2 LTI Systems
§ 2,4.3 Block Diagram Representations of First-Order System
一阶系统的方框图表示
1,Discrete-time systems
nbxnayny 1
Three basic
operations
Addition
Multiplication
by a coefficient
Delay
nx1
nx2
nxnx 21?
nxnax
a
nx1?nxD
nbxnayny 1
Recursive Equation
nx b
D
ny
a?
38
Chapter 2 LTI Systems
2,Continuous-time systems
tbxtayty
taytbxty
daybxty t
Three basic
operations
Addition
Multiplication
by a coefficient
Integrator
tx1
tx2
txtx 21?
tx
a
tax
∫tx dxt
39
Chapter 2 LTI Systems
daybxty t
b ∫
a?
tytx
dtaytbxdtaytbxty ttt
0
0
dtaytbxtyty tt 0
0
Initial Condition
40
Chapter 2 LTI Systems
§ 2.5 Singularity Functions (奇异函数）
§ 2.5.1 The Unit Impulse as an Idealized Short Pulse
0 △ t
t
1
tt 0l i m
Example 2.16tueth t2
1
Example 2.17tueth t20
2
41
Chapter 2 LTI Systems
Example 2.16tueth t2
1
42
Chapter 2 LTI Systems
Example 2.17tueth t20
2
43
Define
Chapter 2 LTI Systems
§ 2.5.2 Defining the Unit Impulse through Convolution
单位冲激的卷积定义
For any x(t)txttx
For any normal,which is continuous at time t=0.t?
0 dtttgttg
tt
44
Chapter 2 LTI Systems
§ 2.5.3 Unit Doublets and Other Singularity Functions
单位冲激偶和其它奇异函数
dt tdtu11,
In general,
k
k
k dt
tdtu
2,
01 dttut For anyt?
01 tkkkk dt tddttut
45
Chapter 2 LTI Systems
3,Properties of Unit Doublets （冲激偶的性质）
01 dttu①
tutu 11②
③ttutut 00
11
4,Derivatives of different orders of the unit impulse
单位冲激的各阶积分
tutututu k 111
tututu rkrk

t i m e s k
46
Chapter 2 LTI Systems
作业：
2.1 2.5 2.7
2.10 2.11 2.12
2.22 (a) (c)
2.20 2.23 2.40 2.46 2.47
47
图解法步骤：
㈠khnhkh?
反折
㈡ 平移knh n
㈢ 求乘积knhkx?
㈣ 对每一个 n求和
knhkxnhnx
k



循环
knkxnx
k



knhkxny
k



nhnx
0,h n L n
48
dthxthtxty
tytx
th
Chapter 2 LTI Systems
thtxthtxty nmmnmn,5
00,4 ttxtttx
ththtxththtx 2121,3
thtxthtxththtx 2121,2
txththtx,1
49
卷积的微分、积分性质
thtxthtxty nmmnmn
Chapter 2 LTI Systems
Properties of LTI systems
An LTI system without memory
nknh
tkth An LTI system without memory
Invertibility of LTI Systems
tthth 1nnhnh 1
50
Chapter 2 LTI Systems
Discrete-time System
Continuous-time System
Causal
system 0,0 tth
Causal
system 0,0 nnh
Stable
System

nh
n
Stable
System dtth
51
Chapter 2 LTI Systems
txttxtt
0 dtttgttg
Defining the Unit Impulse through Convolution
52
Chapter 2 LTI Systems

k
kM
k
kk
kN
k
k dt
txdb
dt
tyda

00
ty
th
tx
hpy t y t y t
Homogeneous Solution
Natural Response
Particular Solution
Forced Response
xfy t y t y t
Zero-input Response Zero-state Response