1
Chapter 2 Problems Solution
4?N
2.5

e l s e w h e r e 0,
90 1,nnx

e l s e w h e r e 0,
0 1,Nnnh
014,54 yynhnxny
Determine the value of N.
2
Chapter 2 Problems Solution
2.7 A linear system S has the relationship
kngkxny
k
2


between its input and outputnynx4 nunung
a 1x n n26y n u n u n
b 2x n n48y n u n u n
(c) S is time-varying.
d x n u n2y n u n u n
21u n n n
3
Chapter 2 Problems Solution
2.10 Suppose that


e l s e w h e r e 0,
10 1,ttx
10,/ aatxth
thtxty
dttyd
(a) Determine
(b) If contains only three
discontinuities,what is the value of a?
Solution,
ty
a
0 a 1 1+a t
4
Chapter 2 Problems Solution
2.12 Let


k
t kttuety 3?
Show that for tAety 30 t
Determine the value of A.
3 3
0
1 0 t 3
1
t k t
k
y t e e ee



3
1
1A e
5
Chapter 2 Problems Solution
20 1
2
3
tx
t
t?sin
th
20 1
1
t
2.22(c)
one period of



0 t 1
2
1 c o s 1 t 3
2
- 1 c o s 3 t 5
0 t 5
t
yt
t





6
Chapter 2 Problems Solution
2.23
tT? 0 T T2T2?
1

1? 0 1
1
t
th
Determine and sketch for the following value of T:thtxty
(a) T=4 (b) T=2 (c) T=3/2 (d) T=1



k
kTttx?

kk
y t t k T h t h t k T?




7
Chapter 2 Problems Solution
1
ty
(a) T=4
-5 -4 -3 -1 0 1 3 5 t

(d) T=1
ty
1
0 t
1ty
-3 -1 0 1 3 t
(b) T=2

1
ty
0 t
(c) T=3/2

8
Chapter 2 Problems Solution
2.40 (a) an LTI system,
dxety t t 2
What is the impulse response for this systemth
(b) Determine the response oftx
1? 0 2
1
tx
t
2a 2th t e u t
14b 1 1 1 4tty t e u t e u t
9
Chapter 2 Problems Solution
2.46 Consider an LTI system S and a signal
12 3 tuetx t
tytx? andtuety
dt
tdx t23If
Determine the impulse response of S.th
2131 12 th t e e u t
10
Chapter 2 Problems Solution
2.47 An LTI system with impulse responseth
0
tytx 00?
0 2 t
1
ty0
In each of these cases,determine
whether or not we have enough
Information to determine the outputty
00d x t x t h t h t
We have not enough information to determine the output
00e x t x t h t h t
0 y t y t
11
例 1 已知一 LTI系统在输入信号 的作用产生的输出为试求该系统在信号 的作用产生的输出 。
1ft1yt
2ft2yt
时域分析例题
0 2 t
tf1
1
0 1 2 t
ty1
1L
0 4 t
2ft
1
0 1 2 3 t
3ft
1
-1
12
时域分析例题
-1/2 0 1 t
tf11
-1 0 1 t
ty11
L
-1/2 0 1/2 1 3/2 t
2ft
1
1/2
3/2
2yt
L
-1 -1/2 0 1/2 1 3/2 t
1
1/2
3/2
11 1 / 2f t f t11 1 / 2y t y t
13
时域分析例题
0 1 2 t
tf1
1
-1
0 1 2 3 4 t
1yt
1
-1
2ft
0 1 2 3 t
1
14
1? 0 1
1
23
th
t
2
例 2 已知一 LTI系统的单位若输入信号为单位阶跃冲激响应 如图所示，th
信号,试求其输出tu
23|?tty
3 23
2
11|
4ty t h d
时域分析例题
15
时域分析例题例 3 图 1所示的 LTI的单位冲激响应如图 2所示，若子系统
，试求子系统 的单位冲激响应。2
2 nununh1hn
nh2nh1nh2nynx
图 1

nh
n50
1
2 31 4
5
6
4
1
7
10 8
11
0,1,2,3,4n,1,2,3,3,11nh
16
时域分析例题例 4 已知离散时间信号,要求：nf n a u n?
1.画出 的波形；1g n f n a f n
2.求出满足 的序列 。1 112 nh n f n u n u n

g n n
hn
17
时域分析例题
nh2nh1nynx
图 1
例 5 某离散时间 LTI系统的系统框图如图 1所示，已知两个子系统的单位冲激响应分别为
12s i n 8,nh n n h n a u n
若输入信号,试求系统的输出yn1x n n a n
s in 8y n n?
18
时域分析例题
1htxt
3ht
2ht
yt
例 6 某连续时间系统由三个子系统组成，其中总系统及子系统的单位冲激响应如图所示，试求
3ht
0 7
1
1ht
t0 1 7 8
2
ht
t 0 1 2 3
1
2ht
t