1
Chapter 6 Problem Solution
6.23 Shown in Figure 6.23 is for a lowpass filter,Determine
and sketch the impulse response of the filter for each of the
following phase characteristics:
jH
a 0Hj
0
1
c
jH
c?
t tth cs i n?
Tt Ttth cs in
TjH(b),where T is a constant.



0 2/
0 2/

- πjH
(c)
t tth c 2/s in2 2
2
Chapter 7 Problem Solution
7.3 Determine the Nyquist rate corresponding to each of the
following signals:
t,t,tx 0004s i n0002co s1 a
t t,tx 0004s i n b

2
0 0 04s in c?


t
t,tx
000,4?M 0 0 0,82 Ms
000,4?M 0 0 0,82 Ms
0 0 0,8?M 0 0 0,162 Ms
3
Chapter 7 Problem Solution
7.6
11,0jX
tx1
tx2
tw
nTttp
n


twp
22,0jX
twp
twDetermine the maximum sampling interval T such that
is recoverable from through the use of an ideal LPF.
maximum
sampling interval
21
m a x
2

sT
4
Chapter 7 Problem Solution
7.9 Consider the signal

2
50s i n?


t
ttx
which we wish to sample with a sampling frequency of
to obtain a signal with Fourier transform,Determine
the maximum value of for which it is guaranteed that
1 5 0?s
0 75 jXjG
tgjG
0?
100? 0
50
jX
100
0?
50
jG
100100
150?150?
100
500?
5
Chapter 8 Problem Solution
tm
0?ty
ttxttgtm 000,4s i n21000,2c o s





000,2 0
000,2 2
jH
8.3 Determine,
000,2,0jX
ttxtg?0 0 0,2s i n?
t?2000c o s
tg
ty
LPF
jH
ty
Solution
Be out of the passband of LPF
6
8.22
In Figure (a),a system is shown with input and output
The input signal has the Fourier transform shown in Figure (b)
Determine and sketch,
Chapter 8 Problem Solution
jX
jY
txty
0
1
W3W3?
jH2
txtr1tr2
W3? W3
1
jH1
W5W5?
tytr3
Wt3cosWt5cos
jX
W2? 0 W2
1Figure (a)
Figure (b)
7
Problems for Fourier Analysis
22 2 jejF
tf
20 1
1
t
t?sinA period of
Example 1
Determine the Fourier
transform oftf
8
Example 2 A real continuous-time signal with
Fourier transform,and
1,If is even,determine,
2,If is odd,determine,
tf
tf
tf
tf
tf
jFjFln
Problems for Fourier Analysis
211,1ft t
22,1 tft t
9
例 3 试计算下列无穷积分
s ina?t dtt

2sin
b?t dtt




3sin
c?t dtt




2sin
d c o s?t t d tt



Problems for Fourier Analysis
10
例 4 试计算下列卷积积分
4a c o stx t e u t t
223 c o s 3 s in 3b c o s 23t t t tt
其中,θ为任意常数。
Problems for Fourier Analysis
11
Problems for Fourier Analysis
Example5 Consider the following LTI systems
with impulse response:
1,
t tth4s i n?
2 8s i n4s i n t ttth2.
t ttth 8c o s4s i n?3.
tttx 6s i n2co s
ty
If the input is
Determine the output
12
5 0?
Xj
5?
5htxt
pt
pytyt
例 在如图 1所示系统中,输入信号 的频谱如图 2所示。
已知,要求：
xt
sin 2? tht t
⑴ 画出图中信号 和 的频谱；
⑵确定 T的取值范围，以使信号 能从 中恢复。
ytpyt
ytpyt
1pt
-2T -T -T1 0 T1 T 2T t

Problems for Fourier Analysis
13
例,在如图所示的系统中
f t F jh t H j?


,2
n
s t t nT T?
⑴ 取何值时能够无失真地从 中恢复出？
⑵ 在上述条件下 通过滤波器 时产生的能量损失为多少？（用百分比表示）
pBytAyt
ftHj?
Ayt
Bytft
st
ht 0
1
Hj?
P P?
0
2
Fj?
2? 2
Problems for Fourier Analysis
14
Problems for Fourier Analysis
Example6 In Figure (a),a system is shown with input signal
and output signal,If the following information are given.
t tdtdth c2s in1 cjejH /22 t tth c3s i n3?
tuth?41,DeterminejH1
2,Determine the impulse response of the whole system,th
3,If the input signal,
determine the output signal
2/co s2s i n tttx cc
ty
th1th4th3
jH 2
tytx +
-
ty
tx
Figure (a)
15
Problems for Fourier Analysis
s i n2,2/ c
cc
tht
tt


1
21,
0
c
c
j ω
Hj





3,c o s / 2cy t t
16
例 7 研究图 1所示的连续时间系统，其中,
1
sin 3 tht
t?
和 的波形如图 2所示。?1Hj
2Hj?
1 htxt
yt
2?Hj
3?Hj 3 0?
3Hj?
3?
1
3 0?
2Hj?
3?
1
图 1 图 2
1.试求整个系统的频率响应；
2.若输入信号,其中
1 kx t x t t k?



1 0,2 5 0,2 5x t u t u t
试求系统的输出 。?yt
Problems for Fourier Analysis
17
Problems for Fourier Analysis
例 8 如图所示系统中，已知,cM cMc 1
概略画出,,和21Mctytr1tr3tr2
的频谱。说明整个系统等价于一个带通滤波器，
并用 和 确定带通滤波器的上、下截止频率
1,c 2?
jH 1ty
tf
tc?cos
jH 2
M 0 M?
jF
tr1tr2tr3
0
1
2?2
jH 2
0
1
1?1
jH1
1