1
Chapter 4 Problem Solution
Homework:
4.3 4.4 4.10 4.11 4.14 4.15 4.24
4.25 4.32 4.35 4.36 4.37 4.43
2
Chapter 4 Problem Solution
4.4 Determine the inverse Fourier transform:
442 a 1jX
πttx 4c o s11



2 0
02 2-
20 2
b 2
jX -2 0 2 ω
2
-2
jX 2
t tjtx? 22 s i n4?
3
Chapter 4 Problem Solution
4.10,(a) Determine the Fourier transform of
(b) Determine the numerical value of

2si n


t
tttx
dtt ttA
4
2 s i n?


Solution

4
22
3
s i n 1 1b
22
tA t d t X j d
t




jX
-2 0 2 ω
/2j?
/2j
4
Chapter 4 Problem Solution
tytg 331? 3 31 BA
4.11 Given the relationship
and
and given
Show that
thtxty
thtxtg 33
BtAytg?
Determine the value of A and B.
jXtx FjHth F
Solution
5
Chapter 4 Problem Solution
4.12 Consider the Fourier transform pair
(a) Find the Fourier transform of
(b) Determine the Fourier transform of
21
2

Fte
tte?
221
4
t
t
221
4
jte Ft
21
4
22

ej
t
t F
6
Chapter 4 Problem Solution
4.14 Consider a signal be a signal with Fourier transform,
Suppose we are given the following facts:
(1),is real and nonnegative.
(2),Where A is independent of t,
(3).
Determine a closed-form expression for,
txjX
tx
tx
tuAejXjF t21 1
22 djX
tueetx tt 12 2
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Chapter 4 Problem Solution
4.15 Let be a signal with Fourier transform,Suppose
we are given the following facts:
(1),is real.
(2),for,
(3).
Determine a closed-form expression for,
txjX
tx
0?tx 0?t
1 Re2 tjtX j e d t e
tx
tutetx t 2
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Chapter 4 Problem Solution
4.24 (a) Determine which,if any,of the real signals depicted in
Figure P4.24 have Fourier transform that satisfy each of the
following conditions:
1, 0RejX
2, 0ImjX
0 djX4.
0 djX5.
3,There exists a real such that is real. jXe j?
jX6,is periodic
(a),(d)
(e),(f)
(a),(b),(e),(f)
(a),(b),(c),(d),(f)
(b),(c),(e),(f)
(b)
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Chapter 4 Problem Solution
(b) Construct a signal that has properties (1),(4),and (5) and
does not have the others.
1, 0RejXtxtx
0 djX4, 00ttx
0 djX5, 0
0

tdt
tdx
3,There does not exist a real such that is real. jXe j?
is not even.tx
jX6,is not periodic
For example,753,,ttttx?
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Chapter 4 Problem Solution
jXtx4.25 Let denote the Fourier transform of the signal,
(a) FindjX?
1? 0 1 2 3
1
2tx
t
(b) Find0jX
(c) Find ; djX

(d) Evaluate
dejX j 2s i n2
(e) Evaluate;2 djX
(f) Sketch the inverse Fourier transform ofjXRe
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Chapter 4 Problem Solution
4.32 Consider an LTI system S with impulse response
Determine the output of S for each of
the following inputs:
1 14s i n t tth?
2/6co s a 1 ttx
kttx
k
k
3s i n21 b
0
2


1a 0yt?
2 1b s in 3 12y t t
12
Chapter 4 Problem Solution
4.35 In this problem,we provide example of the effects of
nonlinear changes in phase.
a 0ajH j aaj
thjHjH
1Hj



atgjH
12
tuaetth at 2?
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Chapter 4 Problem Solution
3/23c o s2/c o s3/3/c o s tttty
txty
t0
(b) When a=1,the input is
Roughly sketch both the input and the output,
ttt 3c o sc o s3/c o s
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Chapter 4 Problem Solution
4.36 Consider an LTI system whose response to the input
tueetx tt 3tueety tt 422is
(a) Find the frequency response of this system.
(b) Determine the system’s impulse response.
(c) Find the differential equation relating the input and
the output of this system.
3924Yj jHj X j j j
tueeth tt 23 42
6 8 3 9y t y t y t x t x t
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Chapter 4 Problem Solution
4.37,Consider the signal
1
10 t
tx
1?
tx
(a) Find the Fourier transformjX

2
2
dt
txd
-1 0 1 t
(1)
(-2)
(1)
Solution 1

22/s i n2


jX
Solution 2

22/s i n2


jX
16
Chapter 4 Problem Solution
(b) Sketch the signal
kttxtx
k
4~


1
10 t
tx~
1?4? 4

(c) Find another signal such that is not
the same as andtx
tgtg
kttgtx
k
4~


(d) Argue that,
jXjG



22
kjXkjG for all integers k.
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Chapter 4 Problem Solution
4.43,Let
t
tttxtg
s i nc o s 2
Assuming that is real andtx
1,0jX
Show that there exists an LTI system S such that
tgtx S


1 0
1
2
1

jX
jG
S can be
,2s in,21 πt ttth
18
19
Problems for Fourier Analysis
22 2 jejF
tf
20 1
1
t
t?sinA period of
Example 1
Determine the Fourier
transform oftf
20
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
21
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
22
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
23
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