1
The Continuous-Time Fourier Transform
Chapter 4
2
Chapter 4 Fourier Transform
§ 4.1 Representation of Aperiodic Signals,
The Continuous-Time Fourier Transform
1
tx
-T -T/2 –T1 0 T1 T/2 T t
101 s i n2 TkcTTa k
011 s i n2 kk TcTTa
3
Chapter 4 Fourier Transform
谱线变密?T T/2
0
14 a TT?
18 b TT?
116 c TT?
Figure 4.2
4
Chapter 4 Fourier Transform
–T1 0 T1 t
tx
1,0 Tttx
-T –T1 0 T1 T t
tx~
Consider an aperiodic Signals
5
Chapter 4 Fourier Transform
jtX j x t e d t
1 2 jtx t X j e d
Synthesis equation
Analysis equation
Fourier Transform Pair
1,A linear combination of complex exponentials.
factor
txjX ——Spectrum（频谱） of 2.
jXtx F
6
Chapter 4 Fourier Transform
011
0
jkXTjXTa kk
The Fourier coefficients of are proportional
to samples of the Fourier transform of one period of
katx~
tx~
Consider a periodic signal
txTtx ~~
Defining
o t h e r s 0
~ 00 Tttttx
tx
7
Chapter 4 Fourier Transform
§ 4.1.2 Convergence of Fourier Transforms
2,Dirichlet Conditions:
dttx 2
1,is square integrablext
8
Chapter 4 Fourier Transform
§ 4.1.3 Fourier Transforms of Typical Signals
Example 4.1
0 atuetx at
jatue Fat 1
2/2a
a/1
aa?
jX
aa?
jX?
2/?
2/
4/?
4/
9
Chapter 4 Fourier Transform
Example 4.2 0 aetx ta
22 20 a aae Fta
Example 4.3ttx
1 Ft?
21 F
10
Chapter 4 Fourier Transform
jtX j x t e d t
1 2 jtx t X j e d
Synthesis equation
Analysis equation
jXtx F
11,0Fate u t aaj
2222,0at F aea a
3,1
1 2
F
F
t?
11
Chapter 4 Fourier Transform
–T1 0 T1 t
1
tx
Example 4.4
t 0
t 1
1
1
T
T
tx
1 0 c
c
Xj
F
s i n c txt t
11 s i n2 TcT
t 0
t 1
1
1
T
T
tx
F
12
Chapter 4 Fourier Transform
Example 1
0 1-
0 0
0 1
s g n
t
t
t
t
t
1?
1
tsgn
jt 2s g n F
tue ata0lim
tue ata 0lim
Example 2
ut 1F
j
13
Chapter 4 Fourier Transform
Example 3 tjetx 0
F
02 tjetx 0
0tt 0F j te
14
Chapter 4 Fourier Transform
§ 4.2 The Fourier Transforms for Periodic Signals
00F0c o st
0?0
00F0s i n jjt
j?
0?0
j
tjkk eatx 0
02 kajX k
k
15
Chapter 4 Fourier Transform
T
2
-ω0 0 ω0 2ω0
jX
0?0
jX
Periodic impulses train
kTttx
k
Periodic square wave
16
Chapter 4 Fourier Transform
§ 4.3 Properties of the Fourier Transforms
§ 4.3.1 Linearity
jYtyjXtx FF
jbYjaXtbytax F
-2 -1 0 1 2 t
1?
1tx
-2 0 2 t
1tx2
tx12
-1 0 1 t
2 s in 24 s inFtx
17
Chapter 4 Fourier Transform
§ 4.3.2 Time Shifting
jXtx F
0 F0 tjejXttx
Example 4.9
0 1 2 3 4 t
2/3
1
tx
18
Chapter 4 Fourier Transform
§ 4.3.3 Conjugation and Conjugate Symmetry
jXtx F
jXjXtxtx
txtx
jXjX ImIm
jXjX ReRe
txtx
jXjX
jXjX
19
Chapter 4 Fourier Transform
real eventx real evenjX
real oddtx Purely imaginary oddjX
jXtxEv ReFjXjtxOd ImF
Example
tuetuee atatta
20
jXj
dt
txd nF
n
n
Chapter 4 Fourier Transform
§ 4.3.4 Differentiation and Integration
1,Differentiation
jXj
dt
tdx F
2,Integration
jXjXdx Ft 10
21
Chapter 4 Fourier Transform
dt
tdx
-2 0 2 t
1
-1
2
2
dt
txd
-2 0 2 t
(1)
(-2)
(1)
224 s inXj
2? 0 2
2
tx
t
Example
Example
jtu F 1
22
Chapter 4 Fourier Transform
§ 4.3.5 Time and Frequency Scaling
a
jX
aatx
1F
jX1
jX 2
2
2?
1
2xt
t 0?
11xt
t
2
0
2
23
Chapter 4 Fourier Transform
Example
jXtx F26 F tx
62xt?
More generally
ajbeajXabatx /F /1
/F 31 22 jX j e
24
Chapter 4 Fourier Transform
jXtx F
xjtX 2F
§ 4.3.6 Duality（对偶性）
Examples
1,t?
1
1
1 t
2,
0 t
T
xt
T
223,1xt t
214,xt t?
25
Chapter 4 Fourier Transform
§ 4.3.7 Differentiation in Frequency Domain
d jdXtjtx F
jXtx F
Examples
nnFn jt 2
1,nx t t?
2,x t t?
2
F 2
t
26
Chapter 4 Fourier Transform
-2 -1 0 1 2 t
jX1F3,?x t X j
ttttx 2c o sc o s1 2
0 1! 1 1F ajatuetn natn?
0 1 2 ajatute Fat?
F 114,? 0nx t a
aj
27
Chapter 4 Fourier Transform
§ 4.3.8 Frequency Shifting
0F0 jXetx tjjXtx F
jX
0?0?
0jX
0
jX
tx
tje 0?
0jX
tjetx 0?
tje 0
jX
tx
28
Chapter 4 Fourier Transform
§ 4.3.9 Parseval’s Relation
djXdttx 22 2 1
2?jX
——Energy-density spectrum
（能量谱密度）
Solution of Infinite Differentiation
dttxX0
djXx 2 10
29
Chapter 4 Fourier Transform
dttxE 2
0?
tdt
tdxD
Example 4.14 Evaluate the following time-domain expressions:
-1 -0.5 0 0.5 1 ω
2/?
jX
Figure (a)
j
j?
-1 0 1 ω
jX
Figure (b)
30
Chapter 4 Fourier Transform
Examples
sin t dt
t?
2
12,?
1 dtt
2
1
1 dtt?
si n1,?t dt
t
22
13,?
1
dt
t
22
1
21
dt
t
31
Chapter 4 Fourier Transform
-1 0 1 2 3 t
1tx
Example 4
Consider a signal with Fourier Transform
Evaluate the following frequency-domain expressions:
txjX
a X j d？
0 b X j？
c X j？
32
Chapter 4 Fourier Transform
§ 4.4 The Convolution Property
y t x t h tY j X j H j
Fx t X jFh t H j
tx
jX
th
jH
thtxty
jHjXjY?
33
Chapter 4 Fourier Transform
LPF filter
1
0
c
c
Hj
0
1
c
jH
c?
Series Systems
th1
jH1
th2
jH 2
jHth
34
Chapter 4 Fourier Transform
Example 4.15
0tt
jH
0
1
jH?
0?
Example 4.16
0
jHjH?
2
2
t
35
Chapter 4 Fourier Transform
0?
jH
jH?
2
2
This is an unstable system.
Example 4.17
tu
h t d t
36
Chapter 4 Fourier Transform
Example 4.18
1,Ideal LPF filter
0
1
c
jH
c?
/c
t0
c/c/?
tcth cc s i n?
37
Chapter 4 Fourier Transform
2,Causal LPF filter
2/1
1
11?
jH
tueth t
1
1
jjH
t0
1th
38
Chapter 4 Fourier Transform
The usefulness of the Convolution Property
Example 4.19,0 ;btx t e u t b
,0 ;ath t e u t a
y t x t h t
Solution
1 a b?
2 ab?
aty t t e u t
1 a t b ty t e e u tba
39
Chapter 4 Fourier Transform
Example 4.20
12s in s in?tt
tt
0s i n tyt t
1
12
sin t
t
2
21
sin t
t
40
Chapter 4 Fourier Transform
The Fourier Analysis of periodic signals
0j k tk
k
x t a e?
00 j k tk
k
y t a H j k e
41
例 图 1所示的 LTI系统中，子系统的频率响应分别如图 2和图 3
所示。
⑴ 试求整个系统的频率响应；
⑶ 若输入信号,试求系统的输出。? tttx 6s in
2
13c o s1
jH 2
jH1
tytx +
-
图 1
0
1
jH1
2? 2
图 2
0
1
jH 2
4? 4图 3
Chapter 4 Fourier Transform
42
Chapter 4 Fourier Transform
§ 4.5 The Multiplication property (modulation property)
（调制特性）
r t s t p t1
2R j S j P j
0 F 0 jts t e S j
F0 0 011c o s 22x t t X j X j
43
Chapter 4 Fourier Transform
Example 4.23
Determine the Fourier transform of the signal
2s in s in / 2ttxt t
2/1jX
2
1?
2
3?
2
3
2
1
0
44
Chapter 4 Fourier Transform
例 某连续时间 LTI系统的单位冲激响应为若输入信号,
试求整个系统的输出。
sin c os 6th t tt
k
x t t k
45
Chapter 4 Fourier Transform
1ht2htxt
cos 4?tcos 6?t
A B C
例 如图所示系统中,1 sin 4 th t t t2 sin,dtht d t t
若输入信号 si n 2?
txt
t
，试分别 A,B,C各点信号的频谱。
46
Chapter 4 Fourier Transform
§ 4.5.1 Frequency-Selective Filtering with Variable Center Frequency
中心频率可调的频率选择性滤波器
ty
tj ce
tx
tj ce?
tf
0
1
WW?
jH
tr
jX
0?
jY
0?c?
jH
WW?
0?WW?
jR
W2
c
0?
W2
jF
c
jG
47
Chapter 4 Fourier Transform
§ 4.6 Frequency-domain analysis of LTI systems
LTI系统的频域分析
thtx
jHjX
ty
jY
thtxty
jHjXjY?
1,Stable System
2,Linear Constant-Coefficients Differential Equations
k
kM
k
kk
kN
k
k dt
txdb
dt
tyda
00
48
Chapter 4 Fourier Transform
3,Partial-Fraction Expansion
01
1
1
01
1
1
asasasa
bsbsbsbsH
n
n
n
n
m
m
m
m
① ——Strictly proper rational function
真分式sHnm,?
②sHnm,?
sD sNcscscsH kk 01?
真分式
01F01 cjcjctctctc kkkk
49
Chapter 4 Fourier Transform
真分式的部分分式展开
① 分母具有 n个不同的单根
npspsps
sNsH
21
Example 4.25 Consider a stable LTI system
tx
dt
tdxty
dt
tdy
dt
tyd 234
2
2
Determine the unit impulse response
50
Chapter 4 Fourier Transform
② 分母具有 1个 r阶重根，其余均为单根
11
r
NsHs
s p D s
可用数学归纳法证明：
1
11 !
1
ps
sHps
ds
d
kr
A rkr
kr
k
1 1 1 1 11
1111
rr
rr
NsA A AHs
s p D ss p s p
51
Chapter 4 Fourier Transform
tuetuetutety ttt 3414121
Example 4.25
Consider the LTI system
tx
dt
tdxty
dt
tdy
dt
tyd 234
2
2
tuetx tIf the input,determine the outputyt
52
Chapter 4 Fourier Transform
Example
Determine the unit impulse response of the LTI system
tx
dt
txd
dt
txdty
dt
tdy
dt
tyd
2
2
3
3
2
2
23
tuetuettth tt 232
53
Homework:
4.3 4.4 4.10 4.11 4.12 4.14 4.15 4.24
4.25 4.32 4.35 4.36 4.37 4.43
Chapter 4 Fourier Transform
54
Chapter 4 Fourier Transform
ttx 2c o s1
t
ttttx 6c o s32 4c o s 2c o s2
t
tttx 4co s 2co s2
t
4?
2
jX 2
2?
4
F
2?
2
jX1
F
4?
2
jX 3
2?
4
32
6
32
6?
F
55
Chapter 4 Fourier Transform
jtX j x t e d t
1 2 jtx t X j e d
Synthesis equation
Analysis equation
jXtx F
11,0Fate u t aaj
2222,0at F aea a
3,1
1 2
F
F
t?
56
Chapter 4 Fourier Transform
0
1
W
W
jX
F
tWttx?s i n?
11 s i n2 TcT
t 0
t 1
1
1
T
T
tx
F
F 25,s g n t j
16,Fut j
0 07,2jt Fe
4.
57
Chapter 4 Fourier Transform
jYtyjXtx FF
jbYjaXtbytax F
jXtx F
0 F0 tjejXttx
jXtx F
Fx t X j
58
real oddtx Purely imaginary oddjX
real eventx real evenjX
jXtxEv ReFjXjtxOd ImF
jXj
dt
txd nF
n
n
jXjdt tdx F
Chapter 4 Fourier Transform
59
Chapter 4 Fourier Transform
a
jX
aatx
1F
jXtx F
xjtX 2F
d jdXtjtx F
jXtx F
60
Chapter 4 Fourier Transform
y t x t h tY j X j H j
Fx t X jFh t H j
djXdttx 22 2 1
0F0 jXetx tj
The Continuous-Time Fourier Transform
Chapter 4
2
Chapter 4 Fourier Transform
§ 4.1 Representation of Aperiodic Signals,
The Continuous-Time Fourier Transform
1
tx
-T -T/2 –T1 0 T1 T/2 T t
101 s i n2 TkcTTa k
011 s i n2 kk TcTTa
3
Chapter 4 Fourier Transform
谱线变密?T T/2
0
14 a TT?
18 b TT?
116 c TT?
Figure 4.2
4
Chapter 4 Fourier Transform
–T1 0 T1 t
tx
1,0 Tttx
-T –T1 0 T1 T t
tx~
Consider an aperiodic Signals
5
Chapter 4 Fourier Transform
jtX j x t e d t
1 2 jtx t X j e d
Synthesis equation
Analysis equation
Fourier Transform Pair
1,A linear combination of complex exponentials.
factor
txjX ——Spectrum（频谱） of 2.
jXtx F
6
Chapter 4 Fourier Transform
011
0
jkXTjXTa kk
The Fourier coefficients of are proportional
to samples of the Fourier transform of one period of
katx~
tx~
Consider a periodic signal
txTtx ~~
Defining
o t h e r s 0
~ 00 Tttttx
tx
7
Chapter 4 Fourier Transform
§ 4.1.2 Convergence of Fourier Transforms
2,Dirichlet Conditions:
dttx 2
1,is square integrablext
8
Chapter 4 Fourier Transform
§ 4.1.3 Fourier Transforms of Typical Signals
Example 4.1
0 atuetx at
jatue Fat 1
2/2a
a/1
aa?
jX
aa?
jX?
2/?
2/
4/?
4/
9
Chapter 4 Fourier Transform
Example 4.2 0 aetx ta
22 20 a aae Fta
Example 4.3ttx
1 Ft?
21 F
10
Chapter 4 Fourier Transform
jtX j x t e d t
1 2 jtx t X j e d
Synthesis equation
Analysis equation
jXtx F
11,0Fate u t aaj
2222,0at F aea a
3,1
1 2
F
F
t?
11
Chapter 4 Fourier Transform
–T1 0 T1 t
1
tx
Example 4.4
t 0
t 1
1
1
T
T
tx
1 0 c
c
Xj
F
s i n c txt t
11 s i n2 TcT
t 0
t 1
1
1
T
T
tx
F
12
Chapter 4 Fourier Transform
Example 1
0 1-
0 0
0 1
s g n
t
t
t
t
t
1?
1
tsgn
jt 2s g n F
tue ata0lim
tue ata 0lim
Example 2
ut 1F
j
13
Chapter 4 Fourier Transform
Example 3 tjetx 0
F
02 tjetx 0
0tt 0F j te
14
Chapter 4 Fourier Transform
§ 4.2 The Fourier Transforms for Periodic Signals
00F0c o st
0?0
00F0s i n jjt
j?
0?0
j
tjkk eatx 0
02 kajX k
k
15
Chapter 4 Fourier Transform
T
2
-ω0 0 ω0 2ω0
jX
0?0
jX
Periodic impulses train
kTttx
k
Periodic square wave
16
Chapter 4 Fourier Transform
§ 4.3 Properties of the Fourier Transforms
§ 4.3.1 Linearity
jYtyjXtx FF
jbYjaXtbytax F
-2 -1 0 1 2 t
1?
1tx
-2 0 2 t
1tx2
tx12
-1 0 1 t
2 s in 24 s inFtx
17
Chapter 4 Fourier Transform
§ 4.3.2 Time Shifting
jXtx F
0 F0 tjejXttx
Example 4.9
0 1 2 3 4 t
2/3
1
tx
18
Chapter 4 Fourier Transform
§ 4.3.3 Conjugation and Conjugate Symmetry
jXtx F
jXjXtxtx
txtx
jXjX ImIm
jXjX ReRe
txtx
jXjX
jXjX
19
Chapter 4 Fourier Transform
real eventx real evenjX
real oddtx Purely imaginary oddjX
jXtxEv ReFjXjtxOd ImF
Example
tuetuee atatta
20
jXj
dt
txd nF
n
n
Chapter 4 Fourier Transform
§ 4.3.4 Differentiation and Integration
1,Differentiation
jXj
dt
tdx F
2,Integration
jXjXdx Ft 10
21
Chapter 4 Fourier Transform
dt
tdx
-2 0 2 t
1
-1
2
2
dt
txd
-2 0 2 t
(1)
(-2)
(1)
224 s inXj
2? 0 2
2
tx
t
Example
Example
jtu F 1
22
Chapter 4 Fourier Transform
§ 4.3.5 Time and Frequency Scaling
a
jX
aatx
1F
jX1
jX 2
2
2?
1
2xt
t 0?
11xt
t
2
0
2
23
Chapter 4 Fourier Transform
Example
jXtx F26 F tx
62xt?
More generally
ajbeajXabatx /F /1
/F 31 22 jX j e
24
Chapter 4 Fourier Transform
jXtx F
xjtX 2F
§ 4.3.6 Duality（对偶性）
Examples
1,t?
1
1
1 t
2,
0 t
T
xt
T
223,1xt t
214,xt t?
25
Chapter 4 Fourier Transform
§ 4.3.7 Differentiation in Frequency Domain
d jdXtjtx F
jXtx F
Examples
nnFn jt 2
1,nx t t?
2,x t t?
2
F 2
t
26
Chapter 4 Fourier Transform
-2 -1 0 1 2 t
jX1F3,?x t X j
ttttx 2c o sc o s1 2
0 1! 1 1F ajatuetn natn?
0 1 2 ajatute Fat?
F 114,? 0nx t a
aj
27
Chapter 4 Fourier Transform
§ 4.3.8 Frequency Shifting
0F0 jXetx tjjXtx F
jX
0?0?
0jX
0
jX
tx
tje 0?
0jX
tjetx 0?
tje 0
jX
tx
28
Chapter 4 Fourier Transform
§ 4.3.9 Parseval’s Relation
djXdttx 22 2 1
2?jX
——Energy-density spectrum
（能量谱密度）
Solution of Infinite Differentiation
dttxX0
djXx 2 10
29
Chapter 4 Fourier Transform
dttxE 2
0?
tdt
tdxD
Example 4.14 Evaluate the following time-domain expressions:
-1 -0.5 0 0.5 1 ω
2/?
jX
Figure (a)
j
j?
-1 0 1 ω
jX
Figure (b)
30
Chapter 4 Fourier Transform
Examples
sin t dt
t?
2
12,?
1 dtt
2
1
1 dtt?
si n1,?t dt
t
22
13,?
1
dt
t
22
1
21
dt
t
31
Chapter 4 Fourier Transform
-1 0 1 2 3 t
1tx
Example 4
Consider a signal with Fourier Transform
Evaluate the following frequency-domain expressions:
txjX
a X j d？
0 b X j？
c X j？
32
Chapter 4 Fourier Transform
§ 4.4 The Convolution Property
y t x t h tY j X j H j
Fx t X jFh t H j
tx
jX
th
jH
thtxty
jHjXjY?
33
Chapter 4 Fourier Transform
LPF filter
1
0
c
c
Hj
0
1
c
jH
c?
Series Systems
th1
jH1
th2
jH 2
jHth
34
Chapter 4 Fourier Transform
Example 4.15
0tt
jH
0
1
jH?
0?
Example 4.16
0
jHjH?
2
2
t
35
Chapter 4 Fourier Transform
0?
jH
jH?
2
2
This is an unstable system.
Example 4.17
tu
h t d t
36
Chapter 4 Fourier Transform
Example 4.18
1,Ideal LPF filter
0
1
c
jH
c?
/c
t0
c/c/?
tcth cc s i n?
37
Chapter 4 Fourier Transform
2,Causal LPF filter
2/1
1
11?
jH
tueth t
1
1
jjH
t0
1th
38
Chapter 4 Fourier Transform
The usefulness of the Convolution Property
Example 4.19,0 ;btx t e u t b
,0 ;ath t e u t a
y t x t h t
Solution
1 a b?
2 ab?
aty t t e u t
1 a t b ty t e e u tba
39
Chapter 4 Fourier Transform
Example 4.20
12s in s in?tt
tt
0s i n tyt t
1
12
sin t
t
2
21
sin t
t
40
Chapter 4 Fourier Transform
The Fourier Analysis of periodic signals
0j k tk
k
x t a e?
00 j k tk
k
y t a H j k e
41
例 图 1所示的 LTI系统中，子系统的频率响应分别如图 2和图 3
所示。
⑴ 试求整个系统的频率响应；
⑶ 若输入信号,试求系统的输出。? tttx 6s in
2
13c o s1
jH 2
jH1
tytx +
-
图 1
0
1
jH1
2? 2
图 2
0
1
jH 2
4? 4图 3
Chapter 4 Fourier Transform
42
Chapter 4 Fourier Transform
§ 4.5 The Multiplication property (modulation property)
（调制特性）
r t s t p t1
2R j S j P j
0 F 0 jts t e S j
F0 0 011c o s 22x t t X j X j
43
Chapter 4 Fourier Transform
Example 4.23
Determine the Fourier transform of the signal
2s in s in / 2ttxt t
2/1jX
2
1?
2
3?
2
3
2
1
0
44
Chapter 4 Fourier Transform
例 某连续时间 LTI系统的单位冲激响应为若输入信号,
试求整个系统的输出。
sin c os 6th t tt
k
x t t k
45
Chapter 4 Fourier Transform
1ht2htxt
cos 4?tcos 6?t
A B C
例 如图所示系统中,1 sin 4 th t t t2 sin,dtht d t t
若输入信号 si n 2?
txt
t
，试分别 A,B,C各点信号的频谱。
46
Chapter 4 Fourier Transform
§ 4.5.1 Frequency-Selective Filtering with Variable Center Frequency
中心频率可调的频率选择性滤波器
ty
tj ce
tx
tj ce?
tf
0
1
WW?
jH
tr
jX
0?
jY
0?c?
jH
WW?
0?WW?
jR
W2
c
0?
W2
jF
c
jG
47
Chapter 4 Fourier Transform
§ 4.6 Frequency-domain analysis of LTI systems
LTI系统的频域分析
thtx
jHjX
ty
jY
thtxty
jHjXjY?
1,Stable System
2,Linear Constant-Coefficients Differential Equations
k
kM
k
kk
kN
k
k dt
txdb
dt
tyda
00
48
Chapter 4 Fourier Transform
3,Partial-Fraction Expansion
01
1
1
01
1
1
asasasa
bsbsbsbsH
n
n
n
n
m
m
m
m
① ——Strictly proper rational function
真分式sHnm,?
②sHnm,?
sD sNcscscsH kk 01?
真分式
01F01 cjcjctctctc kkkk
49
Chapter 4 Fourier Transform
真分式的部分分式展开
① 分母具有 n个不同的单根
npspsps
sNsH
21
Example 4.25 Consider a stable LTI system
tx
dt
tdxty
dt
tdy
dt
tyd 234
2
2
Determine the unit impulse response
50
Chapter 4 Fourier Transform
② 分母具有 1个 r阶重根，其余均为单根
11
r
NsHs
s p D s
可用数学归纳法证明：
1
11 !
1
ps
sHps
ds
d
kr
A rkr
kr
k
1 1 1 1 11
1111
rr
rr
NsA A AHs
s p D ss p s p
51
Chapter 4 Fourier Transform
tuetuetutety ttt 3414121
Example 4.25
Consider the LTI system
tx
dt
tdxty
dt
tdy
dt
tyd 234
2
2
tuetx tIf the input,determine the outputyt
52
Chapter 4 Fourier Transform
Example
Determine the unit impulse response of the LTI system
tx
dt
txd
dt
txdty
dt
tdy
dt
tyd
2
2
3
3
2
2
23
tuetuettth tt 232
53
Homework:
4.3 4.4 4.10 4.11 4.12 4.14 4.15 4.24
4.25 4.32 4.35 4.36 4.37 4.43
Chapter 4 Fourier Transform
54
Chapter 4 Fourier Transform
ttx 2c o s1
t
ttttx 6c o s32 4c o s 2c o s2
t
tttx 4co s 2co s2
t
4?
2
jX 2
2?
4
F
2?
2
jX1
F
4?
2
jX 3
2?
4
32
6
32
6?
F
55
Chapter 4 Fourier Transform
jtX j x t e d t
1 2 jtx t X j e d
Synthesis equation
Analysis equation
jXtx F
11,0Fate u t aaj
2222,0at F aea a
3,1
1 2
F
F
t?
56
Chapter 4 Fourier Transform
0
1
W
W
jX
F
tWttx?s i n?
11 s i n2 TcT
t 0
t 1
1
1
T
T
tx
F
F 25,s g n t j
16,Fut j
0 07,2jt Fe
4.
57
Chapter 4 Fourier Transform
jYtyjXtx FF
jbYjaXtbytax F
jXtx F
0 F0 tjejXttx
jXtx F
Fx t X j
58
real oddtx Purely imaginary oddjX
real eventx real evenjX
jXtxEv ReFjXjtxOd ImF
jXj
dt
txd nF
n
n
jXjdt tdx F
Chapter 4 Fourier Transform
59
Chapter 4 Fourier Transform
a
jX
aatx
1F
jXtx F
xjtX 2F
d jdXtjtx F
jXtx F
60
Chapter 4 Fourier Transform
y t x t h tY j X j H j
Fx t X jFh t H j
djXdttx 22 2 1
0F0 jXetx tj
