4 The continuous time Fourier transform
4.1 Representation of Aperiodic signals:
The Continuous time Fourier Transform
4.1.1 Development of the Fourier transform
representation of the continuous time Fourier
transform
4,The Continuous time Fourier Transform
4 The continuous time Fourier transform
(1) Example ( From Fourier series to Fourier transform )
4 The continuous time Fourier transform
(2) Fourier transform representation of Aperiodic
signal



T
tjk
k
k
tjk
k
dtetx
T
a
eatx
0
0
)(~
1
)(~
For periodic signal,)(~ tx
For aperiodic signal x(t),
)()(~)(~)( lim txtxtxtx T
T


或
4 The continuous time Fourier transform
T
4 The continuous time Fourier transform
When T,







T
T
T
k
d
T
txtx
0
0
2
)()(~
So )()( jXdtetxTa tj
k














dejX
ejkX
e
T
jkX
eatx
tj
k
tjk
k
tjk
T
k
tjk
k
T
)(
2
1
2
)(lim
)(
lim
lim)(
0
0
0
0
0
0
0
4 The continuous time Fourier transform
Fourier transform:





dejXtx
dtetxjX
tj
tj
)(
2
1
)(
)()(
Relation between Fourier series and
Fourier transform:


)(
0
0
|)(
|)(
1
s i g n a lA p er i o d i c
s i g n a l )( P er i o d i c
kk
kk
aTjX
jX
T
a


or
)()(?jXtx F
4 The continuous time Fourier transform
4 The continuous time Fourier transform
4.1.2 Convergence of Fourier transform
Dirichlet conditions:
(1) x(t) is absolutely integrable.
(2) x(t) have a finite number of maxima and minima
within any finite interval.
(3) x(t) have a finite number of discontinuity within
any finite interval,Furthermore,each of these
discontinuities must be finite.
dttx |)(|
4 The continuous time Fourier transform
4.1.3 Examples of Continuous time Fourier Transform
Example 4.1 4.2 4.3 4.4 4.5
Example (1)
)(2)()( 00 jXetx Ftj
Example (2)
)()()(c o s)( 000 jXttx F








dee
dejXtxSo l ut i on
tjtj
tj
)(2
2
1
)(
2
1
)(:
0
0
4 The continuous time Fourier transform
4.2 The Fourier Transform for Periodic Signal
Periodic signal:
)(2 00 ke Ftjk
thus


k
tjk
k eatx 0)(





k
k
F
k
tjk
k kajXeatx )(2)()( 00
Example 4.6 4.7 4.8
4 The continuous time Fourier transform
4.3 Properties of the Continuous time Fourier
Transform
then
)()(
)()(
jYty
jXtx
F
F


If
4.3.1 Linearity
)()()()( jbYjaXtbytax F
4 The continuous time Fourier transform
Example 4.9
then
)()(?jXtx FIf
4.3.2 Time Shifting
)()( 00 jXettx tjF
)()(
)(
2
1
)(
2
1
)(
)(
2
1
)(:Pr
0
0
0
0
)(
0




jXettxor
deejX
dejXttx
dejXtxoof
tjF
tjtj
ttj
tj








4 The continuous time Fourier transform
4.3.3 Conjugation and Conjugate Symmetry
then
)()(?jXtx F(1) If
)()( **?jXtx F







dejX
dejXtxoof
tj
tj
)(
2
1
)(
2
1
)(:Pr
*
**
then
)()( * txtx?(2) If
)()( * jXjX
)()(
)(
2
1)(:Pr
*

jXjXy i e l d
dejXtxF r o moof tj




4 The continuous time Fourier transform
then
)()()()( * txtxtxtx oe(3) If
)()()(
)()()(
)()()()()(



jXjjXtx
jXjXtxand
jXjXjjXjXjX
oI
F
o
eR
F
e
oeIR



)()()()(
s in)(c o s)(
]s in) ] [ c o s()([
)()(:Pr



jXjXjjXjX
t d ttxjt d ttx
dttjttxtx
dtetxjXoof
oeIR
oe
oe
tj












4 The continuous time Fourier transform
4.3.4 Differentiation and Integration
then
)()(?jXtx F(1) If
)()(' jXjtx F
dejXjtxoof tj)(2 1)(':Pr
then
(2) If
)()0()(1)(1 XjXjtx F
)()0()(
1
)(*)(
)(
1
)(
)()(
)(*)()(:Pr
1


XjX
j
tutx
j
tu
jXtx
tutxtxoof
F
F
F



)()(?jXtx F
Example 4.12
4 The continuous time Fourier transform
4.3.5 Time and Frequency Scaling
then
)()(?jXtx FIf
)(|| 1)( ajXaatx F
)/(
||
1
)(
)/(
||
1
2
1
)(
2
1
)(:Pr
ajX
a
atx
deajX
a
dejXatxoof
F
tj
atj







)()(?jXtx FEspecially,
4 The continuous time Fourier transform
4.3.6 Duality
then
)()(?jXtx FIf
)(2)( xjtX F
)(2)(
)()(2
)(
2
1
)(
:
)(
2
1
)(:Pr











xjtX
dtejtXx
dtejtXx
andte x c hange
dejXtxoof
F
tj
tj
tj
Example 4.13
4 The continuous time Fourier transform
4 The continuous time Fourier transform
4.3.7 Parseval’s Relation
then
)()(?jXtx FIf























djX
djXjX
ddtetxjX
dtdejXtx
dttxtxdttxoof
tj
tj
2
*
*
*
*2
|)(|
2
1
)()(
2
1
])()[(
2
1
])(
2
1
)[(
)()(|)(|:Pr
Example 4.14
djXdttx 22 |)(|2 1|)(|
4 The continuous time Fourier transform
4.4 The Convolution Property
h(t)
H(j?)
x(t) y(t)=x(t)*h(t)
X(j? ) Y(j?)=X(j?)H(j?)
Consider a LTI system:
Example 4.15 4.16 4.17 4.19 4.20
4.4.1 Examples
)()()()(*)()( jHjXjYthtxty F
4 The continuous time Fourier transform
4 The continuous time Fourier transform
4.5 The Multiplication Property
The multiplication(modulation) property:
Example 4.21 4.22 4.23
)(*)(2 1)()()()( jPjSjRtptstr F
s(t)
p(t)
r(t)
4 The continuous time Fourier transform
4 The continuous time Fourier transform
4.5.1 Frequency-Selective Filtering with
Variable Center Frequency
A Bandpass Filter,
4 The continuous time Fourier transform
4 The continuous time Fourier transform
4.6 Tables of Fourier Properties and
of Basic Fourier Transform Pairs
Table 4.1
Table 4.2
4 The continuous time Fourier transform
4.7 System Characterized by Linear Constant-
Coefficient Differential Equation
Constant-coefficient differential equation:


M
k
k
k
k
N
k
k
k
k dt
tdxb
dt
tyda
00
)()(
Fourier transform:


M
k
k
k
N
k
k
k jXjbjYja
00
)()()()(
Define,)(
)(
)(
)(
)(
)(
0
0 f u n c t i o ns y s t e m
ja
jb
jX
jY
jH
N
k
k
k
M
k
k
k

Example 4.24 4.25 4.26
4 The continuous time Fourier transform
Problems:
4.3 4.4(a) 4.10 4.11 4.14 4.15 4.24
4.25 4.32(a)(b) 4.35 4.36