연속신호 푸리에 변환의 성질
1
Properties of continuous-time Fourier transform
Linearity
Time shifting
• Example: Find the Fourier transform of )
( )
( )
( )
(t by t aX j
bX j
ax
) (
)
( t t
0e
0X j x
j t) ) (
( 0
0 ( ) ( ) j X j t
t
j X j X j e
e
) 5 . 2 ( )5 . 2 ( )
(t x1 t x1 t x
Properties of continuous-time Fourier transform
Conjugation
Conjugate symmetry
• x(t) is real
• Example: check the conjugate symmetry for Fourier transform of )
(
* )
(
* t X j
x
) (
* )
( jw X jw
X
) ( )
(
| ) (
|
| ) (
|
)}
( Im{
)}
( Im{
)}
( Re{
)}
( Re{
jw X jw
X
jw X jw
X
jw X jw
X
jw X jw
X
) ( )
(t e u t x at
Properties of continuous-time Fourier transform
Differentiation and Integration
• Example. Find the Fourier transform of )
) (
( j
X j
dtt
dx 1 ( ) (0) ( )
)
(
X j Xd j
t x
) ( )
(t u t
x
) 1 (
)
(
j t
u
Properties of continuous-time Fourier transform
Time and frequency scaling
a X j at a
x
|
| ) 1 (
j
X t
x( )
Properties of continuous-time Fourier transform
Duality
d e j X t
x ( ) j t
2 ) 1 (
x t e dt j
X(
) ( ) jtSynthesis and analysis equations have similar forms
X jt e dt
x jt
( )2 ) 1
(
) ( 2 )
( jt x
X
New property
Properties of continuous-time Fourier transform
Example
• Find the Fourier transform of using duality property.2
1 ) 2
( t t
g
Properties of continuous-time Fourier transform
Other properties of Fourier transform can be derived using the duality property
dj t dX
jtx ( )
)
(
)) (
( )
( 0
0
x t X j
ej t Frequency-shifting property
Properties of continuous-time Fourier transform
Parseval’s relation
X j d dtt
x 2 ( ) 2
2 ) 1
(
Energy density spectrum
Properties of continuous-time Fourier transform
Convolution property
) (
) (
) (
) (
* ) ( )
( t h t x t Y j H j X j
y
Frequency response
) (
) ) (
(
j X
j j Y
H
LTI system can be characterized by the frequency response .
Properties of continuous-time Fourier transform
Frequency response of an LTI system
0
Low pass filter
Properties of continuous-time Fourier transform
Example
• Find the frequency response of
• Find the frequency response of
• Find the frequency response of
) (
)
(t t t0 h
dt t t dx
y ( )
) (
c
j c
H
|
| 0
|
| ) 1
(
Properties of continuous-time Fourier transform
Example
• Consider the an LTI system with the impulse response , 0, when the input is given by x , 0
• Consider the output of an ideal lowpass filter when the input and impulse response are given by
t t t
h
t t t
x
c i
) sin( ) sin (
Multiplication property
Use duality in the convolution property
• Example. find the Fourier transform of following signals
s t p t R j S j P j d
t
r ( ) ( ( ))
2 ) 1 ( )
( ) ( )
(
t t
p( ) cos
0) ( ) cos(
)
(t 0t s t
r
Amplitude modulation) ( ) cos(
)
(t 0t r t
z
Frequency response of continuous-time system
Continuous-time filter
• Frequency shaping filter
• Frequency selective filter: some frequencies are undistorted while other frequencies are eliminated or attenuated.
• Lowpass filter
• Highpass filer
• Bandpass filter
Systems characterized by linear constant-coefficient differential equations
Differential equation that describes an LTI system
From the differential equation, we can obtain the frequency response of the LTI system
M
k
k k k N
k
k k
k dt
t x b d
dt t y a d
0 0
) ( )
(
N
k
k k M
k
k k
a j
b j
j X
j j Y
H
0 0
) (
) ( )
( ) ) (
(
Frequency response of continuous-time system
First order RC lowpass filter
R
C vc(t) )
(t vs
) (t
vr ( ) ( ) ( )
t v t
dt v t
RC dvc c s output
input
RCj j
V j j V
H
s c
1
1 )
( ) ) (
(
) 1 (
)
( e u t
t RC
h RC
t
Frequency response of continuous-time system
First order RC highpass filter
R
C vc(t) )
(t vs
) (t vr
dt t RC v t
dt v t
RC dvr r s( )
) ) (
(
output
input
RCj RCj j
V j j V
H
s r
( ) 1
) ) (
(
) ( )
( )
(t e u t t
h RC
t
Frequency response of continuous-time system
Example
• Find the impulse response using the partial fraction method )
( ) 2
) ( ( ) 3
4 ( ) (
2 2
t dt x
t t dx
dt y t dy dt
t y
d
Basic Fourier transform pairs
Formula
1
,Re( ) 0) (
0 ) Re(
1 , )
(
) 1 (
) (
1 )
(
|
| 0
|
| ) 1
sin (
sin 2
|
| 0
|
| ) 1
(
) (
2 ) ( 2 1
2
1 1
1
0 0
j a t a
u te
j a t a
u e
t j u
t
W j W
t X Wt
T T
t T t t
x e
at at
t j
