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Chengbin Ma UM-SJTU Joint Institute
Class#11
- Linearity and symmetry properties (3.9)
- Time- and frequency-shift properties (3.12)
- Scaling properties (3.15)
- Convolution property (3.10)
- Differentiation and integration properties (3.11)
Slide 1
Chengbin Ma UM-SJTU Joint Institute
Review of Previous Lecture
Slide 2
)()(
)()(
)(2
1)(
jXtx
dtetxjX
dejXtx
FT
tj
tj
A non-periodic signal can be
considered as a periodic
signal whose period is
infinite.
Extend FS to solve FT
Three Dirichlet Conditions
Chengbin Ma UM-SJTU Joint Institute
Typical FTs
Slide 3
2222
10),(
a
j
a
a
jaatue
FTat
1)(FT
t
0)( 0tj
FT
ett
)(2 00
FTtj
e
0
00
00
0
)sin()sinc(
sinc2otherwise ,0
|| ,1
T
TT
TTTt FT
Does not converge when a=0!
Chengbin Ma UM-SJTU Joint Institute
How about u(t) ?!
Slide 4
jaatue
FTat
10),(
)(1
)(
jtu
FT
Why not let a=0, then
jtu
FT 1)(
WRONG!
CORRECT
Because the real part of 1/(a+j)
has a spike at origin when a approaches 0,
whose integral is independent of a:
0
0 2222
22
arctan2
2
1Re
a
da
ad
a
a
a
a
ja
Strictly speaking u(t) has no FT
representation because u(t) is not absolutely
integrable. The FT here is actually an
approximation.
and
0 ,
0 ,0lim
220
a
a
a
Chengbin Ma UM-SJTU Joint Institute
Class#11
- Linearity and symmetry properties (3.9)
- Time- and frequency-shift properties (3.12)
- Scaling properties (3.15)
- Convolution property (3.10)
- Differentiation and integration properties (3.11)
Slide 5
Chengbin Ma UM-SJTU Joint Institute
Summary of Properties (1)
Slide 6
)}(Im{)()},(Re{)( decomp.) odd-(Even
)()()( signals) odd and (Real
)()()( signals)even and (Real
)()( signals) (Real
:symmeteryn Conjugatio
)()( :nConjugatio
)()( :Reversal Time
)()()()()()( :Linearity
*
*
*
**
jXjtxjXtx
jXjXjX
jXjXjX
jXjX
jXtx
jXtx
jbYjaXjZtbytaxtz
FT
o
FT
e
FT
FT
FT
Purposes:
1) understand better the
nature of FT and FS
2) Simplify the calculation
of FT and FS
Chengbin Ma UM-SJTU Joint Institute
Summary of Properties (2)
Slide 7
)(||
1)( :Scaling
))(()( :Shift-Frequency
)()( :Shift-Time 00
ajX
aatx
jXtxe
jXettx
FT
FTtj
tjFT
Note: only FTs are shown as examples because they are more
general. For FSs, these properties obviously exist too.
Chengbin Ma UM-SJTU Joint Institute
Examples (1)
Problem 3.16 (a)
2 3
2 3
2
2
( ) 2 ( ) 3 ( ) ( ) ?
Solution:
1 1( ) , ( )
2 3
1 12 ( ) 3 ( ) 2 3
2 3 5 6
t t
t t
t t
x t e u t e u t X j
e u t e u tj j
je u t e u t
j j j j
jatue
FTat
1)(
Chengbin Ma UM-SJTU Joint Institute
Examples (2)
2 2
2 2
( ) ( ) ?
Solution:
( ) ( )
( ) ( ) ( )
( ) ( ) ( ) ( )2 2 2 ( )
2 2
1( ) Re ( ) Re
2( ) 2 ( )
a t
at
at at
at at
e
e
e
x t e X j
y t e u t
x t e u t e u t
e u t e u t y t y ty t
ay t Y j
a j a
ax t y t
a
)}(Re{)( jXtxe
jajX
atuetx at
1)(
0),()(
Note: u(t) is undefined when
t=0. This representation may
not be accurate.
Chengbin Ma UM-SJTU Joint Institute
Example (3)
Slide 10
2)cos( ,3
2 :sinusoidscomplex theof FT 2,
propertyLiearity 1,
:Hints
)3sin(3)cos(2)(
oftion representa FT theDerive
00
jtjt
FTtj
eet
)-(e
tttx
Chengbin Ma UM-SJTU Joint Institute
Linearity Property
FT ( ) ( ) ( ) ( ) ( ) ( )
FS ( ) ( ) ( ) [ ] [ ] [ ]
z t ax t by t Z j aX j bY j
z t ax t by t Z k aX k bY k
)()(
)()(
)(2
1)(
jXtx
dtetxjX
dejXtx
FT
tj
tj
T
tjk
k
tjk
dtetxT
kX
ekXtx
0
0
)(1
][
][)(
Chengbin Ma UM-SJTU Joint Institute
Time Reversal Property
FT ( ) ( )
FS ( ) [ ]
x t X j
x t X k
T
tjk
k
tjk
dtetxT
kX
ekXtx
0
0
)(1
][
][)(
)()(
)()(
)(2
1)(
jXtx
dtetxjX
dejXtx
FT
tj
tj
( )
( ) ( )
( ) ( )
tj t j
j
x t e dt x e d
x e d X j
Chengbin Ma UM-SJTU Joint Institute
Conjugation Property
* *
* *
FT ( ) ( )
FS ( ) [ ]
x t X j
x t X k
*
**
** *
* * *
( ) ( )
( ) ( ) ( )
( ) ( )
( ) ( ) FT{ ( )}
j t
j t j t
j t j t
j t
X j x t e dt
X j x t e dt x t e dt
x t e dt x t e dt
X j x t e dt x t
)()(
)()(
)(2
1)(
jXtx
dtetxjX
dejXtx
FT
tj
tj
Chengbin Ma UM-SJTU Joint Institute
Conjugate Symmetry
For real signals
*
*
If the signal is real, then the Fourier representation
is complex-conjugate symmetric.
FT ( ) ( )
FS [ ] [ ]
X j X j
X k X k
*
* *
*
*
( ) ( )
( ) ( ), ( ) ( )
( ) ( )
( ) ( )
x t x t
x t X j x t X j
X j X j
X j X j
* *
* *
FT ( ) ( )
FS ( ) [ ]
x t X j
x t X k
Chengbin Ma UM-SJTU Joint Institute
Symmetry Properties
For even signals
*
*
If the signal is real and even, then the Fourier
representation is real and even.
FT ( ) ( ) ( )
FS [ ] [ ] [ ]
X j X j X j
X k X k X k
* *
*
*
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( )
( ) ( )
( ) ( )
x t x t X j X j
x t x t X j X j
X j X j X j
X j X j
X j X j
Chengbin Ma UM-SJTU Joint Institute
Symmetry Properties
For odd signals
*
*
If the signal is real and odd, then the Fourier
representation is purely imaginary and odd.
FT ( ) ( ) ( )
FS [ ] [ ] [ ]
X j X j X j
X k X k X k
Fig. 3.51, P259
Chengbin Ma UM-SJTU Joint Institute
Symmetry Properties (Proof)
For real and odd signals
* *
*
* *
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( )
x t x t X j X j
x t x t X j X j
X j X j X j
X j X j X j X j
X j X j
Chengbin Ma UM-SJTU Joint Institute
Symmetry Properties
Even-odd decomposition of real signals
If the signal is real, then
FT ( ) Re{ ( )}, ( ) Im{ ( )}
FS ( ) Re{ [ ]}, ( ) Im{ [ ]}
e o
e o
x t X j x t j X j
x t X k x t j X k
Chengbin Ma UM-SJTU Joint Institute
Symmetry Properties (Proof)
Even-odd decomposition for real signals
* * *
*
*
( ) ( ) ( ) ( )( )
2 2
( ) ( ) ( ) ( )( )
2 2
( ) ( ) ( ) ( ) ( ) ( )
( ) ( )( ) Re{ ( )}
2
( ) ( )( ) Im{ ( )}
2
e
o
e
o
x t x t X j X jx t
x t x t X j X jx t
x t x t X j X j X j X j
X j X jx t X j
X j X jx t j X j
)()(* jXjx
Chengbin Ma UM-SJTU Joint Institute
Time-Shift property
0
0 0
0
0
FT ( ) ( )
FS ( ) [ ]
j t
jk t
x t t e X j
x t t e X k
0
0
0 0
( )
0( ) ( )
( ) ( )
t tj tj t
j t j tj
x t t e dt x e d
e x e d e X j
)()(
)()(
)(2
1)(
jXtx
dtetxjX
dejXtx
FT
tj
tj
Chengbin Ma UM-SJTU Joint Institute
Frequency-Shift Property
0 0
0
FT ( ) ( ( ))
FS ( ) [ ]
j t
jk t
e x t X j
e x t X k k
0 0
0
FT ( ) ( ( ))
FS ( ) [ ]
j t
jk t
e x t X j
e x t X k k
( )( ) ( ) ( ( ))j t j t j te x t e dt x t e d X j
Chengbin Ma UM-SJTU Joint Institute
Scaling Property
Time and frequency scaling property
1FT ( ) ( )
FS ( ), 0 [ ]
x at X ja a
x at a X k
Chengbin Ma UM-SJTU Joint Institute
Scaling Property (Proof)
Proof of time and frequency scaling property: FT
1 1( ) , 0 ( ) , 0
( )1 1
( ) , 0 ( ) , 0
1 1( ) ( )
j ja a
atj t
j ja a
ja
x e d a x e d aa a
x at e dt
x e d a x e d aa a
x e d X ja a a
)()(
)()(
)(2
1)(
jXtx
dtetxjX
dejXtx
FT
tj
tj
Chengbin Ma UM-SJTU Joint Institute
Scaling Property (Proof)
Proof of time and frequency scaling property: FS
0
0 0( ) ( )
0
( ) [ ]
( ) [ ] [ ]
2( 0, ( ) is periodic with period )
jk t
k
jk at jk a t
k k
x t X k e
x at X k e X k e
a x ata
The FS coefficients of x(t) and x(at) are identical;
the scaling operation simply changes the harmonic spacing from 0 to a0.
Chengbin Ma UM-SJTU Joint Institute
Class#11
- Linearity and symmetry properties (3.9)
- Time- and frequency-shift properties (3.12)
- Scaling properties (3.15)
- Convolution property (3.10)
- Differentiation and integration properties (3.11)
Slide 25
Chengbin Ma UM-SJTU Joint Institute
Convolution Property
Perhaps the most important property of Fourier representations is the convolution property.
FT ( ) ( )* ( ) ( ) ( ) ( )
FS ( ) ( ) ( ) [ ] [ ] [ ]
where ( ) ( ) ( ) ( )
(periodic convolution)
T
y t h t x t Y j H j X j
y t h t x t Y k TH k X k
h t x t h x t d
= Very important!
Chengbin Ma UM-SJTU Joint Institute
Ratio of FT
The convolution property implies that the
frequency response of a system may be expressed
as the ratio of the Fourier transform of the output
to that of the input (does not need to depend on
any specific signal any more).
( ) ( )* ( ) ( ) ( ) ( )
( )( )
( )
y t x t h t Y j X j H j
Y jH j
X j
Chengbin Ma UM-SJTU Joint Institute
Proof (FT)
Proof of convolution property: FT
)()(
)()()()(
)()(
)()()()(
jHjX
dehjXdjXeh
ddtetxh
dtedtxhdtetyjY
jj
tj
tjtj
)()(
)()(
0
0
jXettx
dtetxjX
tj
tj
Chengbin Ma UM-SJTU Joint Institute
Proof (FS)
Proof of convolution property: FS
0 0
0 0
0
0 0
( )
( )
( ) ( ) ( ) ( ) ( )
[ ] [ ]
[ ] [ ]
[ ] [ ] [ ]
[ ] [ ] [ ]
[ ] [ ] [ ]
T
jk jr t
Tk r
jr t j k r
Tk r
jr t
k r
jk t jk t
k k
y t h t x t h x t d
H k e X r e d
H k X r e e d
H k X r e T k r
TH k X k e Y k e
Y k TH k X k
][
][)(
0
)( 0
0
rkTdte
ekXtx
Ttrkj
k
tjk
Chengbin Ma UM-SJTU Joint Institute
Example
)2()(2)2()(*)()(
)1()1()(
)()()sin(2
)sin(2
)(
:Solution
?)(
)(sin4
)( 22
trtrtrtztztx
tututz
jZjZjX
tx
jX
)sin(2
)()(
)()()(
)()()(
000 Tttuttu
jXjHjY
txthty
Chengbin Ma UM-SJTU Joint Institute
Filtering
The multiplication that occurs in the frequency-domain representation gives rise to the notion of filtering.
A system performs filtering on the input signal by presenting a different response to components of the input that are at different frequencies.
Typically, the term filtering implies that some frequency components of the input are eliminated while others are passed by the system unchanged.
-40
-30
-20
-10
0
Magnitu
de (
dB
)
101
102
103
104
105
-90
-45
0
Phase (
deg)
Bode Diagram
Frequency (rad/sec)
)(
)()()(
jXA
jXjHjY
Chengbin Ma UM-SJTU Joint Institute
Bode Plots
Slide 32
-40
-30
-20
-10
0M
agnitu
de (
dB
)
101
102
103
104
105
-90
-45
0
Phase (
deg)
Bode Diagram
Frequency (rad/sec)
Chengbin Ma UM-SJTU Joint Institute
Ideal Filters
Continuous-time Discrete-time
Ideal low-pass/high-pass/band-pass filters
Chengbin Ma UM-SJTU Joint Institute
Bands of Filters
The passband of a filter is the band of frequencies that are
passed by the system, while the stopband refers to the
range of frequencies that are attenuated by the system.
Realistic filters always have a gradual transition from the
passband to the stopband. The range of frequencies over
which this occurs is known as the transition band.
-40
-30
-20
-10
0
Magnitu
de (
dB
)
100
101
102
103
104
-90
-45
0
Phase (
deg)
Bode Diagram
Frequency (rad/sec)
1100
1
j
passband stopband
transition band
Chengbin Ma UM-SJTU Joint Institute
Class#11
- Linearity and symmetry properties (3.9)
- Time- and frequency-shift properties (3.12)
- Scaling properties (3.15)
- Convolution property (3.10)
- Differentiation and integration properties (3.11)
Slide 35
Chengbin Ma UM-SJTU Joint Institute
Differentiation in time
0
FT ( ) ( )
FS ( ) [ ]
dx t j X j
dt
dx t jk X k
dt
Very important!
Chengbin Ma UM-SJTU Joint Institute
Proof (FT)
Proof of differentiation-in-time property: FT
1( ) ( )
2
1 1( ) ( ) ( )
2 2
1 1( ) ( )
2 2
( ) ( )
j t
j t j t
j t j t
x t X j e d
d d dx t X j e d X j e d
dt dt dt
X j j e d j X j e d
dx t j X j
dt
Chengbin Ma UM-SJTU Joint Institute
Proof (FS)
Proof of differentiation-in-time property: FS
0
0 0
0 0
0
0
( ) [ ]
( ) [ ] [ ]
[ ] [ ]
( ) [ ]
jk t
k
jk t jk t
k k
jk t jk t
k k
x t X k e
d d dx t X k e X k e
dt dt dt
dX k e jk X k e
dt
dx t jk X k
dt
0
FT ( ) ( )
FS ( ) [ ]
dx t j X j
dt
dx t jk X k
dt
Chengbin Ma UM-SJTU Joint Institute
Differential Equations
The differentiation property may be used to find
the frequency response of a continuous-time
system described by the differential equation
M
kk
k
k
N
kk
k
k txdt
dbty
dt
da
00
)()(
Chengbin Ma UM-SJTU Joint Institute
System Frequency Response
Find the frequency response of a continuous-time system
described by differential equation.
0 0
0 0
0
0
FT ( ) FT ( )
( ) ( ) ( ) ( )
( )( )
( )( )
( )
k kN M
k kk kk k
N Mk k
k k
k k
Mk
k
k
Nk
k
k
d da y t b x t
dt dt
a j Y j b j X j
b jY j
H jX j
a j
Chengbin Ma UM-SJTU Joint Institute
For a differential equation (n=10,000rad/s)
Slide 41
Example
)()()()(2
2
2
txtytydt
d
Qty
dt
dn
n
Its frequency response
22 )()(
1)(
nj
Qj
jHn
102
103
104
105
106
-250
-200
-150
-100
Magnitu
de (
dB
)
Bode Diagram
Frequency (rad/sec)
Q=200, 1, 2/5
In-class problem:
Passband?
0
FT ( ) ( )
FS ( ) [ ]
dx t j X j
dt
dx t jk X k
dt
Chengbin Ma UM-SJTU Joint Institute
Differentiation in Frequency
FT ( ) ( )d
jtx t X jd
Proof of differentiation-in-frequency: FT
( ) ( )
( ) ( ) ( )
( ) ( )
j t
j t j t
X j x t e dt
d dX j x t e dt jtx t e dt
d d
djtx t X j
d
Chengbin Ma UM-SJTU Joint Institute
Example
2
2
1)(
1)(
1)(
:Solution
?)()()(
jatute
ja
j
jad
dtujte
jatue
jXtutetx
at
at
at
at
)()(
)()(
jXd
dtjtx
dtetxjX tj
Chengbin Ma UM-SJTU Joint Institute
Integration Property
0
1FT ( ) ( ) ( ) ( ) ( 0) ( )
1FS ( ) ( ) [ ] [ ]
( ( ) is finite valued and periodic only if [0] 0)
t
t
y t x d Y j X j X jj
y t x d Y k X kjk
y t X
0
FT ( ) ( )
FS ( ) [ ]
dx t j X j
dt
dx t jk X k
dt
If not, the integration of x(t), i.e., y(t), will be infinite!
Chengbin Ma UM-SJTU Joint Institute
Proof (FT)
Proof of integration property: FT
( )* ( ) ( ) ( ) ( )
1( ) ( ) ( ) ( ) ( )
1( ) ( ) ( )
1( ) ( 0) ( )
t
t
x t u t x u t d x d
x d X j U j X jj
X j X jj
X j X jj
)(1
)(
)()()(
)(*)()(
jtu
jXjHjY
txthty
Chengbin Ma UM-SJTU Joint Institute
Proof (FS)
Proof of integration property: FS
0
0
( ) ( ) ( ) ( )
1[ ] [ ] [ ] [ ]
td
y t x d x t y tdt
X k jk Y k Y k X kjk
][)( 0 kXjktxdt
d
Chengbin Ma UM-SJTU Joint Institute
Homework
3.48(d), 3.49(d), 3.50(d) (f)
Prove Fourier Transform using IFT:
3.54(a)(c)(e)
3.58(a)(c)(e)
3.59(a)(c)(e)
3.67(c)
3.68(a)
3.70(b)
Due 2:00PM, Thursday of next week
Slide 47
dtetxjX tj )()(