EE3054
Signals and Systems
Fourier Transform:
Important Properties
Yao Wang
Polytechnic University
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McClellan and Schafer
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4/4/2008
© 2003, JH McClellan & RW Schafer
2
LECTURE OBJECTIVES
Basic properties of Fourier transforms
Duality, Delay, Freq. Shifting, Scaling
Convolution property
Multiplication property
Differentiation property
Freq. Response of Differential Equation System
Fourier Transform Defined
For non-periodic signals
Fourier Synthesis
∞
x (t ) =
1
2π
X
(
j
ω
)
e
∫
jω t
dω
−∞
Fourier Analysis
∞
X ( jω ) =
x
(
t
)
e
∫
− jω t
dt
−∞
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© 2003, JH McClellan & RW Schafer
4
Table of Fourier Transforms
x ( t ) = e −t u ( t ) ⇔
1
x(t ) = 
0
X ( jω ) =
t < T /2
⇔
X ( jω ) =
t > T /2
sin(ωbt )
x (t ) =
πt
x(t ) = δ (t ) ⇔
⇔
⇔
sin(ωT / 2)
ω /2
1 ω < ωb
X ( jω ) = 
0 ω > ω b
X ( jω ) = 1
x (t ) = δ (t − t0 ) ⇔
x(t ) = e jωc t
1
1 + jω
X ( jω ) = e − jω t0
X ( jω ) = 2πδ (ω − ωc )
x(t ) = cos(ω c t ) ⇔
X ( jω ) = πδ (ω − ω c ) + πδ (ω + ω c )
x(t ) = sin(ω c t ) ⇔
X ( jω ) = − jπδ (ω − ωc ) + jπδ (ω + ωc )
Duality of FT Pairs
∞
x (t ) =
1
2π
∞
jω t
X
(
j
ω
)
e
dω
∫
−∞
4/4/2008
− jω t
x
(
t
)
e
dt
∫
X ( jω ) =
−∞
If
x(t ) ⇔ g (ω )
Then
g (t ) ⇔ 2πx(−ω )
© 2003, JH McClellan & RW Schafer
6
Fourier Transform of a
General Periodic Signal
If x(t) is periodic with period T0 ,
∞
x (t ) =
∑ ak e
jkω 0 t
k = −∞
1
ak =
T0
Therefore, since e
jkω0t
T0
∫ x (t )e
− jkω 0 t
dt
0
⇔ 2πδ (ω − kω0 )
∞
X ( jω ) =
∑ 2π akδ (ω − kω0 )
k = −∞
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© 2003, JH McClellan & RW Schafer
7
Square Wave Signal
x(t) = x(t + T0 )
−2T0
1
ak =
T0
−T0
T0 / 2
1
− jω 0 kt
dt +
(−1)e
dt
∫
T0 T0 / 2
− jω0 kt
0
e
ak =
− jω 0 kT0
2T0 t
T0
T0
∫ (1)e
− jω0 kt
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0
T0 / 2
− jω 0 kt
T0
− jπ k
e
1− e
−
=
− j ω 0 kT0 T0 /2
j πk
0 © 2003, JH McClellan & RW Schafer
8
Square Wave Fourier Transform
x(t) = x(t + T0 )
−2T0
−T0
2T0
T0
0
t
∞
X( jω ) =
∑ 2π a δ (ω − k ω
k
k =−∞
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© 2003, JH McClellan & RW Schafer
9
0
)
FT of Impulse Train
The periodic impulse train is
∞
∞
n=−∞
n=−∞
p(t) = ∑ δ (t − nT0 ) = ∑ ak e
ak =
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1
T0
T0 /2
∫ δ (t)e
− jω 0 t
dt
−T0 /2
jkω 0 t
ω 0 = 2π / T0
1
=
T0
for all k
∞
2π 

∴ P( jω ) =
δ (ω − kω 0 )
∑
 T0 
k = −∞
© 2003, JH McClellan & RW Schafer
10
Plot of impulse train in time
and frequency
Table of Easy FT Properties
Linearity Property
ax1 (t) + bx2 (t) ⇔ aX1 ( jω ) + bX2 ( jω )
Delay Property
x(t − td ) ⇔ e
− jω t d
X( jω )
Duality
Frequency Shifting
x(t)e
Scaling
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jω 0 t
x(at) ⇔
⇔ X( j(ω − ω 0 ))
1
|a|
ω
X( j( a ))
© 2003, JH McClellan & RW Schafer
12
Delay Property
x(t − td ) ⇔ e
∞
∫ x(t − td )e
− jω t
− jω t d
∞
dt
∫ x(τ )e
=
−∞
−∞
=
For example, e
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X( jω )
−a(t−5)
e
− j ω (τ +t d )
− jω td
dτ
X( jω )
− jω 5
e
u(t − 5) ⇔
a + jω
© 2003, JH McClellan & RW Schafer
13
Multiply by e^jw0
x(t)e
jω 0 t
⇔ X( j(ω − ω 0 ))
∞
e
∫
−∞
∞
jω 0t
x (t )e
− jω t
dt =
x
(
t
)
e
∫
− j (ω −ω 0 ) t
dt
−∞
= X ( j (ω − ω0 ))
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© 2003, JH McClellan & RW Schafer
14
Multiply by cos(w0)?
x(t )e
jω 0 t
x(t )e
− jω 0 t
⇔ X ( j (ω − ω0 ))
⇔ X ( j (ω + ω0 ))
(
)
1
jω 0 t
− jω 0 t
x(t ) cos(ω0 t ) = x(t )e
+ x(t )e
⇔
2
1
( X ( j (ω − ω0 )) + X ( j (ω + ω0 )))
2
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© 2003, JH McClellan & RW Schafer
15
Shifting in frequency by
multiply by cos()
= (Amplitude Modulation)
Illustrate the spectrum in class
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© 2003, JH McClellan & RW Schafer
16
y(t) = x(t)cos(ω 0 t) ⇔
1
2
1
2
Y( jω ) = X( j(ω − ω 0 )) + X( j(ω + ω 0 ))
x(t)
1
x(t) = 
0
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t <T /2
t >T /2
⇔
sin(ωT / 2)
X( jω ) =
(ω / 2 )
© 2003, JH McClellan & RW Schafer
17
Another example
x(t)=cos (w0 t)
What is y(t)=x(t) * cos (w1 t)
Consider w1 >w0 and w1<w0
Verify by trigonometric identities
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© 2003, JH McClellan & RW Schafer
18
What about multiply by sin( )?
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© 2003, JH McClellan & RW Schafer
19
Scaling Property
1
a
x (at ) ⇔
∞
x
(
at
)
e
∫
a
∞
− jω t
dt =
−∞
=
x ( 2t ) shrinks;
4/4/2008
X(
ω
j )
∫
− jω ( λ / a ) dλ
x ( λ )e
a
−∞
1
a
1
2
X ( j ωa )
X(
ω
j )
© 2003, JH McClellan & RW Schafer
2
expands
20
Scaling Property
x (at ) ⇔
1
a
X ( j ωa )
x2 (t ) = x1 (2t )
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© 2003, JH McClellan & RW Schafer
21
Uncertainty Principle
Try to make x(t) shorter
Then X(jω
ω) will get wider
Narrow pulses have wide bandwidth
Try to make X(jω
ω) narrower
Then x(t) will have longer duration
Cannot simultaneously reduce time
duration and bandwidth
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© 2003, JH McClellan & RW Schafer
22
Table of Easy FT Properties
Linearity Property
ax1 (t) + bx2 (t) ⇔ aX1 ( jω ) + bX2 ( jω )
Delay Property
x(t − td ) ⇔ e
− jω t d
X( jω )
Frequency Shifting
x(t)e
Scaling
4/4/2008
jω 0 t
x(at) ⇔
⇔ X( j(ω − ω 0 ))
1
|a|
ω
X( j( a ))
© 2003, JH McClellan & RW Schafer
23
Significant FT Properties
x(t) ∗ h(t) ⇔ H( jω )X( jω )
Duality
1
x(t)p(t) ⇔ X( jω )∗ P( jω )
2π
jω 0 t
x(t)e
⇔ X( j(ω − ω 0 ))
Differentiation Property
4/4/2008
dx(t)
⇔ ( jω )X( jω )
dt
© 2003, JH McClellan & RW Schafer
24
Convolution Property
y(t) = h(t) ∗ x(t)
x(t)
X( jω )
Y( jω ) = H( jω )X( jω )
Convolution in the time-domain
∞
y(t) = h(t) ∗ x(t) = ∫ h(τ )x(t − τ )dτ
−∞
corresponds to MULTIPLICATION in the
frequency-domain
Y( jω ) = H( jω )X( jω )
4/4/2008
© 2003, JH McClellan & RW Schafer
25
Proof (in class)
4/4/2008
© 2003, JH McClellan & RW Schafer
26
Convolution Example
Bandlimited Input Signal
“sinc” function
Ideal LPF (Lowpass Filter)
h(t) is a “sinc”
Output is Bandlimited
Convolve “sincs”
4/4/2008
© 2003, JH McClellan & RW Schafer
27
Ideally Bandlimited Signal
sin(100π t )
x (t ) =
πt
⇔
1 ω < 100π
X ( jω ) = 
0 ω > 100π
ωb = 100π
4/4/2008
© 2003, JH McClellan & RW Schafer
28
Ex: x(t) and y(t) are both sinc
x(t) ∗ h(t) ⇔ H( jω )X( jω )
sin(100π t) sin(200π t) sin(100π t)
∗
=
πt
πt
πt
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© 2003, JH McClellan & RW Schafer
29
Ex. x(t) and y(t) are both rect.
pulse
Y( jω ) = H( jω )X( jω )
sin(ω / 2)

Y( jω ) =
 ω /2 
2
y(t) = x(t) ∗ h(t)
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© 2003, JH McClellan & RW Schafer
30
Cosine Input to LTI System
y (t ) = h(t ) * cos(ω0 t )
Y (j ω ) = H( jω )X(j ω )
= H( jω )[πδ (ω − ω 0 ) + πδ (ω + ω 0 )]
= H( jω 0 )πδ (ω − ω 0 ) + H(− jω 0 )πδ (ω + ω 0 )
y(t) =
=
=
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H (j ω 0 ) 12 e jω0 t + H(− j ω 0 ) 12 e − jω 0t
1
2
jω 0 t
*
1
2
− jω 0 t
H( jω 0 ) e + H ( jω 0 ) e
H( jω 0 ) cos(ω 0t + ∠H( jω 0 ))
© 2003, JH McClellan & RW Schafer
31
Ideal Lowpass Filter
Hlp ( jω )
−ω co
ω co
y(t) = x(t)
y(t) = 0
4/4/2008
if ω 0 < ω co
if ω 0 > ω co
© 2003, JH McClellan & RW Schafer
32
Ideal Lowpass Filter
1
H( jω ) = 
0
ω < ω co
ω > ω co
f co "cutoff freq."
y(t) =
4/4/2008
4
π
sin(50πt ) +
4
sin (150πt )
3π
© 2003, JH McClellan & RW Schafer
33
Multiplier
y(t) = p(t)x(t)
x(t)
X( jω )
Y( jω ) =
p(t)
1
2π
X( jω ) ∗ P( jω )
Multiplication in the time-domain corresponds to convolution in
the frequency-domain.
1 ∞
Y( jω ) =
∫ X( jθ )P( j(ω − θ ))dθ
2π −∞
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© 2003, JH McClellan & RW Schafer
34
Multiply by cos(w0 t)
p(t) = cos(ω 0 t) ⇔ P( jω ) = πδ (ω − ω 0 )
+ πδ (ω + ω 0 )
y(t) = x(t)p(t) ⇔ Y( jω ) =
1
2π
X( jω )∗ P( jω )
y(t) = x(t)cos(ω 0 t) ⇔
Y( jω ) =
1
2π
Y( jω ) =
4/4/2008
X( jω ) ∗[πδ (ω − ω 0 ) + πδ (ω + ω 0 )]
1
X ( j(ω
2
− ω 0 )) +
© 2003, JH McClellan & RW Schafer
1
X ( j(ω
2
+ ω 0 ))
35
Differentiation Property

dx (t )
d  1 ∞
jω t
=
dω 
∫ X ( jω )e

dt
dt  2 π −∞

1
=
2π
∞
∫ ( j ω ) X( j ω )e
jω t
−∞
dx(t)
⇔ ( jω )X( jω )
dt
4/4/2008
dω
© 2003, JH McClellan & RW Schafer
Multiply by jω
36
Example
e
− at
1
u (t ) ⇔
a + jω
d −at
e u(t) = −ae−at u(t) + e −atδ (t)
dt
(
)
= δ (t) − ae −at u(t)
a
jω
Y ( jω ) = 1 −
=
= j ωX ( j ω )
a + jω a + jω
4/4/2008
© 2003, JH McClellan & RW Schafer
37
High order differentiation?
dx(t)
⇔ ( jω )X( jω )
dt
k
d x(t )
dx
k
⇔ ( jω ) X ( jω )
k
Proof in class
4/4/2008
© 2003, JH McClellan & RW Schafer
38
System of Differential
Equation
N
∑a
k =0
d k y (t )
k
dt
k
M
=
∑b
d k x(t )
k
dt k
k =0
c
N
∑
M
ak ( jω )k Y ( jω ) =
k =0
Y ( jω )
H ( jω ) =
=
X ( jω )
∑
bk ( jω )k X ( jω )
k =0
M
∑
bk ( jω )k
k =0
N
∑
k =0
ak ( jω )k
Recall Difference Equation?
Discrete time system
(Difference equation)
N
∑a
M
k
y[n − k ] =
k =0
Continuous time system
(Differentiation equation)
N
∑ b x[n − k ]
∑a
k =0
k =0
k
d k y (t )
k
dt k
c
∑
ak z − k Y ( z ) =
k =0
Y ( z)
H ( z) =
=
X ( z)
=
∑b
d k x(t )
k
dt k
k =0
c
N
M
N
M
∑
bk z − k X ( z )
k =0
M
∑b z
−k
k
k =0
N
∑a z
M
∑ a ( jω ) Y ( jω ) = ∑ b ( jω )
k
k
k
k =0
Y ( jω )
H ( jω ) =
=
X ( jω )
k
X ( jω )
k =0
M
∑
bk ( jω )k
k =0
N
∑ a ( jω )
k
k
−k
k =0
k
k =0
4/4/2008
© 2003, JH McClellan & RW Schafer
40
Example systems
Example systems described by low order
differential equations
How to determine the frequency response
How to determine the impulse response
Strategy for using the FT
Develop a set of known Fourier transform
pairs.
Develop a set of “theorems” or properties
of the Fourier transform.
Develop skill in formulating the problem in
either the time-domain or the frequencydomain, which ever leads to the simplest
solution.
4/4/2008
© 2003, JH McClellan & RW Schafer
42
Table of Fourier Transforms
x ( t ) = e −t u ( t ) ⇔
1
x(t ) = 
0
X ( jω ) =
t < T /2
⇔
X ( jω ) =
t > T /2
sin(ωbt )
x (t ) =
πt
x(t ) = δ (t ) ⇔
⇔
⇔
sin(ωT / 2)
ω /2
1 ω < ωb
X ( jω ) = 
0 ω > ω b
X ( jω ) = 1
x (t ) = δ (t − t0 ) ⇔
x(t ) = e jωc t
1
1 + jω
X ( jω ) = e − jω t0
X ( jω ) = 2πδ (ω − ωc )
x(t ) = cos(ω c t ) ⇔
X ( jω ) = πδ (ω − ω c ) + πδ (ω + ω c )
x(t ) = sin(ω c t ) ⇔
X ( jω ) = − jπδ (ω − ωc ) + jπδ (ω + ωc )
Table of Easy FT Properties
Linearity Property
ax1 (t) + bx2 (t) ⇔ aX1 ( jω ) + bX2 ( jω )
Delay Property
x(t − td ) ⇔ e
− jω t d
X( jω )
Frequency Shifting
x(t)e
Scaling
4/4/2008
jω 0 t
x(at) ⇔
⇔ X( j(ω − ω 0 ))
1
|a|
ω
X( j( a ))
© 2003, JH McClellan & RW Schafer
44
Significant FT Properties
x(t) ∗ h(t) ⇔ H( jω )X( jω )
1
x(t)p(t) ⇔ X( jω )∗ P( jω )
Duality
2π
jω 0 t
x(t)e
⇔ X( j(ω − ω 0 ))
dx(t)
⇔ ( jω )X( jω )
dt
k
d x(t )
dx
k
⇔ ( jω ) X ( jω )
k
READING ASSIGNMENTS
This Lecture:
Chapter 11, Sects. 11-5 to 11-10
Tables in Section 11-9
Other Reading:
Entire chap 11