Fourier Transform: Important Propertiesyao/EE3054/Chap11.5... · 2008. 4. 4. · Fourier Transform:...
Transcript of Fourier Transform: Important Propertiesyao/EE3054/Chap11.5... · 2008. 4. 4. · Fourier Transform:...
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EE3054
Signals and Systems
Fourier Transform: Important Properties
Yao Wang
Polytechnic University
Some slides included are extracted from lecture presentations prepared by McClellan and Schafer
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4/4/2008 © 2003, JH McClellan & RW Schafer 2
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LECTURE OBJECTIVES
� Basic properties of Fourier transforms
� Duality, Delay, Freq. Shifting, Scaling
� Convolution property
� Multiplication property
� Differentiation property
� Freq. Response of Differential Equation System
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4/4/2008 © 2003, JH McClellan & RW Schafer 4
Fourier Transform Defined
� For non-periodic signals
Fourier Synthesis
Fourier Analysis
∫∞
∞−
−= dtetxjX tjωω )()(
∫∞
∞−
= ωω ωπ
dejXtxtj)()(
21
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Table of Fourier Transforms
)()()()cos()( ccc jXttx ωωπδωωπδωω ++−=⇔=
1)()()( =⇔= ωδ jXttx
>
<=⇔=
b
bb jX
t
ttx
ωω
ωωω
π
ω
0
1)(
)sin()(
2/
)2/sin()(
2/0
2/1)(
ω
ωω
TjX
Tt
Tttx =⇔
>
<=
ωω
jjXtuetx
t
+=⇔= −
1
1)()()(
)(2)()( ctj
jXetx c ωωπδωω −=⇔=
)()()()sin()( ccc jjjXttx ωωπδωωπδωω ++−−=⇔=
0)()()( 0tj
ejXtttxωωδ −=⇔−=
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4/4/2008 © 2003, JH McClellan & RW Schafer 6
Duality of FT Pairs
∫∞
∞−
−= dtetxjX tjωω )()(∫∞
∞−
= ωω ωπ
dejXtxtj)()(
21
( ))(2)( Then
)( If
ωπ
ω
−⇔
⇔
xtg
gtx
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4/4/2008 © 2003, JH McClellan & RW Schafer 7
Fourier Transform of a
General Periodic Signal
� If x(t) is periodic with period T0 ,
∫∑−
∞
−∞=
==0
00
00
)(1
)(
T
tjkk
k
tjkk dtetx
Taeatx
ωω
)(2 since Therefore, 00 ωωπδω ke tjk −⇔
∑∞
−∞=
−=k
k kajX )(2)( 0ωωδπω
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4/4/2008 © 2003, JH McClellan & RW Schafer 8
Square Wave Signal
x(t) = x(t + T0 )
T0−2T0 −T0 2T00t
ak =e
− jω0kt
− jω0kT0 0
T0 / 2
−e
− jω 0kt
− jω0kT0 T0 /2
T0
=1− e− jπk
jπk
ak =1
T0(1)e
− jω0 ktdt +1
T0(−1)e
− jω 0ktdtT0 / 2
T0
∫0
T0 / 2
∫
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4/4/2008 © 2003, JH McClellan & RW Schafer 9
Square Wave Fourier Transform
X( jω ) = 2π akδ(ω − kω0 )k =−∞
∞
∑
x(t) = x(t + T0 )
T0−2T0 −T0 2T00 t
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4/4/2008 © 2003, JH McClellan & RW Schafer 10
FT of Impulse Train
� The periodic impulse train is
p(t) = δ (t − nT0 ) =n=−∞
∞
∑ akejkω0t
n=−∞
∞
∑
ak =1
T0δ (t)e
− jω0tdt =−T0 /2
T0 /2
∫1
T0 for all k
∴ P( jω) =2π
T0
δ (k = −∞
∞∑ ω − kω0 )
ω0 = 2π / T0
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Plot of impulse train in time
and frequency
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4/4/2008 © 2003, JH McClellan & RW Schafer 12
Table of Easy FT Properties
ax1(t) + bx2 (t) ⇔ aX1( jω) + bX2 ( jω )
x(t − td ) ⇔ e− jωtd X( jω )
x(t)ejω0t ⇔ X( j(ω − ω0 ))
Delay Property
Frequency Shifting
Linearity Property
x(at) ⇔ 1|a | X( j(ωa ))
Scaling
Duality
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Delay Property
x(t − td ) ⇔ e− jωtd X( jω )
x(t − td )e− jωtdt
−∞
∞
∫ = x(τ )e− jω(τ +td )dτ
−∞
∞
∫
= e− jω td X( jω )
For example, e−a(t−5)
u(t − 5) ⇔e− jω 5
a + jω
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Multiply by e^jw0
x(t)ejω0t ⇔ X( j(ω − ω0 ))
))((
)()(
0
)( 00
ωω
ωωωω
−=
= ∫∫∞
∞−
−−∞
∞−
−
jX
dtetxdtetxetjtjtj
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Multiply by cos(w0)?
( )
( )))(())((2
1
)()(2
1)cos()(
))(()(
))(()(
00
0
0
0
00
0
0
ωωωω
ω
ωω
ωω
ωω
ω
ω
++−
⇔+=
+⇔
−⇔
−
−
jXjX
etxetxttx
jXetx
jXetx
tjtj
tj
tj
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4/4/2008 © 2003, JH McClellan & RW Schafer 16
Shifting in frequency by
multiply by cos()
= (Amplitude Modulation)
� Illustrate the spectrum in class
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y(t) = x(t)cos(ω0t) ⇔
Y( jω ) =1
2X( j(ω − ω0 )) +
1
2X( j(ω + ω0 ))
x(t)
x(t) =1 t < T / 2
0 t > T / 2
⇔ X( jω ) =sin(ωT / 2)
ω / 2( )
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4/4/2008 © 2003, JH McClellan & RW Schafer 18
Another example
� x(t)=cos (w0 t)
� What is y(t)=x(t) * cos (w1 t)
� Consider w1 >w0 and w1
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What about multiply by sin( )?
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Scaling Property
expands)(shrinks;)2(22
1 ωjXtx
)(
)()(
1
)/(
aa
adajtj
jX
exdteatx
ω
λλωω λ
=
= ∫∫∞
∞−
−∞
∞−
−
)()( 1aa
jXatx ω⇔
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4/4/2008 © 2003, JH McClellan & RW Schafer 21
Scaling Property
)()( 1aa
jXatx ω⇔
)2()( 12 txtx =
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4/4/2008 © 2003, JH McClellan & RW Schafer 22
Uncertainty Principle
� Try to make x(t) shorter
� Then X(jωωωω) will get wider
� Narrow pulses have wide bandwidth
� Try to make X(jωωωω) narrower
� Then x(t) will have longer duration
�� Cannot simultaneously reduce time Cannot simultaneously reduce time
duration and bandwidthduration and bandwidth
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Table of Easy FT Properties
ax1(t) + bx2 (t) ⇔ aX1( jω) + bX2 ( jω )
x(t − td ) ⇔ e− jωtd X( jω )
x(t)ejω0t ⇔ X( j(ω − ω0 ))
Delay Property
Frequency Shifting
Linearity Property
x(at) ⇔ 1|a | X( j(ωa ))
Scaling
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Significant FT Properties
x(t) ∗h(t) ⇔ H( jω )X( jω )
x(t)ejω0t ⇔ X( j(ω − ω0 ))
x(t)p(t) ⇔1
2πX( jω )∗ P( jω )
dx(t)
dt⇔ ( jω)X( jω)
Differentiation Property
Duality
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4/4/2008 © 2003, JH McClellan & RW Schafer 25
Convolution Property
� Convolution in the time-domain
corresponds to MULTIPLICATIONMULTIPLICATION in the
frequency-domain
y(t) = h(t) ∗ x(t) = h(τ )−∞
∞
∫ x(t − τ )dτ
Y( jω ) = H( jω )X( jω )
y(t) = h(t) ∗ x(t)x(t)
Y( jω ) = H( jω )X( jω )X( jω)
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Proof (in class)
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Convolution Example
� Bandlimited Input Signal� “sinc” function
� Ideal LPF (Lowpass Filter)� h(t) is a “sinc”
� Output is Bandlimited� Convolve “sincs”
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4/4/2008 © 2003, JH McClellan & RW Schafer 28
Ideally Bandlimited Signal
>
<=⇔=
πω
πωω
π
π
1000
1001)(
)100sin()( jX
t
ttx
πω 100=b
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4/4/2008 © 2003, JH McClellan & RW Schafer 29
Ex: x(t) and y(t) are both sinc
sin(100π t)
πt∗
sin(200πt)
π t=
x(t) ∗h(t) ⇔ H( jω )X( jω )
sin(100π t)
πt
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4/4/2008 © 2003, JH McClellan & RW Schafer 30
Ex. x(t) and y(t) are both rect.
pulse
Y( jω ) =sin(ω / 2)
ω / 2
2
y(t) = x(t) ∗ h(t)
Y( jω ) = H( jω )X( jω )
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4/4/2008 © 2003, JH McClellan & RW Schafer 31
Cosine Input to LTI System
Y (jω) = H( jω )X( jω)
= H( jω )[πδ(ω − ω0 ) +πδ(ω +ω 0)]
= H( jω0 )πδ (ω −ω0 ) + H(− jω0 )πδ (ω +ω0 )
y(t) = H (jω0 )12 e
jω0t + H(− jω0 )12 e
− jω 0t
= H( jω0 )12 e
jω0t + H*( jω 0)
12 e
− jω0t
= H( jω0 ) cos(ω 0t +∠H( jω0 ))
)cos(*)()( 0tthty ω=
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4/4/2008 © 2003, JH McClellan & RW Schafer 32
Ideal Lowpass Filter
Hlp( jω)
ωco−ωco
y(t) = x(t) if ω0 < ωco
y(t) = 0 if ω0 > ωco
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Ideal Lowpass Filter
y(t) =4
πsin 50πt( ) +
4
3πsin 150πt( )
fco "cutoff freq."
H( jω ) =1 ω < ωco
0 ω > ωco
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4/4/2008 © 2003, JH McClellan & RW Schafer 34
Multiplier
� Multiplication in the time-domain corresponds to convolution in the frequency-domain.
Y( jω ) =1
2πX( jω) ∗ P( jω )
y(t) = p(t)x(t)
X( jω )
x(t)
p(t)
Y( jω ) =1
2πX( jθ )
−∞
∞
∫ P( j(ω −θ ))dθ
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4/4/2008 © 2003, JH McClellan & RW Schafer 35
p(t) = cos(ω0t) ⇔ P( jω) = πδ (ω − ω0 )
+ πδ (ω + ω0 )
y(t) = x(t)p(t) ⇔ Y( jω ) =1
2πX( jω )∗ P( jω)
y(t) = x(t)cos(ω0t) ⇔
Y( jω ) =1
2πX( jω ) ∗[πδ (ω − ω0 ) + πδ (ω + ω0 )]
Y( jω ) =1
2X( j(ω − ω0 )) +
1
2X( j(ω + ω0 ))
Multiply by cos(w0 t)
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4/4/2008 © 2003, JH McClellan & RW Schafer 36
Differentiation Property
dx (t )
dt=
d
dt
1
2πX ( jω )e jω t dω
−∞
∞
∫
=1
2π( jω ) X( jω )e jω tdω
−∞
∞
∫
Multiply by jωωωωdx(t)
dt⇔ ( jω)X( jω)
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4/4/2008 © 2003, JH McClellan & RW Schafer 37
Example
d
dte
−atu(t)( )= −ae−atu(t) + e−atδ (t)
= δ (t) − ae−atu(t)
ωjatue
at
+⇔−
1)(
)(1)( ωωω
ω
ωω jXj
ja
j
ja
ajY =
+=
+−=
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4/4/2008 © 2003, JH McClellan & RW Schafer 38
High order differentiation?
dx(t)
dt⇔ ( jω)X( jω ) ( ) ( )ωω jXj
dx
txd kk
k
⇔)(
Proof in class
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System of Differential
Equation
( ) ( )
( )
( )∑
∑
∑∑
∑∑
=
=
==
==
==
=
=
N
k
kk
M
k
kk
M
k
kk
N
k
kk
M
kk
k
k
N
kk
k
k
ja
jb
jX
jYjH
jXjbjYja
dt
txdb
dt
tyda
0
0
00
00
)(
)()(
)()(
)()(
ω
ω
ω
ωω
ωωωω
c
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4/4/2008 © 2003, JH McClellan & RW Schafer 40
Recall Difference Equation?
( ) ( )
( )
( )∑
∑
∑∑
∑∑
=
=
==
==
==
=
=
N
k
kk
M
k
kk
M
k
kk
N
k
kk
M
kk
k
k
N
kk
k
k
ja
jb
jX
jYjH
jXjbjYja
dt
txdb
dt
tyda
0
0
00
00
)(
)()(
)()(
)()(
ω
ω
ω
ωω
ωωωω
c
∑
∑
∑∑
∑∑
=
−
=
−
=
−
=
−
==
==
=
−=−
N
k
kk
M
k
kk
M
k
kk
N
k
kk
M
k
k
N
k
k
za
zb
zX
zYzH
zXzbzYza
knxbknya
0
0
00
00
)(
)()(
)()(
][][
c
Discrete time system
(Difference equation)
Continuous time system
(Differentiation equation)
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Example systems
� Example systems described by low order differential equations
� How to determine the frequency response
� How to determine the impulse response
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4/4/2008 © 2003, JH McClellan & RW Schafer 42
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 frequency-domain, which ever leads to the simplest solution.
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Table of Fourier Transforms
)()()()cos()( ccc jXttx ωωπδωωπδωω ++−=⇔=
1)()()( =⇔= ωδ jXttx
>
<=⇔=
b
bb jX
t
ttx
ωω
ωωω
π
ω
0
1)(
)sin()(
2/
)2/sin()(
2/0
2/1)(
ω
ωω
TjX
Tt
Tttx =⇔
>
<=
ωω
jjXtuetx
t
+=⇔= −
1
1)()()(
)(2)()( ctj
jXetx c ωωπδωω −=⇔=
)()()()sin()( ccc jjjXttx ωωπδωωπδωω ++−−=⇔=
0)()()( 0tj
ejXtttxωωδ −=⇔−=
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4/4/2008 © 2003, JH McClellan & RW Schafer 44
Table of Easy FT Properties
ax1(t) + bx2 (t) ⇔ aX1( jω) + bX2 ( jω )
x(t − td ) ⇔ e− jωtd X( jω )
x(t)ejω0t ⇔ X( j(ω − ω0 ))
Delay Property
Frequency Shifting
Linearity Property
x(at) ⇔ 1|a | X( j(ωa ))
Scaling
-
Significant FT Properties
x(t) ∗h(t) ⇔ H( jω )X( jω )
x(t)ejω0t ⇔ X( j(ω − ω0 ))
x(t)p(t) ⇔1
2πX( jω )∗ P( jω )
dx(t)
dt⇔ ( jω)X( jω)
Duality
( ) ( )ωω jXjdx
txd kk
k
⇔)(
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READING ASSIGNMENTS
� This Lecture:
� Chapter 11, Sects. 11-5 to 11-10
� Tables in Section 11-9
� Other Reading:
� Entire chap 11