leo lam © 2010-2012 signals and systems ee235. leo lam © 2010-2011 x squared equals 9 x squared...
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Leo Lam © 2010-2012
Signals and SystemsEE235
Leo Lam © 2010-2011
x squared equals 9x squared plus 1 equals yFind value of y
Leo Lam © 2010-2011
Today’s menu
• Fourier Transform Properties (cont’)• Duality• Loads of examples
Leo Lam © 2010-2011
Fourier Transform:
4
• Fourier Transform
• Inverse Fourier Transform:
Leo Lam © 2010-2012
FT Properties Example:
5
• Find FT for:
• We know the pair:
• So:
-8 0 8
G()
Leo Lam © 2010-2012
More Transform Pairs:
6
• More pairs:
time domain Fourier transform
21
aj 1
Leo Lam © 2010-2012
Periodic signals: Transform from Series
7
• Integral does not converge for periodic fns:
• We can get it from Fourier Series:• How? Find x(t) if• Using Inverse Fourier:
• So
)(2)( 0 X
tjtj edetx 0)(22
1)( 0
tje 0)(2 0
Leo Lam © 2010-2012
Periodic signals: Transform from Series
8
• We see this pair:
• More generally, if X(w) has equally spaced impulses:
• Then:
tje 0)(2 0
k
k kdX )(2)( 0
k
tjkk
tj eddeXtx 0)(2
1)(
Fourier Series!!!
Leo Lam © 2010-2012
Periodic signals: Transform from Series
9
• If we know Series, we know Transform
• Then:
• Example: • We know:
• We can write:
k
k kdX )(2)( 0
k
tjkkedtx 0)(
)8cos()( ttf 8,5.0,5.0 011 dd
)8()8()( F
Leo Lam © 2010-2012
Summary
• Fourier Transform Pairs• FT Properties
Leo Lam © 2010-2011
Duality of Fourier Transform
11
• Duality (very neat):
• Duality of the Fourier transform: If time domain signal f(t) has Fourier transform F(), then F(t) has Fourier transform 2 f(-)
• i.e. if:
• Then:
)(2
)(
0
0
0
0
tj
tj
e
ett
Changed sign
)(2)(
)()(
ftF
Ftf
Leo Lam © 2010-2011
Duality of Fourier Transform (Example)
12
• Using this pair:
• Find the FT of– Where T=5
2
TsincT
T
trect
)(5
5)( tF2
tsinctg
52
52)(
rectrectG
)(2)( fG
Leo Lam © 2010-2011
Duality of Fourier Transform (Example)
13
• Using this pair:
• Find the FT of
)(2)(
)(2)( )(
ueF
ueFa
a
jaatue at
10),(
jtatf
1)(
)(2)( fG
Leo Lam © 2010-2011
Convolution/Multiplication Example
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• Given f(t)=cos(t)e–tu(t) what is F()
)()(2
1)()( 2121
FFtftf
)1()1()()cos()( 11 Fttf
jFtuetf t
1
1)()()( 22
j
FFF
1
1)1()1(
2
1)()(
2
1)( 21
11
1
11
1
2
1)(
jjF
Leo Lam © 2010-2011
More Fourier Transform Properties
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Duality
Time-scaling
Multiplication
Differentiation
Integration
Conjugation
time domain Fourier transform
Dual of convolution
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( )n
n
d f t
dt ( )
nj F
( )t
f d
1( ) (0) ( )F F
j
*( )f t *( )F
Leo Lam © 2010-2011
Fourier Transform Pairs (Recap)
16 16
0cos( )t 0 0( ) ( )
0j te 02 ( )
0( )t t 0j te
1 2 ( )
1
j a
• Review:
Leo Lam © 2010-2011
Fourier Transform and LTI System
17
• Back to the Convolution Duality:
• And remember:
• And in frequency domain
)()()()( 2121 FFtftf
Convolution in time
h(t) x(t)*h(t)x(t)Time domain
Multiplication in frequency
H(w) X(w)H(w)X(w)
Frequency domain
input signal’sFourier transform
output signal’sFourier transform
Leo Lam © 2010-2011
Fourier Transform and LTI (Example)
• Delay:
LTIh(t)
Time domain:
( )* ( 3) ( 3)x t t x t Frequency domain (FT):
3
3
( ) ( 3)
( ) ( ) ( ) ( )
j t j
j
H t e dt e
Y X H X e
Shift in time Add linear phase in frequency
( )
( ) 3
[ ( ) 3 ]
( ) ( )
( ) ( )
( )
j X
j X j
j X
X X e
Y X e e
X e
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Leo Lam © 2010-2011
Fourier Transform and LTI (Example)
• Delay:
• Exponential response
LTIh(t)
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Delay 3)3()3( tjtj ete
)3(3)( tjjtjtj eeeHe
Using Convolution Properties
Using FT Duality
Leo Lam © 2010-2011
Fourier Transform and LTI (Example)
• Delay:
• Exponential response
• Responding to Fourier Series
LTIh(t)
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Delay 3)3()( tjtj eHe
Delay 3jtjt ee
ttz
5.05.0
)cos()(
)3cos(
5.05.0 33
t
eeee jjtjjt
Leo Lam © 2010-2011
Another LTI (Example)
• Given Exponential response
• What does this system do? What is h(t)?
• And y(t) if
• Echo with amplification
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LTI532)(
)(
j
tj
eH
He
)5(3)(2)( ttth jtjt eettz 5.05.0)cos()(
)5cos(3)cos(2
)5.0(3)5.0(2)( )5()5(
tt
eeeety tjtjjtjt
Leo Lam © 2010-2011
Another angle of LTI (Example)
• Given graphical H(w), find h(t)
• What does this system do? What is h(t)?
• Linear phase constant delay
22
)5()( tth
magnitude
w
w
phase
0
0
1
Slope=-5
5)( jeH
Leo Lam © 2010-2011
Another angle of LTI (Example)
• Given graphical H(w), find h(t)
• What does this system do (qualitatively
• Low-pass filter. No delay.
23
magnitude
w
w
phase
0
0
1
Leo Lam © 2010-2011
Another angle of LTI (Example)
• Given graphical H(w), find h(t)
• What does this system do qualitatively?
• Bandpass filter. Slight delay.
24
magnitude
w
w
phase
0
1
Leo Lam © 2010-2011
Another angle of LTI (Example)
• Given graphical H(w), find h(t)
• What does this system do qualitatively?
• Bandpass filter. Slight delay.
25
magnitude
w
w
phase
0
1
Leo Lam © 2010-2011
Summary
• Fourier Transforms and examples