Fourier Series, Fourier Transforms, and Periodic Response to ...
04 Fourier Transforms
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Sep 13, 2005 CS477: Analog and Digital Communications 1
Fourier Transforms
Analog and Digital
CommunicationsAutumn 2005-2006
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Sep 13, 2005 CS477: Analog and Digital Communications 2
Fourier Series: Example
cn = T01 R
T0v(t)e j 2nf 0tdt
= T0Asinc( nf 0)
2 2
T0 T0
Av(t)
t
= 4Asinc( 4
n) for = 4T0
-3 -2 -1 0 1 2 30
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
nf 0= 4n
Amplitude norm
alizedby
A/4
Find Fourier Series of ?v(t)
Fourier series expansion is:v(t) =
P 1
1
4Asinc( 4
n)e j 2nf 0t
is sum of rotating phasorsv(t)
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Sep 13, 2005 CS477: Analog and Digital Communications 3
Sinusoidal Fourier SeriesFor real signals
and can be expressed in terms of
= c0 +P
n=1
1(cne j 2nf 0t + cne j 2nf 0t)
cn = c n
v(t) =P
1
1cne j 2nf 0t
Pn=1
1 jcn je j (2nf 0t+
6 cn)
= c0 + 2P
n=1
1cn j j cos(2nf 0t + 6 cn)
= a 0 + 2Pn=1
1[a n cos(2 nf 0t) + bn sin(2 nf 0t )]
a n bn cn
= c0 +Pn=1
1 jcn je j (2nf 0t+
6 cn) +
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Sep 13, 2005 CS477: Analog and Digital Communications 4
Parsevals theoremAverage power of a periodic signal can beobtained from its Fourier coefficients
To prove, write the Fourier series expansion
P = hj x(t)j2i = T1 R
T=2
T=2 jx(t)j2dt
= T1 R
T=2
T=2
x(t)x(t)dt
=P 1
1cncn =
P 1
1 jcn j2
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Sep 13, 2005 CS477: Analog and Digital Communications 5
Fourier Transform
2 2 T0 T0
Av(t)
t
-3 -2 -1 0 1 2 30
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
nf 0= 4n
Amplitude normalizedby
A/4
What if the x-axis in the Fourierseries represents frequency?
The spectral lines appear at integer
multiples of fundamental frequency! Separation between two consecutivespectral lines is equal to f 0 = T
01
How does increasing fundamental period affect theseparation between two consecutive spectral lines? Fourier transform
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Sep 13, 2005 CS477: Analog and Digital Communications 6
Fourier TransformFourier series in the limitRepresents spectral components of a
signalSignal uniquely represented in time orfrequency domain
X(f ) =R
1
1x(t)e j2f t dt
x(t) =R
1
1X(f )e j2ft dt
x(t) $ X(f )
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Shifting and Scaling
Review
a t
x(t)A
ab t
x(t=b)A
a=b t
x(tb)A
a + c t
x(t c)A
a c t
x(t + c)A
a t
x( t)A
d a t
x(d t)A
d + ak t
x( kt ) A
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Fourier Transform:
Example
21 2
1
1
t
x(t) = rect( t)
X(f ) = sinc( f )
2 2
A
t
x(t) = Arect( t)
X(f ) = Asinc( f )
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Properties of FTLinearityScalingDuality
Time shifting
Frequency shifting (modulation)
x(at ) $ a j j1 X(a
f )
X(t) $ x( f )
x(t ) $ X(f )e j2f
x(t)e j 2f ct $ X(f f c)
x1(t) + x2(t ) $ X 1(f ) + X 2(f )
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Properties of FTModulation
Product and convolution
More on this under LTI systems
x(t)y(t) $ X(f ) Y(f )
x(t) y(t) $ X(f )Y(f )
x(t) cos(2 f ct + ) $ 2e j X(f f c) + 2
e j X(f + f c)
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Properties of FTDifferentiation
Integration
dtd x(t) $ j2f X (f )
R 1
x()d $ j 2f1
X(f )
dt ndn x(t) $ ( j2f )nX(f )
R 1
1x()d = X(0)
R 1
1X(f )df = x(0)
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The Dirac Delta FunctionA generalized function
R 1
1
(t )dt =R
(t )dt = 1R
1
1x(t) (t t 1)dt = x(t 1)
R 1
1
x(t t 2) (t t 1)dt = x(t 1 t 2)
x(t) (t ) =R
1
1x() (t )d = x(t)
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FT of 1 and exponentialsR
1
1e j2f t (f f 0)df = e j 2f 0t
) e j 2f 0t $ (f f 0)
1 $ (f ) and (t) $ 1
Fourier transform of RF Pulse:
x(t) = Arect( t) cos2 f ct
X(f ) = 2Asinc( f f c) + 2
Asinc( f + f c)
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Other FunctionsSignum Function:
sgn( t) = 1 t0
n$ jf
1t
1
1
Unit step Function:
u(t) = 21
(1+ sgn( t))u(t) $ 2
1h jf
1 + (f )i
t
1
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Sep 13, 2005 CS477: Analog and Digital Communications 15
FT of Periodic Signalsxp(t) =
P 1
1cne j 2nf 0t; f 0 = T0
1
Let x(t) = 0 elsex p(t) jt j
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Sep 13, 2005 CS477: Analog and Digital Communications 16
FT of Periodic Signalsxp(t ) =
Pm= 1
1 (t mT 0)
xp(t ) $ f 0P
n= 1
1X(nf 0) (f nf 0)
Let x(t) = 0 elsex p(t) jt j