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11/5/2014
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FourierAnalysis and Synthesis Tool
Nikesh [email protected]
Digital Signal Processing
School of Electronics and Communication
Lovely Professional University
Pierre-Simon Laplace (1749-1827)
Sir Isaac Newton (4 January 1643 – 31 March 1727
Jean Baptiste Joseph Fourier (21 March 1768 – 16 May 1830)
Guess….?
2 By: Nikesh Bajaj
Question???
What do you mean by Transform?
What is Fourier Series/Transform?
Different between Fourier Series and
Fourier Transform
Different between Fourier Transform and
Laplace Transform
3 By: Nikesh Bajaj
History
In 1748
L. Euler: Vibrating String
Vertical deflation at any time is the linear
combination of ‘normal modes’, he showed it
In 1753
D. Bernouli argued same on physical ground (No math)
but was not accepted.
Eular discarded ‘Trigonometric series’
In 1759
J. L. Langrange strongly criticized Trig. Series.
4 By: Nikesh Bajaj
Fourier
Jean Baptiste Joseph Fourier-21 March 1768 – 16 May 1830,
Auxerre, France,
Mathematician, Egyptologist,
Revolutionary Discovery (1822). Scientific Advisor
He claimed that any periodic signal can be represented by
a series of harmonically related sinusoidal.
–FOURIER SERIES
He also obtained a representation of Aperiodic signal, not
as weighted sum of harmonically related sinusoidal, but
as weighted Integral of sinusoidal that are not at all
harmonically related.
-FOURIER INTEGRAL or TRANSFORM5 By: Nikesh Bajaj
Fourier Series
“Fourier Series is a Great
Mathematical Poem” Lord Kelvin (William Thompson)
Laplace Rejected Fourier’s Thesis
Laplace Generalized Fourier’s one
By: Nikesh Bajaj6
Nike
sh B
ajaj
Signals & Systems
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11/5/2014
2
Fourier Series
x(t) periodic signal
Where coefficients are
k
tjk
ketx a
)(
T
tjk
kdtetx
Ta
)(1
2, Tperiod
7 By: Nikesh Bajaj
Fourier Series
Periodic signal x(t)
k
tjk
ketx a
)( -To/2 -T1 T1 To/2
1
k
Tkak
1sin
ok k
Tk
To
o
a 1sin21
2T1
w
Line spectra, discrete lines
/T1
8 By: Nikesh Bajaj
A Sum of Sine Waves
xAsin(
9 By: Nikesh Bajaj
Fourier Series
What is the Use of Fourier Series?
Use in Analysis :LTI system
HOW?
Use in Synthesis
By: Nikesh Bajaj10
Fourier Series to
Fourier Transform
-To/2 -T1 T1 To/2
1
2T1
If T0∞ ???
will become Aperiodic
will become Continuese
Fourier Transform comes in the picture11 By: Nikesh Bajaj
Fourier Transform
For Aperiodic signal x(t), FT
Inverse Fourier Transform
dtetxjX tj )()(
dejXtx tj)(2
1)(
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Nike
sh B
ajaj
Signals & Systems
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11/5/2014
3
Fourier Series
Does FS always exist for any periodic signal?
Convergence Criteria: Dirichlet
1. Absolutely Integrable
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11/5/2014
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Properties of Fourier
Properties of
Fourier Transform
Notation
)()( jFtf F
)()]([ jFtfF
)()]([1 tfjF- F
Linearity
)()()()( 22112211 jFajFatfatfaF
Time Scaling
ajF
aatf
||
1)( F
Time Reversal
jFtf F)(
Pf)dtetftf tj
)()]([F dtetft
t
tj
)(
)()( tdetft
t
tj
)()( tdetft
t
tj
dtetft
t
tj
)( dtetft
t
tj
)(
dtetf tj
)( )( jF
Time Shifting
0)( 0tj
ejFttf
F
Pf)dtettfttf tj
)()]([ 00F dtettft
t
tj
)( 0
)()( 0)(0
0
0 ttdetftt
tt
ttj
dtetfet
t
tjtj
)(0
dtetfe tjtj
)(0tj
ejF 0)(
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ajaj
Signals & Systems
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11/5/2014
5
Frequency Shifting (Modulation)
)()( 00
jFetfj F
Pf)dteetfetf tj
tjtj
00 )(])([F
dtetftj
)( 0)(
)( 0 jF
Symmetry Property
)(2)]([ fjtFF
Pf)
dejFtf tj)()(2
dejFtf tj)()(2
dtejtFf tj
)()(2
Interchange symbols and t
)]([ jtFF
Example:
)()]([ jFtfF ?]cos)([ 0 ttfF
Sol)
))((2
1cos)( 000
tjtjeetfttf
])([2
1])([
2
1]cos)([ 000
tjtjetfetfttf
FFF
)]([2
1)]([
2
100 jFjF
Example:
d/2d/2
1
t
wd(t)
d/2d/2t
f(t)=wd(t)cos0t
2sin
2)]([)(
2/
2/
ddtetwjW
d
d
tj
dd F
]cos)([)( 0ttwjF d F0
0
0
0 )(2
sin)(2
sin
dd
Example:
d/2d/2
1
t
wd(t)
d/2d/2t
f(t)=wd(t)cos0t
2sin
2)]([)(
2/
2/
ddtetwjW
d
d
tj
dd F
]cos)([)( 0ttwjF d F0
0
0
0 )(2
sin)(2
sin
dd
-60 -40 -20 0 20 40 60-0.5
0
0.5
1
1.5
F(j
)
d=2
0=5
Example:
t
attf
sin)( ?)( jF
Sol)
d/2d/2
1
t
wd(t)
2sin
2)(
djWd
)(22
sin2
)]([
dd w
td
tjtW FF
)(sin
)]([ 2
aw
t
attf FF
||1
||0
a
a
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ajaj
Signals & Systems
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11/5/2014
6
Fourier Transform of f’(t)
0)(lim and )(
tfjFtft
F
Pf)dtetftf tj
)(')]('[F
dtetfjetf tjtj )()(
)()(' jFjtf F
)( jFj
Fourier Transform of Integral
00)( and )(
FdttfjFtf F
jFj
dxxft 1
)(F
Let dxxftt
)()( 0)(lim tt)()()]([)]('[ jjjFtft FF
)(1
)(
jFj
j
The Derivative of Fourier
Transform
d
jdFtjtf FF )]([
Pf)
dtetfjF tj
)()(
dtetfd
d
d
jdF tj
)(
)(dtetf tj
)(
dtetjtf tj
)]([ )]([ tjtfF
By: Nikesh Bajaj34
By: Nikesh Bajaj35
Properties of Fourier Transform
Spatial Domain (x) Frequency Domain (u)
Linearity xgcxfc 21 uGcuFc 21
Scaling axf
a
uF
a
1
Shifting 0xxf uFeuxi 02
Symmetry xF uf
Conjugation xf uF
Convolution xgxf uGuF
Differentiation n
n
dx
xfd uFui n2
frequency ( )uxie 2Note that these are derived using
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ajaj
Signals & Systems
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11/5/2014
7
Fourier Transform Pairs
By: Nikesh Bajaj37
Proof of Convolution Property
Taking Fourier transforms gives:
Interchanging the order of integration, we have
By the time shift property, the bracketed term is
e-jtH(j), so
ttt dthxty )()()(
dtedthxjY tjttt )()()(
ttt ddtethxjY tj)()()(
)()(
)()(
)()()(
tt
tt
t
t
jXjH
dexjH
djHexjY
j
j
Fourier Series
By: Nikesh Bajaj39 By: Nikesh Bajaj40
Final
By: Nikesh Bajaj41
Fourier Transform Pairs (I)
angular frequency ( )iuxeNote that these are derived using
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ajaj
Signals & Systems
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11/5/2014
8
Problems
System Connections
FT, FS
By: Nikesh Bajaj43 angular frequency ( )iuxeNote that these are
derived using
Fourier Transform Pairs (I)
DTFT
By: Nikesh Bajaj45
DFT
By: Nikesh Bajaj46
Summery- Fourier Series
By: Nikesh Bajaj47 By: Nikesh Bajaj48
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ajaj
Signals & Systems
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11/5/2014
9
By: Nikesh Bajaj49 By: Nikesh Bajaj50
By: Nikesh Bajaj51 By: Nikesh Bajaj52
Fourier Family
By: Nikesh Bajaj53
Transform Time Frequency
Fourier seriesContinuous,
Periodic
Discrete,
Aperiodic
Discrete Fourier
transform
Discrete,
Periodic
Discrete,
Periodic
Continuous Fourier
transform
Continuous,
Aperiodic
Continuous,
Aperiodic
Discrete-time Fourier
transform
Discrete,
Aperiodic
Continuous,
Periodic
Fourier Family
By: Nikesh Bajaj54
Nike
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ajaj
Signals & Systems
