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FOURIER TRANSFORMS
JELMAAN FOURIER:• Definition of the Fourier transforms
• Relationship between Laplace Transforms and Fourier Transforms
• Fourier transforms in the limit
• Properties of the Fourier Transforms
• Circuit applications using Fourier Transforms
• Parseval’s theorem
• Energy calculation in magnitude spectrum
CIRCUIT APPLICATION USING FOURIER TRANSFORMS
• Circuit element in frequency domain:
CjC
LjL
RR
1
Example 1:
• Obtain vo(t) if vi(t)=2e-3tu(t)
Solution:
• Fourier Transforms for vi
jVi
3
2)(
Transfer function:
21
1
/12
/1
)(
)()(
j
j
j
V
VH
i
o
Thus,
)5.0)(3(
1
)21)(3(
2
)()()(
jj
jj
HVV io
From partial fraction:
• Inverse Fourier Transforms:
jjVo
5.0
4.0
3
4.0)(
)()(4.0)( 35.0 tueetv tto
Example 2:
• Determine vo(t) if vi(t)=2sgn(t)=-2+4u(t)
Solution:
jVtv ii
4)()sgn(2
jH
4
4)(
j
B
j
A
jj
VHV io
4
)4(
16
)()()(
jjVo
4
44)(
)(4)sgn(2)( 4 tuettv to
JELMAAN FOURIER:• Definition of the Fourier transforms
• Relationship between Laplace Transforms and Fourier Transforms
• Fourier transforms in the limit
• Properties of the Fourier Transforms
• Circuit applications using Fourier Transforms
• Parseval’s theorem
• Energy calculation in magnitude spectrum
PARSEVAL’S THEOREM
Energy absorbed by a function f(t)
dttfW )(21
Parseval’s theorem stated that energy also can be calculate using,
dFdttf2
2
2
1)(
• Parseval’s theorem also can be written as:
dFdttf2
0
2 1)(
PARSEVAL’S THEOREM DEMONSTRATION
• If a function,
taetf )(
• Integral left-hand side:
aaa
a
e
a
e
dtedtedte
atat
atatta
1
2
1
2
1
220
202
0
20 22
• Integral right-hand side:
aa
aaaa
a
da
a
100
20
2
tan1
2
14
)(
41
0
1222
2
0 222
2
JELMAAN FOURIER:• Definition of the Fourier transforms
• Relationship between Laplace Transforms and Fourier Transforms
• Fourier transforms in the limit
• Properties of the Fourier Transforms
• Circuit applications using Fourier Transforms
• Parseval’s theorem
• Energy calculation in magnitude spectrum
ENERGY CALCULATION IN MAGNITUDE SPECTRUM
• Magnitude of the Fourier Transforms squared is an energy density (J/Hz)
0 0
222222
1dffFdffF
• Energy in the frequency band from ω1 and ω2:
2
1
1
2
2
1
22
2
1
2
1
2
1
1
dFdF
dFW
Example 1:
• The current in a 40Ω resistor is:
• What is the percentage of the total energy dissipated in the resistor can be associated with the frequency band 0 ≤ ω ≤ 2√3 rad/s?
Atuei t )(20 2
Solution:
• Total energy dissipated in the resistor:
J
e
dteW
t
t
4000
416000
40040
0
4
0
440
Check the answer with parseval’s theorem:
• Fourier Transform of the current:
jF
2
20)(
• Magnitude of the current:
24
20)(
F
J
dW
40002
8000
2tan
2
116000
4
40040
0
1
0 240
• Energy associated with the frequency band:
J
dW
3
8000
3
8000
2tan
2
116000
4
40040
32
0
1
32
0 240
• Percentage of the total energy associated:
%67.661004000
3/8000
Example 2:
• Calculate the percentage of output energy to input energy for the filter below:
k10
F10
iv
ov
Vtuev ti )(15 5
• Energy at the input filter:
Je
dteW
t
ti
5.2210
225
)15(
0
10
0
25
• Fourier Transforms for the output voltage: )()()( HVV io
jjRC
RCH
jVi
10
10
/1
/1)(
5
15)(
• Thus,
)100)(25(
22500)(
)10)(5(
150)(
22
2
o
o
V
jjV
• Energy at the output filter:
J
dWo
15210
1
25
1300
100
300
25
3001
)100)(25(
225001
0 20 2
0 22
• Thus the percentage:
%67.66)100(5.22
15
