FOURIER TRANSFORMS. JELMAAN FOURIER: Definition of the Fourier transforms Relationship between Laplace Transforms and Fourier Transforms Fourier transforms.

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