FOURIER TRANSFORMS 1 FOURIER TRANSFORM Definition of the

FOURIER TRANSFORMS 1

FOURIER TRANSFORM: • 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 2

CIRCUIT APPLICATION USING FOURIER TRANSFORMS • Circuit element in frequency domain: 3

Example 1: • Obtain vo(t) if vi(t)=2 e-3 tu(t) 4

Solution: • Fourier Transforms for vi 5

Transfer function: 6

Thus, 7

From partial fraction: • Inverse Fourier Transforms: 8

Example 2: • Determine vo(t) if vi(t)=2 sgn(t) 9

Solution: 10

11

12

FOURIER TRANSFORM: • 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 13

PARSEVAL’S THEOREM Energy absorbed by a function f(t) 14

Parseval’s theorem stated that energy also can be calculate using, 15

• Parseval’s theorem also can be written as: 16

PARSEVAL’S THEOREM DEMONSTRATION • If a function, 17

• Integral left-hand side: 18

• Integral right-hand side: 19

FOURIER TRANSFORM: • 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 20

ENERGY CALCULATION IN MAGNITUDE SPECTRUM • Magnitude of the Fourier Transforms squared is an energy density (J/Hz) 21

• Energy in the frequency band from ω1 and ω2: 22

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? 23

Solution: • Total energy dissipated in the resistor: 24

Check the answer with parseval’s theorem: • Fourier Transform of the current: 25

• Magnitude of the current: 26

27

• Energy associated with the frequency band: 28

• Percentage of the total energy associated: 29

Example 2: • Calculate the percentage of output energy to input energy for the filter below: 30

• Energy at the input filter: 31

• Fourier Transforms for the output voltage: 32

• Thus, 33

• Energy at the output filter: 34

• Thus the percentage: 35