Chapter 5 Fourier Transform 1 FOURIER TRANSFORM Definition

Chapter 5: Fourier Transform 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

Definition of Fourier Transforms: 3

Inverse Fourier Transforms: 4

Example 1: Obtain the Fourier Transform for the function below: 5

Solution: Given function is: 6

Fourier Transforms: 7

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 8

Relationship between Fourier Transforms and Laplace Transforms There are 3 rules apply to the use of Laplace transforms to find Fourier Transforms of such functions. 9

Rule 1: If f(t)=0 for t<=0 Replace s=jω 10

Example: 11

Replace s=jω 12

Rule 2: Inverse negative function 13

Example: Negative 14

Fourier Transforms 15

Rule 3: Add the positive and negative function 16

Thus, 17

Example 1: 18

Fourier transforms: 19

Example 2: Obtain the Fourier Transforms for the function below: 20

Solution: 21

Example 3: 22

Solution: 23

Example 4: 24

Solution: 25

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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 27

Fourier Transforms in the limit Fourier transform for signum function (sgn(t)) 28

29

30

assume ε→ 0, 31

Fourier Transforms for step function: 32

Fourier Transforms for cosine function 33

34

Thus, 35

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 36

Properties of Fourier Transforms Multiplication by a constant 37

Addition and subtraction 38

Differentiation 39

Integration 40

Scaling 41

Time shift 42

Frequency shift 43

Modulation 44

Convolution in time domain 45

Convolution in frequency domain: 46

Example 1: Determine the inverse Fourier Transforms for the function below: 47

Solution: LAPLACE TRANSFORMS 48

A and B value: 49