*Chapter 5: Fourier Transform

FOURIER TRANSFORM:*Definition of the Fourier transformsRelationship between Laplace Transforms and Fourier TransformsFourier transforms in the limitProperties of the Fourier TransformsCircuit applications using Fourier TransformsParsevals theoremEnergy calculation in magnitude spectrum

Definition of Fourier Transforms*Fourier Transforms:

Inverse Fourier Transforms:*

Example 1:Obtain the Fourier Transform for thefunction below:*

Solution:Given function is:*

Fourier Transforms:*

FOURIER TRANSFORM:*Definition of the Fourier transformsRelationship between Laplace Transforms and Fourier TransformsFourier transforms in the limitProperties of the Fourier TransformsCircuit applications using Fourier TransformsParsevals theoremEnergy calculation in magnitude spectrum

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.

Rule 1:If f(t)=0 for t

Example:*

Replace s=j*

Rule 2: Inverse negative function*

Example:*Negative

Fourier Transforms*

Rule 3:Add the positive and negative function*

Thus,*

Example 1:*

Fourier transforms:*

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

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Solution:*

Example 3:*

Solution:*

Example 4:*

Solution:*

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FOURIER TRANSFORM:*Definition of the Fourier transformsRelationship between Laplace Transforms and Fourier TransformsFourier transforms in the limitProperties of the Fourier TransformsCircuit applications using Fourier TransformsParsevals theoremEnergy calculation in magnitude spectrum

Fourier Transforms in the limitFourier transform for signum function (sgn(t))*

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*

assume 0,*

Fourier Transforms for step function:*

Fourier Transforms for cosine function*

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Thus,*

FOURIER TRANSFORM:*Definition of the Fourier transformsRelationship between Laplace Transforms and Fourier TransformsFourier transforms in the limitProperties of the Fourier TransformsCircuit applications using Fourier TransformsParsevals theoremEnergy calculation in magnitude spectrum

Properties of Fourier TransformsMultiplication by a constant*

Addition and subtraction *

Differentiation*

Integration*

Scaling*

Time shift*

Frequency shift*

Modulation

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Convolution in time domain*

Convolution in frequency domain:*

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

Solution:*LAPLACETRANSFORMS

A and B value:*

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