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1 Chapter 5: Chapter 5: Fourier Fourier Transform Transform

# 1 Chapter 5: Fourier Transform. FOURIER TRANSFORM: 2 Definition of the Fourier transforms Definition of the Fourier transforms Relationship between Laplace

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• *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:

*

• Solution:*

• Example 3:*

• Solution:*

• Example 4:*

• Solution:*

• *

• 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))*

• *

• *

• assume 0,*

• Fourier Transforms for step function:*

• Fourier Transforms for cosine function*

• *

• 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*

• Differentiation*

• Integration*

• Scaling*

• Time shift*

• Frequency shift*

• Modulation

*

• 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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