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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*

  • Addition and subtraction *

  • 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:*

    *