Chapter 5: Fourier Transform

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

Definition of Fourier Transforms

Inverse Fourier Transforms:

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

Solution: Given function is:

Fourier Transforms:

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

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<=0- Replace s=jω

Example:

Replace s=jω

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 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

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

assume ε→0,

Fourier Transforms for step function:

Fourier Transforms for cosine function

Thus,

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

Properties of Fourier Transforms Multiplication 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: LAPLACE TRANSFORMS

A and B value: