Fourier Transform Pair

FOURIER TRANSFORM PAIR:

a. Fourier Transform: [Complex Fourier Transform]d.s.A Definition: The complex (or infinite) Fourier Transform

The complex (or infinite) Fourier Transform of f (x) is given by

Then the function f(x) is the inverse Fourier Transform of F (s) and is given by

Note: Whatever definitions or format we use, there will be a difference in constant factor while finding F (s) = F [f(x)]. But this will F (S) = FV be adjusted while expressing f (x) as a Fourier integral.

For example,  definitions or format we use.

INVERSION FORMULA FOR FOURIER TRANSFORM

Let f(x) be a function satisfying Dirichlet's conditions in every finite interval (-1,7). Let F(s) denote the Fourier transform of f(x). point of continuity of f(x)  Then at every  point of  countinuity  of f(x), we have

PROPERTIES- TRANSFORMS OF SIMPLE FUNCTIONS

4.2.c. PROPERTIES OF FOURIER TRANSFORMS :

1. Linear property

F [af(x)+bg(x)] a F [f(x)] + b F [g(x)]

where a and b are real numbers.

Proof :

2. Change of scale property For any non-zero real a, 

3. Shifting property

Proof: (i) We know that,

4. Modulation Property:

Modulation theorem :

d. CONVOLUTION THEOREM - PARSEVAL'S IDENTITY

Definition: Convolution

The convolution of two functions f(x) and g(x) is defined as

Convolution Theorem :

The Fourier transform of the convolution of f (x) and g (x) is the product of their Fourier transforms.

PARSEVAL'S IDENTITY:

If F (s) is the Fourier transform of f (x), then

Note: In the same way, we can prove Parseval's identity for Fourier sine and cosine transforms.