Fourier Transform Pair

FOURIER TRANSFORM PAIR :

a. Fourier Transform: [Complex Fourier Transform] 

Definition : The complex (or infinite) Fourier Transform

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

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

The above (1) & (2) are jointly called Fourier transform pair.

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 be adjusted while expressing f(x) as a Fourier integral.

For example, whatever definitions or format we use.

b. INVERSION FORMULA FOR FOURIER TRANSFORM

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

Proof : By Fourier integral theorem,

PROPERTIES - TRANSFORMS OF SIMPLE FUNCTIONS

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

Proof : 

Similarly if a < 0

Combining (1) & (2), we get

3. Shifting property 

(i) F [f(x − a)] = eias F (s) (ii) F [eiax f (x)] = F [s+a]

Proof : (i) We know that,

(ii) We know that,

4. Modulation Property :

Modulation theorem :

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

F [f(x) cos ax] = 1/2 [F (s + a) + F (s − a)]

Proof: We know that,

5.

Proof : We know that,

Differentiating both sides n times w.r.to s, we get

6.

Proof : We know that,

7.

Proof : 

8.

Proof: We know that,

Taking complex conjugate on both sides

9. F [f(-x)] = F(-s)

Proof: We know that,

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.

(i.e.,) F [f(x) * g (x) ] = F (s) G (s) = F [ƒ (x)] F [g (x)]

Proof: 

by changing the order of integration, we get

PARSEVAL'S IDENTITY:

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

Proof : By convolution theorem,

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

II. (a) Problems based on Fourier Transform

[Complex Fourier Transform]

Formula:

Example 4.2.a(1) : Find the Fourier Transform of

Solution: The given function can be written as

Example 4.2.a(2): Find the Fourier Transform of

Solution : The given function can be written as

Example 4.2.a(3): Find the Fourier transform of f(x) given by

Solution: The given function can be written as

Definition: Self reciprocal :

If a transformation of a function f(x) is equal to ƒ(s) then the function f(x) is called self reciprocal.

Example 4.2.a(4): Show that the Fourier Transform of

(OR)

Show that is self-reciprocal with respect to Fourier Transform.

Solution :

Example 4.2.a(5): Find the Fourier transform of f(x) defined by

Solution: 

Example 4.2.a(6) : Show that the Fourier transform of

Solution: Given function can be written as 

Example 4.2.a(7) : Find the (complex) Fourier transform of

Solution: 

Example 4.2.a(8): Find the Fourier transform of

Solution: We know that,

Example 4.2.a(9) : Find the Fourier transform of

Solution: We know that,

Example 4.2.a(10) : Find the Fourier transform of Hence, show that is self reciprocal under Fourier transform.

Solution: 

Example 4.2.a(11) Find the Fourier transform of Dirac delta function δ (t− a).

Solution: The Dirac delta function is defined as

The Fourier transform of δ (t - a) is

Il.(b) Problems based on Fourier transform and its inversion formula

Example 4.2.b(1) : Find the Fourier transform of the function f(x) defined by

Hence, prove that

Solution: The given function can be written as

By Fourier inversion formula, we have

Example 4.2.b(2) : Find the Fourier Transform of

Solution: We know that,

By inversion formula

Note :

Even function :

If f (-x): f(x) in (−l,l) then f(x) is an even function.

Odd function :

If ƒ (-x) = -f(x) in (−l,l) then f(x) is an odd function

In the above problem, a cos sx is an even function,

a sin sx is an odd function

|x| is an even function,

|x| cos sx is an even function

|x| sin sx is an odd function.

Example 4.2.b(3): Find Fourier transform of e-a|x| and hence deduce that

Solution: We know that,

Using inversion formula, we get

Example 4.2.b(4): Find the Fourier transform of e-|x|  and hence find the Fourier transform of e-|x| cos 2x.

Solution: 

To find : F [e-|x| cos 2x]

By Modulation theorem,

Il.(c) Problems based on inversion formula, Parseval's identity and Convolution theorem

Example 4.2(1): Find the Fourier transform of where a is a positive real number.

Hence deduce that

Solution :

(i) Now, by Fourier inversion formula, we have

(ii) Using Parseval's identity

Example 4.2.c(2) Find the Fourier Transform of

Solution: 

The Fourier transform of f(x) is

Example 4.2.c(3): Show that the Fourier transform of

Solution: 

(i) Using inverse Fourier Transform, we get

Put x = 0, we get

(ii) Using Parseval's identity,

Example 4.2.c(4) : Find the Fourier transform of e-a|x| if a > 0

Deduce that

Solution: Given : f(x) = e-a|x|

See Example 4.2.b(3) in page no. 4.53

By Parseval's identity,

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

Example 4.2.c(5) Find the Fourier transform of

Solution:

See Example 4.2.b(1) in page no. 4.48 for problems based on Fourier transform and its inversion formula

Example 4.2.c(6) : Find the Fourier transform of f(x) given by

Solution : The given equation can be written as

f(x) = 1 if -2 < x < 2

= 0 otherwise

F (s) = F [f(x)]

(i) Now, by Fourier inversion formula, we have

Example 4.2.c(7): Find the Fourier transform of e-|x|, using Parseval's identity show that

Solution: 

Example 4.2.c(8): If f(x) = then find the Fourier transform of f (x) and hence, evaluate using Parseval's identity.

Solution: We know that,

By Parseval's identity,

Example 4.2.c(9) : Verify convolution theorem for

Definition: Convolution theorem for Fourier transforms. The Fourier transform of the convolution of ƒ (x) and g(x) is the product of their Fourier transforms.

F [f(x) *g(x)] = F {f(x)} F {g (x)}

Hence, convolution theorem is verified.