1

CHAPTER 4 FOURIER TRANSFORMS

INTEGRAL TRANSFORM

The integral transform of a function is defined by where

k(s , x) is a known function of s and x and it is called the kernel of the transform. When k(s , x) is a sine or cosine function, we get transforms called Fourier sine or

cosine transforms.

FOURIER INTEGRAL THEOREM If is a given function defined in (-l , l) and satisfies Dirichlet’s conditions, then

At a point of discontinuity the value of the integral on the left of above equation is

EXAMPLES

1. Express the function as a Fourier Integral. Hence evaluate

and find the value of

Solution:

We know that the Fourier Integral formula for is

……………….(1)

Here = 1 for i.e., f(t) = 1 in -1 < t < 1

= 0 for = 0 in and

Equation (1)

2

.………………(2)

[Using sin (A+B) + sin (A-B) = 2 sin A cos B] This is Fourier Integral of the given function. From (2) we get

= ……………….(3

But ………………..(4)

Substituting (4) in (3) we get

=

Putting x = 0 we get

2. Find the Fourier Integral of the function

Verify the representation directly at the point x = 0. Solution:

The Fourier integral of is

……………….(1)

……….………(2)

3

Putting x = 0 in (2), we get

The value of the given function at x = 0 is . Hence verified.

FOURIER SINE AND COSINE INTEGRALS

The integral of the form

is known as Fourier sine integral. The integral of the form

is known as Fourier cosine integral. PROBLEMS 1. Using Fourier integral formula, prove that

Solution:

The presence of in the integral suggests that the Fourier sine integral formula has been used.

Fourier sine integral representation is given by

4

2. Using Fourier integral formula, prove that

Solution:

The presence of in the integral suggests that the Fourier cosine integral formula for has been used.

Fourier cosine integral representation is given by

5

COMPLEX FORM OF FOURIER INTEGRALS The integral of the form

is known as Complex form of Fourier Integral. FOURIER TRANSFORMS COMPLEX FOURIER TRANSFORMS

The function is called the Complex Fourier transform

of .

INVERSION FORMULA FOR THE COMPLEX FOURIER TRANSFORM

The function is called the inversion formula for the

Complex Fourier transform of and it is denoted by FOURIER SINE TRANSFORMS

The function is called the Fourier Sine Transform of

the function .

The function is called the inversion formula for the

Fourier sine transform and it is denoted by FOURIER COSINE TRANSFORMS

The function is called the Fourier Cosine

Transform of .

The function is called the inversion formula for the

Fourier Cosine Transform and it is denoted by PROBLEMS 1. Find the Fourier Transform of

6

Hence prove that

Solution:

We know that the Fourier transform of is given by

By using inverse Fourier Transform we get

7

The second integral is odd and hence its values is zero.

i.e.,

Putting , we get

2. Find the Fourier sine transform of , (or) , x > 0. Hence evaluate

Solution: The Fourier sine transform of f(x) is given by

Here = for x > 0

Using inverse Fourier sine transform we get

8

Replacing x by m we get

[since s is dummy variable, we can replace it by x]

3. Find the Fourier cosine transform of

Solution:

We know that

Here

Let

Then ………………(1)

Differentiating on both sides w.r.t. ‘s’ we get,

Integrating w.r.t. ‘s’ we get

9

4. Find the Fourier cosine transform of

Solution: We know that the Fourier cosine transform of f(x) is

Here

5. Find , if its sine transform is Hence deduce that the inverse sine

transform of

Solution: We know that the inverse Fourier sine transform of is given by

Here

10

Differentiating w.r.t. ‘x’ on both sides, we get,

To find the inverse Fourier sine transform of

Put a = 0, in (1), we get

PROPERTIES

1. Linearity Property If F(s) and G(s) are the Fourier transform of and respectively then

Proof:

11

2. Change of Scale Property

If F(s) is the Fourier transform of then

Proof:

Put ax = y

a dx = dy i.e., dx =

When

3. Shifting Property ( Shifting in x )

If F(s) is the Fourier transform of then Proof:

Put x-a = y dx = dy When

4. Shifting in respect of s

If F(s) is the Fourier transform of then Proof:

12

5. Modulation Theorem

If F(s) is the Fourier transform of then

Proof:

COROLLARIES

6. Conjugate Symmetry Property

If F(s) is the Fourier transform of then Proof:

We know that

Taking complex conjugate on both sides we get

13

Put x = -y dx = -dy When

7. Transform of Derivatives

If F(s) is the Fourier transform of and if is continuous, is piecewise continuously differentiable, and are absolutely integrable in and

, then

Proof:

By the first three conditions given, and exist.

The theorem can be extended as follows.

If are continuous, is piecewise continuous, are absolutely integrable in and , then

14

8. Derivatives of the Transform

If F(s) is the Fourier transform of then

Proof:

Extending, we get,

DEFINITION

is called the convolution product or simply the convolution

of the functions and and is denoted by .

9. Convolution Theorem

If F(s) and G(s) are the Fourier transform of and respectively then the Fourier transform of the convolution of f(x) and g(x) is the product of their Fourier transforms.

i.e., Proof

15

Inverting, we get

10. Parseval’s Identity (or) Energy Theorem

If is a given function defined in then it satisfy the identity,

where F(s) is the Fourier transform of . Proof:

We know that

Putting x = 0, we get

………………..(1)

Let .……………….(2) i.e., ………………..(3) by property (9) i.e., ………………..(4) Substituting (2) and (4) in (1) we get

11. If and are given functions of x and and are their Fourier cosine transforms and and are their Fourier sine transforms then

16

(i)

(ii) ,

which is Parseval’s identity for Fourier cosine and sine transforms. Proof:

(i)

Changing the order of integration

Similarly we can prove the other part of the result. (ii) Replacing in (i) and noting that and

, we get

i.e.,

12. If , then

(i) and

(ii)

Proof:

Similarly the result (ii) follows.

17

PROBLEMS

1. Show that the Fourier transform of is

. Hence deduce that Using Parseval’s

identity show that

Solution:

We know that

When a = 1, ………………..(A)

Using inverse Fourier Transform, we get

18

[The second integral is odd and hence its value is zero]

[since the integrand is an even function of s] Putting a = 1, we get

Putting x = 0, in the given function we get

Using Parseval’s identity, [Using (A)]

19

2. Find the Fourier Transform of if

Hence deduce that

Solution: We know that

Since

The second integral becomes zero since it is an odd function.

20

Using Parseval’s identity

3. Evaluate using transforms.

Solution:

We know that the Fourier cosine transform of

Similarly the Fourier cosine transform of

21

We know that

4. Find the Fourier transform of and hence deduce that

(i)

(ii)

Solution:

22

Using inversion formula, we get

Putting a = 1, we get,

FINITE FOURIER TRANSFORMS

If is a function defined in the interval (0 , l) then the finite Fourier sine transform of in 0

23

where ‘n’ is an integer

The inverse finite Fourier sine transform of is and is given by

The finite Fourier cosine transform of in 0 < x < l is defined as

where ‘n’ is an integer

The inverse finite Fourier cosine transform of is and is given by

PROBLEMS

1. Find the finite Fourier sine and cosine transforms of in 0 < x < l. Solution:

The finite Fourier sine transform is

Here

The finite Fourier cosine transform is

24

Here

2. Find the finite Fourier sine and cosine transforms of . Solution:

The finite Fourier sine transform of is

Here

The finite Fourier cosine transform of is

Here

25

3. Find if its finite sine transform is given by where p is positive

integer and . Solution:

We know that the inverse Fourier sine transform is given by

………………..(1)

Here = ………………..(2)

Substituting (2) in (1), we get

4. If find if 0 < x

26

UNIT-4 PART A

1. State the Fourier integral theorem. Ans:

If is a given function defined in (-l , l) and satisfies Dirichlet’s conditions, then

2. State the convolution theorem of the Fourier transform. Ans:

If F(s) and G(s) are the Fourier transform of and respectively then the Fourier transform of the convolution of f(x) and g(x) is the product of their Fourier transforms.

i.e., 3. Write the Fourier transform pair. Ans:

and are Fourier transform pairs. 4. Find the Fourier sine transform of (a > 0). Ans:

5. If the Fourier transform of is F(s) then prove that . Ans:

Put x-a = y dx = dy When

6. State the Fourier transforms of the derivatives of a function.

27

Ans: 7. Find the Fourier sine transform of . Ans:

Here for x > 0

8. Prove that

Proof:

Put ax = y

a dx = dy i.e., dx =

When

9. If F(s) is the Fourier transform of then prove that

Proof:

28

10. Find the Fourier sine transform of Ans:

11. Find Fourier sine transform of

Ans:

12. Find Fourier cosine transform of Ans:

13. If F(s) is the Fourier transform of then

Proof:

29

14. If F(s) is the Fourier transform of then

Proof:

15. If F(s) is the Fourier transform of then

Proof:

Put ax = y

a dx = dy i.e., dx =

When

30

PART B 1. Find the Fourier Transform of

Hence prove that

2. Find the Fourier cosine transform of

3. Find the Fourier Transform of if

Hence deduce that

4. Evaluate using transforms

5. Find the Fourier transform of and hence deduce that

(i)

(ii)

6. Show that the Fourier transform of is

. Hence deduce that Using Parseval’s identity

show that

7. . Find the Fourier transform of if

Hence deduce that

8. Derive the parseval’s identity for Fourier transforms.

31

9. Find the Fourier sine transform of

10. Find the Fourier transform of and hence deduce that

(i)

(ii)

11. State and prove convolution theorem for Fourier transforms. 12. Using Parseval’s identity calculate

(i) (ii) if a > 0.

13. Find the Fourier cosine transform of 14. (i) Find the Fourier cosine transform of (ii) Find the Fourier sine transform of

15. Find Fourier sine and cosine transform of and hence find the Fourier sine transform

of and Fourier cosine transform of .