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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)
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.………………(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)
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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
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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
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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
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Hence prove that
Solution:
We know that the Fourier transform of is given by
By using inverse Fourier Transform we get
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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
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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
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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
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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:
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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:
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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
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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
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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
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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
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(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.
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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
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[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)]
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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.
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Using Parseval’s identity
3. Evaluate using transforms.
Solution:
We know that the Fourier cosine transform of
Similarly the Fourier cosine transform of
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We know that
4. Find the Fourier transform of and hence deduce that
(i)
(ii)
Solution:
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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
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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
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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
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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
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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.
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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:
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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:
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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
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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.
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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 .