Important Questions and Answers: Fourier Transforms

FOURIER TRANSFORMS

PART –A

1. State Fourier integral theorem.

If  f(x) is piece-wise continuously differentiable and absolutely integrable in (- ¥, ¥) then

This is known as Fourier integral theorem or Fourier integral formula.

2. Define Fourier transform pair (or) Define Fourier transform and its inverse 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

its also called Fourier Transform Pairs.

3.     Show that f(x) = 1, 0 < x < ¥ cannot be represented by a Fourier integral.

4. State and prove the linear property of FT.

5. State and prove the Shifting property of FT. 

Stt:

6. State and prove the Change of scale property of FT. 

Stt:

8. State and prove the Modulation property of FT. (OR) If Fourier transform of f(x) is F(s).

Prove that the Fourier transform of f (x)cos ax is

9. What is meant by self-reciprocal with respect to FT?

If the Fourier transform of  f (x)is obtained just by replacing x by s, then  f (x)is called

self-reciprocal with respect to FT.

12. Define Fourier cosine transform (FCT) pair.

The infinite Fourier cosine transform of f(x) is defined by

13.                        Find the Fourier Cosine transform of f(x) =

14. Find the Fourier Cosine transform of e^-ax  , a > 0.

Given  f (x)=e^-ax

15. Find the Fourier Cosine transform of e^-x .

We know that

16. Define Fourier sine transform (FST) pair.

The infinite Fourier sine transform of f(x) is defined by

17.                        Find the Fourier Sine transform of e^-3x .

18.                        Find the Fourier Sine transform of f(x)= e^-x .

19. Find the Fourier Sine transform of 3e^{-2 x} .

Let  f (x)=3e^{-2 x}

20.                        Find the Fourier Sine transform of 1/x.

We know that

21.                        State the Convolution theorem on Fourier transform.

22.State the Parseval’s formula or identity

If F  s  is the Fourier transform of

PART B

1. State and prove the convolution theorem for Fourier Transforms.

Statement:

PROOF: By convolution of two functions:

F[( f *g )(x)]=F(s)G(s)

3. Show that e ^x2/2 is reciprocal with respect to Fourier transforms

Solution:

Fourier transform:

6. Find the Fourier cosine transform of e^{-a2 x2}

Solution: