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Unit IV Fourier Trans

1. This document outlines problems involving Fourier transforms. It covers topics like the Fourier integral theorem, convolution theorem, Parseval's identity, and finding the Fourier transforms of various functions. 2. Part A involves stating properties of Fourier transforms and calculating the transforms of functions like rectangular pulses and exponentials. Part B focuses on using Fourier transforms to evaluate definite integrals involving trigonometric and exponential functions. 3. Solutions involve applying properties like translating the argument of a function in the Fourier domain and using Fourier transforms to rewrite integrals in terms of simpler forms.

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Nithin Janardhanan
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146 views2 pages

Unit IV Fourier Trans

1. This document outlines problems involving Fourier transforms. It covers topics like the Fourier integral theorem, convolution theorem, Parseval's identity, and finding the Fourier transforms of various functions. 2. Part A involves stating properties of Fourier transforms and calculating the transforms of functions like rectangular pulses and exponentials. Part B focuses on using Fourier transforms to evaluate definite integrals involving trigonometric and exponential functions. 3. Solutions involve applying properties like translating the argument of a function in the Fourier domain and using Fourier transforms to rewrite integrals in terms of simpler forms.

Uploaded by

Nithin Janardhanan
Copyright
© Attribution Non-Commercial (BY-NC)
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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FOURIER TRANSFORMS

PART –A

1. State the Fourier integral theorem.


2. Define Fourier transform pair.
3. State the convolution theorem on Fourier transform.
4. Write the Parseval’s identity for Fourier transform.
5. State Modulation theorem on Fourier transform.
6. Prove that if F(s) is the Fourier transform of f(x), then F{ f(x –a) } = eisa F(s).
7. State and prove change of scale property of Fourier transform.
8. Find the F.T of f(x) defined by f(x) = 1 if a <x < b
0 otherwise
9. Find the F.T of f(x) defined by f(x) = 1 if |x| < a
0 if |x| > a >0
10. Find the F.T of f(x) defined by f(x) = eimx if a <x < b
0 otherwise
11. Find the F.T of f(x) defined by f(x) = √2 / 2t if -t <x < t
0 otherwise
-|x|
12. Find the F.T of f(x) defined by f(x) = e .
13. Find F.C.T and F.S.T of f(x) = e-ax.
14. Find the sine transform of 1/x.
15. If F c { e-ax } = √(2/ ) [ a / s2 +a2], find Fs [ x e-ax ].
16. Find F{ xnf(x) } & F{ fn (x) }.
17. Find F.C.T of e-ax cos ax.
18. Using transform methods, evaluate dx / ( x2 +a2)2. in 0 < x< .
19. Using transform methods, evaluate x2dx / ( x2 +a2)2. in 0 < x< .
20. Find F.S.T of (e-ax- e-bx ) / x.
PART –B
1. A function f(x) is defined as f(x) = 1 if |x| < 1
0 , otherwise . Using Fourier

integral representation of f(x). Hence evaluate [sin s/ s] cos sx ds in 0 < x< .


2. A function f(x) is defined as f(x) = 1 if |x| < 1
0 , otherwise . Using Fourier
cosine integral representation of f(x). Hence evaluate [sin s/ s] cos sx ds in
0<x< .
3. Find the F.T of f(x) is defined as f(x) = 1 if |x| < 1
0 , otherwise . Hence evaluate
(i) [sin / ] d (ii) [sin2 / 2
] d in (0 , ).
4. Find the F.T of f(x) is defined as f(x) = a -|x| if |x| < a
0 , otherwise . Hence evaluate
(i) [sin / ] 2 d in (0 , ). (ii) [sin / ] 4 d in (0 , ).

5. Find the F.T of f(x) is defined as f(x) = 1 –x2 if |x| < 1


0 , otherwise . Hence evaluate
(i) [sin t – t cos t/ t 3] dt in (0 , ). (ii) [x cos x - sin x / x 3] cos (x /2)dx in (0 , ).

6. Find the F.T of f(x) is defined as f(x) = e-a2x2 ,a >0. Hence S.T e-x2 / 2 is self reciprocal
under F.T.
7. Find the F.T of e-|x| and hence find the F.T of e-|x| cos 2x.
8. Obtain the F.S.T of f(x) = x if 0 <x < 1
2 – x if 1<x<2
0 , otherwise
9. Find the F.C.T of f(x) is defined as f(x) = cos x if 0 <x < a
0 , otherwise .
10. State and Prove Parseval’s Identity.
11. Find the F.S.T and F.C.T of x n-1, where 0 < n< 1, x >0 . Deduce that 1/ x is self-
reciprocal under both F.S.T and F.C.T.
12. Find the F.S.T of e-ax / x . Hence find F.S.T of 1 / x.
13. Evaluate [dx / (a2 + x2 ) (b2 + x2) ] dx in (0 , ).
14. Find F c {f ’(x)}.
15. Solve the integral equation [f(x) cos x] dx in (0 , ) and also [cos x / ( 1 + 2)] d
in (0 , ).

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