MODEL PAPER FOR FOURIER TRANSFORMS
Marks:100 Time:3hours
PART—A (16marks)
ATTEMPT ANY 8 OF THE FOLLOWING:(8*2=16):
01.State the fourier integral theorem. Also state the sine and the cosine integral theorems of it with the necessary conditions and expressions.
02.Find the fourier cosine integral of e-ax. Hence deduce the value of the following integral
∞
∫ cosλx dλ.
0 (1+λ2)
03.Derive the fourier cosine and sine transforms.
04.Find the fourier sine transforms of f(x) = sinx for 0≤x<a
=0 for x>a
05.Find the fourier cosine transforms of (e-ax – e-bx)
x .
06.Deduce whether xe-x2/2 is self reciprocal with respect to FST . Given it is self reciprocal with respect to FCT. Use the appropriate property.
07.State and prove the shifting property of the fourier transforms.
08.Show that the fourier transforms obey the modulation theorems.
09.Relate the fourier cosine and sine transforms when the function is first subjected to differentiation and them respective transformations.
11.Evaluate the following using transforms:
∞
∫ dx
0 (x2+a2)(x2+b2)
12.State and prove the change of scale property for a function f(x) in a fourier transformation.
PART—B:(84 marks):
ANSWER ANY 12 FROM THE FOLLOWING:(12*7=84):
13.State and prove the following theorems for F-transforms
(a)Convolution theorem
(b)Rayleigh's theorem
14.Find the FCT of the following functions:
(a) f(x) = e-ax cosax
(b) h(x) = e-ax sinax
17.Find the Fourier Sine transformation of f(x) using the property if
Marks:100 Time:3hours
PART—A (16marks)
ATTEMPT ANY 8 OF THE FOLLOWING:(8*2=16):
01.State the fourier integral theorem. Also state the sine and the cosine integral theorems of it with the necessary conditions and expressions.
02.Find the fourier cosine integral of e-ax. Hence deduce the value of the following integral
∞
∫ cosλx dλ.
0 (1+λ2)
03.Derive the fourier cosine and sine transforms.
04.Find the fourier sine transforms of f(x) = sinx for 0≤x<a
=0 for x>a
05.Find the fourier cosine transforms of (e-ax – e-bx)
x .
06.Deduce whether xe-x2/2 is self reciprocal with respect to FST . Given it is self reciprocal with respect to FCT. Use the appropriate property.
07.State and prove the shifting property of the fourier transforms.
08.Show that the fourier transforms obey the modulation theorems.
09.Relate the fourier cosine and sine transforms when the function is first subjected to differentiation and them respective transformations.
10.Find the fourier transformation of f(x) if
f(x) = 1, for |x|<a
= 0, for |x|>a>0
11.Evaluate the following using transforms:
∞
∫ dx
0 (x2+a2)(x2+b2)
12.State and prove the change of scale property for a function f(x) in a fourier transformation.
PART—B:(84 marks):
ANSWER ANY 12 FROM THE FOLLOWING:(12*7=84):
13.State and prove the following theorems for F-transforms
(a)Convolution theorem
(b)Rayleigh's theorem
14.Find the FCT of the following functions:
(a) f(x) = e-ax cosax
(b) h(x) = e-ax sinax
15.Find Fs[ xn-1] and Fc[xn-1] for 0<n<1. Hence show that 1/√x is self reciprocal under both. Hence deduce the value of F[ 1/√|x|].
16.Evaluate the Fourier transform of
f(x) = 1-x2 in |x| ≤ 1
= 0 in |x| > 1
Hence deduce the value of
∞ ∫ (sin s – s cos s)(cos(s/2)) ds
0 s3
f(x) = 1
x(a2+x2)
18.Find the fourier cosine transformation of f(x)=e -a2x2 and
hence find Fs[ x f(x) ].
19.Find the fourier transformation of
f(x) = a2 – x2 |x|<a
= 0 |x|>a>0
Hence deduce the value of
∞
∫ (sin t – t cost )dt
0 t3
20.Find the F-transform of f(x) if
f(x) = 1-|x| for |x|<1
= 0 for |x|>1
Hence deduce the value of
∞ ∫ sin 4t dt
0 t 4
21.Find the F-transform of f(x) for
f(x) = 1, |x|<a
= 0, |x|>a>0
Hence deduce the value of
∞
(i) ∫ sin t dt
0 t
∞
(ii) ∫ sin 2t dt
0 t2
22.Find the F-transform of f(x)= e-a|x| and hence deduce the following:
∞
(i) ∫ cos xt dt
0 (a2 + t2)
(ii) F[x f(x)]
23.Using fourier integral show that
∞
∫ (1- cosΠλ )sin xλ dλ = П/2 in 0<x<П and 0 in x>П.
0 λ
24.Using fourier integral find the integral value for e-ax - e-bx .
25.Use the Parseval's identity to calculate the following integrals:
∞
(i) ∫ dx for a>0 for both the problems.
0 (a2+x2)2
(ii) ∞
∫ x2 dx
0 (a2+x2)2
26.Find the F-transforms for f(x) provided:
(a) f(x) is “x” when |x|<a and “0” when |x|>a.
(b)f(x) is “x2” when |x|≤a and “0” when |x|>a.
27.Evaluate the following:
(a)The fourier Sine transform of f(x) = e-ax
x
(b)The fourier Cosine transform of f(x) when
f(x) is “cos x” when 0<x<a and “0” when x>a.
ALL THE BEST AND BEST OF EFFORTS!!!
set by,
RANGARAJAN T.R.
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