Fourier series AM

 Laplace And Fourier Transform objective questions (mcq) and answers - MechanicalTutorial

6. For the given periodic function    ≤ ≤ ≤ ≤ = 4 for 2 6 2 for 0 2 ( ) t t t f t with a period T = 6 as shown in Problem 5. The complex form of the Fourier series can be expressed as ∑ ∞ =−∞ = k ikw t k f t C e 0 ~ ( ) . The complex coefficient 1 ~ C can be expressed as (A) 0.4560 + 0.3734i (B) 0.4560 − 0.3734i (C) − 0.4560 + 0.3734i (D) 0.3734 − 0.4560i Solution The correct answer is (C).

1. In Fourier transform f(p)=∫∞−∞e(ipx)F(x)dx,e(ipx) is said to be Kernel function.
a) True
b) False
View Answer

Answer: a
Explanation: In any transform, apart from function given, the other function is said to be Kernel function. So, here in Fourier transform, e(ipx) is said to be the Kernel function.

2. Fourier Transform of e−|x|is 21+p2. Then what is the fourier transform of e−2|x|?
a) 4(4+p2)
b) 2(4+p2)
c) 2(2+p2)
d) 4(2+p2)
View Answer

Answer: a
Explanation: e−2|x|=e−|2x|=F(2x)
F{F(2x)}=12f(p2)
=122(1+p24)
=4(4+p2).

3. What is the fourier sine transform of e-ax?
a) 4(4+p2)
b) 4a(4a2+p2)
c) p(a2+p2)
d) 2p(a2+p2)
View Answer

Answer: c
Explanation: Fourier sine transform of F(x)=∫∞0e−axsin(px)dx
=e−ax(a2+p2)(−asin(px)−pcos(px)) from 0 to ∞
=p(a2+p2).

4. Find the fourier sine transform of x(a2+x2).
a) 2πe−ap
b) π2e−ap
c) 2πe−ap
d) πe−ap
View Answer

Answer: b
Explanation: Fourier transform of e−axisp(a2+p2)
Substitute x=m and p=x.
π2e−am=∫∞0xx2+a2sin(mx)dx
Therefore, fourier sine transform of x(a2+x2)isπ2e−ap.

5. Find the fourier transform of F(x) = 1, |x|<a0, otherwise.
a) 2sin(ap)p
b) 2asin(ap)p
c) 4sin(ap)p
d) 4asin(ap)p
View Answer

Answer: a
Explanation: f(p)=∫a−aeipxdx
=eipxip from -a to a
=eiap−e−iapip
=2sin(ap)p.

6. In Finite Fourier Cosine Transform, if the upper limit l = π, then its inverse is given by ________
a) F(x)=2π∑∞p=1fc(p)cos(px)+1πfc(0)
b) F(x)=2π∑∞p=1fc(p)cos(px)
c) F(x)=2π∑∞p=1fc(p)cos(pxπ)
d) F(x)=2π∑∞p=0fc(p)cos(px)+1πfc(0)
View Answer

Answer: a
Explanation: Now since we have fourier cosine transform, we have to use the constant 2π. And since while writing as sum of series it also has a term if p=0. Hence, F(x)=2π∑∞p=1fc(p)cos(px)+1πfc(0)

7. Find the Fourier Cosine Transform of F(x) = 2x for 0<x<4.
a) fc(p)=32(p2π2)(cos(pπ)−1)p not equal to 0 and if equal to 0 fc(p)=16
b) fc(p)=32(p2π2)(cos(pπ)−1)p not equal to 0 and if equal to 0 fc(p)=32
c) fc(p)=64(pπ2)(cos(pπ)−1)p not equal to 0 and if equal to 0 fc(p)=16
d) fc(p)=32(pπ2)(cos(pπ)−1)p not equal to 0 and if equal to 0 fc(p)=64
View Answer

Answer: a
Explanation: fc(p)=∫402xcos(pπx4)dx
=2[4xsin(pπx4)pπ+16cos(pπx4)p2π2] from 0 to 4
=32(p2π2)(cos(pπ)−1)
When p=0,fc(p)=∫402xdx=16.

8. If Fourier transform of e−|x|=21+p2, then find the fourier transform of t2e−|x|.
a) 41+p2
b) −21+p2
c) 21+p2
d) −41+p2
View Answer

Answer: b
Explanation: F{e−|x|}=21+p2
F{t2e−|x|}=(−i)221+p2=−21+p2.

9. If Fc{e−ax}=pa2+p2, find the Fs{−ae−ax}.
a) 4pa2+p2
b) −p2a2+p2
c) 4p2a2+p2
d) pa2+p2
View Answer

Answer: b
Explanation:−ae−ax=ddx(e−ax)=F′(x)
Fs{F′(x)}=−pfc(p)
=−p2a2+p2.

10. Find the fourier transform of ∂2u∂x2 . (u’(p,t) denotes the fourier transform of u(x,t)).
a) (ip)2 u’(p,t)
b) (-ip)2 u’(p,t)
c) (-ip)2 u(p,t)
d) (ip)2 u(p,t)
View Answer

Answer: a
Explanation: F{∂2u∂x2}=∫∞−∞∂2u∂x2eipxdx
=eipx∂u∂x from (-infinity to infinity) –∫∞−∞ipeipxu
=(ip)2u′(p,t)

11. What is the fourier transform of e-a|x| * e-b|x|?
a) 4ab(a2+p2)(b2+p2)
b) 2ab(a2+p2)(b2+p2)
c) 4(a2+p2)(b2+p2)
d) a2b2(a2+p2)(b2+p2)
View Answer

Answer: a
Explanation: Fourier transform of e−a|x|=2aa2+p2
Fourier transform of e−b|x|=2bb2+p2
fourier transform of e−a|x|∗e−b|x|=2aa2+p2.2bb2+p2
=4ab(a2+p2)(b2+p2).

12. What is the Fourier transform of eax? (a>0)
a) pa2+p2
b) 2aa2+p2
c) −2aa2+p2
d) cant’t be found
View Answer

Answer: d
Explanation: Fourier transform of eax, does not exist because the function does not converge. The function is divergent.

13. F(x)=x(−12)is self reciprocal under Fourier cosine transform.
a) True
b) False
View Answer

Answer: a
Explanation: Fc{x(−12)}=∫∞0x(−12)cos(px)dx=constant∗p(−12)
Inverse fourier transform of p(−12)=constant∗x(−12)
Hence the function x(−12)is self reciprocal.

14. Find the fourier cosine transform of e-ax * e-ax.
a) p2a2+p2
b) p2(a2+p2)2
c) 4p2(a2+p2)2
d) −p2(a2+p2)2
View Answer

Answer: b
Explanation = fourier cosine transform of e−ax=pa2+p2
fourier cosine transform of e−ax∗e−ax=pa2+p2.pa2+p2
=p2(a2+p2)2.

15. Find the fourier sine transform of F(x) = -x when x<c and (π – x) when x>c and 0≤c≤π.
a) πccos(pc)
b) πpcos(pc)
c) πccos(pπ)
d) pπccos(pc)Answer:b

Explanation: fs(p)=–∫c0xsin(px)dx+∫πc(π−x)sin(px)dx
=πpcos(pc).\