Signals & Systems Questions and Answers – Properties of Fourier Transforms

This set of Signals & Systems Multiple Choice Questions & Answers (MCQs) focuses on “Properties of Fourier Transforms”.

1. The Fourier transform of a function x(t) is X(ω). What will be the Fourier transform of \(\frac{dX(t)}{dt}\)?
a) \(\frac{X(f)}{jf}\)
b) j2πfX(f)
c) \(\frac{dX(f)}{dt}\)
d) jfX(f)
View Answer

Answer: b
Explanation: We know that x(t) = \(\frac{1}{2π} \int_{-∞}^∞ X(ω) e^{jωt} \,dω\)
\( \frac{d}{dt} \,x(t) = \frac{1}{2π} \int_{-∞}^∞ X(ω) \frac{d}{dt} e^{jωt} \,dω = \frac{1}{2π} jω X(ω) \int_{-∞}^∞ e^{jωt} \,dω\)
= jω X(ω) = j2πfX(f).

2. Find the Fourier transform of \(\frac{j}{πt}\).
a) sinc(ω)
b) sa(ω)
c) δ(ω)
d) sgn(ω)
View Answer

Answer: d
Explanation: Let x(t) = sgn(t)
The Fourier transform of sgn(t) is X(ω) = F[sgn(t)] = \(\frac{2}{jω}\)
Replacing ω with t
–> X(t) = \(\frac{2}{jt}\)
As per duality property X(t) ↔ 2πx(-ω), we have
F\(\Big[\frac{2}{jt}\Big]\) = 2πsgn(-ω) = -2πsgn(ω)
\(\frac{2}{jt}\) ↔ -2πsgn(ω)
\(\frac{2}{πt}\) ↔ sgn(ω).

3. The Fourier transform of a Gaussian pulse is also a Gaussian pulse.
a) True
b) False
View Answer

Answer: a
Explanation: Gaussian pulse, x(t) = e-πt2
Its Fourier transform is X(f) = e-πf2
Hence, the Fourier transform of a Gaussian pulse is also a Gaussian pulse.
advertisement

4. Find the Fourier transform of f(t)=te-at u(t).
a) \(\frac{1}{(a-jω)^2} \)
b) \(\frac{1}{(a+jω)^2} \)
c) \(\frac{a}{(a-jω)^2} \)
d) \(\frac{ω}{(a-jω)^2} \)
View Answer

Answer: b
Explanation: Using frequency differentiation property, \(tx(t) \leftrightarrow j \frac{d}{dω} \,X(ω)\)
\(F[te^{-at} u(t)] = j \frac{d}{dω} F[te^{-at} \,u(t)] = j \frac{d}{dω} \frac{1}{a+jω} = j \frac{-1(j)}{(a+jω)^2} = \frac{1}{(a+jω)^2} \)
\(te^{-at} \,u(t) \leftrightarrow \frac{1}{(a+jω)^2} \).

5. Find the Fourier transform of e0t.
a) δ(ω + ω0)
b) 2πδ(ω + ω0)
c) δ(ω – ω0)
d) 2πδ(ω – ω0)
View Answer

Answer: d
Explanation: We know that F[1] = 2πδ(ω)
By using the frequency shifting property, e0t x(t) ↔ X(ω – ω0)
We have F[e0t] = F[e0t (1)] = 2πδ(ω – ω0).
Free 30-Day Python Certification Bootcamp is Live. Join Now!

6. Find the Fourier transform of u(-t).
a) πδ(ω) + \(\frac{1}{ω}\)
b) πδ(ω) + \(\frac{1}{jω}\)
c) πδ(ω) – \(\frac{1}{jω}\)
d) δ(ω) + \(\frac{1}{jω}\)
View Answer

Answer: c
Explanation: We know that F[u(t)] = πδ(ω) + \(\frac{1}{jω}\)
Using time reversal property, x(-t) ↔ X(-ω)
We have F[u(-t)] = πδ(ω) – \(\frac{1}{jω}\).

7. Find the Fourier transform of x(t) = f(t – 2) + f(t + 2).
a) 2F(ω)cos⁡2ω
b) F(ω)cos⁡2ω
c) 2F(ω)sin⁡2ω
d) F(ω)sin⁡2ω
View Answer

Answer: a
Explanation: Using linearity property, ax(t) + by(t) ↔ aX(ω) + bY(ω) and
Time shifting property, x(t-t0) ↔ e-jω0t X(ω),
We have F[x(t)] = F[f(t)] e-j2ω + F[f(t)] ej2ω = F(ω)e-j2ω + F(ω)ej2ω = 2F(ω)cos⁡2ω.

8. Find the Fourier transform of \(\frac{1}{a+jt}\).
a) 2πe u(ω)
b) 2πe u(-ω)
c) 2πe-aω u(ω)
d) 2πe-aω u(-ω)
View Answer

Answer: b
Explanation: Let X(t) = \(\frac{1}{a+jt}\)
Replacing t with ω
X(ω) = \(\frac{1}{a+jw}\)
x(t )= e-at u(t)
As per duality property X(t) ↔ 2πx(-ω), we have
\(F[X(t)] = F\Big[\frac{1}{a+jt}\Big]\) = 2πx(-ω) = 2πe u(-ω).

9. Find the Fourier transform of e-2t u(t-1).
a) \(e^{-2} [e^{-jω} \frac{1}{2-jω}]\)
b) \(e^2 [e^{-jω} \frac{1}{2-jω}]\)
c) \(e^{-2} [e^{jω} \frac{1}{2-jω}]\)
d) \(e^{-2} [e^{-jω} \frac{1}{2+jω}]\)
View Answer

Answer: d
Explanation: We know that e-at u(t) ↔ \(\frac{1}{a+jw}\)
Using time shifting property, x(t-t0) ↔ e-jω0t X(ω) we have
f[e-2t u(t-1)] = \(e^{-2} [e^{-jω} \frac{1}{2+jω}]\).
advertisement

10. Find the Fourier transform of sinc(t).
a) Gπ (ω)
b) G (ω)
c) \(G_{\frac{π}{2}}\) (ω)
d) Gπ (-ω)
View Answer

Answer: b
Explanation: Using duality property, X(t) ↔ 2πx(-ω)
We get sinc(t) ↔ G (ω).

11. If the Fourier transform of g(t) is G(ω), then match the following and choose the right answer.

(i) The Fourier transform of g(t-2) is            (A) G(ω)e^-j2ω
(ii) The Fourier transform of g(t/2) is           (B) G(2ω)  
                                                  (C) 2G(2ω)  
                                                  (D) G(ω-2)

a) (i)-B, (ii)-A
b) (i)-A, (ii)-C
c) (i)-D, (ii)-C
d) (i)-C, (ii)-A
View Answer

Answer: b
Explanation: Using time shifting property, x(t – t0) ↔ e-jω0 t X(ω)
g(t – 2) ↔ e-j2ω G(ω)
Time scaling property, x(at) ↔ \( \frac{1}{a} X(\frac{w}{a})\)
g(t/2) ↔ 2G(2ω).

Sanfoundry Global Education & Learning Series – Signals & Systems.

To practice all areas of Signals & Systems, here is complete set of 1000+ Multiple Choice Questions and Answers.

advertisement
advertisement
Subscribe to our Newsletters (Subject-wise). Participate in the Sanfoundry Certification contest to get free Certificate of Merit. Join our social networks below and stay updated with latest contests, videos, internships and jobs!

Youtube | Telegram | LinkedIn | Instagram | Facebook | Twitter | Pinterest
Manish Bhojasia - Founder & CTO at Sanfoundry
I’m Manish - Founder and CTO at Sanfoundry. I’ve been working in tech for over 25 years, with deep focus on Linux kernel, SAN technologies, Advanced C, Full Stack and Scalable website designs.

You can connect with me on LinkedIn, watch my Youtube Masterclasses, or join my Telegram tech discussions.

If you’re in your 40s–60s and exploring new directions in your career, I also offer mentoring. Learn more here.