250+ TOP MCQs on Properties of Fourier Transforms and Answers

Signals & Systems Multiple Choice Questions 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)
Answer: B
Clarification: 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)
Answer: D
Clarification: 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
Answer: A
Clarification: 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.

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} )
Answer: B
Clarification: 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 e^jω₀t.
A. δ(ω + ω₀)
B. 2πδ(ω + ω₀)
C. δ(ω – ω₀)
D. 2πδ(ω – ω₀)
Answer: D
Clarification: We know that F[1] = 2πδ(ω)
By using the frequency shifting property, e^jω₀t x(t) ↔ X(ω – ω₀)
We have F[e^jω₀t] = F[e^jω₀t (1)] = 2πδ(ω – ω₀).

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})
Answer: C
Clarification: 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
Answer: A
Clarification: 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)] e^j2ω = F(ω)e^-j2ω + F(ω)e^j2ω = 2F(ω)cos⁡2ω.

8. Find the Fourier transform of (frac{1}{a+jt}).
A. 2πe^aω u(ω)
B. 2πe^aω u(-ω)
C. 2πe^-aω u(ω)
D. 2πe^-aω u(-ω)
Answer: B
Clarification: 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)] = FBig[frac{1}{a+jt}Big]) = 2πx(-ω) = 2πe^aω 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}])
Answer: D
Clarification: 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}]).

10. Find the Fourier transform of sinc(t).
A. G_π (ω)
B. G_2π (ω)
C. (G_{frac{π}{2}}) (ω)
D. G_π (-ω)
Answer: B
Clarification: Using duality property, X(t) ↔ 2πx(-ω)
We get sinc(t) ↔ G_2π (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