Signals & Systems Multiple Choice Questions on “Inverse Fourier Transform”.
1. Find the inverse Fourier transform of X(ω) = e-2ω u(ω).
A. (frac{1}{2π(2+jt)})
B. (frac{1}{2π(2-jt)})
C. (frac{1}{2(2+jt)})
D. (frac{1}{π(2+jt)})
Answer: B
Clarification: We know that x(t) = (frac{1}{2π} int_{-∞}^∞ X(ω) e^{jωt} ,dω)
x(t) = (frac{1}{2π} int_{-∞}^∞ e^{-2ω} ,u(ω) e^{jωt} ,dω = frac{1}{2π} int_{-∞}^∞ e^{-2ω} e^{jωt} , dω = frac{1}{2π(2-jt)}).
2. Find the inverse Fourier transform of X(ω) = (frac{1+3(jω)}{(3+jω)^2}).
A. 3e-3t u(t) + 8e-3t u(t)
B. 3te-3t u(t) – 8e-8t u(t)
C. 3e-3t u(t) + 8te8t u(t)
D. 3e-3t u(t) – 8te-3t u(t)
Answer: D
Clarification: Given X(ω) = (frac{1+3(jω)}{(3+jω)^2} = frac{A}{3+jω} + frac{B}{(3+jω)^2} = frac{3}{3+jω} – frac{8}{(3+jω)^2})
Applying inverse Fourier transform, we get
x(t) = 3e-3t u(t) – 8te-3t u(t).
3. Find the inverse Fourier transform of δ(ω).
A. (frac{1}{2π})
B. 2π
C. (frac{1}{π})
D. π
Answer: D
Clarification: We know that x(t) = (frac{1}{2π} int_{-∞}^∞ X(ω) e^{jωt} ,dω)
= (frac{1}{2π} int_{-∞}^∞ δ(ω) e^{jωt} ,dω = frac{1}{2π}).
4. Find the inverse Fourier transform of u(ω).
A. (frac{1}{2} δ(t) + frac{j}{2πt})
B. (frac{1}{2} δ(t) – frac{j}{2πt})
C. δ(t) + (frac{j}{2πt})
D. δ(t) – (frac{j}{2πt})
Answer: A
Clarification: We know that u(ω) = (frac{1}{2})[1+sgn(ω)].
Applying linearity property,
u(ω) = -1 ([frac{1}{2}] + F^{-1} [frac{1}{2} sgn(ω)])
u(ω) = (frac{1}{2} δ(t) + frac{j}{2πt}).
5. Find the inverse Fourier transform of ej2t.
A. 2πδ(ω-2)
B. πδ(ω-2)
C. πδ(ω+2)
D. 2πδ(ω+2)
Answer: A
Clarification: We know that ejω0 t ↔ 2πδ(ω-ω0)
∴ ej2t ↔ 2πδ(ω-2).
6. Find the inverse Fourier transform of jω.
A. δ(t)
B. (frac{d}{dt}) δ(t)
C. (frac{1}{δ(t)})
D. ∫δ(t)
Answer: B
Clarification: Time differentiation property, (frac{d}{dt}) x(t) ↔ jωX(ω) and we know that δ(t) ↔ 1
∴ (frac{d}{dt}) δ(t) ↔ jω.
7. Find the inverse Fourier transform of (X(ω) = frac{6+4(jω)}{(jω)^2 + 6(jω) + 8}).
A. e-2t u(t) – 5e-4t u(t)
B. e-2t u(t) + 5e-4t u(t)
C. -e-2t u(t) – 5e-4t u(t)
D. -e-2t u(t) + 5e-4t u(t)
Answer: D
Clarification: (X(ω) = frac{6+4(jω)}{(jω)^2+6(jω)+8} = frac{A}{jω+2} + frac{B}{jω+4} = -frac{1}{jω+2} + frac{5}{jω+4})
Applying inverse Fourier transform, we get
x(t) = -e-2t u(t) + 5e-4t u(t).
8. Find the convolution of the signals x1 (t) = e-2t u(t) and x2 (t) = e-3t u(t).
A. e-2t u(t) – e-3t u(t)
B. e-2t u(t) + e-3t u(t)
C. e2t u(t) – e3t u(t)
D. e2t u(t) – e-3t u(t)
Answer: A
Clarification: Convolution property, x1 (t)*x2 (t) ↔ X1 (ω) X2 (ω)
∴ x1 (t)*x2 (t) = F-1 [X1 (ω) X2 (ω)]
Given x1 (t) = e-2t u(t)
∴ X1 (ω) = (frac{1}{jω+2})
Given x2 (t) = e-3t u(t)
∴ X1 (ω) = (frac{1}{jω+3})
x1 (t)*x2 (t) = F-1 [X1 (ω) X2 (ω)] = F-1 ([frac{1}{jω+2} frac{1}{jω+3}] = F^{-1} [frac{1}{jω+2} – frac{1}{jω+3}] )
∴ x1 (t)*x2 (t) = e-2t u(t)-e-3t u(t).
9. Find the inverse Fourier transform of f(t)=1.
A. u(t)
B. δ(t)
C. e-t
D. (frac{1}{jω})
Answer: B
Clarification: We know that the Fourier transform of f(t) = 1 is F(ω) = 2πδ(ω).
Replacing ω with t
F(t) = 2πδ(t)
As per duality property F(t) ↔ 2πf(-ω), we have
2πδ(t) ↔ 2π(1)
δ(t) ↔ 1
Hence, the inverse Fourier transform of 1 is δ(t).
10. Find the inverse Fourier transform of sgn(ω).
A. (frac{1}{πt})
B. (frac{j}{πt})
C. (frac{j}{t})
D. (frac{1}{t})