Signals & Systems Questions and Answers – Inverse Fourier Transform

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

1. Find the inverse Fourier transform of u(ω).
a) 12δ(t)+j2πt
b) 12δ(t)j2πt
c) δ(t) + j2πt
d) δ(t) – j2πt

Explanation: We know that u(ω) = 12[1+sgn(ω)].
Applying linearity property,
u(ω) = -1 [12]+F1[12sgn(ω)]
u(ω) = 12δ(t)+j2πt.

2. Find the inverse Fourier transform of ej2t.
a) 2πδ(ω-2)
b) πδ(ω-2)
c) πδ(ω+2)
d) 2πδ(ω+2)

Explanation: We know that e0 t ↔ 2πδ(ω-ω0)
∴ ej2t ↔ 2πδ(ω-2).

3. Find the inverse Fourier transform of jω.
a) δ(t)
b) ddt δ(t)
c) 1δ(t)
d) ∫δ(t)

Explanation: Time differentiation property, ddt x(t) ↔ jωX(ω) and we know that δ(t) ↔ 1
∴ ddt δ(t) ↔ jω.

4. Find the inverse Fourier transform of X(ω)=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)

Explanation: X(ω)=6+4(jω)(jω)2+6(jω)+8=Ajω+2+Bjω+4=1jω+2+5jω+4
Applying inverse Fourier transform, we get
x(t) = -e-2t u(t) + 5e-4t u(t).

5. 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)

Explanation: 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) = 1jω+2
Given x2 (t) = e-3t u(t)
∴ X1 (ω) = 1jω+3
x1 (t)*x2 (t) = F-1 [X1 (ω) X2 (ω)] = F-1 [1jω+21jω+3]=F1[1jω+21jω+3]
∴ x1 (t)*x2 (t) = e-2t u(t)-e-3t u(t).

6. Find the inverse Fourier transform of f(t)=1.
a) u(t)
b) δ(t)
c) e-t
d) 1jω

Explanation: 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).

7. Find the inverse Fourier transform of sgn(ω).
a) 1πt
b) jπt
c) jt
d) 1t

Explanation: Given the function F(ω)=sgn(ω). The Fourier transform of a Signum function is sgn(ω) = 2jω.
Applying the duality property F(t) ↔ 2πf(-ω), we get
F(2jt) = 2πsgn(-ω).
As sgn(ω) is an odd function, sgn(-ω)=-sgn(ω).
Hence, 2jt ↔ -sgn(ω)
Or 2πt ↔ sgn(ω)
Therefore, the inverse Fourier transform of sgn(ω) is jπt.

8. Find the inverse Fourier transform of X(ω) = e-2ω u(ω).
a) 12π(2+jt)
b) 12π(2jt)
c) 12(2+jt)
d) 1π(2+jt)

Explanation: We know that x(t) = 12πX(ω)ejωtdω
x(t) = 12πe2ωu(ω)ejωtdω=12πe2ωejωtdω=12π(2jt).

9. Find the inverse Fourier transform of X(ω) = 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)

Explanation: Given X(ω) = 1+3(jω)(3+jω)2=A3+jω+B(3+jω)2=33+jω8(3+jω)2
Applying inverse Fourier transform, we get
x(t) = 3e-3t u(t) – 8te-3t u(t).

10. Find the inverse Fourier transform of δ(ω).
a) 12π
b) 2π
c) 1π
d) π

Explanation: We know that x(t) = 12πX(ω)ejωtdω
12πδ(ω)ejωtdω=12π.

The frequency-domain equivalent of the short-time Fourier transform is the short-frequency inverse Fourier transform (SFIFT) (STFT). We examined the frequency parameters of a signal around a time instant using the STFT approach. The Inverse Fast Fourier Transform (IDFT) algorithm reverses the DFT process. Backward Fourier transform is another name for it. It converts a time or space signal to a frequency domain signal. It involves the transformation of geographical or temporal data into frequency domain data.

By Team Pa

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