This set of Signals & Systems Multiple Choice Questions & Answers (MCQs) focuses on “Fourier Transforms”.
1. Which of the following is the Analysis equation of Fourier Transform?
a) \(F(ω) = \int_{-∞}^∞ f(t)e^{jωt} \,dt\)
b) \(F(ω) = \int_0^∞ f(t)e^{-jωt} \,dt\)
c) \(F(ω) = \int_0^∞ f(t)e^{jωt} \,dt\)
d) \(F(ω) = \int_{-∞}^∞ f(t)e^{-jωt} \,dt\)
View Answer
Explanation: For converting time domain to frequency domain, we use analysis equation. The Analysis equation of Fourier Transform is \(F(ω) = \int_{-∞}^∞ f(t)e^{-jωt} \,dt\).
2. Choose the correct synthesis equation.
a) \(f(t) = \frac{1}{2π} \int_{-∞}^∞ F(ω) e^{-jωt} \,dω\)
b) \(f(t) = \frac{1}{2π} \int_{-∞}^∞ F(ω) e^{jωt} \,dω\)
c) \(f(t) = \frac{1}{2π} \int_0^∞ F(ω) e^{-jωt} \,dω\)
d) \(f(t) = \frac{1}{2π} \int_0^∞ F(ω) e^{jωt} \,dω\)
View Answer
Explanation: Synthesis equation converts from frequency domain to time domain. The synthesis equation of fourier transform is \(f(t) = \frac{1}{2π} \int_{-∞}^∞ F(ω) e^{jωt} \,dω\).
3. Find the fourier transform of an exponential signal f(t) = e-at u(t), a>0.
a) \(\frac{1}{a+jω}\)
b) \(\frac{1}{a-jω}\)
c) \(\frac{1}{-a+jω}\)
d) \(\frac{1}{-a-jω}\)
View Answer
Explanation: Given f(t)= e-at u(t)
We know that \( u(t)=\begin{cases}
0 &\text{\(t<0\)} \\ 1 &\text{\(t>0\)} \\
\end{cases}\)
Fourier transform,
\(F(ω) = \int_{-∞}^∞ f(t)e^{-jωt} \,dt = \int_{-∞}^∞ e^{-at} u(t)e^{-jωt} \,dt = \int_0^∞ e^{-(a+jω)t} \,dt\)
F(ω) = \(\frac{1}{a+jω}\), a>0.
4. Find the fourier transform of the function f(t) = e-a|t|, a>0.
a) \(\frac{2a}{a^2-ω^2}\)
b) \(\frac{2a}{a^2+ω^2}\)
c) \(\frac{2a}{ω^2-a^2}\)
d) \(\frac{a}{a^2+ω^2}\)
View Answer
Explanation: The given two-sided exponential function f(t) = e-a|t|, a>0 can be expressed as
\( f(t)=\begin{cases}
e^{-at} &\text{\(t≥0\)} \\
e^{at} &\text{\(t≤0\)} \\
\end{cases}\)
The Fourier transform is
\(F(ω) = \int_{-∞}^∞ f(t)e^{-jωt} \,dt = \int_{-∞}^0 f(t)e^{-jωt} \,dt + \int_0^∞ f(t)e^{-jωt} \,dt\)
\(F(ω) = \frac{1}{a+jω} + \frac{1}{a-jω} = \frac{2a}{a^2+ω^2}\).
5. Gate function is defined as ______________
a) \(
G(t)=\begin{cases}
1 &\text{\(|t|<\frac{τ}{2}\)} \\
0 &\text{elsewhere} \\
\end{cases} \)
b) \(
G(t)=\begin{cases}
1 &\text{\(|t|>\frac{τ}{2}\)} \\
0 &\text{elsewhere} \\
\end{cases}\)
c) \(
G(t)=\begin{cases}
1 &\text{\(|t|≤\frac{τ}{2}\)} \\
0 &\text{elsewhere} \\
\end{cases}\)
d) \(
G(t)=\begin{cases}
1 &\text{\(|t|≥\frac{τ}{2}\)} \\
0 &\text{elsewhere} \\
\end{cases}\)
View Answer
Explanation: A gate function is a rectangular function defined as
\( G(t) = rect(\frac{t}{τ}) = \begin{cases}
1 &\text{\(|t|<\frac{τ}{2}\)} \\
0 &\text{elsewhere} \\
\end{cases} \)
Where τ is pulse width.
6. Find the fourier transform of the gate function.
a) \(\frac{1}{ω} sin(\frac{ωτ}{2})\)
b) \(\frac{1}{ω} cos(\frac{ωτ}{2})\)
c) \(\frac{2}{ω} sin(\frac{ωτ}{2})\)
d) \(\frac{2}{ω} cos(\frac{ωτ}{2})\)
View Answer
Explanation: Gate function is defined as
\( G(t)=\begin{cases}
1 &\text{\(|t|<\frac{τ}{2}\)} \\
0 &\text{elsewhere} \\
\end{cases} \)
The fourier transform is \(F(ω) = \int_{-∞}^∞ f(t)e^{-jωt} \,dt = \int_{-τ/2}^{τ/2} e^{-jωt} \,dt = \frac{2}{ω} sin(\frac{ωτ}{2})\).
7. Choose the wrong option.
a) G(t) = rect(\(\frac{t}{τ}\))
b) G(t) = u(t + \(\frac{τ}{2}\)) – u(t-\(\frac{τ}{2}\))
c) G(ω) = τ sa(\(\frac{wτ}{2}\))
d) G(f) = τ sinc(f)
View Answer
Explanation: Fourier transform of gate function, G(ω) = \(\frac{2}{ω} sin(\frac{wτ}{2})\)
Multiplying and dividing by τ we get
\(G(ω) = τ \frac{sin(\frac{wτ}{2})}{\frac{wτ}{2}} = τ \frac{sin(\frac{2πfτ}{2})}{\frac{2πfτ}{2}}= τ \frac{sin(πτf)}{πτf} = τ sinc(τf)\).
8. Bandwidth of the gate function is __________
a) τ Hz
b) \(\frac{1}{τ}\) Hz
c) 2τ Hz
d) \(\frac{2}{τ}\) Hz
View Answer
Explanation: The practical bandwidth of the gate function corresponds to the first zero crossing in the spectrum. Therefore, the bandwidth of the pulse or gate function is \(\frac{2π}{τ}\) or \(\frac{1}{τ}\) Hz.
9. Which of the following is not a fourier transform pair?
a) \(u(t) \leftrightarrow πδ(ω) + \frac{1}{jω}\)
b) \(sgn(t) \leftrightarrow \frac{2}{jω}\)
c) \(A \leftrightarrow 2πδ(\frac{ω}{2})\)
d) \(G(t)\leftrightarrow sa(\frac{ωτ}{2})\)
View Answer
Explanation: \(G(t)\leftrightarrow sa(\frac{ωτ}{2})\) is not a fourier transform pair.
\(G(t)\leftrightarrow τsa(\frac{ωτ}{2})\) (or) \(G(t)\leftrightarrow G(t) τ sinc(τf)\).
10. Find the fourier transform of the unit step function.
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
Explanation: We know that sgn(t) = 2u(t) – 1.
u(t) = \(\frac{1}{2}\)[sgn(t)+1] Its Fourier transform is F[u(t)] = \(\frac{1}{2}\) F[sgn(t)] + \(\frac{1}{2}\) F[1]
As the Fourier transforms F[1] = 2πδ(ω) and [sgn(t)] = \(\frac{2}{jω}\), hence
F[u(t)] = πδ(ω) + \(\frac{1}{jω}\).
Sanfoundry Global Education & Learning Series – Signals & Systems.
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