# Signals & Systems Questions and Answers – Fourier Transforms

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

Answer: d
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

Answer: b
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

Answer: a
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.
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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

Answer: b
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

Answer: a
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.
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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

Answer: c
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

Answer: d
Explanation: Fourier transform of gate function, G(ω) = $$\frac{2}{ω} sin⁡(\frac{wτ}{2})$$
Multiplying and dividing by τ we get
$$G(ω) = τ \frac{sin⁡(\frac{wτ}{2})}{\frac{wτ}{2}} = τ \frac{sin⁡(\frac{2πfτ}{2})}{\frac{2πfτ}{2}}= τ \frac{sin⁡(πτf)}{πτf} = τ sinc(τf)$$.
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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

Answer: b
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

Answer: d
Explanation: $$G(t)\leftrightarrow sa(\frac{ωτ}{2})$$ is not a fourier transform pair.
$$G(t)\leftrightarrow τsa(\frac{ωτ}{2})$$ (or) $$G(t)\leftrightarrow G(t) τ sinc(τf)$$.
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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

Answer: b
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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