1
GATE ECE 2023
MCQ (Single Correct Answer)
+1
-0.33

The Fourier transform $$x(\omega )$$ of $$x(t) = {e^{ - {t^2}}}$$ is

Note : $$\int\limits_{ - \infty }^\infty {{e^{ - {y^2}}}dy = \sqrt \pi } $$

A
$$\sqrt \pi {e^{{{{\omega ^2}} \over 2}}}$$
B
$${{{e^{ - {{{\omega ^2}} \over 4}}}} \over {2\sqrt \pi }}$$
C
$$\sqrt \pi {e^{ - {{{\omega ^2}} \over 4}}}$$
D
$$\sqrt \pi {e^{ - {{{\omega ^2}} \over 2}}}$$
2
GATE ECE 2023
MCQ (Single Correct Answer)
+1
-0.33

In the table shown below, match the signal type with its spectral characteristics.

Signal type Spectral characteristics
(i) Continuous, aperiodic (a) Continuous, aperiodic
(ii) Continuous, periodic (b) Continuous, periodic
(iii) Discrete, aperiodic (c) Discrete, aperiodic
(iv) Discrete, periodic (d) Discrete, periodic

A
$$(i)\to(a),(ii)\to(b),(iii)\to(c),(iv)\to(d)$$
B
$$(i)\to(a),(ii)\to(c),(iii)\to(b),(iv)\to(d)$$
C
$$(i)\to(d),(ii)\to(b),(iii)\to(c),(iv)\to(a)$$
D
$$(i)\to(a),(ii)\to(c),(iii)\to(d),(iv)\to(b)$$
3
GATE ECE 2023
MCQ (Single Correct Answer)
+2
-0.67

Consider a discrete-time periodic signal with period N = 5. Let the discrete-time Fourier series (DTFS) representation be $$x[n] = \sum\limits_{k = 0}^4 {{a_k}{e^{{{jk2\pi m} \over 5}}}} $$, where $${a_0} = 1,{a_1} = 3j,{a_2} = 2j,{a_3} = - 2j$$ and $${a_4} = - 3j$$. The value of the sum $$\sum\limits_{n = 0}^4 {x[n]\sin {{4\pi n} \over 5}} $$ is

A
$$-$$10
B
10
C
$$-$$2
D
2
4
GATE ECE 2023
MCQ (Single Correct Answer)
+2
-0.67

Let an input $$x[n]$$ having discrete time Fourier transform $$x({e^{j\Omega }}) = 1 - {e^{ - j\Omega }} + 2{e^{ - 3j\Omega }}$$ be passed through an LTI system. The frequency response of the LTI system is $$H({e^{j\Omega }}) = 1 - {1 \over 2}{e^{ - j2\Omega }}$$. The output $$y[n]$$ of the system is

A
$$\delta [n] + \delta [n - 1] - {1 \over 2}\delta [n - 2] - {5 \over 2}\delta [n - 3] + \delta [n - 5]$$
B
$$\delta [n] - \delta [n - 1] - {1 \over 2}\delta [n - 2] - {5 \over 2}\delta [n - 3] + \delta [n - 5]$$
C
$$\delta [n] - \delta [n - 1] - {1 \over 2}\delta [n - 2] + {5 \over 2}\delta [n - 3] - \delta [n - 5]$$
D
$$\delta [n] + \delta [n - 1] + {1 \over 2}\delta [n - 2] + {5 \over 2}\delta [n - 3] + \delta [n - 5]$$
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