Transforms and Sampling - Signals and Systems - Electrical Engineering Previous Year Questions

1

2007 · Electrical Engineering · Signals and Systems · Transforms and Sampling

Electrical Engineering (EE) 2007

The frequency spectrum of a signal is shown in the figure. If this signal is ideally sampled at intervals of 1 ms, then the frequency spectrum of the sampled signal will be

A

B

C

D

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2

2008 · Electrical Engineering · Signals and Systems · Transforms and Sampling

Electrical Engineering (EE) 2008

Given \( X(z) = \frac{z}{(z - a)^2} \) with \( |z| > a \), the residue of \( X(z) z^{n-1} \) at \( z = a \) for \( n \geq 0 \) will be

A

\(a^{n-1} \)

B

\(a^n \)

C

\(n a^n \)

D

\(n a^{n-1} \)

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3

2008 · Electrical Engineering · Signals and Systems · Transforms and Sampling

Electrical Engineering (EE) 2008

Let \( x(t) = rect(t - \frac{1}{2}) \) (where \( rect(t) = 1 \) for \( -\frac{1}{2} \leq x \leq \frac{1}{2} \) and zero otherwise). Then if \( sinc(x) = \frac{\sin(\pi x)}{\pi x} \), the Fourier Transform of \( x(t) + x(-t) \) will be given by

A

\(sinc\left(\frac{\omega}{2\pi}\right) \)

B

\(2 sinc\left(\frac{\omega}{2\pi}\right) \)

C

\(2 sinc\left(\frac{\omega}{2\pi}\right) \cos\left(\frac{\omega}{2}\right) \)

D

\(sinc\left(\frac{\omega}{2\pi}\right) \sin\left(\frac{\omega}{2}\right) \)

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4

2009 · Electrical Engineering · Signals and Systems · Transforms and Sampling

Electrical Engineering (EE) 2009

The z-transform of a signal \( x[n] \) is given by \( 4z^{-3} + 3z^{-1} + 2 - 6z^2 + 2z^3 \). It is applied to a system, with a transfer function \( H(z) = 3z^{-1} - 2 \). Let the output be \( y(n) \). Which of the following is true?

A

\(y(n) \) is non causal with finite support

B

\(y(n) \) is causal with infinite support

C

\(y(n) = 0 ; |n| > 3 \)

D

\(\text{Re}[Y(z)]_{z=e^{j\theta}} = -\text{Re}[Y(z)]_{z=e^{-j\theta}} ; \text{Im}[Y(z)]_{z=e^{j\theta}} = \text{Im}[Y(z)]_{z=e^{-j\theta}} ; -\pi \leq \theta < \pi \)

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5

2010 · Electrical Engineering · Signals and Systems · Transforms and Sampling

Electrical Engineering (EE) 2010

x(t) is a positive rectangular pulse from t = -1 to t = +1 with unit height as shown in the figure. The value of \int_{-\infty}^{\infty} |X(\omega)|^2 d\omega {where X(\omega) is the Fourier transform of x(t)} is

A

2

B

2π

C

4

D

4π

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6

2010 · Electrical Engineering · Signals and Systems · Transforms and Sampling

Electrical Engineering (EE) 2010

The Laplace transform of g(t) is

A

(1/s)(e^{-3s} - e^{-5s})

B

(1/s)(e^{-5s} - e^{-3s})

C

(e^{-3s}/s)(1 - e^{-2s})

D

(1/s)(e^{-3s} + e^{-5s})

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7

2011 · Electrical Engineering · Signals and Systems · Transforms and Sampling

Electrical Engineering (EE) 2011

Let the Laplace transform of a function \(f(t)\) which exists for \(t > 0\) be \(F_1(s)\) and the Laplace transform of its delayed version \(f(t - \tau)\) be \(F_2(s)\). Let \(F_1^*(s)\) be the complex conjugate of \(F_1(s)\) with the Laplace variable set as \(s = \sigma + j\omega\). If \(G(s) = \frac{F_2(s)F_1^*(s)}{|F_1(s)|^2}\), then the inverse Laplace transform of \(G(s)\) is

A

an ideal impulse \(\delta(t)\)

B

an ideal delayed impulse \(\delta(t - \tau)\)

C

an ideal step function \(u(t)\)

D

an ideal delayed step function \(u(t - \tau)\)

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8

2012 · Electrical Engineering · Signals and Systems · Transforms and Sampling

Electrical Engineering (EE) 2012

If \(x[n] = (1/3)^{|n|} - (1/2)^n u[n]\), then the region of convergence (ROC) of its Z-transform in the Z-plane will be

A

\(\frac{1}{3} < |z| < 3\)

B

\(\frac{1}{3} < |z| < \frac{1}{2}\)

C

\(\frac{1}{2} < |z| < 3\)

D

\(\frac{1}{3} < |z|\)

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9

2012 · Electrical Engineering · Signals and Systems · Transforms and Sampling

Electrical Engineering (EE) 2012

The unilateral Laplace transform of \(f(t)\) is \(\frac{1}{s^2+s+1}\). The unilateral Laplace transform of \(t f(t)\) is

A

\(-\frac{s}{(s^2+s+1)^2}\)

B

\(-\frac{2s+1}{(s^2+s+1)^2}\)

C

\(\frac{s}{(s^2+s+1)^2}\)

D

\(\frac{2s+1}{(s^2+s+1)^2}\)

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10

2012 · Electrical Engineering · Signals and Systems · Transforms and Sampling

Electrical Engineering (EE) 2012

The Fourier transform of a signal h(t) is H(jω) = (2 cos ω)(sin 2ω)/ω. The value of h(0) is

A

1/4

B

1/2

C

1

D

2

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11

2014 · Electrical Engineering · Signals and Systems · Transforms and Sampling

Electrical Engineering (EE) 2014 [Session 1]

Let \(X(z) = \frac{1}{1-z^{-3}}\) be the Z-transform of a causal signal \(x[n]\). Then, the values of \(x[2]\) and \(x[3]\) are

A

0 and 0

B

0 and 1

C

1 and 0

D

1 and 1

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12

2014 · Electrical Engineering · Signals and Systems · Transforms and Sampling

Electrical Engineering (EE) 2014 [Session 1]

Let \( X(s) = \frac{3s+5}{s^2+10s+21} \) be the Laplace Transform of a signal \( x(t) \). Then, \( x(0^+) \) is

A

0

B

3

C

5

D

21

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13

2014 · Electrical Engineering · Signals and Systems · Transforms and Sampling

Electrical Engineering (EE) 2014 [Session 3]

A function f(t) is shown in the figure.
The Fourier transform F(ω) of f(t) is

A

real and even function of ω

B

real and odd function of ω

C

imaginary and odd function of ω

D

imaginary and even function of ω

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14

2014 · Electrical Engineering · Signals and Systems · Transforms and Sampling

Electrical Engineering (EE) 2014 [Session 3]

A signal is represented by \(x(t) = \begin{cases} 1 & |t| < 1 \\ 0 & |t| > 1 \end{cases}\) The Fourier transform of the convolved signal \(y(t) = x(2t) * x(t/2)\) is

A

\(\frac{4}{\omega^2} \sin\left(\frac{\omega}{2}\right) \sin(2\omega)\)

B

\(\frac{4}{\omega^2} \sin\left(\frac{\omega}{2}\right)\)

C

\(\frac{4}{\omega^2} \sin(2\omega)\)

D

\(\frac{4}{\omega^2} \sin^2 \omega\)

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15

2016 · Electrical Engineering · Signals and Systems · Transforms and Sampling

Electrical Engineering (EE) 2016 [Session 1]

The Laplace Transform of \(f(t) = e^{2t} \sin(5t) u(t)\) is

A

\(\frac{5}{s^2 - 4s + 29}\)

B

\(\frac{5}{s^2 + 5}\)

C

\(\frac{s - 2}{s^2 - 4s + 29}\)

D

\(\frac{5}{s + 5}\)

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16

2017 · Electrical Engineering · Signals and Systems · Transforms and Sampling

Electrical Engineering (EE) 2017 [Session 1]

Consider a causal and stable LTI system with rational transfer function \(H(z)\), whose corresponding impulse response begins at \(n = 0\). Furthermore, \(H(1) = \frac{5}{4}\). The poles of \(H(z)\) are \(p_k = \frac{1}{\sqrt{2}} \exp(j\frac{(2k-1)\pi}{4})\) for \(k = 1,2,3,4\). The zeros of \(H(z)\) are all at \(z = 0\). Let \(g[n] = j^n h[n]\). The value of \(g[8]\) equals ______. (Give the answer up to three decimal places.)

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17

2017 · Electrical Engineering · Signals and Systems · Transforms and Sampling

Electrical Engineering (EE) 2017 [Session 2]

The pole-zero plots of three discrete-time systems P, Q and R on the z-plane are shown below. Which one of the following is TRUE about the frequency selectivity of these systems?

A

All three are high-pass filters.

B

All three are band-pass filters.

C

All three are low-pass filters.

D

P is a low-pass filter, Q is a band-pass filter and R is a high-pass filter.

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18

2017 · Electrical Engineering · Signals and Systems · Transforms and Sampling

Electrical Engineering (EE) 2017 [Session 2]

The output $y(t)$ of the following system is to be sampled, so as to reconstruct it from its samples uniquely. The required minimum sampling rate is

A

1000 samples/s

B

1500 samples/s

C

2000 samples/s

D

3000 samples/s

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19

2018 · Electrical Engineering · Signals and Systems · Transforms and Sampling

Electrical Engineering (EE) 2018

The Fourier transform of a continuous-time signal \(x(t)\) is given by \(X(\omega)=\frac{1}{(10+j\omega)^2}\), \(-\infty<\omega<\infty\), where \(j=\sqrt{-1}\) and \(\omega\) denotes frequency. Then the value of \(|\ln x(t)|\) at \(t=1\) is ______ (up to 1 decimal place). (\(\ln\) denotes the logarithm to base \(e\))

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20

2019 · Electrical Engineering · Signals and Systems · Transforms and Sampling

Electrical Engineering (EE) 2019

The inverse Laplace transform of \(H(s) = \frac{s+3}{s^2+2s+1}\) for \(t \geq 0\) is

A

\(3te^{-t} + e^{-t}\)

B

\(3e^{-t}\)

C

\(2te^{-t} + e^{-t}\)

D

\(4te^{-t} + e^{-t}\)

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