Signals and Systems Test 1

Question 1
2 / -0.33

For a finite support signal x[n] = r[n] (u[n] – u[n – 11]) where r[n] is the discrete-time ramp function, the energy of x[n] is

A
385-385

B
3854-3851

C
3857-3856

D
3859-3852

Question 2
2 / -0.33

Which of the following systems are linear time-invariant and causal systems.

A
\(\frac{{dy}}{{dy}} + 6y\left( t \right) = 4x\left( t \right)\)

B
y[n] + 2y [n – 1] = x [n + 1]

C
y[n+1] + 4 y[n] = 3x [n+1] = x[n]

D
\(\frac{{dy}}{{dy}} + {y^2}\left( t \right) = x\left( t \right)\)

Question 3
2 / -0.33

The z-transform of a sequence x[n] is given as x(z) = 3 + 4z – 6z^-1 + 2z^-2
If y[n] is the first difference of x[n], then y(z) is given by

A
1 – 4z + 9z^-1 – 8z^-2 + 2z^-3

B
– 1 + 4z – 9z^-1 + 8z^-2 – 2z^-3

C
– 1 – 4z + 9z^-1 – 8z^-2 – 2z^-3

D
1 + 4z – 9z^-1 + 8z^-2 – 2z^-3

Question 4
2 / -0.33

The value of \(\mathop \sum \limits_{n = 0}^\infty n{\left( {\frac{1}{2}} \right)^n}\) is _________.

A
2-2

B
24-21

C
27-26

D
29-22

Question 5
2 / -0.33

The DTFT of a signal f(n) = {a, b, c, d} is F(ω). The inverse DTFT of F(ω - π) is:

A
{a, b, c, d}

B
{-a, b, -c, d}

C
{a, -b, c, -d}

D
{-a, -b, -c, -d}

Question 6
2 / -0.33

The discrete Fourier series representation for the following sequence:
\(x\left( n \right) = \cos \frac{\pi }{4}n\) is

A
\(\frac{1}{2}{e^{j{{\rm{\Omega }}_0}n}} + \frac{1}{2}{e^{ - j{{\rm{\Omega }}_0}n}}\) and \({{\rm{\Omega }}_0} = \frac{\pi }{8}\)

B
\(\frac{1}{2}{e^{ - j{{\rm{\Omega }}_0}n}} + \frac{1}{2}{e^{ - j2{{\rm{\Omega }}_0}n}}\) and \({{\rm{\Omega }}_0} = \frac{\pi }{4}\)

C
\(\frac{1}{2}{e^{ - j{{\rm{\Omega }}_0}n}} + \frac{1}{2}{e^{ - j{{\rm{\Omega }}_0}n}}\) and \({{\rm{\Omega }}_0} = \frac{\pi }{6}\)

D
\(\frac{1}{2}{e^{j{{\rm{\Omega }}_0}n}} + \frac{1}{2}{e^{j7{{\rm{\Omega }}_0}n}}\) and \({{\rm{\Omega }}_0} = \frac{\pi }{4}\)

Question 7
2 / -0.33

Which of the following statements is false.

A
The Laplace transform of t²u(t) does not converge anywhere on the s-plane.

B
The Laplace transform of \({e^{{t^2}}}u\left( t \right)\) does not converge anywhere on the s-plane.

C
The Laplace transform of \({e^{j{\omega _0}t}}\) does not converge anywhere on the s-plane

D
The Laplace transform of |t| does not converge anywhere on the s-plane.

Question 8
2 / -0.33

The Fourier series coefficients of a periodic signal x(t), with fundamental frequency \({\omega _0} = \frac{\pi }{4}\) are \({X_1} = X_{ - 1}^* = j,\;{X_5} = X_{ - 5}^* = 2\) and the rest are zero. Suppose x(t) is the input to a band-pass filter with the following magnitude and phase responses.
\(H\left( {j\omega } \right) = \left\{ {\begin{array}{*{20}{c}}{\begin{array}{*{20}{c}}{1,\;\;\pi \le \omega \le 1.5\pi }\\{1,\;\; - 1.5\pi \le \omega \le - \pi }\end{array}}\\{0,\;\;otherwise\;\;\;\;\;\;\;\;\;\;\;\;}\end{array}} \right.\)
∠H(jω) = -ω
Let y(t) be the output of the filter and the Fourier series of x(t) in the trigonometric form
\(x\left( t \right) = \mathop \sum \limits_{k = 0}^\infty {X_k}\cos \left( {k{\omega _0}t + \angle {X_k}} \right)\)
x(t) and the steady state response of y(t) are

A
\(x\left( t \right) = - 2\sin \left( {\frac{{\pi t}}{4}} \right) + 4\cos \left( {\frac{{5\pi t}}{4}} \right),\;y\left( t \right) = 4\cos \left( {\frac{{5\pi t}}{4} - \frac{\pi }{4}} \right)\)

B
\(x\left( t \right) = - 2\sin \left( {\frac{{5\pi t}}{4}} \right) + 4\cos \left( {\frac{{\pi t}}{4}} \right),\;y\left( t \right) = 4\cos \left( {\frac{{5\pi t}}{4} - \frac{{5\pi }}{4}} \right)\)

C
\(x\left( t \right) = - 2\sin \left( {\frac{{\pi t}}{4}} \right) + 4\cos \left( {\frac{{5\pi t}}{4}} \right),\;y\left( t \right) = 4\cos \left( {\frac{{5\pi t}}{4} - \frac{{5\pi }}{4}} \right)\)

D
\(x\left( t \right) = - 2\sin \left( {\frac{{5\pi t}}{4}} \right) + 4\cos \left( {\frac{{\pi t}}{4}} \right),\;y\left( t \right) = 4\cos \left( {\frac{{5\pi t}}{4} - \frac{\pi }{4}} \right)\)

Question 9
2 / -0.33

The percentage of the total energy dissipated by a 1 Ω resistor in the frequency band 0 < ω < 10 rad/s when the voltage across it is v(t) = e^-2t u(t) is __________

A
87-88

B
874-881

C
877-886

D
879-882

Question 10
2 / -0.33

Consider the following difference equation:
y[n] + 3y[n – 1] + 2y[n – 2] = 2x[n] – x[n – 1]
If y[-1] = 0, y[-2] = 1, x[n] = u[n], then y[n] can be represented as

A
\(\left[ { - \frac{1}{2}{{\left( 1 \right)}^{n - 1}} + \frac{4}{3}{{\left( { - 2} \right)}^{n - 1}} + \frac{1}{6}} \right]u\left( { - n - 1} \right)\)

B
\(\left[ { - \frac{1}{2}{{\left( 1 \right)}^{n - 1}} + \frac{4}{3}{{\left( { - 2} \right)}^{n - 1}} + \frac{1}{6}} \right]u\left( {n - 1} \right)\)

C
\(\left[ { - \frac{1}{2}{{\left( 1 \right)}^{n - 1}} + \frac{4}{3}{{\left( { - 2} \right)}^{n - 1}}} \right]u\left( {n - 1} \right) + \frac{1}{6}u\left( n \right)\)

D
\(\left[ { - \frac{1}{2}{{\left( 1 \right)}^{n - 1}} + \frac{4}{3}{{\left( { - 2} \right)}^{n - 1}}} \right]u\left( { - n - 1} \right) + \frac{1}{6}u\left( n \right)\)

Question 11
2 / -0.33

A causal LTI system is described by the difference equation
2 y[n] = α y[n - 2] – 2 x[n] + β x[n - 1]
The system is stable only if

A
|α | = 2, |β | < 2

B
|α | > 2, |β | > 2

C
|α | < 2, any value of β

D
|β | < 2, any value of α

Question 12
2 / -0.33

For a given periodic sequence x (n) = {1, 1, 0, 0} with period N = 4, the Fourier coefficient is denoted by c_k. The Value of c₃* is:

A
1/2

B
\(\frac{1}{4}\left( 1+j \right)\)

C
\(\frac{1}{4}\left( 1-j \right)\)

D
0

Submit Test

Signals and Sys...

Question 1

385-385

3854-3851

3857-3856

3859-3852

Question 2

\(\frac{{dy}}{{dy}} + 6y\left( t \right) = 4x\left( t \right)\)

y[n] + 2y [n – 1] = x [n + 1]

y[n+1] + 4 y[n] = 3x [n+1] = x[n]

\(\frac{{dy}}{{dy}} + {y^2}\left( t \right) = x\left( t \right)\)

Question 3

1 – 4z + 9z-1 – 8z-2 + 2z-3

– 1 + 4z – 9z-1 + 8z-2 – 2z-3

– 1 – 4z + 9z-1 – 8z-2 – 2z-3

1 + 4z – 9z-1 + 8z-2 – 2z-3

Question 4

2-2

24-21

27-26

29-22

Question 5

{a, b, c, d}

{-a, b, -c, d}

{a, -b, c, -d}

{-a, -b, -c, -d}

Question 6

\(\frac{1}{2}{e^{j{{\rm{\Omega }}_0}n}} + \frac{1}{2}{e^{ - j{{\rm{\Omega }}_0}n}}\) and \({{\rm{\Omega }}_0} = \frac{\pi }{8}\)

\(\frac{1}{2}{e^{ - j{{\rm{\Omega }}_0}n}} + \frac{1}{2}{e^{ - j2{{\rm{\Omega }}_0}n}}\) and \({{\rm{\Omega }}_0} = \frac{\pi }{4}\)

\(\frac{1}{2}{e^{ - j{{\rm{\Omega }}_0}n}} + \frac{1}{2}{e^{ - j{{\rm{\Omega }}_0}n}}\) and \({{\rm{\Omega }}_0} = \frac{\pi }{6}\)

\(\frac{1}{2}{e^{j{{\rm{\Omega }}_0}n}} + \frac{1}{2}{e^{j7{{\rm{\Omega }}_0}n}}\) and \({{\rm{\Omega }}_0} = \frac{\pi }{4}\)

Question 7

The Laplace transform of t2u(t) does not converge anywhere on the s-plane.

The Laplace transform of \({e^{{t^2}}}u\left( t \right)\) does not converge anywhere on the s-plane.

The Laplace transform of \({e^{j{\omega _0}t}}\) does not converge anywhere on the s-plane

The Laplace transform of |t| does not converge anywhere on the s-plane.

Question 8

\(x\left( t \right) = - 2\sin \left( {\frac{{\pi t}}{4}} \right) + 4\cos \left( {\frac{{5\pi t}}{4}} \right),\;y\left( t \right) = 4\cos \left( {\frac{{5\pi t}}{4} - \frac{\pi }{4}} \right)\)

\(x\left( t \right) = - 2\sin \left( {\frac{{5\pi t}}{4}} \right) + 4\cos \left( {\frac{{\pi t}}{4}} \right),\;y\left( t \right) = 4\cos \left( {\frac{{5\pi t}}{4} - \frac{{5\pi }}{4}} \right)\)

\(x\left( t \right) = - 2\sin \left( {\frac{{\pi t}}{4}} \right) + 4\cos \left( {\frac{{5\pi t}}{4}} \right),\;y\left( t \right) = 4\cos \left( {\frac{{5\pi t}}{4} - \frac{{5\pi }}{4}} \right)\)

\(x\left( t \right) = - 2\sin \left( {\frac{{5\pi t}}{4}} \right) + 4\cos \left( {\frac{{\pi t}}{4}} \right),\;y\left( t \right) = 4\cos \left( {\frac{{5\pi t}}{4} - \frac{\pi }{4}} \right)\)

Question 9

87-88

874-881

877-886

879-882

Question 10

\(\left[ { - \frac{1}{2}{{\left( 1 \right)}^{n - 1}} + \frac{4}{3}{{\left( { - 2} \right)}^{n - 1}} + \frac{1}{6}} \right]u\left( { - n - 1} \right)\)

\(\left[ { - \frac{1}{2}{{\left( 1 \right)}^{n - 1}} + \frac{4}{3}{{\left( { - 2} \right)}^{n - 1}} + \frac{1}{6}} \right]u\left( {n - 1} \right)\)

\(\left[ { - \frac{1}{2}{{\left( 1 \right)}^{n - 1}} + \frac{4}{3}{{\left( { - 2} \right)}^{n - 1}}} \right]u\left( {n - 1} \right) + \frac{1}{6}u\left( n \right)\)

\(\left[ { - \frac{1}{2}{{\left( 1 \right)}^{n - 1}} + \frac{4}{3}{{\left( { - 2} \right)}^{n - 1}}} \right]u\left( { - n - 1} \right) + \frac{1}{6}u\left( n \right)\)

Question 11

|α | = 2, |β | < 2

|α | > 2, |β | > 2

|α | < 2, any value of β

|β | < 2, any value of α

Question 12

1/2

\(\frac{1}{4}\left( 1+j \right)\)

\(\frac{1}{4}\left( 1-j \right)\)

0

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Question Analysis

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1 – 4z + 9z^-1 – 8z^-2 + 2z^-3

– 1 + 4z – 9z^-1 + 8z^-2 – 2z^-3

– 1 – 4z + 9z^-1 – 8z^-2 – 2z^-3

1 + 4z – 9z^-1 + 8z^-2 – 2z^-3

The Laplace transform of t²u(t) does not converge anywhere on the s-plane.