MU Signals & Systems - May 2016 Exam Question Paper

MU Electronics and Telecom Engineering (Semester 4)
Signals & Systems
May 2016

Total marks: --
Total time: --

INSTRUCTIONS
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary

Answer the following

1(a) Determine if the following system is memoryless, causal, linear, time invariant
y(t) = x² (t-t₀) + 2

4 M

1(b) Explain in brief ROC (Region of Convergence) conditions of laplace transform.

4 M

1(c) Consider two LTI systems connected in series. Their impulse response are h₁[n] and h₂[n] respectively. Find the output of the systems if x[n] is the Input being to one of the systems.
x[n]={→ 1, 2} h₁[n]={1,0,-1 ←} h₂[n]={→ 2, 1,-1}

4 M

1(d) State and prove time reversal property of Continuous time Fourier Series.

4 M

1(e) Find energy of a causal exponential pulse x(t) = e^-at u(t)

4 M

2(a) A DT sifnal is given by the following expression. Find its Z transform

x [n] = n (- \frac{1}{2})^{n} u [n] * (\frac{1}{4})^{- n} u [- n]

$x[n]=n(-\dfrac{1}{2})^n u[n]*(\dfrac{1}{4})^{-n}u[-n]$

10 M

2(b) A CT signal x(t) is applied to the input of a CT LTI systems with unit impluse response h(t). Find out y(t) using Convolution integral.

\begin{aligned} x (t) & = e^{- a t} u (t) a > 0 \\ h (t) & = u (t) \end{aligned}

$\begin {align*} x(t)&=e^{-at}u(t)\ \ a>0 \ \\\ h(t)&=u(t) \end{align*}$

10 M

3(a) Consider a causal LTI system with

H (j ω) = \frac{1}{j ω + 2} .

$H(j \omega )=\dfrac{1}{j\omega+2}.$ For a particular input x(t), this system produces output y(t) = e^-2t u(t) - e^-3t u(t). Find out x(t) using Fourier Transform.

10 M

3(b) Obtain Inverse Laplace Transform of the function

X (s) = \frac{3 S + 7}{S^{2} - S - 12}

$X(s)=\dfrac{3S+7}{S^2-S-12}$ for following ROCs. Also comment on the stability and causality of the system for each of the ROC conditions. Support your answer with appropriate sketches of ROCs.
i) Rs(S)>4
ii) Re(S)<-3
iii) -3

10 M

4(a) A DT signal has been shown. Sketch the following signals. i) x[n-4] ii) x[4-n iii) x[-2n+2] iv) x[n]u[3-n]

8 M

Find out DEFT of the following

4(b)(i)

x [n] = {1, - 1, 2, 2}

$x[n]=\left \{ 1,-1,2,2 \right \}$

3 M

4(b)(ii)

x [n] = \sin [\frac{π n}{2}] u [n]

$x[n]=\sin \left [ \dfrac{\pi n}{2} \right ]u[n]$

3 M

4(c) Determine inverse Z Transform of

X (Z) = \frac{3}{(1 - Z^{- 1}) (1 + Z^{- 1}) (1 - 0.5 Z^{- 1}) (1 - 0.2 Z^{- 1})}

$X(Z)=\dfrac{3}{(1-Z^{-1})(1+Z^{-1})(1-0.5Z^{-1})(1-0.2Z^{-1})}$

6 M

5(a) Find the trigonometric Fourier Series for the waveform shown in the following figure.

10 M

5(b) Determine impluse response of h[n] for the system described by the second order difference equation.
y[n] - 4y[n-2] + 4y[n-2] = x[r] - x[n-1] when y[-1] = y[-2] = 0

10 M

6(a) A Lti system has the following transfer function

H (Z) = \frac{Z}{(Z - \frac{1}{4}) (Z + \frac{1}{4}) (Z - \frac{1}{2})}

$H(Z)=\dfrac{Z}{(Z-\dfrac{1}{4})(Z+\dfrac{1}{4})(Z-\dfrac{1}{2})}$

10 M

6(b)(i) Give app possible ROC conditions

5 M

6(b)(ii) Show pole-zero diagram of a system

5 M

6(b)(iii) Find impluse response of system

5 M

6(b)(iv) Comment on the system stability and causality for all possible RoCs

5 M

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