MU Signal and Systems - May 2015 Exam Question Paper

MU Electronics Engineering (Semester 5)
Signal and Systems
May 2015

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

1 (a) Prove convolution property of Fourier Transform.

5 M

1 (b) State and prove final value Theorem of Laplace Transform.

5 M

1 (c) Prove shifting property of Z transform.

5 M

1 (d) Determine energy and / or power of following signals.

i) x (n) = {(\frac{3}{5})}^{n} u (n) - (4)^{n} u (- n - 1) i i) x (t) = 4 e^{- 2 t} u (t)

$i) \ x(n) = \left ( \dfrac {3}{5} \right )^n u(n) - (4)^n u(-n-1) \\ ii) \ x(t) = 4e^{-2t}u (t)$

5 M

2 (a) Obtain output y(t)=x(t)*h(t) using graphical convolution.

\begin{aligned} x (t) & = 1 + t f o r - 1 \leq t \leq 0 \\ = 1 - t f o r 0 \leq t \leq 1 \end{aligned}

$\begin {align*} x(t)& =1 + t \ for \ -1\le t \le 0 \\ & = 1-t \ for \ 0\le t \le 1 \end{align*}$

\begin{aligned} h (t) & = 1 f o r 0 \leq t \leq 2 \\ = 0 e l s e w h e r e \end{aligned}

$\begin {align*} h(t)& =1 \ for \ 0\le t \le 2 \\ &=0 \ elsewhere \end {align*}$

10 M

2 (b) Obtain h(n) for all possible ROC conditions. Also plot the ROC comments on causality and stability at the system.

H (z) = \frac{4 z (z^{2} - 8 z + 9)}{(z - \frac{1}{3}) (z - 3) (z + 4)}

$H(z)= \dfrac {4z (z^2 - 8z +9)}{ \left ( z - \frac {1}{3} \right ) (z-3)(z+4)}$

10 M

3 (a) A.C.T. LTI system has

\frac{d^{2} y (t)}{d t^{2}} + \frac{5 d y (t)}{d t} + 6 y (t) = \frac{7 d x (t)}{d t} - 3 x (t)

$\dfrac {d^2y (t)}{dt^2} + \dfrac {5dy(t)}{dt} + 6y (t) = \dfrac {7dx (t)}{dt} - 3x(t)$ i) Determine Transfer function.
ii) Obtain impulse response
iii) Obtain unit Ramp response.

8 M

3 (b) Plot the magnitude and phase spectrum of the periodic signal. Show below.

8 M

3 (c) Obtain initial and final value:

i f X (z) = \frac{3 z^{2}}{4 z^{2} + 5 z + 1}

$if \ X(z) = \dfrac {3z^2} {4z^2 + 5z +1}$

4 M

4 (a) If two subsystem are connected in cascade
h₁(n)=(0.9)ⁿ u(n) ? 0.5(0.9)^n-1 u(n-1)
h₂= (0.5)ⁿ u(n) ? (0.5)^n-1 u(n-1)
Determine overall impulse response of the interconnected system.

8 M

4 (b) Obtain z transform of the following signal using properties of z transform.

x (n) = {(\frac{3}{4})}^{n - 1} \sin (\frac{π}{6} n) u (n)

$x(n) = \left ( \dfrac {3}{4} \right )^{n-1} \sin \left ( \dfrac {\pi}{6} n \right ) u(n)$

6 M

4 (c) Prove Parsevals theorem of Fourier series.

6 M

5 (a) Obtain circular convolution of
x₁(n)= [3 2 1 4]
x₂ (n) = [ 5 7 -8 2]

5 M

5 (b) Obtain Laplace Transform of following waveforms using its properties.
i)

ii)

5 M

5 (c) Obtain zero input response, zero state response and total response of a D.T.L.T.I. system.

\begin{aligned} y (n) + 7 y (n - 1) + 12 y (n - 2) = 4 x (n) - 11 x (n - 1) \\ i f & y (- 1) = 1 y (- 2) x (- 1) = 0 \\ i f i n p u t & x (n) = u (n) = u n i t s t e p s i g n a l \end{aligned}

$\begin {align*} \ \ \ \ \ &y(n)+7y (n-1) +12 y (n-2)=4 \ x(n)-11 x(n-1)\\ if \ \ \ \ \ \ \ \ \ \ \ \ \ & y(-1)=1 \ y(-2) \ x(-1)=0 \\ if \ input \ \ \ & x(n) = u(n)= \ unit \ step \ signal \end{align*}$

10 M

6 (a) Obtain Fourier transform of the following signal.

6 M

6 (b) Plot even and odd parts of following signals.

6 M

6 (c) Obtain h(t) for causal and stable system If

H (s) = \frac{s^{2} ? 3 s + 11}{(s - 1) (s + 2) (s + 3)}

$H(s) = \dfrac { s^2 ? 3s +11} {(s-1) (s+2) (s+3)}$ Plot the ROC and pole's and zero's of the system.

8 M

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