MU Signal and Systems - May 2016 Exam Question Paper

MU Electronics Engineering (Semester 5)
Signal and 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

1(a) Check for periodicity of the following signals. Also find the new period.

(i) x (t) = 3 \cos (15 π t) + 4 \cos (35 π t - \frac{π}{4}) + 8 \sin (55 π t)

$(i)\ x(t)=3\cos(15\pi t)+4\cos\left ( 35\pi t-\dfrac{\pi}{4} \right )+8\sin(55\pi t)$

(i i) x (n) = 3 \cos^{2} (\frac{π}{6} n) + 2 \cos^{2} (\frac{π}{4} n)

$(ii)\ x(n)=3\cos^2\left ( \dfrac{\pi}{6}n \right )+2\cos^2\left ( \dfrac{\pi}{4}n \right )$

4 M

1(b) Determine whether the given signal is energy or power signal. Hence obtain its energy power accordingly.
\( (i)\ x(t)=4\sin t-\infty

(i i) x (n) = {(\frac{3}{7})}^{n} u (n)

$(ii)\ x(n)=\left ( \dfrac{3}{7} \right )^n u(n)$

4 M

1(c) Plot x(t) = u(t) ' r(t) + r(t-1). Hence plot its even and odd parts also.

4 M

1(d) Prove time shifting of Z-transform.

4 M

1(e) Check for Dynamicity, Linearity, Time variance, causality of the system.
(i) y(t) = t x(t) + x (t-1)
(ii) y(n) = 3x (-n) + 4.

4 M

2(a) Obtain inverse Z-transform for all possible ROC's. Also comment on Causality and Stability in each case.

H (z) = \frac{z (3 z - 7)}{(z - \frac{1}{4}) (z + 2)}

$H(z)=\dfrac{z(3z-7)}{\left ( z-\dfrac{1}{4} \right )(z+2)}$

10 M

2(b) State and prove Time Shifting and Convolution property of Continous Time Fourier Transform.

10 M

3(a) Obtain graphical concolution of following two signals.

10 M

3(b) Obtain exponential Fourier series of the following signal.

10 M

4(a) Determine h(t) for all possible ROC's.
If

T . F . = H (s) = \frac{2 s + 7}{(s + 2) (s - 3)}

$T.F.=H(s)=\dfrac{2s+7}{(s+2)(s-3)}$
Also comment on Causality and Stability of the system for each case.

10 M

4(b) A causal DT LTI system is described as
y(n) = 3y(n-2) + 4y(n-1) +x(n)
Obtain:
(1) T.F. of system
(2) Obtain step response
(3) Obtain response if input

x (n) = {(\frac{1}{2})}^{n} u (n)

$x(n)=\left ( \dfrac{1}{2} \right )^n u(n)$
(4) Also plot pole's and zeros of the T.F. and comment on causality and stability.

10 M

5(a) Determine Impluse response and step response of a CT LTI system.

\frac{d^{2} y (t)}{d t^{2}} + \frac{7 d y (t)}{d t} + 12 y (t) = x (t)

$\dfrac{d^2y(t)}{dt^2}+\dfrac{7dy(t)}{dt}+12y(t)=x(t)$

10 M

5(b) Obtain auto-correlation of following signals

(i) x (t) = 3 e^{- 2 t} u (t)

$(i)\ \ x(t)=3e^{-2t}u(t)$

(i i) x (n) = {(\frac{3}{4})}^{n} u (n)

$(ii)\ \ x(n)=\left ( \dfrac{3}{4} \right )^n u(n)$

10 M

6(a) Obtain DT Fourier Transform of following signal h(n) = [2 1 2] plot its magnitude and phase spectrum.

10 M

6(b) Obtain :
(i) Z- transform of

x (n) = n {(\frac{3}{4})}^{n} u (n) + u (n - 1)

$x(n)=n\left ( \dfrac{3}{4} \right )^n u(n)+u(n-1)$
(ii) Laplace transform of x(t) = t . e^-3t u(t) + t u (t-1)
Use properties of transform only.

10 M

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