MU Signals & Systems - December 2014 Exam Question Paper

MU Electronics and Telecom Engineering (Semester 4)
Signals & Systems
December 2014

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) Determine the fundamental period of the following signals:-

i) x (t) = \cos \frac{π}{3} t + \sin \frac{π}{4} t i i) x [n] = \cos^{2} \frac{π}{8} n

$i) \ x(t) = \cos \dfrac {\pi}{3} t+ \sin \dfrac {\pi}{4} t \\ ii) \ x[n] = \cos ^ 2 \dfrac {\pi}{8}n$

4 M

1 (b) State and prove Time Shifting and Time Scaling property of continuous time Fourier Transform.

4 M

1 (c) For the following system, determine whether it is. (i) memory less, (ii) causal, (iii) linear, (iv) time-invariant y[n]=x[n²]

4 M

1 (d) Find out even and odd component of the following two signals:

i) x (t) = t^{3} + 3 t i i) x [n] = \cos n + \sin n + \cos (n) \sin (n)

$i) \ x(t) = t^3 +3t \\ ii) x[n] = \cos n+ \sin n + \cos (n) \sin (n)$

4 M

1 (e) Determine whether the signals are power of energy signals. Calculate energy/power accordingly:
i) x(t) = 0.9 e^-3t u(t)
ii) x[n]=u[n]

4 M

2 (a)

Find the inverse Laplace Transform of $\dfrac {s-2}{s(s+1)^3}$

5 M

2 (b) Let x(t)=1.......0 ≤ t≤ 2T and; h(t)=e^-at.... 0≤ t≤ T. Compute y(t) using graphical convolution approach.

10 M

2 (c) State and discuss the properties of the region of convergence for z-transform.

5 M

3 (a) An LTI system is characterized by the system function:

h (z) = \frac{z}{(z - \frac{1}{4}) (z + \frac{1}{4}) (z - \frac{1}{2})}

$h(z) = \dfrac {z} { \left ( z- \dfrac {1}{4} \right ) \left ( z+ \dfrac {1}{4} \right ) \left ( z- \dfrac {1}{2} \right )}$ write down possible ROCs. For different possible ROCs, determine causality and stability and impulse response of the system.

10 M

3 (b) Calculate Z transform of the following signals:

i) x [n] = n {(- \frac{1}{4})}^{n} u [n] \times {(- \frac{1}{6})}^{- n} u [- n] i i) x [n] = u [n - 6] - u [n - 10]

$i) \ x[n] =n \left ( - \dfrac {1}{4} \right )^n u[n]\times \left ( - \dfrac {1}{6} \right )^{-n} u [-n] \\ ii) \ x[n] = u[n-6] -u [n-10]$

10 M

4 (a) For the periodic signal x(t)=e^-t with a fundamental period T₀=1 second. Find the exponential form of Fourier Series. Also plot the Fourier spectrum (Magnitude and phase spectrum).

10 M

4 (b) Consider a continuous time LTI system described by

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

$\dfrac {dy(t)}{dt} + 2y(t)=x(t)$ Using the transform, find out output to each of the following input signals.
i) x(t)=e^-t u(t)
ii) x(t)=u(t)

10 M

5 (a) Convolute

x [n] = {(\frac{1}{3})}^{n} u [n] w i t h h [n] = {(\frac{1}{2})}^{n} u [n]

$x[n] = \left ( \dfrac {1}{3} \right )^n u[n] \ with \ h[n]= \left ( \dfrac {1}{2} \right )^n u[n]$ using convolute sum formula and verify your answer using z transform.

10 M

5 (b) Explain Gibb's phenomenon. Also explain conditions necessary for the convergence of Fourier Series.

5 M

5 (c) A system is described by the following difference equation. Find out its transfer function H(z).

y [n] = \frac{3}{4} y [n - 1] - \frac{1}{8} y [n - 2] + x [n] + \frac{1}{2} x [n - 1]

$y[n]= \dfrac {3}{4} y [n-1] - \dfrac {1}{8} y[n-2]+ x[n]+ \dfrac {1}{2} x [n-1]$

5 M

6 (a) For the signal x(t) depicted in the figure given below, sketch the signals:

i) x(-t)
ii) x(t+6)
iii) x(3t)
x(t/2)

10 M

6 (b) For the periodic signal x[n] given below, find out Fourier Series coefficient:

x [n] = 1 + \sin (\frac{2 π}{N}) n + 3 \cos [\frac{2 π}{N}] n + \cos (\frac{4 π}{N} n + \frac{π}{2})

$x[n]= 1+ \sin \left ( \dfrac {2 \pi } { N} \right )n + 3 \cos \left [ \dfrac {2 \pi}{N} \right ]n+ \cos \left ( \dfrac {4\pi}{ N} n + \dfrac{\pi}{2} \right )$

10 M

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