MU Signals & Systems - December 2015 Exam Question Paper

MU Electronics and Telecom Engineering (Semester 4)
Signals & Systems
December 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) Determine the fundamental period of the following signals.

i) x (t) = 2 \cos \frac{2 π t}{3} + 3 \cos \frac{2 π t}{7} i i) x [n] = \cos^{2} [\frac{π}{4} n]

$i) \ x(t) =2\cos \dfrac {2 \pi t}{3} + 3 \cos \dfrac {2\pi t}{7} \\ ii) x[n] = \cos^2 \left [ \dfrac {\pi}{4}n \right]$

5 M

1 (b) Prove and explain time scaling and amplitude scaling property of Continuous time Fourier Transform.

5 M

1 (c) For the given system, determine whether it is, i) memory less, ii) causal, iii) time-invariant y[n]=nx[n].

5 M

1 (d) Find out even and odd component of the following signal.

x (t) = \cos^{2} (\frac{π t}{2})

$x(t)=\cos^2 \left ( \dfrac {\pi t}{2} \right )$

5 M

2 (a) Determine the trigonometric form of Fourier Series of the waveform shown below.

10 M

2 (b) State duality property of Fourier Transform. If Fourier Transform of

e^{- t} u (t) is \frac{1}{1 + j Ω},

$e^{-t} u(t) \text{ is } \dfrac {1}{1+j\Omega},$ then find the Fourier Transform of

\frac{1}{1 + t}

$\dfrac {1}{1+t}$ using duality property.

10 M

3 (a) Obtain inverse Laplace transform of the function. Write down and sketch possible ROCs.

x (s) = \frac{8}{(s + 2)^{3} (s + 4)}

$x(s) = \dfrac {8} {(s+2)^3 (s+4)}$

10 M

3 (b) Using the z transform, solve the difference equation and find out impulse response. y[n]-2y[n-1]+y[n-2]=x[n]+3x[n-3]

10 M

4 (a) State and explain different properties of ROC of Z transform.

5 M

4 (b) Convolve the sequences shown in the following figure using circular convolution.

5 M

4 (c) A continuous time signal is shown below. Sketch the following transformed versions of the signal.

i) x (t - 3) i i) - 2 x (t) i i i) x (t - 3) - 2 x (t) i v) \frac{d x (t)}{d t}

$i)\ x(t-3) \\ ii) \ -2x(t) \\ iii)\ x(t-3)-2x(t) \\ iv) \ \dfrac {dx(t)}{dt}$

10 M

5 (a) Convolve

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

$x[n] = \left ( \dfrac {1}{3} \right )^n \ u[n] \text { with }h[n]= \left ( \dfrac {1}{2} \right )^n \ u[n]$ using convolution integral.

10 M

5 (b) A second order LTI system is described by

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

$\dfrac {d^2 y(t)}{dt^2}+ 5 \dfrac {dy(t)}{dt}+6y(t)=x(t).$ Determine the transfer function and poles and zeros of the systems. Evaluate zero-state response to x(t)=u(t).

10 M

6 (a) 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

6 (b) The input and impulse response of continuous time system are given below. Find out output of the continuous time systems using appropriate method. x(t)=u(t) & h(t)=e^-2tu(t).

10 M

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