MU Signals & Systems - December 2015 Exam Question Paper

MU Electronics and Telecom Engineering (Semester 5)
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) Check whether the following signal is energy or power signal, find its energy and power x(t)=3u(t).

5 M

1 (b) Find the single side and double side spectrum of the signal-sketch the magnitude and phase spectrum.
x(t)=1+3sin (6πt+π/3)+6cos (8πt+π/6)+4 sin (12 πt).

5 M

1 (c) Find the Laplace transform of the signal x(t)=e^-2|t|.

5 M

1 (d) Find h[n] if

H (z) = \frac{5 z^{- 3}}{(z - 0.1) (z - 0.2)}

$H(z) = \dfrac {5z^{-3}}{(z-0.1)(z-0.2)}$ and if the system is stable.

5 M

2 (a) Check whether the following signal is period or not if periodic find the period.
i) x(t)=7cos(2t+π/6) ii) x[n]=e^i7π.

4 M

2 (b) Find the even and odd part of the signal,
i) x(t)=cos(t)+sin(t)+sin(t) cos(t)

x [n] = [1, 2, 8, 7 ↑, - 2, 2, 5, 6]

$x[n]= [1, 2, 8, \underset{\uparrow}{7}, -2,2,5,6]$

6 M

2 (c) Check whether the following system is linear, causal and stable or not.
i) y(t)=cos [x(t)],
x(t) is the input & y(t) is the output
ii) y[n]=nx[n]
where x[n] is the input and y[n] is the output.

10 M

3 (a) Find the output y(t)=e^-2tu(t)*u(t). sketch the output.

8 M

3 (b) Find the convoluted out put of the signals.

$x[n]= \big [7, 6, \underset{\uparrow}{2}, 3, -1, 4 \big ] \ and \\ h[n]=\big [ 1,\underset{\uparrow}{-1}, 0, 2\big]$

6 M

3 (c) Check whether the following system given by the impulse response h(t)=u(t+1)-u(t-2) is stable or not, memory less or not, justify.

6 M

4 (a) Find the Fourier series co-efficients (exponential) of the half wave rectified sine wave of Amplitude A and period 4 seconds. Sketch the amplitude and phase spectrum.

10 M

4 (b) Explain Gibb's phenomenon.

5 M

4 (c) Find Fourier transform of x(t)=u(t).

5 M

5 (a) Find the inverse Fourier transform of

$\begin {align*} X(j\omega) &=2 \cos (\omega) &|\omega| \le \pi \\ &=0 &|\omega |>\pi \end{align*}$

6 M

5 (b) Find x[n] if

$X(z) = \dfrac {z}{z^3+3z+2}$ for all possible ROC.

8 M

5 (c) Find Fourier transform of the signals x(t)=te^-3t u(t) using frequency differentiation property. Prove the property also.

6 M

6 (a) Find the z-transform of the signal

$x[n]=u[n-2]*\left (\dfrac {2}{3} \right )^n u[n]$

6 M

6 (b) Determine Forced response, natural response and total response of the system described by

$\dfrac {dy(t)}{dt} + 3y(t) = 4 x(t) \text { where } x(t)=\cos(2t) \ u(t)$ and initial condition y(o^*)=-2.

8 M

6 (c) Find the Laplace transform of

$\dfrac {d\big [e^{-3(t-2)}u(t-2)\big]}{dt}$ state the properties used.

6 M

7 (a) Explain Gain margin and phase margin.

6 M

7 (b) Sketch the root locus for unity feed back system.

$G(s) = \dfrac {k}{s(s^2 + 2s+2)}$

10 M

7 (c) Examine the stability using Routh Criteria.
s⁶+2s⁵+18s³+20s²+16s+16=0.

4 M

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