Question Paper: Signals & Systems : Question Paper May 2013 - Electronics & Telecomm. (Semester 4) | Mumbai University (MU)
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## Signals & Systems - May 2013

### Electronics & Telecomm. (Semester 4)

TOTAL MARKS: 80
TOTAL TIME: 3 HOURS
(1) Question 1 is compulsory.
(2) Attempt any three from the remaining questions.
(3) Assume data if required.
(4) Figures to the right indicate full marks.
1 (a) Prove differentiation in Z-domain property of Z-transform.(4 marks) 1 (b) The Impulse response of a LTI system is h[n] = {1, 2, 3}. Find the input x[n] for output response which is given by
y[n] = {1, 1, 2, -1, 3}.
(4 marks)
1 (c) Determine whether each of the following signals are periodic. If so find its fundamental period -
(i) cos[πn/20] + cos[πn/10]
(ii) 2cos(100πt) + 5sin(50t)
(4 marks)
1 (d) Sketch even and odd parts of the following signal -
(4 marks)
1 (e) (i) Check dynamicity, linearity, time variance, causality, stability of y(t) = x(t)sin(ωt)
(ii) Determine whether the signal is an energy signal or power signal x[n] = u[n].
(4 marks)
2 (a) Convolve
(10 marks)
2 (b) A periodic square wave is defined as
.
The signal is periodic with fundamental period T. Determine its exponential Fourier series.
(10 marks)
3 (a) (i) Sketch
x(t)=u(t)-r(t-1)+2r(t-2)-r(t-3)+u(t-4)-2u(t-5).
(ii) If x[n]= {-1, 1, 1, 1, 1},
Plot (1) x[n]
(2) x[2-n]
(3) x[n-3]
(4) x[1-n]
(5) 7x[n-1].
(10 marks)
3 (b) The analog signal x(t)=3cos(100πt)
(i) Determine the Nyquist sampling rate.
(ii) If the given x(t) is sampled at a rate of 200 Hz, what is the signal obtained after sampling?
(iii) If the given x(t) is sampled at a rate of 75 Hz, what is the signal obtained after sampling?
(iv) What is the analog signal y(t) we can reconstruct from the samples if ideal interpolation is used and Fs=200Hz?
(10 marks)
4 (a) Find the ZT along with its ROC of-
(i) x[n]=(-0.2)nu[n] + 5(0.5)nu[-n-1]
(ii) x[n] = 2nu[n-2]
(iii) x[n] = (n+1) u[n]
(10 marks)
4 (b) A causal LTI system has a Transfer Function H(z) = H1(z) H2(z) where

(i) If the system is stable give its ROC condition.
(ii) Show cascade and parallel realization.
(iii) Find impulse response of the system.
(iv) Find system response if X(z) = 1/(1 - 0.21z-1)
(10 marks)
5 (a) Find the inverse Laplace transform of -
(10 marks)
5 (b) Find the inverse ZT of X(z) for ROC |Z|> 2 using PFE -
(10 marks)
6 (a) Solve the difference equation
x[n-2] - 9x[n-1] + 18x[n] = 0 with IC's
x[-1] = 1 and x[-2] = 9.
(10 marks)
6 (b) Obtain the DTFT and plot the magnitude and phase response of h[n] = {0, 1, 1, 1}.(5 marks) 6 (c) For the given signal -
y''(t) - y'(t) - 2y(t) = x(t), find
(i) Impulse response and (ii) Draw all possible ROC's for the system to be causal and stable.
(5 marks)
7 (a) Find response of the system for unit step input. Assume zero initial conditions -
(10 marks)
7 (b) Determine state variable model of
y[n] = -2y[n-1] + 3y[n-2] + 0.5y[n-3] + 2x[n].
(10 marks)