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Signals & Systems : Question Paper Dec 2012 - Electronics & Telecomm. (Semester 4) | Mumbai University (MU)
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## Signals & Systems - Dec 2012

### 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) Determine whether the signals are power or energy signals.
(i) x(t)=0.9e-3t u(t)
(ii) x[n]=u[n]
(4 marks)
1 (b) Convolve h[n] = n+1; 0 ≤ n ≤ 3 with
x[n] = n2; 0 ≤ n ≤ 2.
(4 marks)
1 (c) Given Equation: y(t)=2r(t)-2r(t-1)-2u(t-3)
Sketch y(t) and odd part of y(t).
(4 marks)
1 (d) Determine whether each of the signals is periodic. If so find its fundamental period-
(4 marks)
1 (e) Check Dynamicity, Linearity, Time variance and Causality of
y[n] = x[n] + x[n+2]
(4 marks)
2 (a) Determine the exponential Fourier Series of the signal x(t).
(10 marks)
2 (b) Perform convolution of :
(i) 2u(t) with u(t) (2 marks)
(ii) e2t u(t) with e-5t u(t) (4 marks)
(iii) tu(t) with e-5t u(t) (4 marks)
(10 marks)
3 (a) Sketch
x(t) = t ; 0<t<1<br>= 1 ; 1<t<2<br>= 3-t ; 2<t<3<br>Then sketch
(i) x(2-t)
(ii) x(t-3)
(iii) x(2t)
(iv) 0.5x(-t)</t<3<br></t<2<br></t<1<br>
(10 marks)
3 (b) Consider an analog signal:
x(t) = 5cos(50πt) + 2sin(200πt) - 2cos(100πt)
(i) Determine Nyquist Sampling Rate. (2 marks)
(ii) If the given x(t) is sampled at the rate of 200Hz, what is the discrete time signal obtained after sampling? (3 marks)
(5 marks)
3 (c) Find the DTFT of x[n] = {2,1,2} and compute its magnitude at ω = 0 and ω = π/2. (5 marks) 4 (a) Find the Z-Transform
(i) x[n] = (0.1)nu[n] + (0.3)nu[-n-1]
(ii) x[n]=(0.5)n[u[n] - u[n-2]].
(5 marks)
4 (b) Prove convolution property of Z-Transform.(5 marks) 4 (c) Determine the response of an LTI discrete time system governed by the difference equation
y[n] - 2y[n-1] - 3y[n-2] = x[n] + 4x[n-1] for the input
x[n]=2nu[n] with initial conditions
y[-2] = 0, y[-1] = 5.
(10 marks)
5 (a) (i) Using Laplace Transform, determine the total response of the system described by the equation
y''(t) + 5y'(t) + 4y(t) = x'(t).
The initial conditions are y(0)=0 and y'(0)=1. The input to the system is x(t)=e-2tu(t). (6 marks)
(ii) Also find the Impulse Response of the above system assuming initial conditions = 0. (4 marks)
(10 marks)
5 (b) Realize Direct Form I, Direct Form II first order cascade and first order parallel structures
(10 marks)
6 (a) Find x[n] if

(i) ROC: |z|> 1/3
(ii) ROC: 1/4 < |z| < 1/3
(iii) ROC: |z|< 1/4
(10 marks)
6 (b) Prove time shifting property of Fourier Transform.(5 marks) 6 (c) Determine the unit step response of the system whose impulse response is given as h(t) = 3u(t). (5 marks) 7 (a) The state space representation of a discrete time system is given as-

Derive the transfer function H(z) of the system.
(10 marks)
7 (b) Using suitable method obtain the state transition matrix φ(t) for the system matrix.(10 marks)