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 F_s=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)