## 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)^{n}u[n] + 5(0.5)^{n}u[-n-1]

(ii) x[n] = 2^{n}u[n-2]

(iii) x[n] = (n+1) u[n](10 marks)
**4 (b)** A causal LTI system has a Transfer Function H(z) = H_{1}(z) H_{2}(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)