## Signals & Systems - May 2015

### 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 the fundamental period of the following signals. $$ x(t)=14 + 40 \cos (60 \pi t) \\ ii) x[n] = \cos^2 \left [\dfrac {\pi}{4}n \right ] $$ (4 marks)

** 1 (b) ** Compare the nature of ROC of Z transform and Laplace transform. (4 marks)

** 1 (c) ** For the given system, determine whether it is,

i) memory less

ii) causal

linear

iv) time-invariant.

y[n] = x[-n]. (4 marks)

** 1 (d) ** "Find out even and odd component of the following two signals. $$ i) \ x(t) = \cos^2 \dfrac {\pi t}{2} \\ ii) x(t) = \left\{\begin{matrix}t\cdots \cdots &0 \le t \le 1 \\ 2-t \cdots \cdots
& 1< 2\le 2 \end{matrix}\right. $$" (4 marks)

** 1 (e) ** Determine whether the signals are power or energy signals. Calculate energy / power accordingly.

i) x(t)=Ae^{-αt}u(t)........... α>0.

ii) x[n]=u[n]. (4 marks)

** 2 (a) ** Expand the periodic gate function as shown in the figure by the exponential Fourier Series. Also plot the Fourier spectrum (Magnitude and phase spectrum).

(10 marks)

** 3 (a) ** Obtain inverse Laplace transform of the function $$ X(s) = \dfrac {3S+7}{s^2 -2s-3} $$ Write down and sketch possible ROCs. Find out inverse Laplace for all the possible ROCs. (10 marks)

** 3 (b) ** Using the z transform method, solve the difference equation

y[n]-4y[n-1]+4y[n-2]=x[n]-x[n-1]

When y(-1)=y(-2)=0. (10 marks)

** 4 (a) ** Explain Gibbs phenomenon. Also explain conditions necessary for the convergence of Fourier Series. (5 marks)

** 4 (b) ** Find out Fourier Transform of f(t)=10 δ(t-2). Sketch its amplitude and phase spectrum. (5 marks)

** 4 (c) ** Perform convolution of

i) 2u(t) with u(t)

ii) e^{-2t} u(t) with e^{-5t} u(t)

iii) tu(t) with e^{-5t} u(t). (10 marks)

** 5 (a) ** Convolve $$x[n]= \left ( \dfrac {1}{3} \right )^n u [n] \ with \ h[n]= \left ( \dfrac {1}{2} \right )^n u[n] $$ using Fourier transform. (10 marks)

** 5 (b) ** A system is described by the following difference equation. $$ y[n] = \dfrac {3}{4} y [n-1] - \dfrac {1}{8} y [n-2]+ x[n] $$ Determine the following.

i) The system Transfer function H(2)

ii) Impulse response of the system h[n]

iii) Step response of the system s[n]. (10 marks)

** 6 (a) ** A discrete time signal is given by $$ x[n] = \{ \underset {1}, 1, 1, 1, 2\} $$ . Sketch the following signals.

i) x[n]

ii) x[n-2]

iii) x[n] ⋅ u[n-1]

iv) x[3-n]

v) x[n-1]⋅δ[n-1] (10 marks)

** 6 (b) ** For the periodic signal x[n] given below, find out Fourier series coefficient. $$ x[n] = 1 +\sin \left ( \dfrac {2\pi} { N} \right ) n+3 \cos \left ( \dfrac {2\pi}{N} \right ) n+\cos \left ( \dfrac {4\pi}{N} + \dfrac {\pi}{2} \right ). $$ (10 marks)
**2 (b)** Find the inverse Laplace Transform of the following: $$ i) \ X(S) = \dfrac {s-3}{s^2+4s+13} \\ ii) \ X(S)= \dfrac {5s^2 - 15 - 11}{(s+1)(s-2)^3} $$(10 marks)