## Signals & Systems - Dec 2014

### 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:- $$ i) \ x(t) = \cos \dfrac {\pi}{3} t+ \sin \dfrac {\pi}{4} t \\ ii) \ x[n] = \cos ^ 2 \dfrac {\pi}{8}n $$
(4 marks)
**1 (b)** State and prove Time Shifting and Time Scaling property of continuous time Fourier Transform.(4 marks)
**1 (c)** For the following system, determine whether it is. (i) memory less, (ii) causal, (iii) linear, (iv) time-invariant y[n]=x[n^{2}](4 marks)
**1 (d)** Find out even and odd component of the following two signals:

$$ i) \ x(t) = t^3 +3t \\ ii) x[n] = \cos n+ \sin n + \cos (n) \sin (n) $$
(4 marks)
**1 (e)** Determine whether the signals are power of energy signals. Calculate energy/power accordingly:

i) x(t) = 0.9 e^{-3t} u(t)

ii) x[n]=u[n](4 marks)
**2 (a)**

Find the inverse Laplace Transform of $\dfrac {s-2}{s(s+1)^3}$

(5 marks)**2 (b)**Let x(t)=1.......0 ≤ t≤ 2T and; h(t)=e

^{-at}.... 0≤ t≤ T. Compute y(t) using graphical convolution approach. (10 marks)

**2 (c)**State and discuss the properties of the region of convergence for z-transform.(5 marks)

**3 (a)**An LTI system is characterized by the system function: $$ h(z) = \dfrac {z} { \left ( z- \dfrac {1}{4} \right ) \left ( z+ \dfrac {1}{4} \right ) \left ( z- \dfrac {1}{2} \right )} $$ write down possible ROCs. For different possible ROCs, determine causality and stability and impulse response of the system. (10 marks)

**3 (b)**Calculate Z transform of the following signals: $$ i) \ x[n] =n \left ( - \dfrac {1}{4} \right )^n u[n]\times \left ( - \dfrac {1}{6} \right )^{-n} u [-n] \\ ii) \ x[n] = u[n-6] -u [n-10] $$ (10 marks)

**4 (a)**For the periodic signal x(t)=e

^{-t}with a fundamental period T

_{0}=1 second. Find the exponential form of Fourier Series. Also plot the Fourier spectrum (Magnitude and phase spectrum).(10 marks)

**4 (b)**Consider a continuous time LTI system described by $$ \dfrac {dy(t)}{dt} + 2y(t)=x(t) $$ Using the transform, find out output to each of the following input signals.

i) x(t)=e

^{-t}u(t)

ii) x(t)=u(t) (10 marks)

**5 (a)**Convolute $$ x[n] = \left ( \dfrac {1}{3} \right )^n u[n] \ with \ h[n]= \left ( \dfrac {1}{2} \right )^n u[n] $$ using convolute sum formula and verify your answer using z transform. (10 marks)

**5 (b)**Explain Gibb's phenomenon. Also explain conditions necessary for the convergence of Fourier Series.(5 marks)

**5 (c)**A system is described by the following difference equation. Find out its transfer function H(z). $$ y[n]= \dfrac {3}{4} y [n-1] - \dfrac {1}{8} y[n-2]+ x[n]+ \dfrac {1}{2} x [n-1] $$ (5 marks)

**6 (a)**For the signal x(t) depicted in the figure given below, sketch the signals:

i) x(-t)

ii) x(t+6)

iii) x(3t)

x(t/2)(10 marks)

**6 (b)**For the periodic signal x[n] given below, find out Fourier Series coefficient: $$ x[n]= 1+ \sin \left ( \dfrac {2 \pi } { N} \right )n + 3 \cos \left [ \dfrac {2 \pi}{N} \right ]n+ \cos \left ( \dfrac {4\pi}{ N} n + \dfrac{\pi}{2} \right ) $$ (10 marks)