## Signals & Systems - Dec 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. $$ i) \ x(t) =2\cos \dfrac {2 \pi t}{3} + 3 \cos \dfrac {2\pi t}{7} \\ ii) x[n] = \cos^2 \left [ \dfrac {\pi}{4}n \right] $$(5 marks)
**1 (b)** Prove and explain time scaling and amplitude scaling property of Continuous time Fourier Transform.(5 marks)
**1 (c)** For the given system, determine whether it is, i) memory less, ii) causal, iii) time-invariant y[n]=nx[n].(5 marks)
**1 (d)** Find out even and odd component of the following signal. $$ x(t)=\cos^2 \left ( \dfrac {\pi t}{2} \right ) $$(5 marks)
**2 (a)** Determine the trigonometric form of Fourier Series of the waveform shown below.

(10 marks)
**2 (b)** State duality property of Fourier Transform. If Fourier Transform of $ e^{-t} u(t) \text{ is } \dfrac {1}{1+j\Omega}, $ then find the Fourier Transform of $ \dfrac {1}{1+t} $ using duality property.(10 marks)
**3 (a)** Obtain inverse Laplace transform of the function. Write down and sketch possible ROCs. $ x(s) = \dfrac {8} {(s+2)^3 (s+4)} $(10 marks)
**3 (b)** Using the z transform, solve the difference equation and find out impulse response. y[n]-2y[n-1]+y[n-2]=x[n]+3x[n-3](10 marks)
**4 (a)** State and explain different properties of ROC of Z transform.(5 marks)
**4 (b)** Convolve the sequences shown in the following figure using circular convolution.

(5 marks)
**4 (c)** A continuous time signal is shown below. Sketch the following transformed versions of the signal. $$ i)\ x(t-3) \\ ii) \ -2x(t) \\ iii)\ x(t-3)-2x(t) \\ iv) \ \dfrac {dx(t)}{dt} $$

(10 marks)
**5 (a)** Convolve $ x[n] = \left ( \dfrac {1}{3} \right )^n \ u[n] \text { with }h[n]= \left ( \dfrac {1}{2} \right )^n \ u[n] $ using convolution integral.(10 marks)
**5 (b)** A second order LTI system is described by $ \dfrac {d^2 y(t)}{dt^2}+ 5 \dfrac {dy(t)}{dt}+6y(t)=x(t). $ Determine the transfer function and poles and zeros of the systems. Evaluate zero-state response to x(t)=u(t).(10 marks)
**6 (a)** 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)
**6 (b)** The input and impulse response of continuous time system are given below. Find out output of the continuous time systems using appropriate method. x(t)=u(t) & h(t)=e^{-2t}u(t).(10 marks)