Question Paper: Signals & Systems : Question Paper Dec 2014 - Electronics & Telecomm. (Semester 4) | Mumbai University (MU)
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## 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[n2](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 T0=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)