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Signals and Systems Question Paper - Jun 18 - Electronics And Telecomm (Semester 3) - Pune University (PU)
written 6.9 years ago by gravatar for teamques10 teamques10 ★ 70k
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written 6.9 years ago by gravatar for teamques10 teamques10 ★ 70k

Signals and Systems - Jun 18

Electronics And Telecomm (Semester 3)

Total marks: 80
Total time: 3 Hours INSTRUCTIONS
(1) Question 1 is compulsory.
(2) Attempt any three from the remaining questions.
(3) Draw neat diagrams wherever necessary.

1.a. Perform the following operations and sketch the signals :

  1. $y(t) = r(t+1) = r(t) +u(t-2)$
  2. $y[n] = u[n+3] = 2u[n-1]+u[n-4]$.

(6 marks) 00

1.b. Using impulse response properties, determine whether the following systems are :

  1. Static/Dynamic
  2. Casual/non-casual.
  3. Stable/non-stable : (i)$h(t) = e^{-2|t|}$ (ii)$h(n) = 2\delta[n] - 3\delta[n-1]$.

(6 marks) 00

OR

2.a. Find even and odd componemts of the following signals :

  1. $x(t) = 3t+tcost+t^{2}sin^{2}4t$
  2. $x[n] = \{1,1,-1,-1\}$.

(6 marks) 00

2.b. Find convolution of the following, using graphical method :

  1. $x[n] = u[n]$
  2. $h[n] = a^{n}u[n]$ $0\lt a\lt1$.

(6 marks) 00

3.a. Find Fourier transform of the following signals using appropriate properties :

  1. $x(t) = \frac{d}{dt} \{e^{-at}.u(t)\}$
  2. $x(t) = e^{-2t} u(t+2)$.

(6 marks) 00

3.b. Find and sketch exponential Fourier series of the given signal : enter image description here
(6 marks) 00

OR

4.a. Find and sketch the trigonometric Fourier series of train of impulse defined as : $x(t)=\sum_{k=-\infty}^{\infty} \delta\left(t-kTs\right)$
(6 marks) 00

4.b. Find Fourier transform of the following signals :

  1. $u(t)$
  2. $sgn(t)$.

(6 marks) 00

5.a. Find laplace transform of the following :

  1. $x(t) = \frac {d}{dt} t e^{-t} u(t)$
  2. $x(t) = e^{-3t} y(t)*cos(t-2)u(t-2)$.

(7 marks) 00

5.b. Find initial and final values of the signal $x(t)$ having unilateral laplace transform :

  1. $X(s) = \frac {7s+10}{s(s+2)}$
  2. $X(s) = \frac {5s+4}{s^{3}+3s^{2}+2s}$.

(6 marks) 00

OR

6.a. Find inverse laplace transform of : $X(s) = \frac{3s+7}{s^{2}-2s-3}$.

for :

  1. $s \gt 3$
  2. $s \lt -1$
  3. $-1 \gt s \lt 3$.

(7 marks) 00

6.b. Find transfer function and impulse response of the casual system described by the differential equation : $\frac{d^{2}}{dt^{2}}y(t)+5\frac{d}{dt}y(t)+y(t) = 2 \frac{d}{dt}x(t)-3x(t)$.
(6 marks) 00

7.a. Find auto-correlation function of the signal given, using graphical method : enter image description here
(6 marks) 00

7.b. The probability density function of a random variable X is given by: $f_{X}(x) = e^{-x}u(x)$

Determine:

  1. $CDF$
  2. $P(X \le 1)$
  3. $P(1 \lt X \le 2)$
  4. $P(X \gt 2)$.

(7 marks) 00

8.a. The probability density function of a random variable X is given by : $f_{\mathrm{X}}(x)=\left\{\begin{array}{ll}{\frac{1}{a}} & {|x| \leq a \rangle} \\ {0} & {\text { otherwise }}\end{array}\right.$

Determine :

  1. Mean $E[X]$
  2. Mean square value $E[X^{2}]$
  3. Standard deviation.

(7 marks) 00

8.b. State and prove the relationship between auto-correlation and energy spectral density density of energy signal.
(6 marks) 00

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