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(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary

Solve any one question from Q.1(a,b,c) &Q.2(a,b,c)
1(a) Find whether the following signals are energy or power and find the corresponding value:
x(t) = cos (t).
4 M
1(b) Determine wether the following LTI system described by impulse response \( h(t)= e^{-t}u(t +1) \)/ is stable and causal.
4 M
1(c) Find odd and even components of the following signals:
x[n] = {1, 0, -1, 2, 3}
4 M

2(a) An analog signal is given by the equation: \( x(t)= 2\sin 400\pi t+10\cos\ 1000\pi t. \)/ It is sampled at sampling frequency 1000 Hz:
i) What is the Nyquist rate for the above signal?
ii) What is the Nyquist interval of the signal?
2 M
2(b) Determine the convolution sum of the following sequence using equation of convolution sum: \[x(n)=\delta (n)+2\delta \left ( n-2 \right )\\ h(n)=2\delta (n)-\delta \left ( n-2 \right )\]
4 M
2(c) Check whether the following signal is periodic or non-periodic. If periodic, find period of the signal: \[x(t)=10\sin 12\pi t+4\sin 18\pi t\]
4 M

Solve any one question from Q.3(a,b) &Q.4(a,b)
3(a) State and prove the following properties of CTFT:
i) Time scaling
ii) Time shifting.
6 M
3(b) Obtain the trigonometric Fourier series of the rectangular pulse shown in Fig. 1:
6 M

4(a) State the Dirichlet conditions for existence of Fourier series.
4 M
4(b) For the sinc function shown in Fig.2, obtain Fourier transform and plots its spectrum:
8 M

Solve any one question from Q.5(a,b) &Q.6(a,b)
5(a) Find the initial and final value of a signal: \[X(s)=\left ( s+10 \right )/(s^{2}+2s+2)\].
6 M
5(b) Find the inverse Laplace transform of: \[X(s)=-5s-7/\left ( s+1 \right )\left ( s-1 \right )\left ( s+2 \right ).\]
7 M

6(a) Findthe Laplace transform of the following with ROC:\[\begin{align*}& i)x(t)=u\left ( t-5 \right )\\ &ii)x(t)=e^{-at}\sin (\omega t)u(t).\end{align*}\]
7 M
6(b) The differential equation of the system is given by: \[dy(t)/dt+2y(t)=x(t).\]
Determine the output of system for \(x(t)=e^{-3t}u(t). \)/ Assume zero initial condition.
6 M

Solve any one question from Q.7(a,b) &Q.8(a,b)
7(a) What is correlation? Explain the two types of correlations with a practical application for each.
6 M
7(b) The PDF of a random variable x is given by: \[\begin{align*}&f_{x}(x)=1/2\pi \ \ \text{for}\ 0\leq x\leq 2\pi \\ &=0\ \ \ \ \ \ \text{otherwise} \end{align*}\] Calculate mean value, mean square value, variance and standard deviation.
7 M

8(a) In a pack of cards, 2cards are drawn simultaneously. What is the probability of getting of queen, jack combination?
6 M
8(b) Suppose that a certain random variable has a CDF:\( \begin{align*}F_{x}(X)&=0\ \ \ \ \ \ \text{for}\ x\leq 0\\ & = kx^{2}\ \ \ \ \text{for}\ 0\leq x\leq 10\\ & =50k \ \ \ \ \ \text{for}\ x> 10 \end{align*} \)/
i) Determine the value of k
ii) \[P\left ( 4\leq x\leq 7 \right )\]
iii) Find and sketch PDF.
7 M

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