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

1(a) Find whether the following signals are energy or power and find the corresponding value:
\[x(t)=\left ( 1/2 \right )^n.u[n]\]
4 M
1(b) Find the convolution between:
\[\begin{align*} x[n]=\left \{ 1, 1, 1, 1 \right \}\text{and}\\ h[n]=\left \{ 1, 1, 1,1 \right \}\end{align*}\]
4 M
1(c) Find odd and even components of the signal:
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) Find the convolution between:
x(t) = u(t) and h(t) = u(t-2)
6 M
2(c) Check whether the following signal is periodic or non-periodic. If periodic, find period of the signal:
\[x(t)=\cos \left ( n/8 \right ).\cos \left ( n\pi /8 \right ).\]
4 M

Solve any one question fromQ3(a,b) and Q.4(a,b)
3(a) State and explain the properties of Continuous Time Fourier Series.
6 M
3(b) Determine the transfer function and impulse response for the system described by the differential equation shown below for zero initial condtitions: \[d/dt[y(t)]+3 y (t)=x(t).\]
6 M

4(a) Draw the magnitude and phase spectrum of the signal:
\[x(t)= 5\cos \left ( 2\pi 10t +30\right )-10\cos \left ( 2\pi 20t+60 \right ).\]
6 M
4(b) Find the Fourier transform of the signal:
\[x(t)= \sin \omega _c t.u(t).\]
6 M

Solve any one question fromQ5(a,b) and Q.6(a,b)
5(a) State and prove convolution property of Laplace transform.
6 M
5(b) Find the initial and final value of:
\[X(s)= 5s+50/s\left ( s+5 \right ).\]
7 M

6(a) Find the Laplace transform of the given signal and draw its ROC:
\[X(t)=-e^{at}u\left ( -t \right ).\]
6 M
6(b) Find the inverse Laplace transform of:
\[X(s)= 3s+7/\left ( s^2-2s-3 \right ).\]
7 M

Solve any one question fromQ.7(a,b) and Q.8(a,b)
7(a) List the properties of auto correlation and cross correlation for energy signals.
6 M
7(b) A perfect die is thrown. Find the probability that:
i) You get even number.
ii) You get a perfect square.
7 M

8(a) List the properties of probability. Explain conditional probability with an example and formula.
6 M
8(b) A three digit message is transmitted over a noisy channel having a probability of error as P(E) = 2/5 per digit. Find and draw the CDF.
7 M

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