Answer any one question from Q1 and Q2

1 (a)
Perform the following operations on the given signal x(t) which is defined as:

i) Sketch the signal x(t)

ii) Sketch z(t)= x(-t -1).

i) Sketch the signal x(t)

ii) Sketch z(t)= x(-t -1).

4 M

1 (b)
Determine whether the following signals are periodic or not, if periodic find the fundamental period of the signal:

i) x(t) = cos (2t) + sin (2t) \[ x[n]= \cos \left ( \dfrac {8 \pi n}{15} \right ) \]

i) x(t) = cos (2t) + sin (2t) \[ x[n]= \cos \left ( \dfrac {8 \pi n}{15} \right ) \]

4 M

1 (c)
Determine the step response of the following systems whose impulse response is:

h(t)=e

h(t)=e

^{-5t}u(t)
4 M

2 (a)
Compute the convolution integral by graphical method and sketch the output for

x1(t)=1 | 0 ≤ t ≤ 2 |

0, | otherwise |

x2(t)=e-2t u(t) |

6 M

2 (b)
Find even and odd component of

i) x(t) = u(t)

x(t) = sgn (t).

i) x(t) = u(t)

x(t) = sgn (t).

4 M

2 (c)
Determine the whether following signal is periodic or not, if periodic find the the fundamental periodic of the signal.

x(t)=cos

x(t)=cos

^{2}(2 πt).
2 M

Answer any one question from Q3 and Q4

3 (a)
Find the trigonometric Fourier series for the periodic signal x(t). Sketch the amplitude and phase spectra.

6 M

3 (b)
A signal x(t) has Laplace transform \[ X(s)= \dfrac {s+1}{s^2 + 4s + 5 } \] Find the Laplace transform of the following signals:

i) y

ii) y

i) y

_{1}(t)=t x(t)ii) y

_{2}(t) = e^{-t}x (t)
6 M

4 (a)
Find the Fourier transform of \[ x(t) =rect \ \left ( \dfrac {t} {\tau} \right ) \] and sketch the magnitude and phase spectrum.

6 M

4 (b)
Find the transfer function of the following:

i) An ideal differentiator

ii) An ideal integrator

iii) An ideal delay of T second.

i) An ideal differentiator

ii) An ideal integrator

iii) An ideal delay of T second.

6 M

Answer any one question from Q5 and Q6

5 (a)
Find the following for the give signal x(t):

i) Autocorrelation

ii) Energy from Autocorrelation

iii) Energy Spectral Density:

x(t)=Ae

i) Autocorrelation

ii) Energy from Autocorrelation

iii) Energy Spectral Density:

x(t)=Ae

^{-at}u(t).
6 M

5 (b)
Determine the corss correlation between two sequences which are given below:

x

x

x

_{1}(n)={1 2 3 4}x

_{2}(n)= {3 2 1 0}
4 M

5 (c)
State and describe any three properties of Energy Spectral Density (ESD).

3 M

6 (a)
Prove that autocorrelation and energy spectral density form Fourier transform pair of each other and verify the same for x(t) = e

^{-2t}u(t).
9 M

6 (b)
State and explain any four properties of Power Spectral Density (PSD).

4 M

Answer any one question from Q7 and Q8

7 (a)
Explain Gaussian probability model with respect to its density and distribution function.

4 M

7 (b)
Two cards drawn from a 52 card deck successively without replacing the first:

i) Given the first one is heart, what is the probability that second is also a heart?

ii) What is the probability that both cards will be hearts?

i) Given the first one is heart, what is the probability that second is also a heart?

ii) What is the probability that both cards will be hearts?

4 M

7 (c)
A coin is tossed three times. Write the sample space which gives all possible outcomes. A random variable X, which represents the number of heads obtained on any double toss. Draw the mapping of S on to real line. Also find the probabilities of X and plot the C.D.F.

5 M

8 (a)
A random variable X is

Find E[X], E[3X-2], E[X

f_{x}(X) = 5X^{2} ; |
0 ≤ x ≤ 1 |

= 0 ; | elsewhere |

Find E[X], E[3X-2], E[X

^{2}].
6 M

8 (b)
A student arrives late for a class 40% of the time. Class meets five times each week. Find:

i) Probability of students being late for at three classes in a given week.

ii) Probability of students will not be late at all during a given week.

i) Probability of students being late for at three classes in a given week.

ii) Probability of students will not be late at all during a given week.

4 M

8 (c)
State the properties of Probability Density Function (PDF).

3 M

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