STUPIDSID
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-5t u(t)
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)=cos2 (2 πt).
x(t)=cos2 (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) y1(t)=t x(t)
ii) y2(t) = e-tx (t)
i) y1(t)=t x(t)
ii) y2(t) = e-tx (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-at u(t).
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:
x1(n)={1 2 3 4}
x2(n)= {3 2 1 0}
x1(n)={1 2 3 4}
x2(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[X2].
| fx(X) = 5X2 ; | 0 ≤ x ≤ 1 |
| = 0 ; | elsewhere |
Find E[X], E[3X-2], E[X2].
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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