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:
x(t) = u(t) u(t-4)
i) Sketch z(t)=x(-t -1)
ii) Sketch y(t)=x(t)+ z(t).
x(t) = u(t) u(t-4)
i) Sketch z(t)=x(-t -1)
ii) Sketch y(t)=x(t)+ z(t).
4 M
1 (b)
Determine whether the following signals are Energy or Power, and find energy or time averaged power of the signal:
i) x(t)=5 cos (πt) + sin (5πt); - ∞ ≤ t ≤ ∞
i) x(t)=5 cos (πt) + sin (5πt); - ∞ ≤ t ≤ ∞
| ii) | x[n] =n, | 0 ≤ n ≤ 5 |
| = 10 -n, | 5 ≤ n ≤ 10 | |
| = 0, | otherwise |
4 M
1 (c)
Determine whether the following system in Static/Dynamic, Causal/Non-causal and Stable/Unstable and justify:
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 the following signals:
x(t)=u(t)-u(t-2)
h(t)=e-2t u(t)
x(t)=u(t)-u(t-2)
h(t)=e-2t u(t)
6 M
2 (b)
Evaluate the following integrals: \[ i) \ \ \int^\infty _0 t^2 \delta (t-10)dt \\ ii) \ \int^{10}_0 \delta (t) \sin 2 (\pi t) dt \]
4 M
2 (c)
Determine whether the following signal is periodic or not, if periodic, find the fundamental period of the signal:
x(t)=cos2 (2π t).
x(t)=cos2 (2π t).
2 M
Answer any one question from Q3 and Q4
3 (a)
Find the trigonometric Fourier series for the periodic signal x(t) shown in the following figure and sketch the amplitude and phase spectra:
6 M
3 (b)
Find the inverse Laplace transform of:
\[ X(s) = \dfrac {2} {(s+4)(s-1)} \] If the Region of convergence is:
i) -4 ≤ Re(s) <1
ii) Re(s) > 1
iii) Re(s) < - 4.
\[ X(s) = \dfrac {2} {(s+4)(s-1)} \] If the Region of convergence is:
i) -4 ≤ Re(s) <1
ii) Re(s) > 1
iii) Re(s) < - 4.
6 M
4 (a)
Find the Fourier transform of the following signals:
i) x(t) = sng(t)
ii) x(t)=u(t)
iii) x(t)=e-at sin (ω0 t) u(t).
i) x(t) = sng(t)
ii) x(t)=u(t)
iii) x(t)=e-at sin (ω0 t) u(t).
6 M
4 (b)
Find the initial and final value of the following signal: \[ X(s) = \dfrac { 2s+3} {s^2 + 5s -7} \]
4 M
4 (c)
State the relationship between Fourier transform and Laplace transform.
2 M
Answer any one question from Q5 and Q6
5 (a)
Find the following for the given signal x(t):
i) Autocorrelation
ii) Energy from Autocorrelation
iii) Energy Spectral Density.
x(t) e-10 u(t).
i) Autocorrelation
ii) Energy from Autocorrelation
iii) Energy Spectral Density.
x(t) e-10 u(t).
6 M
5 (b)
Determine the cross-correlation between two sequence 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 Power Spectral Density (PSD).
3 M
6 (a)
Prove that autocorrelation function and energy spectral density from Fourier transform pair of each other and verify the same for:
x(t) e-10 u(t).
x(t) e-10 u(t).
9 M
6 (b)
State and describe any four properties of Energy Spectral Density (ESD).
4 M
Answer any one question from Q7 and Q8
7 (a)
Explain Exponential probability model with respect to its
density and distribution function.
4 M
7 (b)
Two cards are 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)
PDF of a random variable X is: fx(x)= ke-10x, x>0 and
fx(x)=0 x≥0.
Find: i) value of k
ii) P(1 ≤ X ≤ 2)
iii) P(X ≥ 3).
fx(x)=0 x≥0.
Find: i) value of k
ii) P(1 ≤ X ≤ 2)
iii) P(X ≥ 3).
6 M
8 (b)
State the properties of Cumulative probability distribution function.
3 M
8 (b)
Find the mean standard deviation and variance of the uniform random variable.
4 M
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