JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD
JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech II Year SIGNALS AND STOCHASTIC PROCESS Previous Question Paper (May/June - 2019 - R16)
Note: This question paper contains two parts A and B. Part A is compulsory which carries 25 marks. Answer all questions in Part A. Part B consists of 5 Units. Answer any one full question from each unit.
Part A
Give the condition for the physical reliability of a system.
Answer
A system is physically reliable (causal) if its impulse response h(t) satisfies the condition h(t) = 0 for all t < 0.
What are the properties of convolution?
Answer
The key properties of convolution are:
- Commutative: x(t) * h(t) = h(t) * x(t)
- Associative: [x(t) * h1(t)] * h2(t) = x(t) * [h1(t) * h2(t)]
- Distributive: x(t) * [h1(t) + h2(t)] = x(t) * h1(t) + x(t) * h2(t)
State any two properties of Fourier series.
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Find the Fourier transform of the signal x(t) = 20 sinc (20t).
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Explain the concept of region of convergence for Laplace transforms.
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Write the differentiation in time property of Laplace transform.
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Define random process.
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Give the relation between correlation and Convolution.
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Verify that the cross spectral density of two uncorrelated stationary random processes is an impulse function.
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Define cross –spectral density and its examples.
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Part B
Graphically convolve the signals X1(t) = 1 for -T <= t <= T, 0 elsewhere and X2(t) = 1 for -2T <= t <= 2T, 0 elsewhere.
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What is an LTI system? Explain the properties of it.
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Find whether x(t) = A e-α|t| u(t), α > 0 is an energy signal or not.
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Obtain the Fourier series coefficients for x(t) = A Sin(ω₀t).
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What is the Significance of Hilbert Transform? Explain.
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Define Fourier transform. Explain the properties of Fourier transform.
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Find the Laplace transform of x(t) = -t2 e-at u(-t) and indicate its ROC.
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Find the inverse Laplace transform of x(s) = 5(s+5) / [s(s+3)(s+7)]; Re(s) > -3.
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Find the inverse Z-transform of X(z) = [1 + z-1] / [1 + 3z-1 + 2z-2] for different possible ROCs.
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Give the relationship between z-transform and Laplace Transform.
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A Random Process X(t) = A Cos(2πfct), where A is a Gaussian Random Variable with zero mean and unity variance, is applied to an ideal integrator, that integrates with respect to ‘t’, over (0,t). Check the output of integrator for stationarity.
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A random Process is defined as X(t) = 3 Cos(2πt+Y), where Y is a random Variable with p(Y=0) = p(Y=π) = 1/2. Find the mean and Variance of the Random Variable X(2).
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State and prove properties of cross correlation function.
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If the PSD of X(t) is Sxx(ω). Find the PSD of dx(t)/dt.
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Find and plot the Autocorrelation function of (i) Wide band White noise (ii) Band Pass White noise.
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Derive the expression for the Cross Spectral Density of the input Process X(t) and the output process Y(t) of an LTI system in terms of its Transfer function.
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The auto correlation function of a random process X(t) is Rxx(τ) = 3 + 2 exp(−4τ2).
Evaluate the power spectrum and average power of X(t).
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Calculate the power in the frequency band -1/√2 < ω < 1/√2.
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