JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD
JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech II Year SIGNALS AND STOCHASTIC PROCESS Previous Question Paper (August/September - 2022 - R16)
Answer any five questions. All questions carry equal marks.
Main
Explain how functions can be approximated using orthogonal functions.
Answer
The approximation of a signal f(t) by a set of orthogonal functions {φn(t)} is based on the principle of minimizing the mean square error. Given a set of orthogonal functions over an interval (t1, t2), the function f(t) can be approximated as:
where the summation runs from n=1 to N. The condition for orthogonality is:
To minimize the error ε, defined as the integral of the squared difference between f(t) and its approximation, we set the partial derivative of the error with respect to the coefficients cn to zero. This leads to the orthogonality principle, where the coefficients are calculated as:
As N approaches infinity, the approximation error approaches zero, provided the set of functions is complete, allowing the signal to be represented exactly as a generalized Fourier series.
Consider a stable LTI system characterized by the differential equation:
Find its impulse response.
Answer
To find the impulse response h(t) of the LTI system, we take the Laplace Transform of the given differential equation:
Applying the Laplace transform property for differentiation, assuming zero initial conditions:
The transfer function H(s) is defined as the ratio of the output transform to the input transform:
The impulse response h(t) is the inverse Laplace transform of H(s). Using the standard transform pair:
Substituting a = 2, we obtain the impulse response:
Explain the concepts of unit step, impulse and Signum function.
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Derive the relationship between rise time and bandwidth.
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State and prove sampling theorem for band limited signals.
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What is the Significance of Hilbert Transform? Explain.
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Derive the expression for trigonometric Fourier series coefficients.
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Find the Fourier transform of symmetrical gate pulse and sketch the spectrum.
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What is the relation between Fourier transform and Laplace transform?
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What is meant by ROC? Prove that the signals
have the same X(s) and differ only in ROC.
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Find the Laplace transform of x(t) = -t^2 e^{-at} u(-t) and indicate its ROC.
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Find the inverse Z-transform of
for different possible ROCs.
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Explain about Auto-correlation function with their properties.
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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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Derive the relationship between cross-power spectrum and cross correlation function.
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Explain about cross power spectrum density and its properties with proofs.
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