JNTU
JNTU B.Tech II SIGNALS SYSTEMS AND SIGNAL PROCESSING Previous Question Paper (March - 2022 - R18)
Answer any five questions. All questions carry equal marks.
Main
A rectangular function is given as x(t) = 1 for 0 < t < π and x(t) = -1 for -π < t < 0. Approximate this function by a sinusoidal waveform over the interval [0, 2π] such that the mean square error is minimum.
Answer
To approximate the function x(t) with a sinusoidal waveform of the form f(t) = A sin(ωt) such that the mean square error is minimized, we use the principle of orthogonal projection. Given the interval [0, 2π], the fundamental frequency is ω = 1. We seek to minimize the error E = ∫[x(t) - A sin(t)]^2 dt from 0 to 2π.
Setting the derivative of E with respect to A to zero yields the condition for the optimal coefficient A:
Evaluating the integral over the interval [0, 2π]:
The denominator is:
Thus, the optimal amplitude is A = 2/π. The approximating sinusoidal waveform is f(t) = (2/π) sin(t).
Represent the following functions graphically and mathematically: i) Impulse function ii) Unit Step function iii) Signum function.
Answer
i) Impulse function δ(t): Defined as 0 for t ≠ 0 and having an area of 1 at t = 0. Mathematically, δ(t) = 0 for t ≠ 0 and ∫ δ(t) dt = 1. Graphically, it is represented as an arrow pointing upwards at the origin with a height labeled as 1.
ii) Unit Step function u(t): Defined as 1 for t ≥ 0 and 0 for t < 0. Mathematically, u(t) = 1, t ≥ 0; 0, t < 0. Graphically, it is a horizontal line at 0 for t < 0 and a horizontal line at 1 for t > 0, with a step jump at the origin.
iii) Signum function sgn(t): Defined as 1 for t > 0, -1 for t < 0, and 0 at t = 0. Mathematically, sgn(t) = 1, t > 0; -1, t < 0; 0, t = 0. Graphically, it shows a horizontal line at -1 for t < 0 and a horizontal line at 1 for t > 0, with a discontinuity at the origin.
What are Orthogonal vectors? Give example.
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Obtain the expression for Mean Square Error used for Evaluation.
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Define System bandwidth.
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Obtain the trigonometric Fourier series for the given wave form.
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What are Dirichlet’s Conditions? State them.
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Find the Fourier transform of e-2t cos 5t u(t).
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Find the Fourier transform of an impulse function.
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Highlight the relation between convolution and correlation.
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The transfer function of an LTI system is H(jw) = 16 / (4 + jw). Find the response y(t) for an input x(t) = u(t).
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Find the graphical convolution between two signals x(t)=u(t+2) and h(t)=u(t-3).
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Describe Causality and Paley Wiener criterion for physical realization of system.
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Obtain the relationship between Bandwidth and rise time.
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Find the Laplace transform and ROC of x(t) = e-t u(t) + e-2t u(-t).
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Using properties of Z-Transform find the Z-Transform of x(n) = n2 u(n).
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Find the Z-Transform of the signal x(n) = n[(1/2)n u(n) * (1/3)n u(n)].
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Determine the inverse Z-transform of X(Z) = (z3 + z2 + 3z - 4) / (z2 + 3z + 2) given ROC |Z| > 0.
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With the help of derivation explain the procedure to compute DFT using Radix-2 DIT FFT.
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Design an analog Butterworth low pass filter that has a dB pass band attenuation at a frequency of 20 rad/sec and at least -10 dB stop band attenuation at 30 rad /sec.
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