JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD
JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech II SIGNALS AND SYSTEMS Previous Question Paper (October - 2020 - R15)
Answer any five questions. All questions carry equal marks.
Main
Define the error function while approximating signals and hence derive the expression for condition for orthogonality between two waveforms f1(t) and f2(t).
Answer
The error function in signal approximation represents the difference between a signal f(t) and its approximation f̂(t), defined as e(t) = f(t) - f̂(t). To minimize the mean square error (MSE), we define the error energy E = ∫ |f(t) - ∑ cᵢφᵢ(t)|² dt. For two signals f₁(t) and f₂(t) to be orthogonal over an interval (t₁, t₂), the inner product must be zero.
The condition for orthogonality is derived by setting the integral of the product of the two functions to zero:
This condition implies that the signals do not share any common components, meaning the projection of one signal onto the other is zero, effectively making them linearly independent in the Hilbert space.
Obtain the Fourier series coefficients for x(t) = A sin(ω0t) and B cos(ω0t).
Answer
Using the exponential form of trigonometric functions, x(t) = A sin(ω₀t) can be expressed as:
Comparing this with the Fourier series definition x(t) = ∑ cₙ e^(jnω₀t), the coefficients for A sin(ω₀t) are:
- c₁ = A/2j = -jA/2
- c₋₁ = -A/2j = jA/2
- cₙ = 0 for |n| ≠ 1
For x(t) = B cos(ω₀t), the expression is:
The coefficients for B cos(ω₀t) are:
- c₁ = B/2
- c₋₁ = B/2
- cₙ = 0 for |n| ≠ 1
Define orthogonal signal space and bring out clearly its application in representing a signal.
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Obtain Fourier series representation of a periodic signal x(t) = 2 for -1 ≤ t ≤ 1 and 0 otherwise. Take T = 4s.
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Determine the Nyquist sampling rate and Nyquist sampling interval for the signals:
i) x(t) = sinc2(200πt)
ii) x(t) = 20cos(500πt) + 40sin(1000πt)
Answer
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Find the Fourier Transform of f(t) = t cos(2t) using properties.
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State and prove Differentiation and integration properties of Fourier Transform.
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What is the Significance of Hilbert Transform? Explain.
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Define Impulse Response, Transfer function, system band width, signal Band width, and Rise time.
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Illustrate the characteristics of ideal LPF, HPF and BPF.
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Explain about LTI system by giving examples for Linear and Time invariant systems.
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Discuss about the Causality and physical realizability of a system, with examples.
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Explain the method of detection of periodic signals in the presence of noise by correlation.
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Explain the relation between auto correlation function and energy/power spectral density function.
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Find the Z-transform of the given signal x(n) = sin(w0n)u(n) and find ROC.
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State and prove the initial and final value theorem of Laplace transform.
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