JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD
JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech III DIGITAL SIGNAL PROCESSING Previous Question Paper (October - 2020 - R15)
Answer any five questions. All questions carry equal marks.
Main
Determine the impulse response and the unit step response of the systems described by the difference equation:
Answer
To find the impulse response h(n), we take the Z-transform of the difference equation y(n) - 0.7y(n-1) + 0.1y(n-2) = 2x(n) - x(n-2). Assuming zero initial conditions, we get Y(z)(1 - 0.7z^-1 + 0.1z^-2) = X(z)(2 - z^-2). The transfer function is H(z) = Y(z)/X(z) = (2 - z^-2) / (1 - 0.7z^-1 + 0.1z^-2). Factoring the denominator, we have (1 - 0.2z^-1)(1 - 0.5z^-1). Using partial fraction expansion: H(z) = A/(1 - 0.2z^-1) + B/(1 - 0.5z^-1). Solving for constants, we find A = -10 and B = 12. Thus, the impulse response is h(n) = -10(0.2)^n u(n) + 12(0.5)^n u(n).
The unit step response s(n) is the response to x(n) = u(n), which is the convolution of h(n) with u(n), or the inverse Z-transform of S(z) = H(z) / (1 - z^-1). Performing partial fraction expansion on S(z) = (2 - z^-2) / [(1 - 0.2z^-1)(1 - 0.5z^-1)(1 - z^-1)], we obtain coefficients for terms (1-z^-1)^-1, (1-0.2z^-1)^-1, and (1-0.5z^-1)^-1, resulting in s(n) = [2.5 - 2.5(0.2)^n + 1(0.5)^n] u(n).
Determine the canonic form realization of the LTI system:
Answer
The given difference equation is y(n) = x(n) - x(n-1) + 2x(n-2) - 3x(n-4). The transfer function is H(z) = 1 - z^-1 + 2z^-2 + 0z^-3 - 3z^-4. In the canonic form (Direct Form II), the realization uses a single delay chain to represent the system. Since the denominator coefficients are all zero except for the leading coefficient (a0 = 1), the feedback path is absent.
- The input x(n) enters the delay line consisting of four unit delays (z^-1).
- The output y(n) is formed by the weighted sum of the input and the outputs of the delay elements: y(n) = 1*x(n) - 1*x(n-1) + 2*x(n-2) + 0*x(n-3) - 3*x(n-4).
- The structure consists of a feedforward path where the input x(n) is passed through four delay elements in series. Multipliers with coefficients 1, -1, 2, 0, and -3 are applied to the input and the taps after each delay, respectively, and summed to produce y(n).
Consider a system with impulse response:
Determine the input x(n) for 0 ≤ n ≤ 8 that will generate the output sequence x(n) = {1, 2, 2.5, 3, 3, 3, 2, 1, 0, ...}
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The step response of an LTI system is s(n) = (1/3)^(n-2) u(n+2). Find the system function H(z) and sketch the pole-zero plot.
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For the sequences x1(n) = cos(2πn/N) and x2(n) = sin(2πn/N) for 0 ≤ n ≤ N-1. Determine the N-point circular convolution of x1(n) and x2(n).
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Compute eight-point DFT using DIT and DIF algorithm for the sequence:
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Determine the system function H(z) of the lowest-order Chebyshev digital filter that meets the following specifications:
a) 1-dB ripple in the passband 0 < |ω| < 0.3π
b) At least 60 dB attenuation in the stopband 0.35π < |ω| < π. Use the bilinear transformation.
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Design a Chebyshev lowpass filter with the specifications αp = 1dB ripple in the pass band 0 ≤ ω ≤ 0.2π, αs = 15dB in the stop band 0.3π ≤ ω ≤ π. Using Impulse Invariance and Bilinear transformation.
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Explain the type I and type II frequency sampling method of designing an FIR digital filter.
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Consider a signal x(n) with Fourier transform X(ω) = 0 for ωm < ω ≤ π. Show that the signal x(n) can be recovered from its samples x(mD) if the sampling frequency ωs = 2π/D ≥ 2ωm (f = 1/D ≥ 2fm).
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Explain about the round-off noise in digital filters.
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