JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD
JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech III DIGITAL SIGNAL PROCESSING Previous Question Paper (August - 2022 - R18)
Answer any five questions. All questions carry equal marks.
Main
Find the impulse response β[π] of the system described by the difference equation 8π¦[π] + 6π¦[π β 1] = π₯[π].
Answer
The given difference equation is 8y[n] + 6y[n-1] = x[n]. To find the impulse response h[n], set x[n] = Ξ΄[n] and assume the system is causal (y[n] = 0 for n < 0). The equation becomes 8h[n] + 6h[n-1] = Ξ΄[n].
For n = 0: 8h[0] + 6h[-1] = Ξ΄[0]. Since h[-1] = 0 and Ξ΄[0] = 1, we have 8h[0] = 1, so h[0] = 1/8.
For n > 0: 8h[n] + 6h[n-1] = 0, which gives h[n] = -(6/8)h[n-1] = -(3/4)h[n-1].
This is a first-order recursive relation. Solving iteratively: h[1] = -(3/4)(1/8), h[2] = (3/4)^2(1/8), and in general, h[n] = (1/8)(-3/4)^n u[n].
Discuss the sampling rate conversion by a factor I with the help of a neat block diagram.
Answer
Sampling rate conversion by a factor I, known as interpolation, increases the sampling rate of a signal x[n] by inserting I-1 zeros between consecutive samples. The process consists of two stages:
- Upsampling: The input signal x[n] is passed through an expander which inserts I-1 zeros between samples to produce x_e[m] = x[m/I] if m is a multiple of I, and 0 otherwise.
- Low-pass Filtering: The upsampled signal contains images of the original spectrum. An anti-imaging low-pass filter with a cutoff frequency of Ο/I is used to remove these images and interpolate the values between the original samples.
The block diagram consists of the input x[n] feeding into an expander block (βI), followed by a digital low-pass filter h[m], resulting in the output y[m].
Define time invariant system. Show that the interpolator is a time-variant system.
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Check the following filter for time invariant, causal and linear:
(i) π¦(π) = (π β 1)π₯Β²(π + 1)
(ii) π¦(π) = πΒ²π₯(π β 2)
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Explain the frequency domain representation of discrete time signals.
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Explain the terms: i) Upβsampling ii) Down-sampling.
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Determine the Inverse Z-Transform of π(π) = 1 / [(1βπβ»ΒΉ)(1βπβ»ΒΉ)Β²].
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Find the linear convolution of the sequences π₯[π] = {1, 4, 0, 9, -1} and β[π] = {-3, -4, 0, 7}.
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Compute the DFT of the sequence π₯(π) = sin[πΟ/4], where N=8 using DIT FFT algorithm.
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Write five properties of DFS.
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Find the Laplace transform of the following function π(π‘) = π‘πΒ²α΅Sin(3π‘).
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Given π₯(π) = {1, 2, 3, 4, 4, 3, 2, 1}, find π(π) using DIF FFT algorithm.
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Design a Chebyshev filter with a maximum passband attenuation of 2 dB; at Ξ©p=20 rad/sec and the stopband attenuation of 35 dB at Ξ©s=50 rad/sec.
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Obtain the impulse response of digital filter to correspond to an analog filter with impulse response βπ(π‘) = 0.5 πβ»Β²α΅ and with a sampling rate of 1.0kHz using impulse invariant method.
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Differentiate βmaximally flat magnitude responseβ and βequiripple magnitude responseβ filters.
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Convert the analog filter to a digital filter whose system function is π»(π) = 1 / [(π+2)Β²(π+1)], Use bilinear transformation.
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What is a Kaiser window? In what way is it superior to other window functions?
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Using a rectangular window technique, design a low pass filter with pass band gain of unity, cut-off frequency of 1000Hz and working at a sampling frequency of 5 KHz. The length of the impulse response should be 7.
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Using the z-transform, find the total solution to the following difference equation with initial conditions, for discrete time π β₯ 0:
5π¦[π + 2] β 3π¦[π + 1] + π¦[π] = (0.8)βΏπ’[π], π¦[0] = β1, π¦[1] = 10
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Determine direct form I and cascade realization of the following system:
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