JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD
JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech IV DIGITAL SIGNAL PROCESSING Previous Question Paper (February/March - 2022 - R18)
Answer any Five Questions. All Questions Carry Equal Marks.
Main
Check whether following systems are linear, causal, time invariant, stable, static: y(n) = x(n) cos (x(n))
Answer
The system is defined by y(n) = x(n) cos(x(n)).
- Static: The system is static (memoryless) because the output y(n) depends only on the current input x(n) and not on past or future values.
- Linear: The system is non-linear. Testing for additivity: y1(n) + y2(n) = x1(n)cos(x1(n)) + x2(n)cos(x2(n)), whereas the output for x1(n) + x2(n) is (x1(n) + x2(n))cos(x1(n) + x2(n)). Since these are not equal, the system is non-linear.
- Time Invariant: The system is time-invariant. If the input is delayed by k, x(n-k), the output is x(n-k)cos(x(n-k)). Shifting the output y(n) by k also yields x(n-k)cos(x(n-k)). Since both are equal, the system is time-invariant.
- Causal: The system is causal because the output at any time n depends only on the input at time n.
- Stable: The system is stable. If the input x(n) is bounded such that |x(n)| ≤ Mx, then |y(n)| = |x(n)cos(x(n))| ≤ |x(n)| |cos(x(n))|. Since |cos(x(n))| ≤ 1, |y(n)| ≤ Mx. Thus, the output is bounded for any bounded input.
Explain in detail about conversion of analog signal to digital signal with suitable block diagram and also explain about reconstructing the signal from its samples.
Answer
The conversion of an analog signal to a digital signal involves three main stages: Sampling, Quantization, and Coding.
- Sampling: The continuous-time signal x(t) is sampled at discrete intervals T to produce x(nT). According to the Nyquist-Shannon sampling theorem, the sampling frequency fs must be at least twice the maximum frequency component of the signal (fs ≥ 2fm) to avoid aliasing.
- Quantization: The sampled values, which are continuous in amplitude, are mapped to a finite set of discrete levels. This introduces quantization error (noise).
- Coding: Each quantized level is represented by a binary code word.
Reconstruction: The original analog signal can be perfectly reconstructed from its samples if the sampling theorem is satisfied. This is achieved using an ideal low-pass filter (reconstruction filter) with a cutoff frequency equal to the Nyquist frequency. The process involves passing the sequence of impulses through the filter, which performs interpolation to fill the gaps between samples, effectively recovering the continuous-time signal x(t) = Σ x(nT) sinc((t-nT)/T).
Consider causal and stable LTI system whose I/Ps and O/Ps are related through second order difference equation y(n) - 1/6 y(n-1) - 1/6 y(n-2) = x(n), determine system impulse response h(n) for the system.
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Describe the decimation process with a factor of M. Obtain necessary expression and sketch frequency response. Also discuss aliasing effect.
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Prove the following properties of DFT when H(K) is the DFT of an N-point sequence h(n):
- H(K) is real and even when h(n) is real and even.
- H(K) is imaginary and odd when h(n) is real and odd.
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Two finite duration sequence are given by x(n) = sin (nπ/2) for n = 0,1,2,3 and h(n) = 2n for n = 0,1,2,3. Determine circular convolution using DFT & IDFT method.
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State and prove the properties of Discrete Fourier Transform.
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Compute IDFT of the sequence x(n) = { 7, -0.707-j0.707, -j, 0.707-j0.707, 1, 0.707+j0.707, j, -0.707+j0.707} using FFT algorithm.
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Determine system function H(z) for a Butterworth filter using Bilinear transformation for the constraints: 0.8 ≤ |H(ejω)| ≤ 1, 0 ≤ ω ≤ 0.2π and |H(ejω)| ≤ 0.2, 0.6π ≤ ω ≤ π with T= 1 sec.
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Discuss the location of poles for Chebyshev filter.
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Discuss in detail about Butterworth filter. What is the effect of varying order of N on magnitude and phase response?
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The specifications of desired low pass filter is 0.8 ≤ |H(ω)| ≤ 1.0; 0 ≤ ω ≤ 0.2π and |H(ω)| ≤ 0.2; 0.6π ≤ ω ≤ π. Design a Chebyshev digital filter using impulse invariant Transformation.
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Determine the filter coefficients of h(n) of length M=15 obtained by sampling method with its frequency response as H(2πK/15) = 1; K=0,1,2,3,4; = 0.4; K=5; = 0; K=6,7.
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Design a HPF of length 8 with cut off frequency of 4 rad/sec using Hamming window. Plot the magnitude and phase response.
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What is the need for signal scaling? How the overflow error scaling is performed?
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Realize system with following difference equation y(n) = (3/4)y(n-1) – (1/8)y(n-2) + x(n) + (1/3)x(n-1) using Direct form-I and Direct form-II.
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