SPPU Electronics and Telecom Engineering (Semester 5)
Digital Signal Processing
June 2015
Total marks: --
Total time: --
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary

Answer any one question from Q1 and Q2
1 (a) Consider the analog signal xa(t) as xa(t)=6 cos 50 πt+3 sin 200 πt-3 cos 100 πt.
i) Determine the minimum sampling frequency.
ii) Determine x(n) at minimum sampling frequency.
iii) Sketch the waveform and show the sampling points.
5 M
1 (b) Determine the transfer function and impulse response of the LTI system given by the difference equation. \[ y(n)+ \dfrac {3}{4} y (n-1) + \dfrac {1}{8} y(n-2)= x(n)+ x (n-1) \]
5 M

2 (a) State and prove convolution property of Z transform.
5 M
2 (b) Compute 4-point DFT of the sequence given by x(n)=(-1)n using DIT FFT algorithm.
5 M

Answer any one question from Q3 and Q4
3 (a) State four important advantages of digital signal processing over analog signal processing.
4 M
3 (b) For the following sequences, \[ x_1 (n) = \left\{\begin{matrix} 1 &0 \le n \le 2 \\ 0 & otherwise \end{matrix}\right. \\ x_2 (n) = \left\{\begin{matrix} 1 & 0 \le n \le 2 \\ 0 & otherwise \end{matrix}\right. \] Compute linear convolution using circular convolution.
6 M

4 (a) Using partial fraction expansion, find inverse Z-Transform of following system function and verify it using long division method, \[ H(Z) = \dfrac {1+2 Z^{-1}}{1-0.4Z^{-1}-0.12 Z^{-2}} \] if h(n) is causal.
5 M
4 (b) State and prove circular time shift property of DFT.
5 M

Answer any one question from Q5 and Q6
5 (a) Design a Butterworth digital IIR lowpass filter using bilinear transformation to satisfy following specifications: \[ \begin {align*} 0.6 \le & \big \vert H(e^{j \omega}) \big \vert \le 1.0 & 0 \le w \le 0.35 \pi \\ & \big \vert H (e^{j \omega}) \big \vert \le 0.1 & 0.7\pi \le w < \pi \end{align*} \] Use T=0.1 seconds.
10 M
5 (b) Compare between Bilinear transformation method and impulse invariant method.
3 M
5 (c) Draw direct form I & direct form II realisations for the second order system given by:
y(n)=2b cos w0 y(n-1) ? b2 y(n+2) + x(n) ? b cos w0 x(n-1)
4 M

6 (a) The system function of an analog filter is given by \[ H(s) = \dfrac {s+0.2} {(s+0.2)^2+9} \] Convert it to digital filter using Impulse Invariant technique. Assume T = 1 second.
4 M
6 (b) Given \[ H(s) = \dfrac {1} {s+1} \] Apply impulse invariant method to obtain digital filter transfer function and difference equation. Assume T=1 second.
4 M
6 (c) For the system given by following equation \[ H(z) = \dfrac {1-z^{-1}} {1-0.2 z^{-1}- 0.15 z^{-2}} \] Draw cascade and parallel realisation.
9 M

Answer any one question from Q7 and Q8
7 (a) Design a linear phase FIR band pass filter using hamming window with cut off frequencies 0.2 rad/sec & 0.3 rad/sec. M = 7.
9 M
7 (b) Explain the characteristics of window function.
4 M
7 (c) Distinguish between FIR and IIR filter.
4 M

8 (a) Design a linear phase FIR lowpass filter with a cutoff frequency of 0.5 rad/sample by taking 11 samples of ideal frequency response.
9 M
8 (b) What is Gibb's phenomenon? How it is reduced?
4 M
8 (c) Show that the filter with symmetric impulse response has linear phase response.
4 M

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