Question Paper: Digital Signal Processing : Question Paper Jun 2015 - Electronics & Telecomm (Semester 5) | Pune University (PU)
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## Digital Signal Processing - Jun 2015

### Electronics & Telecom Engineering (Semester 5)

TOTAL MARKS: 100
TOTAL TIME: 3 HOURS
(1) Question 1 is compulsory.
(2) Attempt any four from the remaining questions.
(3) Assume data wherever required.
(4) Figures to the right indicate full marks.

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

### Answer any one question from Q3 and Q4

3 (a) State four important advantages of digital signal processing over analog signal processing.(4 marks) 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 marks) 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 marks) 4 (b) State and prove circular time shift property of DFT.(5 marks)

### 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 marks) 5 (b) Compare between Bilinear transformation method and impulse invariant method.(3 marks) 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 marks)
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 marks) 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 marks) 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 marks)

### 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 marks) 7 (b) Explain the characteristics of window function.(4 marks) 7 (c) Distinguish between FIR and IIR filter.(4 marks) 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 marks) 8 (b) What is Gibb's phenomenon? How it is reduced?(4 marks) 8 (c) Show that the filter with symmetric impulse response has linear phase response.(4 marks)