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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 x_{a}(t) as x_{a}(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 w_{0} y(n-1) ? b^{2} y(n+2) + x(n) ? b cos w_{0} 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)