Answer any one question from Q1 and Q2

1 (a)
Consider the analog signal x

i) Determine the minimum sampling frequency.

ii) Determine x(n) at minimum sampling frequency.

iii) Sketch the waveform and show the sampling points.

_{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 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 w

y(n)=2b cos w

_{0}y(n-1) ? b^{2}y(n+2) + x(n) ? b cos w_{0}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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