GTU Digital Signal Processing - June 2014 Exam Question Paper

GTU Electronics and Communication Engineering (Semester 7)
Digital Signal Processing
June 2014

Total marks: --
Total time: --

INSTRUCTIONS
(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 the following

1 (a) i) Check whether the following signal is periodic or not. If a signal is periodic, find its fundamental period.

X [n] = \cos^{2} (\frac{π}{8}) 8

$X[n]=\cos^{2}\left ( \dfrac{\pi}{8} \right )8$

2 M

1 (a) ii) X (n) = (-0.5)ⁿ u (n). State whether it is energy or power signal. Justify

2 M

1 (a) iii) Define and explain the convolution and correlation.

2 M

1 (a) iv) Determine if the following system describe by,
Y(t) = Sin [ x (t+2) ] ; is memory less, causal ,linear ,time invariant, stable.

2 M

Answer the following

1 (b) i) Compute the convolution y(n) = x(n) * h(n), Where

X (n) = {1, 1, \underset{↑}{0}, 1, 1} a n d h (n) = {1, \underset{↑}{- 2}, - 3, 4}

$X (n) = \{1, 1,\underset {\uparrow }{0}, 1, 1\} \ and \ h(n) = \{1, \underset {\uparrow }{-2}, -3, 4\}$

3 M

1 (b) ii) Determine if the systems described by following equations are causal or non causal and stable or not.
(1) T[x(n)] = e^x(n)
(2) Y(n) = x(2n)
(3) Y(n) = x(n² ).

3 M

2 (a) State and prove the properties of Z- transform
(i) Convolution of two sequence.
(ii) Differentiation in Z domain.

7 M

2 (b) Given the two sequence of the length 4 are:
X(n) = {0, 1, 2, 3} h(n) = {2, 1, 1, 2}
Find the circular convolution.

7 M

2 (c) Using graphical method, obtain a 5- point circular convolution of two DT signals defined as,
X(n) = (1.5)ⁿ , 2 ≥ n≥ 0
Y(n) = 2n-3, 3 ≥ n ≥ 0.

7 M

3 (a) Find Z transform and ROC of the following sequence.
(i) X₁ (n) = [ 3[2ⁿ ] - 4 [3ⁿ ] ] u[n]
(ii)

X_{2} (n) = [(0.5)^{n} \sin \frac{π n}{4}] u (n)

$X_{2} (n) = [ (0.5)^{n}\sin\dfrac{\pi n}{4}]u(n)$
(iii) X₃(n)=n²-2n+3 for n≥0.

7 M

3 (b) State and prove the properties of DFT (I) Periodicity (II) Time shifting.

7 M

3 (c) Determine the response of the system,

Y (n) = \frac{5}{6} y (n - 1) - \frac{1}{6} y (n - 2) + x (n)

$Y(n)=\dfrac{5}{6}y(n-1)-\dfrac{1}{6}y(n-2)+x(n)$ to the input signals.

X (n) = δ (n) - \frac{1}{3} δ (n - 1)

$X(n)=\delta (n)-\dfrac{1}{3}\delta(n-1)$

7 M

3 (d) State and prove the properties of DFT
(I) Circular convolution (II) Multiplication of two sequences

7 M

4 (a) The transfer function of discrete time causal system is given below

H (z) = \frac{1 - z^{- 1}}{1 - 0.2 z^{- 1} + 0.15 z^{- 2}}

$H(z)=\dfrac{1-z^{-1}}{1-0.2z^{-1}+0.15z^{-2}}$
Find the difference equation and draw cascade and parallel realization.

7 M

4 (b) Derive DIT FFT flow graph for N = 4 hence find DFT of
x(n) = {1, 2, 3, 4}

7 M

4 (c) Consider discrete time linear causal system defined by difference equation.

y (n) - \frac{3}{4} y (n - 1) + \frac{1}{8} y (n - 2) = x (n) + \frac{1}{3} x (n - 1)

$y(n)-\dfrac{3}{4}y(n-1)+\dfrac{1}{8}y(n-2)=x(n)+\dfrac{1}{3}x(n-1)$
Obtain cascade realization of the same.

7 M

4 (d) Determine Inverse Z-transform of the following :

X (z) = \frac{3 z^{- 3}}{{(1 - \frac{1}{4} z - 1)}^{2}}

$X(z)=\dfrac{3z^{-3}}{\left ( 1-\dfrac{1}{4}z-1 \right )^{2}}$ x(n) left handed system.

7 M

5 (a) Compute the eight point DFT of a sequence

X (n) = (\frac{1}{2}, \frac{1}{2}, \frac{1}{2}, \frac{1}{2}, 0, 0, 0, 0)

$X(n)=\left ( \dfrac{1}{2},\dfrac{1}{2},\dfrac{1}{2},\dfrac{1}{2} ,0,0,0,0\right )$
Using decimation in time FFT algorithm.

7 M

5 (b) Write short note on multiplier-accumulator (MAC) hardware of DSP processor

7 M

5 (c) Determine the response [y(n)] of FIR filter. Input x(n) is (1,2,2,1) and h(n) is (1,2,3). Use DFT and IDFT formula.

7 M

5 (d) Write short notes on:
1. Harvard architecture of DSP processor.
2. Hanning Window and Kaiser Window Functions

7 M

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