GTU Digital Signal Processing - December 2014 Exam Question Paper

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

1 (a) Draw the block diagram of a typical Digital Signal Processing system and explain.

7 M

1 (b) A discrete-time signal x(n) is given below :

X (n) = {1, 1, \underset{↑}{1}, 1, 1, 1 / 2}

$X (n) = \{1, 1,\underset {\uparrow }{1}, 1, 1,1/2\}$
Sketch and label carefully each of the following signals: (i)x(n-2)
(ii) x(4-n)
(iii) x(2n)
(iv) x(n)u(2-n)
(v)x(n-1)δ(n-3)

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2 (a) Perform the liner convolution of the following sequences:

X_{1} (n) = {1, \underset{↑}{2}, 3, 4, 5}, X_{2} (n) = {- 1, 0, \underset{↑}{1}}

$X_{1}(n) = \{1, \underset {\uparrow }{2}, 3, 4,5\},\ \ X_{2}(n) = \{-1,0, \underset {\uparrow }{1}\}$

7 M

2 (b) (i) For the following system, determine whether the system is stable,causal,linear,time invariant,memoryless:

T {x (n)} = \sum_{k = n_{0}}^{n} x (k)

$T\left \{ x(n) \right \}=\sum_{k=n_{0}}^{n}x(k)$
What are the advantages of digital signal processing over analog signal processing>

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2 (c) Let X(e^jw ) denote the fourier transform of the signal x(n) .Perform the following calculations without explicitly evaluating X(e^jw )

X_{1} (n) = {- 1, 0, 1, \underset{↑}{2}, 1, 2, 1, 0, - 1}

$X_{1}(n) = \{-1,0,1, \underset {\uparrow }{2}, 1, 2,1,0,-1\}$
i) Evaluate X(e^jw ) ?w=0
ii) Evaluate X(e^jw ) ?w=π
iii) Find θ X((e^jw))
iv) Evaluate

\int_{- π}^{π} X (e^{j w}) d w

$\int_{-\pi}^{\pi}\limits X(e^{jw})dw$
v) Determine and sketch the signal whose fourier transform is X(e- ^jw )
vi) Determine and sketch the signal whose fourier transform is Re{X(e^jw )

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3 (a) Determine the z-transform of the following sequences. Sketch ROC and pole zero plot :
(i)x₁ (n) = α^|n|, 0 < ?α ? < 1
(ii)x₂ (n) = (-1/3)ⁿ u(n) - (1/2)ⁿ u(-n-1)

7 M

3 (b) Suppose the z-transform of x(n) is

X (z) = \frac{z^{10}}{(z - (1 / 2)) (z - (3 / 2))^{10} (z + (3 / 2))^{2} (z + (5 / 2)) (z + (7 / 2))}

$X(z)=\dfrac{z^{10}}{(z-(1/2))(z-(3/2))^{10}(z+(3/2))^{2}(z+(5/2))(z+(7/2))}$
It is also known that x(n) is a stable sequence.
(i)Determine the region of convergence of X(z).
(ii) Determine x(n) at n = -8.

7 M

3 (c) Consider the discrete time system with an ideal low pass filter with cutoff frequency π/8 radian/s.
IMAGE
(i)If x_c (t) is bandlimited to 5 kHz , what is the maximum value of T that will avoid aliasing?
(ii)If 1/T = 10 kHz , what will the cutoff frequency of the continuous-time filter be?
(iii) Repeat part
(iv) for 1/T=20kHz.

7 M

3 (d) Draw the structures of the following discrete time system:

H (z) = \frac{(1 + z^{- 1})^{2}}{1 - 0.75 z^{- 1} + 0.125 z^{2}}

$H(z)=\dfrac{(1+z^{-1})^{2}}{1-0.75z^{-1}+0.125z^{2}}$
(i)Direct form - I
(ii)Direct Form - II
(iii)Cascade form
(iv)Parallel form.

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4 (a) Discuss the following transformation methods to design digital filters: (i)Impulse invariance (ii)Bilinear transformation

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4 (b) Find the circular convolution of the following sequences:

X_{1} (n) = {\underset{↑}{1}, 2, 3, 4} X_{2} (n) {\underset{↑}{2}, 1, 2, 1}

$X_{1}(n) = \{ \underset {\uparrow }{1}, 2,3,4\}\ \ \ \ \ X_{2}(n) \{ \underset {\uparrow }{2}, 1,2,1\}$

7 M

4 (c) Design a Digital low pass FIR filter using Kaiser window to meet the following specifications:
0.99 ≤?H(e^jw )≤ 1.01 , 0 ≤w ≤ 0.4π
?H(e^jw )?≤ 0.001 , 0.6π ≤ w ?π

7 M

4 (d) Consider the real finite-length sequence x(n).

X (n) = {\underset{↑}{4}, 3, 2, 1}

$X(n) = \{ \underset {\uparrow }{4}, 3,2,1\}$
(i)Sketch the finite length sequence y(n) whose six-point DFT is Y(k) = W₆^4k X(k) , Where X(k) is the six-point DFT of x(n).
(ii) Sketch the finite length sequence w(n) whose six-point DFT is W(k) = Re{ X(k) }
(iii) Sketch the finite length sequence q(n) whose three-point DFT is Q(k) = X(2k) , k=0,1,2

7 M

5 (a) Explain the Decimation in Time FFT algorithm.

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5 (b) Discuss the applications of digital signal processing with suitable examples.

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5 (c) Discuss the key features of the architecture of DSP Processors.

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5 (d) Write a short note on coefficient quantization in IIR filters.

7 M

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