VTU Digital Signal Processing - June 2013 Exam Question Paper

VTU Electronics and Communication Engineering (Semester 5)
Digital Signal Processing
June 2013

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) Define DFT. Derive the relationship of DFT to the z-transform.

5 M

1 (b) An analog signal is sampled at 10 KHz and the DFT of 512 samples is computed. Determine the frequency spacing between the spectral samples of DFT.

3 M

1 (c) Consider the finite length sequence x(n)=?(n)-2?(n-5): Find (i) The 10 point DFT of x(n) (ii) The sequence y(n) that has a DFT

y (K) = e^{\frac{J 4 π}{10} K}

$y(K) = e ^{\frac {J4\pi}{10}K}$ X(K) where X(K) is the 10 point DFT of x(n) and W(K) is the 10 point DFT of u(n)-u(n-6).

12 M

2 (a) Determine the circular convolution of the sequence x(n)={1,2,3,1} and h(n)={4,3,2,2} using DFT and IDFT equations.

8 M

2 (b) Let X(K) be a 14 point DFT of a length 14 real sequence x(n). The first 8 samples of X(K) are given by: X(0)=12, X(1)=1+J3, X(2)=3+j4, x(3)=1-J5, X(4)=-2+J2, X(5)=6+J3, X(6)=2-J3, X(7)=10. Determine the remaining samples of X(K). Also evaluate the following functions without computing the IDFT.

i) x (0) i i) x (7) (i i i) \sum_{n = 0}^{13} x (n) (i v) \sum_{n = 0}^{15} | x (n) |^{2}

$i) \ x(0) \ \ \ ii) \ x(7) \ \ \ (iii) \ \sum^{13}_{n=0} x(n) \ \ \ (iv) \ \sum^{15}_{n=0}\big\vert x (n) \big\vert^2$

12 M

3 (a) Consider a FIR filter with impulse response. h(n)={3,2,1,1}. If the input is x(n)={1,2,3,3,2,1,-1,-2,-3,5,6,1,2,1}. Using the overlap save method and 8 point circular convolution.

10 M

3 (b) What are FFT algorithms? Prove the (i) Symmetry and (ii) Periodicity property of the twiddle factor W_n.

6 M

3 (c) How many complex multiplications and additions are required for computing 256 point DFT using FFT algorithms?

4 M

4 (a) Find the DFT of the sequence x(n)={1,2,3,4,4,3,2,1} using the decimation in frequency FFT algorithm and draw the signal flow graph. Show the outputs for each stage.

10 M

4 (b) Given x(n)={1,0,1,0}, find x(2) using the Geortzel algorithm.

5 M

4 (c) Write a note on Chirp z-transform algorithm.

5 M

5 (a) Given that

| H (e^{7 Ω}) |^{2} = \frac{1}{1 + 64 Ω^{6}}

$\bigg \vert H(e^{7 \Omega})\bigg \vert ^{2} = \dfrac {1}{1+64 \Omega^6}$ determine the analog Butterworth low pass filter transfer function.

6 M

5 (b) Design an analog Chebyshev filter with a maximum passband attenuation of 2.5 dB at Ω_p=20 rad/sec and the stopband attenuation of 30 dB at Ω_s=50 rad/sec.

10 M

5 (c) Compare Butterworth and Chebyshev filters.

4 M

6 (a) What are the conditions to be satisfied while transforming an analog filter to a digital HR filter? Explain how this is achieved in Bilinear transformation technique.

5 M

6 (b) Design a Butterworth filter using the impulse invariance method for the following specifications: Take T=1 sec,

\begin{aligned} 0.8 \leq & | H (e^{j W}) | \leq 1 & 0 \leq W \leq 0.2 π \\ | H (j W) | \leq 0.2 & 0.6 π \leq W \leq π \end{aligned}

$\begin {align*}0.8 \le & \bigg \vert H(e^{jW}) \bigg \vert \le 1 & 0 \le W \le 0.2 \pi \\ & \bigg \vert H (jW) \bigg \vert \le 0.2 & 0.6 \pi \le W \le \pi \end{align*}$

10 M

6 (c) Determine H(z) for the given analog system function

H (s) = \frac{(s + a)}{(s + a)^{2} + b^{2}}

$H(s) = \dfrac {(s+a)}{(s+a)^2 +b^2}$ by using Matched z-transform.

5 M

7 (a)

A z-plane pole zero plot for a certain digital filter shown in Fig. Q7 (a). Determine the system function in the $H(z)= \dfrac {(1+a_1 z^{-1})(1+b_1 z^{-1} + b_2 z^{-2})}{(1+c_1 z^{-1})(1+d_1 z^{-1}+d_2 z^{-2})}$ giving the numerical values for parameters a₁, b₁, b₂, c₁, d₁ and d₂. Sketch the direct form II and Cascade realizations of the system.

10 M

7 (b) A FIR filter is given by,

y (n) = x (n) + \frac{2}{5} x (n - 1) + \frac{3}{4} x (n - 2) - \frac{1}{3} x (n - 3)

$y(n)= x(n) + \dfrac {2}{5}x(n-1) + \dfrac {3}{4} x (n-2) - \dfrac {1}{3}x (n-3)$ Draw the direct form I and lattice structure.

10 M

8 (a) Design a FIR filter (low pass) with a desired frequency response,

\begin{aligned} H_{d} (e^{j W}) & = e^{- j 3 w}; & - \frac{3 π}{4} \leq ω \leq \frac{3 π}{4} \\ = 0; & \frac{3 π}{4} < | ω | < π \end{aligned}

$\begin {align*} H_d (e^{jW}) &= e^{-j3w}; &-\dfrac {3 \pi}{4} \le \omega \le \dfrac {3 \pi}{4} \\ &=0; &\dfrac {3 \pi}{4} < \vert \omega \vert < \pi \end{align*}$ Use Hamming window with M=7. Also obtain the frequency response.

10 M

8 (b) Design a linear phase low pass FIR filter with 7 taps and cut off frequency of Ω_c=0.3Π rad. Using the frequency sampling method.

10 M

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