MU Discrete Time Signal Processing - May 2014 Exam Question Paper

MU Electronics and Telecom Engineering (Semester 6)
Discrete Time Signal Processing
May 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) Obtain a digital filter transfer function H(?) by applying inputs invariance transformation on the analog TF.

H_{a} (s) = \frac{s}{s^{2} + 3 s + 2}

$H_a\left(s\right)=\frac{s}{s^2+3s+2}$ Use f_s=1 K samples/sec.

5 M

1 (b) Consider a filter with TF: H(z)= (z^-1-a)/(1-a z^-1) Identify the type of filter and justify it.

5 M

1 (c) Find the number of complex multiplication and complex additions required to find DFT for 32 point sequence. Compare them with the number of computations required if FFT algorithm is used.

5 M

1 (d) Consider the sequence x(n)= ?(n)+2?(n-2) + ?(n-3). Find DFT of x(n).

5 M

2 (a) A sequence is given as x(n)= {1+2j, 1+3j, 2+4j, 2+2j}
(i) Find X(k) using DIT-FFT algorithm.
(ii) Using the result in (i) and not otherwise find DFT of p(n) and q(n) where
p(n)={1,1,2,2}
q(n)={2,3,4,2}

6 M

2 (b) X(k)= {36, -4+j9.656, -4+j4, -4+j 1.656, -4, -4-j 1.656, -4 -j4, -4-j9.656} Find x(n) using IFFT algorithm (use DIT IFFT).

10 M

2 (c) Explain the properties of symmetricity and periodicity of phase factor.

4 M

3 (a) By means of FFT-IFFT method (DIT algo) compute Circular convolution of x(n)={2,1,2,1} h(n) = { 1,2,3,4}

8 M

3 (b) An 8 point sequence x(n) = {1,2,3,4,5,6,7,8}
(i) Find X(K) using DIF FFT algorithm.
(ii) Let x₁(n) = {5,6,7,8,1,2,3,4} Using appropriate DFT property and answer of previous part, determine X₁(K).
(iii) Again use DFT property and find X₂(K) where x₂ (n) =x(n)+x₁(n).

12 M

4 (a) Draw the Lattice filter realization for the all pole filter

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

$H\left(z\right)=\frac{1}{1+\frac{3}{4}z^{-1}+\ \frac{1}{2}z^{-2}+\frac{1}{4}z^{-3}}$

10 M

4 (b) Obtain DF-I , DF-II, cascade (first order sections) and parallel (first order sections) Structures for the system described by
y(n) =-0.1 y(n-1) + 0.72 y(n-2) + 0.7 x(n) - 0.252 x(n-1).

10 M

5 (a) Design a FIR low pass digital filter using hamming window for N=7

\begin{aligned} H_{d} (e^{j ω}) & = e^{- 3 j ω} & 0.75 π \leq ω \leq 0.75 π \\ = 0 & 0.75 π \leq | ω | \leq π \end{aligned}

$\begin {align*}H_d(e^{j\omega})&= e^{- \ 3j\omega} \ &0.75\pi \le \omega\le0.75 \pi \\&=0 \ &0.75\pi\le|\omega|\le\pi\end {align*}$

10 M

5 (b) A LPF has following specification :-

\begin{aligned} 0.8 \leq | H (ω | \leq 1 & f o r 0 \leq ω \leq 0.2 π \\ | H (ω | \leq 0.2 & f o r 0.6 π \leq ω \leq π \end{aligned}

$\begin {align*}&0.8\le|H(\omega|\le 1 \ &for\ 0\le \omega \le0.2\pi\\&|H(\omega|\le 0.2 \ &for \ 0.6 \pi \le \omega \le \pi\end {align*}$
Find filter order and analog cut off frequency if
(i) Bilinear transformation is used for designing.
(ii) Impulse invariance for designing.

10 M

6 (a) Explain up sampling by an integer factor with neat diagram and waveforms.

10 M

6 (b) Explain the need of a low pass filter with a decimator and mathematically prove that ?_y = ?_xD.

10 M

Write notes on any four

7 (a) Frequency sampling realization of FIR filters

5 M

7 (b) Goertzel algorithm

5 M

7 (c) Set top box for digital TV reception

5 M

7 (d) Adaptive echo cancellation

5 M

7 (e) Filter bank

5 M

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