## Digital Signal Processing & Processors - May 2014

### Electronics Engineering (Semester 6)

TOTAL MARKS: 80

TOTAL TIME: 3 HOURS
(1) Question 1 is compulsory.

(2) Attempt any **three** from the remaining questions.

(3) Assume data if required.

(4) Figures to the right indicate full marks.
**1 (a)** Obtain a digital filter transfer function H(ω) by applying inputs invariance transformation on the analog TF.

$$H_a\left(s\right)=\frac{s}{s^2+3s+2}$$ Use f_{s}=1 K samples/sec. (5 marks)
**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 marks)
**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 marks)
**1 (d)** Consider the sequence x(n)= δ(n)+2δ(n-2) + δ(n-3). Find DFT of x(n). (5 marks)
**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 marks)
**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 marks)
**2 (c) ** Explain the properties of symmetricity and periodicity of phase factor. (4 marks)
**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 marks)
**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 x_{1}(n) = {5,6,7,8,1,2,3,4} Using appropriate DFT property and answer of previous part, determine X_{1}(K).

(iii) Again use DFT property and find X_{2}(K) where x_{2} (n) =x(n)+x_{1}(n).(12 marks)
**4 (a)** Draw the Lattice filter realization for the all pole filter

$$ H\left(z\right)=\frac{1}{1+\frac{3}{4}z^{-1}+\ \frac{1}{2}z^{-2}+\frac{1}{4}z^{-3}}$$(10 marks)
**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 marks)
**5 (a)** Design a FIR low pass digital filter using hamming window for N=7

$$ \begin {align*}
H_d(e^{j\omega})&= e^{- \ 3j\omega} \ &0.75\pi \le \omega\le0.75 \pi \\
&=0 \ &0.75\pi\le|\omega|\le\pi
\end {align*} $$(10 marks)
**5 (b)** A LPF has following specification :-

$$ \begin {align*}
&0.8\le|H(\omega|\le 1 \ &for\ 0\le \omega \le0.2\pi\\
&|H(\omega|\le 0.2 \ &for \ 0.6 \pi \le \omega \le \pi
\end {align*} $$

Find filter order and analog cut off frequency if

(i) Bilinear transformation is used for designing.

(ii) Impulse invariance for designing.(10 marks)
**6 (a) ** Explain up sampling by an integer factor with neat diagram and waveforms. (10 marks)
**6 (b)** Explain the need of a low pass filter with a decimator and mathematically prove that ω_{y} = ω_{x}D.(10 marks)

### Write notes on any four

**7 (a)** Frequency sampling realization of FIR filters (5 marks)
**7 (b)** Goertzel algorithm (5 marks)
**7 (c) ** Set top box for digital TV reception (5 marks)
**7 (d) ** Adaptive echo cancellation (5 marks)
**7 (e) ** Filter bank (5 marks)