## Discrete Time Signal Processing - Dec 2014

### Electronics & Telecomm. (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)** Compare impulse invariant and Bilinear transformation techniques.(5 marks)
**1 (b)** A two pole low pass filter has the system function $$ H(z) = \dfrac {b_0}{(1-pz^{-1})^2) $$ Determine the value of b_{0} and p such that the frequency response H(ω) satisfies the conditions $$ H(0)=1 \ and \ \bigg \vert H \left ( \dfrac {\pi }{4} \right ) \bigg \vert^2 = \dfrac {1}{2} $$(5 marks)
**1 (c)** Explain Multi rate sampling? What are the basic methods? List the advantages of it.(5 marks)
**1 (d)** Explain the sub band coding of speech signal as an application of multi rate signal processing.(5 marks)
**2 (a)** If the impulse response of a FIR filters has the property h(n)=±;h(N-1-n), find the expression for magnitude response and phase and show that filters will have linear phase response.(10 marks)
**2 (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]={5,6,7,81,2,3,4} using appropriate DFT property and result of part (i) determine X_{1}[k].(10 marks)
**3 (a)** Draw a lattice filter implementation for the all pole filter, $$ H(z) = \dfrac {1}{1-0.2z^{-1} + 0.4z^{-2}+ 0.6z^{-3}} $$ and determine the number of multiplications, additions and delays required to implement the filter.(10 marks)
**3 (b)** Compare minimum phase, maximum phase and mixed system. Determine the zeros of the following FIR systems and indicate whether the system is minimum phase, maximum phase or mixed phase, H(z)=6+z^{-1}=z^{-2}.(10 marks)
**4 (a)** Develop DIT-FFT algorithm for decomposing the DFT for N=6 and draw the flow diagrams for (i) N=2×3 (ii) N=3×2(10 marks)
**4 (b)** i) Convert the following analog filter system function into digital IIR filter by means of Bilinear transformation. The digital filter should have resonant frequency of ω_{t}=Π/4 $$ Ha(S)= \dfrac {(s+0.1)}{[(s+0.1)^2 + 9]} \ltbr\gt \ltbr\gt ii) For the analog transfer function $$ H(s) = \dfrac {1} {(s+1)(s+2)} $$ Determine H(z) using impulse invariant technique. Assume T=1sec.\lt/span\gt\ltspan class='paper-ques-marks'\gt(10 marks)\lt/span\gt
\lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt5 (a)\lt/b\gt The transfer function discrete time causal system is given below. $$ H(z) = \dfrac {(1-z^{-1})}{(1-0.2 z^{-1}+0.15z^{-2})} $$ i) Find the difference equation. \ltbr\gt ii) DF-I and DF-II \ltbr\gt iii) Draw Parallel and Cascade realization \ltbr\gt iv) Show pole and zero diagram and find magnitude at ω=0 and ω=Π\lt/span\gt\ltspan class='paper-ques-marks'\gt(10 marks)\lt/span\gt
\lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt5 (b)\lt/b\gt A filter is to be designed with the following desired frequency response $$ \begin {align*}
H(e^{i\omega})&=0 &; -\pi/4|\omega | \le \pi /4 \ \
&= e^{-j2\omega} &; -\pi/4 \le |\omega | \le \pi \end{align*} $$ Determine the filter coefficient h(n) if the window function is defined as $$ \begin {align*}
w(n) &=1, &0\le n \le 4 \
&=0, &otherwise
\end{align*} $$ Also determine the frequency response H(e\ltsup\gtiω\lt/sup\gt) of the designed filter.\lt/span\gt\ltspan class='paper-ques-marks'\gt(10 marks)\lt/span\gt
\lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt6 (a)\lt/b\gt Determine H(z) for a digital Butterworth filter that satisfying the following constraints $$\begin {align*}
\sqrt{0.5} \le & \big \vert H_d (e^{j\omega}) \big \vert \le 1 &; 0 \le \omega \le \pi /2 \ \ \
&\bigg \vert H_d (e^{j \omega}) \bigg \vert \le 0.2 &; 3 \pi/4 \le \omega \le \pi
\end{align*} $$ With T=1 sec. Apply impulse invariant transformation.(10 marks)
**6 (b)** i) A sequence is given as x(n)={1+2j, 1+3j, 2+4j, 2+2j}, from the basic definition, find X(k). If x_{1}(n)={1,1,2,2} and x_{1} and x_{1}(n)={1,1,2,2}. Find X_{1}(k) and X_{2}(k) by using DFT only.

ii) Sequence x_{p}(n) is a periodic repletion of sequence x(n). What is the relationship between C_{k} of discrete time Fourier series of x_{p}(n) and X(k) of x(n)?(10 marks)

### Write short note on (any three):

**7 (a)** Adaptive television echo cancellation(7 marks)
**7 (b)** Goertzel algorithm(7 marks)
**7 (c)** Decimation by integer factor (M) and interpolation by integer factor (L).(7 marks)
**7 (d)** Overlap add and overlap save method for long data sequence.(7 marks)