## Digital Signal Processing - Dec 2013

### Electronics & Communication (Semester 5)

TOTAL MARKS: 100

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

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

(3) Assume data wherever required.

(4) Figures to the right indicate full marks.
**1 (a)** What is zero padding? What are its uses?(3 marks)
**1 (b)** Find the DFT of a sequence x(n)={1,1,0,0} and the IDFT of y(k)={1,0,1,0}.(6 marks)
**1 (c)** Find the DFT of a sequence $$ \begin {align*}x(n)&=1 & for \ n \le n \le 1 \\
&=0 & Otherwise \ \ \ \ \ \ \end{align*} $$ $$ for \ N=4 \ plot \ |x(k)| \ and \ \lfloor x(k) $$(8 marks)
**2 (a)** State and prove time shifting property of DFT.(5 marks)
**2 (b)** Find the output y(n) of a filter whose impulse response is h(n)={1,1,1} and the input signal x(n)= {3, -1, 0, 1, 3, 2, 0, 1, 2, 1} using overlap save method.(8 marks)
**2 (c)** Obtain the 8-point circular of the following sequences:

x_{1}(n)={2,3,6,8,2,1,7,5}

x_{2}(n)={0,0,0,0,0,1,0,0}.(7 marks)
**3 (a)** Find the DFT of a sequence x(n)={1,2,3,4,4,3,2,1} using DIT-FFT algorithm.(10 marks)
**3 (b)** Compute the IDFT of the sequence

x(k)={7, 0.707 - j0.707, -j, 0.707 - j0.707, 1, 0.707 + j0, 707, j, -0.707 + j0.707} using DIF-FFT algorithm.(10 marks)
**4 (a)** Discuss Chirp Z-transformation algorithm.(6 marks)
**4 (b)** Explain the following properties of twiddle factor W_{N}.

i) Symmetric property

ii) Periodicity property(4 marks)
**4 (c)** Find x(k) for the input sequence x(n)=n+1 and N=8 using DIF-FFT algorithm.(10 marks)
**5 (a)** Design an analog Chebyshev filter for which the squared magnitude response |H_{a}(jΩ)|^{2}| satisfies the condition $$ 20 \log_{10} |H_a (j\Omega)|_{\Omega=0.2 n} \ge -1; \ \ 20\log_{10} |H_a (j\Omega)|_{\Omega=0.3 n} \le -15 $$(8 marks)
**5 (b)** Distinguish between Butterworth and Chebyshev filters.(4 marks)
**5 (c)** $$ Let \ H(s) = \dfrac {1}{s^2 + \sqrt{2s+1}} $$ represent the transfer function of a lowpass filter with a passband of 1 rad/sec. Use frequency transformation to find the transfer functions the following analog filters:

i) A lowpass filter with passband of 10 rad/sec.

ii) A highpass filter with cut off frequency of 10 rad/sec.(8 marks)
**6 (a)** What is a rectangular window function? Obtain its frequency domain characteristics.(5 marks)
**6 (b)** Design FIR low pass filter using Hamming window (M=7) and also obtain frequency response for $$ \begin {align*}
H_d (e^{j\omega })&=e^{-3\omega }& 3\pi 4 < \omega < 3\pi /4 \\ &=0; & 3\pi/4 < |\omega| \le \pi \ \ \end{align*} $$(12 marks)
**6 (c)** Explain Gibb's phenomenon.(3 marks)
**7 (a)** What is bilinear transformation? Obtain the transformation formula for bilinear transformation.(10 marks)
**7 (b)** Convert the following transfer function. $$ H(s)=\dfrac {s+a}{(s+a)^2 + b^2} $$ into a digital filter with infinite impulse response by the use of impulse invariance mapping technique.(10 marks)
**8 (a)** i) Obtain the cascade realization of the system function

H(z)=(1+2z^{-1}-z^{-2})(1+z^{-1}-z^{-2}).

ii) Determine the direct from realization of the system function

H(z)=1+2z^{-1}-3z^{-2}-4z^{-3}+5z^{-4}(6 marks)
**8 (b)** Find the impulse response of an FIR lattice filter with coefficient k_{1}=0.65, k_{2}=0.34, k_{3}=0.8.(9 marks)
**8 (c)** Obtain the direct form-I and direct form-II structure for the filters given by system function $$ H(z) = \dfrac {1+0.4z^{-1}}{1-0.5z^{-1}+0.06z^{-2}} $$(5 marks)