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## Digital Signal Processing - Dec 2015

### 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)** Define N-point DFT and IDFT of a sequence.(3 marks)
**1 (b)** Find the 8-point DFT of the sequence x(n)= {1, 1, 1, 1, 1, 1, 0, 0}.(8 marks)
**1 (c)** Find the IDFT of X(K) = {4, -2j, 0, 2j}.(6 marks)
**1 (d)** Obtain the relation between DFT and Z-transform.(3 marks)
**2 (a)** State and prove circular convolution property.(6 marks)
**2 (b)** For x(n) = {7, 0, 8, 0}, find y(n), if Y(K)= X((K-2)).(6 marks)
**2 (c)** Let x(n) = {1, 2, 0, 3, -2, 4, 7, 5}. Evaluate the following: $$ i) \ X(0) \\ ii) X(4) \\ iii) \sum_{K=0} X(K) \\ \sum_{K=0}|X(K)|^2 $$(8 marks)
**3 (a)** In the direct computation of N-point DFT of x(n), how many

i) Complex multiplications,

ii) Complex additions

iii) Real multiplications

iv) Real additions and

v) Trigonometric function evaluations are required.(10 marks)
**3 (b)** Find the output y(n) of a filter whose impulse response h(n)={1, 2} and input signal x(n)= {1, 2, -1, 2, 3, -2, -3, -1, 1, 1, 2, -1} using overlap save method.(10 marks)
**4 (a)** Develop 8-point DIF-FFT radix-2 algorithm and draw the signal flow graph.(10 marks)
**4 (b)** Find 8-point DFT of a sequence x(n)= {1, 1, 1, 1, 0, 0, 0, 0} using DIT-FFT radix-2 algorithm. Use butterfly diagram.(10 marks)
**5 (a)** Given $ |H_a(j\Omega)|^2 = \dfrac {1}{(1+4\Omega^2)}, $ determine the analog filter system function H_{a}(s).(8 marks)
**5 (b)** Let $ H(s) = \dfrac {1}{(s^2 + \sqrt{2s}+1)} $ respect transfer function of a low pass filter with a pass band 1 rad/sec. Use frequency transformation to find the transfer function of the analog filters.

i) A LPF with pass band of 10 rad/sec.

ii) A HPF with cut-off frequency of 5 rad/sec.(8 marks)
**5 (c)** Compare Butterworth and Chebyshev filters.(4 marks)
**6 (a)** Realize the FIR filter $ H(z)= \dfrac {1}{2}+ \dfrac {1}{3}z^{-1}+ z^{-2}+ \dfrac {1}{4}z^{-3}+ z^{-4}+ \dfrac {1}{3}z^{-5} + \dfrac {1}{2}z^{-6} $ in direct form.(4 marks)
**6 (b)** Obtain direct form-I, direct form-II, cascade and parallel form realization for the following systems: y(n)=0.75 y(n-1)-0.125y(n-2)+6x(n)+7x(n-1)+x(n-2).(16 marks)
**7 (a)** A LPF is to be designed with frequency response. $$ H_d(e^{j\omega}) = H_d(\omega) = \left\{\begin{matrix}
e^{-j2\omega}& |\omega| < \frac {\pi}{4} \ \ \ \ \ \ \\0, & \frac {\pi}{4} <|\omega|<\pi
\end{matrix}\right. $$ Determine h_{d}(n) and h(n) if ω(n) is rectangular window, $$ \omega_R(n)= \left\{\begin{matrix}
1 &0\le n \le 4 \\0
& \text {Otherwise}
\end{matrix}\right. $$ Also, find the frequency response. H(ω) of the resulting FIR filter.(10 marks)
**7 (b)** Explain the design of linear phase FIR filter using frequency sampling technique.(10 marks)
**8 (a)** Explain the design of IIR filter by using Impulse Invariance Method (IIM) technique also explain mapping of analog to digital filter by IIM.(10 marks)
**8 (b)** Convert the analog filter with system function, $ H_a(s)= \dfrac {s+0.1}{(s+0.1)^2+16} $ into a digital IIR filter by means of bilinear transformation (BJT). The digital filter is to have a resonant frequency of $ \omega_r = \dfrac {\pi}{2} $.(10 marks)