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Digital Signal Processing : Question Paper Dec 2015 - Electronics & Telecomm (Semester 5) | Visveswaraya Technological University (VTU)

## 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,
iii) Real multiplications
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 Ha(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.
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 hd(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)