## Digital Signal Processing - Dec 2014

### 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)** State and prove the relationship between z-transform and DFT.(6 marks)
**1(b)** Determine N point DFT of $$x(n)=cos\frac{2\pi K_{0}n}{N},0\leq K\leq N-1$$(6 marks)
**1(c)** Find the IDFT of x(k) = {255, 48.63 + j166.05,-51+j102, -78.63+j46.05, -85, -78.63-j46.05,-51-j102, 48.63-166j}.(8 marks)
**2(a)** State and prove the relationship between z-t and prove the following properties :

i) Symmetry property

ii) Parseval's theorem(8 marks)
**2(b)** Prove :i) Symmetry and ii) Periodicity property of a twiddle factor.(8 marks)
**2(c)** Find the output y(n) of a filter whose impulse response is h (n) -{1,2,3,4} and the input signal to the filter is x(n) - {1,2,1,-1,3,0,5,6,2,-2,-5,6,7,1,2,0,1} using overlap add method [Use 6 point circular convolution.](8 marks)
**3(a)** Determine y(n)$$x_{1}\circledast x_{2}(n),-n+1,0\leq n\leq 5\ and \ x_{2}(n)=cos\pi0\ n\leqn\leq 5 $$ Using stockhalm's method.(10 marks)
**3(b)** Develop DIT FFT algorithm and write signal flow graph for N=8.(8 marks)
**3(c)** Explain in-place computation of FET.(2 marks)
**4(a)** Explain bit reversal property used in FFT algorithm for N = 16(3 marks)
**4(b)** Develop DIT - FFT algorithm for N = 9.(7 marks)
**4(c)** Find IDFT of x(k) -{36,-4+j9.7,-4+j4,-4+j1.7,-4,-4-j1.7,-4-j4,-4-j9.7}. Using DIF FFT algorithm.Show clearly all the intermediate results.(10 marks)
**5(a)** Design a Chebyshev filter to meet the following specifications :

i) Pass band ripple $$\leq $$ db

ii) Stop band attenuation $$\geq $$ 20 db

iii) Pass band edge : 1 rad/sec

iv) stop band edge : 1;3 rad/sec(10 marks)
**5(b)** Distinguish between IIR and FIR filters.(4 marks)
**5(c)** Derive an expression for order of a a low pass Butterworth filter.(6 marks)
**6(a)** Realize FIR linear phase filter for N to be even.(8 marks)
**6(b)** Evaluate the impulse response for input x(n) $$^{-}\delta (n)$$ of three stage lattice structure having coefficients $$K_{1}=0.65,K_{2}=-0.34\ and \ K_{3}- 0.8.$$ Also draw its direct form - I structure.(12 marks)
**7(a)** Explain how an analog filter is mapped on to digital filter using impulse invariance method. What are the limitations of the method?(10 marks)
**7(b)** Obtain direct form - I and lattices structure for the system described by the difference equation[y(n)=x(n)+frac{2}{5} imes(n-2)+frac{1}{3}x(n-3).](10 marks)
**8(a)** for the desired frequency response

$$H(\omega )=\left\{\begin{matrix}
e^{j3\omega } -\frac{3\pi }{4}& < \omega < \frac{3\pi }{4}\\0,&\frac{3\pi }{4} < |\omega |< \pi \\
&
\end{matrix}\right.$$

Find H(\omega$$ for N = 7 using Hanning window.\lt/span\gt\ltspan class='paper-ques-marks'\gt(10 marks)\lt/span\gt
\lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt8(b)\lt/b\gt Show that for $$\beta |]- 0,Kaiser window becomes a rectangular window.(5 marks)
**8(c)** Mention few advantages and disadvantages of windowing technique.(5 marks)