## Digital Signal Processing - Jun 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)** Define DFT. Derive the relationship of DFT to the z-transform.(5 marks)
**1 (b)** An analog signal is sampled at 10 KHz and the DFT of 512 samples is computed. Determine the frequency spacing between the spectral samples of DFT.(3 marks)
**1 (c)** Consider the finite length sequence x(n)=?(n)-2?(n-5): Find (i) The 10 point DFT of x(n) (ii) The sequence y(n) that has a DFT $$ y(K) = e ^{\frac {J4\pi}{10}K} $$ X(K) where X(K) is the 10 point DFT of x(n) and W(K) is the 10 point DFT of u(n)-u(n-6).(12 marks)
**2 (a)** Determine the circular convolution of the sequence x(n)={1,2,3,1} and h(n)={4,3,2,2} using DFT and IDFT equations.(8 marks)
**2 (b)** Let X(K) be a 14 point DFT of a length 14 real sequence x(n). The first 8 samples of X(K) are given by: X(0)=12, X(1)=1+J3, X(2)=3+j4, x(3)=1-J5, X(4)=-2+J2, X(5)=6+J3, X(6)=2-J3, X(7)=10. Determine the remaining samples of X(K). Also evaluate the following functions without computing the IDFT. $$ i) \ x(0) \ \ \ ii) \ x(7) \ \ \ (iii) \ \sum^{13}_{n=0} x(n) \ \ \ (iv) \ \sum^{15}_{n=0}\big\vert x (n) \big\vert^2 $$(12 marks)
**3 (a)** Consider a FIR filter with impulse response. h(n)={3,2,1,1}. If the input is x(n)={1,2,3,3,2,1,-1,-2,-3,5,6,1,2,1}. Using the overlap save method and 8 point circular convolution.(10 marks)
**3 (b)** What are FFT algorithms? Prove the (i) Symmetry and (ii) Periodicity property of the twiddle factor W_{n}.(6 marks)
**3 (c)** How many complex multiplications and additions are required for computing 256 point DFT using FFT algorithms?(4 marks)
**4 (a)** Find the DFT of the sequence x(n)={1,2,3,4,4,3,2,1} using the decimation in frequency FFT algorithm and draw the signal flow graph. Show the outputs for each stage.(10 marks)
**4 (b)** Given x(n)={1,0,1,0}, find x(2) using the Geortzel algorithm.(5 marks)
**4 (c)** Write a note on Chirp z-transform algorithm.(5 marks)
**5 (a)** Given that $$ \bigg \vert H(e^{7 \Omega})\bigg \vert ^{2} = \dfrac {1}{1+64 \Omega^6} $$ determine the analog Butterworth low pass filter transfer function.(6 marks)
**5 (b)** Design an analog Chebyshev filter with a maximum passband attenuation of 2.5 dB at Ω_{p}=20 rad/sec and the stopband attenuation of 30 dB at Ω_{s}=50 rad/sec.(10 marks)
**5 (c)** Compare Butterworth and Chebyshev filters.(4 marks)
**6 (a)** What are the conditions to be satisfied while transforming an analog filter to a digital HR filter? Explain how this is achieved in Bilinear transformation technique.(5 marks)
**6 (b)** Design a Butterworth filter using the impulse invariance method for the following specifications: Take T=1 sec, $$\begin {align*}0.8 \le & \bigg \vert H(e^{jW}) \bigg \vert \le 1 & 0 \le W \le 0.2 \pi \\ & \bigg \vert H (jW) \bigg \vert \le 0.2 & 0.6 \pi \le W \le \pi \end{align*} $$(10 marks)
**6 (c)** Determine H(z) for the given analog system function $$ H(s) = \dfrac {(s+a)}{(s+a)^2 +b^2} $$ by using Matched z-transform.(5 marks)
**7 (a)**

A z-plane pole zero plot for a certain digital filter shown in Fig. Q7 (a). Determine the system function in the $$ H(z)= \dfrac {(1+a_1 z^{-1})(1+b_1 z^{-1} + b_2 z^{-2})}{(1+c_1 z^{-1})(1+d_1 z^{-1}+d_2 z^{-2})} $$ giving the numerical values for parameters a_{1}, b_{1}, b_{2}, c_{1}, d_{1} and d_{2}. Sketch the direct form II and Cascade realizations of the system.

(10 marks)
**7 (b)** A FIR filter is given by, $$ y(n)= x(n) + \dfrac {2}{5}x(n-1) + \dfrac {3}{4} x (n-2) - \dfrac {1}{3}x (n-3) $$ Draw the direct form I and lattice structure.(10 marks)
**8 (a)** Design a FIR filter (low pass) with a desired frequency response, $$ \begin {align*}
H_d (e^{jW}) &= e^{-j3w}; &-\dfrac {3 \pi}{4} \le \omega \le \dfrac {3 \pi}{4} \\ &=0; &\dfrac {3 \pi}{4} < \vert \omega \vert < \pi \end{align*} $$ Use Hamming window with M=7. Also obtain the frequency response.(10 marks)
**8 (b)** Design a linear phase low pass FIR filter with 7 taps and cut off frequency of Ω_{c}=0.3Π rad. Using the frequency sampling method.(10 marks)