Digital Signal Processing - Jun 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) Compute the DFT of the sequence $$x(n)=\cos\left ( \dfrac{n\pi}{4} \right )$$ for N=4, plot |x(k)| and ∠x(k).(9 marks) 1 (b) Find the DFT of the sequence x(n)=0.5ⁿ u(n) for 0 < n <3 by evaluating x(n)=aⁿ for 0< n < N - 1.(7 marks) 1 (c) Find the relation between DFT and Z transform.(4 marks) 2 (a) State and prove the linearity property of DFT and symmetrical property.(5 marks) 2 (b) The five samples of the 8 point DFT x(k) are given as
x(0)=0.25, x(1)=1.25 - j0.3018, x(6)=x(4)=0, x(5)=0.125 - j0.0518.(5 marks) 2 (c) For x(n)={1,-2,3,-4,5-6}, without computing its DFT, find the following.
$$i)\ x(o)\ ii)\ \sum_{k=0}^{5}\ iii)\ X(3)\ iv)\ \sum_{k=0}^{5}1 \times(k)|^{2}\ v)\ \sum_{k=0}^{5}(-1)^{k}\times (k)$$(10 marks) 3 (a) Consider a FIR filter with impulse response
h(n)={1,1,1}, if the input is
X(n)={1,2,0,-3,4,2,-1,1,-2,3,2,1,-3}. Find the output y(a) using overlap add method.(12 marks) 3 (b) What is an plane computation? What is total number of complex additions and multiplication required for N=256 point, if DFT is computed directly and if FFT is used?(3 marks) 3 (c) For sequence x(n)={2,0,2,0} determine x(2) using Goertzel Filter. Assume the zero initial conditions.(5 marks) 4 (a) Find the circular convolution of x(n)={1,1,1,1} and h(n)={1,0,1,0} using DIF-FFT algorithm.(12 marks) 4 (b) Derive DIT-FFT algorithm for N=4. Draw the complete signal How graph?(8 marks) 5 (a) Design a Chebyshev analog filter (low pass) that has a-3dB cut-off frequency of 100 rad/sce and a stopband attenuation 25dB or greater for all radian frequencies past 250 rad/sec.(14 marks) 5 (b) Compare Butterworth and Chebyshev filters.(3 marks) 5 (c) Let $$H(s)=\dfrac{1}{s^{2}+s+1}$$ represent the transfer function of LPF with a passband of 1 rad/sec. Use frequency transformation (Analog to Analog) to find the transfer function of a band pass filter with passband 10 rad/sec and a centre frequency of 100 rad/sec.(3 marks) 6 (a) Obtain block diagram of the direct form I and direct from II realization for a digital IIR filter described by the system function.
$$H(z)=\dfrac{8z^{3}-4z^{2}+11z-2}{\left ( z-\dfrac{1}{4} \right )\left ( z^{2}-z+\dfrac{1}{2} \right )}$$(10 marks) 6 (b) Find the transfer function and difference equation realization shown in fig Q6(b)

(6 marks) 6 (c) Obtain the direct form realization of liner phase FIR system given by
$$H(z)=1+\dfrac{2}{3}z^{-1}+\dfrac{15}{8}z^{-2}$$(4 marks) 7 (a) "The desired frequency response of low pass filter is given by
$$H_{d}(e^{jw})=H_{d}(\infty )=\left\{\begin{matrix} e^{-j3w} &|\infty|\dfrac{3 \pi}{4} \\0 &\dfrac{3\pi}{4}<|\infty|<\pi \end{matrix}\right$$
Determine the frequency response of the FIR if Hamming window is used with N=7."(10 marks) 7 (b) Compare IIR filter and FIR filters.(6 marks) 7 (c) Consider the pole-zero plot as shown in FigQ7(c). i) Does it represent an FIR filter? ii) Is it linear phase system?

(4 marks) 8 (a) Design a digital filter H(z) that when used in an A/D-H(z)-D/A structure gives an equivalent analog filter with the following specification:
Passband ripple:≤3.01dB
Passband edge : 500Hz
Stopband attenuation : ≥ 15dB
Stopband edge : 750 Hz
Sample Rate : 2 KHz
Use Bilinear transformation to design the filter on an analog system function. Use Butterworth filter prototype. Also obtain the difference equation.(14 marks) 8 (b) Transform the analog filter
$$H_{a}(s)=\dfrac{s+1}{s^{2}+5s+6}$$
Into H(z) using impulse invariant transformation Take T=0.1 Sec.(6 marks)