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Digital Signal Processing & Processors : Question Paper May 2013 - Electronics Engineering (Semester 6) | Mumbai University (MU)

## Digital Signal Processing & Processors - May 2013

### Electronics Engineering (Semester 6)

TOTAL MARKS: 80
TOTAL TIME: 3 HOURS
(1) Question 1 is compulsory.
(2) Attempt any three from the remaining questions.
(3) Assume data if required.
(4) Figures to the right indicate full marks.
1 (a) Assume that a complex multiplier takes 1 micro sec to perform one multiplication and that the amount of time to compute a DFT is determined by the amount of time to perform all the multiplications.
(i) How much time does it take to compute a 1024 point DFT directly?
(ii) How much time is required is FFT is used?
(5 marks)
1 (b) Let h[n] be the unit impulse response of a Low Pass filter with a cutoff frequency ωc, what type of filter has a unit sample response g[n]= (-1)n h[n]. (5 marks) 1 (c) A two pole low pass filter has the system function $$H\left(z\right)=\frac{b_0}{{\left(1-pz^{-1}\right)}^z}$$ Determine the values of b0 and P(5 marks) 1 (d) Consider filter with transfer function. Identify the type of filter and justify it.
$$H\left(z\right)=\frac{z^{-1}-a}{1-az^{-1}}$$
(5 marks)
2 (a) The unit sample response of a system is h(n)={3,2,1} use overlap-add method of linear filtering to determine output sequence for the repeating input sequence x[n]= {2,0,-2,0,2,1,0,-2,-1,0}(10 marks) 2 (b) For a given sequence x(n)= {2,0,0,1}, perform following operation :
(i) Find out the 4 point DFT of x(n)
(ii) Plot x(n), its periodic extension xp(n) and xp(n-3)
(iii) Find out 4 point DFT of xp(n-3)
(iv) Add phase angel in (i) with factor - $$-\ \ [\frac{2\pi{}rk}{N}]$$ where N=4, r=3, k=0,1,2,3
(v) Comment on the result you had in point (i) and (ii)
(10 marks)
3 (a) The transfer function of discrete time causal system is given below :
$$H\left(z\right)=\frac{1-z^{-1}}{1-0.2z^{-1}-0.15z^{-2}}$$
(i) Find the difference equation
(ii) Draw cascade and parallel realization
(iii) Show pole-zero diagram and then find magnitude at ω = 0 and ω= π
(iv) Calculate the impulse response of the system.
(10 marks)
3 (b) Obtain the lattice realization for the system :
$$H\left(z\right)=\frac{1+3z^{-1}+3z^{-2}+z^{-3}}{1+\frac{3}{4}z^{-1}+\frac{1}{2}z^{-2}+\frac{1}{4}z^{-3}}$$
(10 marks)
4 (a) What is a linear phase filter? What conditions are to be satisfied by the impulse response of an FIR system in order to have a linear phase? Plot and justify compulsory zero locations for symmetric even antisymmetric even and antisymmetric odd FIR filters. (10 marks) 4 (b) Determine the zero of be following FIR system and indicate whether the system is minimum phase, maximum phase, or mixed phase
$$H_1\left(z\right)=6+z^{-1}-z^{-2}$$
$$H_2\left(z\right)=1-z^{-1}-6z^{-2}$$
$$H_1\left(z\right)=1-\frac{5}{2}z^{-1}-\frac{3}{2}z^{-2}$$
$$H_1\left(z\right)=1+\frac{5}{3}z^{-1}-\frac{2}{3}z^{-2}$$
Comment on the stability of the minimum and maximum phase system
(10 marks)
5 (a) A digital low pass filter is required to meet the following specification:
Pass band ripple : ≤1dB
Pass band edge : 4KHz
Stop band attenuation : ≥ 40dB
Stop band edge : 8KHz
Sampling rate : 24KHz
Find order, cutoff frequency and pole locations for Butterworth filter using bilinear transformation.
(10 marks)
5 (b) Design an FIR digital filter to approximate an ideal low pass filter with passband gain of unity, cut-off frequency of 950Hz and working at a sampling frequency of Fs=5000 Hz. The length of the impulse response should be 5. Use a rectangular window. (10 marks) 6 (a) Explain the need of a low pass filter with a decimator and mathematically prove that ωy = ωxD.(10 marks) 6 (b) Why is the direct form FIR structure for a multirate system inefficient? Explain with neat diagrams how this inefficiency is overdone in implementing a decimator and an interpolator.(10 marks)

### Write short notes (any four ) :

7 (a) DTMF detection using Geortzel algorithm (5 marks) 7 (b) Filter bank (5 marks) 7 (c) Comparison of FIR and IIR filters (5 marks) 7 (d) Split radix FFT (5 marks) 7 (e) Optimum Equiripple Linear phase FIR filter design (5 marks)

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