MU Discrete Time Signal Processing - May 2013 Exam Question Paper

MU Electronics and Telecom Engineering (Semester 6)
Discrete Time Signal Processing
May 2013

Total marks: --
Total time: --

INSTRUCTIONS
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary

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 M

1 (b) Let h[n] be the unit impulse response of a Low Pass filter with a cutoff frequency ?_c, what type of filter has a unit sample response g[n]= (-1)ⁿ h[n].

5 M

1 (c) A two pole low pass filter has the system function

H (z) = \frac{b_{0}}{{(1 - p z^{- 1})}^{z}}

$H\left(z\right)=\frac{b_0}{{\left(1-pz^{-1}\right)}^z}$ Determine the values of b₀ and P

5 M

1 (d) Consider filter with transfer function. Identify the type of filter and justify it.

H (z) = \frac{z^{- 1} - a}{1 - a z^{- 1}}

$H\left(z\right)=\frac{z^{-1}-a}{1-az^{-1}}$

5 M

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 M

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 x_p(n) and x_p(n-3)
(iii) Find out 4 point DFT of x_p(n-3)
(iv) Add phase angel in (i) with factor -

- [\frac{2 π r k}{N}]

$-\ \ [\frac{2\pi{}rk}{N}]$ where N=4, r=3, k=0,1,2,3
(v) Comment on the result you had in point (i) and (ii)

10 M

3 (a) The transfer function of discrete time causal system is given below :

H (z) = \frac{1 - z^{- 1}}{1 - 0.2 z^{- 1} - 0.15 z^{- 2}}

$H\left(z\right)=\frac{1-z^{-1}}{1-0.2z^{-1}-0.15z^{-2}}$
(i) Find the difference equation
(ii) Draw cascade and parallel realization
(iii) Show pole-zero diagram and then find magnitude at ? = 0 and ?= ?
(iv) Calculate the impulse response of the system.

10 M

3 (b) Obtain the lattice realization for the system :

H (z) = \frac{1 + 3 z^{- 1} + 3 z^{- 2} + z^{- 3}}{1 + \frac{3}{4} z^{- 1} + \frac{1}{2} z^{- 2} + \frac{1}{4} z^{- 3}}

$H\left(z\right)=\frac{1+3z^{-1}+3z^{-2}+z^{-3}}{1+\frac{3}{4}z^{-1}+\frac{1}{2}z^{-2}+\frac{1}{4}z^{-3}}$

10 M

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 M

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} (z) = 6 + z^{- 1} - z^{- 2}

$H_1\left(z\right)=6+z^{-1}-z^{-2}$

H_{2} (z) = 1 - z^{- 1} - 6 z^{- 2}

$H_2\left(z\right)=1-z^{-1}-6z^{-2}$

H_{1} (z) = 1 - \frac{5}{2} z^{- 1} - \frac{3}{2} z^{- 2}

$H_1\left(z\right)=1-\frac{5}{2}z^{-1}-\frac{3}{2}z^{-2}$

H_{1} (z) = 1 + \frac{5}{3} z^{- 1} - \frac{2}{3} 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 M

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 M

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 M

6 (a) Explain the need of a low pass filter with a decimator and mathematically prove that ?_y = ?_xD.

10 M

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 M

Write short notes (any four ) :

7 (a) DTMF detection using Geortzel algorithm

5 M

7 (b) Filter bank

5 M

7 (c) Comparison of FIR and IIR filters

5 M

7 (d) Split radix FFT

5 M

7 (e) Optimum Equiripple Linear phase FIR filter design

5 M

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