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?

(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 ω

_{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 \[H\left(z\right)=\frac{b_0}{{\left(1-pz^{-1}\right)}^z}\] Determine the values of b

_{0}and P
5 M

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

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

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

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

(iii) Find out 4 point DFT of x

(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)

(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 - \[-\ \ [\frac{2\pi{}rk}{N}]\] 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 :

\[H\left(z\right)=\frac{1-z^{-1}}{1-0.2z^{-1}-0.15z^{-2}}\]

(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.

\[H\left(z\right)=\frac{1-z^{-1}}{1-0.2z^{-1}-0.15z^{-2}}\]

(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 M

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

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

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

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

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

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

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

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

Comment on the stability of the minimum and maximum phase system

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

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

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

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

Comment on the stability of the minimum and maximum phase system

10 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.

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 ω

_{y}= ω_{x}D.
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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