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DIGITAL SIGNAL PROCESSING (ETIT-308)
Question 1.
(a)
Let
x n
2d n 2 d n 1
3d n d n 1
2d n 2 .
Evaluate
(i)
X e jw w 0
(ii)
p
X e jw d w (iii) X e jw w 2p without explicitly finding X e jw .
6
p
(b)
Find the state variable matrix A,B,C,D for the input output relation given by the equation :
y n 2 y n 1 3 y n 2 x n 2 x n 1 3x n 2 .
5
(c)
Derive the relationship between DFT (i) Z transform (ii) Fourier Transform of a periodic
sequence.
4
(d)
Discuss the design of digital resonator.
(e)
Derive the input and output relation for a system function h(t) if the input signal x n is
random.
5
5
Solution.
UNIT-1
Question 2.
(a)
Perform the convolution of the following two sequence graphically
h n
1/ 2
n
0 n 2
0
(b)
,x n
d n
d n 1
4d n 2 .
Find the 'Z' transform the following sequence:
x n
6.5
otherwise
6
n
n 1/ 2 u n 2
Question 3.
(a)
Find the inverse 'Z' transform.
(i) X z
1 1/ 4Z
1 1/ 2Z
(ii) X z
(b)
6
1
1 2
,Z
2
1 1/ 4Z 1
1 5 / 6Z 1 1/ 6 Z
2
,Z
1/ 2
Compute the DTFT of the following :
(i) x n
(iii) y n
u n
(ii) x n
3/ 4 y n 1 1/ 8 y n 2
6.5
Cosw 0 n with w 0 2p / 5.
x n.
UNIT-II
Question 4.
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IPU/DSP/1
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(a)
Perform the circular convolution and linear convolution for the input sequence
x1 n
1, 2,3,1, 2 , x2 1, 2,3, 4 . Discuss the result.
(b)
Determine the DFT of the given data sequence x n
algorithm.
1, 2, 3, 4,9, 20,12,6
using DIT
Question 5.
(a)
Prove the following properties of DFT when X k is the N point DFT:
(i) If x n is real and odd.
(ii) If x n is purely imaginary and odd.
(b)
Calculate the IDFT for the given coefficient
X k
{38, 5.828
j 6.07, j 6, 0.172
j8.07, j 6, 5.828
j8.07, 10, 0.172
j 6.07}
using DIF structure.
6.5
UNIT-III
Question 6.
(a)
For the following causal linear
H Z
H Z
in form
H min Z H ap Z .
(i) H z
(ii) H Z
(b)
shift invariant system factorize
1 3Z
2
1 0.5Z
1 0.75Z
5 Z
1
1
7 5Z
1 1/1Z
1
1
1
.
Consider the random process X t
variables i.e. E Y , Z
EY E Z
ySin n0t Z cos n0t Y and Z are uncorrelated random
0, with zero mean and variance s 2 . (i) Find mean of X t
(ii) Auto correlation of X t .
6.5
Question 7.
(a)
Explain the sampling theorem in the frequency domain. Discuss the design of zero order hold
circuit.
8.5
(b)
Consider a linear shift invariant system with system function H Z
Z 1 a*
|a | 1.
1 aZ 1
(i) Find a linear difference equation to implement this system.
(ii) Show that this system is an all pass system.
Solution.
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UNIT-IV
Question 8.
Z2 1
realize H Z as (i)
Z 1/ 2 Z 1/ 3 Z 1/ 4
parallel form (ii) cascade form (iii) cononic form.
(a)
Give the system transfer function H Z
(b)
Design the digital butter worth filter satisfying the constraint given below T=1 using bilinear
transformation 1/ 2 | H e jw | 1 for 0 w p / 2
| H e jw | 0.2 for 3p / 4 w p
Question 9.
(a)
Discuss the properties of Chebyshev polynomial.
(b)
Design a high pass filter where the desired frequency response is given below with t 3 and
wC 2
rad/S
using
a
window
with
stop
band
attenuation
53dB, H d e jw
e
jw z
w C |w | p
0 otherwise
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