Electronics And Communication Engineering (Semester 5)
Total marks: 80
Total time: 3 Hours
Note: Answer any FIVE full questions, choosing ONE full question from each module.
1.a.
Define DFT and IDFT of a signal obtain the relationship between of DFT and z - transform
(6 marks)
00
1.b.
Compute circular convolution using DFT and IDFT for the following sequences,
$x_{1}(n)=\{2,3,1,1\}$ and $x_{2}(n)=\{1,3,5,3\}$
(10 marks)
00
OR
2.a.
The first five samples of the 8-point DFT $x(k)$ are given as follows:
$x(0) = 0.25, x(1) = 0.125 - j0.3018, x(4) = x(6) = 0, x(5) = 0.125 - j0.0518.$ Determine the remaining samples, if the $x(n)$ is real valued sequence.
(4 marks)
00
2.b.
State and prove the circular time shift and circular frequency shift properties.
(6 marks)
00
2.c.
If $x(n)=\{1,2,0,3,-2,4,7,5\}$, evaluate the folowing
i. $x(0)$
ii. $x(4)$
iii. $\sum_{n=0}^{7} x(k)$
(6 marks)
00
Module-2
3.a.
State and prove the following properties of phase factor $\omega_{N}$
i. periodicity
ii. symmetry.
(4 marks)
00
3.b.
Find the output of a filter whose impulse suppose $h(n) = \{1, 2, 3, 4\}$ and input signal to the filter is $x(n) = \{1, 2, 1, -1, 3, 0, 5, 6, 2, -2, -5, -6, 7, 1, 2, 0, 1\}$ using overlap - add method with 6-point circular convolution.
(12 marks)
00
OR
4.a.
In the direct computation of N-point DFT of $x(n)$, how many:
i. Complex addition
ii. Complex multiplications
iii. Real multiplication
iv. Real additions
v. Trigonometric functions
Evaluations are required?
(6 marks)
00
4.b.
Explain the linear filtering of long data sequences using overlap - save method.
(10 marks)
00
Module-3
5.a.
Given $x(n) = \{1, 0, 1, 0\}$, find $x(2)$ using Goertzel algorithm.
(6 marks)
00
5.b.
Find the 8-point DFT of the sequence $x(n) = \{1, 2, 3, 4, 4, 3, 2, 1\}$ using DIT - FFT radix - 2 algorithm.
(10 marks)
00
OR
6.a.
What is chirp-z transform? Mention its applications?
(6 marks)
00
6.b.
Find the 4-point circular convolution of $x(n)$ and $h(n)$ give below, using radix-2. DIF-FFT algorithm.
$x(n) = \{1, 1, 1, 1\}$
$h(x) = \{1, 0, 1, 0\}$
(10 marks)
00
Module-4
7.a.
Derive an expression for the order, cut of frequency and poles of the low pass Butterworth filter.
(8 marks)
00
7.b.
A Butterworth low pass filter has to meet the following s specifications.
i. Pass band gain, $k_{p}=-1 d B$ at $\Omega_{p}=4$ rad/sec
ii. Step band alterations greater than or equal to 20 $\mathrm{dB}$ at $\Omega \mathrm{s}=8 \mathrm{rad} / \mathrm{sec}$
Determine the transfer function $\mathrm{H}_{\mathrm{a}}(\mathrm{s})$ of the Butterworth filter to meet the above specfications.
(8 marks)
00
8.a.
A third - order Butterworth low pass filter has the transfer function:
$H(s)=\frac{1}{(s+1)\left(s^{2}+s+1\right)}$
Design H(z) using impulse invariant technique.
(10 marks)
00
8.b.
List the advantages and disadvantages of IIR filters
(6 marks)
00
Module-5
9.a.
A linear time - invariant digital IIR filter is specified by the following transfer function:
$H(z) = \frac{(z-1)(z-2)(z+1) z}{[z-(1/2+1/2j)] [z-(1 / 2-j 1/ 2)][z-j 1/ 4][z+j 1 / 4]}$
Realize the system in the following forms:
i. direct form - I
ii. Direct form - II
(12 marks)
00
9.b.
Obtain a cascade realization for the system function given below:
$H(z)=\frac{\left(1+z^{-1}\right)^{3}}{\left(1-1 / 4 z^{-1}\right)\left(1-z^{-1}+1 / 2 z^{-2}\right)}$
(4 marks)
00
OR
10.a.
Explain the following terms:
i. Rectangular window
ii. Bartlett window
iii. Hamming window
(8 marks)
00
10.b.
A filter is to be designed with the following desired frequency response:
$\mathrm{H}_{\mathrm{d}}(\omega)=\left\{\begin{array}{ll}{0,} & {-\pi / 4\lt\omega\lt\pi / 4} \\ {\mathrm{e}^{-\mathrm{j} 2 \omega},} & {\pi / 4\lt|\omega|\lt\pi}\end{array}\right.$
Find the frequency response of the FIR filter designed using rectangular window defined below:
$\omega_{R}(n)=\left\{\begin{array}{ll}{1,} & {0 \leq n \leq 4} \\ {0,} & {\text { otherwise }}\end{array}\right.$
(8 marks)
00
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