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Signals and Systems Question Paper - Dec 17 - Electrical And Electronics (Semester 5) - Visvesvaraya Technological University (VTU)
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Signals and Systems - Dec 17
Electrical And Electronics (Semester 5)
Total marks: 80
Total time: 3 Hours
INSTRUCTIONS
(1) Answer any FIVE full questions, choosing ONE full question from each module
(3) Draw neat diagrams wherever necessary.
Module- 1
1.a.
Explain the classification of signals.
(6 marks)
00
1.b.
Find the even and odd components of the signal x(t) = (l+t2)cos3(10t)
(4 marks)
00
1.c.
Sketch the signal $y(t)=[x(t)+x(2-t)] u(1-t)$, where x(t) is shown in the figure.
(6 marks)
00
OR
2.a.
Find the overall operator the system $y(n)=\frac{1}{3}[x(n+1)+x(n)+x(n-1)]$
(4 marks)
00
2.b.
Find the average power of square wave shown in figure.
(7 marks)
00
2.c.
Determine whether the system $y(t)=x\left(\frac{t}{2}\right)$ is
(5 marks)
00
i)Liner
ii) Time- invariant
iii) Memory
iv) Casual
v) Stable
Module-2
3.a.
A continuous time LTI system with unit impulse response $h(t)=u(t)$ and input $x(t)=e^{-a t} u(t)$ ; a > 0. Find the output of the system
(8 marks)
00
3.b.
Find the step response for the LTI system represented by the impulse response $h(n)=\left(\frac{3}{2}\right)^{n} b(i)$
(4 marks)
00
3.c.
Cosider a continuous time LTI system is represented by the impulse response h(t) = e-3tu(t-1). Determine whether it is
(4 marks)
00
i) Stable
ii)Casual.
OR
4.a.
Solve the differential equation:
(8 marks)
00
$\frac{d^{2} y(t)}{d t^{2}}+3 \frac{d y(t)}{d t}+2 y(t)=2 x(t)$ with $y(0)=-1 ;\left.\frac{d y(t)}{d t}\right|_{t=0}=1$ and $x(t)=\operatorname{cost} u(t)$
4.b.
Draw the direct form I and II implementation for the difference equation:
(8 marks)
00
$y(n)+\frac{1}{5} y(n-1)-y(n-3)=2 x(n-1)+7 x(n-2)$
Module-3
5.a.
Find the Fourier transform of $x(t)=\sum_{k=0}^{\infty} \alpha^{k} f(t-k T) ;|\alpha|\lt1$
(6 marks)
00
5.b.
Find the inverse Fourier transform of $k(j \omega)=\frac{j \omega}{(2+j \omega)^{2}}$
(4 marks)
00
5..
The impulse response of a continuous time LTI system is given by $h(t)=\frac{1}{R a} e^{-t / R C}u(t)$. Find the frequency response and draw its spectrum.
(6 marks)
00
OR
6.a.
Find the frequency response and impulse response of the system having $y(t)=e^{-2 t} u(t)+e^{-3} u(t),$ for the input $x(t)=e^{-1} u(t)$
(8 marks)
00
6.b.
Find the frequency response and impulse response of the system described by differential equation : $\frac{d^{2} y(t)}{d t^{2}}+3 \frac{d y(t)}{d t}+2 y(t)=4 \frac{d x(t)}{d t}+x(t)$
(8 marks)
00
Module-4
7.a.
State and prove Parseval's theorem in discrete time domain.
(6 marks)
00
7.b.
Find the DTFT of the signal $x(n)=a^{|n|} ;|a|\lt1$
(5 marks)
00
7.c.
Find the inverse DTFT of the signal $x\left(e^{j \Omega}\right)=\frac{3 \frac{3}{4} e^{-j \Omega}}{-\frac{1}{16} e^{-j 2 Q}+1}$
(5 marks)
00
OR
8.a.
Find the impulse response of the system having output $y(n)=\frac{1}{4}\left(\frac{1}{2}\right)^{n} u(n)+\left(\frac{1}{4}\right)^{n} u(n)$ for the input $x(n)=\left(\frac{1}{2}\right)^{n} u(n)$
(8 marks)
00
8.b.
obtain the difference equation for the system with frequency response:
(8 marks)
00
$\mathrm{H}\left(\mathrm{e}^{j \Omega}\right)=1+\frac{\mathrm{e}^{-\mathrm{j} \Omega}}{\left(1-\frac{1}{2} \mathrm{e}^{-\mathrm{j} \Omega}\right)\left(1+\frac{1}{4} \mathrm{e}^{-\mathrm{j} \Omega}\right)}$
Module-5
9.a.
Determine the z -transformation of $x(n)=-u(-n-1)+\left(\frac{1}{2}\right)^{n} u(n)$ . Find the RoC and poles -zeros locations of x(z)
(6 marks)
00
9.b.
Find the z - transform of $x(n)=n^{2}\left(\frac{1}{2}\right)^{n} u(n-3)$ using appropriate properties.
(4 marks)
00
9.c.
Find the inverse z- transform of x(z) using partial fraction method.
(6 marks)
00
$x(z)=\frac{1+2 z^{-1}+z^{-2}}{1-\frac{3}{2} x^{-1}+\frac{1}{2} z^{-2}} ; |z|\gt1$ as RoC
OR
10.a.
A system has impulse response $h(n)=\left(\frac{1}{2}\right)^{n} u(n)$ . Determine the input to the system if the output is given by , $y(n)=\left(\frac{1}{3}\right)^{n} u(n)+\frac{2}{3}\left(-\frac{1}{2}\right)^{n} u(n)$
(8 marks)
00
10.b.
Solve the following difference equation using z- transform,
(8 marks)
00
$y(n)-\frac{3}{2} y(n-1)+\frac{1}{2} y(n-2)=x(n)$ for $\mathrm{n} \geq 0$ , with y(-1)=4, y(-2)=10 and $x(n)=\left(\frac{1}{4}\right)^{a} u(n)$
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