VTU Signals & Systems - December 2012 Exam Question Paper

VTU Electronics and Communication Engineering (Semester 4)
Signals & Systems
December 2012

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) Determine whether the following systems are:
i) Memoryless ii) Stable iii) Causal iv) Linear and v) Time-invariant.
• y(n)=nx(n)
• y(t)=e^x(t)

10 M

1 (b) Distinguish between;
i) Determine and random signals and
ii) Energy and periodic signals

6 M

1 (c) For any arbitrary signal x(t) which is an even signal, how that

\int_{- \infty}^{\infty} x (t) d t = 2 \int_{0}^{+ \infty} x (t) d t

$\int^{\infty}_{-\infty}x(t)dt=2 \ \int^{+\infty}_{0}x(t)dt$

4 M

2 (a) Find the convolution integral of x(t) and h(t), and sketch the convolved signal, x(t)=(t-1){u(t-1)-u(t-3)} and h(t)=[(t+1)-2u(t-2)].

12 M

2 (b) Determine the discrete time convolution sum of the given sequences.
x(n)={1, 2 , 3, 4} and h(n)={1, 5 , 1}

8 M

3 (a) Determine the conditions of the impulse response of the system if system is
i) Memoryless
ii) Stable

6 M

3 (b) Find the total response of the system given by,

\frac{d^{2} y (t)}{d t^{2}} + 3 \frac{d}{d t} y (t) + 2 y (t) = 2 x (t) w i t h y (0) = - 1; \frac{\frac{d}{d t} y (t)}{t = 0} = 1 a n d x (t) = \cos (t) u (t)

$\dfrac {d^2y(t)}{dt^2}+3 \dfrac {d}{dt}y(t)+2y(t)=2x(t) \ with \ y(0)=-1; \ \dfrac {\frac {d}{dt}y(t)}{t=0}=1 \ and \ x(t)=\cos (t)u(t)$

14 M

4 (a) One period of the DTFS coefficients of a signal is given by, x(k)=(1/2)^k, on 0?K?9. Find the time domain signal x(n) assuming N=10

6 M

4 (b) Prove the following properties of DTFs; i) Convolution ii) Parseval relationship iii) Duality
iv) Symmetry

14 M

5 (a) Find the DTFT of the sequence x(n)=?ⁿ u(n) and determine magnitude and phase spectrum.

4 M

5 (b) Plot the magnitude and phase spectrum of x(t)=e^-j4 u(t).

8 M

5 (c) Find the inverse Fourier transform of the spectra,

x (j ω) = {\begin{matrix} 2 \cos (ω), & | ω | < π \\ 0, & | ω | > 0 \end{matrix}

$x(j\omega)= \left\{\begin{matrix}2\cos (\omega) ,&|\omega|<\pi \\0, &|\omega|>0 \end{matrix}\right.$

8 M

6 (a) Find the frequency response and impulse response of the system described by the differential equation.

\frac{d^{2}}{d t^{2}} y (t) + 5 \frac{d}{d t} y (t) + 6 y (t) = - \frac{d}{d t} x (t)

$\dfrac{d^2}{dt^2}y(t)+5 \dfrac {d}{dt} y(t) + 6y(t)= - \dfrac {d}{dt}x(t)$

8 M

6 (b) State sampling theorem. Explain sampling of continuous time signals with relevant expressions and features.

6 M

6 (c) Find the Nyquist rate for each of the following signals:
i) x₁(t)=sin e(200t)
ii) x₂ = sin e² (500t)

6 M

7 (a) Prove the complex conjugation and time advance properties.

6 M

7 (b) Find the Z-transform of the signal along with ROC.

x (n) = n s i n (\frac{π}{2} n) u (n)

$x(n)=n sin \left ( \dfrac {\pi} {2} n \right ) u(n)$

6 M

7 (c) Determine the inverse z-transform of the following x(z) by partial fraction expansion method,

x (z) = \frac{z + 2}{2 z^{2} - 7 z + 3} i f t h t R O C s a r e i) | z | > 3 i i) | z | < \frac{1}{2} a n d i i i) \frac{1}{2} < | z | < 3

$x(z)= \dfrac {z+2}{2z^2-7z+3} \\ if \ tht \ ROCs \ are \\ i) \ |z|>3 \\ ii) \ |z|<\dfrac {1}{2} \ and \\ iii) \ \dfrac {1}{2}<|z|<3$

8 M

8 (a) A system has impulse response

h (n) = {(\frac{1}{2})}^{n} u (n)

$h(n)= \left ( \dfrac {1}{2} \right )^n u(n)$ determine the input to the system if the output is given by,

y (n) = \frac{1}{3} u (n) + \frac{2}{3} {(- \frac{1}{2})}^{n} u (n)

$y(n)=\dfrac {1}{3}u(n)+ \dfrac {2}{3} \left ( - \dfrac {1}{2} \right )^n u(n)$

8 M

8 (b) Solve the following difference equation using unilateral z-transform.

y (n) - \frac{3}{2} y (n - 1) + \frac{1}{2} y (n - 2) = x (n), f o r n \geq 0, w i t h i n i t i a l c o n d i t i o n s y (- 1) = 4, y (- 2) = 10, a n d x (n) = {(\frac{1}{4})}^{n} u (n)

$y(n) - \dfrac {3}{2} y(n-1)+ \dfrac {1}{2}y(n-2)=x(n), \ for \ n\ge 0, \ with \ initial \ conditions \ y(-1)=4, y(-2)=10, \ and \ x(n)= \left ( \dfrac {1}{4} \right )^n u(n)$

12 M

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