VTU Signals & Systems - June 2012 Exam Question Paper

VTU Electronics and Communication Engineering (Semester 4)
Signals & Systems
June 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) Give a brief classification of signals.

4 M

1 (b) Check whether the following systems are linear, causal and time invariant not

i) \frac{d^{2} y (t)}{d t^{2}} + 2 y (t) \frac{d y (t)}{d t} + 3 t y (t) = x (t) i i) y (n) = x^{2} (n) + \frac{1}{x^{2} (n - 1)}

$i) \ \dfrac {d^2 y(t)}{dt^2}+ 2y(t) \dfrac {dy (t)}{dt} +3ty(t)=x(t) \\ ii) \ y(n)=x^2(n)+ \dfrac{1}{x^2(n-1)}$

8 M

1 (c) Classify the following signals or energy signals or power signals:
i) x(n)=2ⁿ u(-n)
ii) x(n)=(j)ⁿ+(j)ⁿ

5 M

1 (d) A system consists of several sub-systems connected as shown in Fig. Q1 (d). Find the operator H relating x(t) to y(t) for the following sub system operators:
H₁:y₁(t)=x₁(t) x₁(t-1)
H₁: y(t)=1+2x₄(t)
H₂: y₂ (t)=|x₂ (1)|
H₄: y₄(t)=cos (x₄(t))

3 M

2 (a) Find the continuous-time convolution integral given below:
Y(t)=cos (?t)|u(t+1)-u(t-3)|*u(t).

6 M

2 (b) Consider the i/p signal x(n) and impulse response (n) given below:

x (n) = {\begin{matrix} 1, & 0 \leq n \leq 4 \\ 0, & o t h e r w i s e \end{matrix}, h (n) {\begin{matrix} α^{t} & 0 \leq n \leq 6. | α | < 1 \\ 0, & o t h e r w i s e \end{matrix}

$x(n)= \left\{\begin{matrix} 1, &0\le n\le 4 \\0, & otherwise \end{matrix}\right., \\ h(n)\left\{\begin{matrix}\alpha^t &0\le n \le 6. |\alpha|<1 \\0, &otherwise \end{matrix}\right.$ Obtain the convolution sum y(n)=x(n)⁸h(n)

8 M

2(c) Derive the following properties:
i) x(n)×h(n)=h(n)×x(n)
ii) x(n)×|h(n)×g(n)|=[x(n)×h(n)]×g(n).

6 M

3 (a) For each impulse response listed below. Determine whether the corresponding system is memoryless causal and stable;
i) h(n)=(0.99)ⁿ u(n+3)
ii) h(t)=e^-31 u(t-1).

8 M

3 (b) Evaluate the step response for the LTI system represented by the following impulse response: h(t)=u(t+1)-u(t-1).

4 M

3 (c) Draw direct form I implementation of the corresponding systems:

\frac{d^{2} y (t)}{d t^{2}} + 5 \frac{d}{d t} y (t) + 4 y (t) = x (t) + 3 \frac{d}{d t} x (t)

$\dfrac {d^2 y(t)}{dt^2}+ 5 \dfrac{d}{dt} y(t) + 4y(t)= x(t) + 3 \dfrac {d}{dt} x(t)$

4 M

3 (d) Determine the forced response for the system given by:

5 \frac{d y (t)}{d t} + 10 y (t) = 2 x (t)

$5\dfrac {dy(t)}{dt} + 10y(t)=2x(t)$ with input x(t)=2u(t).

4 M

4 (a) State and prove time shift and periodic time convolution properties of DTFS.

6 M

4 (b) Evaluate th e DTFS representation for the signal x(n) shown in Fig. Q4(b) and sketch the spectra.

8 M

4 (c) Determine the time signal corresponding to the magnitude and phase spectra shown in Fig. Q4(c), with W_o=?.

6 M

5 (a) State and prove the frequency differentiation property of DTFT.

6 M

5 (b) Find the time domain signal corresponding to the DTFT shown in Fig. Q5(b).

5 M

5 (c) For the signal x(t) shown in Fig. Q5(c), evaluate the following quantities without explicitly computing x(w).

i) \int_{- \infty}^{\infty} x (ω) d ω i i) \int_{- \infty}^{\infty} | x (ω) |^{2} d ω i i i) \int_{- \infty}^{\infty} x (ω) e^{j 2 ω} d ω

$\displaystyle i) \ \int^{\infty}_{-\infty}x(\omega)d\omega \\ ii) \ \int^{\infty}_{- \infty}|x(\omega)|^2 d\omega \\ iii) \ \int^{\infty}_{-\infty}x(\omega)e^{j2\omega}d\omega$

9 M

6 (a) The input and output of causal LTI system are described by the differential equation

\frac{d^{2} y (t)}{d t^{2}} - 3 \frac{d y (t)}{d t} - 2 y (t) = x (t)

$\dfrac {d^2 y(t)}{dt^2}-3 \dfrac {dy(t)}{dt}-2 y(t)=x(t)$
i) Find the frequency response of the system
ii) Find impulse response of the system
iii) What is the response of the system if x(t)=tc^-1 u(t).

10 M

6 (b) Find the frequency response of the RC circuit shown in Fig. Q6(b). Also find the impulse response of the circuit.

10 M

7 (a) Briefly list the properties of Z-transform.

4 M

7 (b) Using appropriate properties, find the Z-transform

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

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

6 M

7 (c) Determine the inverse Z-transform of

x (z) = \frac{1}{2 - 4 z^{- 1} + 2 z^{- 2}}

$x(z)= \dfrac {1}{2-4 z^{-1}+2z^{-2}}$ by long division method of:
i) ROC; |z|>1.

4 M

7 (d) Determine all possible signals x(n) associated with Z-transform,

x (z) = \frac{(1 / 4) z^{- 1}}{[1 - (1 / 2) z^{- 2}] [1 - (1 / 4) z^{- 2}]}

$x(z)= \dfrac {\left (1/4 \right )z^{-1}}{\left [ 1-\left (1/2 \right )z^{-2} \right ] \left [ 1-\left (1/4 \right ) z^{-2} \right ]}$

6 M

8 (a) An LTI system is described by the equation
y(n)=x(n)+0.81 x(n-1)-0.81 x(n-2)-0.45 y(n-2). Determine the transfer function of the system. Sketch the poles and zeros on the Z-plane. Assess the stability.

5 M

8 (b) A system has impulse response h(n) (1/3)ⁿ n(n). Determine the transfer function. Also determine the input to the system if the output is given by:

y (n) = \frac{1}{2} u (n) + \frac{1}{4} {(- \frac{1}{3})}^{n} u (n)

$y(n)= \dfrac {1}{2} u(n)+ \dfrac {1}{4} \left ( -\dfrac {1}{3} \right )^n u(n)$

5 M

8 (c) A linear shift invariant system is described by the difference equation.

y (n) - \frac{3}{4} y (n - 1) + \frac{1}{8} y (n - 2) = x (n) + x (n - 1)

$y(n) - \dfrac {3}{4} y(n-1)+ \dfrac {1}{8} y(n-2)= x(n)+ x(n-1)$ with y(-1)=0 and y(-2)=-1
Find:
i) The natural response of the system.
ii) The forced response of the system and
iii) The frequency response of the system for a step.

10 M

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