VTU Signals & Systems - December 2013 Exam Question Paper

VTU Electronics and Communication Engineering (Semester 4)
Signals & Systems
December 2013

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) Sketch the even and odd part of the signal shown in Fig. Q1 (a)

6 M

1 (b) Check whether the following signals is periodic or not and if periodic find its fundamental period.
i) x(n)=cos(20?n)+sin(50?n)
ii) x(t)=[cos(2?n)]²

6 M

1 (c) Let x(t) and y(t) as show in Fig Q1(c). Sketch (i) x(t)y(t-1) (ii) x(t)y(-t-1)

8 M

2 (a) Determine the convolution sum of the given sequences
x(n)={1, -2, 3, -3} and h(n)={-2, 2, -2}

4 M

2 (b) Perform the convolution of the following sequences:
x₁(t)=e^-at: 0?t?T
x₂(t)=1: 0?t?2T

10 M

2 (c) An LTI system is characterized by an impulse response,

h(n)=\eft ( \dfrac {1}{2} \right )^n u(n.)

$h(n)=\eft ( \dfrac {1}{2} \right )^n u(n.)$ Find the response of the system for the input

x (n) = {(\frac{1}{4})}^{n} u (n) .

$x(n)= \left ( \dfrac {1}{4} \right )^n u(n).$

6 M

3 (a) Determine the following LTI system characterized by impulse response is memory, casual and stable.
i) h(n)=2u(n)-2u(n-2)
ii) h(n)=(0.99)ⁿu(n+6).

6 M

3 (b) Find the natural response of the system described by a differential equation

$\dfrac {d^2 y(t)}{dt^2}+ 2 \dfrac {dy(t)}{dt}+2y(t)=2x(t), \ with \ y(0)=1, \ and \ \left.\begin{matrix} \dfrac {dy(t)} {dt}\end{matrix}\right|_{t=n}=0$

6 M

3 (c) Find the difference equation description for the system shown in Fig Q3(c).

4 M

3 (d) By converting the differential equation to integral equation draw the direct form-I and direct form-II implementation for the system as

$\dfrac {d^3 y(t)}{dt^3}+ \dfrac {d^2 y(t)}{dt^2}+ 2 \dfrac {dy(t)}{dt}= x(t)+ 6 \dfrac{d^2 x(t)}{dt^2}$

4 M

4 (a) State and prove the following properties of DTES: i) Modulation ii) Parseval's theorem.

10 M

4 (b) Find the Fourier series coefficient of the signal x(t) shown in Fig Q4(b) and also draw its spectra.

10 M

5 (a) Find the DTFT of the following signals:
i) x(n)=a^|n|; |a|<1
ii) x(n)=2ⁿ u(-n)

8 M

5 (b) Determine the signal x(n) if its DTFT is as shown in Fig. Q5(b).

6 M

5 (c) Compute the Fourier transform of the signal

$x(t)= \left\{\begin{matrix} 1+\cos \pi t&;&|t|\le 1 \\0 &; &|t|>1 \end{matrix}\right.$

6 M

6 (a) Find the frequency response of the system describe by the impulse response h(t)=?(t)-2e^-2t u(t) and also draw its magnitude and phase spectra.

8 M

6 (b) Obtain the Fourier transform representation for the periodic signal x(t)=sin w₀t and draw the magnitude and phase.

7 M

6 (c) A signal x(t)=cos(20?t)+1/4 cos(30?t) is sampled with sampling period ?_s. Find the Nyquist rate.

5 M

7 (a) What is region of convergence (ROC)? Mention its properties.

6 M

7 (b) Determine the z-transform and ROC of the sequence

$x(n)=r_1^n u(n)+ r^n_2 u(-n)$

7 M

7 (c) Determine the inverse z-transform of the function,

$x(z)= \dfrac {1+z^{-1}}{1-z^{-1}+0.5z^{-2}}$ using partial fraction expansion.

7 M

8 (a) An LTI system is described by the equation
y(n)=x(n)+0.8 x(n-1)+0.8x(n-2)-0.49y(n-2)

6 M

8 (b) Determine the transfer function H(z) of the system and also sketch the poles and zeros.

8 M

8 (c) Determine whether the system described by the equation y(n)=x(n)+by(n-1) is causal and stable where |b|<1. Find the unilateral z-transform for the sequence y(n)=x(n-2), where x(n)=?ⁿ.

6 M

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