MU Signals and Systems - May 2015 Exam Question Paper

MU Instrumentation Engineering (Semester 5)
Signals and Systems
May 2015

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

Solve the following:

1 (a) Show that x(t)*δ(t-to)=x(t-to)

4 M

1 (b) Obtain the linear convolution of given signals. Also sketch the result. \[ \begin{align*} x(t)&=1 & \text{for }0\le t \le 1 \ &=0 & \text{elsewhere} \ \ \ \ \ \ \ h(t) &=1 &\text{for }0

4 M

1 (c) Find the z-transform of the signal

x (n) = {(\frac{1}{2})}^{n} \cos (w_{0} n) u (n) .

$x(n) = \left ( \dfrac {1}{2} \right )^n \cos(w_0n) \ u(n).$ Specify its ROC.

4 M

1 (d) State and explain Dirichlets conditions for the existence of continuous time Fourier series.

4 M

1 (e) Find the Fourier transform of the signal

x (t) = \frac{d}{d t} [(e^{- 3 t} u (t)) * (e^{- 2 t} u (t - 2))]

$x(t) = \dfrac {d}{dt}[(e^{-3t}u(t))* (e^{-2t}u(t-2))]$

4 M

2 (a) Find if the following sequences are periodic or not. If yes find its fundamental period.
i) x1(n)=e^j(π/4)n
ii) x2(n)=3 sin (1/8)n

6 M

2 (b) Plot the following sequences:
i) x1(n) = (-2)ⁿ u(n)
ii) x2(n)=2+u(t-4)+u(t)
iii) x3(n)=2ⁿ u(-n-1)

10 M

2 (c) Find bilateral z-transform of the signal
x(n)=9 δ(n+2)+3δ(n+1)-4δ(n)+3δ(n-2)+4δ(n-4).

4 M

3 (a) Solve the difference equation

y (n) - \frac{1}{9} y (n - 2) = x (n - 1)

$y(n) - \dfrac {1}{9} y(n-2) = x (n-1)$ with y(-1)=0, y(-2)=1, x(-1)=0 and x(n)=3 u(n).

10 M

3 (b) Classify the following system for memory, linearity, causality, time variance and stability.
y(n)=a x(n)-b x(n-1).

5 M

3 (c) Find x(t) corresponding to FT.

x (j w) = \frac{- j w}{(j w)^{2} + 3 j w + 2}

$x(jw) = \dfrac {-jw}{(jw)^2 + 3jw+2}$

5 M

4 (a) Determine complex exponential Fourier series for the signal x(t) shown below:

10 M

4 (b) Determine z-transform of following function

i) x (n) = {(\frac{2}{3})}^{n} u (n + 2) i i) x (n) = n {(\frac{5}{8})}^{n} u (n) i i i) x (n) = (0.6)^{n} u (n) * (0.9)^{n} u (n)

$i) \ x(n) = \left ( \dfrac {2}{3} \right )^n u(n+2) \ ii) \ x(n) = n \left ( \dfrac {5}{8} \right )^n u(n) \ iii) \ x(n)=(0.6)^n u(n)*(0.9)^n u(n)$

10 M

5 (a) Find Laplace transform of x(t)=te^-3t u(t). Prove the property used.

5 M

5 (b) Find Fourier transform of SINC function.

5 M

5 (c) Find the inverse Laplace transform of

x (s) = \frac{- 3}{(s + 2) (s - 1)}

$x(s) = \dfrac {-3}{(s+2)(s-1)}$ If the ROC is
i) -2 < Re(s) <1
ii) Re(s)>1
iii) Re(s)<-2

10 M

6 (a) If x(t)↔ x(w) is Fourier transform pair then prove that x(t)↔2π x(-w).

5 M

6 (b) Find the initial and final values of the signal

x (z) = \frac{(z - 3) z}{(z - 1) (z - 0.4)}

$x(z) = \dfrac {(z-3)z}{(z-1)(z-0.4)}$

5 M

6 (c) Find inverse z-Transform of

x (z) = \frac{1 - \frac{1}{2} z^{- 1}}{1 + \frac{3}{4} z^{- 1} + \frac{1}{8} z^{- 2}} | z | > \frac{1}{2}

$x(z) = \dfrac {1- \frac {1}{2}z^{-1}}{1+ \frac {3}{4}z^{-1}+ \frac {1}{8}z^{-2}} |z|>\dfrac {1}{2}$

10 M

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