PG – 677

*PG677*

I Semester M.Sc. Degree Examination, January 2015 ELECTRONIC (CBCS) SCIENCE ELS – 107 : Signals and Systems Time : 3 Hours

Max. Marks : 70

Instructions : 1) Section – A : Answer all questions. Each question carries two marks. 2) Section – B : Answer any four full questions. SECTION – A

(5×2=10)

1. Define a Signal. Mention any two methods of classifying signals. 2. Give the mathematical expression for convolution sum and convolution integral. 3. State and prove the linearity property of fourier transform. 4. Define unilateral Laplace transform of a signal x(t). 5. Define the ROC with respect to Z- Transform. SECTION – B 6. a) Determine whether the following signals are periodic. If periodic find its fundamental period.

( )

i) x(t) = cos(t) + sin 2 t

6

⎛ 2 πn ⎞ ⎛ 2 πn ⎞ ⎟ + sin ⎜ ⎟. ii) x(n) = cos ⎜ ⎝ 5 ⎠ ⎝ 7 ⎠

b) Represent the signal x(t) in terms of g(t) :

4

P.T.O.

PG – 677

-2-

*PG677*

c) T{x(n)} = g(n) x(n) determine whether the system is Linear, time invariate , memory, causal and stable. 7. a) Consider the input x(n) and the unit impulse response h(n) as follows x(n) = anu (n); 0 < a < 1 h(n) = u(n). Evaluate and plot the output signal y(n).

5 8

b) Find the forced response for the system described by the difference equation n 1 1 1 ⎛ ⎞ y ( n − 1 ) − y ( n − 2 ) = x ( n ) + x ( n − 1 ) , with forcing function ⎜ ⎟ u(n). y(n) – 4 8 ⎝8⎠

8. a) Obtain the different equation for the corresponding block diagram shown in the figure.

7

4

b) Find the zero input response of the system described by the equation d 2 y( t ) dy( t ) dx (t ) +3 + 2 y( t ) = x( t ) + y(0) = 0 & y′(0 ) = 1 dt dt dt

c) With reference to Fourier series state and prove time shift property.

6 5

9. a) With reference to Fourier Transform state and prove i) Convolution ii) Parseval’s theorem b) Determine the Fourier series represent for x(t) = e–t as shown in the figure, plot the magnitude & phase spectrum.

8 7

*PG677*

PG– 677

-3-

10. a) Find the Laplace Transform of x(t) = eat u(t) mention the ROC, pole zero location in s-plane assume ‘a’ is real value.

6

b) Find the transfer function of LTI system described by the difference equation d 2 y( t ) dy (t ) dx( t ) +3 + 2 y( t ) = 3 x( t ) + 2 . dt dt dt

c) Find the inverse Laplace – Transform of X(S ) =

4 − 5s − 7 with ROC – (s + 1)(s − 1)(s + 2 )

1
5 n

n

2 1 11. a) A causal LTI system has h(n) = ⎛⎜ 1 ⎞⎟ u(n) and input x(n) = ⎛⎜ − ⎞⎟ u(n) 3 ⎝ 2⎠ ⎝2⎠ determine the output (n) using z- transform.

(

)(

)

1 ⎛ ⎞ b) Find the inverse z-transform of x(z) = z 2 ⎜ 1 − z −1 ⎟ 1 + z −1 1 − z −1 . 2 ⎝ ⎠

________________

10 5