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Total Marks: 60 No of Pages: 3 Max. Time: 3 Hour

Attempt all questions.

Property Table for Fourier Transform is on the last page

Section A

Do as directed.

1. Fill in the blanks: (5)

a. A gray scale digital image is an example of ______ dimensional signal.

b. ( ) sin{ ( )}y t x t is a _______ system. (Linear/Non-Linear)

c. ( 5) ______t dt

.

d. ( ) ( ) __________F

x t y t .

e. 0( ) ( ) _______.x t t t

2. Convolve the following signals:

a. ( ) ( )tx t e u t and ( ) ( ) ( 10)h t u t u t (5)

b. ( ) ( 2) ( 2)x t u t u t and ( ) ( 2) ( 2)h t t t (5)

3. Obtain the Fourier series of the signal ( ) ( )s

k

x t t kT

where 5sT s . (4)

4. Obtain the Fourier transform of the signal ( ) ( )x t t . (2)

5. Given ( ) 1F

t use duality property of Fourier transform to obtain the spectrum of a DC

signal. (2)

6. Given 00( )

Fj t

e , obtain the Fourier transform of a cosine signal with frequency

640kHz. (3)

7. Given the transfer function of an ideal lop pass filter below, obtain the impulse response of

the filter. (4)

1,( )

0,

c cH

otherwise

8. Obtain the transfer function of the system given by the impulse response 2( ) ( )th t e u t . (4)

9. Given the signal ( ) 1 2cos100 0.5sin(200 )x t t t :

a. What is the fundamental frequency of the system? (1)

PAF – Karachi Institute of Economics and Technology

Course : Signals and Systems

Faculty : Dr. Ghazanfar Monir Class ID:

Examination: Final

Date: July 22’ 2014

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b. Sketch the two-sided spectrum of the signal. (2)

c. Use Parseval’s power theorem to calculate the power of the signal. (2)

Section B

10. Obtain 1 2( ) ( ). ( )y t x t x t where 21( ) 2 tx t e and 2 ( ) ( )

k

x t t k

(3)

11. A band-limited signal 1( )x t with a bandwidth of 40k radians/second is multiplied with a

signal 2 ( ) ( )s

k

x t t kT

where 16.66sT s . Obtain the spectrum of resulting signal. What

would be the least sampling frequency for which the signal could be reconstructed without

any distortion? (5)

12. An audio signal is to be transmitted over a radio channel at 640kHz. Design a system that

could perform the desired task. Show all the mathematical derivations and calculations

required. Sketch the block diagram of your proposed system. (8)

13. Design a system at the receiving end to translate the signal back to audio frequencies. (5)

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