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**B.E./B.Tech. DEGREE EXAMINATION, APRIL/MAY 2010**

**Third Semester**

**Electronics and Communication Engineering**

**EC2204 - SIGNALS AND SYSTEMS**

**(Regulation 2008)**

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**PART A — (10 × 2 = 20 Marks)**

1. Define unit impulse and unit step signals.

2. When is a system said to be memory less? Give an example.

3. State any two properties of Continuous – Time Fourier Transform.

4. Find the Laplace transform of the signal ) ( ) ( t u e t x at - = .

5. State the convolution integral for continuous time LTI systems.

6. What is the impulse response of two LTI systems connected in parallel?

7. State the Sampling theorem.

8. State the sufficient condition for the existence of DTFT for an aperiodic sequence ) (n x .

9. Define one sided Z-transform and Two-sided Z-transform.

10. Define the shifting property of the discrete time unit Impulse function.

**PART B — (5 × 16 = 80 Marks)**

11. (a) Distinguish between the following:

(i) Continuous Time Signal and Discrete Time Signal. (4)

(ii) Unit step and Unit Ramp functions. (4)

(iii) Periodic and Aperiodic signals. (4)

(iv) Deterministic and Random signals. (4)

Or

(b) (i) Find whether the signal ) 1 4 ( sin ) 1 10 ( cos 2 ) ( - - + = t t t x is periodic or not. (4)

(ii) Find the summation ?8

(iii) Explain the properties of unit impulse function. (4)

(iv) Find the fundamental period T of the continuous time signal

12. (a) (i) Find the trigonometric Fourier series for the periodic signal ) (t x shown in the figure given below : (10)

(ii) Explain the Fourier spectrum of a periodic signal ) (t x . (6)

Or

(b) (i) Find the Laplace transform of the signal ) ( ) ( ) ( t u e t u e t x bt t a - + = - - . (8)

(ii) Find the Fourier transform of t e t x - = ) ( for 1 1 = = - t otherwise 0 = . (8)

13. (a) (i) Explain the steps to compute the convolution integral. (8)

(ii) Find the convolution of the following signals: (8)

) ( ) ( 2 t u e t x t - =

) 2 ( ) ( + = t u t h .

Or

(b) (i) Using Laplace transform, find the impulse response of an LTI system described by the differential equation

(ii) Explain the properties of convolution integral. (8)

14. (a) (i) Find the Fourier Transform of

(ii) Explain any four properties of DTFT. (8)

Or

(b) (i) Find the Z-transform of the given signal ) (n x and find ROC.

( ) ( ) sin ( ) o

x n w n u n = ? ? ? ? . (10)

(ii) Describe the sampling operation and explain how aliasing error can be prevented. (6)

15. (a) (i) Find the impulse response of the discrete time system described by the difference equation

) 1 ( ) ( 2 ) 1 ( 3 ) 2 ( - = + - - - n x n y n y n y . (8)

(ii) Discuss the block diagram representation for LTI discrete time systems. (8)

Or

(b) (i) Describe the state variable model for discrete time systems. (8)

(ii) Find the state variable matrices A, B, C, D for the equation

) 2 ( 6 ) 1 ( 5 ) ( ) 2 ( 2 ) 1 ( 3 ) ( - + - + = - - - - n x n x n x n y n y n y . (8)

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