Unit 1 Operation on signals

Representation of signals & Operation on signals
(Time Reversal, Time Shifting , Time Scaling, Amplitude scaling, Signal addition, Signal Multiplication)

1. Representation of signals & Operation on signals Dr.K.G.SHANTHI Professor/ECE [email protected] RMK College of Engineering and Technology (Time Reversal, Time Shifting , Time Scaling, Amplitude scaling, Signal addition, Signal Multiplication)

2. 2 Representation of CT Signals & DT Signals

3. 3 Functional representation Representation of CT Signals x(t) = 2 ; -1< t <1 0 ; otherwise -1 0 +1 x(t) 2 t Graphical representation

4. Representation of DT Signals 4 Functional representation x(n) = 2 ; n=-1 1 ; n=0 -1; n=1 n x(n) 2 1 -1 -1 0 +1 n -1 1 x(n) 2 1 -1 x(n) = { 2,1,-1} n=0 Note: If there is no arrow in sequence representation, then first signal value indicates n=0 value Graphical representation Sequence representation Tabular representation

5. Practise Problem 1. x(n) = 2 n ; n ≥ 0 0 ; n < 0 2.x(n) = { 1,-2,3,2,-1,0,3} Write the tabular representation of x(n) 5 Draw the graphical representation of x(n)

6. Operation on Signals 6

7. Operation on Independent variable Time Inversion or Time Folding or Time Reversal Time Shifting Time Scaling Operation on Dependent variable Amplitude scaling Signal addition Signal Multiplication 7 Basic Operation on Signals

8. Time Reversal/TIME FOLDING/TIME Inversion In Time reversal, signal x(t) is reversed with respect to time i.e. y(t) = x(-t) is obtained for the given function Time Folding: By folding the signal x(t) about t=0 (Rotating signal by 180 clockwise direction will give mirror image of signal 8 i.e x(-t)

9. Some Examples 9

10. Some Examples 10

11. 11 Time Shifting

12. Time Shifting The original signal x ( t ) is shifted by an amount tₒ . Signal Delayed Signal Advanced 12 X(t) X(t- t ) Shift right X(t) X(t + t ) Shift left

13. Time Shifting 13 Draw y(t) = x(t-2) and y(t) = x(t+2) Signal delay (Shift right) Signal advance (Shift left) Signal delay Signal advance Consider a signal x(t)

14. 14 Consider a signal x(n) Plot y(n) = x(n-3) and y(n) = x(n+2) Signal delay Signal advance y(n) = x(n-3) y(n) = x(n+2) Time Shifting

15. 15

16. 16

17. Time Scaling 17

18. Consider a signal x(t) Plot x(2t) and x(t/2) 18 a= 2 >1, Time compressed a= 1/2 <1, Time expanded

19. ( i ) x(2t) t= -0.5, x(2 x - 0.5) = x(-1) = 0 t=0 , x(2 x 0) = x(0) = 4 t=0.5 , x(2 x 0.5) = x(1) = 4 t=1, x(2 x 1) = x(2) = 4 t=1.5, x(2 x 1.5) = x(3) = 0 19 t= -1, x(2 x - 1) = x(-2) = 0 t=2, x(2 x 2) = x(4) = 0 - ve + ve

20. ( i ) x(t/2) t=-2 , x(-2/2) = x(-1) = 0 t=-1, x (-1/2) = x(-0.5) = 3 t=0, x(0/2) = x(0) = 4 t= 1, x(1/2) = x(0.5) = 4 t=3 , x (3/2) = x(1.5) = 4 t=4 , x(4/2) = x(2) = 4 t=6 , x (6/2) = x(3) = 0 20

21. Example: Given x ( t ) and we are to find y ( t ) = x (2 t ). 21

22. Consider a signal Plot y(n)=x(2n) and x(n/2) 22 x(-4) = 0 x(-3) = 4 x(-2) = 3 x(-1) = 2 x(0) = 1 x(1) = 2 x(2) = 3 x(3) = 4 x(4) = 0 x(n) = {0,4,3,2,1,2,3,4,0}

23. ( i ) y(n) = x(2n) ; y(0) = x(2 x 0) = x(0) = 1 y(1) = x(2 x 1) = x(2) = 3 y(2) = x(2 x 2) = x(4) = 0 y(-1) = x(2 x -1) = x(-2) = 3 y(-2) = x(2 x -2) = x(-4) = 0 23 y(n) = x(2n) a=2 >1 (time compress) x(n) = {0,4,3,2,1,2,3,4,0}

24. ( i ) y(n) = x(n/2) ; y(-2) = x(-2/2) = x(-1) = 2 y(-4) = x(-4/2) = x(-2) = 3 y(-6) = x(-6/2) = x(-3) = 4 y(-8) = x(-8/2) = x(-4) = 0 y(0) = x(0/2) = x(0) = 1 y(2) = x(2/2) = x(1) = 2 y(4) = x(4/2) = x(2) = 3 y(6) = x(6/2) = x(3) = 4 y(8) = x(8/2) = x(4) = 0 24 y(n) = x(n/2) a=1/2 < 1 (Time expands) x(n) = {0,4,3,2,1,2,3,4,0}

25. Amplitude Scaling Multiplying the signal x(t) with A results in output y(t)=A x(t), where A= Amplitude For A ˃ 1 , the signal is amplified (Amplitude increases) For A < 1 , the signal is attenuated (Amplitude decreases) There is no change in time 25

26. 26 y(n) = 2x(n) Consider the signal x(n). Plot y(n)=2x(n) Consider the signal x(t). Plot y(t)=2x(t)

27. Signal addition The addition of two continuous time signals is obtained by adding the value (amplitude) of two signals at same instant of time. 27 x 1 (t) x 2 (t) x 1 (t)+x 2 (t)

28. Signal addition Find u(t) – u(t-10) 28

29. Signal addition Consider x 1 (n) = {1,2,3,1,5} and x 2 (n) = {2,3,4,1,-2}. Find y(n) = x 1 (n) + x 2 (n) Solution : y(n) = { 1+2, 2+3, 3+4, 1+1, 5-2} y(n) = { 3,5,7,2,3} 29 0 1 2 3 4 x 1 (n) 1 2 3 5 n 1 0 1 2 3 4 x 2 (n) 2 3 4 n 1 -2 0 1 2 3 4 y(n) 3 2 3 5 n 7

30. Consider x 1 (n) = {1,2,3,1,5} and x 2 (n) = {2,3,4,1,-2}. Find y(n) = x 1 (n) + x 2 (n) Solution : y(n) = {0+2,1+3,2+4,3+1,1-2,5-0} 30 -1,n=0,1,2,3 n=0,1,2,3,4 0 1 2 3 4 x 1 (n) 1 2 3 5 n 1 -1 0 1 2 3 4 x 2 (n) 2 3 4 n 1 -2 -1 0 1 2 3 4 y(n) 2 -1 5 4 n 6 4 y(n) = { 2,4,6,4,-1,5}

31. Signal Multiplication Multiplication of two signals is obtained by multiplying the value (amplitude) of two signals at same instant of time. Consider x 1 (n) = {1,2,3,4} and x 2 (n) = {2,1,3,2} Find y(n) = x 1 (n) x 2 (n) y(n) ={ 1x2, 2x1, 3x3, 4x2} y(n) = { 2,2,9,8} 31 0 1 2 3 y(n) 2 2 9 8 n

32. Signal Multiplication Multiply the signal values at all time or specific time 32

33. Precedence Rule Follow the precedence rule, if Time shifting and Time scaling , time reversal and amplitude scaling occurs in same signal. Rule: 33 Amplitude scaling Time shifting Time reversal Time scaling

34.   34 ( i ) x(2t+2) Time shifting Time scaling Left a=2 >1 Compress

35. 35 Time shifting Time scaling Right a=1/2 < 1 Expand

36. (iii) x(-t-2) 36 Time shifting Time reversal

37. THANK YOU 37