1. Transformation of Continuous Time Signals
This module shows how different operations such as amplitude scaling, time scaling, time reversing and time shifting transform a continuous time signal.
Course Name: Signals and Systems
Authors
Amita Shinde
Mentor
Sarvanan Vijaykumaran

2. Learning Objectives
After interacting with this Learning Object, the learner will be able to:
Given a continuous time signal, shifting and/or scaling factors required, plot the output of specified amplitude and/or time transformation on the signal.
Given a transformation equation, identify the type of transformation performed.
Given a continuous time signal and transformation equation, plot the outputs of amplitude and time transformations on the signal.
Given input signal and transformed signal, identify the type of transformation performed.

3. Theory button content x(t) x(– t) x(– (t–1))  x(– t+1); x(t–1)
A continuous time signal can be manipulated by modifying or transforming its dependent (amplitude) or independent (time) variable.
The most common time transformations include shifting, scaling and reversal.
Amplitude scaling is a common type of amplitude transformation.
In time shifting, t is replaced by (t-to), where to can be any real number.
If to is positive, the signal is said to be delayed. e.g. x(t-1), where signal is shifted to the right.
If to is negative, the signal is said to be advanced. e.g. x(t+1), where signal is shifted to the left.
In time scaling, t is replaced by a multiple of t i.e. (Ct), where C can be any real number.
If C>1, the signal is said to be compressed in time e.g. x(2t), where signal is compressed twice about t=0.
If 0<C<1, the signal is said to be expanded in time e.g. x(0.5t) or x(t/2), where signal is expanded twice about t=0.
In time reversal, t is replaced by (-t). The signal is folded about t=0
In amplitude scaling, signal amplitude is multiplied by a number e.g. x(t) is replaced by Ax(t).
If A>1, the signal is said to be amplified e.g. 2x(t), where signal amplitude is doubled about x(t)=0.
If 0<A<1, the signal is said to be attenuated e.g. 0.5x(t), where signal amplitude is halved about x(t)=0.
Multiple transformations can be applied to a signal in a certain sequence to manipulate it in a particular way.
The time transformation affects only time parameter and amplitude transformation affects only amplitude.
The sequence of time transformations is significant.
x(t)
Reversed
x(– t)
x(– (t–1))  x(– t+1);
x(t–1)
x((– t)–1)  x(– t–1)
Delayed by 1
Reversal & Delaying:
x(t)
Scaled by 2
x(2 t)
Delayed by 1
x(2 (t–1))  x(2t–2);
x( t–1)
x((2t)–1)  x(2t –1)
Scaling & Delaying:

4. Continuous Time Signals
Electrical Engineering > Signals and Systems
Transformation of
Continuous Time Signals
Master Layout 1: Linear menu
Input signal expression
x(t) t -1 1 2 -2 3 -3 -4 -5 4 5 6 -6
Types of transformation
Amplitude Scaling
Amplification
Attenuation
Time Scaling
Expansion
Compression
Time Shifting
Delaying
Advancing
Time Reversing
x(t)= 0, for t<1
= t-2, for 1<t<3
= 0, for t>3
i/p signal eqn t -1 1 2 -2 3 -3 -4 -5 4 5 6 -6
x(t-3)
manipulated signal expression
(different for each animation)
Input signal
x(t)= 0, for t<4
= t-5, for 4<t<6
= 0, for t>6
Manipulated signal
o/p signal eqn
(varies for each animation)
s/g before
animation
s/g after
Animation
(Shown here in black colour for animator’s reference.
In actual animation, it is not shown initially.
It is the last stage of the animation itself.
Need not show different colour in actual animation. )
List of basic transformation.
Radio buttons denote selectable menu.
It can also be shown as a drop down list
or some other way.
This design, layout etc is suggestive & may be varied by designer/animator.
Also, the linear & interactive menu may be merged.

5. A continuous time signal x(t) is shown here.
Step 1
Introduction
A continuous time signal x(t) is shown here.
Select a type of basic signal transformation from the list.
Then click on ‘Play’ to view how this transformation is performed.
Also observe the signal representations.
Once you have learnt about the basic transformations,
you can also try various combinations of time transformations
In the interactive section.
Types of transformation
Amplitude Scaling
Amplification
Attenuation
Time Scaling
Expansion
Compression
Time Shifting
Delaying
Advancing
Time Reversing
x(t) t 1 -1 2 -2 3 -3 -4 –1 –2 4
x(t)= 0, for t<1
= t-2, for 1<t<3
= 0, for t>3

6. Selection of Basic transformations to view
Step 2
Selection of Basic transformations to view
Types of transformation
Amplitude Scaling
Amplification
Attenuation
Time Scaling
Expansion
Compression
Time Shifting
Delaying
Advancing
Time Reversing
Interactivity
Boundary Limit
Result
Instructions to learner
Selection of basic transformation from list
Only one option selectable at a time out of 7
User selects an option.
Select a type of signal transformation.
Play button
Activate when values selected
When a s/g transformation option selected & play button clicked, play respective animation.
Refer to following list.
Click on play to view animation
Slide No.
Animation 7
Amplitude scaling: Amplification 9
Amplitude scaling: Attenuation 11
Time reversing 12
Time Shifting: Delaying 13
Time Shifting: Advancing 14
Time Scaling: Compression 15
Time Scaling: Expansion
List of basic transformation.
Radio buttons denote selectable menu.
When a menu selected,
respective animation is played.
Only 1 option selected at a time

7. 1. Amplification x(t) t 2.x(t) t x(t)= 0, for t<1-1 2 -2 3 -3 -4 –1 –2 4
x(t)= 0, for t<1
= t-2, for 1<t<3
= 0, for t>3 t 1 -1 2 -2 3 -3 -4 –1 –2 4
2.x(t)
2.x(t)= 0, for t<1
= 2t-4, for 1<t<3
= 0, for t>3

8. Amplification with Negative Value
x(t) t 1 -1 2 -2 3 -3 -4 –1 –2 4
x(t)= 0, for t<1
= t-2, for 1<t<3
= 0, for t>3 t 1 -1 2 -2 3 -3 -4 –1 –2 4
-2.x(t)
x(t)= 0, for t<1
= -2t+4, for 1<t<3
= 0, for t>3

9. 2. Attenuation x(t) t 0.5x(t) t x(t)= 0, for t<1-1 2 -2 3 -3 -4 –1 –2 4
x(t)= 0, for t<1
= t-2, for 1<t<3
= 0, for t>3 t 1 -1 2 -2 3 -3 -4 –1 –2 4
0.5x(t)
0.5 x(t)= 0, for t<1
= 0.5t-1, for 1<t<3
= 0, for t>3

10. Attenuation with Negative Value
x(t) t 1 -1 2 -2 3 -3 -4 –1 –2 4
x(t)= 0, for t<1
= t-2, for 1<t<3
= 0, for t>3 t 1 -1 2 -2 3 -3 -4 –1 –2 4
-0.5x(t)
-0.5 x(t)= 0, for t<1
= -0.5t+1, for 1<t<3
= 0, for t>3

11. 3. Time Reversal x(t) t x( – t) t x(t)= 0, for t<1-1 1 2 -2 3 -3 -4 -5 4 5 6 -6
x(t)= 0, for t<1
= t-2, for 1<t<3
= 0, for t>3
x( – t) t -1 1 2 -2 3 -3 -4 -5 4 5 6 -6
x(-t)= 0, for t>1
= -t-2, for -1<t<-3
= 0, for t<-3

12. 4. Time Shifting : Delaying
x(t) t -1 1 2 -2 3 -3 -4 -5 4 5 6 -6
x(t)= 0, for t<1
= t-2, for 1<t<3
= 0, for t>3
x ( t – 1) t -1 1 2 -2 3 -3 -4 -5 4 5 6 -6
x(t-1)= 0, for t<2
= t-3, for 2<t<4
= 0, for t>3

13. 5. Time Shifting : Advancing
x(t) t -1 1 2 -2 3 -3 -4 -5 4 5 6 -6
x(t)= 0, for t<1
= t-2, for 1<t<3
= 0, for t>3 t -1 1 2 -2 3 -3 -4 -5 4 5 6 -6
x ( t + 1)
x(t+1)= 0, for t<0
= t-1, for 0<t<2
= 0, for t>2

14. 6. Time Scaling : Compression
x(t) t -1 1 2 -2 3 -3 -4 -5 4 5 6 -6
x(t)= 0, for t<1
= t-2, for 1<t<3
= 0, for t>3 t -1 1 2 -2 3 -3 -4 -5 4 5 6 -6
x ( 2 t )
x(2t)= 0, for t<0.5
= 2t-2, for 0.5<t<1.5
= 0, for t>1.5

15. 7. Time Scaling : Expansion
x(t) t -1 1 2 -2 3 -3 -4 -5 4 5 6 -6
x(t)= 0, for t<1
= t-2, for 1<t<3
= 0, for t>3 t -1 1 2 -2 3 -3 -4 -5 4 5 6 -6
x(0.5 t )
x(0.5t)= 0, for t<2
= 0.5t-2, for 2<t<6
= 0, for t>6

16. Continuous Time Signals
Electrical Engineering > Signals and Systems
Transformation of
Continuous Time Signals
Master Layout 2: Interactive menu
Input signal expression
x(t)
Reverse (-t)
X(t-3) t -1 1 2 -2 3 -3 -4 -5 4 5 6 7 8 -6 -7 -8
Delay by 1(t-1)
Advance by 1(t+1)
Shift
Compress by 2 (2t)
Expand by 2(t/2)
Scale
3 options ( reverse, shift, scale)
2 sub-options in Shift (delay & advance) and Scale ( compress & expand)
Each of 3 option selected only once
As option selected animation is shown and that option is disabled
As next option is selected, its animation is shown.
When reset is selected, the signal is reset and all 3 options enabled
Reset

17. Interactivity Menu Step 1 Step 2 Step 3 Representation Slide Reversing
No.
Step 1
Step 2
Step 3
Representation
Slide 1
Reversing
(-t)
x (-t ) 22 2
Compression
( 2t )
X ( – 2 t ) 23 3
Delaying ( t – 1)
X (– 2 t + 2) 24 4
Advancing ( t + 1)
X (– 2 t – 2) 25 5
Expansion
( 0.5 t )
X (– 0.5 t ) 26 6
X (– 0.5 t ) 27 7
X (– 0.5 t – 0.5 ) 28 8
Delaying
(t - 1)
X ( -t +1 ) 29 9
Compression ( 2 t )
X ( -2t + 1 ) 30 10
Expansion ( t / 2 )
X (– 0.5 t + 1) 31 11
Advancing
(t + 1)
X (–t – 1) 32 12
X (– 2 t – 1) 33 13
X (– 0.5 t – 1) 34
*For animator’s reference

18. Interactivity Menu No. Step 1 Step 2 Step 3 Slide 14 Delaying (t - 1)
Representation
Slide 14
Delaying
(t - 1)
X ( t – 1) 35 15
Reversing
( - t )
X (–t – 1) 36 16
Compression ( 2 t )
X (– 2 t – 1) 37 17
Expansion (0.5 t )
X (– 0.5 t – 1) 38 18
Compression
( 2t )
X ( 2 t – 1) 39 19
Reversing ( - t ) 40 20
Expansion
(0.5 t )
X (0.5 t – 1) 41 21 42 22
Advancing
(t + 1)
X ( t + 1 ) 43 23
X (– t + 1) 44 24
X (– 2 t + 1) 45 25
X (– 0.5 t + 1) 46 26
X ( 2t + 1 ) 47 27
X (– 2 t + 1) 48 28
( t / 2 )
X ( 0.5 t + 1 ) 49 29
X (– 0.5 t + 1) 50
*For animator’s reference

19. Interactivity Menu No. Step 1 Step 2 Step 3 Slide 30 Compression
Representation
Slide 30
Compression
( 2t )
X(2t) 51 31
Reversing
( - t )
X ( – 2 t ) 52 32
Delaying ( t – 1)
X (– 2 t + 2) 53 33
Advancing ( t + 1)
X (– 2 t – 2) 54 34
Delaying
(t - 1)
X ( 2 t – 2) 55 35
Reversing ( - t ) 56 36
Advancing
(t + 1)
X ( 2 t + 2) 57 37 58 38
Expansion
( t / 2 )
X (0.5t) 59 39
X (– 0.5 t ) 60 40
X (– 0.5 t ) 61 41
X (– 0.5 t – 0.5 ) 62 42
X ( 0.5 t – 0.5 ) 63 43 64 44
X ( 0.5 t ) 65 45 66
*For animator’s reference

20. Interactivity: Initial screen
x(t) t -1 1 2 -2 3 -3 -4 -5 4 5 6 7 8 -6 -7 -8
Delay by 1(t-1)
Advance by 1(t+1)
Compress by 2 (2t)
Expand by 2(t/2)
Reverse (-t)
Reset
Shift
Scale

21. Interactivity: On reset
x(t) t -1 1 2 -2 3 -3 -4 -5 4 5 6 7 8 -6 -7 -8
Delay by 1(t-1)
Advance by 1(t+1)
Compress by 2 (2t)
Expand by 2(t/2)
Reverse (-t)
Reset
Shift
Scale

22. Interactivity: Reversal Compress by 2 (2t) Delay by 1(t-1)
x( – t) t -1 1 2 -2 3 -3 -4 -5 4 5 6 7 8 -6 -7 -8
Delay by 1(t-1)
Advance by 1(t+1)
Compress by 2 (2t)
Expand by 2(t/2)
Reverse (-t)
Reset
Shift
Scale

23. Reversal & Compression
Interactivity:
Reversal & Compression
x( – 2 t ) t -1 1 2 -2 3 -3 -4 -5 4 5 6 7 8 -6 -7 -8
Compress by 2 (2t)
Expand by 2(t/2)
Delay by 1(t-1)
Advance by 1(t+1)
Reverse (-t)
Reset
Shift
Scale

24. Reversal & Compression & Delaying
Interactivity:
Reversal & Compression & Delaying
x( – 2 t + 2 ) t -1 1 2 -2 3 -3 -4 -5 4 5 6 7 8 -6 -7 -8
Delay by 1(t-1)
Advance by 1(t+1)
Compress by 2 (2t)
Expand by 2(t/2)
Reverse (-t)
Reset
Scale
Shift

25. Reversal & Compression & Advancing
Interactivity:
Reversal & Compression & Advancing
x( – 2 t – 2 ) t -1 1 2 -2 3 -3 -4 -5 4 5 6 7 8 -6 -7 -8
Delay by 1(t-1)
Advance by 1(t+1)
Compress by 2 (2t)
Expand by 2(t/2)
Reverse (-t)
Reset
Scale
Shift

26. Interactivity: Reversal & Expansion Compress by 2 (2t) Delay by 1(t-1)
x( – 0.5 t ) t -1 1 2 -2 3 -3 -4 -5 4 5 6 7 8 -6 -7 -8
Compress by 2 (2t)
Expand by 2(t/2)
Delay by 1(t-1)
Advance by 1(t+1)
Reverse (-t)
Reset
Shift
Scale

27. Reversal & Expansion & Delaying
Interactivity:
Reversal & Expansion & Delaying t -1 1 2 -2 3 -3 -4 -5 4 5 6 7 8 -6 -7 -8
x( – t )
Delay by 1(t-1)
Advance by 1(t+1)
Compress by 2 (2t)
Expand by 2(t/2)
Reverse (-t)
Reset
Scale
Shift

28. Reversal & Expansion & Advancing
Interactivity:
Reversal & Expansion & Advancing t -1 1 2 -2 3 -3 -4 -5 4 5 6 7 8 -6 -7 -8
x( – 0.5 – 0.5 t )
Delay by 1(t-1)
Advance by 1(t+1)
Compress by 2 (2t)
Expand by 2(t/2)
Reverse (-t)
Reset
Shift
Scale

29. Interactivity: Reversal & Delaying Delay by 1(t-1) Advance by 1(t+1)
x( – t + 1) t -1 1 2 -2 3 -3 -4 -5 4 5 6 7 8 -6 -7 -8
Delay by 1(t-1)
Advance by 1(t+1)
Compress by 2 (2t)
Expand by 2(t/2)
Reverse (-t)
Reset
Shift
Scale

30. Reversal & Delaying & Compression
Interactivity:
Reversal & Delaying & Compression
x( – 2 t + 1) t -1 1 2 -2 3 -3 -4 -5 4 5 6 7 8 -6 -7 -8
Delay by 1(t-1)
Advance by 1(t+1)
Compress by 2 (2t)
Expand by 2(t/2)
Reverse (-t)
Reset
Shift
Scale

31. Reversal & Delaying & Expansion
Interactivity:
Reversal & Delaying & Expansion
x( – 0.5 t + 1) t -1 1 2 -2 3 -3 -4 -5 4 5 6 7 8 -6 -7 -8
Delay by 1(t-1)
Advance by 1(t+1)
Compress by 2 (2t)
Expand by 2(t/2)
Reverse (-t)
Reset
Shift
Scale

32. Interactivity: Reversal & Advancing Delay by 1(t-1) Compress by 2 (2t)
x( – t – 1) t -1 1 2 -2 3 -3 -4 -5 4 5 6 7 8 -6 -7 -8
Delay by 1(t-1)
Advance by 1(t+1)
Compress by 2 (2t)
Expand by 2(t/2)
Reverse (-t)
Reset
Shift
Scale

33. Reversal & Advancing & Compression
Interactivity:
Reversal & Advancing & Compression
x( – 2 t – 1) t -1 1 2 -2 3 -3 -4 -5 4 5 6 7 8 -6 -7 -8
Compress by 2 (2t)
Expand by 2(t/2)
Delay by 1(t-1)
Advance by 1(t+1)
Reverse (-t)
Reset
Scale
Shift

34. Reversal & Advancing & Expansion
Interactivity:
Reversal & Advancing & Expansion
x( – 0.5 t – 1) t -1 1 2 -2 3 -3 -4 -5 4 5 6 7 8 -6 -7 -8
Delay by 1(t-1)
Advance by 1(t+1)
Compress by 2 (2t)
Expand by 2(t/2)
Reverse (-t)
Reset
Shift
Scale

35. Interactivity: Delaying Delay by 1(t-1) Compress by 2 (2t)
x ( t – 1) t -1 1 2 -2 3 -3 -4 -5 4 5 6 7 8 -6 -7 -8
Delay by 1(t-1)
Advance by 1(t+1)
Compress by 2 (2t)
Expand by 2(t/2)
Reverse (-t)
Reset
Shift
Scale

36. Interactivity: Delaying & Reversal Compress by 2 (2t) Delay by 1(t-1)
x ( – t – 1) t -1 1 2 -2 3 -3 -4 -5 4 5 6 7 8 -6 -7 -8
Delay by 1(t-1)
Advance by 1(t+1)
Compress by 2 (2t)
Expand by 2(t/2)
Reverse (-t)
Reset
Shift
Scale

37. Delaying & Reversal & Compression
Interactivity:
Delaying & Reversal & Compression
(same as Reversal & Advancing & Compression)
x( – 2 t – 1) t -1 1 2 -2 3 -3 -4 -5 4 5 6 7 8 -6 -7 -8
Delay by 1(t-1)
Advance by 1(t+1)
Compress by 2 (2t)
Expand by 2(t/2)
Reverse (-t)
Reset
Shift
Scale

38. Delaying & Reversal & Expansion
Interactivity:
Delaying & Reversal & Expansion
(same as Reversal & Advancing & Expansion)
x( – 0.5 t – 1) t -1 1 2 -2 3 -3 -4 -5 4 5 6 7 8 -6 -7 -8
Compress by 2 (2t)
Expand by 2(t/2)
Delay by 1(t-1)
Advance by 1(t+1)
Reverse (-t)
Reset
Shift
Scale

39. Delaying & Compression
Interactivity:
Delaying & Compression
x( 2 t – 1) t -1 1 2 -2 3 -3 -4 -5 4 5 6 7 8 -6 -7 -8
Compress by 2 (2t)
Expand by 2(t/2)
Delay by 1(t-1)
Advance by 1(t+1)
Reverse (-t)
Reset
Shift
Scale

40. Delaying & Compression & Reversal
Interactivity:
Delaying & Compression & Reversal
x( – 2 t – 1) t -1 1 2 -2 3 -3 -4 -5 4 5 6 7 8 -6 -7 -8
Delay by 1(t-1)
Advance by 1(t+1)
Compress by 2 (2t)
Expand by 2(t/2)
Reverse (-t)
Reset
Shift
Scale

41. Delaying & Expansion Interactivity: Compress by 2 (2t) Delay by 1(t-1)
x( 0.5 t – 1) t -1 1 2 -2 3 -3 -4 -5 4 5 6 7 8 -6 -7 -8
Delay by 1(t-1)
Advance by 1(t+1)
Compress by 2 (2t)
Expand by 2(t/2)
Reverse (-t)
Reset
Shift
Scale

42. Delaying & Expansion & Reversal
Interactivity:
Delaying & Expansion & Reversal
x( – 0.5 t – 1) t -1 1 2 -2 3 -3 -4 -5 4 5 6 7 8 -6 -7 -8
Delay by 1(t-1)
Advance by 1(t+1)
Compress by 2 (2t)
Expand by 2(t/2)
Reverse (-t)
Reset
Shift
Scale

43. Interactivity: Advancing Compress by 2 (2t) Delay by 1(t-1)
x ( t + 1) t -1 1 2 -2 3 -3 -4 -5 4 5 6 7 8 -6 -7 -8
Delay by 1(t-1)
Advance by 1(t+1)
Compress by 2 (2t)
Expand by 2(t/2)
Reverse (-t)
Reset
Shift
Scale

44. Advancing & Reversal Interactivity: Compress by 2 (2t) Delay by 1(t-1)
x (– t + 1) t -1 1 2 -2 3 -3 -4 -5 4 5 6 7 8 -6 -7 -8
Delay by 1(t-1)
Advance by 1(t+1)
Compress by 2 (2t)
Expand by 2(t/2)
Reverse (-t)
Reset
Shift
Scale

45. Advancing & Reversal & Compression
Interactivity:
Advancing & Reversal & Compression
(same as Reversal & Delaying & Compression)
x (– 2t + 1) t -1 1 2 -2 3 -3 -4 -5 4 5 6 7 8 -6 -7 -8
Compress by 2 (2t)
Expand by 2(t/2)
Delay by 1(t-1)
Advance by 1(t+1)
Reverse (-t)
Reset
Shift
Scale

46. Advancing & Reversal & Expansion
Interactivity:
Advancing & Reversal & Expansion
(Reversal & Delaying & Expansion)
x( – 0.5 t + 1) t -1 1 2 -2 3 -3 -4 -5 4 5 6 7 8 -6 -7 -8
Delay by 1(t-1)
Advance by 1(t+1)
Compress by 2 (2t)
Expand by 2(t/2)
Reverse (-t)
Reset
Scale
Shift

47. Advancing & Compression
Interactivity:
Advancing & Compression
x (2 t + 1) t -1 1 2 -2 3 -3 -4 -5 4 5 6 7 8 -6 -7 -8
Delay by 1(t-1)
Advance by 1(t+1)
Compress by 2 (2t)
Expand by 2(t/2)
Reverse (-t)
Reset
Shift
Scale

48. Advancing & Compression & Reversal
Interactivity:
Advancing & Compression & Reversal
x (– 2 t + 1) t -1 1 2 -2 3 -3 -4 -5 4 5 6 7 8 -6 -7 -8
Delay by 1(t-1)
Advance by 1(t+1)
Compress by 2 (2t)
Expand by 2(t/2)
Reverse (-t)
Reset
Shift
Scale

49. Interactivity: Advancing & Expansion Delay by 1(t-1)
x (0.5 t + 1) t -1 1 2 -2 3 -3 -4 -5 4 5 6 7 8 -6 -7 -8
Delay by 1(t-1)
Advance by 1(t+1)
Compress by 2 (2t)
Expand by 2(t/2)
Reverse (-t)
Reset
Shift
Scale

50. Advancing & Expansion & Reversal Interactivity:
x (0.5 t + 1) t -1 1 2 -2 3 -3 -4 -5 4 5 6 7 8 -6 -7 -8
Delay by 1(t-1)
Advance by 1(t+1)
Compress by 2 (2t)
Expand by 2(t/2)
Reverse (-t)
Reset
Shift
Scale

51. Compression Interactivity: Delay by 1(t-1) Compress by 2 (2t)
x ( 2 t ) t -1 1 2 -2 3 -3 -4 -5 4 5 6 7 8 -6 -7 -8
Delay by 1(t-1)
Advance by 1(t+1)
Compress by 2 (2t)
Expand by 2(t/2)
Reverse (-t)
Reset
Shift
Scale

52. Compression & Reversal
Interactivity:
Compression & Reversal
x( – 2 t ) t -1 1 2 -2 3 -3 -4 -5 4 5 6 7 8 -6 -7 -8
Delay by 1(t-1)
Advance by 1(t+1)
Compress by 2 (2t)
Expand by 2(t/2)
Reverse (-t)
Reset
Shift
Scale

53. Compression & Reversal & Delaying Interactivity:
(Reversal & Compression & Delaying)
x( – 2 t + 2 ) t -1 1 2 -2 3 -3 -4 -5 4 5 6 7 8 -6 -7 -8
Delay by 1(t-1)
Advance by 1(t+1)
Compress by 2 (2t)
Expand by 2(t/2)
Reverse (-t)
Reset
Shift
Scale

54. Compression & Reversal & Advancing
Interactivity:
Compression & Reversal & Advancing
(Reversal & Compression & Advancing)
x( – 2 t – 2 ) t -1 1 2 -2 3 -3 -4 -5 4 5 6 7 8 -6 -7 -8
Delay by 1(t-1)
Advance by 1(t+1)
Compress by 2 (2t )
Expand by 2(t/2)
Reverse (-t)
Reset
Shift
Scale

55. Compression & Delaying
Interactivity:
Compression & Delaying
x(2 t – 2 ) t -1 1 2 -2 3 -3 -4 -5 4 5 6 7 8 -6 -7 -8
Delay by 1(t-1)
Advance by 1(t+1)
Compress by 2 (2t)
Expand by 2(t/2)
Reverse (-t)
Reset
Shift
Scale

56. Compression & Delaying & Reversal
Interactivity:
Compression & Delaying & Reversal
x (– 2 t + 2 ) t -1 1 2 -2 3 -3 -4 -5 4 5 6 7 8 -6 -7 -8
Delay by 1(t-1)
Advance by 1(t+1)
Compress by 2 (2t)
Expand by 2(t/2)
Reverse (-t)
Reset
Shift
Scale

57. Compression & Advancing
Interactivity:
Compression & Advancing
x(2 t + 2 ) t -1 1 2 -2 3 -3 -4 -5 4 5 6 7 8 -6 -7 -8
Delay by 1(t-1 )
Advance by 1(t+1)
Compress by 2 (2t)
Expand by 2(t/2)
Reverse (-t)
Reset
Shift
Scale

58. Compression & Advancing & Reverse Interactivity:
x(–2 t + 2 ) t -1 1 2 -2 3 -3 -4 -5 4 5 6 7 8 -6 -7 -8
Delay by 1(t-1)
Advance by 1(t+1)
Compress by 2 (2t)
Expand by 2(t/2)
Reverse (-t)
Reset
Shift
Scale

59. Interactivity: Expansion Compress by 2 (2t) Expand by 2(t/2)
x(0.5 t ) t -1 1 2 -2 3 -3 -4 -5 4 5 6 7 8 -6 -7 -8
Compress by 2 (2t)
Expand by 2(t/2)
Delay by 1(t-1)
Advance by 1(t+1)
Reverse (-t)
Reset
Shift
Scale

60. Expansion & Reversal Interactivity: Delay by 1(t-1) Compress by 2 (2t)
x(–0.5 t ) t -1 1 2 -2 3 -3 -4 -5 4 5 6 7 8 -6 -7 -8
Delay by 1(t-1)
Advance by 1(t+1)
Compress by 2 (2t)
Expand by 2(t/2)
Reverse (-t)
Reset
Shift
Scale

61. Expansion & Reversal & Delaying
Interactivity:
Expansion & Reversal & Delaying
(Reversal & Expansion & Delaying)
x(–0.5 t + 0.5) t -1 1 2 -2 3 -3 -4 -5 4 5 6 7 8 -6 -7 -8
Delay by 1(t-1)
Advance by 1(t+1)
Compress by 2 (2t)
Expand by 2(t/2)
Reverse (-t)
Reset
Shift
Scale

62. Expansion & Reversal & Advancing
Interactivity:
Expansion & Reversal & Advancing
(Reversal & Expansion & Advancing)
x(–0.5 t – 0.5) t -1 1 2 -2 3 -3 -4 -5 4 5 6 7 8 -6 -7 -8
Delay by 1(t-1)
Advance by 1(t+1)
Compress by 2 (2t)
Expand by 2(t/2)
Reverse (-t)
Reset
Shift
Scale

63. Interactivity: Expansion & Delaying Delay by 1(t-1) Advance by 1(t+1)
x(0.5 t – 0.5 ) t -1 1 2 -2 3 -3 -4 -5 4 5 6 7 8 -6 -7 -8
Delay by 1(t-1)
Advance by 1(t+1)
Compress by 2 (2t)
Expand by 2(t/2)
Reverse (-t)
Reset
Shift
Scale

64. Expansion & Delaying & Reversal
Interactivity:
Expansion & Delaying & Reversal
x( – 0.5 t –0.5 ) t -1 1 2 -2 3 -3 -4 -5 4 5 6 7 8 -6 -7 -8
Delay by 1(t-1)
Advance by 1(t+1)
Compress by 2 (2t)
Expand by 2(t/2)
Reverse (-t)
Reset
Shift
Scale

65. Interactivity: Expansion & Advancing Delay by 1(t-1) Advance by 1(t+1)
x(0.5 t + 0.5) t -1 1 2 -2 3 -3 -4 -5 4 5 6 7 8 -6 -7 -8
Delay by 1(t-1)
Advance by 1(t+1)
Compress by 2 (2t)
Expand by 2(t/2)
Reverse (-t)
Reset
Shift
Scale

66. Expansion & Advancing & Reversal
Interactivity:
Expansion & Advancing & Reversal
x (– 0.5 t + 0.5) t -1 1 2 -2 3 -3 -4 -5 4 5 6 7 8 -6 -7 -8
Delay by 1(t-1)
Advance by 1(t+1)
Compress by 2 (2t)
Expand by 2(t/2)
Reverse (-t)
Reset
Shift
Scale

67. Questionnaire: t x(t) t y(t) y(t) t t x(t) a b c d 2. x(-0.5 t+3)
1. Identify the correct signal transformations equation that would generate signal y(t) from x(t). t
x(t) -1 1 2 -2 3 -3 t
y(t) -1 1 2 -2 3 -3
2. x(-0.5 t+3)
0.5 x(-2 t +3)
2. x(-2t + 6)
2 x( -0.5 t + 1.5)
2. Identify the correct sequence of signal transformations that would generate signal y(t) from x(t).
y(t) t -1 1 2 -2 3 -3 4 -4 t
x(t) -1 1 2 -2 3 -3
Delaying by 1
Time scaling by 3
Amplitude scaling by ½
Time reversal
Delaying by 1
Time scaling by 3
Amplitude scaling by -½
Delaying by 1
Time scaling by 1/3
Amplitude scaling by ½
Time reversal
Delaying by 1
Time scaling by 1/3
Amplitude scaling by -½ a b c d

68. Questionnaire: a c b d a b c d x(t)-1 1 2 -2 3 -3
3. Which waveform correctly shows manipulated signal y(t) = 2x(2t-1),
where x(t) is as shown here. t
y(t) -1 1 2 -2 3 -3 t
y(t) -1 1 2 -2 3 -3 t
y(t) -1 1 2 -2 3 -3 t
y(t) -1 1 2 -2 3 -3 a c b d
4. Which sequence of signal transformation will not generate, Y(t) = 0.3 x( – 0.2 t – 0.4)
Time reversal
Time advancing by 0.4
Time scaling by 0.2
Amplitude scaling by 0.3
Amplitude scaling by 0.3
Time reversal
Time scaling by 0.2
Time delaying by 0.4
Amplitude scaling by 0.3
Time delaying by 0.4
Time scaling by 0.2
Time reversal
Amplitude scaling by 0.3
Time reversal
Time scaling by 0.2
Time advancing by 2 a b c d
5. When a reversed signal is delayed, it is
d. Moved away from t=0
a. Shifted to left
b. Shifted to right
c. Moved towards t=0

69. Glossary
Signal: Any physical quantity that varies with time space, or any other variable.
In signal processing, a time signal is electrical representation of a time varying quantity.
Signal Transformation: Arithmetic operations performed on signal amplitude of time that manipulate the signal.
Shifting: The shifting of a signal in time.
Delaying: Causing the signal to occur late (shift to right) by subtracting a positive constant from time parameter.
Advancing: Causing the signal to occur late (shift to right) by adding a positive constant to time parameter..
Scaling: Compressing or expanding a signal by multiplying the time variable by some quantity.
Compression: Narrowing the signal in time, by multiplying time parameter by a positive constant grater than 1.
Expansion: Widening the signal in time, by multiplying time parameter by a positive constant lower than 1.
Reversal: Flipping the signal over amplitude axis by multiplying time parameter by -1.

70. References
John G. Proakis and Dimitris G. Manolakis, Digital Signal Processing: Principles, Algorithms, and Applications,
Prentice Hall, 3rd edition, 1996
M. H. Hayes, Digital Signal Processing, McGraw Hill, 2007
S Salivahanan, A Vallavaraj, C Gnanapriya, Digital Signal Processing, McGraw Hill, 2007