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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. Transformation of Continuous Time Signals. Course Name: Signals and Systems. Authors Amita Shinde Mentor Sarvanan Vijaykumaran.

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Transformation of Continuous Time Signals

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

  2. 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. Learning Objectives

  3. x(t) x(– (t–1))  x(– t+1); x(– t) Delayed by 1 Delayed by 1 Reversal & Delaying: x(t–1) x(t) x((– t)–1)  x(– t–1) Reversed Reversed Scaled by 2 Delayed by 1 x(t) x(2 t) x(2 (t–1))  x(2t–2); Scaling & Delaying: Delayed by 1 Scaled by 2 x( t–1) x(t) x((2t)–1)  x(2t –1) Theory button content 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.

  4. Electrical Engineering > Signals and Systems 1 1 0 0 t t -6 -6 1 1 -3 -3 -1 6 6 -1 -4 -4 2 2 5 2 2 5 -5 -2 -5 -2 4 4 3 3 -1 -1 Transformation of Continuous Time Signals Master Layout 1: Linear menu Input signal expression x(t) • 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 manipulated signal expression (different for each animation) x(t-3) Input signal x(t)= 0, for t<4 = t-5, for 4<t<6 = 0, for t>6 o/p signal eqn (varies for each animation) Manipulated signal 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. ) s/g before 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. 2 1 0 1 -3 -1 -4 2 2 4 -2 t 3 –1 –2 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) x(t)= 0, for t<1 = t-2, for 1<t<3 = 0, for t>3

  6. 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 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. 2 2 1 1 0 0 1 1 -3 -3 -1 -1 -4 -4 2 2 2 2 4 4 -2 -2 t t 3 3 –1 –1 –2 –2 1. Amplification x(t) x(t)= 0, for t<1 = t-2, for 1<t<3 = 0, for t>3 2.x(t) 2.x(t)= 0, for t<1 = 2t-4, for 1<t<3 = 0, for t>3

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

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

  10. 2 2 1 1 0 0 1 1 -3 -3 -1 -1 -4 -4 2 2 2 2 4 4 -2 -2 t t 3 3 –1 –1 –2 –2 Attenuation with Negative Value x(t) x(t)= 0, for t<1 = t-2, for 1<t<3 = 0, for t>3 -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) 1 0 t -6 1 -3 6 -1 -4 2 2 5 -5 -2 4 3 -1 1 0 t -6 1 -3 6 -1 -4 2 2 5 -5 -2 4 3 -1 x(t)= 0, for t<1 = t-2, for 1<t<3 = 0, for t>3 x( – t) x(-t)= 0, for t>1 = -t-2, for -1<t<-3 = 0, for t<-3

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

  13. x(t) 1 0 t -6 1 -3 6 -1 -4 2 2 5 -5 -2 4 3 -1 1 0 t -6 1 -3 6 -1 -4 2 2 5 -5 -2 4 3 -1 5. Time Shifting : Advancing x(t)= 0, for t<1 = t-2, for 1<t<3 = 0, for t>3 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) 1 0 t -6 1 -3 6 -1 -4 2 2 5 -5 -2 4 3 -1 1 0 t -6 1 -3 6 -1 -4 2 2 5 -5 -2 4 3 -1 x(t)= 0, for t<1 = t-2, for 1<t<3 = 0, for t>3 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) 1 0 t -6 1 -3 6 -1 -4 2 2 5 -5 -2 4 3 -1 1 0 t -6 1 -3 6 -1 -4 2 2 5 -5 -2 4 3 -1 x(t)= 0, for t<1 = t-2, for 1<t<3 = 0, for t>3 x(0.5 t ) x(0.5t)= 0, for t<2 = 0.5t-2, for 2<t<6 = 0, for t>6

  16. Electrical Engineering > Signals and Systems 1 0 -7 -6 -8 -3 1 6 7 -1 -4 2 2 5 -5 -2 8 4 3 t -1 Transformation of Continuous Time Signals Master Layout 2: Interactive menu Input signal expression x(t) Reverse (-t) X(t-3) • 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 *For animator’s reference

  18. Interactivity Menu *For animator’s reference

  19. Interactivity Menu *For animator’s reference

  20. 1 0 -7 -6 -8 1 -3 6 7 -1 -4 2 2 5 -5 -2 8 4 3 t -1 Interactivity: Initial screen x(t) • Compress by 2 (2t) • Expand by 2(t/2) • Delay by 1(t-1) • Advance by 1(t+1) Reverse (-t) Reset Scale Shift

  21. 1 0 -7 -6 -8 1 -3 6 7 -1 -4 2 2 5 -5 -2 8 4 3 t -1 Interactivity: On reset x(t) • Compress by 2 (2t) • Expand by 2(t/2) • Delay by 1(t-1) • Advance by 1(t+1) Reverse (-t) Reset Scale Shift

  22. 1 0 -7 -6 -8 1 -3 6 7 -1 -4 2 2 5 -5 -2 8 4 3 t -1 Interactivity: Reversal x( – t) • Compress by 2 (2t) • Expand by 2(t/2) • Delay by 1(t-1) • Advance by 1(t+1) Reverse (-t) Reset Scale Shift

  23. Interactivity: Reversal & Compression 1 0 -7 -6 -8 1 -3 6 7 -1 -4 2 2 5 -5 -2 8 4 3 t -1 x( – 2 t ) Compress by 2 (2t) Expand by 2(t/2) • Delay by 1(t-1) • Advance by 1(t+1) Reverse (-t) Reset Scale Shift

  24. 1 0 -7 -6 -8 1 -3 6 7 -1 -4 2 2 5 -5 -2 8 4 3 t -1 Reversal & Compression & Delaying Interactivity: x( – 2 t + 2 ) Compress by 2 (2t) Expand by 2(t/2) Delay by 1(t-1) Advance by 1(t+1) Reverse (-t) Reset Scale Shift

  25. Interactivity: 1 0 -7 -6 -8 1 -3 6 7 -1 -4 2 2 5 -5 -2 8 4 3 t -1 Reversal & Compression & Advancing x( – 2 t – 2 ) 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. 1 0 -7 -6 -8 1 -3 6 7 -1 -4 2 2 5 -5 -2 8 4 3 t -1 Interactivity: Reversal & Expansion x( – 0.5 t ) 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. 1 0 -7 -6 -8 1 -3 6 7 -1 -4 2 2 5 -5 -2 8 4 3 t -1 Interactivity: Reversal & Expansion & Delaying x( – 0.5 + 0.5 t ) Compress by 2 (2t) Expand by 2(t/2) Delay by 1(t-1) Advance by 1(t+1) Reverse (-t) Reset Scale Shift

  28. Interactivity: 1 0 -7 -6 -8 1 -3 6 7 -1 -4 2 2 5 -5 -2 8 4 3 t -1 Reversal & Expansion & Advancing x( – 0.5 – 0.5t ) Compress by 2 (2t) Expand by 2(t/2) Delay by 1(t-1) Advance by 1(t+1) Reverse (-t) Reset Scale Shift

  29. 1 0 -7 -6 -8 1 -3 6 7 -1 -4 2 2 5 -5 -2 8 4 3 t -1 Interactivity: Reversal & Delaying x( – t + 1) Compress by 2 (2t) Expand by 2(t/2) Delay by 1(t-1) Advance by 1(t+1) Reverse (-t) Reset Shift Scale

  30. 1 0 -7 -6 -8 1 -3 6 7 -1 -4 2 2 5 -5 -2 8 4 3 t -1 Reversal & Delaying & Compression Interactivity: x( – 2 t + 1) Compress by 2 (2t) Expand by 2(t/2) Delay by 1(t-1) Advance by 1(t+1) Reverse (-t) Reset Scale Shift

  31. 1 0 -7 -6 -8 1 -3 6 7 -1 -4 2 2 5 -5 -2 8 4 3 t -1 Interactivity: Reversal & Delaying & Expansion x( – 0.5 t + 1) Compress by 2 (2t) Expand by 2(t/2) Delay by 1(t-1) Advance by 1(t+1) Reverse (-t) Reset Scale Shift

  32. 1 0 -7 -6 -8 1 -3 6 7 -1 -4 2 2 5 -5 -2 8 4 3 t -1 Interactivity: Reversal & Advancing x( – t – 1) • Compress by 2 (2t) • Expand by 2(t/2) Delay by 1(t-1) Advance by 1(t+1) Reverse (-t) Reset Scale Shift

  33. 1 0 -7 -6 -8 1 -3 6 7 -1 -4 2 2 5 -5 -2 8 4 3 t -1 Interactivity: Reversal & Advancing & Compression x( – 2 t – 1) 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. 1 0 -7 -6 -8 1 -3 6 7 -1 -4 2 2 5 -5 -2 8 4 3 t -1 Interactivity: Reversal & Advancing & Expansion x( – 0.5 t – 1) Compress by 2 (2t) Expand by 2(t/2) Delay by 1(t-1) Advance by 1(t+1) Reverse (-t) Reset Shift Scale

  35. Interactivity: 1 0 -7 -6 -8 1 -3 6 7 -1 -4 2 2 5 -5 -2 8 4 3 t -1 Delaying x ( t – 1) • Compress by 2 (2t) • Expand by 2(t/2) Delay by 1(t-1) Advance by 1(t+1) Reverse (-t) Reset Scale Shift

  36. 1 0 -7 -6 -8 1 -3 6 7 -1 -4 2 2 5 -5 -2 8 4 3 t -1 Interactivity: Delaying & Reversal x ( – t – 1) • Compress by 2 (2t) • Expand by 2(t/2) Delay by 1(t-1) Advance by 1(t+1) Reverse (-t) Reset Scale Shift

  37. Interactivity: 1 0 -7 -6 -8 1 -3 6 7 -1 -4 2 2 5 -5 -2 8 4 3 t -1 Delaying & Reversal & Compression (same as Reversal & Advancing & Compression) x( – 2 t – 1) Compress by 2 (2t) Expand by 2(t/2) Delay by 1(t-1) Advance by 1(t+1) Reverse (-t) Reset Scale Shift

  38. 1 0 -7 -6 -8 1 -3 6 7 -1 -4 2 2 5 -5 -2 8 4 3 t -1 Interactivity: Delaying & Reversal & Expansion (same as Reversal & Advancing & Expansion) x( – 0.5 t – 1) Compress by 2 (2t) Expand by 2(t/2) Delay by 1(t-1) Advance by 1(t+1) Reverse (-t) Reset Scale Shift

  39. 1 0 -7 -6 -8 1 -3 6 7 -1 -4 2 2 5 -5 -2 8 4 3 t -1 Interactivity: Delaying & Compression x( 2 t – 1) Compress by 2 (2t) Expand by 2(t/2) Delay by 1(t-1) Advance by 1(t+1) Reverse (-t) Reset Scale Shift

  40. 1 0 -7 -6 -8 1 -3 6 7 -1 -4 2 2 5 -5 -2 8 4 3 t -1 Interactivity: Delaying & Compression & Reversal x( –2 t – 1) Compress by 2 (2t) Expand by 2(t/2) Delay by 1(t-1) Advance by 1(t+1) Reverse (-t) Reset Scale Shift

  41. 1 0 -7 -6 -8 1 -3 6 7 -1 -4 2 2 5 -5 -2 8 4 3 t -1 Delaying & Expansion Interactivity: x( 0.5 t – 1) Compress by 2 (2t) Expand by 2(t/2) Delay by 1(t-1) Advance by 1(t+1) Reverse (-t) Reset Scale Shift

  42. 1 0 -7 -6 -8 1 -3 6 7 -1 -4 2 2 5 -5 -2 8 4 3 t -1 Interactivity: Delaying & Expansion & Reversal x( – 0.5 t – 1) Compress by 2 (2t) Expand by 2(t/2) Delay by 1(t-1) Advance by 1(t+1) Reverse (-t) Reset Scale Shift

  43. Interactivity: 1 0 -7 -6 -8 1 -3 6 7 -1 -4 2 2 5 -5 -2 8 4 3 t -1 Advancing x ( t + 1) • Compress by 2 (2t) • Expand by 2(t/2) Delay by 1(t-1) Advance by 1(t+1) Reverse (-t) Reset Scale Shift

  44. 1 0 -7 -6 -8 1 -3 6 7 -1 -4 2 2 5 -5 -2 8 4 3 t -1 Advancing & Reversal Interactivity: x (–t + 1) • Compress by 2 (2t) • Expand by 2(t/2) Delay by 1(t-1) Advance by 1(t+1) Reverse (-t) Reset Scale Shift

  45. 1 0 -7 -6 -8 1 -3 6 7 -1 -4 2 2 5 -5 -2 8 4 3 t -1 Interactivity: Advancing & Reversal & Compression (same as Reversal & Delaying & Compression) x (– 2t + 1) Compress by 2 (2t) Expand by 2(t/2) Delay by 1(t-1) Advance by 1(t+1) Reverse (-t) Reset Scale Shift

  46. 1 0 -7 -6 -8 1 -3 6 7 -1 -4 2 2 5 -5 -2 8 4 3 t -1 Interactivity: Advancing & Reversal & Expansion (Reversal & Delaying & Expansion) x( – 0.5 t + 1) Compress by 2 (2t) Expand by 2(t/2) Delay by 1(t-1) Advance by 1(t+1) Reverse (-t) Reset Scale Shift

  47. 1 0 -7 -6 -8 1 -3 6 7 -1 -4 2 2 5 -5 -2 8 4 3 t -1 Advancing & Compression Interactivity: x (2t + 1) Delay by 1(t-1) Advance by 1(t+1) Compress by 2 (2t) Expand by 2(t/2) Reverse (-t) Reset Scale Shift

  48. 1 0 -7 -6 -8 1 -3 6 7 -1 -4 2 2 5 -5 -2 8 4 3 t -1 Advancing & Compression & Reversal Interactivity: x (–2t + 1) Compress by 2 (2t) Expand by 2(t/2) Delay by 1(t-1) Advance by 1(t+1) Reverse (-t) Reset Scale Shift

  49. 1 0 -7 -6 -8 1 -3 6 7 -1 -4 2 2 5 -5 -2 8 4 3 t -1 Interactivity: Advancing & Expansion x (0.5t + 1) Compress by 2 (2t) Expand by 2(t/2) Delay by 1(t-1) Advance by 1(t+1) Reverse (-t) Reset Scale Shift

  50. 1 0 -7 -6 -8 1 -3 6 7 -1 -4 2 2 5 -5 -2 8 4 3 t -1 Advancing & Expansion & Reversal Interactivity: x (0.5t + 1) Compress by 2 (2t) Expand by 2(t/2) Delay by 1(t-1) Advance by 1(t+1) Reverse (-t) Reset Scale Shift

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