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

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.

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:

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.

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

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

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

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

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

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

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

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

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

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

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

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

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 + 0.5 ) 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

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

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 + 0.5 ) 61 41 X (– 0.5 t – 0.5 ) 62 42 X ( 0.5 t – 0.5 ) 63 43 64 44 X ( 0.5 t + 0.5 ) 65 45 66 *For animator’s reference

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

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

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

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

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

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

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

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( – 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 Scale Shift

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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.

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