Continuous Time Convolution In this animation, the continuous time convolution of signals is discussed. Convolution is the operation to obtain response of a linear system to input x(t). The output y(t) is given as a weighted superposition of impulse responses, time shifted by ‏ Course Name: Signals and Systems Level: UG Author Phani Swathi Chitta Mentor Prof. Saravanan Vijayakumaran

Learning Objectives After interacting with this Learning Object, the learner will be able to: Explain the convolution of two continuous time signals

1 2 3 4 5 Definitions of the components/Keywords: Convolution of two signals: Let x(t) and h(t) are two continuous signals to be convolved. The convolution of two signals is denoted by which means where t is the variable of integration. 2 3 4 5 3

** 1 Master Layout 2 3 4 5 y(t) 1 t -2 2 3 22 2 1 t f(t) --22 1 2 g(t) Signals taken to convolve 2 ** 22 2 1 t f(t) --22 1 2 g(t) 3 Output of the convolution y(t) 1 4 t -2 2 3 5

3 Step 1: 1 2 4 5 22 2 1 t f(t) = 2 --22 1 2 g(t)= -t+1 Instruction for the animator Text to be displayed in the working area (DT) The first point in DT has to appear before the figures. Then the blue figure has to appear. After that the red figure has to appear. After the figures, the next point in DT has to appear. f(t) and g(t) are the two continuous signals to be convolved. The convolution of the signals is denoted by which means where t is a dummy variable. 4 5

3 Step 2: 1 2 4 5 Fig. a Fig. b 2 t f(t) 1 -2 g(-t) 2 t f(t) 1 -1 + t Instruction for the animator Text to be displayed in the working area (DT) The figure in blue in fig. a has to appear then its label should appear. Then the red figure has to appear. After that the labeling of red figure has to appear. In parallel to the fig. the text in DT has to appear. First two sentences in DT has to appear with fig. a The last sentence should appear with fig. b. The signal f(t ) is shown The reversed version of g(t) i.e., g(-t ) is shown The shifted version of g(-t) i.e., g(t-t) is shown 4 5

Step 3: Calculation of y(t) in five stages 1 Stage - I : t < -2 2 t f(t) 1 -1 + t -2 g(t-t) 2 t 3 Instruction for the animator Text to be displayed in the working area (DT) The figure in blue has to appear then its label should appear. Then the red figure has to appear. After that the labeling of red figure has to appear. In parallel to the fig. the text in DT has to appear. After the figures, the 3, 4 lines in DT should appear. The signal f(t ) is shown The reversal and shifted version of g(t) i.e., g(t-t ) is shown Two functions do not overlap Area under the product of the functions is zero 4 5

3 Step 4: 1 2 4 5 Stage - II : -2 ≤ t < -1 2 t f(t) 1 -1 + t -2 g(t-t) 2 t 3 Instruction for the animator Text to be displayed in the working area (DT) The figure in blue has to appear then its label should appear. Then the red figure has to appear. After that the labeling of red figure has to appear. In parallel to the fig. the text in DT has to appear. After the figures, the 3, 4 lines in DT should appear. The signal f(t ) is shown The reversal and shifted version of g(t) i.e., g(t-t ) is shown Part of g(t-t ) overlaps part of f(t ) Area under the product 4 5

3 Step 5: 1 2 4 5 Stage - III : -1 ≤ t < 2 2 t f(t) 1 -1 + t -2 g(t-t) 2 t 3 Instruction for the animator Text to be displayed in the working area (DT) The figure in blue has to appear then its label should appear. Then the red figure has to appear. After that the labeling of red figure has to appear. In parallel to the fig. the text in DT has to appear. After the figures, the 3, 4 lines in DT should appear. The signal f(t ) is shown The reversal and shifted version of g(t) i.e., g(t-t ) is shown g(t-t ) completely overlaps f(t ) Area under the product 4 5

3 Step 6: 1 2 4 5 Stage - IV : 2 ≤ t < 3 2 t f(t) 1 -1 + t -2 g(t-t) 2 2 t 3 Instruction for the animator Text to be displayed in the working area (DT) The figure in blue has to appear then its label should appear. Then the red figure has to appear. After that the labeling of red figure has to appear. In parallel to the fig. the text in DT has to appear. After the figures, the 3, 4 lines in DT should appear. The signal f(t ) is shown The reversal and shifted version of g(t) i.e., g(t-t ) is shown Part of g(t-t ) overlaps part of f(t ) Area under the product 4 5

3 Step 7: 1 2 4 5 Stage - V : t ≥ 3 2 t f(t) 1 -1 + t -2 g(t-t) t Instruction for the animator Text to be displayed in the working area (DT) The figure in blue has to appear then its label should appear. Then the red figure has to appear. After that the labeling of red figure has to appear. In parallel to the fig. the text in DT has to appear. After the figures, the 3, 4 lines in DT should appear. The signal f(t ) is shown The reversal and shifted version of g(t) i.e., g(t-t ) is shown Two functions do not overlap Area under the product of the functions is zero 4 5

3 Step 8: 1 2 4 5 y(t) 1 t -2 2 3 Output of Convolution 2 3 3 Instruction for the animator Text to be displayed in the working area (DT) The figure in green has to appear then its label should appear. In parallel to the fig. the text in DT has to appear. After the figure, the equations in DT should appear . The signal y(t) is shown 4 5

Electrical Engineering Slide 1 Slide 3 Slide 24-26 Slide 28 Slide 27 Introduction Definitions Analogy Test your understanding (questionnaire)‏ Lets Sum up (summary)‏ Want to know more… (Further Reading)‏ f(t ) and g(t-t ) Interactivity: +1 Try it yourself g(t) f(t) +1 +1 t -1 t t -1 -1 f(t ) g(t-t ) Select Select +1 +1 +1 t -1 -1 +1 +1 t -1 13 The four signals must be repeated under select for both f(t) and g(t) Credits 13

Test your understanding The signal selected under f(t) must be shown Electrical Engineering Slide 1 Slide 3 Slide 24-26 Slide 28 Slide 27 Introduction Definitions Analogy Test your understanding (questionnaire)‏ Lets Sum up (summary)‏ Want to know more… (Further Reading)‏ f(t ) and g(t-t ) Interactivity: +1 Try it yourself g(t) f(t) +1 +1 t -1 t t -1 -1 f(t ) g(t-t ) Select Select +1 +1 +1 t -1 -1 +1 +1 t -1 14 The signal selected under f(t) must be shown Credits

Test your understanding The signal selected under g(t) must be shown Electrical Engineering Slide 1 Slide 3 Slide 24-26 Slide 28 Slide 27 Introduction Definitions Analogy Test your understanding (questionnaire)‏ Lets Sum up (summary)‏ Want to know more… (Further Reading)‏ f(t ) and g(t-t ) Interactivity: +1 Try it yourself g(t) f(t) +1 +1 t -1 t t -1 -1 f(t ) g(t-t ) Select Select +1 +1 +1 t -1 -1 +1 +1 t -1 15 The signal selected under g(t) must be shown Credits

Test your understanding Electrical Engineering Slide 1 Slide 3 Slide 24-26 Slide 28 Slide 27 Introduction Definitions Analogy Test your understanding (questionnaire)‏ Lets Sum up (summary)‏ Want to know more… (Further Reading)‏ f(t ) and g(t-t ) Interactivity: +1 Try it yourself g(t) f(t) +1 +1 t -1 t t -1 -1 f(t ) g(t-t ) Select Select +1 +1 +1 t -1 -1 +1 +1 t -1 16 The red figure is the shifted and reversed version of g(t) The slides 16-21 should be shown in a smooth fashion Credits

Test your understanding Electrical Engineering Slide 1 Slide 3 Slide 24-26 Slide 28 Slide 27 Introduction Definitions Analogy Test your understanding (questionnaire)‏ Lets Sum up (summary)‏ Want to know more… (Further Reading)‏ f(t ) and g(t-t ) Interactivity: +1 Try it yourself g(t) f(t) +1 +1 t -1 t t -1 -1 f(t ) g(t-t ) Select Select +1 +1 +1 t -1 -1 +1 +1 t -1 17 Credits

Test your understanding Electrical Engineering Slide 1 Slide 3 Slide 24-26 Slide 28 Slide 27 Introduction Definitions Analogy Test your understanding (questionnaire)‏ Lets Sum up (summary)‏ Want to know more… (Further Reading)‏ f(t ) and g(t-t ) Interactivity: +1 Try it yourself g(t) f(t) +1 +1 t -1 t t -1 -1 f(t ) g(t-t ) Select Select +1 +1 +1 t -1 -1 +1 +1 t -1 18 Credits

Test your understanding Electrical Engineering Slide 1 Slide 3 Slide 24-26 Slide 28 Slide 27 Introduction Definitions Analogy Test your understanding (questionnaire)‏ Lets Sum up (summary)‏ Want to know more… (Further Reading)‏ f(t ) and g(t-t ) Interactivity: +1 Try it yourself g(t) f(t) +1 +1 t -1 t t -1 -1 f(t ) g(t-t ) Select Select +1 +1 +1 t -1 -1 +1 +1 t -1 19 Credits

Test your understanding Electrical Engineering Slide 1 Slide 3 Slide 24-26 Slide 28 Slide 27 Introduction Definitions Analogy Test your understanding (questionnaire)‏ Lets Sum up (summary)‏ Want to know more… (Further Reading)‏ f(t ) and g(t-t ) Interactivity: +1 Try it yourself g(t) f(t) +1 +1 t -1 t t -1 -1 f(t ) g(t-t ) Select Select +1 +1 +1 t -1 -1 +1 +1 t -1 20 Credits

Test your understanding Electrical Engineering Slide 1 Slide 3 Slide 24-26 Slide 28 Slide 27 Introduction Definitions Analogy Test your understanding (questionnaire)‏ Lets Sum up (summary)‏ Want to know more… (Further Reading)‏ f(t ) and g(t-t ) Interactivity: +1 Try it yourself g(t) f(t) +1 +1 t -1 t t -1 -1 f(t ) g(t-t ) Select Select +1 +1 +1 t -1 -1 +1 +1 t -1 21 Credits

* * * Electrical Engineering Slide 1 Slide 3 Slide 24-26 Slide 28 Introduction Definitions Analogy Test your understanding (questionnaire)‏ Lets Sum up (summary)‏ Want to know more… (Further Reading)‏ f(t) +1 +1 +1 * t -1 -1 -1 f(t) +1 +1 * t -1 +1 +1 +1 * -1 -1 -1 22 The same procedure is done to the above given combination of signals Credits

* * Electrical Engineering Slide 1 Slide 3 Slide 24-26 Slide 28 Introduction Definitions Analogy Test your understanding (questionnaire)‏ Lets Sum up (summary)‏ Want to know more… (Further Reading)‏ +1 +1 * -1 +1 +1 * 23 The same procedure is done to the above given combination of signals Credits

Questionnaire 1 1. If the unit-impulse response of an LTI system and the input signal both are rectangular pulses, then the output will be a Answers: a) rectangular pulse b) triangular pulse c) ramp function d)‏ none of the above 2. Find Convolution * Answers: a) b) 2 x(t) δ(t-5) 3 5 4 5 5 5

Questionnaire 1 3. If impulse response and input signal both are unit step responses, then the output will be * Answers: a) Triangular pulse b) Unit step function c) Ramp function d)‏ None of the above 4. The convolution integral is given by i) ii) Hint: let Answers: a) i b) ii c) Both i and ii d)‏either i or ii 2 3 4 5

Questionnaire 1 5. If h(t) is a unit-step function and x(t) is a unit-ramp function, then the output y(t) will be a Answers: a) step function b) ramp function c) Triangular pulse d) Quadratic function‏ 2 3 4 5

Links for further reading Reference websites: Books: Signals & Systems – Alan V. Oppenheim, Alan S. Willsky, S. Hamid Nawab, PHI learning, Second edition. Signals and Systems – Simon Haykin, Barry Van Veen, John Wiley & Sons, Inc. Research papers:

Summary The convolution operation is used to obtain the output of linear time – invariant system in response to an arbitrary input. In continuous time, the representation of signals is taken to be the weighted integrals of shifted unit impulses. The convolution integral of two continuous signals is represented as where The convolution integral provides a concise, mathematical way to express the output of an LTI system based on an arbitrary continuous-time input signal and the system‘s response.