1 / 28

Continuous Time Convolution

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 ‏.

geordi
Download Presentation

Continuous Time Convolution

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 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

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

  3. Definitions of the components/Keywords: 1 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

  4. f(t) g(t) 2 2 1 ** t t --22 22 1 Master Layout 1 Signals taken to convolve 2 3 Output of the convolution y(t) 1 4 t -2 0 2 3 5

  5. f(t) = 2 g(t)= -t+1 2 2 1 t t --22 22 1 Step 1: 1 2 3 4 5

  6. g(t-t) 2 2 g(-t) f(t) f(t) 1 1 t t -1 + t -2 -2 2 2 Step 2: 1 2 -1 t Fig. a Fig. b 3 4 5

  7. g(t-t) 2 f(t) 1 t -1 + t -2 2 Step 3: Calculation of y(t) in five stages 1 Stage - I : t < -2 2 t 3 4 5

  8. 2 g(t-t) f(t) 1 t -1 + t -2 2 Step 4: 1 Stage - II : -2 ≤ t < -1 2 t 3 4 5

  9. 2 g(t-t) f(t) 1 t -2 -1 + t 2 Step 5: 1 Stage - III : -1 ≤ t < 2 2 t 3 4 5

  10. f(t) 2 g(t-t) 1 t -2 -1 + t Step 6: 1 Stage - IV : 2 ≤ t < 3 2 2 t 3 4 5

  11. f(t) 2 g(t-t) 1 t -2 2 -1 + t Step 7: 1 Stage - V : t ≥ 3 2 t 3 4 5

  12. Step 8: Output of Convolution 1 y(t) 1 2 t 3 -2 0 2 3 4 5

  13. 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: Try it yourself +1 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

  14. 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: Try it yourself +1 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

  15. 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: Try it yourself +1 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

  16. 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

  17. 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: Try it yourself +1 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

  18. 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: Try it yourself +1 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

  19. 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: Try it yourself +1 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

  20. 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: Try it yourself +1 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

  21. 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: Try it yourself +1 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

  22. 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) +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

  23. 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)‏ +1 +1 * -1 +1 +1 * 23 The same procedure is done to the above given combination of signals Credits

  24. 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

  25. 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

  26. 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

  27. 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:

  28. 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.

More Related