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. 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

4. ** 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

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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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

12. 3 Step 8: 1 2 4 5 y(t) 1 t -2 2 3 Output of Convolution2 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

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: +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

14. 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

15. 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

16. 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 should be shown in a smooth fashion
Credits

17. 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

18. 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

19. 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

20. 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

21. 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

22. * * * 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

23. * * 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

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
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.