continuous time convolution
DESCRIPTION
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 . - PowerPoint PPT PresentationTRANSCRIPT
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
AuthorPhani Swathi Chitta
MentorProf. Saravanan Vijayakumaran
Course Name: Signals and Systems Level: UG
Learning ObjectivesAfter interacting with this Learning Object, the learner will be able to:• Explain the convolution of two continuous time signals
Definitions of the components/Keywords:
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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 is the variable of integration.
Master Layout
5
3
2
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1
**22
2 1
t
f(t)
--22 1
2
t
g(t)
t
y(t)
0 2 3 -2
1
Signals taken to convolve
Output of the convolution
Step 1: 1
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32
4
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 is a dummy variable.
22
2 1
t
f(t) = 2
--22 1
2
t
g(t)= -t+1
Step 2: 1
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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() is shown• The reversed version of g() i.e., g(-is shown• The shifted version of g(-i.e., g(t-) is shown
2
2
f()1
-1 + t -2
g(t-)
t
2
2
f()1
-2
g(-)
Fig. a Fig. b-1
Step 3: 1
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Calculation of y(t) in five stages
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() is shown• The reversal and shifted version of g(t) i.e., g(t-is shown• Two functions do not overlap• Area under the product of the functions is zero
Stage - I : t < -2
2
2
f()1
-1 + t -2
g(t-)
t
Step 4: 1
5
32
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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() is shown• The reversal and shifted version of g(t) i.e., g(t-is shown• Part of g(t- overlaps part of f()
• Area under the product
Stage - II : -2 ≤ t < -1
2
2
f()1
-1 + t -2
g(t-)
t
Step 5: 1
5
32
4
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() is shown• The reversal and shifted version of g(t) i.e., g(t-is shown• g(t- completely overlaps f()
• Area under the product
Stage - III : -1 ≤ t < 2
2
2
f()1
-1 + t -2
g(t-)
t
Step 6: 1
5
32
4
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() is shown• The reversal and shifted version of g(t) i.e., g(t-is shown• Part of g(t- overlaps part of f()
• Area under the product
Stage - IV : 2 ≤ t < 3
2
f()
1
-1 + t -2
g(t-)
t2
Step 7: 1
5
32
4
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() is shown• The reversal and shifted version of g(t) i.e., g(t-is shown• Two functions do not overlap• Area under the product of the functions is zero
Stage - V : t ≥ 3
2
2
f()
1
-1 + t -2
g(t-)
t
Step 8: 1
5
32
4
Output of Convolution
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
t
y(t)
0 2 3 -2
1
3for 032for 9 6
21for 112for 2
2for 0
)(*)()(2
2
tttttttt
t
tgtfty
The four signals must be repeated under select for both f(t) and g(t)
Introduction
Credits
13
Definitions Test your understanding (questionnaire) Lets Sum up (summary) Want to know more…
(Further Reading)
Try it yourselfInteractivity:
Analogy
Slide 1
Slide 3
Slide 24-26
Slide 28
Slide 27
Electrical Engineering
+1
-1
+1
+1
-1
-1
f() and g(t-
f() g(t-
+1
-1 -1
+1
Select Select
t
t t
f(t) g(t)
+1 +1
+1
-1
The signal selected under f(t) must be shown
Introduction
Credits
14
Definitions Test your understanding (questionnaire) Lets Sum up (summary) Want to know more…
(Further Reading)
Try it yourselfInteractivity:
Analogy
Slide 1
Slide 3
Slide 24-26
Slide 28
Slide 27
Electrical Engineering
+1
-1
+1
+1
-1
-1
f() and g(t-
f() g(t-
+1
-1 -1
+1
Select Select
t
t t
f(t) g(t)
+1 +1
+1
-1
The signal selected under g(t) must be shown
Introduction
Credits
15
Definitions Test your understanding (questionnaire) Lets Sum up (summary) Want to know more…
(Further Reading)
Try it yourselfInteractivity:
Analogy
Slide 1
Slide 3
Slide 24-26
Slide 28
Slide 27
Electrical Engineering
+1
-1
+1
+1
-1
-1
f() and g(t-
f() g(t-
+1
-1 -1
+1
Select Select
t
t t
f(t) g(t)
+1 +1
+1
-1
The red figure is the shifted and reversed version of g(t) The slides 16-21 should be shown in a smooth fashion
Introduction
Credits
16
Definitions Test your understanding (questionnaire) Lets Sum up (summary) Want to know more…
(Further Reading)
Try it yourselfInteractivity:
Analogy
Slide 1
Slide 3
Slide 24-26
Slide 28
Slide 27
Electrical Engineering
+1
-1
+1
+1
-1
-1
f() and g(t-
f() g(t-
+1
-1 -1
+1
Select Select
t
t t
f(t) g(t)
+1 +1
+1
-1
Introduction
Credits
17
Definitions Test your understanding (questionnaire) Lets Sum up (summary) Want to know more…
(Further Reading)
Try it yourselfInteractivity:
Analogy
Slide 1
Slide 3
Slide 24-26
Slide 28
Slide 27
Electrical Engineering
+1
-1
+1
+1
-1
-1
f() and g(t-
f() g(t-
+1
-1 -1
+1
Select Select
t
t t
f(t) g(t)
+1 +1
+1
-1
Introduction
Credits
18
Definitions Test your understanding (questionnaire) Lets Sum up (summary) Want to know more…
(Further Reading)
Try it yourselfInteractivity:
Analogy
Slide 1
Slide 3
Slide 24-26
Slide 28
Slide 27
Electrical Engineering
+1
-1
+1
+1
-1
-1
f() and g(t-
f() g(t-
+1
-1 -1
+1
Select Select
t
t t
f(t) g(t)
+1 +1
+1
-1
Introduction
Credits
19
Definitions Test your understanding (questionnaire) Lets Sum up (summary) Want to know more…
(Further Reading)
Try it yourselfInteractivity:
Analogy
Slide 1
Slide 3
Slide 24-26
Slide 28
Slide 27
Electrical Engineering
+1
-1
+1
+1
-1
-1
f() and g(t-
f() g(t-
+1
-1 -1
+1
Select Select
t
t t
f(t) g(t)
+1 +1
+1
-1
Introduction
Credits
20
Definitions Test your understanding (questionnaire) Lets Sum up (summary) Want to know more…
(Further Reading)
Try it yourselfInteractivity:
Analogy
Slide 1
Slide 3
Slide 24-26
Slide 28
Slide 27
Electrical Engineering
+1
-1
+1
+1
-1
-1
f() and g(t-
f() g(t-
+1
-1 -1
+1
Select Select
t
t t
f(t) g(t)
+1 +1
+1
-1
Introduction
Credits
21
Definitions Test your understanding (questionnaire) Lets Sum up (summary) Want to know more…
(Further Reading)
Try it yourselfInteractivity:
Analogy
Slide 1
Slide 3
Slide 24-26
Slide 28
Slide 27
Electrical Engineering
+1
-1
+1
+1
-1
-1
f() and g(t-
f() g(t-
+1
-1 -1
+1
Select Select
t
t t
f(t) g(t)
+1 +1
+1
-1
The same procedure is done to the above given combination of signals
Introduction
Credits
22
Definitions Test your understanding (questionnaire) Lets Sum up (summary) Want to know more…
(Further Reading)Analogy
Slide 1
Slide 3
Slide 24-26
Slide 28
Slide 27
Electrical Engineering
+1
-1
+1
-1
*
*+1
+1
-1
*+1
-1
+1
-1
+1
-1 t
f(t)
+1
-1 t
f(t)
The same procedure is done to the above given combination of signals
Introduction
Credits
23
Definitions Test your understanding (questionnaire) Lets Sum up (summary) Want to know more…
(Further Reading)Analogy
Slide 1
Slide 3
Slide 24-26
Slide 28
Slide 27
Electrical Engineering
+1
-1*
+1
+1+1
*
Questionnaire1. 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)
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δ(t-5)x(t)
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Questionnaire3. 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 byi) ii) Hint: let Answers: a) i b) ii c) Both i and ii d)either i or ii
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Questionnaire5. 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
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Links for further readingReference 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.