13.1.2 - Definition of Convolution
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Practice Questions
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Define convolution in your own words.
💡 Hint: Think about how we can mix two different functions.
What does the notation \( (f \ast g)(t) \) represent?
💡 Hint: Look at how we express the product of functions in convolution.
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Interactive Quizzes
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What is the definition of convolution?
💡 Hint: Think about integrating the product of two time-based functions.
The Commutative property states that \( (f \ast g)(t) \) equals which of the following?
💡 Hint: Consider if changing the order matters during addition.
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Challenge Problems
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Using the functions \( f(t) = sin(t) \) and \( g(t) = e^{-t} \), compute the convolution \( (f \ast g)(t) \). Show all steps.
💡 Hint: Use the identity of sine and consider integration by parts.
Illustrate the behavior of convolution through a graphical representation of \( (f \ast g)(t) \) for \( f(t) = t \) and \( g(t) = u(t) \) over [0, 5].
💡 Hint: Draw graphs and look at how the areas combine. Use numerical integration if needed.
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