Section 21.9
Convolution Integrals
Multiplication in the s-domain becomes "Convolution" in the time domain. A powerful tool for transfer functions.
1
Introduction
We know .
But .
Instead, it corresponds to a special integral called the Convolution, denoted .
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Definition
Convolution Integral
Theorem
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Visual: Sliding Window
Interactive: Overlap Area
We "flip" one function and "slide" it across the other. The area of overlap is the convolution.
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Worked Examples
Example 1: Calculating Convolution
Calculate directly.
1. Setup:
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2. Integrate:
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3. Verify with Laplace:
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. Matches.
Example 2: Inverse Transform
Find using convolution.
1. Split:
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2. Convolve:
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Trig Identity: .
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3. Solve Integral:
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Practice Quiz
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