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The Game
The idea behind the game is to use linearity (superposition and proportionality) and time invariance to find an output for a given input. An initial input and output are given.
Input
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LTI System
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Output
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Reason
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Given
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Time Invarience
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Proportionality
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Superposition
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With the derived equation, note that you can put in any to find the given output. Just change your t for a lambda and plug n chug.
Example 1
Let
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Let thus
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The order of integration switched due to changing from
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Different notation
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Different notation
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Example 2
Let
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From Example 1
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Different notation
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Questions
- How do eigenfunction and basisfunctions differ?
- Eigenfunctions will "point" in the same direction after going through the LTI system. It may (probably) have a different coefficient however. Very convenient.