10/08 - Mechanics of Convolution & Fourier Transform: Difference between revisions
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*Flip: Flip one about the dependant axis |
*Flip: Flip one about the dependant axis |
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*Shift: The initial flipped function is at <math> t = 0 \,\!</math>. Shift this function for the multiply & integrate |
*Shift: The initial flipped function is at <math> t = 0 \,\!</math>. Shift this function for the multiply & integrate |
||
*Multiply: Multiply the two |
*Multiply: Multiply the two functions |
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*Add/Integrate: You may need to make multiple equations for different intersections |
*Add/Integrate: You may need to make multiple equations for different intersections |
||
[[Image:Convolution3.PNG]] |
[[Image:Convolution3.PNG]] |
Revision as of 17:00, 14 November 2008
Mechanics of the Convolution
Remember from the game:
Input | LTI System | Output | Reason |
Given | |||
Time Invarience | |||
Proportionality | |||
Superposition |
We will also denote the convolution as
Communative Property
Let thus | ||
The order of integration switched due to changing from | ||
Example 1
Example 2
- If isn't involved, then you can plug n chug with the integral. The u(t) will change the limits, which can be impractical to evaulate if you have more than 2.
Convolution: A visual approach
- Flip: Flip one about the dependant axis
- Shift: The initial flipped function is at . Shift this function for the multiply & integrate
- Multiply: Multiply the two functions
- Add/Integrate: You may need to make multiple equations for different intersections