10/08 - Mechanics of Convolution & Fourier Transform: Difference between revisions
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===Example 2=== |
===Example 2=== |
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<math> \overbrace{[u(t)-u(t-1)]}^{x(t)}*\overbrace{[u(t-1)-u(t-3)]}^{h(t)} = |
<math> \overbrace{[u(t)-u(t-1)]}^{x(t)}*\overbrace{[u(t-1)-u(t-3)]}^{h(t)} = |
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\begin{cases} |
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\begin{cases} n/2, & \mbox{if }n\mbox{ is even} \\ 3n+1, & \mbox{if }n\mbox{ is odd} \end{cases} |
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0, & t \le 1 \\ |
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\int_{0}^{t-1} 1 \cdot 2 \,d\lambda, & 1 \le t \le 2\\ |
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\int_{0}^{1} 1 \cdot 2 \,d\lambda, & 2 \le t \le 3\\ |
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\int_{t-3}^{1} 1 \cdot 2 \,d\lambda, & 3 \le t \le 4\\ |
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0, & t > 4 \\ |
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\end{cases} = \begin{cases} |
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0, & t \le 1 \\ |
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2\,t - 2, & 1 \le t \le 2\\ |
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2, & 2 \le t \le 3\\ |
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-2\,t - 4, & 3 \le t \le 4\\ |
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0, & t > 4 \\ |
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\end{cases} |
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</math> |
</math> |
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*In this case, we are doing the FSMI to <math> h(t) \,\!</math> |
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*If <math> u(t) \,\!</math> 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. |
*If <math> u(t) \,\!</math> 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. |
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*? Does it matter which one you FSMI? |
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[[Image:Oct8.PNG]] |
[[Image:Oct8.PNG]] |
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Latest revision as of 01:48, 17 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
- In this case, we are doing the FSMI to
- 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.
- ? Does it matter which one you FSMI?
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