10/08 - Mechanics of Convolution & Fourier Transform: Difference between revisions
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(New page: ==Mechanics of the Convolution== Remember from the game:) |
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==Mechanics of the Convolution== |
==Mechanics of the Convolution== |
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Remember from the game: |
Remember from the game: |
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{| border="1" cellpadding="5" cellspacing="0" |
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|- |
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|Input |
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|LTI System |
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|Output |
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|Reason |
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|- |
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|<math> \delta (t)\,\!</math> |
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|<math> \Longrightarrow </math> |
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|<math> h(t) \,\!</math> |
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|Given |
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|- |
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|<math> \delta (t-\lambda)\,\!</math> |
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|<math> \Longrightarrow </math> |
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|<math> h(t-\lambda) \,\!</math> |
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|Time Invarience |
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|- |
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|<math> x(\lambda) \delta (t-\lambda)\,\!</math> |
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|<math> \Longrightarrow </math> |
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|<math> x(\lambda)h(t-\lambda) \,\!</math> |
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|Proportionality |
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|- |
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|<math> x(t) = \int_{-\infty}^{\infty} x(\lambda) \delta (t-\lambda)\, dx</math> |
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|<math> \Longrightarrow </math> |
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|<math> \underbrace{\int_{-\infty}^{\infty} x(\lambda)h(t-\lambda)\, dx}_{Convolution Integral}</math> |
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|Superposition |
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|} |
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We will also denote the convolution as <math> x(t) * h(t) \equiv \int_{-\infty}^{\infty} x(\lambda)h(t-\lambda)\, dx</math> |
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===Communative Property=== |
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{| border="0" cellpadding="0" cellspacing="0" |
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|- |
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|<math>x(t) * h(t) \,\!</math> |
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|<math>=\int_{-\infty}^{\infty} x(\lambda)h(t-\lambda)\, d\lambda</math> |
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|Let <math> t-\lambda = u \,\!</math> thus <math> du = -d\lambda \,\! </math> |
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|- |
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| |
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|<math>=-\int_{\infty}^{-\infty} x(t-u)h(u)\, du</math> |
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|The order of integration switched due to changing from <math>-\lambda = u\,\!</math> |
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|- |
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|<math>=\int_{-\infty}^{\infty} h(u)x(t-u)\, du</math> |
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|- |
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|<math>=h(t)*x(t) \,\!</math> |
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|} |
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===Example 1=== |
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{| border="0" cellpadding="0" cellspacing="0" |
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|- |
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|<math>\delta(t)*x(t)\,\!</math> |
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|<math>=\int_{-\infty}^{\infty} \delta(\lambda)x(t-\lambda)\,d\lambda</math> |
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|- |
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| |
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|<math>=x(t)\int_{-\infty}^{\infty} \,d\lambda</math> |
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|- |
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| |
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|<math>=x(t)\,\!</math> |
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|} |
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===Example 2=== |
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<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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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> |
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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. |
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*? Does it matter which one you FSMI? |
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[[Image:Oct8.PNG]] |
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==Convolution: A visual approach== |
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*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 |
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*Multiply: Multiply the two functions |
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*Add/Integrate: You may need to make multiple equations for different intersections |
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[[Image:Convolution3.PNG]] |
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