10/08 - Mechanics of Convolution & Fourier Transform

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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?

Oct8.PNG

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

Convolution3.PNG