# 10/08 - Mechanics of Convolution & Fourier Transform

From Class Wiki

## Contents

## 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