A Musician's Intuition on the Fourier Transform (feat. the Inner Product) | 3b1b SoME1

My entry for the 3Blue1Brown Summer of Musical Exposition competition. In this video I describe one way you could derive the Fourier Transform from scratch using trigonometry, numerical simulations, the majesty of the inner product, and no complex numbers. My hope is that this is a somewhat-broadly accessible explanation for why the Fourier Transform works that may serve as a bridge to why variants of the Fourier Transform also work.


Knowledge of calculus is not required, though I do show a few integrals here and there for those who are into that.

I am not good at explaining things succinctly. I have also never made a "video essay" before. This video is messier than I'd like but I have run out of time to whittle it down or clean it up - hopefully it is watchable.

  / wgwgsa  

watergunsky.bandcamp.com

0:00 Introduction
1:18 Basics of the Inner Product
8:25 Inner Products on Sums of Sines
12:47 Phase Problems
14:37 How to Fix Phase Problems
16:35 Two Inner Products to Find One Wave
21:45 Double-Inner Product Examples
25:56 Yep, That's The Fourier Transform
28:13 Concluding Remarks