i and the Fourier Transform; what do they have to do with each other? The answer is the complex exponential. It's called complex because the "i" turns an exponential function into a spiral containing within it a cosine wave and a sine wave. By using convolution, these two functions allow the Fourier Transform to model almost any signal as a collection of sinusoids.

In this video, we look at an intuitive way to understand what "i" is and what it is doing in the Fourier Transform.

Other videos of interest:
Convolution and the Fourier Transform:
   • Convolution and the Fourier Transform...  

Convolution playlist:
   • Convolution and the Fourier Transform  

How Imaginary Numbers were invented:
   • How Imaginary Numbers Were Invented  

0:00 - Introduction
1:15 - Ident
1:20 - Welcome
1:29 - The history of imaginary numbers
3:48 - The origin of my quest to understand imaginary numbers
4:32 - A geometric way of looking at imaginary numbers
9:37 - Looking at a spiral from different angles
10:39 - Why "i" is used in the Fourier Transform
10:44 - Answer to the last video's challenge
11:39 - How "i" enables us to take a convolution shortcut
13:05 - Reversing the Cosine and Sine Waves
15:01 - Finding the Magnitude
15:12 - Finding the Phase
15:20 - Building the Fourier Transform
15:38 - The small matter of a minus sign
16:34 - This video's challenge
17:10 - End Screen