The imaginary number i and the Fourier Transform

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 playlist:

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

[adetoyesealbert]

The most mind blowing intuitive explanation of any idea that I have come across on Internet yet. I 'm holding to this forever.

[ALP]

I just discover your channel thanks to YouTube algorithm (since I like this kind of subjects...), I have to say that your videos are truly awesome, the way you help with the visual representation is incredible, it can explain complex thing is such an intuitive way ...Bravo

[James_Hello]

This man is brilliant and needs his own TV show!!

[ag_rfdesigner]

The most intuitive video on imaginary numbers I have ever seen on the internet. Your videos are just brilliant. Thanks and please keep up the good work.

[plemplem]

Great work ! Keep on going, you shape the world with such outstanding presentation of normally high complex processes

[tashi]

Please make more such videos. . It's extremely useful

[leandrogcosta]

Very good interpreation. Many thanks for your help

[bartlx]

Wow...this was an unexpected fantastic explanation!

[moulinexish]

I did not expect to get this much feeling from a fourier transform explanation video.

[jacobbordelon]

Never heard a video emphasizing the fascinating world of signals and systems. Love to see some passion in educational EE videos. Thank you for sharing!

[andrewcb]

Thanks Mark, well done. I think you will do well with this format.

[teddyspaw]

A visual interpretation of Euler's equation - mind blown.

[oskarfjortoft]

this is on my top 5 list of the greatest math-videos on youtube.

[longluo]

Thank you very much! Eureka! I got it. Really Amazing! Mark， I love you.

[anasssofti]

I m working on image processing in Fourier domaine, and finally i did understand why the formula of FT is like this . Thank you so much !!!

[Saw-qvbl]

I didnt know unhinged maths was what I needed. But it is

[noahsalazar]

Thank you sir. This video is by far the best explanation of i that I have encountered on the Internet yet. Congratulations.

[JohnVKaravitis]

0:17 A spiral has a continuously narrowing or widening radius. What you have there is a helix.

[eg]

I have to say I love the blues clues aesthetic of your videos. Being new to systems and signals and taking a class for it, this is a lifesaver.

[talmidengineeringacademy]

You have done a good work that is going to be remembered by students around the world