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 and the Fo...
Convolution playlist:
• Convolution and the Fo...
How Imaginary Numbers were invented:
• How Imaginary Numbers ...
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

[@MarkNewmanEducation]

Please support the making of these videos: www.patreon.com/MarkNewman

[@pvsheridan]

Mark: How does one make a one-time support donation?

[@MarkNewmanEducation]

Thank you. That's very kind of you. Here's the link for a one time donation: paypal.me/MarkHNewmam

[@James_Hello]

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

[@cai0_o]

This video is absolutely amazing.

[@MarkNewmanEducation]

Really glad you liked it. I wanted to explain it in the way I wish it had been explained to me all those years ago.

[@moulinexish]

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

[@adetoyesealbert2093]

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

[@alexandreperoumal2158]

I just discover your channel thanks to KZbin 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

[@oskarfjortoft]

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

[@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.

[@jacobbordelon3574]

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!

[@tashi2009]

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

[@MarkNewmanEducation]

Thank you. More videos on the way. The next video on how to interpret the output of an FFT is currently in production and I hope to release it at the beginning of May. Please add ideas for videos you would like to see in the comments so I can add it to my list.

[@noahsalazar2738]

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

[@MarkNewmanEducation]

Wow, thanks!

[@141592635eg]

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.

[@k.k.skesava7561]

Blue clues aesthetic? (U speaking of the background?)

[@teddyspaw]

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

[@plemplem94]

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

[@ItsMeTheUser]

The BEST channel explaining Signals and Systems.

[@anasssofti9271]

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 !!!

[@MarkNewmanEducation]

Amazing. I'm so glad I was able to help you. Please share the video with anyone you feel it could help. If you're working with image processing, are you using 2D Fourier Transforms?

[@user-oe3zn7jq3i]

老外对问题的执着,及疑问的态度的确值得我国的老师学习😃😃

[@Isysnation]

We need more of this ...

[@Paulancar]

WOW!!!!! thanks Mark. Please continue and continue!!! now you are the best.

[@MarkNewmanEducation]

Thanks, will do!

[@sakethsaketh750]

This is the worlds best explantion ever I heard. Thank you so much

[@andrewc778b]

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

[@dvorakaudio9046]

best video i've seen on the Fourier transform. makes the subject very accessible:)

[@manueltellez1045]

Excelente explicación, muchas gracias.

[@huyvuquang2041]

Thank you. Your clip is amazing. Keep up your work!

[@sirousmohseni4]

I have to watch all your videos. Thanks for presenting these topics so well that my old brain can follow.

[@talmidengineeringacademy1824]

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

[@MarkNewmanEducation]

If the material I am trying to teach is remembered and understood by students, then there can be no greater compliment to a teacher than that.

[@leandrogcosta]

Very good interpreation. Many thanks for your help

[@jac6255]

these are so good and you've made it really interesting

[@DirkArnez]

I cannot thank you more... this video is the only video i can understand

[@MarkNewmanEducation]

AMAZING. I had such trouble understanding the concept of imaginary numbers back in my days as a student. It was only after working with them in my professional life for several years that the penny finally dropped. This video details my path to understanding.

[@kathrynt.3710]

I love your videos! I haven't learned any maths higher than integral calculus and I understand everything you discuss very well. I wish more people explained the basics as simply as you do; it makes understanding the more advanced things so easy. I feel like your videos are a series of stepping-stones that explain everything gradually. Wonderful channel!

[@bartlx]

Wow...this was an unexpected fantastic explanation!

[@100deep1001]

Love you man! Thanks a lot for the efforts you took ❤ simply the best videos out there on Fourier transform. Even better than 3b1b !

[@sylvesterjonas9141]

Pure gold mine for me!

[@sirajussalekinsami]

OUTSTANDING!!

[@jagatkumartudu]

Thank you sir ,its really amazing .finally I got what I was looking for ...lots of love from India

[@pinguinauta9353]

Eres un profesor realmente impresionante You are a very awesome teacher !

[@curtpiazza1688]

Great stuff! 😊

[@alimoradi2323]

Perfect. Thanks a lot

[@TheMightySponge]

Thank you!

[@joaquingiorgi5809]

Amazing video! Cheers from Argentina :)

[@MarkNewmanEducation]

You're welcome. Please share with anyone you think it could help.

[@grounded9623]

Awesome. Thanks.

[@kshaunish555]

Superb explaination 👍🏻

[@MarkNewmanEducation]

Thank you.

[@Khashayarissi-ob4yj]

so excellent, so beautiful with regards

[@SampleroftheMultiverse]

Very well done 👍 17:26

[@muffinfighter3680]

Thank you!!!

[@longluo1788]

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

[@MarkNewmanEducation]

AMAZING!! I love giving people Eureka moments.

[@muhittinselcukgoksu1327]

super...thank you so much...

[@diegoteceletro]

beautiful

[@mihaleben6051]

2 variables in an exponent is wild

[@subhadipkarmakar2841]

Thank you Sir 😇 I am from India 🇮🇳 This video is really amazing, nice explanation

[@johnnixon1437]

you are the best

[@panduwilantara3070]

amazing

[@chatrughanprasad7778]

Great video

[@MarkNewmanEducation]

Glad you enjoyed it

[@eldi]

Legend

[@MarkNewmanEducation]

Thank you

[@AdityaYadav-is9sl]

Nice sir 👍

[@kevinyang4539]

thanks

[@MlNECRAFT69]

BEST DESCRIPTION OF I EVER

[@maurolawall]

Nice!

[@MarkNewmanEducation]

Thank you

[@shacharstern8670]

Wonderful videos with great explanations. Could you please explain why do the real and imaginary results are the amplitudes of the cos and sine that together give the sinusoid with the maximum score for the given frequency?

[@TheMrhenon]

Pulley or leakage. Leakage of dimensions which are otherwise orthogonal. Water pipe with hole for leakage which can let water both direction.

[@showrovsheikh3562]

Really this video is very helpful to understand Fourier Transform. Thank u so much sir. If you make a video about Laplace Transform, we will be able to understand it clearly.

[@MarkNewmanEducation]

Thank you. Glad it was helpful. Laplace is definitely on my radar, but I need to get the the same level as my understanding of Fourier before I can release a video about it. The research continues.

[@Wheelie1999]

i’m only a minute in and i’m blown away, thank you!

[@JohnVKaravitis]

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

[@MarkNewmanEducation]

Thanks for pointing this out. You are, of course, correct. I must change this in my scripts for the Udemy course I am building on the Fourier Transform. Thanks.

[@jiangxu3895]

Thank you so much for such great explanation of Fourier transformation, that I have never seen a second person explain the algorithm in this way. I started to learn some theory of x-ray crystallography 8 years ago, but never got the idea of the benefit of complexed sinosoid, and why the equation is true. I almost get the idea of how the aim is achieved with sinosoid with a certain phase that produce the maximum value when convolved with the signal, which represent the maximum contribution of the sinosoid with that frequency. However, I want to know why the statement at 14:33 is true, that the sine and cos wave with different amplitude, the same frequency and 0 phase difference, when each convolved with the signal, can produce a sinosoid with a certain phase, at which the maximum of the contribution of the sinosoid to the signal occur? Could you explain this part? 😅👍

[@vex18th]

@Jiang Xu you understood him wrong, first you get the sinusiod with the amplitude and phase by combining pure cosine and pure sine, and then with the resulting sinsuoid with phase shift you convolute with the original signal

[@jiangxu3895]

Hi, thanks for your explanation. I think my expression was not correct. I just want to know why this short cut is true that when using complexed sinosoid calculation shown starting from 14:33 , you dont' have to slide the sinosoid wave with a arbitary frequency against the signal wave to integrate and to get the maxima contribution from that arbitary sinosoid wave. I think this part was not mentioned in the video, Mark just used this method but didn't explain why it is true@@vex18th

[@sureshbingi]

Nice explanation about the concept of i (rotation). May I ask, why only e to the i theta is spiral. What would be any number to the i theta looks like. In natural logarithm why we use the number e as base why it can't be pi i.e., log to the base pi . What is meaning of pi to the i theta

[@MarkNewmanEducation]

Interesting question. The special thing about Euler's number is it's relationship to sine and cosine as shown by Euler's formula which no other number shares. See my video on Euler's identity for more info: kzbin.info/www/bejne/qXzXnaJ4dtqDbZY

[@vex18th]

hi, great lectures, 14:32 shouldn't the real and imaginary parts of fourier transforms be divided by T/2 to get the amplitudes ?? my logic stems from the fact that Fourier Series is a special case (special form) of Fourier Transform , correct me if i am wrong greetings !!!

[@user-rt7op6sq6m]

I came for the math, the kipa was the cherry on top.

[@amansaxena7024]

I can finally see signals from Mathematic equations

[@MarkNewmanEducation]

Amazing! In that case I achieved what I set out to do when I scripted the video. Thanks for the feedback.

[@SAJAN_ECE]

I did not get the idea of shifting the sinusoidal by adding a sine and cosine with varying amplitudes. From where that concept came from ? That idea is not present in the actual Fourier Transform Equation.

[@MarkNewmanEducation]

It is implied in the equation by the fact that the result is a list of complex numbers. Each complex number in the list represents a sinusoid with a different frequency. The complex number, which consists of a real and imaginary part represents the magnitude of a cosine wave (the real part) and a sine wave (the imaginary part). Try adding together a sine wave and a cosine wave with the same frequency but different magnitudes in excel or MATLAB or some other software that allows you to do these sorts of operations. Look what happens to the phase of the resultant wave.

[@MMP1]

beautiful, even my little brother understood it.

[@MarkNewmanEducation]

Wow. Thank you. How old is your little brother?

[@MMP1]

@@MarkNewmanEducation 14 🙂

[@elsharko87]

Tanks!

[@gok_dogan]

16:13 explains why e to the power minus i is used in fourier transform formula .

[@Paulancar]

unnecessary help!!!

[@gok_dogan]

@@Paulancar unnecessary 😒answer

[@skilz8098]

I'm wondering if the same can be said about division by 0. Could it be that division by 0, tan(90) and multiplying by +/-i are related?

[@sunilpal1000]

Sir why you can't your idea patented?

[@MarkNewmanEducation]

What idea? The maths isn't mine. It's hundreds of years old. I'm just trying to explain it in a more intuitive way.

[@sunilpal1000]

@@MarkNewmanEducation sir you explation is excellent.

[@JeanSarfati]

The imaginary number is Not imaginary. It is Unimaginable, or inimaginable. It preserves the symbolic i. Ouf ! 😉

[@SurprisedDivingBoard-vu9rz]

How do you prove that the earth is moving at a speed of 12 times the speed of light. The reason why light is a trapped one. 3 square plus three. 3 square for mass. Light in three directions. Light is a three d vector.

[@SampleroftheMultiverse]

U Shape Wave Thanks for your informative and well produced video. You and your viewers might find my quantum-like analog interesting and or useful. I have been trying to describe the “U” shape wave that is produced in my amateur science mechanical model in the video linked below. I hear if you over-lap wave together using Fournier Transforms, it may make a “U” shape or square wave. Can this be correct representation Feynman Path Integrals? In the model, “U” shape waves are produced as the loading increases and just before the wave-like function shifts to the next higher energy level. Your viewers might be interested in seeing the load verse deflection graph in white paper found elsewhere on my KZbin channel. Actually replicating it with a sheet of clear folder plastic and tape and seeing it first hand is worth the effort. kzbin.info/www/bejne/raOlpKSfepWpfZYsi=waT8lY2iX-wJdjO3

[@jameshopkins3541]

??????????

[@TheDavidlloydjones]

"In my last video" is a meaningless term until you number or date your videos and point us to the relevant menu.

[@MarkNewmanEducation]

In light of your comment, I've added a link to the playlist. However, there is already a link to the video I refer to in the description (as I mentioned in the video) and there is a card, timed with my mention, that links to the video which should appear in the top right hand corner.