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

Пікірлер: 108

  • @MarkNewmanEducation
    @MarkNewmanEducation2 жыл бұрын

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

  • @pvsheridan

    @pvsheridan

    2 жыл бұрын

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

  • @MarkNewmanEducation

    @MarkNewmanEducation

    2 жыл бұрын

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

  • @James_Hello
    @James_Hello20 күн бұрын

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

  • @moulinexish
    @moulinexish4 күн бұрын

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

  • @adetoyesealbert2093
    @adetoyesealbert2093 Жыл бұрын

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

  • @cai0_o
    @cai0_o2 жыл бұрын

    This video is absolutely amazing.

  • @MarkNewmanEducation

    @MarkNewmanEducation

    2 жыл бұрын

    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.

  • @oskarfjortoft
    @oskarfjortoft2 ай бұрын

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

  • @jacobbordelon3574
    @jacobbordelon35744 ай бұрын

    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!

  • @ag_rfdesigner
    @ag_rfdesigner11 ай бұрын

    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.

  • @alexandreperoumal2158
    @alexandreperoumal21585 ай бұрын

    I just discover your channel thanks to KZread 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

  • @teddyspaw
    @teddyspaw7 ай бұрын

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

  • @tashi2009
    @tashi20092 жыл бұрын

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

  • @MarkNewmanEducation

    @MarkNewmanEducation

    2 жыл бұрын

    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.

  • @Isysnation
    @Isysnation4 ай бұрын

    We need more of this ...

  • @ItsMeTheUser
    @ItsMeTheUser7 ай бұрын

    The BEST channel explaining Signals and Systems.

  • @plemplem94
    @plemplem94 Жыл бұрын

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

  • @user-oe3zn7jq3i
    @user-oe3zn7jq3i Жыл бұрын

    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😃😃

  • @noahsalazar2738
    @noahsalazar2738 Жыл бұрын

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

  • @MarkNewmanEducation

    @MarkNewmanEducation

    Жыл бұрын

    Wow, thanks!

  • @Paulancar
    @Paulancar5 ай бұрын

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

  • @MarkNewmanEducation

    @MarkNewmanEducation

    5 ай бұрын

    Thanks, will do!

  • @sirousmohseni4
    @sirousmohseni410 ай бұрын

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

  • @andrewc778b
    @andrewc778b2 жыл бұрын

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

  • @anasssofti9271
    @anasssofti9271 Жыл бұрын

    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

    @MarkNewmanEducation

    Жыл бұрын

    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?

  • @dvorakaudio9046
    @dvorakaudio9046 Жыл бұрын

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

  • @leandrogcosta
    @leandrogcosta7 ай бұрын

    Very good interpreation. Many thanks for your help

  • @huyvuquang2041
    @huyvuquang2041 Жыл бұрын

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

  • @talmidengineeringacademy1824
    @talmidengineeringacademy18242 жыл бұрын

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

  • @MarkNewmanEducation

    @MarkNewmanEducation

    2 жыл бұрын

    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.

  • @sakethsaketh750
    @sakethsaketh750 Жыл бұрын

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

  • @jac6255
    @jac6255 Жыл бұрын

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

  • @141592635eg
    @141592635eg Жыл бұрын

    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

    @k.k.skesava7561

    5 сағат бұрын

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

  • @DirkArnez
    @DirkArnez Жыл бұрын

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

  • @MarkNewmanEducation

    @MarkNewmanEducation

    Жыл бұрын

    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.

  • @bartlx
    @bartlx2 ай бұрын

    Wow...this was an unexpected fantastic explanation!

  • @100deep1001
    @100deep10015 ай бұрын

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

  • @manueltellez1045
    @manueltellez104510 күн бұрын

    Excelente explicación, muchas gracias.

  • @kathrynt.3710
    @kathrynt.37102 жыл бұрын

    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!

  • @jagatkumartudu
    @jagatkumartudu Жыл бұрын

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

  • @curtpiazza1688
    @curtpiazza1688 Жыл бұрын

    Great stuff! 😊

  • @pinguinauta9353
    @pinguinauta93532 ай бұрын

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

  • @alimoradi2323
    @alimoradi2323 Жыл бұрын

    Perfect. Thanks a lot

  • @sylvesterjonas9141
    @sylvesterjonas91414 ай бұрын

    Pure gold mine for me!

  • @TheMightySponge
    @TheMightySponge Жыл бұрын

    Thank you!

  • @grounded9623
    @grounded96235 ай бұрын

    Awesome. Thanks.

  • @sirajussalekinsami
    @sirajussalekinsami7 ай бұрын

    OUTSTANDING!!

  • @Khashayarissi-ob4yj
    @Khashayarissi-ob4yj5 ай бұрын

    so excellent, so beautiful with regards

  • @muffinfighter3680
    @muffinfighter36806 ай бұрын

    Thank you!!!

  • @joaquingiorgi5809
    @joaquingiorgi5809 Жыл бұрын

    Amazing video! Cheers from Argentina :)

  • @MarkNewmanEducation

    @MarkNewmanEducation

    Жыл бұрын

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

  • @muhittinselcukgoksu1327
    @muhittinselcukgoksu1327 Жыл бұрын

    super...thank you so much...

  • @SampleroftheMultiverse
    @SampleroftheMultiverse5 ай бұрын

    Very well done 👍 17:26

  • @longluo1788
    @longluo17882 жыл бұрын

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

  • @MarkNewmanEducation

    @MarkNewmanEducation

    2 жыл бұрын

    AMAZING!! I love giving people Eureka moments.

  • @johnnixon1437
    @johnnixon14379 ай бұрын

    you are the best

  • @kshaunish555
    @kshaunish5552 жыл бұрын

    Superb explaination 👍🏻

  • @MarkNewmanEducation

    @MarkNewmanEducation

    2 жыл бұрын

    Thank you.

  • @panduwilantara3070
    @panduwilantara3070 Жыл бұрын

    amazing

  • @diegoteceletro
    @diegoteceletro9 ай бұрын

    beautiful

  • @subhadipkarmakar2841
    @subhadipkarmakar2841 Жыл бұрын

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

  • @eldi
    @eldi2 жыл бұрын

    Legend

  • @MarkNewmanEducation

    @MarkNewmanEducation

    2 жыл бұрын

    Thank you

  • @AdityaYadav-is9sl
    @AdityaYadav-is9sl Жыл бұрын

    Nice sir 👍

  • @chatrughanprasad7778
    @chatrughanprasad7778 Жыл бұрын

    Great video

  • @MarkNewmanEducation

    @MarkNewmanEducation

    Жыл бұрын

    Glad you enjoyed it

  • @kevinyang4539
    @kevinyang45399 ай бұрын

    thanks

  • @maurolawall
    @maurolawall Жыл бұрын

    Nice!

  • @MarkNewmanEducation

    @MarkNewmanEducation

    Жыл бұрын

    Thank you

  • @shacharstern8670
    @shacharstern86705 ай бұрын

    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?

  • @MlNECRAFT69
    @MlNECRAFT693 ай бұрын

    BEST DESCRIPTION OF I EVER

  • @showrovsheikh3562
    @showrovsheikh3562 Жыл бұрын

    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

    @MarkNewmanEducation

    Жыл бұрын

    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
    @Wheelie19998 ай бұрын

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

  • @TheMrhenon
    @TheMrhenon8 ай бұрын

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

  • @jiangxu3895
    @jiangxu3895 Жыл бұрын

    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

    @vex18th

    Жыл бұрын

    @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

    @jiangxu3895

    8 ай бұрын

    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
    @sureshbingi Жыл бұрын

    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

    @MarkNewmanEducation

    Жыл бұрын

    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: kzread.info/dash/bejne/pX-oztF7cdi2Z5c.html

  • @vex18th
    @vex18th Жыл бұрын

    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 !!!

  • @JohnVKaravitis
    @JohnVKaravitis Жыл бұрын

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

  • @MarkNewmanEducation

    @MarkNewmanEducation

    Жыл бұрын

    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.

  • @MMP1
    @MMP12 жыл бұрын

    beautiful, even my little brother understood it.

  • @MarkNewmanEducation

    @MarkNewmanEducation

    2 жыл бұрын

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

  • @MMP1

    @MMP1

    2 жыл бұрын

    @@MarkNewmanEducation 14 🙂

  • @user-rt7op6sq6m
    @user-rt7op6sq6m11 ай бұрын

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

  • @elsharko87
    @elsharko873 ай бұрын

    Tanks!

  • @SAJAN_ECE
    @SAJAN_ECE2 жыл бұрын

    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

    @MarkNewmanEducation

    2 жыл бұрын

    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.

  • @amansaxena7024
    @amansaxena70242 жыл бұрын

    I can finally see signals from Mathematic equations

  • @MarkNewmanEducation

    @MarkNewmanEducation

    2 жыл бұрын

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

  • @skilz8098
    @skilz80984 ай бұрын

    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?

  • @gok_dogan
    @gok_dogan6 ай бұрын

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

  • @Paulancar

    @Paulancar

    5 ай бұрын

    unnecessary help!!!

  • @gok_dogan

    @gok_dogan

    5 ай бұрын

    @@Paulancar unnecessary 😒answer

  • @sunilpal1000
    @sunilpal1000 Жыл бұрын

    Sir why you can't your idea patented?

  • @MarkNewmanEducation

    @MarkNewmanEducation

    Жыл бұрын

    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

    @sunilpal1000

    Жыл бұрын

    @@MarkNewmanEducation sir you explation is excellent.

  • @JeanSarfati
    @JeanSarfati Жыл бұрын

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

  • @SurprisedDivingBoard-vu9rz
    @SurprisedDivingBoard-vu9rz3 күн бұрын

    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
    @SampleroftheMultiverse5 ай бұрын

    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 KZread channel. Actually replicating it with a sheet of clear folder plastic and tape and seeing it first hand is worth the effort. kzread.info/dash/bejne/qaZ21dOidZPcd5c.htmlsi=waT8lY2iX-wJdjO3

  • @jameshopkins3541
    @jameshopkins35413 ай бұрын

    ??????????

  • @TheDavidlloydjones
    @TheDavidlloydjones2 жыл бұрын

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

  • @MarkNewmanEducation

    @MarkNewmanEducation

    2 жыл бұрын

    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.