What is a Fourier Series? (Explained by drawing circles) - Smarter Every Day 205

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    Doga's a super smart dude who writes a Turkish blog "Bi Lim Ne Güzel Lan" that roughly translates roughly to "Science is Awesome Dude". We had a lot of fun working on this together. He would really appreciate it if you checked out his blog. The fun thing is that most of his articles transcend language.
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    Publicerades den År sedan


    1. Ben Schofield

      Holy cow, Destin how are you today?

      1. Rock Brentwood

        @SmarterEveryDay Here's something that will make everybody's life easier ... and at the same time generalizes what you presented. Since e^{2πi} = 1, then declare by fiat that 1ⁿ = e^{2πin} = cos(2πin) + i sin(2πin), for non-integer and even irrational n, and define it to be so. Then the Fourier Series becomes simply: ∀ν∈ℱ: F(ν) = Σ_{t∈𝒯} f(t) 1^{+νt} Δt, ∀t∈𝒯: f(t) = Σ_{ν∈ℱ} F(ν) 1^{-νt} Δν, where the domains are ℱ ≡ {ν₀ + n Δν: 0 ≤ n < N}, 𝒯 ≡ {t₀ + k Δt: 0 ≤ k < N}, each with equal spacings, with the spacings being subject to the condition that 1/(ΔνΔt) = N ∈ {1,2,3,4,⋯}, the size the Fourier transform, be a positive integer. This is suitable for handling frequencies in the range ν ∈ [ν₀, ν₀ + 1/Δt) over times in the range t ∈ [t₀, t₀ + 1/Δν). It is the discrete, finite version of the Fourier integral: ∀ν∈ℝ: Φ(ν) = ∫ φ(t) 1^{+νt} dt, ∀t∈ℝ: φ(t) = ∫ Φ(ν) 1^{-νt} dν, the correspondence between the two becoming easily visible when written in this way.

      2. Annie Katsu

        @aneyesky In some sense yes (if you're willing to abstract a bit). The Fourier series is really just a basis of a space of functions from [-π,π] with finite squared integral. So, we only need to find the correct space to answer your question. Consider a factal (such a Julia set aka "mandelbot image") in some Euclidean space. Consider any Borel measure with full support on that fractal, and consider the space of all functions from that fractal to R which have finite square integral wrt that measure. Then, you can find an orthonormal basis of that space which "does the same job" as the Fourier series. Moreover, you can show that you can map the Fourier series to and from this basis of functions since both constructions describe separable Hilbert spaces (infinite-dimensional) and all infinite-dimensional separable Hilbert spaces are isometric. So in particular, their orthornomal basises can always be mapped from one another.

      3. Joshua Charlery

        I love Math!

      4. onafixedincome

        @SmarterEveryDay I'm always a bit shy of addressing you Prominent Busy People...you've a lot better things to do than put up with nosy folk like me. I plan to keep waving at my screen and yelling "HI DESTIN!" every time I see you pop up? Does that count? Ps....Anyone who hustles you for more uploads, etc, should have to wait an extra week to see the next one. :P

      5. Eli Spizzichino

        @SmarterEveryDay Destin4president! I've loved your podcast (and I'm not an American). China's social ranking system is really scaring and the world is going straight in that direction. I agree on the nose-dive problem (that could be addressed by a weighting algorithm) but I think the core problem is that we're getting used to the idea that machines can be better then humans in judging humans. I'm not saying that AI has no good potential in this field (there are many example of this already), I'm saying that we're inherently superior to do so, it's built in our brain for example we have entire set of neurons just dedicated to judge mouth and facial muscles movements to get an intuitive "feeling" of the other person/animal intention/truthiness. The smell test you were speaking about, it came by evolution. The dystopian future is reaching our present time faster then one might think, and is very important to speak about this stuff to hopefully prevent the inevitable nefarious outcome these issues presents.

    2. Hansang Bae

      @0:48. Ahhhh the green grid paper. While I was visiting a book store for my son's college tour, I *had* to pick up a pack. It's been thirty years, but boy did it bring back memories!!!!! Everyone else thought it was weird, but I bet every engineering can commiserate and understand the nostalgia.

    3. Nicolai Feldthaus

      4:49 Ancient Astronomers: Hold my dinosaur

    4. Julian Paez

      I hope that, one day, I'll be smart enough to appear on one of Destin's videos

    5. Zichang Wang

      Visualizing math helped us to understand it really well

    6. sam urai

      For-ee-err series is how i always pronounced it in uni 😅 for-ee-yay sounds so odd to me

    7. Joseph Coon

      And that’s how your brain functions. The waveform produced by the oscillating activation patterns based on the ever evolving synaptic connections perfectly mirrors the ever increasing number of waves used to bring an approximation of reality closer to reality.

    8. Yousri Bouamara

      how do you animate it?

    9. Alex Walton

      Hi could you help me as I am struggling with a maths assamenmt, would it be possible to email you the PDF?

    10. Raj Patel

      I am blown away!

    11. Nic Tai

      Great representation and video!

    12. Jason Perry

      "We can approximate anything as long you have enough terms." That right there is what makes mathematics so beautiful!!

    13. Jason Perry

      Love the GIF joke :)

    14. Jason Perry

      Honestly - had now clue you were this smart. Even smarter then you were yesterday!

    15. Deleted Account

      Yes, any wave can be created by a sine wave. That is the basic principal of FM synthesis. Different algorithms give different waves. This video is awesome.

    16. Jim Devilbiss

      In one class in college engineering our final was to reverse mathematically a Fourier analysis. I didn’t get the right answer but the brag yes to brag I got the closest wrong answer. It would’ve been wonderful to have his program. Fourier analysis and FORTRAN computer language were absolutely my favorite topics in college. Because you could do almost anything with both of them.

    17. Dipu Biswas


    18. LOLz

      yo! u remember that Fourier series from 2nd year engineering!! Im in my 4th year already forgot it !!!

    19. Mario Mendez

      am i the only one who gets anciety to see that Destin only uses the first face of the paper?

    20. b ro

      This was great. Maybe this can be applied as part of auto segmentation as used in medical imaging !

    21. Fredashay Klavierstein

      Oh wow! I wish I had a big brain that could understand the math behind this!

    22. Có ý thức đi

      this guy so smart not you destin

    23. Cem Deletioglu

      turk beee

    24. Demiaz

      There is this game that can no longer be found called Principia, by Bithack.

    25. Kenneth Duncan

      This is going out to my audio engineering buddies. It's super interesting to see a whole different visualization of waves we like to mess around with in synthesis.

    26. joe koplar

      Man, I need sleep. Why you give me this recommendation yt?

    27. StructuredBASIC

      Destin, nobody cares if it's pronounced "jif"

    28. Mohit Bhalotia

      thanks you guys thinks a lot i also learned it but was unable to apply that much

    29. Nitreon

      this is just an average Tuesday for a music producer

    30. urk arsh

      but how did he knew which combination of circles to use to create the picture

    31. snowissj

      Just rewatching this, my first reaction was the yin/yang, the two opposing forces of nature that created everything.

    32. Oscar Valencia

      brrrrr my youtube recommend is weird one second a video is making fun of something on some stupid game on steam the next i’m learning the existence of the universe (ik the vid doesn’t talk about that but i’m just Exaggerating)

    33. Joel Rosas

      Wt would be the writen equation or formula of a circle within a circle ?

    34. Valn3d

      3:10 this made my day

    35. Andres Castañeda

      Play a synthesizer and you will get a new different understanding playing with the oscillators and getting sound out of the waves

    36. gacha Nguyen


    37. DanXtian

      Ojalá estuviese en Español

    38. Howard A. Roark

      Very good explanation.

    39. Buck The Banjo Player

      this proves this whole series is BS

    40. Buck The Banjo Player

      JFC......SAS-GRAPH was in 1980

    41. Nic De Houwer

      But that's how timbre works in audio..

    42. Armando M.

      its totally amazing .. hats off

    43. Cuco Fiat Punto

      What a pity that you can only send one like. I would have sent one billion otherwise. If maths were teached with this approach and enthousiasm they were not the black sheep of the family.

    44. Harold L Potts

      I'm wondering why the term ' epicycle ' was not used [ or if it was I missed it ]. As an interesting aside : a few decades ago there was a lot of interest in how simple systems could produce chaos. There was an adage coined at the time " three makes chaos ". It refers to a system in which a second pendulum is added to a simple pendulum, then a third pendulum is added to the second. If a two pendulum system is driven at constant speed the second pendulum will behave in a harmonic manner. If however a three P system is driven at constant speed the behaviour of the system will become chaotic. Science novelty stores used to sell nifty models reproducing this effect. Applying a thought experiment to the graphic simulations can we imagine that they would show chaos as in a real dynamic system ? If no chaos is observed then why not ? Conversely we can ponder the question why in the real world do systems descend into chaos very soon ? Another thought experiment : can an individual in the act of drawing a continuous line but with a shaky hand be said to drawing an analogue of a Fourier system ?

    45. Manny Khan

      Dogas a bit cocky for a mathematician

    46. 王錫九

      Smart n cool guys

    47. Black Clover

      look what i learned today, anti-aliasing in games is just Fourier series expansion lol

    48. Udo Matthias drums

      love it!!

    49. SykkeX

      Hello, do you know some good app for fourier series and fourier transform for Android?

    50. Dinamy

      No, it's not jif. You don't say "jraphics" interface format, you say GRAPHICS interface format

    51. MD Toukir Hasan

      what is the software name

    52. ليطمئن قلبي

      تم الاشتراك 👍

    53. Hamza Shittu

      6:30 then nothing is truly infinite

    54. Ron Sonntag

      Loved this! Great presentation. Thank You.

    55. Nawadeep Atreya

      What software he used?... 😅😅😅

    56. Peeyush Awasthi

      Great intuitive explanation for beginners

    57. Patrick Bell, Sr.

      Mathematica Rules!!

    58. Saurav Kumar

      Awesome man

    59. Shaun Eisner

      Actually a math and physics student at Georgia Tech doin research in PDEs which uses a lot of Fourier analysis. There is actually a cool relationship between what functions can be represented by a Fourier series. Such functions are called L2 functions and essentially they are the set of functions whose square is integrable over some domain. So most everyday functions can be represented via Fourier series, but if the function is not square integrable, the traditional Fourier series will not always converge. (Note: there are ways to extend the Fourier series to non-L2 functions, but this is a lot more complicated and general than just a sum of sine waves) This is really cool and related to a field of math called functional analysis which describes how different functions can be represented in terms of other functions.

    60. Rob Marzden

      That is sooo cool but is the SmarterEveryDay Icon technically a function or just a set of x&y points that are digitized. Well-deserved mathswagger indeed.

    61. gavquinny

      It's actually Gif, with a g, not a J - the inventor wanted it to be J, but was overthrown by overwhelming public demand

    62. Ahamed Kamil

      Mind blowing

    63. Jackson Morris-Thring

      Something that really made me appreciate the power of the Fourier Series was my 9 hour Fourier optics practical I did this year. Essentially, you collomate a laser beam and put it through a hexagonal matrix-hole’d piece of plastic. You’d pass this through a Fourier lens and it’d leave you with an effect that essentially allowed you to seperate higher order frequencies from lower order frequencies going from the centre outward. Through using another filter, you can then block those said frequencies, and make “false” images. You can change the shape of a sticker-star’s shadow cast onto a ccd camera. Blew.my.mind!

    64. q zorn

      Geee, this is what we need in the white-house and i do not why, it just seems important..........:)

    65. Hemani Gajjar


    66. B. M.

      We call it "gif" not "jif" anyway :)))

    67. sean

      ....what?.... haha..... wow

    68. user 109

      As a professor at MIT said, an idea that you didn't think of sounds genius until you hear the explanation. Same happened cuz after watching this video I'm like, "ya that's what a Fourier series does, transforming functions into sine and cosine. Pretty much like Mc Lauren series transforms a function into polynomial."

    69. Emrah E

      You approach or attitude against Doga was not really kind really. Shame

    70. ANCHOR

      You'd make the greatest teacher

    71. Jarrod C

      Wish you would have explained more in depth and the n terms,,, and how the circle relates , i just wanted more depth

    72. Pro Tem

      It’s pronounced “gif”. Not “gif”

    73. science in time

      Very nice animation explanation

    74. Ivan Windsor

      This is without doubt the best way to teach the Fourier series. I saw this and it clicked immediately after hours of confusion studying books

    75. Prince Baba

      Wish I have the tools and gadgets that you possess.. Includes a lot of investment...But sorry, Can't afford $10/day...well-done

    76. Prince Baba

      Ur videos force us to hit the LIKE button...lol

    77. shakthidhar G A

      Today i actually understood Fourier functions. U are my professor

    78. M A

      4th year electrical engineering here👋

      1. escControl deleteCommand

        You've wasted more than any money can amount with this nonsense. Hyperbolioid and torus keys to the universe.

    79. Ashish Devassy

      This is brilliant

    80. Marcelo Mellado

      How would that face wave sound? Can you do that in Max/MSP?

    81. Victor Wheeler

      **jaw on floor**

    82. Isabel Martín

      Me parece buen video y también el sponsor de lo último. Gracias

    83. Arthur Henrique D. A. Ribeiro

      Nyquist-Shannon too, that's nice

    84. RonG

      It's gif with a g not gif with a j... Unlike the peanut butter

    85. Akul Jamwal

      Dr. Doga hasn't looked happy since his pronunciation of GIF was corrected

    86. Mike Daniels

      Isn’t this how old auto-pens used for signatures operated?

    87. Matt Phillips

      Oh snap! It's the good ol' epicycles from Ptolemaic Astronomy :D

    88. Zelle Sev

      not much of a math wiz but def felt mathswagger after watching and understanding this video😂

    89. Curtis Densmore

      How can it be jif if it's gif?

    90. David Zare

      Check out dr feynmann path integral! You will find the origin of these complex circles he used to invent QED my friend!

    91. Rusty Shackleford

      free man beyond the wall donnie gilbert episode brought mne here today great video

    92. Jamie Poppins

      Are you J-Roc's dad from Trailer Park Boys? Asking for a friend. Haha

    93. Harim Castillo

      You can aproximate anything aslong as you have enough terms🤯

    94. Raimond Spadaro

      Wow!!! Soooo Cool! I Love the way Fourier was visualized. It clicked. Why, oh why, couldn't it have clicked so many years ago....

    95. Javier Flores

      What is the part you used to connect the IPhone wave generator ap to your oscilloscope?

    96. うめぇファンタ


    97. JD Saravaiya

      Flat earthers can really use those kits for understanding basic physics.

    98. Sounak Sinha

      Great explanation. Btw, you look like Marty McFLy :)

    99. Nano Seeker

      Can you match a wave to a planet perfect lower conversation ?

    100. Adinix

      The output on the graphs look alot like the behavior of a properly setup control circuit designed to match a process value like a moving temperature setpoint. This might be an alternative to PID loops.