from Hacker News

An animated introduction to the Fourier Transform [video]

by e0m on 1/26/18, 7:24 PM with 31 comments

  • by knolan on 1/28/18, 10:43 AM

    3blue1brown’s videos are excellent. They build intuition in a calm and friendly way with an appropriate amount of useful animation. This is how we make mathematics accessible.

    I’m currently considering moving back into academia and there are a lot of topics in my field that I know students often struggle with that would be greatly helped by some simple animations. Fortunately I’m pretty competent with blender and I relish the idea of developing something worthwhile.

  • by magnat on 1/28/18, 10:01 AM

  • by Patient0 on 1/28/18, 11:15 AM

    He touches on it - but I’d love to see an intuitive explanation of why the response of each frequency to the input function is linearly independent. i.e the fact that Fourier transform of the sum is equal to the sum of the Fourier transforms. This is “why it works” - it’s what makes the frequency space an orthonormal basis - but it’s never been intuitively obvious to me. Otherwise, there would be more than one way of decomposing a function into a superposition. e.g. what would be useful is to give an example of a set of functions which are not linearly independent.

  • by nayuki on 1/28/18, 4:20 PM

    BetterExplained (Kalid Azad) has a good written article that covers the Fourier transform in a similar manner to the 3Blue1Brown video: https://betterexplained.com/articles/an-interactive-guide-to...

    I have an article explaining step by step how to implement code for the discrete version of the Fourier transform: https://www.nayuki.io/page/how-to-implement-the-discrete-fou...

  • by adamnemecek on 1/28/18, 9:28 AM

    I’ll just leave this here

    http://tomlr.free.fr/Math%E9matiques/Math%20Complete/Analysi...

    Mathematics of the discrete Fourier Transform by Julius O. Smith. (O stands for Orange I hope)

  • by nsb1 on 1/28/18, 2:37 PM

    I really wish this stuff existed when I was learning about FFTs - this video describes the theory far better and in far less time than my broken-english college professors ever could.

  • by madez on 1/28/18, 5:34 PM

    Sound waves don't add up linearly. However, it is a good enough idealization for many uses.

    Fourier analysis is also approachable from the discrete setting of finite vectors instead of functions, where the fourier analysis is just an orthogonal (orthonomal when sanely defined) linear function, i.e. it acts by matrix multiplication and is represented as that matrix.

    This appropriately extended to the continous setting leads to the fourier transform on functions, and also gives intuition why the fourier transform uses integrals.

  • by kranner on 1/28/18, 11:09 AM

    This one is related and (I think) quite good:

    https://www.youtube.com/watch?v=r18Gi8lSkfM

  • by nwatson on 1/28/18, 9:05 AM

    I think it's much easier and more direct to visualize the time-domain as superposition of helical components and the transform as an exploration of what happens when you twist the "cylinder" with varying "intensities". You avoid the vague center-of-mass spike depicted here and start from the get-go with the terms of the transform.

  • by probinso on 1/28/18, 6:11 PM

    In fMRI data, we refer to frequency space of volumetric image data as K-Space.

    I would like a general term for frequency space of a signal, without the use of the word `frequency` . This is because `frequency` is also used when describing histograms in general image processing, and is in general an overloaded term.

    Any established words or phrases in the corpus? any tips?

  • by ablaba on 1/28/18, 1:30 PM

    This fourier transform simulation example from shadertoy is good. https://www.shadertoy.com/view/ltKSWD

  • by ablaba on 1/28/18, 1:33 PM

    This fourier transform simulation example from shadertoy is good.

    https://www.shadertoy.com/view/ltKSWD

  • by wendyjreichert on 1/28/18, 4:39 PM

    How do you animate something like this?

  • by tambourine_man on 1/28/18, 11:48 AM