9 ms·

An Interactive Introduction to Fourier Transforms

421 points by bpierre 6y ago

phab 6y ago
This very elegantly explains how passband-style frequency based smoothing algorithms work... without trying!
hawktheslayer 6y ago
Man where was this site when I was studying for my Linear Systems and Signals class in my EE program?
corytheboyd 6y ago
This is remarkable, very well put together OP!
miccah 6y ago
This interactive intro to DSP is also great for gaining an intuition. https://jackschaedler.github.io/circles-sines-signals/index.html https://jackschaedler.github.io/circles-sines-signals/index....
adamnemecek 6y ago
Fourier is legit magic. A physical manifestation of Fourier is light being split into it's component frequencies when passing through a prism. I think that this can be used to solve NP in P. All current thinking happens in the context of Turing machines, however if you can factorize a number in O(1) (not that a prism has any concept of Big O) things are different.
clashmeifyoucan 6y ago
Those interactive animations are fantastic. I could just keep looking and looking.
bonoboTP 6y ago
Even after taking a college level signal processing course, I didn't really grok why we use sines and cosines. Any explanation of Fourier theory just starts with the idea to break up a signal into sine and cosine components. Perhaps they'd say those are the simplest, purest instantiation of a frequency. But the idea still just falls out of the sky. I believe I know now: it's because of the convolution theorem. When performing a Fourier transform, we represent the signal in a new basis, where each component will independently get transformed by linear shift invariant systems. Basically the Fourier basis diagonalizes the convolution operator. And the even deeper reason for that is that a complex exponential function can be shifted by simply multiplying it with a constant. In simpler words, it comes down to the barber pole illusion. If you rotate a spring-shaped 3d curve, it looks as if it was traveling upwards. And the 2d projection of the spring are sines and cosines. And it turns out that linear, shift-invariant systems are really common (or are at least a good approximation of many common natural phenomena), so it's very helpful to break up a signal into pieces that each get independently transformed, without any interaction effects.
- sbeller 6y ago
  > Perhaps they'd say those are the simplest, purest instantiation of a frequency. Actually I thought it is the other way round: A frequency is defined as a sine basically, it has an * amplitude * frequency * phase shift The phase shift is why we need cosines and sines in the Fourier transform as cosine is just a shifted sine. The frequency is the constant in the argument of the sine. And here is where the tail might chase the dog: it's the definition, not an observation I'd think. > When performing a Fourier transform, we represent the signal in a new basis, where each component ... each component is a sine, that is described by the 3 constants above. > In simpler words, it comes down to the barber pole illusion. If you rotate a spring-shaped 3d curve, it looks as if it was traveling upwards. And the 2d projection of the spring are Sines and cosine. Exactly, a complex exponential function is just that spring. And if the absolute value of the argument is not exactly one, the spring spirals outwards or inwards.
  - bonoboTP 6y ago
    > Actually I thought it is the other way round: A frequency is defined as a sine basically, it has an * amplitude * frequency * phase shift But this is also true of any periodic wave. The beginner question is, why don't we use triangle waves of square waves? Those also have amplitude, frequency and phase. Frequency just means the reciprocal of the period. To which my answer is that the magical property of complex exponential functions is that they can be shifted by constant (pointwise) multiplication. Which is a really non-obvious fact at first but is crucial in the machinery. The complex exponentials constitute an orthogonal basis which diagonalizes the convolution.
    - kmill 6y ago
      I think it's that exponentials simultaneously diagonalize time shifts, derivatives, and integrals. Convolution follows from these -- though there's something to be said for putting convolution above derivatives and integrals in importance. (A deeper thing going on is that a periodic function can be thought of as a function whose domain is a circle. Circles have obvious rotational symmetry, and when you have symmetry you can use representation theory to decompose things into an (orthogonal) basis. In this case, rotations commute with each other so by some theory the decomposition is going to be entirely through eigenvectors of rotation, which happen to precisely be the exponential functions e^(n theta i) for n an integer. This decomposition is also an isomorphism that carries convolutions to point-wise products in both directions. Also: if you make it so the circle is the complex unit circle, a Fourier transform is the idea that you can create a Laurent polynomial that extends the function to the complex plane minus the origin.)
greggman3 6y ago
Borrowed the data from one of the illustrations on that site and dropped it into a vertex shader https://www.vertexshaderart.com/art/XwLcGCStsrbhX6jFY https://www.vertexshaderart.com/art/XwLcGCStsrbhX6jFY
- djmips 6y ago
  haha. nice!
dukeofdoom 6y ago
Is the source code provided for the demo...where you can type and it will convert it to that animation.
- GormHouj 6y ago
  It's linked at the bottom of the article: https://github.com/Jezzamonn/fourier https://github.com/Jezzamonn/fourier
jbverschoor 6y ago
That was a lot better than most "FFT translates waves to frequencies" I enjoyed looking at the square sound, as it now makes sense why a square sound is a lot fuller. It simply has a lot more waves for your ears. Also the short JPG intro was really good!
LockAndLol 6y ago
> We're going to leave the mathematics and equations out of it for now. There's a bunch of interesting maths behind it, but it's better to start with what it actually does, and why you'd want to use it first. If you want to know more about the how, there's some further reading suggestions below! Yes, yes, yes, yes! Anybody teaching really has to understand that if you need a formula to explain a concept, you haven't understood the concept yourself OR you simply don't know how to explain it yet. Sure, there are exceptions (there always are), but in my experience, lecturers, teachers and academics reach for formulas to "explain" things too quickly. Formulas are not explanations. They serve as PROOF that what you explained actually works.
itsmrwaverly 6y ago
Taking a signals & systems course, currently, and will be sharing this during our next lecture. So helpful!
daza 6y ago
Amazing work!
deleted 6y ago
[deleted]
diedyesterday 6y ago
Fourier transform is so fundamental to understanding the world and mastering Fourier transform (and linear algebra too) is so sooo important before entering anything "quantum". Quantum field theory (QFT), especially, is essentially a way to look at the world through its Fourier transform. In a nutshell Fourier transform, the derivative operator, exponential function and CAUSALITY are related at a very deep level. And that's why it's such an important mathematical tool. Because it is related to causality.
- alok-g 6y ago
  I understand all the noted mathematics. Any suggestions for references on learning quantum theories? For most materials, I get stuck as they use notations unfamiliar to me in spite of me being familiar with the underlying mathematics. Thanks.
  - sannee 6y ago
    This lecture series by Frederic Schuller is pretty amazing: https://www.youtube.com/watch?v=GbqA9Xn_iM0&list=PLPH7f_7ZlzxQVx5jRjbfRGEzWY_upS5K6 https://www.youtube.com/watch?v=GbqA9Xn_iM0&list=PLPH7f_7Zlz... Notably, it treats QM with much more rigor than is usual in introductory courses, so having a nontrivial mathematical background is not an obstacle.
    - alok-g 6y ago
      Thanks! Will watch. :-)
  - diedyesterday 6y ago
    You could greatly benefit from Prof. Leonard Susskind's lectures on quantum mechanics and QFT on Youtube (Standford university's channel), which are very accessible. A great physicist who happens to be a great teacher and who gets all the deep interconnections involved. This is one playlist (he has many more): https://www.youtube.com/watch?v=JzhlfbWBuQ8&list=PL84C10A9CB1D13841 https://www.youtube.com/watch?v=JzhlfbWBuQ8&list=PL84C10A9CB...
kzrdude 6y ago
I think I need an intutive explanation for the Laplace tranform instead
intricatedetail 6y ago
Wow this is very cool! Do you know how people achieve something that works like FFT but without much latency? Go example an EQ that you can draw any shape of.
secondcoming 6y ago
This is awesome. Great effort.
mensetmanusman 6y ago
Fun fact: the human auditory system works by splitting incoming sounds into frequencies. Each hair has a different characteristic resonant mode, so our brain processes information in f-space. https://www.discovermagazine.com/mind/the-brain-ringing-in-the-ears-actually-goes-much-deeper-than-that https://www.discovermagazine.com/mind/the-brain-ringing-in-t...
interestica 6y ago
Another cool FFT + image use: https://www.youtube.com/watch?v=yyox358zIRw https://www.youtube.com/watch?v=yyox358zIRw You can use FFT principles to isolate and remove repeated patterns in images (such as the patterns created by those textured papers of old photographs). https://www.3d4x.ch/Swift's-Reality/FFT-Photoshop-plugin-by-Alex-Chirokov/16,35 https://www.3d4x.ch/Swift's-Reality/FFT-Photoshop-plugin-by-... I've personally used this in a project and the results are impressive.

421 points by bpierre 6y ago

phab 6y ago

This very elegantly explains how passband-style frequency based smoothing algorithms work... without trying!

hawktheslayer 6y ago

Man where was this site when I was studying for my Linear Systems and Signals class in my EE program?

corytheboyd 6y ago

This is remarkable, very well put together OP!

miccah 6y ago

This interactive intro to DSP is also great for gaining an intuition. https://jackschaedler.github.io/circles-sines-signals/index.html https://jackschaedler.github.io/circles-sines-signals/index....

adamnemecek 6y ago

Fourier is legit magic. A physical manifestation of Fourier is light being split into it's component frequencies when passing through a prism. I think that this can be used to solve NP in P. All current thinking happens in the context of Turing machines, however if you can factorize a number in O(1) (not that a prism has any concept of Big O) things are different.

clashmeifyoucan 6y ago

Those interactive animations are fantastic. I could just keep looking and looking.

bonoboTP 6y ago

Even after taking a college level signal processing course, I didn't really grok why we use sines and cosines. Any explanation of Fourier theory just starts with the idea to break up a signal into sine and cosine components. Perhaps they'd say those are the simplest, purest instantiation of a frequency. But the idea still just falls out of the sky. I believe I know now: it's because of the convolution theorem. When performing a Fourier transform, we represent the signal in a new basis, where each component will independently get transformed by linear shift invariant systems. Basically the Fourier basis diagonalizes the convolution operator. And the even deeper reason for that is that a complex exponential function can be shifted by simply multiplying it with a constant. In simpler words, it comes down to the barber pole illusion. If you rotate a spring-shaped 3d curve, it looks as if it was traveling upwards. And the 2d projection of the spring are sines and cosines. And it turns out that linear, shift-invariant systems are really common (or are at least a good approximation of many common natural phenomena), so it's very helpful to break up a signal into pieces that each get independently transformed, without any interaction effects.

sbeller 6y ago
> Perhaps they'd say those are the simplest, purest instantiation of a frequency. Actually I thought it is the other way round: A frequency is defined as a sine basically, it has an * amplitude * frequency * phase shift The phase shift is why we need cosines and sines in the Fourier transform as cosine is just a shifted sine. The frequency is the constant in the argument of the sine. And here is where the tail might chase the dog: it's the definition, not an observation I'd think. > When performing a Fourier transform, we represent the signal in a new basis, where each component ... each component is a sine, that is described by the 3 constants above. > In simpler words, it comes down to the barber pole illusion. If you rotate a spring-shaped 3d curve, it looks as if it was traveling upwards. And the 2d projection of the spring are Sines and cosine. Exactly, a complex exponential function is just that spring. And if the absolute value of the argument is not exactly one, the spring spirals outwards or inwards.
- bonoboTP 6y ago
  > Actually I thought it is the other way round: A frequency is defined as a sine basically, it has an * amplitude * frequency * phase shift But this is also true of any periodic wave. The beginner question is, why don't we use triangle waves of square waves? Those also have amplitude, frequency and phase. Frequency just means the reciprocal of the period. To which my answer is that the magical property of complex exponential functions is that they can be shifted by constant (pointwise) multiplication. Which is a really non-obvious fact at first but is crucial in the machinery. The complex exponentials constitute an orthogonal basis which diagonalizes the convolution.
  - kmill 6y ago
    I think it's that exponentials simultaneously diagonalize time shifts, derivatives, and integrals. Convolution follows from these -- though there's something to be said for putting convolution above derivatives and integrals in importance. (A deeper thing going on is that a periodic function can be thought of as a function whose domain is a circle. Circles have obvious rotational symmetry, and when you have symmetry you can use representation theory to decompose things into an (orthogonal) basis. In this case, rotations commute with each other so by some theory the decomposition is going to be entirely through eigenvectors of rotation, which happen to precisely be the exponential functions e^(n theta i) for n an integer. This decomposition is also an isomorphism that carries convolutions to point-wise products in both directions. Also: if you make it so the circle is the complex unit circle, a Fourier transform is the idea that you can create a Laurent polynomial that extends the function to the complex plane minus the origin.)

greggman3 6y ago

Borrowed the data from one of the illustrations on that site and dropped it into a vertex shader https://www.vertexshaderart.com/art/XwLcGCStsrbhX6jFY https://www.vertexshaderart.com/art/XwLcGCStsrbhX6jFY

djmips 6y ago
haha. nice!

dukeofdoom 6y ago

Is the source code provided for the demo...where you can type and it will convert it to that animation.

GormHouj 6y ago
It's linked at the bottom of the article: https://github.com/Jezzamonn/fourier https://github.com/Jezzamonn/fourier

jbverschoor 6y ago

That was a lot better than most "FFT translates waves to frequencies" I enjoyed looking at the square sound, as it now makes sense why a square sound is a lot fuller. It simply has a lot more waves for your ears. Also the short JPG intro was really good!

LockAndLol 6y ago

> We're going to leave the mathematics and equations out of it for now. There's a bunch of interesting maths behind it, but it's better to start with what it actually does, and why you'd want to use it first. If you want to know more about the how, there's some further reading suggestions below! Yes, yes, yes, yes! Anybody teaching really has to understand that if you need a formula to explain a concept, you haven't understood the concept yourself OR you simply don't know how to explain it yet. Sure, there are exceptions (there always are), but in my experience, lecturers, teachers and academics reach for formulas to "explain" things too quickly. Formulas are not explanations. They serve as PROOF that what you explained actually works.

itsmrwaverly 6y ago

Taking a signals & systems course, currently, and will be sharing this during our next lecture. So helpful!

daza 6y ago

Amazing work!

deleted 6y ago

[deleted]

diedyesterday 6y ago

Fourier transform is so fundamental to understanding the world and mastering Fourier transform (and linear algebra too) is so sooo important before entering anything "quantum". Quantum field theory (QFT), especially, is essentially a way to look at the world through its Fourier transform. In a nutshell Fourier transform, the derivative operator, exponential function and CAUSALITY are related at a very deep level. And that's why it's such an important mathematical tool. Because it is related to causality.

alok-g 6y ago
I understand all the noted mathematics. Any suggestions for references on learning quantum theories? For most materials, I get stuck as they use notations unfamiliar to me in spite of me being familiar with the underlying mathematics. Thanks.
- sannee 6y ago
  This lecture series by Frederic Schuller is pretty amazing: https://www.youtube.com/watch?v=GbqA9Xn_iM0&list=PLPH7f_7ZlzxQVx5jRjbfRGEzWY_upS5K6 https://www.youtube.com/watch?v=GbqA9Xn_iM0&list=PLPH7f_7Zlz... Notably, it treats QM with much more rigor than is usual in introductory courses, so having a nontrivial mathematical background is not an obstacle.
  - alok-g 6y ago
    Thanks! Will watch. :-)
- diedyesterday 6y ago
  You could greatly benefit from Prof. Leonard Susskind's lectures on quantum mechanics and QFT on Youtube (Standford university's channel), which are very accessible. A great physicist who happens to be a great teacher and who gets all the deep interconnections involved. This is one playlist (he has many more): https://www.youtube.com/watch?v=JzhlfbWBuQ8&list=PL84C10A9CB1D13841 https://www.youtube.com/watch?v=JzhlfbWBuQ8&list=PL84C10A9CB...

kzrdude 6y ago

I think I need an intutive explanation for the Laplace tranform instead

intricatedetail 6y ago

Wow this is very cool! Do you know how people achieve something that works like FFT but without much latency? Go example an EQ that you can draw any shape of.

secondcoming 6y ago

This is awesome. Great effort.

mensetmanusman 6y ago

Fun fact: the human auditory system works by splitting incoming sounds into frequencies. Each hair has a different characteristic resonant mode, so our brain processes information in f-space. https://www.discovermagazine.com/mind/the-brain-ringing-in-the-ears-actually-goes-much-deeper-than-that https://www.discovermagazine.com/mind/the-brain-ringing-in-t...

interestica 6y ago

Another cool FFT + image use: https://www.youtube.com/watch?v=yyox358zIRw https://www.youtube.com/watch?v=yyox358zIRw You can use FFT principles to isolate and remove repeated patterns in images (such as the patterns created by those textured papers of old photographs). https://www.3d4x.ch/Swift's-Reality/FFT-Photoshop-plugin-by-Alex-Chirokov/16,35 https://www.3d4x.ch/Swift's-Reality/FFT-Photoshop-plugin-by-... I've personally used this in a project and the results are impressive.