11 ms·

The Fourier Transform, explained in one sentence

364 points by ggonweb 12y ago

ggonweb 12y ago
http://betterexplained.com/articles/an-interactive-guide-to-the-fourier-transform/ http://betterexplained.com/articles/an-interactive-guide-to-... The animations are here
j2kun 12y ago
This is an extremely confusing sentence. My sentence would be: > Every "nice" function can be uniquely decomposed into complex exponentials, which in some contexts represent physical frequencies.
- fizx 12y ago
  It's confusing because it explains "how." Your sentence explains "what," which misses the original point.
  - j2kun 12y ago
    If you understand the "what" deeply, then the "how" is trivial. The decomposition is in the sense of an orthogonal basis, and so to compute the transform you just do a dot product in the relevant vector space.
- tspiteri 12y ago
  Your sentence describes what the Fourier transform is, whereas the sentence in the article tries to describe how the Fourier transform does it. What the article does is translate the mathematical notation to a sentence in English, attempting to retain all the information in the mathematical notation. It can be helpful if you are trying to visualize what each component of the mathematical notation is doing, but if you separate the sentence from the equation, it is not very helpful. For example, how would you interpret "spin your signal around a circle"?
- ChuckMcM 12y ago
  The key of course it that it is matching the colors of the words with the notation. It is a helpful tool for people who cannot read the notation.
platz 12y ago
i.e. color coding is necessary because math notation is typically expressed with ambiguous assumptions in service to absolute terseness. As far as I can tell the only time it's attempted to combat this is in proofs. Note that in a real paper or textbook, such a detailed description explaining each term would most definitely be absent, even without the color coding.
- bkcooper 12y ago
  Note that in a real paper or textbook, such a detailed description explaining each term would most definitely be absent, even without the color coding. I disagree. In a textbook, it is very unlikely (and would be a sign of a badly written book) that there are terms in a displayed equation that are genuinely undefined. The worst case scenario is that they are not defined immediately next to the equation and presume that you have read the preceding material.
  - cjslep 12y ago
    > The worst case scenario is that they are not defined immediately next to the equation and presume that you have read the preceding material. The textbooks I have used written by Russian mathematicians are almost entirely equation after equation of "explanation", derived from previous equations with the exact transformations in between left as an exercise for the reader. There's a very distinct cultural difference for sure, but that doesn't make these textbooks less real. Edit, to clarify: A lot of terms are not explained in natural language either as their very definition is an equation of other terms (feel free to apply recursively).
  - platz 12y ago
    Yes that's fair - I guess I was thinking more of homework materials distributed as a part of educational content, but all good teaching should try to address this, yes.
crimsonalucard 12y ago
Maybe an animation would serve to be a better vehicle for understanding.
- j_s 12y ago
  https://en.wikipedia.org/wiki/File:Fourier_transform_time_and_frequency_domains.gif https://en.wikipedia.org/wiki/File:Fourier_transform_time_an...
  - Houshalter 12y ago
    All of these explanations remind me of this: http://www.joe-ks.com/archives_may2011/HowToDrawAnOwl.jpg http://www.joe-ks.com/archives_may2011/HowToDrawAnOwl.jpg
deleted 12y ago
[deleted]
marmaduke 12y ago
Neat idea but this is a discrete Fourier transform (the original is continuous, with an integral), and it's only "explained" in the context of signal processing. And even then, the explanation is imperative. I agree with another commenter that the more useful explanation is in terms of a change of basis.
- lisper 12y ago
  And it also relies on the rather fuzzily defined concept of "a bunch of points." Which bunch? Some particular bunch, or will any old bunch do?
  - dnautics 12y ago
    yeah, I don't like the idea of 'spin your signal around a circle', and prefer 'compare your signal to a perfect wave' (multiplication).
karpathy 12y ago
Fourier Transform became trivial to me when I noticed that it's just a basis transform, as you would do with ANY other basis. Except this basis happens to be sine and cos waves of different frequencies. i.e. you view the entire signal in the original domain as a single point in space of dimensionality equal to number of measurements, and then you project it onto the new (fourier) basis. For example, one of the new basis vectors could be sine signal of some frequency k. You find the component of the original signal along this new basis just as you would do for any other basis vector, with a dot product. The only slight complication is that to get the phase information you have to actually dot with a cosine at that frequency as well. The confusing e^ix complex number multiply "hides" the fact that we are actually doing two simultaneous dot products (since e^ix = sinx + icosx): one with the sine and the other with the cos, to get both the frequency and phase. this is the intuition for DFT but for FT you just have infinite-dimensional vector space, done. This is much more intuitive to me than "spinning a signal in circle", which I personally find very to be a very confusing statement. EDIT: granted, basic linear algebra concepts (vectors, basis transform) needed.
- eridius 12y ago
  The isolated phrase "spinning a signal in a circle" is confusing, but the accompanying visualization of tracking a point on a spinning speaker cone is something that can be visualized. More generally, your explanation may make sense to you, but it makes no sense to a layperson (i.e. me). But the linked short description, along with the speaker cone visualization, does make a reasonable amount of sense. I've also seen a similar description in the past with accompanying animated GIFs illustrating the point, instead of the prose about a spinning speaker cone. I thought the animated GIFs made it even easier to understand. Sadly, I don't remember where I saw that.
  - jesuslop 12y ago
    Agreed a layperson could benefit, but karpathy insight is also very nice, it allows to leapfrog from elementary linear methods to a good intuition of the real thing.
    - 12y ago
dlwj 12y ago
Would thinking of the fourier transform as breaking a signal down into basic elements (like atoms) and figuring out how much of each atom is in the signal? Each different value of "K" then is like using a Ph strip to measure approximate energy b/c that particular strip reacts most strongly to a certain k.
ggonweb 12y ago
Here is another visualized Fourier http://blog.matthen.com/post/42112703604/the-smooth-motion-of-rotating-circles-can-be-used http://blog.matthen.com/post/42112703604/the-smooth-motion-o...
Lambdanaut 12y ago
For a `slightly` longer explanation: http://betterexplained.com/articles/an-interactive-guide-to-the-fourier-transform/ http://betterexplained.com/articles/an-interactive-guide-to-...
ggonweb 12y ago
Understanding the fourier transform (from archive) https://archive.today/ulPFk https://archive.today/ulPFk (original link(dead) http://www.altdev.co/2011/05/17/understanding-the-fourier-transform/ http://www.altdev.co/2011/05/17/understanding-the-fourier-tr...)
Terr_ 12y ago
I like to think of it like a vertical line of individual LEDs that sequentially shine to show current amplitude. (Like a Cylon visor or KITT the car.) Sweep it sideways across a dark area, and you will see the waveform as an afterimage. But start to spin it around the center, and you get a blob of light, one that becomes tighter and more symmetric whenever the spin-rate approaches the rate of a repeating subpattern.
- ams6110 12y ago
  THIS makes the most sense to me of any of the explanations on this page.
burtonator 12y ago
you know, it should be possible to generate some of these sentences from the mathematical notation directly to make concepts more understandable to the lay person (and people in general).
madengr 12y ago
It just correlates a signal against harmonically related complex sinusoids.
- cm127 12y ago
  Yeah, and don't forget to sum it all up and divide it by the number of samples.
natch 12y ago
>then there is frequency corresponding the pole's rotational frequency is represented in the sound. Parse error.
- callum85 12y ago
  I can't actually work out what this was meant to say. Anyone?
zwieback 12y ago
I think all the efforts to make FT more intuitive are necessary because representation of signals in terms of complex numbers is a tricky abstraction. It's definitely worth taking the time to get comfortable with the idea of representing a signal in the complex plane without trying to understand the FT at the same time. Then all the spinning-around-a-pole verbiage makes a lot more sense.
- percentcer 12y ago
  I'm glad to hear that, as I didn't understand the pole analogy one bit.
tylerneylon 12y ago
The Fourier transform is the map between sound waves and music as written on a staff. That's an informal statement with some hand-waving since, for example, music notation is discrete and sound waves aren't, but that's the main idea.
- quarterwave 12y ago
  Saw this after posting my comment. I agree this is the simplest way to think of an FFT.
quarterwave 12y ago
Think of FFT like music notation - a separation into notes, dynamics, and tempo. Alexander Graham Bell came up with the idea of increasing the capacity of telegraph lines by combining several 'dit-da-dit' streams at different pitches. He called it a 'harmonic telegraph'. The idea was that a mechanical arrangement of reeds tuned to different pitches would pluck out each stream of data. This insight eventually led to the invention of the telephone. Finally, if 'orthogonal basis set' is difficult to understand, think of giving someone 'left/right' driving directions. An FFT would then (roughly) be the mathematical equivalent of 'driving directions' for a clip of music or speech.
shubhamjain 12y ago
One thing I don't fathom is why FT is always explained in terms of circles? For me it was always confusing this way; the concept was much more graspable when visualized in the terms superposition of sinusoidal waves.
- waldir 12y ago
  Because those sinusoidal waves are merely projections (in the real and complex plane) of the true elemental component of the FT, the complex exponential, which is really a helix: http://m.eet.com/media/1068017/lyons_pt2_3.gif http://m.eet.com/media/1068017/lyons_pt2_3.gif A while ago I made some drafts for a series of diagrams to help visualize this (http://imgur.com/vEcnVdn http://imgur.com/vEcnVdn) but unfortunately never got around to finish it (https://en.wikipedia.org/wiki/Wikipedia:Graphics_Lab/Illustration_workshop/Archive/Sep_2013#RfC_-_Visualization_of_exponential_function https://en.wikipedia.org/wiki/Wikipedia:Graphics_Lab/Illustr...). Anyway, for those who find it easier to think of sinusoidal curves, the animation in the Wikipedia article (https://commons.wikimedia.org/wiki/File:Fourier_transform_time_and_frequency_domains_(small).gif https://commons.wikimedia.org/wiki/File:Fourier_transform_ti...) is a very good visualization (also, BetterExplained's rant on sines being explained as circles may resonate: http://betterexplained.com/articles/intuitive-understanding-of-sine-waves/ http://betterexplained.com/articles/intuitive-understanding-...)
- smosher_ 12y ago
  To be fair, the expression does contain (edit: ) Euler's Formula, which is basically drawing circles in the complex plane, so people are going to draw circles. I started thinking about sinusoids this way after seeing stuff like this: https://3.bp.blogspot.com/_6t_ZmJSkbL4/TJLIDoBQMzI/AAAAAAAACb8/WEbPI4KjoJQ/s1600/sinewave_spiral.gif https://3.bp.blogspot.com/_6t_ZmJSkbL4/TJLIDoBQMzI/AAAAAAAAC... — if you look down at the complex plane, you will see a circle.
- bluecalm 12y ago
  Sine an cosine are functions which tells you what your coordinates are when you travel around a circle (of diameter 1) so anything containing those (here hidden in Euler's formula) will provoke thinking in terms of circles. I find it very useful btw. Once you realize sin(x) represents how high you are on a circle and cos(x) how far to the right (if you start walking on it from [0,1] counter clockwise) then a lot of stuff start making intuitive sense.
jokoon 12y ago
isn't this theorem used to transmit signals across DSL connexions ? isn't it also used for wireless internet ?
__m 12y ago
What about phase?
- dsego 12y ago
  The complex number contains both cos and sin components. Sine is the y axis, cos is the x. From that you know the angle, i.e. phase, right?
Adrock 12y ago
Here's how to make this with MathJax: http://adereth.github.io/blog/2013/11/29/colorful-equations/ http://adereth.github.io/blog/2013/11/29/colorful-equations/
vpeters25 12y ago
I remember fondly a college lab experiment where we had to design a circuit that modulated a signal on a carrier (basic FM modulator). The prep for the lab included a paper running all the math predicting the outcome, that included fourier transforms. The moment we hooked the circuit's output to an spectral analyzer was the first time I saw years of theoretical math resulting on something "tangible": the carrier signal's peak and +/- modulated peaks around it.
bluecalm 12y ago
As I switch off the moment I see "signal" or "wave" let alone spinning it around the circle here is my take without any engineering intuitions: -you can represent any (nice enough) function as sum of sines and cosines of given frequencies and amplitudes (frequency is how often the function hits the whole period, amplitude is by how much you multiple its value). The functions look this way: Amplitude* sine(x* freq) -So you have your sines and cosines at various frequencies and you wonder what the amplitudes should be for every one o them so that if we add them all up we end up with the original function -The idea to answer this question is to see how values of your sine or cosine correlate with values of original function. If it turns out that the original function is high in value when your sine/cosine is high in value then sine/cosine at this frequency needs to have higher weight than sine/cosine at other frequency where it doesn't correlate well with original function. Intuitively if you take functions which correlate well with original one with higher weight and those which doesn't with lower weight then you will have good approximation of the original function. -Euler formula tells you what sine and cosine at given point are representing it as one complex number; to get value of sine/cosine at frequency different than 1 you need to multiple the argument by k*2PI (as frequency is given in beats/period and period is 2PI it's quite obvious why); So e^(xi) is value of cosx and sinx and e^(xik2PI) is value of those functions at frequencies different than 1. So now let's see what this formula is: -it's an average of things (1/N and a sum of N elements) -those things are multplications of values of sine/cosine at given frequency at given point with value of original function at this point -this average is going to be high if places where sine/cosine at given frequency is high and original function is high overlap (you multiple big number by big number) and low if described correlation is low; that's the concept of correlation -the points are evenly spaced from 0 to n-1 (that's why n/N in the formula) and the sum goes over them. That's it. No circles and spinning. The only trick is to realize that: e^(ix) represent value of sine and cosine at x and e^(ixk2PI) represents value of sine/cosine at frequency k (if k is 1, then it's standard sine/cosine as everything simplifies). One sentence explanation could be: the more sine/cosine at given frequency correlates with original function the more that sine/cosine contribute to the original function.
bbrennan 12y ago
Felt inspired after reading this this morning and spent my Saturday hacking up a little demo in d3: http://bbrennan.info/fourier-work.html http://bbrennan.info/fourier-work.html

364 points by ggonweb 12y ago

http://betterexplained.com/articles/an-interactive-guide-to-the-fourier-transform/ http://betterexplained.com/articles/an-interactive-guide-to-... The animations are here

j2kun 12y ago

This is an extremely confusing sentence. My sentence would be: > Every "nice" function can be uniquely decomposed into complex exponentials, which in some contexts represent physical frequencies.

fizx 12y ago
It's confusing because it explains "how." Your sentence explains "what," which misses the original point.
- j2kun 12y ago
  If you understand the "what" deeply, then the "how" is trivial. The decomposition is in the sense of an orthogonal basis, and so to compute the transform you just do a dot product in the relevant vector space.
tspiteri 12y ago
Your sentence describes what the Fourier transform is, whereas the sentence in the article tries to describe how the Fourier transform does it. What the article does is translate the mathematical notation to a sentence in English, attempting to retain all the information in the mathematical notation. It can be helpful if you are trying to visualize what each component of the mathematical notation is doing, but if you separate the sentence from the equation, it is not very helpful. For example, how would you interpret "spin your signal around a circle"?
ChuckMcM 12y ago
The key of course it that it is matching the colors of the words with the notation. It is a helpful tool for people who cannot read the notation.

platz 12y ago

i.e. color coding is necessary because math notation is typically expressed with ambiguous assumptions in service to absolute terseness. As far as I can tell the only time it's attempted to combat this is in proofs. Note that in a real paper or textbook, such a detailed description explaining each term would most definitely be absent, even without the color coding.

bkcooper 12y ago
Note that in a real paper or textbook, such a detailed description explaining each term would most definitely be absent, even without the color coding. I disagree. In a textbook, it is very unlikely (and would be a sign of a badly written book) that there are terms in a displayed equation that are genuinely undefined. The worst case scenario is that they are not defined immediately next to the equation and presume that you have read the preceding material.
- cjslep 12y ago
  > The worst case scenario is that they are not defined immediately next to the equation and presume that you have read the preceding material. The textbooks I have used written by Russian mathematicians are almost entirely equation after equation of "explanation", derived from previous equations with the exact transformations in between left as an exercise for the reader. There's a very distinct cultural difference for sure, but that doesn't make these textbooks less real. Edit, to clarify: A lot of terms are not explained in natural language either as their very definition is an equation of other terms (feel free to apply recursively).
- platz 12y ago
  Yes that's fair - I guess I was thinking more of homework materials distributed as a part of educational content, but all good teaching should try to address this, yes.

crimsonalucard 12y ago

Maybe an animation would serve to be a better vehicle for understanding.

j_s 12y ago
https://en.wikipedia.org/wiki/File:Fourier_transform_time_and_frequency_domains.gif https://en.wikipedia.org/wiki/File:Fourier_transform_time_an...
- Houshalter 12y ago
  All of these explanations remind me of this: http://www.joe-ks.com/archives_may2011/HowToDrawAnOwl.jpg http://www.joe-ks.com/archives_may2011/HowToDrawAnOwl.jpg

deleted 12y ago

[deleted]

marmaduke 12y ago

Neat idea but this is a discrete Fourier transform (the original is continuous, with an integral), and it's only "explained" in the context of signal processing. And even then, the explanation is imperative. I agree with another commenter that the more useful explanation is in terms of a change of basis.

lisper 12y ago
And it also relies on the rather fuzzily defined concept of "a bunch of points." Which bunch? Some particular bunch, or will any old bunch do?
- dnautics 12y ago
  yeah, I don't like the idea of 'spin your signal around a circle', and prefer 'compare your signal to a perfect wave' (multiplication).

karpathy 12y ago

Fourier Transform became trivial to me when I noticed that it's just a basis transform, as you would do with ANY other basis. Except this basis happens to be sine and cos waves of different frequencies. i.e. you view the entire signal in the original domain as a single point in space of dimensionality equal to number of measurements, and then you project it onto the new (fourier) basis. For example, one of the new basis vectors could be sine signal of some frequency k. You find the component of the original signal along this new basis just as you would do for any other basis vector, with a dot product. The only slight complication is that to get the phase information you have to actually dot with a cosine at that frequency as well. The confusing e^ix complex number multiply "hides" the fact that we are actually doing two simultaneous dot products (since e^ix = sinx + icosx): one with the sine and the other with the cos, to get both the frequency and phase. this is the intuition for DFT but for FT you just have infinite-dimensional vector space, done. This is much more intuitive to me than "spinning a signal in circle", which I personally find very to be a very confusing statement. EDIT: granted, basic linear algebra concepts (vectors, basis transform) needed.

eridius 12y ago
The isolated phrase "spinning a signal in a circle" is confusing, but the accompanying visualization of tracking a point on a spinning speaker cone is something that can be visualized. More generally, your explanation may make sense to you, but it makes no sense to a layperson (i.e. me). But the linked short description, along with the speaker cone visualization, does make a reasonable amount of sense. I've also seen a similar description in the past with accompanying animated GIFs illustrating the point, instead of the prose about a spinning speaker cone. I thought the animated GIFs made it even easier to understand. Sadly, I don't remember where I saw that.
- jesuslop 12y ago
  Agreed a layperson could benefit, but karpathy insight is also very nice, it allows to leapfrog from elementary linear methods to a good intuition of the real thing.
  - 12y ago

dlwj 12y ago

Would thinking of the fourier transform as breaking a signal down into basic elements (like atoms) and figuring out how much of each atom is in the signal? Each different value of "K" then is like using a Ph strip to measure approximate energy b/c that particular strip reacts most strongly to a certain k.

ggonweb 12y ago

Here is another visualized Fourier http://blog.matthen.com/post/42112703604/the-smooth-motion-of-rotating-circles-can-be-used http://blog.matthen.com/post/42112703604/the-smooth-motion-o...

Lambdanaut 12y ago

For a `slightly` longer explanation: http://betterexplained.com/articles/an-interactive-guide-to-the-fourier-transform/ http://betterexplained.com/articles/an-interactive-guide-to-...

ggonweb 12y ago

Understanding the fourier transform (from archive) https://archive.today/ulPFk https://archive.today/ulPFk (original link(dead) http://www.altdev.co/2011/05/17/understanding-the-fourier-transform/ http://www.altdev.co/2011/05/17/understanding-the-fourier-tr...)

Terr_ 12y ago

I like to think of it like a vertical line of individual LEDs that sequentially shine to show current amplitude. (Like a Cylon visor or KITT the car.) Sweep it sideways across a dark area, and you will see the waveform as an afterimage. But start to spin it around the center, and you get a blob of light, one that becomes tighter and more symmetric whenever the spin-rate approaches the rate of a repeating subpattern.

ams6110 12y ago
THIS makes the most sense to me of any of the explanations on this page.

burtonator 12y ago

you know, it should be possible to generate some of these sentences from the mathematical notation directly to make concepts more understandable to the lay person (and people in general).

madengr 12y ago

It just correlates a signal against harmonically related complex sinusoids.

cm127 12y ago
Yeah, and don't forget to sum it all up and divide it by the number of samples.

natch 12y ago

>then there is frequency corresponding the pole's rotational frequency is represented in the sound. Parse error.

callum85 12y ago
I can't actually work out what this was meant to say. Anyone?

zwieback 12y ago

I think all the efforts to make FT more intuitive are necessary because representation of signals in terms of complex numbers is a tricky abstraction. It's definitely worth taking the time to get comfortable with the idea of representing a signal in the complex plane without trying to understand the FT at the same time. Then all the spinning-around-a-pole verbiage makes a lot more sense.

percentcer 12y ago
I'm glad to hear that, as I didn't understand the pole analogy one bit.

tylerneylon 12y ago

The Fourier transform is the map between sound waves and music as written on a staff. That's an informal statement with some hand-waving since, for example, music notation is discrete and sound waves aren't, but that's the main idea.

quarterwave 12y ago
Saw this after posting my comment. I agree this is the simplest way to think of an FFT.

quarterwave 12y ago

Think of FFT like music notation - a separation into notes, dynamics, and tempo. Alexander Graham Bell came up with the idea of increasing the capacity of telegraph lines by combining several 'dit-da-dit' streams at different pitches. He called it a 'harmonic telegraph'. The idea was that a mechanical arrangement of reeds tuned to different pitches would pluck out each stream of data. This insight eventually led to the invention of the telephone. Finally, if 'orthogonal basis set' is difficult to understand, think of giving someone 'left/right' driving directions. An FFT would then (roughly) be the mathematical equivalent of 'driving directions' for a clip of music or speech.

shubhamjain 12y ago

One thing I don't fathom is why FT is always explained in terms of circles? For me it was always confusing this way; the concept was much more graspable when visualized in the terms superposition of sinusoidal waves.

waldir 12y ago
Because those sinusoidal waves are merely projections (in the real and complex plane) of the true elemental component of the FT, the complex exponential, which is really a helix: http://m.eet.com/media/1068017/lyons_pt2_3.gif http://m.eet.com/media/1068017/lyons_pt2_3.gif A while ago I made some drafts for a series of diagrams to help visualize this (http://imgur.com/vEcnVdn http://imgur.com/vEcnVdn) but unfortunately never got around to finish it (https://en.wikipedia.org/wiki/Wikipedia:Graphics_Lab/Illustration_workshop/Archive/Sep_2013#RfC_-_Visualization_of_exponential_function https://en.wikipedia.org/wiki/Wikipedia:Graphics_Lab/Illustr...). Anyway, for those who find it easier to think of sinusoidal curves, the animation in the Wikipedia article (https://commons.wikimedia.org/wiki/File:Fourier_transform_time_and_frequency_domains_(small).gif https://commons.wikimedia.org/wiki/File:Fourier_transform_ti...) is a very good visualization (also, BetterExplained's rant on sines being explained as circles may resonate: http://betterexplained.com/articles/intuitive-understanding-of-sine-waves/ http://betterexplained.com/articles/intuitive-understanding-...)
smosher_ 12y ago
To be fair, the expression does contain (edit: ) Euler's Formula, which is basically drawing circles in the complex plane, so people are going to draw circles. I started thinking about sinusoids this way after seeing stuff like this: https://3.bp.blogspot.com/_6t_ZmJSkbL4/TJLIDoBQMzI/AAAAAAAACb8/WEbPI4KjoJQ/s1600/sinewave_spiral.gif https://3.bp.blogspot.com/_6t_ZmJSkbL4/TJLIDoBQMzI/AAAAAAAAC... — if you look down at the complex plane, you will see a circle.
bluecalm 12y ago
Sine an cosine are functions which tells you what your coordinates are when you travel around a circle (of diameter 1) so anything containing those (here hidden in Euler's formula) will provoke thinking in terms of circles. I find it very useful btw. Once you realize sin(x) represents how high you are on a circle and cos(x) how far to the right (if you start walking on it from [0,1] counter clockwise) then a lot of stuff start making intuitive sense.

jokoon 12y ago

isn't this theorem used to transmit signals across DSL connexions ? isn't it also used for wireless internet ?

__m 12y ago

What about phase?

dsego 12y ago
The complex number contains both cos and sin components. Sine is the y axis, cos is the x. From that you know the angle, i.e. phase, right?

Adrock 12y ago

Here's how to make this with MathJax: http://adereth.github.io/blog/2013/11/29/colorful-equations/ http://adereth.github.io/blog/2013/11/29/colorful-equations/

vpeters25 12y ago

I remember fondly a college lab experiment where we had to design a circuit that modulated a signal on a carrier (basic FM modulator). The prep for the lab included a paper running all the math predicting the outcome, that included fourier transforms. The moment we hooked the circuit's output to an spectral analyzer was the first time I saw years of theoretical math resulting on something "tangible": the carrier signal's peak and +/- modulated peaks around it.

bluecalm 12y ago

As I switch off the moment I see "signal" or "wave" let alone spinning it around the circle here is my take without any engineering intuitions: -you can represent any (nice enough) function as sum of sines and cosines of given frequencies and amplitudes (frequency is how often the function hits the whole period, amplitude is by how much you multiple its value). The functions look this way: Amplitude* sine(x* freq) -So you have your sines and cosines at various frequencies and you wonder what the amplitudes should be for every one o them so that if we add them all up we end up with the original function -The idea to answer this question is to see how values of your sine or cosine correlate with values of original function. If it turns out that the original function is high in value when your sine/cosine is high in value then sine/cosine at this frequency needs to have higher weight than sine/cosine at other frequency where it doesn't correlate well with original function. Intuitively if you take functions which correlate well with original one with higher weight and those which doesn't with lower weight then you will have good approximation of the original function. -Euler formula tells you what sine and cosine at given point are representing it as one complex number; to get value of sine/cosine at frequency different than 1 you need to multiple the argument by k*2PI (as frequency is given in beats/period and period is 2PI it's quite obvious why); So e^(xi) is value of cosx and sinx and e^(xik2PI) is value of those functions at frequencies different than 1. So now let's see what this formula is: -it's an average of things (1/N and a sum of N elements) -those things are multplications of values of sine/cosine at given frequency at given point with value of original function at this point -this average is going to be high if places where sine/cosine at given frequency is high and original function is high overlap (you multiple big number by big number) and low if described correlation is low; that's the concept of correlation -the points are evenly spaced from 0 to n-1 (that's why n/N in the formula) and the sum goes over them. That's it. No circles and spinning. The only trick is to realize that: e^(ix) represent value of sine and cosine at x and e^(ixk2PI) represents value of sine/cosine at frequency k (if k is 1, then it's standard sine/cosine as everything simplifies). One sentence explanation could be: the more sine/cosine at given frequency correlates with original function the more that sine/cosine contribute to the original function.

bbrennan 12y ago

Felt inspired after reading this this morning and spent my Saturday hacking up a little demo in d3: http://bbrennan.info/fourier-work.html http://bbrennan.info/fourier-work.html