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.<p>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.<p>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.<p>EDIT: granted, basic linear algebra concepts (vectors, basis transform) needed.
<a href="http://betterexplained.com/articles/an-interactive-guide-to-the-fourier-transform/" rel="nofollow">http://betterexplained.com/articles/an-interactive-guide-to-...</a><p>The animations are here
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.)<p>Sweep it sideways across a dark area, and you will see the waveform as an afterimage.<p>But start to <i>spin</i> 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.
i.e. color coding is necessary because math notation is typically expressed with ambiguous assumptions in service to absolute terseness.<p>As far as I can tell the only time it's attempted to combat this is in proofs.<p>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.
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.
Think of FFT like music notation - a separation into notes, dynamics, and tempo.<p>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.<p>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.
Here is another visualized Fourier
<a href="http://blog.matthen.com/post/42112703604/the-smooth-motion-of-rotating-circles-can-be-used" rel="nofollow">http://blog.matthen.com/post/42112703604/the-smooth-motion-o...</a>
This is an extremely confusing sentence. My sentence would be:<p>> Every "nice" function can be uniquely decomposed into complex exponentials, which in some contexts represent physical frequencies.
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.<p>I agree with another commenter that the more useful explanation is in terms of a change of basis.
The Fourier transform is the map between sound waves and music as written on a staff.<p>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.
I remember fondly a college lab experiment where we had to design a circuit that modulated a signal on a carrier (basic FM modulator).<p>The prep for the lab included a paper running all the math predicting the outcome, that included fourier transforms.<p>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.
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.
For a `slightly` longer explanation: <a href="http://betterexplained.com/articles/an-interactive-guide-to-the-fourier-transform/" rel="nofollow">http://betterexplained.com/articles/an-interactive-guide-to-...</a>
Here's how to make this with MathJax:<p><a href="http://adereth.github.io/blog/2013/11/29/colorful-equations/" rel="nofollow">http://adereth.github.io/blog/2013/11/29/colorful-equations/</a>
Felt inspired after reading this this morning and spent my Saturday hacking up a little demo in d3:<p><a href="http://bbrennan.info/fourier-work.html" rel="nofollow">http://bbrennan.info/fourier-work.html</a>
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.
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:<p>-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)<p>-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<p>-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.<p>-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.<p>So now let's see what this formula is:<p>-it's an average of things (1/N and a sum of N elements)<p>-those things are multplications of values of sine/cosine at given frequency at given point with value of original function at this point<p>-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<p>-the points are evenly spaced from 0 to n-1 (that's why n/N in the formula) and the sum goes over them.<p>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).<p>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.
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).