Kalid, your article appeared in my mailbox and I had no intention of reading it at that moment, but read the first bit and I was totally, totally hooked! I was flat out excellent and we all certainly appreciate the seriously hard work and thought that you obviously put into this. It was riveting, and helped me understand in a much different and wholly more satisfying way, than my college math days, The Fourier Transform. Full marks, Kalid! Full marks!
the only way I know to ‘abort’ the system as we know it (prostitutes like bankers…) is to give it away. Best things in life are not free-they just make you happy to think they are.
Engineering is the art(science?) of fudge factors…
Mother Earth News solved the global energy crisis in the 70’s- nobody listened…
do not share those secrets.
if you know the shortcut maths - hide it. use it for your own profit.
it’s the only power you have. if you share it to everyine - you will lose your advantage. they won’t pay you back, won’t share their secrets.
does bankeers share the shortcut maths for integral sums? no. they just convert it into caviar and gold. leaving others complex as hell ineffective school methods.
@James: Glad you liked it! Thanks for the link, I’ll check it out (I’m planning on doing a more math-focused follow-up, so that’ll come in handy).
@Francisco: More than welcome, glad you enjoyed it.
@Rene: Thanks!
@Yves: No problem :). Beauty (a clear explanation?) is in the eye of the beholder.
@Kwazai: Neat stuff! I believe heat transfer was the original use case for the transform. I’m a physics newbie but would like to get into more applications.
@NeilPost: The Fourier Transform was a brutal mistress for most of us ;).
@Dan: Thanks, really glad it’s helping!
@Peter: D’oh! Of course there’d be a typo in the first line. Fixed now.
@Zaine: Awesome. If there are any parts that are confusing after the 2nd reading, I probably need to reword them :).
Loved this! I’m going to reread it about four more times until I’ve memorized it.
I got stuck at the first formula, where there is an x missing in the exponent.
thank you very much for your help!. I want some solved problems
What a great article - I’ve been playing on and off with Fourier transforms for years… I don’t think I’ve ever seen anyone elucidate on the subject quite as well as this…
Sorry, Fourier transform’s rotted my brain an University, and I accepted a career in Computing and not Electrical and Electronic Engineering 
I know my place.
Thank you for the effort in making this .
Hope everybody who gets swamped in this domain comes here.
I learned it as a way to approximate the solution to conductive heat transfer integral math. The length of the series that substitutes for the equation has diminishing changes after only a few members of the series are calculated. when the material constants are only measureable to within 10%, the answer would be good. the integral math would normally be very hard to solve- the series is easy comparatively speaking (accurate to how many places…).
It was at that time taught for computerized solutions as finite difference method (rather than finite element method). letting the computer grind out the solution to ‘complicated’ math. Not quite Rayleigh’s ‘shooting’ method (superposition methods solved with a first guess on computer-ie natural frequencies of rotating shafts) but at least as effective. Bessel functions ultimately similar- just more obtuse (3rd order nonlinear partial diff eq…) approximations.
[…] http://betterexplained.com/articles/an-interactive-guide-to-the-fourier-transform/ Diversos none […]
[…] felt this way with learning many mathematical concepts. The depth of insight in something like Fourier analysis is so large it’s impossible to fully grasp it with a first pass. New exposures, particularly […]
Rarely seen such a complicated and confusing presentation of the Fourier transform. Sorry, I didn’t like it.
Nicely done!
@Pi: Sure thing. I’m using Javascript and the HTML5 Canvas tag to make the animations. You can open http://betterexplained.com/examples/fourier/?cycles=0,1 and do View Source to see the code. The details of how to do web programming will probably need a few more articles though!
I think the best way to intuit why a spike can be built in the way you describe is by going back to the circle. I am going to speak extremely loosely in the spirit of your blog.
Instead of thinking of a sum of sines, let’s go back to your circle analogy. Imagine N evenly spaced “slots” around the circle, in which we can place some number of “dots” which represent the presence of a frequency. As you said, at time 1, the dots will be all evently spaced out throughout the circle, spaced by 1 unit representing each frequency increment. Then at time 2 they will be spaced out by 2, at time 3, 3, etc. However, very often the dots will “wrap around”. In fact, this will always happen unless the dots are placed with a one unit spacing (N slots, N dots, so either you go in steps of one or you have to loop around). What I want to convince you of now is that whether or not there is looping, the resulting occupied slots form a regular polygon on the unit circle, and further, every occupied slot is occupied equally with dots. If the “step size” is not “divisible” into N, then you will hit every slot exactly once. This is an N-sided regular polygon, each slot gets hit the same number of times. (Good!) When there is “divisibility”, this means that you will eventually hit a space you already occupied, and therefore will begin repeating. We are left with a regular polygon, and it also must have each vertex occupied evenly, since repetition happens after a number of times divisible into N. So the result is always expressible as the sum of the vertices of a regular polygon (times some integer to account for some integer number of layers). But it is visually obvious that the sum of vertices of a regular polygon sum to the center of the polygon (zero).
There is one exception to this rule: a “one-gon” is the only shape with a “bias”. This is what will be responsible for our “spike”. Actually, it’s not surprising that there is a weird exception somewhere, after all, if there were destructive interference everythere, then we woud have found a nontrivial sum of sines that add to zero, proving they are not linearly independent!
Everything I’ve said above is very loose, but I think the purpose of this blog is not to prove things rigorously, but to get a really good intuition for them. And I hope this is what I have done.
Simply, Brilliant
“Click graph the graph to pause/unpause.” --> “Click the graph to pause/unpause.”