Thanks Frank! Really glad you enjoyed it :).
“…decomponse each element into components with sin and cosine as function. …” I mean sin and cosine as basis of course 
Sorry if this is a dumb question, but I am confused about one thing, what is a time spike exactly?
Like what is the difference btw (,) and [] in your notation?
No worries, let me see if I can clarify. Imagine our signal has 4 datapoints. At t=0 the signal has value a, at t=1 it has value b, and so on:
(a b c d)
We can see this signal as 4 separate signals:
(a 0 0 0) + (0 b 0 0) + (0 0 c 0) + (0 0 0 d)
That is, the signal we’re looking at is really made up of a “spike” that shows up for an instant and disappears. (Usually we think of a signal as a single, smooth entity but what if we “pixelate” it and look at it piece by piece?)
The Fourier Transform helps us create an individual spike, i.e. how to make (a 0 0 0) only using circular components. Next, we can recreate (0 b 0 0) with circles, and then (0 0 c 0), and finally (0 0 0 d).
If we know how to create each instant of the signal (each spike), we can combine the recipes to generate the entire signal.
Notation-wise, I used () when talking about the signal’s original data points in time, and [] when talking about the circular pattern that would help us model the signal. It might not be clear enough in the article though.
Got lost when you started talking about “time spikes”. I still don’t really understand anything tbh. Not really sure I understand “time points” either from the first simulation.
@P: Good feedback, thanks. The “time spike” is my informal name for a delta function, which is a burst of activity at a single instant and 0 activity otherwise (for example: http://en.wikipedia.org/wiki/Kronecker_delta). I clarified the post a bit.
that was fascinating, keep writing! i would appreciate it, if you could post something about hilbert transform as well; and thanks for the great explanation, it really worked for me 
Kalid
I continue to be fascinated, dare I say MESMERIZED by your website. Today I discovered your excellent piece on Fourier Transforms which brought back memories. Nearly 20 years ago I was writing my dissertation and had occasion to use the FT. I confess that I struggled to learn only sort of how it worked (my dissertation was not in mathematics). Would that your site had existed then.
That got me thinking about how you have many components that can be combined into much more complex ideas, essentially the beauty of math itself. You might be interested in just one example near to my area of interest which is extreme value theory.
Your site touches on the stock market. Elsewhere it deals with probability. Through the concept of risk these are related. But the story gets better. Most of the headlines in the popular press are about extreme values, mostly headed as “crashes” or “meltdowns” in order to sell more page views. Indeed, the risk manager who fails to allow for extreme values is at more risk than he perceives, especially if his view of risk is through the famous (and simple) normal distribution.
It turns out that the normal distribution (which appeals to the Central Limit Theorem) is a special case of a more general class of distributions known as Stable-Paretian (from Pareto, which is discussed in your 80/20 article). The S-P distribution appeals to the Generalized Central Limit Theorem. S-P distributions allow for extreme values, popularized as “long tailed” or “black swan-type” therefore more accurately describing the actual risk in the real world. The practical problem is that, except for the normal and two others, the pdf for Stable distributions does not exist. BUT a solution is found in the fact that the S-P characteristic function (which always exists) and its pdf is a Fourier Transform pair! As you would say: Whablamo! We have a way to model rare events in a conventional fashion. Once you have a pdf a lot of things get easier. There is a lot of detail about this at www.mathestate.com for those interested.
Anyway, thanks for not only bringing back fond memories but deepening my understanding of my own research.
My best
RJB
Hi Kalid,
Fantastic site, and great explanation of the Fourier Transform! I wrote up my own explanation of the Discrete Fourier Transform which is more focused on signal processing. May be interesting for your readers!
Hi Roger, thanks so much for the note. I really want to get into stats this year as it’s a giant gap in my intuition. Glad the article was helpful for your research!
It’s largely just a vector space of function where you can decomponse each element into components with sin and cosine as function. And the Fourier transform is just a change of basis in this vector space.
Hey, the section “Making A Spike In Time” could be improved, by not presenting the “position of each cycle” table just like that, before explaining it. Because people will try to understand things the moment they read them, and not know to put them “on hold” for a coming explanation. But even putting them“on hold” is a serious problem, because it takes away one of only 6-9 active memory slots, and makes processing anything that builds upon that blocker impossible.
I was stuck for way too long, until I saw the explanation below, and still felt very uncomfortable, having to first read below, then jump up and now be able to understand the table, and then jump back down behind the explanation again.
In fact, this is a general concept: Never mention anything before its explanation. Inside its explanation is acceptable, if the reader is aware he’s inside at that point. (In the above example, one isn’t.)
If you could make that a principle of all articles on this site, it would be a big factor in making sure the articles are understandable.
Also: Could you make every headline and embedded thing (like tables, those graph animations, images, etc) linkable via a anchor, that would be nice.
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Fantastic article!
Kalid… amazing work! your intuitions link to natural truths…and explains another dimension.
Thank you for sharing this.
This is a peice of artwork ! Great work . Consider it a charity at Warren Buffet level to make this so intutitve to understand.
This is a peice of artwork ! Great work . Consider it a charity as Warren Buffet level to make this so intutitve to understand.
great storytelling…
can u plz tell me why do we have to multiply an exponential raised to a complex number meaning that IT HAS TO BE rotated to get frequency information?
Great article Kalid. This will help to intuitive learners.
Fourier, in addition to his work in theoretical physics and math, he was also the first to discover the greenhouse effect. http://geosci.uchicago.edu/~rtp1/papers/NatureFourier.pdf
Great article. Well done!