I have desperately wanted to understand the Fourier Transform and signal processing for a long time. In high school I read every tutorial or book I could get my hands on, hoping that one would finally "click." The very first thing I did when I got to college was approach the CS faculty and ask for recommendations for literature that would help me understand it. My college didn't have a signals and systems class, but a few years later I heard that Richard Lyons's book "Understanding Digital Signal Processing" was supposed to be the friendliest book on the subject so I bought and read it.

I say all of this just to impress upon you that I was serious about wanting to understand this stuff.

Despite all the effort, the FFT and DFT were never more than opaque blobs of math to me, mysterious boxes that you put inputs into and could extract outputs from (albeit in a very obscure format that seemed to involve complex numbers for no apparent reason). I sort of gave up and forgot about it for several years.

Recently I came upon Stuart Riffle's article (http://www.altdevblogaday.com/2011/05/17/understanding-the-f...) which explained it in terms of the circular approach that this article adopts. Rotate the waveform about the origin at the frequency of interest and average the samples, simple as that.

I am not exaggerating when I say that after 10 minutes of reading that article, I had an understanding that years of highly motivated knowledge-seeking had not given me.

I take a few things from this story. One is that, given a specific learner (ie. me), some ways of explaining things are infinitely better than others. I mean that literally, because this new, circle-based explanation gave me an understanding that I literally was not capable of achieving with all of the symbolic and algebraic explanations I had studied previously. Judging from other comments I've read, I'm not alone in this, which says to me that there could be exciting advances ahead of us in the way we learn. Just as the Khan Academy is making learning more accessible, I hope that it could also help discover and widely disseminate the best explanations for things. Sal Khan is very good at explaining things but we can't expect him to think of everything. This fantastic DFT explanation was created by a random systems/graphics programmer; I hope it will percolate into DFT curricula until everyone who is learning about the subject is at least exposed to it.

My other takeaway, though, is that hard-core math people really think in a fundamentally different way than I do. I'm a highly intuitive thinker, and formulas are a sea of meaningless symbols to me without an intuitive understanding of what is going on. That someone could understand the DFT without thinking in terms of the circular interpretation is amazing to me. I now know that I am indeed capable of understanding the concept, but only by thinking about it in a different way than most math people do. I suspect, however, that my intuitive way of thinking about it would be more difficult to formalize and make rigorous, so in the end I am dependent on the mathematicians and their way of thinking, even if I can't as easily understand things in their terms.

(One other example of this: I think calculus is simpler to understand in terms of infinitesimals rather than limits, but this is another example where the infinitesimals are more difficult to make rigorous).