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      First time I’ve ever understood how this works, rather than just the end result.

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        Wouldn’t it be simpler not to even talk about energy and spinning etc? Here is a simpler one sentence:

        FT is an operation for calculating which frequencies your signal contains and how much.

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          You are explaining what it is but not how it works.

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            I can’t quite picture “spinning a signal around a circle” in my mind’s eye, can you? It is just stating in English what e^(i*2*pi) is, which is a circle.

            How about this:

            “Probing your signal with a unit vector spinning at all angles between 0 and 2π to calculate how much of it lies along each angle”

            This I can picture..

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              Maybe just “wrap” might be better than “spin”.

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                Sure I can. It’s like an oscilloscope where the horizontal sweep is a polar sweep instead.

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              I think that you’re half-right. Your approach is “simpler” in the sense that it’s more specific to signals, and that’s the context that we’ll usually care about. I wonder whether we could also be “simpler” in the sense that we would talk about conjugates instead of signals in particular. Maybe we could adapt the article’s sentence to something like:

              To find the conjugate variable for a particular measurement, differentiate that variable with respect to its conjugate at that measurement, and average the values that the variable can take.

              Let’s try it for position and (linear) momentum:

              To find the momentum of a particle given our ability to measure its position, take the finite differences between each measured position, and average them.

              Technically correct! The missing ingredient is time; we normally don’t want to talk about time because we are using time as one of our two domains for signal analysis, and this is a blind spot for the original one-sentence approach.

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              When I finished my bachelor thesis (that dealt with wavelets, a continuation of the Fourier Kingdom) I was trying to understand the process, the “intuition”, behind the Fourier Transform through blogs, gifs and other easy materials such as this.

              The end result, like others are also stating here, is that I still did not manage to understand it. Not even a simple one color coded sentence such as the one posted here. This only frustrated me further.

              Today, 15 years later, I find that, for me at least, it was just a waste of time and energy that I could have put into what I ended up doing anyway: further study. The more I read about applied math, signal processing, optimization, calculus and abstract algebra things often and unintended came back to the Fourier transform or one of its many derivations or abstractions such as the DCT or the Z transform.

              So my advice to people reading this type of blogs is to continue to do so for fun, but to mostly stick to in-depth constant and consistent reading of papers and books on the subject while trying to solve different practical or, later on, teoretical problems. Things will come around after a while and they will keep on coming around :)

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                I found a link to a snapshot of Stuart’s blog article that’s referenced in there: https://archive.ph/ulPFk Annoyingly, that animation seems to be gone (or at least it’s not working in Firefox)…

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                  Interesting. I approach it via curve fitting, and choosing the set of curves I fit to have specific interesting properties. This also generalizes nicely to things like Legendre and Laguerre polynomials and other orthonomal function bases, even as far out as things like Gegenbauer polynomials.

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                    That is another great way to look at it. The sine and cosine waves form a basis for a certain set of functions - and infinite dimensional space. and indeed other useful spaces and bases’ exist.