This probably won't be a very popular take.. but to me Fourier Transforms are the perfect example of the opposite problem - where the educators are so hellbound on constructing a visual explanation that they end up brain damaging the students. I've seen this come up several times in my math education.
For the Fourier transform there are primarily two issue:
1. The "complex plane" - this tries to make complex numbers somehow less scary and more intuitive? But it's actually a useless crutch that gives people a false sense of understanding. The core issue is that you can multiply two complex numbers and get another complex number. If I give you two vectors or two points on a map and tell you to multiply them.. you can't - b/c that is not a defined operation. We have no intuition about how to multiple 2D numbers (you might think, oh dot products and cross products! but those are something unfortunately unrelated)
2. "Frequency Space" and an the constant suggestion that your signal is somehow being broken up into it's magical innate hidden frequencies. This is also very deceptive (or doesn't hold up in the discrete case). The basis is selected to be mathematically convenient (ie. orthogonal) and is based on your sampling window and properties of complex numbers. These may correspond to some underlying natural frequency that's occurring - or it may not. But it's not helpful to try to confound the two
If you instead approach the whole problem from a purely mathematical perspective of projection on an orthogonal basis and how to construct complex values that are convenient - then the whole setup is less "fun" but it actually becomes a lot more understandable. You can then move on from there and start asking yourself much more interesting questions about aliasing, ringing, fm/am, phase etc.
I feel it took be ~5 attempts of learning Fourier Analysis to unwire my brain and unlearn these bad visual intuitions that send you down the wrong path
I gotta say I disagree. I got the space change bit quickly (piece-wise linear space into frequency space). Complex plane's aren't a crutch, really they are the firm theoretical basis in which to express the detailed discussion properly.
You could always just switch to the Fourier sine and cosine forms, and avoid all of the other theoretical basis baggage. Sort of like physics for poets (I'm a computational theoretical physicist by training), leaving out the more detailed derivations and background, for a more straightforward approach.
Moreover, the DFT is not the FT. In the limit as your sampled points get very large, it will approach FT. There's a great book covering lots of these things in depth[1], with a pragmatic as well as theoretical approach. I think I gave my daughter this one (math phd student) last year.
FTs aren't merely a change of basis, there is quite a bit more to them than that. For DFT you can look at the process as a sequence of operator applications, but in the FT case this becomes a continuous sequence. Hence the space bits.
Can't speak to Fourier specifically -- but heartily agree that a lot of times in math, teachers try to make it "easy" and provide visuals that are actually a hindrance. Instead of understanding _the material_ you understand the _crutch_.
I think there is a special trick that happens many places in math. When you are given a visual, and then some integral or derivative is applied to it, you need to pause and make sure to truly include that in the mental model. Often I find math videos go far too fast through this section and I have to re-watch that specific spot and even pause and just ponder on how does the integral or derivative change what I was working with.
There's an old joke in math/physics student circles that covers this.
Student goes to see the prof in their office hours to try to understand something the prof said is seen to be trivial. They spend 4 hours working on it, and the student returns to their friends in class. They ask how it went. The student says "yeah, it was trivial."
For those who don't quite get the joke, 4 hours to show something is trivial, tends to not support that the thing is trivial to comprehend.
To be fair there is a grain of truth to the joke. E.g. Cayley’s theorem is sort of like that — once you get it, it feels like it barely qualifies as a theorem, it’s just a natural consequence of the definitions involved.
That doesn’t provide the full understanding you might think it does. For example: define ‘right now’.
Instantaneously, music is a single sound pressure measurement. That doesn’t have a Fourier transform. It doesn’t have a frequency. It’s just a single sample.
Fourier transforms work on functions. Typically functions in the time domain. And typically (but not always) on that function within a bounded range of time. And the result is another function, this one of frequency.
A spectrum analyzer, though, is showing the Fourier transform of a short snippet of some music. Then a moment later it’s showing you the transform for the next snippet.
Looking at a spectrum analyzer makes you think a Fourier transform is itself a function of time (to some vector of numbers perhaps?). That is not the case. So looking at a spectrum analyzer can give you an incorrect intuition for what Fourier does.
But you can do a Fourier transform on the whole of a piece of music. You’ll pick up frequency components like the overall beat, the bar structure, the verse/chorus alternation.
I think I get what you are trying to say... but the intuition about "this moment in time" is perfectly reasonable for a spectrum analyzer, since it's actually doing a DFT (not continuous from +-infinity) with the last sample (i.e. "now") defining the end of the window.
The thing that makes a DFT discrete is that it is over individual samples rather than a continuous function - not that it is over a finite domain.
A Fourier transform applied to a brief window of an underlying continuous function is called a ‘short-time Fourier transform’.
And the frequency information a STFT can pick up is bounded on the low end (think, like the opposite of the Nyquist limit) by the length of the window - this is called the ‘Rayleigh frequency’ - if your window is of length t, you can not detect frequencies lower than 1/t. Which is why your ‘instantaneous’ spectrum analyzer (looking at a short burst of maybe 0.05s of samples) for your 120bpm EDM doesn’t pick up a frequency component at 2Hz - even though that component is there in a Fourier analysis of the whole piece. It can only measure down to 20Hz. Which is fine because that’s also roughly the limit of the part of the song ‘function’ that we hear as ‘tone’ rather than ‘rhythm’.
Respectfully, your characterization of a DFT's infinite domain conflicts with the definition of the DFT -- it is defined as a finite sequence, and that's how it's used in common industry usage. Case in point:
> In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples [...]
In practice it’s hard to store an infinite number of discrete samples, let alone process them. So I assume that’s why people don’t try.
A spectrogram remains a visualization of a short time Fourier transform at a number of points in time. In practice usually produced using a DFT because discrete samples are what you have to work with.
Pedantics aside: Spectrum analyzers are computing DFTs over a finite window, and it's perfectly reasonable to think of these as (an approximation of) power spectral density changing over time.
Right. But if you think Fourier transforms produce a function of ‘power spectral density over time’ you are on a road to misunderstanding. Or even if you think that it makes sense to talk about the Fourier transform ‘at a moment in time’.
The thing I am railing against here is the idea that you can just look at a spectrogram to grasp Fourier. You can’t. It is an advanced application of Fourier transforms that creates a visualization of power spectral density over time but it is not a (simple) Fourier transform of the underlying data.
That’s a good place to start building intuition, but it can also distract you actual understanding.
What that visualization really is is a bunch of arbitrarily bounded Fourier transforms of little windowed slices of a piece of music. The true Fourier transform of a time bounded piece of music is a single unbounded function over an infinite frequency spectrum.
Well, yes. That part is obvious enough. The interesting part is how you build the mapping from time domain to frequency domain. That is the part that never clicked for me.
Maybe that's because a sinusoid of any frequence has its "roundness" and that is a matter of "hunting" which that formula performs every infinite-small moment to an x/y graph of your favourite music composition.
