It was pretty clear at the R1 institution I attended (MIT) that it is really a huge research lab with a small school attached, and that the educational part, for undergrads at least, was definitely not a priority. The continuing stream of advertising (err, alumni updates) that I’ve received over the subsequent decades don’t seem to contradict this either.
Still glad I went and learned a ton, but it’s not for everyone.
I used MIT OCW for maths mostly and the teaching seems pretty amazing. But I could be cherry picking ... also they were Gilbert Strang's classes
My kids all ended up at various small private colleges depending on what they found as a fit for their interests and I got to see through them the difference between going to a school with 20,000 students (USC) vs one with 1,500 students (Reed). In hindsight I suspect I would have done better at a more intimate school (harder to have one's slacking off ignored by the administration right?)
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.
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’.
https://en.wikipedia.org/wiki/Discrete_Fourier_transform
> 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 [...]
Related: Showing energy content (i.e. DFT) versus time -- aka spectrograms: https://en.wikipedia.org/wiki/Spectrogram
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.
https://en.wikipedia.org/wiki/Discrete-time_Fourier_transfor...
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.
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.
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
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.
[1] https://epubs.siam.org/doi/book/10.1137/1.9781611971514 I highly recommend this book.
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.
There is SO much good content on there though. I can only guess "the algorithm" was hiding it from you. Sadly we often have to rely on sites like reddit or HN to find these creators.
Entertaining is hard to resist, so just create a second account and curate the history of one of them.
I also find my feed doesn't change that much, so the downside of being picky is that you might miss out on some good content.
It is part of our job to lecture, but that's not our only job as an instructor, and I wouldn't even say it's the most important one. Our job is to assemble a coherent course -- typically involving lectures, homeworks, exams, etc. where everything is tied together, and where I try to chart a coherent path through the material, so that the students have learned a subject by the end.
Once I was teaching vector calculus, a subject which 3b1b covers, and I showed a couple short clips in class. I also shared additional links with my students and encouraged them to watch. As instructor I try to make good use of any and all helpful resources which are available, and 3b1b is certainly one of them!
> This video is not an outlier.
I would argue: this video is an outlier in that it's "best of the best". I've seen other videos. While they're usually "good", they're also usually very dry.
So good videos aren't a new thing. But I think there's MUCH more decent and excellent content coming out these days.
Also there's the part where they can't assume their target audience is a bunch of students who have to learn a subject so they really do try to make it as layman as possible.
I wish there was a big list that had links to the best lectures and the accompanying book for so many things.
I suppose the idea that we should look to the results rather than some Ivory Tower education professor's ideas about what constitutes good teaching is just too foreign for the establishment as a whole. I've got armies of PhDs A-B testing whether they can extract .001 cent more per ad from me, but it's too much to ask of society as a whole to check into the question of what curriculum works better. Grant shouldn't have to be all but forcing his way uphill into this space.
raising e to something
> is converted into another function where it's not nonsensical to do such a thing.)
That function is the power series expansion, taught in your calculus class. https://en.wikipedia.org/wiki/Matrix_exponential
> CLEARLY superior to the instruction I received at a tier 1 research institution.
3B1B is a tier 1 educational institution.
But also, videos often feel better than classes, because videos don't have homework, so you don't actually have your feeling of understanding tested.
They are complementary, an entertaining YouTube video may first give you motivation to get into a seemingly boring subject. It may also help you after you took your classes, to help make elements fall into place, but it only works if you have learned the elements beforehand.
As educative these videos are, they are entertainment, and they are driven by the rules of entertainment, they wouldn't get much views otherwise. So it leans heavily on accessible and clever explanations, surprising results and relatively short form, the boring stuff is often left out, and unfortunately, some of it is necessary to progress. Most science YouTubers admit it, and urge everyone to take school seriously.
I have never learned a field just from YouTube videos. It either left me with superficial knowledge (which is better than nothing) or acted as a way to consolidate knowledge I already had.
In about 10 minutes he explains something my math methods instructor struggled with over two weeks (4 classes). I was resistant to the view that YouTube could host "decent" instructional content for a long time. That video was one of the first that was CLEARLY superior to the instruction I received at a tier 1 research institution.