Fourier Visualizations
bgrawi.com
bgrawi.com
The first day, the teacher walked in, wrote an equation (FFT) on the board, and spent the next 90 minutes manipulating it into different forms (from his in-hand notes), with no explanation of what or why we were doing so. Occasionally he would stop, say something along the lines of "and you can convert what we have here to this..." and write another equation on the board, and continue "which is one of your homework problems for next time."
The next class was identical. When the third class started the same way, I walked out 40 minutes into it and dropped it. I had zero incentive to continue, the teacher hadn't explained at all what the purpose of what we were doing yet. It had literally been us writing down his math notes for three straight hours while he explained how the mathematical manipulations worked, but not why it mattered or how it was relevant.
I've vacillated over the years between being upset with myself for abandoning that class, and not "getting" fourier related things, and being mad at the teacher at being so horrible at teaching, at least for the portion I was there for.
This is simple. This is obvious. Something like this would have gone a long way towards explaining what the fuck I was doing. I'm no longer upset at myself. I am doubly upset at that teacher now.
You shouldn't be. Wireless Technology is an advanced class for Electrical Engineering majors. Electrical Engineering majors would have gone through a class of Linear systems, where they would have learnt the intuition behind Fourier transforms as shown in OP's demo. I think the teacher might have done you a disservice by not mentioning the prerequisites, but thats the only thing you can be upset about.
It was offered as a double elective for CS and CEE, which meant it was supposed to be accessible to both majors (if admittedly harder for CS majors). I can accept the class being hard, I can't accept the first three hours of class time being spent copying notes of his math notes as he copied them out of his binder onto the board, with a total of class discussion being less than 10 minutes in total for those two class periods. There wasn't even and explanation of what our goal in this was, just "Here's an equation. It's useful. Now we are going to derive some other useful forms of it."
Never before or since have I ever been in a class where the teacher walks in, introduces himself, says what we'll be covering (FFT), turns to the board and starts copying his math notes onto the board, and doesn't stop until the class is over, keeping in mind, it wasn't how to solve an equation, it was how to transform a given equation into another given form with no explanation as to why. Things have come close in some of the more dry history classes I've had, but even then you generally get some additional context from the teacher expanding on the notes presented.
I'm mad about it, but in the weird way multiple years can change that into into something that's not quite the same as the word usually implies.
It makes me curious about one point. It sounds as if the class you took covered what my engineering program called "Communication Theory". We were expected to have two terms of differential equations and one each of linear algebra and statistics under our belts as prerequisites. (I think. This was a long time ago...)
Were you missing, at least, having at strong basis in differential equations? If you didn't, that would have made the class double-tough.
To be clear, my problem wasn't that there was a lot of math. My problem was that there was no context, and what we were copying off the board could easily have been copied from a book or pamphlet. I'm not sure what the practical reason is to spend multiple classes deriving one version of the FFT from another with zero context.
One set of two small phasors (one rotating CW relative to the main phasor, and one rotating CCW to the main phasor) is useful for thinking about amplitude and phase modulation. But I find it hard to believe that having four phasors stacked on top of each other strengthens anyone's intuition about Fourier transforms.
Never assume something is evident for everyone else.
I had similar classes in robotics (the lower-level kind: control theory, path planning, joint configurations, robot dynamics, inverse kinematics, stability), where we often had zero idea what was going on. The professor showed some slides dense with walls of text and badly typeset large equations, where we couldn't even keep track of what letter stood for what, the abstraction levels were mixed like spaghetti - sometimes talking about exact voltages, about mathematical tricks and shortcuts - but things weren't glued together with an overarching motivation.
So we learned stuff like Lyapunov-stability, the K-step ahead prediction, etc. but basically nobody had a clue what we were actually doing. And there was lots and lots of in-detail stuff.
This was in Hungary by the way. When I spent a semester in Germany and had a robotics course (slightly different focus), I thought it was so easy. There was a lot less material and we progressed much slower, but people actually followed what was happening. When you came out of a lecture, you could summarize in a few sentences what you've been doing. In my home university robotics class, we had no idea: we just saw a bunch of slides of incomprehensible math with no time to digest it.
When I see MIT lectures online, it also seems very "easy", in the sense that all the intuitions and motivations are laid out for you. I think this also has disadvantages. I can remember having many Heureka moments when disentangling the mess and finally putting puzzle pieces together, when learning from not so high-quality materials. I was proud about a lot of these "connections that I discovered", only to learn that some universities start with these high-level overviews and they don't count as exciting insight at elite universities, it's just part of the "boring, dry curriculum" as seen by students there.
http://betterexplained.com/articles/an-interactive-guide-to-...
Here is how I see this:
-Sinus (and cosinus etc) is defined using a right triangle, unit length hypotenuse simplifies things, so they are shown on a right triangle inside a unit circle.
-Time dependent sinus function, sin(wt), can be represented using a uniform circular motion and y coordinate of position.
-Fourier says that you can represent (may be almost all) functions using sinus functions with proper frequencies and amplitudes. There is a way to find those coefficients using some mathematics: infinite or finite sums or integrals depending on application.
-Oh well, I have a wonderful idea, great analogy, listen: why not show Fourier transform/series using a circle on circle on circles, uniformly moving (constant speeds) particles and projections of their position to sideways as a function of time.
I don't think understanding its mathematical form is more complicated than understanding what a function is. It is already very simple, too much enforced simplification makes it complicated.
I appreciate your feedback, my eventual goal of this site is to expand to concepts like directly using FFTs with arbitrary waves where your suggestion might be constructive, as yes circles on circles does make it seem more complicated than it is. One of the goals of this specific visualization was to demonstrate how fourier series relate to oscillators and sound, in which case a circle representing a frequency or set of tones added together made more sense.
For reference, I did not personally come up with this particular style of representing Fourier Series, I based it on this: https://commons.wikimedia.org/wiki/File:Fourier_series_squar... which is used on the main Wikipedia page about Fourier Series themselves.
The github link is here: https://github.com/bgrawi/Fourier-Visualizations
f(x) := a * sin(b + x) + c * sin(d + x)
there are some constants m, n so that:
f(x) == m * sin (n + x)
It's easy to prove --- but it allowed me to see why one coefficient plus phase was enough for each frequency.