zlacker

>>qqqqqu+(OP)
Fourier Transforms. I'd wish I had a intuitive understanding of how they work. Until then I'm stuck with just believing that the magic works out.

>>rambol+mj
The best way to understand the Fourier transformations is to think of them as change-of-basis operations, like we do in linear algebra. Specifically a change from the "time basis" (normal functions) to the "frequency basis" (consisting of a family of orthonormal functions).

Here is the chapter on Fourier transforms from my linear algebra book that goes into more details: https://minireference.com/static/excerpts/fourier_transforma...

As for the math, there really is no other way to convince yourself that sin(x) and sin(2x) are orthogonal with respect to the product int(f,g,[0,2pi]) other than to try it out https://live.sympy.org/?evaluate=integrate(%20sin(x)*sin(2*x... Try also with sin(3x) etc. and cos(n*x) etc.

>>ivan_a+6o
Sure but the hard part to understand and accept is that ANY squiggle can be represented by a weighted sum of sinusoids...I mean, that's really an amazing insight, and I don't think it's obvious even after-the-fact. Forget about the details of computing coefficients - just the fact that it works at all remains counter-intuitive to me. (A neat visualization would ask the user to wiggle their mouse, producing a sparkline of some finite length, and then, in real-time, update a frequency domain representation of the motion, perhaps represented as a bunch of connected circles that rotate steadily but at different rates to produce an equivalent graph)