I don't know who you mean by "he" as I was talking about the machine built by Kelvin. As far as the difference goes, my uncertainty comes from knowing a tiny bit about wavelets. A decomposition means that you can reconstruct the signal by adding all the (independent) components, whereas a projection gives you more of a spectrum picture involving non-orthogonal bases? I guess this doesn't apply to the Fourier basis.

Algebra is definitely necessary and useful here, but the concepts can be learnt without knowing algebra, which means they are open to a lot more people.