4 ms·

You could learn about it from looking at and understanding a harmonic analyser: http://www.sciencemuseum.org.uk/objects/mathematics/1946-343.aspx http://www.sci

by theoh 12y ago

You could learn about it from looking at and understanding a harmonic analyser: http://www.sciencemuseum.org.uk/objects/mathematics/1946-343.aspx http://www.sciencemuseum.org.uk/objects/mathematics/1946-343... Or more likely the reverse mechanism, Kelvin's tide predictor being a good example: http://www.sciencemuseum.org.uk/objects/oceanography/1876-1129.aspx http://www.sciencemuseum.org.uk/objects/oceanography/1876-11... Honestly that is more important to my mental model of the 1D Fourier transform than the algebraic description. It may be slightly inaccurate (are we talking projection or decomposition when calculating the settings on that tide predictor? for example). 2D Fourier transform is more interesting to me as it has applications in graphics. For example, you can synthesise ocean wave-like patterns in a height field by generating a noise image, shaping the noise into a slightly modified disc around the origin, and doing an IFFT. Intuitively you get random wavelength components within a given range, with random orientations. I couldn't write you a formula for that. You can imagine that a 12 year old could understand it, thought. Old paper is here: http://www.computer.org/csdl/mags/cg/1987/03/mcg1987030016.pdf http://www.computer.org/csdl/mags/cg/1987/03/mcg1987030016.p... Another intuitively comprehensible 2D trick is to use an FFT to implement a simulation of fluid dynamics. That's a more complex one but again, it can be understood without algebra. http://www.dgp.toronto.edu/people/stam/reality/Research/pdf/jgt01.pdf http://www.dgp.toronto.edu/people/stam/reality/Research/pdf/...

darkmighty 12y ago
I think he meant the 'decomposition' definition. You're decomposing your signal into other signals, that's the fundamental aspect. This decomposition is a linear, so you can also make it so your new components have certain nice properties -- they can be orthogonal, unitary, etc. Then you naturally get waves (complex exponentials) as the components. And then you realize -- hey, those are exponentials -- so the rule that integrating/differentiating each component is trivial allows you to solve linear ODE really quickly. I don't think algebra gets in the way at all -- it just makes this decomposition crystal clear. It makes generalizing easy also -- there are other nice basis like cosine basis, and so on. You can choose any N functions to decompose your N-point function, as long as they are linearly independent, you're guaranteed perfect reconstruction.
- theoh 12y ago
  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.

by theoh 12y ago

darkmighty 12y ago

I think he meant the 'decomposition' definition. You're decomposing your signal into other signals, that's the fundamental aspect. This decomposition is a linear, so you can also make it so your new components have certain nice properties -- they can be orthogonal, unitary, etc. Then you naturally get waves (complex exponentials) as the components. And then you realize -- hey, those are exponentials -- so the rule that integrating/differentiating each component is trivial allows you to solve linear ODE really quickly. I don't think algebra gets in the way at all -- it just makes this decomposition crystal clear. It makes generalizing easy also -- there are other nice basis like cosine basis, and so on. You can choose any N functions to decompose your N-point function, as long as they are linearly independent, you're guaranteed perfect reconstruction.

theoh 12y ago
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.