The Fourier Transform, explained in one sentence

Re:The Fourier Transform, explained in one sentence by darkmighty on Saturday October 25, 2014 @01:36AM (#8507319)

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.

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Re:The Fourier Transform, explained in one sentence by theoh on Saturday October 25, 2014 @08:21AM (#8507987)

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.

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