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# explanation for Fourier transform

boinsterman
Does anyone have a good explanation for how a Fourier transform works? or a good, free resource that explains it well? (I reviewed my precalculus on complex numbers, but get lost when it gets to the transform formula and chart. Most importantly, what does it actually do?

For those who are unaware of it, a Fourier transform is the mathematical means by which wave functions are described, or some wave functions for some types of radiation. It seems to require an understanding of a formula with the name Euler, which states

e^ix = cos(x) + isin(x)

and is much better explained on Wikipedia at http://en.wikipedia.org/wiki/Fourier_transform.

I tried to post this yesterday with the actual name of the formula, but that contains a single apostrophe, and that supposedly is what caused the text to not display (and for the post to be deleted?).
Bikerman
Try:
http://www.thefouriertransform.com/
boinsterman
Thanks, Bikerman. I'll work through it.
Pippo90
The Fourier transform is a powerful tool to transfer the domain of a function from a space (or time) representation to a frequency one. In other words, given a certain function with two variables, space (or time) and frequency, while usually you consider the former as the true variable and the latter as a fixed parameter, after having "Fourier transformed" this function you consider the frequency as the true variable, the one varying along the X-axis, hence enabling you to observe the frequency components of a function.

For example, the Fourier transform of a cosine returns two Dirac deltas, which actually corresponds to the only frequency of the cosine (which, for $\cos(\omega x)$, is $\omega$) and to its inverse, $-\omega$ (given that the cosine is an even function). For other, more complicated functions, the Fourier transform returns a not so obvious result.

For more introductory details, I found a "Fourier transform for dummies" tutorial here: http://math.stackexchange.com/questions/1002/fourier-transform-for-dummies
boinsterman
I checked out the site. It looks good, and the siteexchange.com looks very useful.

Still confused. I still don't know where the data comes from and what the inputs and ouputs are, but I didn't expect it to be easy.

Thanks, Pippo90.