[karpathy]

My preferred way to think about it is in linear algebra terms: your signal of length n is an n-dimensional vector. Now you just need to change the coordinate system into fourier basis, which is made up of vectors of length n who's entries are sin/cos waves of different frequencies. The way you change basis in linear alebra is by doing a dot product with the basis vectors you want to transform to, and it's no different here... it's just a projection of your single point into fourier basis and that's all the formula says.

The fancy e^i stuff is just for mathematical beauty and compactness and should be avoided when explaining the fourier transform, imo. There's absolutely no need for it as far as the idea goes, just do the sins and coses separately.

[shmageggy]

If, however, one wanted a similarly intuitive explanation of the e^i part of the story, I recommend these two articles in order. They explain the hand wavy part about complex numbers and rotation: http://betterexplained.com/articles/a-visual-intuitive-guide... http://betterexplained.com/articles/intuitive-understanding-...