I got it by first thinking about the *inverse* Fourier transform, which is just ...

I got it by first thinking about the inverse Fourier transform, which is just saying that I want to write a signal as a weighted sum of sinusoids:

`x(t) = sum_k w_k sin(2pik*t)`

Then you just ask the question: given x, solve for w.

The next question is "why sinusoids", and the answer is that because for any linear, time invariant system acting on x:

y(t) = F[x](t)

that system diagonalizes over (complex) sinusoids:

y(f) = F(f)x(f)