The Fourier transform
The Fourier transform, in its raw form, is an operation performed on mathematical functions across a continuous band of frequencies. If you haven’t worked with complex exponentials or even with integrals before, or you simply haven’t worked with them recently as it often turns out, then don’t worry. We won’t be doing the calculations by hand. The following formula is how to apply the Fourier transform to a continuous function. This is used for analog data; any continuous periodic function can be perfectly represented as a sum of complex exponentials, or equivalent sine and cosine waves.
Formula 8.1 – Fourier transform
In effect, the Fourier transform sweeps across all possible frequency values, 
, outputting a high value when the frequency in question correlates strongly to f(x) and a low value when it does not. Entire books have been written on the Fourier transform, when it works, why it works,...
