The online Fourier transform
The online Fourier transform tracks a sliding window of the recent history of the time series. This window is updated by advancing one time step, and the FFT is used to get the power spectrum.
One can speed up computation by making updates recursive: on each time step, first subtract the contribution of the oldest timepoint, then add the contribution from the current timepoint. We can generalize this recursive approach by interpreting the online FT as a bank of band-pass filters.
