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By making use of periodicities in the sines that are multiplied to do the transforms, the FFT greatly reduces the amount of calculation required. Here's a little overview.

Functionally, the FFT decomposes the set of data to be transformed into a series of smaller data sets to be transformed. Then, it decomposes those smaller sets into even smaller sets. At each stage of processing, the results of the previous stage are combined in special way. Finally, it calculates the DFT of each small data set.

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