, which arises if one simply applies the definition of DFT, to , where is the length of the sequence. The difference in speed can be enormous, especially for long sequences where may be in the thousands or millions.

As the FFT is merely an algebraic refactoring of terms within the DFT, the DFT and the FFT both perform mathematically equivalent and interchangeable operations, assuming that all terms are computed with infinite precision. However, in the presence of round-off error, many FFT algorithms