Roundoff noise in floating point fast Fourier transform computation

A statistical model for roundoff errors is used to predict output noise-to-signal ratio when a fast Fourier transform is computed using floating point arithmetic. The result, derived for the case of white input signal, is that the ratio of mean-squared output noise to mean-squared output signal varies essentially as where is the number of points transformed. This predicted result is significantly lower than bounds previously derived on mean-squared output noise-to-signal ratio, which are proportional to ν². The predictions are verified experimentally, with excellent agreement. The model applies to rounded arithmetic, and it is found experimentally that if one truncates, rather than rounds, the results of floating point additions and multiplications, the output noise increases significantly (for a given ν). Also, for truncation, a greater than linear increase with ν of the output noise-to-signal ratio is observed; the empirical results seem to be proportional to ν², rather than to ν.