Error propagation in fourier transforms

The Fourier transform has long been of great use in simulating mathematical or physical phenomena, especially in signal theory. However the finite length representation of numbers introduces round-off errors in computing. Here, developing a new point of view on the topic, we give an evaluation of the total relative mean square error in the computation of direct and fast Fourier transforms using floating point artihmetic. Thus we show that in direct Fourier transforms the output noise-to-signal ratio is equivalent to N or N² according to whether the arithmetic is a rounding or a chopping one, whereas for fast Fourier transforms it is equivalent to log₂(N) or log₂(N)]², with N being the number of points of the signal. Good agreement with numerical results is observed.