In billiards the signal consists of an irregular series of spikes (corresponding to the collisions with the walls--they are infinitely narrow in the hard-wall limit), with the average value subtracted (see Section 3.1.1).
In order to use the FFT as in the previous section, is needed on a regular time grid. The problem arises: how to represent irregular -functions on a regular grid? Is the limit of grid spacing needed? The answer is no. The spikes can be replaced by (i.e. convolved with) Gaussians of width , if is only desired up to frequency . This modified signal is then windowed to give and resampled on regular grid of spacing . For efficiency of the FFT, is chosen such that the number of samples, , is an integer power of 2. In frequency-space, the convolution has the effect of multiplying by a high-frequency `roll-off' function . Since this roll-off is known, it can be exactly corrected for by division of by . This correction allowed accurate estimation up to .
Note that the -function nature of the signal might suggest that its FT could best be performed by direct summation of the terms arising from each collision . However this turns out to scale like where is the number of collisions needed ( ), i.e. it is a `slow' FT. Resampling onto an even grid and using the FFT, which scales like , is vastly superior.