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.