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.