Considerations in a billiard system

In billiards the signal ${\mathcal{F}}(t)$ 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, ${\mathcal{F}}(t)$ is needed on a regular time grid. The problem arises: how to represent irregular $\delta$-functions on a regular grid? Is the limit of grid spacing $\Delta t\rightarrow0$ needed? The answer is no. The spikes can be replaced by (i.e. convolved with) Gaussians of width $t_r$, if $\tilde{C}(\omega)$ is only desired up to frequency $\omega_0$. This modified signal is then windowed to give $A(t)$ and resampled on regular grid of spacing $\Delta t < t_r$. For efficiency of the FFT, $\Delta t$ is chosen such that the number of samples, $T_0 \Delta t$, is an integer power of 2. In frequency-space, the convolution has the effect of multiplying $\tilde{A}(\omega)$ by a high-frequency `roll-off' function $R(\omega) = e^{-t_r^2 \omega^2 / 2}$. Since this roll-off is known, it can be exactly corrected for by division of $\tilde{A}(\omega)$ by $R(\omega)$. This correction allowed accurate $\tilde{C}(\omega)$ estimation up to $\omega \approx 2 / t_r$.

Note that the $\delta$-function nature of the signal might suggest that its FT could best be performed by direct summation of the terms $e^{i \omega t_i}$ arising from each collision $i$. However this turns out to scale like $N^2$ where $N$ is the number of collisions needed ( $\approx T_0/\tau_{{\mbox{\tiny bl}}}$), i.e. it is a `slow' FT. Resampling onto an even grid and using the FFT, which scales like $N \log_2 N$, is vastly superior.