Figure 1

The number of filters as a function of minimal match, which increases with the density of the template bank. The total number of filters in the template bank, $N$, is shown by the dashed line. The number of filters needed to reconstruct the template matrix such that $⟨\delta \rho /\rho ⟩={10}^{-7}$, $N^{\prime }$, is shown by the solid line. We find that the number of filters needed to reconstruct $\mathbf{H}$ saturates when the minimal match reaches $\sim 89.9\%$.Reuse & Permissions

Figure 2

A visual representation showing the two types of points of $\mathcal{P}$ that we can choose to reconstruct. The left panel shows an example point that is in $\mathbf{H}$ and thus in $\mathcal{P}$. The right panel shows an example point that is not part of $\mathbf{H}$ but is in $\mathcal{P}$.Reuse & Permissions

Figure 3

Histograms of $(\delta {\rho }_{\alpha }/{\rho }_{\alpha }{)}_{\perp }$, $(\delta {\rho }_{\alpha }/{\rho }_{\alpha }{)}_{\parallel }$, and $\delta {\rho }_{\alpha }/{\rho }_{\alpha }$ for the waveforms that went into the construction of $\mathbf{H}$. As expected, $(\delta {\rho }_{\alpha }/{\rho }_{\alpha }{)}_{\perp }$ is around single-precision floating-point roundoff error, and the average $\delta {\rho }_{\alpha }/{\rho }_{\alpha }$ (solid vertical line) matches the expected fractional SNR loss (dashed vertical line).Reuse & Permissions

Figure 4

(a) Histograms of $(\delta {\rho }_{\alpha }/{\rho }_{\alpha }{)}_{\perp }$, $(\delta {\rho }_{\alpha }/{\rho }_{\alpha }{)}_{\parallel }$, and $\delta {\rho }_{\alpha }/{\rho }_{\alpha }$ for waveforms from $\mathcal{P}$ that were not in $\mathbf{H}$. $(\delta {\rho }_{\alpha }/{\rho }_{\alpha }{)}_{\perp }$ is seen to be of the same order of magnitude as $(\delta {\rho }_{\alpha }/{\rho }_{\alpha }{)}_{\parallel }$. The peak of $\delta {\rho }_{\alpha }/{\rho }_{\alpha }$ is above the expected fractional SNR loss for waveforms from $\mathbf{H}$ (vertical dashed line); however, the average $\delta {\rho }_{\alpha }/{\rho }_{\alpha }$ is skewed to large values by a tail in the distribution. (b) How these mismatches vary across $\mathcal{P}$, averaged over the ${\tau }_3$ direction. The largest mismatches come from near the borders of the template bank in the ${\tau }_0$ direction. Figure 5 restricts our attention to the central 75% of the domain of $\mathcal{P}$, whose boundaries are shown by the vertical lines here in Fig. 4b.Reuse & Permissions

Figure 5

Histograms of $(\delta {\rho }_{\alpha }/{\rho }_{\alpha }{)}_{\perp }$, $(\delta {\rho }_{\alpha }/{\rho }_{\alpha }{)}_{\parallel }$, and $\delta {\rho }_{\alpha }/{\rho }_{\alpha }$ for waveforms from $\mathcal{P}$ that were not in $\mathbf{H}$, similar to Fig. 4b, eliminating, however, test points near the boundaries in the ${\tau }_0$ direction. The average $\delta {\rho }_{\alpha }/{\rho }_{\alpha }$ (solid vertical line) accuracy is slightly worse than the expected fractional SNR loss (dashed vertical line).Reuse & Permissions

Figure 6

(a) A plot showing a region of $\mathcal{P}$. The circles are points whose waveforms go into $\mathbf{H}$. The line segment $\overline{AB}$ connects one of those points, $A$, with the point $B$, which is the central point of $A$ and two of its nearest neighbors. Point $B$ is situated such that it should have a 89.9% fitting factor with each of the surrounding points. (b) How well a waveform from a given point of $\overline{AB}$, ${\overrightarrow{h}}_p$, can be “matched.” The lowest, light-grey curve shows the match, which is the normalized inner product between ${\overrightarrow{h}}_A$ and the waveform from the corresponding point along $\overline{AB}$. The next-lowest (dark-grey) curve shows the fitting factor, which is the match maximized over phase and time. Since we are using a 89.9% minimal match bank, it is expected that the fitting factor falls to around that value. The three upper curves are associated with the SVD projection. As before, the solid curve is $1-\delta {\rho }_{\alpha }/{\rho }_{\alpha }$, the dashed curve is $1-(\delta {\rho }_{\alpha }/{\rho }_{\alpha }{)}_{\parallel }$, and the dotted curve is $1-(\delta {\rho }_{\alpha }/{\rho }_{\alpha }{)}_{\perp }$. The SVD-basis vectors are able to reconstruct to high accuracy all points along the line.Reuse & Permissions