Efficient Reconstruction of Matched-Filter SNR Time Series from Nearby Templates

TL;DR

The paper tackles the computational and storage burden of matched filtering for long-duration, low-mass CBC signals by introducing a ratio-filter that reconstructs nearby templates' SNR time series from a reference template via a short ratio waveform. By exploiting smooth frequency-domain template ratios and applying windowing and truncation, the method reduces waveform length and memory usage while preserving accuracy, demonstrated to $\mathcal{O}(10^{-4})$ in SNR difference and over 25% faster performance with about 60x storage savings for the ratio waveforms. Validation uses mock non-spinning injections in Gaussian noise within the $1$–$3~M_\odot$ range, showing robust agreement with standard matched filtering across multiple chirp masses and mass ratios. The approach enables efficient searches for long-duration CBC signals in current and future detectors, and can be extended to include spins and hierarchical search strategies, offering a scalable path for low-frequency sensitivity enhancements.

Abstract

We present a method for efficiently searching long-duration gravitational wave signals from compact binary coalescences (CBCs). The approach exploits the smooth frequency-domain behavior of ratios between neighboring waveform templates. The matched-filter signal-to-noise ratio (SNR) time series of a data segment is first computed for a reference template, and the SNRs of nearby templates are then reconstructed by convolving this reference SNR time series with the ratio waveforms, defined as the frequency-domain ratios between the reference and neighboring templates. The computational speedup arises because the ratio waveforms can be safely truncated: they are significant only over a short interval approximately equal to the duration difference between the templates. Storing these truncated ratio waveforms is practical and enables additional efficiency gains, in contrast to storing full templates, which is generally infeasible for long-duration, low-mass signals. We demonstrate the efficacy of the method with mock non-spinning CBC injections in the $1-3~M_\odot$ range. The reconstructed SNR time series agrees with that obtained from standard matched filtering to an accuracy of $O(10^{-4})$, while the relative computational cost is reduced by $\gtrsim 25\%$. With a truncation threshold of $10^{-3}$ applied to the ratio waveform amplitudes, the storage requirement is reduced by a factor of $\sim 60$ relative to storing the full template bank.

Figures (5)

• Figure 1: Left Panel: Time-domain plots of the reference (top) and target (middle) waveforms generated using the TaylorF2 model, and their ratio (bottom), all shown as a function of time relative to the waveform peak $t_\mathrm{peak}$. The reference waveform has a chirp mass $\sim0.875~M_\odot$ with a mass ratio of unity, while the target waveform has a chirp mass $\sim0.870~M_\odot$ with the same mass ratio.
• Figure 2: Absolute value of the ratio waveform $r(t)$ as a function of time relative to the waveform peak, $t-t_{\rm peak}$. The blue and orange curves corresponding to the ratio waveforms with the window function Eq.\ref{['eq:window']} for $\Delta f_{\rm low}=0~{\rm Hz}, \Delta f_{\rm high}=0~{\rm Hz}$, and $\Delta f_{\rm low}=1.92~{\rm Hz}, \Delta f_{\rm high}=53~{\rm Hz}$, respectively. The ratio waveform is computed by Eq.\ref{['eq:ratio_waveform']} using the same parameters for the reference and target waveform as in Fig. \ref{['fig:waveforms']}. The cyan, green, and red curves represent the analytic expressions expanded for each frequency, as derived in Appendix \ref{['sec:app1']}.
• Figure 3: Relative SNR loss as a function of the retained side duration of the ratio waveform $r(t)$. Each curve corresponds to a different mass ratio, while keeping the chirp mass fixed at $1.219~M_\odot$. The horizontal axis represents the duration of the ratio waveforms $\tau_\mathrm{ratio}$, normalized by the duration of the target waveform $\tau_\mathrm{target}$, i.e., the duration of the injected signal. The ratio waveform length is varied by changing the threshold amplitude applied to $\abs{r(t)}$ using analytical solutions (Appendix \ref{['sec:app1']}). The vertical dashed lines indicate the value of the horizontal axis when the threshold amplitude is set to $10^{-3}~[\mathrm{s^{-1}}]$.
• Figure 4: Top Panel: SNR time series of the reference waveform associated with the group containing the loudest template among its dependent templates. Bottom Panel: Comparison of the SNR time series from standard and ratio filter.
• Figure 5: Comparison of the relative SNR loss (top) and improvement in computation time (bottom) between standard and ratio filter.