${\chi }^2$ time-frequency discriminator for gravitational wave detection

Searches for known waveforms in gravitational wave detector data are often done using matched filtering. When used on real instrumental data, matched filtering often does not perform as well as might be expected, because nonstationary and non-Gaussian detector noise produces large spurious filter outputs (events). This paper describes a ${\chi }^2$ time-frequency test which is one way to discriminate such spurious events from the events that would be produced by genuine signals. The method works well only for broadband signals. The case where the filter-template does not exactly match the signal waveform is also considered, and upper bounds are found for the expected value of ${\chi }^2$.

Figure 1

A typical set of frequency intervals $\Delta f_j$ for the case $p=4$. These intervals are narrowest where the detector is the most sensitive, and broadest where it is least sensitive.

Figure 2

The output of $p=4$ single-phase filters for a simulated chirp signal added into a stream of detector noise (left set of figures) and a transient burst present in detector noise (right set of figures). For the simulated chirp, the filters in the different frequency bands all peak at the same time offset $t_0$: the time offset which maximizes the SNR. At this instant in time, all of the $z_j$ are about the same value. However when the filter was triggered by the transient burst, the filters in the different frequency bands peak at different times. At time $t_0$ they have very different values (some large, some small, and so on).

Figure 3

The expected value of ${\chi }^2$ for a signal that is not matched to the template satisfies $⟨{\chi }^2⟩=p-1+\kappa ⟨z{⟩}^2$ where $⟨z⟩$ is the expected SNR. If signal and template share the same set of frequency bands, then below the curve $ϵ=2/p$ one has $\kappa <(p-1){ϵ}^2$. Above the curve $\kappa <2ϵ$. If signal and template do not share the same frequency bands, then $\kappa <2ϵ$ is the only limit that applies.

Figure 4

A comparison of different thresholds ${\chi }_{*}^2$ for the ${\chi }^2$ time-frequency discriminator, for the LIGO S1 two-phase case with $p=8$, $ϵ=0.03$, and worst-case $\kappa =2ϵ$. The bottom solid line shows the expected value of ${\chi }^2$ for stationary Gaussian noise given by (8.1). The top solid line shows a threshold (8.4) set $4.9\sigma$ above this expected value. The upper dashed line shows the heuristic threshold (8.3) used for the LIGO S1 analysis, which was determined from Monte-Carlo studies. The lower dashed line shows the threshold which would be exceeded with probability 0.025% in stationary Gaussian noise.

Figure 5

A comparison of different thresholds for the ${\chi }^2$ time-frequency discriminator, at large SNR. The upper solid curve is the LIGO S1 threshold (8.3). The lower (dashed) curve is the threshold that would be exceeded with probability 0.025% in stationary Gaussian noise. (On the scale of this graph the approximation (8.4) to the dashed curve is difficult to distinguish from the exact result obtained from the inverse of the noncentral ${\chi }^2$ distribution.)