Best chirplet chain: Near-optimal detection of gravitational wave chirps

The list of putative sources of gravitational waves possibly detected by the ongoing worldwide network of large scale interferometers has been continuously growing in the last years. For some of them, the detection is made difficult by the lack of a complete information about the expected signal. We concentrate on the case where the expected gravitational wave (GW) is a quasiperiodic frequency modulated signal i.e., a chirp. In this article, we address the question of detecting an a priori unknown GW chirp. We introduce a general chirp model and claim that it includes all physically realistic GW chirps. We produce a finite grid of template waveforms which samples the resulting set of possible chirps. If we follow the classical approach (used for the detection of inspiralling binary chirps, for instance), we would build a bank of quadrature matched filters comparing the data to each of the templates of this grid. The detection would then be achieved by thresholding the output, the maximum giving the individual which best fits the data. In the present case, this exhaustive search is not tractable because of the very large number of templates in the grid. We show that the exhaustive search can be reformulated (using approximations) as a pattern search in the time-frequency plane. This motivates an approximate but feasible alternative solution which is clearly linked to the optimal one. The time-frequency representation and pattern search algorithm are fully determined by the reformulation. This contrasts with the other time-frequency based methods presented in the literature for the same problem, where these choices are justified by “ad hoc” arguments. In particular, the time-frequency representation has to be unitary. Finally, we assess the performance, robustness and computational cost of the proposed method with several benchmarks using simulated data.

Figure 1

Chirplet chains—The TF domain of interest $\mathcal{D}$ is tiled by a regular TF grid of $N_t$ time intervals and $N_f$ frequency bins. A chirplet is a short piece of signal whose frequency varies linearly between two successive nodes of the grid. It is thus represented by a line joining the grid nodes. The slope of the chirplet frequency is limited (triangular region in light gray, here $N_r^{\prime }=1$). We chain the chirplets, imposing the continuity of the chain and limiting the difference between the slopes of two consecutive chirplets (triangular region with dark gray stripes, here $N_r^{\prime \prime }=1$). Admissible chirplets in time interval $j$ belong to the intersection of these two regions associated to the regularity constraints. Clearly, a chirplet chain is represented by a broken line in the TF plane.

Figure 2

Smooth chirp (solid curve) and its geometrically close CC (dotted broken line).

Figure 3

Discrete Wigner-Ville of two chirp signals—The signals are normalized to unit ${\ell }_2$ norm and we show the contour at the level 1/8. Left: when the chirp frequency is a linear function of time, its WV is almost Dirac along the corresponding TF line in the TF half plane associated to positive frequencies. For the negative frequencies, the WV distribution exhibits aliasing terms (we highlight them by a background in light gray) which we neglect in the simplified model in Eq. (58). Right: when the chirp frequency is not linear (here, it is a parabolic chirp), interference terms appear (evidenced by a dark gray background). Their contribution are also disregarded in the simplified model. Note that the WV of the nonlinear chirp chosen for this illustration do present aliasing terms (with a light gray background), but they have a smaller amplitude than in the linear case.

Figure 4

Principle of optimality of DP

Figure 5

Newtonian chirp in white Gaussian noise—Left: WV distribution of the signal. Only positive contributions are displayed (negative ones are set to zero) with a gray-scaled color map going from white (minimum i.e, zero) to black (maximum). Right: the best CC in dashed/red closely matches the actual instantaneous frequency in solid/green in the region where the regularity constraints are satisfied. We indicate the instant when the chirp reaches the chirping rate limits with the dotted vertical lines and the frequency at LSCO with dashed-dotted horizontal line.

Figure 6

Newtonian chirp in white Gaussian noise—Left: WV distribution of the signal (displayed similarly as in Fig. 5). Right: actual chirp frequency in solid/green and best CC in dashed/red. We indicate where the chirp reaches the chirping rate limits with dotted vertical lines and the frequency at LSCO with dashed-dotted horizontal line.

Figure 7

Newtonian chirp in white Gaussian noise—Comparison of ROCs of the best CC search (dashed/blue, with error bars in solid/red) with STS (dotted/magenta) and TFC (dashed-dotted/cyan). The computation of each ROCs is perfomed over $2\times {10}^5$ trials (half for the false alarm probability and half for the detection probability). The diagram also includes the ROC of the clairvoyant quadrature matched filter (bold dashed/green) shown here with the SNR ${\rho }_c=6.15$ adjusted to reasonably fit the ROC of the best CC search.

Figure 8

Random CC in white Gaussian noise—In these plots, we arbitrarily set $f_s=1$. Top left: example of a random CC in white Gaussian noise. Top right: noise free random CC. Bottom left: WV distribution of the signal (displayed similarly as in Fig. 5). Bottom right: actual chirp frequency in solid/green and best CC in dashed/red. It is worthwhile to note that, although the best CC can lose track for some time because of noise fluctuations, it is able to recover the exact TF path.

Figure 9

Random CC in white Gaussian noise—This diagram displays the ROC of the best CC search (dashed/blue, obtained from $2\times {10}^5$ trials) compared with the analytical ROC of the clairvoyant matched filter (bold dashed/green) with the SNR ${\rho }_c=4.55$, adjusted to produce a reasonable fit.