Wavelet-based tools to analyze, filter, and reconstruct transient gravitational-wave signals

The analysis of gravitational-wave signals is one of the most challenging application areas of signal processing, because of the extreme weakness of these signals and of the great complexity of gravitational-wave detectors. Wavelet transforms are specially helpful in detecting and analyzing gravitational-wave transients and several analysis pipelines are based on these transforms, both continuous and discrete. While discrete wavelet transforms have distinct advantages in terms of computing efficiency, continuous wavelet transforms (CWT) produce smooth and visually stunning time-frequency maps where the wavelet energy is displayed in terms of time and frequency. In addition to wavelets, short-time Fourier transforms (STFT) and Stockwell transforms (ST) are also used, or the $Q$-transform, which is a Morlet waveletlike transform where the width of the Gaussian envelope is parametrized by a parameter denoted by $Q$ [Chatterji et al., Classical Quantum Gravity 21, S1809 (2004)]. To date, the use of CWTs in gravitational-wave data analysis has been limited by the higher computational load when compared with discrete wavelets, and also by the lack of an inversion formula for wavelet families that do not satisfy the admissibility condition. In this paper we consider Morlet wavelets parametrized in the same way as the Q-transform (hence the name wavelet Q-transform) which have all the advantages of the Morlet wavelets and where the wavelet transform can be inverted with a computationally efficient specialization of the nonstandard inversion formula of Lebedeva and Postnikov [Lebedeva and Postnikov, R. Soc. Open Sci. 1, 140124 (2014)]. We also introduce a two-parameter extension (the wavelet Qp-transform) which is well adapted to chirping signals like those originating from compact binary coalescences (CBC), and show that it is also invertible just like the wavelet Q-transform. The inversion formulas of both transforms allow for effective noise filtering and produce very clean reconstructions of gravitational-wave signals. Our preliminary results indicate that the method could be well suited to perform accurate tests of general relativity by comparing modeled and unmodeled reconstructions of CBC gravitational-wave signals.