Sparse Time-Frequency Representation of Gravitational-Wave signals in Unions of Wilson Bases

We investigate the question of obtaining a reduced time-frequency description of a chirp type signal that can be used as reference pattern in time-frequency searches. This is particularly relevant for searches of transient gravitational waves from astrophysical sources such as the mergers of neutron stars and/or black holes, the main area of this study. Sparse approximation algorithms that allow constraints on the approximation error do not perform well when the decomposition bases are redundant. This study puts in evidence some of the shortcomings of sparse approximation algorithms when dealing with unions of highly correlated bases, a case that currently lacks of a comprehensive mathematical analysis, and proposes solutions to mitigate them. We propose a variation of the matching pursuit algorithm that improves its robustness in the context of gravitational waves patterns construction. We also compare this algorithm to standard sparse approximation methods.