In this talk, we consider the problem of detecting and recovering chirps from noisy data. Chirps are signals which are neither smoothly varying nor stationary but rather, which exhibit rapid oscillations and rapid changes in their frequency content. This behavior is very different than that assumed in the standard literature which typically assumes smoothness and homogeneity. One particular application of note in conjunction with this line of research is the detection of gravitational waves.
Building on recent advances in computational harmonic analysis, we design libraries of multiscale chirplets, and introduce detection strategies which are more sensitive than existing feature detectors. The idea is to use structured algorithms which exploit information in the chirplet dictionary to chain chirplets together adaptively as to form chirps with polygonal instantaneous frequency; these structured algorithms are so sensitive that they allow to detect signals whenever their strength makes them detectable by any method, no matter how intractable. Formally, we propose a test statistic which provably attains near-optimal decision bounds over a wide range of meaningful classes of chirps. In addition, there is a way to invoke dynamic programming and network flows algorithms to rapidly compute our test statistics. Similar strategies and results extend to the estimation problem. We hope to report on early numerical experiments.
This is joint work with Hannes Helgason.
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