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Detecting high oscillatory signals by chirplet path pursuit

Candès, Emmanuel J. and Charlton, Philip R. and Helgason, Hannes (2008) Detecting high oscillatory signals by chirplet path pursuit. Applied and Computational Harmonic Analysis, 24 (1). pp. 14-40. ISSN 1063-5203. doi:10.1016/j.acha.2007.04.003.

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This paper considers the problem of detecting nonstationary phenomena, and chirps in particular, from very noisy data. Chirps are waveforms of the very general form A(t)exp(iλφ(t)), where λ is a (large) base frequency, the phase φ(t) is time-varying and the amplitude A(t) is slowly varying. Given a set of noisy measurements, we would like to test whether there is signal or whether the data is just noise. One particular application of note in conjunction with this problem is the detection of gravitational waves predicted by Einstein's Theory of General Relativity. We introduce detection strategies which are very sensitive and more flexible than existing feature detectors. The idea is to use structured algorithms which exploit information in the so-called chirplet graph to chain chirplets together adaptively as to form chirps with polygonal instantaneous frequency. We then search for the path in the graph which provides the best trade-off between complexity and goodness of fit. Underlying our methodology is the idea that while the signal may be extremely weak so that none of the individual empirical coefficients is statistically significant, one can still reliably detect by combining several coefficients into a coherent chain. This strategy is general and may be applied in many other detection problems. We complement our study with numerical experiments showing that our algorithms are so sensitive that they seem to detect signals whenever their strength makes them detectable.

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Candès, Emmanuel J.0000-0001-9234-924X
Additional Information:© 2007 Elsevier Inc. Received 25 July 2006; revised 20 March 2007; accepted 10 April 2007. Communicated by Radu Balan. Available online 4 May 2007. E. C. was partially supported by National Science Foundation grants DMS 01-40698 (FRG) and ITR ACI-0204932. P. C. was partially supported by NSF grant PHY-0107417. Many thanks to David Donoho, Houman Ohwadi, Justin Romberg and Chiara Sabatti for fruitful conversations. We would also like to thank Ery Arias-Castro for references. The results in this paper were first presented at “Regularization in Statistics,” Banff, Canada, September 2003 and at “Multiscale Geometric Analysis in High Dimensions,” UCLA, Los Angeles, California, November 2004 [9].
Funding AgencyGrant Number
NSFDMS 01-40698
NSFITR ACI-0204932
Subject Keywords:Signal detection; Nonparametric testing; Likelihood ratios; Adaptivity; Chirps; Chirplets; Time–frequency analysis; Gravitational waves; Graphs; Shortest path in a graph; Dynamic programming
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Record Number:CaltechAUTHORS:20091016-160452647
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:16377
Deposited On:19 Oct 2009 22:09
Last Modified:08 Nov 2021 23:26

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