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Adaptive tests for periodic signal detection with applications to laser vibrometry

Published online by Cambridge University Press:  31 January 2006

Magalie Fromont  and
Céline Lévy-leduc
Magalie Fromont
Affiliation:
Université Rennes II, Place du Recteur H. Le Moal, CS 24307, 35043 Rennes cedex, France; magalie.fromont@uhb.fr
Céline Lévy-leduc
Affiliation:
Université Paris-Sud, Bât. 425, 91405 Orsay; France; celine.levy-leduc@math.u-psud.fr
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Abstract

Initially motivated by a practical issue in target detection vialaser vibrometry, we are interested in the problem of periodicsignal detection in a Gaussian fixed design regression framework.Assuming that the signal belongs to some periodic Sobolev ball andthat the variance of the noise is known, we first consider theproblem from a minimax point of view: we evaluate the so-calledminimax separation rate which corresponds to the minimall 2-distance between the signal and zero so that the detection ispossible with prescribed probabilities of error. Then, we propose atesting procedure which is available when the variance of the noiseis unknown and which does not use any prior information about thesmoothness degree or the period of the signal. We prove that it isadaptive in the sense that it achieves, up to a possible logarithmicfactor, the minimax separation rate over various periodic Sobolevballs simultaneously. The originality of our approach as compared torelated works on the topic of signal detection is that our testingprocedure is sensitive to the periodicity assumption on the signal.A simulation study is performed in order to evaluate the effect ofthis prior assumption on the power of the test. We do observe thegains that we could expect from the theory. At last, we turn to theapplication to target detection by laser vibrometry that we had inview.

Keywords

Periodic signal detectionadaptive testminimax separation ratesnonparametric regression.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 10 , February 2006 , pp. 46 - 75
Copyright
© EDP Sciences, SMAI, 2006

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