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May 2015 On detecting harmonic oscillations
Anatoli Juditsky, Arkadi Nemirovski
Bernoulli 21(2): 1134-1165 (May 2015). DOI: 10.3150/14-BEJ600


In this paper, we focus on the following testing problem: assume that we are given observations of a real-valued signal along the grid $0,1,\ldots,N-1$, corrupted by white Gaussian noise. We want to distinguish between two hypotheses: (a) the signal is a nuisance – a linear combination of $d_{n}$ harmonic oscillations of known frequencies, and (b) signal is the sum of a nuisance and a linear combination of a given number $d_{s}$ of harmonic oscillations with unknown frequencies, and such that the distance (measured in the uniform norm on the grid) between the signal and the set of nuisances is at least $\rho>0$. We propose a computationally efficient test for distinguishing between (a) and (b) and show that its “resolution” (the smallest value of $\rho$ for which (a) and (b) are distinguished with a given confidence $1-\alpha$) is $\mathrm{O}(\sqrt{\ln(N/\alpha)/N})$, with the hidden factor depending solely on $d_{n}$ and $d_{s}$ and independent of the frequencies in question. We show that this resolution, up to a factor which is polynomial in $d_{n}$, $d_{s}$ and logarithmic in $N$, is the best possible under circumstances. We further extend the outlined results to the case of nuisances and signals close to linear combinations of harmonic oscillations, and provide illustrative numerical results.


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Anatoli Juditsky. Arkadi Nemirovski. "On detecting harmonic oscillations." Bernoulli 21 (2) 1134 - 1165, May 2015.


Published: May 2015
First available in Project Euclid: 21 April 2015

zbMATH: 06445970
MathSciNet: MR3338659
Digital Object Identifier: 10.3150/14-BEJ600

Rights: Copyright © 2015 Bernoulli Society for Mathematical Statistics and Probability


Vol.21 • No. 2 • May 2015
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