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

Abstract

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.

Citation

Download Citation

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

Information

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

JOURNAL ARTICLE
32 PAGES


SHARE
Vol.21 • No. 2 • May 2015
Back to Top