• Bernoulli
  • Volume 21, Number 2 (2015), 1134-1165.

On detecting harmonic oscillations

Anatoli Juditsky and Arkadi Nemirovski

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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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Bernoulli, Volume 21, Number 2 (2015), 1134-1165.

First available in Project Euclid: 21 April 2015

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detection by convex optimization detection in the presence of nuisance harmonic oscillations detection multiple hypothesis testing


Juditsky, Anatoli; Nemirovski, Arkadi. On detecting harmonic oscillations. Bernoulli 21 (2015), no. 2, 1134--1165. doi:10.3150/14-BEJ600.

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