On detecting harmonic oscillations

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, . . . , N − 1, corrupted by white Gaussian noise. We want to distinguish between two hypotheses: (a) the signal is a nuisance – a linear combination of dn harmonic oscillations of known frequencies, and (b) signal is the sum of a nuisance and a linear combination of a given number ds 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 ρ > 0. We propose a computationally efficient test for distinguishing between (a) and (b) and show that its “resolution” (the smallest value of ρ for which (a) and (b) are distinguished with a given confidence 1 − α) is O(p ln(N/α)/N), with the hidden factor depending solely on dn and ds and independent of the frequencies in question. We show that this resolution, up to a factor which is polynomial in dn, ds 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.