On the asymptotical separation of linear signals from harmonics by singular spectrum analysis

Abstract

The general theoretical approach to the asymptotic extraction of the signal series from the additively perturbed signal with the help of singular spectrum analysis (briefly, SSA) was already outlined in Nekrutkin (2010), SII, vol. 3, 297–319. In this paper we consider the example of such an analysis applied to the linear signal and the additive sinusoidal noise. It is proved that in this case the so-called reconstruction errors ri(N) of SSA uniformly tend to zero as the series length N tends to infinity. More precisely, we demonstrate that maxi |ri(N)| = O(N−1) if N → ∞ and the “window length” L equals (N + 1)/2. It is important to mention, that the completely different result is valid for the increasing exponential signal and the same noise. As it is proved in Ivanova, Nekrutkin (2019), SII, vol. 12, 1, 49–59, in this case any finite number of last terms of the error series does not tend to any finite or infinite values.