Two asymptotic approaches for the exponential signal and harmonic noise in Singular Spectrum Analysis

TL;DR

This paper investigates the asymptotic behavior of Singular Spectrum Analysis when extracting exponential signals amid harmonic noise, revealing that unbounded signals cause non-vanishing errors, while bounded signals allow errors to diminish.

Contribution

It introduces a discretization scheme for SSA that effectively handles increasing exponential signals, providing new insights into error behavior for unbounded versus bounded signals.

Findings

Unbounded exponential signals cause SSA reconstruction errors not to tend to zero.

Bounded exponential signals allow all error elements to tend to zero.

Discretization scheme is effective for theoretical SSA analysis with increasing signals.

Abstract

The general theoretical approach to the asymptotic extraction of the signal series from the perturbed signal with the help of Singular Spectrum Analysis (briefly, SSA) was already outlined in Nekrutkin 2010, SII, v. 3, 297--319. In this paper we consider the example of such an analysis applied to the increasing exponential signal and the sinusoidal noise. It is proved that if the signal rapidly tends to infinity, then the so-called reconstruction errors of SSA do not uniformly tend to zero as the series length tends to infinity. More precisely, in this case any finite number of last terms of the error series do not tend to any finite or infinite values. On the contrary, for the "discretization" scheme with the bounded from above exponential signal, all elements of the error series tend to zero. This effect shows that the discretization model can be an effective tool in the theoretical SSA considerations with increasing signals.