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Abstract <p>General approach to asymptotic signal extraction from an additively perturbed series with the help of Singular Spectrum Analysis (SSA) has already been described in [5]. This paper considers an example of this analysis applied to a polynomial signal and additive noise as linear combination of harmonics. In this case, the so-called reconstruction errors <i>r</i><sub><i>i</i></sub>(<i>N</i>) of SSA have been found to uniformly tend to zero as the series length <i>N</i> tends to infinity. More precisely, we have proved that max<sub><i>i</i></sub> |<i>r</i><sub><i>i</i></sub>(<i>N</i>)| =&#xa0;<i>O</i>(<i>N</i><sup>–1</sup>) as <i>N</i> → ∞ and for a “window length” <i>L</i> ~ α<i>N</i>, α ∈ (0, 1). This solves the problem of the accuracy of asymptotic separation of a polynomial signal from the seasonal component. The results on discrete Chebyshev polynomials in equally spaced points are essentially used to go to a polynomial of an arbitrary order from a linear signal, which was addressed for the case <i>L</i> = (<i>N</i> + 1)/2 in [7].</p>

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Asymptotical Separation of Polynomial Signals from Harmonics by Singular Spectrum Analysis

  • 期刊论文

  • 发表日期：

    2026年4月19日

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  • V. V. Nekrutkin,
  • D. M. Yakovlev

摘要

Abstract

General approach to asymptotic signal extraction from an additively perturbed series with the help of Singular Spectrum Analysis (SSA) has already been described in [5]. This paper considers an example of this analysis applied to a polynomial signal and additive noise as linear combination of harmonics. In this case, the so-called reconstruction errors ri(N) of SSA have been found to uniformly tend to zero as the series length N tends to infinity. More precisely, we have proved that maxi |ri(N)| = O(N–1) as N → ∞ and for a “window length” L ~ αN, α ∈ (0, 1). This solves the problem of the accuracy of asymptotic separation of a polynomial signal from the seasonal component. The results on discrete Chebyshev polynomials in equally spaced points are essentially used to go to a polynomial of an arbitrary order from a linear signal, which was addressed for the case L = (N + 1)/2 in [7].