Asymptotical Separation of Harmonics by Singular Spectrum Analysis

Abstract

The paper is devoted to studying the sufficient conditions for the asymptotical separability of distinct terms in the linear combination of harmonics by singular spectrum analysis (SSA). Namely, the series x₀, …, \({{x}_{{N - 1}}}\) with x_n = \(\sum\nolimits_{i = 1}^r {{{f}_{{i,n}}}} \) , where \({{f}_{{i,n}}}\) = \({{b}_{i}}\cos ({{\omega }_{i}}n + {{\gamma }_{i}})\) and both amplitudes |b_i| and frequencies ω_i ∈ (0, 1/2) are pairwise different, are considered. Then, as is proved in this study, under some relationship between amplitudes |b_i| and the choice of standard SSA parameters, the so-called reconstructed values \({{\tilde {f}}_{{i,n}}}\) prove to be very close to \({{f}_{{i,n}}}\) for large N. Moreover, \({{\max }_{n}}\left( {\left| {{{{\tilde {f}}}_{{i,n}}} - {{f}_{{i,n}}}} \right|} \right)\) =  \(O({{N}^{{ - 1}}})\) for any i, if N → ∞.