Perturbation expansions of signal subspaces for long signals

TL;DR

This paper develops perturbation expansion techniques for analyzing the stability of signal subspaces in long signals, providing bounds and convergence conditions for subspace proximity in signal processing.

Contribution

It introduces a perturbation expansion framework for signal subspace analysis, especially for long signals, with new bounds and convergence insights.

Findings

Derived upper bounds for subspace perturbations

Analyzed asymptotic behavior as signal length increases

Provided conditions for convergence and rate of convergence

Abstract

Singular Spectrum Analysis and many other subspace-based methods of signal processing are implicitly relying on the assumption of close proximity of unperturbed and perturbed signal subspaces extracted by the Singular Value Decomposition of special "signal" and "perturbed signal" matrices. In this paper, the analysis of the main principal angle between these subspaces is performed in terms of the perturbation expansions of the corresponding orthogonal projectors. Applicable upper bounds are derived. The main attention is paid to the asymptotical case when the length of the time series tends to infinity. Results concerning conditions for convergence, rate of convergence, and the main terms of proximity are presented.