Perturbation expansions of signal subspaces for long signals

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.