A unified convergence analysis of Normalized PAST algorithms for estimating principal and minor components

We present a unified convergence analysis, based on a deterministic discrete time (DDT) approach, of the normalized projection approximation subspace tracking (Normalized PAST) algorithms for estimating principal and minor components of an input signal. The proposed analysis shows that the DDT system of the Normalized PAST algorithm (for PCA/MCA), with any forgetting factor in a certain range, converges to a desired eigenvector. This eigenvector is completely characterized as the normalized version of the orthogonal projection of the initial estimate onto the eigensubspace corresponding to the largest/smallest eigenvalue of the autocorrelation matrix of the input signal. This characterization holds in general case where the eigenvalues are not necessarily distinct. Numerical examples show that the proposed analysis demonstrates very well the convergence behavior of the Normalized PAST algorithms which uses a rank-1 instantaneous approximation of the autocorrelation matrix.

► A unified convergence analysis is given for Normalized PAST via DDT approach. ► We present a complete characterization of the limit of DDT systems. ► The proposed analysis holds even if the eigenvalues are not distinct. ► The DDT systems demonstrate well the convergence behavior of Normalized PAST.