Necessary and sufficient conditions for convergence of the DDT systems of the Normalized PAST algorithms

• We propose a refined DDT analysis for Normalized PAST algorithms.
• The DDT analysis clarified necessary and sufficient conditions for the convergence.
• The range of the forgetting factor, for MCA, is doubled from the first DDT analysis.
• The range of the forgetting factor, for PCA, is extended to the full range (0, 1].
• With the forgetting factor in extended ranges, the convergence can be accelerated.

Recently, we presented a first deterministic discrete time (DDT) analysis of the normalized normalized projection approximation subspace tracking   (Normalized PAST) algorithms, for estimating principal and minor components of an input signal. The analysis shows that the DDT systems of the Normalized PAST algorithms converge to the desired eigenvectors under certain sufficient conditions on the forgetting factor β∈(0,1]β∈(0,1]. However, it has not yet been clarified whether the sufficient conditions can be relaxed or not for guaranteed convergence. In this paper, by characterizing the maximal ranges of the forgetting factor, we establish the necessary and sufficient conditions for convergence of the DDT systems of the Normalized PAST algorithms. The proposed maximal range of the forgetting factor, for the minor component estimation, is doubled from the range assumed in the first DDT analysis, while the proposed maximal range of the forgetting factor, for principal component estimation, achieves the full range (0, 1]. Numerical examples further confirm the results.