Convergence, convergence point and convergence rate for Steiglitz-McBride method; a unified approach

Abstract

This paper presents a unified approach to analyze the convergence properties of Steiglitz-McBride(1966) method (SMM) in general environments. SMM is formulated as a successive substitution equation. Using results from fixed point theory enables a unified analysis of SMM in both white and colored noise, and sufficient and insufficient order cases. This analysis provides us with several new results. Specifically, for sufficient order filters in white noise environments, the convergence rate of SMM can be predicted by the signal-power to noise-power ratio (SNR) at plant output. For sufficient order filters in colored noise, SMM may diverge or converge depending on the initial estimate and SNR at plant output. If SMM converges, the convergence point is near the unbiased solution. SNR again determines the bias magnitude. For insufficient order filters, in addition to the possible multiple convergence points, we also demonstrate the existence of diverging fixed points of SMM. These diverging fixed points can be used to separate the convergence region, and identify the convergence points for each initial estimate.<>