A common approach to the analysis of cumulant-based AR and ARMA identification

Abstract

This paper introduces a common approach for analysing cumulant-based AR and ARMA identification methods. This particular derivation applies to methods using 1-D cumulant slices in linear algebraic types of solutions. It is based on the replacement of poles of identified systems by expansions to infinite geometric series. Using this method, a flexible and comprehensible z-domain transform is obtained for each identification scheme of the mentioned type. Explicit expressions for the cumulant-induced poles and zeros prove that the failures in the identification methods result from the zero-pole cancellations between these poles or zeros and the system poles or zeros. In addition we show that a non-identifiable class of systems can always be defined for the relationship between the system poles and zeros. Three different approaches were analysed, i.e., higher-order Yule–Walker type of solutions, q-slices, and w-slices, deriving at necessary and sufficient conditions for identifiability which confirm previously known results for the former two approaches. The conditions are presented analytically by the location of the system zeros referring to the system poles. The basis of our unified analytical approach is derived at initially for single system poles and for single cumulants. In further discussion, it is extended to multiple poles and linear combinations of multiple 1-D cumulant slices. Extensive simulations using the third- and fourth-order cumulants completely confirm our theoretical expectations.