Extended Levinson and Chandrasekhar equations for general discrete-time linear estimation problems

Recursive algorithrms for the solution of linear least-squares estimation problems have been based mainly on state-space models. It has been known, however, that recursive Levinson-Whittle-Wiggins-Robinson (LWR) algorithms exist for stationary time-series, using only input-output information (i.e, covariance matrices). By introducing a way of classifying stochastic processes in terms of an "index of nonstationarity" we derive extended LWR algorithms for nonstationary processes We show also how adding state-space structure to the covariance matrix allows us to specialize these general results to state-space type estimation algorithms. In particular, the Chandrasekhar equations are shown to be natural descendants of the extended LWR algorithm.