The shift-invariance approach to continuous-time fast estimation algorithms

We show here how fast estimation algorithms can be obtained by developing innovations decompositions that exploit certain shift-invariance properties of the involved processes. Such invariances are quite natural for stationary processes and can be employed through the use of backwards innovations processes to develop a set of identities for the solutions of the associated Wiener-Hopf equations. These results provide continuous-parameter counterparts of the well-known discrete recursions given by Levinson in 1947, and have many connections to the multi-faceted works of M.G. Krein. For nonstationary processes, there is no natural shift-invariance. However, we show how state-space descriptions can be employed to expose suitable invariances for nonstationary processes generated as the response to white noise of constant-parameter state models.