Kalman filtering for self-similar processes

In our earlier work, we introduced a class of stochastic processes obeying a structure of the form, E[X(t)X(t/spl lambda/)]=R(/spl lambda/), t, /spl lambda/>0, and outlined a mathematical framework for the modeling and analysis for these processes. We referred to this class of processes as scale stationary processes. We demonstrated that the scale stationarity framework leads to engineering oriented mathematical tools and concepts, such as autocorrelation and spectral density function and finite parameter ARMA models for modeling and analysis of statistically self-similar signals. In this work, we introduce a state space representation for self-similar signals and systems based on scale stationary ARMA models. Such a representation provides a complete description of the inner and outer dynamics of a self-similar system or signal that can not be obtained from transfer function representation. We introduce Kalman filtering techniques and Riccati equations for smoothing and prediction of self-similar processes.