Realizations of stationary stochastic processes: applications of passive system theory

Abstract

In the paper, we investigate realizations of a $p$-dimensionalregular weak stationary discrete time stochastic process $y(t)$ asthe output data of a passive linear bi-stable discrete timedynamical system. The state $x(t)$ is assumed to tend to zero as ttends to $-\infty$, and the input data is the $m$-dimensional whitenoise. The results are based on author's development of the Darlington method for passive impedance systems with losses of thescattering channels. Here we establish that considering realizationfor a discrete time process is possible, if the spectral density $\rho(e^{i\mu})$ of the process is a nontangential boundary value ofa matrix valued meromorphic function $\rho(z)$ of rank $m$ withbounded Nevanlinna characteristic in the open unitdisk. A parameterization of all such realizations is given and minimal,optimal minimal, and *-optimal minimal realizations areobtained. The last two coincide with those which are obtained by Kalman filters. This is a further development of the Lindquist-Picci realization theory.