Topics in classical and quantum linear stochastic systems

This thesis considers two topics in the area of linear stochastic systems. The first topic is the construction of approximate finite dimensional linear time invariant (LTI) models for classical wide sense stationary stochastic signals with a non-coercive and non-rational spectral density, utilizing the recently developed theory of degree constrained rational interpolation. Non-coercive means that the spectral density has zeros on the unit circle or the imaginary axis (depending on whether the stochastic process is in discrete or continuous time, respectively), while non-rationality implies that the underlying system generating such a signal is infinite dimensional. As one example, spectral densities of this type appear when measurements are taken of signals that have traversed through the earth’s turbulent atmosphere, such as light from a distant star captured by astronomical telescopes on the ground. The operation of obtaining an LTI model from a spectral density is known in the literature as spectral factorization and has played an important role in both deterministic and stochastic linear systems theory. The non-rational and non-coercive spectral densities which are considered herein are known to be difficult to factorize numerically. The most general algorithms for spectral factorization, such as the maximum entropy method, converge slowly for these spectral densities, and can lead to approximate models of degree higher than is necessary. The first part of this thesis establishes some new results in the theory of degree constrained rational interpolation and then proposes and analyzes a new approach to spectral factorization, based on so-called rational covariance extensions. A new spectral factorization algorithm is then introduced. In a number of simulations, which include some physically motivated spectral densities, it is demonstrated that the new algorithm gives lower degree approximations than the well-known maximum entropy method. The second topic is the issue of physical realizability of a given system represented by linear quantum stochastic differential equations (QSDEs) on a quantum probability space. Physical realizability here means that the QSDEs should rep-