Partial Covariance Sequence Research Articles

From Finite Covariance Windows to Modeling Filters: A Convex Optimization Approach

The trigonometric moment problem is a classical moment problem with numerous applications in mathematics, physics, and engineering. The rational covariance extension problem is a constrained version of this problem, with the constraints arising from the physical realizability of the corresponding solutions. Although the maximum entropy method gives one well-known solution, in several applications a wider class of solutions is desired. In a seminal paper, Georgiou derived an existence result for a broad class of models. In this paper, we review the history of this problem, going back to Carath{éodory, as well as applications to stochastic systems and signal processing. In particular, we present a convex optimization problem for solving the rational covariance extension problem with degree constraint. Given a partial covariance sequence and the desired zeros of the shaping filter, the poles are uniquely determined from the unique minimum of the corresponding optimization problem. In this way we obtain an algorithm for solving the covariance extension problem, as well as a constructive proof of Georgiou's existence result and his conjecture, a generalized version of which we have recently resolved using geometric methods. We also survey recent related results on constrained Nevanlinna--Pick interpolation in the context of a variational formulation of the general moment problem.

A covariance extension approach to identification of time series

In this paper we consider a three-step procedure for identification of time series, based on covariance extension and model reduction, and we present a complete analysis of its statistical convergence properties. A partial covariance sequence is estimated from statistical data. Then a high-order maximum-entropy model is determined, which is finally approximated by a lower-order model by stochastically balanced model reduction. Such procedures have been studied before, in various combinations, but an overall convergence analysis comprising all three steps has been lacking. Supposing the data is generated from a true finite-dimensional system which is minimum phase, it is shown that the transfer function of the estimated system tends in H ∞ to the true transfer function as the data length tends to infinity, if the covariance extension and the model reduction is done properly. The proposed identification procedure, and some variations of it, are evaluated by simulations.

On an Alternative Parameterization of the Solutions of the Partial Realization Problem

The solutions of the partial realization problem have to satisfy a finite number of interpolation conditions at ∞. The minimal degree of an interpolating deterministic system is called the algebraic degree or McMillan degree of the partial covariance sequence and is easy to compute. The solutions of the partial stochastic realization problem have to satisfy the same interpolation conditions and have to fulfill a positive realness constraint. The minimal degree of a stochastic realization is called the positive degree. In the literature, solutions of the partial realization problem are parameterized by the Kimura–Georgiou parameterization. Solutions of the partial stochastic realization problem are then obtained by checking the positive realness constraint for the interpolating solutions of the corresponding partial realization problem. In this paper, an alternative parameterization is developed for the solutions of the partial realization problems. Both the solutions of the partial and partial stochastic realization problem are analyzed in this parameterization, while the main concerns are the minimality and the uniqueness of the solutions. Based on the structure of the parameterization, a lower bound for the positive degree is derived.

A Convex Optimization Approach to the Rational Covariance Extension Problem

In this paper we present a convex optimization problem for solving the rational covariance extension problem. Given a partial covariance sequence and the desired zeros of the modeling filter, the poles are uniquely determined from the unique minimum of the corresponding optimization problem. In this way we obtain an algorithm for solving the covariance extension problem, as well as a constructive proof of Georgiou's seminal existence result and his conjecture, a stronger version of which we have resolved in [Byrnes et al., IEEE Trans. Automat. Control, AC-40 (1995), pp. 1841--1857].

A complete parameterization of all positive rational extensions of a covariance sequence

In this paper we formalize the observation that filtering and interpolation induce complementary, or dual, decompositions of the space of positive real rational functions of degree less than or equal to n. From this basic result about the geometry of the space of positive real functions, we are able to deduce two complementary sets of conclusions about positive rational extensions of a given partial covariance sequence. On the one hand, by viewing a certain fast filtering algorithm as a nonlinear dynamical system defined on this space, we are able to develop estimates on the asymptotic behavior of the Schur parameters (1918) of positive rational extensions. On the other hand we are also able to provide a characterization of all positive rational extensions of a given partial covariance sequence. Indeed, motivated by its application to signal processing, speech processing, and stochastic realization theory, this characterization is in terms of a complete parameterization using familiar objects from systems theory and proves a conjecture made by Georgiou (1983, 1987). Our basic result, however, also enables us to analyze the robustness of this parameterization with respect to variations in the problem data. The methodology employed is a combination of complex analysis, geometry, linear systems, and nonlinear dynamics. >

Minimax control under a bound on the partial covariance sequence of the disturbance

A discrete-time linear SISO plant that is acted on by a stochastic disturbance is considered. It is assumed that a restriction on the partial covariance sequence of the disturbance is given. The cost function is a maximum of the closed-loop system output variance over the class of disturbances. The problem is to find the optimal linear stabilizing regulator. The transfer function of the optimal closed-loop system is constructed in explicit form. It is shown that the minimax regulator is intermediate between the H ∞ and H 2 optimal regulators. As a result of the problem solution, generalizations of the Caratheodory-Fejer and Nevanlinna-Pick theorems are obtained.

On the realization of band-limited power spectra from partial covariance sequence

The set M( Omega ) of all (discrete-time) power spectra supported on a known interval (- Omega , Omega ), (with Omega < pi ) and whose first N+1 correlation coefficients coincide with a given sequence ( gamma n)n=0,N is characterized. By using the results related to the trigonometric moment problem, it is shown that the positive real functions associated with the elements of M( Omega ) are given by a linear fractional map of Schur functions satisfying certain properties.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>