A Kalman filter solution of the inverse scattering problem with a rational reflection coefficient

Bibliography: leaves 16-17."March 1984""ECS-83-12921" "AFOSR-82-0135A"Bernard C. Levy

Linear stochastic systems: a white noise approach

Using the white noise setting, in particular the Wick product, the Hermite transform, and the Kondratiev space, we present a new approach to study linear stochastic systems, where randomness is also included in the transfer function. We prove BIBO type stability theorems for these systems, both in the discrete and continuous time cases. We also consider the case of dissipative systems for both discrete and continuous time systems. We further study $\ell_1$-$\ell_2$ stability in the discrete time case, and ${\mathbf L}_2$-${\mathbf L}_\infty$ stability in the continuous time case

Indefinite metric spaces in estimation, control and adaptive filtering

The goal of this thesis is two-fold: first to present a unified mathematical framework (based upon optimization in indefinite metric spaces) for a wide range of problems in estimation and control, and second, to motivate and introduce the problem of robust estimation and control, and to study its implications to the area of adaptive signal processing. Robust estimation (and control) is concerned with the design of estimators (and controllers that have acceptable performance in the face of model uncertainties and lack of statistical information, and can be considered an outgrowth and extension of (the now classical) LQG theory, developed in the 1950's and 1960's which assumed perfect models and complete statistical knowledge. It has particular significance in adaptive signal processing where one needs to cope with time-variations of system parameters and to compensate for lack of a priori knowledge of the statistics of the input data and disturbances. One method of addressing the above problem is the so-called H∞ approach, which was introduced by G. Zames in 1980 and that has been recently solved by various authors. Despite the "fundamental differences" between the philosophies of the H∞ and LQG approaches to control and estimation, there are striking "formal similarities" between the controllers and estimators obtained from these two methodologies. In an attempt to explain these similarities, we shall describe a new approach to H∞ estimation (and control), different from the existing (e.g., interpolation-theoretic-based, game-theoretic-based, etc) approaches, that is based upon setting up estimation (and control problems) not in the usual Hilbert space of random variables, but in an indefinite (so-called Krein) space

Remarks on orthogonal polynomials with respect to varying measures and related problems

We point out the relation between the orthogonal polynomials with respect to (w.r.t.) varying measures and the so-called orthogonal rationals on the unit circle in the complex plane. This observation enables us to combine different techniques in the study of these polynomials and rationals. As an example, we present a simple and short proof for a known result on the weak-star convergence of orthogonal polynomials w.r.t, varying measures. Some related problems are also considered

Convergence of modified approximants associated with orthogonal rational functions

AbstractLet {αn} be a sequence in the unit disk D = {z ∈ C: ¦z¦ < 1} consisting of a finite number of points cyclically repeated, and let L be the linear space generated by the functions Bn(z) = Πk=0n − αk(z − αk)¦αk¦(1 − αkz). Let {ϕn(z)} be orthogonal rational functions obtained from the sequence {Bn(z)} (orthogonalization with respect to a given functional on L), and let {ψn(z)} be the corresponding functions of the second kind (with superstar transforms ϕn∗(z) and ψn∗(z) respectively). Interpolation and convergence properties of the modified approximants Rn(z, un, vn) = (unψn(z) − vnψn∗(z))(unϕn(z) + vnϕn∗(z)) that satisfy ¦un¦ = ¦vn¦ are discussed

The Schur algorithm and its applications

Includes bibliographical references (p. 47-50).Research supported by the Air Force Office of Scientific Research AFOSR-82-0135A Research supported by the Exxon Education Foundation.Andrew E. Yagle and Bernard C. Levy

Connections between three-dimensional inverse scattering and linear least-squares estimation of random fields

The three-dimensional Schrödinger equation inverse scattering problem with a nonspherically-symmetric potential is related to the filtering problem of computing the linear leastsquares estimate of the three-dimensional random field on the surface of a sphere from noisy observations inside the sphere. The relation consists of associating an estimation problem with the inverse scattering problem, and vice-versa. This association allows equations and quantities for one problem to be given interpretations in terms of the other problem. A new fast algorithm is obtained for the estimation of random fields using this association. The present work is an extension of the connections between estimation and inverse scattering already known to exist for stationary random processes and one-dimensional scattering potentials, and isotropic random fields and radial scattering protentials.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/41642/1/10440_2004_Article_BF00046966.pd

The Transformation of Issai Schur and Related Topics in an Indefinite Setting

We review our recent work on the Schur transformation for scalar generalized Schur and Nevanlinna functions. The Schur transformation is defined for these classes of functions in several situations, and it is used to solve corresponding basic interpolation problems and problems of factorization of rational J-unitary matrix functions into elementary factors. A key role is played by the theory of reproducing kernel Pontryagin spaces and linear relations in these spaces