The Schur algorithm and its applications

Abstract

The Schur algorithm and its times-domain counterpart, the fast Cholseky recursions, are some efficient signal processing algorithms which are well adapted to the study of inverse scattering problems.These algorithms use a layer stripping approach to reconstruct a lossless scattering medium described by symmetric two-component wave equations which model the interaction of right and left propagating waves.In this paper, the Schur and fast Cholesky recursions are presented and are used to study several inverse problems such as the reconstruction of nonuniform lossless transmission lines, the inverse problem for a layered acoustic medium, and the linear least-squares estimation of stationary stochastic processes. The inverse scattering problem for asymmetric two-component wave equations corresponding to lossy media is also examined and solved by using two coupled sets of Schur recursions. This procedure is then applied to the inverse problem for lossy transmission lines.-3-

On the GBDT Version of the Bäcklund-Darboux Transformation and its Applications to Linear and Nonlinear Equations and Weyl Theory

“…Transmission and reflection coefficients for system (5.1) are defined [22,33,82] in terms of special solutions of (5.1), that is, m × p 1 and m × p 2 solutions Y and Z, respectively, which we determine by the relations …”

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

“…Such a reflector appears also in the inverse scattering formulation of the linear filtering problem for a stationary stochastic process, as shown in Dewilde et al [11] and [14] (see also Fig. 2).…”

“…Yagle and Levy [14]. In this approach it is assumed that the medium is originally at rest, and is probed from the left by a known rightward propagating wave p(0,t) = 6(t) + p(0,t)u(t) (9) which is incident on the medium at t = 0.…”

“…Thus, assume that the waves p(x,t) and q(x,t) at depth x have been computed. The reflectivity function r(x) is obtained from (14) and is used in (13) to compute the waves p(x+A,t) and q(x+A,t) at depth x+A, as shown in Fig. 1.…”

“…These inverse scattering methods can be divided roughly in two categories, depending on whether they use integral equations [1] - [8] or a differential formulation [9] - [14] to reconstruct the potential of the Schrodinger equation, or the reflectivity function of the two-component wave system. However, even the most efficient of these techniques such as the fast Cholesky recursions described in [11] - [14] require a large volume of computations.…”