Synthesis of RLC Ladder Networks by Matrix Tridiagonalization

Marshall, Jr. T.

doi:10.1109/tct.1969.1082885

Cited by 12 publications

(4 citation statements)

References 7 publications

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“…As is well known, this problem can be solved via the Lanczos algorithm. Specifically, with orthogonalization in the transpose bilinear form [14,16], vector y/ y T y as a starting vector, and the matrix of eigenvalues Λ as iteration matrix, the Lanczos algorithm produces the desired tridiagonal matrix. The algorithm is given in algorithm 1 and as soon as the tridiagonal matrix is obtained, the coefficients γ j and γj can be extracted using the recursive scheme shown in algorithm 2.…”

Section: Building a Rom From Spectral Datamentioning

confidence: 99%

Solving inverse scattering problems via reduced-order model embedding procedures

Zimmerling,

Druskin,

Guddati

et al. 2023

Inverse Problems

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We present a reduced-order model (ROM) methodology for inverse scattering problems in which the ROMs are data-driven, i.e. they are constructed directly from data gathered by sensors. Moreover, the entries of the ROM contain localised information about the coefficients of the wave equation. We solve the inverse problem by embedding the ROM in physical space. Such an approach is also followed in the theory of ‘optimal grids,’ where the ROMs are interpreted as two-point finite-difference discretisations of an underlying set of equations of a first-order continuous system on this special grid. Here, we extend this line of work to wave equations and introduce a new embedding technique, which we call Krein embedding, since it is inspired by Krein’s seminal work on vibrations of a string. In this embedding approach, an adaptive grid and a set of medium parameters can be directly extracted from a ROM and we show that several limitations of optimal grid embeddings can be avoided. Furthermore, we show how Krein embedding is connected to classical optimal grid embedding and that convergence results for optimal grids can be extended to this novel embedding approach. Finally, we also briefly discuss Krein embedding for open domains, that is, semi-infinite domains that extend to infinity in one direction.

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Section: Building a Rom From Spectral Datamentioning

confidence: 99%

Solving inverse scattering problems via reduced-order model embedding procedures

Zimmerling,

Druskin,

Guddati

et al. 2023

Inverse Problems

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“…mk Matrix M is of order 2k. The vector v chosen will be the coordinate vector of so = 1 for the basis L. From (6) and (9) and by choosing a ( s ) = so, it is seen that…”

Section: The Block Representationmentioning

confidence: 99%

“…Several .polynomial operation are necessary for filter design. It is assumed that all the high degree polynomials at various stages of design are known only by their coordinate vectors as given by (9). Therefore, these operations will be different from those developed for the natural representation.…”

Section: Polynomial Operations For the Block Representationmentioning

confidence: 99%

Filter realization using a generalized lagrange polynomial representation

Liu¹,

Marshall²

1981

Circuit Theory & Apps

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SUMMARYA polynomial representation using a set of generalized Lagrange polynomials for a basis is considered as an alternative to the natural representation. This new representation, which is employed in a previously described APL filter design package,* is considered in detail, here, with regard to its algebraic and numerical properties for solving the realization problem of electrical filter synthesis. The necessary polynomial operations for filter realization are described in this representation. Then vector space algorithms are presented for these operations. The accuracy obtained from this representation is discussed. Advantages in both accuracy and simplicity of the new algorithms are noted.

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“…Again, the ROM is obtained via direct layer stripping, implemented with the J-symmetric Lanczos algorithm [23,27]. However, there are two notable differences from the lossless case: (1) The Lanczos algorithm may break down, although it is unlikely to do so as explained in [19].…”

mentioning

confidence: 99%

Reduced Order Model Approach to Inverse Scattering

Borcea¹,

Druskin²,

Mamonov³

et al. 2020

SIAM J. Imaging Sci.

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We study an inverse scattering problem for a generic hyperbolic system of equations with an unknown coefficient called the reflectivity. The solution of the system models waves (sound, electromagnetic or elastic), and the reflectivity models unknown scatterers embedded in a smooth and known medium. The inverse problem is to determine the reflectivity from the time resolved scattering matrix (the data) measured by an array of sensors. We introduce a novel inversion method, based on a reduced order model (ROM) of an operator called wave propagator, because it maps the wave from one time instant to the next, at interval corresponding to the discrete time sampling of the data. The wave propagator is unknown in the inverse problem, but the ROM can be computed directly from the data. By construction, the ROM inherits key properties of the wave propagator, which facilitate the estimation of the reflectivity. The ROM was introduced previously and was used for two purposes: (1) to map the scattering matrix to that corresponding to the single scattering (Born) approximation and (2) to image i.e., obtain a qualitative estimate of the support of the reflectivity. Here we study further the ROM and show that it corresponds to a Galerkin projection of the wave propagator. The Galerkin framework is useful for proving properties of the ROM that are used in the new inversion method which seeks a quantitative estimate of the reflectivity. * One can also consider truncation of the whole space, as long as the medium is known and non-scattering on one side of the array of sensors. P(q) = cos τ L(q)L(q) T .( 1.9) The ROM is defined by a pair of matrices P ROM (q) ∈ R nm×nm and b ROM ∈ R nm×m , which are proxies of P(q) and the initial wave b(x). These matrices are calculated from the data (1.8) (i.e., the ROM is data-driven) and they capture physical aspects of the wave propagation that are needed for inversion. The ROM was introduced in [12,6] and was used in [13] for imaging, and in [5] for transforming the data (1.8) to that corresponding to the single scattering (Born) approximation. The new results in this paper are:1. We show that the ROM propagator P ROM (q) is a Galerkin projection of the operator (1.9), and use the Galerkin framework to prove properties of the ROM that facilitate the solution of the inverse scattering problem. † The kinematic model (smooth wave speed) appears in the coefficients of the operators L(q) and L(q) T (see [6] and sections 3-4). We suppress the dependence on the known kinematic model in our notation.3

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Synthesis of RLC Ladder Networks by Matrix Tridiagonalization

Cited by 12 publications

References 7 publications

Solving inverse scattering problems via reduced-order model embedding procedures

Solving inverse scattering problems via reduced-order model embedding procedures

Filter realization using a generalized lagrange polynomial representation

Reduced Order Model Approach to Inverse Scattering

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