Inverse scattering for star-shaped nonuniform lossless electrical networks

TL;DR

This paper investigates the use of Frequency Domain Reflectometry to solve inverse scattering problems in star-shaped electrical networks, enabling detection of faults and inhomogeneities with minimal experimental setup.

Contribution

It provides a method to identify the network's geometry and inhomogeneities using high-frequency asymptotics and two boundary measurements, extending inverse scattering theory to star-shaped graphs.

Findings

Identifies the number and lengths of edges in the network.

Establishes unique determination of inhomogeneities from two boundary measurements.

Proves the asymptotic behavior of the reflection coefficient for geometry detection.

Abstract

The Frequency Domain Reflectometry (FDR) is studied as a powerful tool to detect hard or soft faults in star-shaped networks of nonuniform lossless transmission lines. Processing the FDR measurements leads to solve an inverse scattering problem for a Schrodinger operator on a star-shaped graph. Throughout this paper, we restrict ourselves to the case of minimal experimental setup corresponding to only one diagnostic port plug. First, by studying the asymptotic behavior of the reflection coefficient in the high-frequency limit, we prove the identifiability of the geometry of this star-shaped graph: the number of edges and their lengths. The proof being rather constructive, it provides a method to detect the hard faults in the network. Next, we study the potential identification problem by inverse scattering, noting that the potentials represent the inhomogeneities due to the soft faults in the network wirings. Here, the main result states that the measurement of two reflection coefficients, associated to two different sets of boundary conditions at the extremities of the tree, determines uniquely the potentials; it is a generalization of the theorem of the two boundary spectra on an interval.