星形非均匀无耗电网络的逆散射
Inverse scattering for star-shaped nonuniform lossless electrical networks
摘要：
频域反射技术是检测星型非均匀无损耗传输线网络中硬故障和软故障的有力工具。处理的FDR测量导致解决一个逆散射问题的薛定谔算子的星形图。在本文中，我们限制自己的情况下，最小的实验设置对应于只有一个诊断端口插头。首先，通过研究反射系数在高频极限下的渐近行为，我们证明了这个星形图的几何形状：边的数量和它们的长度的可识别性。该证明具有较强的建设性，为检测网络中的硬故障提供了一种方法。接下来，我们研究了潜在的识别问题的逆散射，注意到的电位表示的不均匀性，由于软故障的网络布线。在这里，主要的结果指出，测量的两个反射系数，与两个不同的边界条件在树的两端，唯一地确定的潜力，它是一个推广的定理的两个边界谱的间隔。
英文摘要：
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.
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