Minimal symmetric Darlington synthesis

We consider the symmetric Darlington synthesis of a p x p rational symmetric Schur function S with the constraint that the extension is of size 2p x 2p. Under the assumption that S is strictly contractive in at least one point of the imaginary axis, we determine the minimal McMillan degree of the extension. In particular, we show that it is generically given by the number of zeros of odd multiplicity of I-SS*. A constructive characterization of all such extensions is provided in terms of a symmetric realization of S and of the outer spectral factor of I-SS*. The authors's motivation for the problem stems from Surface Acoustic Wave filters where physical constraints on the electro-acoustic scattering matrix naturally raise this mathematical issue

Relative entropy and the multi-variable multi-dimensional moment problem

Entropy-like functionals on operator algebras have been studied since the pioneering work of von Neumann, Umegaki, Lindblad, and Lieb. The most well-known are the von Neumann entropy $trace (\rho\log \rho)$ and a generalization of the Kullback-Leibler distance $trace (\rho \log \rho - \rho \log \sigma)$, refered to as quantum relative entropy and used to quantify distance between states of a quantum system. The purpose of this paper is to explore these as regularizing functionals in seeking solutions to multi-variable and multi-dimensional moment problems. It will be shown that extrema can be effectively constructed via a suitable homotopy. The homotopy approach leads naturally to a further generalization and a description of all the solutions to such moment problems. This is accomplished by a renormalization of a Riemannian metric induced by entropy functionals. As an application we discuss the inverse problem of describing power spectra which are consistent with second-order statistics, which has been the main motivation behind the present work.Comment: 24 pages, 3 figure

A Dirichlet-to-Neumann approach for the exact computation of guided modes in photonic crystal waveguides

This works deals with one dimensional infinite perturbation - namely line defects - in periodic media. In optics, such defects are created to construct an (open) waveguide that concentrates light. The existence and the computation of the eigenmodes is a crucial issue. This is related to a self-adjoint eigenvalue problem associated to a PDE in an unbounded domain (in the directions orthogonal to the line defect), which makes both the analysis and the computations more complex. Using a Dirichlet-to-Neumann (DtN) approach, we show that this problem is equivalent to one set on a small neighborhood of the defect. On contrary to existing methods, this one is exact but there is a price to be paid : the reduction of the problem leads to a nonlinear eigenvalue problem of a fixed point nature

A Study of Adobe Wall Moisture Profiles and the Resulting Effects on Matched Illumination Waveforms in Through-The-Wall Radar Applications

In this dissertation, methods utilizing matched illumination theory to optimally design waveforms for enhanced target detection and identification in the context of through-the-wall radar (TWR) are explored. The accuracy of assumptions made in the waveform design process is evaluated through simulation. Additionally, the moisture profile of an adobe wall is investigated, and it is shown that the moisture profile of the wall will introduce significant variations in the matched illumination waveforms and subsequently, affect the resulting ability of the radar system to correctly identify and detect a target behind the wall. Experimental measurements of adobe wall moisture and corresponding dielectric properties confirms the need for accurate moisture profile information when designing radar waveforms which enhance signal-to-interference-plus-noise ratio (SINR) through use of matched illumination waveforms on the wall/target scenario. Furthermore, an evaluation of the ability to produce an optimal, matched illumination waveform for transmission using simple, common radar systems is undertaken and radar performance is evaluated

Discrete analytic Schur functions

We introduce the Schur class of functions, discrete analytic on the integer lattice in the complex plane. As a special case, we derive the explicit form of discrete analytic Blaschke factors and solve the related basic interpolation problem