High-order, Dispersionless "Fast-Hybrid" Wave Equation Solver. Part I: O(1)\mathcal{O}(1) Sampling Cost via Incident-Field Windowing and Recentering

This paper proposes a frequency/time hybrid integral-equation method for the time dependent wave equation in two and three-dimensional spatial domains. Relying on Fourier Transformation in time, the method utilizes a fixed (time-independent) number of frequency-domain integral-equation solutions to evaluate, with superalgebraically-small errors, time domain solutions for arbitrarily long times. The approach relies on two main elements, namely, 1) A smooth time-windowing methodology that enables accurate band-limited representations for arbitrarily-long time signals, and 2) A novel Fourier transform approach which, in a time-parallel manner and without causing spurious periodicity effects, delivers numerically dispersionless spectrally-accurate solutions. A similar hybrid technique can be obtained on the basis of Laplace transforms instead of Fourier transforms, but we do not consider the Laplace-based method in the present contribution. The algorithm can handle dispersive media, it can tackle complex physical structures, it enables parallelization in time in a straightforward manner, and it allows for time leaping---that is, solution sampling at any given time $T$ at $\mathcal{O}(1)$-bounded sampling cost, for arbitrarily large values of $T$, and without requirement of evaluation of the solution at intermediate times. The proposed frequency-time hybridization strategy, which generalizes to any linear partial differential equation in the time domain for which frequency-domain solutions can be obtained (including e.g. the time-domain Maxwell equations), and which is applicable in a wide range of scientific and engineering contexts, provides significant advantages over other available alternatives such as volumetric discretization, time-domain integral equations, and convolution-quadrature approaches.Comment: 33 pages, 8 figures, revised and extended manuscript (and now including direct comparisons to existing CQ and TDIE solver implementations) (Part I of II

Uniqueness results for the phase retrieval problem of fractional Fourier transforms of variable order

In this paper, we investigate the uniqueness of the phase retrieval problem for the fractional Fourier transform (FrFT) of variable order. This problem occurs naturally in optics and quantum physics. More precisely, we show that if $u$ and $v$ are such that fractional Fourier transforms of order $\alpha$ have same modulus $|F_\alpha u|=|F_\alpha v|$ for some set $\tau$ of $\alpha$'s, then $v$ is equal to $u$ up to a constant phase factor. The set $\tau$ depends on some extra assumptions either on $u$ or on both $u$ and $v$. Cases considered here are $u$, $v$ of compact support, pulse trains, Hermite functions or linear combinations of translates and dilates of Gaussians. In this last case, the set $\tau$ may even be reduced to a single point (i.e. one fractional Fourier transform may suffice for uniqueness in the problem)

Windowed Integral Equation Methods for Problems of Scattering by Defects and Obstacles in Layered Media

This thesis concerns development of efficient high-order boundary integral equation methods for the numerical solution of problems of acoustic and electromagnetic scattering in the presence of planar layered media in two and three spatial dimensions. The interest in such problems arises from application areas that benefit from accurate numerical modeling of the layered media scattering phenomena, such as electronics, near-field optics, plasmonics and photonics as well as communications, radar and remote sensing. A number of efficient algorithms applicable to various problems in these areas are pre- sented in this thesis, including (i) A Sommerfeld integral based high-order integral equation method for problems of scattering by defects in presence of infinite ground and other layered media, (ii) Studies of resonances and near resonances and their impact on the absorptive properties of rough surfaces, and (iii) A novel Window Green Function Method (WGF) for problems of scattering by obstacles and defects in the presence of layered media. The WGF approach makes it possible to completely avoid use of expensive Sommerfeld integrals that are typically utilized in layer-media simulations. In fact, the methods and studies referred in points (i) and (ii) above motivated the development of the markedly more efficient WGF alternative.</p

Time domain reflectometry in time variant plasmas

The effects of time-dependent electron density fluctuations on a synthesized time domain reflectometry response of a one-dimensional cold plasma sheath are considered. Numerical solutions of the Helmholtz wave equation, which describes the electric field of a normally incident plane wave in a specified static electron density profile, are used. A study of the effects of Doppler shifts resulting from moving density fluctuations in the electron density profile of the sheath is included. Varying electron density levels corrupt time domain and distance measurements. Reducing or modulating the electron density levels of a given electron density profile affects the time domain response of a plasma and results in motion of the turning point, and the effective motion has a significant effect on measuring electron density locations

Application of cepstral techniques to the automated determination of the sound power absorption coefficient

Includes bibliographical references.This thesis builds on research by Bolton and Gold, who developed the theory of using cepstral analysis to determine the absorption coefficient of elastic porous materials. Jongens, in his Masters thesis, applied this technique to determine the absorption coefficient of asphalt samples mounted in a sample holder at the end of a tube. Jongens and others identified numerous factors that introduced uncertainties into the measurement. These uncertainties fall into two main categories. The first deals with the influences that the links of the measurement chain have on the ability to separate the incident and reflected signal. The second deals with the influence of the air leakage between the tube and the surface under measurement in-situ. This thesis deals with the first category. The objectives of this project are to continue the work of Jongens, to produce an apparatus that can rapidly determine the sound power absorption coefficient by a non-skilled operator in a noisy environment. The results should correlate closely with the standardised impedance tube method, within 0.05 over the range 200 Hz to 2000 Hz. The constraint that the apparatus be usable by a non-skilled operator means that little or no calibration should be required, nor should the microphone need to be handled. This thesis presents a survey of related methods used to determine the sound power absorption coefficient. Theory of the cepstral technique is discussed, along with methods that could be used to improve the accuracy of the technique. Excitation signals that could be used with the cepstral method are put forward. The Inverse Repeat Sequence (IRS) was used to excite the system. It was chosen for its high noise immunity, as well as its complete odd-order non-linearity immunity. Sources of uncertainties from the links of the measurement chain are considered and methods to overcome them are presented. Issues that arise from liftering - cepstral equivalent of windowing - are then highlighted. The apparatus for the cepstral technique and method of standing wave ratios used to determine the absorption coefficient is given. The results obtained using the cepstral technique are correlated with the impedance tube results. It was found that the cepstral method correlates closely with the impedance tube over the range of 200 Hz to 2000 Hz for a wide variety of samples. The apparatus was developed to be used by a non-skilled operator, only requiring the press of a button to perform the measurement. With the high noise immunity of the IRS signal, the measurement could be carried out in a noisy environment