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

On Improved Accuracy Chirp Parameter Estimation using the DFRFT with Application to SAR-based Vibrometry

The Discrete Fractional Fourier Transform (DFRFT) has in recent years, become a useful tool for multicomponent chirp signal analysis. Chirp signals are transformed into spectral peaks in the chirp rate versus center frequency representation, whose coordinates are related to the underlying chirp parameters via a computed empirical peak to parameter mapping incorporated into the Santhanam-Peacock algorithm. In this thesis, we attempt to quantify the accuracy of the DFRFT approach by first studying the discretization error sources that arise from the transitioning of the continuous FRFT to DFRFT. Then, we refine prior work by Ishwor Bhatta to develop analytical expressions for the chirp rate and center frequency parameters instead of the empirical mapping approach. We further study the extensions of this refined DFRFT approach using zero padding, spectral peak interpolation, and chirp-z-transform based zooming. The performance of the refined estimators is compared versus the Cramer-Rao lower bound and shown to asymptotically approach the bound. This refined DFRFT approach is then applied to Synthetic Aperture Radar Vibrometry data from several vibrating targets and the estimated acceleration information and vibration frequencies are shown to be very close to the corresponding ground-truth accelerometer measurements

Synchrophasors Determination Based on Interpolated FFT Algorithm

Within the standard IEEE C37.118 applications and proposed hardware structure of a phasor measurement unit (PMU) are described. This paper presents the concept of the system for measuring and transferring synchrophasors from a theoretical aspect. Synchrophasor algorithms are developed in MATLAB/Simulink for the purpose of easier verification and hardware deployment on today’s market available and affordable real time development kits. Analysis of the synchrophasor measurement process is performed gradually. Firstly, by defining the synchrophasor based on three-phase to αβ-transformation and then introducing a discrete Fourier transform (DFT) based on synchrophasor estimation algorithm. Later, accompanying adverse effects resulting from its application are analyzed by means of simulation. To increase accuracy and improve estimation algorithm interpolated discrete Fourier transform (IpDFT) with and without windowing technique is used. To further optimize algorithm performance convolution sum in recursive form has been implemented instead of classical DFT approach. This study was carried out in order to validate described measurement system for the monitoring of transients during island operation of a local power electric system. Finally, simulation and experimental results including error analysis are also presented

Synchrophasors Determination Based on Interpolated FFT Algorithm

Within the standard IEEE C37.118 applications and proposed hardware structure of a phasor measurement unit (PMU) are described. This paper presents the concept of the system for measuring and transferring synchrophasors from a theoretical aspect. Synchrophasor algorithms are developed in MATLAB/Simulink for the purpose of easier verification and hardware deployment on today’s market available and affordable real time development kits. Analysis of the synchrophasor measurement process is performed gradually. Firstly, by defining the synchrophasor based on three-phase to αβ-transformation and then introducing a discrete Fourier transform (DFT) based on synchrophasor estimation algorithm. Later, accompanying adverse effects resulting from its application are analyzed by means of simulation. To increase accuracy and improve estimation algorithm interpolated discrete Fourier transform (IpDFT) with and without windowing technique is used. To further optimize algorithm performance convolution sum in recursive form has been implemented instead of classical DFT approach. This study was carried out in order to validate described measurement system for the monitoring of transients during island operation of a local power electric system. Finally, simulation and experimental results including error analysis are also presented

ShearLab: A Rational Design of a Digital Parabolic Scaling Algorithm

Multivariate problems are typically governed by anisotropic features such as edges in images. A common bracket of most of the various directional representation systems which have been proposed to deliver sparse approximations of such features is the utilization of parabolic scaling. One prominent example is the shearlet system. Our objective in this paper is three-fold: We firstly develop a digital shearlet theory which is rationally designed in the sense that it is the digitization of the existing shearlet theory for continuous data. This implicates that shearlet theory provides a unified treatment of both the continuum and digital realm. Secondly, we analyze the utilization of pseudo-polar grids and the pseudo-polar Fourier transform for digital implementations of parabolic scaling algorithms. We derive an isometric pseudo-polar Fourier transform by careful weighting of the pseudo-polar grid, allowing exploitation of its adjoint for the inverse transform. This leads to a digital implementation of the shearlet transform; an accompanying Matlab toolbox called ShearLab is provided. And, thirdly, we introduce various quantitative measures for digital parabolic scaling algorithms in general, allowing one to tune parameters and objectively improve the implementation as well as compare different directional transform implementations. The usefulness of such measures is exemplarily demonstrated for the digital shearlet transform.Comment: submitted to SIAM J. Multiscale Model. Simu

The SFXC software correlator for Very Long Baseline Interferometry: Algorithms and Implementation

In this paper a description is given of the SFXC software correlator, developed and maintained at the Joint Institute for VLBI in Europe (JIVE). The software is designed to run on generic Linux-based computing clusters. The correlation algorithm is explained in detail, as are some of the novel modes that software correlation has enabled, such as wide-field VLBI imaging through the use of multiple phase centres and pulsar gating and binning. This is followed by an overview of the software architecture. Finally, the performance of the correlator as a function of number of CPU cores, telescopes and spectral channels is shown.Comment: Accepted by Experimental Astronom

High-order, Dispersionless “Fast-Hybrid” Wave Equation Solver. Part I: 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 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

Sampling of the diffraction field

Cataloged from PDF version of article.When optical signals, like diffraction patterns, are processed by digital means the choice of sampling density and geometry is important during analog-to-digital conversion. Continuous band-limited signals can be sampled and recovered from their samples in accord with the Nyquist sampling criteria. The specific form of the convolution kernel that describes the Fresnel diffraction allows another, alternative, full-reconstruction procedure of an object from the samples of its diffraction pattern when the object is space limited. This alternative procedure is applicable and yields full reconstruction even when the diffraction pattern is undersampled and the Nyquist criteria are severely violated. Application of the new procedure to practical diffraction-related phenomena, like in-line holography, improves the processing efficiency without creating any associated artifacts on the reconstructed-object pattern. (C) 2000 Optical Society of America. OCIS codes: 050.1940, 070.6020, 090.1760, 100.2000