Discrete Gaussian Research Articles

A Phase-Space Beam Summation Formulation for Ultrawide-Band Radiation

A new discrete phase space Gaussian beam (GB) summation representation for ultrawide-band (UWB) radiation from an aperture source distribution is presented. The formulation is based on the theory of the windowed Fourier transform (WFT) frames, wherein we introduce a novel relation between the frequency and the frame overcompleteness. With this procedure, the discrete lattice of beams that are emitted by the aperture satisfies the main requirement of being frequency independent, so that only a single set of beams needs to be traced through the medium for all the frequencies in the band. It is also shown that a properly tuned class of iso-diffracting (ID) Gaussian-windows provides the "snuggest" frame representation for all frequencies, thus generating stable and localized expansion coefficients. Furthermore, due to the ID property, the resulting GBs propagators are fully described by frequency independent matrices whose calculation in the ambient environment need to be done only once for all frequencies. Consequently, the theory may also be expressed directly in the time-domain as will be presented elsewhere. The localization implied by the new formulation is demonstrated numerically for an UWB focused aperture. It is shown that the algorithm extracts the local radiation properties of the aperture source and enhances only those beams that conform with these properties, i.e., those residing near the phase space Lagrange manifold. Further localization is due to the fact the algorithm accounts only for beams that pass within a few beamwidths vicinity of the observation point. It is thus shown that the total number of beams is much smaller than the Landau Pollak bound on the aperture's degrees of freedom.

Explicit Form of the Time Operator of a Gaussian Stationary Process

We present the time operator theory in the framework of stationary stochastic processes. The main results of the paper is the derivation of the time operator acting on the Fock space associated with a discrete time gaussian stationary process.

AN EFFICIENT SOLUTION SCHEME FOR APPLYING THE INTEGRAL-TYPE DYNAMIC LOCALIZATION SUBGRID-SCALE MODEL IN TURBULENCE WITH HOMOGENEOUS DIRECTIONS

The conventional integral-type dynamic localization subgrid-scale stress model is a Fredholm integral equation of the second kind. This model is mathematically consistent, but it has not been widely used in the large-eddy simulation community due to the relatively high computational cost of solving the Fredholm integral equation using an iterative scheme. In this article, a direct solution scheme based on a discrete Gaussian filter is developed to solve the integral system for turbulent flow with homogeneous dimensions. The proposed direct solution scheme is computationally efficient and makes the integral-type dynamic localization subgrid-scale stress model affordable. Turbulent Couette flows with Reynolds numbers of 2,600 and 4,762 are used in numerical simulations to validate the proposed approach.

On the equilibrium of shell membranes under normal loading. Hidden integrability

Restricted accessMoreSectionsView PDF ToolsAdd to favoritesDownload CitationsTrack Citations ShareShare onFacebookTwitterLinked InRedditEmail Cite this article Rogers C. and Schief W. K. 2003On the equilibrium of shell membranes under normal loading. Hidden integrabilityProc. R. Soc. Lond. A.4592449–2462http://doi.org/10.1098/rspa.2003.1135SectionRestricted accessOn the equilibrium of shell membranes under normal loading. Hidden integrability C. Rogers C. Rogers School of Mathematics, University of New South Wales, Sydney, NSW 2052, Australia Google Scholar Find this author on PubMed Search for more papers by this author and W. K. Schief W. K. Schief School of Mathematics, University of New South Wales, Sydney, NSW 2052, Australia Google Scholar Find this author on PubMed Search for more papers by this author C. Rogers C. Rogers School of Mathematics, University of New South Wales, Sydney, NSW 2052, Australia Google Scholar Find this author on PubMed Search for more papers by this author and W. K. Schief W. K. Schief School of Mathematics, University of New South Wales, Sydney, NSW 2052, Australia Google Scholar Find this author on PubMed Search for more papers by this author Published:08 October 2003https://doi.org/10.1098/rspa.2003.1135AbstractIt is established that a well–known system of classical shell theory descriptive of membranes in equilibrium is, in fact, integrable. The membranes are shown to have geometries within the integrable class of so–called O surfaces. The membrane O surfaces include inter alia minimal, constant mean curvature, constant Gaussian curvature and, more generally, linear Weingarten surfaces, as well as canal surfaces and Dupin cyclides. Previous ArticleNext Article VIEW FULL TEXT DOWNLOAD PDF FiguresRelatedReferencesDetailsCited by Tellier X, Douthe C, Baverel O and Hauswirth L (2023) Designing funicular grids with planar quads using isotropic Linear-Weingarten surfaces, International Journal of Solids and Structures, 10.1016/j.ijsolstr.2022.112028, 264, (112028), Online publication date: 1-Mar-2023. Hayashi K, Jikumaru Y, Ohsaki M, Kagaya T and Yokosuka Y (2021) Discrete Gaussian Curvature Flow for Piecewise Constant Gaussian Curvature Surface, Computer-Aided Design, 10.1016/j.cad.2021.102992, 134, (102992), Online publication date: 1-May-2021. Tellier X, Douthe C, Hauswirth L and Baverel O (2020) Linear‐Weingarten membranes with funicular boundaries, Structural Concrete, 10.1002/suco.202000030, 21:6, (2293-2306), Online publication date: 1-Dec-2020. Mesnil R, Douthe C, Baverel O and Léger B (2018) Morphogenesis of surfaces with planar lines of curvature and application to architectural design, Automation in Construction, 10.1016/j.autcon.2018.08.007, 95, (129-141), Online publication date: 1-Nov-2018. Douthe C, Mesnil R, Orts H and Baverel O (2017) Isoradial meshes: Covering elastic gridshells with planar facets, Automation in Construction, 10.1016/j.autcon.2017.08.015, 83, (222-236), Online publication date: 1-Nov-2017. Bobenko A, Hertrich-Jeromin U and Lukyanenko I (2014) Discrete constant mean curvature nets in space forms: Steiner’s formula and Christoffel duality, Discrete & Computational Geometry, 10.1007/s00454-014-9622-5, 52:4, (612-629), Online publication date: 1-Dec-2014. Schief W, Szereszewski A and Rogers C (2009) On shell membranes of Enneper type: generalized Dupin cyclides, Journal of Physics A: Mathematical and Theoretical, 10.1088/1751-8113/42/40/404016, 42:40, (404016), Online publication date: 9-Oct-2009. Rogers C and Szereszewski A (2009) A Bäcklund transformation for L-isothermic surfaces, Journal of Physics A: Mathematical and Theoretical, 10.1088/1751-8113/42/40/404015, 42:40, (404015), Online publication date: 9-Oct-2009. Guha P and Shipman P (2009) Polarized Hessian covariant: Contribution to pattern formation in the Föppl–von Kármán shell equations, Chaos, Solitons & Fractals, 10.1016/j.chaos.2008.10.025, 41:5, (2828-2837), Online publication date: 1-Sep-2009. Szereszewski A (2009) L-isothermic and L-minimal surfaces, Journal of Physics A: Mathematical and Theoretical, 10.1088/1751-8113/42/11/115203, 42:11, (115203), Online publication date: 20-Mar-2009. Schief W, Szereszewski A and Rogers C (2007) The Lamé equation in shell membrane theory, Journal of Mathematical Physics, 10.1063/1.2747721, 48:7, (073510), Online publication date: 1-Jul-2007. Schief W (2005) Integrable discrete differential geometry of ‘plated’ membranes in equilibrium, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 461:2062, (3213-3229), Online publication date: 8-Oct-2005.Schief W, Kléman M and Rogers C (2005) On a nonlinear elastic shell system in liquid crystal theory: generalized Willmore surfaces and Dupin cyclides, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 461:2061, (2817-2837), Online publication date: 8-Sep-2005. This Issue08 October 2003Volume 459Issue 2038 Article InformationDOI:https://doi.org/10.1098/rspa.2003.1135Published by:Royal SocietyPrint ISSN:1364-5021Online ISSN:1471-2946History: Published online08/10/2003Published in print08/10/2003 License: Citations and impact KeywordsDupin cyclidesshell membraneintegrabilityO surfacesBäcklund transformation

Design and control of a thermal stabilizing system for a MEMS optomechanical uncooled infrared imaging camera

In this paper, the design and control of a thermal stabilizing system for an optomechanical uncooled infrared (IR) imaging camera is presented, which uses an array of MEMS bimaterial cantilever beams to sense an IR image source. A one-dimensional lumped parameter model of the thermal stabilization system was derived and experimentally validated. A model-based discrete time linear quadratic gaussian regulator (LQGR) control scheme, with a stochastic ambient noise model, was implemented. The control system incorporates a reference model, which generates desired reference temperature trajectory, and integral action to respectively diminish overshoots and achieve zero steady state error in closed loop. Simulation results show that the designed LQGR is able to enhance ambient temperature low frequency disturbance attenuation by more than 50 dB. The control system is able to regulate the focal-plane array (FPA) temperature with a standard deviation of about 100 μ K, in spite of the fact that the temperature measurement noise has a standard deviation of 1 mK. Noise analysis results for the present stage of the optomechanical IR imaging system are summarized. The noise equivalent temperature difference (NETD) of the current stage of the IR camera system can achieve about 200 mK.

On the unification of classical and novel integrable surfaces. II. Difference geometry

A novel class of discrete integrable surfaces is recorded. This class of discrete O surfaces is shown to include discrete analogues of classical surfaces such as isothermic, ‘linear’ Weingarten, Guichard and Petot surfaces. Moreover, natural discrete analogues of the Gaussian and mean curvatures for surfaces parametrized in terms of curvature coordinates are used to define surfaces of constant discrete Gaussian and mean curvatures and discrete minimal surfaces. Remarkably, these turn out to be prototypical examples of discrete O surfaces. It is demonstrated that the construction of a Backlund transformation for discrete O surfaces leads in a natural manner to an associated parameter–dependent linear representation. Canonical discretizations of the classical pseudosphere and breather pseudospherical surfaces are generated. Connections with pioneering work by Bobenko and Pinkall are established.

A Generative Probabilistic Oriented Wavelet Model for Texture Segmentation

This Letter addresses image segmentation via a generative model approach. A Bayesian network (BNT) in the space of dyadic wavelet transform coefficients is introduced to model texture images. The model is similar to a Hidden Markov model (HMM), but with non-stationary transitive conditional probability distributions. It is composed of discrete hidden variables and observable Gaussian outputs for wavelet coefficients. In particular, the Gabor wavelet transform is considered. The introduced model is compared with the simplest joint Gaussian probabilistic model for Gabor wavelet coefficients for several textures from the Brodatz album [1]. The comparison is based on cross-validation and includes probabilistic model ensembles instead of single models. In addition, the robustness of the models to cope with additive Gaussian noise is investigated. We further study the feasibility of the introduced generative model for image segmentation in the novelty detection framework [2]. Two examples are considered: (i) sea surface pollution detection from intensity images and (ii) image segmentation of the still images with varying illumination across the scene.

The spectra and periodograms of anti-correlated discrete fractional Gaussian noise

Discrete fractional Gaussian noise (dFGN) has been proposed as a model for interpreting a wide variety of physiological data. The form of actual spectra of dFGN for frequencies near zero varies as f 1−2 H , where 0< H<1 is the Hurst coefficient; however, this form for the spectra need not be a good approximation at other frequencies. When H approaches zero, dFGN spectra exhibit the 1−2 H power-law behavior only over a range of low frequencies that is vanishingly small. When dealing with a time series of finite length drawn from a dFGN process with unknown H, practitioners must deal with estimated spectra in lieu of actual spectra. The most basic spectral estimator is the periodogram. The expected value of the periodogram for dFGN with small H also exhibits non-power-law behavior. At the lowest Fourier frequencies associated with a time series of N values sampled from a dFGN process, the expected value of the periodogram for H approaching zero varies as f 0 rather than f 1−2 H . For finite N and small H, the expected value of the periodogram can in fact exhibit a local power-law behavior with a spectral exponent of 1−2 H at only two distinct frequencies.

Optimal control of a flywheel energy storage system with a radial flux hybrid magnetic bearing

This paper describes the design and implementation of digital controllers for a flywheel energy storage device that incorporates a radial flux hybrid permanent magnetic bearing. Although the uncontrolled device is asymptotically stable, active control is required to: (i) ensure that a finite radial air gap is maintained at all times, and (ii) attenuate the oscillations of the flywheel which reduce the efficiency of the motor generator. The paper presents the design of gain scheduled discrete time linear quadratic regulator (LQR) and linear quadratic Gaussian (LQG) controllers for this rotordynamic system. Real time experiments are conducted to investigate the performance of the controllers. The result indicates that the LQR controller with approximate system velocities is easier to implement than the LQG controller, and also provides superior performance.

DESIGN AND CONTROL OF A THERMAL STABILIZING SYSTEM FOR AN OPTOMECHANICAL UNCOOLED INFRARED IMAGING CAMERA

In this paper, the design and control of a thermal stabilizing system for an Optomechanical Uncooled Infrared Imaging camera is presented, which uses an arra y of MEMS bimaterial can tilev erbeams to sense an infrared image source. A one-dimensional lumped parameter model of the thermal stabilization system was derived and experimentally validated. A model-based discrete time Linear Quadratic Gaussian Regulator(LQGR) control scheme, with a stochastic ambien t noise model, was implemented. The control system incorporates a reference model, which generates desired reference temperature trajectory, and integral action to respectively diminish overshoots and achieve zero steady state error in closed loop. Simulation results show that the designed LQGR controller is able to enhance ambient temperature low frequency disturbance attenuation by more than 50dB. The control system is able to regulate the focal-plane array (FPA) temperature with a standard deviation of about 100μK, in spite of the fact that the temperature measurement noise has a standard deviation of 1mK.

Effective paging schemes with delay bounds as QoS constraints in wireless systems

In this paper new paging schemes are presented for locating mobile users in wireless networks. Paging costs and delay bounds are considered since paging costs are associated with bandwidth utilization and delay bounds influence call setup time. In general, location tracking schemes require intensive computation to search for a mobile terminal in current PCS networks. To reduce the paging costs, three new paging schemes, reverse, semi-reverse and uniform, are introduced to provide a simple way of partitioning the service areas and decrease the paging costs based on each mobile terminal's location probability distribution. Numerical results demonstrate that our approaches significantly reduce the paging costs for various probability distributions such as uniform, truncated discrete Gaussian, and irregular distributions.

Upper bounds on the capacity of discrete-time blockwise white Gaussian channels with feedback

Although it is well known that feedback does not increase the capacity of an additive white Gaussian channel, Yanagi (1992) gave the necessary and sufficient condition under which the capacity C/sub n,FB/(P) of a discrete time nonwhite Gaussian channel is increased by feedback. In this correspondence we show that the capacity C/sub n,FB/(P) of the Gaussian channel with feedback is a concave function of P, and give two types of inequalities: both 1//spl alpha/ C/sub n,FB/(/spl alpha/P) and C/sub n,FB/(/spl alpha/P)+ 1/2 ln 1//spl alpha/ are decreasing functions of /spl alpha/>0. As their application we can obtain two upper bounds on the capacity of the discrete-time blockwise white Gaussian channel with feedback. The results are quite useful when power constraint P is relatively not large.

CHAOS COMMUNICATION OVER NOISY CHANNELS

The problem of transmitting digital information using chaotic signals over a channel with Gaussian white noise perturbation is introduced rigorously. It is shown that discrete time base-band chaotic communication systems with discrete time Gaussian white noise in the channel are sufficiently general in this context. The optimal receiver is given explicitly in terms of conditional probabilities. For the example of chaos shift keying using iterations of the tent map, the optimal classifier is constructed explicitly. Finally, it is shown how previously published methods, in particular those based on chaos synchronization, fit into this framework.

Deriving dispersional and scaled windowed variance analyses using the correlation function of discrete fractional Gaussian noise

Methods for estimating the fractal dimension, D, or the related Hurst coefficient, H, for a one-dimensional fractal series include Hurst's method of rescaled range analysis, spectral analysis, dispersional analysis, and scaled windowed variance analysis (which is related to detrended fluctuation analysis). Dispersional analysis estimates H by using the variance of the grouped means of discrete fractional Gaussian noise series (DfGn). Scaled windowed variance analysis estimates H using the mean of grouped variances of discrete fractional Brownian motion (DfBm) series. Both dispersional analysis and scaled windowed variance analysis have small bias and variance in their estimates of the Hurst coefficient. This study demonstrates that both methods derive their accuracy from their strict mathematical relationship to the expected value of the correlation function of DfGn. The expected values of the variance of the grouped means for dispersional analysis on DfGn and the mean of the grouped variance for scaled windowed variance analysis on DfBm are calculated. An improved formulation for scaled windowed variance analysis is given. The expected values using these analyses on the wrong kind of series (dispersional analysis on DfBm and scaled windowed variance analysis on DfGn) are also calculated.

Refinements of the half-bit and factor-of-two bounds for capacity in Gaussian channel with feedback

We consider the upper bounds of the finite blocklength capacity C/sub n,FB/(P) of the discrete time Gaussian channel with feedback. We also let C/sub n/(p) be the nonfeedback capacity. We prove the relations C/sub n/(P)/spl les/C/sub n,FB/(P)/spl les/C/sub n/(/spl alpha/P)+ 1/2 ln(1+1//spl alpha/) and C/sub n/(P)/spl les/C/sub n,FB/(P)/spl les/(1+1//spl alpha/)C/sub n/(/spl alpha/P) for any P>0 and any /spl alpha/>0, which induce the half-bit and factor-of-two bounds given by Cover and Pombra (1989) in the special case of /spl alpha/=1.

Surface width of the Solid-On-Solid models

The low-temperature series for the surface width of the Absolute value Solid-On-Solid model and the Discrete Gaussian model both on the square lattice and on the triangular lattice are generated to high orders using the improved finite-lattice method. The series are analyzed to give the critical points of the roughening phase transition for each model.

Quantitative criteria for the design of dither-based quantizing systems

This paper addresses the problem of designing quantizing systems employing dither. These are composed of a dither generator, a quantizer, a subtracting block and a low-pass digital filter. Quantitative criteria for choosing the parameters that influence the resolution and accuracy of the quantizing system are given, and the system performance is determined under worst, best, and average conditions for discrete binary, uniform, and Gaussian dither signals. The results presented also can be applied when the signal plus input-referred noise are quantized.

Aq-Series Approach to Deblurring the Discrete Gaussian

A method is presented for deblurring an image blurred by the discrete Gaussian. The method, based on classical theorems of Jacobi and Ramanujan, not only provides exact formulas for the deblurring, but also condition numbers and error bounds estimating the agreement between the original and reconstructed image. The use of the Jacobi Triple Product Theorem provides a convenient factorization of the formulas used in the inversion process into a product of three infinite series. These three series correspond to a constant together with a Toeplitz operator and its transpose. In the finite setting this factorization corresponds to the factorization of a matrix into the product of Toeplitz matrices, where the entries can be computed using simple recursion formulas. For selected choices of σ, condition numbers are calculated for these operators. The results are similar to a method developed by Kimia and Zucker.

The Joint Moment Generating Function of Quadratic Forms in Multivariate Autoregressive Series

Let (X1) be a discrete multivariate Gaussian autoregressive process of order 1. The paper derives the exact finite-sample joint moment generating function (m.g.f.) of the three quadratic forms constituting the sufficient statistic of the process. The formula is then specialized to some cases of interest, including the m.g.f. of functional of multivariate Ornstein-Uhlenbeck processes that arise asymptotically from more general (X1) processes as well.

Deblurring images blurred by the discrete Gaussian

Gaussian blur is frequently used to model the degradation of images or signals. We will show how to clean the blurred image. We shall obtain exact formulas. To achieve this goal, we will make use of some formulas from the theory of basic hypergeometric series. We will also show how to deblur images blurred by a modified Gaussian blur.