Order Fourier Research Articles

Integrated optical wave analyzer using the discrete fractional Fourier transform

In this work, we detail a proposal for optical signals to be represented and analyzed in phase-space. Our proposal aims to integrate a series of operations in waveguide realization, as a compact all-together platform that takes an initial wavefield and returns a two-dimensional representation of the information. We show, step by step, that the quantum harmonic oscillator can be considered as a propagator of initial fields, and when a discretized version is implemented, the fractional order Fourier transform emerges. This last is crucial, since the Wigner-Radon theorem is used to establish a path between the propagated wavefield and the phase-space representation. We show by example that this integration offers a direct and efficient method for characterizing optical signals by reconstructing their Wigner phase-space in the scope of integrated optics.

Suppressing DFB Lasers Relaxation Oscillations by High-Order Continuous Shaping Current With CNN Prediction

A technique of continuous shaping current waveform to suppress relaxation oscillations (ROs) of distributed feedback (DFB) laser for a high-performance optic system is demonstrated. To effectively suppress ROs, expressions for the shaping current waveform are theoretically derived based on the rate equations and different polynomials for the 3 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">rd</sup> , 5 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">th</sup> , and 8 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">th</sup> order Fourier basis functions are introduced. The convolutional neural network (CNN) is employed to predict the multi-parameter values that determine the results of the shaping input current, which exempt from the difficult and time-consuming process of parameter selection. Prior to training, preprocessing of the data obtained from DFB laser forward simulation using min-max normalization aims to improve the training efficiency of the CNN. The shaping current signals obtained from the CNN predicted parameters are put into the equivalent circuit model for the DFB laser to verify the effectiveness of the shaping current technique and CNN parameter optimization. Afterwards, the shaping current waveform is verified in a time division multiplex passive optical network (TDM-PON) utilizing the DFB laser model as a directly modulated source achieving remarkable performance with low cost. The results show that the high-order continuous shaping current modulated technique can successfully suppress the ROs and enhance the performance of the optic system.

Univariate versus multivariate spectrophotometric data analysis of triamterene and xipamide; a quantitative and qualitative greenly profiled comparative study

Triamterene (TRI) and xipamide (XIP) mixture is used as a binary medication of antihypertension which is considered as a major cause of premature death worldwide. The purpose of this research is the quantitative and qualitative analysis of this binary mixture by green univariate and multivariate spectrophotometric methods. Univariate methods were zero order absorption spectra method (D0) and Fourier self-deconvolution (FSD), as TRI was directly determined by D0 at 367.0 nm in the range (2.00–10.00 µg/mL), where XIP show no interference. While XIP was determined by FSD at 261.0 nm in the range (2.00–8.00 µg/mL), where TRI show zero crossing. Multivariate methods were Partial Least Squares, Principal Component Regression, Artificial Neural Networks, and Multivariate Curve Resolution-Alternating Least Squares. A training set of 25 mixtures with different quantities of the tested components was used to construct and evaluate them, 3 latent variables were displayed using an experimental design. A set of 18 synthetic mixtures with concentrations ranging from (3.00–7.00 µg/mL) for TRI and (2.00–6.00 µg/mL) for XIP, were used to construct the calibration models. A collection of seven synthetic mixtures with various quantities was applied to build the validation models. All the proposed approaches quantitative analyses were evaluated using recoveries as a percentage, root mean square error of prediction, and standard error of prediction. Strong multivariate statistical tools were presented by these models, and they were used to analyze the combined dosage form available on the Egyptian market. The proposed techniques were evaluated in accordance with ICH recommendations, where they are capable of overcoming challenges including spectral overlaps and collinearity. When the suggested approaches and the published one were statistically compared, there was no discernible difference between them. The green analytical method index and eco-scale tools were applied for assessment of the established models greenness. The suggested techniques can be used in product testing laboratories for standard pharmaceutical analysis of the substances being studied.

FRACTIONAL MODEL OF BRINKMAN-TYPE NANOFLUID FLOW WITH FRACTIONAL ORDER FOURIER’S AND FICK’S LAWS

Nanofluids are used to achieve maximum thermal performance with the smallest concentration of nanoparticles and stable suspension in conventional fluids. The effectiveness of nanofluids in convection processes is significantly influenced by their increased thermophysical characteristics. Based on the characteristics of nanofluids, this study examines generalized Brinkman-type nanofluid flow in a vertical channel. Three different types of ultrafine solid nanoparticles such as GO, [Formula: see text], and [Formula: see text] are dispersed uniformly in regular water to form nanofluid. Partial differential equations (PDEs) are used to model the phenomena. Fick’s and Fourier’s laws of fractional order were then used to formulate the generalized mathematical model. The exact solution of the generalized mathematical model has been obtained by the joint use of Fourier sine and the Laplace transform (LT) techniques. The obtained solution is represented in Mittag-Leffler function. To analyze the behavior of fluid flow, heat and mass distribution in fluid, the obtained solution was computed numerically and then plotted in response to different physical parameters. It is worth noting from the analysis that the heat transfer efficiency of regular water has been improved by 25% by using GO nanoparticles, 23.98% by using [Formula: see text], and 20.88% by using [Formula: see text].

Airborne SAR Suppression of Blanket Jamming Based on Second Order Blind Identification and Fractional Order Fourier Transform

Airborne SAR Suppression of Blanket Jamming Based on Second Order Blind Identification and Fractional Order Fourier Transform

Second-Order Bounds on Correlations Between Increasing Families

Harris’s correlation inequality states that any two monotone functions on the Boolean hy-percube are positively correlated. Talagrand [11] started a line of works in search of quantitative versions of this fact by providing a lower bound on the correlation in terms of the influences of the functions. A famous conjecture of Chvátal [2] was found by Friedgut, Kahn, Kalai and Keller [5] to be equivalent to a certain strengthening of Talagrand’s bound, conjectured to hold true when one of the functions is antipodal (i.e., g(x) = 1 − g(−x). Motivated by this conjecture, we strengthen some of those bounds by giving estimates that also involve the second order Fourier coefficients of the functions. In particular, we show that the bounds due to Talagrand and due to Keller, Mossel and Sen [8] can be improved when one of the functions is antipodal. Our proofs follow a different route than the ones in the literature, and the analysis is carried out in the Gaussian setting.

A robust and secure watermarking algorithm based on DWT and SVD in the fractional order fourier transform domain

The present paper aims to develop and validate a robust and secure watermarking algorithm. Firstly, the algorithm transforms the cover image and the watermark image separately by FRFT to obtain the amplitudes of both the cover image and watermark image, and then the two-level DWT transformation only carries out for the amplitude of the cover image. Secondly, the algorithm applies the SVD to the low-frequency sub-band of the second level DWT of the cover image and the amplitude of the watermark image. Thirdly, a new matrix is constructed based on the singular values from both the cover and watermark images to embed watermark information. Lastly, the algorithm takes the preliminary numerical calculation to determine an appropriate transformation order of FRFT for the cover image and to provide an outstanding balance between the imperceptibility and robustness in the proposed watermarking scheme. Moreover, it guarantees the security of watermarking scheme by the transformation order of FRFT. Experimental results demonstrate that the proposed watermarking scheme provides better performance in imperceptibility and resistance against traditional signal processing and geometric attacks, especially in image rotation, image cropping, average filtering, median filtering, and Gaussian filtering in comparison with the existing methods in the literature.

Sensitivity and reproducibility of transverse magneto-optical Kerr effect (T-MOKE) ellipsometry

We report a comprehensive experimental study to analyze the limiting factors and physical mechanisms that determine the achievable performance of transverse magneto-optical Kerr effect (T-MOKE) ellipsometry. Specifically, we explore different approaches to achieve high sensitivity and reduced acquisition times. The best sensitivity is observed for an incident light polarization with balanced s-p components. We also verify experimentally that the method’s theoretical description is accurately describing data for any s-p combination of the incoming light. Furthermore, two alternative measurement strategies are explored by using different measurement sequences for the polarization sensitive optics, which both achieve a very comparable, high quality of results. Signal-to-noise ratios and systematic deviations are measured and analyzed based on a large number of nominally identical measurement repeats, both for entire signal sequences as well as for individual Fourier components of the magneto-optical signal generated by a sinusoidal magnetic field sequence. Hereby, we observe that while higher order Fourier components have a significantly reduced signal amplitude and correspondingly exhibit reduced signal-to-noise and repeatability performance, signal-to-noise ratios always exceed values of 100 even for the lowest signal Fourier component and the lowest signal sample that we investigated, illustrating the extremely precise nature of T-MOKE ellipsometry.

Nonlocal interaction engineering of 2D roton-like dispersion relations in acoustic and mechanical metamaterials

The interior of the synthetic unit cells and their interactions determine the wave properties of metamaterials composed of periodic lattices of these cells. While local interactions with the nearest neighbors are well appreciated, nonlocal beyond-nearest-neighbor interactions are often considered as a nuisance. Here, by introducing a versatile effectively two-dimensional metamaterial platform for airborne sound and elastic waves, we exploit nonlocal effects as a powerful design tool. Within a simplified discrete model, we analytically show that the lowest band can be engineered by Fourier synthesis, where the N-th order Fourier coefficient is determined by the N-th nearest-neighbor interaction strength. Roton-like dispersion relations are an example. The results of the discrete model agree well with a refined model and with numerical calculations. In addition, we engineer the passage of waves from a local metamaterial into a nonlocal metamaterial by carefully tailoring the transition region between the two.

A Robust and Secure Watermarking Algorithm Based on Dwt and Svd in the Fractional Order Fourier Transform Domain

A Robust and Secure Watermarking Algorithm Based on Dwt and Svd in the Fractional Order Fourier Transform Domain

A Robust and Secure Watermarking Algorithm Based on Dwt and Svd in the Fractional Order Fourier Transform Domain

A Robust and Secure Watermarking Algorithm Based on Dwt and Svd in the Fractional Order Fourier Transform Domain

Gravitational instability of gas-dust circumnuclear disks in nearby galaxies

Using the two-dimensional multi-component hydrodynamical simulations, we study the effect of dust on the development of instability in the gas-dust mini-disks observed in the central regions of galaxies. The dust is coupled with the gas gravitationally and by means of a friction force depending on the velocity difference between the two components. The presence of dust leads to the instability of a mini-disk and the development of a spiral structure observed in the gas-dust mini-disks. The spiral structure is multi-armed, which is caused by a dominance of the high order Fourier harmonics in the unstable modes. The instability of gas-dust circumnuclear disks is an important mechanism for explaining the activity of galactic nuclei associated with the accretion of matter onto a central black hole. We find that the admixture of dust with a dust-to-gas ratio of 10-20 % significantly destabilizes the gas-dust disk.

Reconstructing small perturbations of an obstacle for acoustic waves from boundary measurements on the perturbed shape itself

We derive relationships between the shape deformation of an impenetrable obstacle and boundary measurements of scattering fields on the perturbed shape itself. Our derivation is rigorous by using a systematic way, based on layer potential techniques and the field expansion (FE) method (formal derivation). We extend these techniques to derive asymptotic expansions of the Dirichlet‐to‐Neumann (DNO) and Neumann‐to‐Dirichlet (NDO) operators in terms of the small perturbations of the obstacle as well as relationships between the shape deformation of an obstacle and boundary measurements of DNO or NDO on the perturbed shape itself. All relationships lead us to very effective algorithms for determining lower order Fourier coefficients of the shape perturbation of the obstacle.

A retrospective in-depth analysis of continuous glucose monitoring datasets for patients with hepatic glycogen storage disease: Recommended outcome parameters for glucose management.

Continuous glucose monitoring (CGM) systems have great potential for real‐time assessment of glycemic variation in patients with hepatic glycogen storage disease (GSD). However, detailed descriptions and in‐depth analysis of CGM data from hepatic GSD patients during interventions are scarce. This is a retrospective in‐depth analysis of CGM parameters, acquired in a continuous, real‐time fashion describing glucose management in 15 individual GSD patients. CGM subsets are obtained both in‐hospital and at home, upon nocturnal dietary intervention (n = 1), starch loads (n = 11) and treatment of GSD Ib patients with empagliflozin (n = 3). Descriptive CGM parameters, and parameters reflecting glycemic variation and glycemic control are considered useful CGM outcome parameters. Furthermore, the combination of first and second order derivatives, cumulative sum and Fourier analysis identified both subtle and sudden changes in glucose management; hence, aiding assessment of dietary and medical interventions. CGM data interpolation for nocturnal intervals reduced confounding by physical activity and diet. Based on these analyses, we conclude that in‐depth CGM analysis can be a powerful tool to assess glucose management and optimize treatment in individual hepatic GSD patients.

Improved generalized periods estimates over curves on Riemannian surfaces with nonpositive curvature

Abstract We show that, on compact Riemannian surfaces of nonpositive curvature, the generalized periods, i.e. the 𝜈-th order Fourier coefficients of eigenfunctions e λ e_{\lambda} over a closed smooth curve 𝛾 which satisfies a natural curvature condition, go to 0 at the rate of O ⁢ ( ( log ⁡ λ ) - 1 2 ) O((\log\lambda)^{-\frac{1}{2}}) in the high energy limit λ → ∞ \lambda\to\infty if 0 &lt; | ν | λ &lt; 1 - δ 0&lt;\frac{\lvert\nu\rvert}{\lambda}&lt;1-\delta for any fixed 0 &lt; δ &lt; 1 0&lt;\delta&lt;1 . Our result implies, for instance, that the generalized periods over geodesic circles on any surfaces with nonpositive curvature would converge to zero at the rate of O ⁢ ( ( log ⁡ λ ) - 1 2 ) O((\log\lambda)^{-\frac{1}{2}}) .

Time domain simulations of piston-like resonant flow in the gap of an oscillating twin-hull using inviscid and viscous flow solvers

A numerical study of a semi-circular twin-hull section under heave oscillation is presented. Two different time domain simulation methods were used: Firstly, a boundary element method based on potential theory which incorporates fully nonlinear free surface boundary conditions by using a mixed Eulerian Lagrangian scheme. Secondly, a finite volume method in combination with a volume of fluid scheme for capturing the free surface. In order to evaluate the effects of viscosity, simulations with the finite volume method were carried out using inviscid as well as viscous flow assumptions. Hydrodynamic mass and damping coefficients were derived from first order Fourier coefficients and validated against results from linear theory and experiments. A detailed comparison of nonlinear results from both simulation methods was carried out in vicinity of the piston-mode resonance frequency and a very good agreement was found. The phase angles corresponding to the Fourier coefficients varied strongly with frequency, which went along with a phase shift of the fluid motion in the gap. Influence of viscosity on the flow was found to be present but had relatively low impact on the hydrodynamic forces.

Corneal Analysis with Swept Source Optical Coherence Tomography in Patients with Coexisting Cataract and Fuchs Endothelial Corneal Dystrophy

This study focused on defining the characteristic features of keratometry and pachymetry elevation maps based on swept source optical coherence tomography (SS OCT) in Fuchs endothelial corneal dystrophy (FECD) eyes with a coexisting cataract. 70 eyes of 35 patients diagnosed with FECD and a coexisting cataract and 70 control eyes were included in this prospective, controlled, observational, cross-sectional study. Features characteristic of intermediately affected eyes included an increased corneal thinnest thickness (CTT) (p = 0.01), 3 and 6 mm asymmetry (p < 0.0001), higher order Fourier indices (p < 0.05 and p ≤ 0.0001, respectively), chord µ, and a posterior Ectasia Screening Index (pESI) (p < 0.01). The lack of agreement between the anterior and posterior elevation map and a significant area of negative values in the posterior map were detected. In advanced FECD eyes, our study additionally revealed decreased posterior keratometry steep (Ks), keratometry flat (Kf), keratometry average (AvgK), eccentricity (Ecc), an increased corneal apex thickness (CAT), and decreased 3 and 6 mm posterior spherical indices (p < 0.0001 for all of the above). Characteristic features of subclinical FECD, independent of the corneal thickness, can be detected by SS OCT and should be considered during the preoperative assessment of patients with a coexisting cataract.

A machining accuracy improvement approach for a horizontal machining center based on analysis of geometric error characteristics

In the actual production, it is expected to find several sensitivity geometric errors which have great influence on machining accuracy, so as to provide references for the manufacturing, assembly, and other links of the machine tool, and fundamentally improve the working performance. To study this problem, a novel global sensitivity analysis (GSA) method is proposed. Based on the MBS theory, the spatial error model is established to analyze the local influence of geometric error on machining accuracy. An improved second-order partial correlation coefficient based on the Pearson product moment is proposed to analyze the correlation between geometric errors. In addition, the error parameters will change greatly in the stroke of the moving parts. However, the rapid fluctuation of geometric error value will have a dynamic impact on the machining accuracy of machine tools, which is rarely noticed in the previous sensitivity analysis. This change is defined as the fluctuation of error. The high fitting degree function of error-displacement is obtained by using the high order Fourier series and sine series. Then, the fluctuation of the error in the machining stroke is analyzed by using the derivative of the function to the displacement. Through geometric error characteristics (including the local influence, correlation, and fluctuation) are studied comprehensively, the GSA of error is carried out. Finally, taking a machining center as an example and combining the sensitivity analysis results, the improvement measures are proposed to verify the correctness of the method.

Emergence and stability of periodic two-cluster states for ensembles of excitable units.

We study dynamics in ensembles of identical excitable units with global repulsive interaction. Starting from active rotators with additional higher order Fourier modes in on-site dynamics, we observe, at sufficiently strong repulsive coupling, large-scale collective oscillations in which the elements form two separate clusters. Transitions from quiescence to clustered oscillations are caused by global bifurcations involving the unstable clustered steady states. For clusters of equal size, the scenarios evolve either through simultaneous formation of two heteroclinic trajectories or through two simultaneous saddle-node bifurcations on invariant circles. If the sizes of clusters differ, two global bifurcations are separated in the parameter space. Stability of clusters with respect to splitting perturbations depends on the kind of higher order corrections to on-site dynamics; we show that for periodic oscillations of two equal clusters the Watanabe-Strogatz integrability marks a change of stability. By extending our studies to ensembles of voltage-coupled Morris-Lecar neurons, we demonstrate that similar bifurcations and switches in stability occur also for more elaborate models in higher dimensions.

Higher Order Fourier Finite Element Methods for Hodge Laplacian Problems on Axisymmetric Domains

Higher Order Fourier Finite Element Methods for Hodge Laplacian Problems on Axisymmetric Domains