Fourier Series Research Articles

Seepage-induced passive earth pressure modification of retaining structures considering composite failure surface

Waterfront and submarine retaining structures are normally exposed to catastrophic seepage conditions under the effect of tidal and occasionally heavy rainfall effect, resulting in a decreased passive earth thrust and thus the higher risk of instability of retaining structures. To examine the effect of seepage flow on the magnitude and distribution of passive earth thrust, this paper assumes a composite curved-planar failure surface and presents a modified method of passive earth pressure considering the seepage flow effect. The flow field and pore pressure are firstly solved by the two-dimensional (2D) Laplace equation using the Fourier series expansion. The effective reaction force acting on the composite failure surface is then obtained using a modified Kötter equation. Compared to conventional methods based on limit equilibrium, the present method facilitates a straightforward assessment of both the magnitude and distribution of passive earth thrust without the prior assumption of the application point. The outcomes highlight that the passive earth thrust decreases with the ratios of permeability coefficients. The greater effective friction angle and a smaller ratio of permeability coefficients result in the lower application point of the passive earth thrust.

When grey model meets deep learning: A new hazard classification model

The classification of hazard is critical in industrial informatics, as it enhances early safety alerts, supports decision-making, and facilitates policy assessment. However, previous studies have generally neglected the temporal attributes of hazards, thereby constraining the effectiveness of models. This paper introduces a new model for hazard classification termed DLGM. DLGM represents a deep learning framework, with the structural parameters of grey models to encapsulate the hazard temporal attributes. To better accommodate the fluctuations of hazard series, a new grey model termed FSGM(1,1) equipped with Fourier series is proposed. Moreover, DLGM leverages a novel hierarchical feature fusion neural network (HFFNN) to optimize feature processing. Extensive experiments across three hazard themes involving 18 large-scale industrial processes have demonstrated the competitiveness of DLGM (e.g., it surpasses benchmark models such as Random Forest and BERT by approximately 2% in both accuracy and F1), the suitability of FSGM(1,1) (e.g., its mean absolute percentage error is less than 10% and mean squared error is below 0.02) and the effectiveness of HFFNN (e.g., it enhances the accuracy and F1 of DLGM by about 1%).

On the stability analysis of a restrained functionally graded nanobeam in an elastic matrix with neutral axis effects

Abstract In this work, a general eigenvalue solution of an arbitrarily constrained nonlocal strain gradient nanobeam made of functionally graded material is presented for the first time for the stability response by the effect of the Winkler foundation. Elastic springs at the ends of the nanobeam are considered in the formulation, which have not been considered in most studies. In order to analyze deformable boundary conditions, linear equation systems are derived in terms of infinite power series by using the Fourier sine series together with the Stokes’ transform. The higher-order force boundary conditions are used to obtain a coefficient matrix including different end conditions, power-law index, elastic medium, and small-scale parameters. A general eigenvalue problem of technical interest, associated with nonlocal strain gradient theory, is mathematically evaluated and presented in detail. Parametric results are obtained to investigate the effects of material length scale parameter, Winkler stiffness, power-law index, nonlocal parameter, and elastic springs at the ends. In addition, the effects of the other higher-order elasticity theories simplified from nonlocal strain gradient theory are also investigated and some benchmark results are presented.

Interpolation problems for random fields on Sierpinski’s carpet

The prediction of stochastic processes and the estimation of random fields of different natures is becoming an increasingly common field of research among scientists of various specialties. However, an analysis of papers across different estimating problems shows that a dynamic approach over an iterative and recursive interpolation of random fields on fractal is still an open area of investigation. There are many papers related to the interpolation problems of stationary sequences, estimation of random fields, even on the perforated planes, but all of this still provides a place for an investigation of a more complicated structure like a fractal, which might be more beneficial in appliances of certain industry fields. For example, there has been a development of mobile phone and WiFi fractal antennas based on a first few iterations of the Sierpinski carpet. In this paper, we introduce an estimation for random fields on the Sierpinski carpet, based on the usage of the known spectral density, and calculation of the spectral characteristic that allows an estimation of the optimal linear functional of the omitted points in the field. We give coverage of an idea of stationary sequence estimating that is necessary to provide a basic understanding of the approach of the interpolation of one or a set of omitted values. After that, the expansion to random fields allows us to deduce a dynamic approach on the iteration steps of the Sierpinski carpet. We describe the numerical results of the initial iteration steps and demonstrate a recurring pattern in both the matrix of Fourier series coefficients of the spectral density and the result of the optimal linear functional estimation. So that it provides a dependency between formulas of the different initial sizes of the field as well as a possible generalizing of the solution for N-steps in the Sierpinski carpet. We expect that further evaluation of the mean squared error of this estimation can be used to identify the possible iteration step when further estimation becomes irrelevant, hence allowing us to reduce the cost of calculations and make the process viable.

Invariant distributions and the transport twistor space of closed surfaces

AbstractWe study transport equations on the unit tangent bundle of a closed oriented Riemannian surface and their links to the transport twistor space of the surface (a complex surface naturally tailored to the geodesic vector field). We show that fibrewise holomorphic distributions invariant under the geodesic flow — which play an important role in tensor tomography on surfaces — form a unital algebra, that is, multiplication of such distributions is well defined and continuous. We also exhibit a natural bijective correspondence between fibrewise holomorphic invariant distributions and genuine holomorphic functions on twistor space with polynomial blowup on the boundary of the twistor space. Additionally, when the surface is Anosov, we classify holomorphic line bundles over twistor space which are smooth up to the boundary. As a byproduct of our analysis, we obtain a quantitative version of a result of Flaminio [C. R. Acad. Sci. Paris Sér. I Math. 315 (1992) no. 6, 735–738] asserting that invariant distributions of the geodesic flow of a positively curved metric on are determined by their zeroth and first Fourier modes.

The Modeling Method for Vibration Characteristics Analysis of Composite-Laminated Rotationally Stiffened Plate

The composite-laminated rotationally stiffened plate is widely applied in aviation, aerospace, ship, machinery, and other fields. For structural design and optimization, to investigate the vibration characteristics is important. In this paper, a modeling method of composite-laminated rotationally plate is established. The first-order shear deformation theory (FSDT) and the modified Fourier series are applied to construct the admissible displacement function of the stiffened plate-coupled systems. On this basis, the energy function of composite-laminated rotationally stiffened plate is established. Combined with the artificial virtual spring technology, the proposed theory could be used to analyze the vibration characteristics of composite-stiffened plate-coupled systems with various classical boundary conditions or arbitrary elastic boundary conditions. The Rayleigh–Ritz method is used to solve the energy function. Thus, the vibration characteristics of the composite-laminated rotationally stiffened plate are obtained and analyzed. The correctness of the theoretical analysis model was verified through modal experiments. On this basis, the effect of some important parameters on the vibration characteristics of stiffened plate structures is studied, such as the number, thickness, and width of the laminated stiffener, varying structural parameters, and different boundary conditions. This study can provide the theoretical basis for the vibration and noise reduction of such structures.

Was the interest burden of the Ottoman public debt sustainable? A cliometric analysis

Purpose The purpose of this study is to determine whether the Ottoman Empire’s net interest payments and foreign debt were sustainable or not in terms of their burden on budget revenues, using the method of historical econometric analysis. Design/methodology/approach In this study, the period between 1847 and 1882 of the Ottoman Empire is analyzed for sustainability analysis. Within the framework of the study, unit root tests and econometric analysis methods frequently used in the literature were used to analyze the sustainability of public debt. In the econometric analysis, in addition to various unit root tests, current econometric analysis methods, in particular Fourier expansion, were also used. Findings The results of econometric analyses showed that the burden of interest payments and foreign debt on the budget of the Ottoman state was unsustainable. This situation clearly shows the reason for the official bankruptcy of the Ottoman Empire, which was declared in 1875. Practical implications Although this study reveals the bankruptcy process of an important structure such as the Ottoman Empire in the historical process through econometric analyses, it also gives a very important message to today’s states. Accordingly, today’s state policies and decision-making mechanisms should take these results into account and strive to make the burden of public interest payments sustainable. It is believed that the study will shed light on the public finance policies of today’s states by drawing lessons from the collapse process of the Ottoman state. Originality/value Unlike the historical assessments in the literature on the decline of the Ottoman Empire, this study presents a cliometric approach by applying current econometric analysis techniques to past historical data. The study explains the unsustainability of the Ottoman Empire’s interest payments and external debt burden in the period under consideration in a way that, to the best of the authors’ knowledge, has not been done before.

On approximation of functions, conjugate to the functions of several variables belonging to weighted Lipschitz and Zygmund classes

In the present paper, we introduce a new weighted Lipschitz class W(Lp(TN), ξ1(s1), . . . , ξN(sN)) and Zygmund class Z(Lp(TN), ξ1(s1), . . . , ξN(sN)) for N ∈ N, which generalizes the classes given in [7, 11]. We prove two theorems about
 the degree (error) of approximation of functions, conjugate to the N-variable functions (2π-periodic in each variable) belonging to these classes using the N-multiple matrix means of their N-multiple conjugate Fourier series. We improve the results of Móricz and Rhoades [6] and Móricz and Shi [7], which are given in the form of corollaries.

Experimental assessment of square-wave spatial spanwise forcing of a turbulent boundary layer

We present an experimental realisation of spatial spanwise forcing in a turbulent boundary layer flow, aimed at reducing the frictional drag. The forcing is achieved by a series of spanwise running belts, running in alternating spanwise direction, thereby generating a steady spatial square-wave forcing. Stereoscopic particle image velocimetry in the streamwise–wall-normal plane is used to investigate the impact of actuation on the flow in terms of turbulence statistics, drag performance characteristics, and spanwise velocity profiles, for a non-dimensional wavelength of λx+=397\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda _x^+ = 397$$\\end{document}. In line with reported numerical studies, we confirm that a significant flow control effect can be realised with this type of forcing. The scalar fields of the higher-order turbulence statistics show a strong attenuation of stresses and production of turbulence kinetic energy over the first belt already, followed by a more gradual decrease to a steady-state energy response over the second belt. The streamwise velocity in the near-wall region is reduced, indicative of a drag-reduced flow state. The profiles of the higher-order turbulence statistics are attenuated up to a wall-normal height of y+≈100\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$y^+ \\approx 100$$\\end{document}, with a maximum streamwise stress reduction of 45% and a reduction of integral turbulence kinetic energy production of 39%, for a non-dimensional actuation amplitude of A+=12.7\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A^+ = 12.7$$\\end{document}. An extension of the classical laminar Stokes layer theory is introduced, based on the linear superposition of Fourier modes, to describe the non-sinusoidal boundary condition that corresponds to the current case. The experimentally obtained spanwise velocity profiles show good agreement with this extended theoretical model. The drag reduction was estimated from a linear fit in the viscous sublayer in the range 2≤y+≤5\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2 \\le y^+\\le 5$$\\end{document}. The results are found to be in good qualitative agreement with the numerical implementations of Viotti et al. (Phys Fluids 21, 2009), matching the drag reduction trend with A+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A^+$$\\end{document}, and reaching a maximum of 20%.Graphical abstract

Renormalisable Non-Local Quark–Gluon Interaction: Mass Gap, Chiral Symmetry Breaking and Scale Invariance

We derive a Nambu–Jona-Lasinio (NJL) model from a non-local gauge theory and show that it has confining properties at low energies. In particular, we present an extended approach to non-local QCD and a complete revision of the technique of Bender, Milton and Savage applied to non-local theories, providing a set of Dyson–Schwinger equations in differential form. In the local case, we obtain closed-form solutions in the simplest case of the scalar field and extend it to the Yang–Mills field. In general, for non-local theories, we use a perturbative technique and a Fourier series and show how higher-order harmonics are heavily damped due to the presence of the non-local factor. The spectrum of the theory is analysed for the non-local Yang–Mills sector and found to be in agreement with the local results on the lattice in the limit of the non-locality mass parameter running to infinity. In the non-local case, we confine ourselves to a non-locality mass that is sufficiently large compared to the mass scale arising from the integration of the Dyson–Schwinger equations. Such a choice results in good agreement, in the proper limit, with the spectrum of the local theory. We derive a gap equation for the fermions in the theory that gives some indication of quark confinement in the non-local NJL case as well. Confinement seems to be a rather ubiquitous effect that removes some degrees of freedom in the original action, favouring the appearance of new observable states, as seen, e.g., for quantum chromodynamics at lower energies.

Performance and harmonic detection algorithm of phase locked Loop for parallel APF

As the boost of power electronics technology, the harmonic problem in the power system is becoming increasingly prominent. Fourier decomposition is performed on the load current in the power system, and components with a frequency that is an integer multiple of the fundamental wave are referred to as harmonic components. Harmonic control is essential to establish a safe and reliable power grid environment and provide high-quality and clean electricity to power users. The study focused on parallel active power filters, proposed a specific harmonic detection method on the grounds of synchronous harmonic rotating coordinate system, and developed a phase-locked loop design on the grounds of order generalized integrator. Meanwhile, a compensation current control method on the grounds of space vector pulse width modulation was introduced. The results showed that in the full compensation simulation experiment, the compensated A-phase grid side current waveform was significantly improved and presented a sinusoidal shape. After 0.05 s, the actual output compensation current closely followed the command current. Meanwhile, after compensation, the total harmonic distortion rate decreased from 26.58 to 3.06%. In specific harmonic compensation simulation experiments, when the sum of 5th, 7th, and 11th harmonic components was used as the command current for compensation, the distortion of the current waveform was improved after the load undergoes a sudden change. After compensation, the 5th, 7th, and 11th harmonic content significantly decreased, and the total harmonic distortion rate decreased to 4.08%. This indicated that the proposed phase-locked loop design and harmonic detection method for active power filters had high stability and effectiveness. The study’s primary contribution is to enhance the utilization efficiency of DC voltage and improve the dynamic response ability of current. Additionally, it offers a new method for reducing the impact of harmonics on the power grid and improving power quality. It provided an effective method reference for technological progress in related fields such as power electronics and control engineering.

A novel algorithm for open switch fault detection and fault tolerant control of interleaved DC‐DC boost converters

AbstractThis paper presents a novel algorithm for fault diagnosis and fault tolerant control of interleaved boost converters (IBCs) in the presence of single or double simultaneous open‐circuit faults (OCFs) in power switches. An innovative diagnosis signal (recorded by a cheap current sensor) will be introduced whose waveform mainly depends on the healthy or faulty condition of the converter. Since the diagnosis signal is periodic in the steady‐state operation of the converters, its Fourier Series coefficients, together with the duty cycle of the converter are used as distinguishing features for fault diagnosis of the OCFs in the converter. The well‐known Kalman filter is utilized for robust estimation of the Fourier‐based features. Finally, the modulation of the remaining healthy phases is rearranged in a way that the pre‐fault performance recovers. The proposed algorithm is verified in a laboratory four‐phase IBC setup in which the experimental results show its satisfactory performance. Also, the structure of the proposed method is straightforward and can be implemented in the same microcontroller which is used for voltage and current regulation of the converter.

Unfolding and injectivity of the Kudla–Millson lift of genus 1

We unfold the theta integrals defining the Kudla–Millson lift of genus 1 associated to even lattices of signature (b, 2), where b>2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$b>2$$\\end{document}. This enables us to compute the Fourier expansion of such defining integrals and prove the injectivity of the Kudla–Millson lift. Although the latter result has been already proved in [5], our new procedure has the advantage of paving the ground for a strategy to prove the injectivity of the lift also for the cases of general signature and of genus greater than 1.

THE ANALYSIS OF THE FLOW OF BLOOD IN A STENOSED ARTERY THROUGH SIMULATION: A COMPARISON AMONG VARIOUS NON-NEWTONIAN MODELS

This paper focuses on the dual quality of blood, Newtonian and non-Newtonian, in particular by exploring the energy curves. Careful investigation of the dual property of blood has been made by considering two different geometries to represent a stenosed arterial segment. We present a cautious assessment of non-Newtonian blood rheology impacts in arterial stream simulations by coupling the Newtonian and non-Newtonian models. The flow of energy through the two flow dimensions is meticulously investigated using velocity (kinetic energy), pressure, and wall shear stress (pressure energy). Besides, the proper implementation of an interface boundary condition (IBC) was emphasized to ensure consistency with the flow conditions downstream of a backward-facing step. The integration of the Newtonian and non-Newtonian models adjoins the novelty of the current research. The energy curves are obtained by implementing five different non-Newtonian models to designate a suitable non-Newtonian model for blood flow investigations. The combination of the non-Newtonian models enforced in this research is novel and particular attention is paid to the energy curves obtained. The conclusion was to elect the Carreau model as a suitable non-Newtonian rheological model for the blood flow study. This study was able to finalize the fact that the coupling of Newtonian and non-Newtonian models is necessary to obtain accurate results. For the sinusoidal waveform considered for the velocity, Carreau and the Power law models yield better results, eliminating the other non-Newtonian models from the list. With a better inlet condition imposed in the form of the Fourier series for pressure and velocity, the Carreau model yields the best results.

Optimal decay rates in Sobolev norms for singular values of integral operators

The regularity of integration kernels forces decay rates of singular values of associated integral operators. This is well-known for symmetric operators with kernels defined on (a,b)×(a,b), where (a,b) is an interval. Over time, many authors have studied this case in detail [9,10,4,15,12]. The case of spheres has also been resolved [3]. A few authors have examined the higher dimensional case or the case of manifolds [1,6]. Typically, these authors have provided decay estimates of singular values in lp norms, p≥1 or in case of faster decay due to regularity, lp quasi-norms, 0<p<1. With that approach, it is straightforward to show that their estimates are optimal using periodic kernels obtained from Fourier series. Our new approach for deriving decay estimates of these singular values uses Weyl's asymptotic formula for Neumann eigenvalues [21] that we combine to an appropriately defined inverse Laplacian. We obtain decay estimates in the form nα where for the n -th singular value where α depends on dimension and on the Sobolev regularity of the kernel. Since we are interested in optimal estimates in case of regular kernels, instead of writing an upper bound by a constant times nα, we use the singular values of the kernel obtained by differentiation. While in [1,6]lp estimates were proven to be optimal by simply considering periodic Fourier series, these Fourier series do not provide sharp results for our estimates. Instead, series of Neumann eigenfunctions for the Laplacian that are specific to the domain of interest are used to prove that our decay estimates are optimal. Finally, we cover the case of real analytic kernels where we are also able to derive optimal estimates.

Sums of Independent Circular Random Variables and Maximum Likelihood Circular Uniformity Tests Based on Nonnegative Trigonometric Sums Distributions

The sum of independent circular uniformly distributed random variables is also circular uniformly distributed. In this study, it is shown that a family of circular distributions based on nonnegative trigonometric sums (NNTS) is also closed under summation. Given the flexibility of NNTS circular distributions to model multimodality and skewness, these are good candidates for use as alternative models to test for circular uniformity to detect different deviations from the null hypothesis of circular uniformity. The circular uniform distribution is a member of the NNTS family, but in the NNTS parameter space, it corresponds to a point on the boundary of the parameter space, implying that the regularity conditions are not satisfied when the parameters are estimated by using the maximum likelihood method. Two NNTS tests for circular uniformity were developed by considering the standardised maximum likelihood estimator and the generalised likelihood ratio. Given the nonregularity condition, the critical values of the proposed NNTS circular uniformity tests were obtained via simulation and interpolated for any sample size by the fitting of regression models. The validity of the proposed NNTS circular uniformity tests was evaluated by generating NNTS models close to the circular uniformity null hypothesis.

A geometric modelling approach to estimate apple fruit size by means of LiDAR 3D point clouds

ABSTRACT Remote sensing in agriculture aims to search new methods to monitor fruit at the tree, thus improving the estimation of yield-related variables. Light detection and range (LiDAR) scanning was introduced to obtain geometric and radiometric information from fruit surfaces by means of 3D point clouds. A geometric model to estimate apple size by means of segmented 3D point cloud of fruit is proposed in the present study. The model consists in the approximation of 3D point clouds to a reference shape given by 2D Fourier series expansion. Each point cloud was approximated to its reference shape using an iterative error minimization routine. The geometric model was applied to laboratory and field data of spheres and apples during fruit growth, ranging from 60 to 151 days after full bloom (DAFB). An overall RMSE between measured and predicted fruit radius of 20.1, 76.8, and 119.1% was found for the geometric model, mean, and maximum Euclidean distance approaches, respectively, including all studied growth periods in field conditions. Moreover, the linear regression on measured and predicted values showed considerably improved coefficient of determination ( R 2 ) of the geometric model in comparison to Euclidean distance calculations with R 2 values of 0.76 and 0.49 for laboratory and field scanned apples, respectively. The data processing method enables fruit monitoring and their application of terrestrial LiDAR sensing in precise orchard management.

Integrability of series with respect to multiplicative systems and generalized derivatives

We give some necessary and sufficient conditions for the convergence of generalized derivatives of sums of series with respect to multiplicative systems and the corresponding Fourier series. These conditions are counterparts of trigonometric results of S. Sheng, W.O. Bray and C\v .V. Stanojevi´c and extend some results of F. M´oricz proved for Walsh–Fourier series

Image Processing in MATLAB Enhancement, Encryption and Decryption

Photography was invented in 1826 by French inventor Joseph Nicéphore Niépce. Since then, it has revolutionized the modern world. From medical evidence in court &amp; drawing images of industrial machines to personal photos for memory, not a single field is left untouched by photography. Nowadays when there is so many photo editing software, not a single one ensures true privacy and provides open-source code to ensure that no data is being collected. In six months of project making, we have developed an Image processing app that not only provides enhancement tools but also displays the power of steganography, a user-friendly environment to encrypt your enhanced images before sending them forward, and a magnificent display of Fourier series epicycles which creates an image using Fourier series, an intuitive way to learn and understand the beauty of mathematics. Mathematics could be called the language of science. Our tool proves why it is said so. And what could be a better platform to make all this other than MATLAB which not only provides numerous libraries but also is a powerful debugging tool that can display all the things happening inside the code. This app is built using the Guide function in MATLAB and has the potential to influence this generation toward data security as well as mathematics. Keyword: Enhancement, Cryptography, Epicycles, and Steganography

An improved BRDF hotspot model and its use in VLIDORT for studying the impact of atmospheric scattering on hotspot directional signatures in the atmosphere

Abstract. The term “hotspot” refers to the sharp increase in the reflectance occurring when incident (solar) and reflected (viewing) directions almost coincide in the backscatter direction. The accurate simulation of hotspot directional signatures is important for many remote sensing applications. The RossThick–LiSparse–Reciprocal (RTLSR) bidirectional reflectance distribution function (BRDF) model is widely used in radiative transfer simulations, and the hotspot model mostly used is from Maignan–Bréon, but it typically requires large values of numerical quadrature and Fourier expansion terms in order to represent the hotspot accurately for its use coupled with atmospheric radiative transfer modeling (RTM). In this paper, we have developed a modified version based on the Maignan–Bréon's hotspot BRDF model that converges much faster numerically, making it more practical for use in the RTMs that require Fourier expansion of BRDF to simulate the top-of-atmosphere (TOA) hotspot signatures, such as in the RTM models using the Doubling–Adding or discrete ordinate method. Using the vector linearized discrete ordinate radiative transfer model (VLIDORT), we found that reasonable TOA–hotspot accuracy can be obtained with just 23 Fourier terms for clear atmosphere and 63 Fourier terms for atmosphere with aerosol scattering. In order to study the impact of molecular and aerosol scattering on hotspot signatures, we carried out a number of hotspot signature simulations with VLIDORT. We confirmed that (1) atmospheric molecule scattering and the existence of aerosol tend to smooth out the hotspot signature at the TOA and that (2) the hotspot signature at the TOA in the near-infrared is larger than in the visible, and its impact by surface reflectance is more significant. As the hotspot amplitude at the TOA with aerosol scattering included is smaller than that with molecular scattering only, the amplitude of hotspot signature at the surface is likely underestimated in the previous analysis based on the POLDER measurements, where the atmospheric correction was based on a single-scatter Rayleigh-only calculation. This modified model can calculate the amplitude of the hotspot accurately, and, as it agrees very well with the original RossThick model away from the hotspot region, this model can be simply used in conditions with and without hotspots. However, there are some differences in this modified model compared to the original Maignan–Bréon model for the scattering angles close to the hotspot point; thus, it may not be appropriate for those who need an exact representation of the hotspot angular signature.