Fourier Series Approximations Research Articles

A Generative adversarial network model for estimating temporal frequency variation of vehicle-bridge interaction using modified Stockwell transform

The presence of massive and directional vehicles moving on a railroad bridge causes temporal fluctuations in the fundamental frequencies of both bridge and vehicle systems, making traditional structural inspections difficult to employ effectively. Thus, this paper proposes an image-to-image Generative Adversarial Network (GAN) that estimates temporal frequency variation for the vehicle-bridge interaction system. To address the challenges associated with conventional approaches, eigenvalue analysis, numerical substitution, and Fourier series approximations are examined using a simple degree of freedom and a more complex model. Thus, the GAN model is proposed to overcome those limitations. In the framework, based on the properties of a real bridge and vehicle model, the modified Stockwell transform of bridge acceleration from the dynamic simulation is used as input data with the paired data from the numerical substitution approach. Then, the model is validated through quantitative measures and laboratory scale experiments. Results showed that the model has strong performances across the various scenarios, demonstrating the potential for bridge condition assessment under operational conditions.

Pointwise and Uniform Convergence of Fourier Extensions

Fourier series approximations of continuous but nonperiodic functions on an interval suffer the Gibbs phenomenon, which means there is a permanent oscillatory overshoot in the neighborhoods of the endpoints. Fourier extensions circumvent this issue by approximating the function using a Fourier series that is periodic on a larger interval. Previous results on the convergence of Fourier extensions have focused on the error in the L^2 norm, but in this paper we analyze pointwise and uniform convergence of Fourier extensions (formulated as the best approximation in the L^2 norm). We show that the pointwise convergence of Fourier extensions is more similar to Legendre series than classical Fourier series. In particular, unlike classical Fourier series, Fourier extensions yield pointwise convergence at the endpoints of the interval. Similar to Legendre series, pointwise convergence at the endpoints is slower by an algebraic order of a half compared to that in the interior. The proof is conducted by an analysis of the associated Lebesgue function, and Jackson- and Bernstein-type theorems for Fourier extensions. Numerical experiments are provided. We conclude the paper with open questions regarding the regularized and oversampled least squares interpolation versions of Fourier extensions.

Elastic waves in periodically heterogeneous two-dimensional media: locally periodic and anti-periodic modes

Propagation of anti-plane waves through a discrete square lattice and through a continuous fibrous medium is studied. In the long-wave limit, for periodically heterogeneous structures the solution can be periodic or anti-periodic across the unit cell. It is shown that combining periodicity and anti-periodicity conditions in different directions of the translational symmetry allows one to detect different types of modes that do not arise in the purely periodic case. Such modes may be interpreted as counterparts of non-classical waves appearing in phenomenological theories. Dispersion diagrams of the discrete square lattice are evaluated in a closed analytical from. Dispersion properties of the fibrous medium are determined using Floquet–Bloch theory and Fourier series approximations. Influence of a viscous damping is taken into account.

Limitations and Accuracy of a Continuous Reduced-Order Model for Modular Multilevel Converters

This paper analyzes the limitations of a reduced-order model for modular multilevel converters (MMCs) by elucidating the relation between its accuracy, operating frequency, and converter parameters. A reduced-order simplifies the analysis of the MMC and thereby may provide additional information about the converter behavior. However, the accuracy of such model depends on several factors. In this paper, the effect of approximating the converter as a continuous system by neglecting quantization issues associated with having a finite number of modules is studied in detail. The analysis is done based on Fourier-series approximations with which it is possible to elucidate the relationship between the resonant frequencies of the MMC and the error of the reduced-order model. With the Fourier approximation, it is also possible to characterize resonant frequencies of the converter, both numerically and analytically, in terms of the converter parameters. The results can serve as a tool to identify situations when the reduced-order model produces good and also less accurate approximations especially when a low number of modules is available.

Hip and Trunk Kinematics Estimation in Gait Through Kalman Filter Using IMU Data at the Ankle

The purpose of this paper is to provide a new method of estimating the hip acceleration and trunk posture in the sagittal plane during a walking task using an extended Kalman filter (EKF) and an unscented Kalman filter (UKF). A comparison between these two estimation techniques is also provided. Considering the periodic nature of gait, a modified biomechanical model with Fourier series approximations are utilized as a priori knowledge. Inertial measurement units (IMUs) are placed on the right side of the ankle, hip, and middle of the trunk of twenty recruited participants, as input, a posteriori data, and the ground truth for the model, separately. The results show a better performance of the EKF in estimating the hip acceleration (6.5% error) and the trunk posture (3.12% error). Moreover, both the EKF and the UKF provide low error rates for the trunk posture in comparison to the hip acceleration. This paper provides an inexpensive and novel method to estimate and filter the kinematics of motion for different body locations from a single accurate IMU attached to the ankle considering the periodic nature of gait that can be extended to other activities as well as real-time applications.

A Conceptual Numerical Model of the Wave Equation Using the Complex Variable Boundary Element Method

In this work, a conceptual numerical solution of the two-dimensional wave partial differential equation (PDE) is developed by coupling the Complex Variable Boundary Element Method (CVBEM) and a generalized Fourier series. The technique described in this work is suitable for modeling initial-boundary value problems governed by the wave equation on a rectangular domain with Dirichlet boundary conditions and an initial condition that is equal on the boundary to the boundary conditions. The new numerical scheme is based on the standard approach of decomposing the global initial-boundary value problem into a steady-state component and a time-dependent component. The steady-state component is governed by the Laplace PDE and is modeled with the CVBEM. The time-dependent component is governed by the wave PDE and is modeled using a generalized Fourier series. The approximate global solution is the sum of the CVBEM and generalized Fourier series approximations. The boundary conditions of the steady-state component are specified as the boundary conditions from the global BVP. The boundary conditions of the time-dependent component are specified to be identically zero. The initial condition of the time-dependent component is calculated as the difference between the global initial condition and the CVBEM approximation of the steady-state solution. Additionally, the generalized Fourier series approximation of the time-dependent component is fitted so as to approximately satisfy the derivative of the initial condition. It is shown that the strong formulation of the wave PDE is satisfied by the superposed approximate solutions of the time-dependent and steady-state components.

Spectrally Accurate Causality Enforcement Using SVD-Based Fourier Continuations for High-Speed Digital Interconnects

We introduce an accurate and robust technique for accessing the causality of network transfer functions given in the form of band-limited discrete frequency responses. These transfer functions are commonly used to represent the electrical response of high-speed digital interconnects used on chip and in electronic package assemblies. In some cases, small errors in the model development lead to noncausal behavior that does not accurately represent the electrical response and may lead to a lack of convergence in simulations that utilize these models. The approach is based on Hilbert transform relations or Kramers–Kronig dispersion relations and a construction of causal Fourier continuations using a regularized singular value decomposition method. Given a transfer function, nonperiodic in general, this procedure constructs highly accurate Fourier series approximations on the given frequency interval by allowing the function to be periodic in an extended domain. The causality dispersion relations are enforced spectrally and exactly. This eliminates the necessity of approximating the transfer function behavior at infinity and explicit computation of the Hilbert transform. We perform the error analysis of the method and take into account a possible presence of a noise or approximation errors in data. The developed error estimates can be used in verifying the causality of the given data. The performance of the method is tested on several analytic and simulated examples that demonstrate an excellent accuracy and reliability of the proposed technique in agreement with the obtained error estimates. The method is capable of detecting very small localized causality violations with amplitudes close to the machine precision.

Strong ionization asymmetry in a geometrically symmetric radio frequency capacitively coupled plasma induced by sawtooth voltage waveforms.

The ionization dynamics in geometrically symmetric parallel plate capacitively coupled plasmas driven by radio frequency tailored voltage waveforms is investigated using phase resolved optical emission spectroscopy (PROES) and particle-in-cell (PIC) simulations. Temporally asymmetric waveforms induce spatial asymmetries and offer control of the spatiotemporal dynamics of electron heating and associated ionization structures. Sawtooth waveforms with different rise and fall rates are employed using truncated Fourier series approximations of an ideal sawtooth. Experimental PROES results obtained in argon plasmas are compared with PIC simulations, showing excellent agreement. With waveforms comprising a fast voltage drop followed by a slower rise, the faster sheath expansion in front of the powered electrode causes strongly enhanced ionization in this region. The complementary waveform causes an analogous effect in front of the grounded electrode.

Fourier Series Approximations toJ2-Bounded Equatorial Orbits

The current paper offers a comprehensive dynamical analysis and Fourier series approximations ofJ2-bounded equatorial orbits. The initial conditions of heterogeneous families ofJ2-perturbed equatorial orbits are determined first. Then the characteristics of two types ofJ2-bounded orbits, namely, pseudo-elliptic orbit and critical circular orbit, are studied. Due to the ambiguity of the closed-form solutions which comprise the elliptic integrals and Jacobian elliptic functions, showing little physical insight into the problem, a new scheme, termed Fourier series expansion, is adopted for approximation herein. Based on least-squares fitting to the coefficients, the solutions are expressed with arbitrary high-order Fourier series, since the radius and the flight time vary periodically as a function of the polar angle. As a consequence, the solutions can be written in terms of elementary functions such as cosines, rather than complex mathematical functions. Simulations enhance the proposed approximation method, showing bounded and negligible deviations. The approximation results show a promising prospect in preliminary orbits design, determination, and transfers for low-altitude spacecrafts.

On the Efficacy of Fourier Series Approximations for Pricing European Options

This paper investigates several competing procedures for computing the prices of vanilla European options, such as puts, calls and binaries, in which the underlying model has a characteristic function that is known in semi-closed form. The algorithms investigated here are the half-range Fourier cosine series, the half-range Fourier sine series and the full-range Fourier series. Their performance is assessed in simulation experiments in which an analytical solution is available and also for a simple affine model of stochastic volatility in which there is no closed-form solution. The results suggest that the half-range sine series approximation is the least effective of the three proposed algorithms. It is rather more difficult to distinguish between the performance of the half-range cosine series and the full-range Fourier series. However there are two clear differences. First, when the interval over which the density is approximated is relatively large, the full-range Fourier series is at least as good as the half-range Fourier cosine series, and outperforms the latter in pricing out-of-the-money call options, in particular with maturities of three months or less. Second, the computational time required by the half-range Fourier cosine series is uniformly longer than that required by the full-range Fourier series for an interval of fixed length. Taken together, these two conclusions make a case for pricing options using a full-range range Fourier series as opposed to a half-range Fourier cosine series if a large number of options are to be priced in as short a time as possible.

Clustering of time-course gene expression profiles using normal mixture models with autoregressive random effects

BackgroundTime-course gene expression data such as yeast cell cycle data may be periodically expressed. To cluster such data, currently used Fourier series approximations of periodic gene expressions have been found not to be sufficiently adequate to model the complexity of the time-course data, partly due to their ignoring the dependence between the expression measurements over time and the correlation among gene expression profiles. We further investigate the advantages and limitations of available models in the literature and propose a new mixture model with autoregressive random effects of the first order for the clustering of time-course gene-expression profiles. Some simulations and real examples are given to demonstrate the usefulness of the proposed models.ResultsWe illustrate the applicability of our new model using synthetic and real time-course datasets. We show that our model outperforms existing models to provide more reliable and robust clustering of time-course data. Our model provides superior results when genetic profiles are correlated. It also gives comparable results when the correlation between the gene profiles is weak. In the applications to real time-course data, relevant clusters of coregulated genes are obtained, which are supported by gene-function annotation databases.ConclusionsOur new model under our extension of the EMMIX-WIRE procedure is more reliable and robust for clustering time-course data because it adopts a random effects model that allows for the correlation among observations at different time points. It postulates gene-specific random effects with an autocorrelation variance structure that models coregulation within the clusters. The developed R package is flexible in its specification of the random effects through user-input parameters that enables improved modelling and consequent clustering of time-course data.

Sobolev smoothing of SVD-based Fourier continuations

A method for calculating Sobolev smoothed Fourier continuations is presented. The method is based on the recently introduced singular value decomposition based Fourier continuation approach. This approach allows for highly accurate Fourier series approximations of non-periodic functions. These super-algebraically convergent approximations can be highly oscillatory in an extended region, contaminating the Fourier coefficients. It is shown that through solving a subsequent least squares problem, a Fourier continuation can be produced which has been dramatically smoothed in that the Fourier coefficients exhibit a prescribed rate of decay as the wave number increases. While the smoothing procedure has no significant negative effect on the accuracy of the Fourier series approximation, in some situations the smoothed continuations can actually yield increased accuracy in the approximation of the function and its derivatives.

Optimally Sparse Representations of 3D Data with $C^2$ Surface Singularities Using Parseval Frames of Shearlets

This paper introduces a Parseval frame of shearlets for the representation of three-dimensional (3D) data, which is especially designed to handle geometric features such as discontinuous boundaries with very high efficiency. This system of 3D shearlets forms a multiscale pyramid of well-localized waveforms at various locations and orientations, which become increasingly thin and elongated at fine scales. We prove that this 3D shearlet construction provides essentially optimal sparse representations for functions on $\mathbb{R}^3$ which are $C^2$-regular away from discontinuities along $C^2$ surfaces. As a consequence, we show that, within this class of functions, the N-term approximation $f_N^S$ obtained by selecting the N largest coefficients of the shearlet expansion of f satisfies the asymptotic estimate $\|f-f_N^S\|_2^2 \asymp N^{-1} (\log N)^2$ as $N \to \infty$. This asymptotic behavior significantly outperforms wavelet and Fourier series approximations, which yield an approximation rate of only $O(...

A Fast Algorithm for Fourier Continuation

A new algorithm is presented which provides a fast method for the computation of recently developed Fourier continuations (a particular type of Fourier extension method) that yield superalgebraically convergent Fourier series approximations of nonperiodic functions. Previously, the coefficients of an approximating Fourier series have been obtained by means of a regularized singular value decomposition (SVD)-based least-squares solution to an overdetermined linear system of equations. These SVD methods are effective when the size of the system does not become too large, but they quickly become unwieldy as the number of unknowns in the system grows. We demonstrate a novel decoupling of the least-squares problem which results in two systems of equations, one of which may be solved quickly by means of fast Fourier transforms (FFTs) and another that is demonstrated to be well approximated by a low-rank system. Utilizing randomized algorithms, the low-rank system is reduced to a significantly smaller system of equations. This new system is then efficiently solved with drastically reduced computational cost and memory requirements while still benefiting from the advantages of using a regularized SVD. The computational cost of the new algorithm in on the order of the cost of a single FFT multiplied by a slowly increasing factor that grows only logarithmically with the size of the system.

Spectral properties for the vibrational response of parametrically excited system

The response spectra are helpful to understand the vibrational characteristics of linear system, and further could be used as important indications for the failure diagnosis and state detecting of the system. Due to the parametric excitation, the vibrational spectra of linear periodic system differ distinctly from that of the linear time-invariant system. Thus, utilizing the Sylvester’s theorem and Fourier series approximations, commonly used in the matrix decomposition, the spectral components for both the free and forced responses of linear parametrically excited system are obtained theoretically in the first part of the paper. The effects of system stability and damping on the response spectra are discussed in detail. Moreover, the external resonant condition for periodic system is found in the analysis. In the following part of the paper, spectral simulations for a spur gear pair under idle load (modeled as the free response) and fluctuating load (modeled as forced response) are done respectively to validate the theoretical results and illustrate the special frequency distributing features of periodic system intuitively.

A Computational Approach to the Functional Clustering of Periodic Gene-Expression Profiles

DNA microarray analysis has emerged as a leading technology to enhance our understanding of gene regulation and function in cellular mechanism controls on a genomic scale. This technology has advanced to unravel the genetic machinery of biological rhythms by collecting massive gene-expression data in a time course. Here, we present a statistical model for clustering periodic patterns of gene expression in terms of different transcriptional profiles. The model incorporates biologically meaningful Fourier series approximations of gene periodic expression into a mixture-model-based likelihood function, thus producing results that are likely to be closer to biological relevance, as compared to those from existing models. Also because the structures of the time-dependent means and covariance matrix are modeled, the new approach displays increased statistical power and precision of parameter estimation. The approach was used to reanalyze a real example with 800 periodically expressed transcriptional genes in yeast, leading to the identification of 13 distinct patterns of gene-expression cycles. The model proposed can be useful for characterizing the complex biological effects of gene expression and generate testable hypotheses about the workings of developmental systems in a more precise quantitative way.

SIX-PORT MODEL FOR SOUND PROPAGATION IN A POROUS MEDIA WITH APPLICATION TO DIESEL PARTICULATE FILTERS

ABSTRACTThis work presents a 2-D field model for the study of sound propagation in a dieselparticulate filter (DPF) unit. The 2-D model is formulated using the field Navier-stocks,energy, and continuity equations and retains the normal as well as transversecomponent of gas velocity. Temperature, pressure, density, and velocities are taken tobe as plane and time harmonic variations. By substituting the differentials of thesequantities with respect to both plane and time in field equations, a set of three coupledlinear 2-D field variation equations for pressure, axial and transverse velocities isobtained. The obtained model is solved analytically using Fourier seriesapproximations. The approximate solution is used to build a 2D acoustic model for theexhaust gases emission, with the existence of the diesel particulate filter, whichaccounts for both attenuation and phase shift defining the propagation wave constant.In addition the obtained approximate solution is used to determine the acousticsimpedance of the DPF unit, comparing between different types of DPF based on soundtransmission losses performance, soot loading, noise and vibration damping, in additionto calculating the noise reduction factor (NRF). In the present study, unlike previousones, six, rather than four, roots for wave propagation constant are obtainedcorresponding to the obtained six port acoustic DPF model. The results obtained usingthe present six -port model, for selected system parameters are graphically displayedand compared with those available in the open literature using four- port models. Thepresent model results show, in general, similar qualitative behavior and a significantquantitative improvement of the available results in the open literature obtained using afour port model.

Fourier series for zeta function via Sinc

In this paper we derive some Fourier series and Fourier polynomial approximations to a function F which has the same zeros as the zeta function, ζ(z) on the strip {z∈C:0<Rz<1}. These approximations depend on an arbritrary positive parameter h, and which for arbitrary ε∈(0,1/2), converge uniformly to ζ(z) on the rectangle {z∈C:ε<Rz<1-ε,-π/h<Iz<π/h}.

Fourier series of half-range functions by smooth extension

This paper considers Fourier series approximations of one- and two-dimensional functions over the half-range, that is, over the sub-interval [0, L ] of the interval [− L, L ] in one-dimensional problems and over the sub-domain [0, L x ] × [0, L y ] of the domain [− L x , L x ] × [− L y , L y ] in two-dimensional problems. It is shown how to represent these functions using a Fourier series that employs a smooth extension. The purpose of the smooth extension is to improve the convergence characteristics otherwise obtained using the even and odd extensions. Significantly improved convergence characteristics are illustrated in one-dimensional and two-dimensional problems.

Accurate, high-order representation of complex three-dimensional surfaces via Fourier continuation analysis

We present a new method for construction of high-order parametrizations of surfaces: starting from point clouds, the method we propose can be used to produce full surface parametrizations (by sets of local charts, each one representing a large surface patch – which, typically, contains thousands of the points in the original point-cloud) for complex surfaces of scientific and engineering relevance. The proposed approach accurately renders both smooth and non-smooth portions of a surface: it yields super-algebraically convergent Fourier series approximations to a given surface up to and including all points of geometric singularity, such as corners, edges, conical points, etc. In view of their C ∞ smoothness (except at true geometric singularities) and their properties of high-order approximation, the surfaces produced by this method are suitable for use in conjunction with high-order numerical methods for boundary value problems in domains with complex boundaries, including PDE solvers, integral equation solvers, etc. Our approach is based on a very simple concept: use of Fourier analysis to continue smooth portions of a piecewise smooth function into new functions which, defined on larger domains, are both smooth and periodic. The “continuation functions” arising from a function f converge super-algebraically to f in its domain of definition as discretizations are refined. We demonstrate the capabilities of the proposed approach for a number of surfaces of engineering relevance.