Fourier Basis Functions Research Articles

Nonstationary Signal Denoising Using an Envelope-Tracking Filter

Signal denoising is crucial to condition monitoring of mechanical systems as fault characteristics may be hidden by strong noise in some cases. In this paper, we focus on the denoising of nonstationary vibration signals which contain multiple complex components. To this end, we introduce a time-frequency filter called envelope-tracking filter (ETF). The basic idea of the ETF is to characterize the envelope of a signal using Fourier basis functions and estimate the envelope by solving a least-squares problem. We show that the ETF is similar to the Vold–Kalman filter, but it has several promising advantages, i.e., easier to determine the bandwidth expression, more flexible to handle signals at different noise levels, and less sensitive to errors of instantaneous frequency (IF) estimations. Our denoising method includes two steps: 1) extract IFs of each nonstationary component by optimizing a spectrum concentration index, and 2) recover each component by using the ETF. Our method works very well in heavy noise and is effective for signals with overlapped components. Some examples including simulated and real-life vibration signals are offered to show the validity of the method.

Analysis of a Wireless Power Transfer System by the Impedance Expansion Method Using Fourier Basis Functions

Analysis of a Wireless Power Transfer System by the Impedance Expansion Method Using Fourier Basis Functions

Receding Horizon Pseudospectral Control for Energy Maximization With Application to Wave Energy Devices

This paper addresses the issue of real-time control for applications, subject to physical constraints, involving an energy maximization objective. Typical application areas include renewable energy systems where, in spite of the fact that the raw energy resource is free, the capital and operational costs associated with the energy conversion process are not. In addition, economic energy delivery can only be achieved if the conversion device is operated efficiently. Previous approaches to this problem include model predictive control (MPC), but the computational cost associated with MPC can be high. Pseudospectral solutions show considerable promise in achieving a good balance between performance and computation, but currently available solutions deal with fixed-period optimization. In this paper, a receding horizon real-time pseudospectral control is developed, based on half-range Chebyshev Fourier basis functions, which can accurately represent harmonic signals in the application domain, while also efficiently dealing with the signal truncation effects associated with a receding horizon formulation. An application example, based on a wave energy converter, is used to illustrate the new control algorithm.

Exploring functional data analysis and wavelet principal component analysis on ecstasy (MDMA) wastewater data.

BackgroundWastewater-based epidemiology (WBE) is a novel approach in drug use epidemiology which aims to monitor the extent of use of various drugs in a community. In this study, we investigate functional principal component analysis (FPCA) as a tool for analysing WBE data and compare it to traditional principal component analysis (PCA) and to wavelet principal component analysis (WPCA) which is more flexible temporally.MethodsWe analysed temporal wastewater data from 42 European cities collected daily over one week in March 2013. The main temporal features of ecstasy (MDMA) were extracted using FPCA using both Fourier and B-spline basis functions with three different smoothing parameters, along with PCA and WPCA with different mother wavelets and shrinkage rules. The stability of FPCA was explored through bootstrapping and analysis of sensitivity to missing data.ResultsThe first three principal components (PCs), functional principal components (FPCs) and wavelet principal components (WPCs) explained 87.5-99.6 % of the temporal variation between cities, depending on the choice of basis and smoothing. The extracted temporal features from PCA, FPCA and WPCA were consistent. FPCA using Fourier basis and common-optimal smoothing was the most stable and least sensitive to missing data.ConclusionFPCA is a flexible and analytically tractable method for analysing temporal changes in wastewater data, and is robust to missing data. WPCA did not reveal any rapid temporal changes in the data not captured by FPCA. Overall the results suggest FPCA with Fourier basis functions and common-optimal smoothing parameter as the most accurate approach when analysing WBE data.Electronic supplementary materialThe online version of this article (doi:10.1186/s12874-016-0179-2) contains supplementary material, which is available to authorized users.

Closed fringe demodulation using phase decomposition by Fourier basis functions.

We report a new technique for the demodulation of a closed fringe pattern by representing the phase as a weighted linear combination of a certain number of linearly independent Fourier basis functions in a given row/column at a time. A state space model is developed with the weights of the basis functions as the elements of the state vector. The iterative extended Kalman filter is effectively utilized for the robust estimation of the weights. A coarse estimate of the fringe density based on the fringe frequency map is used to determine the initial row/column to start with and subsequently the optimal number of basis functions. The performance of the proposed method is evaluated with several noisy fringe patterns. Experimental results are also reported to support the practical applicability of the proposed method.

A Legendre–Fourier spectral method with exact conservation laws for the Vlasov–Poisson system

We present the design and implementation of an L2-stable spectral method for the discretization of the Vlasov- Poisson model of a collisionless plasma in one space and velocity dimension. The velocity and space dependence of the Vlasov equation are resolved through a truncated spectral expansion based on Legendre and Fourier basis functions, respectively. The Poisson equation, which is coupled to the Vlasov equation, is also resolved through a Fourier expansion. The resulting system of ordinary differential equation is discretized by the implicit second-order accurate Crank-Nicolson time discretization. The non-linear dependence between the Vlasov and Poisson equations is iteratively solved at any time cycle by a Jacobian-Free Newton-Krylov method. In this work we analyze the structure of the main conservation laws of the resulting Legendre-Fourier model, e.g., mass, momentum, and energy, and prove that they are exactly satisfied in the semi-discrete and discrete setting. The L2-stability of the method is ensured by discretizing the boundary conditions of the distribution function at the boundaries of the velocity domain by a suitable penalty term. The impact of the penalty term on the conservation properties is investigated theoretically and numerically. An implementation of the penalty term that does not affect the conservation of mass, momentum and energy, is also proposed and studied. A collisional term is introduced in the discrete model to control the filamentation effect, but does not affect the conservation properties of the system. Numerical results on a set of standard test problems illustrate the performance of the method.

Simultaneous estimation of phase and its pth order derivatives

One unaddressed challenge in optical metrology has been the measurement of higher order derivatives of rough specimens subjected to loading. In this paper, we investigate an approach that allows for the simultaneous estimation of the phase and its higher order derivatives from a noisy interference field. The interference phase is represented as a weighted linear combination of linearly independent Fourier basis functions. The interference field is represented as a state space model with the weights of the basis functions as the elements of the state vector. These weights are accurately estimated by employing the extended Kalman filter. The interference phase and phase derivatives are subsequently computed using the estimated weights. Since the Fourier basis functions are infinitely differentiable, phase derivatives of any arbitrary order can be estimated. Simulation and experimental results are provided to substantiate the effectiveness of the proposed method in the presence of high noise.

Hermiticity of Hamiltonian Matrix using the Fourier Basis Sets in Bond-Bond-Angle and Radau Coordinates

In quantum calculations a transformed Hamiltonian is often used to avoid singularities in a certain basis set or to reduce computation time. We demonstrate for the Fourier basis set that the Hamiltonian can not be arbitrarily transformed. Otherwise, the Hamiltonian matrix becomes non-hermitian, which may lead to numerical problems. Methods for correctly constructing the Hamiltonian operators are discussed. Specific examples involving the Fourier basis functions for a triatomic molecular Hamiltonian (J=0) in bond-bond angle and Radau coordinates are presented. For illustration, absorption spectra are calculated for the OClO molecule using the time-dependent wavepacket method. Numerical results indicate that the non-hermiticity of the Hamiltonian matrix may also result from integration errors. The conclusion drawn here is generally useful for quantum calculation using basis expansion method using quadrature scheme.

Efficient algorithms for constraining orientation tensors in Galerkin methods for the Fokker–Planck equation

This paper deals with the problem of positivity preservation in numerical algorithms for simulating fiber suspension flows. In contrast to fiber orientation models based on the Advani–Tucker evolution equations for even-order orientation tensors, the probability distribution function of fiber orientation is approximated using the Galerkin discretization of the Fokker–Planck equation with Fourier basis functions or spherical harmonics. This procedure leads to a natural generalization of orientation tensor models replacing ad hoc closure approximations by Galerkin equations for the fine-scale components. After introducing an operator splitting approach to solving the discretized Fokker–Planck equation, we present conditions and correction techniques that guarantee physically correct distribution functions. As the reader will see, the derivation of these conditions is independent of the space dimension and their applicability is not limited to the simulation of fiber suspensions.

Design and implementation of a robust FNN-based adaptive sliding-mode controller for pneumatic actuator systems

An adaptive Fourier neural network sliding mode controller with H∞ tracking performance (AFNN-SMC+ H∞) is applied for a Pneumatic actuator system (PAS) to overcome time-varying nonlinear dynamics and external disturbances. Benefiting from the use of orthogonal Fourier basis function, the proposed AFNN has fast estimated convergence speed; also, because AFNN has unique solution, it can avoid falling into the local minimum. The architecture of AFNN can also easily be determined by its clear physical meaning of the neurons. To attenuate the vibration of proportional directional control valve and the adaptive approximation error, the H∞ tracking design technique is incorporated into the proposed AFNN-SMC. Finally, practical experiments are successfully implemented in position regulation, trajectory tracking, and velocity control of the PAS, which illustrates the effectiveness of the proposed controller.

Iterative learning control for flexible manipulator using fourier basis function

Perfect tracking of the tip position of a flexible-link manipulator (FLM) is unable to be achieved by causal control because it is a typical non-minimum phase system. Combined with non-causal stable inversion, an adaptive iterative learning control scheme based on Fourier basis function is presented for the tip trajectory tracking of FLM performing repetitive tasks. In this method, an iterative identification algorithm is used to construct the Fourier basis function space model of the manipulator, and a pseudoinverse type iterative learning law is designed to approximate the stable inversion of the non-minimum phase system, which guarantees the convergence and robustness of the control system. Simulation results show the performance and effectiveness of the proposed scheme.

Multi-dimensional, fully-implicit, spectral method for the Vlasov–Maxwell equations with exact conservation laws in discrete form

A spectral method for the numerical solution of the multi-dimensional Vlasov–Maxwell equations is presented. The plasma distribution function is expanded in Fourier (for the spatial part) and Hermite (for the velocity part) basis functions, leading to a truncated system of ordinary differential equations for the expansion coefficients (moments) that is discretized with an implicit, second order accurate Crank–Nicolson time discretization. The discrete non-linear system is solved with a preconditioned Jacobian–Free Newton–Krylov method.It is shown analytically that the Fourier–Hermite method features exact conservation laws for total mass, momentum and energy in discrete form. Standard tests involving plasma waves and the whistler instability confirm the validity of the conservation laws numerically. The whistler instability test also shows that we can step over the fastest time scale in the system without incurring in numerical instabilities. Some preconditioning strategies are presented, showing that the number of linear iterations of the Krylov solver can be drastically reduced and a significant gain in performance can be obtained.

Fourier based methodology for simulating 2D-random shapes in heterogeneous materials

Gaining insights into the effects of microstructural details on materials behavior may be achieved by incorporating their attributes into numerical modeling. This requires us to make considerable efforts to feature heterogeneity morphology distributions and their spatial arrangement. This paper focuses on modeling the scatter observed in materials heterogeneity geometry. The proposed strategy is based on the development of a 1D-shape signature function representing the 2D-section of a given shape, on Fourier basis functions. The Fourier coefficients are then considered as random variables. This methodology has been applied to flax fibers which are gradually introduced into composite materials as a potential alternative to synthetic reinforcements. In this contribution, the influence of some underlying assumptions regarding the choice of one 1D-shape signature function, its discretization scheme and truncation level, and the best way of modeling the associated random variables is also investigated. Some configurations coming from the combination of these tuning parameters are found to be sufficiently relevant to render efficiently the morphometric factors of the observed fibers statistically speaking.

An Efficient Transceiver Scheme for SC-FDMA Systems Based on Discrete Wavelet Transform and Discrete Cosine Transform

In this paper, the discrete cosine transform (DCT) is utilized for single carrier frequency division multiple access (SC-FDMA) transmission. Firstly, an improved DCT-based SC-FDMA (DCT SC-FDMA) system is introduced. Then, the paper presents a new transceiver architecture design for the uplink SC-FDMA system that implements a cosine basis function or a complex exponential Fourier basis function. The proposed architecture uses the discrete wavelet transform and a hybrid companding and clipping method for peak-to-average power ratio (PAPR) reduction. From the simulation results, it is shown that the DCT SC-FDMA system achieves a superior bit error rate (BER) performance than that of the discrete Fourier transform based SC-FDMA (DFT SC-FDMA) system. The results also show that the proposed wavelet based transceiver architectures for the DFT SC-FDMA system and the DCT SC-FDMA system can provide a better BER performance and a lower PAPR than the conventional DFT SC-FDMA system.

Learning to Walk Fast: Optimized Hip Height Movement for Simulated and Real Humanoid Robots

The linear inverted pendulum model has been used predominantly to generate balanced humanoid walking. This model assumes that the hip height is fixed during the walk. In this paper, generating a fast walk is studied with the main focus on the effect of hip height movement. Our approach is based on modeling the hip height movement and learning its parameters in order to generate a fast walk. The hip height trajectory is generated using Fourier basis functions. The generated trajectory is the input to programmable Central Pattern Generators (CPGs) in order to modulate generated trajectories smoothly. The inverted pendulum model is utilized to model a balanced walking. A numerical approach is presented to control inverted pendulum dynamics. Covariance Matrix Adaptation Evolution Strategy (CMA-ES) is employed to search for appropriate hip height trajectory and walking parameters that optimize walking speed. This approach has been tested not only to obtain fast forward walk but also a fast side walk. Experiments are conducted on both simulated and real NAO robots. The results show that the change from the learned forward walk to learned side walk is performed stably, which confirm the important role of using CPGs. The comparison of the results of the proposed gait model (and development approach) with those obtained using fixed hip height also shows that fixed height walking is slower than variable height walking.

Quaternion-based polarimetric array manifold interpolation

In order to do direction of arrival (DoA) estimation the array manifold is needed. However, in practical scenarios the available calibration data is sampled on a finite grid only. Hence, a suitable beam pattern interpolation scheme is essential to perform high precision DoA estimation. A reasonable interpolation algorithm is present via the Effective Aperture Distribution Function (EADF). The EADF basically relies on a two-dimensional Discrete Fourier Transform of each beam pattern.Since the EADF does not inherently incorporate polarization two separate EADFs are needed to describe a single array element. Hence, we propose to use a Quaternion EADF (QEADF) that jointly interpolates both patterns. Moreover, since the QEADF is based on a Discrete Quaternion Fourier Transform it opens up some degrees of freedom with respect to the Fourier basis functions.

A Spectrum Analysis Method Based on Neural Network

This paper presents a spectrum analysis method using recursive least square algorithm to train the weights of Fourier Basis Functions (FBF) neural network, according to the weight to obtain the signal amplitude spectrum and phase spectrum. The method does not involve complex multiplication and addition operations, convenient for software and hardware, especially suitable for DSP software and hardware implementation. The simulation results show that, this method is not only high precision, fast calculation speed, but also has the noise filtering function, is a kind of effective method for spectrum analysis.

A Fast Fourier--Galerkin Method Solving a Boundary Integral Equation for the Biharmonic Equation

We develop a fast Fourier--Galerkin method for solving a boundary integral equation which is a reformulation of the Dirichlet problem of the biharmonic equation. The proposed method is based on a splitting of the resulting boundary integral operator. That is, we write the operator as a sum of two integral operators, one having the Fourier basis functions as eigenfunctions and the other whose representation matrix in the Fourier basis can be compressed to a sparse matrix having only $\mathcal{O}(n\log n)$ number of nonzero entries, where $n$ is the order of the Fourier basis functions used in the method. We then project the solution of the boundary integral equation onto the space spanned by the Fourier basis functions. This leads to a system of linear equations. A fast solver for the system is based on a compression of its coefficient matrix. We show that the method has the optimal convergence order $\mathcal{O}(n^{-q})$, where $q$ denotes the degree of regularity of the exact solution of the boundary integral equation and requires computing only $\mathcal{O}(n \log n)$ number of entries of the coefficient matrix. Numerical examples are presented to confirm the theoretical results for the approximation accuracy and computational complexity of the proposed method.

A Novel Algorithm of Network Trade Customer Classification Based on Fourier Basis Functions

Learning algorithm of neural network is always an important research contents in neural network theory research and application field, learning algorithm about the feed-forward neural network has no satisfactory solution in particular for its defects in calculation speed. The paper presents a new Fourier basis functions neural network algorithm and applied it to classify network trade customer. First, 21 customer classification indicators are designed, based on characteristics and behaviors analysis of network trade customer, including customer characteristics type variables and customer behaviors type variables,; Second, Fourier basis functions is used to improve the calculation flow and algorithm structure of original BP neural network algorithm to speed up its convergence and then a new Fourier basis neural network model is constructed. Finally the experimental results show that the problem of convergence speed can been solved, and the accuracy of the customer classification are ensured when the new algorithm is used in network trade customer classification practically. DOI: http://dx.doi.org/10.11591/telkomnika.v11i11.2978 Full Text: PDF

Parity symmetry with respect to both [formula omitted] and [formula omitted] requires periodicity with period [formula omitted]: Connections between computer graphics, group theory and spectral methods for solving partial differential equations

A function is symmetric with respect to a point x=L if f(x+L)=f(-x+L) for all x and similarly is antisymmetric if f(x+L)=-f(-x+L). A function which is either symmetric or antisymmetric is said to be of “definite parity” with respect to L. The sines and cosines of a Fourier series have definite parity with respect to two points; all cosines are symmetric with respect to the origin while all sines are antisymmetric with respect to x=0;cos(2nx) and sin([2n+1]x) for integral n are also symmetric with respect to x=π/2 while all other Fourier basis functions are antisymmetric with respect to the same point. Such symmetries can be exploited in numerical calculations; for example, computing the angular Mathieu functions using N basis functions can be split into four uncoupled eigenproblems each of dimension N/4. It is natural to ask: Are there other classes of functions with similar symmetries? Using concepts from computer graphics, we prove that all functions which are symmetric with respect to two points separated by a distance L must be spatially periodic with period 4L. We also prove that the only function which is of definite parity with respect to three distinct points must be a constant. These theorems define parity in the usual sense of a global property such that even parity with respect to the origin means f(x)=f(-x) for all x∈[-∞,∞]. We construct counterexamples to both theorems that are functions with local parity, that is, symmetry which applies only for a finite interval in x.