Number Of Fourier Coefficients Research Articles

Malliavin--Mancino Estimators Implemented with Nonuniform Fast Fourier Transforms

We implement and test kernel averaging Non-Uniform Fast Fourier Transform (NUFFT) methods to enhance the performance of correlation and covariance estimation on asynchronously sampled event-data using the Malliavin-Mancino Fourier estimator. The methods are benchmarked for Dirichlet and Fej\'{e}r Fourier basis kernels. We consider test cases formed from Geometric Brownian motions to replicate synchronous and asynchronous data for benchmarking purposes. We consider three standard averaging kernels to convolve the event-data for synchronisation via over-sampling for use with the Fast Fourier Transform (FFT): the Gaussian kernel, the Kaiser-Bessel kernel, and the exponential of semi-circle kernel. First, this allows us to demonstrate the performance of the estimator with different combinations of basis kernels and averaging kernels. Second, we investigate and compare the impact of the averaging scales explicit in each averaging kernel and its relationship between the time-scale averaging implicit in the Malliavin-Mancino estimator. Third, we demonstrate the relationship between time-scale averaging based on the number of Fourier coefficients used in the estimator to a theoretical model of the Epps effect. We briefly demonstrate the methods on Trade-and-Quote (TAQ) data from the Johannesburg Stock Exchange to make an initial visualisation of the correlation dynamics for various time-scales under market microstructure.

Continuation of quasi-periodic solutions with two-frequency Harmonic Balance Method

The continuation of quasi-periodic solutions for autonomous or forced nonlinear systems is presented in this paper. The association of the Asymptotic Numerical Method, a robust continuation method, and a two-frequency Harmonic Balance Method, is performed thanks to a quadratic formalism. There is no need for a priori knowledge of the solution: the two pulsations can be unknown and can vary along the solution branch, and the double Fourier series are computed without needing a harmonic selection. A norm criterion on Fourier coefficients can confirm a posteriori the accuracy of the solution branch. On a forced system, frequency-locking regions are approximated, without blocking the continuation process. The continuation of these periodic solutions can be done independently. On an autonomous system an example of solution is shown where the number of Fourier coefficients is increased to improve the accuracy of the solution.

A patch near-field acoustical holography procedure based on a generalized discrete Fourier series

Planar near-field acoustical holography (NAH) can be used to reconstruct a three-dimensional sound field from sound pressure data measured by a planar microphone array. The conventional planar NAH makes use of the discrete Fourier transform (DFT) to process the measured data, yielding a low computational cost. However, if the measurement aperture does not fully cover the sound source extension, the spatial windowing will lead to severe reconstruction errors. Many patch NAH methods have been proposed to allow measurement apertures smaller than the source size, such as the statistically optimized NAH (SONAH), which is not based on the DFT. These methods have proven to outperform the conventional NAH for small measurement apertures, but with an increased computation time and more complex implementation. This paper introduces an alternative patch procedure for planar NAH that replaces the DFT with a so-called “generalized discrete Fourier series” (GDFS). Unlike the DFT, the periods of the two-dimensional GDFS and the number of Fourier coefficients are made larger than the measurement aperture and the number of microphones, respectively. Then, the Fourier coefficients are evaluated in the least-norm sense. This reduces the spectral leakage due to the spatial windowing, improving the NAH results. As a numerical example, a simply supported plate driven by a point force is considered, and patches of the plate normal velocity are estimated from simulations of the radiated sound pressure on a small microphone array. It is shown that the GDFS-based method might lead to reconstructed velocity fields as accurate as SONAH, or even more accurate. However, unlike SONAH, the proposed method presents a low computational cost and a straightforward implementation. Therefore, it is a worthy alternative to the currently available patch procedures for planar NAH.

Программная реализация оценивания коэффициентов Фурье при ограниченных вычислительных ресурсах

Carrying out a scientific research in different areas requires huge amounts of computing power. Moreo-ver, computationally demanding tasks are assumed to have access to powerful servers and computing clusters and are expected to support handling big data sets. On the other hand, mobile devices have become increasingly powerful, that makes it possible to employ them in large-scale data processing, particularly, for volunteer computing like BOINC and Folding@home. Nevertheless, despite the advances in hardware, it is important to improve computational algorithms taking into account the following constraints which are determined by mobile devices characteristics: long response time, limited memory and battery life. The main purpose of this research is to create the algorithms to estimate Fourier coefficients according to these minimal requirements. To attain the aim, we employed connection coefficients method to work out specific relations for continuous Laguerre functions. In comparison with the corresponding recurrence re-lations, which entail an enormous computational cost, the proposed analytical relations require less computational re-sources to produce results (in particular, time and space). To support the theoretical results, we conducted a series of computational experiments using MATLAB Profiler. The findings of this research present the software implementa-tion of Fourier coefficients estimation to operate with limited computational resources and the results of the mobile application tested on a device. Based on these tests, we analysed the elapsed running time to process big data sets vary-ing the volume of data and the number of Fourier coefficients.

Fourier Analysis of Milling Force for General Helical Cutters via Spacetime Convolution, Part 1: Model Development

A space-time convolution approach to analyze the milling force for a general helical cutter is proposed. The single flute cutting force is firstly established through an integrated space-time convolution process. Subsequently, multi-flute milling forces are obtained through convolution integration in time domain (i.e., angular domain). In this convolution force model, convolution theorem does not apply directly and a modified convolution theorem is presented to find the Fourier coefficients of the total milling force for any analytically definable helical cutter. From the Fourier analysis, the magnitudes of higher order Fourier coefficients are shown to drop off quickly. Therefore, the capability to extract a small number of Fourier coefficients as characteristic values of milling forces is an important advantage of this convolution force model. Furthermore, this model provides a convenient means to calculate the Fourier coefficients of the milling forces if the profile of cutter/workpiece can not be analytically definable and only the discrete values of cutter/workpiece profile data are given from scanning. Also, the general effects of cutter geometry and cutting parameters on the spectra characteristics of the milling forces are extracted and discussed. Based on the spectra characteristics, the strategy of selecting cutters for rough and finish machining in slot milling is presented.

Modelling of random vehicle loading histories for fatigue analysis

The concise description of one dimensional vehicle loading histories for fatigue analysis using stochastic process theory is presented in this study. For fatigue analysis using stochastic process theory is presented in this study. The load history is considered to have stationary random and nonstationary mean content. The stationary variations are modelled by an Autoregressive Moving Average (ARMA) model, while a Fourier series is used to model the variation of the mean. Due to the use of random phase angles in the Fourier series an ensemble of mean variations is obtained. The methods of nonparametric statistics are used to determine the success of the modelling of nonstationarity. Justification of the method is obtained through comparison of rainflow cycle distributions and resulting fatigue lives of original and simulated loadings. Due to the relatively small number of Fourier coefficients needed together with the use of ARMA models, a concise description of complex loadings is achieved. The overall frequency content and sequential information of the load history is statistically preserved. An ensemble of load histories can be constructed on–line with minimal computer storage capacity as used in testing equipment.

Effective Parameter Dimension via Bayesian Model Selection in the Inverse Acoustic Scattering Problem

We address a prototype inverse scattering problem in the interface of applied mathematics, statistics, and scientific computing. We pose the acoustic inverse scattering problem in a Bayesian inference perspective and simulate from the posterior distribution using MCMC. The PDE forward map is implemented using high performance computing methods. We implement a standard Bayesian model selection method to estimate an effective number of Fourier coefficients that may be retrieved from noisy data within a standard formulation.

Optimal design of Fourier estimator in the presence of microstructure noise

The Fourier estimator of Malliavin and Mancino depends on both sample size and a so-called cutting frequency. The latter controls the number of Fourier coefficients to be included, and it also determines how the Fourier estimator responds to market microstructure noise. By examining the finite sample properties of the Fourier estimator, an easy-to-implement procedure is developed for the optimal cutting frequency which minimizes the mean squared error in the presence of the microstructure noise, along with a modified Whittle likelihood approach for the estimation of the signal-to-noise ratio.

Estimation of conditional moment restrictions without assuming parameter identifiability in the implied unconditional moments

A well-known difficulty in estimating conditional moment restrictions is that the parameters of interest need not be globally identified by the implied unconditional moments. In this paper, we propose an approach to constructing a continuum of unconditional moments that can ensure parameter identifiability. These unconditional moments depend on the “instruments” generated from a “generically comprehensively revealing” function, and they are further projected along the exponential Fourier series. The objective function is based on the resulting Fourier coefficients, from which an estimator can be easily computed. A novel feature of our method is that the full continuum of unconditional moments is incorporated into each Fourier coefficient. We show that, when the number of Fourier coefficients in the objective function grows at a proper rate, the proposed estimator is consistent and asymptotically normally distributed. An efficient estimator is also readily obtained via the conventional two-step GMM method. Our simulations confirm that the proposed estimator compares favorably with that of Domínguez and Lobato (2004, Econometrica) in terms of bias, standard error, and mean squared error.

Improved rigorous coupled-wave analysis for polar magnetic gratings

The rigorous coupled-wave analysis method with Airy-like Internal-Reflection Series for the polar gyrotropic grating is improved with the Fourier factorization rule. In the process of derivation, Maxwell's equation is reformulated in order to avoid type 3 product and, either Laurent's rule or the inverse rule is used for the product of two Fourier series with discontinuities. The number of Fourier coefficients or the size of the coefficient matrix required for the proper convergence is reduced by more than 50% compared to our previous works for the calculation of the diffracted Kerr rotation from the polar magnetic gratings.

Estimation of Conditional Moment Restrictions Without Assuming Parameter Identifiability in the Implied Unconditional Moments

A well known difficulty in estimating conditional moment restrictions is that the parameters of interest need not be globally identified by the implied unconditional moments. In this paper, we propose an approach to constructing a continuum of unconditional moments that can ensure parameter identifiability. These unconditional moments depend on the instruments generated from a generically comprehensively revealing function, and they are further projected along the exponential Fourier series. The objective function is based on the resulting Fourier coefficients, from which an estimator can be easily computed. A novel feature of our method is that the full continuum of unconditional moments is incorporated into each Fourier coefficient. We show that, when the number of Fourier coefficients in the objective function grows at a proper rate, the proposed estimator is consistent and asymptotically normally distributed. An efficient estimator is also readily obtained via the conventional two-step GMM method. Our simulations confirm that the proposed estimator compares favorably with that of Dominguez and Lobato (2004, Econometrica) in terms of bias, standard error, and mean squared error.

Component covariance analysis for periodically correlated random processes

The paper is dedicated to the component method for estimating the periodically correlated random processes (PCRP) mean and covariance functions, when number of harmonics is finite. This method is based on the decomposition of these time periodic functions into trigonometric polynomials and the estimation of their Fourier coefficients. Then the component estimates of the PCRP mean and covariance functions are constructed on the basis of the coefficient estimates. The properties of the PCRP mean and covariance functions component estimates are investigated, asymptotical unbiasedness and mean square consistency for these estimates, and the corresponding formulae for their biases and variances, which depend on the record length and number of Fourier coefficients, are expressed. Comparison for the component and coherent method estimates is carried out for the case of amplitude and phase modulated signals.

A global solution curve for a class of periodic problems, including the pendulum equation

Using continuation methods and bifurcation theory, we study the exact multiplicity of periodic solutions, and the global solution structure, for a class of periodically forced pendulum-like equations. Our results apply also to the first order equations. We also show that by choosing a forcing term, one can produce periodic solutions with any number of Fourier coefficients arbitrarily prescribed.

Efficient visual homing based on Fourier transformed panoramic images

We present a fast and efficient homing algorithm based on Fourier transformed panoramic images. By continuously comparing Fourier coefficients calculated from the current view with coefficients representing the goal location, a mobile robot is able to find its way back to known locations. No prior knowledge about the orientation with respect to the goal location is required, since the Fourier phase is used for a fast sub-pixel orientation estimation. We present homing runs performed by an autonomous mobile robot in an office environment. In a more comprehensive investigation the algorithm is tested on an image data base recorded by a small mobile robot in a toy house arena. Catchment areas for the proposed algorithm are calculated and compared to results of a homing scheme described in [M. Franz, B. Schölkopf, H. Mallot, H. Bülthoff, Where did I take that snapshot? Scene based homing by image matching, Biological Cybernetics 79 (1998) 191–202] and a simple homing strategy using neighbouring views. The results show that a small number of coefficients is sufficient to achieve a good homing performance. Also, a coarse-to-fine homing strategy is proposed in order to achieve both a large catchment area and a high homing accuracy: the number of Fourier coefficients used is increased during the homing run.

Inequalities on time-concentrated or frequency-concentrated functions

We obtain an inequality on a measure of the spread in time of periodic functions that are e-concentrated in frequency, i.e. all but a fixed finite number of Fourier coefficients vanish with mean-squared error up to e. We characterize an extremal function and give an asymptotic formula for the measure of spread of this extremal function as e approaches 0. We also consider the corresponding problem for functions on the real line that are e-concentrated in time or frequency. When e=0, the above reduce to inequalities on time-limited or band-limited functions and these are discussed in more detail.

The Interacting Galaxy Pair NGC 5394/95: Near-Infrared Photometry, Structure, and Morphology

We present near-infrared observations in the J, H, and K' passbands of the interacting pair of galaxies NGC 5394/95 (KPG 404). The total magnitudes, colors, surface brightnesses, and color profiles are calculated. In addition, aperture magnitudes are compared against previous determinations. We also perform a structural (disk + bulge) analysis, as well as a two-dimensional Fourier analysis, to gain insight into the morphology of the pair. The disk + bulge fit shows that NGC 5394 (KPG 404A) is more compact than normal galaxies, while NGC 5395 (KPG 404B) is less concentrated. The two-dimensional Fourier analysis shows that NGC 5394 is an H2β galaxy in the dust-penetrated (DP) classification by Block & Puerari. NGC 5395, in contrast, displays a very complex structure that needs a number of Fourier coefficients to be explained. A tightly wound m = 1 coefficient (DP class H1α) is the main structure, but other m = 1 and m = 2 coefficients (suggesting modulation) are also present in the Fourier spectra. The m = 1 coefficients represent a pseudo-ring-type structure, indicative of a collision rather than a passage. Based on our results we are able to assert that the scenario of the interaction between the galaxy members of KPG 404 should take into account a crossing of NGC 5394 through the disk of NGC 5395 in a Cartwheel-like encounter rather than a passage as in M51-type pairs. Numerical simulations could help to unravel the structural and morphological evolution of this interacting pair.

Partial averaging and the centroid virial estimator for stereographic projection path-integral simulations in curved spaces

We develop and test three different partial averaging theories for the stereographic projection path integral in curved spaces. Additionally, we adapt and test the centroid virial estimator for the kinetic energy in curved spaces. We tested both a confining as well as a nonconfining potential for the particle in a ring. All three partial averaging theories are demonstrated to converge linearly in the asymptotic region with k(-2)max, the number of Fourier coefficients. We use an error estimator to determine the optimal parameters for an extrapolation to infinite kmax. We verify that the centroid virial estimator (derived from the primitive DeWitt path-integral formula) converges to the kinetic energy for both confining and nonconfining potentials.

Towards the resolution of the Gibbs phenomena

It is well known that the expansion of an analytic nonperiodic function on a finite interval in a Fourier series leads to spurious oscillations at the interval boundaries. This result is known as the Gibbs phenomenon. The present paper introduces a new method for the resolution of the Gibbs phenomenon which follows on the reconstruction method of Gottlieb and coworkers (SIAM Rev. 39 (1997) 644) based on Gegenbauer polynomials orthogonal with respect to weight function (1− x 2) λ−1/2 . We refer to their approach as the direct method and to the new methodology as the inverse method. Both methods use the finite set of Fourier coefficients of some given function as input data in the re-expansion of the function in Gegenbauer polynomials or in other orthogonal basis sets. The finite partial sum of the new expansion provides a spectrally accurate approximation to the function. In the direct method, this requires that certain conditions are met concerning the parameter λ in the weight function, the number of Fourier coefficients, N and the number of Gegenbauer polynomials, m. We show that the new inverse method can give exact results for polynomials independent of λ and with m= N. The paper presents several numerical examples applied to a single domain or to subdomains of the main domain so as to illustrate the superiority of the inverse method in comparison with the direct method.

The Bernstein-von Mises theorem and spectral asymptotics of Bayes estimators for parabolic SPDEs

AbstractThe Bernstein-von Mises theorem, concerning the convergence of suitably normalized and centred posterior density to normal density, is proved for a certain class of linearly parametrized parabolic stochastic partial differential equations (SPDEs) as the number of Fourier coefficients in the expansion of the solution increases to infinity. As a consequence, the Bayes estimators of the drift parameter, for smooth loss functions and priors, are shown to be strongly consistent, asymptotically normal and locally asymptotically minimax (in the Hajek-Le Cam sense), and asymptotically equivalent to the maximum likelihood estimator as the number of Fourier coefficients become large. Unlike in the classical finite dimensional SDEs, here the total observation time and the intensity of noise remain fixed.

Calculating shape descriptors from Fourier analysis: shape analysis of Asterionella (Heterokontophyta, Bacillariophyceae)

In our study, shape descriptors were calculated for specimens from the diatom genus Asterionella, using the method of arc lengths and tangent angles in Fourier analysis. Asterionella species are character-poor, populations being distinguished mostly by subtle shape differences. Fourier analysis has been used in the past as an aid in taxonomy for many organisms. Fourier coefficients are least-squares estimators of best-fit planar closed curves that provide a quantitative measure of shape and can be useful shape descriptors in character-poor organisms. We determined the relation between number of x,y coordinates and the number of Fourier coefficients used in shape analysis for Asterionella. In a worked example, using a single extracted specimen outline, the method of calculating Fourier coefficients is demonstrated; coefficients were calculated for 100, 140, and 200 coordinates, in order to determine how many coordinates are necessary for effective shape analysis. Then, for each in a size range of Asterionella specimens, Fourier coefficients were calculated, using the same number of coordinates for each. Reconstructed outlines were compared graphically with the original. Statistical measures of average difference, variance, standard deviation, and coefficient of variation were calculated between x's, y's and Euclidean distance for the original and reconstructed outlines. From this, using 100 x,y coordinates, the number of Fourier coefficients necessary to give the best-fit outline over a size range from 30 to 95 μm is 22. Although we used Asterionella as an example, the method may be applied to any diatom valve outline.