A robust method for accurately representing nonperiodic functions given Fourier coefficient information

When a function is smooth but not smoothly periodic with a particular period, and nonetheless is represented by partial sums of a Fourier series calculated using that period, the well-known Gibbs phenomenon defeats uniform convergence of the sums, and convergence is slow. In recent years, several workers have developed methods for recovering accurate and fast converging representations for functions in this situation. These efforts have not concentrated on bounds for the operators corresponding to the methods, and thus have not explicitly proven robustness in the presence of noise. In this paper we present a method for which explicit bounds are established for the operator. The method is, in effect, least-squares fitting of the given Fourier coefficients by the coefficients of polynomial splines with appropriate discontinuities. We obtain bounds by exact calculations of projections in spline spaces, using a computer algebra system. We give examples of the method and two other published methods working with noisy data.     

Local approximation on surfaces with discontinuities,given limited order Fourier coefficients

We present a spline approximation method for a piece of a surface where jump discontinuities occur along curves. The data for the surface is assumed to be Fourier coefficients which are limited in order and possibly contaminated with noise. The support of the approximation is bounded by three sides of a rectangle with a fourth boundary possibly curved. Discontinuities of the surface may occur across the curved side and linear sides adjacent to it. The approximation uses a small number of lines through the support and parallel to the straight boundary lines that are adjacent to the curve. Along each line a one-dimensional spline approximation is done for a section of the surface over the line. This approximation uses two-dimensional Fourier coefficient data, localizing spline functions, and a technique which we developed earlier for one-dimensional analogues of the problem. We use a spline quasi-interpolation scheme to create a surface approximation from the section approximations. The result is accurate even when the surface is discontinuous across the curved boundary and adjacent side boundaries.     

Iterative methods based on spline approximations to detect discontinuities from Fourier data

Recently, spline approximations have been proposed for the reconstruction of piecewise smooth functions from Fourier data. That approach makes possible to retrieve the functions from their Fourier coefficients for any given degree of accuracy when the discontinuity points are known. In this paper we present iterative methods based on those spline approximations, for several degrees, to find locations and amplitudes of the jumps of a piecewise smooth function, given its Fourier coefficients. We also present numerical experiments comparing with different previous approaches.     

Vertical and oblique fault detection in explicit surfaces

The purpose of this paper is to study the problem of detection of vertical and oblique faults in explicit surfaces. First, we characterize the finite jump discontinuities of a univariate function in terms of the divergence of sequences related to the slopes of least-squares polynomial approximations of the function. Then, we propose an algorithm to locate the finite jump discontinuities of a univariate function and its first derivative from a finite set of scattered data values of the function. As a consequence, we derive a method to detect vertical and oblique faults in explicit surfaces when the data sets are distributed along lines. We finally present some numerical and graphical examples.     

Detecting derivative discontinuity locations in piecewise continuous functions from Fourier spectral data

We present a method that uses Fourier spectral data to locate jump discontinuities in the first derivatives of functions that are continuous with piecewise smooth derivatives. Since Fourier spectral methods yield strong oscillations near jump discontinuities, it is often difficult to distinguish true discontinuities from artificial oscillations. In this paper we show that by incorporating a local difference method into the global derivative jump function approximation, we can reduce oscillations near the derivative jump discontinuities without losing the ability to locate them. We also present an algorithm that successfully locates both simple and derivative jump discontinuities. This work was partially supported by NSF grants CNS 0324957 and DMS 0510813, and NIH grant EB 02553301 (AG).     

Detecting discontinuity points from spectral data with the quotient-difference (qd) algorithm

This paper introduces a new technique for the localization of discontinuity points from spectral data. Through this work, we will be able to detect discontinuity points of a 2π-periodic piecewise smooth function from its Fourier coefficients. This could be useful in detecting edges and reducing the effects of the Gibbs phenomenon which appears near discontinuities and affects signal restitution. Our approach consists in moving from a discontinuity point detection problem to a pole detection problem, then adapting the quotient-difference (qd) algorithm in order to detect those discontinuity points.     

Numerical solution of linear integro-differential equation by using sine–cosine wavelets

We use the continuous sine–cosine wavelets on the interval [0, 1] to solve the linear integro-differential equation. To do so, we construct the quadrature formulae for the calculation of inner products of any functions, which are required in the approximation for the integro-differential equation. Then, we reduced the integro-differential equation to the solution of linear algebraic equations.     

The fault detection problem in nonlinear systems using residual generators

In this work we study the fault detection problem using residualgenerators based upon high gain nonlinear observers in a differentialalgebraic framework. We analyse the stability of the residualgenerator when a fault occurs. We also consider two faults types:constant and time-varying faults. It is shown that under somemild conditions over the aforementioned faults the residualis different from zero.     

Initial transient detection in simulations using the second-order cumulant spectrum

Procedures for detecting an initial transient in simulation output data are developed. The tests use the second-order cumulant spectrum which differs from the power spectrum in that the stationarity constraint is not required for the former. The second-order cumulant spectrum can be interpreted as the nonstationary power spectrum and is an orthogonal decomposition of the variance of a nonstationary process. The null hypothesis is that the simulation output data series is a covariance stationary process. Equivalently, all estimates of the second-order cumulant spectrum in the region which excludes the estimates of the power spectrum will have an expected value of zero. The test procedures are designed to detect initialization bias in the estimation of the mean and the variance. These procedures can be extended to detect bias in the moments of cumulants of ordern, wheren>2. Results are presented from the application of the test to simulated processes with superimposed mean and variance transients and anM/M/1 queue example.     

Multisignal histogram‐based islanding detection using neuro‐fuzzy algorithm

Islanding is an important concern for grid‐connected distributed resources due to personnel and equipment safety issues. Several techniques based on passive and active detection schemes have been proposed previously. Although passive schemes have a large nondetection zone (NDZ), concerns have been raised about active methods because of their degrading effect on power quality. Reliably detecting this condition is regarded by many as an ongoing challenge because existing methods are not entirely satisfactory. This article proposes a new integrated histogram analysis method using a neuro‐fuzzy approach for islanding detection in grid‐connected wind turbines. The main objective of the proposed approach is to reduce the NDZ to as close as possible to zero and to maintain the output power quality unchanged. In addition, this technique can also overcome the problem of setting detection thresholds which is inherent in existing techniques. The method proposed in this study has a small NDZ and is capable of detecting islanding accurately within the minimum standard time. Moreover, for those regions which require better visualization, the proposed approach can serve as an efficient aid for better detecting grid‐power disconnection. © 2014 Wiley Periodicals, Inc. Complexity 21: 195–205, 2015     

Some comparisons for damage detection on structures using genetic algorithms and modal sensitivity method

The last decade witnessed the development of a large number of non-destructive tests for structural integrity evaluation. This growth is due to attracted interest to reduce time and costs to perform damage monitoring and predictive maintenance. In this way, several methods intended to detect structural damage based on sensitivity and statistical methods were proposed. However, some of these methods present some practical problems in measuring structural dynamic characteristics such as dynamic mode shapes. Some methods based exclusively on structural responses show disadvantages in finding the damage position on structures.     

Signal analysis based on complex wavelet signs

We propose a signal analysis tool based on the sign (or the phase) of complex wavelet coefficients, which we call a signature. The signature is defined as the fine-scale limit of the signs of a signal's complex wavelet coefficients. We show that the signature equals zero at sufficiently regular points of a signal whereas at salient features, such as jumps or cusps, it is non-zero. At such feature points, the orientation of the signature in the complex plane can be interpreted as an indicator of local symmetry and antisymmetry. We establish that the signature rotates in the complex plane under fractional Hilbert transforms. We show that certain random signals, such as white Gaussian noise and Brownian motions, have a vanishing signature. We derive an appropriate discretization and show the applicability to signal analysis.