Comments on Paper by Kurtz,Link, Tukey and Wallace

One way that a process can be out of statistical control is that the process mean can follow a cyclical pattern over time. This out-of-control condition leads to increased variability in the variable of interest over any production interval. Recently, a control chart based on a common time series technique, the periodogram, was proposed as a means of detecting cyclical behavior of the process mean. We show that this control chart is not able to detect certain periodicities and that it does not make use of all available information when multiple observations are taken at each sampling time. Using a multiple regression approach, we present modifications of the periodogram approach that produce control charts sensitive to a wider range of periodicities and use all available information. We compare the average run lengths of the procedures.     

The role of Fourier theory and of modulation in the prediction of visible moiré effects

Modern moiré theory is largely based on the Fourier approach. And yet, it seems that the Fourier theory by itself cannot answer all questions related to moiré effects. We present some visible phenomena in the superposition of periodic structures (such as line gratings) that are not captured by the Fourier approach. We discuss their significance, and provide possible explanations. In particular, we introduce the infinite module spanned by the frequencies of the original individual layers as a main tool in the moiré theory. We discuss its significance on the visible periodicities in the layer superposition–either real moiré effects, or pseudo-moiré (modulation) effects having no corresponding impulses in the Fourier spectral domain.     

Image spectral amplitude distributions

Models for the probability density functions of the Fourier amplitude of images are derived. The densities are based on a simple model of an image made up of independent objects and incorporates the observed behavior of the circularly averaged power spectrum versus spatial frequency. The density function over all spatial frequencies gives a good fit to spectral amplitude data from a variety of images.     

Control Charts for the Detection of a Periodic Component

Control charts are widely used in manufacturing to distinguish between variation that is inherent to the process and variation that signals a special event or problem. Several control charts have recently been proposed for the detection of a periodic component, but their analysis is limited to a moving span of recent observations. This article develops tests for periodicity that make use of independent information about the variability of the time series. These tests are then used to construct control charts that search for periodicity in a span of recent observations but estimate the variability of the process from a longer span of earlier observations. By employing more of the available information, the new charts are able to detect cyclic disturbances more quickly than the existing charts.     

Adaptive spectral doppler estimation

In this paper, 2 adaptive spectral estimation techniques are analyzed for spectral Doppler ultrasound. The purpose is to minimize the observation window needed to estimate the spectrogram to provide a better temporal resolution and gain more flexibility when designing the data acquisition sequence. The methods can also provide better quality of the estimated power spectral density (PSD) of the blood signal. Adaptive spectral estimation techniques are known to provide good spectral resolution and contrast even when the observation window is very short. The 2 adaptive techniques are tested and compared with the averaged periodogram (Welch's method). The blood power spectral capon (BPC) method is based on a standard minimum variance technique adapted to account for both averaging over slow-time and depth. The blood amplitude and phase estimation technique (BAPES) is based on finding a set of matched filters (one for each velocity component of interest) and filtering the blood process over slow-time and averaging over depth to find the PSD. The methods are tested using various experiments and simulations. First, controlled flow-rig experiments with steady laminar flow are carried out. Simulations in Field II for pulsating flow resembling the femoral artery are also analyzed. The simulations are followed by in vivo measurement on the common carotid artery. In all simulations and experiments it was concluded that the adaptive methods display superior performance for short observation windows compared with the averaged periodogram. Computational costs and implementation details are also discussed.     

Historical Misconceptions in Autocorrelation Estimation

The sample autocovariance function, which is estimated as mean lagged products (LPs) of random observations, can also be obtained as the inverse Fourier transform of the periodogram of the data that are augmented with zeros. Hence, the quality of the sample autocovariance as a representation of stochastic data is the same as that of a raw periodogram. The LP estimate is not based on any efficient estimation principle for random data. The spectral density and the autocovariance function can be estimated much more accurately with parametric time series models. A recent development in time series analysis gives the possibility to select automatically the type and the order of the best time series model for any set of random observations. The spectral accuracy of the selected model is better than the accuracy of modified periodograms. In addition, the accuracy of the parametric estimate of the autocovariance function is the same or better for every individual lag than what can be achieved by the nonparametric mean-LP estimates. Perhaps even more important, the estimated time series parameters define the autocovariance as a complete function, for all lags together.     

Analysis and reproduction of a signal's periodic components by means of an extended block-adaptive Fourier analyzer

This paper presents some results of developing a signal analyzer for performing spectral analysis and simultaneous reproduction/filtering of the input signal's periodic components of different (in general, not harmonically related) frequencies and waveforms/spectra. The proposed analyzer accomplishes this task by applying adaptive recursive Fourier analysis.     

Incremental harmonic balance method with multiple time variables for dynamical systems with cubic non‐linearities

The incremental harmonic balance method with multiple time variables is developed for analysis of almost periodic oscillations in multi‐degree‐of‐freedom dynamical systems with cubic non‐linearities, subjected to the external multi‐tone excitation. The method is formulated to treat non‐autonomous as well as autonomous dynamical systems. The almost periodic oscillations, which coexist with periodic oscillations in a rotating system model with cubic restoring force and an electromagnetic eddy‐current damper are analysed. The closed form solutions based on generalized Fourier series containing two incommensurate frequencies are obtained in the case of small non‐dimensional stiffness ratio. Almost periodic oscillations of a rotating system model in dependence on variable parameters are also analysed, where solutions are computed through an augmentation process including a greater number of harmonics and combination frequencies involved. Copyright © 2003 John Wiley & Sons, Ltd.     

Electromagnetic field computation in semiconductor laser resonators

An electromagnetic method based on rigorous diffraction theory of gratings is applied to the analysis of fields in semiconductor laser cavities. The method is based on the Fourier modal method; it is fully rigorous for infinitely periodic resonators and highly accurate for single resonators when absorbing boundary conditions are applied. Fundamental-mode intracavity and near-field distributions are evaluated for some selected geometries, and resonance frequencies are predicted.     

The sequence Fourier transform method for the analysis of a periodic viaduct subjected to non-uniform seismic waves

This study is concerned with the analysis of the dynamic response of an infinite periodic viaduct to non-uniform seismic waves. The periodic viaduct is assumed to be composed of an infinite number of spans, and each span is supposed to consist of a pier, two longitudinal beams (left and right beams) and three linking springs. The sequence Fourier transform method is used to decompose a non-uniform seismic wave into a set of spatially harmonic waves. By using the periodic condition of the viaduct and the transfer matrix method, the wavenumber domain response of the viaduct to each component of the non-uniform seismic wave is obtained. The space-domain response of the periodic viaduct is retrieved by applying the inverse sequence Fourier transform to the wavenumber domain solutions. Numerical results for energy bands of the periodic viaduct and those for the dynamic responses of the periodic viaduct to spatially harmonic waves as well as to non-uniform seismic waves are presented. It is found that there exist three kinds of characteristic waves for both the in-plane and out-of-plane wave vibrations of the periodic viaduct. Also, proposed numerical results show that when the periodic viaduct is subjected to traveling waves, serious resonance may occur. Resonance may occur at the bounding frequencies of the passbands for the characteristic waves. Also, due to the coincidence effect between traveling seismic waves and the characteristic waves, resonance may occur at frequencies in the passbands of the characteristic waves.     

An almost leakage free method for assessing Σ-Δmodulator spectra

This paper addresses the problem of spectral analysis of Σ-Δ modulator output spectra. Often, data containing a periodic signal are evaluated using fast-Fourier-transform (FFT) based spectral analysis. If the data set is not sampled coherently, long- and short-range leakage can hamper or prevent the frequency domain analysis, particularly if frequencies near the excitation sine frequency are of interest. Techniques such as windowing or coherent sampling help, but unfortunately may add their own effects to the spectra under scrutiny. Where coherent sampling is not easily possible, another approach to reliable spectrum analysis is a hybrid computation of the spectrum using the output bit stream from a Σ-Δ modulator. This will yield spectral information and an assessment of modulator performance. The method is demonstrated on MATLAB simulated Σ-Δ modulator bit-stream data from a single-loop, first order modulator as well as a single-loop, third order modulator, both excited by a multitone sinusoidal excitation     

EWMA control charts with variable sample sizes and variable sampling intervals

Traditional control charts for process monitoring are based on taking samples of fixed size from the process using a fixed sampling interval. Variable Sample Size (VSS) and Variable Sampling Interval (VSI) control charts vary the sampling rate from the process as a function of the data from the process. VSS and VSI control charts sample at a higher rate when there is evidence of a change in the process, and are thus able to detect process changes faster than traditional control charts. This paper considers general VSS and VSI control charts and develops integral equation and Markov chain methods for finding the statistical properties of these charts. EWMA charts with the VSS and/or the VSI features are studied in detail, and different ways of defining the EWMA control statistic are investigated. It is shown that using either the VSS or VSI feature in an EWMA control chart substantially improves the ability to detect all but very large shifts in the process mean. The VSI feature usually gives more improvement in detection ability than the VSS feature, and using both features together sometimes gives more improvement than using either one separately. Guidelines are given for choosing the possible sample sizes and the possible sampling intervals for these charts. EWMA charts with the VSS and/or VSI feature are compared to CUSUM charts and Shewhart X¯ charts with the VSS and/or VSI features.     

Study on vibration characteristics in periodic plate structures using the spectral element method

Periodic structures have great scientific potential because of their superior structural dynamic properties. The aim of this study is to analyze the dynamic behavior of periodic plate structures using the spectral element method (SEM). The spectral equations of the plate elements with two parallel sides that are simply supported are established. Then, the spectral dynamic stiffness matrix of the whole periodic plate structure is assembled. The frequency responses are obtained by the calculation of the spectral equations to illustrate the characteristics of the band gaps. From the results, it is seen that the SEM can be effectively applied to study the vibration properties of the periodic structures. Compared with the results calculated by the finite element method, it can be observed that more accurate results in high-frequency ranges can be achieved by the SEM. These results indicate that the band gap characteristics depend on both the material properties and unit cell numbers. Furthermore, the effects of the structure damping and type on the frequency responses are investigated.     

On a class of mixed EWMA-CUSUM median control charts for process monitoring

Monitoring of any manufacturing, production, or industrial process can be controlled and improved by removing these special cause of variations using control charts. Shewhart-type control charts are effective to control a large amount of special variations, whereas, cumulative sum (CUSUM) and exponentially weighted moving average (EWMA) charts detect small and moderate variations efficiently in the process parameters. Monitoring of location parameter can be done with mean control charts under the assumption that the parameters are known or correctly estimated from in-control samples and data are free from outliers (but in practice data occasionally have outliers). In this study, we have proposed generalized mixed EWMA-CUSUM median control charts structures for known and unknown parameters based on auxiliary variables for detecting shifts in process location parameter. The proposed charts are compared with the corresponding charts for the mean, based on contaminated and uncontaminated data. Different performance measures are used to evaluate the performance of proposed control charts and revealed through results that the median-based charts are more sensitive to detect a shift in process location parameter in the presence of outliers. An illustrative example using real data is also shown for practical consideration.     

Experimental comparison of fourier transformation and model-fitting methods of spectral analysis

A summary of the results of an extensive comparative experimental study of Fourier transformation and model-fitting methods of spectral analysis of random time-series data is presented. It is illustrated that Fourier transformation methods can be an essential companion to model-fitting methods even for short data segments with underlying sharp spectral peaks. The best spectrum estimates can be obtained by taking advantage of the strengths of both types of methods. For example, it is shown that detection and estimation of the frequencies of spectral lines for short data segments can be best accomplished using certain parametric methods in conjunction with Fourier transformation methods to aid in model-order selection and identification of spurious peaks in the parametric spectrum estimate, and that estimation of amplitude and phase for sine-wave removal, given frequency estimates, and spectrum estimation after sine-wave removal can often be best accomplished with Fourier transformation methods alone.     

非高斯背景下的频域广义匹配检测

采用广义匹配滤波器在功率谱未知的非高斯噪声中检测宽带信号。在估计功率谱时,为了减小周期图法谱估计产生的误差,采用平滑周期图法来估计非高斯噪声的功率谱,并在此基础上对信号进行恒虚警概率损失下的广义匹配检测。实验表明,用平滑周期图法估计功率谱并进行的广义匹配检测,其检测性能稍弱于功率谱取真值时的检测,但远远优于传统的匹配滤波器。     

A Reappraisal of the Period Spectral Analysis

The purpose of this paper is to revive interest in the periodogram approach to time series analysis which, at present, is only of historical interest and is seldom used. During the late 1940's, when it was realized that the smoothed periodogram could be used to estimate the spectral density of a stationary time series, the method was impractical because of the amount of computations. This is no longer the case, but not realized by many applied workers. In this paper the smoothed periodogram is considered from the point of view of its spectral window. The results are compared with two standard spectral windows by a method which avoids defining a bandwidth. The spectral windows are normalized so that they have the same variance, and plotted. The user can then choose the window which best suits his needs. Rejection filtering, trigonometric regression and cross-spectra analysis are discussed. An example is given in which the spectra and cross-spectrum of a bivariate time series are estimated.     

Multivariate EWMA control charts using individual observations for process mean and variance monitoring and diagnosis

Most multivariate control charts in the literature are designed to detect either mean or variation shifts rather than both. A simultaneous use of the Hotelling T ² and |S| control charts has been proposed but the Hotelling T ² reacts to mean shifts, dispersion changes, and changes of correlations among responses. The combination of two multivariate control charts into one chart sometimes loses the ability to provide detailed diagnostic information when a process is out-of-control. In this research a new multivariate control chart procedure based on exponentially weighted moving average (EWMA) statistics is proposed to monitor process mean and variance simultaneously to identify proper sources of variations. Two multivariate EWMA control charts using individual observations are proposed to achieve a quick detection of mean or variance shifts or both. Simulation studies show that the proposed charts are capable of identifying appropriate types of shifts in terms of correct detection percentages. A manufacturing example is used to demonstrate how the proposed charts can be properly set-up based on average run length values via simulations. In addition, correct detection rates of the proposed charts are explored.     

A New Method for Rolling Element Bearing Fault Diagnosis Based on Cyclostationary Theory

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Facts and fiction in spectral analysis

This analysis is limited to the spectral analysis of stationary stochastic processes with unknown spectral density. The main spectral estimation methods are: parametric with time series models, or nonparametric with a windowed periodogram. A single time series model will be chosen with a statistical criterion from three previously estimated and selected models: the best autoregressive (AR) model, the best moving average (MA) model, and the best combined ARMA model. The accuracy of the spectrum, computed from this single selected time series model, is compared with the accuracy of some windowed periodogram estimates. The time series model generally gives a spectrum that is better than the best possible windowed periodogram. It is a fact that a single good time series model can be selected automatically for statistical data with unknown spectral density. It is fiction that objective choices between windowed periodograms can be made