Multichannel arma spectral estimation by the modified Yule-Walker method

: This report describes some interpretations and uses of eigenvalues and eigenvectors of spectral and sample spectral density matrices of multiple stationary time series. The spectral density matrix of a zero-mean multiple stationary time series is defined. Eigenvalues and eigenvectors of the spectral density matrix are discussed and principal component theory is presented. Statistical distribution theory and related results are used to investigate the eigenvalues of a sample spectral density matrix. This investigation gives methods for obtaining simultaneous confidence bounds on the elements of the true spectral density matrix and its inverse, and also methods for obtaining confidence bounds on the eigenvalues of the true spectral density matrix.

Transfer function estimation of turboalternator from practical operating data by spectral analysis

The aim of this article is to introduce new resampling scheme for nonstationary time series, called generalized resampling scheme (GRS). The proposed procedure is a generalization of well known in the literature subsampling procedure and is simply related to existing block bootstrap techniques. To document the usefulness of GRS, we consider the example of model with almost periodic phenomena in mean and variance function, where the consistency of the proposed procedure was examined. Finally, we prove the consistency of GRS for the spectral density matrix for nonstationary, multivariate almost periodically correlated time series. We consider both zero mean and non‐zero mean case. The consistency holds under general assumptions concerning moment and α‐mixing conditions for multivariate almost periodically correlated time series. Proving the consistency in this case poses a difficulty since the estimator of the spectral density matrix can be interpreted as a sum of random matrixes whose dependence grow with the sample size.

A technique is presented for generating a modified cross- spectral density matrix (CSDM) that is sensitive to fluctuations in the frequency bin amplitudes and phases of sinusoidal signals (plus noise) and noise. The beamformed output of such a fluctuation-based CSDM provides signal-to- noise ratio (SNR) gains in excess of those achieved using a conventional CSDM. The additional gains are inversely proportional to the beamformed ratio of the signal bin fluctuations to the noise bin fluctuations. An unalerted auto-detection capability is another advantage of this new technique.

SUMMARY The problem of testing for periodicities in multiple time series is considered. The test statistic proposed is based on the Euclidean norm of the matrix of periodograms. We first obtain the exact distribution of our test statistic under the hypothesis that observations are taken from an n-vector time series consisting of independent series of independent normal variables. We extend the theory to normal linear series with a specified spectral density matrix and then show that a suitable estimate can be used to replace this matrix in our test statistic. These results are asymptotic in nature. Extreme value theorems are stated for the case of normal variates and it is shown that the distributions for nonnormal series can be approximated adequately by those for normal series. An exact test for periodic components in an univariate time series was first proposed by Fisher (1929) who considered the null hypothesis of normal white noise. The test was based upon the maximum of a set of periodogram ordinates each divided by their sum, the division making it unnecessary to know in advance the variance of the components of the series. The use of the rth largest of these ordinates, r > 1, in testing for periodicities was also discussed by Fisher (1940). Whittle (1951, p. 101) showed how one could apply the Fisher tests to normal linear series provided the number of observations was large and one could specify the spectral density for the series. Hannan (1961) proposed a modification to this test that removed the necessity of knowing beforehand the spectral density function. Nicholls (1967) proposed a variation on Hannan's test statistic that improves its power under certain alternatives. Some extreme value results for periodogram ordinates were given by Walker (1965), who also demonstrated that dropping the normality condition has little effect on the large sample distributions, at least in the upper tails. In the sequel we consider the problem of testing for periodicities in multiple time series and pursue a course that parallels the historical development for univariate series as outlined in the paragraph above. Our tests are based on the Euclidean norm of the matrix of periodo- grams. In this connection Nicholls (1969) has discussed a test for discrete mass in a cross- spectrum. We consider real, discrete time parameter, n-vector, time series, Z(t) (t = 0, + 1, + 2, ...), where Z(t) = It+m(t) +X(t), with ,u+m(t) being a mean value function and X( ) being a weakly stationary, zero mean, time series possessing a spectral density matrix fxx(A) with tr{fxx(A)} <1 M < oo (JAI < 7r). We also assume the existence of a moving average representation 00 wo,t) , CO v - )

The conventional method for identifying the transfer function of an unknown linear system consists of a least squares fit of its input to its output. It is equivalent to identifying the frequency response of the system by calculating the empirical cross-spectrum between the system's input and output, divided by the empirical auto-spectrum of the input process. However, if the additive noise at the system's output is correlated with the input process, e.g., in case of environmental noise that affects both system's input and output, the method may suffer from a severe bias effect. We present a modification of the cross-spectral method that exploits nonstationary features in the data in order to circumvent bias effects caused by correlated stationary noise. The proposed method is particularly attractive to problems of multichannel signal enhancement and noise cancellation, when the desired signal is nonstationary in nature, e.g., speech or image.

In the transfer function with zeros restricted on the imaginary axis of S‐plane the phase characteristic is essentially determined by that of the denominator polynomial. As long as the denominator polynomial is a Hurwitz polynomial, the tangent of its phase characteristic, i.e., the ratio of the imaginary part (odd part) to the real part (even part), is a reactance function. This paper proposes a waveform approximation method which approximates the phase tangent of the Fourier transform of the object waveform by a reactance function using an optimization technique. The Hurwitz polynomial, which is the sum of the numerator and denominator of the reactance function, is adopted as the denominator polynomial of the approximate transfer function. The numerator polynomial of the approximate transfer function is derived by adjusting the zeros on the imaginary axis by an optimization technique to approximate the amplitude of the Fourier transform of the object waveform. An evaluation measure in the proposed approximation method is presented, describing its problems and physical implications. It is shown that the proposed method can provide a satisfactory approximation for symmetrical waveforms. For asymmetrical waveforms it is shown that the approximation can be improved by adding a delay time. An appropriate weighting function is proposed for the evaluation measure. Lastly, the proposed method is compared with other methods of approximation.

The paper presents how to identify natural frequencies and mode shapes of structures by Operational Modal Analysis (OMA) technique, in which the Frequency Domain Decomposition (FDD) method is used. This method is an experimental method only base on the data of measuring the dynamic response of the structures under the excitation due to ambient forces and operational loads to determine the vibration characteristics. Measure vibration (acceleration) and determine spectral density matrix, using the singular values decomposition method of spectral density matrix to determine the natural frequencies and mode shapes of structures. The calculation results show that the natural frequencies, the mode shapes form determined by the OMA technique is consistent with the calculation results according to the theory and show the reliability of the method.KeywordsNatural frequencyMode shapeIdentifyEMAOMAFDD

A new relative-entropy method is presented for estimating the power spectral density matrix for multichannel data, given correlation values for linear combinations of the channels, and given an initial estimate of the spectral density matrix. A derivation of the method from the relative-entropy-principle is given. The basic approach is similar in spirit to the multisignal relative-entropy spectrum analysis of Johnson and shore, but the results differ significantly because the present method does not arbitrarily require the final distributions of the various channels to be independent. For the special case of separately estimating the spectra of a signal and noise, given the correlations of their sum, multichannel relative-entropy spectrum analysis turns into a two-stage procedure. First, a smooth power spectrum model is fitted to the correlations of the signal plus noise. Then, final estimates of the spectra and cross spectra are obtained through linear filtering. For the special case where p uniformly spaced correlations are known, and where the initial estimate of the signal-plus-noise spectrum is all-pole with order p or less, this method fits a standard maximum-entropy autoregressive spectrum to the noisy correlations, then linearly filters to calculate the signal and noise spectra and cross spectra. Consideration is given to the case where only an initial estimate of the noise power spectrum is available. An illustrative example is given.

A new method is presented for estimating the power spectral density matrix for multichannel data, given correlation values for linear combinations of the channels and given an initial estimate of the spectral density matrix. A derivation of the method from the relative-entropy principle is given. The results differ significantly from the Multisignal Relative-Entropy (Minimum-Cross-Entropy) Spectrum Analysis of Johnson and Shore because the present method does not arbitrarily force the final distributions of the various channels to be independent. For the special case when correlation values are given only for the sum of the channels, Multichannel Relative-Entropy Spectrum Analysis is shown to reduce to a two-stage procedure: first a smooth power-spectrum model is fitted to the correlations of the sum; then final estimates of the spectra and cross spectra are obtained through linear filtering. Illustrative numerical examples are presented.

Frequency Domain Principal Components Estimation of Fractionally Cointegrated Processes

Testing nonparametric and semiparametric hypotheses in vector stationary processes

Vector Autoregressive Modal Analysis With Application To Ship Structures

Analyzing time series in the frequency domain enables the development of powerful tools for investigating the second-order characteristics of multivariate processes. Parameters like the spectral density matrix and its inverse, the coherence or the partial coherence, encode comprehensively the complex linear relations between the component processes of the multivariate system. In this article, we develop inference procedures for such parameters in a high-dimensional, time series setup. Toward this goal, we first focus on the derivation of consistent estimators of the coherence and, more importantly, of the partial coherence which possess manageable limiting distributions that are suitable for testing purposes. Statistical tests of the hypothesis that the maximum over frequencies of the coherence, respectively, of the partial coherence, do not exceed a prespecified threshold value are developed. Our approach allows for testing hypotheses for individual coherences and/or partial coherences as well as for multiple testing of large sets of such parameters. In the latter case, a consistent procedure to control the false discovery rate is developed. The finite sample performance of the inference procedures introduced is investigated by means of simulations and applications to the construction of graphical interaction models for brain connectivity based on EEG data are presented.Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.

The output controllability condition of the class of time-invariant multivariable linear delay-differential systems is investigated by using an algebraic approach. The controllability condition is given in terms of the kernel of a map, which determines the behaviour of output after time t = 0 due to input up to t = 0. This kernel is then completely characterized by a module generated by the columns of a matrix directly computable from the system transfer function. It is shown that every reachable output of the system considered can be controlled to zero in finite time. This paper is intended to form a part of the first stage in the development of an operator transfer function approach to the design of certain class of infinite dimensional systems which commonly occur in industry and other fields.