Spectral Density Matrices Research Articles

Testing Equality of Multiple Power Spectral Density Matrices

This paper studies the existence of optimal invariant detectors for determining whether $P$ multivariate processes have the same power spectral density. This problem finds application in multiple fields, including physical layer security and cognitive radio. For Gaussian observations, we prove that the optimal invariant detector, i.e., the uniformly most powerful invariant test, does not exist. Additionally, we consider the challenging case of close hypotheses, where we study the existence of the locally most powerful invariant test (LMPIT). The LMPIT is obtained in the closed form only for univariate signals. In the multivariate case, it is shown that the LMPIT does not exist. However, the corresponding proof naturally suggests an LMPIT-inspired detector that outperforms previously proposed detectors.

Nonparametric change point detection in multivariate piecewise stationary time series

ABSTRACTDetecting change points in multivariate time series is an important problem with numerous applications. We develop a nonparametric method to detect multiple change points in multivariate piecewise stationary processes when the locations and number of change points are unknown. Based on a test statistic that measures differences in the spectral density matrices through the norm, we sequentially identify points of local maxima in the test statistic and test for the significance of each of them being change points. In addition, the components responsible for the change in the covariance structure at each detected change point are identified. The asymptotic properties of the test for significant change points under the null and alternative hypothesis are derived. We illustrate the better performance of our method in comparison to some of the recent methods through a few simulation examples and discuss applications of our method in seismology and finance.

On the Existence and Uniqueness of the Eigenvalue Decomposition of a Parahermitian Matrix

This paper addresses the extension of the factorization of a Hermitian matrix by an eigenvalue decomposition (EVD) to the case of a parahermitian matrix that is analytic at least on an annulus containing the unit circle. Such parahermitian matrices contain polynomials or rational functions in the complex variable $z$ and arise, e.g., as cross spectral density matrices in broadband array problems. Specifically, conditions for the existence and uniqueness of eigenvalues and eigenvectors of a parahermitian matrix EVD are given, such that these can be represented by a power or Laurent series that is absolutely convergent, at least on the unit circle, permitting a direct realization in the time domain. Based on an analysis of the unit circle, we prove that eigenvalues exist as unique and convergent but likely infinite-length Laurent series. The eigenvectors can have an arbitrary phase response and are shown to exist as convergent Laurent series if eigenvalues are selected as analytic functions on the unit circle, and if the phase response is selected such that the eigenvectors are Holder continuous with $\alpha >\frac{1}{2}$ on the unit circle. In the case of a discontinuous phase response or if spectral majorisation is enforced for intersecting eigenvalues, an absolutely convergent Laurent series solution for the eigenvectors of a parahermitian EVD does not exist. We provide some examples, comment on the approximation of a parahermitian matrix EVD by Laurent polynomial factors, and compare our findings to the solutions provided by polynomial matrix EVD algorithms.

An effective feature extraction method by power spectral density of EEG signal for 2-class motor imagery-based BCI.

EEG signals have weak intensity, low signal-to-noise ratio, non-stationary, non-linear, time-frequency-spatial characteristics. Therefore, it is important to extract adaptive and robust features that reflect time, frequency and spatial characteristics. This paper proposes an effective feature extraction method WDPSD (feature extraction from the Weighted Difference of Power Spectral Density in an optimal channel couple) that can reflect time, frequency and spatial characteristics for 2-class motor imagery-based BCI system. In the WDPSD method, firstly, Power Spectral Density (PSD) matrices of EEG signals are calculated in all channels, and an optimal channel couple is selected from all possible channel couples by checking non-stationary and class separability, and then a weight matrix which reflects non-stationary of PSD difference matrix in selected channel couple is calculated; finally, the robust and adaptive features are extracted from the PSD difference matrix weighted by the weight matrix. The proposed method is evaluated from EEG signals of BCI Competition IV Dataset 2a and Dataset 2b. The experimental results show a good classification accuracy in single session, session-to-session, and the different types of 2-class motor imagery for different subjects.

АНАЛИТИЧЕСКОЕ КОНСТРУИРОВАНИЕ РАЗОМКНУТЫХ СИСТЕМ СТОХАСТИЧЕСКОГО УПРАВЛЕНИЯ МНОГОМЕРНЫМ НЕЛИНЕЙНЫМ ОБЪЕКТОМ

This article is devoted to the development of a new method of synthesis a regulators’ transfer functions matrix of an optimal multivariable open-loop control system. The regulator is designed to maximize an accuracy of a nonlinear multivariable control object transition from one steady state to another. It is assumed that disturbances act on the control object and sensors measuring data has inertia and noise. Both disturbances and noises are an additive combination of regular and random components. Random components belong to a class of interconnected stationary processes with rational spectral density matrices. A substantiated method differs from the known one by the fact that during formulation and solution of the problem somebody uses a new block diagram of the control system, which takes into account the results of metrological certification of a sensor dynamics. Synthesis of the regulator is carried out in the frequency domain by the Wiener-Kolmogorov method. A new algorithm, which is obtained as a result of synthesis problem solution, allows you to find the matrix of regulator transfer functions , which provides a minimum of corresponding quadratic quality criteria’s. The first of them is equal to the sum of certain way weighted squared deviations regular repetition errors of the object path and control signals. The second criterion is equal to the sum of the weighted variance of the random error components and the control signals. To execute the proposed algorithm it is necessary to perform the operations of Wiener factorization and separation of rational matrices. The corresponding functions are contained in the freely distributed software package SciLab.

Generation of spectral density matrices for multichannel continuous systems under white noise action

Generation of spectral density matrices for multichannel continuous systems under white noise action

Multi-damage identification based on joint approximate diagonalisation and robust distance measure

Mode shapes or operational deflection shapes are highly sensitive to damage and can be used for multi-damage identification. Nevertheless, one drawback of this kind of methods is that the extracted spatial shape features tend to be compromised by noise, which degrades their damage identification accuracy, especially for incipient damage. To overcome this, joint approximate diagonalisation (JAD) also known as simultaneous diagonalisation is investigated to estimate mode shapes (MS’s) statistically. The major advantage of JAD method is that it efficiently provides the common Eigen-structure of a set of power spectral density matrices. In this paper, a new criterion in terms of coefficient of variation (CV) is utilised to numerically demonstrate the better noise robustness and accuracy of JAD method over traditional frequency domain decomposition method (FDD). Another original contribution is that a new robust damage index (DI) is proposed, which is comprised of local MS distortions of several modes weighted by their associated vibration participation factors. The advantage of doing this is to include fair contributions from changes of all modes concerned. Moreover, the proposed DI provides a measure of damage-induced changes in ‘modal vibration energy’ in terms of the selected mode shapes. Finally, an experimental study is presented to verify the efficiency and noise robustness of JAD method and the proposed DI. The results show that the proposed DI is effective and robust under random vibration situations, which indicates that it has the potential to be applied to practical engineering structures with ambient excitations.

Dynamic Amplification of Bridge-Expansion-Joints considering Roughness induced Vehicle Vibrations

The response of bridge expansion joints to the transient impact induced by crossing vehicles always represent the dynamic reaction of the former to the dynamic wheel contact forces at that particular moment. Separating the dynamic components in the vertical wheel contact forces, the dynamic amplification of the bridge expansion joints response, which is dominated by the driving velocity of the vehicle, its crossing position in lateral direction, the dynamic behavior of the expansion joint itself and most importantly by the interaction between the vehicle tire and the expansion joint’s lamellas can be discussed. Due to the random distribution of the roughness amplitudes a stochastic characterization of the vehicle vibrations is necessary. In addition to the classical frequency approach based on complex transfer matrices completely describing the dynamic behavior of the vehicles and spectral density matrices characterizing the pavement roughness, the numerically efficient covariance analysis is applied. Therefore the spectral characteristics of the roughness amplitudes, which are described on the basis of multi-correlated random processes, are considered by introducing shaping filters in combination with a state space augmentation. The dynamic behavior of the considered bridge expansion joint is dominated by the properties of the discrete elastomeric spring elements. Hence a a non-proportional damping approach has to be applied to analyze the three dimensional finite element model of the structure using a modal superposition.

Robust structural damage detection and localization based on joint approximate diagonalization technique in frequency domain

The structural characteristic deflection shapes (CDS’s) such as mode shapes and operational deflection shapes are highly sensitive to structural damage in beam- or plate-type structures. Nevertheless, they are vulnerable to measurement noise and could result in unacceptable identification errors. In order to increase the accuracy and noise robustness of damage identification based on CDS’s using vibration responses of random excitation, joint approximate diagonalization (JAD) technique and gapped smoothing method (GSM) are combined to form a sensitive and robust damage index (DI), which can simultaneously detect the existence of damage and localize its position. In addition, it is possible to apply this approach to damage identification of structures under ambient excitation. First, JAD method which is an essential technique of blind source separation is investigated to simultaneously diagonalize a set of power spectral density matrices corresponding to frequencies near a certain natural frequency to estimate a joint unitary diagonalizer. The columns of this joint diagonalizer contain dominant CDS’s. With the identified dominant CDS’s around different natural frequencies, GSM is used to extract damage features and a robust damage identification index is then proposed. Numerical and experimental examples of beams with cracks are used to verify the validity and noise robustness of JAD based CDS estimation and the proposed DI. Furthermore, damage identification using dominant CDS’s estimated by JAD method is demonstrated to be more accurate and noise robust than by the commonly used singular value decomposition method.

Spotforming: Spatial Filtering With Distributed Arrays for Position-Selective Sound Acquisition

Hands-free capture of speech often requires extraction of sources from a certain spot of interest (SOI), while reducing interferers and background noise. Although state-of-the-art spatial filters are fully data-dependent and computed using the power spectral density (PSD) matrices of the desired and the undesired signals, the existing solutions to extract sources from a SOI are only partially data-dependent. Estimating the time-varying PSD matrices from the data is a challenging problem, especially in dynamic and quickly time-varying acoustic scenes. Hence, the spot signal statistics are often pre-computed based on a near-field propagation model, resulting in suboptimal filters. In this work, we propose a fully data-dependent spatial filtering framework for extraction of speech signals that originate from a SOI. To achieve position-based spatial selectivity, distributed arrays are used, which offer larger spatial diversity compared to arrays of closely spaced microphones. The PSD matrices of the desired and the undesired signals are updated at each time-frequency bin by using a minimum Bayes risk detector that is based on a probabilistic model of narrowband position estimates. The proposed framework is applicable in challenging multitalk situations, without requiring any prior information, except the geometry, location, and orientation of the arrays.

Testing equality of spectral densities using randomization techniques

In this paper, we investigate the testing problem that the spectral density matrices of several, not necessarily independent, stationary processes are equal. Based on an $L_2$-type test statistic, we propose a new nonparametric approach, where the critical values of the tests are calculated with the help of randomization methods. We analyze asymptotic exactness and consistency of these randomization tests and show in simulation studies that the new procedures posses very good size and power characteristics.

Multi-channel post-filtering based on spatial coherence measure

A multi-channel post-filtering algorithm using the proposed spatial coherence measure is derived. The spatial coherence measure evaluates the similarity between the measured signal fields using power spectral density matrices. In the proposed post-filter, the assumption of homogeneous sound fields is relaxed. Besides, multi-rank signal models can be easily adopted. Under this measure, the bias term due to the similarity of the desired signal field and the noise field is further investigated and a solution based on bias compensation is proposed. It can be shown that the compensated solution is equivalent to the optimal Wiener filter if the bias or the noise power spectral density matrix is perfectly measured. Simulations with incoherent, diffuse, and coherent noise fields and a local scattered desired source were conducted to evaluate the algorithms. The results demonstrate the superiority of the proposed bias compensated post-filter across different types of noise fields with a more accurate signal model.

Spectro-Temporal Filtering for Multichannel Speech Enhancement in Short-Time Fourier Transform Domain

In this letter, we propose a spectro-temporal filtering algorithm for multichannel speech enhancement in the short-time Fourier transform (STFT) domain. Compared with the traditional multiplicative filtering technique, the proposed method takes account of interdependencies between components in adjacent frames and frequency bins. For spectro-temporal filtering, speech and noise power spectral density (PSD) matrices are estimated based on an extended formulation utilizing temporal and spectral correlations, and the parametric noise reduction filter based on these PSD matrices is applied to the input microphone array signal. Moreover, multichannel speech presence probabilities are also estimated within a unified framework. A number of experimental results show that the proposed spectro-temporal filtering method improves the performance of multichannel speech enhancement.

Spectral estimation in frequency-domain by subspace techniques

In this paper, frequency-domain subspace-based algorithms are proposed to estimate discrete-time cross-power spectral density (cross-PSD) and auto-power spectral density (auto-PSD) matrices from samples of the Welch cross-PSD and auto-PSD estimators on uniform grids of frequencies. The proposed algorithms are shown to be strongly consistent. A link between the well-known time-domain covariance-based estimation methods and the frequency-domain realization-based methods of this paper is also established. The consistency of the proposed algorithms is not expected a priori since the periodograms are not consistent spectrum estimators even if the true power spectrum is rational. The algorithms are tested on numerical examples and a real-life application example concerned with the modeling of acoustic spectra for fault detection in induction motors.

Data-driven shrinkage of the spectral density matrix of a high-dimensional time series

Time series data obtained from neurophysiological signals is often high-dimensional and the length of the time series is often short relative to the number of dimensions. Thus, it is difficult or sometimes impossible to compute statistics that are based on the spectral density matrix because estimates of these matrices are often numerically unstable. In this work, we discuss the importance of regularization for spectral analysis of high-dimensional time series and propose shrinkage estimation for estimating high-dimensional spectral density matrices. We use and develop the multivariate Time-frequency Toggle (TFT) bootstrap procedure for multivariate time series to estimate the shrinkage parameters, and show that the multivariate TFT bootstrap is theoretically valid. We show via simulations and an fMRI data set that failure to regularize the estimates of the spectral density matrix can yield unstable statistics, and that this can be alleviated by shrinkage estimation.

基于发射方向图合成的低PAPR 宽带MIMO雷达波形设计

Transmit beam-pattern synthesis is discussed regarding unidirectional and wide-beam signals for wideband multi-input multi-output radars. First, a unified optimal framework, which generalizes some existing criteria, is proposed to match a given unidirectional transmit beam-pattern. A new method, which satisfies the unit-modulus or low peak-to-average power ratio (PAPR) constraints, is proposed to synthesize transmit waveforms that match the above beam-pattern design. Finally, another approach is proposed to generate power spectral density matrices using mixed norms that match a given wider main-beam beam-pattern. The designed transmit beam-pattern not only matches the desired wideband transmit beam-pattern in the mainlobe region but also maintains the sidelobes below a specified threshold value. The two approaches to beam-pattern synthesis and waveform design are converted into convex optimization problems and solved using the modeling system Concurrent Versions System. Simulations show that the designed transmit beam-pattern with unidirectional signal approximates the given transmit beam-pattern and the signals obtained also satisfy the PAPR constraint. Similarly, we demonstrate that the designed wide-beam transmit beam-pattern approximates the desired beam- pattern in the mainlobe region and maintains sidelobes-to-peak ratio below -40 dB.

Application of convolved blind separation based on second-order cyclic statistics in rolling element bearing feature extraction

In condition monitoring of mechanical equipment, the collected signals are mostly mixtures of several different sources. It is very important to recover the individual effects of each source for exact feature extraction and further fault diagnosis. A blind separation method based on second-order cyclic statistics is presented for convolved cyclostationary processes such as those observed in rotating machinery. The convolutive mixtures in time domain are transformed into instantaneous mixtures in frequency domain by discrete Fourier transform (DFT). Then the aim is to find a separating matrix that simultaneously and jointly diagonalizes the set of cyclic spectral density (CSD) matrices of the observed signals. Examples of successful separation are provided on both synthetic convolutive mixed bearing signals and real data tested from a gearbox.

Distances and Riemannian Metrics for Multivariate Spectral Densities

We first introduce a class of divergence measures between power spectral density matrices. These are derived by comparing the suitability of different models in the context of optimal prediction. Distances between "infinitesimally close" power spectra are quadratic, and hence, they induce a differential-geometric structure. We study the corresponding Riemannian metrics and, for a particular case, provide explicit formulae for the corresponding geodesics and geodesic distances. The close connection between the geometry of power spectra and the geometry of the Fisher-Rao metric is noted.

Electroencephalogram signals classification for sleep-state decision – a Riemannian geometry approach

In this work, the authors study the classification of electroencephalogram (EEG) signals for the determination of the state of sleep of a patient. They employ the power spectral density (PSD) matrices as the feature for the distinction between different classes of EEG signals. This not only allows us to examine the power spectrum contents of each signal as well as the correlation between the multi-channel signals, but also complies with what clinical experts use in their visual judgement of EEG signals. To establish a metric facilitating the classification, the authors exploit the specific geometric properties, and develop, with the aid of fibre bundle theory, an appropriate metric in the Riemannian manifold described by the PSD matrices. To use this new metric effectively for the EEG signal classification, the authors further need to find a weighting for the PSD matrices so that the distances of similar features are minimised whereas those for dissimilar features are maximised. A closed form of this weighting matrix is obtained by solving an equivalent convex optimisation problem. The effectiveness of using these new metrics is examined by applying them to a collection of recorded EEG signals for sleep pattern classification based on the k-nearest neighbour decision algorithm with excellent outcome.

A Spectral Graphical Model Approach for Learning Brain Connectivity Network of Children's Narrative Comprehension

Narrative comprehension is a fundamental cognitive skill that involves the coordination of different functional brain regions. We develop a spectral graphical model with model averaging to study the connectivity networks underlying these brain regions using fMRI data collected from a story comprehension task. Based on the spectral density matrices in the frequency domain, this model captures the temporal dependency of the entire fMRI time series between brain regions. A Bayesian model averaging procedure is then applied to select the best directional links that constitute the brain network. Using this model, brain networks of three distinct age groups are constructed to assess the dynamic change of network connectivity with respect to age.