Cross-spectral Density Matrices Research Articles

Difference of two sinc Schell-model cross-spectral density matrices

We propose a convenient method for modeling what we believe to be novel classes of cross-spectral density (CSD), in which the difference between two CSD matrices belonging to the same class represents a supposed novel valid CSD matrix under sufficient conditions. Using such a method, we introduce what we think is a new class of CSD matrix relating to the difference of two sinc Schell-model CSD matrices, and derive the sufficient conditions for the non-negative definiteness of the resulting CSD matrix. The statistical characteristics of such electromagnetic stochastic beams propagating in free space were investigated using weighted superposition and numerical simulation methods. It is demonstrated that this new light fields with any admissible choice of source parameters possess very peculiar evolution characteristics. The results illustrate new opportunities for study new classes of electromagnetic stochastic beams.

Multifrequency matched-field source localization based on Wasserstein metric for probability measures.

Matched-field processing (MFP) for underwater source localization serves as a generalized beamforming approach that assesses the correlation between the received array data and a dictionary of replica vectors. In this study, the processing scheme of MFP is reformulated by computing a statistical metric between two Gaussian probability measures with the cross-spectral density matrices (CSDMs). To achieve this, the Wasserstein metric, a widely used notion of metric in the space of probability measures, is employed for developing the processor to attach the intrinsic properties of CSDMs, expressing the underlying optimal value of the statistic. The Wasserstein processor uses the embedded metric structure to suppress ambiguities, resulting in the ability to distinguish between multiple sources. In this foundation, a multifrequency processor that combines the information at different frequencies is derived, providing improved localization statistics with deficient snapshots. The effectiveness and robustness of the Wasserstein processor are demonstrated using acoustic simulation and the event S5 of the SWellEx-96 experiment data, exhibiting correct localization statistics and a notable reduction in ambiguity. Additionally, this paper presents an approach to derive the averaged Bartlett processor by evaluating the Wasserstein metric between two Dirac measures, providing an innovative perspective for MFP.

Functional connectivity and power spectral density analysis of EEG signals in trained practitioners of Bhramari pranayama

Bhramari Pranayama, which belongs to the group of pranayama, is a calming breadth practice. In the present study, we aim to investigate the cortical dynamics associated with Bhramari pranayama. We evaluated the power spectral density and cross-spectral density matrices in thirty trained practitioners before, during, and after they performed Bhramari pranayama. A significant increase (p < 0.05) in overall theta (12.64 µV2/Hz and 17.51 µV2/Hz) and alpha spectral power (19.36 µV2/Hz and 26.11 µV2/Hz) in the right temporal region of the brain was found during and after the pranayama. Additionally, a significant increase (p < 0.05) in gamma spectral power in the right (1.99 µV2/Hz) and left temporal (1.66 µV2/Hz) and parietal (1.62 µV2/Hz) regions, was observed during and after the pranayama. The cross-spectral density significantly increases after performing the Bhramari pranayama compared to before the pranayama in the alpha band (p < 0.05). Subsequently, all-to-all functional connectivity between EEG channels in sensor space is presented using circular graphs. Further graph theory analysis is employed to assess the effects of Bhramari pranayama on functional brain networks. These findings reveal that the alpha band's functional network topology of Bhramari meditators is more integrated during and after the meditation. Also, the power spectral changes are more prominent in the temporal region of the brain due to the generation of humming sound. By measuring these changes, meditators can determine the effectiveness of their pranayama practice inducing relaxation. Therefore, these results are valuable for enhancing pranayama practice and help in understanding the cortical mechanisms underlying pranayama.

Underwater source localization based on a matrix distance measurement

In this paper, a matrix distance measurement is introduced to underwater source localization. Matched field processing (MFP) and beamforming(BF) are widely used methods that both make the correlation between data and copy vectors. We proposed statistical matched field processing (SMFP) and statistical beamforming (SBF) in this paper that measure the distance between data and copy cross-spectral density matrices (CSDMs) in place of the correlation of the two vectors. As CSDM is Hermitian and positive semidefinite, Riemann distance is suitable. Matrix-matrix comparison increases the dimension of the system compared with vector-vector correlation, so as to include environmental uncertainty. SMFP and SBF are compared with conventional matched field processing (CMFP) and conventional beamforming (CBF) respectively in simulations. The results show that SMFP and SBF achieve satisfactory localization performances.

Singular-value decomposition and electromagnetic coherence of optical beams.

We investigate the implications of the singular-value decomposition of the cross-spectral density (CSD) matrix to the description of electromagnetic spectral spatial coherence of stationary light beams. We show that in a transverse plane any CSD matrix can be expressed as a mixture of two CSD matrices corresponding to beams which are fully polarized but in general spatially partially coherent. The polarization and coherence structures of these constituent beams are specified, respectively, by the singular vectors and singular values of the full CSD matrix. It follows that vector-beam coherence, including the coherence Stokes parameters and the degree of coherence, can be formulated in terms of only two correlation functions. We further establish two-point analogs of the spectral and characteristic decompositions of the polarization matrix and show that in the case of a Hermitian CSD matrix their composition is specified by the so-called degree of cross-polarization.

Non-Euclidean estimation of acoustic cross-spectral density matrices: Evaluation by matched-field localization in an ocean waveguide.

A set of non-Euclidean cross-spectral density matrix (CSDM) estimators based on the Frechet mean and median is applied to spatially localize an acoustic source in a stochastic shallow water environment driven by internal gravity waves. The resulting geometric-based matrix estimates are incorporated into matched-field processors defined through steering matrices using intrinsic distance measures between pairs of CSDMs on a Riemannian manifold for each replica source location. Their performance is evaluated using the probability of correct localization to determine any improvement in source localization over the conventional (maximum likelihood) statistical approach for CSDM estimation.

A non-Euclidean approach to source localization in an ocean waveguide

In recent years, there has been significant interest in the natural extension of signal processing techniques based on Euclidean distances to non-Euclidean metrics. Covariance or cross-spectral density matrices (CSDMs), often the basis for localization and detection processors, can be interpreted as “points” in a Riemannian manifold on which a distance metric can be defined. This mathematical structure has proven quite useful in a number of processing schemes involving detection, source localization and classification of data. In this presentation, I consider a fully geometric, non-Euclidean approach to source localization in a simulated shallow water ocean waveguide where the sound speed field is stochastic due to the presence of internal gravity waves. A number of different CSDM estimators are constructed by formulas describing the mean or median of a set of sample dyads obtained from Green function realizations of the propagating field from a source point to a vertical array of acoustic sensors. These CSDM estimates are not derived by statistics, but through the geometric structure of the CSDM. The resulting estimates are incorporated into matched-field localization processors whose measure of similarity or distance between CSDMs is determined on the manifold. Work supported by funds from the Office of Naval Research.

Use of the Cross-Spectral Density Matrix for Enhanced Passive Ultrasound Imaging of Cavitation.

Passive ultrasound imaging is of great interest for cavitation monitoring. Spatiotemporal monitoring of cavitation bubbles in therapeutic applications is possible using an ultrasound imaging probe to passively receive the acoustic signals from the bubbles. Fourier-domain (FD) beamformers have been proposed to process the signals received into maps of the spatial localization of cavitation activity, with reduced computing times with respect to the time-domain approach, and to take advantage of frequency selectivity for cavitation regime characterization. The approaches proposed have been mainly nonadaptive, and these have suffered from low resolution and contrast, due to the many reconstruction artifacts. Inspired by the array-processing literature and in the context of passive ultrasound imaging of cavitation, we propose here a robust estimation of the second-order statistics of data through spatial covariance matrices in the FD or cross-spectral density matrices (CSMs). The benefits of such formalism are illustrated using advanced reconstruction algorithms, such as the robust Capon beamformer, the Pisarenko class beamformer, and the multiple signal classification approach. Through both simulations and experiments in a water tank, we demonstrate that enhanced localization of cavitation activity (i.e., improved resolution and contrast with respect to nonadaptive approaches) is compatible with the rapid and frequency-selective approaches of the FD. Robust estimation of the CSM and the derived adaptive beamformers paves the way to the development of powerful passive ultrasound imaging tools.

Non-linear embedding of acoustic cross-spectral density matrices through diffusion maps.

Based on the notion of similarity or "distance" between cross-spectral density matrices (CSDMs), a recent analysis of matched-field source localization in a stochastic ocean waveguide provided evidence that geodesic distances between CSDMs could be employed to estimate the source location in range and depth. For M acoustic sensors configured as a vertical array, these M×M matrices were estimated from source and replica fields propagated to the array and interpreted as points in a Riemannian manifold whose dimension is M2. Because they serve as fundamental constructs for many source localization algorithms, visualizations of CSDM manifolds are illustrated here in an attempt to gain insight into this geometric approach by using simulated acoustic fields propagated through an ocean waveguide with internal wave-induced variability. The manifold is treated as an undirected, weighted graph whose nodes are CSDMs with edges (weights) describing a measure of similarity between nodes. A non-linear dimensionality reduction technique, diffusion maps, is applied to project these high-dimensional matrices onto a three-dimensional subspace using a spectral decomposition of the graph in an attempt to grasp relationships among such matrices. The mapping is designed to preserve the notion of distance between matrices, allowing for a meaningful visualization of the high-dimensional manifold.

Dimensionality reduction of cross-spectral density matrices using diffusion map projections

Cross-spectral density matrices (CSDMs) constructed for matched-field source localization in an ocean waveguide are both Hermitian and positive semi-definite, allowing them to be interpreted in a non-Euclidean geometric framework as “points” in a Riemannian manifold. This abstract matrix representation can be used to construct distance metrics on the manifold, and distances between pairs of such matrices provide a measure of similarity between the true source CSDM and various replica CSDMs. With this geometric interpretation, the shortest “distance” between pairs of source/replica CSDMs represents an estimate of the source location obtained from acoustic field amplitude and phase acquired on a vertical array [Finette and Mignerey, JASA 143 (2018)]. In this presentation, visualizations of CSDM manifolds obtained from simulated acoustic fields propagating in an ocean waveguide with internal wave-induced variability are illustrated to gain insight into this approach to passive source localization. Since the original manifold resides in a high-dimensional space determined by the number of sensors, manifold learning using diffusion maps is employed to reduce the dimensionality but constrained to retain the relative distance relationships among the CSDMs. The original manifold is projected down to three dimensions for visualization. [Work supported by the Office of Naval Research.]

Generating electromagnetic Schell-model sources using complex screens with spatially varying auto- and cross-correlation functions

We present a method to generate any physically realizable electromagnetic Schell-model source. Our technique can be directly implemented on existing vector-beam generators that utilize spatial light modulators for coherence control, beam shaping, and relative phasing. This work significantly extends published research on the subject, where control over the partially coherent source’s cross-spectral density matrix was limited.We begin by presenting the statistical optics theory necessary to derive and implement our method. We then apply our technique, both analytically and in simulation, to produce two electromagnetic Schell-model sources from the literature. We demonstrate control over the full cross-spectral density matrices of both partially coherent beams. We compare the simulated results of these two sources to the corresponding theoretical or designed quantities to validate our approach. We find, through examination of the two-dimensional correlation coefficients, that both sources converge to their desired, ensemble (or by ergodicity, “long-time”) statistics within 500 random field instances.Our method and subsequent findings will be useful in any application where control over beam shape, polarization, and spatial coherence are important. These include but are not limited to free-space/underwater optical communications, directed energy, medicine, atomic optics, and optical tweezers.

Resolvent-based modeling of coherent wave packets in a turbulent jet

Coherent turbulent wave-packet structures in a jet at Reynolds number 460000 and Mach number 0.4 are extracted from experimental measurements and are modeled as linear fluctuations around the mean flow. The linear model is based on harmonic optimal forcing structures and their associated flow response at individual Strouhal numbers, obtained from analysis of the global linear resolvent operator. These forcing-response wave packets ("resolvent modes") are first discussed with regard to relevant physical mechanisms that provide energy gain of flow perturbations in the jet. Modal shear instability and the nonmodal Orr mechanism are identified as dominant elements, cleanly separated between the optimal and suboptimal forcing-response pairs. A theoretical development in the framework of spectral covariance dynamics then explicates the link between linear harmonic forcing-response structures and the cross-spectral density (CSD) of stochastic turbulent fluctuations. A low-rank model of the CSD at given Strouhal number is formulated from a truncated set of linear resolvent modes. Corresponding experimental CSD matrices are constructed from extensive two-point velocity measurements. Their eigenmodes (spectral proper orthogonal or SPOD modes) represent coherent wave-packet structures, and these are compared to their counterparts obtained from the linear model. Close agreement is demonstrated in the range of "preferred mode" Strouhal numbers, around a value of 0.4, between the leading coherent wave-packet structures as educed from the experiment and from the linear resolvent-based model.

Analysis of Functional Connectivity and Oscillatory Power Using DICS: From Raw MEG Data to Group-Level Statistics in Python.

Communication between brain regions is thought to be facilitated by the synchronization of oscillatory activity. Hence, large-scale functional networks within the brain may be estimated by measuring synchronicity between regions. Neurophysiological recordings, such as magnetoencephalography (MEG) and electroencephalography (EEG), provide a direct measure of oscillatory neural activity with millisecond temporal resolution. In this paper, we describe a full data analysis pipeline for functional connectivity analysis based on dynamic imaging of coherent sources (DICS) of MEG data. DICS is a beamforming technique in the frequency-domain that enables the study of the cortical sources of oscillatory activity and synchronization between brain regions. All the analysis steps, starting from the raw MEG data up to publication-ready group-level statistics and visualization, are discussed in depth, including methodological considerations, rules of thumb and tradeoffs. We start by computing cross-spectral density (CSD) matrices using a wavelet approach in several frequency bands (alpha, theta, beta, gamma). We then provide a way to create comparable source spaces across subjects and discuss the cortical mapping of spectral power. For connectivity analysis, we present a canonical computation of coherence that facilitates a stable estimation of all-to-all connectivity. Finally, we use group-level statistics to limit the network to cortical regions for which significant differences between experimental conditions are detected and produce vertex- and parcel-level visualizations of the different brain networks. Code examples using the MNE-Python package are provided at each step, guiding the reader through a complete analysis of the freely available openfMRI ds000117 “familiar vs. unfamiliar vs. scrambled faces” dataset. The goal is to educate both novice and experienced data analysts with the “tricks of the trade” necessary to successfully perform this type of analysis on their own data.

Implications of Riemannian geometry in underwater acoustics

Signal processing involving source localization and detection in underwater acoustics often depends on comparing covariance or cross-spectral density matrices (CSDMs) estimated for data, replica or noise source data from passively sensed acoustic fields incident on sensor arrays. Such comparisons often depend on a measure of similarity between matrix pairs and this typically involves a Euclidean metric. While Euclidean-based metrics are ubiquitous, convenient and useful, they are not fundamental to this task. By exploiting the facts that a CSDM is Hermitian and positive semi-definite, one can interpret such matrices as points constrained to a Riemannian, rather than a Euclidean manifold, with the implication that similarity between matrix pairs should be measured using a metric consistent with the manifold’s intrinsic structure. This geometric interpretation leads to alternative matched-field processors for source localization involving the Riemannian distance as a measure of similarity and can be used to solve this inverse problem. Some implications of this non-Euclidean approach in underwater acoustics, as well as a possible extension to source detection are discussed. [Work supported by the Office of Naval Research.]

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.

Generation of non-uniformly correlated stochastic electromagnetic beams

Until now, there have been many reports concerning the generation and propagation of partially coherent beams due to their less influencing ability in turbulent atmosphere and random media. Of particular interest, a Gaussian-Schell model beam has been widely chosen as a special example of partially coherent beam, since its spatial coherence degree is dependent on position only through the difference between the two position vectors. In the scalar domain, many coherent models have been well studied such as Gaussian and multi-Gaussian Schell-model sources, Bessel-Gaussian and Laguerre-Gaussian Schell-model sources and so on. Based on the theory for devising genuine cross-spectral density matrices for a stochastic electromagnetic beam, several scalar models have been also extended to the electromagnetic domain. In recent years, the propagation of partially coherent beams with spatially varying and non-uniform correlations has become a hot topic, because of their interesting characteristics such as locally sharpened and laterally shifted intensity maxima. In one of our previous studies, we have experimentally investigated the generation of non-uniformly correlated partially coherent beams. However, to the best of our knowledge, so far, there has been no investigation on the generation of non-uniformly correlated stochastic electromagnetic beams. In this paper, we theoretically and experimentally investigate the generation of non-uniformly correlated stochastic electromagnetic beams. Based on the relation between phase correlation and optical coherence, we investigate the 22 cross-spectral density matrix and the coherence distribution of the non-uniformly correlated stochastic electromagnetic beam we generated. It is shown that the coherence degree between two points in the generated beam depends not only on the distance between them, but also on the distances between the points and the center of the beam. In experiment, we use the Matlab rand function to generate a random phase pattern with uniform distribution. The modulation magnitudes of different positions are different and follow an inverse Gaussian distribution in position. Dynamic phase patterns are created from a series of random grey-scale images. Two phase-only liquid crystal spatial light modulators are employed to display computer-generated dynamic phase patterns and modulate the two orthogonally polarized components of the incident coherent light, respectively, and generate a stochastic electromagnetic beam. We measure the correlation distribution of the generated beam in Young's two-pinhole experiment. It is shown that the experimental observations are in agreement with our theoretical analyses. Other kinds of non-uniformly correlated stochastic electromagnetic beams can also be obtained by this approach. Non-uniformly correlated stochastic electromagnetic beams may have some applications in optical manipulation and free-space optical communication.

Acoustic source localization in ocean waveguides using principles of Riemannian geometry

Source localization in underwater acoustics entails a comparison between acoustic fields involving some measure of correlation, looking for similarity between the acoustic field propagated from the true source location and replica fields propagated from different locations in the waveguide. The uniqueness of the deterministic Green function between pairs of source-receiver positions forms the basis for the solution to this inverse problem. We consider a novel approach to source localization based on non-Euclidean geometry, where the “distance” between cross-spectral density matrices (CSDMs) is used to estimate the spatial location of the source. The traditional Euclidean distance is not necessarily appropriate because CSDMs are not arbitrary points in space; rather, they form a manifold constrained by the facts that CSDMs are both Hermitian and positive definite. These properties naturally lead to the interpretation of geodesic distance between CSDMs as a measure of similarity between acoustic fields with this minimum distance, parametrized by replica source location, establishing an estimate of the source position. We discuss the underlying concepts and present simulation results for a waveguide with internal wave-induced ocean variability. Several Riemannian metrics are considered and compared to more traditional approaches to matched-field localization. [Work supported by the Office of Naval Research.]

Source localization in underwater waveguides using machine learning

The problem of underwater acoustic source localization is solved using an artificial neural network (ANN) under the machine learning framework. Source localization in a waveguide is posed as a classification problem by training a feed-forward neural network (FFN) with one hidden layer. In this paper, the acoustic pressure signals received by a vertical linear array are preprocessed by constructing the normalized cross-spectral density matrices (CSDM), which are used as input of the FFN. Each unit of the output layer represents one possible range (Here, for simplicity, source range is the only parameter that needs to be determined). Different from model-based localization methods such as matched field processing (MFP), ANN as a data-driven method can learn features from real acoustic data, thereby bypassing the sound propagation modeling step completely. Simulations show that FFN achieves a good performance in determining the source ranges even with deficient training data samples and low signal-to-noise ratios (SNRs). The validity of FFN in source localization is further demonstrated with vertical array data from Noise09 experiment where the ship is located successfully using FFN.

Stochastic-basis matched-field processing

Matched-field processing (MFP) suffers serious degradation due to environmental mismatch between received acoustic-field vectors and modeled replica vectors. Physical reasons for degradation include uncertainty caused by incomplete descriptions of the parameters and fields necessary for correct specification of the acoustic waveguide (i.e., environmental uncertainty), and system uncertainty associated with incomplete knowledge of the array configuration, source depth, etc. Recent research in the theory of stochastic-basis expansions (polynomial chaos) provides a mathematically consistent way of incorporating both types of uncertainty into MFP. Such expansions are used efficiently to construct replica matrices that steer high-rank subspaces capable of capturing signals with uncertain wavefronts. When combined with cross-spectral density matrices, stochastic-basis steering matrices enable the design of new processors with properties not previously evaluated in a MFP context. A maximum likelihood processor is developed which incorporates environmental uncertainty through polynomial chaos expansions. The processor can be written as a Frobenius product between an estimated cross-spectral density matrix and the inverse of a stochastic-basis replica matrix. This talk will outline the theoretical foundation of stochastic-basis MFP, and illustrate the method with some simulations. [Work supported by the Office of Naval Research.]

Van Cittert-Zernike theorem and symmetry properties of the normalized cross-spectral density matrix.

The van Cittert-Zernike theorem is formulated for the normalized cross-spectral density matrix of fields emitted by spatially incoherent sources. This formulation shows that sources with homogeneous polarization characteristics at second-order lead in the far field to normalized cross-spectral density matrices that possess high-symmetry properties. These symmetry properties are lost with polarization inhomogeneities of the source. These results are illustrated with ring-shaped sources.