Augmented Covariance Matrix Research Articles

Classification of BCI-EEG Based on the Augmented Covariance Matrix.

Electroencephalography signals are recorded as multidimensional datasets. We propose a new framework based on the augmented covariance that stems from an autoregressive model to improve motor imagery classification. From the autoregressive model can be derived the Yule-Walker equations, which show the emergence of a symmetric positive definite matrix: the augmented covariance matrix. The state-of the art for classifying covariance matrices is based on Riemannian Geometry. A fairly natural idea is therefore to apply this Riemannian Geometry based approach to these augmented covariance matrices. The methodology for creating the augmented covariance matrix shows a natural connection with the delay embedding theorem proposed by Takens for dynamical systems. Such an embedding method is based on the knowledge of two parameters: the delay and the embedding dimension, respectively related to the lag and the order of the autoregressive model. This approach provides new methods to compute the hyper-parameters in addition to standard grid search. The augmented covariance matrix performed ACMs better than any state-of-the-art methods. We will test our approach on several datasets and several subjects using the MOABB framework, using both within-session and cross-session evaluation. The improvement in results is due to the fact that the augmented covariance matrix incorporates not only spatial but also temporal information. As such, it contains information on the nonlinear components of the signal through the embedding procedure, which allows the leveraging of dynamical systems algorithms. These results extend the concepts and the results of the Riemannian distance based classification algorithm.

Source Localization Based on Generalized Augmented Covariance Matrix Reconstruction in Microwave Interferometric Radiometry

Source Localization Based on Generalized Augmented Covariance Matrix Reconstruction in Microwave Interferometric Radiometry

Structured Nyquist Correlation Reconstruction for DOA Estimation With Sparse Arrays

Sparse arrays are known to achieve an increased number of degrees-of-freedom (DOFs) for direction-of-arrival (DOA) estimation, where an augmented virtual uniform array calculated from the correlations of sub-Nyquist spatial samples is processed to retrieve the angles unambiguously. Nevertheless, the geometry of the derived virtual array is dominated by the specific physical array configurations, as well as the deviation caused by the practical unforeseen circumstances such as detection malfunction and missing data, resulting in a quite sensitive model for virtual array signal processing. In this paper, we propose a novel sparse array DOA estimation algorithm via structured correlation reconstruction, where the Nyquist spatial filling is implemented on the physical array with a compressed transformation related to its equivalent filled array to guarantee the general applicability. While the unknown correlations located in the whole rows and columns of the augmented covariance matrix lead to the fact that strong incoherence property is no longer satisfied for matrix completion, the structural information is introduced as <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a prior</i> to formulate the structured correlation reconstruction problem for matrix reconstruction. As such, the reconstructed covariance matrix can be effectively processed with full utilization of the achievable DOFs from the virtual array, but with a more flexible constraint on the array configuration. The described estimation problem is theoretically analyzed by deriving the corresponding Cram´er-Rao bound (CRB). Moreover, we compare the derived CRB with the performance of the virtual array interpolation-based algorithm. Extensive simulations are conducted to demonstrate the effectiveness of the proposed DOA estimation algorithm in terms of achievable DOFs, resolution, and estimation accuracy.

DOA Estimation of Mixed Circular and Noncircular Sources Using Nonuniform Linear Array

In this article, an algorithm for accurately estimating the directions-of-arrival (DOAs) of mixed circular and noncircular signals with a nonuniform linear array is devised. Via applying coarray interpolation on the difference and sum coarrays, we first formulate two nuclear norm minimization problems to reconstruct an augmented covariance matrix and an augmented unconjugated covariance matrix, respectively. Based on the reconstructed matrices, we further build the extended covariance matrix in the coarray domain. Finally, we construct several spectral functions to successively estimate the DOAs of the nonstrictly noncircular, strictly noncircular, and circular signals. Unlike the existing schemes, the proposed algorithm can identify accurately circular, strictly noncircular, and nonstrictly noncircular signals. Numerical results demonstrate the superiority of the proposed algorithm over existing techniques.

Direction-of-Arrival Estimation for Nested Array Using Mixed-Resolution ADCs

In this letter, an effective method for estimating direction-of-arrival (DOA) using the nested array via mixed-resolution (MR) quantization is developed. First, we derive the output signal model of the nested array under MR quantization, and utilize MR quantized data to construct a quantized covariance matrix. Subsequently, we recover an unquantized augmented covariance matrix by formulating a rank-minimization problem in the coarray domain. Finally, we perform DOA estimation using the recovered covariance matrix. The proposed method can achieve excellent performance close to that of the existing methods using high-resolution data. Simulation results demonstrate the validity of our approach.

RFI Localization via Reweighted Nuclear Norm Minimization in Microwave Interferometric Radiometry

Radio frequency interference (RFI) has become an increasing and challenging problem in microwave interferometric radiometry (MIR). Accurate localization of RFI sources is helpful to provide location information for switching off unauthorized transmitters causing RFI and mitigating the impact of these RFI sources. In this article, we propose a new RFI localization method based on reweighted nuclear norm minimization (RNNM). This method exploits the low-rank property of augmented covariance matrix (ACM) collecting visibility samples in MIR and introduces a singular value weighting strategy to consider different contributions of ACM components. First, ACM is constructed from the original covariance matrix of sparse array, which increases the degree of freedom (DOF) for array processing and hence improves the angular resolution performance. Second, we present a fixed point iteration (FPI)-based RNNM Algorithm, named FRA, to achieve low-rank approximation of ACM involving contribution degrees of ACM components. In this way, the ACM components corresponding to RFI signals are retained well and ones corresponding to background noises are suppressed. Third, we use a subspace-based direction-of-arrival (DOA) estimation approach, i.e., MUSIC algorithm, on the weighted completed ACM (WCACM) (obtained by FRA in the second stage) to locate the potential RFI sources. Retrieved results using synthetic data and real soil moisture and ocean salinity (SMOS) satellite data demonstrate that the proposed RNNM-based method not only has the superiority on improved detection performance, especially for identifying weak sources, but also shows better or competitive localization accuracy and angular resolution, compared with the existing commonly used RFI localization methods in MIR.

Mitigating baryonic effects with a theoretical error covariance

ABSTRACTOne of the primary sources of uncertainties in modelling the cosmic-shear power spectrum on small scales is the effect of baryonic physics. Accurate cosmology for stage-IV surveys requires knowledge of the matter power spectrum deep in the non-linear regime at the per cent level. Therefore, it is important to develop reliable mitigation techniques to take into account baryonic uncertainties if information from small scales is to be considered in the cosmological analysis. In this work, we develop a new mitigation method for dealing with baryonic physics for the case of the shear angular power spectrum. The method is based on an augmented covariance matrix that incorporates baryonic uncertainties informed by hydrodynamical simulations. We use the results from 13 hydrodynamical simulations and the residual errors arising from a fit to a ΛCDM model using the extended halo model code HMCode to account for baryonic physics. These residual errors are used to model a so-called theoretical error covariance matrix that is added to the original covariance matrix. In order to assess the performance of the method, we use the 2D tomographic shear from four hydrodynamical simulations that have different extremes of baryonic parameters as mock data and run a likelihood analysis comparing the residual bias on Ωm and σ8 of our method and the HMCode for an LSST-like survey. We use different modelling of the theoretical error covariance matrix to test the robustness of the method. We show that it is possible to reduce the bias in the determination of the tested cosmological parameters at the price of a modest decrease in the precision.

Augmented Covariance Matrix Reconstruction for DOA Estimation Using Difference Coarray

As is well known, nonuniform linear arrays have significant advantages in array aperture and degrees of freedom over uniform linear arrays. Using their difference coarrays, subspace-based approaches can be utilized to perform underdetermined and high-resolution direction-of-arrival (DOA) estimation. However, the subspace-based approaches depend on the covariance matrix reconstruction in the coarray domain, which are not statistically efficient when the number of sources is more than one and less than the number of sensors. In this paper, to overcome this drawback, we devise an augmented covariance matrix reconstruction algorithm for DOA estimation in the coarray domain. The proposed algorithm recovers the complete augmented covariance matrix by solving a rank-minimization problem. But unlike the conventional schemes, it exploits the estimation error distribution of the incomplete augmented covariance matrix to derive the constraint condition of the rank-minimization problem. Based on the reconstructed augmented covariance matrix, we can enhance the DOA estimation performance for multiple source scenario at high signal-to-noise ratio. Although our algorithm is developed based on the non-consecutive coarray, it is also suitable for the consecutive coarray. Numerical results demonstrate the superiority of the proposed algorithm over several existing approaches.

Improved “Augmented MVDR Spectrum-Based Frequency Estimation Algorithm” for Unbalanced Power Systems

This article proposes mymargin an improved augmented iterative minimum variance distortionless response (AI-MVDR) frequency estimation algorithm which resolves a significant performance degradation of the AI-MVDR algorithm at off-nominal frequency of unbalanced power signals. This problem of the AI-MVDR algorithm is caused by the omitted terms in computing the augmented covariance matrix which is suggested to compensate the unbalance effect. In this article, a new frequency estimator is analytically derived considering the effect of the omitted terms by replacing a fixed nominal frequency parameter with a variable frequency parameter in obtaining the augmented covariance matrix. The performance of the derived equation is also verified through computer simulations.

A Matrix Completion Based Method for RFI Source Localization in Microwave Interferometric Radiometry

The Soil Moisture and Ocean Salinity (SMOS) mission led by the European Space Agency (ESA) is aimed to globally monitor the Earth surface moisture and ocean salinity. As the single payload of SMOS satellite, the Microwave Interferometric Radiometer with Aperture Synthesis (MIRAS) operates in the protected L-band. Nonetheless, the artificial sources emitting close to or/and fully in this band are contaminating the collected remote sensing data and deteriorating the performance of the SMOS mission. Identifying and localizing such sources is crucial to improve the quality of SMOS scientific products. In this article, we propose a method based on matrix completion (MC) for localization of radio frequency interference (RFI) sources. This method mainly exploits the low-rank property of the augmented covariance matrix (ACM) of the sparse array and addresses the ACM incompleteness (i.e., sampling data loss) due to inherent array geometry (e.g., the SMOS Y-shaped array) or potential hardware malfunction (e.g., the correlator failure). Validation results show that, compared with existing RFI localization approaches such as the discrete Fourier transformation (DFT) inversion and covariance-based direction-of-arrival (DOA) estimation, the proposed MC method possesses competitive localization accuracy, superior spatial resolution, and reduced artifacts.

Two-dimensional direction-of-arrival estimation for cylindrical nested conformal arrays

Recently, nested arrays have received considerable interest because such array configurations can be systematically designed and their degrees of freedom (DOFs) can be expressed in closed form. Compared with uniform linear arrays, they have obvious advantages in array aperture and DOFs. In this paper, we introduce nested subarray structure to cylindrical conformal array to form a nested conformal array, and develop a corresponding algorithm for two-dimensional (2-D) direction-of-arrival (DOA) estimation. By vectorizing the covariance matrices of the nested subarray outputs, we firstly generate the three-parallel difference coarray signal and derive the rotational invariance structures between three coarray manifolds. An augmented covariance matrix of the coarray outputs is then constructed via three-dimensional spatial smoothing processing. Finally, the 2-D DOAs of the sources are estimated by the total least squares (TLS)-ESPRIT method. Compared with the traditional schemes with uniform conformal arrays, our algorithm fully exploits the extended aperture of the nested subarrays, and thus enhances the DOA estimation accuracy. Numerical results demonstrate the superiority of the proposed algorithm in comparison to the existing methods.

Efficient Two-Dimensional Direction Finding Algorithm for Rectilinear Sources Under Unknown Mutual Coupling.

Digital communication signals in wireless systems may possess noncircularity, which can be used to enhance the degrees of freedom for direction-of-arrival (DOA) estimation in sensor array signal processing. On the other hand, the electromagnetic characteristics between sensors in uniform rectangular arrays (URAs), such as mutual coupling, may significantly deteriorate the estimation performance. To deal with this problem, a robust real-valued estimator for rectilinear sources was developed to alleviate unknown mutual coupling in URAs. An augmented covariance matrix was built up by extracting the real and imaginary parts of observations containing the circularity and noncircularity of signals. Then, the actual steering vector considering mutual coupling was reparameterized to make the rank reduction (RARE) property available. To reduce the computational complexity of two-dimensional (2D) spectral search, we individually estimated y-axis and x-axis direction-cosines in two stages following the principle of RARE. Finally, azimuth and elevation angle estimates were determined from the corresponding direction-cosines respectively. Compared with existing solutions, the proposed method is more computationally efficient, involving real-valued operations and decoupled 2D spectral searches into twice those of one-dimensional searches. Simulation results verified that the proposed method provides satisfactory estimation performance that is robust to unknown mutual coupling and close to the counterparts based on 2D spectral searches, but at the cost of much fewer calculations.

Two-Dimensional DOA Estimation Using Two Parallel Nested Arrays

In this letter, we propose a new method for two-dimensional (2-D) direction-of-arrival (DOA) estimation using two parallel nested arrays. In this method, an augmented covariance matrix is firstly constructed using the outputs of two parallel difference coarrays. Based on the augmented covariance matrix, the 2-D DOA estimation problem is then converted into two one-dimensional estimation problems. Finally, the azimuth and elevation angle estimates are derived using the estimated direction cosines. Unlike the traditional methods, our algorithm exploits the difference coarray to increase the array aperture and degrees of freedom. Moreover, it does not require any peak searching and can achieve parameter automatic pairing. Numerical simulations are conducted to verify the performance of the proposed method.

Grid-Less DOA Estimation Using Sparse Linear Arrays Based on Wasserstein Distance

Sparse linear arrays, such as nested and co-prime arrays, are capable of resolving $O(M^2)$ sources using only $O(M)$ sensors by exploiting their so-called difference coarray model. One popular approach to exploit the difference coarray model is to construct an augmented covariance matrix from the sample covariance matrix. By applying common direction-of-arrival (DOA) estimation algorithms to this augmented covariance matrix, more sources than the number of sensors can be identified. In this letter, inspired by the optimal transport theory, we develop a new approach to construct this augmented covariance matrix. We formulate a structured covariance estimation problem that minimizes the Bures–Wasserstein distance between the sample covariance matrix and the subsampled augmented covariance matrix, which can be either casted to a semi-definite programming problem, or directly solved using gradient-based methods. Our approach contributes to a new grid-less DOA estimation algorithm for sparse linear arrays. Numerical examples show that our approach achieves state-of-art estimation performance.

Direction-of-Arrival Estimation of Circular and Noncircular Wideband Source Signals via Augmented Envelope Alignment

In practical applications of array system, the incident signals may come from either circular or noncircular sources, or both. The problem of direction-of-arrival (DOA) estimation is herein addressed for both circular and noncircular wideband source signals. Based on the concept of effective rank, the aligned augmented effective noise subspace, spanned by the subordinate eigenvectors of the aligned array output augmented covariance matrix, is constructed for each scanning angle. It is shown that the aligned augmented effective noise subspace corresponding to a true DOA of signal is: 1) approximately orthogonal to a frequency-invariant one-dimensional subspace spanned by the aligned augmented transfer vector of the signal in the presence of multiple strictly noncircular sources; and 2) approximately orthogonal to a frequency-invariant two-dimensional subspace spanned by the aligned augmented transfer matrix of the signal in the presence of multiple circular or partially noncircular sources. This enables the noncircularity exploited wideband DOA estimation based on rank-reduction and norm-nulling. The performance of the proposed method has been evaluated and compared with some existing popular wideband DOA estimators by extensive simulations.

Underdetermined Wideband DOA Estimation for Off-Grid Sources with Coprime Array Using Sparse Bayesian Learning.

Sparse Bayesian learning (SBL) is applied to the coprime array for underdetermined wideband direction of arrival (DOA) estimation. Using the augmented covariance matrix, the coprime array can achieve a higher number of degrees of freedom (DOFs) to resolve more sources than the number of physical sensors. The sparse-based DOA estimation can deteriorate the detection and estimation performance because the sources may be off the search grid no matter how fine the grid is. This dictionary mismatch problem can be well resolved by the SBL using fixed point updates. The SBL can automatically choose sparsity and approximately resolve the non-convex optimizaton problem. Numerical simulations are conducted to validate the effectiveness of the underdetermined wideband DOA estimation via SBL based on coprime array. It is clear that SBL can obtain good performance in detection and estimation compared to least absolute shrinkage and selection operator (LASSO), simultaneous orthogonal matching pursuit least squares (SOMP-LS) , simultaneous orthogonal matching pursuit total least squares (SOMP-TLS) and off-grid sparse Bayesian inference (OGSBI).

Regularized Extended Estimation With Stabilized Exponential Forgetting

This technical note concerns the problem of variable regularized estimation of time-varying nonlinear systems from the Bayesian viewpoint. The questions of how to impose the posterior information being variably regularized and how to forget this information are carefully discussed. The estimator design adopts the strategy of the iterated Kalman filter but differs in that, instead of the separated moments of the linearized system, only the augmented covariance matrix is updated. To suppress obsolete information, a decision problem involving the Kullback-Leibler divergence is solved. The decision provides the best combination of a pair of time-evolution model hypotheses in terms of the geometric mean. As a result, exponential forgetting with the adaptively tuned factor is inserted into the estimation process. The regularization of the investigated statistics is induced through the processing of externally supplied information. The presented estimator allows for absolute discarding or, conversely, retention of external information produced in terms of the Normal-Wishart distribution.

Facilitating an expectation-maximization (EM) algorithm to solve an integrated choice and latent variable (ICLV) model with fully correlated latent variables

It is well known that estimating the parameters of an integrated choice and latent variable (ICLV) model is not a trivial undertaking. The log-likelihood of an ICLV model cannot be evaluated analytically, and can only be evaluated by a simulation that requires large numbers of sample draws. While conducting simulation-based model estimations, researchers often encounter an estimation failure. Sohn (2017) suggests a novel estimation method to circumvent the problem by using an expectation-maximization algorithm (EM). However, a drawback of this method continues to be the requirement of a huge amount of computer memory to deal with an augmented covariance matrix. In the present study, this problem was overcome by connecting each latent variable in a structural equation to all individual specific variables. This restriction did not hamper the utility of an ICLV model during empirical experimentation. The main contribution of this study is to introduce a simple method devised to solve large-scale ICLV models.

Wideband source enumeration and direction-of-arrival estimation using sparse array periodogram averaging in low snapshot scenarios

Sparse arrays, such as minimum redundancy arrays and coprime arrays, often exploit the second-order statistics of the propagating field to localize more sources than sensors by constructing an augmented covariance matrix (ACM) from the estimated spatial correlation. The source localization performance largely depends on the number of snapshots available, which might be limited in many acoustical environments due to the propagation speed, large array aperture, and non-stationary field. This paper proposes a new approach for wideband source enumeration and direction-of-arrival (DOA) estimation on any sparse array geometry. The proposed algorithm decomposes the wideband signals into multiple disjoint frequency bands, computes the narrowband spatial periodograms and averages them to reinforce the sources’ spatial spectral information. The spatial correlation estimated from the wideband periodogram populates the diagonals of a Hermitian Toeplitz ACM. This ACM then goes through eigenvalue decomposition, where i...

Stokes parameters and DOAs estimation of partially polarized sources using a EM vector sensor

In this paper, the problem of polarization and direction-of-arrival (DOA) estimation using an electromagnetic (EM) vector sensor is addressed based on the augmented second-order statistics of quaternions. The augmented quaternion model of an EM vector sensor is introduced by employing the augmented second-order statistics of quaternions, which include the information in both the standard covariance and the pseudo-covariance. The fact is revealed that the Stokes parameters of an EM wave can be extracted from an augmented covariance matrix. An augmented quaternion-based approach is proposed for the Stokes parameters and DOA estimation of polarized EM waves. The proposed approach uses only information in three imaginary parts of both the standard covariance and the pseudo-covariance matrix. In theory, it is guaranteed to remove the effect of the additive noise which is uncorrelated between measurement channels. The proposed approach is applicable to both completely polarized and partially polarized EM waves. Simulations are used to verify the performance of the proposed approach.