Mainlobe Interference Suppression for Radar Network via RPCA-Based Covariance Matrix Reconstruction

Traditional adaptive beamformers are sensitive to model mismatch and they have poor performances when the input signal-to-noise ratio (SNR) is close to or more than interference-to-noise ratio (INR). In this letter, a novel robust adaptive beamforming (RAB) algorithm based on twice interference-plus-noise covariance (INC) matrix reconstruction is proposed. Firstly, the desired signal (DS) steering vector is estimated by the alternating projection algorithm based on the antenna array geometry and angular sector prior information. Then, the proposed method obtains the eigenvalue corresponding to DS by correlation coefficient and takes the place of it by noise power to obtain the first INC matrix. Furthermore, the second precise covariance matrix is reconstructed based on the theoretical INC matrix expression. The interferences powers are estimated by the theoretical formulation of the interference covariance matrix (ICM), which is derived from the first INC matrix. The simulation results demonstrate that the proposed algorithm achieves good performance especially when the input SNR is high, and its performance is close to the optimal beamformer in the case of look direction error.

This paper develops an improved covariance matrix estimator in the mean-variance optimisation setting. Well-known problems with the sample covariance matrix are that it is singular when the number of observations is less than the number of assets, and can be nearly singular when the number of observations exceeds the number of assets. Therefore, using the sample covariance matrix as an input in mean-variance optimisation can result in unreasonable optimal portfolios and badly biased estimates of Sharpe ratios. We address this problem by imposing structure on the estimated covariance matrix by putting constraints on the Sharpe ratio, asset return variances, and the variance of the global minimum variance portfolio. We show that the constrained maximum likelihood estimator (CMLE) performs better than the sample covariance matrix. Moreover, when the shrinkage approach is applied to the CMLE and single index covariance matrix, it performs better than the shrinkage estimator of Ledoit and Wolf (2004).

Summary The primary purpose of this paper is to propose empirical measures of the structural differences between two communities of plants or animals composed of the same species. Structure is defined to consist of; 1) the species in the community, 2) the pattern of interactions as represented by the covariance or correlation matrix of successive observations on each species, and 3) the mean abundances of each species in each of the two communities. Statistical tests are proposed for testing whether the covariance matrices and the vectors of mean densities for each community are equal and empirical measures of the differences between the covariance matrices and mean vectors are proposed. Given unequal covariance or correlation matrices the factor analysis model is used to derive empirical measures of the degree to which each variable of the ecosystem is responsible for the observed defferences in the pattern of interactions in each community. These tests and measures were applied to data gathered by Hunter (1966) on the abundances of six species of Drosophila censused monthly over a period of approximately two and a half years in two adjacent, but different habitats near Bogota, colombia. The two covariance matrices were significantly different indicating different patterns of interactions in the two Drosophila communities.

A novel semi-blind source separation framework towards maximum signal-to-interference ratio

A mainlobe interference suppression method is proposed based on eigen-projection processing and covariance matrix reconstruction. In the proposed method, the eigen-projection matrix is calculated based on the eigenvector of mainlobe interference to suppress the mainlobe interference in received echo data. The interference covariance matrix is reconstructed after the eigenvalue of mainlobe interference is replaced by the average value of noise eigenvalues to eliminate the effect of mainlobe interference to adaptive weight vector. Finally, the output of adaptive beamforming is calculated by processed echo data and adaptive weight vector obtained based on reconstructed covariance matrix. Simulation results show that the proposed method can suppress the mainlobe interference effectively and achieve excellent performance .

Estimating large covariance and precision (inverse covariance) matrices has become increasingly important in high dimensional statistics because of its wide applications. The estimation problem is challenging not only theoretically due to the constraint of its positive definiteness, but also computationally because of the curse of dimensionality. Many types of estimators have been proposed such as thresholding under the sparsity assumption of the target matrix, banding and tapering the sample covariance matrix. However, these estimators are not always guaranteed to be positive-definite, especially, for finite samples, and the sparsity assumption is rather restrictive. We propose a novel two-stage adaptive method based on the Cholesky decomposition of a general covariance matrix. By banding the precision matrix in the first stage and adapting the estimates to the second stage estimation, we develop a computationally efficient and statistically accurate method for estimating high dimensional precision matrices. We demonstrate the finite-sample performance of the proposed method by simulations from autoregressive, moving average, and long-range dependent processes. We illustrate its wide applicability by analyzing financial data such S&P 500 index and IBM stock returns, and electric power consumption of individual households. The theoretical properties of the proposed method are also investigated within a large class of covariance matrices.

Many applications require precise estimates of high-dimensional covariance matrices. The standard estimator is the sample covariance matrix, which is conceptually simple, fast to compute and has favorable properties in the limit of infinitely many observations. The picture changes when the dimensionality is of the same order as the number of observations. In such cases, the eigenvalues of the sample covariance matrix are highly biased, the condition number becomes large and the inversion of the matrix gets numerically unstable.
\nA number of alternative estimators are superior in the high-dimensional setting, which include as subcategories structured estimators, regularized estimators and spectrum correction methods. In this thesis I contribute to all three areas. 
\nIn the area of structured estimation, I focus on models with low intrinsic dimensionality. I analyze the bias in Factor Analysis, the state-of-the-art factor model and propose Directional Variance Adjustment (DVA) Factor Analysis, which reduces bias and yields improved estimates of the covariance matrix.
\nAnalytical shrinkage of Ledoit and Wolf (LW-Shrinkage) is the most popular regularized estimator. I contribute in three aspects: first, I provide a theoretical analysis of the behavior of LW-Shrinkage in the presence of pronounced eigendirections, a case of great practical relevance. I show that LW-Shrinkage does not perform well in this setting and propose aoc-Shrinkage which yields significant improvements. Second, I discuss the effect of autocorrelation on LW-Shrinkage and review the Sancetta-Estimator, an extension of LW-Shrinkage to autocorrelated data. I show that the Sancetta-Estimator is biased and propose a theoretically and empirically superior estimator with reduced bias. Third, I propose an extension of shrinkage to multiple shrinkage targets. Multi-Target Shrinkage is not restricted to covariance estimation and allows for many interesting applications which go beyond regularization, including transfer learning. I provide a detailed theoretical and empirical analysis.
\nSpectrum correction approaches the problem of covariance estimation by improving the estimates of the eigenvalues of the sample covariance matrix. I discuss the state-of-the-art approach, Nonlinear Shrinkage, and propose a cross-validation based covariance (CVC) estimator which yields competitive performance at increased numerical stability and greatly reduced complexity and computational cost. On all data sets considered, CVC is on par or superior in comparison to the regularized and structured estimators.
\nIn the last chapter, I conclude with a discussion of the advantages and disadvantages of all covariance estimators presented in this thesis and give situation-specific recommendations. In addition, the appendix contains a systematic analysis of Linear Discriminant Analysis as a model application, which sheds light on the interdependency between the generative model of the data and various covariance estimators.

Robust adaptive beamforming (RAB) based on interference-plus-noise covariance (INC) matrix reconstruction can experience performance degradation when model mismatch errors exist, particularly when the input signal-to-noise ratio (SNR) is large. In this work, we devise an efficient RAB technique for dealing with covariance matrix reconstruction issues. The proposed method involves INC matrix reconstruction using an idea in which the power and the steering vector of the interferences are estimated based on the power method. Furthermore, spatial match processing is computed to reconstruct the desired signal-plus-noise covariance matrix. Then, the noise components are excluded to retain the desired signal (DS) covariance matrix. A key feature of the proposed technique is to avoid eigenvalue decomposition of the INC matrix to obtain the dominant power of the interference-plus-noise region. Moreover, the INC reconstruction is carried out according to the definition of the theoretical INC matrix. Simulation results are shown and discussed to verify the effectiveness of the proposed method against existing approaches.

SummarySample covariance matrices play a central role in numerous popular statistical methodologies, for example principal components analysis, Kalman filtering and independent component analysis. However, modern random matrix theory indicates that, when the dimension of a random vector is not negligible with respect to the sample size, the sample covariance matrix demonstrates significant deviations from the underlying population covariance matrix. There is an urgent need to develop new estimation tools in such cases with high‐dimensional data to recover the characteristics of the population covariance matrix from the observed sample covariance matrix. We propose a novel solution to this problem based on the method of moments. When the parametric dimension of the population spectrum is finite and known, we prove that the proposed estimator is strongly consistent and asymptotically Gaussian. Otherwise, we combine the first estimation method with a cross‐validation procedure to select the unknown model dimension. Simulation experiments demonstrate the consistency of the proposed procedure. We also indicate possible extensions of the proposed estimator to the case where the population spectrum has a density.

Without the response variation constraint, the main‐lobe responses of wideband beamforming usually vary with the frequencies of interest, and the beam pointing may deviate from the incident angle of the desired signal. If the interference‐plus‐noise covariance matrix (INCM) is directly replaced by the sample covariance matrix, the desired signal in the sample data may result in self‐null problem, which can lead to the degradation of the performance of wideband beamformers. Through frequency division, a novel wideband adaptive beamforming is proposed by reconstructing the INCM and solving the convex optimisation problem. Using fast Fourier transform (FFT), the time‐domain samples are transformed into the frequency field, and the covariance matrix of each subband can be obtained from the frequency‐domain data. In the spatial‐frequency domain, the INCM and signal‐plus‐noise covariance matrix can be respectively reconstructed by integrating the stacked steering vector and the Capon spatial spectrum over the corresponding angular region. Then, we establish the optimisation criteria, that is, minimising the output interference‐plus‐noise power, minimising the main‐lobe spatial response variation, and constraining the response deviation of the desired signal. A novel convex optimisation model can be established, and then the weight coefficients for time‐domain wideband beamforming can be obtained by solving the optimisation problem. The simulation and experimental results demonstrate that the proposed beamformer can provide better performance compared with other tested beamformers and maintain good robustness in various applications.

We address the problem of estimating high-dimensional covariance matrices (CM) for the explicit purpose of supervised anomaly detection, in the case when the number n of data points is lower than their dimensionality p. This is increasingly common with the emergence of the Internet of Things that makes it possible to collect data from many sensors simultaneously, resulting in very high-dimensional data points. When we attempt to perform anomaly detection for such data by modeling the normal behavior of the system by means of a multivariate Gaussian distribution, and n < p, the sample CM is singular, and cannot be used directly without some form of regularization. In contrast to existing methods for CM regularization that aim to fit the training data accurately, we propose a regularization algorithm for CM estimation that directly aims to maximize the area under the resulting receiver-operator characteristic (AUROC) for the ultimate decision problem that needs to be solved: anomaly detection. Experiments on test problems demonstrate the ability of the proposed algorithm to find CM estimates significantly better at anomaly detection than existing estimation methods that are unaware of the decision task that the CMs they produce will be used in.

Acoustic environment has a low signal-to-noise ratio (SNR); hence, array signal processing is widely used for noise reduction and signal enhancement. The actual ambient noise includes uncorrelated noise and correlated noise. The received noises of the two arbitrary array elements are correlated. Consequently, the performance of array signal processing method decreases obviously. Aiming at this problem, the real-part elimination of covariance matrix method is proposed. Firstly, from a physical point of view, the noise signals can be generated by using a number of uncorrelated noise sources: the more the noise sources, the less the error between the noise from the model and the actual noise will be. Theoretically, the noise field is decomposed into the symmetrical noise field and the asymmetrical noise field. A number of noise sources generate the symmetrical noise fields; the directions of these noise sources are symmetric, and the powers of two arbitrary symmetric sources are the same. Secondly, the symmetry of the ambient noise is analyzed, as a result, the symmetrical noise can only affect the real part of the covariance matrix. Thirdly, the real part of covariance matrix is eliminated in order to reduce the noise, and then the delay-and-sum beamforming is achieved by using only the imaginary part. The advantages are that the output signal-to-noise ratio is increased and the noise output power is reduced obviously; the disadvantage is that it produces a false target. The azimuth of the actual target differs from that of the false target by 180, and the false target cannot be distinguished. Finally, to eliminate the false target, the real part of the signal covariance matrix is reconstructed by establishing a constrained optimization problem, which is solved by using the particle swarm algorithm. Then, the reconstructed covariance matrix composed of the imaginary part and the reconstruction of real part is applied to delay-and-sum beamforming, as a result, the false target is eliminated. The simulation results show that the real-part elimination of covariance matrix method reduces the symmetrical ambient noise, the noise output power is reduced, the output signal-to-noise ratio is increased, and this method improves the performance of array signal processing. The experimental results show that the output SNRs of two targets with using the imaginary part of covariance matrix are increased by 3.57 dB and 3.149 dB, respectively, and the output SNRs of two targets with using the reconstructed covariance matrix are increased by 7.027 dB and 6.985 dB, respectively. The real-part elimination of covariance matrix method is easy to implemente, and has a definite value for engineering application.

A new beamformer based on covariance matrix reconstruction is introduced. The essence of the new approach is to eliminate the desired signal component in the sample covariance matrix and thus complex integral operation is avoided in the procession of covariance matrix reconstruction. Besides, the actual array steering vector is estimated by a new technique. Contrary to other reference beamformers, simulation results demonstrate the effectiveness of our proposed method.

Journal Article Nonparametric estimation of large covariance matrices of longitudinal data Get access Wei Biao Wu, Wei Biao Wu Search for other works by this author on: Oxford Academic Google Scholar Mohsen Pourahmadi Mohsen Pourahmadi Search for other works by this author on: Oxford Academic Google Scholar Biometrika, Volume 90, Issue 4, December 2003, Pages 831–844, https://doi.org/10.1093/biomet/90.4.831 Published: 01 December 2003

In this paper the model of experiment in which several characteristics are measured or the measurements are repeated in time, is considered. It is assumed that the covariance matrix of the characteristics/time points is unknown, but has the banded Toeplitz structure. The problem of estimating such a covariance structure was considered by e.g. Cui et al. (2016), Filipiak et al. (2018). We discuss two methods that aim to deal with singularity and numerical ill-conditioning of the estimators of covariance matrix under high-dimensional regime. The method of estimation of banded Toeplitz covariance matrix based on a shrinking, described by Ledoit and Wolf (2004) for unstructured covariance matrices is proposed. The estimators obtained with the use of the proposed method are then compared with the estimators obtained by the projection of sample covariance matrix onto the asymptotic cone of the nonnegative definite Toeplitz matrices (Filipiak et al. 2018). For this purpose simulation studies concerning bias and risk are conducted for several sets of parameters.