Signal Covariance Matrices Research Articles

A KKT Conditions Based Transceiver Optimization Framework for RIS-Aided Multiuser MIMO Networks

In many core problems of signal processing and wireless communications, Karush-Kuhn-Tucker (KKT) conditions based optimization plays a fundamental role. Hence we investigate the KKT conditions in the context of optimizing positive semidefinite matrix variables under nonconvex rank constraints. More explicitly, based on the properties of KKT conditions, we optimize a reconfigurable intelligent surface (RIS) aided multi-user multi-input multi-output (MU-MIMO) network. Specifically, we consider the capacity maximization and sum mean square error (MSE) minimization problems of both the RIS-aided MU-MIMO uplink (UL) and downlink (DL) under multiple weighted power constraints and rank constraints. As for the RIS-aided MU-MIMO UL, the optimal structures of the signal covariance matrices are derived based on the KKT conditions. Furthermore, an efficient procedure is designed for solving the capacity maximization and sum mean square error (MSE) minimization problems. Then the UL-DL dualities are exploited for solving the capacity maximization and MSE minimization problems of the RIS-aided MU-MIMO DL based on the results of the UL optimization. Hence in the proposed framework, the phase shifting matrix of the RIS is jointly optimized with the signal covariance matrices for both the UL and DL. Our simulation results demonstrate the performance advantages of the proposed framework.

Multi-source signal alignment and efficient multi-dimensional feature classification in the application of EEG-based subject-independent drowsiness detection

Electroencephalography (EEG)-based driving drowsiness detection in brain computer interface (BCI) systems is an effective way to prevent traffic accidents. Usually a pre-calibration process is required for a specific target because of significant inter-subject variability in EEG signals. Recently transfer learning methods including domain adaptation and deep neural networks are proposed to be applied in subject-independent applications. However, feature extraction in domain adaptation does not consider the multimodal characteristics of EEG signals whereas deep neural networks consume massive computing resources with tremendous parameters. To overcome the limitations, we propose a multi-source signal alignment (MSSA) and multi-dimensional feature classification framework. EEG signals from multiple source subjects are directly aligned with the target subject via one-versus-one minimization of signal covariance matrices. Then the generalized multi-dimensional features are extracted and classified via tensor network (TN). Extensive experiments were conducted in a recently published EEG dataset during a sustained-attention driving task for subject-independent drowsiness detection. Compared with state-of-the-art transfer learning methods, MSSA-TN improves classification accuracy by at least 3.71%, which is promising in developing practical drowsiness detection systems.

Suppression of motion noise based on a linear‐homomorphic filtering algorithm in airborne electromagnetic survey

ABSTRACTThe movement of the receiver coil in a magnetic field produces motion noise in the airborne electromagnetic field. With an amplitude that is several orders of magnitude higher than that of other noise sources, this motion noise is one of the main low‐frequency (<1 kHz) noise sources. Especially in the late decay period, the residual motion noise produces false anomalous information, which is a key problem of low‐base‐frequency airborne electromagnetic surveys. In this paper, a method for calculating the motion noise is proposed. This method takes into account both the earth's magnetic field and the active secondary magnetic field and introduces additive and multiplicative motion noise. Then, a linear‐homomorphic filtering algorithm is proposed to handle the additive and multiplicative motion noises. In the linear filtering part of our algorithm, the minimum noise fraction transform is applied to suppress additive motion noise. It provides sets of components ordered by the signal‐to‐noise ratio by deriving the eigenvalues and eigenvectors for the signal and uncorrelated noise covariance matrices. In the homomorphic filtering part, the multiplicative motion noise is removed. To do this, the adaptive‐width smoothing algorithm is applied to the logarithm of the filtered with the linear filtering process profile data. The results of synthetic and field data demonstrate that the linear filtering process based on the minimum noise fraction transform is effective for very low signal‐to‐noise ratios of the late‐time data, and the homomorphic filtering process can help distinguish small anomalies of these channels.

Superpixel-Based Minimum Noise Fraction Feature Extraction for Classification of Hyperspectral Images

In this paper, a novel feature extraction technique called SuperMNF is proposed, which is an extension of the minimum noise fraction (MNF) transformation. In SuperMNF, each superpixel has its own transformation matrix and MNF transformation is performed on each superpixel individually. The basic idea behind the SuperMNF is that each superpixel contains its specific signal and noise covariance matrices which are different from the adjacent superpixels. The extracted features, owning spatial-spectral content and provided in the lower dimension, are classified by maximum likelihood classifier and support vector machines. Experiments that are conducted on two real hyperspectral images, named Indian Pines and Pavia University, demonstrate the efficiency of SuperMNF since it yielded more promising results than some other feature extraction methods (MNF, PCA, SuperPCA, KPCA, and MMP).

Maximum Entropy-Based Interference-Plus-Noise Covariance Matrix Reconstruction for Robust Adaptive Beamforming

To ensure signal receiving quality, robust adaptive beamforming (RAB) is of vital importance in modern communications. In this letter, we propose a new low-complexity RAB approach based on interference-plus-noise covariance matrix (IPNC) reconstruction and steering vector (SV) estimation. In this method, the IPNC and desired signal covariance matrices are reconstructed by estimating all interference powers as well as the desired signal power using the principle of maximum entropy power spectrum (MEPS). Numerical simulations demonstrate that the proposed method can provide superior performance to several previously proposed beamformers.

Residual least-squares collocation: use of covariance matrices from high-resolution global geopotential models

The paper presents a modified formulation of least-squares collocation. This residual least-squares collocation (RLSC) includes a remove--compute--restore procedure with a high-resolution global geopotential model (GGM) and a topographic gravitational potential model. In contrast to previous approaches, in RLSC, the remaining input residuals are modeled with error covariance matrices instead of signal covariance matrices. Therefore, we include the full variance--covariance information of a high-resolution GGM, namely the XGM2016, to the procedure. The included covariance matrices are anisotropic and location-dependent and enable a realistic error modeling of a target area. This fact represents an advantage over covariance matrices derived from signal degree variances or empirical covariance fitting. Additionally, due to the stochastic modeling of all involved components, RLSC provides realistic accuracy estimates. In a synthetic closed-loop test case with a realistic data distribution in the Andes we demonstrate the advantages of RLSC for regional geoid modeling and quantify the benefit which results mainly from a rigorously handled high-resolution GGM. In terms of root mean square deviations from the true reference solution, RLSC delivers an improvement of about 30% compared to a standard LSC approach, where the benefit is particularly pronounced in areas with a sparse data distribution. This improved performance, together with the fact that the resulting stochastic error estimates better reflect the true errors, might be an important aspect for the application of RLSC to derive gravity potential values and their uncertainties at reference stations of the international height reference system.

Decorrelation of GRACE Time Variable Gravity Field Solutions Using Full Covariance Information

In this study the feasibility and performance of time variable decorrelation (VADER) filters derived from covariance information on decadal Gravity Recovery and Climate Experiment (GRACE) time series are investigated. The VADER filter is based on publicly available data that are provided by several GRACE processing centers, and does not need its own Level-2 processing chain. Numerical closed loop simulations, incorporating stochastic and deterministic error budgets, serve as basis for the design of the filter setup, and the resulting filters are subsequently applied for real data processing. The closed loop experiments demonstrate the impact of temporally varying error and signal covariance matrices that are used for the design of decorrelation filters. The results indicate an average reduction of cumulative geoid height errors of 15% using time-variable instead of static decorrelation. Based on the simulation experience, a real data filtering procedure is designed and set up. It is applied to the ITSG-Grace2014 time variable gravity field time series with its associated full monthly covariance matrices. To assess the validity of the approach, linear mass trend estimates for the Antarctic Peninsula are computed using VADER filters and compared to previous estimates from both, GRACE and other mass balance estimation techniques. The mass change results obtained show very good agreement with other estimates and are robust against variations of the filter strength. The DDK decorrelation filter serves as main benchmark for the assessment of the VADER filter. For comparable filter strengths the VADER filters achieve a better de-striping and deliver smaller formal errors than static filters like the DDK.

Joint Source-Relay Secure Precoding for MIMO Relay Networks With Direct Links

In this paper, we propose a joint source-relay precoding scheme to secure an amplify-and-forward multiple-input multiple-output wireless relay network in the existence of a multi-antenna eavesdropper. Different from existing works that only consider some specific signal design to simplify the problem, we take both the direct links from the source to the destination and to the eavesdropper into account, and investigate the general joint signal covariance matrices optimization problem to maximize the secrecy rate, which leads to a difficult non-convex optimization problem. To handle it, we propose a group alternating optimization algorithm to find a solution, which alternately optimizes the signal covariance matrix and the linear precoding matrix at the source and the relay, respectively. For optimizing the linear precoding matrix at the relay, the problem is still non-convex, and we propose a minorization–maximization (MM) method to handle it. The MM method transforms the original non-convex problem into a series of convex problems and guarantees the convergence of a local optimum. For optimizing the signal covariance matrix at the source, we reveal the convex–concave property of the problem, and find its global optimum using a barrier method combined with the Newton iteration. We also provide an initialization method to trigger the algorithm and analyze the convergence and complexity. The numerical results show the computational efficiency and the prominent performance of the proposed algorithm.

Supplementations to the Higher Order Subspace Algorithm for Suppression of Spatially Colored Noise

Recently, researchers have proposed to represent the observed multichannel speech data as a 3-D tensor and then directly reduce the noise level in the time domain. For example, a higher order subspace algorithm (HOSA) was proposed for the reduction of spatially white noise (i.e., the noise added to different sensors is mutually uncorrelated) and yielded an excellent performance. Nevertheless, for spatially colored noise (i.e., the noise added to different sensors is correlated or even perfectly coherent), HOSA suffers from serious performance degradation because the ${{\mathbf T}}$ -weighted noise covariance matrix is not strictly diagonal any more. In this letter, we present three efficient supplementations to HOSA to improve its performance on spatially colored noise. First, we directly estimate the weighted signal covariance matrices by speech subtraction because noise statistics are unknown and have to be estimated in the case. Second, we clear all the off-diagonal elements of the whitened ${{\mathbf T}}$ -weighted noise covariance matrix as they generally have negligible absolute values. Third, we process speech frames in an overlap-free manner to better meet the white noise assumption. In both simulations and real experiments, these supplementations can greatly improve the performance of HOSA on spatially colored noise while not affecting its performance on spatially white noise.

Friendly Jamming in a MIMO Wiretap Interference Network: A Nonconvex Game Approach

We consider joint optimization of artificial noise (AN) and information signals in a MIMO wiretap interference network, wherein the transmission of each link may be overheard by several MIMO-capable eavesdroppers. Each information signal is accompanied with AN, generated by the same user to confuse nearby eavesdroppers. Using a noncooperative game, a distributed optimization mechanism is proposed to maximize the secrecy rate of each link. The decision variables here are the covariance matrices for the information signals and ANs. However, the nonconvexity of each link's optimization problem (i.e., best response) makes conventional convex games inapplicable, even to find whether a Nash Equilibrium (NE) exists. To tackle this issue, we analyze the proposed game using a relaxed equilibrium concept, called quasi-Nash equilibrium (QNE). Under a constraint qualification condition for each player's problem, the set of QNEs includes the NE of the proposed game. We also derive the conditions for the existence and uniqueness of the resulting QNE. It turns out that the uniqueness conditions are too restrictive, and do not always hold in typical network scenarios. Thus, the proposed game often has multiple QNEs, and convergence to a QNE is not always guaranteed. To overcome these issues, we modify the utility functions of the players by adding several specific terms to each utility function. The modified game converges to a QNE even when multiple QNEs exist. Furthermore, players have the ability to select a desired QNE that optimizes a given social objective (e.g., sum-rate or secrecy sum-rate). Depending on the chosen objective, the amount of signaling overhead as well as the performance of resulting QNE can be controlled. Simulations show that due to the QNE selection mechanism, we can achieve a significant improvement in terms of secrecy sum-rate and power efficiency.

A high resolution model of linear trend in mass variations from DMT-2: Added value of accounting for coloured noise in GRACE data

We present a high resolution model of the linear trend in the Earth's mass variations based on DMT-2 (Delft Mass Transport model, release 2). DMT-2 was produced primarily from K-Band Ranging (KBR) data of the Gravity Recovery And Climate Experiment (GRACE). It comprises a time series of monthly solutions complete to spherical harmonic degree 120. A novel feature in its production was the accurate computation and incorporation of stochastic properties of coloured noise when processing KBR data. The unconstrained DMT-2 monthly solutions are used to estimate the linear trend together with a bias, as well as annual and semi-annual sinusoidal terms. The linear term is further processed with an anisotropic Wiener filter, which uses full noise and signal covariance matrices. Given the fact that noise in an unconstrained model of the trend is reduced substantially as compared to monthly solutions, the Wiener filter associated with the trend is much less aggressive compared to a Wiener filter applied to monthly solutions. Consequently, the trend estimate shows an enhanced spatial resolution. It allows signals in relatively small water bodies, such as Aral sea and Ladoga lake, to be detected. Over the ice sheets, it allows for a clear identification of signals associated with some outlet glaciers or their groups. We compare the obtained trend estimate with the ones from the CSR-RL05 model using (i) the same approach based on monthly noise covariance matrices and (ii) a commonly-used approach based on the DDK-filtered monthly solutions. We use satellite altimetry data as independent control data. The comparison demonstrates a high spatial resolution of the DMT-2 linear trend. We link this to the usage of high-accuracy monthly noise covariance matrices, which is due to an accurate computation and incorporation of coloured noise when processing KBR data. A preliminary comparison of the linear trend based on DMT-2 with that computed from GSFC_global_mascons_v01 reveals, among other, a high concentration of the signal along the coast for both models in areas like the ice sheets, Gulf of Alaska, and Iceland.

Low complexity robust adaptive beamforming for general-rank signal model with positive semidefinite constraint

We propose a low complexity robust beamforming method for the general-rank signal model, to combat against mismatches of the desired signal array response and the received signal covariance matrix. The proposed beamformer not only considers the norm bounded uncertainties in the desired and received signal covariance matrices, but also includes an additional positive semidefinite constraint on the desired signal covariance matrix. Based on the worst-case performance optimization criterion, a computationally simple closed-form weight vector is obtained. Simulation results verify the validity and robustness of the proposed beamforming method.

Optimal design of modal beamformers for circular arrays.

An approach to optimal design of circular harmonics based modal beamformers for circular arrays is presented. Theoretical analysis shows the equivalence between the element-space and circular-harmonics-domain beamformers. By deriving the expression for the signal and noise covariance matrices, array manifold, and the array response in terms of the circular-harmonics-domain beamforming weights, the weights design problem is formulated as one of minimum variance distortionless response beamforming in the circular harmonics domain. It is found that the well-known phase-mode beamformer for circular arrays can be viewed as an ideal minimum variance distortionless response beamformer against the planarly isotropic noise in the circular harmonics domain. This interesting result is useful for analyzing and evaluating the performance of the phase-mode circular arrays. In this circular-harmonics-domain beamformer, additional constraints can be imposed to improve its robustness and control the sidelobes, which is important in practical applications. The developed approach can include both the delay-and-sum and phase-mode beamformers as special cases, which leads to very flexible designs. Simulation and experimental results show excellent performance of the proposed beamforming approach.

Blind Channel Estimation for MIMO-OFDM Systems With Low Order Signal Constellation

In this paper, we design a new blind channel estimation method for multi-input multi-output orthogonal frequency division multiplexing (MIMO-OFDM) systems to cope with possible rank deficient signal covariance matrices when symbols are drawn from low order constellations, e.g., BPSK, QPSK, etc. By collecting the signal covariance matrices from a group of carefully selected subcarriers, the proposed method could identify the channel information from a small amount of received blocks. Moreover, the proposed method also outperforms the existing competitor in terms of both estimation performance and computational complexity. The numeral results are provided to corroborate the proposed studies.

A Comment on “Pair-Matching Method by Signal Covariance Matrices for 2D-DOA Estimation”

A recent two-dimensional (2-D) direction-of-arrival (DOA) estimation method proposed by Wei and Guo has a flaw that the existence of sources with the same or nearly the same impinging angles along $x$ -axis subarray or $y$ -axis subarray will result in incorrect DOA estimation. This comment proposes a feasible solution to patch the original method.

Distributed adaptive generalized eigenvector estimation of a sensor signal covariance matrix pair in a fully connected sensor network

The generalized eigenvalue decomposition (GEVD) of a pair of matrices generalizes the concept of the eigenvalue decomposition (EVD) of a single matrix. It is a widely used tool in signal processing applications, in particular in a context of spatial filtering and subspace estimation. In this paper, we describe a distributed adaptive algorithm to estimate generalized eigenvectors (GEVCs) of a pair of sensor signal covariance matrices in a fully connected wireless sensor network. The algorithm computes these GEVCs in an iterative fashion without explicitly constructing the full network-wide covariance matrices. Instead, the nodes only exchange compressed sensor signal observations, providing a significant reduction in per-node communication and computational cost compared to the scenario where all the raw sensor signal observations are collected and processed in a fusion center.

Pair-Matching Method by Signal Covariance Matrices for 2D-DOA Estimation

The two-dimensional direction-of-arrival (2D-DOA) estimation for the L-shaped array is usually divided into two one-dimensional DOA (1D-DOA) estimation for the two linear subarrays, and then the elevation and azimuth angles are obtained from the two correctly matched 1D-DOAs. This letter presents a novel method for pair-matching of the two 1D-DOAs. The key points of the proposed method are: 1) construct two signal covariance matrices corresponding to the two estimated 1D-DOAs; 2) introduce an objective function, defined over the two signal source covariance matrices, to find the matched 1D-DOAs; 3) by introducing a permutation matrix for the signal covariance matrices, the optimal pair-matching is obtained with an efficient method. Simulations under different scenarios, with low signal-to-noise ratio (SNR), snapshot deficiency, and uncorrelated or coherent signal settings, show the robustness of the proposed method.

Sparse covariance fitting method for direction of arrival estimation of uncorrelated wideband signals

We propose a new direction of arrival estimation method for wideband uncorrelated signals. The wideband signals are first decomposed into narrowband signals. A group sparse Lasso formulation is proposed that jointly fits the powers of the signals using overcomplete dictionaries of the directions of arrival, into the estimated covariance matrices for all narrowband signals. Then, we propose a new formulation that determines the regularization parameters and becomes the group Lasso formulation. Additionally, we propose a modified algorithm for direction of arrival estimation with lower complexity that uses conventional based methods in the preprocessing stage to reduce the number of variables in the optimization task. We compare the performance of the proposed method to the conventional methods for a circular antenna array.

Estimation for the Linear Model With Uncertain Covariance Matrices

We derive a maximum a posteriori estimator for the linear observation model, where the signal and noise covariance matrices are both uncertain. The uncertainties are treated probabilistically by modeling the covariance matrices with prior inverse-Wishart distributions. The nonconvex problem of jointly estimating the signal of interest and the covariance matrices is tackled by a computationally efficient fixed-point iteration as well as an approximate variational Bayes solution. The statistical performance of estimators is compared numerically to state-of-the-art estimators from the literature and shown to perform favorably.

Covariance Eigenvector Sparsity for Compression and Denoising

Sparsity in the eigenvectors of signal covariance matrices is exploited in this paper for compression and denoising. Dimensionality reduction (DR) and quantization modules present in many practical compression schemes such as transform codecs, are designed to capitalize on this form of sparsity and achieve improved reconstruction performance compared to existing sparsity-agnostic codecs. Using training data that may be noisy a novel sparsity-aware linear DR scheme is developed to fully exploit sparsity in the covariance eigenvectors and form noise-resilient estimates of the principal covariance eigenbasis. Sparsity is effected via norm-one regularization, and the associated minimization problems are solved using computationally efficient coordinate descent iterations. The resulting eigenspace estimator is shown capable of identifying a subset of the unknown support of the eigenspace basis vectors even when the observation noise covariance matrix is unknown, as long as the noise power is sufficiently low. It is proved that the sparsity-aware estimator is asymptotically normal, and the probability to correctly identify the signal subspace basis support approaches one, as the number of training data grows large. Simulations using synthetic data and images, corroborate that the proposed algorithms achieve improved reconstruction quality relative to alternatives.