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The Symmetric Informationally Complete Positive Operator-Valued Measures (SIC-POVMs) are known to exist in all dimensions ⩽ 151 and many larger dimensions as high as 39 604 . All known solutions with the exception of the Hoggar solutions are covariant with respect to the Weyl-Heisenberg group and in the case of dimension 3, it has been proven that all SIC-POVMs are Weyl-Heisenberg group covariant. In this work, we explore this by SIC-POVMs in dimensions 4–7 without the assumption of group covariance. We introduce two functions with which SIC-POVM Gram matrices can be generated without the group covariance constraint, and analytically show that the SIC-POVM Gram matrices exist on critical points of the surfaces defined by the two functions on a subspace of Hermitian matrices. In dimensions 4–7, all known SIC-POVM Gram matrices lie in disjoint continuous sets of solution. Thus, we define an equivalent class that relates Gram matrices based on the trivial symmetries of the two functions. In dimensions 4–7, we generated { 1.7 × 10 6 , 1.1 × 10 5 , 169 , 50 } Gram matrices, respectively. For each of the Gram matrices, we generate the symmetry group of their respective disjoint sets. In all cases, the symmetry group was isomorphic to a subgroup of the Clifford group containing the Weyl-Heisenberg group matrices and the order-3 unitaries. In dimensions 4–6, All constructed solutions belonged to a single equivalence class of Gram matrices whereas in dimension 7, we find two distinct families of SIC-POVMs. In dimensions 4 and 5, the absence of new classes of SIC-POVM strongly suggests that the functions do not have a non-trivial symmetry. Furthermore, in all the dimensions tested, the only equivalence classes found correspond to the distinct stabilizers of fiducial vectors of each dimension. [ABSTRACT FROM AUTHOR]

Aiming at solving the problem that the capability of direction of arrival method for estimation of nested arrays decreases sharply under unknown mutual coupling, an iterative weighted low‐rank matrix reconstruction algorithm is proposed here. First, the received data covariance matrix is extended, and a weighted low‐rank matrix recovery problem is formulated for joint estimations of the Toeplitz matrix and mutual coupling coefficients. Next, the direction of arrival from recovered Toeplitz matrix is retrieved by using the root multiple signal classification algorithm. Simulation results show that iterative weighted low‐rank matrix reconstruction algorithm can effectively avoid the influence of mutual coupling while solving the grid‐mismatch problem so that the accuracy and the resolution of estimation are improved. [ABSTRACT FROM AUTHOR]

Coprime arrays (CPA) are extensively applied to estimate the direction of arrival (DOA). Yet, the effectiveness of existing DOA estimation algorithms for coprime arrays is predicated on the precise calibration of the array. Gain-phase errors in the sensors, when present, are magnified by the co-array, greatly impacting the performance of DOA estimation algorithms. To address the issue, this paper proposes a joint DOA estimation algorithm combining auxiliary arrays with atomic norm minimization. Exploiting the sub-array decomposition feature of coprime array, gain-phase error estimation is accomplished with the addition of precisely calibrated auxiliary arrays. Besides, the estimated gain-phase errors are used to correct the covariance of the signals received by the array. By combining array interpolation, all virtual sensors are fully utilized, and the interpolated virtual received signals are reconstructed into a Toeplitz matrix. The covariance matrix of these signals is then refined using atomic norm minimization, preparing the data for the final DOA estimation performed by subspace algorithms. In addition, for comparative analysis, the Cramér–Rao bound has also been derived under the condition of gain-phase errors, providing a benchmark for the performance of the proposed algorithm. [ABSTRACT FROM AUTHOR]

We estimated convex-structured covariance/correlation matrices by minimizing the entropy loss corresponding to the given matrix. We first considered the estimation of the Weighted sum of known Rank-one matrices with unknownWeights (W-Rank1-W) structural covariance matrices, which appeared commonly in array signal processing tasks, e.g., direction-of-arrival (DOA) estimation. The associated minimization problem is convex and can be solved using the primal-dual interior-point algorithm. However, the objective functions (the entropy loss function) can be bounded above by a sequence of separable functions--we proposed a novel estimation algorithm based on this property under the Majorization-Minimization (MM) algorithmic framework. The proposed MM algorithm exhibited very low computational complexity in each iteration, and its convergence was demonstrated theoretically. Subsequently, we focused on the estimation of Toeplitz autocorrelation matrices, which appeared frequently in time-series analysis. In particular, we considered cases in which the autocorrelation coefficient decreased as the time lag increased. We transformed the Toeplitz structure into aW-Rank1-W structure via special variable substitution, and proposed anMMalgorithm similar to that for the W-Rank1-W covariance estimation. However, each MM iteration involved a second-order cone programming SOCP problem that must be resolved. Our numerical experiments demonstrated the high computational efficiency and satisfactory estimation accuracy of the proposedMMalgorithms in DOA and autocorrelation matrix estimation. [ABSTRACT FROM AUTHOR]

When uncorrelated signals are incident on a uniform linear array, the array covariance matrix is of the Toeplitz form. An adaptive beamforming method is proposed based on the signal-plus-interference (SI) subspace via the Toeplitz rectification of the sample matrix. The rectified matrix is shown to be more accurate in a norm sense than the modified matrix according to the centro-Hermitian property. Since the former also is centro-Hermitian we can efficiently obtain its eigen-decomposition from a real matrix and then the weight vector in the estimated SI subspace. The proposed method, showing robustness to pointing errors, is not only computationally efficient but also very quickly converges to the optimum performance as demonstrated in the simulation. [ABSTRACT FROM AUTHOR]

In this paper, a Toeplitz construction method based on eigenvalues and eigenvectors is proposed to combine with traditional denoising algorithms, including fractional low-order moment (FLOM), phased fractional low-order moment (PFLOM), and correntropy-based correlation (CRCO) methods. It can improve the direction of arrival (DOA) estimation of signals in impulsive noise. Firstly, the algorithm performs eigenvalue decomposition on the received covariance matrix to obtain eigenvectors and eigenvalues, and then the Toeplitz matrix is created according to the eigenvectors corresponding to its eigenvalues. Secondly, the spatial averaging method is used to obtain an unbiased estimate of the Toeplitz matrix, which is then weighted and added based on the corresponding eigenvalues. Next, the noise subspace of the Toeplitz matrix is reconstructed to obtain the one that has less angle information. Finally, the DOA of the coherent signal is estimated using the Multiple Signal Classification (MUSIC) algorithm. The improved method based on the Toeplitz matrix can not only suppress the effect of impulsive noise but can also solve the problem of aperture loss due to its decoherence. A series of simulations have shown that they have better performances than other algorithms. [ABSTRACT FROM AUTHOR]

Covariance matrices with a linear structure are widely used in multivariate analysis. The choice of covariance structure can be made from a set of possible linear structures. As a result, the most appropriate structure is determined by minimizing the discrepancy function. This paper is a continuation of previous work on identifying linear structures with an entropy loss function as a discrepancy function. We present extensive simulation studies on the correctness of identification with the assumed pentagonal banded Toeplitz structure. [ABSTRACT FROM AUTHOR]

This paper proposes a fast direction of arrival (DOA) estimation method based on positive incremental modified Cholesky decomposition atomic norm minimization (PI-CANM) for augmented coprime array sensors. The approach incorporates coprime sampling on the augmented array to generate a non-uniform, discontinuous virtual array. It then utilizes interpolation to convert this into a uniform, continuous virtual array. Based on this, the problem of DOA estimation is equivalently formulated as a gridless optimization problem, which is solved via atomic norm minimization to reconstruct a Hermitian Toeplitz covariance matrix. Furthermore, by positive incremental modified Cholesky decomposition, the covariance matrix is transformed from positive semi-definite to positive definite, which simplifies the constraint of optimization problem and reduces the complexity of the solution. Finally, the Multiple Signal Classification method is utilized to carry out statistical signal processing on the reconstructed covariance matrix, yielding initial DOA angle estimates. Experimental outcomes highlight that the PI-CANM algorithm surpasses other algorithms in estimation accuracy, demonstrating stability in difficult circumstances such as low signal-to-noise ratios and limited snapshots. Additionally, it boasts an impressive computational speed. This method enhances both the accuracy and computational efficiency of DOA estimation, showing potential for broad applicability. [ABSTRACT FROM AUTHOR]

In this article, an effective algorithm is proposed for direction‐of‐arrival (DOA) estimation of coherent signals using the l2‐norm with moving coprime arrays. The proposed algorithm exhibits high accuracy and robustness in decorrelation of coherent signals compared to existing methods. Firstly, the received signal is filtered on the moving coprime array and a signal model is constructed that is compatible with the moving coprime array. Next, virtual sensors is inserted into the holes of the moving coprime array and the received signals are initialised. Then, the Toeplitz matrix is reconstructed based on the covariance matrix of the initial signal and perform low‐rank matrix recovery to eliminate signal coherence by compensating the matrix for zero. Finally, the DOA estimation for the signal is performed using spectral peak search. Simulation results verify that the estimation accuracy of our proposed algorithm outperforms various existing algorithms. [ABSTRACT FROM AUTHOR]

Copyright of Telecommunication Engineering is the property of Telecommunication Engineering and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Systems Engineering & Electronics is the property of Journal of Systems Engineering & Electronics Editorial Department and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this paper, we consider an integrated sensing and communications system assisted by a reconfigurable intelligent surface (RIS). A small number of sparsely placed active sensors are applied in the RIS to perform effective channels and, thereby, enable optimized beamforming for both communications and sensing objectives, namely, establishing reliable communication links with communication users (CUs) and effectively localize targets. The time-varying multipath channels between the RIS and the CU as well as the time-varying channel between the RIS and the targets are estimated by exploiting an interpolated Hermitian and Toeplitz covariance matrix followed by direction-of-arrival estimation using the MUSIC algorithm. Based on such results, we jointly optimize the transmit beamformer at the base station and the unit-modulus RIS passive beamformer. The RIS beamformer is optimized to maximize its minimum beam-pattern gain towards the desired sensing angles subject to the minimum signal-to-noise ratio requirement at the CU. Simulation results verify the effectiveness of the proposed approach, and the performance of different sparse array configurations is compared. [ABSTRACT FROM AUTHOR]

Copyright of Systems Engineering & Electronics is the property of Journal of Systems Engineering & Electronics Editorial Department and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

This letter proposes an efficient tensor model for Direction of Departure (DOD) and Direction of Arrival (DOA) estimation based on bistatic nested Multiple‐Input Multiple‐Output (MIMO) radar. Tensor offers a robust solution in characterizing multi‐dimensional data structures, ensuring data integrity, and has demonstrated remarkable effectiveness in the realm of multi‐dimensional parameter estimation. However, the complexity of traditional methods for constructing tensors is a major obstacle to practical applications. In response, this research introduces a novel approach that utilizes an iterative two‐dimensional Toeplitz matrix reconstruction method to rapidly derive expanded virtual covariance matrices prior to tensor construction. This streamlines the algorithm and improves efficiency. Numerical simulations have verified the superiority of this algorithm. [ABSTRACT FROM AUTHOR]

Linearly structured covariance matrices are widely used in multivariate analysis. The covariance structure can be chosen from a class of linear structures. Therefore, the optimal structure is identified in terms of minimizing the discrepancy function. In this research, the entropy loss function is used as the discrepancy function. We give a methodology and algorithm for determining the optimal structure from the class of structures under consideration. The accuracy of the proposed method is checked using a simulation study. [ABSTRACT FROM AUTHOR]

The problem of estimating a large covariance matrix arises in various statistical applications. This paper develops new covariance matrix estimators based on shrinkage regularization. Individually, we consider two kinds of Toeplitz-structured target matrices as the data come from the complex Gaussian distribution. We derive the optimal tuning parameter under the mean squared error criterion in closed form by discovering the mathematical properties of the two target matrices. We get some vital moment properties of the complex Wishart distribution, then simplify the optimal tuning parameter. By unbiasedly estimating the unknown scalar quantities involved in the optimal tuning parameter, we propose two shrinkage estimators available in the large-dimensional setting. For verifying the performance of the proposed covariance matrix estimators, we provide some numerical simulations and applications to array signal processing compared to some existing estimators. [ABSTRACT FROM AUTHOR]

To reduce the complexity of direction-of-arrival (DOA) algorithms in polarization-sensitive arrays and maintain high estimation accuracy while decreasing runtime, this paper proposes a channel compression model based on orthogonal dipole arrays. In addition, a DOA estimation algorithm using atomic norm minimization (ANM) is introduced for this model. This algorithm performs eigenvalue decomposition on the covariance matrix of the data received from the polarization-sensitive channel compressed array. A new observation vector is constructed for the ANM problem under the channel compression model using the obtained eigenvalues and eigenvectors. Subsequently, a Toeplitz matrix is constructed from the optimal solution of the semi-positive definite programming (SDP) problem. The Vandermonde decomposition of the Toeplitz matrix provides estimates of the signal DOA parameters. By combining the vectorization result of the covariance matrix and the least-squares calculation, the polarization auxiliary angle and polarization phase angle information of the incident source are also obtained. Simulation experiments compare the root-mean-squared error (RMSE) of each algorithm at different angular intervals and signal-to-noise ratios (SNR) to confirm the validity of estimating DOA and polarization parameters. [ABSTRACT FROM AUTHOR]

We consider the maximum likelihood estimation of a covariance matrix that is known to be Toeplitz. The corresponding optimization problem is nonconvex, and to compute a local minimizer we propose an efficient implementation of Newton's method. The cost per iteration of a straightforward implementation is of order O (n 4) , where n + 1 is the dimension of the covariance matrix. By exploiting fast Fourier transforms to assemble the gradient and Hessian we reduce the iteration cost to order O (n 3). Through extensive simulations we demonstrate that for direction-of-arrival estimation, the performance of MUSIC is significantly enhanced by using it together with the maximum likelihood Toeplitz covariance estimate. The simulations also demonstrate that our open-source implementation is orders of magnitude faster than two existing Toeplitz covariance estimators. • Maximum likelihood estimation of a Toeplitz covariance matrix can be done efficiently. • The Toeplitz maximum likelihood estimate significantly enhances MUSIC. • We provide an open-source implementation of our method. [ABSTRACT FROM AUTHOR]

To diminish the performance degradation of coprime array beamforming arising from sensor gain and phase uncertainties, we propose a robust beamformer based on covariance matrix modification with augmented instrumental sensors. Initially, the received array data are split into two subarrays. Utilizing the instrumental sensors, we estimate the gain-phase errors of these two subarrays and then recombine them. Subsequently, error compensation is implemented to calibrate the received covariance matrix into a Toeplitz matrix relevant to the coprime array structure, in which the holes need to be further filled by means of some matrix completion operation. Then, the above covariance matrix modification is integrated into the classical beamforming procedure including MUSIC decomposition, spectral peak searching, power estimation, INCM (Interference-plus-Noise Covariance Matrix) reconstruction, thus yielding the final beamformer weight vector. Both theoretical analysis and simulations demonstrate that this method not only can accurately estimate the gain and phase errors, but also can effectively improve the beamformer's output SINR (i.e., the robustness is enhanced), which presents the proposed beamformer with vast potentials in practical sparse-array involved applications. • The designed beamformer can improve the robustness of a coprime array under gain-phase error. • Gain-phase error calibration via setting four instrumental sensors ahead of the coprime array. • Enlarging the continuous range of virtual array and performing matrix completion. [ABSTRACT FROM AUTHOR]

Abstract Coprime arrays (CPAs) have been found to be an effective configuration for adaptive beamforming (ABF). However, most ABF methods for CPAs are embedded with the spatial spectrum estimation technique, and therefore they usually deteriorate in the case of low signal‐to‐noise ratio (SNR), especially without prior information of interferences. To address this issue, a robust ABF method for CPAs via atomic‐norm‐based sparse recovery is proposed. This method begins with initialised virtual array interpolation to avoid aperture loss caused by ‘holes’, which is subsequently transformed to a matrix completion and denoising issue via a Toeplitz step. With the matrix’s Hermitian Toeplitz structure and the intrinsic sparsity of interferences, sparse recovery for the interference covariance matrix is introduced by subtracting the target signal and noise from the received signal and implemented by atomic‐norm minimisation. Concretely, this non‐convex process is decomposed into two convex steps and solved iteratively. Unlike previous methods, this method can interpolate the virtual array and estimate the steering vector of the target and the noise power simultaneously; consequently, the interference‐plus‐noise covariance matrix is reconstructed with merely some trivial entry selections without the need for the spatial spectrum. Furthermore, the optimality conditions and boundedness in this method are proven theoretically. Simulation results demonstrate the effectiveness of the proposed method and its advantages over previous methods under those non‐ideal conditions.

Under the background of spatially correlated color noise, the incidence angle of a jamming signal in a high-speed moving platform rapidly changes, which leads to the degradation of the anti-interference performance and the waveform distortion of the adaptive beamformer. In this paper, a projection-constrained null broadening beamforming algorithm based on the Toeplitz matrix structure is proposed. The algorithm first extracts the subspace of the covariance matrix of the steering vector of the pre-determined extended angle interval and constructs the constraint matrix and the projection transformation matrix. The received signal covariance matrix with a Toeplitz structure is then constructed using the correlation number between each array element and the pre-set reference array element. Finally, the constructed covariance matrix is transformed through projection, and the weight of each array element is constrained by the constraint matrix. The theoretical optimal solution of adaptive wide null beamforming in spatially correlated color noise is obtained. The simulation results show that, compared with the existing robust adaptive beamforming algorithms, the proposed algorithm can efficiently improve the distortion of adaptive anti-jamming beams, and can achieve null broadening in the jamming direction under the condition of spatially correlated color noise, which improves the output signal to the interference-plus-noise ratio (SINR) of the adaptive beamformer. [ABSTRACT FROM AUTHOR]

A principal singular vector based on multiple Toeplitz matrices is proposed to solve the accuracy problem of direction-of-arrival (DOA) estimation for coherent signals. First, the data matrix received by uniform linear array (ULA) is transformed into a Toeplitz matrix. An equivalent covariance matrix is obtained by square weighted summation method using the Toeplitz matrix. Then, a polynomial containing DOA information is constructed because the signal space and the steering matrix have the same column space; the Toeplitz matrix is built using polynomial coefficients. The problem is transformed into solving linear equations by establishing the relationship between the Toeplitz matrix and the signal subspace. Furthermore, the weighted least square method is used to obtain multiple candidates for linear equations. Finally, the maximum likelihood (ML) rule is used to select source signal candidates from multiple candidates. In comparison with currently known algorithm, the proposed algorithm has the characteristics of high estimation accuracy, low-complexity, and strong anti-interference ability and resolution. Even when the signal-to-noise ratio (SNR) is low, the snapshot number is small, and multiple signals exist; this method can still provide good estimation performance and resolution, which is more than 90% in most cases. Simulation experiments verify the superiority of the algorithm. [ABSTRACT FROM AUTHOR]

Recently, deep unfolding networks with interpretable parameters have been widely utilized in direction of arrival (DOA) estimation due to the faster convergence speed and better generalization ability. However, few consider the nested array for gridless DOA estimation. In this letter, we propose a deep alternating projection network to address the problem. We first convert the covariance matrix into a measurement vector in the form of atomic norm, which can reduce the matrix dimension during projection. We then train the proposed network to alternately obtain the positive semi-definite matrix and the corresponding irregular Hermitian Toeplitz matrix, where the loss function is derived by employing the trace of network output. Finally, we apply the irregular root Multiple Signal Classification (MUSIC) method to obtain gridless DOA via nested array. We demonstrate that the proposed networks can accelerate the convergence rate and reduce computational cost. Simulations verify the performance of proposed networks in comparison with the existing methods. [ABSTRACT FROM AUTHOR]

In this letter, we develop an effective algorithm for direction-of-arrival (DOA) estimation of coherent sources with coprime arrays. Firstly, we generate a virtual uniform linear array (ULA) through coprime array interpolation. Subsequently, we derive an augmented noise-free covariance matrix that is constructed using the entries of the covariance matrix of the virtual ULA outputs, and recover the Hermitian Toeplitz matrix by solving an atomic norm minimization problem with multiple measurement vectors. Finally, the DOAs of coherent sources are estimated via MUSIC spectral search. Unlike the state-of-the-art approach, our algorithm is insensitive to the phase differences among sources. Numerical results demonstrate the superiority of the proposed algorithm over several existing techniques. [ABSTRACT FROM AUTHOR]

In this paper, we design an effective algorithm for direction-of-arrival (DOA) estimation using the coprime array in the presence of unknown nonuniform noise. Firstly, we replace the diagonal entries of the array covariance matrix with zeros and generate the coarray output signal by vectorization and removing redundancy. Subsequently, we build a Toeplitz matrix via virtual array interpolation, and utilize matrix completion and linear prediction to recover the missing interpolated and diagonal entries, respectively. Finally, we acquire the source DOAs by applying the MUSIC algorithm on the reconstructed covariance matrix. Numerical results demonstrate the superiority of the proposed algorithm in comparison with several existing approaches. [ABSTRACT FROM AUTHOR]

Sparse Bayesian learning (SBL) is a popular and robust algorithm for sparse signal reconstruction (SSR). Unfortunately, the SBL algorithm suffers from heavy computational complexity when it is implemented directly since the inversion and multiplying operations of the estimation of the covariance matrix are involved in each iteration, which is proportional to the cube of the observed data length and thus prevents it solving large-scale problems. In many applications, such as radar imaging and array signal processing, the signal to be recovered is sparse in the Fourier dictionary. In this article, we propose an efficient implementation method for the Fourier dictionary-based SBL (FD-SBL). In the case that the Fourier dictionary is adopted, the estimation of the covariance matrix is a Toeplitz matrix for 1-D data or a Toeplitz-block-Toeplitz (TBT) matrix for 2-D data during the FD-SBL iterations. By utilizing this property, we employ the Gohberg–Semencul (G-S)-type factorization to accelerate the implementation of FD-SBL. To be noted, there is no approximation in our proposed method, and the computational cost is reduced by several orders of magnitude compared with the direct implementation of SBL. Finally, the experimental results verify the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

In this work, we apply deep learning to estimate the direction-of-arrival (DoA) of multiple narrowband signals with a uniform linear array in a coherent environment. First, the logarithmic eigenvalue-based classification network (LogECNet) is introduced to enhance signal number detection accuracy in challenging scenarios, such as the low signal-to-noise (SNR) regime and limited snapshots. Next, a multi-label classification model called the root-spectrum network (RSNet) is devised to estimate the DoAs using the signal number inferred by LogECNet. In the proposed architecture, the full-row Toeplitz matrices reconstruction (FTMR), which exploits all rows of the signal covariance matrix (SCM), is combined with LogECNet and RSNet to inversely map the SCM to the numerical DoAs in the coherent scenario. It is shown that the eigenvalues factorized from the FTMR output matrix become more robust sources for signal enumeration than those of the forward/backward spatial smoothing (FBSS) algorithm. Furthermore, the logarithmic scaling of the eigenvalues of the FTMR results in LogECNet outperforming other detectors. The simulation results show our proposed method not only improves the signal number detection and angular estimation performance, but also achieves the complexity reduction with respect to the prior schemes. [ABSTRACT FROM AUTHOR]

Copyright of Systems Engineering & Electronics is the property of Journal of Systems Engineering & Electronics Editorial Department and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

• The spectral symmetry of sea clutter is validated in numerous measured data. • A specific matrix structure constraint is proposed based on the spectral symmetry. • Alternating optimization is adopted to solve the constrained ML estimation. Speckle covariance matrix estimation plays an important role in adaptive target detection of maritime radars. Spatial heterogeneity of sea clutter incurs insufficient secondary data in estimation, which degrades detection performance. It is an effective way of estimation improvement to exploit the Doppler spectrum knowledge of sea clutter for structural dimension reduction of the speckle covariance matrix. This paper proposes a speckle covariance matrix estimation method based on the spectral symmetry knowledge of sea clutter, indicating that the spectrum of sea clutter is symmetric with respect to the Doppler offset. First of all, the spectral symmetry of sea clutter is analyzed by using the measured data from several opened databases and the results show that most of the data excluding sea spikes have symmetric Doppler spectra. Secondly, the spectral symmetry constrains the speckle covariance matrix into the Hadamard product of a special rank-one Toeplitz matrix and a real positive definite symmetric matrix. The degrees of freedom are reduced to almost half of that of a Hermitian matrix and thus fewer secondary data are required. Thirdly, a robust iterative maximum likelihood estimator is proposed to estimate the speckle covariance matrix with structural constraint and is free of the clutter texture distribution. The performance of the estimator is verified by the measured data and the results show that it is superior to traditional estimators, particularly in fewer secondary data. [ABSTRACT FROM AUTHOR]

Direction-Of-Arrival (DOA) estimation in environments with coherent sources is a challenging problem, when the number of narrowband sources is more than elements and current coherent algorithms cannot accurately overcome such underdetermined problems. In this paper, we propose the Sparse-based Toeplitz Covariance Reconstruction (STCR) algorithm, which obtains a hole-free extended-aperture array with increased Degrees of Freedom (DOF) by exploiting a minimum redundancy array. This method provides superiority in estimating coherent sources by Toeplitz covariance matrix reconstruction. Also, using a Sparse Signal Representation (SSR) framework makes the DOA estimator stronger in source detection and robust to the noise power. Numerical results show the effectiveness of the proposed method even at low SNRs for over- and underdetermined DOA estimators compared to well-known methods. Also, it has high accuracy in DOA estimation of closely-located coherent sources using a small number of snapshots. [ABSTRACT FROM AUTHOR]

Purpose: The existing dimensionality reduction algorithms suffer serious performance degradation under low signal-to-noise ratio (SNR) owing to the presence of noise. To address these problems, an enhanced spatial smoothing scheme is proposed that exploits the subarray time-space correlation matrices to reconstruct the data matrix to overcome this weakness. This method uses the strong correlation of signal and the weak correlation of noise in time and space domains, which improves the noise suppression ability. Design/methodology/approach: In this paper, an enhanced spatial smoothing method is proposed. By using the strong correlation of signal and the weak correlation of noise, the time-space smoothed array covariance matrix based on the subarray time-space correlation matrices is constructed to improve the noise suppression ability. Compared with the existing Toeplitz matrix reconstruction and spatial smoothing methods, the proposed method improves the DOA estimation performance at low SNR. Findings: Theoretical analysis and simulation results show that compared with the existing dimensionality reduction processing algorithms, the proposed method improves the DOA estimation performance in cases with a low SNR. Furthermore, in cases where the DOAs between the coherent sources are closely spaced and the snapshot number is low, our proposed method significantly improves the performance of the DOA estimation performance. Originality/value: The proposed method improves the DOA estimation performance at low SNR. In particular, for the cases with a low SNR, the proposed method provides a better RMSE. The convergence of the proposed method is also faster than other methods for the low number of snapshots. Our analysis also confirms that in cases where the DOAs between the coherent sources are closely spaced, the proposed method achieves a much higher angular resolution than that of the other methods. [ABSTRACT FROM AUTHOR]

Compared with the Nyquist sampling scheme, the coprime sampling scheme reduces the requirement of sampling rate significantly and simultaneously shows high degrees of freedom. Actually, the information received by the sampler is not fully utilized due to the non-uniformity of derived second-order virtual coarray. In addition, the virtual signals corresponding to the non-uniform virtual coarray may be contaminated by noise, which could affect the accuracy of spectrum estimation. In this letter, we propose a novel virtual coarray interpolation method for the inexact Toeplitz covariance matrix based on truncated nuclear norm regularization. The missing elements and exact covariance matrix can be obtained simultaneously by solving the covariance matrix completion problem that contains the error term and non-Toeplitz errors propagation control term. Simulation results demonstrate the superiorities of the proposed spectrum estimation method in terms of accuracy and computational complexity. [ABSTRACT FROM AUTHOR]

Aiming at the problem that the performance of under determined direction of arrival (DOA) estimation method based on coprime array decreases sharply under the condition of array element gain and phase error, a coprime array DOA estimation method based on correction array element is proposed. Firstly, the received data of the array is decomposed into two sub array data, the gain and phase error of the sub array are estimated based on the correction array elements, and the gain and phase error of the sub array are sorted and reorganized. Then, the error of the received data covariance matrix is compensated and extended to a high- dimensional Toeplitz matrix. Finally, the high-dimensional covariance matrix is filled with holes based on the matrix filling theory, and the root multiple signal classification (root-MUSIC) algorithm is used for DOA estimation. Theoretical analysis and simulation results show that this method can realize the gain and phase error estimation of Coprime array, effectively recover the DOA estimation performance of Coprime array under the condition of gain and phase error through error compensation, and improve the estimation accuracy. [ABSTRACT FROM AUTHOR]

Coprime arrays (CPAs) have been found to be an effective configuration for adaptive beamforming (ABF). However, most ABF methods for CPAs are embedded with the spatial spectrum estimation technique, and therefore they usually deteriorate in the case of low signal‐to‐noise ratio (SNR), especially without prior information of interferences. To address this issue, a robust ABF method for CPAs via atomic‐norm‐based sparse recovery is proposed. This method begins with initialised virtual array interpolation to avoid aperture loss caused by 'holes', which is subsequently transformed to a matrix completion and denoising issue via a Toeplitz step. With the matrix's Hermitian Toeplitz structure and the intrinsic sparsity of interferences, sparse recovery for the interference covariance matrix is introduced by subtracting the target signal and noise from the received signal and implemented by atomic‐norm minimisation. Concretely, this non‐convex process is decomposed into two convex steps and solved iteratively. Unlike previous methods, this method can interpolate the virtual array and estimate the steering vector of the target and the noise power simultaneously; consequently, the interference‐plus‐noise covariance matrix is reconstructed with merely some trivial entry selections without the need for the spatial spectrum. Furthermore, the optimality conditions and boundedness in this method are proven theoretically. Simulation results demonstrate the effectiveness of the proposed method and its advantages over previous methods under those non‐ideal conditions. [ABSTRACT FROM AUTHOR]

The Toeplitz matrix reconstruction algorithms exploit the row vector of an array output covariance matrix to reconstruct Toeplitz matrix, which provide the directionof-arrival (DOA) estimation of coherent signals. However, the Toeplitz matrix reconstruction method based on any row vector of the array output covariance matrix suffers from signal correlation, it results in poor robustness. The methods based on multi-row vectors suffer serious performance degradation when in the low signal-to-noise ratio (SNR) owing to the noise energy is the square of the input noise energy. To solve the above problems, we propose an improved method that exploits all rows of the time-space correlation matrix to reconstruct the Toeplitz matrix, namely TS-MTOEP. This method firstly uses the coherence of the narrowband signal and the uncorrelated noise at different snapshots to construct the time-space correlation matrix, it effectively eliminates the influence of noise. Then, the Toeplitz matrix is reconstructed via all rows of the time-space correlation matrix, which effectively improves the energy of the signal, and further results in the improvement of the SNR. Finally, the DOAs can be obtained by combining it with the subspace-based methods. The theoretical analysis and simulation results indicate that compared with the existing Toeplitz and spatial smoothing methods, the proposed method in this paper provides good performance on estimation and resolution in cases with low input signal-to-noise due to time-space correlation matrix processing. Furthermore, in cases where the DOAs between the coherent sources are closely spaced and the snapshot number is low, our proposed method significantly improves the performance of the DOA estimation. We also provide the code to realize the reproducibility of the proposed method. [ABSTRACT FROM AUTHOR]

Acoustical properties of snoring signal have been widely studied as a potentially cost-effective and reliable alternative to diagnosing obstructive sleep apnea hypopnea syndrome, with a common recognition that the diagnostic accuracy depends heavily on the snoring signal quality. In the paper, generalized subspace noise reduction based on noise covariance matrix estimate is proposed. The noise covariance matrix is the Toeplitz matrix of the unbiased autocorrelation sequence which is estimated by recursive averaging its past value adjusted by a time-varying smoothing parameter controlled by the snoring signal presence probability, and the signal presence is determined by the ratio of temporal frame autocorrelation value to its minimum absolute value. The proposed method has a better estimate of noise covariance matrix, and the results of objective quality measurements and spectrograms of snoring signal show obvious improvement in terms of noise reduction and signal distortion under different non-stationary noise environments compared with conventional subspace enhancement algorithm. [ABSTRACT FROM AUTHOR]

In order to realize the high-precision direction of arrival (DOA) estimation of the coherent source of two-dimensional multiple-input and multiple-output (MIMO) radar, a solution is given by combining Toeplitz matrix set reconstruction. MIMO radar obtains a larger aperture with fewer arrays. Traditional two-dimensional reconstruction Toeplitz-like algorithms use part of the information in the construction of two correlation matrices or covariance matrices to construct the Toeplitz matrix when performing two-dimensional coherent source DOA estimation, which makes the information utilization incomplete and requires additional denoising processing. To solve the above problems, this paper proposes an improved Toeplitz matrix set reconstruction algorithm based on the two-dimensional reconstruction Toeplitz class algorithm. The complete array element receiving signal vector is used to construct two Toeplitz matrix sets containing complete information, and then their conjugate transposes. Multiply and sum to correct the matrix to obtain a full-rank matrix, so as to achieve the purpose of decoherence and combine the traditional ESPRIT algorithm to perform two one-dimensional reconstruction processing through rotation invariance and then perform angle matching to achieve two-dimensional coherent signal angle estimation, while avoiding additional denoising processing. Finally, the simulation results of the cross array and the L-shaped array verify the effectiveness of the algorithm in this paper and further extend it to the two-dimensional MIMO radar array model and compare it with the traditional ESPRIT-like algorithm and the REC-FBSS-ESPRIT algorithm. In comparison, the algorithm in this paper has better performance under the conditions of successful resolution, DOA estimation accuracy, and low signal-to-noise ratio. [ABSTRACT FROM AUTHOR]

We consider the problem of clutter covariance matrix (CCM) estimation for space-time adaptive processing (STAP) radar in the small sample. In this paper, a fast efficient algorithm for CCM reconstruction is proposed to overcome this shortcoming for the linear structure. Particularly, we present a low-rank matrix recovery (LRMR) question about CCM estimation based on the Toeplitz structure of CCM and the prior knowledge of the noise. The closed-form solution is obtained by relaxing the nonconvex LRMR problem that the trace norm replaces the rank norm. The target can then be efficiently detected by using the recovered CCM according to the STAP theorem. We also analyze the algorithm model under the linear structure in the presence of unknown mutual coupling. It is shown that our method can obtain accurate CCM in the small sample, with even higher accuracy than the traditional algorithms in the same number of samples. It also can reduce the coupling effect and obtain more degrees of freedom (DOF) with limited sensors and pulses by utilizing sparse linear structure (SLS) and improve angle and Doppler resolutions. Finally, numerical simulations have verified the effectiveness of the proposed method in comparison with some of the existing methods. [ABSTRACT FROM AUTHOR]

A gridless direction-of-arrival (DOA) estimation method to improve the estimation accuracy and resolution in nonuniform noise is proposed in this paper. This algorithm adopts the structure of minimum-redundancy linear array (MRA) and can be composed of two stages. In the first stage, by minimizing the rank of the covariance matrix of the true signal, the covariance matrix that filters out nonuniform noise is obtained, and then a gridless residual energy constraint scheme is designed to reconstruct the signal covariance matrix of the Hermitian Toeplitz structure. Finally, the unknown DOAs can be determined from the recovered covariance matrix, and the number of sources can be acquired as a byproduct. The proposed algorithm can be regarded as a gridless version method based on sparsity. Simulation results indicate that the proposed method has higher estimation accuracy and resolution compared with existing algorithms. [ABSTRACT FROM AUTHOR]

This paper studies sensor calibration in spectral estimation where the true frequencies are located on a continuous domain. We consider a uniform array of sensors that collects measurements whose spectrum is composed of a finite number of frequencies, where each sensor has an unknown calibration parameter. Our goal is to recover the spectrum and the calibration parameters simultaneously from multiple snapshots of the measurements. In the noiseless case with an infinite number of snapshots, we prove uniqueness of this problem up to certain trivial, inevitable ambiguities based on an algebraic method, as long as there are more sensors than frequencies. We then analyze the sensitivity of this algebraic technique with respect to the number of snapshots and noise. We next propose an optimization approach that makes full use of the measurements by minimizing a non-convex objective which is non-negative and continuously differentiable over all calibration parameters and Toeplitz matrices. We prove that, in the case of infinite snapshots and noiseless measurements, the objective vanishes only at equivalent solutions to the true calibration parameters and the measurement covariance matrix. The objective is minimized using Wirtinger gradient descent which is proven to converge to a critical point. We show empirically that this critical point provides a good approximation of the true calibration parameters and the underlying frequencies. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

A novel two-stage gridless sparsity-based method for sparse linear array direction-of-arrival (DOA) estimation is presented. First, the covariance matrix gridless sparse representation method based on atomic norm minimisation is proposed for corresponding structured Toeplitz matrix construction and source number detection. Then, the conventional MUSIC algorithm can be employed for DOA estimation. Compared with conventional subspace-based algorithms, the proposed two-stage method can be carried out without knowing source number and directly detect more source signals than sensors. Numerical simulations demonstrate the effectiveness and outperformance of the proposed method. [ABSTRACT FROM AUTHOR]

A novel two-stage gridless sparsity-based method for sparse linear array direction-of-arrival (DOA) estimation is presented. First, the covariance matrix gridless sparse representation method based on atomic norm minimisation is proposed for corresponding structured Toeplitz matrix construction and source number detection. Then, the conventional MUSIC algorithm can be employed for DOA estimation. Compared with conventional subspace-based algorithms, the proposed two-stage method can be carried out without knowing source number and directly detect more source signals than sensors. Numerical simulations demonstrate the effectiveness and outperformance of the proposed method.

This letter addresses the problem of signal detection in additive correlated noise whose covariance matrix is Toeplitz. Particularly, we design a novel detection approach in the framework of generalized likelihood ratio test, in which the maximum likelihood (ML) estimate of the Toeplitz covariance matrix is needed. Since there are no closed-form expressions for this ML estimate, we resort to the inverse iterative algorithm. The proposed detector surpasses existing methods in detection power and enjoys the constant false-alarm rate property. Besides, accurate asymptotic null and non-null distributions of the test statistic are derived. Numerical results are presented to validate our theoretical findings. [ABSTRACT FROM AUTHOR]

Practical applicability of many statistical algorithms is limited by large sizes of corresponding covariance matrices. These limitations can be significantly weakened due to effective use of the structure of covariance matrices, properties of the autocorrelation function, and advantages of the architecture of modern GPUs. This paper presents GPU implementations of the algorithms for inversion of a covariance matrix and solution of a system of linear equations whose coefficient matrix is a covariance matrix. Inversion of close to sparse covariance matrices is also considered in the work. For all the cases considered, significant accelerations were obtained in comparison with Octave mathematical software and ViennaCL computational library. For example, implemented algorithm of solution of a linear system was 6 times faster as compared with the implementation of Octave on the CPU and 3 times faster as compared with the ViennaCL implementation on the GPU for general matrices. The performance of inversion of a covariance matrix was 14 times faster than inversion algorithm of Octave on the CPU and 6 times faster than ViennaCL inversion algorithm on GPU. [ABSTRACT FROM AUTHOR]

Covariance matrix estimation concerns the problem of estimating the covariance matrix from a collection of samples, which is of extreme importance in many applications. Classical results have shown that $O(n)$ samples are sufficient to accurately estimate the covariance matrix from $n$ -dimensional independent Gaussian samples. However, in many practical applications, the received signal samples might be correlated, which makes the classical analysis inapplicable. In this paper, we develop a nonasymptotic analysis for the covariance matrix estimation from linearly-correlated Gaussian samples. Our theoretical results show that the error bounds are determined by the signal dimension $n$ , the sample size $m$ , and the shape parameter of the distribution of the correlated sample covariance matrix. Particularly, when the shape parameter is a class of Toeplitz matrices (which is of great practical interest), $O(n)$ samples are also sufficient to faithfully estimate the covariance matrix from correlated samples. Simulations are provided to verify the correctness of the theoretical results. [ABSTRACT FROM AUTHOR]

Highlights • Persymmetric and Toeplitz structures of covariance matrix are exploited for target detection in MIMO radar. • Analytical expressions for the probability of false alarm and detection probability are derived for the proposed detector that takes the persymmetry into account. • The proposed detectors outperform their counterparts. Abstract We aim to address the problem of target detection in collocated multiple-input multiple-output (MIMO) radar where the disturbance covariance matrix is unknown. In practice, the disturbance covariance matrix has a persymmetric or Toeplitz structure, when the MIMO radar receiver is equipped with a symmetrically or uniformly spaced linear array. We exploit the persymmetric or Toeplitz structure of the disturbance covariance matrix to design two adaptive detectors that do not require training data. Analytical expressions for the probability of false alarm and detection probability are derived for the proposed detector that takes the persymmetry into account. Numerical examples are provided to show that the proposed detectors outperform the conventional counterparts. [ABSTRACT FROM AUTHOR]

In this work, we propose a novel method for quantifying distances between Toeplitz structured covariance matrices. By exploiting the spectral representation of Toeplitz matrices, the proposed distance measure is defined based on an optimal mass transport problem in the spectral domain. This may then be interpreted in the covariance domain, suggesting a natural way of interpolating and extrapolating Toeplitz matrices, such that the positive semidefiniteness and the Toeplitz structure of these matrices are preserved. The proposed distance measure is also shown to be contractive with respect to both additive and multiplicative noise and thereby allows for a quantification of the decreased distance between signals when these are corrupted by noise. Finally, we illustrate how this approach can be used for several applications in signal processing. In particular, we consider interpolation and extrapolation of Toeplitz matrices, as well as clustering problems and tracking of slowly varying stochastic processes. [ABSTRACT FROM AUTHOR]