Coprime Array Research Articles

Rank Minimization-Based Toeplitz Reconstruction for DoA Estimation Using Coprime Array

In this paper, we address the problem of direction finding using coprime array, which is one of the most preferred sparse array configurations. Motivated by the fact that non-uniform element spacing hinders full utilization of the underlying information in the receive signals, we propose a direction-of-arrival (DoA) estimation algorithm based on low-rank reconstruction of the Toeplitz covariance matrix. The atomic-norm representation of the measurements from the interpolated virtual array is considered, and the equivalent dual-variable rank minimization problem is formulated and solved using a cyclic optimization approach. The recovered covariance matrix enables the application of conventional subspace-based spectral estimation algorithms, such as MUSIC, to achieve enhanced DoA estimation performance. The estimation performance of the proposed approach, in terms of the degrees-of-freedom and spatial resolution, is examined. We also show the superiority of the proposed method over the competitive approaches in the root-mean-square error sense.

Improved DFT Method for DOA Estimation with Extended Coprime Array:Based on Large Difference CoArray

In this paper, Direction of arrival estimation (DOA) in coprime array is investigated, and an advanced method for DOA estimation is presented with extended coprime array (ECA). Specifically, we use Discrete Fourier Transform (DFT) to obtain initial DOA estimation with large difference co-arrays generated by ECA, then perform Taylor series approximation and total least squares (TLS) method around the initial estimation to compensate the estimation offset. Compared to conventional DFT method, the proposed method enjoys better estimation accuracy with fewer computational burden since it avoids extra searches and benefits more from large difference co-array. Besides, in contrast to spatial smoothing based methods which halve array aperture, the proposed method avoids the process of spatial smoothing and fully utilizes uniform degrees of freedom (DOFs). In this case, the proposed method has advantages on estimation accuracy and computational efficiency with multiple impinging signals, especially under the condition of large difference co-array. Furthermore, the proposed method is applicable to other types of advanced coprime array. The superiority of the proposed method has been proved by numerical simulations.

Coprime array‐adaptive beamforming via atomic‐norm‐based sparse recovery

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.

A Two-Dimensional DOA Estimation Method Based on Virtual Extension of Sparse Array

The interelement spacing of a coprime array breaks through the half-wavelength limitation, so that a larger array aperture can be obtained with a fixed number of array elements. In this paper, the symmetry of the noncircular signal is used to virtually expand the L-shaped array into an orthogonal cross array. Furthermore, the virtual received signal of the augmented array is obtained by the second-order statistic of the received data. Decoupling and dimension reduction of elevation and azimuth are realized by a z-axis subarray and y-axis subarray. Finally, the sparse reconstruction of the signal is realized by the minimum absolute convergence and selection operator method. This method can enlarge the aperture and freedom of array and has higher accuracy and resolution of DOA estimation. It has the advantages of automatic parameter pairing without additional pairing operation and is effective for coherent and incoherent signals. The final numerical simulation results prove the effectiveness of the method in this paper.

A Propagator Method for Bistatic Coprime EMVS-MIMO Radar

In this paper, a novel two-dimensional (2D) direction-of-departure (DOD) and 2D direction-of-arrival (DOA) estimate algorithm is proposed for bistatic multiple-input multiple-output (MIMO) radar system equipped with coprime electromagnetic vector sensors (EMVS) arrays. Firstly, we construct the propagator to obtain the signal subspace. Then, the ambiguous angles are estimated by using rotation invariant technique. Based on the characteristic of coprime array, the unambiguous angles estimation is achieved. Finally, all azimuth angles estimation is followed via vector cross product. Compared to the existing uniform linear array, coprime MIMO radar is occupying large array aperture, and the proposed algorithm does not need to obtain signal subspace by eigendecomposition. In contrast to the state-of-the-art algorithms, the proposed algorithm shows better estimation performance and simpler computation performance. The proposed algorithm’s effectiveness is proved by simulation results.

Difference Coarray-Based Direction Finding via Covariance Fitting

Sparse linear arrays like nested arrays and coprime arrays, can offer more accurate direction-of-arrival estimation than uniform linear arrays by exploiting their difference coarrays. In this letter, we present a new algorithm for direction finding using difference coarray. The proposed estimator firstly formulates a recovery problem of a column vector based on the covariance fitting criterion and the probability distribution of the vectorized estimation error in the coarray covariance matrix. Subsequently, we reconstruct the expected coarray covariance matrix by recovering the column vector. Based on the reconstructed covariance matrix, the MUSIC algorithm is adopted to perform direction finding. Numerical results demonstrate the superiority of our scheme over several existing approaches.

Generalized parallel coprime array for two-dimensional DOA estimation: A perspective from maximizing degree of freedom

Parallel arrays with coprime subarrays have shown its potential advantages for two dimensional direction of arrival (DOA) estimation. In this paper, by introducing two flexible coprime factors to enlarge the inter-element spacing of parallel uniform subarrays, we propose a generalized parallel coprime array (GPCA) geometry. The proposed geometry enjoys flexible array layouts by the coprime factors and enables to extend the array aperture to achieve great improvement of estimation performance. Meanwhile, we verify that GPCA always can obtain M <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> degrees of freedom (DOFs) in co-array domain via 2M sensors after optimization, which outperforms sparse parallel array geometries, such as parallel coprime array (PCA) and parallel augmented coprime array (PACA), and is the same as parallel nested array (PNA) with extended aperture. The superiority of GPCA geometry has been proved by numerical simulations with sparse representation methods.

Minimum Mean-Squared-Error autocorrelation processing in coprime arrays

Coprime arrays enable Direction-of-Arrival (DoA) estimation of an increased number of sources. To that end, the receiver estimates the autocorrelation matrix of a larger virtual uniform linear array (coarray), by applying selection or averaging to the physical array's autocorrelation estimates, followed by spatial-smoothing. Both selection and averaging have been designed under no optimality criterion and attain arbitrary (suboptimal) Mean-Squared-Error (MSE) estimation performance. In this work, we design a novel coprime array receiver that estimates the coarray autocorrelation with Minimum-MSE (MMSE), for any probability distribution of the source DoAs. Our extensive numerical evaluation illustrates that the proposed MMSE approach returns superior autocorrelation estimates which, in turn, enable higher DoA estimation performance compared to standard counterparts.

2D Direction Finding of Coherent Sources Using Three Parallel Sparse Arrays with Less Computational Complexity

This paper addresses the problem of two-dimensional (2D) direction of arrival estimation of coherent sources with less computational complexity using sparse arrays. The proposed method incorporates the unitary ESPRIT method into three parallel nested and coprime arrays configuration. The advantages of the proposed method are computationally efficient due to real-valued operations, automatic pairing of 2D angles, resolves coherent targets, large array aperture and less mutual coupling. Simulation results show the effectiveness of the proposed method. Cramer–Rao lower bound (CRLB) has been given for the reference. The performance of the proposed method is compared with the methods available in the existing literature. The results achieved by the proposed method were found to be significant in terms of estimation accuracy, resolution, and the computational complexity.

An Efficient Strategy for Accurate Detection and Localization of UAV Swarms

Unmanned aerial vehicle (UAV) swarms have shown great potential for Internet of Things (IoT). Meantime, its malicious use may cause huge threat to the national security. UAV swarms show the characteristic of high density which poses formidable challenges to radar resolution in the defense of critical areas. In this article, we consider a radar equipped with the coprime array, and then, use the coherent long-time integration (LTI) technique and gridless sparse technique to detect and localize UAVs in a swarm. This strategy takes full account of advantages of the coprime array, coherent LTI technique, and gridless sparse technique, i.e.: 1) the coprime array can provide a larger array aperture than the uniform linear array with the same number of array elements to relieve the stress of the gridless sparse technique and 2) the combination of coherent LTI technique and gridless sparse technique can maximize their advantages and make up for their shortcomings. By mathematical analyses and extensive numerical examples, we show the superiority of the proposed strategy in terms of accurate detection and localization of UAV swarms.

Efficient direction-of-arrival estimation via annihilating-based denoising with coprime array

Although the coprime array provides the possibility for performing underdetermined direction-of-arrival (DOA) estimation, its applicable high-performance algorithm is still in demand. In this paper, we propose a new DOA estimation method for coprime array. This method is motivated by the possibility of applying the annihilating filter theory to the non-uniform samples of the complex exponential sum. The non-uniform coarray array is transformed into a uniform linear array by introducing the concept of interpolating via denoising. The model noise caused by finite snapshots is taken into account, and the array noise variance is included as an unknown variable. The annihilating filter is designed by forming a nonlinear annihilating relation constrained least squares problem, which is solved by sequential quadratic programming. The final DOA can be obtained by the rooting method or spectral method, based on the designed filter. This method makes full use of the degree of freedom and signal information and is proved to be an asymptotic conditional maximum likelihood estimator, which leads to high accuracy under both underdetermined and overdetermined scenarios. It is gridless, free of regularization parameter, and even more computationally efficient than the coarray MUSIC method. Its performances are verified by simulations and compared with representative algorithms.

Hybrid Reweighted Optimization Method for Gridless Direction of Arrival Estimation in Heteroscedastic Noise Environment

In this paper, we present a hybrid optimization framework for gridless sparse Direction of Arrival (DoA) estimation under the consideration of heteroscedastic noise scenarios. The key idea of the proposed framework is to combine global and local minima search techniques that offer a sparser optimizer with boosted immunity to noise variation. In particular, we enforce sparsity by means of reformulating the Atomic Norm Minimization (ANM) problem through applying the nonconvex Schatten-p quasi-norm (0&lt;p&lt;1) relaxation. In addition, to enhance the adaptability of the relaxed ANM in more practical noise scenarios, it is combined with a covariance fitting (CF) criterion resulting in a locally convergent reweighted iterative approach. This combination forms a hybrid optimization framework and offers the advantages of both optimization approaches while balancing their drawbacks. Numerical simulations are performed taking into account the configuration of co-prime array (CPA). The simulations have demonstrated that the proposed method can maintain a high estimation resolution even in heteroscedastic noise environments, a low number of snapshots, and correlated sources.

Manifold Ambiguity-Free Low Complexity DOA Estimation Method for Unfolded Co-Prime Arrays

The problem of manifold ambiguity-free direction-of-arrival (DOA) estimation for an unfolded co-prime array is of prime research interest. The manifold ambiguity problem is resolved by using beamforming-like methods. However, the performance of this method is limited by the searching step, computational complexity and also fails to resolve the closely spaced sources. In order to overcome the above limitations, in this letter, the DOA estimation is viewed as a function approximation problem. The unknown mapping function that relates the received signals and its DOAs is approximated by using the support vector regression (SVR). The proposed method resolves the ambiguity problem completely with low computational complexity. The simulation results are provided to validate the superiority and effectiveness of DOA estimation in terms of estimation accuracy, computational complexity and reliability.

Rearranged coprime array to increase degrees of freedom and reduce mutual coupling

Coprime array for direction of arrivals finding has attracted increasing interest since it can provide enhanced degrees of freedom. However, due to the holes in the difference coarray, the available information is not fully exploited. In this paper, we propose a rearranged coprime array configuration to fill the holes in the coprime difference coarray. The proposed new array allows filling the holes and reducing the mutual coupling effect at the same time. We classify all the holes in a holes-triangle as a design guideline for the proposed rearranged coprime array. Few redundant sensors of the generalized coprime array are rearranged at sparse positions, which are determined by the position of holes. By doing so, the consecutive part of the conventional coprime difference coarray can be significantly increased. Meanwhile, the mutual coupling can also be dramatically reduced because the redundant sensors are relocated at remote positions. Simulation results verify the superiority of the proposed array compared with other sparse arrays.

CAP-3 Coprime Array for DOA Estimation With Enhanced Uniform Degrees of Freedom and Reduced Mutual Coupling

Compared with nested array, coprime array can significantly alleviate the mutual coupling between array sensors due to the sparser array structure. However, plenty of holes in its difference co-array extremely reduce the achievable number of uniform degrees of freedom(uDOFs). In this letter, a coprime array with prime integer 3(CAP-3) is proposed, in contrast to several well-known nested arrays, which can offer comparable number of uDOFs, and alleviate mutual coupling simultaneously. Specifically, the CAP-3 consists of three appropriately placed subarrays, and it can eradicate most of the holes in difference co-array except two at both ends. Hence a large number of uDOFs can be obtained. Furthermore, CAP-3 possesses robustness against heavy level of mutual coupling due to the reduced number of sensor pairs with small inter-element spacing. Simulations reveal the advantage of the CAP-3 in uDOF, mutual coupling robustness, and direction of arrival estimation accuracy.

Structured autocorrelation matrix estimation for coprime arrays

A coprime array receiver processes a collection of received-signal snapshots to estimate the autocorrelation matrix of a larger (virtual) uniform linear array, known as coarray. By the received-signal model, this matrix has to be (i) Positive Definite, (ii) Hermitian, (iii) Toeplitz, and (iv) its noise-subspace eigenvalues have to be equal. Existing coarray autocorrelation matrix estimates satisfy a subset of the above conditions. In this work, we propose an optimization framework which offers a novel estimate satisfying all four conditions: we propose to iteratively solve a sequence of distinct structure-optimization problems and show that, upon convergence, we provably obtain a single estimate satisfying (i)-(iv). Numerical studies illustrate that the proposed estimate outperforms standard counterparts, both in autocorrelation matrix estimation error and Direction-of-Arrival estimation.

Spatial‐Frequency Estimation for OFDM System with Coprime Array

Orthogonal frequency division multiplexing (OFDM) is the key technique of the communication system. In this paper, coprime array is designed in OFDM system to obtain joint direction of arrival (DOA) and carrier frequency offset (CFO) estimates. The proposed system can achieve beamforming and adaptive signal processing, reduce the multipath fading, and attain angle diversity. Meanwhile, coprime array has the advantages of extended array aperture, increased degrees of freedom, and reduced mutual coupling effect, which is adaptable in OFDM system. Moreover, this paper proposes a low‐complexity parameter algorithm with superior performance, which first exploits propagator method (PM) as the initialization and then parallel factor (PARAFAC) method is employed for the accurate DOA and CFO estimation. Simulation results verify the effectiveness of the OFDM coprime array system and the proposed low‐complexity multiparameter algorithm.

Coherent Target Direction‐of‐Arrival Estimation for Coprime Arrays: From Spatial Smoothing Perspective

In this paper, we investigate the issue of direction‐of‐arrival (DOA) estimation of multiple signals in coprime arrays. An algorithm based on multiple signal classification (MUSIC) and forward and backward spatial smoothing (FBSS) is used for DOA estimation of this signal caused by multipath and interference. The large distance between adjacent elements of each subarray in the coprime arrays will bring phase ambiguity issues. According to the feature of the coprime number, the ambiguity problem can be eliminated. The correct DOA estimation can be obtained by searching for the common peak of the spatial spectrum and finding the overlapping peaks in the MUSIC spectrum of the two subarrays. For the rank deficit problem caused by the coherent signal, the FBSS algorithm is used for signal preprocessing before the MUSIC algorithm. Theoretical analysis and simulation results show that the algorithm can effectively solve the rank deficiency and phase ambiguity problems caused by coherent signals and sparse arrays in the coprime arrays.

A Transformed Coprime Array With Reduced Mutual Coupling for DOA Estimation of Non-Circular Signals

Recently, the design of sparse linear array for direction of arrival (DOA) estimation of non-circular (NC) signals has attracted great attention because the difference and sum co-array offered by non-circularity increases the aperture of virtual linear array. In this paper, we present a coprime array with shifted and flipped sub-array for the DOA of non-circular signals. By shifting one sub-array, the proposed array structure achieves a higher number of consecutive lags than the prototype coprime array with the same number of sensors. Then, through flipping the shifted sub-array with the zero point as the symmetry point, the number of sensor pairs with small separation is significantly reduced, making the resulting structure much sparser. For the proposed array structure, we derive the closed-form expression for the number of consecutive lags, the optimal distribution of two sub-arrays with a given number of sensors and the weight function. Numerical simulations are conducted to verify the superiority of the proposed array over the existing sparse arrays.

Fractional Low-Order Moments Based DOA Estimation With Co-Prime Array in Presence of Impulsive Noise

Direction of arrival (DOA) estimation with co-prime array is a hot issue in array signal processing. By using co-prime array structure, the degrees of freedom has been greatly improved, which can estimate much more sources than that of the conventional array structure. In the scenario of impulsive noise, the high-order (2-order or 4-order) moments of received signal do not exist, fractional low-order moments (FLOM) can be applied. In this paper, the concept of the co-prime array is extended to be applied on FLOM of received signal in presence of impulsive noise. And then MUSIC algorithm is used for DOA estimation. The proposed method is tested on the numerical data. The simulation results prove the effectiveness of the proposed method in impulsive noise scheme.