Coprime Array Research Articles

Sequential Estimation of Frequency and Direction-of-Arrival Based on the Relaxed Coprime Array for Multiple Targets

Recently, extensive researches focus on the joint estimation of carrier frequency and direction-of-arrival, whereas the heavy computation burden and affordable hardware cost are always required therein. To alleviate the problem, we propose a relaxed coprime array based two-stage estimator, which can sequentially detect the frequencies and direction-of-arrival for multiple targets. Guided by the closed-form robust algorithm, this relaxation yields a higher array sparsity compared with existing sparse arrays. Specifically, the first stage aims to achieve the classification of the frequency and angle remainders arising from temporal-spatial undersampling, which is realized by combining spectrum correction with pattern clustering. By incorporating the closed-form robust Chinese Remainder Theorem into the remainder classification result, we obtain all the frequencies and direction-of-arrival in the second stage. Both performance analysis and numerical results were conducted to verify the effectiveness of the proposed estimator in the multi-target case.

Robust Two-Stage Reduced-Dimension Sparsity-Aware STAP for Airborne Radar With Coprime Arrays

Space-time adaptive processing (STAP) algorithms with coprime arrays can provide good clutter suppression potential with low cost in airborne radar systems as compared with their uniform linear arrays counterparts. However, the performance of these algorithms is limited by the training samples support in practical applications. To address this issue, a robust two-stage reduced-dimension (RD) sparsity-aware STAP algorithm is proposed in this work. In the first stage, an RD virtual snapshot is constructed using all spatial channels but only $m$ adjacent Doppler channels around the target Doppler frequency to reduce the slow-time dimension of the signal. In the second stage, an RD sparse measurement modeling is formulated based on the constructed RD virtual snapshot, where the sparsity of clutter and the prior knowledge of the clutter ridge are exploited to formulate an RD overcomplete dictionary. Moreover, an orthogonal matching pursuit (OMP)-like method is proposed to recover the clutter subspace. In order to set the stopping parameter of the OMP-like method, a robust clutter rank estimation approach is developed. Compared with recently developed sparsity-aware STAP algorithms, the size of the proposed sparse representation dictionary is much smaller, resulting in low complexity. Simulation results show that the proposed algorithm is robust to prior knowledge errors and can provide good clutter suppression performance in low sample support.

Improved two-dimensional DOA estimation using parallel coprime arrays

The conventional coprime array consists of two uniform linear subarrays to construct an effective difference coarray with desirable characteristics. Such linear coprime arrays only provide one-dimensional (1-D) direction-of-arrival (DOA) estimation. In this paper, we propose a novel coprime array configuration with parallel subarrays, along with an effective method for two-dimensional (2-D) DOA estimation. The 2-D DOA estimation problem is cast as two separate 1-D problems for reduced complexity and is solved using one of the two mechanisms based on the number of sensors and that of sources. When there are less sources than the number of sensors, subspace-based and rank-reduction estimation (RARE) techniques are sequentially applied to the physical array output. On the other hand, when the number of sources is equal to or larger than that of sensors, a virtual difference coarray is formed and group sparse reconstruction and least squares operations are then applied. In both scenarios, the proposed methods automatically pair the corresponding azimuth and elevation angles. The proposed methods resolve up to MN sources using 2M+N−1 sensors, which are the same as in the 1-D DOA estimation using conventional coprime arrays. Simulations results are presented delineating both the accuracy and resolution capability of the proposed method.

Patch and monopole antennas in linear coprime arrays for direction of arrival estimation using compressed sensing

Original work on direction of arrival (DOA) estimation relied on uniform linear arrays (ULAs) of antennas. Most of the work focused on improving the algorithms and the configuration of the antenna array and overlooked the effects of practical antennas on the algorithm performance. Very limited work studied DOA estimation within the physical limitations of handheld devices. In this work, we introduce three nonuniform linear coprime arrays based on patch and monopole antenna elements operating in 2.1 and 5.8 GHz bands and assess their behaviour in DOA estimation. The complex radiation patterns of the arrays were incorporated in the DOA estimation algorithm using compressed sensing (CS). Estimation accuracy is quantified by the root mean square error (RMSE) and the results are compared with those obtained by using isotropic antennas, showing that physical antennas can introduce up to 8° of error. Simulations were also carried out using the multiple signal classification (MUSIC) algorithm to demonstrate the advantage of CS in coprime arrays. The MUSIC algorithm failed to detect all sources even at maximum SNR. The impact of reducing the fundamental inter-element spacing in coprime arrays below 0.5 λ to achieve smaller array sizes is investigated as well.

Expanded coprime array for DOA estimation: augmented consecutive co-array and reduced mutual coupling

Generalized coprime structure decomposes the interleaved subarrays in the conventional coprime array by introducing a displacement and the resulting CADiS, i.e. coprime array with displaced subarrays, configuration can enlarge the minimum adjacent spacing between elements to multiples of half-wavelength, which is considerably attractive in alleviating mutual coupling effect. However, the difference co-array that CADiS yields is fractured, which greatly deteriorates direction of arrival (DOA) estimation performance and achievable degrees of freedom of the algorithms based on consecutive co-array, e.g. spatial smoothing technique and Toeplitz matrix method. In this paper, from the mutual coupling effect and difference co-array perspective, we propose an expanded coprime array (ECA) structure by two steps. The first step is to further augment the displacement between the subarrays of CADiS to suppress the mutual coupling effect. The second step is to relocate a proper number of rightmost elements to concatenate the dominant consecutive co-array generated by CADiS to enhance the consecutive difference co-array. Specifically, we provide the closed-form expressions of resulting consecutive difference co-array, the number of relocated elements and their positions. Furthermore, different from the spatial smoothing based methods, we employ Toeplitz matrix property to directly construct the full-rank covariance matrix of the received data from the consecutive co-array with a lower computational cost and present the Toeplitz-MUSIC algorithm to testify the effectiveness and superiority of the proposed ECA structure.

Robust adaptive beamforming via coprime coarray interpolation

Recently, the coprime array has aroused wide concern due to its capability of achieving enhanced degrees of freedom (DOF) and increased array aperture compared to uniform linear arrays (ULAs). However, it experiences a certain DOF and aperture loss due to the existence of “holes” in the difference coarray. These drawbacks are adverse to enhance the performance of direction-of-arrival (DOA) estimation and adaptive beamforming. In this paper, we propose a new robust adaptive beamforming algorithm with the coprime array. The proposed algorithm yields the beamforming coefficients by estimating the steering vector of the signal-of-interest (SOI) and reconstructing the interference-plus-noise covariance matrix (INCM). Unlike previous techniques, we estimate the DOAs of the SOI and interferences using an interpolated virtual ULA, and further derive their steering vectors and powers. With the high-precision DOA estimates, the steering vectors and powers can be more accurately estimated. The proposed algorithm avoids the coarray aperture loss and enhances the accuracy of INCM reconstruction. Numerical results indicate that the proposed algorithm outperforms the existing approaches in high SNR regions.

Generalized Three Dimensional Coprime Array for Two-Dimensional DOA Estimation

In this paper, we extend the coprime planar array to coprime cubic array and propose a three dimensional coprime array (TDCA) for two-dimensional direction of arrival (DOA) estimation, which possesses larger inter-element spacing so that the mutual coupling effects can be effectively relieved. Moreover, due to the larger array aperture, better DOA estimation performance can be achieved. To obtain more degrees of freedom (DOFs), a generalized TDCA (G-TDCA) configuration is proposed, which has a flexible array geometry and can extend the array aperture to obtain performance improvement. The principle for optimal array design of G-TDCA is derived to achieve the maximum DOF. And the Cramer–Rao Bounds which denotes as a theoretical benchmark for the lower bound of unbiased estimate are derived. Then we propose Khatri–Rao estimation signal parameters via rotational invariance techniques (KR-ESPRIT) algorithm and obtain the close-form expression. The algorithm can achieve angles with automatic pairing and doesn’t need spectrum peak searching. In order to decrease the computational complexity, we propose KR-Unitary ESPRIT algorithm, which can achieve approximate DOA estimation performance of the KR-ESPRIT algorithm with the lower computational complexity. Numerical simulation results verify the superiority of G-TDCA with proposed algorithms.

Performance analysis of sparse and small arrays

Horizontal arrays are often used to detect/separate a weak signal and estimate its direction of arrival among many loud interfering sources and ambient noise. Conventional beamforming (CBF) is robust but suffers from fat beams and high level sidelobes. To improve performance, either high resolution beamforming is used, which has its own problems, or one needs to increase the array aperture, hence requiring more elements on the array. To reduce the cost, sparse and/or nested arrays have been proposed, such as the coprime arrays, to achieve approximately the same aperture, and hence the same beam resolution with a less number of elements. For sonar arrays, deconvolution is shown to yield an unique solution as opposed to, say, image processing. For coprime array, deconvolution algorithm is shown to yield a narrower beam width, lower sidelobe levels and higher array gain than the product or min processing, Deconvolution is shown to yield superdirectivity and supergain for a small array. This raises new research issues about optimal array configuration design.

Bi‐level JOMP for DOA estimation in coprime array with off‐grid targets

The direction of arrival (DOA) estimation for off-grid targets in the coprime array, which can extend degrees of freedom, is studied. Among existing research results, joint orthogonal matching pursuit (JOMP) has obvious advantages in complexity, but drawbacks in accuracy. Therefore, a bi-level iterative algorithm is proposed to improve estimation performance of JOMP in this study. First, JOMP is executed to obtain the initial estimations of DOAs and mismatch parameters. Then, an iterative proximity searching is implemented to update target vectors to decline the residual further. After that, on the strength of results of the secondary iteration, the signal spectrum and mismatch parameters can be obtained through exploiting LS algorithm. Simulation results verify effectiveness of the proposed algorithm.

Array gain of coprime arrays.

A coprime array of M + N - 1 elements is designed, using product processing (PP), to yield the same beam width as that of a full array of MN elements using conventional beamforming (CBF), while at the same time suppressing the grating lobes. The array gain (AG) for isotropic noise using conventional PP is slightly less than that obtained using CBF, namely, 10log(M + N - 1). It is shown that using the coherent product processing, the coprime array yields an AG equal to that of the full array of MN elements, 10log(MN). The superior gain is often referred to as supergain.

A Robust DOA Estimator Based on Compressive Sensing for Coprime Array in the Presence of Miscalibrated Sensors.

Coprime array with sensors can achieve an increased degrees-of-freedom (DOF) of for direction-of-arrival (DOA) estimation. Utilizing the compressive sensing (CS)-based DOA estimation methods, the increased DOF offered by the coprime array can be fully exploited. However, when some sensors in the array are miscalibrated, these DOA estimation methods suffer from degraded performance or even failed operation. Besides, the key to the success of CS-based DOA estimation is that every target falls on the predefined grid. Thus, a coarse grid may cause the mismatch problem, whereas a fine grid requires great computational cost. In this paper, a robust CS-based DOA estimation algorithm is proposed for coprime array with miscalibrated sensors. In the proposed algorithm, signals received by the miscalibrated sensors are viewed as outliers, and correntropy is introduced as the similarity measurement to distinguish these outliers. Incorporated with maximum correntropy criterion (MCC), an iterative sparse reconstruction-based algorithm is then developed to give the DOA estimation while mitigating the influence of the outliers. A multiresolution grid refinement strategy is also incorporated to reconcile the contradiction between computational cost and the mismatch problem. The numerical simulation results verify the effectiveness and robustness of the proposed method.

Low‐complexity DOA estimation method for a co‐prime linear array

The problem of direction-of-arrival (DOA) estimation for a co-prime linear array is researched. A low-complexity DOA estimation method is proposed. The co-prime array is decomposed into two sparse uniform linear sub-arrays whose inter-element spacing meets the co-prime condition. Then the received signals of the two sub-arrays are processed for statistical signal processing. Although the DOA estimation based on the under-sampled signal will produce phase ambiguity, the sparse array element spacing that satisfies the co-prime condition makes the phase ambiguity have certain periodic rules. Finally, the unique solution corresponding to the true DOA can be obtained in the phase ambiguities of the two sub-arrays through the application of the co-prime nature. Simulation results validate the effectiveness of the proposed method.

Extended Coprime Array Configuration Generating Large-Scale Antenna Co-Array in Massive MIMO System

Generally, massive multiple-input multiple-output (MIMO) system incorporates hundreds of antennas at the base station for various attractive merits, whereas severe mutual coupling arises in the dense structure. In this paper, we introduce the sparse array configuration, coprime array, into the massive MIMO system to alleviate mutual coupling, increase the degrees of freedom (DOFs) and enhance the spatial resolution. Specifically, we propose the extended coprime array structures through two systematical schemes. One scheme is to slide one subarray of the typical augmented coprime array to decompose the interleaved subarrays and the resulting sliding extended coprime array (SECA) produces weaker mutual coupling than the traditional dense uniform linear array, especially the newly proposed super nested array and augmented nested array. Meanwhile, SECA can produce an augmented co-array with more antennas than the physical array and resultantly increase the available DOFs as well as the array aperture. The other scheme is to relocate a proper number of antennas in the generalized coprime array composed of two subarrays with a displacement, namely coprime array with displaced subarrays (CADiS), and the resulting relocating extended coprime array can further alleviate mutual coupling and enhance the consecutive co-array. Furthermore, we provide the closed-form expressions of the co-array, sliding factor, the minimum number of relocating sensors, and their corresponding positions.

CFAR Block-Sparse Bayesian Learning Algorithm for the Off-grid DOA Estimation with Coprime Array

Constant false-alarm rate Block-sparse Bayesian learning (CFAR-BSBL) algorithm is proposed to reduce the computational complexity and improve Direction of arrival (DOA) estimation accuracy of offgrid signals with coprime array. Firstly, a signal model with normalized noise is built to avoid the learning procedure of noise parameter. Secondly, a block sparse Bayesian framework is built with the introduction of a temporary correlation matrix in order to use t he temporal structure of incident signals. Then the algorithm uses CFAR detection to detect the grids close to the real DOA and relieve the dependence on the number of signals. Finally, an off-grid process based on the closest grids is adopted to deal with the off-grid problem. The proposed CFAR-BSBL algorithm can obtain high accuracy and low complexity DOA estimation of off-grid signals with coprime array.

Reduced‐dimension space‐time adaptive processing for airborne radar with co‐prime array

Space-time adaptive processing (STAP) for airborne radar with co-prime arrays is shown to have excellent superiority compared to traditional STAP with uniform linear array radar. However, high arithmetic computational complexity and large amount of training data are required in this approach. This motivates the authors to present a new approach which is relatively low computational load and fast convergence with satisfactory performance. Specifically, a reduced-dimension transformation in Doppler domain is incorporated into the existing approach. The reduced-dimension interference covariance matrix and target steering vector are then achieved by performing the reduced-dimension process, and hence two reduced-dimension STAP filters are designed using the derived interference covariance matrix and target-steering vector. Numerical simulations are carried out to reveal the superiority of the proposed approach.

Robustness of Difference Coarrays of Sparse Arrays to Sensor Failures—Part II: Array Geometries

In array processing, sparse arrays are capable of resolving O(N^2) uncorrelated sources with N sensors. Sparse arrays have this property because they possess uniform linear array (ULA) segments of size O(N^2) in the difference coarray, defined as the differences between sensor locations. However, the coarray structure of sparse arrays is susceptible to sensor failures and the reliability of sparse arrays remains a significant but challenging topic for investigation. In the companion paper, a theory of the k -essential family, the k -fragility, and the k -essential Sperner family were presented not only to characterize the patterns of k faulty sensors that shrink the difference coarray, but also to provide a number of insights into the robustness of arrays. This paper derives closed-form characterizations of the k -essential Sperner family for several commonly used array geometries, such as ULA, minimum redundancy arrays (MRA), minimum holes arrays (MHA), Cantor arrays, nested arrays, and coprime arrays. These results lead to many insights into the relative importance of each sensor, the robustness of these arrays, and the DOA estimation performance in the presence of sensor failure. Broadly speaking, ULAs are more robust than coprime arrays, while coprime arrays are more robust than maximally economic sparse arrays, such as MRA, MHA, Cantor arrays, and nested arrays.

Sparsity-Based DOA Estimation of Coherent and Uncorrelated Targets With Flexible MIMO Radar

This paper considers the direction of arrival (DOA) estimation of coherent and uncorrelated targets by exploiting a flexible multiple input multiple output radar. First, the proposed radar is generalized with two coprime expansion factors for enlarging the inter-element spacing of transmit and receive arrays, referred to as sparse arrays with flexible inter-element spacing (SA-FIS), which shows that the conventional nested or coprime ones are the special cases. The range of consecutive lags and the number of unique lags in the sum-difference coarray are derived in closed form. It is verified that SA-FIS can obtain the maximum unique lags and also suppress the mutual coupling effects. We then extend the FIS viewpoint to sparse nonuniform linear arrays, where the two-level nested and coprime arrays are employed to achieve a significant increase in degrees of freedom (DOFs). Furthermore, to fully utilize these unique lags, we propose a reduced-complexity two-step sparse representation algorithm. By modifying and removing the off-diagonal elements of the estimated target covariance matrix, the proposed method can just identify the diagonal ones, thereby leading to further improved performance with much lower computational complexity. Finally, the Cramer-Rao bound on DOAs and correlated coefficients for SA-FIS is derived. Numerical simulations demonstrate the superiority of the proposed method with SA-FIS in terms of DOFs, computations, estimation accuracy, and resolution capability compared with previous ones.

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

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

L-shaped coprime array structures for DOA estimation

This paper proposes a new sparse array geometry for 2-D (azimuth and elevation) direction-of-arrival (DOA) estimation based on coprime sampling. The proposed array structure is L-shaped coprime array (LCA) whose each portion is one dimensional coprime linear arrays in y- and z-dimensions. Each portion of the array is used separately for 1-D azimuth and elevation angle estimation. In order to obtain the paired DOA estimates the cross-covariance matrix of two portion of the array is utilized and the paired DOA angles are estimated. LCA provides to estimate $$K \le MN$$ source directions with $$2M+N-1$$ sensors in each portion and totally $$4M+2N-3$$ sensor elements. The proposed method is evaluated through numerical simulations and its performance is compared with other coprime planar array structures. It is shown that LCA has less computational complexity together with less real sensor elements and it provides superior performance as compared to the conventional 2-D coprime planar arrays.

Underdetermined DOA Estimation of Wideband LFM Signals Based on Gridless Sparse Reconstruction in the FRF Domain.

An underdetermined direction of arrival (DOA) estimation method of wideband linear frequency modulated (LFM) signals is proposed without grid mismatch. According to the concentration property of LFM signal in the fractional Fourier (FRF) domain, the received sparse model of wideband signals with time-variant steering vector is firstly derived based on a coprime array. Afterwards, by interpolating virtual sensors, a virtual extended uniform linear array (ULA) is constructed with more degrees of freedom, and its covariance matrix in the FRF domain is recovered by employing sparse matrix reconstruction. Meanwhile, in order to avoid the grid mismatch problem, the modified atomic norm minimization is used to retrieve the covariance matrix with the consecutive basis. Different from the existing methods that approximately assume the frequency and the steering vector of the wideband signals are time-invariant in every narrowband frequency bin, the proposed method not only can directly solve more DOAs of LFM signals than the number of physical sensors with time-variant frequency and steering vector, but also obtain higher resolution and more accurate DOA estimation performance by the gridless sparse reconstruction. Simulation results demonstrate the effectiveness of the proposed method.