Coprime Planar Array Research Articles

Symmetric Displaced Coprime Planar Array for Two-Dimensional Direction-of-Arrival Estimation

Recently, sparse planar arrays have been developed to improve the 2-D direction-of-arrival (DOA) estimation performance by their superiorities in the degree of freedom (DOF), array aperture, and mutual coupling over uniform planar arrays. In this article, we devise a new sparse planar array, termed symmetric displaced coprime planar array (SDCPA), for estimating the 2-D DOAs. The SDCPA configuration consists of three sparse uniform rectangular arrays (URAs) with a certain horizontal and vertical displacement, implying that there are two bilaterally SDCPAs with one shared subarray. Under any number of sensors, the sensor positions of an SDCPA can be expressed in a closed form. We also derive the optimal displacement factors to achieve the maximum number of consecutive coarray lags, which can be analytically calculated. Compared with the conventional CPAs, the SDCPA configuration possesses a higher number of uniform DOFs and fewer mutual coupling effects. Numerical results demonstrate its superiority over several existing sparse planar arrays.

Sparsity-Based Two-Dimensional DOA Estimation for Co-Prime Planar Array via Enhanced Matrix Completion

In this paper, the two-dimensional (2-D) direction-of-arrival (DOA) estimation problem is explored for the sum-difference co-array (SDCA) generated by the virtual aperture expansion of co-prime planar arrays (CPPA). Since the SDCA has holes, this usually causes the maximum virtual aperture of CPPA to be unavailable. To address this issue, we propose a complex-valued, sparse matrix recovery-based 2-D DOA estimation algorithm for CPPA via enhanced matrix completion. First, we extract the difference co-arrays (DCA) from SDCA and construct the co-array interpolation model via nuclear norm minimization to initialize the virtual uniform rectangular array (URA) that does not contain the entire rows and columns of holes. Then, we utilize the shift-invariance structure of the virtual URA to construct the enhanced matrix with a two-fold Hankel structure to fill the remaining empty elements. More importantly, we apply the alternating direction method of the multipliers (ADMM) framework to solve the enhanced matrix completion model. To reduce the computational complexity of the traditional vector-form, sparse recovery algorithm caused by the Kronecker product operation between dictionary matrices, we derive a complex-valued sparse matrix-recovery model based on the fast iterative shrinkage-thresholding (FISTA) method. Finally, simulation results demonstrate the effectiveness of the proposed algorithm.

Blind signal separation for coprime planar arrays: An improved coupled trilinear decomposition method

AbstractIn this study, the problem of blind signal separation for coprime planar arrays is investigated. For coprime planar arrays comprising two uniform rectangular subarrays, we link the signal separation to the tensor‐based model called coupled canonical polyadic decomposition (CPD) and propose an improved coupled trilinear decomposition approach. The output data of coprime planar arrays are modeled as a coupled tensor set that can be further interpreted as a coupled CPD model, allowing a signal separation to be achieved using coupled trilinear alternating least squares (TALS). Furthermore, in the procedure of the coupled TALS, a Vandermonde structure enforcing approach is explicitly applied, which is shown to ensure fast convergence. The results of Monto Carlo simulations show that our proposed algorithm has the same separation accuracy as the basic coupled TALS but with a faster convergence speed.

Two-Dimensional DOA Estimation for Coprime Planar Arrays Based on Self-Correlation Tensor

In the coprime planar array (CPA), the existing tensor DOA estimation has the problem that the statistics are not fully utilized. We propose a two-dimensional DOA estimation method based on tensor self-correlation, which realizes the high-resolution and high-precision joint estimation of elevation angle and azimuth angle. Firstly, we represent the received signals of two subarrays with tensors and then obtain the self-correlation covariance tensor of the subarrays themselves and the cross-correlation covariance tensor of the two subarrays. Then, we extract the covariance tensor corresponding to the maximum continuous virtual array and prove the expression of the maximum continuous virtual array aperture of the proposed method. Compared with the existing methods, the proposed method effectively improves the maximum aperture of the continuous virtual array. Finally, the signal subspace is solved by tensor expansion and tensor decomposition. Simulation results show that under the same conditions, the proposed method has higher estimation accuracy and degree of freedom than the cross-correlation tensor method, and the resolution is also improved significantly.

Generalized and symmetric coprime planar array for direction of arrival estimation

Degrees of freedom and mutual coupling are two important factors to determine the ability of direction of arrival (DOA) estimation. Although coprime planar array (CPPA) has less mutual coupling than hourglass array (HA) and ladder array (LA), its positive and negative covariance corresponding lags are discontinuous, resulting in low degrees of freedom. In this study, we propose a generalized and symmetric coprime planar array (GSCA) by employing the symmetric property of the difference co-array. By moving one subarray and introducing a supplementary subarray, we construct a connected virtual uniform rectangular array (URA). Through further optimizing the sensor number, the redundant lags are reduced and the array structure is more general. We show that the degrees of freedom are much higher than those of CPPA, HA and LA. Besides, the mutual coupling is close to CPPA, but less than HA and LA. The superiority of GSCA is verified by simulations.

A Novel Approach of 2-D DOA Estimation by Employing Coprime Linear Array Motion

In this letter, we explore the feature of coprime linear array (CLA) motion for enhanced array aperture and higher degree of freedom (DOF) by implementing synthetic aperture processing. Since the traditional fixed CLA cannot achieve two-dimensional (2-D) DOA estimation, we propose a novel approach to construct the virtual coprime planar array (VCPA) just by the CLA motion. Specifically, the CLA moves along the vertical direction at a constant velocity and the two subarrays of CLA are controlled to snap at different time delays related to the coprime number of sensors, by which the received signals at different time delays are synthetically processed to construct a VCPA. Simulation results demonstrate that: a) our proposed approach outperforms the synthetic CLA moving along the direction of layout of the initial arrays on DOF and the dimension of DOA estimation; b) the proposed VCPA can achieve 2-D DOA estimation just by the motive CLA and achieve no worse performance than the practical coprime planar array but by using fewer sensors.

FDA-MIMO radar for 3D localization: Virtual coprime planar array with unfolded coprime frequency offset framework and TRD-MUSIC algorithm

The frequency diverse array multiple-input multiple-output (FDA-MIMO) radar can detect target range by exploiting a small frequency offset across the transmit sensors, which can be utilized to jointly estimate angle and range. Nevertheless, the detection performance is basically restricted by the array aperture and signal bandwidth. In this paper, we propose a joint optimization design scheme for FDA-MIMO radar, i.e., the virtual coprime planar array with ‘unfolded’ coprime frequency offsets (VCPA-UCFO) framework, for 3D localization without ambiguity. The VCPA-UCFO framework can significantly save sensors and physical space while remarkably extend array aperture and signal bandwidth, which brings about remarkably enhanced economic benefits and estimation performance. Additionally, we construct the 3D localization problem as 3D-MUSIC spatial spectrum function and transform the 3D total spectrum search of the conventional 3D-MUSIC algorithm into 1D local spectrum search by cooperating ESPRIT algorithm and twice reduce dimension MUSIC (TRD-MUSIC) algorithm. The TRD-MUSIC algorithm can significantly relieve computational burden but with no performance degradation. The CRBs are given as performance benchmark. The analysis and simulations have verified the effectiveness and advantages of VCPA-UCFO framework and TRD-MUSIC algorithm in system cost, localization accuracy, resolution and computational complexity.

Hole Locations and a Filling Method for Coprime Planar Arrays for DOA Estimation

Coprime linear arrays (CLAs) have drawn lots of attention. Efforts have been made to reveal the hole locations in the difference coarrays of CLAs, based on which many methods have been proposed to fill the holes, lengthening the consecutive difference coarray part and increasing the effective degrees of freedom (DOFs). Compared with CLAs, two dimensional (2D) coprime planar arrays (CPAs) are more relevant to real applications. However, no closed-form expressions for the hole locations in the difference coarray of a CPA have been found in the open literature, and the advantage in terms of DOFs of CPAs has not been well explored. In this letter, the structure of the difference coarrays of CPAs is investigated and the exact expressions of all hole positions are provided. Then, a holes-filling method is proposed, by which the most critical holes in the difference coarray can be filled such that a difference coarray with more consecutive lags as well as higher effective DOFs can be obtained.

Computationally Efficient Ambiguity-Free Two-Dimensional DOA Estimation Method for Coprime Planar Array: RD-Root-MUSIC Algorithm

While the two-dimensional (2D) spectral peak search suffers from expensive computational burden in direction of arrival (DOA) estimation, we propose a reduced-dimensional root-MUSIC (RD-Root-MUSIC) algorithm for 2D DOA estimation with coprime planar array (CPA), which is computationally efficient and ambiguity-free. Different from the conventional 2D DOA estimation algorithms based on subarray decomposition, we exploit the received data of the two subarrays jointly by mapping CPA to the full array of the CPA (FCPA), which contributes to the enhanced degrees of freedom (DOFs) and improved estimation performance. In addition, due to the ambiguity-free characteristic of the FCPA, the extra ambiguity elimination operation can be avoided. Furthermore, we convert the 2D spectral search process into 1D polynomial rooting via reduced-dimension transformation, which substantially reduces the computational complexity while preserving the estimation accuracy. Finally, numerical simulations demonstrate the superiority of the proposed algorithm.

Efficient Gridless 2-D Direction-of-Arrival Estimation for Coprime Array Based on Decoupled Atomic Norm Minimization

This paper presents an efficient gridless sparse reconstruction algorithm for the coprime planar array in two-dimensional (2-D) direction-of-arrival (DOA) estimation problem. According to the equivalent second-order statistic signals derived from the covariance matrix of the coprime planar array, we construct a virtual 2-D difference co-array extended from the coprime line arrays along two directions. The virtual array has a double-sized array aperture leading to an increased number of degree-of-freedoms (DOFs). To address the discontinuity of the virtual planar array, and to reduce the computation complexity for the increased array size, decoupled atomic norm minimization approach is investigated to interpolate the missing sensors without discarding any virtual sensors. The problem of decoupled atomic norm minimization can be solved by semidefinite programming with significantly lower computational cost. Besides, the ratio of the number of missing sensors to full sensors in the interpolated virtual uniform array is smaller than that of the physical coprime array, which further improves the recovery accuracy of decoupled atomic norm minimization algorithm. The numerical examples are provided to demonstrate the practical ability of the proposed method in terms of DOF, computational complexity, and DOA estimation error.

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.

Fast Two-Dimensional Direction-of-Arrival Estimation of Multiple Signals in Coprime Planar Array

Two-dimensional direction-of-arrival (DOA) estimation in coprime planar array involves problems that the complexity of spectral peak search is huge and the noncircular feature of signals is not considered. Considering that unitary estimating signal parameters via rotational invariance techniques (Unitary-ESPRIT) is a low complexity subspace algorithm, an approach to estimate DOA fast for multiple signals is proposed in this paper. We first apply Unitary-ESPRIT to solve one possible value of each signal. Given the relationship between the ambiguous values and real values, we then have all possible values belonging to each subarray. Through finding the common values of two subarrays, we finally obtain the highly precise true DOAs. Moreover, when the signals are noncircular, we present an improved method using noncircular Unitary-ESPRIT, which is favorable in terms of accuracy and degree of freedom. Simulation results demonstrate the effectiveness of our proposed methods.

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.

Two-Dimensional Direction-of-Arrival Estimation of Noncircular Signals in Coprime Planar Array with High Degree of Freedom

In estimating the two-dimensional (2D) direction-of-arrival (DOA) using a coprime planar array, there are problems of the limited degree of freedom (DOF) and high complexity caused by the spectral peak search. We utilize the time-domain characteristics of signals and present a high DOF algorithm with low complexity based on the noncircular signals. The paper first analyzes the covariance matrix and ellipse covariance matrix of the received signals, vectorizes these matrices, and then constructs the received data of a virtual uniform rectangular array (URA). 2D spatial smoothing processing is applied to calculate the covariance of the virtual URA. Finally, the paper presents an algorithm using 2D multiple signal classification and an improved algorithm using unitary estimating signal parameters via rotational invariance techniques, where the latter solves the closed-form solutions of DOAs replacing the spectral peak search to reduce the complexity. The simulation experiments demonstrate that the proposed algorithms obtain the high DOF and enable to estimate the underdetermined signals. Furthermore, both two proposed algorithms can acquire the high accuracy.

Direction-of-Arrival Estimation for Both Uncorrelated and Coherent Signals in Coprime Array

Direction-of-arrival (DOA) estimation in coprime array involves the problem that there is the coexistence of both uncorrelated signals and coherent signals in the multipath environment. An approach to estimate DOA of uncorrelated and coherent signals separately, based on the coprime linear array and coprime planar array, is proposed in this paper. The uncorrelated signals are estimated first, where the root multiple signal classification (root-MUSIC) is applied in the coprime linear array and unitary estimating signal parameters via rotational invariance techniques (Unitary-ESPRIT) is utilized in the coprime planar array. Those subspace algorithms are of low complexity compared with the spectral peak search method. We then eliminate the components of noises and uncorrelated signals and reconstruct a covariance matrix of coherent signals. Finally, we use the root-MUSIC or Unitary-ESPRIT to resolve the one-dimensional or two-dimensional DOAs of coherent signals, respectively. The simulation results demonstrate the computational efficiency and high accuracy of the proposed algorithm. The results also prove that this algorithm can estimate the number of signals more than that of subarray sensors and separate two signals from the same angle.

An Improved Trilinear Model-Based Angle Estimation Method for Co-Prime Planar Arrays.

Angle estimation methods in two-dimensional co-prime planar arrays have been discussed mainly based on peak searching and sparse recovery. Peak searching methods suffer from heavy computational complexity and sparse recovery methods face some problems in selecting the regularization parameters. In this paper, we propose an improved trilinear model-based method for angle estimation for co-prime planar arrays in the view of trilinear decomposition, namely parallel factor analysis. Due to the principle of trilinear decomposition, our method does not require peak searching and can conduct auto-pairing easily, which can reduce the computational loads and avoid parameter selection problems. Furthermore, we exploit the virtual array concept of the whole co-prime planar array through the cross-correlation matrix obtained from the received signal data and present a matrix reconstruction method using the Khatri–Rao product to tackle the matrix rank deficiency problem in the virtual array condition. The simulation results show that our proposed method can not only achieve high estimation accuracy with low complexity compared to other similar approaches, but also utilize limited sensor number to implement the angle estimation tasks.

V-Shaped Sparse Arrays for 2-D DOA Estimation

This paper proposes a new sparse array geometry for 2-D (azimuth and elevation) direction-of-arrival (DOA) estimation. The proposed array geometry is V-shaped sparse array, and it is composed of two linear portions which are crossing each other. The degrees of freedom of the sparse array are enhanced by sparse sampling property. In this respect, V-shaped coprime (VCA) and V-shaped nested array (VNA) structures are developed. VCA can resolve both azimuth and elevation angles up to MN sources with $$2M + N -1$$ sensors in each portion, and the total number of sensors is $$4M+2N-3$$ . VNA can resolve $$O(N^2)$$ sources with 2N sensors. Instead of 2-D grid search, the proposed method computes 1-D search for azimuth and elevation angle estimation in a computational efficient way. In order to solve the pairing problem in 2-D scenario, the cross-covariance matrix of two portion is utilized and 2-D paired DOA estimation is performed. The performance of the proposed method is evaluated with numerical simulations, and it is shown that the proposed array geometries VCA and VNA can provide much less sensors as compared to the conventional coprime planar arrays.

Two-Dimensional Direction-of-Arrival Fast Estimation of Multiple Signals with Matrix Completion Theory in Coprime Planar Array.

In estimating the two-dimensional (2D) direction-of-arrival (DOA) using a coprime planar array, the main issues are the high complexity of spectral peak search and the limited degree of freedom imposed by the number of sensors. In this paper, we present an algorithm based on the matrix completion theory in coprime planar array that reduces the computational complexity and obtains a high degree of freedom. The algorithm first analyzes the covariance matrix of received signals to estimate the covariance matrix of a virtual uniform rectangular array, which has the same aperture as the coprime planar array. Matrix completion theory is then applied to estimate the missing elements of the virtual array covariance matrix. Finally, a closed-form DOA solution is obtained using the unitary estimation signal parameters via rotational invariance techniques (Unitary-ESPRIT). Simulation results show that the proposed algorithm has a high degree of freedom, enabling the estimation of more signal DOAs than the number of sensors. The proposed algorithm has reduced computational complexity because the spectral peak search is replaced by Unitary-ESPRIT, but attains similarly high levels accuracy to those of the 2D multiple signal classification algorithm.

Reduced Dimension Based Two-Dimensional DOA Estimation with Full DOFs for Generalized Co-Prime Planar Arrays.

In this paper, we investigate the problem of two-dimensional (2D) direction-of-arrival (DOA) estimation for generalized co-prime planar arrays. The classic multiple signal classification (MUSIC)-based methods can provide a superior estimation performance, but suffer from a tremendous computational burden caused by the 2D spectral search. To this end, we reduce the 2D problem into a one-dimensional (1D) one and propose a reduced dimension partial spectral search estimation method, which can compress the search region into a small 1D sector. Moreover, the proposed method can utilize the full information of the entire array without degrees-of-freedom loss. Furthermore, an iterative approach is also proposed to reduce complexity and improve performance. Simulation results show that the proposed methods can provide improved performance with substantially reduced complexity, as compared to other state-of-the-art methods.

Two-dimensional DOA estimation for generalized coprime planar arrays: a fast-convergence trilinear decomposition approach

In this paper, we investigate the problem of two-dimensional (2D) direction of arrival (DOA) estimation of multiple signals for generalized coprime planar arrays consisting of two rectangular uniform planar subarrays. We propose a fast-convergence trilinear decomposition approach, which uses propagator method (PM) as the initialization of the angle estimation to speed the convergence of trilinear decomposition. The received signal of each subarray can be fitted into a trilinear model or parallel factor (PARAFAC) model so that the trilinear alternating least square algorithm can be used to estimate the angle information. Meanwhile, the necessary initialization of DOA estimates can be achieved via PM, which endows the proposed approach a fast convergence and subsequently results in a low complexity. Specifically, we eliminate the ambiguous estimates by utilizing the coprime property and the true DOA estimates can be achieved by selecting the nearest ones of all DOA estimates. The proposed approach can obtain the same estimation performance as the conventional PARAFAC algorithm, but with a low computational cost. Numerical simulation results are provided to validate the effectiveness and superiority of the proposed algorithm.