Coprime Array Research Articles

Pseudo-beam-Forming for Direction-of-Arrival Estimation with Difference Co-array of Co-prime Array

This paper focuses on direction finding using a co-prime array from the view of a difference co-array. According to the corresponding relationship between the correlation lag and virtual element position of the difference co-array, from the correlation matrix of the co-prime array, the desired correlation units are extracted as single-snapshot data of the virtual co-array elements of the co-prime array and are then coherently accumulated into a pseudo-beam pattern. Because the difference co-array of a co-prime array consists of a group of contiguous virtual elements and multiple non-uniform virtual elements, this paper considers pseudo-beam-forming using only contiguous virtual elements, as in existing studies, and all of the virtual elements. Compared with the existing sub-beam multiplication method, pseudo-beam-forming reduces the negative effect from grating lobes and resolves more uncorrelated sources than the number of physical elements. Moreover, application of non-uniform virtual elements improves the resolvable source number, angle resolution and noise immunity, which are analyzed quantitatively based on the proposed distribution characteristic of virtual elements. Finally, to suppress side-lobe interference caused by the non-uniform virtual elements, we introduce and evaluate three coherence weighting factors, namely coherence factor (CF), phase coherence factor (PCF) and sign coherence factor (SCF), where CF is proved to be ineffective and SCF is optimal in suppression and computation performance.

A Fast and Robust DOA Estimation Method Based on JSVD for Co-Prime Array

Compared with the uniform linear array, the co-prime array can obtain a large array aperture with fewer array elements, which is good to improve the accuracy of direction-of-arrival (DOA) estimation. However, most of existing DOA estimation methods is not suitable for co-prime array because of high computational complexity and low adaptability. Therefore, a fast and robust DOA estimation method for the co-prime array based on the joint singular value decomposition (JSVD) is proposed in this paper. In the proposed method, the co-prime array is equivalently divided into two uniform sparse linear subarrays according to the definition of co-prime array geometry firstly. Then, utilizing the JSVD algorithm and the periodic repeatability of uniform sparse linear array, two series of DOAs, including ambiguity angles can be obtained individually by each subarray. By analysis of these two series of DOAs, the correct DOAs can be obtained based on the coprime property of two equivalent uniform sparse linear subarrays of the co-prime array. The spectral search is not required in the proposed method, so that the computational complexity is lower relative to the MUSIC-based DOA estimation methods. Moreover, the angle pairing is achieved automatically in JSVD processing, thus the adaptability of proposed method is better than the traditional DOA estimation methods. Finally, the performance and advantages of proposed method are verified by numerical simulations.

Direct Position Determination of Non-Circular Sources Based on a Doppler-Extended Aperture With a Moving Coprime Array

Direct position determination (DPD) is a new promising technique in wireless location. Compared with conventional two-step localization methods, DPD achieves a higher accuracy by directly estimating the source location without computing the intermediate parameters. However, all the existing angle-based DPD algorithms for non-circular sources use uniform linear arrays, which lead to low degrees of freedom and poor estimation precision, and do not make use of the Doppler characteristics of the moving station. To improve the DPD performance, this paper proposes a novel DPD algorithm for non-circular sources based on a Doppler-extended aperture with a moving coprime array. First, the coprime array is introduced to angle-based DPD, and an extended array model is established by exploiting the Doppler characteristics of the moving array. The characteristics of non-circular sources are used to further improve the positioning accuracy. Finally, merging the subspace data for each measuring position, the target positions can be obtained. The Cramer-Rao lower bound of the DPD algorithm is presented for non-circular sources in a coprime array combined with Doppler-extended aperture. Performance analysis and the simulation experiments show that compared with the conventional two-step localization method and DPD based on a uniform array, including subspace data fusion and weighted subspace fitting algorithms, the proposed algorithm effectively improves the location accuracy at the expense of a slight complexity increase and can determine the location of multiple targets in underdetermined conditions.

Novel Rank Enhancing Algorithm for 2D Sparse Arrays with Contiguous Coarrays

Sparse array is an irregular array with a high degree of freedom in its difference coarray domain. Nested array and coprime array are two important one dimensional sparse arrays. For fixed number of sensors, nested array has a high degree of freedom, resolution and accuracy than coprime array. 1D nested arrays do not provide 2D scanning. 2D nested array which is an extension of 1D nested array provides both 2D scanning as well as high degree of freedom, however direction of arrival (DoA) estimation techniques used for 2D nested array such as 2D unitary ESPRIT and 2D spatial smoothing MUSIC has high computational complexity. This paper shows that spatial smoothing is not necessary to apply 2D MUSIC in the difference coarray domain of 2D nested array. A new Hermitian Toeplitz matrix is constructed instead of spatial smoothing. 2D MUSIC on this Hermitian Toeplitz matrix can resolve the DoA of sources with less computations.

Space-Time Adaptive Processing for Airborne Radars With Space-Time Coprime Sampling Structure

Coprime arrays and coprime time samplers have been receiving attractive attention recently due to their advantages of large apertures and achievable degrees of freedom with low cost. In this paper, different from traditional airborne radar with uniform transmitting pulses, the considered radar is configured with a coprime array in receiver and transmits the pulses with coprime repetition intervals. Because this sampling structure violates the conditions of the conventional clutter rank estimation rules, we first propose a novel method to estimate the clutter rank based on effective aperture-bandwidth product. Then, two space-time adaptive processing (STAP) algorithms, namely, the sample matrix inverse- and principal components-STAP, are proposed by using the reconstructed virtual space-time snapshots. Additionally, a thoroughly theoretical convergence analysis of the proposed algorithms is conducted. Exclusive simulation results are provided to demonstrate the effectiveness of the proposed STAP algorithms.

Interpolating Coprime Arrays With Translocated and Axis Rotated Compressed Subarrays by Iterative Power Factorization for DOA Estimation

In this paper, a novel array structure exploiting coprime arrays is proposed, which comprises the translocation of one subarray and axis rotation with a compression of another subarray to produce a larger number of consecutive lags. This composition is named CATARCS, which stands for coprime array with translocated and axis rotated compressed subarrays. The scheme offers consecutive co-array lags in remarkable number $4MN-1$ by using only $2M+N-1$ number of sensors attaining higher degrees of freedom than the state of the art, where $M$ and $N$ are coprime with congenial interelement spacings. However, a very few holes still exist in the co-array, which can be filled up through interpolation. In this paper, the iterative power factorization (IPF) algorithm has been demonstrated as a tool for incremented-rank-IPF matrix recovery which is unostentatious. No extra tuning parameter is required for this approach and several undesired issues of other techniques can be reconciled by this. The main contribution of this paper is suitable modification of array geometry arrangements to increase consecutive lags and to interpolate via IPF, which is used first time for direction of arrival (DOA) estimation in order to exhibit better performances in terms of detecting and resolving a larger number of sources by enhancing resolution. Later, DOA estimation can be achieved by performing multiple signal classification (MUSIC) algorithm. Simulation results validate its effectiveness by achieving lower cost, faster, and accurate DOA estimation even in lower angular separation of sources compared with other techniques like spatial smoothing MUSIC, co-array the least absolute shrinkage and selection operator, spline, iterative co-array array interpolation, and Toeplitz completion.

Off-Grid DOA Estimation Aiding Virtual Extension of Coprime Arrays Exploiting Fourth Order Difference Co-Array With Interpolation

In this paper, a novel array structure exploiting coprime arrays is proposed which can be very proficient to determine the number of consecutive lags in proportion with the number of array elements. The proposed method comprises novel array structure by configuring three subarrays positioned in alignment with some prescribed values. By increasing array elements in third subarray while keeping other subarrays fixed, explicit number of consecutive lags could be obtained proportionately. The proposed method offers maximization of consecutive lags in remarkable number by calculating the fourth order difference co-array unifying interpolation. The forth order difference co-array is achieved by exploiting the second order difference co-array twice. The consideration of third subarray in addition with two coprime subarrays leads to a novel array structure which can significantly enhance degrees of freedom. An effective interpolation technique nuclear norm minimization is considered to fill the holes subsisting in the virtual co-array in order to exploit full virtual co-array length. This interpolation method uses convex framework which is trackable and very simple to implement yielding a freedom of fixing any predefined extra tuning parameter. The proposed method in this paper is named as VEFODCI which stands for virtual extension of coprime arrays by exploiting fourth order difference co-array with interpolation. The sparse Bayesian learning is used for direction of arrival (DOA) estimation exploiting the proposed novel array structure by imposing interpolation to fill the holes. The array geometry of sensor distributions proves that the proposed method is less susceptible from mutual coupling effect. The simulation results stipulate that the proposed method is performing DOA estimation accurately even in lower angular separation of sources by achieving larger number of consecutive lags than the state of the art.

Robust Generalized Chinese-Remainder- Theorem-Based DOA Estimation for a Coprime Array

In this paper, we consider the problem of determining the direction of arrival (DOA) of two sources for a coprime array. The novelty of this paper is that we present an efficient algorithm to estimate two DOAs from their wrapped phases with errors. The proposed robust generalized Chinese remainder theorem is considerably less computational complex than the searching method while maintaining comparable estimation precision. Moreover, the largest range of DOA that leads to an unambiguous determination is obtained for a given coprime array. Numerical simulations are presented to verify the efficiency and the robustness of the proposed algorithm. The simulation results show that the proposed algorithm is much faster than the searching method while achieving a similar root-mean-square error.

DOA Estimation for Generalized Three Dimensional Coprime Arrays: A Fast-Convergence Quadrilinear Decomposition Algorithm

In this paper, we construct the generalized 3-D coprime array (G-TDCA) configuration, which consists of two 2-D uniform subarrays, by selecting three pairs of coprime integers to obtain the extension of inter-element spacing. Meanwhile, we derive the analytical expression of Cramer–Rao bound for G-TDCA and verify that G-TDCA outperforms the conventional 3-D uniform array (TDUA) in the direction of arrival (DOA) (i.e., azimuth and elevation angles) estimation performance and resolution with numerical simulations. In addition, we propose a fast convergence quadrilinear decomposition (FC-QD) algorithm to extract the DOA estimates for G-TDCA. In the FC-QD algorithm, we first employ propagator method (PM) to achieve rough DOA estimates that are utilized to initialize the steering matrices. Subsequently, the received signal is constructed as two quadrilinear models and quadrilinear alternating least square algorithm is operated to attain fine DOA estimates. Moreover, phase ambiguity problem, caused by large adjacent distance between array sensors, is eliminated based on coprime property, where the uniqueness of DOA estimates can be achieved. Specifically, the proposed FC-QD algorithm has a fast convergence due to initialization via PM, and hence, FC-QD can significantly lower the computational cost without degrading the DOA estimation performance, which is demonstrated by extensive simulation results.

Cyclic block coordinate minimization algorithms for DOA estimation in co-prime arrays

We derive several closed-form expressions that generalize co-prime array system model and study a nonnegative gridless compressive sensing formulation of the problem of estimating direction-of-arrival (DOA) based on the derived model. To solve the problem, two computationally efficient cyclic block coordinate minimization algorithms are proposed; the algorithms perform atomic norm minimization of an objective function through a sequence of computationally efficient atom merging and atom activation steps conducted in subdomains of a continuous atom search space. The convergence properties of the developed algorithms are analyzed. Numerical simulations demonstrate that the proposed techniques outperform the joint sparsity reconstruction method (JLASSO) and the ESPRIT method with spatial smoothing (SS-ESPRIT) in terms of various criteria. It is also demonstrated that our methods are significantly faster and yield competitive performance in terms of root mean square error (RMSE), detection probability, and false alarms when compared to the recent convex optimization based methods, i.e. the gridless SPICE with ESPRIT (GLS-ESPRIT), the atomic norm minimization with dimension reduction and ESPRIT (ANM-ESPRIT), and the nuclear norm minimization with ESPRIT (NNM-ESPRIT).

Unified coprime array with multi-period subarrays for direction-of-arrival estimation

The original coprime array is a sparse geometry with a pair of one-period subarrays, where the interelement spacing and number of sensors are coprime so that no elements overlap except at the zeroth position. In this paper, we unify the coprime array by extending two subarrays from one period to multiple periods. Each subarray can have an arbitrary number of periods and therefore more than one element of the unified structure coincide. The resulting configuration treats the nested array as well as the existing coprime structures, such as the original coprime array, the extended one and the coprime array with compressed inter-element spacing (CACIS) as its special cases. The new coprime array has a lower peak side-lobe level than the nested array and the CACIS. An improved structure based on the multi-period coprime array is further proposed to improve the degrees-of-freedom (DOF) of the unified geometry with one of the subarrays displaced by a certain number of spacings. Although the improved configuration does not reach the same DOF as the CACIS and the coprime array with displaced subarrays (CADiS), it can acquire a low peak side-lobe level with the DOF not being decreased too much. Different from CADiS, we focus on the analysis and proper choice of the displacement to acquire more consecutive lags. The properties of coarray of the proposed array design strategy are provided as well. Finally, the direction-of-arrival estimation, physical aperture and peak side-lobe level performance of the unified structures are examined through numerical simulations.

N-tuple coprime sensor arrays.

Until now, coprime sensor arrays have used two sparsely spaced subarrays to emulate the performance of a single uniform array with many more sensors (generally on the order of the product of each subarrays' number of sensors). This allows for similar results with fewer sensors, or the observation of higher frequencies (above the Nyquist limit) with a similar number of sensors. The theory rests on the cross-referencing (using directional filter banks) or cancellation (using product processing) of the M grating lobes in one subarray's beampattern and N grating lobes in the other, where M and N are coprime integers. Sets of coprime integers can consist of more than two integers, however, and introducing another coprime factor theoretically multiplies observable frequency (or further decreases the number of array elements needed for the same frequency). Any amount, n, of coprime integers and corresponding subarrays may be used. In this work, "n-tuple coprime sensor array" theory is expounded and implemented. Experimentally measured beampattern results of a triple coprime sensor array (with three subarrays) are shown, using an extension of the authors' previously established product processing. Results also confirm that the usable range of an n-tuple coprime array extends below its design frequency.

2D-DOA Estimation for Coprime L-shaped Arrays with MUSIC Algorithm

In this paper, a two-dimensional direction of arrival (2D-DOA) estimation algorithm for coprime L-shaped arrays with multiple signal classification (MUSIC) was proposed. The coprime Lshaped arrays consists of two uniform L-shaped subarrays with the minimum inter-element spacing larger than the half-wavelength of signal, which helps increase the degree of freedom and enhance the resolution of the array, but meanwhile causes phase ambiguity. In the proposed algorithm, we exploited the coprime property between the two single subarrays to eliminate ambiguity. And, MUSIC is one of the most classical and traditional algorithm, which has a great estimation performance and accuracy. Comparing to the typical one with the same total number of sensor elements, the proposed algorithm for coprime arrays obtain a better performance and a lower complexity. And the simulation results on MATLAB verify the algorithm’s effectiveness and advantages.

Detection performance analysis of product processing of colinear arrays

Non-uniform linear arrays (NULAs) often achieve better resolution than standard uniform linear arrays (ULAs) with equal numbers of sensors. However, conventional beamforming (CBF) of an NULA received signal leads to the same detection statistic PDF as a ULA with the equal number of sensors, undercutting NULA's improved resolution. Nested and coprime arrays partition an NULA into two subarrays and multiply the subarrays' CBF outputs. This research compares this product processor's detection performance against a CBF detector with an equal number of sensors for a narrowband Gaussian signal in spatially white additive Gaussian noise. The product processor's detection PDF is a scaled product of the detection statistic with modified Bessel functions. Receiver operation characteristics (ROC) curves illustrate that the product processor's performance is inferior to the CBF detector with an equal number of sensors. The detection performance of a product processor matches the CBF detector only for high SNRs and large numbers of sensors. However, in the presence of interferers, the product processor for coprime arrays can outperform both the CBF detector and product processing nested arrays with an equal number of sensors. [Work supported by ONR.]

Correlation Subspaces: Generalizations and Connection to Difference Coarrays

Direction-of-arrival (DOA) estimation finds applications in many areas of science and engineering. In these applications, sparse arrays such as minimum redundancy arrays, nested arrays, and coprime arrays can be exploited to resolve $O(N^2)$ uncorrelated sources using $N$ physical sensors. Recently, it has been shown that correlation subspaces , which reveal the structure of the covariance matrix, help to improve some existing DOA estimators. However, the bases, the dimension, and other theoretical properties of correlation subspaces remain to be investigated. This paper proposes generalized correlation subspaces in one and multiple dimensions. This leads to new insights into correlation subspaces and DOA estimation with prior knowledge. First, it is shown that the bases and the dimension of correlation subspaces are fundamentally related to difference coarrays , which were previously found to be important in the study of sparse arrays. Furthermore, generalized correlation subspaces can handle certain forms of prior knowledge about source directions. These results allow one to derive a broad class of DOA estimators with improved performance. It is demonstrated through examples that using sparse arrays and generalized correlation subspaces, DOA estimators with source priors exhibit better estimation performance than those without priors, in extreme cases like low SNR and limited snapshots.

Subarray processing with coprime and minimum redundancy arrays

A coprime sensor array (CSA) refers to a non-uniform linear array consisting of two undersampled uniform linear arrays (ULAs) with coprime numbers of elements, respectively. Its beam pattern can be enhanced by multiplying beampatterns of individual subarrays. This technique, called product processing, provides improved resolution because its sum coarray length is extended from the product of beampatterns. Nevertheless, the extended coarray inevitably includes holes that result in higher peak sidelobes than the full ULA case. In this paper, we propose a technique to reduce peak sidelobes through the combination with minimum redundancy arrays (MRAs) and an additional subarray. Two ULAs comprising the coprime array are thinned by the MRA technique, and the reduced number of sensors is then utilized to construct the third subarray. The third array suppresses peak sidelobes by placing nulls at the peak sidelobe locations. The characteristics of the proposed technique are analyzed in terms of white noise gain (WNG) and maximum sidelobe level (MSL). It is demonstrated that the proposed technique can improve the MSL with similar number of sensors, thereby reducing the estimation error in the presence of multiple sources.

A Direct Coarray Interpolation Approach for Direction Finding.

Sparse arrays have gained considerable attention in recent years because they can resolve more sources than the number of sensors. The coprime array can resolve sources with only sensors, and is a popular sparse array structure due to its closed-form expressions for array configuration and the reduction of the mutual coupling effect. However, because of the existence of holes in its coarray, the performance of subspace-based direction of arrival (DOA) estimation algorithms such as MUSIC and ESPRIT is limited. Several coarray interpolation approaches have been proposed to address this issue. In this paper, a novel DOA estimation approach via direct coarray interpolation is proposed. By using the direct coarray interpolation, the reshaping and spatial smoothing operations in coarray-based DOA estimation are not needed. Compared with existing approaches, the proposed approach can achieve a better accuracy with lower complexity. In addition, an improved angular resolution capability is obtained by using the proposed approach. Numerical simulations are conducted to validate the effectiveness of the proposed approach.

A Super-Resolution Direction of Arrival Estimation Algorithm for Coprime Array via Sparse Bayesian Learning Inference

In this paper, we address the problem of direction of arrival (DOA) estimation with coprime array in the context of sparse signal reconstruction to fully exploit the enhanced degrees of freedom (DOF) offered by the difference coarray. The proposed method is based on the framework of sparse Bayesian learning and can jointly refine the unknown DOAs and the sparse signals in a gradual and interweaved manner. Specifically, the proposed approach is constructed by iteratively decreasing a surrogate function majorizing a given objective function, which results in accelerating the speed to converge to the global minimum. Furthermore, for facilitating a noise-free sparse representation, a customized linear transformation is judiciously incorporated in our sparsity-inducing DOA estimator to eliminate the unknown noise variance, and in the mean time, the sample covariance matrix perturbation can be normalized to an identity matrix as a by-product. Extensive simulation experiments under different conditions finally demonstrate the superiority of our suggested algorithm in terms of mean-squared DOA estimation error, DOF and resolution ability over state-of-the-art techniques.

Direction-of-Arrival Estimation with Coarray ESPRIT for Coprime Array.

A coprime array is capable of achieving more degrees-of-freedom for direction-of-arrival (DOA) estimation than a uniform linear array when utilizing the same number of sensors. However, existing algorithms exploiting coprime array usually adopt predefined spatial sampling grids for optimization problem design or include spectrum peak search process for DOA estimation, resulting in the contradiction between estimation performance and computational complexity. To address this problem, we introduce the Estimation of Signal Parameters via Rotational Invariance Techniques (ESPRIT) to the coprime coarray domain, and propose a novel coarray ESPRIT-based DOA estimation algorithm to efficiently retrieve the off-grid DOAs. Specifically, the coprime coarray statistics are derived according to the received signals from a coprime array to ensure the degrees-of-freedom (DOF) superiority, where a pair of shift invariant uniform linear subarrays is extracted. The rotational invariance of the signal subspaces corresponding to the underlying subarrays is then investigated based on the coprime coarray covariance matrix, and the incorporation of ESPRIT in the coarray domain makes it feasible to formulate the closed-form solution for DOA estimation. Theoretical analyses and simulation results verify the efficiency and the effectiveness of the proposed DOA estimation algorithm.

Compressive sensing‐based coprime array direction‐of‐arrival estimation

A coprime array has a larger array aperture as well as increased degrees-of-freedom (DOFs), compared with a uniform linear array with the same number of physical sensors. Therefore, in a practical wireless communication system, it is capable to provide desirable performance with a low-computational complexity. In this study, the authors focus on the problem of efficient direction-of-arrival (DOA) estimation, where a coprime array is incorporated with the idea of compressive sensing. Specifically, the authors first generate a random compressive sensing kernel to compress the received signals of coprime array to lower-dimensional measurements, which can be viewed as a sketch of the original received signals. The compressed measurements are subsequently utilised to perform high-resolution DOA estimation, where the large array aperture of the coprime array is maintained. Moreover, the authors also utilise the derived equivalent virtual array signal of the compressed measurements for DOA estimation, where the superiority of coprime array in achieving a higher number of DOFs can be retained. Theoretical analyses and simulation results verify the effectiveness of the proposed methods in terms of computational complexity, resolution, and the number of DOFs.