Difference Coarray Research Articles

Fast Underdetermined DOA Estimation Based on Generalized MRA via Original Covariance Vector Sparse Reconstruction

Minimum redundancy array (MRA) has the maximum aperture with continuous difference co-array among various sparse arrays with same number of physical sensors, but it is hard to calculate the sensor position of MRA and realize array design by using MRA. To solve those problem, generalized MRA is proposed with mutual coupling limitation and easy calculation method of sensor position. Based on proposed array configuration, a high-precision underdetermined direction of arrival (DOA) estimation method is proposed with reduced computational complexity. In this method, fast covariance matrix reconstruction is achieved by trace norm minimization with more accurate covariance estimation. Based on the Toeplitz property of covariance matrix in uniform array, a new sparse representation model is established with reduced dimension of covariance vector and faster DOA estimation is achieved via convex optimization. In addition, the proposed method can also be used for underdetermined DOA estimation of other sparse arrays. Using simulation experiments, we demonstrate that the proposed sparse array configuration has superiority over other sparse arrays and the proposed method can outperform most existing methods in terms of underdetermined DOA estimation accuracy and efficiency.

DOA Estimation With Nonuniform Moving Sampling Scheme Based on a Moving Platform

The generalized linear moving sampling scheme (MSS) exploiting the second-order statistics and also the high-order cumulants is studied, where the set of MSS is defined as the shifted distance offsets involved in estimation based on a moving platform. Then, sparse physical arrays (SPAs) with nonuniform linear moving sampling schemes (NL-MSS), referred to as SPA-NL-MSS, are proposed to optimize the consecutive difference co-arrays. For the same number of sensors and data samples, better performance in terms of both the number of degrees of freedom (DOFs) and estimation accuracy can be achieved by SPA-NL-MSS than existing array structures exploiting array motions at the second order level.

Dilated Arrays: A Family of Sparse Arrays With Increased Uniform Degrees of Freedom and Reduced Mutual Coupling on a Moving Platform

Recently, dilated nested arrays have been proposed on a moving platform to increase the uniform degrees of freedom (uDOF) by a factor of three by exploiting array motion. However, no literature addresses the issue whether the same dilation method still performs well for other array geometries such as coprime arrays, augmented nested arrays and minimum redundancy arrays. Compared with nested arrays, these arrays either achieve higher uDOF or exhibit more robustness to mutual coupling among sensors. In this paper, we propose a novel sparse array geometry named dilated arrays (DAs) on a moving platform by applying the dilation method to other array geometries. First, by exploiting the relationship between the element positions in the difference coarrays of the original linear array and the synthetic array after motion, we prove that, for a DA on a moving platform, the maximum uDOF can be tripled compared to that of its original array regardless of the array geometry. Therefore, the number of sources that can be resolved for direction-of-arrival (DOA) estimation is increased threefold. Second, we prove that a DA reduces mutual coupling compared with its original array. As a result, the DA is more robust to mutual coupling than its original array. Third, we extend one-dimensional DAs to the two-dimensional (2-D) case, yielding a new 2-D sparse array geometry named two-parallel DAs. We show that by exploiting array motion, two-parallel DAs can increase the number of detectable sources threefold. Numerical simulations demonstrate the superior performance of the proposed array geometries.

On the Size and Redundancy of the Fourth-Order Difference Co-Array

In array processing, the fourth-order difference co-array of sparse arrays with $N$ sensors admits to resolve $\mathcal {O}(N^4)$ source directions with proper assumptions. This $\mathcal {O}(N^4)$ property is relevant to the size of the central uniform linear array (ULA) segment of the fourth-order difference co-array. However, among the existing sparse arrays, the fundamental limits for the sizes of the fourth-order difference co-array remain a topic for further study. This paper characterizes the lower and upper bounds of the size of the fourth-order difference co-array. It is proved that the ULA and the exponential array achieve the lower and upper bounds, respectively. We also propose the fourth-order redundancy to quantify the efficiency of the ULA segment in the fourth-order difference co-array. The fourth-order redundancy owns a provable lower bound depending only on $N$ , providing further insights into the size of the central ULA segment of the fourth-order difference co-array. These bounds are validated through numerical examples.

2D DOA Estimation Exploiting Vertical Synthetic Planar Arrays

In this paper, vertical motions of sparse linear arrays (SLAs) are utilized to generate equivalent synthetic planar arrays for two-dimensional (2D) direction-of-arrival (DOA) estimation. The proposed array geometry named vertical synthetic planar array (VSPA) consists of an arbitrary SLA on the x-axis and a series of its time-shifted arrays in the vertical orientation. With the original linear array and the moving trail known, the difference coarray of VSPA can be easily obtained. By utilizing both the synthetic aperture processing and the coarray technique, VSPA has the ability to construct a synthetic planar array with only an SLA used. Compared with the traditional sparse planar arrays (SPAs), such as 2D nested array and 2D coprime array, VSPA can achieve higher degree-of-freedom and improved 2D DOA estimation performance with the same number of sensors. Moreover, as different original linear arrays and moving trails can be chosen for different applications, the construction of VSPA is of high flexibility. Numerical simulations are presented to verify the superiority of the proposed VSPA geometry over other typical SPAs.

Design of novel sparse arrays for DOA estimation of noncircular sources

ABSTRACT In this paper, we first propose a novel sparse array (NSA) composed of three adjoined sparse subarrays, among which the first and third ones can form the prototype coprime array (CA), and the second one is nested to them. When deploying NSA for direction of arrival (DOA) estimation, two virtual CAs with larger apertures than the physical ones can be generated by vectorising the covariance matrices corresponding to specific subarray pairs. Accordingly, improved DOA estimation performance can be achieved. Afterwards, to enhance the DOA estimation of noncircular sources, four improved NSAs are designed, which have minimised redundancy between the difference and sum coarrays. Hence, they have larger utilised apertures than NSA. In addition, due to the larger physical aperture and inter-element spacing, all the five proposed arrays have less mutual coupling and better DOA estimation performance than the prototype CA. Numerical simulations confirm their effectiveness and superiority.

Hole-Free Coprime Array for DOA Estimation: Augmented Uniform Co-Array

The coprime array owns a sparse array structure, which can effectively alleviate mutual coupling, while the holes existing in the difference co-array(DCA) greatly reduce the number of uniform degrees of freedom(uDOFs). In this letter, we propose a new coprime array structure called hole-free coprime array (HFCA) by carefully assembling the subarrays, and the resulting HFCA can generate a hole-free DCA. Furthermore, we derive the optimal HFCA that can produce the largest hole-free DCA with a given number of sensors, which contributes to an increase of uDOFs compared with existing coprime array structures. In addition, the simulations are provided to demonstrate the superiority of HFCA in terms of uDOFs, direction of arrival estimation performance, and spatial resolution.

Improved unfolded coprime array subject to motion for DOA estimation: augmented consecutive synthetic difference co‐array

Direction of arrival (DOA) estimation using the improved unfolded coprime array (IUFCA) subject to array motion isdiscussed in this study. Unfolded coprime array (UFCA) consists of two uniform linear subarrays, and the two subarrays are arranged at different sides of theaxis, which leads to a large number of holes in the difference co-array (DCA). With array motion and DCA synthesis,part of the holes can be filled, but there are still holes in the center which lead to the virtual arrays separated.By analyzing the hole positions in the synthetic DCA generated by UFCA motion, the authors improve the originalUFCA by relocating some physical elements, then the two dominantconsecutive DCA segments in the positive and negative sides can be connected. The expression of synthetic DCA is analyzed, and the closed-form expression of theuniform degree of freedom (uDOF) subject to IUFCA motion is studied. Simulation results show that IUFCA motion can obtain a largenumber of uDOFs, which leads to better DOA estimation performance and more identifiable signals compared with existingcoprime array configurations.

Increasing Utilization of Redundant Virtual Array for DOA Estimation Based on Coprime Array

In this paper, we initiated a method to estimate the direction of arrival (DOA) of far-field, narrowband, and incoherent targets using coprime array. First, we proposed a coprime array structure and analysed the distribution of difference coarray (DCA). The degrees of freedom (DOF) of the proposed coprime array became clearer by referring to the DCA conception. However, previous algorithm only uses the continuous virtual array, which causes the virtual array elements in the repeated position being abandoned. Therefore, the paper analyses the distribution of virtual array based on DCA conception and averages the receiving signal on these redundant virtual array elements to increase the utilization of receiving data. As a result, the algorithm has high precision in parameter estimation. Simulation results have shown the superiority of the proposed algorithm.

Extensions of Co-Prime Array for Improved DOA Estimation With Hole Filling Strategy

Although co-prime array is more attractive than nested array in reducing mutual coupling benefiting from a sparser structure, it generates a difference co-array with multiple holes that can considerably decrease the number of uniform degrees of freedom (uDOFs) when spatial smoothing methods are employed. In this paper, we present a hole filling strategy for co-prime array to improve its difference co-array. Specifically, we propose a novel co-prime array with a filled difference co-array (CAFDC) by introducing two extra subarrays, which can significantly increase the number of achievable uDOFs. However, CAFDC is sensitive to mutual coupling because a densely assembled subarray is involved. To tackle this problem, we first derive the location expressions of the sensors in CAFDC for the largest difference co-array with a fixed number of sensors and then enlarge the inter-element spacing of its dense subarray. The resulting co-prime array can effectively mitigate mutual coupling effect by restricting the number of sensor pairs with small separations and produce a two-hole difference co-array with no loss of uDOFs, termed as co-prime array with a two-hole difference co-array (CATHDC). In addition, simulation results are provided to demonstrate the superiority of the newly proposed co-prime arrays over existing sparse arrays in DOF, mutual coupling and direction-of-arrival (DOA) accuracy.

Structured sparse array design exploiting two uniform subarrays for DOA estimation on moving platform

Array motion can efficiently enhance the achievable number of degrees-of-freedom (DOFs) by the virtue of filling the neighboring holes of each lag in the difference co-array of a linear sparse array. In this paper, a new structured sparse array design exploiting two uniform subarrays on a moving platform is proposed for direction-of arrival (DOA) estimation problems. The proposed array design yields a fully filled co-array. It compresses sensor spacing of one subarray to three times the unit sensor spacing while dilating that of the other subarray. The numbers of sensors in the two subarrays are chosen as arbitrary integers without the limitation of coprimality. The conditions on the array parameters to achieve full augmentability under array motion are provided. Closed-form expressions of maximum DOFs in the difference co-array of the synthetic array, which comprises the sensor positions before and after motion, are delineated. The non-coprimality of the sensor numbers is analyzed when the two subarrays are aligned at the reference sensor, and shown to offer higher DOFs than the coprimality. Numerical results of DOA estimation using the proposed array design are provided for performance comparison and validations of analysis.

A Novel ULA-Difference-Coarray-Based DOA Estimation Method for General Coherent Signals

In this article, a difference-coarray-based direction of arrival (DOA) method is introduced, which utilizes the uniform linear array (ULA) in a novel fashion to address the problem of DOA estimation for coherent signals. Inspired by the coarray-based estimators employed in cases of sparse arrays, we convert the sample covariance matrix of the observed signals into the difference coarray domain and process the signals using a spatial smoothing technique. The proposed method exhibits good accuracy and robustness in both the uncorrelated and coherent cases. Numerical simulations verify that the ULA difference coarray- (UDC-) based method can achieve good DOA estimation accuracy even when the SNR is very low. In addition, the UDC-based method is insensitive to the number of snapshots. Under extremely challenging conditions, the proposed UDC-ES-DOA method is preferred because of its outstanding robustness, while the UDC-MUSIC method is suitable for most moderate cases of lower complexity. Due to its demonstrated advantages, the proposed method is a promising and competitive solution for practical DOA estimation, especially for low-SNR or snapshot-limited applications.

Two-dimensional direction-of-arrival estimation for cylindrical nested conformal arrays

Recently, nested arrays have received considerable interest because such array configurations can be systematically designed and their degrees of freedom (DOFs) can be expressed in closed form. Compared with uniform linear arrays, they have obvious advantages in array aperture and DOFs. In this paper, we introduce nested subarray structure to cylindrical conformal array to form a nested conformal array, and develop a corresponding algorithm for two-dimensional (2-D) direction-of-arrival (DOA) estimation. By vectorizing the covariance matrices of the nested subarray outputs, we firstly generate the three-parallel difference coarray signal and derive the rotational invariance structures between three coarray manifolds. An augmented covariance matrix of the coarray outputs is then constructed via three-dimensional spatial smoothing processing. Finally, the 2-D DOAs of the sources are estimated by the total least squares (TLS)-ESPRIT method. Compared with the traditional schemes with uniform conformal arrays, our algorithm fully exploits the extended aperture of the nested subarrays, and thus enhances the DOA estimation accuracy. Numerical results demonstrate the superiority of the proposed algorithm in comparison to the existing methods.

Improved Coprime Linear Array Configuration for Moving Platform in DOA Estimation

Moving platform based coprime linear arrays (CLAs) have been studied in recent years. It is shown that by shifting a CLA a half wavelength, the resulting difference coarray can be regarded as the union of the difference coarray of the original CLA and its two shifted versions with one lag to the left and one lag to the right, respectively, filling the majority or all holes in the difference coarray. However, only the lags which are neighbors of holes are actually useful, while the shifts of the other lags generate lags which already exist, contributing then negligibly to the increase of the degrees of freedom (DOFs). In this letter, an improved CLA configuration for moving platform is proposed. By judiciously designing the sensor positions, all lags can be utilized to fill holes, consequently a difference coarray with more consecutive lags can be obtained. Compared with the original configuration, the proposed configuration can detect more sources with the same number of sensors and same length of array motion.

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.

2-D direction finding using parallel nested arrays with full co-array aperture extension

Sparse arrays, such as nested arrays, are able to resolve more sources than sensors because their difference co-arrays can provide O(L2) degrees-of-freedom (DOFs) with L physical sensors. Most of the existing nested array direction-finding algorithms apply the spatial smoothing technique to realize this DOFs enhancement. One shortcoming of this type of methods is the efficient co-array aperture is reduced in processing the spatial smoothing, resulting in the DOFs are not fully utilized. In this paper, we propose a new approach using two parallel linear nested arrays to contribute full DOFs for two-dimensional direction-finding. To exploit the entire DOFs, we perform the vectorization of multiple fourth-order cumulant matrices and take the average of their co-array covariance matrices, instead of the spatial smoothing of the vectorization of the data covariance matrix. Based on a well-posed identification analysis, we show that the proposed approach can identify the number of sources approximately three times than the algorithms using the spatial smoothing technique. For example, for a two-level parallel nested array of 2+2 sensors in each subarray, the maximum number of sources that can be resolved by the proposed approach is 32, whereas, for most of the existing algorithms, the number reduces to 10.

2-D DOA Estimation for L-Shaped Sparse Array via Joint Use of Spatial and Temporal Information

In this letter, we propose a joint processing method of spatial and temporal information from L-shaped sparse array (LSA) for two-dimensional (2-D) direction of arrival (DOA) estimation. In particular, either of the two subarrays in LSA can be any type of linear sparse array. With the proposed method, we can construct a virtual planar sparse array first, whose corresponding 2-D difference coarray owns significantly increased degrees of freedom, which can help to enhance the 2-D DOA estimation performance. Thereafter, subspace-based method with 2-D spatial smoothing can be applied to achieve the automatically paired and unambiguous 2-D DOA estimation. Simulation results verify the superiority of the proposed method.

Improved DFT Algorithm for 2D DOA Estimation Based on 1D Nested Array Motion

In this letter, two-dimensional (2D) direction of arrival (DOA) estimation using one-dimensional (1D) nested array motion is studied, and an improved Discrete Fourier transform (DFT) algorithm is proposed. Exploiting nested array to move vertically along its axis enables 2D DOA estimation with large-scale and hole-free difference co-array. DFT algorithm has lower complexity than multiple signal classifications(MUSIC) method, but traditional DFT algorithm still requires multiple extra searches to ensure estimation accuracy. We use DFT to obtain the initial angle estimations, and then compensate the estimation offsets via total least squares (TLS) constructed from Taylor expansion approximation. Compared with the traditional DFT search method, the proposed method has lower complexity and better performance. Multiple simulations are conducted to verify the effectiveness of our approach.

Near-Field Source Localization With Two-Level Nested Arrays

The concept of the nested array has been used widely for underdetermined far-field source localization. In this case, the enhanced degrees-of-freedom (DOFs) is often achieved by using spatial smoothing because the difference co-array of a two-level nested array behaves like a uniformly-spaced linear array (ULA). This ingenious operation cannot be used for near-field localization because the ULA behavior of the co-array is violated by the near-field's curved wavefronts and range-dependent attenuation. Thus, the application of nested arrays for near-field localization becomes challenging. In this letter, we propose to concatenate the vectorization of multiple fourth-order cumulant matrices to increase the DOFs offered by the nested arrays. We also show that when the sources are in the near-field, the maximum DOFs provided by a two-level nested array with L physical sensors can mathematically be up to exactly L <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> . The well-posed identification conditions are discussed as well. Numerical examples are conducted to verify our analysis. Comparisons with existing ULA-based methods are also provided.

Symmetric Displaced Coprime Array Configurations for Mixed Near- and Far-Field Source Localization

Conventionally, mixed near-field (NF) and far-field (FF) source localization is performed using a symmetric uniform linear array (ULA). Recently, two symmetric nonuniform linear arrays (NLAs), symmetric nested array (SNA) and compressed SNA (CSNA), have been developed to enhance the positioning performance by their advantages in the degrees of freedom and the array aperture over symmetric ULAs. In this article, we devise a new symmetric NLA, termed symmetric displaced coprime array (SDCA), to locate the NF and FF sources simultaneously. The SDCA configuration consists of three sparse ULAs with a certain displacement, implying there are two displaced coprime arrays with one common subarray. For a given number of sensors, the SDCA configuration is solely determined by a closed-form expression and its consecutive coarray ranges can also be analytically computed. In addition, we derive two optimum SDCA configurations by maximizing the number of the unique and consecutive lags in the difference coarray. Compared with the SNA and the CSNA, the SDCA provides more unique and consecutive virtual sensors, as well as a larger physical array aperture. Numerical experiments are presented to verify the superiorities of the SDCA configuration over the existing symmetric NLAs.