Sparse Linear Array Research Articles

Matrix Completion for Downward-Looking 3-D SAR Imaging With a Random Sparse Linear Array

Downward-looking linear array 3-D synthetic aperture radar (SAR) has attracted increasing attention in the field of radar imaging. As widely reported, the volume of data can be significantly reduced by a random sparse linear array. However, the 2-D under-sampled azimuth-cross-track data brought by the sparse linear array will produce high-level side-lobes, as well as the aliasing and the false-alarm targets. To deal with those problems, this paper introduces a recently developed theory, matrix completion (MC). The new theory could recover a matrix with a small subset of known elements of the matrix. It is founded on the assumption that the matrix is essentially low rank. For downward-looking 3-D SAR with a random sparse linear array, the received 3-D data can be treated as a series of uncorrelated 2-D matrices by the separated channel process. First, range compression can be realized by means of pulse compression. Then, the sets of the 2-D under-sampled azimuth-cross-track matrix can be completed into a full-sampled one via MC trick. The resulting 3-D images can be focused by synthetic aperture technique along the azimuth direction and beamforming operation along the cross-track direction, with the recovered full-sampled matrix. The proposed algorithm achieves high resolution and low-level side-lobes with the acceptable computational cost and memory consumption. It is verified by several numerical simulations and multiple comparative studies on real data. The experimental results clearly demonstrate the imaging performance across different under-sampling rates and signal-noise rates.

Localisation of mixed near‐field and far‐field sources using the largest aperture sparse linear array

In some applications, the signals received by an array are a mixture of signals emitted by both far-field and near-field sources. This study develops a new cumulant-based multiple signal classification (MUSIC) algorithm for source localisation using a new structural sparse array for scenarios where both far-field and near-field sources coexist. The key feature of this algorithm is that it utilises fourth-order cumulants to compute the virtual covariance matrix and constructs a new special cumulant matrix to acquire the largest number of virtual sensors and the largest array aperture for a given number of sensors. The authors provide a geometric proof to justify the utilisation of the proposed sparse linear array and compute the effective aperture of the array. The proposed algorithm increases resolution ability, direction of arrival (DOA) and range estimation accuracy, and the number of sources to be localised. Moreover, the new method has the main advantage that it does not use the information of all sensors; so that it provides somewhat low computational complexity while it uses many actual and virtual sensors. Monte Carlo simulations are provided to demonstrate the effectiveness of the proposed method.

Sampling with semi-coprime arrays

This research introduces a new array geometry called Semi-Coprime Array (SCA) that has the potential to significantly increase the degrees of freedom of a Uniform Linear Array (ULA). An SCA interleaves three ULAs (Subarray 1, Subarray 2, and Subarray 3). Each SCA has two underlying coprime integers (M and N). Subarray 1 and Subarray 2 have undersampling factors based on the coprime integers M and N, while Subarray 3 is a full ULA with the number of sensors determined by the undersampling factors of Subarray 1 and Subarray 2. Interleaving the three subarrays results in a highly sparse non-uniform linear array. Taking the minimum of the three subarrays' outputs produces a pattern that is devoid of aliasing, yet offers the resolution of a full ULA with equal aperture. An SCA offers closed form expressions for sensor locations which many existing sparse arrays do not. Compared to other existing sparse arrays that offer closed form expressions for sensor locations, coprime arrays and nested arrays, an SCA saves more sensors and matches the peak side lobe height of a full ULA with less extension.

An Effective Approach for the Synthesis of Uniformly Excited Large Linear Sparse Array

A novel approach for the synthesis of large sparse linear arrays featuring a minimum sidelobe level is presented. The approach consists of two stages and has an efficient way in dealing with multiple constraints. An analytical approximation method, where a local optimum can be obtained, is carried out to generate the initial populations in the first step. Then, a population renewal strategy based on differential evolution and Levy flight is applied to search the global optimal solutions. Numerical experiments are provided to validate the effectiveness and outperformance of the proposed method.

Azimuth-elevation direction finding, using three uni-axial velocity sensors as an arbitrarily sparse linear array

A tri-axial velocity sensor measures the acoustic particle-velocity vector, by all three of its Cartesian components. This particle-velocity vector equals the spatial gradient of the acoustic pressure field. Song & Wong [1] has advanced an algorithm to estimate an incident source's azimuth-elevation (θ, Φ) direction-of-arrival using three uni-axial velocity sensors that are orthogonally oriented among themselves and placed arbitrarily in three-dimensional space. This work will focus on a particular array geometry whereby the three uni-axial velocity sensors are placed on a straight line in three-dimensional space but the inter-element spacings may be entirely arbitrary. This work will show how to estimate a far-field emitter's azimuth-elevation direction-of-arrival, despite the three elements’ spatial separation, and despite the separation's arbitrary length and possible sparseness. Furthermore, the Cramer-Rao Bounds (CRB) will be presented analytically for the case of the linear array being aligned along the x-axis. This work differs from [2], which requires an additional pressure-sensor. [1] Y. Song & K. T. Wong, “Acoustic direction finding using a spatially spread tri-axial velocity sensor,” IEEE Trans. Aerosp. Electron. Syst. 51(2), 834–842 (2015). [2] Y. Song & K. T. Wong, “Azimuth-elevation direction finding, using one four-component acoustic vector-sensor spread spatially along a straight line,” Proc. of Meetings on Acoustics—Meeting of the ASA, Pittsburgh, U.S.A., May 18–22, 2015.

A Spatial Filtering Based Gridless DOA Estimation Method for Coherent Sources

In this paper, we investigate a covariance matrix reconstruction approach (CMRA) for direction-of-arrival (DOA) estimation in correlated/coherent sources scenario. We incorporate a spatial filtering (SF) model into our recently developed method CMRA in order to enhance its adaptation ability. In particular, a sliding window scheme is proposed to estimate the number of sources, and an iterative procedure is provided to estimate the DOAs of the signals. Since the original CMRA provides inaccurate estimate of the noise power which is undesirable during iterations, a new update rule for the noise power is proposed. Moreover, we derive a fast implementation of the SF-CMRA to accelerate the DOA estimation in each iteration. The proposed methods are suitable for both uniform and sparse linear arrays and are able to provide accurate estimates of DOAs, signal powers, and noise power. We also show that the proposed algorithmic framework can be easily extended to other gridless DOA estimation methods for accuracy improvement. Simulation results are provided to illustrate the superiority of our proposed methods.

Fast DOA Estimation Algorithms for Sparse Uniform Linear Array With Multiple Integer Frequencies

The sparse uniform linear array (SULA) with larger array aperture is adopted to improve the accuracy of direction of arrival (DOA) estimation with a limited number of physical antennas. In the case of single working frequency configuration, spatial aliasing problem occurs due to the spatial under-sampling. However, with a sufficient anti-aliasing condition for multiple frequencies, the unique DOA estimations can be obtained without ambiguity by using multiple integer frequencies. The multiple equivalent array structures of the multiple integer frequencies are analyzed. To perform DOA estimation, in the first part of the paper, a pair of fast DOA estimation methods are proposed. Based on the fact that a real DOA and its ambiguous positions are periodic distributed in the sine domain, the two proposed methods are applied to each equivalent array separately to obtain the real DOA positions as well as their replica positions. Then, an effective matching algorithm is proposed to pick out the real DOAs from the ambiguous ones. The first algorithm is a partial spectral search-based algorithm, and the second algorithm is a search-free-based method. These two algorithms are referred to as matching partial spectral search-based algorithm (matching PPS) and matching root-multiple signal classification algorithm (matching root-MUSIC), respectively. Our third proposed algorithm is a combined root-MUSIC which is applied to the multiple equivalent arrays by taking them as sub-arrays of a common filled ULA. The combined root-MUSIC algorithm utilizes the MUSIC null-spectrum functions of all equivalent arrays to form a new polynomial, and the DOA estimations can be obtained efficiently without any spatial ambiguity by applying polynomial rooting to the new polynomial. Simulations are provided to verify the validity of the proposed algorithms.

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.

Mixed Far-Field and Near-Field Source Localization Algorithm via Sparse Subarrays

Based on a dual-size shift invariance sparse linear array, this paper presents a novel algorithm for the localization of mixed far-field and near-field sources. First, by constructing a cumulant matrix with only direction-of-arrival (DOA) information, the proposed algorithm decouples the DOA estimation from the range estimation. The cumulant-domain quarter-wavelength invariance yields unambiguous estimates of DOAs, which are then used as coarse references to disambiguate the phase ambiguities in fine estimates induced from the larger spatial invariance. Then, based on the estimated DOAs, another cumulant matrix is derived and decoupled to generate unambiguous and cyclically ambiguous estimates of range parameter. According to the coarse range estimation, the types of sources can be identified and the unambiguous fine range estimates of NF sources are obtained after disambiguation. Compared with some existing algorithms, the proposed algorithm enjoys extended array aperture and higher estimation accuracy. Simulation results are given to validate the performance of the proposed algorithm.

Density Tapering of Linear Arrays Radiating Pencil Beams: A New Extremely Fast Gaussian Approach

In this communication, a very simple and extremely fast algorithm is proposed for the pencil beam synthesis of linear sparse arrays having uniform distribution of the excitations. The key idea is that of selecting, as a desired pattern, a Gaussian function having small standard deviation, so as to obtain a narrow beam. This immediately provides the excitation density of the corresponding continuous array of infinite length. Starting from this result and considering a linear array of length $L$ with $N$ elements having equal excitations, an extremely fast and accurate algorithm based on a density tapering approach is proposed that yields suitable positions of the elements, in such a way as to provide an array factor that well approximates the desired pattern. Numerical examples are presented to show the effectiveness of the developed procedure, also when compared with state-of-the-art algorithms. The proposed approach does not consider the mutual coupling between the array elements, but it is numerically shown that this effect produces quite acceptable degradation on the synthesized patterns. Finally, it is shown that also problems involving thousands of elements can be solved in a very accurate way in few milliseconds.

Two-dimensional direction estimation of multiple signals using two parallel sparse linear arrays

In this paper, we focus on the problem of estimating the two-dimensional (2-D) direction-of-arrival (DOA) of multiple signals and propose a novel approach for 2-D DOA estimation using two parallel sparse arrays structured by two sparse uniform linear arrays (ULAs) and an auxiliary sensor. Firstly, we obtain automatically paired direction cosine estimates along the x and y axes by using the mutual and internal shift-invariance properties of two sparse ULAs and the propagator method, where the direction cosine estimates along the y-axis are cyclically ambiguous due to the fact that the inter-element spacings of two sparse ULAs exceed a half-wavelength. Then, we extract unambiguous direction cosine estimates by means of the auxiliary sensor and a novel disambiguation procedure. Finally, we use the unambiguous direction cosine estimates to calculate the elevation and azimuth angles. The proposed method does not require extra pair-matching process and can achieve automatic pairing for 2-D DOA estimation. Moreover, it has a low computational complexity without any spectral search. Simulation results demonstrate that the proposed method outperforms the existing methods based on the two-parallel ULA, especially for small number of sensors case.

A Toeplitz Covariance Matrix Reconstruction Approach for Direction-of-Arrival Estimation

It is known that there exist two kinds of methods for direction-of-arrival (DOA) estimation in the literature: the subspace-based method and the sparsity-based method. However, pervious works reveal that the former method cannot address the case in which the number of signals is larger than that of sensors, whereas the latter one always suffers from the influence of basis mismatch. In this paper, to overcome these two shortcomings, we propose a new method called covariance matrix reconstruction approach (CMRA) for both uniform linear array and sparse linear array. In particular, by exploiting the Toeplitz structure of the covariance matrix of the array output, we formulate a low-rank matrix reconstruction (LRMR) problem for covariance matrix recovery. The nonconvex LRMR problem is then relaxed by replacing the rank norm with the nuclear norm and solved using the optimization toolbox. Next, we retrieve the DOAs from the recovered covariance matrix by using the subspace-based methods and obtain an estimated number of signals as a byproduct. We also provide two algorithm implementations for the LRMR problem based on duality and alternating direction method of multipliers, respectively. It is shown that CMRA can be regarded as an atomic norm minimization model or a gridless version of the sparsity-based methods and can recover more signals than sensors with a well-designed array. Numerical experiments are provided to validate the effectiveness of the proposed method, in comparison with some of the existing methods.

An Effective Method for Synthesizing Multiple-Pattern Linear Arrays With a Reduced Number of Antenna Elements

An innovative technique based on the enhanced unitary matrix pencil (MP) method is presented for the design of sparse multiple-pattern linear arrays. By virtue of the equivalent MP obtained with a unitary transformation, the relation between the element positions and the generalized eigenvalues is achieved in this method, which contributes to the real solutions of the common element positions for all multiple patterns. Owing to the utilization of a unitary transformation that can convert a complex matrix to a real one, the computational complexity can be significantly reduced since only the real computations are involved in the singular value decomposition and eigenvalue decomposition procedures. Consequently, by only varying the obtained excitation distributions, different patterns are generated with a higher matching accuracy. Representative experiments are provided to validate the effectiveness and advantages of the proposed method.

Hourglass arrays: Planar sparse arrays with hole-free coarrays and reduced mutual coupling

Linear (1D) sparse arrays such as nested arrays have hole-free difference coarrays with O(N 2) virtual sensor elements, where N is the number of physical sensors. This property implies that O(N 2) monochromatic and uncorrelated sources can be identified. For the 2D case, planar sparse arrays with hole-free coarrays having O(N 2) elements have also been known for a long time. These include billboard arrays, open box arrays (OBA), and 2D nested arrays. Their merits are similar to those of the 1D sparse arrays mentioned above, although identifiability claims regarding O(N 2) sources have to be handled with more care in 2D. In this presentation, we propose hourglass arrays, which have closed-form 2D sensor locations and hole-free coarrays with O(N 2) elements just like the OBA. Furthermore, the mutual coupling effect, which is the undesired interaction between sensors, is reduced since the number of sensor pairs with small spacings such as λ/2 decreases. Among the planar arrays mentioned above, simulations show that hourglass arrays have the best estimation performance in the presence of mutual coupling.

Joint sparsity‐driven three‐dimensional imaging method for multiple‐input multiple‐output radar with sparse antenna array

Three-dimensional (3D) imaging of a moving target can be achieved via a wideband multiple-input multiple-output (MIMO) radar with planar array in a single snapshot. The sparse recovery (SR) method has been introduced to deal with the MIMO imaging problem with sparse array recently. For sparse linear array, SR method can successfully achieve the high cross-range resolution. For sparse planar array, however, it is not effective to use a SR algorithm directly in 2D cross-range direction, because the final 2D imagery will be blurred by the SR-induced range cell migration (RCM) in the former cross-range direction. To remove the RCM, a joint sparsity-driven method is proposed in this study by exploiting the joint sparsity of echo signal. An improved imagery can be obtained with fewer antennas and higher calculation efficiency. Comparative experiments demonstrate the superiority of the proposed method.

DOA Estimation With Enhanced DOFs by Exploiting Cyclostationarity

Many modulated signals exhibit a cyclostationarity property, which has been applied in direction of arrival estimation due to its immunity to interference and noise. In this paper, we focus on how this cyclostationarity can be effectively integrated with the spatial dimension to enhance both the degrees of freedom and the accuracy of DOA estimation. First, for narrowband signal case, by vectorizing the second-order cyclic correlation matrix (instead of the conventional zero-lag covariance matrix), one can directly generate an augmented virtual array with sensors at positions defined by the difference coarray of the physical array. Then for wideband signal case, two additional fractional factors are introduced so that the vectorized cyclic correlation matrix can be viewed as a single snapshot received signal from an array with sensors at positions defined by the fractional weighted difference coarray. Due to the fact that cyclic correlation matrices, as special second-order statistics, allow us to obtain more degrees of freedom, the proposed two virtual array models can resolve $O(N^2)$ sources by using a sparse linear array consisting of only $N$ physical sensors (e.g., nested/coprime array). Furthermore, a scheme of multipseudosampling is proposed in order to reduce the sensitivity to noise and parameter. The proposed models can be considered as an extension of difference coarray perspective via the combination of cyclic frequency and temporal lag. In the end, numerical simulation results validate the effectiveness of the proposed models.

Coherent unambiguous transmit for sparse linear array with geography constraint

This study considers the sparse linear array (SLA) design problem of coherent unambiguous transmit for the distributed aperture coherence-synthetic radar, accounting for the antenna size constraint and geography constraint where the elements cannot be placed in a given region. The authors develop an objective function considering the sum of the peak sidelobe level (PSL) for different beam shifting directions. Then, based on a processing strategy for considered constraints, they derive the antenna element locations by minimising the objective function through particle swarm optimisation algorithm. Finally, several numerical simulations verify the effectiveness of the proposed SLA design method.

Evaluating Spatial Resolution and Channel Capacity of Sparse Cylindrical Arrays for Massive MIMO

How to design antenna arrays plays an important role in wireless communications, especially when there are hundreds of antennas. In massive multiple-input and multiple-output (MIMO), if the number of antennas and RF chains are increasing, the channel capacity and transmission efficiency could be obviously improved as well. Since in massive MIMO, antennas at the base station (BS) usually scale up greater than 100, the complexity and hardware requirement of the system are also increased. Many studies about massive MIMO are focused on the analysis of channel capacity, precoding, and so on, and very few are about the sparse antenna array deployment. Based on the sparse linear array structures as indicated by previous studies, three new arrays, namely co-prime cylindrical array (CCA), nested cylindrical array (NCA), and sparse nested cylindrical array (SNCA), are proposed, which are based on co-prime linear array, nested linear array, and sparse circular array, respectively. Compared with the traditional uniform cylindrical array (UCA), the proposed arrays vastly reduce the number of antennas used at the BS. In addition, the performance of spatial resolution and channel capacity of CCA, NCA, and SNCA are discussed in detail. The results show that our proposed sparse cylindrical arrays can obtain a higher resolution with much fewer antennas. Besides, the uplink channel capacity of all the three sparse cylindrical arrays is larger than the UCA with the same number of antennas at the BS.

Shaped-Beam Pattern Synthesis of Sparse Linear Arrays Using the Unitary Matrix Pencil Method

In this letter, the unitary matrix pencil (UMP) method is utilized for the shaped-beam pattern synthesis of maximally sparse linear arrays. Taking advantage of the equivalent matrix pencil obtained by a unitary transformation, the UMP-based synthesis method constructs a novel relation between the element positions and the generalized eigenvalues, which contributes to the real solutions for all element positions and thereby improves the matching accuracy of shaped-beam patterns. In addition, complex computations in both the singular value decomposition and the eigenvalue decomposition procedures are converted into real ones through a unitary transformation, which significantly reduces the computational complexity. Numerical experiments assess the effectiveness and advantages of the proposed method in comparison to the state-of-the-art synthesis approaches.

OPTIMAL DESIGN OF SPARSE ARRAYS BASED ON MODIFIED QUANTUM GENETIC ALGORITHM

In order to reduce the peak sidelobe level (PSLL) of the linear array pattern, algorithm of linear array pattern optimization method is proposed based on margin space coding and quantum genetic algorithm (MS_QGA). Using the quantum genetic algorithm to optimize the position margin of array elements in the case that constrain the array aperture, the number of array elements and the minimum array elements space of the uniform linear array, and the optimization goal is to reduce the PSLL. The convergence speed of the proposed method is accelerated, the local search ability and the diversity of the population are improved to solve the premature convergence problem, the complexity of the method and the computation are reduced, the performance of PSLL of the sparse linear arrays is improved, and the array element is better distributed in the array aperture.