Sparse Linear Array Research Articles

Sparse MIMO linear array imaging based on Localized Low-rank Promoting algorithm

In order to improve the image quality of (Multiple-Input Multiple-Output, MIMO) sparse linear array, this paper deeply explored the block sparse characteristics of imaging targets. By introducing LLP algorithm, firstly, the elements of echo signal are divided into 4-length blocks, and each block matrix is transformed into 2-2-size matrix. By using a logarithmic determinant function which has the ability to promote the block-sparsity and local smoothness simultaneously, the low rank of these 2 × 2 matrices is improved. Finally, the iterative weighting method can efficiently reconstruct the objective image by iteratively minimizing a surrogate function of the original objective function. Simulation results show that the proposed method can get higher quality target image than the traditional algorithm.

L-Shaped Sparse Array Structure for 2-D DOA Estimation

In order to improve the performance of two-dimensional (2-D) direction-of-arrival (DOA) estimation, an L-shaped sparse array structured by two new sparse linear arrays is proposed. Each part of the proposed L-shaped array is used for one-dimensional (1-D) azimuth and elevation estimation, respectively. The new sparse linear array configuration is consisted of two subarrays which have N and M physical sensors, respectively. Owing to the advantages of the proposed sparse linear array configuration, higher degrees-of-freedom (DOF) and larger array aperture can be achieved when the second-order statistics of the received data is used. To match the azimuth and elevation angles automatically, the cross-covariance matrix of the two parts of the proposed L-shaped array is used to estimate the paired DOA angles. Based on the proposed L-shaped array configuration, (M +1)(N -[M/2]) signal directions can be estimated with M +N sensors in each part and totally 2M + 2N - 1 sensors. Through the comparative analysis of the parameters with other sparse planar arrays, the proposed L-shaped array can achieve better performance due to its higher DOF and larger array aperture. Finally, numerical simulation results verify the superiority of the 2-D DOA method based on the proposed L-shaped array.

Gridless Direction of Arrival Estimation Exploiting Sparse Linear Array

In this letter, a dual formulation of atomic norm minimization (ANM) approach is proposed by exploiting the vectorized covariance data of the signals received by the extended virtual array. Compared with the traditional ANM-based gridless direction of arrival (DOA) estimation algorithms for sparse linear array (SLA), the proposed algorithm not only realizes the parameter continuity and data completion, but also reduces the size of optimization model and the influence of noise on the estimation accuracy, which can significantly improve the performance of the algorithm. In addition, after establishing an ANM model based on complete vectorized covariance data, the corresponding dual problem is formulated to make the subsequent DOA recovery process no longer need the number of sources as a priori information. Numerial simulations demonstrate the superiority of the proposed algorithm over state of the art gridless DOA estimation algorithms.

Extended Nested Arrays for Consecutive Virtual Aperture Enhancement

In this letter, we propose an extended nested array (ENA) strategy, which relaxes the constraints on the two-side subarrays of the augmented nested array and allows the sensors further placed from the sparse uniform linear array. To guarantee the continuity of the difference co-array (DCA), several conditions are provided for specific design. Based on these rules, two ENAs, referred to as one-side extended nested array (OS-ENA) and two-side extended nested array (TS-ENA), are developed. The DCAs of both the OS-ENA and TS-ENA are hole-free and possess more consecutive lags than the existing nested-like arrays. Simulations verify the superiorities of the proposed ENAs with the spacial smoothing MUSIC algorithm used.

A Robust Multi Sample Compressive Sensing Technique for DOA Estimation Using Sparse Antenna Array

In this paper, a multi sample compressive sensing (CS) technique is presented for the direction of arrival (DOA) estimation using sparse antenna array that has applications in several fields including radars and sonars. Two different types of sparse antenna arrays are considered. One is linear sparse array for DOA estimation in one dimension and other is L shaped sparse array for DOA estimation in two dimensions. To make the algorithm robust against impulsive and Gaussian noise, a preprocessing stage is introduced. First, in the preprocessing stage median difference correntropy is used that combines median difference and the generalized correntropy. This suppresses the amplitude of impulsive noise. Second, the strength of weighted moving average filter is exploited before applying the CS technique to make the algorithm more robust. In the CS techniques, the source energy is distributed among the adjacent grid due to grid mismatch. Therefore, a fitness function based on the difference of the source energy among the adjacent grid is introduced. This provides the best discretization value through iterative grid refinement for the grid. The effectiveness and robustness of the proposed method is verified through exhaustive simulations for different number of sources and noise scenarios using one dimensional and two-dimensional sparse array structures.

Sparse linear array in the estimation of AoA and AoD with high resolution and low complexity

AbstractHigh resolution with low complexity angle of arrival and departure estimation in real‐time radar and wireless communication has been a topic of intensive research. In this paper, two novel algorithms are proposed, namely, minimum redundancy and minimum redundancy–reduced correlation algorithms, for finding the angles of arrival and departure with high resolution and low complexity. The first of the two proposed algorithms is meant for the reduction in the active number of antennas (minimum redundancy) without generating virtual angles. The second involves a reduction in subspace leakage to enable improvement of the peak power of the estimated true angles. The authors have assumed the users as coherent and the channel following Saleh‐Valenzuela model. This has been done for the validation of the performance of the proposed algorithms. Numerical simulations demonstrate the out performance of the existing sparse array subspace‐based methods in the estimation of angles by the proposed methods.

An Asymptotically Efficient Weighted Least Squares Estimator for Co-Array-Based DoA Estimation

Co-array-based Direction of Arrival (DoA) estimation using Sparse Linear Arrays (SLAs) has recently gained considerable interest in array processing thanks to its capability of providing enhanced degrees of freedom. Although the literature presents a variety of estimators in this context, none of them are proven to be statistically efficient. This work introduces a novel estimator for the co-array-based DoA estimation employing the Weighted Least Squares (WLS) method. An analytical expression for the large sample performance of the proposed estimator is derived. Then, an optimal weighting is obtained so that the asymptotic performance of the proposed WLS estimator coincides with the Cramer-Rao Bound (CRB), thereby ensuring asymptotic statistical efficiency of resulting WLS estimator. This implies that the proposed WLS estimator has a significantly better performance compared to existing methods. Numerical simulations are provided to validate the analytical derivations and corroborate the improved performance.

Sparse nested linear array for direction of arrival estimation

In this paper, we provide an approach to perform direction of arrival (DOA) estimation with the hole-free part of difference coarray, including the physical sparse uniform linear array (ULA) itself. The sparse two-level nested linear array (NLA), whose inter-element spacings are all enlarged by the identical rate compared with the conventional two-level NLA, is selected as the example to present our work due to the typicalness, under-determinedness and the relatively obvious mutual coupling effect. The difference coarray of the sparse two-level NLA is a sparse ULA and the existing methods can not be exploited for it. We derive the sparse uniform linear array discrete Fourier transform (SULA-DFT) algorithm based on the original DFT method to extend its application from only conventional ULA to sparse ULA to solve this problem. With the SULA-DFT, reliable estimation for positive angle within a limited range under a constraint condition between angle and sparsity of array is able to be obtained directly without any ambiguous angle. Compared with the conventional two-level NLA and many other state of the art configurations, the sparse two-level NLA is able to get least mutual coupling effect and largest array aperture and the SULA-DFT often obtain better estimation accuracy with the sparse two-level NLA. On the whole, sparse two-level NLA with more sparsity gets better estimation accuracy. The numerical simulations verify the superiority of the SULA-DFT with the sparse two-level NLA.

An Enhanced Time-Reversal Imaging Algorithm-Driven Sparse Linear Array for Progressive and Quantitative Monitoring of Cracks

A Lamb wave and linear piezoelectric lead zirconate titanate (PZT) array-based monitoring method for the detection and quantification of crack damage is presented in this paper. Because existing PZT array arrangements are not suitable for quantitative monitoring of crack damage both in orientation and in length, a sparse linear PZT array is introduced and applied to collect crack reflections. Based on this new array, a method for estimating crack orientation is proposed. An amplitude spectrum as a function of angle is mapped using time delayed and summed signals. By finding the peaks in the spectra, the central actuator element and corresponding orientation angle are determined. Furthermore, the time of flight imaging method is modified to display and evaluate cracks quantitatively. Validating experiments are conducted on a T6061 aluminum plate, monitoring and evaluating single and connected cracks with various orientations in different locations. As suggested by the experiments, the orientation of most cracks can be well recognized and all cracks can be quantitatively displayed by the proposed methods.

Multi‐function array SAR system on a spaceborne dual platform

This study proposes a sparse linear array synthetic aperture radar (LASAR) on a spaceborne dual platform. The transmitting elements are placed at the LASAR tips that form a satellite's dual platform, and the receiving elements are placed on the large deformation composite pine which is between the satellite's dual platform. Based on the principle of phase centre of approximation and multiple input multiple output, LASAR with the sparse array could obtain a high-resolution imaging in the cross-along direction, which is crucial to a three-dimensional SAR system. Furthermore, the LASAR system has different modes, such as interferometry and ground moving target indication.

Gridless sparsity‐based DOA estimation for sparse linear array

A novel two-stage gridless sparsity-based method for sparse linear array direction-of-arrival (DOA) estimation is presented. First, the covariance matrix gridless sparse representation method based on atomic norm minimisation is proposed for corresponding structured Toeplitz matrix construction and source number detection. Then, the conventional MUSIC algorithm can be employed for DOA estimation. Compared with conventional subspace-based algorithms, the proposed two-stage method can be carried out without knowing source number and directly detect more source signals than sensors. Numerical simulations demonstrate the effectiveness and outperformance of the proposed method.

Research on Side Lobe Suppression of Time-Modulated Sparse Linear Array Based on Particle Swarm Optimization

An efficient pattern synthesis approach is proposed for the synthesis of a time-modulated sparse linear array (TMSLA) in this paper. Due to the introduction of time modulation, the low/ultralow side lobe level can be obtained with a low amplitude dynamic range ratio. Besides, it helps reduce the difficulty of antenna feeding system effectively. Based on particle swarm optimization (PSO) and convex (CVX) optimization, this paper proposes a hybrid optimization method to suppress the grating lobes of the sparse arrays, peak side lobe level (PSLL), and peak sideband level (PSBL). Firstly, the paper utilizes the CVX optimization as a local optimization algorithm to optimize the elements’ switch-on duration time, which reduces the side lobe of the array. Secondly, with the PSBL as the objective function, the paper adopts the PSO as a global optimization algorithm to optimize the elements’ positions and switch-on time instant, which helps reduce the loss of sideband power caused by time modulation. With respect to the time modulation model, variable aperture sizes (VAS) and more flexible pulse-shifting (PS) schemes are used in this paper. Owing to the introduction of time modulation and CVX optimization, the proposed method is much more feasible and efficient than conventional approaches. Furthermore, it has better array pattern synthesis performance. Numerical examples of the TMSLA and comparisons with the reference are presented to demonstrate the effectiveness of the proposed method.

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

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

DOA Estimation Exploiting Sparse Array Motions

This paper utilizes sparse array motion to increase the numbers of achievable both degrees of freedom (DOFs) and consecutive lags in direction-of-arrival (DOA) estimation problems. We use commonly employed environment-independent sparse array configurations. The design of these arrays is not dependent on the sources in the field of view, but rather aims at achieving desirable difference co-arrays. They include structured coprime and nested arrays, minimum redundancy array (MRA), minimum hole array (MHA), and sparse uniform linear array (SULA). Array motion can fill the holes in the spatial autocorrelation lags associated with a fixed platform and, therefore, increases the number of sources detectable by the same number of array sensors. Quasi-stationarity of the environment is assumed where the source locations and waveforms are considered invariant over array motion of half wavelength. Closed-form expressions of the number of DOFs and consecutive spatial correlation lags for coprime and nested arrays as well as SULA, due to array translation motion, are derived. The number of DOFs and consecutive lags for the specific cases of MRA an 5 avaluated. We show the respective DOA estimation performance based on sparse reconstruction techniques.

Synthesis of linear sparse arrays based on dynamic parameters differential evolution algorithm

Synthesis of antenna arrays is a momentous technology possessing many advantages and broad prospects. Dynamic parameters differential evolution (DPDE) with two modifications is proposed for the synthesis of linear sparse arrays (LSA) which aims at reducing peak side-lobe level. One modification is that the scaling factor decreases monotonically over iterations, making exploration in the later period more efficient and accuracy. The other modification is that the crossover probability changes dynamically during iterative process through a novel self-adaptive strategy. Experiments of checking on benchmark functions and simulations of synthesising LSA validate that DPDE performs better than reference methods convincingly.

Two-Dimensional DOA Estimation via Shifted Sparse Arrays with Higher Degrees of Freedom

Sparse antenna arrays provide larger virtual arrays to estimate the direction of arrivals (DOAs) of more sources than the number of physical antennas in the array. While the degrees of freedom (DOF) can be increased by the special structure of the antenna array, a shift in the antenna positions can generate new lags in the difference co-array; hence, more sources can be resolved. In this paper, we propose shifted sparse array structures composed of two overlapping arrays shifted by one lag. It is shown that the shifting property fills the holes in the co-array, which yields larger virtual arrays. We derive stationary and moving array models where overlapping sparse arrays can be realized. The proposed shifting property is applied to coprime, nested and sparse linear arrays, and we show that the proposed technique guaranteed to increase the DOF. Using the proposed sparse array structures, we also propose a 2-D DOA estimation algorithm by utilizing the cross-covariance matrix of an L-shaped sparse array. The performance of the proposed approach is evaluated through numerical simulations, and we show that it can resolve more sources than the conventional sparse arrays with the same number of physical antennas, providing less computational complexity.

Further Results on the Cramér–Rao Bound for Sparse Linear Arrays

Sparse linear arrays, such as co-prime and nested arrays, can identify up $O(M^2)$ uncorrelated sources with only $O(M)$ sensors by utilizing their difference coarray model. In our previous work, we derived closed-form asymptotic mean-squared error (MSE) expressions for two coarray based MUSIC algorithms and analyzed the Cramer-Rao bound (CRB) in high signal-to-noise ratio (SNR) regions, under the assumption of uncorrelated sources. In this paper, we provide further analysis of the CRB presented in our previous work. We first establish the connection between two CRBs, the CRB derived with the assumption that the sources are uncorrelated, and the classical stochastic CRB derived without this assumption. We show that they are asymptotically equal in high SNR regions for uncorrelated sources. Next, we analyze the behavior of the former CRB for co-prime and nested arrays with a large number of sensors. We investigate the effect of configuration parameters on this CRB. We show that, for co-prime and nested arrays, this CRB can decrease at a rate of $O(M^{-5})$ for large values of $M$ , while this rate is only $O(M^{-3})$ for an $M$ -sensor ULA. This finding analytically demonstrates that co-prime and nested arrays can achieve better asymptotic estimation performance when the number of sensors is a limiting factor. We also show that for a fixed aperture, co-prime and nested arrays require more snapshots to achieve the same performance as ULAs, showing the trade-off between the number of spatial samples and the number of temporal samples. Finally, we demonstrate our analytical results with numerical experiments.

Steerable sparse linear array design based on compressive sensing with multiple measurement vectors.

This paper presents the synthesis of a non-uniform linear array for two-dimensional Range-Bearing underwater acoustic imaging based on the multiple beam, measurement vectors (MMV) Compressive Sensing (CS) method. This method considers multiple steering directions in the mathematical model and exploits the sparsity of the oversampled linear aperture to obtain a thinned array structure. In this paper, a simulation study to synthesise the sparse linear array (SLA) for a Forward Looking Sonar system considering multiple beams in the field of view (FOV) was conducted. The comparison of the array synthesised using single measurement vector compressive sensing (SMV CS), which considers only the broadside beam in the mathematical model, is also presented. From the imaging point of view, when the SMV CS synthesised SLA is steered in a direction other than broadside, spurious/unwanted lobes appear in the beam pattern (BP). To overcome this problem, the MMV method and demonstrate its performance efficacy with a simulation study are proposed. The design of SLA for 37.5, 75, 150, 300, and 600 kHz with angular resolution 6.4°-0.4° with 11-149 beams in the FOV keeping the aperture fixed are presented, as is a design keeping angular resolution fixed at 1.5° to generalise a design of SLA for the above mentioned interested frequencies.

Deep-learning framework and acoustic reflector for improved limited-view and sparse photoacoustic tomography

Photoacoustic imaging is a hybrid imaging modality that relies upon optical absorption of pulsed light and subsequent thermoelastic generation of ultrasound. The detection of the induced acoustic waves outside the tissue enables image reconstruction. A major challenge encountered in photoacoustic tomography (PAT) lies in the inability to acquire complete projection data from the region of interest due to both the limited view and sparsity of available ultrasonic sensors. The resulting images are characterized by severe artifacts and poor quality. In this work, we examined the utility of incorporating an acoustic reflector to address the limited view problem and to train a convolutional neural network (CNN) to improve PAT image reconstruction from sparsely sampled data. Photoacoustic wave propagation was simulated in MATLAB using the k-Wave toolbox. We compared the performance of a sparse linear transducer array (with and without reflector) to that of a circular transducer array. The structural similarity index (SSI) was used as a metric for evaluating image quality. The combination of a curved reflector and artifact-removal using a CNN improved the quality of PAT images from the linear configuration. The resulting mean SSI value (0.859) was comparable to that achieved using the circular transducer array (0.926).

Incoherent Synthesis of Sparse Arrays for Frequency-Invariant Beamforming

Frequency-invariant beamformers are used to prevent signal waveform distortions in real world applications like audio, underwater acoustics, and radar. Most of existing methods assume uniform arrays, and only few consider sparse designs, which may lead to higher performance in terms of robustness and directivity factor. We propose an incoherent approach that first determines for each frequency bin a sparse set of sensors positions. Subsequently, by using tools of dimensionality reduction and clustering, these selections are merged together yielding the optimal sensors on a sparse array layout. We present design examples of sparse linear and planar superdirective array designs. We show that the proposed incoherent sparse design obtains superior performance in terms of white noise gain, directivity factor, and computational load compared to a uniform array design and compared to a coherent sparse approach, where the sensors’ locations and the beamformer coefficients are optimized simultaneously for all frequencies.