Off-grid Direction-of-arrival Research Articles

Deep Unfolding Sparse Bayesian Learning Network for Off-Grid DOA Estimation with Nested Array

Recently, deep unfolding networks have been widely used in direction of arrival (DOA) estimation because of their improved estimation accuracy and reduced computational cost. However, few have considered the existence of a nested array (NA) with off-grid DOA estimation. In this study, we present a deep sparse Bayesian learning (DSBL) network to solve this problem. We first establish the signal model for off-grid DOA with NA. Then, we transform the array output into a real domain for neural networks. Finally, we construct and train the DSBL network to determine the on-grid spatial spectrum and off-grid value, where the loss function is calculated using reconstruction error and the sparsity of network output, and the layers correspond to the steps of the sparse Bayesian learning algorithm. We demonstrate that the DSBL network can achieve better generalization ability without training labels and large-scale training data. The simulation results validate the effectiveness of the DSBL network when compared with those of existing methods.

Non-Circular Signal DOA Estimation with Nested Array via Off-Grid Sparse Bayesian Learning.

For the traditional uniform linear array (ULA) direction of arrival (DOA) estimation method with a limited array aperture, a non-circular signal off-grid sparse Bayesian DOA estimation method based on nested arrays is proposed. Firstly, the extended matrix of the received data is constructed by taking advantage of the fact that the statistical properties of non-circular signals are not rotationally invariant. Secondly, we use the difference and sum co-arrays for the nested array technique, thus increasing the array aperture and improving the estimation accuracy. Finally, we take the noise as part of the interest signal and iteratively update the grid points using the sparse Bayesian learning (SBL) method to eliminate the modeling errors caused by off-grid gaps. The simulation results show that the proposed algorithm can improve the accuracy of DOA estimation compared with the existing algorithms.

Real-Valued Deep Unfolded Networks for Off-Grid DOA Estimation via Nested Array

Recently, deep unfolded networks have been widely utilized in direction of arrival (DOA) estimation due to the reduced computational complexity and improved estimation accuracy. However, few consider the nested array for off-grid DOA estimation, where the estimated DOAs are not on the prespecified grids. In this paper, we propose a deep unfolded FOCal Underdetermined System Solver (FOCUSS) network and a deep unfolded Alternating Direction Method of Multiplies (ADMM) to address the problem, which respectively aim to improve estimation accuracy and further reduce computational complexity. We first apply first-order Taylor expansion and vectorize the covariance matrix into a real-valued single snapshot for network input. We then train the proposed networks to obtain on-grid DOA spatial spectrum and off-grid values, where the off-grid DOA estimation is calculated by the peaks of on-grid DOA spatial spectrum and corresponding off-grid values. We demonstrate that the proposed networks with interpretable parameters can accelerate the convergence rate and achieve better generalization. Simulations verify the performance of proposed networks in comparison with the existing methods.

An efficient off-grid direction-of-arrival estimation method based on inverse-free sparse Bayesian learning

The direction-of-arrival (DOA) estimation problem can be formulated in a Bayesian framework and solved well through sparse Bayesian learning (SBL). However, SBL-based methods often suffer from high computational complexity due to the matrix inversion of a large matrix in the Bayesian inference procedure. To address this issue, we propose an efficient DOA estimation method based on inverse-free sparse Bayesian learning (IFSBL). A real-valued vectorized array covariance model and a hierarchical Bayesian model are established. Then, a relaxed lower bound on the likelihood, namely relaxed evidence lower bound (relaxed-ELBO), is obtained as the optimization objective of the Bayesian inference. By utilizing the variational expectation–maximization method for the Bayesian inference, the estimation of the statistics of interest can be achieved, which is free of matrix inversion as the matrix to be inverted is diagonal. Moreover, when we take the off-grid DOAs into account, an efficient grid refinement is developed to iteratively modify the DOA estimates derived from the estimated spatial spectrum. The off-grid error obtained by the root of a simple polynomial is used to compensate for the corresponding DOA estimate. Simulations show good accuracy and improved computational efficiency of the proposed method compared with other state-of-art methods. Further experimental results verify the advantage in the runtime for the proposed method.

Adaptive Grid Refinement Method for DOA Estimation via Sparse Bayesian Learning

In sparse signal recovery methods for direction of arrival (DOA) estimation, a set of uniform angular grid points is usually predefined. Dense grid points will improve the resolution and precision, but increase computational workload distinctly. To improve the efficiency and performance when using coarse initial grid points, an adaptive grid refinement (AGR) sparse Bayesian learning (SBL) method is proposed. The key idea of the proposed method is to adaptively insert new grid points based on the spatial spectrum learned from SBL iterations, as a result, grid points become denser and denser around the potential DOAs. The number of total grid points in the AGR process is much smaller than that of traditional uniform grid points, which enhances the computation efficiency. After the improved on-grid estimation of the AGR process, a post-processing DOA search procedure is implemented to reduce the off-grid DOA error. Furthermore, the proposed method is extended into the wideband case. Simulation results demonstrate that the proposed method has higher computation efficiency and precision than the classical off-grid SBL methods in scenarios of low SNR and limited snapshots. The effectiveness of the proposed method is also validated using the data of the SWellEx-96 ocean acoustic experiment.

A Joint Angle and Frequency Spectrum Estimation Algorithm Using Difference Coarray

The spectrum sensing that jointly estimates direction-of-arrival (DOA) and frequency spectrum is an important issue for cognitive radio. Existing off-grid DOA estimation algorithms using difference coarray require a large number of snapshots to guarantee the estimation accuracy. Meanwhile, inaccurate DOA estimation renders inaccurate frequency spectrum estimation due to coupled estimation process. In order to overcome these disadvantages, we propose a joint angle and frequency spectrum estimation algorithm using difference coarray in this work. The proposed algorithm first transforms the received signal into the coarray domain and then adopts an “estimate and subtract” method to separate multiple signals. Subsequently, the DOA estimate of each source is corrected iteratively based on the proposed simple interpolation method. Finally, the frequency spectra are obtained by estimating the frequency response matrix based on the previous estimated DOAs. The shortage of requiring a large number of snapshots are overcome by utilizing the deterministically orthogonal signal model in which the signals are completely uncorrelated. Simulation results demonstrate the effectiveness of the proposed algorithm for off-grid DOA and frequency spectrum estimation using difference coarray with few snapshots.

Off-grid DOA estimation using improved root sparse Bayesian learning for non-uniform linear arrays

This paper concerns direction of arrival (DOA) estimation based on a sparse Bayesian learning (SBL) approach. We address two inherent problems of this class of DOA estimation methods: (i) a predefined dictionary can generate off-grid problems to a SBL DOA estimator; (ii) a parametric prior generally enforces the solution to be sparse, but the existence of noise can greatly affect the sparsity of the solution. Both of these issues may have a negative impact on the estimation accuracy. In this paper, we propose an improved root SBL (IRSBL) method for off-grid DOA estimation that adopts a coarse grid to generate an initial dictionary. To reduce the bias caused by dictionary mismatch, we integrate the polynomial rooting approach into the SBL method to refine the spatial angle grid. Then, we integrate a constant false alarm rate rule in the SBL framework to enforce sparsity and improve computational efficiency. Finally, we generalize the IRSBL method to the case of non-uniform linear arrays. Numerical analysis demonstrates that the proposed IRSBL method provides improved performance in terms of both estimation accuracy and computational complexity over the most relevant existing method.

DOA Estimation in Impulsive Noise Based on FISTA Algorithm

This paper investigates the challenging problem of direction-of-arrival (DOA) estimation in impulsive noise and presents a fast iterative shrinkage-thresholding algorithm (FISTA)-based approach to tackle the difficulty. More specifically, the underlying noise is modelled as the superposition of outliers in the white Gaussian noise. Leveraging on the spot-sparse characteristic of the outlier matrix, the FISTA is conducted on each snapshot of the array output. With the estimated outlier matrix and the coarse on-grid DOA estimates, an alternating optimization method is developed to retrieve the final off-grid DOA estimates. Simulation results show that the proposed method outperforms existing methods in terms of resolution capability and estimation accuracy especially in severe noise environments.

Off-Grid DOA Estimation for Noncircular Signals via Block Sparse Representation Using Extended Transformed Nested Array

An off-grid direction-of-arrival (DOA) estimation method based on block sparse representation is proposed to localize the strictly noncircular (NC) sources utilizing an extended transformed nested array (ETNA). This novel off-grid DOA estimation algorithm effectively promotes spatial distribution information mining. Furthermore, it is conducive to providing stable signal recovery, which refines the DOA estimation precision with interpolation over a coarse grid. We then combine the above algorithm with the designed ETNA to improve the detection performance. The ETNA is an optimal displacement on the existing TNA, which enlarges the degree of freedom (DOF) and lengthens the maximum contiguous segment from the derived virtual array. Simulation results demonstrate its superiority in estimation performance and DOF.

Off-grid sparse based two-dimensional direction of arrival estimation of acoustic vector sensor array in impulse noise

In order to estimate the direction of arrival (DOA) of acoustic vector sensor, the phased fractional low-order moment (PFLOM) is used to suppress impulse noise. First, based on PFLOM, an off-gird sparse based two-dimensional (2D) DOA estimation model for acoustic vector sensor array is proposed. Then, a three-step alternating iterative Orthogonal Matching Pursuit (OMP) algorithm is introduced to realize the off-grid 2D DOA estimation, and the self-pairing of azimuth and elevation is achieved. Finally, the effectiveness and accuracy of the proposed algorithm under impulse noise and small snapshots are verified by simulation.

Off-Grid DOA Estimation Using Sparse Bayesian Learning for MIMO Radar under Impulsive Noise.

Direction of arrival (DOA) estimation is an essential and fundamental part of array signal processing, which has been widely used in radio monitoring, autonomous driving of vehicles, intelligent navigation, etc. However, it remains a challenge to accurately estimate DOA for multiple-input multiple-output (MIMO) radar in impulsive noise environments. To address this problem, an off-grid DOA estimation method for monostatic MIMO radar is proposed to deal with non-circular signals under impulsive noise. In the proposed method, firstly, based on the property of non-circular signal and array structure, a virtual array output was built and a real-valued sparse representation for the signal model was constructed. Then, an off-grid sparse Bayesian learning (SBL) framework is proposed and further applied to the virtual array to construct novel off-grid sparse model. Finally, off-grid DOA estimation was realized through the solution of the sparse reconstruction with high accuracy even in impulsive noise. Numerous simulations were performed to compare the algorithm with existing methods. Simulation results verify that the proposed off-grid DOA method enables evident performance improvement in terms of accuracy and robustness compared with other works on impulsive noise.

A novel variational SBL approach for off-grid DOA detection under nonuniform noise

In this paper, a novel variational sparse Bayesian learning (SBL) approach is proposed for off-grid direction of arrival (DOA) estimation in the presence of nonuniform noise. In the proposed approach, a variational off-grid sparse model over the array covariance vector is first constructed, where the nonuniform noise covariance is modeled into a variational signal vector. At the same time, the grid refining procedure based on the variational off-grid sparse model is re-derived for reducing off-grid error, in which the spatial grid points are linear approximated and iteratively updated. Benefit from the constructed variational off-grid sparse model, the proposed approach does not require a separate estimation for the nonuniform noise variance. Numerous simulation experiments illustrate that the proposed approach has strong robustness to both off-grid error and nonuniform noise, and has superior performance on DOA detection with nonuniform noise.

Complex Multisnapshot Sparse Bayesian Learning for Offgrid DOA Estimation

Direction of arrival (DOA) estimation has recently been developed based on sparse signal reconstruction (SSR). Sparse Bayesian learning (SBL) is a typical method of SSR. In SBL, the two-layer hierarchical model in Gaussian scale mixtures (GSMs) has been used to model sparsity-inducing priors. However, this model is mainly applied to real-valued signal models. In order to apply SBL to complex-valued signal models, a general class of sparsity-inducing priors is proposed for complex-valued signal models by complex Gaussian scale mixtures (CGSMs), and the special cases correspond to complex versions of several classical priors are provided, which is helpful to analyze the connections with different modeling methods. In addition, the expression of the SBL form of the real- and complex-valued model is unified by parameter values, which makes it possible to generalize and improve the properties of the SBL methods. Finally, the SBL complex-valued form is applied to the offgrid DOA estimation complex-valued model, and the performance between different sparsity-inducing priors is compared. Theoretical analysis and simulation results show that the proposed algorithm can effectively process complex-valued signal models and has lower algorithm complexity.

Gain and Phase Calibration of Uniform Rectangular Arrays Based on Convex Optimization and Neural Networks

A calibration method based on convex optimization (CVX) and neural networks is proposed for the large planar arrays of phased array three-dimensional imaging sonar systems. The method only needs an acoustic calibration source at an unknown position in the far field, and the direction of arrival (DOA) and gain and phase error are jointly estimated. The method uses a CVX algorithm to solve an optimization problem and initially estimates the DOA of the calibration source robustly. Subsequently, according to the estimation results, a neural network is used for fitting to obtain off-grid DOA estimation of the calibration source. Thereafter, spatial matched filtering is performed to obtain the gain and phase residual estimations. The root mean square error (RMSE) of the beam pattern calibrated by the method for uniform planar arrays can reach a value of 4.9542 × 10−5. The experimental results demonstrate the efficiency of the proposed method for gain and phase calibration.

Off-Grid DOA Estimation With Mutual Coupling via Block Log-Sum Minimization and Iterative Gradient Descent

Mutual coupling (MC) effect between antennas can deteriorate the direction of arrival (DOA) estimation performance. Block sparse signal representation (BSSR) is an effective solution to mitigate the MC effect. Nevertheless, continuous angle space has to be discretized into a finite set of grid points, which lead to the grid mismatch. In this paper, we present an iterative block sparsity reconstruction algorithm for off-grid DOA estimation with MC. First, by exploiting the banded symmetric Toeplitz structure of MC matrix, a reduced BSSR model is formulated to preferably mitigate the MC effect. Then, an iterative block sparsity reconstruction algorithm with multiple measurement vectors (MMV) is proposed. Due to the NP-hard property of block l0 norm minimization, a block log-sum minimization method is developed to encourage sparsity for such a block sparse recovery problem, and a surrogate objective function is derived via the majorization-minimization approach. Finally, an iterative gradient descent algorithm is explored to refine the DOAs. Simulations verify that the proposed off-grid algorithm has robustness under limited snapshots and coherent sources.

An off-grid wideband DOA estimation method with the variational Bayes expectation-maximization framework

Wideband direction-of-arrival (DOA) estimation has important applications in many fields. To improve the performance of wideband DOA estimation, an off-grid approach with the variational Bayes Expectation-Maximization (VBEM) framework is proposed. In this work, a hierarchical Bernoulli-Gaussian model with several latent variables and parameters is formulated to describe the array observation. A latent indicator vector is assigned to describe the existence of the potential sources as well as promote sparsity in the angle domain. The VBEM framework is utilized to solve the latent variables and parameters. By modeling the existence and non-existence of potential sources separately and utilizing the wideband data jointly, our method performs better than the existing methods. The proposed method does not require any prior knowledge of the number of sources nor the initial DOAs, and it can solve both coherent and incoherent wideband sources. Simulation results show that the proposed method is superior to the existing wideband DOA estimation methods with few snapshots in frequency.

Real-valued sparse Bayesian learning algorithm for off-grid DOA estimation in the beamspace

In recent years, sparse Bayesian learning (SBL) algorithms in the array element space have become very popular for off-grid direction-of-arrival (DOA) estimation. However, the SBL algorithm in beamspace has not been researched. Therefore, this paper proposes a real-valued SBL algorithm in the beamspace. First, the received data in the array element space is transformed to the beamspace using the receive beamspace matrix. Subsequently, singular value decomposition (SVD) is introduced to reduce the computational burden further. Finally, the SBL algorithm is combined to estimate the off-grid DOA. Compared with the SBL algorithms in the array element space, the proposed algorithm has less computational complexity, better robustness to grid interval, and better DOA estimation performance under certain conditions. In addition, we also show the DOA estimation performance of the proposed algorithm for correlated signals and large-scale arrays. Numerical simulations verify the effectiveness of the proposed algorithm.

Multiobjective bilevel evolutionary approach for off-grid direction-of-arrival estimation

The source number identification is an essential step in direction-of-arrival (DOA) estimation. Existing methods may provide a wrong source number due to modeling errors caused by relaxing sparse penalties, especially in impulsive noise. This paper proposes a novel idea of simultaneous source number identification and DOA estimation to address this issue. We formulate a multiobjective off-grid DOA estimation model to realize this idea, by which the source number can be automatically identified together with DOA estimation. In particular, the source number is correctly exploited by the l0 norm of impinging signals without relaxations, guaranteeing accuracy. We further design a multiobjective bilevel evolutionary algorithm to solve this model. The source number identification and sparse recovery are simultaneously optimized at the on-grid (lower) level. A forward search strategy is developed to further refine the grid at the off-grid (upper) level. This strategy does not need linear approximations and can eliminate the off-grid gap with low computational complexity. Simulation results demonstrate the outperformance of our method in terms of source number and root mean square error.

1-Bit direction of arrival estimation via improved complex-valued binary iterative hard thresholding

Aiming to estimate direction-of-arrival (DOA) using 1-bit quantized observation of sensor arrays, an improved complex-valued binary iterative hard thresholding (iCBIHT) algorithm is proposed in this research. In this work, an error function of signal reconstruction is defined. The signals are estimated by gradient descending. A refined hard threshold function and a novel stopping criterion are designed to judge the convergence. A bases-updating strategy is introduced to solve off-grid DOAs, which improves the accuracy of DOA estimation. A backtracking strategy is utilized to refine the estimated signals processed by the hard thresholding function as well as promote convergence. Simulations and analyses show that the proposed algorithm is superior to 1-bit multiple signal classification (MUSIC) and complex-valued binary iterative hard thresholding (CBIHT) with few snapshots and low signal-to-noise ratio (SNR).

An Improved OMP-GA Combined Algorithm for Off-grid DOA Estimation

DOA (Direction of Arrival) is used to determine the direction of the targets and is one of the important technologies in array signal processing. To ensure the accuracy of traditional OMP-based DOA estimation algorithm, the over-complete dictionary needs to contain grids that match the real incident angles. However, for off-grid targets, the estimation accuracy will decrease. In this paper, we propose an improved OMP-based DOA estimation algorithm with genetic algorithm (GA), in which GA is introduced to search incident angle of the off-grid target directly, the subsequent traditional OMP steps are remained and the result got by which is used to filter out the false estimation results that deviate from the true target obtained by GA. Compared with the traditional OMP-based DOA estimation algorithm, this proposed algorithm could achieve higher accuracy. Simulation results verify the effectiveness of this proposed algorithm in the case of low signal-to-noise ratio (SNR) and low number of snapshots.