Off-grid Direction-of-arrival Estimation 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.

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.

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.

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.

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.

Block Newtonised orthogonal matching pursuit for off‐grid DOA estimation in the presence of unknown mutual coupling

Mutual coupling (MC) between array elements can lead to severe accuracy degradation in direction-of-arrival (DOA) estimation. In this study, an off-grid compressed sensing (CS) approach for DOA estimation in the presence of unknown MC is proposed using block Newtonised orthogonal matching pursuit (NOMP). First, the model of DOA estimation with MC is block sparsely represented to fully utilise the whole array aperture. Then, a block NOMP algorithm with multiple measurement vectors (MMV) is developed to achieve the estimates of DOA, in which the Newton refinement procedure is presented by deriving the first- and second-order derivatives of the block steering matrix with respect to the angle grid. Compared with the subspace-based and sparse recovery-based methods with MC, the proposed block NOMP approach has robustness in a small number of snapshots and owns no grid-mismatch effect. Meanwhile, in comparison to several existing off-grid DOA estimation methods with MC, it can achieve higher estimation accuracy and similarly low computational complexity. The superior performance of the proposed approach is shown via numerical simulations.

Multi-source off-grid DOA estimation with single snapshot using non-uniform linear arrays

Direction of arrival (DOA) estimation methods suffer from the well-known off-grid problem, that shows up when the true DOAs are not located exactly on the discretized sampling grid points. Existing estimation algorithms should be designed by taking into account the trade-off between density of sampling grids and computational complexity. Moreover, most of the computationally efficient DOA methods cannot be applied to non-uniform linear arrays (NLAs). In order to overcome these drawbacks, we propose a simple but effective off-grid DOA (OGDOA) estimation method that adopts an “estimate and subtract” strategy and then iteratively corrects the DOA estimate of each source based on a closed-form estimator that does not suffer from the off-grid problem. We derived the analytical expression of the mean square error (MSE) of the proposed method and verified that the MSE approaches the Crame´r-Rao lower Bound (CRB) when the signal-to-noise ratio (SNR) increases. We also analyzed the convergence of the proposed OGDOA algorithm. Numerical analyses demonstrated the goodness of the proposed method for OGDOA estimation for single snapshot and NLAs over some existing methods.

RxCV-based unitary SBL algorithm for off-grid DOA estimation with MIMO radar in unknown non-uniform noise

As an indispensable part of array signal processing, direction-of-arrival (DOA) estimation has been well investigated over the past few decades, and many excellent DOA estimation methods have been proposed. In this paper, a receiving domain covariance vector (RxCV) based unitary SBL algorithm is proposed for the off-grid DOA estimation of monostatic multiple-input multiple-output (MIMO) radar in unknown non-uniform noise environment. In the proposed algorithm, the data received by MIMO radar is firstly transformed into receiving domain by a reshape operation. Then the RxCV-based unitary off-grid sparse model without non-uniform noise is constructed through unitary transformation and first-order linear approximation, where the unknown non-uniform noise is got rid off by a linear transformation. Based on the RxCV-based unitary off-grid sparse model, the sparse Bayesian learning (SBL) criterion is adopted to estimate the parameters, where signal variance and off-gird error are estimated by using expectation-maximization (EM) strategy. The DOA estimation is ultimately realized through 1-dimensional spectrum search of the received data. Results of the simulation experiments have provided the evidence of that the proposed algorithm is robustness against nonuniform noise and off-grid error, and it can maintain superior DOA estimation performance compared with other reported sparse signal representation based methods.

Off-Grid DOA Estimation Based on Circularly Fully Convolutional Networks (CFCN) Using Space-Frequency Pseudo-Spectrum.

Low-frequency multi-source direction-of-arrival (DOA) estimation has been challenging for micro-aperture arrays. Deep learning (DL)-based models have been introduced to this problem. Generally, existing DL-based methods formulate DOA estimation as a multi-label multi-classification problem. However, the accuracy of these methods is limited by the number of grids, and the performance is overly dependent on the training data set. In this paper, we propose an off-grid DL-based DOA estimation. The backbone is based on circularly fully convolutional networks (CFCN), trained by the data set labeled by space-frequency pseudo-spectra, and provides on-grid DOA proposals. Then, the regressor is developed to estimate the precise DOAs according to corresponding proposals and features. In this framework, spatial phase features are extracted by the circular convolution calculation. The improvement in spatial resolution is converted to increasing the dimensionality of features by rotating convolutional networks. This model ensures that the DOA estimations at different sub-bands have the same interpretation ability and effectively reduce network model parameters. The simulation and semi-anechoic chamber experiment results show that CFCN-based DOA is superior to existing methods in terms of generalization ability, resolution, and accuracy.

Multi-source off-grid DOA estimation using iterative phase offset correction in coarray domain

Direction-of-arrival (DOA) estimation performance degrades when arrival angles of incident signals do not locate exactly on the discretized grid points. Existing off-grid DOA estimation algorithms either suffer from high computational complexity or are restricted to uniform linear arrays (ULAs). In order to overcome these disadvantages, we propose a simple but effective off-grid DOA estimation method, referred to as iterative phase offset correction (IPOC), for non-uniform linear arrays (NLAs) in this work. The proposed IPOC first transforms the received signal into the coarray domain and then iteratively corrects the phase offset between the coarray data and presumed model caused by angle biases according to a closed-form formula. In the case of multiple sources, we separate these sources using generalized inner product and then utilize IPOC to sequentially estimate the off-grid angle of one source in each iteration. The spectral leakage between multiple sources is reduced as well by subtracting cross-correlation terms, thus overcoming the shortage of requiring a large number of training snapshots. We calculate the mean squared error (MSE) of the proposed method theoretically. Simulation results demonstrate the effectiveness and efficacy of the proposed method for off-grid DOA estimation using NLAs with very few snapshots.