Underdetermined Direction Of Arrival Estimation Research Articles

Gridless Underdetermined DOA Estimation for Mobile Agents with Limited Snapshots Based on Deep Convolutional Generative Adversarial Network

Deep learning techniques have made certain breakthroughs in direction-of-arrival (DOA) estimation in recent years. However, most of the current deep-learning-based DOA estimation methods view the direction finding problem as a grid-based multi-label classification task and require multiple samplings with a uniform linear array (ULA), which leads to grid mismatch issues and difficulty in ensuring accurate DOA estimation with insufficient sampling and in underdetermined scenarios. In order to solve these challenges, we propose a new DOA estimation method based on a deep convolutional generative adversarial network (DCGAN) with a coprime array. By employing virtual interpolation, the difference co-array derived from the coprime array is extended to a virtual ULA with more degrees of freedom (DOFs). Then, combining with the Hermitian and Toeplitz prior knowledge, the covariance matrix is retrieved by the DCGAN. A backtracking method is employed to ensure that the reconstructed covariance matrix has a low-rank characteristic. We performed DOA estimation using the MUSIC algorithm. Simulation results demonstrate that the proposed method can not only distinguish more sources than the number of physical sensors but can also quickly and accurately solve DOA, especially with limited snapshots, which is suitable for fast estimation in mobile agent localization.

Exploration on 2D DOA estimation of linear array motion: Uniform linear motion

Generally, due to the limitation of the dimension of the array aperture, linear arrays cannot achieve two-dimensional (2D) direction of arrival (DOA) estimation. But the emergence of array motion provides a chance for that. In this paper, a generalized motion scheme and a novel method of 2D DOA estimation are proposed by exploring the linear array motion. To be specific, the linear arrays are controlled to move along an arbitrary direction at a constant velocity and snap per fixed time delay. All the received signals are processed to synthesize the comprehensive observation vector for an extended 2D virtual aperture. Subsequently, since most of 2D DOA estimation methods are not universal to our proposed motion scheme and the reduced-dimensional (RD) method fails to handle the case of the coupled parameters, a decoupled reduced-complexity multiple signals classification (DRC MUSIC) algorithm is designed specifically. Simulation results demonstrate that: a) our proposed scheme can achieve underdetermined 2D DOA estimation just by the linear arrays; b) our designed DRC MUSIC algorithm has the good properties of high accuracy and low complexity; c) our proposed motion scheme with the DRC method has better universality in the motion direction.

Underdetermined direction of arrival estimation for multiple input and multiple outputs sparse channel based on Bayesian learning framework

Direction of arrival (DOA) estimation for a sparse channel has attracted serious attention recently. Better signal analysis and denoising achieve accuracy in DOA determination. This paper proposes an underdetermined DOA estimation for multiple input and multiple outputs (MIMO) sparse channels. A novel multi-kernel-based non-negative sparse Bayesian learning (MK NNSBL) framework is implemented using the multiplied form of basis vector within the manifold matrix for a defined grid. Meanwhile, virtual antenna locations are reconfigured by exploiting the conventional cuckoo search algorithm (CCSA) for the fine reception of incoming signals on a nonuniform linear array (NULA). The simulated results reveal that the novel approach outperforms in its optimal root mean square error (RMSE) for various signal-to-noise ratio (SNR) limits and the compilation time. The convergence comparative graph indicates the improved performance in the proposed framework over existing algorithms.

Underdetermined direction of arrival estimation with nonuniform linear motion sampling based on a small unmanned aerial vehicle platform

Uniform linear array (ULA) radars are widely used in the collision-avoidance radar systems of small unmanned aerial vehicles (UAVs). In practice, a ULA’s multi-target direction of arrival (DOA) estimation performance suffers from significant performance degradation owing to the limited number of physical elements. To improve the underdetermined DOA estimation performance of a ULA radar mounted on a small UAV platform, we propose a nonuniform linear motion sampling underdetermined DOA estimation method. Using the motion of the UAV platform, the echo signal is sampled at different positions. Then, according to the concept of difference co-array, a virtual ULA with multiple array elements and a large aperture is synthesized to increase the degrees of freedom (DOFs). Through position analysis of the original and motion arrays, we propose a nonuniform linear motion sampling method based on ULA for determining the optimal DOFs. Under the condition of no increase in the aperture of the physical array, the proposed method obtains a high DOF with fewer sampling runs and greatly improves the underdetermined DOA estimation performance of ULA. The results of numerical simulations conducted herein verify the superior performance of the proposed method.

Ziv-Zakai Bound for DOAs Estimation

Lower bounds on the mean square error (MSE) play an important role in evaluating the direction-of-arrival (DOA) estimation performance. Among numerous bounds for DOA estimation, the local Cram <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathrm{\acute{e}}$</tex-math></inline-formula> r-Rao bound (CRB) is only tight asymptotically. By contrast, the existing global tight Ziv-Zakai bound (ZZB) is appropriate for evaluating the single source estimation only. In this paper, we derive an explicit ZZB applicable for evaluating hybrid coherent/incoherent multiple sources DOA estimation. It is first shown that, a straightforward generalization of ZZB from single source estimation to multiple sources estimation cannot keep the bound valid in the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a priori</i> performance region. To derive a global tight ZZB, we then introduce order statistics to describe the change of the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a priori</i> distribution of DOAs caused by ordering process during the MSE calculation. The derived ZZB is for the first time formulated as a function of coherent coefficients between coherent sources, and reveals the relationship between the MSE convergency in the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a priori</i> performance region and the number of sources. Moreover, the derived ZZB also provides a unified tight bound for both overdetermined and underdetermined DOA estimation. Simulation results demonstrate the obvious advantages of the derived ZZB over the CRB on evaluating and predicting the estimation performance for multiple sources DOA.

Underdetermined direction of arrival estimation of wideband signal based on sparse array

To solve the problem of the mismatch of the wideband underdetermined direction of arrival (DOA) estimation under the condition of the On-grid model, this paper extends the narrowband Off-Grid model to wideband and proposes a new DOA estimation algorithm for Off-Grid sources based on group sparsity. The proposed algorithm first obtains the preliminary estimation result under the current predefined discrete grid through the group sparsity wideband DOA estimation algorithm. Then, the Off-Grid optimisation problem is adopted to calculate the Off-Grid deviation vector. It is also assumed that the off-grid deviation vectors of each frequency subband are exactly the same, thereby reducing the number of parameters to be estimated. Therefore, the proposed algorithm can not only maintain similar or even better estimation accuracy but also greatly reduce the computational complexity. Finally, simulation is conducted and the results verify the effectiveness and performance of the proposed method.

Underdetermined DOA estimation for moving array with reduced mutual coupling in unknown nonuniform noise environment

In practice, direction of arrival (DOA) estimation methods will suffer from significant performance degradation due to limited physical sensors, mutual coupling effect and unknown nonuniform noise. To improve the performance of underdetermined DOA estimation in the case, this paper proposes a moving sparse uniform linear array (ULA) for more degrees of freedom (DOF) and lower mutual coupling, and then provides an estimation algorithm with unknown nonuniform noise mitigation. There is a systematic procedure to determine the inter-sensor spacing of the original array and number of shifted arrays for the synthetic ULA, for achieving increased DOF. The construction and analysis of mutual coupling matrix are exploited to show the reduced mutual coupling in the synthetic array. A procedure of nonuniform noise suppression for the synthetic array is given by averaging the noise power from the original array, and then root-MUSIC is employed for DOA estimation with low-complexity. Simulation results are given to demonstrate the superiority of the proposed array and algorithm in terms of DOF enhancement, mutual coupling reduction and nonuniform noise mitigation.

Underdetermined DOA estimation exploiting the higher‐order cumulants of harmonic steering vector with time‐modulated arrays

Time-modulated arrays (TMAs) have been widely studied owing to their convenient control mode and simple structure. Although direction-of-arrival (DOA) estimation based on TMA has garnered considerable attention, underdetermined DOA estimation in TMA is yet to be studied. In this study, a novel method is proposed for underdetermined DOA estimation based on the higher-order cumulants of harmonic signal in a TMA. The periodic modulation of radio-frequency switches leads to a TMA generating harmonics in space. Cumulants of harmonic statistics can be used for DOA estimation, and with higher-order harmonics cumulants, underdetermined DOA estimation can be realised with a TMA using the multiple signal classification method, and the degree of freedom can be improved further. Another advantage of the proposed method is its simple structure, which only requires one receiver with several filters. Numerical simulations show that the proposed method can achieve good resolution and precision performance.

A Passive DOA Estimation Algorithm of Underwater Multipath Signals via Spatial Time-Frequency Distributions

Direction of arrival (DOA) estimation of underwater multipath signals has become an indispensable part of many military and civil applications in the oceanic field. However, DOAs of different paths of the multipath signals may be the same or too close to be separated in space, which brings a great challenge to the DOA estimation of multipath signals. Besides, the number of multipath is generally unknown and time-varying in passive scenarios which generally lead to an underdetermined case. Therefore, it is essential to study an efficient algorithm to realize the underdetermined DOA estimation of underwater multipath signals in passive scenarios. This paper proposes a passive DOA estimation algorithm of underwater multipath signals via spatial time-frequency distributions (STFDs). Firstly, we establish a multipath signals model and utilize the quadratic time-frequency (TF) analysis method to reduce cross-terms while keeping a good resolution. Then, we estimate the number of multipath and introduce a novel multi-ridge detection technique to extract the TF trajectories of multipath signals, which adopts the idea of smoothing forward and backward, and the edge TF point detection strategy. Lastly, we estimate the DOA of each path of the multipath signals separately and realize source association based on the clustering-based method. Simulation results and analysis demonstrate that this algorithm can effectively improve the estimation accuracy of instantaneous frequency (IF) and DOA estimation of underwater multipath signals in underdetermined cases and passive situations.

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

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

Cramér-Rao Bound Analysis of Underdetermined Wideband DOA Estimation Under the Subband Model via Frequency Decomposition

A class of Cramér-Rao bounds (CRBs) for wideband direction-of-arrival (DOA) estimation under the subband (or frequency bin) model is studied for the underdetermined case, where the number of sources is no less than that of physical sensors. A unified framework is proposed to encompass the closed-form CRB expressions for DOAs in four cases where the sources are known a priori to 1) have flat spectra/cross spectra, 2) be spatially uncorrelated, 3) be spatially uncorrelated and have proportional spectra up to unknown factors, 4) be spatially uncorrelated and have flat spectra. The relationship between the wideband CRBs and the subband ones is investigated, and the order relationship among the derived CRBs are provided. The asymptotic behavior of the CRBs with respect to the number of snapshots and the signal-to-noise ratio (SNR) is discussed. Two asymptotic expressions for sufficiently large SNR are derived in both overdetermined and underdetermined cases. Existence of the derived CRBs is examined through rank conditions of the introduced matrices, which yields upper bounds on the resolution capacities. Different from the narrowband scenario, underdetermined wideband DOA estimation is feasible even if a sparse array is not used given different a priori knowledge about the source spectra. It is possible to resolve more wideband Gaussian sources than the number of DOFs offered by the difference co-array. Finally, further interpretations of the subband model are provided, revealing the underlying connections with the multi-frequency co-array augmentation concept and the non-coherent subarray system.

Direction of Arrival Estimation of Wideband Sources Using Sparse Linear Arrays

In this paper, we study the problem of wideband direction of arrival (DoA) estimation with sparse linear arrays (SLAs), where a number of uncorrelated wideband signals impinge on an SLA and the data is collected from multiple frequency bins. To boost the performance and perform underdetermined DoA estimation, the difference co-array response matrices for all frequency bins are constructed first. Then, to merge the data from different frequency bins, we resort to the Jacobi-Anger approximation to transform the co-array response matrices of all frequency bins into a single virtual uniform linear array (ULA) response matrix. The major advantage of this approach is that the transformation matrices are all signal independent. For the special case where all sources share an identical distribution of the power spectrum, we develop two super-resolution off-the-grid DoA estimation approaches based on atomic norm minimization (ANM), one with and one without prior knowledge of the power spectrum. Our solution is able to resolve more sources than the number of antennas but also more than the number of degrees of freedom (DoF) of the difference co-array of the SLA. For the general case where each source has an arbitrary power spectrum, we propose a multi-task ANM method to exploit the joint sparsity from all frequency bins. Simulation results show that our proposed methods present a clear performance advantage over existing methods, and achieve an estimation accuracy close to the associated Cramér-Rao bounds (CRBs).

Underdetermined wideband DOA estimation for off‐grid targets: a computationally efficient sparse Bayesian learning approach

Underdetermined wideband direction of arrival (DOA) estimation based on the sparse array is studied here and a novel algorithm is developed to improve the estimation performance of off-grid targets in the framework of sparse Bayesian learning. First, the narrowband off-grid model is extended to a wideband case and the sparse Bayesian model containing off-grid biases is deduced. Then, a sequential solution is proposed to obtain the estimation, where the fast sparse Bayesian learning strategy is employed to improve the computational efficiency. The estimation accuracy is improved significantly through off-grid compensation and the computational complexity is reduced remarkably. Simulation results verify the effectiveness of the proposed method.

Calibration Nested Arrays for Underdetermined DOA Estimation Using Fourth-Order Cumulant

The application of a nonuniform nested array provides a simple way for underdetermined direction of arrival (DOA) estimation, enabling us to develop a suited self-calibration algorithm for perturbed nested array. The key points of this algorithm contain two steps: a first step is to obtain gain-phase errors estimation by utilizing partly calibrated array and properly choosing the elements of fourth-order cumulant matrix; a second step to form a small perturbed model after compensating the gain-phase errors as well as obtain DOA estimation using the sparse total east squares (STLS) technique. The proposed algorithm is very suited for underdetermined DOA estimation in the presence of gain-phase errors. Numerical simulations validate the effectiveness of the proposed algorithm.

Underdetermined Wideband DOA Estimation Utilizing Noncircular Complex Gaussian Distribution

With the emerging of sparsely spaced sensor arrays, the study on underdetermined direction-of-arrival (DOA) estimation methods has drawn much attention. In most of the existing methods, the vectorized sample covariance matrix was considered as circularly symmetric Gaussian by default. However, the sample covariance vector is in fact noncircular, and its pseudo covariance matrix has not yet been utilized for underdetermined DOA estimation. On account of the wide usage of wideband signals nowadays, in this letter, the underdetermined DOA estimation problem for wideband signals is addressed, where the noncircularity of the sample covariance vectors is exploited. Moreover, a hierarchical Bayesian model is established, modeling the noncircular sample covariance vectors, the nonnegative group-sparse signal variance vectors, the auto-estimated regularization parameter and the off-grid difference, which were not considered simultaneously in the existing methods. A sparse Bayesian learning algorithm is derived via expectation-maximization, and numerical simulations show that the proposed method outperforms the methods that do not consider the noncircularity of sample covariance vectors.

Underdetermined DOA Estimation for Wideband Signals via Focused Atomic Norm Minimization.

In underwater acoustic signal processing, direction of arrival (DOA) estimation can provide important information for target tracking and localization. To address underdetermined wideband signal processing in underwater passive detection system, this paper proposes a novel underdetermined wideband DOA estimation method equipped with the nested array (NA) using focused atomic norm minimization (ANM), where the signal source number detection is accomplished by information theory criteria. In the proposed DOA estimation method, especially, after vectoring the covariance matrix of each frequency bin, each corresponding obtained vector is focused into the predefined frequency bin by focused matrix. Then, the collected averaged vector is considered as virtual array model, whose steering vector exhibits the Vandermonde structure in terms of the obtained virtual array geometries. Further, the new covariance matrix is recovered based on ANM by semi-definite programming (SDP), which utilizes the information of the Toeplitz structure. Finally, the Root-MUSIC algorithm is applied to estimate the DOAs. Simulation results show that the proposed method outperforms other underdetermined DOA estimation methods based on information theory in term of higher estimation accuracy.

Underdetermined DOA Estimation Algorithm Based on an Improved Nested Array

A nonuniform array group is proposed to solve the inflexibility problem of existing nested arrays. The array structure in the group is not required to be composed of two or more uniform linear sub-arrays on the premise of maintaining the same degree of freedom as that of nested arrays. Compared with nested arrays, a nonuniform array group is easier to arrange in airborne radar or limited physical space. For the nonuniform array group proposed in this study, an array is randomly selected for deployment, and the covariance matrix of the data received by the array is quantised. Accordingly, an algorithm for data de-redundancy and sorting is proposed, and the virtual array receiving data are obtained. Lastly, the target direction of arrival (DOA) is estimated using a sparse reconstruction method without losing array aperture. Simulation results show that the proposed method can achieve accurate DOA estimations at low signal-to-noise ratio and snapshot number.

Coprime sparse circular array with little angular dependence and reduced mutual coupling

Sparse circular arrays (SCAs) have many advantages, e.g., providing 360° azimuthal coverage and reducing mutual coupling. This paper presents a novel coprime sparse circular array (CSCA) for significantly improving underdetermined direction of arrival (DOA) estimation performance compared to the existing SCAs, e.g., nested sparse circular array (NSCA) and super nested sparse circular array (S-NSCA). The CSCA is obtained by connecting two sparse subarrays concircularly with two displacements, and the sensor positions are determined by a systematic procedure. The array geometry has closed-form expressions and the inter-sensor spacings of the two sparse subarrays are coprime. The CSCA contains all the good properties of the SCAs. Simultaneously, the CSCA has less angular dependence due to its nearly uniform sensor distribution and has much smaller mutual coupling effect by enlarging its minimum inter-sensor spacing, when compared to the NSCA and S-NSCA. Due to its merits, it can be applied to sonar, communication, and navigation, to name a few. Then an l1-based method is used for underdetermined DOA estimation. Theoretical analysis and simulation results show the superiority of the CSCA.

Extended Transformed Nested Arrays for DOA Estimation of Non-Circular Sources

Recently, direction of arrival (DOA) estimation for non-circular (NC) sources has attracted great attention in array signal processing since NC signals can provide more information to generate the virtual difference and sum (diff-sum) coarray. In this paper, we propose a generalized vectorized noncircular MUSIC (GVNCM) method, which can solve the underdetermined DOA estimation problem when the difference coarray and the sum coarray are discontinuous. Based on the GVNCM algorithm, an extended transformed nested array strategy is proposed by splitting the dense subarray of transformed nested array (TNA) into several parts and shifting them to the right to reduce the redundant elements between the difference coarray and the sum coarray. Following this strategy, two novel array configurations are developed. The analytical expressions of the new structures and the consecutive range of their diff-sum coarrays are provided. The two proposed arrays can achieve higher degrees of freedom and maximum number of detectable signals compared with TNA. In order to evaluate the underdetermined DOA estimation performance of NC signals, we derive the corresponding expression of Cramer-Rao bound. The effectiveness of the proposed arrays is verified through numerical simulations.

Robust DOA Estimation Against Mutual Coupling With Nested Array

In this letter, we focus on the problem of direction-of-arrival (DOA) estimation with a nested array in the presence of unknown mutual coupling. Firstly, the measurement model of the nested array with mutual coupling is established, and the properties of the corresponding array covariance matrix are derived. Based on the obtained properties, the coarray output signal is then reconstructed with reduced mutual coupling. Finally, joint sparse recovery technique is employed to extract the DOAs. Our algorithm can achieve underdetermined DOA estimation with satisfactory performance under unknown mutual coupling. Numerical results demonstrate the effectiveness of the proposed algorithm.