A substantial body of work has focused on two-dimensional (2-D) self-localization of linear arrays using wireless sensor networks (WSN). However, tackling the higher-dimensional three-dimensional (3-D) case remains an open research question. This paper explores the use of one-dimensional (1-D) angle-of-arrival (AOA) measurements, also referred to as space angles (SA), to achieve 3-D self-localization of a linear array in the presence of anchor position errors. Unlike traditional 3-D source localization using SAs, 3-D self-localization requires simultaneous estimation of the array’s position and orientation (direction vector). First, we acquire a coarse solution to the weighted least-squares (WLS) problem via semidefinite relaxation (SDR), and then we enhance it through perturbation analysis. We then address the maximum likelihood (ML) estimation problem using block majorization-minimization (block-MM), which guarantees convergence to the Karush-Kuhn-Tucker (KKT) point. In addition, we propose an improved and tighter quadratic upper bound for a Rayleigh-quotient-like function introduced in our previous work. Simulations confirm that the proposed algorithm reaches the Cramér-Rao lower bound (CRLB) performance in low-noise regimes, with the block-MM-based ML method exhibiting minimal bias.

Direct localization (DL) of high-order quadrature amplitude modulation (QAM) sources is a pivotal challenge in wireless communications, particularly in environments characterized by complex multipath propagation and the presence of multiple sensor array-based anchors. This paper introduces a novel solution based on dual atomic norm minimization (DANM) framework that capitalizes on the fourth-order cumulant property of QAM signals to suppress Gaussian noise and expand the effective array aperture. Unlike traditional DL frameworks based on discrete Fourier transform (DFT) and spatial smoothing pre-processing (SSP) techniques, the proposed framework enhances localization accuracy and improves robustness against multipath effects. By framing the localization problem as a semidefinite program that utilizes dual atomic norm properties, our solution eliminates the need for prior knowledge of the number of sources and achieves a favorable balance between computational complexity and localization performance. Simulation results reveal that the DANM-based DL algorithm outperforms existing DFT- and SSP-based DL methods in terms of localization accuracy, with its root mean square error (RMSE) closely approaching the Cramér-Rao bound (CRB) even under challenging conditions. These findings underscore the potential of DANM in advancing high-precision DL for high-order QAM sources, thereby paving the way for more reliable and precise wireless communication systems.

It is challenging to ensure both high accuracy and low complexity when localizing radiation sources. To address this challenge, we propose two novel methods leveraging time difference of arrival (TDOA) measurements. Specifically, we introduce a TDOA estimation method and a direct position determination (DPD) method based on parallel factor (PARAFAC) analysis in the cross-correlation domain. Initially, multiple sensors synchronously capture the source signal, and the cross-correlation function between signals received from a reference sensor and other sensors is calculated. Then, the primary cross-spectrum data undergoes an expansion and integration process to establish the PARAFAC model. Through cross-spectrum expansion, virtual nodes are formed, which further improves the estimation performance. The TDOA estimates for each sensor are obtained by normalizing and extracting the phase from this matrix. Additionally, we introduce a novel DPD method tailored for multipath propagation scenarios. Simulations and real-world measurements demonstrate the superiority and effectiveness of our proposed methods compared with cutting-edge methods.

Array self-position determination methods based on multiple emitter data can avoid significant deviations of vehicle satellite navigation in harsh environments. However, existing array self-position determination methods show decrease in performance under multipath environments. To deal with this problem, we propose an array self-position determination method based on orthogonal grid matching with the spatial differencing method. Specifically, the direction of arrival (DOA) of direct path and multipath signals are respectively estimated by array spatial differencing method. The matching accuracy is enhanced by utilizing the prior information of direct path signal. After calculating correlation coefficients of different sources, estimated angles with high correlation are then classified into the same set. Then, the noise subspace of each angle set is reconstructed and the position is estimated by grid matching with the orthogonal property between the noise subspaces and the characteristic steering vectors. The matching results of redundant angle sets are removed as non-matching items, thus averting positioning deviations. The simulation results demonstrate that the computational complexity of the proposed method is comparable to that of the signal subspace fitting (SSF). Moreover, in terms of positioning precision, the proposed method outperforms multiple signal classification with enhanced spatial smoothing (ESSMUSIC), initial signal fitting (ISF), and SSF.

The global navigation satellite systems (GNSS) are crucial for applications such as traffic monitoring, intelligent transportation systems and advanced driver assistance systems; however, they are prone to large deviations or even failures in hidden scenarios such as urban canyons and tunnels. In this paper, we investigate a system based on distributed arrays for positioning and tracking of multiple vehicles as a supplement to the GNSS. Conventional tracking methods often assume a constant signal-to-noise ratio (SNR) across arrays; however, this assumption does not hold in dynamic, dense traffic or signal-obstructed areas where variable SNR conditions frequently arise. To address this, this paper proposes two adaptive passive position tracking methods utilizing distributed sparse arrays, which are robust to variations in SNR and capable of maintaining reliable tracking performance. The first method relies on the extended Kalman filter (EKF) algorithm, while the second employs the unscented Kalman filter (UKF) algorithm. A novel tracking system model is developed, including a multi-target state transfer equation and a fused signal equation. The unknown noise covariance and signal covariance of the fused signal are estimated using vectorization and least squares methods, respectively. Prior state information and minimum mean square error estimation are incorporated to address unknown signal challenges within the EKF and UKF frameworks. Additionally, the posterior Cramér-Rao bound provides a performance benchmark, underscoring the robustness of the proposed methods. Computer simulation studies and practical tests show that the proposed methods significantly improve computational efficiency and tracking performance in vehicle tracking, demonstrating high adaptability to noise variations compared to traditional methods.

Direct position determination (DPD) (a.k.a. direct localization) offers enhanced precision over traditional two-step approaches. This technique, however, involves considerable communication overhead for transmitting raw data. Low-bit direct localization methods have recently been introduced to address this issue. In this letter, we present a gridless, one-bit maximum likelihood (ML) approach for the direct localization of an orthogonal frequency division multiplexing (OFDM) signal source. A recent majorization-minimization (MM) algorithm introduced a surrogate function for the log-likelihood function, which lacks a closed-form optimal solution and requires exhaustive searches at each iteration. Our method improves upon this algorithm by developing a refined surrogate function that yields a closed-form optimal solution, thereby eliminating the need for exhaustive searches. Accordingly, the proposed MM approach can eliminate grid quantization errors (GQE) by eliminating the search process. Simulation results validate the proposed method's efficacy in mitigating GQE and its efficiency in scenarios with densely populated search grids.

Coprime array possesses a larger array aperture and element spacing compared with the conventional uniform linear array (ULA) for the equivalent number of sensors, attracting considerable scholarly attention. However, the direction of arrival (DOA) estimation of coherent signals has been a major challenge for the practical application of the coprime array. In this paper, based upon the perspective of signal subspace reconstruction, we propose two effective approaches to resolve the DOA of completely correlated signals with a coprime array. For the first method, we exploit two selection matrices to separate the signal subspace into two parts and rearrange the elements within them to construct two Hankel matrices. By applying the MUSIC method to these Hankel matrices and finding common solutions, we can determine the DOA of coherent signals. In the second method, we first convert the signal subspace of the coprime array into the signal subspace of ULA using a mapping matrix and the total least squares method. We then construct a Hankel matrix and restore its rank by solving a rank minimization problem. Finally, by applying the MUSIC method to the rank-restored Hankel matrix, we can obtain the angles of the coherent signals. Finally, simulation results are presented to demonstrate the efficiency and superiority of our proposed methods.

The global positioning system (GPS), which provides ubiquitous location-awareness with a constellation of satellites, has become an instrumental function of multiple mass-market applications. Satellite signals, however, may not be capable of penetrating obstacles in harsh environments (e.g., urban canyons, tree canopies, and flyovers). Hence, GPS may not provide adequate localization accuracy for applications like autonomous vehicles. Resorting to data fusion of heterogeneous signals emanating from multiple anchors, we advocate a self-localization method that provides accurate estimates of the vehicle position. To be more specific, several heterogenous emitters whose positions are known are used as anchors to determine the vehicle’s position based on the weighted direct position determination (DPD) method that eliminates nonhomogeneity among different emitters. However, the weighted DPD method requires an exhaustive search of the parameter search space and is thus time-consuming. To reduce the computational burden, we propose a weighted cascade compensation estimator (WCCE) that is tailored for real-time tracking and self-localization. The proposed WCCE outperforms traditional DPD methods in terms of computational complexity while achieving nearly comparable localization accuracy. The effectiveness of the proposed method is corroborated by extensive simulated examples.

Existing path planning schemes designed for wireless sensor networks generally account for abstract payload sensors, rendering them inapplicable to concrete sensor-array-based localization systems due to differences in measurement models. In this paper, we establish a general framework for path planning of a practical sensor array and factor in an oft-neglected degree of freedom regarding optimality, i.e., the array’s orientation/attitude. The optimization problem is formulated based on the A-optimality criterion under constraints arising from the maximum distance between consecutive waypoints, maximal heading change, and forbidden regions. To facilitate semidefinite relaxation (SDR), we recast the optimization function into a fractional nonhomogeneous quadratic structure and transform the constraints into a bilinear form. By applying SDR and replacing the bilinear terms with a matrix variable, the problem is relaxed into a single-ratio fractional program. By leveraging the Charnes-Cooper variable transformation, we transform the single-ratio fractional program into a mixed semidefinite/second-order cone program (SD/SOCP) that can be solved in polynomial time. Finally, we apply the results to angle-of-arrival (AOA) and direct localization. Simulation results demonstrate that the proposed path planning scheme attains near-optimal performance.

In this article, source localization exploiting time difference of arrival (TDOA) information is discussed, and a parallel factor (PARAFAC) analysis-based method for multi-TDOA estimation and direct position determination (DPD) is proposed. First, signals from the radiation source are received by multiple antennas through synchronous sensing. Then, the data from multiple antennas is fused by extracting the time-delay matrix to construct a PARAFAC model. Thereafter, the time-delay matrix is obtained by fast iterative decomposition using the trilinear alternating least square (TALS) algorithm. In the case of no multipath or weak multipath, where the typical scenario is the antennas being deployed on the airborne platform, the multi-TDOA estimation can be obtained simultaneously according to the time-delay matrix to improve the processing speed. In the case of multipath, such as the antennas being located on the ground, the DPD cost function directly related to the source position can be established based on the time-delay matrix, and the source position estimation can be achieved through grid search. Compared with the other DPD methods, the proposed DPD method has better positioning performance and lower complexity. The effectiveness and superiority of the proposed methods are verified by both simulations and actual scenario tests.

An unmanned aerial vehicle (UAV) array is composed of multiple UAVs carrying array elements, which can sense signals synchronously through a global positioning system (GPS) trigger. However, gain-phase errors caused by synchronization errors and the inconsistent complex gains of receiving array channels result in array manifold perturbation, which makes the performance of traditional localization methods degrade or even fail. In this article, two efficient algorithms for source direction finding and direct localization using a UAV array are, respectively, presented. First, the array manifold changes with the movement of UAVs, which contributes to multiposition fusion, thus avoiding the infinite solutions of underdetermined equations. Then, the quadratic optimization problem can be constructed by using the orthogonal relation between noise subspaces and contaminated steering vectors obtained from multiple observations. Thereafter, we can construct the cost function and obtain the spectral function, from which the source directions and positions can be all estimated by grid search. Meanwhile, the gain-phase error values can be solved successively. Moreover, in order to avoid the influence of heteroscedasticity of different observation positions in direct localization, we carry out a blind weighting operation. Simulation results demonstrate the effectiveness and superiority of the proposed algorithms.

Arrays deployed in antenna-array-based direct position determination (DPD) systems are conventionally placed parallel to each other with invariant orientations, which may be inappropriate. This paper proposes a novel array orientation adjustment method to achieve optimal DPD performance, in which we seek optimal array orientations that maximize the trace of the Fisher information matrix (FIM) and then minimize localization uncertainty. The optimization problem is unconstrained, multivariable, and can be converted into multiple tractable single-variable optimization problems. An exact solution can be obtained by a straightforward but computationally tedious one-dimensional exhaustive search. To reduce the computational load, we then derive an asymptotic solution in cases where the arrays consist of massive antennas. Numerical examples demonstrate the effectiveness and feasibility of the proposed method in resolution and precision enhancement, especially in a low signal-to-noise ratio (SNR).

Millimeter-wave (mmWave) fifth-generation (5G) and beyond 5G localization enables the provisioning of extremely accurate positioning information, a feature that has attracted substantial research efforts. In this paper, we contribute to this effort by exploring optimal array geometric structures of direct position determination (DPD) systems, inspired by sensor placement problems that predominantly focus on two-step localization and have not yet been extended to DPD. Specifically, we research an optimal array placement and orientation strategy for two-dimensional (2-D) DPD systems that use sensors equipped with uniform linear arrays (ULA) to localize an agent. The A-optimality criterion in Bayesian optimal (experimental) design theory is invoked to formulate this problem. We use an optimization subproblem that optimizes array orientations when array locations are arbitrary but fixed to tackle this high-dimensional optimization problem. Then the optimization problem is converted into a typical optimal angular separation problem in two-step localization. Experimental results show that judiciously designed array geometric structures can lead to significant performance improvements.

The coprime array owns a sparse array structure, which can effectively alleviate mutual coupling, while the holes existing in the difference co-array(DCA) greatly reduce the number of uniform degrees of freedom(uDOFs). In this letter, we propose a new coprime array structure called hole-free coprime array (HFCA) by carefully assembling the subarrays, and the resulting HFCA can generate a hole-free DCA. Furthermore, we derive the optimal HFCA that can produce the largest hole-free DCA with a given number of sensors, which contributes to an increase of uDOFs compared with existing coprime array structures. In addition, the simulations are provided to demonstrate the superiority of HFCA in terms of uDOFs, direction of arrival estimation performance, and spatial resolution.

The increasing of illegal radiations, which are either artificial or unintentional, has seriously influenced the reliable communication and operation of industrial facilities. In this article, we discuss the simultaneous localization of multiple emitters based on big data monitored by a moving unmanned aerial vehicle. Conventional direct position determination (DPD) method suffers from the non-homogeneity of the observation error and is sensitive to the environment, so we develop the weight DPD methods. First, we consider to strengthen the spectrums obtained at slots with higher signal-to-noise ratio, which is blindly calculated. Thereafter, an improved weight is designed to further enhance the localization accuracy, and it can obtain the asymptotically optimal performance under the general Gaussian noise model, which is proved theoretically. Numerical simulations demonstrate that the proposed weight DPD methods outperform the conventional two-step methods and subspace data fusion DPD method in terms of localization accuracy and resolution.

A partial dictionary based direction of arrival (DOA) estimation method which addresses the off‐grid problem and exploits combined coprime and nested array (CCNA) is proposed. Compared to general coprime array, CCNA yields two sparse coprime subarrays in the coarray domain by adding a third subarray in the physical‐array domain. To ensure the DOA estimation performance, the subarray with larger aperture is chosen, and the cyclic phase ambiguity caused by the sparse subarray allows partial dictionary covering arbitrary cycle to represent the whole atoms, and then, the off‐grid sparse reconstruction method is developed to amend the grid mismatch. After the sparse recovery and off‐grid compensation, ambiguous DOA estimations can be eliminated by substituting the estimations into the whole virtual array. Multiple simulations verify that the proposed algorithm outperforms the other state‐of‐the‐art methods in terms of DOA estimation accuracy and angular resolution.

A joint processing of direction of arrival (DOA) and signal separation for planar array is proposed in this paper. Through sensor array processing theory, the output data of a planar array can be reconstructed as a parallel factor (PARAFAC) model, which can be decomposed with the trilinear alternating least square (TALS) algorithm. Aiming at the problem of slow speed on convergence for the standard PARAFAC method, we introduce the propagator method (PM) to accelerate the convergence of the TALS method and propose a novel method to jointly separate signals and estimate the corresponding DOAs. Given the initial angle estimates with PM, the number of iterations of TALS can be reduced considerably. The experiments indicate that our method can carry out signal separation and DOA estimation for typical modulated signals well and remain the same performance as the standard PARAFAC method with lower computational complexity, which verifies that our algorithm is effective.

Traditional two‐step passive localization methods need to extract the parameters like the direction of arrival (DOA), time of arrival (TOA), and time difference of arrival (TDOA) from the original data to determine the source position, which causes the poor positioning accuracy due to error accumulation. In this paper, a direct position determination (DPD) method is proposed to improve the positioning accuracy and robustness, which is based on a correlation algorithm. Firstly, the cost function directly related to the location of the source can be established by synthesizing the data received by multiantenna in the frequency domain. Then, the position of the source is estimated by the correlation DPD method to search the monitoring area. Compared to the improved TDOA algorithm and Least Squares DPD algorithm, the proposed method shows better localization accuracy of different SNRs. Finally, based on real measured data, it can be seen that the results of the proposed algorithm are better than the improved TDOA algorithm.

Direction of arrival (DOA) estimation using the improved unfolded coprime array (IUFCA) subject to array motion isdiscussed in this study. Unfolded coprime array (UFCA) consists of two uniform linear subarrays, and the two subarrays are arranged at different sides of theaxis, which leads to a large number of holes in the difference co‐array (DCA). With array motion and DCA synthesis,part of the holes can be filled, but there are still holes in the center which lead to the virtual arrays separated.By analyzing the hole positions in the synthetic DCA generated by UFCA motion, the authors improve the originalUFCA by relocating some physical elements, then the two dominantconsecutive DCA segments in the positive and negative sides can be connected. The expression of synthetic DCA is analyzed, and the closed‐form expression of theuniform degree of freedom (uDOF) subject to IUFCA motion is studied. Simulation results show that IUFCA motion can obtain a largenumber of uDOFs, which leads to better DOA estimation performance and more identifiable signals compared with existingcoprime array configurations.

In massive multiple-input multiple-output (MIMO) systems, it is critical to obtain the accurate direction of arrival (DOA) estimation. Conventional three-dimensional array mainly focuses on the uniform array. Due to the dense arrangement of the sensors, the array aperture is limited and severe mutual coupling effects arise. In this paper, a coprime cubic array (CCA) configuration design is presented, which is composed of two uniform cubic subarrays and can extend the interelement spacing with a selection of three pairs of coprime integers. Compared with uniform cubic array (UCA), CCA achieves the larger array aperture and less MC effects. And the analytical expression of Cramer–Rao Bound (CRB) for CCA is derived which verifies that the proposed CCA geometry outperforms the conventional UCA in two-dimensional (2D) DOA estimation performance in massive MIMO systems. Meanwhile, we propose a computationally efficient 2D DOA estimation algorithm with high accuracy for CCA. Specifically, we utilize array mapping to extract two uniform arrays from the nonuniform array by exploiting the relation derived from the signal subspace and the two directional matrices. Then, we operate a reduced dimension process on the uniform arrays and convert the 2D spectrum peak searching (SPS) problem into one-dimensional (1D) one, which significantly reduces the computational complexity. In addition, we employ the polynomial root finding technique with a lower complexity instead of 1D SPS to further relieve the computational complexity. Simultaneously, with coprime property, the phase ambiguity problem is solved, which results from the large interelement spacing. Numerical simulation results demonstrate that the proposed algorithm is very computationally efficient without degradation of DOA estimation performance.

In this paper, the topic of coherent two-dimensional direction of arrival (2D-DOA) estimation is investigated. Our study jointly utilizes the compressed sensing (CS) technique and the parallel profiles with linear dependencies (PARALIND) model and presents a 2D-DOA estimation algorithm for coherent sources with the uniform rectangular array. Compared to the traditional PARALIND decomposition, the proposed algorithm owns lower computational complexity and smaller data storage capacity due to the process of compression. Besides, the proposed algorithm can obtain autopaired azimuth angles and elevation angles and can achieve the same estimation performance as the traditional PARALIND, which outperforms some familiar algorithms presented for coherent sources such as the forward backward spatial smoothing-estimating signal parameters via rotational invariance techniques (FBSS-ESPRIT) and forward backward spatial smoothing-propagator method (FBSS-PM). Extensive simulations are provided to validate the effectiveness of the proposed CS-PARALIND algorithm.

Compared with the standard estimation of signal parameters via rotational invariance technique (ESPRIT) for direction of arrival estimation, the propagator‐based ESPRIT is a more computationally efficient method since it requires no eigenvalue decomposition when calculating the signal subspace. In this study, the propagator‐based ESPRIT method is analysed by deriving its theoretical asymptotic error. With first‐order perturbation analysis, it is shown that both propagator‐based ESPRIT and standard ESPRIT have the same asymptotic error with high signal‐to‐noise ratio. Simulation results of both the propagator‐based ESPRIT and the standard ESPRIT are presented, and they are shown to be consistent with the theoretical derivations.

The issue of joint angle and array gain‐phase error estimation for a bistatic multiple‐input multiple‐output array is discussed in this study, and an algorithm for the joint estimation with non‐linear arrays is proposed. First, the estimations of the transmit and receive direction matrices can be obtained via trilinear decomposition, then the relationship between the columns of the direction matrices is utilised to eliminate the influence of the gain‐phase errors, and the angles can be estimated two by two via least squares. Finally, the array gain‐phase error vectors can be estimated for calibration according to the estimated angles and direction matrices. The proposed algorithm requires no eigenvalue decomposition of the received data, and can achieve automatically paired estimations of the angles. Furthermore, no information of the gain‐phase error is needed. The simulation results verify the algorithmic effectiveness of the proposed algorithm.

In this letter, we investigate the subject of joint angle and array gain-phase error estimation for bistatic multiple-input multiple-output radar, and propose a joint scheme for angle and array gain-phase error estimation based on trilinear decomposition. The estimations of transmit and receive direction matrices are primarily obtained via trilinear decomposition, after which the optimization problem for estimating array gain-phase errors and angles can be constructed. Array gain-phase error vectors are obtained by Lagrange multipliers, and the angles are estimated according to the estimated gain-phase errors. In contrast to the ESPRIT-like algorithm, the proposed method not only obtains automatically paired estimations of the angles but also has much better performance for angle and array gain-phase error estimation. Simulations verify the effectiveness of our approach.

This letter discusses the problem of direction of departure (DOD) and direction of arrival (DOA) estimation for multi-input multi-output (MIMO) array system, which is configured with multiple transmit sensors and multiple velocity receive sensors. Reference presented a successive multiple signal classification MUSIC algorithm for joint estimation of DOD and DOA for the bistatic MIMO array system with velocity receive sensors. In this letter, we propose an improved joint DOD and DOA estimation algorithm, which has much better angle estimation performance and lower complexity than the successive MUSIC algorithm in . Furthermore, the proposed algorithm can be suitable for irregular array geometry, obtain automatically paired 2-D angle estimation, and avoid 2-D searching. Simulation results verify the usefulness of the proposed algorithm.