Accurate direction-of-arrival (DOA) estimation and localization are essential for navigation and positioning. However, most existing model-based approaches are restricted to specific scenarios, which results in poor scalability. In addition, deep learning–based methods require extensive data training. To overcome these challenges, we propose a multistation direction-finding (DF) and localization method based on a multi-agent deep deterministic policy gradient (MADDPG) algorithm (DF-MADDPG). In the proposed method, agents monitor target positions by accurately selecting signals from the time–frequency spectrum. First, interference is removed from the received signals through space–time high-resolution processing, and DOA is estimated. Specifically, agents are deployed on the time–frequency spectra to establish an efficient collaborative communication policy, thereby improving the signal-to-noise ratio of signal selection. In addition, we design a reward function that employs a multistation cross-localization strategy to guide agents in quickly and accurately selecting signals for precise localization. To further improve environmental awareness under uncertainty, the time–frequency spectra are treated as states and integrated with the MADDPG algorithm to train agents for target localization. Compared to single-agent reinforcement learning approaches, the proposed DF-MADDPG method improves localization efficiency by 9.5% while maintaining consistent localization accuracy.

To address the phase ambiguity problem arising from baseline extension in phase interferometer for improving direction-finding accuracy, this paper proposes a CNN-SE-LSTM model-based phase ambiguity resolution method. By analyzing the mapping relationship between the phase differences and phase ambiguity values, a multidimensional phase differences dataset is constructed. The method employs a Convolutional Neural Network (CNN) to extract spatially localized features, a Squeeze-and-Excitation (SE) block for adaptive feature weight recalibration, and a Long Short-Term Memory (LSTM) network to capture temporal dependencies within features. This approach effectively overcomes the reliance of conventional methods on specific baseline configurations. Experimental results demonstrate a 72% improvement in the success rate of ambiguity resolution compared to existing methods at 0 dB signal-to-noise ratio (SNR) across varying array geometries. The proposed method provides an innovative solution for phase interferometer ambiguity resolution challenge.

Abstract This article proposes a direct position determination (DPD) algorithm for non‐circular sources observed by a moving array using the self‐calibration technique in the presence of mutual coupling. The method first utilises the symmetric Toeplitz property of uniform linear array matrices with mutual coupling and cyclic Toeplitz property of uniform circular array coupling matrix, realising the decoupled estimations of target position parameters and sensor error parameters. Then the position parameters of multiple non‐circular are directly determined based on the subspace data fusion criterion in a decoupled manner, where the subspaces are obtained using the extended array data model with the non‐circular properties of the sources. This results in a significant improvement in the accuracy of the target position estimation and the number of distinguishable sources compared to the traditional mutual coupling calibration algorithm. In addition, the theoretical mean square error expression for the position estimations of the proposed algorithm under the influence of finite sampling is derived based on the matrix perturbation analysis theory, and the corresponding Cramér‐Rao bound is given. Finally, the correctness of the theoretical derivation and the superiority of the method is verified by simulation experiments.

Abstract It is well known that the Doppler frequency shift contains the emitter location information. The problem of emitter location based on frequency of arrival (FOA) measurements of multiple moving observers is focussed. In contrast to most relevant works, the newly proposed positioning methods consider the influence of signal frequency deviation due to source instability and present two mathematical models of the frequency deviation. The first model assumes that the frequency offset is a small random parameter. A two‐stage closed‐form location method (called method a) is proposed. In stage 1, a set of pseudo‐linear equations are established according to the geometric relationship between the observers and emitter, so as to obtain an intermediate estimate of source position. Stage 2 returns to the original FOA measurement model and uses the estimation result in stage 1 to construct the second set of pseudo‐linear equations, yielding the final positioning result. The second model assumes that the signal frequency deviation is a large deterministic parameter. We propose an alternative two‐stage localisation method (called method b). In stage 1, a decoupling optimisation algorithm is developed based on the pseudo‐linear equations derived in stage 1 of method a, and intermediate estimates of the source position and frequency deviation are obtained successively. Stage 2 returns to the original FOA measurement model again and forms the second set of pseudo‐linear equations, yielding the final closed‐form solutions of emitter position and frequency offset. Both methods are proved analytically to achieve the Cramér–Rao bound (CRB) under the moderate noise level. Simulation results demonstrate the superior performance of the new methods.

Abstract In this paper, we focus on the radiation source localization problem based on time‐difference‐of‐arrival (time difference of arrival (TDOA)) measurements in the presence of sensor position errors. In order to alleviate the estimation degradation in locating a radiation source caused by the sensor position uncertainty, we propose to use a single calibration source and develop a novel two‐stage localization algorithm based on the constrained scoring algorithm (CSA). In the first stage, we use the information of the calibration source to reduce the sensor position deviation, and here the CSA method is utilised to solve a quadratic constrained maximum likelihood (ML) objective function which is formulated from a set of underdetermined pseudo‐linear equations. In the second stage, using the updated sensor positions and the TDOA measurements of the radiation source, another ML estimation problem with quadratic constraints is formulated, and we again invoke the CSA method to jointly estimate the source and sensor positions. Moreover, the asymptotically optimal performance of the proposed algorithm is mathematically analysed and proved based on the first‐order error analysis. The simulation results verify the excellent performance of our algorithm and also demonstrate its robustness of resisting large measurement noise and sensor position errors.

Abstract This study addresses the problem of geolocating a strictly non‐circular source on the surface of Earth by a cluster of passive satellites. The known satellite positions and velocities are subject to random errors. The authors propose a single‐step satellite geolocation algorithm that directly localises the transmitter from sensor outputs using the information of time delays and Doppler shifts but without explicitly estimating them. It exploits the non‐circular property of signals and a priori information of satellite orbit error distribution to jointly calibrate orbit errors and determine the longitude and latitude of the transmitter based on the ellipsoidal Earth model, which integrates an alternating iteration scheme for the estimation of various unknowns instead of the exhaustive grid search. Additionally, a detailed Cramér‐Rao bound (CRB) derivation is presented for the single‐step satellite geolocation of a non‐circular source on Earth with and without satellite orbit perturbations, and it is proved that these CRBs are lower than the associated CRBs for a circular source. The simulation results illustrate that the proposed method asymptotically attains the associated CRB, and shows greater performance robustness to signal‐to‐noise ratio (SNR) and satellite orbit errors compared with conventional two‐step satellite geolocation approaches.

Source localisation withtime‐difference‐of‐arrival (TDOA) and frequency‐difference‐of‐arrival (FDOA)measurements is of great interest since it can provide the location information with highaccuracy.Although the maximumlikelihood (ML) estimator exhibits excellent asymptotic properties, the non‐linearity and non‐convexity of ML estimator requiremuch computation resources.In this study, source localisation with TDOA and FDOA measurementsis developed viaMonteCarlo importance sampling (IS).In particular, the optimalperformance can be guaranteed by constructing an optimalimportance function whosecovariance is equivalent to the inverse of Fisher information matrix.The derived variance of the proposed estimator showsgood consistency with the theoretical lowerbound. The improved performance of the proposed method is due to the optimal selection ofimportance function and it canconverge to the global optimum with a large number of samples. Although an initial estimate of source localisation information isrequired, the proposedmethod is robust to this a priori knowledge via IS. Moreover, the scenario ofconsidering sensor location uncertainties is analysed and the corresponding IS based solution is derived. Simulation results show that the proposed methods can achieve the Cramér–Rao lower bound at moderate level noises and is superior to several existing methods.

Single-station passive localization technology avoids the complex time synchronization and information exchange between multiple observatories, and is increasingly important in electronic warfare. Based on a single moving station localization system, a new method with high localization precision and numerical stability is proposed when the measurements from multiple disjoint sources are subject to the same station position and velocity displacement. According to the available measurements including the angle-of-arrival (AOA), time-of-arrival (TOA), and frequency-of-arrival (FOA), the corresponding pseudo linear equations are deduced. Based on this, a structural total least squares (STLS) optimization model is developed and the inverse iteration algorithm is used to obtain the stationary target location. The localization performance of the STLS localization algorithm is derived, and it is strictly proved that the theoretical performance of the STLS method is consistent with that of the constrained total least squares method under first-order error analysis, both of which can achieve the Cramér-Rao lower bound accuracy. Simulation results show the validity of the theoretical derivation and superiority of the new algorithm.

Multiple sources localization based on time difference of arrival (TDOA) measurements is investigated in this paper. Different from the traditional methods, a novel and practical multisource localization algorithm is proposed by adopting a priori information of relative distance among emitting sources. Since the maximum likelihood (ML) cost function for multisource estimation is highly nonconvex, the semidefinite relaxation (SDR) is utilized to reformulate the ML cost function. A robust estimator is obtained, which can be solved by semidefinite programming (SDP). Moreover, the constrained Cramér‐Rao bound is also derived as a benchmark by considering the range constraints between sources. Simulation results verify the superior performance of the proposed algorithm over the traditional methods.

Passive source localization via a maximum likelihood (ML) estimator can achieve a high accuracy but involves high calculation burdens, especially when based on time-of-arrival and frequency-of-arrival measurements for its internal nonlinearity and nonconvex nature. In this paper, we use the Pincus theorem and Monte Carlo importance sampling (MCIS) to achieve an approximate global solution to the ML problem in a computationally efficient manner. The main contribution is that we construct a probability density function (PDF) of Gaussian distribution, which is called an important function for efficient sampling, to approximate the ML estimation related to complicated distributions. The improved performance of the proposed method is attributed to the optimal selection of the important function and also the guaranteed convergence to a global maximum. This process greatly reduces the amount of calculation, but an initial solution estimation is required resulting from Taylor series expansion. However, the MCIS method is robust to this prior knowledge for point sampling and correction of importance weights. Simulation results show that the proposed method can achieve the Cramér-Rao lower bound at a moderate Gaussian noise level and outperforms the existing methods.

Determining the position of an emitter on Earth by using a satellite cluster has many important applications, such as in navigation, surveillance, and remote sensing. However, in realistic situations, a number of factors, such as errors in the measurement of signal parameters, uncertainties regarding the position of satellites, and errors in the location of calibration sources, are known to degrade the accuracy of target localization in satellite geolocation systems. We systematically analyze the performance of multi-satellite joint geolocation based on time difference of arrival (TDOA) measurements. The theoretical analysis starts with Cramér–Rao bound (CRB) derivations for four localization scenarios under an altitude constraint and Gaussian noise assumption. In scenario 1, only the TDOA measurement errors of the emitting source are considered and the satellite positions are assumed to be perfectly estimated. In scenario 2, both the TDOA measurement errors and satellite position uncertainties are taken into account. Scenario 3 assumes that some calibration sources with accurate position information are used to mitigate the influence of satellite position perturbations. In scenario 4, several calibration sources at inaccurate locations are used to alleviate satellite position errors in target localization. Through comparing the CRBs of the four localization scenarios, some valuable’s insights are gained into the effects of various error sources on the estimation performance. Two kinds of location mean-square errors (MSE) expressions under the altitude constraint are derived through first-order perturbation analysis and the Lagrange method. The first location MSE provides the theoretical prediction when an estimator assumes that the satellite locations are accurate but in fact have errors. The second location MSE provides the localization accuracy if an estimator assumes that the known calibration source locations are precise while in fact erroneous. Simulation results are included to verify the theoretical analysis.

In this paper, the active calibration method for multiplicative modelling errors using phase‐modulated (also called constant modulus) auxiliary sources is presented. Compared with some existing calibration methods, the proposed approach can exploit the constant modulus characteristic of the sources and has significantly better performance. For the purpose of incorporating the constant modulus information into the procedure for finding array error parameters, the maximum likelihood criterion is chosen as the optimisation function and a concentrated alternating iteration algorithm is developed, which has rapid convergence rate. In addition, to reduce the effects of azimuth deviations of the auxiliary sources, the study proceeds to extend the novel algorithm to the scenario where the true azimuths of the sources deviate slightly from the nominal values with a prior known Gaussian distribution. The Cramér–Rao bound (CRB) expressions for the unknowns are derived for the case when it is known that the sources are phase‐modulated. Simulation results show that the performance of the proposed algorithms are considerably better than that of subspace‐based calibration methods and closely follows the CRB for array error estimation.