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We focus on the Direct position determination (DPD) of a moving narrowband source based on Doppler frequency shifts of signals with known waveforms. Two common motion models are considered: the Constant velocity (CV) model and the Constant acceleration (CA) model. The DPD cost function is obtained after some algebraic manipulations using the Maximum likelihood (ML) criterion. To develop a computationally efficient optimization algorithm, we present a preliminary mathematical result that plays a fundamental role in development of the proposed algorithm. Subsequently, a Newton iterative algorithm is tailored to the two motion models to determine the moving transmitter's trajectory. When compared with multidimensional grid searching, the proposed algorithm is more computationally attractive without compromising its estimation accuracy.Simulation results confirm the superiority of the proposed algorithm.

Direct position determination (DPD) methods are known to outperform classical two-step localization methods under the condition of low signal-to-noise ratios and/or small number of snapshots. Additionally, they can directly exploit the prior knowledge of signal waveforms to achieve higher estimation accuracy. In this paper, we concentrate on the DPD method for locating multiple constant modulus (CM) sources based on a single moving receiving station. Both direction of arrival and Doppler frequency shift are used for source localization. First, a received array signal model with a small time delay that can incorporate the Doppler frequency information is formed. Subsequently, the corresponding maximum likelihood estimation criterion is established. An effective alternating minimization algorithm is developed to solve the optimization problem. Subsequently, we also derive the Cramer–Rao bound expressions for parameter estimation. It is proved that the CM property of the signals is useful in reducing the lower bound for location estimation. Simulation results demonstrate that the proposed method is asymptotically efficient. Moreover, it is also shown that the estimation accuracy of direct localization can be greatly increased if the CM characteristics of the radiated signals are adequately exploited.

This paper investigates the geolocation for an over-the-horizon (OTH) transmitter observed by widely separated arrays. We propose a maximum likelihood (ML) based direct position determination (DPD) method to directly locate the transmitter in a single step by exploiting the position information embedded in azimuth angles. The Monte Carlo importance sampling (IS) technique is employed to find an approximate global solution to this DPD problem, where the importance function analogous to Gaussian distribution is derived. This enables the transmitter to be precisely located with low complexity in a noniterative manner. Additionally, we derive the Cramér–Rao bound (CRB) expression for the investigated problem. The simulation results corroborate the superior localization performance of the proposed method with respect to the conventional two-step approaches and the iterative DPD method.

It is well known that the Two-step Weighted Least-Squares (TWLS) is a widely used method for source localization and sensor position refinement. For this reason, we propose a unified framework of the TWLS method for joint estimation of multiple disjoint sources and sensor locations in this paper. Unlike some existing works, the presented method is based on more general measurement model, and therefore it can be applied to many different localization scenarios. Besides, it does not have the initialization and local convergence problem. The closed-form expression for the covariance matrix of the proposed TWLS estimator is also derived by exploiting the first-order perturbation analysis. Moreover, the estimation accuracy of the TWLS method is shown analytically to achieve the Cramér-Rao Bound (CRB) before the threshold effect takes place. The theoretical analysis is also performed in a common mathematical framework, rather than aiming at some specific signal metrics. Finally, two numerical experiments are performed to support the theoretical development in this paper. Keywords: Cramér-Rao Bound (CRB), Disjoint sources, General framework, Performance analysis, Sensor position refinement, Source localization, Weighted least-squares

Time difference of arrival (TDOA) positioning is one of the widely used techniques for locating an emitter. Besides TDOA measurement errors, clock synchronization bias is an important factor that can degrade the localization accuracy. This paper focuses on TDOA localization using a set of receivers, where timing synchronization offsets exist among different receiver groups. A theoretical analysis is conducted and a new localization solution is developed for the case of imperfect time synchronization. The analysis starts with the Cramer–Rao bound (CRB) for the problem and derives the estimation bias and mean square error (MSE) using the classical quadratic constraint weighted least-squares (QCWLS) estimator, which does not consider synchronization errors. Additionally, an alternative performance measure, namely the localization success probability (SP), is introduced to evaluate the location accuracy. An explicit formula for calculating the localization success probability is presented. In addition, an improved quadratic constraint weighted least-squares estimator that accounts for synchronization errors is proposed to reduce the positioning errors. The Lagrange multiplier technique is used to solve this estimator. As a byproduct, a closed-form solution for the estimation of clock bias is also provided. First-order perturbation analysis reveals that the performance of the proposed estimates achieves the Cramer–Rao bound. Simulations corroborate the theoretical results and the good performance of the proposed method.

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 Cramer-Rao lower bound accuracy. Simulation results show the validity of the theoretical derivation and superiority of the new algorithm.

Direct position determination (DPD) is a single-step method that directly localizes transmitters from sensor outputs without estimating intermediate parameters. With a moving array, DPDs are usually performed by implicitly using only angle information. This letter proposes a DPD algorithm for multiple stationary transmitters by exploiting location information embedded in angles and Doppler shifts. We fit all signal subspaces with the space-time steering vectors at all positions of the moving array using the weighted subspace fitting method, and then perform an alternating projection procedure to directly localize transmitters. The simulation results corroborate the good localization performance of the proposed algorithm.

This paper concentrates on the location methods for strictly noncircular sources by widely separated arrays. The conventional two-step methods extract measurement parameters and then, estimate the positions from them. Compared with the conventional two-step methods, direct position determination (DPD) is a promising technique, which locates transmitters directly from original sensor outputs without estimating intermediate parameters in a single step, and thus, improves the location accuracy and avoids the data association problem. However, existing DPD methods mainly focus on complex circular sources without considering noncircular signals, which can be exploited to enhance the localization accuracy. This paper proposes a maximum likelihood (ML)-based DPD algorithm for strictly noncircular sources whose waveforms are unknown. By exploiting the noncircularity of sources, we establish an ML-based function in time domain under the constraint on the waveforms of signals. A decoupled iterative method is developed to solve the prescribed ML estimator with a moderate complexity. In addition, we derive the deterministic Cramer–Rao Bound (CRB) for strictly noncircular sources, and prove that this CRB is upper bounded by the associated CRB for circular signals. Simulation results demonstrate that the proposed algorithm has a fast convergence rate, and outperforms the other location methods in a wide range of scenarios.

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 Cramer-Rao lower bound at a moderate Gaussian noise level and outperforms the existing methods.

Compared with conventional two-step localization methods, direct position determination (DPD) is a promising technique that offers superior performance under low signal-to-noise ratio conditions. However, existing DPD methods mainly focus on complex circular sources without considering noncircular signals, which can be exploited to enhance the localization accuracy. This study proposes an improved subspace data fusion (SDF)-based DPD algorithm for multiple noncircular sources with a moving array. By constructing and decomposing the extended covariance matrices, extended noise subspaces are obtained for all positions of the moving array. The source positions are then directly estimated by fusing the extended noise subspaces without computing the intermediate parameters, thereby avoiding the data association problem inherent in two-step methods. Our proposed DPD algorithm combines the low complexity of SDF with the high robustness to noise and sensor errors that comes from exploiting signal noncircularity. Specifically, a closed-form expression for the localization mean square error (MSE) of the algorithm and the stochastic Cramer–Rao bound for strict-sense noncircular signals are derived. Simulation results validate our theoretical prediction for MSE and also demonstrate that the proposed algorithm outperforms other localization methods in terms of accuracy and capacity to resolve noncircular sources.

It is well known that sensor location uncertainties can significantly deteriorate the source positioning accuracy. Therefore, improving the sensor locations is necessary in order to achieve better localization performance. In this paper, a constrained-total-least-squares (CTLS) method for simultaneously locating multiple disjoint sources and refining the erroneous sensor positions is presented. Unlike conventional localization techniques, an important feature of the proposed method is that it establishes a general framework that is suitable for many different location measurements. First, a modified CTLS optimization problem is formulated after some algebraic manipulations and then the corresponding Newton iterative algorithm is derived to give the numerical solution. Subsequently, by using the first-order perturbation analysis, the explicit expression for the covariance matrix of the proposed CTLS estimator is deduced under the Gaussian assumption. Moreover, the estimation accuracy of the CTLS method is shown to achieve the Cramér-Rao bound (CRB) before the thresholding effect occurs by a rigorous proof. Finally, two kinds of numerical examples are given to corroborate the theoretical development in this paper. One uses the TDOAs/GROAs measurements and the other is based on the TOAs/FOAs parameters.

Angularly dependent gain and phase uncertainties are produced by the combined effects of multiple sensor errors. This paper proposes a direction-finding method for noncircular signals in the presence of angularly dependent gain/phase errors, which utilizes instrumental sensors to achieve auto-calibration and relies on an improved alternating projection procedure. By applying the principle of the extended 2-sided instrumental variable signal subspace fitting algorithm, the proposed method is effective for separating spatially and temporally correlated noncircular sources from the unknown colored (i.e., spatially correlated) noise. Considering that modeling errors of instrumental sensors are frequently encountered in practice, this paper also presents a theoretical derivation for the closed-form expression of the mean square error of the estimation under the influence of modeling errors of instrumental sensors in the first-order analysis. Finally, the results of two series of simulations are demonstrated. The first series of simulations verifies the effectiveness of the proposed auto-calibration method, and shows that noncircularity and temporal correlation of sources are informative for enhancing the calibration performance of our method. The results also prove that the proposed method performs better than the instrumental sensor method when applied to spatially and temporally correlated noncircular sources. Moreover, this performance advantage of our method is more prominent when signal-to-noise ratio is low, or in spatially correlated noise fields. The second series of simulations validates the theoretical prediction, and thus our statistical analysis has a high predictive value for calibration performance of the proposed method under the influence of modeling errors.

Direct position determination (DPD) is a single-step method which directly localizes transmitters from sensor outputs without computing intermediate parameters. Compared with conventional two-step localization methods, DPD achieves higher accuracy especially under low signal to noise ratio (SNR) conditions. This paper proposes an improved subspace data fusion (SDF)-based DPD algorithm with a moving array. Different from the existing SDF-based DPD which uses only the arrival angle information, the proposed algorithm exploits the location information embedded in both arrival angles and Doppler shifts. It relies on a unitary space-time SDF, where multiple stationary transmitters are directly localized by fusing all the unitary (real-valued) space-time subspaces at all positions of the moving array. Therefore, our algorithm realizes the localization by real-valued computations. The simulation results show the superior localization performance of the proposed DPD compared to the existing SDF-based DPD and two-step localization algorithms.

Satellite position errors are known to deteriorate geolocation accuracy significantly. Based on a time difference of arrival (TDOA) satellite localization system, this paper investigates the use of calibration sources to reduce the loss of localization accuracy due to satellite position errors. First, we deduce pseudo-linear equations according to the available TDOA measurements from the target and calibration sources. Based on these equations, a total least squares optimization model with an equality constraint is formulated. To solve this constrained optimization problem, we propose an augmented Lagrange geolocation algorithm by which the target position and refined satellite positions can be obtained. Subsequently, the theoretical performance of the augmented Lagrange geolocation algorithm is derived, and it is proved that the mean square error of the proposed algorithm coincides with the corresponding Cramer–Rao lower bound under first-order perturbation analysis. Finally, simulation results corroborate the theoretical developments and demonstrate the superior performance of the proposed algorithm compared with existing algorithms.

The rank reduction estimator (RARE) is one kind of autocalibration method used in the presence of sensor errors. It demonstrates high accuracy of direction-of-arrival (DOA) estimation in the absence of multidimensional search or iteration. However, its estimation performance is affected by “unexpected modeling errors.” There is a lack of research regarding the performance of 2-D RARE estimation, although 2-D RARE estimation is extensively employed in applications. This paper presents a theoretical derivation for the closed-form expression of the mean square error (MSE) of 2-D RARE estimation under the influence of small unexpected modeling errors in the first order analysis. First, three definitions of 2-D joint direction-finding success are introduced, in order to establish the criterion for estimate performance. Then corresponding theoretical formulas for three probabilities of direction-finding success are given with the circularly Gaussian assumption of unexpected modeling errors, and their relations are discussed. Finally, the results of simulations utilizing our analysis method are demonstrated, verifying the effectiveness of the MSE expression and the formulas for probabilities of success. Therefore, our first order approximation provides a good prediction of the necessary calibration accuracy in the presence of unexpected modeling errors in order to help RARE meet an expected performance specification.