Direct Position Determination Methods Research Articles

Self-Position Awareness Based on Cascade Direct Localization Over Multiple Source Data

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.

Joint Direct Estimation of Emitter Location and Antenna Beam Direction Based on Baseband Reconstruction

This paper addresses the problem of locating a stationary noncooperative emitter using multiple moving receivers. A signal baseband reconstruction based algorithm is proposed to locate the emitter directly from the received signals, without assuming static delay and Doppler shift in any observing interval. Besides the information of time-differences-of arrival (TDOA) and frequency-differences-of-arrival (FDOA), the received signal strength (RSS) information is further utilized when the source emits signals in free space with an omni antenna. In a more complex scenario where the emitter adopts a directional antenna, an iterative algorithm is proposed to jointly estimate the emitter location and its antenna beam direction, where a traditional direct position determination (DPD) method is used to initiate the iteration. The Cramér-Rao lower bounds (CRLBs) of the emitter position and the beam direction are derived. Three numerical cases are included to investigate the performance of the proposed algorithms. The signal baseband reconstruction based DPD algorithm shows improved applicability when the assumption of static time delay and Doppler shift is inappropriate and performs similarly with the classical DPD methods when the assumption holds. In case of free space propagation, it shows that the localization accuracy can be improved when RSS is further utilized. When the transmitting beam is directional, however, simply ignoring the directivity characteristics would lead to nonnegligible localization biases. By jointly estimating the emitter location and antenna beam direction, the RSS information can be effectively utilized to improve the localization accuracy, and a much more accurate estimation of beam direction can be obtained simultaneously.

Multi-TDOA Estimation and Source Direct Position Determination Based on Parallel Factor Analysis

In this paper, 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. Firstly, 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 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.

Direct Position Determination Using Compressive Sensing Measurements Without Reconstruction

This paper proposes a direct position determination (DPD) method for stationary emitters with compressed signal measurements. Employing Hadamard matrix properties, the method estimates the source location directly in the compressing measurement domain, without the step of signal reconstruction and time-differences-of-arrival (TDOA) parameter extraction. In case of free space propagation, an augmented DPD method is proposed to further utilize the position related received signal strength (RSS) information. In addition, the Cramér-Rao lower bounds (CRLBs) for emitter position of both DPD methods are derived for performance evaluation.

A Direct Position Determination Method under Unknown Multi-Perturbation with Moving Distributed Arrays

Distributed array manifold perturbation, which includes synchronization errors, amplitude-phase errors, and path attenuation, has seriously degraded the accuracy of existing direct position determination (DPD) methods. In this paper, a DPD method under unknown multi-perturbation with moving distributed coprime arrays is advocated for. Firstly, by means of array position interchange, the integrated signals received from distributed arrays can be fused, which contributes to multi-position fusion. Subsequently, by resorting to the orthogonality between the noise subspace and steering vector received via distributed arrays, a quadratic optimization problem is constructed. Finally, we realize multi-parameter decoupling and achieve localization regardless of unknown perturbations. The superiority of the advocated method is substantiated from simulation examples.

Experimental Study on the Target–Receiver Formation Problem with the Exploitation of Coherent and Non-Coherent Bearing Information

Localization of emitting sources is a fundamental task in sonar applications. One of the most important factors that affect the localization performance is the sensor–target geometry. The sensor formation problem is usually addressed in related work assuming that the target is static and the location is known to a certain degree, but this is not the case for many underwater surveillance problems. In this paper, we deal with the target–receiver formation problem from a different perspective, and propose to investigate the effect of target–receiver geometry on localization performance by exploiting the spatial spectrum of the direct position determination (DPD) methods. For a given multi-array system, the transformation of geometrical patterns can be explicitly demonstrated as the target moves along the track. Meaningful characteristics of the DPD methods are obtained from the experimental results, where coherent and non-coherent bearing information is used and compared. The feasibility of the DPD approaches in the ocean environments is also investigated by comparing with a matched filter processing (MFP)-based multi-array processor in order to validate the credibility of the results in this paper.

A Direct Position Determination Method Based on Subspace Orthogonality in Cross-Spectra under Multipath Environments.

Without the estimation of the intermediate parameters, the direct position determination (DPD) method can achieve higher localization accuracy than conventional two-step methods. However, multipath environments are still a key problem, and complex high-dimensional matrix operations are required in most DPD methods. In this paper, a time-difference-of-arrival-based (TDOA-based) DPD method is proposed based on the subspace orthogonality in the cross-spectra between the different sensors. Firstly, the cross-spectrum between the segmented received signal and reference signal is calculated and eigenvalue decomposition is performed to obtain the subspaces. Then, the cost functions are constructed by using the orthogonality of subspace. Finally, the location of the radiation source is obtained by searching the superposition of these cost functions in the target area. Compared with other DPD methods, our proposed DPD method leads to better localization accuracy with less complexity. The superiority of this method is verified by both simulated and real measured data when compared to other TDOA and DPD algorithms.

Direct position determination with phase error compensation for multiple targets using spatial sparsity

Measuring the positions of multiple targets is key in many positioning applications. Utilizing compressed sensing theory, measurements using the sparse direct position determination (DPD) methods are more accurate than the two-step methods and the traditional DPD methods. Position measurement with mobile access points (APs) expands the network coverage, increases information throughput and reduces implementation costs, which motivates us to extend sparse DPD (SDPD) to the mobility-assisted positioning system. However, one of the most critical problems in mobility-assisted positioning is the model errors coming from the AP position and velocity uncertainty. The paper proposes a phase error compensation algorithm for the SDPD algorithm based on mobile APs. The algorithm makes two key technical contributions. First, a double iteration for -norm optimization is designed to compensate phase errors and recover target distribution maps simultaneously. Second, when the data collection is incomplete or irregular, the algorithm jointly integrates the global information of target distribution maps at different APs to improve the measurement accuracy. Experimental results indicate the effectiveness of the algorithm.

Simultaneous Localization of Multiple Unknown Emitters Based on UAV Monitoring Big Data

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 Direct Position Determination Approach for Underwater Acoustic Sensor Networks

In this paper, we consider the source localization problem with an underwater acoustic sensor network (UASN) consisting of a number of platforms each equipped with an array, such as an autonomous underwater vehicle (AUV). Direct position determination (DPD) has been a promising topic in recent years. Existing DPD methods can be divided into two categories: global coherent processing and non-coherent processing. The former is easy to implement in practical tasks but the coherence between the received signals at different nodes are not exploited, while the requirements for coherent processing may not be fulfilled in real environments. A multi-cluster array model is constructed considering the deployment of the nodes with the intention to exploit the available coherence in the UASN and avoiding the side effects of imperfect spatial coherence. A weighted MUSIC direct source localization approach is proposed based on this model. The arrays within a cluster work coherently as a large array and contribute a MUSIC spectrum. The final source position estimate is obtained from a weighted combination of these spectra, and the weights are dependent on the eigenstucture and noise variance of each cluster. We first investigate the theoretical gain of inter-array coherence exploitation on localization accuracy by means of Cramer-Rao bound analysis. Then the model for imperfect spatial coherence is generalized to the multi-cluster system to examine the performance of the proposed approach when the inherent constraints for intra-cluster coherent processing can not be guaranteed. Numerical results demonstrate the superior resolution capability of the proposed approach and its robustness to imperfect spatial coherence.

Self-Calibration direct position determination using a single moving array with sensor gain and phase errors

The problem of direct position determination (DPD) using a single moving array in the presence of deterministic sensor gain and phase errors is considered. To eliminate the localization bias caused by these errors, an eigenstructure based self-calibrating DPD method is first introduced, in which the sensor gain and phase errors and the emitter positions are jointly estimated by an iterative process. Considering the performance deterioration of eigenstructure methods when the signal to noise ratio or the number of samples is not sufficiently large, a maximum likelihood (ML) based two-step self-calibration approach for DPD is subsequently proposed. The sensor gain errors are provided using the diagonal of the covariance matrix of the array output by a closed form solution at the first step. Then, the phase errors and the emitter positions are jointly estimated by an iterative scheme based on ML, in which the phase errors are also determined by a closed form solution in each iteration. Besides, detailed analyses and discussions about the differences between the introduced eigenstructure based and the proposed ML based self-calibration DPD methods are also provided. At last, numerical simulations are involved to examine their performance.

Robust Direct position determination against sensor gain and phase errors with the use of calibration sources

The direct position determination (DPD) method can provide high localization performance than conventional two-step localization methods. However, the existing DPD methods only consider the scenario of parameters of the receiving arrays, and the localization performance decreases dramatically when the array model is inaccurate in practice. This paper studies the problem for positioning a stationary emitter in the presence of sensor gain and phase errors (SGPEs) aided by calibration sources. To remove these negative effects caused by SGPEs, calibration sources with known positions are introduced. The extended relationship between parameters of calibration sources and errors is used to establish a structural objective function based on the maximum likelihood estimate. The calibration parameters are jointly optimized with target-related parameters and an alternating iterative algorithm is then developed to decouple the multidimensional search into several low-dimensional optimizations. We also derive the Cramer–Rao bound (CRB) to evaluate the performance of the proposed method. Simulation results demonstrate that the proposed method outperforms the existing DPD methods and two-step methods, which incorporates the error information, and the accuracy attains the associated CRB.

Enhanced direct position determination using dynamic sensor array response

An improvement in localisation of co-channel transmitters using direct position determination (DPD) is addressed. DPD techniques estimate the locations of signal emitters by finding the peaks of a spatial pseudo-spectral-function (PSF) within a definite area. Actually, different algorithms/beamformers such as Bartlett, minimum variance distortionless response (MVDR), and multiple signal classification (MUSIC) are distinguished by their PSFs. Mathematically, a PSF is a function of location through the array steering/pointing vector whose elements are the channel responses between any arbitrary point in the search area and each sensor antenna for a narrowband system model. Clearly, the amplitude and phase of each complex element of the pointing vector characterise the attenuation and delay of paths between search points and sensors, respectively. Classical DPD methods do not consider the dependency of attenuation on channel path length variation during area scanning, which consequently generates PSFs full of spurious peaks. The authors propose a more realistic channel model that remarks the free-space-path-loss as a deterministic, path-length-dependent factor beside the stochastic ones such as fading; therefore, they call it dynamic sensor array response (DSAR). Simulations show that range dependency of DSAR can effectively enhance the performance of the DPD algorithm for MVDR and MUSIC beamformers.

Low-complexity TDOA and FDOA localization: A compromise between two-step and DPD methods

Conventional two-step passive localization methods have poor localization precision when the input signal-to-noise ratio (SNR) is low, although they are usually computationally attractive. The direct position determination (DPD) methods, on the contrary, exhibit stronger robustness to low SNRs at the cost of much higher computational complexity. In this paper, an expectation-maximization (EM)-based computationally inexpensive localization method is proposed, which serves as a compromise between the two-step methods and DPD methods. During each iteration of the proposed method, the constraint that all intermediate parameters correspond to a common source position is taken into account explicitly, and this constraint is used to refine the time difference of arrival (TDOA) and frequency difference of arrival (FDOA) measurements. Then the source position is updated using these renewed measurements. The proposed method has superior localization performance under low SNR conditions compared to the two-step methods at the cost of limited increment in computational load, and its computational complexity is much lower than DPD methods at the cost of acceptable accuracy degradation. Moreover, the proposed method is applicable to other localization scenarios as well with appropriate modifications. Simulation results demonstrate the effectiveness of the proposed method.

Single-step localization using multiple moving arrays in the presence of observer location errors

Abstract Direct position determination (DPD) is a promising technique that offers superior performance compared with conventional two-step localization methods. Existing DPD methods presume that the observer locations are known exactly, whereas in practical environments, a small error in the observer locations will lead to an erroneous localization. This study considers the localization of a stationary transmitter by separated moving arrays from passive measurements taken at different points along the trajectory. The precise locations and velocities of the observers are not available, but their errors are assumed to be Gaussian distributed. Using this probability distribution, we propose maximum likelihood-based DPD approaches in the presence of observer location errors for both unknown and known signals. The proposed DPDs rely on alternating iteration schemes, which reduce the multidimensional nonlinear optimization problem to optimizations of dimensions that are much smaller than the number of unknowns. As opposed to the conventional two-step methods that extract measurement parameters and then estimate the positions from them, the proposed DPDs achieve the localization in a single step by exploiting the information of angles, time delays, and Doppler frequency shifts, but without computing them. Additionally, we derive the Cramer–Rao bound (CRB) formula for this DPD problem in the presence of observer location errors. The simulation results prove that the performance of our methods attains the associated CRB. Moreover, they are more robust than the conventional two-step approaches with respect to observer location errors. We demonstrate our methods for the scenario of multiple moving arrays, but these methods can easily be extended to DPD problems accounting for observer location errors in different scenarios.

Robust direct position determination methods in the presence of array model errors

Direct position determination (DPD) methods are known to have many advantages over the traditional two-step localization method, especially for low signal-to-noise ratios (SNR) and/or short data records. However, similar to conventional direction-of-arrival (DOA) estimation methods, the performance of DPD estimators can be dramatically degraded by inaccuracies in the array model. In this paper, we present robust DPD methods that can mitigate the effects of these uncertainties in the array manifold. The proposed technique is related to the classical auto-calibration procedure under the assumption that prior knowledge of the array response errors is available. Localization is considered for the cases of both unknown and a priori known transmitted signals. The corresponding maximum a posteriori (MAP) estimators for these two cases are formulated, and two alternating minimization algorithms are derived to determine the source location directly from the observed signals. The Cramér-Rao bounds (CRBs) for position estimation are derived for both unknown and known signal waveforms. Simulation results demonstrate that the proposed algorithms are asymptotically efficient and very robust to array model errors.

Performance analysis and improvement of direct position determination based on Doppler frequency shifts in presence of model errors: case of known waveforms

The direct position determination (DPD) method based on Doppler frequency shifts for signals with known waveforms is first proposed by Amar and Weiss. Although this method exhibits excellent asymptotic performance, but does not account for the effects of uncertainties in the receiver positions and velocities. These uncertainties may result in a considerable reduction of localization accuracy. In this paper, the statistical performance of the DPD estimator is investigated under receiver position and velocity errors (also called model errors). We derive an analytical expression for the mean square error in the estimated source location in the case where the estimator assumes that the receiver positions and velocities are accurate, but they in fact contain errors. The main difficulty in the mathematical analysis is that the DPD cost function is not explicit with respect to the emitter position. Consequently, some algebraic manipulation is required to derive closed-form expressions for the first- and second-order partial derivatives of the cost function. The Cramer–Rao bounds (CRBs) for the target position estimation are also deduced in the presence and absence of model errors. These CRBs provide insights into the effects of model errors on the localization accuracy. Two improved DPD methods are developed based on our analysis results. The first has lower complexity than the common grid search, and the second exhibits increased robustness to model errors. Simulation results support and corroborate the theoretical developments in this paper.

ML-based single-step estimation of the locations of strictly noncircular sources

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 Cramér–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.

Direct Position Determination of Multiple Noncircular Sources with a Moving Array

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.

SPARSITY-BASED MULTI-TARGET DIRECT POSITIONING ALGORITHM BASED ON JOINT-SPARSE RECOVERY

The direct position determination (DPD) method can improve the location accuracy compared with the traditional two- step location methods due to omitting the intermediate procedure of estimating the measurement parameters. However, the DPD methods presented so far are signiflcantly more complex than the two-step approach. To overcome the shortcomings of the published DPD algorithms, a novel multi-target direct localization approach is flrstly proposed by exploiting the jointly sparse property in the discrete spatial domain. The main idea of this paper is that the location estimation can be obtained by flnding the sparsest solution according to the predeflned overcomplete basis. Furthermore, the locations of targets can be obtained from noisy signals, even if the number of targets is not known a priori. Experimental results demonstrate that the proposed algorithm has superior positioning accuracy to other DPD methods and improves computational e-ciency greatly.