Efficient Bi-Iterative Method for Source Position and Propagation Speed Estimation Using TDOA Measurements

This paper describes two new techniques for the joint estimation of source location and propagation speed using measured time difference of arrival (TDOA) for a sensor array. Previous methods for source location either assumed the array consisted of widely separated subarrays, or used an iterative procedure that required a good initial estimate. The first method directly estimates the source location and propagation speed by converting the solution of a system of nonlinear equations to an overdetermined system of linear equations with two supplemental variables. The second method provides improved estimates by using the solution of the first method as the initial condition for further iteration. The Cramer-Rao Bound (CRB) on the joint estimation is derived, and simulations show the new methods compare favorably to the bound.

The problem of source localization from arrival time delay estimates requires a computationally costly iterative solution of a set of nonlinear equations. Most known methods assume that the propagation speed is known. In this paper, we provide several effective source localization and propagation velocity estimation methods which only use measurements of the relative arrival time delays between sensors. The formulae for source localization and propagation speed estimation are derived based on least squares, total least squares, bounded data uncertainty, and constrained least squares methods. Statistical performance of these methods are compared via computer simulation. In addition, in order to avoid time delay ambiguity problems and obtain smoother time delays, two time delay smoothing methods based on the forward backward algorithm and the Viterbi algorithm are also proposed. Field experiment results based on these techniques are also presented.

Accurate three-step algorithm for joint source position and propagation speed estimation

Uncertainty in sensor (receiver) locations can seriously deteriorate the source localization accuracy. A localization algorithm needs to take the sensor location errors into account in order to improve source location estimate. This paper considers the localization for the position and velocity of a moving source using the time differences of arrival (TDOAs) and frequency differences of arrival (FDOAs) of a signal received at a number of sensors that have random location errors. The proposed method is based on the Taylor-series method that jointly estimates both the source and sensor locations simultaneously. Simulation results show that with a reasonable initial guess close to the true solution to begin with, the proposed method converges to the true solution and achieves the CRLB accuracy for both near-field and far-field sources.

One main drawback of using time difference of arrival (TDOA) methods for source localization and navigation is that they require centralization of multiple copies of a signal. This paper considers blindly estimating the location of a cyclic prefix (CP) in an orthogonal frequency division multiplexing (OFDM) signal, enabling distributed TDOA computation up to an integer ambiguity. This ambiguity can be resolved using integer least-squares methods, if enough TDOAs are available, requiring only minimal cooperation between receivers. The contributions of this paper are development of an algorithm for simultaneously resolving the integer ambiguities and obtaining a position estimate; and derivation of the Cramer-Rao lower bound (CRLB) on locating the CP, and hence, on the underlying source localization and navigation problems.

Time difference of arrival (TDOA)-based methods usually achieve the most accurate source localization in passive systems. In these methods, one of the sensors is assigned as the reference and the TDOA of the others with respect to the reference is measured. The source location is estimated by processing these TDOAs. In this paper, a multireference maximum likelihood (ML)-Taylor algorithm is proposed in order to decrease the source localization error of the TDOA methods. In the proposed algorithm, TDOAs are measured assuming 2 different reference sensors, and then the ML objective function of this multireference TDOA algorithm is derived and solved using Taylor series linearization. The developed method is evaluated regarding a 2-dimensional space with some different but deterministic sensor locations. Monte Carlo simulation is used to evaluate the proposed solution. The simulation results show that the proposed method has better performance relative to the other traditional TDOA-based algorithms.

The maximum likelihood estimator (MLE) and its performance for the localization of a stationary emitter using a network of spatially separated passive stationary sensors is presented. The conventional approach for localization using multiple sensors is to first estimate the time differences of arrival (TDOAs) independently between pairs of sensors and then find the location of the emitter using the intersection point of the hyperbolas defined by these TDOAs. It has recently been shown that this two-step approach is suboptimal and an alternate direct position determination (DPD) approach has been proposed. In the work presented here we take the DPD approach to derive the MLE and show that the MLE outperforms the conventional two-step approach.We analyze the two commonly occurring cases of signal waveform unknown and signal waveform known with unknown transmission time. This paper covers a wide variety of transmitted signals such as narrowband or wideband, lowpass or bandpass, etc. Sampling of the received signals has a quantization-like effect on the location estimate and so a continuous time model is used instead.We derive the Fisher information matrix (FIM) and show that the proposed MLE attains the Cramer-Rao lower bound (CRLB) for high signal-to-noise ratios (SNRs).

In this study, we propose an algorithm for selecting nodes with the minimum dilution of precision (DOP) to improve the accuracy of estimating the location of an unknown signal source in a swarm drone environment. The pulse radar emits a narrow beam, and the swarm drones receive the emitted pulses at different times. The ground control station derives the time difference between the received signals based on the PRI. For accurate location estimation of the signal source using time difference of arrival (TDOA), a plane perpendicular to the vector connecting the estimated location coordinates of the signal source and center of gravity of the swarm drone is determined. The swarm drones are orthogonally projected on the determined plane, and a reference node that can reduce the DOP is selected by dividing the plane. Then, the location of the TDOA-based signal source is estimated by selecting a set of nodes that forms the minimum DOP with the reference node. The receiving nodes with the minimum DOP were selected among the deployed swarm drones, and it was confirmed using MATLAB simulation to effectively estimate the location of the signal source.

Electromagnetic, acoustic, or seismic sources can be positioned using direction, time, frequency or time/frequency difference of their transmitted signals. Source coordinates are found using time difference of arrival (TDOA) method that depends on the time differences of spatially separated receivers. In these methods, two different TDOA sets are used, including independent/spherical and full. In this study, positioning with particle swarm optimization (PSO) is achieved using independent set and full sets and positioning error is compared with Cramer-Rao lower bound (CRLB). In addition, it is proven that CRLB can be exceeded using the PSO algorithm that uses the full TDOA set.

This paper presents an automatic sound source localization approach based on a combination of the basic time delay estimation sub-methods namely, Time Difference of Arrival (TDOA), and Steered Response Power (SRP) methods. The TDOA method is a fast but vulnerable approach for finding the sound source location in long distances and reverberant environments and is so sensitive in noisy situations. On the other hand, the conventional SRP method is time consuming, but a successful approach to accurately find sound source location in noisy and reverberant environments. Also, another SRP-based method, SRP Phase Transform (SRP-PHAT), has been suggested for the better noise robustness and more accuracy of sound source localization. In this paper, based on the combination of TDOA and SRP based methods, two approaches were proposed for sound source localization. In the first proposed approach called Classical TDOA-SRP, the TDOA method is used to find the approximate sound source direction and then SRP based methods were used to find the accurate location of sound source in the Field of View (FOV) which is obtained by the TDOA method. In the second proposed approach which called Optimal TDOA-SRP, for more reduction of computational processing time of SRP-based methods and better noise robustness, a new criterion has been proposed for finding the effective FOV which is obtained through the TDOA method. Experiments were carried out under different conditions confirming the validity of the purposed approaches.

Receiver location uncertainty plays an important role in passive source localization since it can significantly deteriorate the accuracy of the source location. In this paper, we first derive a joint position estimation model for source location and receiver positions with erroneous receiver positions, and then, based on this model, an efficient algorithm for estimation the location of source and the positions of receivers using Time Difference of Arrival (TDOA) measurements is proposed. The proposed method employs modified Genetic Algorithm (MGA) and jointly estimates both source and receiver positions simultaneously. Simulation results show that when receiver position or TDOA measurement errors are large, the proposed method can achieve better performance than the modified Taylor method and the original Taylor method. It is more suitable for adverse measurement conditions.

We investigate the problem of sensor and source joint localization using time-difference of arrivals (TDOAs) of an ad-hoc array. A major challenge is that the TDOAs contain unknown time offsets between asynchronous sensors. To address this problem, we propose a low-rank approximation method that does not need any prior knowledge of sensor and source locations or timing information. At first, we construct a pseudo time of arrival (TOA) matrix by introducing two sets of unknown timing parameters (source onset times and device capture times) into the current TDOA matrix. Then we propose a Gauss-Newton low-rank approximation algorithm to jointly identify the two sets of unknown timing parameters, exploiting the low-rank property embedded in the pseudo TOA matrix. We derive the boundaries of the timing parameters to reduce the initialization space and employ a multi-initialization scheme. Finally, we use the estimated timing parameters to correct the pseudo TOA matrix, which is further applied to sensor and source localization. Experimental results show that the proposed approach outperforms state-of-the-art algorithms.

Efficient semidefinite solutions for TDOA-based source localization under unknown PS

In this paper, we address the motion parameter estimation problem using time difference of arrival (TDOA) measurements, where a mobile source starts from an initial position with constant velocity. The problem is formulated as three non-convex constrained weighted least squares (CWLS) problems. Owing to their non-convex nature, local convergence may occur when solving them by an iterative algorithm, implying good estimates are not guaranteed. We propose to solve these CWLS problems by applying semidefinite relaxation (SDR) to generate three different semidefinite programming (SDP) problems, namely, the motion unconstrained semidefinite programming (MU-SDP), motion constrained semidefinite programming (MC-SDP), and motion parameter direct semidefinite programming (MPD-SDP). The MU-SDP, MC-SDP, and MPD-SDP methods are then extended to the scenario of unknown propagation speed (PS), which is common in underwater and underground localizations. Specifically, for this scenario, the two-step MU-SDP, two-step MC-SDP, and two-step MPD-SDP methods are designed to keep the relaxed SDP problems tight by introducing the penalty function. Simulation results show that MU-SDP performs worse than MC-SDP and MPD-SDP owing to the ignorance of the motion equation, and MC-SDP and MPD-SDP are able to reach the Cramér-Rao lower bound (CRLB) accuracy.

Source localisation using time difference of arrival (TDOA) measurements has drawn considerable attention in the past few decades. The presence of outliers in TDOAs could deteriorate the localisation performance significantly. Under the reasonable assumption that outliers are sparse and they do not dominate in the raw measurement set, a computationally efficient outlier‐robust TDOA localisation method is proposed in this study. It integrates the half‐quadratic minimisation and reweighted least absolute shrinkage and selection operator to iteratively identify the outliers and find the source location estimate using the TDOA inliers only. Both additive and multiplicative forms of the proposed method are established. Simulation results demonstrate the computational efficiency and effectiveness of the developed algorithms.