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TDOA versus ATDOA for wide area multilateration system
 Jacek Stefanski^{1}Email authorView ORCID ID profile and
 Jaroslaw Sadowski^{1}
https://doi.org/10.1186/s1363801811915
© The Author(s). 2018
 Received: 27 July 2017
 Accepted: 26 June 2018
 Published: 16 July 2018
Abstract
This paper outlines a new method of a location service (LCS) in the asynchronous wireless networks (AWNs) where the nodes (base stations) operate asynchronously in relation to one another. This method, called asynchronous time difference of arrival (ATDOA), enables the calculation of the position of the mobile object (MO) through the measurements taken by a set of nonsynchronized fixed nodes and is based on the measurement of the virtual distance difference between the reference nodes and the several MO positions (more than two), as well as on the solution of a nonlinear system of equations. The novelty of the proposed solution is using the measurements taken by at least five ground sensors without time synchronization between them to estimate the position of the tracked MO transmitting four or more sounding signals in random time.
The new method significantly simplifies the localization process in reallife AWNs. It can be used on its own or to complement the traditional synchronous method. The paper focuses on the description of the proposed ATDOA method, two algorithms TSLS (Taylor series leastsquares) and GA (genetic algorithm) for solving the nonlinear system of equations, example application of the new method for a threedimensional space, and presentation of the simulation models and simulation results. An important part of the paper is the comparison of the efficiency between the asynchronous method and the synchronous one for wide area multilateration (WAM) system. In addition, the CramérRao lower bound (CRLB) is derived for this problem as a benchmark. The preliminary measurement results obtained by applying the proposed ATDOA method against the background of the synchronous one are presented at the end of the paper. As it could be expected, the synchronous solution gives better results. The synchronous method allows to locate the aircraft within 15 m in about 80% of the time, while the ATDOA method in 74% of the time for the base stations clocked from the reference clocks with the stability equal to 10^{−9}, and in 58% of the time for the base stations clocked from the reference clocks with the stability equal to 10^{−8}. The new method therefore should not be treated as the improvement of the existing synchronous positioning systems but as a backup solution which allows to keep the LCS systems running even during ground stations synchronization failure.
Keywords
 Asynchronous mode
 Radio navigation
 Wireless sensor networks
 TDOA
 ATDOA
 WAM
1 Introduction
Radio positioning can be defined as a method of determining the coordinates of a radio device (object) using the properties of radio waves. Various methods have been developed over the years, including the measurements of angle of arrival (AOA), time of arrival (TOA), time difference of arrival (TDOA), and received signal strength (RSS).
The architecture of positioning systems is based on fixed nodes (base stations) and mobile objects (mobile terminals), the location of which is required [1]. In a wireless network, it is often interesting to determine the position of an object by its emission. In that case, the wireless network carries out the measurements and makes position calculations (networkbased positioning) [2]. A critical aspect of a networkbased positioning system is precise synchronization of the fixed nodes between one another. Synchronization systems in wireless networks are rather expensive and complicated in their architectures. Moreover, bad synchronization leads to significant errors in the positioning of objects. Therefore, the paper presents a comparison of two methods: a synchronous TDOA and an asynchronous one, where the nodes operate asynchronously in relation to one another.
The proposed asynchronous method [3], which was called asynchronous time difference of arrival (ATDOA), is based on the measurement of the time difference of arrival between the mobile object (MO) and the same set of fixed nodes at different times and on the solution of a nonlinear system of equations.
Several research groups have been working to develop asynchronous localization systems. In [4], the location system consists of distributed and autonomous sensors at some fixed and known position. The position of the object which emits some designed and known signal was estimated in that system. Sensors process the received signal (pulses) independently and send the observation results to a master station to estimate the position of that object. The master station knows the expected interval between the successive pulses and considers only pairs of pulses received from each sensor. Vaghefi [5] described asynchronous wireless source localization using TOA measurements where the source transmit time is unknown. The TOA measurements have a positive bias due to the synchronization error which could lead to a large localization error. This work presents asynchronous TOAbased source localization using a semidefinite programming (SDP) technique. The SDP is a form of convex optimization which, unlike the nonconvex maximum likelihood estimator, does not have convergence problems [6, 7]. We can find another approach in [8]. The asynchronous TDOA used time difference of arrival from a set of base stations and the interval of radar scanning between the master station and slave stations to determine the location of the target. In that system, one master station and three slave stations constituted a passive surveillance system. In turn, [9] described several referencefree localization estimators based on the TOA measurements for a scenario where the anchor nodes are synchronized and the clock of the target node runs freely. The systems described in [10–15] are a different group of solutions. All these systems rely on a twoway transmission and/or require additional reference (special) node. The ATDOA proposed in this article is a totally passive method, i.e., the transmission takes place only in one direction from the MO to fixed nodes, and all fixed nodes in the wireless network are identical.
The novelty of the paper is that the process of the asynchronous location of a moving object is based on measuring the virtual distance difference between the reference nodes and the several MO positions using four or more sounding signals transmitted by the MO in random time.
This paper is organized as follows: Section 2 describes the ATDOA method, and the next two sections present an algorithm for calculating the position of the mobile object and the simulation results respectively. Section 5 outlines the application of the ATDOA method for the WAM system together with a comparison between the synchronous and asynchronous method. Section 6 derives the CramérRao lower bound (CRLB) for this problem as a benchmark, while Section 7 presents the preliminary measurement results of aircraft position estimation obtained by using the proposed ATDOA method. Finally, the last section concludes the paper.
2 Description of the proposed ATDOA method

The coordinates of fixed nodes S_{i},

The virtual distance differences between S_{i} and the MO (D_{i,k}) at the observation time k and k + 1 which are measured by the fixed nodes,

The coordinates of the tracked object MO (x_{l}, y_{l}, z_{l}) at the observation time l = 1, …, M,

The repetition time of the radio impulses which are transmitted by the mobile object (Δt_{k}).
Each node in Fig. 1 transmits the results of the measurements of the time differences Δt_{i,k} to the computing unit (CU). The transmission between the nodes and the CU which can be based on the wired or wireless link is asynchronous. The computing unit does not make any measurements but uses the results of the measurements taken by the ground sensors; therefore, the data transmission delay between the ground nodes and CU is negligible. The CU can estimate the positions of the mobile object at the observation time l, because the results of the measurements received from the fixed nodes have an MO identifier and are numbered.
In summary, the proposed method leads to establishing the coordinates of the mobile object (x_{l}, y_{l}, z_{l}) and indirectly the repetition times (Δt_{k}). In order to achieve so, assuming that N = 5 and M = 4, one must solve a system of Eq. (1) with 15 unknowns (12 coordinates of the MO in a threedimensional space in 4 consequent measurements and 3 repetition times). Of course, in a twodimensional space (N = 4 and M = 3), four nodes and three observation times are enough and (1) has only eight unknowns. The final part of this section emphasizes the difference between the proposed method and the solution described in [3]. The method presented in [3] requires more measurement nodes than the solution proposed in this paper. The process of asynchronous location of a moving object in the method [3] is based on measuring the virtual distance difference between the reference nodes and the MO in only two distinct positions. In the method proposed herein, these measurements are taken between the reference nodes and several positions of the MO (more than two). By using the measurements obtained from several positions of the MO, we can reduce the minimum number of the required fixed stations in the 3D case to five. Furthermore, in the proposed method, the pulse repetition time may be variable and even unknown, which means that the Δt_{1} does not have to be equal to Δt_{2}, the Δt_{2} does not have to be equal to Δt_{3}, etc.
3 Calculating the position of the mobile object
Solving the abovementioned nonlinear equations is difficult. Classical methods, such as those proposed in [16], do not lead to a correct solution. This paper describes two methods of obtaining a solution of the nonlinear system of equations: an iterative algorithm based on the Taylor series leastsquares (TSLS) and a genetic algorithm (GA).
3.1 Taylor series leastsquares algorithm for calculating the position of the MO
D_{i,k}represents the measured values of the virtual distance differences of arrival between each node and the mobile object during the collection time. The values r_{i,k} and r_{i,k + 1} are computed from (2) with x = x_{0}. In the next iteration, x is then set to x_{0} − Δx. The whole process is repeated again until Δx is sufficiently small.
3.2 Genetic algorithm for calculating the position of the MO
Genetic algorithms are adaptive heuristic search algorithms based on the mechanics of a natural selection and genetics. The concept of GAs is designed to simulate processes in nature that are necessary for evolution. They loosely reflect the phenomena related to chromosomes, genes, and the evolutionary passing of a genetic material from one generation to another. The GAs do not always converge to the true minimum in a search problem. The strength of GAs is that they mostly converge rapidly to a nearoptimal solution [3, 20]. The proposed genetic algorithm, which has been implemented, is as follows:
The results of numerical calculations are presented next. These calculations were carried out in order to estimate the effectiveness of solving nonlinear equations using the above methods for the asynchronous network.
4 Asynchronous multilateration system for aircraft position tracking
The proposed asynchronous method of position estimation using freerunning sensors may be useful to create instances of ubiquitous positioning technology [21]. It can be applied physically to find the coordinates of mobile nodes in radio and nonradio applications. This chapter shows an example application of the ATDOA method in a threedimensional test environment.
Currently, there are many systems available for aircraft navigation and position estimation. One of them is a wide area multilateration (WAM) system. The WAM is a technology for determining the position of an emitter (e.g., an aircraft transponder) by measuring the time difference of arrival of a signal between several known observation points [22]. In general, the majority of the WAM systems in the world utilize signals from the onboard transponders of the secondary surveillance radar (SSR) system. The SSR consists of a ground component (radar) and an airborne component (transponder) on board in an aircraft. The radar emits a signal (at 1030 MHz) which triggers a response from the airborne transponder (at 1090 MHz). A critical aspect of a working WAM system is precise synchronization of the ground stations with each other [22]. The asynchronous system presented in [23] would require the cooperation of the SSR radar to trigger emission from onboard transponders in a defined interval. The ATDOA method proposed in this paper is completely passive. It is based on the reception of broadcast signals from an airborne transponder. The time between transmissions of signals from the aircraft should be assumed as unknown and variable (irregular). Therefore, various sources of radio signals from onboard transmitters may be used for position estimation using the asynchronous multilateration system (MLAT), such as SSR, an automatic dependent surveillancebroadcast (ADSB), or even distance measuring equipment (DME) pulses.
The simulation results for the WAM system are presented in the next section.
5 Simulation results

The result of the computation is a complex number,

The result does not lie in the area of interest.
Comparison of ME, MSE, and RMSE errors in the case under consideration. The ε represents the ε_{ATDOA} or ε_{TDOA} in the adopted simulation model
Type of error  ε = 1 m  ε = 10 m  ε = 100 m  

ATDOA  TDOA  ATDOA  TDOA  ATDOA  TDOA  
IA  GA  IA  GA  IA  GA  IA  GA  IA  GA  IA  GA  
MEx [m]  8.08  − 1.22  − 0.26  − 1.55  147.45  0.94  6.28  − 15.49  –  − 18.12  376.20  − 11.72 
MEy [m]  11.17  1.42  − 0.05  − 0.63  1.14  1.14  2.20  − 7.20  –  10.72  128.81  − 10.15 
MEz [m]  13.37  1.63  − 0.37  − 2.34  2.67  2.67  9.83  − 24.04  –  − 0.48  547.48  − 23.20 
MSEx [m^{2}]  1.60e+3  182.15  209.13  177.90  509.63  509.63  2.09e+4  4.86e+3  –  7.77e+3  2.50e+7  1.44e+4 
MSEy [m^{2}]  3.62e+3  112.92  31.94  26.60  195.05  195.05  3.21e+3  829.71  –  3.94e+3  4.00e+6  7.34e+3 
MSEz [m^{2}]  5.06e+3  87.89  443.68  380.50  402.59  402.59  4.44e+4  1.07e+4  –  477.94  5.21e+7  2.80e+4 
MSE [m^{2}]  1.15e+4  382.96  684.75  585.00  1.11e+3  1.11e+3  6.85e+4  1.64e+4  –  1.22e+4  8.11e+7  4.98e+4 
RMSEx [m]  39.99  13.50  14.46  13.33  22.57  22.57  144.67  69.72  –  88.15  5.00e+3  120.19 
RMSEy [m]  60.15  10.63  5.65  5.16  13.97  13.97  56.62  28.80  –  62.76  2.00e+3  85.67 
RMSEz [m]  71.14  9.37  21.06  19.51  20.06  20.06  210.73  103.33  –  21.86  7.21e+3  167.25 
RMSE [m]  107.10  19.57  26.17  24.19  33.28  33.28  261.81  127.94  –  110.40  9.01e+3  223.06 
In turn, the genetic algorithm always leads to a solution, even if it is not correct (e.g., due to convergence to some local minima of f(x_{0}) instead of a global one). The research results of the test conducted in the threedimensional simulation environment presented here prove that the proposed method is an effective alternative to synchronous solutions.
The accepted values of position estimation errors for WAM systems are not frequently discussed in the literature. However, in [26, 27], the accuracy of 0.1 NM is mentioned as satisfactory for the aircraft surveillance systems with 3 and 5 NM separation between planes, so one can assume that the values of the position estimation error reaching the size of the aircraft (tens of meters) is acceptable in most applications. Hence, the comments about the obtained results are related to this assumption.
To sum up, the proposed method gives quite good results and significantly simplifies the localization process in the multilateration system. In addition, the ATDOA method does not need the round trip time (RTT) measurement, as compared with the solution in [23, 28]. Moreover, the proposed method can be applied to track unmanned aerial vehicles (UAV) [29], commonly known as drones. Tracked drones should only transmit control signals which, to hide their identities, do not have to be repeated at a constant frequency or contain any repetitive identifier. On the other hand, the development of the asynchronous method is not meant to compete with the synchronous one but to ensure the continuity of the location service (LCS) services in the WAM system. Therefore, the new method should be treated as a backup solution which allows access to the LCS service during a failure of the synchronous system.
6 CramérRao lower bound
7 ATDOA measurement experiment
8 Conclusions
The paper presents a new position estimation method called ATDOA, based on the virtual distance differences between the reference nodes and several positions of the mobile object (more than two), which is dedicated to asynchronous wireless networks, especially for the application in the WAM system.
In order to estimate the position of objects, the new method requires at least one more reference node than the synchronous solutions. Simulation studies of the new method in the 3D test environments were conducted providing satisfactory results every time. For example, the aircraft is located within 100 m in about 96% of the time by using the ATDOA method for the genetic algorithm and for the distance measurement error ε = 10 m.
This method can be used on its own or to complement the typical position estimation algorithms in synchronous systems in the case of the node synchronization failure. The disadvantage of this method resulting from the iterative algorithm or the genetic algorithm is the fact that it does not always lead to the best solution due to the possibility of convergence to the incorrect local minimum of the target function. However, studies show that it happens relatively rarely and can be detected by, e.g., tracking the position estimates by using the consecutive measurements.
Declarations
Acknowledgements
The authors would like to thank AVIONIX ENGINEERING sp. z o. o. for providing the measurement results.
Funding
This work is supported by the Polish National Centre for Research and Development under grant number POIR.04.01.04000032/16.
Availability of data and materials
The measurement data, which were used during our investigations, are the property of AVIONIX ENGINEERING sp. z o. o.
Authors’ contributions
Both authors contributed to this work, and they jointly proposed a new localization method. JSt performed the experiments, while JSt and JSa analyzed the results. Both authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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Authors’ Affiliations
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