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Correction to: TDOA versus ATDOA for wide area multilateration system
EURASIP Journal on Wireless Communications and Networking volume 2020, Article number: 43 (2020)
Abstract
The paper (Stefanski and Sadowski, EURASIP J. Wirel. Commun. Netw. 2018, Article 179) introduces a multilateration algorithm for unsynchronized sensor networks. However, a very similar method has been proposed before that is not cited. Furthermore, in the measurement model of Stefanski and Sadowski (EURASIP J. Wirel. Commun. Netw. 2018, Article 179), an incorrect covariance matrix (Eq. (11) in Stefanski and Sadowski (EURASIP J. Wirel. Commun. Netw. 2018, Article 179)) has been used that leads to inferior results. We summarize the context and explain the measurement methodology proposed in Stefanski and Sadowski (EURASIP J. Wirel. Commun. Netw. 2018, Article 179), while referring to the missing citation. Finally, we derive the correct covariance matrix of the measurement error and demonstrate that the covariance matrix proposed in Stefanski and Sadowski (EURASIP J. Wirel. Commun. Netw. 2018, Article 179) is incorrect.
Context
Multilateration algorithms estimate the location of a target. This requires the target to actively emit electromagnetic or acoustic signals that are then received by multiple sensors on the ground. Based on the time of arrival (TOA) of the target’s transmission at each sensor, the target can be located. Because of the high propagation speed of electromagnetic signals in particular, the clocks of all sensors must be accurately synchronized, which can be quite challenging. According to the speed of light, electromagnetic signals travel around 30 cm in 1 ns. Depending on the specific requirements for localization accuracy, the clock offsets therefore must be in the order of 0.1 μs, 10 ns, or even 1 ns. In general, networks with rather low absolute accuracy requirements span larger areas on the other hand, which keeps synchronization difficult.
Key idea and prior work
The paper [1] presents a multilateration method for asynchronous sensor networks. The basic idea is to always use differences of TOA locally on one sensor, so the sensor’s individual clock offset is canceled. This principle has first been proposed in [2], where always two transmissions were considered in one measurement equation. In a followup publication [3], that method was extended to combine an entire window of transmissions into one measurement equation. Target transmission intervals are assumed to be known, but this is just a minor issue as the respective variables can simply be treated as additional unknowns. Otherwise, [3] is in parts identical to [1] and should be cited as an important reference.
Stefanski [4] also published his approach for only two transmissions first. As an extension to that, the paper [1] (under discussion here) now can utilize more than two TOA in one measurement equation. For this purpose, multiple local differences of times of arrival (LDOTA) are calculated by subtracting all pairs of successive TOA. As correctly noted, this leads to a dependency of the error in the individual lines in the measurement equation. However, unfortunately a wrong covariance matrix is stated for these dependencies (Eq. (11) in [1]).
Methods
In order to derive the correct covariance matrix, we first define a distinct notation for all occurring variables, i.e., positions, distances, points in time, and time differences, which is consistent with the notation in [1]. Then we derive the measurement equation that connects a sequence of target positions to LDOTA measurements. These measurements can be obtained by any number of sensors, whereby the clocks of these sensors do not need to be synchronized. Finally, using the definitions of expected value and covariance, we derive the correct covariance matrix for said measurement equation. It states how the individual measurements are correlated. Its inverse should be used as a weighting matrix when solving the system of equations for the target positions.
Notation
We consider a single moving target at different time steps \(t_{k}^{\text {MO}}\), k∈{1,2,…,M}. The respective target positions are denoted by \(\left [ x_{k},y_{k},z_{k} \right ]^{\top } \!\in \mathbb {R}^{3}\). The target emits messages at the time steps \(t_{k}^{\text {MO}}\) that propagate with uniform propagation speed ν. Target transmission time intervals \( \Delta t_{k} = t_{k+1}^{\text {MO}}  t_{k}^{\text {MO}} \) do not need to be known beforehand. Therefore, the target’s clock does not need to be synchronized with any other clock, and transmissions can be periodic or aperiodic.
A number of sensors is placed at known locations [X_{i},Y_{i},Z_{i}]^{⊤}, i∈{1,2,…,N}. After the signal propagation time t_{i,k}, the transmission reaches the sensors. Each sensor records the TOA \(t_{k}^{\mathrm {S},i}\) of the transmission,
in terms of its own local clock. It is important to note that the N receivers are not timesynchronized with each other and also not synchronized with the target.
The goal is to determine the target position [x_{k},y_{k},z_{k}]^{⊤} at time steps k, based on the LDOTA \(\Delta t_{i,k}=t_{k+1}^{\mathrm {S},i} t_{k}^{\mathrm {S},i}\) measured by the N receivers.
Measurement equation
First, we state how the measured LDOTA Δt_{i,k} is obtained from the target transmission interval Δt_{k} and the propagation times of the two transmissions t_{i,k} and t_{i,k+1},
see also Fig. 2 in [1].
The measured LDOTA Δt_{i,k} can be multiplied with the propagation speed ν and then interpreted as virtual distance difference D_{i,k}
Furthermore, the signal propagation times t_{i,k} and t_{i,k+1} by multiplication with ν become the respective Euclidean distances between target and sensor, and their difference then represents the distance difference of arrival d_{i,k}
resulting in
Suppose we have N=5 sensors and M=4 successive transmissions in 3D space (dimension D=3), then there are
unknowns from the M successive target positions and M−1 target transmission intervals. On the other hand, there are also
measurements. We can write (1) in short form as
where the d_{p} correspond to noisy LDOTA measurements Δt_{i,k} multiplied by ν, i.e., a noisy measurement of D_{i,k}, and q_{p} is an additive random error. Looking at this in the context of state estimation, the function f_{p}(x) is the nonlinear measurement function
that maps the unknowns x
i.e., target positions and transmission time intervals, to the LDOTA measurements d_{p}.
Covariance
We will take a closer look into that additive random error q_{p} now. The true expected measurement without any error would be the LDOTA Δt_{i,k}, i.e., the difference of the recorded TOA \(t_{k}^{\mathrm {S},i}\) and \(t_{k+1}^{\mathrm {S},i}\)
In contrast, in the actual realworld LDOTA measurement d_{p}, the underlying realworld TOA measurements \(t_{k}^{\mathrm {S},i}\) and \(t_{k+1}^{\mathrm {S},i}\) have additional random errors \(q_{k}^{\mathrm {S},i}\)
The system of equations from (2) can be written in vector form as
The measurement error q is assumed to be a multivariate random vector with covariance matrix
We assume the additional random errors \(q_{k}^{\mathrm {S},i}\) (and therefore also q_{p}) to have a mean of 0,
Now, we will derive the elements of Q,
Assuming independent and identically distributed errors
we finally get the covariance matrix elements
with the Kronecker delta function
Results
According to (4), the covariance matrix Q is blockwise diagonal as measurements from different sensors \(i_{p_{1}}, i_{p_{2}}\) are uncorrelated. Furthermore, the local covariance matrix Q^{i} of one sensor has the form
Equation (11) in [1] states a different and, hence, incorrect covariance matrix of the form
It would be the correct sensorlocal covariance matrix for a different type of LDOTA, namely, taking time differences to one fixed reference TOA, for example, the earliest one,
instead of the LDOTA between successive TOA (3)
that was also proposed in [1], see [1, Eq. (3)].
Using an incorrect covariance matrix for the solution of the maximum likelihood problem can easily remain undetected, as it does not always cause completely wrong results, but it does lead to inferior results compared to what could have been achieved with the correct covariance matrix. A more detailed comparison between the two LDOTA types and the error caused by incorrect covariance matrices can be found in [5].
Availability of data and materials
Not applicable.
Abbreviations
 LDOTA:

Local differences of times of arrival
 TOA:

Time of arrival
References
 1
J. Stefanski, J. Sadowski, TDOA versus ATDOA for wide area multilateration System. EURASIP J. Wirel. Commun. Netw. (2018). Article 179. https://doi.org/10.1186/s1363801811915.
 2
T. Li, A. Ekpenyong, Y. F. Huang, in IEEE INFOCOM 2004, vol. 1. A location system using asynchronous distributed sensors, (2004), p. 628. https://doi.org/10.1109/INFCOM.2004.1354533.
 3
T. Li, A. Ekpenyong, Y. F. Huang, Source localization and tracking using distributed asynchronous sensors. IEEE Trans. Signal Proc.54(10), 3991–4003 (2006). https://doi.org/10.1109/TSP.2006.880213.
 4
J. Stefanski, Asynchronous time difference of arrival (ATDOA) method. Pervasive Mob. Comput.23:, 80–88 (2015). https://doi.org/10.1016/j.pmcj.2014.10.008.
 5
D. Frisch, U. D. Hanebeck, in Proceedings of the 22nd International Conference on Information Fusion (Fusion 2019). ROTA: Round Trip Times of Arrival for Localization with Unsynchronized Receivers (International Society of Information FusionOttawa, Canada, 2019).
Acknowledgements
We acknowledge support by the KITPublication Fund of the Karlsruhe Institute of Technology.
Funding
Daniel Frisch is funded as a research assistant at the Chair for Intelligent SensorActuatorSystems (ISAS) at the Karlsruhe Institute of Technology.
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DF and UDH contributed equally to the paper. Both authors read and approved the final manuscript.
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Correspondence to Daniel Frisch.
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Frisch, D., Hanebeck, U.D. Correction to: TDOA versus ATDOA for wide area multilateration system. J Wireless Com Network 2020, 43 (2020). https://doi.org/10.1186/s1363802016561
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Keywords
 Correction
 Asynchronous
 Multilateration
 TOA
 TDOA
 ATDOA
 WAM