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Multiple sources localization based on time difference of arrival (TDOA) measurements is investigated in this paper. Different from the traditional methods, a novel and practical multisource localization algorithm is proposed by adopting a priori information of relative distance among emitting sources. Since the maximum likelihood (ML) cost function for multisource estimation is highly nonconvex, the semidefinite relaxation (SDR) is utilized to reformulate the ML cost function. A robust estimator is obtained, which can be solved by semidefinite programming (SDP). Moreover, the constrained Cramér-Rao bound is also derived as a benchmark by considering the range constraints between sources. Simulation results verify the superior performance of the proposed algorithm over the traditional methods.

Multisource localization is an essential task in radar, sonar, navigation, and other applications [

The positioning problem of using TDOA measurements is a nontrivial task due to its high nonlinearity and nonconvexity. In [

Since the convex optimization has been applied to solve localization problems, many researchers prove that this method is attractive to robustly achieve excellent estimate results and accuracy even at high noise levels [

Multisource localization is of great interest for its frequent emergence in practice. Here we address the problem in some common scenarios where emitting sources lie within a certain range or move in group, which can be referred to as group targets. The targets are typically sensors in the wireless sensor network or formation-flight aircraft and aircraft carrier fleet in the open space. Such patterns usually indicate that these group targets will stay within a certain relative distance with each other and keep the resemble velocity for most of the times. For example, aircraft in formation-flight usually keeps a distance around several meters to tens of meters between each companion in the formation [

To date, only several papers have considered the problem of localizing multiple sources [

In this paper, the number of sources to be located is assumed to be already known and the associations between the multiple signal measurements at each sensor and the corresponding sources have been accomplished. Unlike the traditional approaches mentioned above, this paper devises a novel method using SDR techniques to jointly locate the multiple sources by introducing the prior information constraints on relative distances between them. Practically, these relative distance information could be obtained through other observation methods. And it can improve the localization accuracy with such range constraints for group targets. The contributions of this paper are summarized as follows.

The rest of the paper is organized as follows. In Section

In the sequel of this paper, boldface lowercase letters represent column vectors and boldface uppercase letters denote matrices.

Consider that there are

Without loss of generality, the first sensor is chosen as the reference sensor and line-of-sight propagation condition is considered. The TDOA measurement between a sensor pair

Then collect all measurements of

Then the problem of interest is to estimate the unknown vector

In this section, the SDR techniques are employed to approximate the ML problem for multiple sources localization and incorporate the range inequalities as convex constraints. According to (

Then, (

It is proved in [

By applying the basic property of

Using the SDR principle, we relax the constraints

It is necessary to point out that, as shown in [

We consider using the relationship between elements of

Therefore, it is straightforward to obtain the following inequalities:

It is necessary to point out that the inequalities above impose constraints on the product of ranges for each source with different sensor pairs. That is, the correspondent elements that lie in the upper triangular matrix around the principle diagonal in

Then the prior information of relative distance in (

Squaring both sides of (

Note that the quadratic inequality constraint in (

Substituting

This canonical convex optimization problem can be solved by CVX Toolbox in Matlab. By solving the SDP in (

The covariance matrix

Theoretically, in terms of the effect of the range constraints introduced for group targets, the constraints may have little effect to improve the estimation performance for group targets when the measurement noise is small. This is mainly because, at small measurement noise level, the position estimation results without constraints may just slightly deviate from the true target positions; thus they will probably fall into the feasible domain confined by the range constraints and meet the constraints. However, when the measurement noise grows larger, the position estimation results without constraints might deviate from the real target positions by a large amount with a high probability, which are likely to exceed the feasible domain. In this case, the range constraints will definitely confine and reduce the deviations and significantly decrease the estimation errors. Therefore, the improvement of the estimation performance can be more significant by utilizing the range constraints at large measurement noise level. This will be illustrated by the simulation results in Section

Computational complexity is analyzed in this section. Here, we apply the result of [

Complexity comparison of different algorithms.

Algorithm | Iteration Number | Operation per Iteration |
---|---|---|

TS-WLS | 1 | |

SDP-SL | | |

SDP-CL | | |

The CRB is usually seen as a benchmark against which the statistical efficiency of any unbiased estimators can be compared. The CRBs for single source and multisource localization using TDOA measurements have been investigated in [

The probability density function (PDF) of jointly Gaussian distributed measurement noise

The corresponding Fisher information matrix (FIM) is obtained as

The inequalities in (

The constrained parameter space

It is proved in [

It can be seen that

The diagonal elements of

In this section, simulation has been conducted to evaluate the TDOA-based localization performance of the proposed SDR algorithm. The localization algorithm developed in this paper for multiple sources, denoted as “SDP-CL”, is compared with the single target SDR algorithm which is similar to [

In the simulation, six stationary sensors are employed to locate more than two targets in the sensor network. The positions of the sensors are listed in Table

Positions of sensors (m).

Sensor no. | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|

| 300 | 400 | 300 | 350 | -100 | 200 |

| 100 | 150 | 500 | 200 | -100 | -300 |

| 150 | 100 | 200 | 100 | -100 | -200 |

The positioning performance is evaluated by the root mean square errors (RMSE), defined by

In the first test, the impact of

Comparison of RMSE and measurement noise

RMSE of target 1

RMSE of target 2

In the second test, the number of targets is increased to three. Their positions are 654, 803, and 708 m, 652, 808, and 710 m, and 650, 814, and 705 m. Thus the true ranges between the targets are approximately up to 12m. Here the observed upper range bound information

Comparison of RMSE and measurement noise

RMSE of target 1

RMSE of target 2

RMSE of target 3

In the third test, the group targets are assumed to consist of three emitting sources that are farther from the sensors than that in the second test. Their positions are 1260, 1440, and 1193 m, 1256, 1450, and 1195 m, and 1251, 1456, and 1188 m. Thus the maximum relative distance among them is around 20m and the upper range bound

Comparison of RMSE and measurement noise

RMSE of target 1

RMSE of target 2

RMSE of target 3

In this paper, we propose to jointly localize multiple emitting sources by utilizing the prior relative distance information between them based on TDOA measurements. The SDR technique is applied to reformulate the original nonconvex ML problem for multisource to obtain an SDP. Moreover, the constrained CRB, which incorporates the inequality constraints of prior relative distance information, is also derived in this paper. Simulation results show that the proposed method significantly improves the localization accuracy and can achieve the corresponding CCRB when the range information is more precise. In addition, the proposed method performs well even when the measurement error is intensive.

The authors claim that the data used in this article are provided by their simulations according to some real localization scenarios, and this article is developed without using any data in a published article to support their results.

The authors declare that they have no conflicts of interest.

The authors acknowledge support from the National Natural Science Foundation of China (Grants no. 61201381, no. 61401513, and no. 61772548), China Postdoctoral Science Foundation (Grant no. 2016M592989), the Self-Topic Foundation of Information Engineering University (Grant no. 2016600701), and the Outstanding Youth Foundation of Information Engineering University (Grant no. 2016603201).