Source localization using TDOA and FDOA measurements based on semidefinite programming and reformulation linearization

https://doi.org/10.1016/j.jfranklin.2019.10.029Get rights and content

Abstract

The problem of source localization using time-difference-of-arrival (TDOA) and frequency-difference-of-arrival (FDOA) measurements has been widely studied. It is commonly formulated as a weighted least squares (WLS) problem with quadratic equality constraints. Due to the nonconvex nature of this formulation, it is difficult to produce a global solution. To tackle this issue, semidefinite programming (SDP) is utilized to convert the WLS problem to a convex optimization problem. However, the SDP-based methods will suffer obvious performance degradation when the noise level is high. In this paper, we devise a new localization solution using the SDP together with reformulation-linearization technique (RLT). Specifically, we firstly apply the RLT strategy to convert the WLS problem to a convex problem, and then add the SDP constraint to tighten the feasible region of the resultant formulation. Moreover, this solution is also extended for cases when there are sensor position and velocity errors. Numerical results show that our solution has significant accuracy advantages over the existing localization schemes at high noise levels.

Introduction

The problem of source localization has attracted considerable interest owing to its importance in many applications like radar, sonar, surveillance, target tracking, wireless communications and sensor networks [1], [2], [3], [4]. Source localization can be performed using time-difference-of-arrival (TDOA), frequency-difference-of-arrival (FDOA), or angle-of-arrival (AOA). The problem of source localization using TDOA has been intensively discussed [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21] due to its high accuracy and the unnecessity of time synchronization with the source. But in modern localization systems, the receivers are usually mounted on moving platforms like vehicles or aerocrafts. In this scenario, there exists relative moving between the source and receivers, and the FDOA measurements resulted from the Doppler effect can also be utilized to enhance the localization precision.

The problem of positioning one moving source using TDOA and FDOA is challenging owing to its high nonconvexity and nonlinearity. The maximum likelihood (ML) estimator [22] is considered as an optimal solution to this problem and can asymptotically reach the Cramér-Rao lower bound (CRLB). However, this estimator needs to execute multi-dimensional searching in the whole solution space, and thus it involves expensive computation. An possible alternative is to linearize the nonlinear equations via Taylor-series expansion [23]. However, this method requires a proper initial estimate, and it cannot ensure to converge to the global optimum solution. To overcome this defect, several closed-form solutions have been developed. The two-step WLS (TSWLS) method [24] is the most well-known one, that introduces two auxiliary variables to convert the nonlinear TDOA/FDOA equations into linear equations and solved them in two successive steps. Extensions to this approach have been presented in [25], [26], [27], [28]. Specifically, the uncertainty in sensor positions and velocities are addressed in [25], while the localization of multiple moving sources is discussed in [26]. To achieve improved localization performance, the technique using calibration sensors is proposed in [27] and an improvement on the second stage of the TSWLS solution is suggested in [28]. The TSWLS methods have closed-form solutions and are computationally attractive, but they suffer significant nonlinear threshold effect and cannot attain good performance when the noise level is high.

Besides the TSWLS methods, constrained WLS (CWLS) methods are also developed by exploiting the relationship between the position and velocity of source [29], [30], [31]. The CWLS method [29] imposes two second-order equality constraints to the WLS problem and solves the problem by introducing Lagrange multipliers, but its main disadvantage is that it needs to solve high-order polynomial equations to determine Lagrange multipliers. To tackle this problem, Yu et al. [30] proposed a computationally more attractive technique based on Newton’s method, where the computation of Lagrange multipliers is avoided. In [31], an iterative CWLS approach is presented, in which the quadratic constraints are recursively approximated by linear equality constraints and a closed-form solution is obtained in every iteration. However, a proper initial guess and iteration process are required and its convergence to global optimum solution cannot be guaranteed. Moreover, Wei et al. [32] developed a new positioning framework by using the multidimensional scaling (MDS) analysis. The MDS method can provide higher localization accuracy than the TSWLS approach at high noise levels, and it is insensitive to big measurement errors. In [33], a hybrid pseudo linear estimator (PLE) is presented by adding the AOA measurements. The hybrid PLE is not only closed-form but also free from nuisance parameters. The above-mentioned methods are valuable because they do not suffer the divergence problem and intensive computation. Furthermore, they can achieve nearly optimal performance at low noise levels.

Inspired by the convex optimization methods to solve quadratically constrained quadratic program (QCQP) problems, semidefinite programming (SDP) has been recently applied in moving source localization [34], [35], [36], [37]. The method in [34] reformulates this geolocation problem as the ML estimation, and use efficient semidefinite relaxations to solve the nonconvex problem. Wang et al. [35] proposed another SDP-based method for source localization, which firstly approximates the ML problem as a WLS problem, and then solve this problem via SDP relaxation. In [36], the robust LS criterion is exploited to reformulate the localization problem. This method does not require any initial estimate and achieves improved robustness than other SDP-based methods. In [37], Zou et al. proposed an iterative algorithm which utilizes the solution of the SDP problem as initial estimates to enhance the localization precision. This iterative method updates the velocity by the WLS method and updates the position by the SDP method. The major advantages of the SDP methods are that they outperform other existing closed-form solutions in localization accuracy significantly, especially for high noise levels.

It is generally known that the reformulation-linearization technique (RLT) [38] is an effective technique to solve the QCQP problem besides the SDP relaxation. In addition, it has been proved in [39] that the use of SDP and RLT constraints together can yield bounds that are considerably better than when either of them is applied alone.

Motivated by this idea, in this paper, we devise a new localization scheme using SDP and RLT relaxations. In this scheme, the ML localization problem is firstly transformed into an approximate WLS problem, and then the SDP and RLT constraints of the localization problem are created and are together utilized to formulate a convex estimation problem that can be efficiently solved using interior-point methods. Furthermore, our scheme is also extended for cases with sensor position and velocity errors. Numerical examples are provided to verify the superiority of the proposed scheme over various existing approaches.

The rest of this paper is organized as follows. Section 2 introduces the measurement models and problem formulation. Section 3 describes the localization scheme based on SDP and RLT relaxations. Tightness analysis on relaxation algorithms is provided in Section 4, and numerical simulations are exhibited in Section 5. Finally, conclusions are given in Section 6.

Notations: ( · )T and (·)1 represent the transpose and inverse, respectively. tr(A) denotes the trace of A, and diag(a) denotes a diagonal matrix formed by the elements of a. E{ · } is the statistical expectation and ‖ · ‖ means the Euclidean norm. a(i) denotes the ith element of a, and a(i: j) denotes a subvector formed from the ith to jth element of a. A(i, : ) denotes the ith row of A, A(i, j) denotes the (i, j)th element of A, and A(i: j, k: l) stands for a submatrix of A, formed from its ith row, kth column to its jth row, lth column. 1k and Ik represent the k × 1 all-one vector and the k × k identity matrix, respectively. 0k × l means the k × l all-zero matrix. AB stands for that AB is positive semidefinite.

Section snippets

Measurement models

Let us consider a scenario of M passive sensors collaborating with each other to locate one moving source in a three-dimensional (3-D) space. The position and velocity of the ith sensor are accurately known and denoted by si=[xi,yi,zi]T and s˙i=[x˙i,y˙i,z˙i]T (i=1,2,,M), respectively. The position and velocity of the source are unknown and denoted by u=[x0,y0,z0]T and u˙=[x˙0,y˙0,z˙0]T, respectively. In general, we choose the first sensor as the reference. If c denotes the wave propagation

Proposed localization solution

In this section, we devise a new scheme to approximately solve the ML localization problem (15). Specifically, we firstly convert the ML problem (15) to an approximate WLS problem, and then derive a new solution to the WLS problem by using jointly SDP and RLT relaxations. In addition, our approach is extended to this situation with sensor position and velocity errors.

Tightness analysis

Here, we investigate the performance of our proposed solution by tightness analysis. Specifically, we study the tightness of the SDP plus RLT problem (39) in comparison with the RLT problem (38) and the SDP problem [35]. To simplify the analysis, we consider two variables θ1 and θ2 with l1 ≤ θ1 ≤ u1, l2 ≤ θ2 ≤ u2. According to the RLT constraints Eq. (37), the feasible regions of Y11 and Y22 are given by2l1θ1l12Y11l1θ1+u1θ1l1u1,2l2θ2l22Y22l2θ2+u2θ2l2u2.

Now we impose the SDP constraint Y

Numerical simulations

Numerical simulations are executed to investigate the performance of our presented approach. It is compared to the TSWLS method [24], the SDP method [35], and the CRLB [24]. We use the same localization geometry as [24], where there are five moving sensors, and the positions and velocities are shown in Table 1. For the SDP method and our presented approach, the localization problems are solved by the MATLAB CVX toolbox [41]. For our proposed method, the upper and lower bounds of the feasible

Conclusion

In this paper, we have introduced a new source localization scheme with TDOA and FDOA measurements. The introduced scheme firstly reformulates the source localization problem as a WLS problem, and then utilizes SDP and RLT relaxations together to solve this WLS problem. In this scheme, the bounds of the source position and velocity may be set to be large enough. Therefore, our proposed scheme does not require more prior information than the traditional schemes. Since the feasible region of the

Acknowledgment

This work is supported in part by the National Natural Science Foundation of China under Grants 61571081, by the Sichuan Science and Technology Program under Grant 2019YJ0191, by the Key Project of Sichuan Education Department of China under Grant 18ZA0221, and by the Fundamental Research Funds for the Central Universities of China under Grant 2672018ZYGX2018J003.

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