Noise resilient solution and its analysis for multistatic localization using differential arrival times☆
Introduction
The problem of object localization has received a lot of attention, since it has wide applications in various fields, including wireless sensor network, radar, sonar, and many others [1], [2], [3], [4], [5]. Typical measurements used in object localization include time-of-arrival (TOA) [6], [7], [8], [9], [10], time-difference-of-arrival (TDOA) [11], [12], [13], [14], [15], angle-of-arrival (AOA) [16], [17], [18], [19], and many others [20], [21], [22]. In practical applications, the following three aspects may significantly degrade the localization performance of time-based localizations, including: 1) The positions of sensors are not fixed and time-varying, which leads to random and time-varying position errors; 2) The signal propagation speed may not be a constant. Such a situation about the propagation speed often happens in underwater signal propagation [23], [24], [25]. It also appears when the signal goes through some inhomogenous medium of unknown characteristics, examples include underground [26] positioning and in-solid localization [27]. 3) The clock synchronization between the transmitters and receivers is typically costly and may not be sufficiently accurate owing to the different internal clocks used by the transmitters and receivers, and hence, a localization method that does not require time synchronization would be desirable.
It is well-known that multistatic systems have more robust performance and flexible applications than the traditional monostatic systems [28], [29]. In the multistatic systems, each signal emitted by the transmitter will reach the receiver at a different position from two propagation paths: the direct path from the transmitter to the receiver and reflected path through the object. The measurements include time delay, bearing angle, Doppler frequency shift, and their combinations [30], [31], [32], [33], [34], [35]. In this paper, we focus on the multistatic localization problem using time delay measurements. More specifically, the differential arrival time between the signals of direct and reflected paths is used. The differential arrival time from each pair of transmitter and receiver defines an ellipse whose foci are the transmitter and receiver positions, and the object position is the intersection of the ellipses. The highly nonlinear relationship between the object position and the measurements leads to a very difficult localization problem.
The multistatic localization problem has been extensively studied in the literature. Kim et al. [30] compared the object localization performance of the monostatic, bistatic, and multistatic systems, and developed a weighted least squares (WLS) method for localization with multistatic systems. However, the transmitter or receiver position errors are not taken into account. In [31], the authors developed an algebraic closed-form solution for the moving target localization problem using time delay and Doppler shift measurements. The performance of this method may not be sufficient when the noise is large. Rui and Ho [32] developed two efficient four-step closed-form WLS estimators for the object location using differential time and AOA measurements, where the practical factors including unknown signal propagation speed are incorporated in the algorithm developments. Nevertheless, they may not have adequate performance or even fail when the noise is large and/or the sensor configuration is bad. Jia et al. [33] improved the four-step WLS method in [32] in two ways. One is to reduce the four-step method to a two-step solution and thereby decreasing the complexity. The other is to apply the generalized trust region subproblem (GTRS) method to improve the localization performance. However, the GTRS method cannot handle the multiple transmitters case, which limits its applications. Recently, Zhang and Ho [34] extended the multistatic localization problem to the more general scenario where the transmitter position is not available, and when the transmitter can be moving and the measurements have offsets [35]. To jointly estimate the unknown object and transmitter positions, the time measurements from the respective direct and reflective paths are used and time synchronizations between transmitters and receivers are required. Moreover, the signal propagation speed is assumed to be a known constant, which can restrict the method in [34] to be effective for localization in some particular environments only, e.g., the shallow water region.
In this paper, we advance the multistatic localization problem further in practical scenarios [32]. Using the differential arrival time measurements, time synchronization among the transmitters and receivers, and with the object is not needed. As a result, it can greatly improve the applicability of the proposed method in practical localization systems. Moreover, the transmitter and receiver position errors are properly handled to reduce the performance degradation that they cause. Furthermore, owing to the possibility of varying signal propagation speed in some environments, we do not assume a constant speed. Rather, we assume that the signal propagation speed is a random variable of known partial statistics or the propagation speed is an unknown parameter with no additional information available. By transforming the highly nonlinear model that relates the measurements with the unknowns, we formulate separate WLS problems by following [32] for the two cases regarding the propagation speed. The mean and covariance matrix of the transmitter and receiver position errors are included in the WLS formulation to mitigate their effect on the localziation performance. Both WLS problems are recast as non-convex constrained optimization problems, with the relationships among the variables incorporated in the constraints. To solve the difficult non-convex problems, we apply the semidefinite relaxation (SDR) technique [36, Chapters 4–6], [37] to relax them into tractable semidefinite programs (SDPs). The relaxed SDP problems are further tightened by adding a set of second-order cone (SOC) constraints. Our simulation results show that the tightened SDP problems are able to reach the optimal solutions of the original WLS problems when the noise is not considerably large, indicating that the relaxed SDPs are tight. As compared to the four-step WLS method in [32], the proposed completely different method solves the localization problem in one step, where the relations among the variables are transformed into the constraints of the SDP problem. Moreover, the proposed method has much better performance when the localization geometry becomes poor, the noise level is higher or the number of sensors/transmitters are less, as shown in the simulation results. The mean square error (MSE) analysis is provided to show that the WLS solutions are able to reach the hybrid Cramer-Rao lower bound (HCRLB) [38] accuracy when the noise level is not exceedingly high, implying that the proposed SDR methods are also able to reach the HCRLB. Furthermore, the theoretical bias expressions for the two WLS problems are derived. The expressions are used to estimate the biases, which are subtracted from the solutions to achieve the approximately unbiased estimates.
The work here uses multistatic setting with arrival time difference between the direct and reflected paths for localization. It is straightforward to extend the work to other settings such as the object emitting a signal where a transmitter is not needed and TDOA localization.
The SDR method has been studied for localization over the years and algorithms are available from the literature; see e.g., [12], [14], [19], [21], [39], [40], [41], [42]. This paper continues the research of the subject further by investigating the effect and developing the algorithm using the SDR technique when the signal propagation speed in time-based localization is not accurate or not available. We have not come across any work from the literature in addressing the uncertainty in the propagation speed using SDR for localization.
In summary, the contributions of this work include:
- 1.
We formulate two WLS problems and propose two completely new SDR methods for two localization scenarios using differential arrival time measurements, which have the ability of reaching the optimal localization performance.
- 2.
We develop two relaxed SDRs that provide higher robustness and achieve much better results than the existing methods from the literature when the localization geometry is less regular, the noise level is higher, or the number of sensors or transmitters is limited.
- 3.
We show through MSE analysis that the proposed WLS solutions are able to reach the HCRLB accuracy.
- 4.
We derive the theoretical expressions of the biases for the WLS solutions, and perform bias reduction to obtain approximately unbiased estimates.
The rest of the paper is organized as follows. Section 2 describes the localization scenario and gives the measurement model. Section 3 develops the SDR method for localization, where both the cases of known and unknown propagation speed distribution are investigated. Section 4 presents the MSE analysis and analytically validates that the WLS solutions are able to reach the HCRLB. Section 5 derives the theoretical expressions of the solution biases. Section 6 shows the performance of the proposed method by numerical simulations. Finally, Section 7 concludes the paper.
We shall follow the conventional notations that bold lower and upper case letters denote column vectors and matrices, respectively. and represent the all-zero and all-one column vectors of length . is the identity matrix. For vector , and represent the -norm and -norm of , respectively. For matrix , represents the Frobenius norm of . is the th element of and is a subvector produced by the th to the th elements of . represents a diagonal matrix formed by the elements of and is a block diagonal matrix whose diagonal blocks are to . For matrix , and are the trace and rank of , respectively. , , and denote the th element of , the subvector formed by the th to the th rows in the th column of , and the submatrix including the th to the th rows, and the th to the th columns of , respectively. means that is positive semidefinite. The operator between two matrices represents the Kronecker product and between two vectors stands for the element-by-element multiplication. represents the logarithm with base 10 of .
Section snippets
System model
We are interested in locating an object, whose unknown position is denoted by , in a -dimensional space by a multistatic localization system that consists of transmitters and sensors (also called receivers in this paper). The positions of the transmitters and receivers are and , respectively. Transmitter sends a signal and it reaches a receiver by the direct path and the reflected path in the form of an echo from the object. The receiver here is strictly
Semidefinite relaxation method
We shall apply the SDR technique [36, Chapters 4–6], [37] for the multistatic localization problem. Two scenarios, i.e., the scenarios of known partial statistics of and completely not known are separately investigated. In the following, we consider the two-dimensional (2-D) localization scenario, i.e., , and extension to the 3-D scenario is straightforward.
MSE Analysis
In this section, we apply the perturbation analysis to validate that the MSEs of the WLS solutions can reach the HCRLB [32] accuracy under the condition that , , and follow Gaussian distributions, implying that the proposed SDR methods have potential to work well.
Bias analysis
During the transformation of the original measurement model, several approximations are introduced to simplify the estimation task, such as ignoring the second-order error terms and applying the first-order Taylor expansion, which may introduce bias to the object position estimation. In this section, we first derive the bias expressions of the WLS solutions for both cases. We can then obtain a bias estimate by replacing the true value with the object position estimate in the bias expression,
Simulations
Simulations are conducted to verify the performance of the proposed SDR method. We present the results separately for the two scenarios of known statistics of and completely unknown . For performance comparison, we include the Four-Step WLS method in [32] and the GTRS method in [33]. The performance is evaluated in terms of both the MSE and bias. The proposed method without bias reduction is denoted by “SDR” and that with bias reduction is represented by “SDR-BR”. Note that the GTRS method
Conclusion and future work
In this paper, we have presented two SDR methods for two cases of the multistatic localization problem using differential arrival times, where the signal propagation speed is unavailable. The two cases are the partial statistics (first and second order) of the signal propagation speed is known, and the speed is completely not known. Two optimization problems are formulated, which are solved by employing the SDR technique. We also have provided the MSE and bias analyses to show the theoretical
CRediT authorship contribution statement
Shuli Yang: Methodology, Software, Formal analysis. Gang Wang: Conceptualization, Supervision, Writing - original draft. K.C. Ho: Writing - review & editing.
Declaration of Competing Interest
We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted.
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This work was supported in part by the Zhejiang Provincial Natural Science Funds for Distinguished Young Scholars under Grant LR20F010001 and in part the K. C. Wong Magna Fund in Ningbo University.