An algebraic method for moving source localization using TDOA, FDOA, and differential Doppler rate measurements with receiver location errors
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Abstract
To weaken the effect of receiver location error on localization accuracy and make the localization model closer to the practical scenario, this paper considers the receiver location errors, usually neglected in prior studies into the measurement model, and proposes an algebraic method for locating a moving source using time difference of arrival (TDOA), frequency difference of arrival (FDOA), and differential Doppler rate measurements. The proposed method is based on the pseudo-linear set of equations and two-step weighted least square estimator. Only noise values of receiver locations and three types of positioning measurements are available for processing. In addition, a new Cramér-Rao lower bound (CRLB) combining TDOA, FDOA, and differential Doppler rate in the presence of receiver location errors is also derived in this paper. Theoretical analysis and simulation results both indicate that the proposed method can attain CRLB at a moderate noise level, avoid the rank deficiency problem efficiently, and achieve a significant improvement over the existing methods.
Keywords
Moving source localization Time difference of arrival Frequency difference of arrival Differential Doppler rate Receiver location errorsAbbreviations
- 3-D
Three-dimensional
- AOA
Angle of arrival
- CRLB
Cramér-Rao lower bound
- FDOA
Frequency difference of arrival
- RMSE
Root mean square error
- RSS
Received signal strength
- SNR
Signal-to-noise ratio
- TDOA
Time difference of arrival
- TOA
Time of arrival
- TSWLS
Two-step weighted least square
- WLS
Weighted least square
1 Introduction
Emitter localization based on multiple passive moving receivers has been receiving growing attention for many years due to its important military and civilian application values [1, 2, 3, 4]. Passive localization usually includes two steps: firstly, several receivers intercept the signal emitted by a source and measure the proper positioning measurements such as received signal strength (RSS), angle of arrival (AOA), time of arrival (TOA), time difference of arrival (TDOA), and frequency difference of arrival (FDOA), and then use them to obtain the source location [5, 6, 7, 8, 9, 10, 11, 12, 13, 14]. The estimation accuracy of RSS is sensitive to the channel environment. If the practical environment exists strong scatters, the signal strength is difficult to estimate which influences on localization accuracy [6]. The TOA-and-AOA-based localization system can provide an accurate source position estimate, at the cost of requiring time synchronization between receivers and source, and complicated antenna hardware [7, 8]. For non-cooperative source localization problem, the TDOA can be utilized to improve localization accuracy. If there is relative motion between the source and receivers, the FDOA can be combined with the TDOA, which can significantly improve the source location accuracy and estimate the position and velocity of the source simultaneously [10, 11, 12, 13, 14].
As for high-dimensional and nonlinear characteristic of TDOA-and-FDOA-based localization problem, many efficient methods have been proposed. Foy put forward a linearization method by using Taylor expansion [11], but a proper initial guess is necessary, or its convergence cannot be guaranteed. To avoid it, the algebraic methods have always been compelling to researchers due to their independence of initial estimate and computational efficiency. Ho and Xu proposed the well-known two-step weighted least square (TSWLS) method [10] that gives an algebraic solution for moving source localization without initial guess. However, this method has poor localization accuracy when a source is near to any coordinate axis of the reference receiver and needs several extra mathematical operations before determining the source location in the second step. After that, the semi-definite relaxation methods are applied in a localization system [12, 13]. Essentially, these methods are also an iteration-based method and always suffer from a high computational complexity. Recently, an improved TSWLS method was presented in [14], which outperforms the original TSWLS method.
As we know, the source localization accuracy depends much on the TDOA and FDOA estimation accuracy. Hence, to improve the performance of radar detection and parameter estimation, some studies aim at the long-time coherent integration [15, 16]. Nevertheless, extending the observation time would improve FDOA estimation accuracy, which has contributed to source localization [17], but it will cause serious relative Doppler companding problem [17, 18]. To overcome it, several efforts considered the differential Doppler rate in the traditional signal model and proposed the joint estimation method, from which TDOA, FDOA, and differential Doppler rate can be obtained simultaneously [18, 19, 20]. It should be noted that this measurement can be obtained jointly with TDOA and FDOA requiring no additional data collection device, which means using the same received data with TDOA and FDOA estimation [21, 22]. Moreover, it is closely associated with the source position and velocity. Hence, in the localization system, ignoring the differential Doppler rate measurement may not be acceptable for many scenarios with high maneuvering or long observation time [20] and is utilized to determine the source location. Quite recently, Hu et al. proposed a gradient method using TDOA, FDOA, and differential Doppler rate measurements [17], but the iteration process of the second step could cause local convergence or even divergence at low signal-to-noise ratio (SNR) conditions.
Generally, the above localization methods are all based on the ideal assumption that the receiver locations are known exactly, which is not realistic for practical scenario [23, 24]. The receiver locations are inevitably contaminated by errors to some extent and are often non-negligible, especially in practical environments. Ho et al. proved that the source localization accuracy is very sensitive to the accuracy of receiver locations and developed a solution with random location errors [25], but neglects the differential Doppler rate measurement in solving step. Moreover, the second step of this method has poor robustness due to its rank deficiency problem, which is analyzed in the simulation part of this paper.
- 1.
The receiver location errors are considered to offset the weakness of the existing methods, which can produce a more reliable estimation result of source location.
- 2.
The second step of the proposed method gives a final algebraic solution, while the existing methods need extra mathematics operations, such as multiple iterations in [17] and square root operations in [25].
- 3.
The proposed method can efficiently avoid rank deficiency problem and is more robust than the TSWLS method in some special localization scenarios where source moves near to any coordinate axis of the reference receiver.
- 4.
The proposed method outperforms the existing methods with respect to localization accuracy as SNR and receiver location error change. Simulation results can support the theoretical analysis and the above contributions.
The rest of paper is organized as follows. Section 2 introduces the method and experimentally used in this paper. Section 3 formulates the problem of source localization using TDOA, FDOA, and differential Doppler rate measurements with receiver location errors. Section 4 gives a detailed derivation of the proposed method. Section 5 derives a new Cramér-Rao lower bound (CRLB) and gives a comparison between the estimation accuracy of the proposed method and CRLB. Section 6 provides numerous simulations to support our theoretical study and evaluate the localization accuracy of the proposed method. Section 8 concludes this paper. Some important derivations are given in Appendix 1 and 2.
This paper contains a number of symbols. Following the convention, we represent the matrixes and vectors as bold italic case letters. The operations [ ⋅ ]^{T} and [ ⋅ ]^{−1} denote the matrix transpose and inverse. E[ ⋅ ], ‖ ⋅ ‖, and ∣ ⋅ ∣ represent the mathematical expectation, 2-norm, and determinant, respectively. The superscript ( ⋅ )^{o} and ( ⋅ ) are used to distinguish between the true value and noisy value. α(i : j) stands for a column sub-vector including the i^{th} to the j^{th} element of the vector α. V_{i × i} is a i square matrix with 1 in the diagonal and 0.5 in all other elements. 0_{i × j} denotes a i × j vector of zeros. O_{i × j} and I_{i × j} represent a i × j zero matrix and identity matrix.
2 Methods/experimental
Despite the rapid development of the localization method, it is still quite a challenge owing to the increasing demand for high localization accuracy. This paper considers the receiver location errors in the traditional measurement model and gives an algebraic solution using TDOA, FDOA, and differential Doppler rate measurements, aiming to produce a robust and accurate source location. The proposed method is designed based on the idea of the TSWLS method. We transform the localization equations into a pseudo-linear set of the equation by introducing three additional parameters, and a rough estimate is obtained in a WLS sense. Then, a final algebraic solution is given by using the additional parameters through another WLS. To evaluate the performance of the proposed method, we derive a new CRLB on position and velocity estimation with receiver location errors and also analysis of the relationship between the covariance of the method and the CRLB.
A set of Monte Carlo simulations is conducted to examine the behavior of the proposed method. The root mean square errors (RMSEs) are used to evaluate the estimation accuracy for source position and velocity, which is acquired from 5000 independent trials. In order to prove the necessity of considering the receiver location errors during the design of methods, we compare the CRLB derived in this paper with the CRLB in [17] as SNR and receiver location errors change. Then, to exhibit the superiority of the proposed method in robustness and performance, we compare our method with Hu’s method, TSWLS method, as well as the three types of CRLBs in three-dimensional (3-D) scenario. Besides, all the experiments are performed in two noise conditions including measurement noise and receiver location error noise.
This paper does not contain any studies with human participants or animals performed by any of the authors.
3 Problem formulation
Define \( \varDelta \boldsymbol{\upalpha} ={\left[\varDelta {\mathbf{r}}^T,\varDelta {\dot{\mathbf{r}}}^T,\varDelta {\ddot{\mathbf{r}}}^T\right]}^T \) as the measurement error vector, which follows the zero mean Gaussian distribution with covariance matrix E[ΔαΔα^{T}] = Q_{α}. Our primary objective is to estimate the source position u and velocity \( \dot{\mathbf{u}} \) as accurately as possible using the noisy measurement vector \( \boldsymbol{\upalpha} ={\left[{\mathbf{r}}^T,{\dot{\mathbf{r}}}^T,{\ddot{\mathbf{r}}}^T\right]}^T \) with noisy receiver location vector β.
4 Localization method
It can be seen that (8) is nonlinear concerning the unknown source position and velocity, which is not a trivial task to solve these nonlinear measurement equations directly. The work in [17] uses the iteration-based method to solve these, which exists the convergence problem. Hence, this paper is inspired by the idea of WLS and introduces two steps to obtain the source location including a rough estimation step and a refined estimation step. The first step solution is introduced in the following:
4.1 First WLS step
4.2 Second WLS step
Hu et al. obtained the source location from Eq. (30) by utilizing the iteration process [17]. However, this method ignores the existence of the receiver location errors and has limited localization accuracy. Therefore, this section focuses on giving an algebraic solution without iteration to upgrade localization accuracy.
- (i)
Set W_{1} = I_{3(M − 1) × 3(M − 1)} and use \( {\left({\mathbf{G}}_1^T{\mathbf{G}}_1\right)}^{-1}{\mathbf{G}}_1^T{\mathbf{h}}_1 \) to obtain an initial estimate of \( {\widehat{\boldsymbol{\uptheta}}}_1 \);
- (ii)
Utilize this initial estimate to form the approximate W_{1} and reuse (26) to update \( {\widehat{\boldsymbol{\uptheta}}}_1 \);
- (iii)
- (iv)
The final source position and velocity estimate can be obtained from \( {\widehat{\boldsymbol{\uptheta}}}_2\left(1:3\right) \) and \( {\widehat{\boldsymbol{\uptheta}}}_2\left(4:6\right) \).
5 CRLB and performance analysis
5.1 CRLB
It is important to emphasize that, in (45), \( {\boldsymbol{R}}_1^{-1} \) is the CRLB of the source position and velocity without receiver location errors derived in [17]. Since \( \left({\boldsymbol{R}}_3-{\boldsymbol{R}}_2^T{\boldsymbol{R}}_1^{-1}{\boldsymbol{R}}_2\right) \) is the positive definite matrix, the second term in (45) stands for the increase in CRLB in the presence of receiver location errors. In the subsequent numerical simulation section, the CRLB with receiver location errors and that without receiver location errors will be compared numerically to verify this analysis.
5.2 Performance analysis
Observe that the covariance matrix of the proposed method in (48) and the CRLB in (45) are of the same form. Under assumptions that the measurement noise and receiver location errors are sufficiently small, we can conclude that the CRLB is achieved.
6 Simulation results
Position and velocity of the receivers
Receiver no. i | x_{i} (m) | y_{i} (m) | z_{i} (m) | \( {\dot{x}}_i \) (m/s) | \( {\dot{y}}_i \) (m/s) | \( {\dot{z}}_i \) (m/s) |
---|---|---|---|---|---|---|
1 | 0 | 0 | − 100 | 0 | 0 | 0 |
2 | 0 | 300 | 0 | − 20 | 0 | 0 |
3 | − 300 | 0 | 0 | 0 | − 20 | 0 |
4 | 0 | − 300 | 0 | 20 | 0 | 0 |
5 | 300 | 0 | 0 | 0 | 20 | 0 |
6.1 CRLB comparison
In this section, we compare the CRLB derived in Section 4.1 with the CRLB in [17] to illustrate how sensitive the localization accuracy is with regard to the receiver location errors and SNR. We set R = 500m and ϕ = π/3 in (52), thereby \( {\mathbf{u}}^o=\left[250,250\sqrt{3},400\right]\mathrm{m} \) and \( {\dot{\mathbf{u}}}^o={\left[15\sqrt{3},15,0\right]}^T\mathrm{m}/\mathrm{s} \).
6.2 Robustness comparison
- 1.
The RMSE curves of source position and velocity with R = 500m and R = 1000m are shown in Fig. 4c, d, e, and f. This is the scenario where the source is in the moderate distance from the receivers. It can be seen that due to the existence of receiver location errors, Hu’s method cannot attain its CRLB and perform lower localization accuracy compared with the proposed method. Moreover, as ϕ changes, the proposed method always performs robustly while the RMSE of TSWLS method increases sharply at some values of ϕ even at high SNR due to its rank deficiency problem in its second step. More specifically, the matrix B_{2} of the TSWLS method in [25], defined as
- 2.
The RMSE curves of source position and velocity with R = 20m are shown in Fig. 4a and b. This is the scenario where the source approaches receivers. It can be observed that the TSWLS method and Hu’s method in [25] cannot give an accurate source location, while the proposed method still works in this near-field scenario and the accuracy curve achieves CRLB very well. The reason why the TSWLS method fails still lies in (58). For example, when R = 20m, \( {\left[{\mathbf{u}}^o-{\mathbf{s}}_1^o\right]}^T \) is [20 cos ϕ, 20 sin ϕ, 500]^{T}. The ratio between the maximum element and the minimum element is at least 25. Therefore, the rank deficiency problem of the TSWLS method still easily occurs in this condition.
- 3.
The RMSE curves of source position and velocity with R = 4000m are shown in Fig. 4g and h. This is the scenario where the source is far from the receivers. With the change of ϕ, the RMSE of the proposed method still always matches its CRLB without any fail point when the source moves in the far-field scenario, while Hu’s method cannot attain its CRLB under the condition of existing receiver location errors. Moreover, the TSWLS method does not work and is not suitable for locating the far-field source, even after considering the receiver location errors in the method.
In summary, when the source is close to or far from the receivers, the estimation accuracy of the TSWLS method is poor. In addition, even if the source moves to a moderate distance from the receivers, this method always suffers the rank deficiency problem in some special points. Moreover, Hu’s method cannot achieve its CRLB and is not suitable for scenarios with receiver location errors. On the contrary, the proposed method always attains its CRLB well and performs a better accuracy wherever the source moves, which exhibits strong robustness and better estimation performance.
6.3 Performance comparison
In order to further evaluate the efficiency, we study the localization performance of the proposed solution in terms of RMSE corresponding to different receiver location errors and SNRs. We compare its performance against Hu’s method in [17], TSWLS method in [25], as well as the three types of CRLBs via Monte Carlo simulations. It should be explained that the improved TSWLS method in [14] is not simulated in this section, because this method used the TDOA and FDOA measurements only and not considered the receiver location errors, which will definitely result in some negative influences on estimation. The source location where R = 500m and ϕ = π/3 are set in this simulation, which is a robust position for the compared methods. SNR is equal to 10 dB when we investigate the estimation performance by varying receiver location error σ_{s}. σ_{s} is set to 10^{0.5} m when SNR changes. Other simulation conditions are the same as that in Section 6.1.
6.4 Running time comparison
The comparison of the method in terms of average running time
It can be seen from Table 2, Hu’s method and the proposed method cost more time than the TSWLS methods because the additional differential Doppler rate measurement is added into the traditional TDOA-and-FDOA-based method. The running time of the proposed method is approximately three times larger than that of the Hu’s method even using the same positioning measurements, which endures the relatively heavy computational burden among the methods. This is hardly surprising since the proposed method takes the receiver location errors into consideration while the existing methods did not. That is to say, it is at the expense of the higher computation cost that the proposed method achieves higher localization accuracy. In consideration of remarkable performance improvement, the increased but acceptable complexity is worthy.
7 Discussion
To summarize, this paper mitigates the effects of receiver location errors in the localization accuracy and gives a final algebraic solution without extra mathematics operations usually used in existing methods. Based on the simulation results, we can observe that the proposed method can effectively avoid the rank deficiency problem when the source moves around. Moreover, the proposed method achieves the corresponding CRLB well at moderate noise and error levels and produces a remarkable gain in estimation accuracy compared with the existing methods for low SNRs and large receiver location errors. However, the complexity of the proposed method is higher than that of the existing methods because the proposed method considers the receiver location errors into the measurement model, while the existing methods did not. And this solution is for the localization of a single source only. We currently focus on extending it to the multiple source localization scenario with lower computational complexity. In addition, the estimation accuracy of TDOA, FDOA, and differential Doppler rate distinctly affect localization accuracy; hence, we intend to propose a high accuracy estimation method in our further study.
8 Conclusions
To improve localization accuracy and make the localization model closer to the practical environment, an algebraic method for moving source localization based on TDOA, FDOA, and differential Doppler rate measurements in the presence of receiver location errors is presented in this paper. The proposed method gives a final algebraic solution without iteration and extra mathematics operations by employing the basic idea of WLS processing, while the existing methods did not. The consideration of receiver location errors makes the proposed method achieve CRLB, no matter whether there exist receiver location errors or not. The proposed method was compared with several existing methods and showed generally better robustness and performance as SNR and receiver location error change both theoretically and numerically.
Notes
Acknowledgements
The authors would like to thank the Editorial board and anonymous Reviewers for their careful reading and constructive comments which provide an important guidance for our paper writing and research work. The authors would also like to thank K. C. Ho for their previous studies, which helped us very much.
Funding
This work was supported by the National Natural Science Foundation of China under Grant 61703433.
Availability of data and materials
The datasets generated during the current study are not publicly available but are available from the corresponding author on reasonable request.
Authors’ contributions
ZL and DH derived the theoretical of the method and wrote the manuscript. YZ and JZ were in charge of the experiment and results. All authors had a significant contribution to the development of early ideas and design of the final methods. All authors read and approved the final manuscript.
Ethics approval and consent to participate
This paper does not contain any studies with human participants or animals performed by any of the authors.
Consent for publication
Informed consent was obtained from all authors included in the study.
Competing interests
The authors declare that they have no competing interests.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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